The Collected Works of Ludwig Wittgenstein
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NOTEBOOKS 1914-1916
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Titlepage
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NOTEBOOKS
1914-1916
LUDWIG WITTGENSTEIN
Second Edition
Edited by
G. H. von WRIGHT
and
G. E. M. ANSCOMBE
with an English translation by
G. E. M. ANSCOMBE
Index prepared by
E. D. KLEMKE
BASIL BLACKWELL
OXFORD
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Copyright Page
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Copyright © Blackwell Publishers 1998
First published 1961
Second edition 1979
Reprinted 1998
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ISBN 0-631-12499-3
Printed in Great Britain by Athenaeum Press Ltd, Gateshead, Tyne & Wear
This book is printed on acid-free paper
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PREFACE TO THE SECOND EDITION
Page 1
THE text of this volume has been completely revised for this edition, and a number of misreadings have been
corrected. These were mostly very small. The most serious one that I have found was the reading of "u.u." ("und
umgekehrt) as "u.U" ("unter Umständen"). The diagram on p. 126 has been corrected in accordance with the MS.
Page 1
The second appendix, Notes on Logic 1913, appears here in a different arrangement from that of the first
edition. That edition used the text published in the Journal of Philosophy (Vol. LIV (1957), p. 484) by J. J.
Costelloe: he reported having got it from Bertrand Russell in 1914. There was a different text which the editors had,
and which they had also got from Russell. It was clear that the Costelloe version was a slightly corrected total
rearrangement of that text under headings, and we assumed that it had been made by Wittgenstein himself.
Page 1
A debt of gratitude is owing to Brian McGuinness, not only for having pointed out some errors of
transcription in the first edition, but also for having proved that the Costelloe version was constructed by Russell.
The other one is therefore closer to Wittgenstein, the first part of it being his own dictation in English and the rest a
translation by Russell of material dictated by Wittgenstein in German. Mr. McGuinness' article giving the evidence
for this can be found in the Revue Internationale de Philosophie, no. 102 (1972).
Page 1
In the first edition a number of passages of symbolism, in one case with accompanying text, were omitted
because nothing could be made of them: they were presumably experimental, but it seemed impossible to interpret
them. Nor would it always have been clear what was an exact transcription of them. Photographs of them are printed
here as a fourth appendix.
Page 1
At the 20th of December 1914 there was a rough line of adjacent crayonned patches, using 7 colours. This
was treated as a mere doodle in the first edition, and so it may be. But, having regard to the subject matter of
meaning and negation, which is the topic of the surrounding text, it is possible that there is here an anticipation of
Philosophical Investigations § 48. A representation of it is printed on the dust cover of this edition.
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[Notebooks: Main Body]
22.8.14.
Page 2
Logic must take care of itself. [See 5.473.]
Page 2
If syntactical rules for functions can be set up at all, then the whole theory of things, properties, etc., is
superfluous. It is also all too obvious that this theory isn't what is in question either in the Grundgesetze, or in
Principia Mathematica. Once more: logic must take care of itself. A possible sign must also be capable of
signifying. Everything that is possible at all, is also legitimate. Let us remember the explanation why "Socrates is
Plato" is nonsense. That is, because we have not made an arbitrary specification, NOT because a sign is, shall we say,
illegitimate in itself! [Cf. 5.473.]
2.9.14.
Page 2
It must in a certain sense be impossible for us to go wrong in logic. This is already partly expressed by
saying: Logic must take care of itself. This is an extremely profound and important insight. [Cf. 5.473.]
Page 2
Frege says: Every well-formed sentence must make sense; and I say: Every possible sentence is well-formed,
and if it does not make sense that can only come of our not having given any meaning to certain of its parts. Even
when we believe we have done so. [Cf. 5.4733.]
3.9.14.
Page 2
How is it reconcilable with the task of philosophy, that logic should take care of itself? If, for example, we
ask: Is such and such a fact of the subject-predicate form?, we must surely know what we mean by
"subject-predicate form". We must know whether there is such a form at all. How can we know this? "From the
signs". But how? For we haven't got any signs of this form. We may indeed say: We have signs that behave like
signs of the subject-predicate form, but does that mean that there really must be facts of this form? That is, when
those signs are completely analysed? And here the question arises again: Does such a complete analysis exist? And
if not: then what is the task of philosophy?!!?
Page 2
Then can we ask ourselves: Does the subject-predicate form exist? Does the relational form exist? Do any of
the forms exist at all that
Page Break 3
Russell and I were always talking about? (Russell would say: "Yes! that's self-evident." Ha!)
Page 3
Then: if everything that needs to be shewn is shewn by the existence of subject-predicate SENTENCES etc.,
the task of philosophy is different from what I originally supposed. But if that is not how it is, then what is lacking
would have to be shewn by means of some kind of experience, and that I regard as out of the question.
Page 3
The obscurity obviously resides in the question: what does the logical identity of sign and thing signified
really consist in? And this question is (once more) a main aspect of the whole philosophical problem.
Page 3
Let some philosophical question be given: e.g., whether "A is good" is a subject-predicate proposition; or
whether "A is brighter than B" is a relational proposition. How can such a question be settled at all? What sort of
evidence can satisfy me that--for example--the first question must be answered in the affirmative? (This is an
extremely important question.) Is the only evidence here once more that extremely dubious "self-evidence"? Let's
take a question quite like that one, which however is simpler and more fundamental, namely the following: Is a point
in our visual field a simple object, a thing? Up to now I have always regarded such questions as the real
philosophical ones: and so for sure they are in some sense--but once more what evidence could settle a question of
this sort at all? Is there not a mistake in formulation here, for it looks as if nothing at all were self-evident to me on
this question; it looks as if I could say definitively that these questions could never be settled at all.
4.9.14.
Page 3
If the existence of the subject-predicate sentence does not show everything needful, then it could surely only
be shewn by the existence of some particular fact of that form. And acquaintance with such a fact cannot be
essential for logic.
Page 3
Suppose we had a sign that actually was of the subject-predicate form, would this be somehow better suited
to express subject-predicate propositions than our subject-predicate sentences are? It seems not! Does this arise
from the signifying relation?
Page 3
If logic can be completed without answering certain questions, then it must be completed without answering
them.
Page 3
The logical identity between sign and thing signified consists in
Page Break 4
its not being permissible to recognize more or less in the sign than in what it signifies.
Page 4
If sign and thing signified were not identical in respect of their total logical content then there would have to
be something still more fundamental than logic.
5.9.14.
Page 4
φ(a). φ(b). aRb = Def φ[aRb]
Page 4
Remember that the words "function", "argument", "sentence" etc. ought not to occur in logic.
Page 4
To say of two classes that they are identical means something. To say it of two things means nothing. This of
itself shews the inadmissibility of Russell's definition.
6.9.14.
Page 4
The last sentence is really nothing but the old old objection against identity in mathematics. Namely the
objection that if 2 × 2 were really the same as 4, then this proposition would say no more than a = a.
Page 4
Could it be said: Logic is not concerned with the analysability of the functions with which it works.
7.9.14.
Page 4
Remember that even an unanalysed subject-predicate proposition is a clear statement of something quite
definite.
Page 4
Can't we say: It all depends, not on our dealing with unanalysable subject-predicate sentences, but on the
fact that our subject-predicate sentences behave in the same way as such sentences in every respect, i.e. that the
logic of our subject-predicate sentences is the same as the logic of those. The point for us is simply to complete
logic, and our objection-in-chief against unanalysed subject-predicate sentences was that we cannot construct their
syntax so long as we do not know their analysis. But must not the logic of an apparent subject-predicate sentence be
the same as the logic of an actual one? If a definition giving the proposition the subject-predicate form is possible at
all...?
8.9.14.
Page 4
The "self-evidence" of which Russell has talked so much can only be dispensed with in logic if language
itself prevents any logical mistake. And it is clear that that "self-evidence" is and always was wholly deceptive. [Cf.
5.4731.]
Page Break 5
19.9.14.
Page 5
A proposition like "this chair is brown" seems to say something enormously complicated, for if we wanted to
express this proposition in such a way that nobody could raise objections to it on grounds of ambiguity, it would
have to be infinitely long.
20.9.14.
Page 5
That a sentence is a logical portrayal of its meaning is obvious to the uncaptive eye.
Page 5
Are there functions of facts? e.g. "It is better for this to be the case than for that to be the case?"
Page 5
What, then, is the connexion between the sign p and the rest of the signs of the sentence "that p is the case, is
good"? What does this connexion consist in?
Page 5
The uncaptive judgement will be: Obviously in the spatial relation of the letter p to the two neighbouring
signs. But suppose the fact "p" were such as to contain no things?
Page 5
"It is good that p" can presumably be analysed into "p. it is good if p".
Page 5
We assume: p is NOT the case: now what does it mean to say "that p, is good"? Quite obviously we can say
that the situation p is good without knowing whether "p" is true or false.
Page 5
This throws light on what we say in grammar: "One word refers to another".
Page 5
That is why the point in the above cases is to say how propositions hang together internally. How the
propositional bond comes into existence. [Cf. 4.221.]
Page 5
How can a function refer to a proposition???? Always the old old questions.
Page 5
Don't let yourself get overwhelmed with questions; just take it easy.
Page 5
"φ(ψx)": Suppose we are given a function of a subject-predicate proposition and we try to explain the way
the function refers to the proposition by saying: The function only relates immediately to the subject of the
subject-predicate proposition, and what signifies is the logical product of this relation and the subject-predicate
propositional sign. Now if we say this, it can be asked: If you can explain the
Page Break 6
proposition like that, then why not give an analogous explanation of what it stands for? Namely: "It is not a function
of a subject-predicate fact but the logical product of such a fact and of a function of its subject"? Must not the
objection to the latter explanation hold against the former too?
21.9.14.
Page 6
Now it suddenly seems to me in some sense clear that a property of a situation must always be internal.
Page 6
φa, ψb, aRb. It could be said that the situation aRb always has a certain property, if the first two propositions
are true.
Page 6
When I say: It is good for p to be the case, then this must be good in itself.
Page 6
It now seems clear to me that there cannot be functions of situations.
23.9.14.
Page 6
It could be asked: How can the situation p have a property if it turns out that the situation does not hold at
all?
24.9.14.
Page 6
The question how a correlation of relations is possible is identical with the problem of truth.
25.9.14.
Page 6
For the latter is identical with the question how the correlation of situations is possible (one that signifies and
one that is signified).
Page 6
It is only possible by means of the correlation of the components; the correlation between names and things
named gives an example. (And it is clear that a correlation of relations too takes place somehow.)
| aRb |; | a b|; p = aRb Def
Here a simple sign is correlated with a situation.
26.9.14.
Page 6
What is the ground of our--certainly well founded--confidence that we shall be able to express any sense we
like in our two-dimensional script?
27.9.14.
Page 6
A proposition can express its sense only by being the logical portrayal of it.
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Page 7
The similarity between these signs is striking:
"aRb"
"aσR.Rσb".
29.9.14.
Page 7
The general concept of the proposition carries with it a quite general concept of the coordination of
proposition and situation: The solution to all my questions must be extremely simple.
Page 7
In the proposition a world is as it were put together experimentally. (As when in the law-court in Paris a
motor-car accident is represented by means of dolls, etc.†1) [Cf. 4.031.]
Page 7
This must yield the nature of truth straight away (if I were not blind).
Page 7
Let us think of hieroglyphic writing in which each word is a representation of what it stands for. Let us think
also of the fact that actual pictures of situations can be right and wrong. [Cf. 4.016.]
Page 7
" ": If the right-hand figure in this picture represents the man A, and the left-hand one
stands for the man B, then the whole might assert, e.g.: "A is fencing with B". The proposition in picture-writing can
be true and false. It has a sense independent of its truth or falsehood. It must be possible to demonstrate everything
essential by considering this case.
Page 7
It can be said that, while we are not certain of being able to turn all situations into pictures on paper, still we
are certain that we can portray all logical properties of situations in a two-dimensional script.
Page 7
This is still very much on the surface, but we are on good ground.
30.9.14.
Page 7
It can be said that in our picture the right-hand figure is a representation of something and also the left-hand
one, but even if this were not the case, their relative position could be a representation of something. (Namely a
relation.)
Page Break 8
Page 8
A picture can present relations that do not exist! How is that possible?
Page 8
Now once more it looks as if all relations must be logical in order for their existence to be guaranteed by that
of the sign.
2.10.14.
Page 8
What connects a and c in "aRb.bSc" is not the sign "." but the occurrence of the same letter "b" in the two
simple sentences.
Page 8
We can say straight away: Instead of: this proposition has such and such a sense: this proposition represents
such and such a situation. [See 4.031.]
Page 8
It portrays it logically.
Page 8
Only in this way can the proposition be true or false: It can only agree or disagree with reality by being a
picture of a situation. [See 4.06.]
3.10.14.
Page 8
The proposition is a picture of a situation only in so far as it is logically articulated. (A
simple-non-articulated-sign can be neither true nor false.) [Cf. 4.032.]
Page 8
The name is not a picture of the thing named!
Page 8
The proposition only says something in so far as far as it is a picture! [See 4.03.]
Page 8
Tautologies say nothing, they are not pictures of situations: they are themselves logically completely neutral.
(The logical product of a tautology and a proposition says neither more nor less than the latter by itself.) [See 4.462
and 4.463.]
4.10.14.
Page 8
It is clear that "xRy" can contain the signifying element of a relation even if "x" and "y" do not stand for
anything. And in that case the relation is the only thing that is signified in that sign.
Page 8
But in that case,†1 how is it possible for "kilo" in a code to mean: "I'm all right"? Here surely a simple sign
does assert something and is used to give information to others.--
Page 8
For can't the word "kilo", with that meaning, be true or false?
Page Break 9
5.10.14.
Page 9
At any rate it is surely possible to correlate a simple sign with the sense of a sentence.--
Page 9
Logic is interested only in reality. And thus in sentences ONLY in so far as they are pictures of reality.
Page 9
But how CAN a SINGLE word be true or false? At any rate it cannot express the thought that agrees or does
not agree with reality. That must be articulated.
Page 9
A single word cannot be true or false in this sense: it cannot agree with reality, or the opposite.
6.10.14.
Page 9
The general concept of two complexes of which the one can be the logical picture of the other, and so in one
sense is so.
Page 9
The agreement of two complexes is obviously internal and for that reason cannot be expressed but can only
be shewn.
Page 9
"p" is true, says nothing else but p.
Page 9
"'p' is true" is--by the above--only a pseudo-proposition like all those connexions of signs which apparently
say something that can only be shewn.
7.10.14.
Page 9
If a proposition φa is given, then all its logical functions (~φa, etc.) are already given with it! [Cf. 5.442.]
8.10.14.
Page 9
Complete and incomplete portrayal of a situation. (Function plus argument is portrayed by function plus
argument.)
Page 9
The expression "not further analysable" too is one of those which, together with "function", "thing" etc. are
on the Index; but how does what we try to express by means of it get shewn?
Page 9
(Of course it cannot be said either of a thing or of a complex that it is not further analysable.)
9.10.14.
Page 9
If there were such a thing as an immediate correlation of relations, the question would be: How are the things
that stand in these relations correlated with one another in this case? Is there such a thing as a direct correlation of
relations without consideration of their direction?
Page Break 10
Page 10
Are we misled into assuming "relations between relations" merely through the apparent analogy between the
expressions:
"relations between things"
and "relations between relations"?
Page 10
In all there considerations I am somewhere making some sort of FUNDAMENTAL MISTAKE.
Page 10
The question about the possibility of existence propositions does not come in the middle but at the very first
beginning of logic.
Page 10
All the problems that go with the Axiom of Infinity have already to be solved in the proposition "(∃x)x = x".
[Cf. 5.535.]
10.10.14.
Page 10
One often makes a remark and only later sees how true it is.
11.10.14.
Page 10
Our difficulty now lies in the fact that to all appearances analysability, or its opposite, is not reflected in
language. That is to say: We can not, as it seems, gather from language alone whether for example there are real
subject-predicate facts or not. But how COULD we express this fact or its opposite? This must be shewn.
Page 10
But suppose that we did not bother at all about the question of analysability? (We should then work with
signs that do not stand for anything but merely help to express by means of their logical properties.) For even the
unanalysed proposition mirrors logical properties of its meaning. Suppose then we were to say: The fact that a
proposition is further analysable is shewn in our further analysing it by means of definitions, and we work with it in
every case exactly as if it were unanalysable.
Page 10
Remember that the "propositions about infinite numbers" are all represented by means of finite signs.
Page 10
But do we not--at least according to Frege's method--need 100 million signs in order to define the number
100,000,000? (Doesn't this depend on whether it is applied to classes or to things?)
Page 10
The propositions dealing with infinite numbers, like all propositions of logic, can be got by calculating the
signs themselves (for at no point does a foreign element get added to the original primitive signs). So
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here, too, the signs must themselves possess all the logical properties of what they represent.
12.10.14.
Page 11
The trivial fact that a completely analysed proposition contains just as many names as there are things
contained in its reference; this fact is an example of the all-embracing representation of the world through language.
Page 11
It would be necessary to investigate the definitions of the cardinal numbers more exactly in order to
understand the real sense of propositions like the Axiom of Infinity.
13.10.14.
Page 11
Logic takes care of itself; all we have to do is to look and see how it does it. [Cf. 5.473.]
Page 11
Let us consider the proposition: "there is a class with only one member". Or, what comes to the same thing,
the proposition:
(∃φ):.(∃x):φx:φy.φz. ⊃y,z.y = z
Page 11
If we take "(∃x)x = x" it might be understood to be tautological since it could not get written down at all if it
were false, but here! This proposition can be investigated in place of the Axiom of Infinity.
Page 11
I know that the following sentences as they stand are nonsensical: Can we speak of numbers if there are only
things? I.e. if for example the world only consisted of one thing and of nothing else, could we say that there was
ONE thing? Russell would probably say: If there is one thing then there is also a function (∃x) = x. But!---
Page 11
If this function does not do it then we can only talk of 1 if there is a material function which is satisfied only
by one argument.
Page 11
How is it with propositions like:
(∃φ).(∃x).φ(x).
and (∃φ).(∃x).~φ(x).
Is one of these a tautology? Are these propositions of some science, i.e. are they propositions at all?
Page 11
But let us remember that it is the variables and not the sign of generality that are characteristic of logic.
14.10.14.
Page 11
For is there such a thing as a science of completely generalized propositions? This sounds extremely
improbable.
Page Break 12
Page 12
This is clear: If there are completely generalized propositions, then their sense does not depend on any
arbitrary formation of signs! In that case, however, such a connexion of signs can represent the world only by means
of its own logical properties, i.e. it can not be false, and not be true. So there are no completely generalized
propositions. But now the application)
Page 12
But now the propositions: "(∃φ,x).φ(x)"
and "~(∃φ,x).φ(x)".
Which of these is tautological, which contradictory?
Page 12
We keep on needing a comparative arrangement of propositions standing in internal relations. This book
might well be equipped with diagrams.
Page 12
(The tautology shews what it appears to say, the contradiction shews the opposite of what it appears to say.)
Page 12
It is clear that we can form all the completely general propositions that are possible at all as soon as we are
merely given a language. And that is why it is scarcely credible that such connexions of signs should really say
anything about the world. On the other hand, however, this gradual transition from the elementary proposition to the
completely general one!
Page 12
We can say: The completely general propositions can all be formed a priori.
15.10.14.
Page 12
Yet it does not look as if the mere existence of the forms contained in "(∃x,φ).φ(x)" could by itself determine
the truth or falsehood of this proposition! So it does not appear unthinkable that, e.g., the negation of no elementary
proposition should be true. But would not this statement itself touch the SENSE of negation?
Page 12
Obviously we can conceive every quite general proposition as the affirmation or negation of the existence of
some kind of facts. But does this not hold of all propositions?
Page 12
Every connexion of signs which appears to say something about its own sense is a pseudo-proposition (like
all propositions of logic).
Page 12
The proposition is supposed to give a logical model of a situation. It can surely only do this, however,
because objects have been arbitrarily correlated with its elements. Now if this is not the case in the
Page Break 13
quite general proposition, then it is difficult to see how it should represent anything outside itself.
Page 13
In the proposition we--so to speak--arrange things experimentally, as they do not have to be in reality; but
we cannot make any unlogical arrangement, for in order to do that we should have to be able to get outside logic in
language.--But if the quite general proposition contains only "logical constants", then it cannot be anything more to
us than--simply--a logical structure, and cannot do anything more than shew us its own logical properties. If there
are quite general propositions--what do we arrange experimentally in them? [Cf. 4.031 and 3.03.]
Page 13
When one is frightened of the truth (as I am now) then it is never the whole truth that one has an inkling of.
Page 13
Here I regarded the relations of the elements of the proposition to their meanings as feelers, so to say, by
means of which the proposition is in contact with the outer world; and the generalization of a proposition is in that
case like the drawing in of feelers; until finally the completely general proposition is quite isolated. But is this picture
right? (Do I really draw a feeler in when I say (∃x).φx instead of φa?) [Cf. 2.1515.]
16.10.14.
Page 13
Now, however, it looks as if exactly the same grounds as those I produced to shew that "(∃x,φ).φx" could not
be false would be an argument shewing that "~(∃x,φ).φx" could not be false; and here a fundamental mistake makes
its appearance. For it is quite impossible to see why just the first proposition and not the second is supposed to be a
tautology. But do not forget that the contradiction "p.~p" etc. etc. cannot be true and is nevertheless itself a logical
structure.
Page 13
Suppose that no negation of an elementary proposition is true, has not "negation" another sense in this case
than in the opposite case?
Page 13
"(∃φ):(x).φx"--of this proposition it appears almost certain that it is neither a tautology nor a contradiction.
Here the problem becomes extremely sharp.
17.10.14.
Page 13
If there are quite general propositions, then it looks as if such propositions were experimental combinations
of "logical constants".(!)
Page Break 14
Page 14
But is it not possible to describe the whole world completely by means of completely general propositions?
(The problem crops up on all sides.)
Page 14
Yes, the world could be completely described by completely general propositions, and hence without using
any sort of names or other denoting signs. And in order to arrive at ordinary language one would only need to
introduce names, etc. by saying, after an "(∃x)", "and this x is A" and so on. [Cf. 5.526.]
Page 14
Thus it is possible to devise a picture of the world without saying what is a representation of what.
Page 14
Let us suppose, e.g., that the world consisted of the things A and B and the property F, and that F(A) were
the case and not F(B). This world could also be described by means of the following propositions:
Page 14
(∃x,y).(∃φ).x ≠ y.φx.~φy:φu.φz. ⊃u,z.u = z
(∃φ).(ψ).ψ = φ
(∃x,y).(z).z = x ∨ z = y
And here one also needs propositions of the type of the last two, only in order to be able to identify the objects.
Page 14
From all this, of course, it follows that there are completely general propositions!
Page 14
But isn't the first proposition above enough: (∃x,y,φ)φx.~φy.x ≠ y? The difficulty of identification can be
done away with by describing the whole world in a single general proposition beginning: "(∃x,y,z... φ,ψ... R,S...)"
and now follows a logical product, etc.
Page 14
If we say "φ is a unit function and (x).φx", that is as much as to say: There is only one thing! (By this means
we have apparently got round the proposition (∃x)(y).y = x".)
18.10.14.
Page 14
My mistake obviously lies in a false conception of logical portrayal by the proposition.
Page 14
A statement cannot be concerned with the logical structure of the world, for in order for a statement to be
possible at all, in order for a proposition to be CAPABLE of making SENSE, the world must already have just the
logical structure that it has. The logic of the world is prior to all truth and falsehood.
Page Break 15
Page 15
Roughly speaking: before any proposition can make sense at all the logical constants must have reference.†1
19.10.14.
Page 15
The description of the world by means of propositions is only possible because what is signified is not its
own sign! Application--.
Page 15
Light on Kant's question "How is pure mathematics possible?" through the theory of tautologies.
Page 15
It is obvious that we must be able to describe the structure of the world without mentioning any names. [Cf.
5.526.]
20.10.14.
Page 15
The proposition must enable us to see the logical structure of the situation that makes it true or false. (As a
picture must shew the spatial relation in which the things represented in it must stand if the picture is correct (true).)
Page 15
The form of a picture might be called that in which the picture MUST agree with reality (in order to be
capable of portraying it at all). [Cf. 2.17 and 2.18.]
Page 15
The first thing that the theory of logical portrayal by means of language gives us is a piece of information
about the nature of the truth-relation.
Page 15
The theory of logical portrayal by means of language says--quite generally: In order for it to be possible that
a proposition should be true or false--agree with reality or not--for this to be possible something in the proposition
must be identical with reality. [Cf. 2.18.]
Page 15
What negates in "p" is not the "~" in front of the "p", but is what is common to all the signs that have the
same meaning as "~p" in this notation; and therefore what is common in
and the same holds for the generality notation, etc.
[Cf. 5.512.]
Page Break 16
Page 16
Pseudo-propositions are such as, when analysed, turn out after all only to shew what they were supposed to
say.
Page 16
Here we have a justification for the feeling that the proposition describes a complex in the kind of way that
Russellian descriptions do: the proposition describes the complex by means of its logical properties.
Page 16
The proposition constructs a world by means of its logical scaffolding, and that is why we can actually see in
the proposition how everything logical would stand if it were true: we can draw conclusions from a false
proposition, etc. (In this way I can see that if "(x,φ).φx" were true, this proposition would contradict a proposition
"ψa".) [Cf 4.023.]
Page 16
The possibility of inferring completely general propositions from material propositions--the fact that the
former are capable of standing in meaning ful [[sic]] internal relations with the latter--shews that the completely
general propositions are logical constructions from situations.
21.10.14.
Page 16
Isn't the Russellian definition of nought nonsensical? Can we speak of a class (x ≠ x) at all?--Can we
speak of a class (x = x) either? For is x ≠ x or x = x a function of x?--Must not 0 be defined by means of the
hypothesis (∃φ):(x)~φx? And something analogous would hold of all other numbers. Now this throws light on the
whole question about the existence of numbers of things.
0 = {(∃φ):(x)~φx.α = û(φu)} Def.
1 = {∃φ)::(∃x).φx:φy.φz ⊃y,z y = z:α = û(φu)} Def.
Page 16
[The sign of equality in the curly brackets could be avoided if we were to write:
0 = {(x)~φx}.†1]
Page 16
The proposition must contain (and in this way shew) the possibility of its truth. But not more than the
possibility. [Cf. 2.203 and 3.02 and 3.13.]
Page 16
By my definition of classes (x).~ (φx) is the assertion that x(φx) is null and the definition of 0 is in that case
0 = [(x).~α] Def.
Page Break 17
Page 17
I thought that the possibility of the truth of the proposition φa was tied up with the fact (∃x,φ).φx. But it is
impossible to see why φa should only be possible if there is another proposition of the same form. φa surely does
not need any precedent. (For suppose that there existed only the two elementary propositions "φa" and "ψa" and
that "φa", were false: Why should this proposition only make sense if "ψa" is true?)
22.10.14.
Page 17
There must be something in the proposition that is identical with its reference, but the proposition cannot be
identical with its reference, and so there must be something in it that is not identical with the reference. (The
proposition is a formation with the logical features of what it represents and with other features besides, but these
will be arbitrary and different in different sign-languages.) So there must be different formations with the same
logical features; what is represented will be one of these, and it will be the business of the representation to
distinguish this one from other formations with the same logical features. (Since otherwise the representation would
not be unambiguous.) This part of the representation (the assignment of names) must take place by means of
arbitrary stipulations. Every proposition must accordingly contain features with arbitrarily determined references.
Page 17
If one tries to apply this to a completely generalized proposition, it appears that there is some fundamental
mistake in it.
Page 17
The generality of the completely general proposition is accidental generality. It deals with all the things that
there chance to be. And that is why it is a material proposition.
23.10.14.
Page 17
On the one hand my theory of logical portrayal seems to be the only possible one, on the other hand there
seems to be an insoluble contradiction in it!
Page 17
If the completely generalized proposition is not completely dematerialized, then a proposition does not get
dematerialized at all through generalization, as I used to think.
Page 17
Whether I assert something of a particular thing or of all the things that there are, the assertion is equally
material.
Page Break 18
Page 18
"All things"; that is, so to speak, a description taking the place of "a and b and c".
Page 18
Suppose our signs were just as indeterminate as the world they terror?
Page 18
In order to recognize the sign in the sign we have to attend to the use. [Cf. 3.326.]
Page 18
If we were to try and express what we express by means of "(x).φx" by prefixing an index to "φx", e.g., like
this: "Gen. φx", it would not be adequate (we should not know what was being generalized).
Page 18
If we tried to shew it by means of an index to the "x", e.g., like this: φ(xG), it would still not be adequate (in
this way we should not know the scope of generality).
Page 18
If we thought of trying to do it by inserting a mark in the empty argument places, e.g., like this:
"(G,G).ψ(G,G)" it would not be adequate (we could not settle the identity of the variables).
Page 18
All these methods of symbolizing are inadequate because they do not have the necessary logical properties.
All those collections of signs lack the power to portray the requisite sense-in the proposed way. [Cf. 4.0411.]
24.10.14.
Page 18
In order to be able to frame a statement at all, we must--in some sense--know how things stand if the
statement is true (and that is just what we portray). [Cf. 4.024.]
Page 18
The proposition expresses what I do not know; but what I must know in order to be able to say it at all, I
shew in it.
Page 18
A definition is a tautology and shews internal relations between its two terms!
25.10.14.
Page 18
But why do you never investigate an individual particular sign in order to find out how it is a logical
portrayal?
Page 18
The completely analysed proposition must image its reference.
Page 18
We might also say that our difficulty starts from the completely generalized proposition's not appearing to be
complex.
Page 18
It does not appear, like all other propositions, to consist of arbitrarily symbolizing component parts which are
united in a logical form. It appears not to HAVE a form but itself to be a form complete in itself.
Page Break 19
Page 19
With the logical constants one need never ask whether they exist, for they can even vanish!
Page 19
Why should "φ( )" not image how (x).φx is the case? Doesn't it all depend here only on how--in what kind
of way--that sign images something?
Page 19
Suppose that I wanted to represent four pairs of men fighting; could I not do so by representing only one and
saying: "That is how they all four look"? (By means of this appendix I determine the kind of representation.)
(Similarly I represent (x).φx by means of "φ( )".)
Page 19
Remember that there are no hypothetical internal relations. If a structure is given and a structural relation to
it, then there must be another structure with that relation to the first one. (This is involved in the nature of structural
relations.)
Page 19
And this speaks for the correctness of the above remark: it stops it from being--an evasion.
26.10.14.
Page 19
So it looks as if the logical identity between sign and things signified were not necessary, but only an
internal, logical, relation between the two. (The holding of such a relation incorporates in a certain sense the holding
of a kind of fundamental--internal--identity.)
Page 19
The point is only that the logical part of what is signified should be completely determined just by the logical
part of the sign and the method of symbolizing: sign and method of symbolizing together must be logically identical
with what is signified.
Page 19
The sense of the proposition is what it images. [Cf. 2.221.]
27.10.14.
Page 19
"x = y" is not a propositional form. (Consequences.)
Page 19
It is clear that "aRa" would have the same reference as "aRb.a = b". So we can make the pseudo-proposition
"a = b" disappear by means of a completely analysed notation. The best proof of the correctness of the above
remark.
Page 19
The difficulty of my theory of logical portrayal was that of finding a connexion between the signs on paper
and a situation outside in the world.
Page Break 20
Page 20
I always said that truth is a relation between the proposition and the situation, but could never pick out such
a relation.
Page 20
The representation of the world by means of completely generalized propositions might be called the
impersonal representation of the world.
Page 20
How does the impersonal representation of the world take place?
Page 20
The proposition is a model of reality as we imagine it. (See 4.01.)
28.10.14.
Page 20
What the pseudo-proposition "There are n things" tries to express shews in language by the presence of n
proper names with different references. (Etc.)
Page 20
What the completely general propositions describe are indeed in a certain sense structural properties of the
world. Nevertheless these propositions can still be true or false. According as they make sense the world still has that
permanent range.
Page 20
In the end the truth or falsehood of every proposition makes some difference to the general structure of the
world. And the range which is left to its structure by the TOTALITY of all elementary propositions is just the one
that is bounded by the completely general propositions. [Cf. 5.5262.]
29.10.14.
Page 20
For, if an elementary proposition is true, then at any rate one more elementary proposition is true, and
conversely. [See 5.5262.]
Page 20
In order for a proposition to be true it must first and foremost be capable of truth, and that is all that
concerns logic.
Page 20
The proposition must shew what it is trying to say.--Its relation to its reference must be like that of a
description to its subject.
Page 20
The logical form of the situation, however, cannot be described.--[Cf. 4.12 and 4.121.]
Page 20
The internal relation between the proposition and its reference, the method of symbolizing is the system of
co-ordinates which projects the situation into the proposition. The proposition corresponds to the fundamental
co-ordinates.
Page 20
We might conceive two co-ordinates aF and bP as a proposition stating that the material point P is to be
found in the place (ab). For
Page Break 21
this statement to be possible the co-ordinates a and b must really determine a place. For a statement to be possible
the logical coordinates must really determine a logical place!
Page 21
(The subject-matter of general propositions is really the world; which makes its appearance in them by
means of a logical description.--And that is why the world does not really occur in them, just as the subject of the
description does not occur in it.)
Page 21
The fact that in a certain sense the logical form of p must be present even if p is not the case, shews
symbolically through the fact that "p" occurs in "~p".
Page 21
This is the difficulty: How can there be such a thing as the form of p if there is no situation of this form? And
in that case, what does this form really consist in?
Page 21
There are no such things as analytic propositions.
30.10.14.
Page 21
Could we say: In "~φ(x)" "φ(x)" images how things are not?
Page 21
Even in a picture we could represent a negative fact by representing what is not the case.
Page 21
If, however, we admit these methods of representation, then what is really characteristic of the relation of
representing?
Page 21
Can't we say: It's just that there are different logical co-ordinate-systems!
Page 21
There are different ways of giving a representation, even by means of a picture, and what represents is not
merely the sign or picture but also the method of representation. What is common to all representation it that they
can be right or wrong, true or false.
Page 21
Then--picture and way of representing are completely outside what is represented!
Page 21
The two together are true or false, namely the picture, in a particular way. (Of course this holds for the
elementary proposition tool)
Page 21
Any proposition can be negated. And this shews that "true" and "false" mean the same for all propositions.
(This is of the greatest possible importance.) (In contrast to Russell.)
Page Break 22
Page 22
The reference of the proposition must be fixed, as conning or contradicting it, through it together with its
method of representation. [Cf. 4.023.]
Page 22
In logic there is no side by side, there cannot be any classification. [See 5.454.]
31.10.14.
Page 22
A proposition like "(∃x,φ).φx" is just as complex as an elementary one. This comes out in our having to
mention "φ" and "x" explicitly in the brackets. The two stand--independently--in symbolizing relations to the world,
just as in the case of an elementary proposition "ψ(a)". [Cf. 5.5261.]
Page 22
Isn't it like this: The logical constants signalise the way in which the elementary forms of the proposition
represent?
Page 22
The reference of the proposition must be fixed as confirming or contradicting it, by means of it and its way
of representing. To this end it must be completely described by the proposition. [Cf. 4.023.]
Page 22
The way of representing does not portray; only the proposition is a picture.
Page 22
The way of representing determines how the reality has to be compared with the picture.
Page 22
First and foremost the elementary propositional form must portray; all portrayal takes place through it.
1.11.14.
Page 22
We readily confuse the representing relation which the proposition has to its reference, and the truth relation.
The former is different for different propositions, the latter is one and the same for all propositions.
Page 22
It looks as if "(x,φ).φx" were the form of a fact φa.ψb.θc etc. (Similarly (∃x).φx would be the form of φa, as I
actually thought.)
Page 22
And this must be where my mistake is.
Page 22
Examine the elementary proposition: What is the form of "φa" and how is it related to "~φ(a)"?
Page 22
That precedent to which we should always like to appeal must be involved in the sign itself. [Cf. 5.525.]
Page Break 23
Page 23
The logical form of the proposition must already be given by the forms of its component parts. (And these
have to do only with the sense of the propositions, not with their truth and falsehood.)
Page 23
In the form of the subject and of the predicate there already lies the possibility of the subject-predicate
proposition, etc.; but fair enough--nothing about its truth or falsehood.
Page 23
The picture has whatever relation to reality it does have. And the point is how it is supposed to represent. The
same picture will agree or fail to agree with reality according to how it is supposed to represent.
Page 23
Analogy between proposition and description: The complex which is congruent with this sign. (Exactly as in
representation in a map.)
Page 23
Only it just cannot be said that this complex is congruent with that (or anything of the kind), but this shews.
And for this reason the description assumes a different character. [Cf. 4.023.]
Page 23
The method of portrayal must be completely determinate before we can compare reality with the proposition
at all in order to see whether it is true of false. The method of comparison must be given me before I can make the
comparison.
Page 23
Whether a proposition is true or false is something that has to appear.
Page 23
We must however know in advance how it will appear.
Page 23
That two people are not fighting can be represented by representing: them as not fighting and also by
representing them as fighting and saying that the picture shews how things are not. We could represent by means of
negative facts just as much as by means of positive ones----. However, all we want is to investigate the principles of
representing as such.
Page 23
The proposition "'p' is true" has the same reference as the logical product of 'p', and a proposition "'p'" which
describes the proposition 'p', and a correlation of the components of the two propositions.--The internal relations
between proposition and reference are portrayed by means of the internal relations between 'p' and "'p'". (Bad
remark.)
Page 23
Don't get involved in partial problems, but always take flight to where there is a free view over the whole
single great problem, even if this view is still not a clear one.
Page Break 24
Page 24
'A situation is thinkable' ('imaginable') means: We can make ourselves a picture of it. [3.001.]
Page 24
The proposition must determine a logical place.
Page 24
The existence of this logical place is guaranteed by the existence of the component parts alone, by the
existence of the significant proposition.
Page 24
Supposing there is no complex in the logical place, there is one then that is: not in that logical place. [Cf. 3.4.]
2.11.14.
Page 24
In the tautology the conditions of agreement with the world (the truth-conditions)--the representing
relations--cancel one another out, so that it does not stand in any representing relation to reality (says nothing). [Cf.
4.462.]
Page 24
a = a is not a tautology in the same sense as p ⊃ p.
Page 24
For a proposition to be true does not consist in its having a particular relation to reality but in its really
having a particular relation.
Page 24
Isn't it like this: the false proposition makes sense like the true and independently of its falsehood or truth,
but it has no reference? (Is there not here a better use of the word "reference"?)
Page 24
Could we say: As soon as I am given subject and predicate I am given a relation which will exist or not exist
between a subject-predicate proposition and its reference. As soon as I really know subject and predicate, I can also
know about the relation, which is an indispensable presupposition even for the case of the subject-predicate
proposition's being false.
3.11.14.
Page 24
In order for it to be possible for a negative situation to exist, the picture of the positive situation must exist.
[Cf. 5.5151.]
Page 24
The knowledge of the representing relation must be founded only on the knowledge of the component parts
of the situation!
Page 24
Then would it be possible to say: the knowledge of the subject predicate proposition and of subject and
predicate gives us the knowledge of an internal relation, etc.?
Page 24
Even this is not strictly correct since we do not need to know any particular subject or predicate.
Page Break 25
Page 25
It is evident that we feel the elementary proposition as the picture of a situation.--How is this? [Cf. 4.012.]
Page 25
Must not the possibility of the representing relation be given by the proposition itself?
Page 25
The proposition itself sunders what is congruent with it from what is not congruent.
Page 25
For example: if the proposition is given, and congruence, then the proposition is true if the situation is
congruent with it. Or: the proposition is given and non-congruence; then the proposition is true if the situation is not
congruent with it.
Page 25
But how is congruence or non-congruence or the like given us?
Page 25
How can I be told how the proposition represents? Or can this not be said to me at all? And if that is so can I
"know" it? If it was supposed to be said to me, then this would have to be done by means of a proposition; but the
proposition could only shew it.
Page 25
What can be said can only be said by means of a proposition, and so nothing that is necessary for the
understanding of all propositions can be said.
Page 25
That arbitrary correlation of sign and thing signified which is a condition of the possibility of the
propositions, and which I found lacking in the completely general propositions, occurs there by means of the
generality notation, just as in the elementary proposition it occurs by means of names. (For the generality notation
does not belong to the picture.) Hence the constant feeling that generality makes its appearance quite like an
argument. [Cf. 5.523.]
Page 25
Only a finished proposition can be negated. (And similarly for all ab-functions.)†1 [Cf. 4.064 and 4.0641.]
Page 25
The proposition is the logical picture of a situation.
Page 25
Negation refers to the finished sense of the negated proposition and not to its way of presenting. [Cf. 4.064
and 4.0641.]
Page 25
If a picture presents what-is-not-the-case in the forementioned way, this only happens through its presenting
that which is not the case.
Page 25
For the picture says, as it were: "This is how it is not", and to the question "How is it not?" just the positive
proposition is the answer.
Page Break 26
Page 26
It might be said: The negation refers to the very logical place which is determined by the negated proposition.
[See 4.0641.]
Page 26
Only don't lose the solid ground on which you have just been standing!
Page 26
The negating proposition determines a different logical place from the negated proposition. [See 4.0641.]
Page 26
The negated proposition not only draws the boundary between the negated domain and the rest; it actually
points to the negated domain.
Page 26
The negating proposition uses the logical place of the negated proposition to determine its own logical place.
By describing the latter as the place that is outside the former. [See 4.0641.]
Page 26
The proposition is true when what it images exists.
4.11.14.
Page 26
How does the proposition determine the logical place?
Page 26
How does the picture present a situation?
Page 26
It is after all itself not the situation, which need not be the case at all.
Page 26
One name is representative of one thing, another of another thing, and they themselves are connected; in this
way--like a tableau vivant--the whole images the situation. [Cf. 4.0311.]
Page 26
The logical connexion must, of course, be one that is possible as between the things that the names are
representatives of, and this will always be the case if the names really are representatives of the things. N.B. that
connexion is not a relation but only the holding of a relation.
5.11.14.
Page 26
In this way the proposition represents the situation--as it were off its own bat.
Page 26
But when I say: the connexion of the propositional components must be possible for the represented
things--does this not contain the whole problem? How can a non-existent connexion between objects be possible?
Page 26
"The connexion must be possible" means: The proposition and the components of the situation must stand
in a particular relation.
Page Break 27
Page 27
Then in order for a proposition to present a situation it is only necessary for its component parts to represent
those of the situation and for the former to stand in a connexion which is possible for the latter.
Page 27
The propositional sign guarantees the possibility of the fact which it presents (not, that this fact is actually the
case)--this holds for the general propositions too.
Page 27
For if the positive fact φa is given then so is the possibility of (x).φx, ~(∃x).φx, ~φa etc. etc. (All logical
constants are already contained in the elementary proposition.) [Cf. 5.47.]
Page 27
That is how the picture arises.--
Page 27
In order to designate a logical place with the picture we must attach a way of symbolizing to it (the positive,
the negative, etc.).
Page 27
We might, e.g., shew how not to fence by means of fencing puppets.
6.11.14.
Page 27
And it is just the same with this case as with ~φa, although the picture deals with what should not happen
instead of with what does not happen.
Page 27
The possibility of negating the negated proposition in its turn shews that what is negated is already a
proposition and not merely the preliminary to a proposition. [See 4.0641.]
Page 27
Could we say; Here is the picture, but we cannot tell whether it is right or not until we know what it is
supposed to say.
Page 27
The picture must now in its turn cast its shadow on the world.
7.11.14.
Page 27
Spatial and logical place agree in both being the possibility of an existence. [Cf. 3.411.]
8.11.14.
Page 27
What can be confirmed by experiment, in propositions about probability, cannot possibly be mathematics.
[Cf. 5.154.]
Page 27
Probability propositions are abstracts of scientific laws. [Cf. 5.156.]
Page 27
They are generalizations and express an incomplete knowledge of those laws. [Cf. 5.156.]
Page Break 28
Page 28
If, e.g., I take black and white balls out of an urn I cannot say before taking one out whether I shall get a
white or a black ball, since I am not well enough acquainted with the natural laws for that, but all the same I do
know that if there are equally many black and white balls there, the numbers of black balls that are drawn will
approach the number of white ones if the drawing is continued; I do know the natural laws as accurately as this. [Cf.
5.154]
9.11.14.
Page 28
Now what I know in probability statements are certain general properties of ungeneralized propositions of
natural science, such as, e.g., their symmetry in certain respects, and asymmetry in others, etc. [Cf. 5.156.]
Page 28
Puzzle pictures and the seeing of situations. [Cf. 5.5423.]
Page 28
It has been what I should like to call my strong scholastic feeling that has occasioned my best discoveries.
Page 28
"Not p" and "p" contradict one another, both cannot be true; but I can surely express both, both pictures
exist. They are to be found side by side.
Page 28
Or rather "p" and "~p" are like a picture and the infinite plane outside this picture. (Logical place.)
Page 28
I can construct the infinite space outside only by using the picture to bound that space.
10.11.14.
Page 28
When I say "p is possible", does that mean that "'p' makes sense"? Is the former proposition about language,
so that the existence of a propositional sign ("p") is essential for its sense? (In that case it would be quite
unimportant.) But does it not rather try to say what "p ∨ ~p" shews?
Page 28
Does not my study of sign language correspond to the study of the processes of thought, which philosophers
have always taken as so essential for philosophy of logic?--Only they always got involved in inessential
psychological investigations, and there is an analogous danger with my method too. [See 4.1121.]
11.11.14.
Page 28
Since "a = b" is not a proposition, nor "x = y" a function, a "class (x = x)" is a chimera, and so equally is
the so-called null class. (One
Page Break 29
did indeed always have the feeling that wherever x = x, a = a, etc. were used in the construction of sentences, in all
such cases one was only getting out of a difficulty by means of a swindle; as though one said "a exists" means
"(∃x)x = a".)
Page 29
This is wrong: since the definition of classes itself guarantees the existence of the real functions.
Page 29
When I appear to assert a function of the null class, I am saying that this function is true of all functions that
are null--and I can say that even if no function is null.
Page 29
Is x ≠ x. ≡x. φx identical with
(x).~φx? Certainly!
Page 29
The proposition points to the possibility that such and such is the case.
12.11.14.
Page 29
The negation is a description in the same sense as the elementary proposition itself.
Page 29
The truth of the proposition might be called possible, that of a tautology certain, and that of a contradiction
impossible. Here we already get the hint of a gradation that we need in the probability calculus. [Cf. 4.464.]
Page 29
In the tautology the elementary proposition does, of course, still portray, but it is so loosely connected with
reality that reality has unlimited freedom. Contradiction in its turn imposes such constraints that no reality can exist
under them.
Page 29
It is as if the logical constants projected the picture of the elementary proposition on to reality--which may
then accord or not accord with this projection.
Page 29
Although all logical constants must already occur in the simple proposition, its own peculiar proto-picture
must surely also occur in it whole and undivided.
Page 29
Then is the picture perhaps not the simple proposition, but rather its prototype which must occur in it?
Page 29
Then, this prototype is not actually a proposition (though it has the Gestalt of a proposition) and it might
correspond to Frege's "assumption".
Page Break 30
Page 30
In that case the proposition would consist of proto-pictures, which were projected on to the world.
13.11.14.
Page 30
In this work more than any other it is rewarding to keep on looking at questions, which one considers solved,
from another quarter, as if they were unsolved.
14.11.14.
Page 30
Think of the representation of negative facts by means of models. E.g.: two railway trains must not stand on
the rails in such-and-such a way. The proposition, the picture, the model are--in the negative sense--like a solid body
restricting the freedom of movement of others; in the positive sense, like the space bounded by solid substance, in
which there is room for a body. [Cf. 4.463.]
This image is very clear and must lead to the solution.
15.11.14.
Page 30
Projection of the picture on to reality.
(Maxwell's method of mechanical models.)
Page 30
Don't worry about what you have already written. Just keep on beginning to think afresh as if nothing at all
had happened yet.
Page 30
That shadow which the picture as it were casts upon the world: How am I to get an exact grasp of it?
Page 30
Here is a deep mystery.
Page 30
It is the mystery of negation: This is not how things are, and yet we can say how things are not.--
Page 30
For the proposition is only the description of a situation. (But this is all still only on the surface.) [Cf. 4.023.]
Page Break 31
Page 31
A single insight at the start is worth more than ever so many somewhere in the middle.
16.11.14
Page 31
The introduction of the sign "0" in order to make the decimal notation possible: the logical significance of this
procedure.
17.11.14.
Page 31
Suppose "φa" is true: what does it mean to say ~φa is possible? (φa is itself equivalent in meaning with
~(~φa).)
18.11.14.
Page 31
It is all simply a matter of the existence of the logical place. But what the devil is this "logical place"?!
19.11.14.
Page 31
The proposition and the logical co-ordinates: that is the logical place. [Cf. 3.41.]
20.11.14.
Page 31
The reality that corresponds to the sense of the proposition can surely be nothing but its component parts,
since we are surely ignorant of everything else.
Page 31
If the reality consists in anything else as well, this can at any rate neither be denoted nor expressed; for in the
first case it would be a further component, in the second the expression would be a proposition, for which the same
problem would exist in turn as for the original one.
21.11.14.
Page 31
What do I really know when I understand the sense of "φa" but do not know whether it is true or false? In
that case I surely know no more than φa ∨ ~φa; and that means I know nothing.
Page 31
As the realities corresponding to the sense of a proposition are only its component parts, the logical
co-ordinates too can only refer to these.
22.11.14.
Page 31
At this point I am again trying to express something that cannot be expressed.
23.11.14.
Page 31
Although the proposition must only point to a region of logical space, still the whole of logical space must
already be given by means of it.--
Page Break 32
Otherwise new elements--and in co-ordination--would keep on being introduced by means of negation, disjunction,
etc.; which, of course, must not happen. [Cf. 3.42.]
24.11.14.
Page 32
Proposition and situation are related to one another like the yardstick and the length to be measured.
Page 32
That the proposition "φa" can be inferred from the proposition "(x).φx" shews how generality is present
even in the sign "(x).φx".
Page 32
And the same thing, of course, holds for any generality notation.
Page 32
In the proposition we hold a proto-picture up against reality.
Page 32
(When investigating negative facts one keeps on feeling as if they presupposed the existence of the
propositional sign.)
Page 32
Must the sign of the negative proposition be constructed by means of the sign of the positive one? (I believe
so.)
Page 32
Why shouldn't one be able to express the negative proposition by means of a negative fact? It's as if one
were to take the space outside the yardstick as the object of comparison instead of the yardstick. [Cf. 5.5151.]
Page 32
How does the proposition "~p" really contradict the proposition "p"? The internal relations of the two signs
must mean contradiction.
Page 32
Of course it must be possible to ask whenever we have a negative proposition: What is it that is not the case?
But the answer to this is, of course, in its turn only a proposition. (This remark incomplete.)
25.11.14.
Page 32
That negative state of affairs that serves as a sign can, of course, perfectly well exist without a proposition
that in turn expresses it.
Page 32
In investigating these problems it's constantly as if they were already solved, an illusion which arises from the
fact that the problems often quite disappear from our view.
Page 32
I can see that ~φa is the case just by observing φzxq and a.
Page 32
The question here is: Is the positive fact primary, the negative secondary, or are they on the same level? And
if so, how is it with the facts p ∨ q, p ⊃ q, etc.? Aren't these on the same level as ~p? But then
Page Break 33
must not all facts be on the same level? The question is really this: fire there facts besides the positive ones? (For it is
difficult not to confuse what is not the case with what is the case instead of it.)
Page 33
It is clear that all the ab-functions are only so many different methods for measuring reality.--And certainly
the methods of measurement by means of p and ~p have some special advantage over all others.--
Page 33
It is the dualism, positive and negative facts, that gives me no peace. For such a dualism can't exist. But how
to get away from it?
Page 33
All this would get solved of itself if we understood the nature of the proposition.
26.11.14.
Page 33
If all the positive statements about a thing are made, aren't all the negative ones already made too? And that
is the whole point.
Page 33
The dualism of positive and negative that I feared does not exist, for (x).φx, etc. etc. are neither positive nor
negative.
Page 33
If the positive proposition does not have to occur in the negative, mustn't at any rate the proto-picture of the
positive proposition occur in the negative one?
Page 33
By making a distinction--as we do in any possible notation--between ~aRb and ~bRa we presuppose in any
notation a particular correlation between argument and argument place in the negative proposition; the correlation
gives the prototype of the negated positive proposition.
Page 33
Then is that correlation of the components of the proposition by means of which nothing is yet said the real
picture in the proposition?
Page 33
Doesn't my lack of clarity rest on a lack of understanding of the nature of relations?
Page 33
Can one negate a picture? No. And in this lies the difference between picture and proposition. The picture
can serve as a proposition. But in that case something gets added to it which brings it about that now it says
something. In short: I can only deny that the picture is right, but the picture I cannot deny.
Page 33
By my correlating the components of the picture with objects, it
Page Break 34
comes to represent a situation and to be right or wrong. (E.g., a picture represents the inside of a room, etc.)
27.11.14.
"~p" is true when p is false. So part of the true proposition "~p" is a false proposition. How can the
mere twiddle "~" bring it into agreement with reality? We have, of course, already said that it is not the
twiddle "~" alone but everything that is common to the different signs of negation. And what is
common to all these must obviously proceed from the meaning of negation itself. And so in this way
the sign of negation must surely mirror its own reference. [Cf. 5.512.]
28.11.14.
Page 34
Negation combines with the ab-functions of the elementary proposition. And the logical functions of the
elementary proposition must mirror their reference, just as much as all the others.
29.11.14.
Page 34
The ab-function does not stop short of the elementary proposition but penetrates it.
Page 34
What can be shewn cannot be said. [4.1212.]
Page 34
I believe that it would be possible wholly to exclude the sign of identity from our notation and always to
indicate identity merely by the identity of the signs (and conversely). In that case, of course, φ(a,a) would not be a
special case of (x,y).φ(x,y), and φa would not be a special case of (∃x,y).φx.φy. But then instead of φx.φy ⊃x,y x = y
one could simply write ~(∃x,y).φx.φy. [Cf. 5.53 and 5.533]
Page 34
By means of this notation the pseudo-proposition (x)x = a or the like would lose all appearances of
justification. [Cf. 5.534.]
1.12.14.
Page 34
The proposition says as it were: This picture cannot (or can) present a situation in this way.
2.12.14.
Page 34
It all depends on settling what distinguishes the proposition from the mere picture.
4.12.14.
Page 34
Let us look at the identity ~~p = p: this, together with others, determines the sign for p, since it says that
there is something that "p"
Page Break 35
and "~~p" have in common. Through this that sign acquires properties which mirror the fact that double negation is
an affirmation.
5.12.14.
Page 35
How does "p ∨ ~p" say nothing?
6.12.14.
Page 35
Newtonian mechanics brings the description of the world into a unitary form. Let us imagine a white surface
with irregular black spots on it. We now say: Whatever sort of picture arises in this way, I shall always be able to
approximate as close as I like to its description by covering the surface with a suitably fine square network and
saying of each square that it is white or is black. In this way I shall have brought the description of this surface into a
unitary form. This form is arbitrary, for I could with equal success have used a triangular or hexagonal net. It may be
that the description by means of a triangular net would have been simpler, i.e. that we could have given a more
accurate description of the surface with a coarser triangular net than with a finer square one (or vice versa), etc.
Different systems of describing the world correspond to the different nets. Mechanics determines the form of
description of the world by saying: All propositions in a description of the world must be capable of being got in a
given way from a number of given propositions--the axioms of mechanics. In this way it supplies the stones for
building up natural science and says: Whatever building you want to erect you must construct it somehow with
these and only these stones.
Page 35
Just as it must be possible to write down any arbitrary number by means of the system of numbers, so it
must be possible to write down any arbitrary proposition of physics by means of the system of mechanics.
Page 35
[6.341.]
Page 35
And here we see the relative position of logic and mechanics.
Page 35
(One might also allow the net to consist of a variety of figures.)
Page 35
The fact that a configuration like that mentioned above can be described by means of a net of a given form
asserts nothing about the configuration (for this holds for any such configuration). What does characterize the
configuration however is that it can be described by means of a particular net of a particular degree of fineness. In
this way too it tells us nothing about the world that it can be described by means of Newtonian mechanics; but it
does tell us something that it can be
Page Break 36
described by means of Newtonian mechanics in the way that it actually can. (This I have felt for a long time.)--It also
asserts something about the world, that it can be described more simply by means of one mechanics than by means
of another.
Page 36
[Cf. 6.342.]
Page 36
Mechanics is one attempt to construct all the propositions that we need for the description of the world
according to a single plan. (Hertz's invisible masses.) [Cf. 6.343.]
Page 36
Hertz's invisible masses are admittedly pseudo-objects.
7.12.14.
Page 36
The logical constants of the proposition are the conditions of its truth.
8.12.14.
Page 36
Behind our thoughts, true and false, there is always to be found a dark background, which we are only later
able to bring into the light and express as a thought.
12.12.14.
Page 36
p. Taut = p, i.e. Taut says nothing. [Cf. 4.465.]
13.12.14.
Page 36
Does it exhaust the nature of negation that it is an operation cancelling itself? In that case χ would have to
stand for negation if χχp = p, assuming that χp ≠ p.
Page 36
This for one thing is certain, that according to these two equations χ can no longer express affirmation.
Page 36
And does not the capacity which these operations have of vanishing shew that they are logical?
15.12.14.
Page 36
It is obvious that we can introduce whatever we like as the written signs of the ab-function, the real sign will
form itself automatically. And when this happens what properties will be formed of themselves?
Page 36
The logical scaffolding surrounding the picture (in the proposition) determines logical space. [Cf. 3.42.]
16.12.14.
Page 36
The proposition must reach out through the whole of logical space. [Cf. 3.42.]
Page Break 37
17.12.14.
Page 37
The signs of the ab-function are not material, otherwise they could not vanish. [Cf. 5.44 and 5.441.]
18.12.14.
Page 37
It must be possible to distinguish just as much in the real propositional sign as can be distinguished in the
situation. This is what their identity consists in. [Cf. 4.04.]
20.12.14.
Page 37
In "p" neither more nor less can be recognized than in "~p".
Page 37
How can a situation agree with "p" and not agree with "~p"?
Page 37
The following question might also be asked: If I were to try to invent Language for the purpose of making
myself understood to someone else, what sort of rules should I have to agree on with him about our expression?
23.12.14.
Page 37
A characteristic example for my theory of the significance of descriptions in physics: The two theories of
heat; heat conceived at one time as a stuff, at another time as a movement.
25.12.14.
Page 37
The proposition says something, is identical with: It has a particular relation to reality, whatever this may be.
And if this reality is given and also that relation, then the sense of the proposition is known. "p∨q" has a different
relation to reality from "p.q", etc.
Page 37
The possibility of the proposition is, of course, founded on the principle of signs as GOING PROXY for
objects. [Cf. 4.0312.]
Page 37
Thus in the proposition something has something else as its proxy.
Page 37
But there is also the common cement.
Page 37
My fundamental thought is that the logical constants are not proxies. That the logic of the fact cannot have
anything as its proxy. [See 4.0312.]
29.12.14.
Page 37
In the proposition the name goes proxy for the object. [3.22.]
11.1.15.
Page 37
A yardstick does not say that an object that is to be measured is one yard long.
Page Break 38
Page 38
Not even when we know that it is supposed to serve for the measurement of this particular object.
Page 38
Could we not ask: What has to be added to that yardstick in order for it to assert something about the length
of the object?
Page 38
(The yardstick without this addition would be the "assumption".)
15.1.15.
Page 38
The propositional sign "p ∨ q" is right if p is the case, if q is the case and if both are the case, otherwise it is
wrong. This seems to be infinitely simple; and the solution will be as simple as this.
16.1.15.
Page 38
The proposition is correlated with a hypothetical situation.
Page 38
This situation is given by means of its description.
Page 38
The proposition is the description of a situation. [See 4.023.]
Page 38
As the description of an object describes it by its external properties, so the proposition describes the fact by
its internal properties. [See 4.023.]
Page 38
The description is right if the object has the asserted property: the proposition is right if the situation has the
internal properties given by the proposition.
17.1.15.
Page 38
The situation p.q falls under the proposition "p ∨ q".
Page 38
On the analogy of the net for physics: although the spots are geometrical figures, all the same geometry can,
of course, say nothing at all about their form and position. The net, however, is purely geometrical, all its properties
can be given a priori. [See 6.35.]
18.1.15.
Page 38
The comparison between proposition and description is purely logical and for that reason must be carried
farther.
20.1.15.
Page 38
How is it that all is a logical concept?
Page 38
How is it that all is a concept of form?
Page 38
How does it come about that all can occur in any proposition?
Page Break 39
Page 39
For that is the characteristic mark of the concept of a form.
Page 39
All APPEARS to be nearer to the content of the proposition than to the form.
Page 39
All: things, All: functions, All: relations: it is as if All were a connecting term between the concept of the
thing, of functions, etc., and the individual thing, the individual functions.
Page 39
Generality is essentially connected with the elementary FORM.
Page 39
The key word--?
21.1.15.
Page 39
The transition from the general consideration of the propositional form: infinitely difficult, fantastic.
22.1.15.
Page 39
My whole task consists in explaining the nature of the proposition.
Page 39
That is to say, in giving the nature of all facts, whose picture the proposition is.
Page 39
In giving the nature of all being.
Page 39
(And here Being does not mean existing--in that case it would be nonsensical.)
23.1.15.
Page 39
Negation is an operation. [Cf. 5.2341.]
Page 39
An operation denotes an operation.
Page 39
Words are probes; some reach very deep; some only to a little depth.
Page 39
An operation, of course, does not say anything, only its result does; and this depends on its object. [Cf.
5.25.]
24.1.15.
Page 39
The logical pseudo-functions are operations.
Page 39
Only operations can vanish!
Page 39
The negative proposition excludes reality.
Page 39
How can the all-embracing world-mirroring logic make use of such special twiddles and manipulations? Only
by all these being linked together to form one infinitely fine network, to form the great mirror. [5.511.]
Page Break 40
25.1.15.
Page 40
We can also say: ~p is false, when p is true.
29.1.15.
Page 40
Language is articulated. [Cf. 3.141.]
7.2.15.
Page 40
Musical themes are in a certain sense propositions. Knowledge of the nature of logic will for this reason lead
to knowledge of the nature of music.
14.2.15.
Page 40
If there were mathematical objects--logical constants--the proposition "I am eating five plums" would be a
proposition of mathematics. And it is not even a proposition of applied mathematics.
Page 40
The proposition must describe its reference completely. [Cf. 4.023.]
4.3.15.
Page 40
A tune is a kind of tautology, it is complete in itself; it satisfies itself.
5.3.15.
Page 40
Mankind has always had an inkling that there must be a sphere of questions where the answers must--a
priori--be arranged symmetrically, and united into a complete regular structure. [See 5.4541.]
Page 40
(The older a word, the deeper it reaches.)
6.3.15.
Page 40
The problems of negation, of disjunction, of true and false, are only reflections of the one great problem in
the variously placed great and small mirrors of philosophy.
7.3.15.
Page 40
Just as ~ξ, ~ξ ∨ ~ξ etc. are the same function, so too are ~η ∨ η, η ⊃ η, etc. the same--that is, the
tautological--function. Just as the others can be investigated, so can it--and perhaps with advantage.
8.3.15.
Page 40
My difficulty is only an--enormous--difficulty of expression.
18.3.15.
Page 40
It is clear that the closest examination of the propositional sign cannot yield what it asserts--what it can yield
is what it is capable of asserting.
Page Break 41
27.3.15.
Page 41
The picture can replace a description.
29.3.15.
Page 41
The law of causality is not a law but the form of a law. [Cf. 6.32.]
Page 41
"Law of causality" is a class name. And just as in mechanics--let us say--there are minimum laws--e.g., that
of least action-so in physics there is A law of causality, a law of the causality form. [Cf. 6.321.]
Page 41
Just as men also had an inkling of the fact that there must be a "law of least action", before precisely knowing
how it ran.
Page 41
(Here, as so often happens, the a priori turns out to be something purely logical.)
Page 41
[Cf. 6.3211.]
3.4.15.
Page 41
The proposition is a measure of the world.
Page 41
This is the picture of a process and is wrong. In that case how can it still be a picture of that process?
Page 41
"a" can go proxy for a and "b" can go proxy for b when "a" stands in the relation "R" to "b": this is what that
POTENTIAL internal relation that we are looking for consists in.
5.4.15.
Page 41
The proposition is not a blend of words. [See 3.141.]
11.4.15.
Page 41
Nor is a tune a blend of notes, as all unmusical people think. [Cf. 3.141.]
12.4.15.
Page 41
I cannot get from the nature of the proposition to the individual logical operations!!!
15.4.15.
Page 41
That is, I cannot bring out how far the proposition is the picture of the situation.
Page 41
I am almost inclined to give up all my efforts.---- ----
16.4.15.
Page 41
Description is also, so to speak, an operation with the means of description as its basis, and with the
described object as its result.
Page Break 42
Page 42
The sign "not" is the class of all negating signs.
17.4.15.
Page 42
The subjective universe.
Page 42
Instead of performing the logical operations in the proposition upon its component propositions, we can
correlate marks with these and operate with them. In that case a single propositional formation has correlated with it
a constellation of marks which is connected with it in a most complicated way.
Page 42
(aRb, cSd, φe) ((p ∨ q).r: ⊃ q.r. ≡.p ∨ r)
Page 42
p q r
18.4.15.
Page 42
The transition from p to ~p is not what is characteristic of the operation of negation. (The best proof of this:
negation also leads from ~p to p.)----------.
19.4.15.
Page 42
What is mirrored in language I cannot use language to express. [Cf. 4.121.]
23.4.15.
Page 42
We do not believe a priori in a law of conservation, we know a priori the possibility of its logical form.
[6.33.]
Page 42
All those propositions which are known a priori, like the principle of sufficient reason, of continuity in
nature, etc., etc., all these are a priori insights relating to the possible ways of forming the propositions of natural
science. [Cf. 6.34.]
Page 42
"Ockham's razor" is, of course, not an arbitrary rule or one justified by its practical success. What it says is
that unnecessary sign-units mean nothing. [See 5.47321.]
Page 42
It is clear that signs fulfilling the same purpose sire logically identical. The purely logical thing just is what all
of these are capable of accomplishing [Cf. 5.47321.]
24.4.15.
Page 42
In logic (mathematics) process and result are equivalent. (Hence no surprises.) [6.1261.]
25.4.15.
Page 42
Since language stands in internal relations to the world, it and these relations determine the logical
possibility of facts. If we have a significant
Page Break 43
sign it must stand in a particular internal relation to a structure. Sign and relation determine unambiguously the
logical form of the thing signified.
Page 43
But cannot any so-called thing be correlated in one and the same way with any other such?
Page 43
It is, for example, quite clear that the separate words of language are--experienced and--used as logically
equivalent units.
Page 43
It always seems as if there were something that one can regard as a thing, and on the other hand real
simple things.
Page 43
It is clear that neither a pencil-stroke nor a steamship is simple. Is there really a logical equivalence between
these two?
Page 43
"Laws" like the law of sufficient reason, etc. deal with the network not with what the network describes. [See
6.35.]
26.4.15.
Page 43
It must be through generality that ordinary propositions get their stamp of simplicity.
Page 43
We must recognize how language takes care of itself.
Page 43
The proposition that is about a complex stands in internal relation to the proposition about its component
part. [See 3.24.]
27.4.15.
Page 43
The freedom of the will consists in the fact that future events cannot be KNOWN now. It would only be
possible for us to know them, if causality were an INNER necessity--like, say, that of logical inference.--The
connexion between knowledge and thing known is the connexion of logical necessity. [See 5.1362.]
Page 43
I cannot need to worry about language.
Page 43
Non-truth is like non-identity.
28.4.15.
Page 43
The operation of negating does not consist in, say, putting down a ~, but in the class of all negating
operations.
Page 43
But in that case what really are the properties of this ideal negating operation?
Page 43
How does it come out that two assertions are compatible?
Page 43
If one puts p instead of q in p ∨ q the statement turns into p.
Page Break 44
Page 44
Does the sign p.q also belong among those which assert p?--Is p one of the signs for p∨q?
Page 44
Can one say the following?: All signs that do not assert p, are not asserted by p and do not contain p as
tautology or contradiction does--all these signs negate p.
29.4.15.
Page 44
That is to say: All signs that are dependent on p and that neither assert p nor are asserted by p.
30.4.15.
Page 44
The occurrence of an operation cannot, of course, have any import by itself.
Page 44
p is asserted by all propositions from which it follows. [5.124.]
Page 44
Every proposition that contradicts p negates p. [See 5.1241.]
1.5.15.
Page 44
The fact that p.~p is a contradiction shews that ~p contradicts p. [Cf. 6.1201.]
Page 44
Scepticism is not irrefutable, but obvious nonsense if it tries to doubt where no question can be asked. [See
6.51.]
Page 44
For doubt can only exist where a question exists; a question can only exist where an answer exists, and this
can only exist where something can be said. [See 6.51.]
Page 44
All theories that say: "This is how it must be, otherwise we could not philosophize" or "otherwise we surely
could not live", etc. etc., must of course disappear.
Page 44
My method is not to sunder the hard from the soft, but to see the hardness of the soft.
Page 44
It is one of the chief skills of the philosopher not to occupy himself with questions which do not concern
him.
Page 44
Russell's method in his "Scientific method in philosophy" is simply a retrogression from the method of
physics.
2.5.15.
Page 44
The class of all signs that assert both p and q is the sign for p.q. The class of all signs which assert either p or
q is the proposition "p ∨ q". [Cf. 5.513.]
Page Break 45
3.5.15.
Page 45
We cannot say that both tautology and contradiction say nothing in the sense that they are both, say, zero
points in a scale of propositions. For at least they are opposite poles.
Page 45
Can we say: two propositions are opposed to one another when there is no sign that asserts them
both--which really means: when they have no common member? [Cf. 5.1241.]
Page 45
Thus propositions are imagined as classes of signs--the propositions "p" and "q" have the member "p.q" in
common--and two propositions are opposed to one another when they lie quite outside one another. [Cf. 5.513.]
4.5.15.
Page 45
The so-called law of induction cannot at any rate be a logical law, for it is evidently a proposition. [See 6.31.]
Page 45
The class of all propositions of the form Fx is the proposition (x) φx.
5.5.15.
Page 45
Does the general form of proposition exist?
Page 45
Yes, if by that is understood the single "logical constant". [Cf. 5.47.]
Page 45
It keeps on looking as if the question "Are there simple things?" made sense. And surely this question must
be nonsense!----
6.5.15.
Page 45
It would be vain to try and express the pseudo-sentence "Are there simple things?" in symbolic notation.
Page 45
And yet it is clear that I have before me a concept of a thing, of simple correlation, when I think about this
matter.
Page 45
But how am I imagining the simple? Here all I can say is always "'x' has reference".--Here is a great riddle!
Page 45
As examples of the simple I always think of points of the visual field (just as parts of the visual field always
come before my mind as typical composite objects).
7.5.15.
Page 45
Is spatial complexity also logical complexity? It surely seems to be.
Page 45
But what is a uniformly coloured part of my visual field composed of? Of minima sensibilia? How should
the place of one such be determined?
Page Break 46
Page 46
Even if the sentences which we ordinarily use all contain generalizations, still there must surely occur in them
the proto-pictures of the component parts of their special cases. Thus the question remains how we arrive at those.
8.5.15.
Page 46
The fact that there is no sign for a particular proto-picture does not show that that proto-picture is not
present. Portrayal by means of sign language does not take place in such a way that a sign of a proto-picture goes
proxy for an object of that proto-picture. The sign and the internal relation to what is signified determine the
proto-picture of the latter; as the fundamental co-ordinates together with the ordinates determine the points of a
figure.
9.5.15.
Page 46
A question: can we manage without simple objects in LOGIC?
Page 46
Obviously propositions are possible which contain no simple signs, i.e. no signs which have an immediate
reference. And these are really propositions making sense, nor do the definitions of their component parts have to
be attached to them.
Page 46
But it is clear that components of our propositions can be analysed by means of a definition, and must be, if
we want to approximate to the real structure of the proposition. At any rate, then, there is a process of analysis.
And can it not now be asked whether this process comes to an end? And if so: What will the end be?
Page 46
If it is true that every defined sign signifies via its definitions then presumably the chain of definitions must
some time have an end. [Cf. 3.261.]
Page 46
The analysed proposition mentions more than the unanalysed.
Page 46
Analysis makes the proposition more complicated than it was, but it cannot and must not make it more
complicated than its meaning was from the first.
Page 46
When the proposition is just as complex as its reference, then it is completely analysed.
Page 46
But the reference of our propositions is not infinitely complicated.
Page 46
The proposition is the picture of the fact. I can devise different pictures of a fact. (The logical operations serve
this purpose.) But
Page Break 47
what is characteristic of the fact will be the same in all of these pictures and will not depend on me.
Page 47
With the class of signs of the proposition "p" the class "~p", etc. etc. is already given. As indeed is necessary.
Page 47
But does not that of itself presuppose that the class of all propositions is given us? And how do we arrive at
it?
11.5.15.
Page 47
Is the logical sum of two tautologies a tautology in the first sense? Is there really such a thing as the duality:
tautology--contradiction?
Page 47
The simple thing for us is: the simplest thing that we are acquainted with.----The simplest thing which our
analysis can attain--it need appear only as a prototype, as a variable in our propositions----that is the simple thing
that we mean and look for.
12.5.15.
Page 47
The general concepts (a) of portrayal and (b) of co-ordinates.
Page 47
Supposing that the expression "~(∃x)x = x" were a proposition, namely (say), this one: "There are no
things", then it would be matter for great wonder that, in order to express this proposition in symbols, we had to
make use of a relation (=) with which it was really not concerned at all.
13.5.15.
Page 47
A singular logical manipulation, the personification of time!
Page 47
Just don't pull the knot tight before being certain that you have got hold of the right end.
Page 47
Can we regard a part of space as a thing? In a certain sense we obviously always do this when we talk of
spatial things.
Page 47
For it seems--at least so far as I can see at present-that the matter is not settled by getting rid of names by
means of definitions: complex spatial objects, for example, seem to me in some sense to be essentially things--I as it
were see them as things.--And the designation of them by means of names seems to be more than a mere trick of
language. Spatial complex objects--for example--really, so it seems, do appear as things.
Page 47
But what does all that signify?
Page Break 48
Page 48
At any rate that we quite instinctively designate those objects by means of names.--
14.5.15.
Page 48
Language is a part of our organism, and no less complicated than it. [Cf. 4.002.]
Page 48
The old problem of complex and fact!
15.5.15.
Page 48
The theory of the complex is expressed in such propositions as: "If a proposition is true then Something
exists"; there seems to be a difference between the fact expressed by the proposition: a stands in the relation R to b,
and the complex: a in the relation R to b, which is just that which "exists" if that proposition is true. It seems as if
we could designate this Something, and what's more with a real "complex sign".--The feelings expressed in these
sentences are quite natural and unartificial, so there must be some truth at the bottom of them. But what truth?
Page 48
What depends on my life?
Page 48
So much is clear, that a complex can only be given by means of its description; and this description will hold
or not hold. [See 3.24.]
Page 48
The proposition dealing with a complex will not be nonsensical if the complex does not exist, but simply
false. [See 3.24.]
16.5.15.
Page 48
When I see space do I see all its points?
Page 48
It is no more possible to present something "contradicting logic" in language than to present a figure
contradicting the laws of space in geometry by means of its coordinates, or, say, to give the coordinates of a point
that does not exist. [3.032.]
Page 48
If there were propositions asserting the existence of proto-pictures they would be unique and would be a
kind of "logical propositions" and the set of these propositions would give logic an impossible reality. There would
be co-ordination in logic.
18.5.15.
Page 48
The possibility of all similes, of the whole pictorial character of our language, is founded in the logic of
portrayal. [4.015.]
Page Break 49
19.5.15.
Page 49
We can even conceive a body apprehended as in movement, and together with its movement, as a thing. So
the moon circling round the earth moves round the sun. Now here it seems clear that this reification is nothing but a
logical manipulation--though the possibility of this may be extremely significant.
Page 49
Or let us consider reifications like: a tune, a spoken sentence.--
Page 49
When I say "'x' has reference" do I have the feeling: "it is impossible that "x" should stand for, say, this knife
or this letter"? Not at all. On the contrary.
20.5.15.
Page 49
A complex just is a thing!
21.5.15.
Page 49
We can quite well give a spatial representation of a set of circumstances which contradict the laws of physics,
but not of one contradicting the laws of geometry. [3.0321.]
22.5.15.
Page 49
The mathematical notation for infinite series like
"1 + x/1! + x2/2! +....."
together with the dots is an example of that extended generality. A law is given and the terms that are written down
serve as an illustration.
Page 49
In this way instead of (x)fx one might write "fx.fy....".
Page 49
Spatial and temporal complexes.
23.5.15.
Page 49
The limits of my language mean the limits of my world. [5.6]
Page 49
There really is only one world soul, which I for preference call my soul and as which alone I conceive what I
call the souls of others.
Page 49
The above remark gives the key for deciding the way in which solipsism is a truth. [See 5.62.]
Page 49
I have long been conscious that it would be possible for me to write a book: "The world I found". [Cf. 5.631.]
Page 49
The feeling of the simple relation which always comes before our mind as the main ground for the
assumption of "simple objects"--
Page Break 50
haven't we got this very same feeling when we think of the relation between name and complex object?
Page 50
Suppose the complex object is this book. Let it be called "A". Then surely the occurrence of "A" in the
proposition shews the occurrence of the book in the fact. For it is not arbitrarily resolved even when it is analysed,
so as, e.g., to make its resolution a completely different one in each propositional formation.----[See 3.3442.]
Page 50
And like the occurrence of the name of a thing in different propositions, the occurrence of the name of
compounded objects shews that there is a form and a content in common.
Page 50
In spite of this the infinitely complex situation seems to be a chimera.
Page 50
But it also seems certain that we do not infer the existence of simple objects from the existence of particular
simple objects, but rather know them--by description, as it were--as the end-product of analysis, by means of a
process that leads to them.
Page 50
For the very reason, that a bit of language is nonsensical, it is still possible to go on using it--see the last
remark.
Page 50
In the book "The world I found" I should also have to report on my body and say which members are
subject to my will, etc. For this is a way of isolating the subject, or rather of skewing that in an important sense there
is no such thing as the subject; for it would be the one thing that could not come into this book. [See 5.631.]
24.5.15.
Page 50
Even though we have no acquaintance with simple objects we do know complex objects by acquaintance,
we know by acquaintance that they are complex.--And that in the end they must consist of simple things?
Page 50
We single out a part of our visual field, for example, and we see that it is always complex, that any part of it
is still complex but is already simpler, and so on----.
Page 50
Is it imaginable that--e.g.--we should see that all the pointy of a surface are yellow, without seeing any
single point of this surface? It almost seems to be so.
Page 50
The way problems arise: the pressure of a tension which then concentrates into a question, and becomes
objective.
Page Break 51
Page 51
How should we describe, e.g., a surface uniformly covered with blue?
25.5.15.
Page 51
Does the visual image of a minimum visibile actually appear to us as indivisible? What has extension is
divisible. Are there parts in our visual image that have no extension? E.g., the images of the fixed stars?--
Page 51
The urge towards the mystical comes of the non-satisfaction of our wishes by science. We feel that even if all
possible scientific questions are answered our problem is still not touched at all. Of course in that case there are no
questions any more; and that is the answer. [Cf. 6.52.]
Page 51
The tautology is asserted, the contradiction denied, by every proposition. (For one could append 'and' and
some tautology to any proposition without altering its sense; and equally the negation of a contradiction.)
Page 51
And "without altering its sense" means: without altering the essential thing about the sign itself. For: the sign
cannot be altered without altering its sense. [Cf. 4.465.]
Page 51
"aRa" must make sense if "aRb" makes sense.
26.5.15.
Page 51
But how am I to explain the general nature of the proposition now? We can indeed say: everything that is
(or is not) the case can be pictured by means of a proposition. But here we have the expression "to be the case"! It is
just as problematic.
Page 51
Objects form the counterpart to the proposition.
Page 51
Objects I can only name. Signs go proxy for them. [See 3.221.]
27.5.15.
Page 51
I can only speak of them, I cannot express them. [See 3.221.]
Page 51
"But might there not be something which cannot be expressed by a proposition (and which is also not an
object)?" In that case this could not be expressed by means of language; and it is also impossible for us to ask about
it.
Page 51
Suppose there is something outside the facts? Which our propositions are impotent to express? But here we
do have, e.g., things and we feel no demand at all to express them in propositions.
Page Break 52
Page 52
What cannot be expressed we do not express----. And how try to ask whether THAT can be expressed
which cannot be EXPRESSED?
Page 52
Is there no domain outside the facts?
28.5.15.
Page 52
"Complex sign" and "proposition" are equivalent.
Page 52
Is it a tautology to say: Language consists of sentences?
Page 52
It seems it is.
29.5.15.
Page 52
But is language: the only language?
Page 52
Why should there not be a mode of expression through which I can talk about language in such a way that it
can appear to me in co-ordination with something else?
Page 52
Suppose that music were such a mode of expression: then it is at any rate characteristic of science that no
musical themes can occur in it.
Page 52
I myself only write sentences down here. And why?
Page 52
How is language unique?
30.5.15.
Page 52
Words are like the film on deep water.
Page 52
It is clear that it comes to the same thing to ask what a sentence is, and to ask what a fact is--or a complex.
Page 52
And why should we not say: "There are complexes; one can use names to name them, or propositions to
portray them"?
Page 52
The name of a complex functions in the proposition like the name of an object that I only know by
description.----The proposition that depicts it functions as a description.
Page 52
But if there are simple objects, is it correct to call both the signs for them and those other signs "names"?
Page 52
Or is "name" so to speak a logical concept?
Page 52
"It signalises what is common to a form and a content."----
Page 52
According to the difference in the structure of the complex its name denotes in a different way and is subject
to different syntactical laws.
Page Break 53
Page 53
The mistake in this conception must lie in its, on the one hand, contrasting complexes and simple objects,
while on the other hand it treats them as akin.
Page 53
And yet: Components and complex seem to be akin, and to be opposed to one another.
Page 53
(Like the plan of a town and the map of a country which we have before us, the same size and on different
scales.)
Page 53
What is the source of the feeling "I can correlate a name with all that I see, with this landscape, with the
dance of motes in the air, with all this; indeed, what should we call a name if not this"?!
Page 53
Names signalise what is common to a single form and a single content.--Only together with their syntactical
use do they signalise one particular logical form. [Cf. 3.327.]
31.5.15.
Page 53
One cannot achieve any more by using names in describing the world than by means of the general
description of the world!!!
Page 53
Could one then manage without names? Surely not.
Page 53
Names are necessary for an assertion that this thing possesses that property and so on.
Page 53
They link the propositional form with quite definite objects.
Page 53
And if the general description of the world is like a stencil of the world, the names pin it to the world so that
the world is wholly covered by it.
1.6.15.
Page 53
The great problem round which everything that I write turns is: Is there an order in the world a priori, and if
so what does it consist in?
Page 53
You are looking into fog and for that reason persuade yourself that the goal is already close. But the fog
disperses and the goal is not yet in sight.
2.6.15.
Page 53
I said: "A tautology is asserted by every proposition"; but that is not enough to tell us why it is not a
proposition. For has it told us why a proposition cannot be asserted by p and ~p?
Page 53
For my theory does not really bring it out that the proposition must have two poles.
Page Break 54
Page 54
For what I should now have to do is to find an expression in the language of this theory for HOW MUCH a
proposition says. And this would have to yield the result that tautologies say NOTHING.
Page 54
But how can we find the measure of amount-that-is-said?
Page 54
At any rate it is there; and our theory must be able to give it expression.
3.6.15.
Page 54
One could certainly say: That proposition says the most from which the most follows.
Page 54
Could one say: "From which the most mutually independent propositions follow"?
Page 54
But doesn't it work like this: If p follows from q but not q from p, then q says more than p?
Page 54
But now nothing at all follows from a tautology.----It however follows from every proposition. [Cf. 5.142.]
Page 54
The analogous thing holds of its opposite.
Page 54
But then! Won't contradiction now be the proposition that says the most? From "p.~p" there follows not
merely "p" but also "~p"! Every proposition follows from them and they follow from none!? But I surely can't infer
anything from a contradiction, just because it is a contradiction.
Page 54
But if contradiction is the class of all propositions, then tautology becomes what is common to any classes
of propositions that have nothing in common and vanishes completely. [Cf. 5.143.]
Page 54
"p ∨ ~p" would then be a sign only in appearance. But in reality the dissolution of the proposition.
Page 54
The tautology as it were vanishes inside all propositions, the contradiction outside all propositions. [See
5.143.]
Page 54
In these investigations I always seem to be unconsciously taking the elementary proposition as my starting
point.--
Page 54
Contradiction is the outer limit of propositions; no proposition asserts it. Tautology is their substanceless
centre. (The middle point of a circle can be conceived as its inner boundary.) [Cf. 5.143.]
Page 54
(The key word still hasn't yet been spoken.)
Page Break 55
Page 55
The thing is that here it is very easy to confuse the logical product and the logical sum.
Page 55
For we come to the apparently remarkable result that two propositions must have something in common in
order to be capable of being asserted by one proposition.
Page 55
(Belonging to a single class, however, is also something that propositions can have in common.)
Page 55
(Here there is still a definite and decisive lack of clarity in my theory. Hence a certain feeling of
dissatisfaction')
4.6.15.
Page 55
"p.q" only makes sense if "p ∨ q" makes sense.
5.6.15.
Page 55
"p.q" asserts "p" and "q" but that surely does not mean that "p.q" is the common component of "p" and "q",
but on the contrary that "p" and also "q" are equally contained in "p.q".
Page 55
In this sense p and ~p would have something in common, for example propositions like ~p ∨ q and p ∨ q.
That is: there are indeed propositions which are asserted by "p" as well as by "~p"--e.g. the above ones--but there are
none that assert p as well as also asserting ~p.
Page 55
In order for a proposition to be capable of being true it must also be capable of being false.
Page 55
Why does tautology say nothing? Because every possibility is admitted in it in advance; because.....
Page 55
It must shew in the proposition itself that it says something and in the tautology that it says nothing.
Page 55
p.~p is that thing--perhaps that nothing--that p and ~p have in common.
Page 55
In the real sign for p there is already contained the sign "p ∨ q". (For it is then possible to form this sign
WITHOUT FURTHER ADO.)
6.6.15.
Page 55
(This theory treats of propositions exclusively, so to speak, as a world on their own and not in connexion
with what they present.)
Page 55
The connexion of the picture-theory with the class-theory†1 will only become quite obvious later.
Page 55
One cannot say of a tautology that it is true, for it is made so as to be true.
Page Break 56
Page 56
It is not a picture of reality, in the sense that it does not PRESENT anything; it is what all--mutually
contradictory--pictures have in common.
Page 56
In the class-theory it is not yet evident why the proposition needs its counter-proposition. Why it is a part of
logical space which is separated from the remaining part of logical space.
Page 56
The proposition says: this is how it is and not: that. It presents a possibility and itself conspicuously forms
one part of a whole,--whose features it bears--and from which it stands out.
Page 56
p ∨ q ∨ ~p is also a tautology.----
Page 56
There are certainly propositions that allow p as well as ~p but none that assert p as well as ~p.
Page 56
The possibility of "p ∨ q" when "p" is given, is a possibility in a different dimension from the impossibility of
"~p".
Page 56
"p ∨ ~p" is a QUITE SPECIAL CASE of "p ∨ q".
Page 56
"p" has nothing in common with "~p ∨ q".
Page 56
By my attaching the "~" to "p" the proposition gets into a different class of propositions.
Page 56
Every proposition has only one negative;... There is only one proposition lying quite outside "p". [Cf. 5.513.]
Page 56
It could also be said: The proposition which asserts p and ~p is negated by all propositions; the proposition
which asserts p or ~p is asserted by all propositions.
Page 56
My mistake must lie in my wanting to use what follows from the nature of negation, etc. In its
definition.--That "p" and "~p" have
Page Break 57
a common boundary is no part of the explanation of negation that I am trying for.
7.6.15.
Page 57
If, e.g., it could be said: All propositions that do not assert p assert ~p, then that would give us an adequate
description.--But that doesn't work.
Page 57
But can't we say that "~p" is what is common only to such propositions as do not assert "p"? And from this
there already follows the impossibility of "p.~p".
Page 57
(All this, of course, already presupposes the existence of the whole world of propositions. Rightly?)
Page 57
IT IS NOT ENOUGH to point to ~p's lying outside p. It will only be possible to derive all the properties of
"~p" if "~p" is introduced essentially as the negative of p.
Page 57
But how to do that?--
Page 57
Or is it like this: We cannot "introduce" the proposition ~p at all, but we encounter it as a fait accompli and
we can only point to its individual formal properties, as, e.g., that it has nothing in common with p, that no
proposition contains it and p, etc. etc.?
8.6.15.
Page 57
Every mathematical proposition is a symbolic representation of a modus ponens. (And it is clear that the
modus ponens cannot be expressed in a proposition.) [Cf. 6.1264.]
Page 57
p and ~p have a common boundary: this is expressed by the fact that the negative of a proposition is only
determined by means of the proposition itself. For we say: The negative of a proposition is a proposition which...
and now follows the relation of ~p to p.----
9.6.15.
Page 57
It will, of course, be possible simply to say: The negation of p is the proposition which has no proposition in
common with p.
Page 57
The expression "tertium non datur" is really a piece of nonsense. (For no third thing is in question in p ∨ ~p.)
Page 57
Should we not be able to use this for our definition of the negative of a proposition?
Page 57
Can't we say: Among all the propositions that are dependent on p alone, there are only such as assert p and
such as deny it?
Page Break 58
Page 58
So I can say that the negative of p is the class of all propositions which are dependent on "p" alone and do
not assert "p".
10.6.15.
Page 58
"p.q ∨ ~q" is NOT dependent on "q"!
Page 58
Whole propositions, to disappear!
Page 58
The very fact that "p.q ∨ ~q" is independent of "q", although it obviously contains the written sign "q",
shews us how signs of the form η ∨ ~η can apparently, but still only apparently, exist.
Page 58
This naturally arises from the fact that this arrangement "p ∨ ~p" is indeed externally possible, but does not
satisfy the conditions for such a complex to say something and so be a proposition.
Page 58
"p.q ∨ ~q" says the same as
"p.r ∨ ~r"
--whatever q and r may say--: All tautologies say the same thing. (Namely nothing.) [Cf. 5.43.]
Page 58
From the last explanation of negation it follows that all propositions which are dependent on p alone and
which do not assert p--and only these--negate p. So "p ∨ ~p" and "p.~p" are not propositions, for the first sign
neither asserts nor denies p and the second would have to affirm both.
Page 58
But since I can after all write down p∨~p and p.~p, particularly in connexion with other sentences, it must be
clearly set forth what role these pseudo-propositions have, especially in such connexions. For they are not, of
course, to be treated as a completely meaningless appendix--like e.g. a meaningless name. Rather do they belong in
the symbolism--like "0" in arithmetic. [Cf. 4.4611.]
Page 58
Here it is clear that p∨~p has the role of a true proposition, which however says nought.
Page 58
So we have again arrived at the quantity of what is said.
11.6.15.
Page 58
The opposite of "p.~p" follows from all propositions; is that as much as to say that "p.~p" says nothing?--By
my earlier rule the contradiction would have to say more than all other propositions.
Page 58
If a proposition saying a great deal is false, it ought to be interesting
Page Break 59
that it is false. It is astonishing that the negative of a proposition that says a great deal should say absolutely nothing.
Page 59
We said: If p follows from q but not q from p, q says more than p. But now, if it follows from p that q is
false, but not from q that p is false, what then?
Page 59
From p there follows ~q, from q not ~p.----?
12.6.15.
Page 59
In connexion with any proposition it could really be asked: what does it come to for it to be true? What does
it come to for it to be false?
Page 59
Now the 'assumption' in p.~p is never anything but false, and so this does not come to anything; and as for
what it would amount to if it were true, of course, that can't be asked at all.
13.6.15.
Page 59
If "p.~p" COULD be true it would indeed say a very great deal. But the assumption that it is true does not
come into consideration in connexion with it, as the 'assumption' in it is always false.
Page 59
Singular, since the words "true" and "false" refer to the relation of the proposition to the world, that these
words can be used in the proposition itself for purposes of representation!
Page 59
We have said: if a proposition depends only on p and it asserts p then it does not negate it, and vice versa: Is
this the picture of that mutual exclusion of p and ~p? Of the fact that ~p is what lies outside p?
Page 59
It seems so! The proposition "~p" is in the same sense what lies outside "p".----(Do not forget either that the
picture may have very complicated co-ordinates to the world.)
Page 59
One might simply say: "p.~p" says nothing in the proper sense of the word. For in advance there is no
possibility left which it can correctly present.
Page 59
Incidentally, if "p follows from q" means: If q is true then p must be true, then it cannot be said at all that
anything follows from "p.~p", since there is no such thing as the hypothesis that "p.~p" is true.
14.6.15.
Page 59
We have become clear, then, that names may and do stand for the most various forms, and that it is only the
syntactical application that signalises the form that is to be presented.
Page 59
Now what is the syntactical application of names of simple objects?
Page 59
What is my fundamental thought when I talk about simple objects? Do not 'complex objects' in the end
satisfy just the demands which I
Page Break 60
apparently make on the simple ones? If I give this book a name "N" and now talk about N, is not the relation of N to
that 'complex object', to those forms and contents, essentially the same as I imagined only between name and
simple object?
Page 60
For N.B.: even if the name "N" vanishes on further analysis, still it indicates a single common thing.
Page 60
But what about the reference of names out of the context of the proposition?
Page 60
The question might however also be presented like this: It seems that the idea of the SIMPLE is already to be
found contained in that of the complex and in the idea of analysis, and in such a way that we come to this idea quite
apart from any examples of simple objects, or of propositions which mention them, and we realize the existence of
the simple object--a priori--as a logical necessity.
Page 60
So it looks as if the existence of the simple objects were related to that of the complex ones as the sense of
~p is to the sense of p: the simple object is prejudged in the complex.
15.6.15.
Page 60
(This is NOT to be confused with the fact that its component is prejudged in the complex.)
Page 60
(One of the most difficult of the philosopher's tasks is to find out where the shoe pinches.)
Page 60
It is quite clear that I can in fact correlate a name with this watch just as it lies here ticking in front of me, and
that this name will have reference outside any proposition in the very sense I have always given that word, and I feel
that that name in a proposition will correspond to all the requirements of the 'names of simple objects'.
16.6.15.
Page 60
Now we just want to see whether this watch in fact corresponds to all the conditions for being a 'simple
object'.----
Page 60
The question is really this: In order to know the syntactical treatment of a name, must I know the
composition of its reference? If so, then the whole composition is already expressed even in the unanalysed
proposition....
Page 60
(One often tries to jump over too wide chasms of thought and then falls in.)
Page Break 61
Page 61
What seems to be given us a priori is the concept: This.--Identical with the concept of the object.
Page 61
Relations and properties, etc. are objects too.
Page 61
My difficulty surely consists in this: In all the propositions that occur to me there occur names, which,
however, must disappear on further analysis. I know that such a further analysis is possible, but am unable to carry it
out completely. In spite of this I certainly seem to know that if the analysis were completely carried out, its result
would have to be a proposition which once more contained names, relations, etc. In brief it looks as if in this way I
knew a form without being acquainted with any single example of it.
Page 61
I see that the analysis can be carried farther, and can, so to speak, not imagine its leading to anything
different from the species of propositions that I am familiar with.
Page 61
When I say this watch is shiny, and what I mean by this watch alters its composition in the smallest
particular, then this means not merely that the sense of the sentence alters in its content, but also what I am saying
about this watch straightway alters its sense. The whole form of the proposition alters.
Page 61
That is to say, the syntactical employment of the names completely characterizes the form of the complex
objects which they denote.
Page 61
Every proposition that has a sense has a COMPLETE sense, and it is a picture of reality in such a way that
what is not yet said in it simply cannot belong to its sense.
Page 61
If the proposition "this watch is shiny" has a sense, it must be explicable HOW THIS proposition has THIS
sense.
Page 61
If a proposition tells us something, then it must be a picture of reality just as it is, and a complete picture at
that.----There will, of course, also be something that it does not say--but what it does say it says completely and it
must be susceptible of SHARP definition.
Page 61
So a proposition may indeed be an incomplete picture of a certain fact, but it is ALWAYS a complete
picture. [Cf. 5.156.]
Page 61
From this it would now seem as if in a certain sense all names were genuine names. Or, as I might also say,
as if all objects were in a certain sense simple objects.
Page Break 62
17.6.15.
Page 62
Let us assume that every spatial object consists of infinitely many points, then it is clear that I cannot
mention all these by name when I speak of that object. Here then would be a case in which I cannot arrive at the
complete analysis in the old sense at all; and perhaps just this is the usual case.
Page 62
But this is surely clear: the propositions which are the only ones that humanity uses will have a sense just as
they are and do not wait upon a future analysis in order to acquire a sense.
Page 62
Now, however, it seems to be a legitimate question: Are--e.g.--spatial objects composed of simple parts; in
analysing them, does one arrive at parts that cannot be further analysed, or is this not the case?
Page 62
--But what kind of question is this?--
Page 62
Is it, A PRIORI, clear that in analysing we must arrive at simple components--is this, e.g., involved in the
concept of analysis--, or is analysis ad infinitum possible?--Or is there in the end even a third possibility?
Page 62
This question is a logical one and the complexity of spatial objects is a logical complexity, for to say that one
thing is part of another is always a tautology.
Page 62
But suppose, for example, that I wanted to say that ONE component of a fact had a particular property?
Then I should have to mention it by name and use a logical sum.
Page 62
And nothing seems to speak against infinite divisibility.
Page 62
And it keeps on forcing itself upon us that there is some simple indivisible, an element of being, in brief a
thing.
Page 62
It does not go against our feeling, that we cannot analyse PROPOSITIONS so far as to mention the elements
by name; no, we feel that the WORLD must consist of elements. And it appears as if that were identical with the
proposition that the world must be what it is, it must be definite. Or in other words, what vacillates is our
determinations, not the world. It looks as if to deny things were as much as to say that the world can, as it were, be
indefinite in some such sense as that in which our knowledge is uncertain and indefinite.
Page 62
The world has a fixed structure.
Page Break 63
Page 63
Is the representation by means of unanalysable names only one system?
Page 63
All I want is only for my meaning to be completely analysed!
Page 63
In other words the proposition must be completely articulated. Everything that its sense has in common with
another sense must be contained separately in the proposition. If generalizations occur, then the forms of the
particular cases must be manifest and it is clear that this demand is justified, otherwise the proposition cannot be a
picture at all, of anything. [Cf. 3.251.]
Page 63
For if possibilities are left open in the proposition, just this must be definite: what is left open. The
generalizations of the form--e.g.--must be definite. What I do not know I do not know, but the proposition must
shew me WHAT I know. And in that case, is not this definite thing at which I must arrive precisely simple in that
sense that I have always had in mind? It is, so to speak, what is hard.
Page 63
In that case, then, what we mean by "complex objects do not exist" is: It must be clear in the proposition
how the object is composed, so far as it is possible for us to speak of its complexity at all.--The sense of the
proposition must appear in the proposition as divided into its simple components--. And these parts are then actually
indivisible, for further divided they just would not be THESE. In other words, the proposition can then no longer be
replaced by one that has more components, but any that has more components also does not have this sense.
Page 63
When the sense of the proposition is completely expressed in the proposition itself, the proposition is always
divided into its simple components--no further division is possible and an apparent one is superfluous--and these are
objects in the original sense.
18.6.15.
Page 63
If the complexity of an object is definitive of the sense of the proposition, then it must be portrayed in the
proposition to the extent that it does determine the sense. And to the extent that its composition is not definitive of
this sense, to this extent the objects of this proposition are simple. THEY cannot be further divided.----
Page 63
The demand for simple things is the demand for definiteness of sense. [Cf. 3.23.]
Page 63
----For if I am talking about, e.g., this watch, and mean something complex by that and nothing depends
upon the way it is compounded,
Page Break 64
then a generalization will make its appearance in the proposition and the fundamental forms of the generalization
will be completely determinate so far as they are given at all.
Page 64
If there is a final sense and a proposition expressing it completely, then there are also names for simple
objects.
Page 64
That is the correct designation.
Page 64
But suppose that a simple name denotes an infinitely complex object? For example, perhaps we assert of a
patch in our visual field that it is to the right of a line, and we assume that every patch in our visual field is infinitely
complex. Then if we say that a point in that patch is to the right of the line, this proposition follows from the
previous one, and if there are infinitely many points in the patch then infinitely many propositions of different
content follow LOGICALLY from that first one. And this of itself shews that the proposition itself was as a matter
of fact infinitely complex. That is, not the propositional sign by itself, but it together with its syntactical
application.
Page 64
Now it seems, of course, perfectly possible that in reality infinitely many different propositions do not follow
from such a proposition, because our visual field perhaps--or probably--does not consist of infinitely many
parts--but continuous visual space is only a subsequent construction--; and in that case only a finite number of
propositions follow from the one known and it itself is finite in every sense.
Page 64
But now, does not this possible infinite complexity of the sense impair its definiteness?
Page 64
We might demand definiteness in this way too!: if a proposition is to make sense then the syntactical
employment of each of its parts must be settled in advance.--It is, e.g., not possible only subsequently to come upon
the fact that a proposition follows from it. But, e.g., what propositions follow from a proposition must be completely
settled before that proposition can have a sense!
Page 64
It seems to me perfectly possible that patches in our visual field are simple objects, in that we do not perceive
any single point of a patch separately; the visual appearances of stars even seem certainly to be so. What I mean is:
if, e.g., I say that this watch is not in the drawer, there is absolutely no need for it to FOLLOW LOGICALLY that a
wheel which is in the watch is not in the drawer, for perhaps I had not the least knowledge that the wheel was in the
watch, and hence
Page Break 65
could not have meant by "this watch" the complex in which the wheel occurs. And it is certain--moreover--that I do
not see all the parts of my theoretical visual field. Who knows whether I see infinitely many points?
Page 65
Let us suppose that we were to see a circular patch: is the circular form its property? Certainly not. It seems
to be a structural "property". And if I notice that a spot is round, am I not noticing an infinitely complicated
structural property? Or I notice only that the spot has finite extension, and this of itself seems to presuppose an
infinitely complex structure.
Page 65
Not: One proposition follows from another, but the truth of the one follows from the truth of the other. (That
is why it follows from "All men are mortal" that "If Socrates is a man, then he is mortal.")
Page 65
A proposition can, however, quite well treat of infinitely many points without being infinitely complex in a
particular sense.
19.6.15.
Page 65
When we see that our visual field is complex we also see that it consists of simpler parts.
Page 65
We can talk of functions of this and that kind without having any particular application in view.
Page 65
For we don't have any examples before our minds when we use Fx and all the other variable form-signs.
Page 65
In short: if we were to apply the prototypes only in connexion with names, there would be the possibility
that we should know the existence of the prototypes from the existence of their special cases. But as it is we use
variables, that is to say we talk, so to speak, of the prototypes by themselves, quite apart from any individual cases.
Page 65
We portray the thing, the relation, the property, by means of variables and so shew that we do not derive
these ideas from particular cases that occur to us, but possess them somehow a priori.
Page 65
For the question arises: If the individual forms are, so to speak, given me in experience, then I surely can't
make use of them in logic; in that case I cannot write down an x or a φy. But this I can surely not avoid at all.
Page Break 66
Page 66
An incidental question: Does logic deal with certain classes of functions and the like? And if not, what then is
the import of Fx, φz, and so on in logic?
Page 66
Then these must be signs of more general import!
Page 66
There doesn't after all seem to be any setting up of a kind of logical inventory as I formerly imagined it.
Page 66
The component parts of the proposition must be simple = The proposition must be completely articulated.
[Cf. 3.251.]
Page 66
But now does this SEEM to contradict the facts?----
Page 66
For in logic we are apparently trying to produce ideal pictures of articulated propositions. But how is that
possible?
Page 66
Or can we deal with a proposition like "The watch is on the table" without further ado according to the rules
of logic? No, here we say, for example, that no date is given in the proposition, that the proposition is only
apparently... etc. etc.
Page 66
So before we can deal with it we must, so it seems, transform it in a particular way.
Page 66
But perhaps this is not conclusive, for could we not just as well apply our usual logical notation to the
special proposition?
20.6.15.
Page 66
Yes, this is the point: Can we justly apply logic just as it stands, say in Principia Mathematica, straightaway
to ordinary propositions?
Page 66
Of course we cannot disregard what is expressed in our propositions by means of endings, prefixes, vowel
changes, etc. etc.
Page 66
But we do apply mathematics, and with the greatest success, to ordinary propositions, namely to those of
physics.
Page 66
But how remarkable: in the familiar theorems of mathematical physics there appear neither things nor
functions nor relations nor any other logical forms of object! Instead of things what we have here is numbers, and
the functions and relations are purely mathematical throughout)
Page 66
But it is surely a fact that these propositions are applied to solid reality.
Page Break 67
Page 67
The variables in those theorems do not--as is often said--stand for lengths, weights, time intervals, etc. at all,
they simply stand for numbers and for nothing else.
Page 67
When, however, I want to apply numbers, I come to relations, things, etc. etc. I say, e.g.: This length is 5
yards and here I am talking of relations and things, and in the completely ordinary sense at that.
Page 67
Here we come to the question about the reference of variables in the propositions of physics. For these are
not tautologies.
Page 67
A proposition of physics is obviously senseless if its application is not given. What sort of sense would it
make to say: "k = m.p"?
Page 67
So the complete physical proposition does after all deal with things, relations and so on. (Which was really to
be expected.)
Page 67
Now everything turns on the fact that I apply numbers to ordinary things, etc., which in fact says no more
than that numbers occur in our quite ordinary sentences.
Page 67
The difficulty is really this: even when we want to express a completely definite sense there is the possibility
of failure. So it seems that we have, so to speak, no guarantee that our proposition is really a picture of reality.
Page 67
The division of the body into material points, as we have it in physics, is nothing more than analysis into
simple components.
Page 67
But could it be possible that the sentences in ordinary use have, as it were, only an incomplete sense (quite
apart from their truth or falsehood), and that the propositions in physics, as it were, approach the stage where a
proposition really has a complete sense?
Page 67
When I say, "The book is lying on the table", does this really have a completely clear sense? (An
EXTREMELY important question.)
Page 67
But the sense must be clear, for after all we mean something by the proposition, and as much as we certainly
mean must surely be clear.
Page 67
If the proposition "The book is on the table" has a clear sense, then I must, whatever is the case, be able to
say whether the proposition is true or false. There could, however, very well occur cases in which I should not be
able to say straight off whether the book is still to be called "lying on the table". Then--?
Page Break 68
Page 68
Then is the case here one of my knowing exactly what I want to say, but then making mistakes in expressing
it?
Page 68
Or can this uncertainty TOO be included in the proposition?
Page 68
But it may also be that the proposition "The book is lying on the table" represents my sense completely, but
that I am using the words, e.g., "lying on", with a special reference here, and that elsewhere they have another
reference. What I mean by the verb is perhaps a quite special relation which the book now actually has to the table.
Page 68
Then are the propositions of physics and the propositions of ordinary life at bottom equally sharp, and does
the difference consist only in the more consistent application of signs in the language of science?
Page 68
Is it or is it not possible to talk of a proposition's having a more or less sharp sense?
Page 68
It seems clear that what we MEAN must always be "sharp".
Page 68
Our expression of what we mean can in its turn only be right or wrong. And further the words can be applied
consistently or inconsistently. There does not seem to be any other possibility.
Page 68
When I say, e.g., that the table is a yard long, it is extremely questionable what I mean by this. But I
presumably mean that the distance between THESE two points is a yard, and that the points belong to the table.
Page 68
We said that mathematics has already been applied with success to ordinary propositions, but in propositions
of physics it treats of completely different objects from those of our ordinary language. Must our propositions
undergo such preparation, to make them capable of being dealt with mathematically? Evidently they must. When
quantities come in question, then an expression like, e.g., "the length of this table" would not be adequate. This
length would have to be defined, say, as the distance between two surfaces, etc. etc.
Page 68
Mathematical sciences are distinguished from non-mathematical ones by treating of things of which ordinary
language does not speak, whereas the latter talk about things that are generally familiar.
21.6.15.
Page 68
Our difficulty was that we kept on speaking of simple objects and were unable to mention a single one.
Page Break 69
Page 69
If a point in space does not exist, then its co-ordinates do not exist either, and if the coordinates exist then
the point exists too.--That's how it is in logic.
Page 69
The simple sign is essentially simple.
Page 69
It functions as a simple object. (What does that mean?)
Page 69
Its composition becomes completely indifferent. It disappears from view.
Page 69
It always looks as if there were complex objects functioning as simples, and then also really simple ones, like
the material points of physics, etc..
Page 69
It can be seen that a name stands for a complex object from an indefiniteness in the proposition in which it
occurs. This comes of the generality of such propositions. We know that not everything is yet determined by this
proposition. For the generality notation contains a proto-picture. [Cf. 3.24.]
Page 69
All invisible masses, etc. etc. must come under the generality notation.
Page 69
What is it for propositions to approximate to the truth?
Page 69
But logic as it stands, e.g., in Principia Mathematica can quite well be applied to our ordinary propositions,
e.g., from "All men are mortal" and "Socrates is a man" there follows according to this logic "Socrates is mortal"
which is obviously correct although I equally obviously do not know what structure is possessed by the thing
Socrates or the property of mortality. Here they just function as simple objects.
Page 69
Obviously the circumstance that makes it possible for certain forms to be projected by means of a definition
into a name, guarantees of itself that this name can then also be treated as a real one.
Page 69
To anyone that sees clearly, it is obvious that a proposition like "This watch is lying on the table" contains a
lot of indefiniteness, in spite of its form's being completely clear and simple in outward appearance. So we see that
this simplicity is only constructed.
22.6.15.
Page 69
It is then also clear to the UNPREJUDICED mind that the sense of the proposition "The watch is lying on
the table" is more complicated than the proposition itself.
Page Break 70
Page 70
The conventions of our language are extraordinarily complicated. There is enormously much added in
thought to each proposition and not said. (These conventions are exactly like Whitehead's 'Conventions'. They are
definitions with a certain generality of form.) [Cf. 4.002.]
Page 70
I only want to justify the vagueness of ordinary sentences, for it can be justified.
Page 70
It is clear that I know what I mean by the vague proposition. But now someone else doesn't understand and
says: "Yes, but if you mean that then you should have added such and such"; and now someone else again will not
understand it and will demand that the proposition should be given in more detail still. I shall then reply: NOW
THAT can surely be taken for granted.
Page 70
I tell someone "The watch is lying on the table" and now he says: "Yes, but if the watch were in
such-and-such a position would you still say it was lying on the table?" And I should become uncertain. This shews
that I did not know what I meant by "lying" in general. If someone were to drive me into a corner in this way in
order to shew that I did not know what I meant, I should say: "I know what I mean; I mean just THIS", pointing to
the appropriate complex with my finger. And in this complex I do actually have the two objects in a relation.----But
all that this really means is: The fact can SOMEHOW be portrayed by means of this form too.
Page 70
Now when I do this and designate the objects by means of names, does that make them simple?
Page 70
All the same, however, this proposition is a picture of that complex.
Page 70
This object is simple for me!
Page 70
If, e.g., I call some rod "A", and a ball "B", I can say that A is leaning against the wall, but not B. Here the
internal nature of A and B comes into view.
Page 70
A name designating an object thereby stands in a relation to it which is wholly determined by the logical kind
of the object and which signalises that logical kind.
Page 70
And it is clear that the object must be of a particular logical kind, it just is as complex, or as simple, as it is.
Page 70
"The watch is sitting on the table" is senseless!
Page Break 71
Page 71
Only the complex part of the proposition can be true or false.
Page 71
The name compresses its whole complex reference into one.
--------------
15.4.16.
Page 71
We can only foresee what we ourselves construct. [See 5.556.]
Page 71
But then where is the concept of a simple object still to be found?
Page 71
This concept does not so far come in here at all.
Page 71
We must be able to construct the simple functions because we must be able to give each sign a meaning.
Page 71
For the only sign which guarantees its meaning is function and argument.
16.4.16.
Page 71
Every simple proposition can be brought into the form φx.
Page 71
That is why we may compose all simple propositions from this form.
Page 71
Suppose that all simple propositions were given me: then it can simply be asked what propositions I can
construct from them. And these are all propositions and this is how they are bounded. [4.51.]
Page 71
(p): p = aRx.xRy... zRb
Page 71
(p): p = aRx
17.4.16.
Page 71
The above definition can in its general form only be a rule for a written notation which has nothing to do with
the sense of the signs.
Page 71
But can there be such a rule?
Page 71
The definition is only possible if it is itself not a proposition.
Page 71
In that case a proposition cannot treat of all propositions, while a definition can.
23.4.16.
Page 71
The above definition, however, just does not deal with all propositions, for it essentially contains real
variables. It is quite analogous to an operation whose own result can be taken as its base.
26.4.16.
Page 71
In this way, and in this way alone, is it possible to proceed from one type to another. [Cf. 5.252.]
Page 71
And we can say that all types stand in hierarchies.
Page Break 72
Page 72
And the hierarchy is only possible by being built up by means of operations.
Page 72
Empirical reality is bounded by the number of objects.
Page 72
The boundary turns up again in the totality of simple propositions. [See 5.5561.]
Page 72
The hierarchies are and must be independent of reality. [See 5.5561.]
Page 72
The meanings of their terms are only determined by the correlation of objects and names.
27.4.16.
Page 72
Say I wanted to represent a function of three non-interchangeable arguments.
φ(x): φ(), x
Page 72
But should there be any mention of non-interchangeable arguments in logic? If so, this surely presupposes
something about the character of reality.
6.5.16.
Page 72
At bottom the whole Weltanschauung of the moderns involves the illusion that the so-called laws of nature
are explanations of natural phenomena. [6.371.]
Page 72
In this way they stop short at the laws of nature as at something impregnable as men of former times did at
God and fate. [See 6.372.]
Page 72
And both are right and wrong. The older ones are indeed clearer in the sense that they acknowledge a clear
terminus, while with the new system it is supposed to look as if everything had a foundation. [See 6.372.]
11.5.16.
Page 72
|p |(a, a)
Page 72
There are also operations with two bases. And the "|"-operation is of this kind.
Page 72
| (ξ,η)... is an arbitrary term of the series of results of as operation.
(∃x).φx
Page 72
Is (∃x) etc. really an operation?
Page 72
But what would be its base?
11.6.16.
Page 72
What do I know about God and the purpose of life?
Page 72
I know that this world exists.
Page Break 73
Page 73
That I am placed in it like my eye in its visual field.
Page 73
That something about it is problematic, which we call its meaning.
Page 73
That this meaning does not lie in it but outside it. [Cf. 6.41.]
Page 73
That life is the world. [Cf. 5.621.]
Page 73
That my will penetrates the world.
Page 73
That my will is good or evil.
Page 73
Therefore that good and evil are somehow connected with the meaning of the world.
Page 73
The meaning of life, i.e. the meaning of the world, we can call God.
And connect with this the comparison of God to a father.
Page 73
To pray is to think about the meaning of life.
Page 73
I cannot bend the happenings of the world to my will: I am completely powerless.
Page 73
I can only make myself independent of the world--and so in a certain sense master it--by renouncing any
influence on happenings.
5.7.16.
Page 73
The world is independent of my will. [6.373.]
Page 73
Even if everything that we want were to happen, this would still only be, so to speak, a grace of fate, for what
would guarantee it is not any logical connexion between will and world, and we could not in turn will the supposed
physical connexion. [6.374.]
Page 73
If good or evil willing affects the world it can only affect the boundaries of the world, not the facts, what
cannot be portrayed by language but can only be shewn in language. [Cf. 6.43.]
Page 73
In short, it must make the world a wholly different one. [See 6.43.]
Page 73
The world must, so to speak, wax or wane as a whole. As if by accession or loss of meaning. [Cf. 6.43.]
Page 73
As in death, too, the world does not change but stops existing. [6.431.]
6.7.16.
Page 73
And in this sense Dostoievsky is right when he says that the man who is happy is fulfilling the purpose of
existence.
Page 73
Or again we could say that the man is fulfilling the purpose of existence who no longer needs to have any
purpose except to live. That is to say, who is content.
Page Break 74
Page 74
The solution of the problem of life is to be seen in the disappearance of this problem. [See 6.521.]
Page 74
But is it possible for one so to live that life stops being problematic? That one is living in eternity and not in
time?
7.7.16.
Page 74
Isn't this the reason why men to whom the meaning of life had become clear after long doubting could not
say what this meaning consisted in? [See 6.521.]
Page 74
If I can imagine a "kind of object" without knowing whether there are such objects, then I must have
constructed their proto-picture for myself.
Page 74
Isn't the method of mechanics based on this?
8.7.16.
Page 74
To believe in a God means to understand the question about the meaning of life.
Page 74
To believe in a God means to see that the facts of the world are not the end of the matter.
Page 74
To believe in God means to see that life has a meaning.
Page 74
The world is given me, i.e. my will enters into the world completely from outside as into something that is
already there.
Page 74
(As for what my will is, I don't know yet.)
Page 74
That is why we have the feeling of being dependent on an alien will.
Page 74
However this may be, at any rate we are in a certain sense dependent, and what we are dependent on we can
call God.
Page 74
In this sense God would simply be fate, or, what is the same thing: The world--which is independent of our
will.
Page 74
I can make myself independent of fate.
Page 74
There are two godheads: the world and my independent I.
Page 74
I am either happy or unhappy, that is all. It can be said: good or evil do not exist.
Page 74
A man who is happy must have no fear. Not even in face of death.
Page 74
Only a man who lives not in time but in the present is happy.
Page Break 75
Page 75
For life in the present there is no death.
Page 75
Death is not an event in life. It is not a fact of the world. [Cf. 6.4311.]
Page 75
If by eternity is understood not infinite temporal duration but non-temporality, then it can be said that a man
lives eternally if he lives in the present. [See 6.4311.]
Page 75
In order to live happily I must be in agreement with the world. And that is what "being happy" means.
Page 75
I am then, so to speak, in agreement with that alien will on which I appear dependent. That is to say: 'I am
doing the will of God'.
Page 75
Fear in face of death is the best sign of a false, i.e. a bad, life.
Page 75
When my conscience upsets my equilibrium, then I am not in agreement with Something. But what is this?
Is it the world?
Page 75
Certainly it is correct to say: Conscience is the voice of God.
Page 75
For example: it makes me unhappy to think that I have offended such and such a man. Is that my
conscience?
Page 75
Can one say: "Act according to your conscience whatever it may be"?
Page 75
Live happy!
9.7.16.
Page 75
If the most general form of proposition could not be given, then there would have to come a moment where
we suddenly had a new experience, so to speak a logical one.
Page 75
That is, of course, impossible.
Page 75
Do not forget that (∃x)fx does not mean: There is an x such that fx, but: There is a true proposition "fx".
Page 75
The proposition fa speaks of particular objects, the general proposition of all objects.
11.7.16.
Page 75
The particular object is a very remarkable phenomenon.
Page 75
Instead of "all objects" we might say: All particular objects.
Page Break 76
Page 76
If all particular objects are given, "all objects" are given.
Page 76
In short with the particular objects all objects are given. [Cf. 5.524.]
Page 76
If there are objects, then that gives us "all objects" too. [Cf. 5.524.]
Page 76
That is why it must be possible to construct the unity of the elementary propositions and of the general
propositions.
Page 76
For if the elementary propositions are given, that gives us all elementary propositions, too, and that gives us
the general proposition.--And with that has not the unity been constructed? [Cf. 5.524.]
13.7.16.
Page 76
One keeps on feeling that even in the elementary proposition mention is made of all objects.
Page 76
(∃x)φx.x = a
Page 76
If two operations are given which cannot be reduced to one, then it must at least be possible to set up a
general form of their combination.
φx, ψy|χz , (∃x)., (x).
Page 76
As obviously it can easily be explained how propositions can be formed by means of these operations and
how propositions are not to be formed, this must also be capable somehow of exact expression.
14.7.16.
Page 76
And this expression must already be given in the general form of the sign of an operation.
Page 76
And mustn't this be the only legitimate expression of the application of an operation? Obviously it must!
Page 76
For if the form of operation can be expressed at all, then it must be expressed in such a way that it can only
be applied correctly.
Page 76
Man cannot make himself happy without more ado.
Page 76
Whoever lives in the present lives without fear and hope.
21.7.16.
Page 76
What really is the situation of the human will? I will call "will" first and foremost the bearer of good and evil.
Page 76
Let us imagine a man who could use none of his limbs and hence could, in the ordinary sense, not exercise
his will. He could, however, think and want and communicate his thoughts to someone else. Could
Page Break 77
therefore do good or evil through the other man. Then it is clear that ethics would have validity for him, too, and that
he in the ethical sense is the bearer of a will.
Page 77
Now is there any difference in principle between this will and that which sets the human body in motion?
Page 77
Or is the mistake here this: even wanting (thinking) is an activity of the will? (And in this sense, indeed, a
man without will would not be alive.)
Page 77
But can we conceive a being that isn't capable of Will at all, but only of Idea (of seeing for example)? In
some sense this seems impossible. But if it were possible then there could also be a world without ethics.
24.7.16.
Page 77
The World and Life are one. [5.621.]
Page 77
Physiological life is of course not "Life". And neither is psychological life. Life is the world.
Page 77
Ethics does not treat of the world. Ethics must be a condition of the world, like logic.
Page 77
Ethics and aesthetics are one. [See 6.421.]
29.7.16.
Page 77
For it is a fact of logic that wanting does not stand in any logical connexion with its own fulfilment. And it is
also clear that the world of the happy is a different world from the world of the unhappy. [Cf. 6.43.]
Page 77
Is seeing an activity?
Page 77
Is it possible to will good, to will evil, and not to will?
Page 77
Or is only he happy who does not will?
Page 77
"To love one's neighbour" would mean to will!
Page 77
But can one want and yet not be unhappy if the want does not attain fulfilment? (And this possibility always
exists.)
Page 77
Is it, according to common conceptions, good to want nothing for one's neighbour, neither good nor evil?
Page 77
And yet in a certain sense it seems that not wanting is the only good.
Page Break 78
Page 78
Here I am still making crude mistakes! No doubt of that!
Page 78
It is generally assumed that it is evil to want someone else to be unfortunate. Can this be correct? Can it be
worse than to want him to be fortunate?
Page 78
Here everything seems to turn, so to speak, on how one wants.
Page 78
It seems one can't say anything more than: Live happily!
Page 78
The world of the happy is a different world from that of the unhappy. [See 6.43.]
Page 78
The world of the happy is a happy world.
Page 78
Then can there be a world that is neither happy nor unhappy?
30.7.16.
Page 78
When a general ethical law of the form "Thou shalt..." is set up, the first thought is: Suppose I do not do it?
Page 78
But it is clear that ethics has nothing to do with punishment and reward. So this question about the
consequences of an action must be unimportant. At least these consequences cannot be events. For there must be
something right about that question after all. There must be a kind of ethical reward and of ethical punishment but
these must be involved in the action itself.
Page 78
And it is also clear that the reward must be something pleasant, the punishment something unpleasant.
Page 78
[6.422.]
Page 78
I keep on coming back to this! simply the happy life is good, the unhappy bad. And if I now ask myself: But
why should I live happily, then this of itself seems to me to be a tautological question; the happy life seems to be
justified, of itself, it seems that it is the only right life.
Page 78
But this is really in some sense deeply mysterious! It is clear that ethics cannot be expressed! [Cf. 6.421.]
Page 78
But we could say: The happy life seems to be in some sense more harmonious than the unhappy. But in
what sense??
Page 78
What is the objective mark of the happy, harmonious life? Here it is again clear that there cannot be any such
mark, that can be described.
Page 78
This mark cannot be a physical one but only a metaphysical one, a transcendental one.
Page Break 79
Page 79
Ethics is transcendental. [See 6.421.]
1.8.16.
Page 79
How things stand, is God.
Page 79
God is, how things stand.
Page 79
Only from the consciousness of the uniqueness of my life arises religion--science--and art.
2.8.16.
Page 79
And this consciousness is life itself.
Page 79
Can there be any ethics if there is no living being but myself?
Page 79
If ethics is supposed to be something fundamental, there can.
Page 79
If I am right, then it is not sufficient for the ethical judgment that a world is given.
Page 79
Then the world in itself is neither good nor evil.
Page 79
For it must be all one, as far as concerns the existence of ethics, whether there is living matter in the world or
not. And it is clear that a world in which there is only dead matter is in itself neither good nor evil, so even the world
of living things can in itself be neither good nor evil.
Page 79
Good and evil only enter through the subject. And the subject is not part of the world, but a boundary of the
world. [Cf. 5.632.]
Page 79
It would be possible to say (à la Schopenhauer): It is not the world of Idea that is either good or evil; but the
willing subject.
Page 79
I am conscious of the complete unclarity of all these sentences.
Page 79
Going by the above, then, the willing subject would have to be happy or unhappy, and happiness and
unhappiness could not be part of the world.
Page 79
As the subject is not a part of the world but a presupposition of its existence, so good and evil which are
predicates of the subject, are not properties in the world.
Page 79
Here the nature of the subject is completely veiled.
Page 79
My work has extended from the foundations of logic to the nature of the world.
Page Break 80
4.8.16.
Page 80
Isn't the thinking subject in the last resort mere superstition?
Page 80
Where in the world is a metaphysical subject to be found? [See 5.633.]
Page 80
You say that it is just as it is for the eye and the visual field. But you do not actually see the eye. [See 5.633.]
Page 80
And I think that nothing in the visual field would enable one to infer that it is seen from an eye. [Cf. 5.633.]
5.8.16.
Page 80
The thinking subject is surely mere illusion. But the willing subject exists. [Cf. 5.631.]
Page 80
If the will did not exist, neither would there be that centre of the world, which we call the I, and which is the
bearer of ethics.
Page 80
What is good and evil is essentially the I, not the world.
Page 80
The I, the I is what is deeply mysterious!
7.8.16.
Page 80
The I is not an object.
11.8.16.
Page 80
I objectively confront every object. But not the I.
Page 80
So there really is a way in which there can and must be mention of the I in a non-psychological sense in
philosophy. [Cf. 5.641.]
12.8.16.
Page 80
The I makes its appearance in philosophy through the world's being my world. [See 5.641.]
Page 80
The visual field has not, e.g., a form like this:
Page 80
[5.6331.]
Page 80
This is connected with the fact that none of our experience is a priori. [See 5.634.]
Page 80
All that we see could also be otherwise.
Page 80
All that we can describe at all could also be otherwise. [See 5.634.]
Page Break 81
13.8.16.
Page 81
Suppose that man could not exercise his will, but had to suffer all the misery of this world, then what could
make him happy?
Page 81
How can man be happy at all, since he cannot ward off the misery of this world?
Page 81
Through the life of knowledge.
Page 81
The good conscience is the happiness that the life of knowledge preserves.
Page 81
The life of knowledge is the life that is happy in spite of the misery of the world.
Page 81
The only life that is happy is the life that can renounce the amenities of the world.
Page 81
To it the amenities of the world are so many graces of fate.
16.8.16.
Page 81
A point cannot be red and green at the same time: at first sight there seems no need for this to be a logical
impossibility. But the very language of physics reduces it to a kinetic impossibility. We see that there is a difference
of structure between red and green.
Page 81
And then physics arranges them in a series. And then we see how here the true structure of the objects is
brought to light.
Page 81
The fact that a particle cannot be in two places at the same time does look more like a logical impossibility.
Page 81
If we ask why, for example, then straight away comes the thought: Well, we should call particles that were in
two places different, and this in its turn all seems to follow from the structure of space and of particles.
Page 81
[Cf. 6.3751.]
17.8.16.
Page 81
An operation is the transition from one term to the neat one in a series of forms.
Page 81
The operation and the series of forms are equivalents.
29.8.16.
Page 81
The question is whether the usual small number of fundamental operations is adequate for the construction
of all possible operations.
Page 81
It looks as if it must be so.
Page Break 82
Page 82
We can also ask whether those fundamental operations enable us to pass from any expression to any related
ones.
2.9.16.
Page 82
Here we can see that solipsism coincides with pure realism, if it is strictly thought out.
Page 82
The I of solipsism shrinks to an extensionless point and what remains is the reality coordinate with it.
Page 82
[5.64.]
Page 82
What has history to do with me? Mine is the first and only world!
Page 82
I want to report how I found the world.
Page 82
What others in the world have told me about the world is a very small and incidental part of my experience
of the world.
Page 82
I have to judge the world, to measure things.
Page 82
The philosophical I is not the human being, not the human body or the human soul with the psychological
properties, but the metaphysical subject, the boundary (not a part) of the world. The human body, however, my
body in particular, is a part of the world among others, among beasts, plants, stones etc., etc. [Cf. 5.641.]
Page 82
Whoever realizes this will not want to procure a pre-eminent place for his own body or for the human body.
Page 82
He will regard humans and beasts quite naïvely as objects which are similar and which belong together.
11.9.16.
Page 82
The way in which language signifies is mirrored in its use.
Page 82
That the colours are not properties is shewn by the analysis of physics, by the internal relations in which
physics displays the colours.
Page 82
Apply this to sounds too.
12.9.16.
Page 82
Now it is becoming clear why I thought that thinking and language were the same. For thinking is a kind of
language. For a thought too is, of course, a logical picture of the proposition, and therefore it just is a kind of
proposition.
Page Break 83
19.9.16.
Page 83
Mankind has always looked for a science in which simplex sigillum veri holds. [Cf. 5.4541.]
Page 83
There cannot be an orderly or a disorderly world, so that one could say that our world is orderly. In every
possible world there is an order even if it is a complicated one, just as in space too there are not orderly and
disorderly distributions of points, but every distribution of points is orderly.
Page 83
(This remark is only material for a thought.)
Page 83
Art is a kind of expression.
Page 83
Good art is complete expression.
7.10.16
Page 83
The work of art is the object seen sub specie aeternitatis; and the good life is the world seen sub specie
aeternitatis. This is the connexion between art and ethics.
Page 83
The usual way of looking at things sees objects as it were from the midst of them, the view sub specie
aeternitatis from outside.
Page 83
In such a way that they have the whole world as background.
Page 83
Is this it perhaps--in this view the object is seen together with space and time instead of in space and time?
Page 83
Each thing modifies the whole logical world, the whole of logical space, so to speak.
Page 83
(The thought forces itself upon one): The thing seen sub specie aeternitatis is the thing seen together with
the whole logical space.
8.10.16.
Page 83
As a thing among things, each thing is equally insignificant; as a world each one equally significant.
Page 83
If I have been contemplating the stove, and then am told: but now all you know is the stove, my result does
indeed seem trivial. For this represents the matter as if I had studied the stove as one among the many things in the
world. But if I was contemplating the stove it was my world, and everything else colourless by contrast with it.
Page 83
(Something good about the whole, but bad in details.)
Page 83
For it is equally possible to take the bare present image as the worthless momentary picture in the whole
temporal world, and as the true world among shadows.
Page Break 84
9.10.16.
Page 84
But now at last the connexion of ethics with the world has to be made clear.
12.10.16.
Page 84
A stone, the body of a beast, the body of a man, my body, all stand on the same level.
Page 84
That is why what happens, whether it comes from a stone or from my body is neither good nor bad.
Page 84
"Time has only one direction" must be a piece of nonsense.
Page 84
Having only one direction is a logical property of time.
Page 84
For if one were to ask someone how he imagines having only one direction he would say: Time would not be
confined to one direction if an event could be repeated.
Page 84
But the impossibility of an event's being repeated, like that of a body's being in two places at once, is
involved in the logical nature of the event.
Page 84
It is true: Man is the microcosm:
Page 84
I am my world. [Cf. 5.63.]
15.10.16.
Page 84
What cannot be imagined cannot even be talked about. [Cf. 5.61.]
Page 84
Things acquire "significance" only through their relation to my will.
Page 84
For "Everything is what it is and not another thing".
Page 84
One conception: As I can infer my spirit (character, will) from my physiognomy, so I can infer the spirit
(will) of each thing from its physiognomy.
Page 84
But can I infer my spirit from my physiognomy?
Page 84
Isn't this relationship purely empirical?
Page 84
Does my body really express anything?
Page 84
Is it itself an internal expression of something?
Page 84
Is, e.g., an angry face angry in itself or merely because it is empirically connected with bad temper?
Page 84
But it is clear that the causal nexus is not a nexus at all. [Cf. 5.136.]
Page Break 85
Page 85
Now is it true (following the psycho-physical conception) that my character is expressed only in the build of
my body or brain and not equally in the build of the whole of the rest of the world?
Page 85
This contains a salient point.
Page 85
This parallelism, then, really exists between my spirit, i.e. spirit, and the world.
Page 85
Only remember that the spirit of the snake, of the lion, is your spirit. For it is only from yourself that you are
acquainted with spirit at all.
Page 85
Now of course the question is why I have given a snake just this spirit.
Page 85
And the answer to this can only lie in the psycho-physical parallelism: If I were to look like the snake and to
do what it does then I should be such-and-such.
Page 85
The same with the elephant, with the fly, with the wasp.
Page 85
But the question arises whether even here, my body is not on the same level with that of the wasp and of the
snake (and surely it is so), so that I have neither inferred from that of the wasp to mine nor from mine to that of the
wasp.
Page 85
Is this the solution of the puzzle why men have always believed that there was one spirit common to the
whole world?
Page 85
And in that case it would, of course, also be common to lifeless things too.
Page 85
This is the way I have travelled: Idealism singles men out from the world as unique, solipsism singles me
alone out, and at last I see that I too belong with the rest of the world, and so on the one side nothing is left over,
and on the other side, as unique, the world. In this way idealism leads to realism if it is strictly thought out. [Cf.
5.64.]
17.10.16.
Page 85
And in this sense I can also speak of a will that is common to the whole world.
Page 85
But this will is in a higher sense my will.
Page 85
As my idea is the world, in the same way my will is the world-will.
20.10.16.
Page 85
It is clear that my visual space is constituted differently in length from breadth.
Page Break 86
Page 86
The situation is not simply that I everywhere notice where I see anything, but I also always find myself at a
particular point of my visual space, so my visual space has as it were a shape.
Page 86
In spite of this, however, it is true that I do not see the subject.
Page 86
It is true that the knowing subject is not in the world, that there is no knowing subject. [Cf. 5.631.]
Page 86
At any rate I can imagine carrying out the act of will for raising my arm, but that my arm does not move.
(E.g., a sinew is torn.) True, but, it will be said, the sinew surely moves and that just shews that the act of will related
to the sinew and not to the arm. But let us go farther and suppose that even the sinew did not move, and so on. We
should then arrive at the position that the act of will does not relate to a body at all, and so that in the ordinary sense
of the word there is no such thing as the act of the will.
Page 86
Aesthetically, the miracle is that the world exists. That there is what there is.
Page 86
Is it the essence of the artistic way of looking at things, that it looks at the world with a happy eye?
Page 86
Life is grave, art is gay.†1
21.10.16.
Page 86
For there is certainly something in the conception that the end of art is the beautiful.
Page 86
And the beautiful is what makes happy.
29.10.16.
Page 86
Could it not be said that generality is no more co-ordinated with the complex than is fact with thing?
Page 86
Both kinds of operation sign must or can occur in the proposition side by side.
4.11.16.
Page 86
Is the will an attitude towards the world?
Page 86
The will seems always to have to relate to an idea. We cannot imagine, e.g., having carried out an act of will
without having detected that we have carried it out.
Page Break 87
Page 87
Otherwise there might arise such a question as whether it had yet been completely carried out.
Page 87
It is clear, so to speak, that we need a foothold for the will in the world.
Page 87
The will is an attitude of the subject to the world.
Page 87
The subject is the willing subject.
Page 87
Have the feelings by which I ascertain that an act of the will takes place any particular characteristic which
distinguishes them from other ideas?
Page 87
It seems not!
Page 87
In that case, however, I might conceivably get the idea that, e.g., this chair was directly obeying my will.
Page 87
Is that possible?
Page 87
In drawing the square in the mirror one notices that one is only able to manage it if one prescinds
completely from the visual datum and relies only on muscular feeling. So here after all there are two quite different
acts of the will in question. The one relates to the visual part of the world, the other to the muscular-feeling part.
Page 87
Have we anything more than empirical evidence that the movement of the same part of the body is in
question in both cases?
Page 87
Then is the situation that I merely accompany my actions with my will?
Page 87
But in that case how can I predict--as in some sense I surely can--that I shall raise my arm in five minutes'
time? That I shall will this?
Page 87
This is clear: it is impossible to will without already performing the act of the will.
Page 87
The act of the will is not the cause of the action but is the action itself.
Page 87
One cannot will without acting.
Page 87
If the will has to have an object in the world, the object can be the intended action itself.
Page 87
And the will does have to have an object.
Page Break 88
Page 88
Otherwise we should have no foothold and could not know what we willed.
Page 88
And could not will different things.
Page 88
Does not the willed movement of the body happen just like any unwilled movement in the world, but that it
is accompanied by the will?
Page 88
Yet it is not accompanied just by a wish! But by will.
Page 88
We feel, so to speak, responsible for the movement.
Page 88
My will fastens on to the world somewhere, and does not fasten on to other things.
Page 88
Wishing is not acting. But willing is acting.
Page 88
(My wish relates, e.g., to the movement of the chair, my will to a muscular feeling.)
Page 88
The fact that I will an action consists in my performing the action, not in my doing something else which
causes the action.
Page 88
When I move something I move.
Page 88
When I perform an action I am in action.
Page 88
But: I cannot will everything.--
Page 88
But what does it mean to say: "I cannot will this"?
Page 88
Can I try to will something?
Page 88
For the consideration of willing makes it look as if one part of the world were closer to me than another
(which would be intolerable).
Page 88
But, of course, it is undeniable that in a popular sense there are things that I do, and other things not done by
me.
Page 88
In this way then the will would not confront the world as its equivalent, which must be impossible.
Page 88
The wish precedes the event, the will accompanies it.
Page 88
Suppose that a process were to accompany my wish. Should I have willed the process?
Page 88
Would not this accompanying appear accidental in contrast to the compelled accompanying of the will?
Page Break 89
9.11.16.
Page 89
Is belief a kind of experience?
Page 89
Is thought a kind of experience?
Page 89
All experience is world and does not need the subject.
Page 89
The act of will is not an experience.
19.11.16.
Page 89
What kind of reason is there for the assumption of a willing subject?
Page 89
Is not my world adequate for individuation?
21.11.16.
Page 89
The fact that it is possible to erect the general form of proposition means nothing but: every possible form of
proposition must be FORESEEABLE.
Page 89
And that means: We can never come upon a form of proposition of which we could say: it could not have
been foreseen that there was such a thing as this.
Page 89
For that would mean that we had had a new experience, and that it took that to make this form of proposition
possible.
Page 89
Thus it must be possible to erect the general form of proposition, because the possible forms of proposition
must be a priori. Because the possible forms of proposition are a priori, the general form of proposition exists.
Page 89
In this connexion it does not matter at all whether the given fundamental operations, through which all
propositions are supposed to arise, change the logical level of the propositions, or whether they remain on the same
logical level.
Page 89
If a sentence were ever going to be constructable it would already be constructable.
Page 89
We now need a clarification of the concept of the atomic function and the concept "and so on".
Page 89
The concept "and so on", symbolized by "...." is one of the most important of all and like all the others
infinitely fundamental.
Page 89
For it alone justifies us in constructing logic and mathematics "so on" from the fundamental laws and
primitive signs.
Page 89
The "and so on" makes its appearance right away at the very beginning of the old logic when it is said that
after the primitive signs have been given we can develop one sign after another "so on".
Page Break 90
Page 90
Without this concept we should be stuck at the primitive signs and could not go "on".
Page 90
The concept "and so on" and the concept of the operation are equivalent. [Cf. 5.2523.]
Page 90
After the operation sign there follows the sign "...." which signifies that the result of the operation can in its
turn be taken as the base of the operation; "and so on".
22.11.16.
Page 90
The concept of the operation is quite generally that according to which signs can be constructed according to
a rule.
23.11.16.
Page 90
What does the possibility of the operation depend on?
Page 90
On the general concept of structural similarity.
Page 90
As I conceive, e.g., the elementary propositions, there must be something common to them; otherwise I
could not speak of them all collectively as the "elementary propositions" at all.
Page 90
In that case, however, they must also be capable of being developed from one another as the results of
operations.
Page 90
For if there really is something common to two elementary propositions which is not common to an
elementary proposition and a complex one, then this common thing must be capable of being given general
expression in some way.
24.11.16.
Page 90
When the general characteristic of an operation is known it will also be clear of what elementary component
parts an operation always consists.
Page 90
When the general form of operations is found we have also found the general form of the occurrence of the
concept "and so on".
26.11.16.
Page 90
All operations are composed of the fundamental operations.
28.11.16.
Page 90
Either a fact is contained in another one, or it is independent of it.
2.12.16.
Page 90
The similarity of the generality notation and the argument appears if we write (ax)φx instead of φa. [Cf.
5.523.]
Page Break 91
Page 91
We could introduce the arguments also in such a way that they only occurred on one side of the sign of
identity, i.e. always on the analogy of "(Ex).φx.x = a" instead of "φa".
Page 91
The correct method in philosophy would really be to say nothing except what can be said, i.e. what belongs
to natural science, i.e. something that has nothing to do with philosophy, and then whenever someone else tried to
say something metaphysical to shew him that he had not given any reference to certain signs in his sentences. [See
6.53.]
Page 91
This method would be unsatisfying for the other person (he would not have the feeling that we were teaching
him philosophy) but it would be the only correct one. [See 6.53.]
7.1.17.
Page 91
In the sense in which there is a hierarchy of propositions there is, of course, also a hierarchy of truths and of
negations, etc.
Page 91
But in the sense in which there are, in the most general sense, such things as propositions, there is only one
truth and one negation.
Page 91
The latter sense is obtained from the former by conceiving the proposition in general as the result of the
single operation which produces all propositions from the first level. Etc.
Page 91
The lowest level and the operation can stand for the whole hierarchy.
8.1.17.
Page 91
It is clear that the logical product of two elementary propositions can never be a tautology. [Cf. 6.3751.]
Page 91
If the logical product of two propositions is a contradiction, and the propositions appear to be elementary
propositions, we can see that in this case the appearance is deceptive. (E.g.: A is red and A is green.)
10.1.17.
Page 91
If suicide is allowed then everything is allowed.
Page 91
If anything is not allowed then suicide is not allowed.
Page 91
This throws a light on the nature of ethics, for suicide is, so to speak, the elementary sin.
Page 91
And when one investigates it it is like investigating mercury vapour in order to comprehend the nature of
vapours.
Page 91
Or is even suicide in itself neither good nor evil?
Page Break 92
Page Break 93
APPENDIX I
NOTES ON LOGIC
by
Ludwig Wittgenstein
1913
SUMMARY
Page 93
ONE reason for thinking the old notation wrong is that it is very unlikely that from every proposition p an infinite
number of other propositions not-not-p, not-not-not-not-p, etc., should follow. [Cf. 5.43.]
Page 93
If only those signs which contain proper names were complex then propositions containing nothing but
apparent variables would be simple. Then what about their denials?
Page 93
The verb of a proposition cannot be "is true" or "is false", but whatever is true or false must already contain
the verb. [See 4.063.]
Page 93
Deductions only proceed according to the laws of deduction but these laws cannot justify the deduction.
Page 93
One reason for supposing that not all propositions which have more than one argument are relational
propositions is that if they were, the relations of judgment and inference would have to hold between an arbitrary
number of things.
Page 93
Every proposition which seems to be about a complex can be analysed into a proposition about its
constituents and about the proposition which describes the complex perfectly; i.e., that proposition which is
equivalent to saying the complex exists. [Cf. 2.0201.]
Page 93
The idea that propositions are names of complexes suggests that whatever is not a proper name is a sign for a
relation. Because spatial complexes†1 consist of Things and Relations only and the idea of a complex is taken from
space.
Page 93
In a proposition convert all its indefinables into variables; there then remains a class of propositions which is
not all propositions but a type [Cf. 3.315.]
Page 93
There are thus two ways in which signs are similar. The names "Socrates" and "Plato" are similar: they are
both names. But whatever they have in common must not be introduced before "Socrates" and "Plato" are
introduced. The same applies to a subject-predicate form etc.. Therefore, thing, proposition, subject-predicate forth,
etc., are not indefinables, i.e., types are not indefinables.
Page Break 94
Page 94
When we say A judges that etc., then we have to mention a whole proposition which A judges. It will not do
either to mention only its constituents, or its constituents and form, but not in the proper order. This shows that a
proposition itself must occur in the statement that it is judged; however, for instance, "not-p" may be explained, the
question what is negated must have a meaning.
Page 94
To understand a proposition p it is not enough to know that p implies '"p" is true', but we must also know
that ~p implies "p is false". This shows the bi-polarity of the proposition.
Page 94
To every molecular function a WF†1 scheme corresponds. Therefore we may use the WF scheme itself
instead of the function. Now what the WF scheme does is, it correlates the letters W and F with each proposition.
These two letters are the poles of atomic propositions. Then the scheme correlates another W and F to these poles.
In this notation all that matters is the correlation of the outside poles to the poles of the atomic propositions.
Therefore not-not-p is the same symbol as p. And therefore we shall never get two symbols for the same molecular
function.
Page 94
The meaning of a proposition is the fact which actually corresponds to it.
Page 94
As the ab functions of atomic propositions are bi-polar propositions again we can perform ab operations on
them. We shall, by doing so, correlate two new outside poles via the old outside poles to the poles of the atomic
propositions.
Page 94
The symbolising fact in a-p-b is that, SAY†2 a is on the left of p and b on the right of p; then the correlation
of new poles is to be transitive, so that for instance if a new pole a in whatever way i.e. via whatever poles is
correlated to the inside a, the symbol is not changed thereby. It is therefore possible to construct all possible ab
functions by performing one ab operation repeatedly, and we can therefore talk of all ab functions as of all those
functions which can be obtained by performing this ab operation repeatedly.
Page 94
Naming is like pointing. A function is like a line dividing points of a plane into right and left ones; then "p or
not-p" has no meaning because it does not divide the plane.
Page 94
But though a particular proposition "p or not-p" has no meaning, a general proposition "for all p's, p or
not-p" has a meaning because this
Page Break 95
does not contain the nonsensical function "p or not-p" but the function "p or not-q" just as "for all x's xRx" contains
the function "xRy".
Page 95
A proposition is a standard to which facts behave,†1 with names it is otherwise; it is thus bi-polarity and
sense comes in; just as one arrow behaves†2 to another arrow by being in the same sense or the opposite, so a fact
behaves to a proposition.
Page 95
The form of a proposition has meaning in the following way. Consider a symbol "xRy". To symbols of this
form correspond couples of things whose names are respectively "x" and "y". The things xy stand to one another in
all sorts of relations, amongst others some stand in the relation R, and some not; just as I single out a particular thing
by a particular name I single out all behaviours of the points x and y with respect to the relation R. I say that if an x
stands in the relation R to a y the sign "xRy" is to be called true to the fact and otherwise false. This is a definition of
sense.
Page 95
In my theory p has the same meaning as not-p but opposite sense. The meaning is the fact. The proper
theory of judgment must make it impossible to judge nonsense. [Cf. 4.0621 and 5.5422.]
Page 95
It is not strictly true to say that we understand a proposition p if we know that p is equivalent to "p is true"
for this would be the case if accidentally both were true or false. What is wanted is the formal equivalence with
respect to the forms of the proposition, i.e., all the general indefinables involved. The sense of an ab function of a
proposition is a function of its sense. There are only unasserted propositions. Assertion is merely psychological. In
not-p, p is exactly the same as if it stands alone; this point is absolutely fundamental. Among the facts which make
"p or q" true there are also facts which make "p and q" true; if propositions have only meaning, we ought, in such a
case, to say that these two propositions are identical, but in fact, their sense is different for we have introduced sense
by talking of all p's and all q's. Consequently the molecular propositions will only be used in cases where their ab
function stands under a generality sign or enters into another function such as "I believe that, etc.", because then the
sense enters. [Cf. 5.2341.]
Page 95
In "a judges p" p cannot be replaced by a proper name. This appears if we substitute "a judges that p is true
and not p is false". The proposition "a judges p" consists of the proper name a, the proposition p with its 2 poles, and
a being related to both of these poles in a certain way. This is obviously not a relation in the ordinary sense.
Page 95
The ab notation makes it clear that not and or are dependent on one another and we can therefore not use
them as simultaneous indefinables.
Page Break 96
Same objections in the case of apparent variables to old indefinables, as in the case of molecular functions. The
application of the ab notation to apparent variable propositions becomes clear if we consider that, for instance, the
proposition "for all x, φx" is to be true when φx is true for all x's and false when φx is false for some x's. We see that
some and all occur simultaneously in the proper apparent variable notation.
Page 96
The notation is
for (x)φx: a-(x)-aφxb-(∃x)-b and
for (∃x)φx: a-(∃x)-aφxb-(x)-b
Page 96
Old definitions now become tautologous.
Page 96
In "aRb" it is not the complex that symbolises but the fact that the symbol "a" stands in a certain relation to
the symbol "b". Thus facts are symbolised by facts, or more correctly: that a certain thing is the case in the symbol
says that a certain thing is the case in the world. [Cf. 3.1432.]
Page 96
Judgment, question and command are all on the same level. What interests logic in them is only the
unasserted proposition. Facts cannot be named.
Page 96
A proposition cannot occur in itself. This is the fundamental truth of the theory of types. [Cf. 3.332.]
Page 96
Every proposition that says something indefinable about one thing is a subject-predicate proposition, and so
on.
Page 96
Therefore we can recognize a subject-predicate proposition if we know it contains only one name and one
form, etc. This gives the construction of types. Hence the type of a proposition can be recognized by its symbol
alone.
Page 96
What is essential in a correct apparent-variable notation is this: (1) it must mention a type of propositions; (2)
it must show which components of a proposition of this type are constants.
Page 96
[Components are forms and constituents.]
Page 96
Take (φ).φ!x. Then if we describe the kind of symbols, for which "φ!" stands and which, by the above, is
enough to determine the type, then automatically "(φ).φ!x" cannot be fitted by this description, because it
CONTAINS "φ!x" and the description is to describe ALL that symbolises in symbols of the φ! kind. If the description
is thus complete vicious circles can just as little occur as for instance (φ).(x)φ (where (x)φ is a subject-predicate
proposition).
FIRST MS
Page 96
Indefinables are of two sorts: names, and forms. Propositions cannot consist of names alone; they cannot be
classes of names. A name can not only occur in two different propositions, but can occur in the same way in both.
Page Break 97
Page 97
Propositions [which are symbols having reference to facts] are themselves facts: that this inkpot is on this
table may express that I sit in this chair. [Cf. 2.141 and 3.14.]
Page 97
It can never express the common characteristic of two objects that we designate them by the same name but
by two different ways of designation, for, since names are arbitrary, we might also choose different names, and
where then would be the common element in the designations? Nevertheless one is always tempted, in a difficulty,
to take refuge in different ways of designation. [Cf. 3.322.]
Page 97
Frege said "propositions are names"; Russell said "propositions correspond to complexes". Both are false;
and especially false is the statement "propositions are names of complexes". [Cf. 3.143.]
Page 97
It is easy to suppose that only such symbols are complex as contain names of objects, and that accordingly
"(∃x,φ).φx" or "(∃x,y).xRy" must be simple. It is then natural to call the first of these the name of a form, the second
the name of a relation. But in that case what is the meaning of (e.g.) "~(∃x,y)xRy? Can we put "not" before a name?
Page 97
The reason why "~Socrates" means nothing is that "~x" does not express a property of x.
Page 97
There are positive and negative facts: if the proposition "this rose is not red" is true, then what it signifies is
negative. But the occurrence of the word "not" does not indicate this unless we know that the signification of the
proposition "this rose is red" (when it is true) is positive. It is only from both, the negation and the negated
proposition, that we can conclude to a characteristic of the significance of the whole proposition. (We are not here
speaking of negations of general propositions i.e. of such as contain apparent variables. Negative facts only justify
the negations of atomic propositions.)
Page 97
Positive and negative facts there are, but not true and false facts.
Page 97
If we overlook the fact that propositions have a sense which is independent of their truth or falsehood, it
easily seems as if true and false were two equally justified relations between the sign and what is signified. (We
might then say e.g. that "q" signifies in the true way what "not-q" signifies in the false way.) But are not true and
false in fact equally justified? Could we not express ourselves by means of false propositions just as well as hitherto
with true ones, so long as we know that they are meant falsely? No! For a proposition is then true when it is as we
assert in this proposition; and accordingly if by "q" we mean "not-q", and it is as we mean to assert, then in the new
interpretation "q" is actually true and not false. But it is important that we can mean the same by "q" as by "not-q",
for it shows that neither to the symbol "not" nor to the manner of its combination with "q" does a
Page Break 98
characteristic of the denotation of "q" correspond. [Cf. 4.061, 4.062, 4.0621.]
SECOND MS
Page 98
We must be able to understand propositions which we have never heard before. But every proposition is a
new symbol. Hence we must have general indefinable symbols; these are unavoidable if propositions are not all
indefinable. [Cf. 4.02, 4.021, 4.027.]
Page 98
Whatever corresponds in reality to compound propositions must not be more than what corresponds to their
several atomic propositions.
Page 98
Not only must logic not deal with [particular] things, but just as little with relations and predicates.
Page 98
There are no propositions containing real variables.
Page 98
What corresponds in reality to a proposition depends upon whether it is true or false. But we must be able to
understand a proposition without knowing if it is true or false.
Page 98
What we know when we understand a proposition is this: We know what is the case if the proposition is true,
and what is the case if it is false. But we do not know (necessarily) whether it is true or false. [Cf. 4.024.]
Page 98
Propositions are not names.
Page 98
We can never distinguish one logical type from another by attributing a property to members of the one
which we deny to members of the other.
Page 98
Symbols are not what they seem to be. In "aRb", "R" looks like a substantive, but is not one. What
symbolizes in "aRb" is that R occurs between a and b. Hence "R" is not the indefinable in "aRb". Similarly in "φx",
"φ" looks like a substantive but is not one; in "~p", "~" looks like "φ" but is not like it. This is the first thing that
indicates that there may not be logical constants. A reason against them is the generality of logic: logic cannot treat a
special set of things. [Cf. 3.1423.]
Page 98
Molecular propositions contain nothing beyond what is contained in their atoms; they add no material
information above that contained in their atoms.
Page 98
All that is essential about molecular functions is their T-F schema (i.e. the statement of the cases when they
are true and the cases when they are false).
Page 98
Alternative indefinability shows that the indefinables have not been reached.
Page 98
Every proposition is essentially true-false: to understand it, we must know both what must be the case if it is
true, and what must be the case if it is false. Thus a proposition has two poles, corresponding to the
Page Break 99
case of its truth and the case of its falsehood. We call this the sense of a proposition.
Page 99
In regard to notation, it is important to note that not every feature of a symbol symbolizes. In two molecular
functions which have the same T-F schema, what symbolizes must be the same. In "not-not-p", "not-p" does not
occur; for "not-not-p" is the same as "p", and therefore, if "not-p" occurred in "not-not-p", it would occur in "p".
Page 99
Logical indefinables cannot be predicates or relations, because propositions, owing to sense, cannot have
predicates or relations. Nor are "not" and "or", like judgment, analogous to predicates or relations, because they do
not introduce anything new.
Page 99
Propositions are always complex even if they contain no names.
Page 99
A proposition must be understood when all its indefinables are understood. The indefinables in "aRb" are
introduced as follows:
Page 99
"a" is indefinable;
Page 99
"b" is indefinable;
Page 99
Whatever "x" and "y" may mean, "xRy" says something indefinable about their meaning. [Cf. 4.024.]
Page 99
A complex symbol must never be introduced as a single indefinable. [Thus e.g. no proposition is
indefinable.] For if one of its parts occurs also in another connection, it must there be re-introduced. And would it
then mean the same?
Page 99
The ways by which we introduce our indefinables must permit us to construct all propositions that have
sense from these indefinables alone. It is easy to introduce "all" and "some" in a way that will make the construction
of (say) "(x,y).xRy" possible from "all" and "xRy" as introduced before.
THIRD MS
Page 99
An analogy for the theory of truth: Consider a black patch on white paper; then we can describe the form of
the patch by mentioning, for each point of the surface, whether it is white or black. To the fact that a point is black
corresponds a positive fact, to the fact that a point is white (not black) corresponds a negative fact. If I designate a
point of the surface (one of Frege's "truth-values"), this is as if I set up an assumption to be decided upon. But in
order to be able to say of a point that it is black or that it is white, I must first know when a point is to be called black
and when it is to be called white. In order to be able to say that "p" is true (or false), I must first have determined
under what circumstances I call a proposition true, and thereby I determine the sense of a proposition. The point in
which the analogy fails is this: I can indicate a point of the paper which is white and black,†1 but to a
Page Break 100
proposition without sense nothing corresponds, for it does not designate a thing (truth-value), whose properties
might be called "false" or "true"; the verb of a proposition is not "is true" or "is false", as Frege believes, but what is
true must already contain the verb. [Cf. 5.132.]
Page 100
The comparison of language and reality is like that of retinal image and visual image: to the blind spot
nothing in the visual image seems to correspond, and thereby the boundaries of the blind spot determine the visual
image--as true negations of atomic propositions determine reality.
Page 100
Logical inferences can, it is true, be made in accordance with Frege's or Russell's laws of deduction, but this
cannot justify the inference; and therefore they are not primitive propositions of logic. If p follows from q, it can also
be inferred from q, and the "manner of deduction" is indifferent.
Page 100
Those symbols which are called propositions in which "variables occur" are in reality not propositions at all,
but only schemes of propositions, which only become propositions when we replace the variables by constants.
There is no proposition which is expressed by "x = x", for "x" has no signification; but there is a proposition "(x).x =
x" and propositions such as "Socrates = Socrates" etc.
Page 100
In books on logic, no variables ought to occur, but only the general propositions which justify the use of
variables. It follows that the so-called definitions of logic are not definitions, but only schemes of definitions, and
instead of these we ought to put general propositions; and similarly the so-called primitive ideas (Urzeichen) of logic
are not primitive ideas, but the schemes of them. The mistaken idea that there are things called facts or complexes
and relations easily leads to the opinion that there must be a relation of questioning to the facts, and then the
question arises whether a relation can hold between an arbitrary number of things, since a fact can follow from
arbitrary cases. It is a fact that the proposition which e.g. expresses that q follows from p and p ⊃ q is this: p.p ⊃ q.
⊃p.q.q.
Page 100
At a pinch, one is tempted to interpret "not-p" as "everything else, only not p". That from a single fact p an
infinity of others, not-not-p etc., follow, is hardly credible. Man possesses an innate capacity for constructing
symbols with which some sense can be expressed, without having the slightest idea what each word signifies. The
best example of this is mathematics, for man has until lately used the symbols for numbers without knowing what
they signify or that they signify nothing. [Cf. 5.43.]
Page 100
Russell's "complexes" were to have the useful property of being compounded, and were to combine with this
the agreeable property that they could be treated like "simples". But this alone made them
Page Break 101
unserviceable as logical types, since there would have been significance in asserting, of a simple, that it was
complex. But a property cannot be a logical type.
Page 101
Every statement about apparent complexes can be resolved into the logical sum of a statement about the
constituents and a statement about the proposition which describes the complex completely. How, in each case, the
resolution is to be made, is an important question, but its answer is not unconditionally necessary for the
construction of logic. [Cf. 2.0201.]
Page 101
That "or" and "not" etc. are not relations in the same sense as "right" and "left" etc., is obvious to the plain
man. The possibility of cross-definitions in the old logical indefinables shows, of itself, that these are not the right
indefinables, and, even more conclusively, that they do not denote relations. [Cf. 5.42.]
Page 101
If we change a constituent a of a proposition φ(a) into a variable, then there is a class
{(∃x).φ(x) = p}
This class in general still depends upon what, by an arbitrary convention, we mean by "φ(x)". But if we change into
variables all those symbols whose significance was arbitrarily determined, there is still such a class. But this is not
dependent upon any convention, but only upon the nature of the symbol "φ(x)". It corresponds to a logical type.
[Cf. 3.315.]
Page 101
Types can never be distinguished from each other by saying (as is often done) that one has these but the
other has those properties, for this presupposes that there is a meaning in asserting all these properties of both types.
But from this it follows that, at best, these properties may be types, but certainly not the objects of which they are
asserted. [Cf. 4.1241.]
Page 101
At a pinch we are always inclined to explanations of logical functions of propositions which aim at
introducing into the function either only the constituents of these propositions, or only their form, etc. etc.; and we
overlook that ordinary language would not contain the whole propositions if it did not need them: However, e.g.,
"not p" may be explained, there must always be a meaning given to the question "what is denied?"
Page 101
The very possibility of Frege's explanations of "not-p" and "if p then q", from which it follows that
"not-not-p" denotes the same as p, makes it probable that there is some method of designation in which "not-not-p"
corresponds to the same symbol as "p". But if this method of designation suffices for logic, it must be the right one.
Page 101
Names are points, propositions arrows--they have sense. The sense
Page Break 102
of a proposition is determined by the two poles true and false. The form of a proposition is like a straight line, which
divides all points of a plane into right and left. The line does this automatically, the form of proposition only by
convention. [Cf. 3.144.]
Page 102
Just as little as we are concerned, in logic, with the relation of a name to its meaning, just so little are we
concerned with the relation of a proposition to reality, but we want to know the meaning of names and the sense of
propositions--as we introduce an indefinable concept "A" by saying: "'A' denotes something indefinable", so we
introduce e.g. the form of propositions aRb by saying: "For all meanings of "x" and "y", "xRy" expresses something
indefinable about x and y".
Page 102
In place of every proposition "p", let us write " ": Let every correlation of propositions to each other or of
names to propositions be effected by a correlation of their poles "a" and "b". Let this correlation be transitive. Then
accordingly is the same symbol as " ". Let n propositions be given. I then call a "class of poles" of
these propositions every class of n members, of which each is a pole of one of the n propositions, so that one
member corresponds to each proposition. I then correlate with each class of poles one of two poles (a and b). The
sense of the symbolizing fact thus constructed I cannot define, but I know it.
Page 102
If p = not-not-p etc., this shows that the traditional method of symbolism is wrong, since it allows a plurality
of symbols with the same sense; and thence it follows that, in analyzing such propositions, we must not be guided
by Russell's method of symbolizing.
Page 102
It is to be remembered that names are not things, but classes: "A" is the same letter as "A". This has the most
important consequences for every symbolic language. [Cf. 3.203].
Page 102
Neither the sense nor the meaning of a proposition is a thing. These words are incomplete symbols.
Page 102
It is impossible to dispense with propositions in which the same argument occurs in different positions. It is
obviously useless to replace φ(a,a) by φ(a,b).a = b.
Page 102
Since the ab-functions of p are again bi-polar propositions, we can form ab-functions of them, and so on. In
this way a series of propositions will arise, in which in general the symbolizing facts will be the same in several
members. If now we find an ab-function of such a kind that by repeated application of it every ab-function can be
generated, then we can introduce the totality of ab-functions as the totality of those that are generated by application
of this function. Such a function is ~p ∨ ~q.
Page 102
It is easy to suppose a contradiction in the fact that on the one hand every possible complex proposition is a
simple ab-function of simple
Page Break 103
propositions, and that on the other hand the repeated application of one ab-function suffices to generate all these
propositions. If e.g. an affirmation can be generated by double negation, is negation in any sense contained in
affirmation? Does "p" deny "not-p" or assert "p", or both? And how do matters stand with the definition of "⊃" by
"∨" and ".", or of "∨" by "." and "⊃"? And how e.g. shall we introduce p|q (i.e. ~p ∨ ~q), if not by saying that this
expression says something indefinable about all arguments p and q? But the ab-functions must be introduced as
follows: The function p|q is merely a mechanical instrument for constructing all possible symbols of ab-functions.
The symbols arising by repeated application of the symbol "|" do not contain the symbol "p|q". We need a rule
according to which we can form all symbols of ab functions, in order to be able to speak of the class of them; and
now we speak of them e.g. as those symbols of functions which can be generated by repeated application of the
operation "|". And we say now: For all p's and q's, "p|q" says something indefinable about the sense of those simple
propositions which are contained in p and q. [Cf. 5.44.]
Page 103
The assertion-sign is logically quite without significance. It only shows, in Frege and Whitehead and Russell,
that these authors hold the propositions so indicated to be true. " " therefore belongs as little to the proposition as
(say) the number of the proposition. A proposition cannot possibly assert of itself that it is true. [Cf. 4.442.]
Page 103
Every right theory of judgment must make it impossible for me to judge that this table penholders the book.
Russell's theory does not satisfy this requirement. [See 5.5422.]
Page 103
It is clear that we understand propositions without knowing whether they are true or false. But we can only
know the meaning of a proposisition [[sic]] when we know if it is true or false. What we understand is the sense of
the proposition. [Cf. 4.024.]
Page 103
The assumption of the existence of logical objects makes it appear remarkable that in the sciences
propositions of the form "p ∨ q", "p ⊃ q", etc. are only then not provisional when "∨" and "⊃" stand within the
scope of a generality-sign [apparent variable].
FOURTH MS
Page 103
If we formed all possible atomic propositions, the world would be completely described if we declared the
truth or falsehood of each. [Cf. 4.26.]
Page 103
The chief characteristic of my theory is that, in it, p has the same meaning as not-p. [Cf. 4.0621.]
Page 103
A false theory of relations makes it easily seem as if the relation of
Page Break 104
fact and constituent were the same as that of fact and fact which follows from it. But the similarity of the two may be
expressed thus:
φa. ⊃φ,a a = a.
Page 104
If a word creates a world so that in it the principles of logic are true, it thereby creates a world in which the
whole of mathematics holds; and similarly it could not create a world in which a proposition was true, without
creating its constituents. [Cf. 5.123.]
Page 104
Signs of the form "p ∨ ~p" are senseless, but not the proposition "(p).p ∨ ~p". If I know that this rose is
either red or not red, I know nothing. The same holds of all ab-functions. [Cf. 4.461.]
Page 104
To understand a proposition means to know what is the case if it is true. Hence we can understand it without
knowing if it is true. We understand it when we understand its constituents and forms. If we know the meaning of
"a" and "b", and if we know what "xRy" means for all x's and y's, then we also understand "aRb". [Cf. 4.024.]
Page 104
I understand the proposition "aRb" when I know that either the fact that aRb or the fact that not aRb
corresponds to it; but this is not to be confused with the false opinion that I understood "aRb" when I know that
"aRb or not aRb" is the case.
Page 104
But the form of a proposition symbolizes in the following way: Let us consider symbols of the form "xRy";
to these correspond primarily pairs of objects, of which one has the name "x", the other the name "y". The x's and
y's stand in various relations to each other, among others the relation R holds between some, but not between
others. I now determine the sense of "xRy" by laying down: when the facts behave in regard to†1 "xRy" so that the
meaning of "x" stands in the relation R to the meaning of "y", then I say that the [the facts] are "of like sense"
["gleichsinnig"] with the proposition "xRy"; otherwise, "of opposite sense" [entgegengesetzt"]; I correlate the facts
to the symbol "xRy" by thus dividing them into those of like sense and those of opposite sense. To this correlation
corresponds the correlation of name and meaning. Both are psychological. Thus I understand the form "xRy" when
I know that it discriminates the behaviour of x and y according as these stand in the relation R or not. In this way I
extract from all possible relations the relation R, as, by a name, I extract its meaning from among all possible things.
Page 104
Strictly speaking, it is incorrect to say: we understand the proposition p when we know that "'p' is true" ≡ p;
for this would naturally always be the case if accidentally the propositions to right and left of the symbol "≡" were
both true or both false. We require not only an
Page Break 105
equivalence, but a formal equivalence, which is bound up with the introduction of the form of p.
Page 105
The sense of an ab-function of p is a function of the sense of p. [Cf. 5.2341.]
Page 105
The ab-functions use the discrimination of facts, which their arguments bring forth, in order to generate new
discriminations.
Page 105
Only facts can express sense, a class of names cannot. This is easily shown.
Page 105
There is no thing which is the form of a proposition, and no name which is the name of a form. Accordingly
we can also not say that a relation which in certain cases holds between things holds sometimes between forms and
things. This goes against Russell's theory of judgment.
Page 105
It is very easy to forget that, though the propositions of a form can be either true or false, each one of these
propositions can only be either true or false, not both.
Page 105
Among the facts which make "p or q" true, there are some which make "p and q" true; but the class which
makes "p or q" true is different from the class which makes "p and q" true; and only this is what matters. For we
introduce this class, as it were, when we introduce ab-functions. [Cf. 5.1241.]
Page 105
A very natural objection to the way in which I have introduced e.g. propositions of the form xRy is that by it
propositions such as (∃.x.y).xRy and similar ones are not explained, which yet obviously have in common with aRb
what cRd has in common with aRb. But when we introduce propositions of the form xRy we mentioned no one
particular proposition of this form; and we only need to introduce (∃x,y).φ(x,y) for all φ's in any way which makes
the sense of these propositions dependent on the sense of all propositions of the form φ(a,b), and thereby the
justness of our procedure is proved.
Page 105
The indefinables of logic must be independent of each other. If an indefinable is introduced, it must be
introduced in all combinations in which it can occur. We cannot therefore introduce it first for one combination, then
for another; e.g., if the form xRy has been introduced it must henceforth be understood in propositions of the form
aRb just in the same way as in propositions such as (∃x,y).xRy and others. We must not introduce it first for one
class of cases, then for the other; for it would remain doubtful if its meaning was the same in both cases, and there
would be no ground for using the same matter of combining symbols in both cases. In short for the introduction of
indefinable symbols and combinations of symbols the same holds, mutatis mutandis, that Frege has said for the
introduction of symbols by definitions. [Cf. 5.451.]
Page Break 106
Page 106
It is a priori likely that the introduction of atomic propositions is fundamental for the understanding of all
other kinds of propositions. In fact the understanding of general propositions obviously depends on that of atomic
propositions.
Page 106
Cross-definability in the realm of general propositions leads to quite similar questions to those in the realm of
ab-functions.
Page 106
When we say "A believes p", this sounds, it is true, as if here we could substitute a proper name for "p"; but
we can see that here a sense, not a meaning, is concerned, if we say "A believes that 'p' is true"; and in order to make
the direction of p even more explicit, we might say "A believes that 'p' is true and 'not-p' is false". Here the bipolarity
of p is expressed, and it seems that we shall only be able to express the proposition "A believes p" correctly by the
ab-notation; say by making "A" have a relation to the poles "a" and "b" of a-p-b. The epistemological questions
concerning the nature of judgment and belief cannot be solved without a correct apprehension of the form of the
proposition.
Page 106
The ab-notation shows the dependence of or and not, and thereby that they are not to be employed as
simultaneous indefinables.
Page 106
Not: "The complex sign 'ago'" says that a stands in the relation R to b; but that 'a' stands in a certain relation
to 'b' says that aRb. [3.1432.]
Page 106
In philosophy there are no deductions: it is purely descriptive.
Page 106
Philosophy gives no pictures of reality.
Page 106
Philosophy can neither confirm nor confute scientific investigation.
Page 106
Philosophy consists of logic and metaphysics: logic is its basis.
Page 106
Epistemology is the philosophy of psychology. [Cf. 4.1121.]
Page 106
Distrust of grammar is the first requisite for philosophizing.
Page 106
Propositions can never be indefinables, for they are always complex. That also words like "ambulo" are
complex appears in the fact that their root with a different termination gives a different sense. [Cf. 4.032.]
Page 106
Only the doctrine of general indefinables permits us to understand the nature of functions. Neglect of this
doctrine leads to an impenetrable thicket.
Page 106
Philosophy is the doctrine of the logical form of scientific propositions (not only of primitive propositions).
Page 106
The word "philosophy" ought always to designate something over or under but not beside, the natural
sciences. [Cf. 4.111.]
Page Break 107
Page 107
Judgment, command and question all stand on the same level; but all have in common the propositional
form, which does interest us.
Page 107
The structure of the proposition must be recognized, the rest comes of itself. But ordinary language conceals
the structure of the proposition: in it, relations look like predicates, predicates like names, etc.
Page 107
Facts cannot be named.
Page 107
It is easy to suppose that "individual", "particular", "complex" etc. are primitive ideas of logic. Russell e.g.
says "individual" and "matrix" are "primitive ideas". This error presumably is to be explained by the fact that, by
employment of variables instead of the generality-sign, it comes to seem as if logic dealt with things which have
been deprived of all properties except thing-hood, and with propositions deprived of all properties except
complexity. We forget that the indefinables of symbols [Urbilder von Zeichen] only occur under the generality-sign,
never outside it.
Page 107
Just as people used to struggle to bring all propositions into the subject-predicate form, so now it is natural to
conceive every proposition as expressing a relation, which is just as incorrect. What is justified in this desire is fully
satisfied by Russell's theory of manufactured relations.
Page 107
One of the most natural attempts at solution consists in regarding "not-p" as "the opposite of p", where then
"opposite" would be the indefinable relation. But it is easy to see that every such attempt to replace the ab-functions
by descriptions must fail.
Page 107
The false assumption that propositions are names leads us to believe that there must be logical objects: for
the meanings of logical propositions will have to be such things.
Page 107
A correct explanation of logical propositions must give them a unique position as against all other
propositions.
Page 107
No proposition can say anything about itself, because the symbol of the proposition cannot be contained in
itself; this must be the basis of the theory of logical types. [Cf. 3.332.]
Page 107
Every proposition which says something indefinable about a thing is a subject-predicate proposition; every
proposition which says something indefinable about two things expresses a dual relation between these things, and
so on. Thus every proposition which contains only one name and one indefinable form is a subject-predicate
proposition, and so on. An indefinable simple symbol can only be a name, and therefore we can know, by the
symbol of an atomic proposition, whether it is a subject-predicate proposition.
Page Break 108
APPENDIX II
NOTES DICTATED TO G. E. MOORE IN NORWAY†1
April 1914
Page 108
LOGICAL so-called propositions shew [the] logical properties of language and therefore of [the] Universe, but say
nothing. [Cf. 6.12.]
Page 108
This means that by merely looking at them you can see these properties; whereas, in a proposition proper,
you cannot see what is true by looking at it. [Cf. 6.113.]
Page 108
It is impossible to say what these properties are, because in order to do so, you would need a language,
which hadn't got the properties in question, and it is impossible that this should be a proper language. Impossible to
construct [an] illogical language.
Page 108
In order that you should have a language which can express or say everything that can be said, this language
must have certain properties; and when this is the case, that it has them can no longer be said in that language or any
language.
Page 108
An illogical language would be one in which, e.g., you could put an event into a hole.
Page 108
Thus a language which can express everything mirrors certain properties of the world by these properties
which it must have; and logical so-called propositions shew in a systematic way those properties.
Page 108
How, usually, logical propositions do shew these properties is this: We give a certain description of a kind of
symbol; we find that other symbols, combined in certain ways, yield a symbol of this description; and that they do
shews something about these symbols.
Page 108
As a rule the description given in ordinary Logic is the description of a tautology; but others might shew
equally well, e.g., a contradiction. [Cf. 6.1202.]
Page 108
Every real proposition shews something, besides what it says, about the Universe: for, if it has no sense, it
can't be used; and if it has a sense, it mirrors some logical property of the Universe.
Page 108
E.g., take φa, φa ⊃ ψa, ψa. By merely looking at these three, I can see that 3 follows from 1 and 2; i.e. I can
see what is called the truth of a logical proposition, namely, of [the] proposition φa.φa ⊃ ψa: ⊃:ψa. But this is not a
proposition; but by seeing that it is a tautology I can
Page Break 109
see what I already saw by looking at the three propositions: the difference is that I now see THAT it is a tautology.
[Cf. 6.1221.]
Page 109
We want to say, in order to understand [the] above, what properties a symbol must have, in order to be a
tautology.
Page 109
Many ways of saying this are possible:
Page 109
One way is to give certain symbols; then to give a set of rules for combining them; and then to say: any
symbol formed from those symbols, by combining them according to one of the given rules, is a tautology. This
obviously says something about the kind of symbol you can get in this way.
Page 109
This is the actual procedure of [the] old Logic: it gives so-called primitive propositions; so-called rules of
deduction; and then says that what you get by applying the rules to the propositions is a logical proposition that you
have proved. The truth is, it tells you something about the kind of proposition you have got, viz that it can be
derived from the first symbols by these rules of combination (= is a tautology).
Page 109
Therefore, if we say one logical proposition follows logically from another, this means something quite
different from saying that a real proposition follows logically from another. For so-called proof of a logical
proposition does not prove its truth (logical propositions are neither true nor false) but proves that it is a logical
proposition = is a tautology. [Cf. 6.1263.]
Page 109
Logical propositions are forms of proofs: they shew that one or more propositions follow from one (or
more). [Cf. 6.1264.]
Page 109
Logical propositions shew something, because the language in which they are expressed can say everything
that can be said.
Page 109
This same distinction between what can be shewn by the language but not said, explains the difficulty that is
felt about types--e.g., as to [the] difference between things, facts, properties, relations. That M is a thing can't be
said; it is nonsense: but something is shewn by the symbol "M". In [the] same way, that a proposition is a
subject-predicate proposition can't be said: but it is shown by the symbol.
Page 109
Therefore a THEORY of types is impossible. It tries to say something about the types, when you can only
talk about the symbols. But what you say about the symbols is not that this symbol has that type, which would be
nonsense for [the] same reason: but you say simply: This is the symbol, to prevent a misunderstanding. E.g., in
"aRb", "R" is not a symbol, but that "R" is between one name and another symbolizes. Here we have not said: this
symbol is not of this type but of that, but only: This symbolizes and not that. This seems again to make the same
Page Break 110
mistake, because "symbolizes" is "typically ambiguous". The true analysis is: "R" is no proper name, and, that "R"
stands between "a" and "b" expresses a relation. Here are two propositions of different type connected by "and".
Page 110
It is obvious that, e.g., with a subject-predicate proposition, if it has any sense at all, you see the form, so
soon as you understand the proposition, in spite of not knowing whether it is true or false. Even if there were
propositions of [the] form "M is a thing" they would be superfluous (tautologous) because what this tries to say is
something which is already seen when you see "M".
Page 110
In the above expression "aRb", we were talking only of this particular "R", whereas what we want to do is to
talk of all similar symbols. We have to say: in any symbol of this form what corresponds to "R" is not a proper
name, and [the] fact that ["R" stands between "a" and "b"] expresses a relation. This is what is sought to be
expressed by the nonsensical assertion: Symbols like this are of a certain type. This you can't say, because in order
to say it you must first know what the symbol is: and in knowing this you see the type and therefore also [the] type
of [what is] symbolized. I.e. in knowing what symbolizes, you know all that is to be known; you can't say anything
about the symbol.
Page 110
For instance: Consider the two propositions (1) "What symbolizes here is a thing", (2) "What symbolizes
here is a relational fact = relation". These are nonsensical for two reasons: (a) because they mention "thing" and
"relation"; (b) because they mention them in propositions of the same form. The two propositions must be
expressed in entirely different forms, if properly analysed; and neither the word "thing" nor "relation" must occur.
Page 110
Now we shall see how properly to analyse propositions in which "thing", "relation", etc., occur.
Page 110
(1) Take φx. We want to explain the meaning of 'In "φx" a thing symbolizes'. The analysis is:--
Page 110
(∃y). y symbolizes. y = "x". "φx"
Page 110
["x" is the name of y: "φx" = '"φ" is at [the] left of "x"' and says φx.]
Page 110
N.B. "x" can't be the name of this actual scratch y, because this isn't a thing: but it can be the name of a
thing; and we must understand that what we are doing is to explain what would be meant by saying of an ideal
symbol, which did actually consist in one thing's being to the left of another, that in it a thing symbolized.
Page Break 111
Page 111
[N.B. In [the] expression (∃y).φy, one is apt to say this means "There is a thing such that... ". But in fact we
should say "There is a y, such that... "; the fact that the y symbolizes expressing what we mean.]
Page 111
In general: When such propositions are analysed, while the words "thing", "fact", etc. will disappear, there
will appear instead of them a new symbol, of the same form as the one of which we are speaking; and hence it will
be at once obvious that we cannot get the one kind of proposition from the other by substitution.
Page 111
In our language names are not things: we don't know what they are: all we know is that they are of a
different type from relations, etc. etc.. The type of a symbol of a relation is partly fixed by [the] type of [a] symbol of
[a] thing, since a symbol of [the] latter type must occur in it.
Page 111
N.B. In any ordinary proposition, e.g., "Moore good", this shews and does not say that "Moore" is to the left
of "good"; and here what is shewn can be said by another proposition. But this only applies to that part of what is
shewn which is arbitrary. The logical properties which it shews are not arbitrary, and that it has these cannot be said
in any proposition.
Page 111
When we say of a proposition of [the] form "aRb" that what symbolizes is that "R" is between "a" and "b", it
must be remembered that in fact the proposition is capable of further analysis because a, R, and b are not simples.
But what seems certain is that when we have analysed it we shall in the end come to propositions of the same form
in respect of the fact that they do consist in one thing being between two others.†1
Page 111
How can we talk of the general form of a proposition, without knowing any unanalysable propositions in
which particular names and relations occur? What justifies us in doing this is that though we don't know any
unanalysable propositions of this kind, yet we can understand what is meant by a proposition of the form (∃x, y,
R).xRy (which is unanalysable), even though we know no proposition of the form xRy.
Page 111
If you had any unanalysable proposition in which particular names and relations occurred (and unanalysable
proposition = one in which only fundamental symbols = ones not capable of definition, occur) then you always can
form from it a proposition of the form (∃x, y, R).xRy, which though it contains no particular names and relations, is
unanalysable.
Page 111
(2) The point can here be brought out as follows. Take φa and φA:
Page Break 112
and ask what is meant by saying, "There is a thing in φa, and a complex in φA"?
(1) means: (∃x).φx.x = a
(2) (∃x,ψξ).φA = ψx.φx.†1
Page 112
Use of logical propositions. You may have one so complicated that you cannot, by looking at it, see that it is
a tautology; but you have shewn that it can be derived by certain operations from certain other propositions
according to our rule for constructing tautologies; and hence you are enabled to see that one thing follows from
another, when you would not have been able to see it otherwise. E.g., if our tautology is of [the] form p ⊃ q you can
see that q follows from p; and so on.
Page 112
The Bedeutung of a proposition is the fact that corresponds to it, e.g., if our proposition be "aRb", if it's true,
the corresponding fact would be the fact aRb, if false, the fact ~aRb. But both "the fact aRb" and "the fact ~aRb" are
incomplete symbols, which must be analysed.
Page 112
That a proposition has a relation (in wide sense) to Reality, other than that of Bedeutung, is shewn by the
fact that you can understand it when you don't know the Bedeutung, i.e. don't know whether it is true or false. Let us
express this by saying "It has sense" (Sinn).
Page 112
In analysing Bedeutung, you come upon Sinn as follows:
Page 112
We want to explain the relation of propositions to reality.
Page 112
The relation is as follows: Its simples have meaning = are names of simples; and its relations have a quite
different relation to relations; and these two facts already establish a sort of correspondence between a proposition
which contains these and only these, and reality: i.e. if all the simples of a proposition are known, we already know
that we CAN describe reality by saying that it behaves†2 in a certain way to the whole proposition. (This amounts to
saying that we can compare reality with the proposition. In the case of two lines we can compare them in respect of
their length without any convention: the comparison is automatic. But in our case the possibility of comparison
depends upon the conventions by which we have given meanings to our simples (names and relations).)
Page 112
It only remains to fix the method of comparison by saying what about our simples is to say what about
reality. E.g., suppose we take two lines of unequal length; and say that the fact that the shorter is of the length it is is
to mean that the longer is of the length it is. We should then have established a convention as to the meaning of the
shorter, of the sort we are now to give.
Page Break 113
Page 113
From this it results that "true" and "false" are not accidental properties of a proposition, such that, when it
has meaning, we can say it is also true or false: on the contrary, to have meaning means to be true or false: the being
true or false actually constitutes the relation of the proposition to reality, which we mean by saying that it has
meaning (Sinn).
Page 113
There seems at first sight to be a certain ambiguity in what is meant by saying that a proposition is "true",
owing to the fact that it seems as if, in the case of different propositions, the way in which they correspond to the
facts to which they correspond is quite different. But what is really common to all cases is that they must have the
general form of a proposition. In giving the general form of a proposition you are explaining what kind of ways of
putting together the symbols of things and relations will correspond to (be analogous to) the things having those
relations in reality. In doing thus you are saying what is meant by saying that a proposition is true; and you must do
it once for all. To say "This proposition has sense" means '"This proposition is true" means....' ("p" is true = "p". p.
Def.: only instead of "p" we must here introduce the general form of a proposition.)†1
-----
Page 113
It seems at first sight as if the ab notation must be wrong, because it seems to treat true and false as on
exactly the same level. It must be possible to see from the symbols themselves that there is some essential difference
between the poles, if the notation is to be right; and it seems as if in fact this was impossible.
Page 113
The interpretation of a symbolism must not depend upon giving a different interpretation to symbols of the
same types.
Page 113
How asymmetry is introduced is by giving a description of a particular form of symbol which we call a
"tautology". The description of the ab-symbol alone is symmetrical with respect to a and b; but this description plus
the fact that what satisfies the description of a tautology is a tautology is asymmetrical with regard to them. (To say
that a description was symmetrical with regard to two symbols, would mean that we could substitute one for the
other, and yet the description remain the same, i.e. mean the same.)
Page 113
Take p.q and q. When you write p.q in the ab notation, it is impossible to see from the symbol alone that q
follows from it, for if you were to interpret the true-pole as the false, the same symbol would stand for p ∨ q, from
which q doesn't follow. But the moment you
Page Break 114
say which symbols are tautologies, it at once becomes possible to see from the fact that they are and the original
symbol that q does follow.
Page 114
Logical propositions, OF COURSE, all shew something different: all of them shew, in the same may, viz by
the fact that they are tautologies, but they are different tautologies and therefore shew each something different.
Page 114
What is unarbitrary about our symbols is not them, nor the rules we give; but the fact that, having given
certain rules, others are fixed = follow logically. [Cf. 3.342.]
Page 114
Thus, though it would be possible to interpret the form which we take as the form of a tautology as that of a
contradiction and vice versa, they are different in logical form because though the apparent form of the symbols is
the same, what symbolizes in them is different, and hence what follows about the symbols from the one
interpretation will be different from what follows from the other. But the difference between a and b is not one of
logical form, so that nothing will follow from this difference alone as to the interpretation of other symbols. Thus,
e.g., p.q., p ∨ q seem symbols of exactly the same logical form in the ab notation. Yet they say something entirely
different; and, if you ask why, the answer seems to be: In the one case the scratch at the top has the shape b, in the
other the shape a. Whereas the interpretation of a tautology as a tautology is an interpretation of a logical form, not
the giving of a meaning to a scratch of a particular shape. The important thing is that the interpretation of the form of
the symbolism must be fixed by giving an interpretation to its logical properties, not by giving interpretations to
particular scratches.
Page 114
Logical constants can't be made into variables: because in them what symbolizes is not the same; all symbols
for which a variable can be substituted symbolize in the same way.
Page 114
We describe a symbol, and say arbitrarily "A symbol of this description is a tautology". And then, it follows
at once, both that any other symbol which answers to the same description is a tautology, and that any symbol
which does not isn't. That is, we have arbitrarily fixed that any symbol of that description is to be a tautology; and
this being fixed it is no longer arbitrary with regard to any other symbol whether it is a tautology or not.
Page 114
Having thus fixed what is a tautology and what is not, we can then, having fixed arbitrarily again that the
relation a-b is transitive get from the two facts together that "p ≡ ~(~p)" is a tautology. For ~(~p) = a-b-a-p-b-a-b.
The point is: that the process of reasoning by which we arrive at the result that a-b-a-p-b-a-b is the same symbol as
a-p-b, is
Page Break 115
exactly the same as that by which we discover that its meaning is the same, viz where we reason if b-a-p-b-a, then
not a-p-b, if a-b-a-p-b-a-b then not b-a-p-b-a, therefore if a-b-a-p-b-a-b, then a-p-b.
Page 115
It follows from the fact that a-b is transitive, that where we have a-b-a, the first a has to the second the same
relation that it has to b. It is just as from the fact that a-true implies b-false, and b-false implies c-true, we get that
a-true implies c-true. And we shall be able to see, having fixed the description of a tautology, that p ≡ ~(~p) is a
tautology.
Page 115
That, when a certain rule is given, a symbol is tautological shews a logical truth.
Page 115
This symbol might be interpreted either as a tautology or a contradiction.
Page 115
In settling that it is to be interpreted as a tautology and not as a contradiction, I am not assigning a meaning
to a and b; i.e. saying that they symbolize different things but in the same way. What I am doing is to say that the
way in which the a-pole is connected with the whole symbol symbolizes in a different way from that in which it
would symbolize if the symbol were interpreted as a contradiction. And I add the scratches a and b merely in order
to shew in which ways the connexion is symbolizing, so that it may be evident that wherever the same scratch
occurs in the corresponding place in another symbol, there also the connexion is symbolizing in the same way.
Page 115
We could, of course, symbolize any ab-function without using two outside poles at all, merely, e.g., omitting
the b-pole; and here what would symbolize would be that the three pairs of inside poles of the propositions were
connected in a certain way with the a-pole, while the other pair was not connected with it. And thus the difference
between the scratches a and b, where we do use them, merely shews that it is a different state of things that is
symbolizing in the one case and the other: in the one case that certain inside poles are connected in a certain way
with an outside pole, in the other that they're not.
Page Break 116
Page 116
The symbol for a tautology, in whatever form we put it, e.g., whether by omitting the a-pole or by omitting
the b, would always be capable of being used as the symbol for a contradiction; only not in the same language.
Page 116
The reason why ~x is meaningless, is simply that we have given no meaning to the symbol ~ξ. I.e. whereas
φx and φp look as if they were of the same type, they are not so because in order to give a meaning to ~x you would
have to have some property ~ξ. What symbolizes in φξ is that φ stands to the left of a proper name and obviously
this is not so in ~p. What is common to all propositions in which the name of a property (to speak loosely) occurs is
that this name stands to the left of a name-form.
Page 116
The reason why, e.g., it seems as if "Plato Socrates" might have a meaning, while "Abracadabra Socrates"
will never be suspected to have one, is because we know that "Plato" has one, and do not observe that in order that
the whole phrase should have one, what is necessary is not that "Plato" should have one, but that the fact that
"Plato" is to the left of a name should.
Page 116
The reason why "The property of not being green is not green" is nonsense, is because we have only given
meaning to the fact that "green" stands to the right of a name; and "the property of not being green" is obviously not
that.
Page 116
φ cannot possibly stand to the left of (or in any other relation to) the symbol of a property. For the symbol of
a property, e.g., ψx is that ψ stands to the left of a name form, and another symbol φ cannot possibly stand to the
left of such a fact: if it could, we should have an illogical language, which is impossible.
p is false = ~(p is true) Def.
Page 116
It is very important that the apparent logical relations ∨, ⊃, etc. need brackets, dots, etc., i.e. have "ranges";
which by itself shews they are not relations. This fact has been overlooked, because it is so universal--the very thing
which makes it so important. [Cf. 5.461.]
Page 116
There are internal relations between one proposition and another; but a proposition cannot have to another
the internal relation which a name has to the proposition of which it is a constituent, and which ought to be meant
by saying it "occurs" in it. In this sense one proposition can't "occur" in another.
Page 116
Internal relations are relations between types, which can't be expressed in propositions, but are all shewn in
the symbols themselves, and can be exhibited systematically in tautologies. Why we come to
Page Break 117
call them "relations" is because logical propositions have an analogous relation to them, to that which properly
relational propositions have to relations.
Page 117
Propositions can have many different internal relations to one another. The one which entitles us to deduce
one from another is that if, say, they are φa and φa ⊃ ψa, then φa.φa ⊃ ψa: ⊃:ψa is a tautology.
Page 117
The symbol of identity expresses the internal relation between a function and its argument: i.e. φa = (∃x).φx.x
= a.
Page 117
The proposition (∃x).φx.x = a: ≡:φa can be seen to be a tautology, if one expresses the conditions of the truth
of (∃x). φx. x = a, successively, e.g., by saying: This is true if so and so; and this again is true if so and so, etc., for
(∃x). φx. x = a; and then also for φa. To express the matter in this way is itself a cumbrous notation, of which the
ab-notation is a neater translation.
Page 117
What symbolizes in a symbol, is that which is common to all the symbols which could in accordance with
the rules of logic = syntactical rules for manipulation of symbols, be substituted for it. [Cf. 3.344.]
Page 117
The question whether a proposition has sense (Sinn) can never depend on the truth of another proposition
about a constituent of the first. E.g., the question whether (x) x = x has meaning (Sinn) can't depend on the question
whether (∃x) x = x is true. It doesn't describe reality at all, and deals therefore solely with symbols; and it says that
they must symbolize, but not what they symbolize.
Page 117
It's obvious that the dots and brackets are symbols, and obvious that they haven't any independent meaning.
You must, therefore, in order to introduce so-called "logical constants" properly, introduce the general notion of all
possible combinations of them = the general form of a proposition. You thus introduce both ab-functions, identity,
and universality (the three fundamental constants) simultaneously.
Page 117
The variable proposition p ⊃ p is not identical with the variable proposition ~(p.~p). The corresponding
universals would be identical. The variable proposition ~(p.~p) shews that out of ~(p.q) you get a tautology by
substituting ~p for q, whereas the other does not shew this.
Page 117
It's very important to realize that when you have two different relations (a,b)R, (c,d)S this does not establish a
correlation between a and c, and b and d, or a and d, and b and c: there is no correlation whatsoever thus established.
Of course, in the case of two pairs of terms united by the same relation, there is a correlation. This shews that the
theory which held that a relational fact contained the terms and
Page Break 118
relations united by a copula (ε2 is untrue; for if this were so there would be a correspondence between the terms of
different relations.
Page 118
The question arises how can one proposition (or function) occur in another proposition? The proposition or
function itself can't possibly stand in relation to the other symbols. For this reason we must introduce functions as
well as names at once in our general form of a proposition; explaining what is meant, by assigning meaning to the
fact that the names stand between the | ,†1 and that the function stands on the left of the names.
Page 118
It is true, in a sense, that logical propositions are "postulates"--something which we "demand"; for we
demand a satisfactory notation. [Cf. 6.1223.]
Page 118
A tautology (not a logical proposition) is not nonsense in the same sense in which, e.g., a proposition in
which words which have no meaning occur is nonsense. What happens in it is that all its simple parts have meaning,
but it is such that the connexions between these paralyse or destroy one another, so that they are all connected only
in some irrelevant manner.
-----
Page 118
Logical functions all presuppose one another. Just as we can see ~p has no sense, if p has none; so we can
also say p has none if ~p has none. The case is quite different with φa, and a; since here a has a meaning
independently of φa, though φa presupposes it.
Page 118
The logical constants seem to be complex-symbols, but on the other hand, they can be interchanged with
one another. They are not therefore really complex; what symbolizes is simply the general way in which they are
combined.
Page 118
The combination of symbols in a tautology cannot possibly correspond to any one particular combination of
their meanings--it corresponds to every possible combination; and therefore what symbolizes can't be the
connexion of the symbols.
Page 118
From the fact that I see that one spot is to the left of another, or that one colour is darker than another, it
seems to follow that it is so; and if so, this can only be if there is an internal relation between the two; and we might
express this by saying that the form of the latter is part of the form of the former. We might thus give a sense to the
assertion that logical laws are forms of thought and space and time forms of intuition.
Page Break 119
Page 119
Different logical types can have nothing whatever in common. But the mere fact that we can talk of the
possibility of a relation of n places, or of an analogy between one with two places and one with four, shews that
relations with different numbers of places have something in common, that therefore the difference is not one of
type, but like the difference between different names--something which depends on experience. This answers the
question how we can know that we have really got the most general form of a proposition. We have only to
introduce what is common to all relations of whatever number of places.
Page 119
The relation of "I believe p" to "p" can be compared to the relation of '"p" says (besagt) p' to p: it is just as
impossible that I should be a simple as that "p" should be. [Cf. 5.542.]
Page Break 120
APPENDIX III
EXTRACTS FROM WITTGENSTEIN'S LETTERS TO
RUSSELL, 1912-20
Cambridge, 22.6.12.
Page 120
... Logic is still in the melting pot but one thing gets more and more obvious to me: The propositions of Logic
contain ONLY apparent variables and whatever may turn out to be the proper explanation of apparent variables, its
consequences must be that there are NO logical constants.
Page 120
Logic must turn out to be a totally different kind than any other science.
1.7.12.
Page 120
... Will you think that I have gone mad if I make the following suggestion?: The sign "(x).φx" is not a complete
symbol but has meaning only in an inference of the kind: from φx⊃xψx.φ(a) follows ψa. Or more general: from
(x).φx.ε0(a) follows φ(a). I am--of course--most uncertain about the matter but something of the sort might really
be true.
Hochreit, Post Hohenberg, Nieder-Österreich. (Summer, 1912.)
Page 120
... What troubles me most at present, is not the apparent-variable-business, but rather the meaning of "∨" "⊃" etc.
This latter problem is--I think--still more fundamental and, if possible, still less recognized as a problem. If "p∨q"
means a complex at all--which is quite doubtful--then, as far as I can see, one must treat "∨" as part of a copula in
the way we have talked over before. I have--I believe--tried all possible ways of solution under that hypothesis and
found that if any one will do it must be something like this:
Page 120
Let us write the proposition "from p and q follows r" that way: "i(p; q; r)". Here "i" is a copula (we
may call it inference) which copulates complexes.
Page 120
Then " (ε1 (x,y).∨.ε1 (u,z)" is to mean:
" (ε1 (x,y), ε1 (z,u), β (x,y,z,u)).i[ε1 (x,y); ε1 (z,u); β (x,y,z,u)]
(ε1 (x,y), ε1 (z,u), β (x,y,z,u)).i[~ε1 (x,y); ε1 (z,u); β (x,y,z,u)]
(ε1 (x,y), ε1 (z,u), β (x,y,z,u)).i[ε1 (x,y); ~ε1 (z,u); β (x,y,z,u)]
(ε1 (x,y), ε1 (z,u), β (x,y,z,u)).i[~ε1 (x,y); ~ε1 (z,u); β (x,y,z,u)]
(x,y,z,u)".
Page Break 121
Page 121
If "p∨q" does not mean a complex, then heaven knows what it means!!
August, 1912.
Page 121
... Now as to 'p∨q' etc: I have thought that possibility--namely that all our troubles could be overcome by assuming
different sorts of Relations of signs to things--over and over and over again! for the last eight weeks!!! But I have
come to the conclusion that this assumption does not help us a bit. In fact if you work out any such theory--I believe
you will see that it does not even touch our problem. I have lately seen a new way out (or perhaps not out) of the
difficulty. It is too long to be explained here, but I tell you so much, that it is based on new forms of propositions.
For instance: ~|~ (p,q), which is to mean 'the complex p has the opposite form of q's form'. That means that ~|~ (p,q)
holds for instance when p is ε1 (a,b) and q is ~ε1 (c,d). Another instance of the new forms is (p,q,r) which
means something like: "The form of the complex r is composed of the forms of p and q in the way 'or'". That means
that (p,q,r) holds for instance when p is ε1 (a,b), q is ε1 (c,d) and r is ε1 (e,f) ∨ ε1 (g,h) etc. etc. The rest I leave to
your imagination.
1912.
Page 121
I believe that our problems can be traced down to the atomic propositions. This you will see if you try to
explain precisely in what way the Copula in such a proposition has meaning.
Page 121
I cannot explain it and I think that as soon as an exact answer to this question is given the problem of "∨"
and of the apparent variable will be brought very near to their solution if not solved. I now think about "Socrates is
human" (Good old Socrates!).
IV Alleegasse 16. Wien. 26.12.12.
Page 121
... I had a long discussion with Frege about our theory of symbolism of which, I think, he roughly understood the
general outline. He said he would think the matter over. The complex-problem is now clearer to me and I hope very
much that I may solve it.
IV Alleegasse 16. Jan. 1913.
Page 121
... I have changed my views on "atomic" complexes: I now think that qualities, relations (like love) etc. are all
copulae! That means I for instance analyse a subject-predicate proposition, say, "Socrates is
Page Break 122
human" into "Socrates" and "something is human", (which I think is not complex). The reason for this is a very
fundamental one: I think that there cannot be different Types of things! In other words whatever can be symbolized
by a simple proper name must belong to one type. And further: every theory of types must be rendered superfluous
by a proper theory of symbolism: For instance if I analyse the proposition Socrates is mortal into Socrates, mortality
and (∃x,y) ε1 (x,y) I want a theory of types to tell me that "mortality is Socrates" is nonsensical, because if I treat
"mortality" as a proper name (as I did) there is nothing to prevent me to make the substitution the wrong way round.
But if I analyse (as I do now) into Socrates and (∃x).x is mortal or generally into x and (∃x)φx it becomes impossible
to substitute the wrong way round because the two symbols are now of a different kind themselves. What I am most
certain of is not however the correctness of my present way of analysis, but of the fact that all theory of types must
be done away with by a theory of symbolism showing that what seem to be different kinds of things are
symbolized by different kinds of symbols which cannot possibly be substituted in one another's places. I hope I
have made this fairly clear!
Page 122
Propositions which I formerly wrote ε2 (a,R,b) I now write R(a,b) and analyse them into a,b and
June, 1913.
Page 122
... I can now express my objection to your theory of judgement exactly: I believe it is obvious that, from the
proposition "A judges that (say) a is in a relation R to b", if correctly analysed, the proposition "a R b.∨.~a R b"
must follow directly without the use of any other premiss. This condition is not fulfilled by your theory.
Hochreit, Post Hohenberg, Nieder-Österreich, 22.7.13.
Page 122
... My work goes on well; every day my problems get clearer now and I feel rather hopeful. All my progress comes
out of the idea that the indefinables of Logic are of the general kind (in the same way as the so-called definitions of
Logic are general) and this again comes from the abolition of the real variable.
Page 122
... I am very sorry to hear that my objection to your theory of judgment paralyses you. I think it can only be
removed by a correct theory of propositions.
Page Break 123
Hochreit, Post Hohenberg, N.-Ö.
Page 123
(This letter seems to have been written sometime near to that of the letter dated 22.7.13.)
Page 123
Your axiom of reducibility is
:(∃f):[f}x≡x f!x;
now is this not all nonsense as this proposition has only then a meaning if we can turn the φ into an apparent
variable. For if we cannot do so no general laws can ever follow from your axiom. The whole axiom seems to me at
present a mere juggling trick. Do let me know if there is more in it. The axiom as you have put it is only a schema
and the real Pp ought to be
:.(φ):(∃f):φ(x)≡x f!x,
and where would be the use of that?--
5.9.13.
Page 123
I am sitting here in a little place inside a beautiful fiord and thinking about the beastly theory of types. There
are still some very difficult problems (and very fundamental ones too) to be solved and I won't begin to write until I
have got some sort of a solution for them. However I don't think that will in any way affect the bipolarity business
which still seems to me to be absolutely untangible.
c/o Draegni, Skjolden, Sogn, Norway. 29.10.13.
Page 123
... Identity is the very Devil and immensely important; very much more so than I thought. It hangs--like everything
else--directly together with the most fundamental questions, especially with the questions concerning the occurrence
of the same argument in different places of a function. I have all sorts of ideas for a solution of the problem but
could not yet arrive at anything definite. However, I don't lose courage and go on thinking.
30.10.
Page 123
I wrote this†1 letter yesterday. Since then quite new ideas have come into my mind; new problems have
arisen in the theory of molecular propositions and the theory of inference has received a new and very important
aspect. One of the consequences of my new ideas will--I think--be that the whole of Logic follows from one Pp
only!! I cannot say mote about it at present.
Page Break 124
1913.
Page 124
Thanks for your letter and the typed stuff!†1 I will begin by answering your questions as well as I can:
Page 124
(1) Your question was--I think due to the misprint (polarity instead of bi-polarity). What I mean to say is that
we only then understand a proposition if we know both what would be the case if it was false and what if it was
true.
Page 124
(2) The symbol for ~p is a-b-p-a-b. The proposition p has two poles and it does not matter a hang where they
stand. You might just as well write ~p like this
or b--a--p--b--a etc. etc. All that is important is that the new a-pole should be correlated to the old b-pole and vice
versa wherever these old poles may stand. If you had only remembered the WF scheme of ~p you would never
have asked this question (I think). In fact all rules of the ab symbolism follow directly from the essence of the WF
scheme.
Page 124
(3) Whether ab-functions and your truth-functions are the same cannot yet be decided.
Page 124
(4) "The correlation of new poles is to be transitive" means that by correlating one pole in the symbolizing
way to another and the other to a third we have thereby correlated the first in the symbolizing way to the third, etc..
For instance in a-b-a-bpa-b-a-b, a and b are correlated to b and a respectively and this means that our symbol is the
same as a-bpa-b.
Page 124
(5) (p) p ∨ ~p is derived from the function p ∨ ~q but the point will only become quite clear when identity is
clear (as you said). I will some other time write to you about this matter at length.
Page 124
(6) Explanation in the typed stuff.
Page 124
(7) You say, you thought that Bedeutung was the "fact", this is quite true, but remember that there are no
such things as facts and that therefore this proposition itself wants analysing. If we speak of "die Bedeutung" we
seem to be speaking of a thing with a proper name. Of course the symbol for "a fact" is a proposition and this is no
incomplete symbol.
Page Break 125
Page 125
(8) The exact a-b indefinable is given in the manuscript.
Page 125
(9) An account of general indefinables? Oh Lord! It is too boring!!! Some other time!--Honestly--I will write
to you about it some time, if by that time you have not found out all about it (because it is all quite clear in the
manuscript I think). But just now I am so troubled with Identity that I really cannot write any long jaw. All sorts of
new logical stuff seems to be growing in me, but I can't yet write about it.
Page 125
... The following is a list of the questions you asked me in your letter of the 25th.10.:
Page 125
(1) "What is the point of 'p.≡. "p" is true'? I mean why is it worth saying?"
Page 125
(2) "If 'apb' is the symbol for p, is 'bpa' the symbol for ~p? and if not, what is?"
Page 125
(3) "What you call ab-functions are what the Principia calls 'truth-functions'. I don't see why you shouldn't
stick to the name 'truth-functions'."
Page 125
(4) "I don't understand your rules about a's and b's, i.e. 'the correlation of new poles is to be transitive'."
Page 125
(5) (Is obvious from my letter) so is (6).
Page 125
(7) "You say 'Weder der Sinn noch die Bedeutung eines Satzes ist ein Ding. Jene Worte sind unvollständige
Zeichen". I understand neither being a thing, but I thought the Bedeutung was the fact, which is surely not indicated
by an incomplete symbol?"
Page 125
I don't know whether I have answered the question (7) clearly. The answer is of course this: The Bedeutung
of a proposition is symbolized by the proposition--which is of course not an incomplete symbol, but the word
"Bedeutung" is an incomplete symbol.
Page 125
(8) and (9) are obvious.
Nov., 1913.
Page 125
... I beg you to notice that, although I shall make use in what follows of my ab notation, the meaning of this notation
is not needed; that is to say, even if this notation should turn out not to be the final correct notation what I am going
to say is valid if you only admit--as I believe you must do--that it is a possible notation. Now listen! I will first talk
about those logical propositions which are or might be contained in the first 8 chapters of Principia Mathematica.
That they all follow from one proposition is clear because one symbolic rule is
Page Break 126
sufficient to recognize each of them as true or false. And this is the one symbolic rule: write the proposition down in
the ab notation, trace all connections (of poles) from the outside to the inside poles: Then if the b-pole is connected
to such groups of inside poles only as contain opposite poles of one proposition, then the whole proposition is a
true, logical proposition. If on the other hand this is the case with the a-pole the proposition is false and logical. If
finally neither is the case the proposition may be true or false, but it is in no case logical. Such for instance (p).~p--p
transmuted to a suitable type, of course--is not a logical proposition at all and its truth can neither be proved nor
disproved from logical propositions alone. The same is the case--by the way--with your axiom of reducibility, it is
not a logical proposition at all and the same applies to the axioms of infinity and the multiplicative axiom. If these
are true propositions they are what I shall call "accidentally" true and not "essentially" true. Whether a
proposition is accidentally or essentially true can be seen by writing it down in the ab notation and applying the
above rule. What I--in stating this rule--called "logical" proposition is a proposition which is either essentially true or
essentially false. This distinction of accidentally and essentially true propositions explains--by the way--the feeling
one always had about the infinity axiom and the axiom of reducibility, the feeling that if they were true they would
be so by a lucky accident.
Page 126
Of course the rule I have given applies first of all only for what you called elementary propositions. But it is
easy to see that it must also apply to all others. For consider your two Pps in the theory of apparent variables *9.1
and *9.11. Put then instead of φx, (∃y).φy.y = x and it becomes obvious that the special cases of these two Pps like
those of all the previous ones become tautologous if you apply the ab notation. The ab Notation for Identity is not
yet clear enough to show this clearly but it is obvious that such a Notation can be made up. I can sum up by saying
that a logical proposition is one the special cases of which are either tautologous--and then the proposition is true--or
self-contradictory (as I shall call it) and then it is false. And the ab notation simply shows directly which of these two
it is (if any).
Page 126
That means that there is one method of proving or disproving all logical propositions and this is: writing them
down in the ab notation and looking at the connections and applying the above rule. But if one symbolic rule will do,
there must also be one Pp that will do. There is much that follows from all this and much that I could only explain
vaguely but if you really think it over you will find that I am right.
Page Break 127
Norway, 1913.
Page 127
... Ich will dasjenige, was ich in meinem letzten Brief über Logik schrieb, noch einmal in anderer Weise wiederholen:
Alle Sätze der Logik sind Verallgemeinerungen von Tautologien and alle Verallgemeinerungen von Tautologien sind
Sätze der Logik. Andere logische Sätze gibt es nicht. (Dies halte ich für definitiv). Ein Satz wie "(∃x).x = x" zum
Beispiel ist eigentlich ein Satz der Physik. Der Satz "(x):x = x.⊃.(∃y).y = y" ist ein Satz der Logik; es ist nun Sache
der Physik zu sagen, ob es ein Ding gibt. Dasselbe gilt vom infinity axiom; ob es ℵ0 Dinge gibt, das zu bestimmen
ist Sache der Erfahrung (und die kann es nicht entscheiden). Nun aber zu Deinem Reductions-Axiom: Stell' Dir vor,
wir leben in einer Welt, worin es nichts als Dinge gäbe and außerdem nur noch eine Relation, welche zwischen
unendlich vielen dieser Dinge bestehe and zwar so, daß sie nicht zwischen jedem Ding and jedem anderen besteht,
and daß sie ferner auch nie zwischen einer endlichen Anzahl von Dingen besteht. Es ist klar, daß das axiom of
reducibility in einer solchen Welt sicher nicht bestünde. Es ist mir aber auch klar, daß es nicht die Sache der Logik
ist darüber zu entscheiden, ob die Welt worin wir leben nun wirklich so ist, oder nicht. Was aber Tautologien
eigentlich sind, das kann ich selber noch nicht ganz klar sagen, will aber trachten es ungefähr zu erklären. Es ist das
eigentümliche (und höchst wichtige) Merkmal der nicht-logischen Sätze, daß man ihre Wahrheit nicht am
Satzzeichen selbst erkennen kann. Wenn ich z. B. sage "Meier ist dumm", so kannst Du dadurch, daß Du diesen
Satz anschaust, nicht sagen ob er wahr oder falsch ist. Die Sätze der Logik aber--und sie allein--haben die
Eigenschaft, daß sich ihre Wahrheit bezw. Falschheit schon in ihrem Zeichen ausdrückt. Es ist mir noch nicht
gelungen, für die Identität eine Bezeichnung zu finden, die dieser Bedingung genügt; aber ich zweifle nicht, daß sich
eine solche Bezeichnungsweise finden lassen muß. Für zusammengesetzte Sätze (elementary propositions) genügt
die ab-Bezeichnungsweise. Es ist mir unangenehm, daß Du die Zeichenregel an meinem letzten Brief nicht
verstanden hast, denn es langweilt mich unsagbar sie zu erklären!! Du könntest sie auch durch ein bißchen
Nachdenken selber finden!
Dies ist das Zeichen für p≡p; es ist tautologisch weil b nur mit solchen Polpaaren verbunden ist, welche aus den
entgegengesetzten Polen eines Satzes (nämlich p) bestehen; wenn Du dies auf Sätze anwendest, die mehr als 2
Argumente haben so erhälst Du die allgemeine Regel,
Page Break 128
wonach Tautologien gebildet werden. Ich bitte Dich denke selbst über die Sache nach, es ist mir schrecklich eine
schriftliche Erklärung zu wiederholen, die ich schon zum ersten Mal mit dem allergrößten Widerstreben gegeben
habe. Die Identität ist mir--wie gesagt--noch gar nicht klar. Also hierüber ein andermal! Wenn Dein Axiom of
Reducibility fällt, so wird manches geändert werden müssen. Warum gebrauchst Du als Definition der Klassen nicht
diese:
F[ (φx)] =:φz≡zψx.⊃ψ.F(ψ) Def. ?
... Die große Frage ist jetzt: Wie muß ein Zeichensystem beschaffen sein, damit es jede Tautologie auf eine and
dieselbe Weise als Tautologie erkennen läßt? Dies ist das Grundproblem der Logik!
Page 128
... Ich will nur noch sagen, daß Deine Theorie der "Descriptions" ganz zweifellos richtig ist, selbst wenn die
einzelnen Urzeichen darin ganz andere sind als Du glaubst.
Page 128
Translation of the above:
I want to repeat what I wrote about logic in my last letter, putting it in a different way: All propositions of
logic are generalizations of tautologies and all generalizations of tautologies are propositions of logic. There are no
logical propositions but these. (I consider this to be definitive.) A proposition like "(∃x)x = x" is for example really a
proposition of physics. The proposition "(x):x = x.⊃.(∃y).y = y" is a proposition of logic: it is for physics to say
whether any thing exists. The same holds of the infinity axiom; whether there are ℵ0 things is for experience to
settle (and experience can't decide it). But now for your reducibility axiom: Imagine our living in a world, where
there is nothing but things, and besides only one relation, which holds between infinitely many of these things, but
does not hold between every one and every other of them: further, it never holds between a finite number of things.
It is clear that the axiom of reducibility would certainly not hold in such a world. But it is also clear to me that it is
not for logic to decide whether the world we live in is actually like this or not. However, I can't myself say quite
clearly yet what tautologies really are, but I'll try to give a rough account. It is the peculiar (and most important)
characteristic of non-logical propositions, that their truth cannot be seen in the propositional sign itself. If I say for
example 'Meier is stupid', you cannot tell whether this proposition is true or false by looking at it. But the
propositions of logic--and they alone--have the property of expressing their truth or falsehood in the very sign itself.
I haven't yet succeeded in getting a notation for identity which satisfies this condition; but I don't doubt that such a
Page Break 129
notation must be discoverable. For compounded propositions (elementary propositions) the ab notation is adequate.
I am upset that you did not understand the rule for the signs in my last letter, since it bores me unspeakably to
explain it! You could get at it for yourself if you would think a bit!
This is the sign for p ≡ p; it is tautological because b is connected only with such pairs of poles as consist of opposed
poles of a proposition (p); if you apply this to propositions with more than 2 arguments, you get the general rule
according to which tautologies are constructed. Please think the matter over yourself, I find it awful to repeat a
written explanation, which I gave the first time with the greatest reluctance. So, another time! If your Axiom of
Reducibility fails, various things will have to be altered. Why don't you use the following as a definition of classes:
F[ (φx)] =:φz≡zψx.⊃ψ.F(ψ) Def.?
... The great question is now: How should a notation be constructed, which will make every tautology recognizable
as a tautology in one and the same way? This is the fundamental problem of logic.
Page 129
... The only other thing I want to say is that your Theory of Descriptions is quite undoubtedly right, even if the
individual primitive signs in it are quite different from what you believe.
Skjolden, 15.12.13.
Page 129
... Die Frage nach dem Wesen der Identität läßt sich nicht beantworten, ehe das Wesen der Tautologies erklärt ist.
Die Frage nach diesem aber ist die Grundfrage aller Logik.
Page 129
... The question of the nature of identity cannot be answered until the nature of tautologies is explained. But that
question is the fundamental question of all logic.
Skjolden/Januar, 1914/.
Page 129
... Jetzt noch eine Frage: Sagt der "Satz vom zureichenden Grunde" (Law of causality) nicht einfach, daß Raum and
Zeit relativ sind? Dies scheint mir jetzt ganz klar zu sein; denn alle die Ereignisse von denen dieser Satz behaupten
soll, daß sie nicht eintreten können, könnten überhaupt nur in einer absoluten Zeit and einem absoluten
Page Break 130
Raum eintreten. (Dies wäre freilich noch kein unbedingter Grund zu meiner Behauptung.) Aber denke an den Fall
des Massenteilchens, das, allein in der Welt existierend, and seit aller Ewigkeit in Ruhe, plötzlich im Zeitpunkt A
anfängt sich zu bewegen; und denke an ähnliche Fälle, so wirst Du--glaube ich--sehen, daß keine Einsicht a priori
uns solche Ereignisse als unmöglich erscheinen läßt, außer eben in dem Fall, da Raum and Zeit relativ sind. Bitte
schreibe mir Deine Meinung in diesem Punkte.
Page 130
... Now another question: Doesn't the "Principle of Sufficient Reason" (Law of Causality) simply say that space and
time are relative? At present this seems to me to be quite clear; for all the events, whose occurrence this principle is
supposed to exclude, could only occur at all in an absolute time and an absolute space. (This would not of course
quite justify my assertion.) But think of the case of a particle, which was the only thing in the world and had been at
rest from all eternity, and then suddenly begins to move at a moment of time A; and of similar cases: then you will
see--or so I believe--that no a priori insight makes such events seem impossible to us, except in the case of space
and time's being relative. Please write me your opinion on this point.
Cassino, 19.8.19.†1
Page 130
(1) "What is the difference between Tatsache and Sachverhalt?" Sachverhalt is, what corresponds to an
Elementarsatz if it is true. Tatsache is what corresponds to the logical product of elementary props when this product
is true. The reason why I introduce Tatsache before introducing Sachverhalt would want a long explanation.
Page 130
(2) "... But a Gedanke is a Tatsache: what are its constituents and components, and what is their relation to
those of the pictured Tatsache?" I don't know what the constituents of a thought are but I know that it must have
such constituents which correspond to the words of Language. Again the kind of relation of the constituents of the
thought and of the pictured fact is irrelevant. It would be a matter of psychology to find out.
Page 130
(3) "The theory of types, in my view, is a theory of correct symbolism: a simple symbol must not be used to
express anything complex: more generally, a symbol must have the same structure as its meaning." That's exactly
what one can't say. You cannot prescribe to a symbol
Page Break 131
what it may be used to express. All that a symbol can express, it may express. This is a short answer but it is true!
Page 131
(4) "Does a Gedanke consist of words?" No! But of psychical constituents that have the same sort of relation
to reality as words. What those constituents are I don't know.
Page 131
(5) "It is awkward to be unable to speak of NccV†1." This touches the cardinal question of what can be
expressed by a prop, and what can't be expressed, but only shown. I can't explain it at length here. Just think that,
what you want to say by the apparent proposition "There are 2 things" is shown by there being two names which
have different meanings (or by there being one name which may have two meanings). A proposition e.g. φ(a,b) or
(∃φ,x,y).φ(x,y) doesn't say that there are two things, it says something quite different; but whether it's true or false,
it shows you what you want to express by saying: "there are 2 things".
Page 131
(6) Of course no elementary props are negative.
Page 131
(7) "It is necessary also to be given the proposition that all elementary propositions are given." This is not
necessary because it is even impossible. There is no such proposition! That all elementary propositions are given is
shown by there being none having an elementary sense which is not given. This is again the same story as in No. 5.
Page 131
(8) I suppose you didn't understand the way how I separate in the old notation of generality what is in it
truth-function and what is purely generality. A general proposition is a truth-function of all propositions of a certain
form.
Page 131
(9) You are quite right in saying that "N( )" may also be made to mean ~p∨~q ∨~r ∨.... But this doesn't
matter! I suppose you don't understand the notation "ξ". It does not mean "for all values of ξ...". But all is said in my
book about it and I feel unable to write it again.
9.4.20.
Page 131
Besten Dank für Dein Manuskript.†2 Ich bin mit so manchem darin nicht ganz einverstanden; sowohl dort,
wo Du mich kritisierst, als auch dort, wo Du bloß meine Ansicht klarlegen willst. Das macht aber nichts. Die Zukunft
wird über uns urteilen. Oder auch nicht--und wenn sie schweigen wird, so wird das auch ein Urteil sein.
Page Break 132
Page 132
Many thanks for your manuscript. I am not quite in agreement with a lot of it: both where you criticize me,
and where you are merely trying to expound my views. But it doesn't matter. The future will judge between us. Or it
won't--and if it is silent, that will be a judgment too.
6.5.20.
Page 132
... Nun wirst Du aber auf mich böse sein, wenn ich Dir etwas erzähle; Deine Einleitung wird nicht gedruckt and
infolgedessen wahrscheinlich auch mein Buch nicht.--Als ich nämlich die deutsche Übersetzung der Einleitung vor
mir hatte, da konnte ich mich doch nicht entschließen, sie mit meiner Arbeit drucken zu lassen. Die Feinheit Deines
englischen Stils war nämlich in der Übersetzung--selbstverständlich--verloren gegangen and was übrig blieb, war
Oberflächlichkeit and Mißverständnis. Ich schickte nun die Abhandlung and Deine Einleitung an Reclam und
schrieb ihm, ich wünschte nicht, daß die Einleitung gedruckt würde, sondern sie solle ihm nur zur Orientierung über
meine Arbeit dienen. Es ist nun höchst wahrscheinlich, daß Reclam meine Arbeit daraufhin nicht nimmt (obwohl ich
noch keine Antwort von ihm habe).
Page 132
... Now, however, you will be angry at what I have to tell you: your introduction will not be printed, and in
consequence probably neither will my book. For when I got the German translation of the introduction, I couldn't
bring myself to have it printed with my work after all. For the fineness of your English style was--of course--quite
lost and what was left was superficiality and misunderstanding. Now I have sent the treatise and your introduction to
Reclam, and have written to him that I did not want the introduction to be printed, but that he should just use it to
orientate himself about my work. It is now extremely likely that in consequence Reclam will not take my work
(although I have not had an answer from him yet).
Page Break 133
APPENDIX IV†*
Page Break 134
Page Break 135
Page Break 136
FOOTNOTE
Page 7
†1 This remark refers to an incident, about which Wittgenstein later told several of his friends. (Cf. G. H. von
Wright, Ludwig Wittgenstein, a Biographical Sketch in the Philosophical Review, Vol. LXIV, 1955, pp. 532-533.)
To judge from the date of the present MS., however, this incident cannot very well have taken place in a trench on
the East Front. (Edd.)
Page 8
†1 Referring back.
Page 15
†1 I render 'Bedeutung', here and elsewhere, by 'reference' in order to bring it especially to the reader's
attention, (a) that Wittgenstein was under the influence of Frege in his use of 'Sinn' ('sense') and 'Bedeutung'
('reference' or 'meaning' in the sense 'what a word or sentence stands for') and (b) that there is a great contrast
between his idea at this stage of the Notebooks and those of the Tractatus, where he denies that logical constants or
sentences have 'Bedeutung'. (Translator.)
Page 16
†1 To be read as: the class of all classes of elements u such that φu, such that nothing is φ. (Edd.)
Page 25
†1 ab-functions are the truth-functions. Cf. Appendix I. [Edd.]
Page 55
†1 I.e. the theory of a proposition as a class. (Edd.)
Page 86
†1 Schiller, Prologue to Wallensteins Lager. [Edd.]
Page 93
†1 Russell for instance imagines every fact as a spatial complex.
Page 94
†1 W-F = Wahr-Falsch--i.e. True-False.
Page 94
†2 This is quite arbitrary but, if we once have fixed on which order the poles have to stand we must of course
stick to our convention. If for instance "apb" says p then bpa says nothing. (It does not say ~p). But a-apb-b is the
same symbol as apb (here the ab-function vanishes automatically) for here the new poles are related to the same side
of p as the old ones. The question is always: how are the new poles correlated to p compared with the way the old
poles are correlated to p.
Page 95
†1 I.e. sich verhalten, are related. Edd.
Page 95
†2 I.e. sich verhält, is related. Edd.
Page 99
†1 Sic in Russell's MS.; but comparison with the Tractatus shows that "without knowing" has fallen out
after 'paper'. Edd.
Page 104
†1 I.e. sich verhalten zu, are related to. Edd.
Page 108
†1 Square brackets round whole sentences or paragraphs are Wittgenstein's; otherwise they mark something
supplied in editing.
Page 111
†1 This paragraph is lightly deleted.
Page 112
†1 ξ is Frege's mark of an Argumentstelle, to show that ψ is a Funktionsbuchstabe. There are several deleted
and partly illegible definitions.
Page 112
†2 Presumably "verhält sich zu", i.e. "stands towards." [Edd.]
Page 113
†1 The reader should remember that according to Wittgenstein '"p"' is not a name but a description of the fact
constituting the proposition. See above, p. 109. [Edd.]
Page 118
†1 Possibly "between the Sheffer-strokes".
Page 123
†1 The foregoing letter; the present extract is a postscript. [Edd.]
Page 124
†1 Presumably the 1913 Notes on Logic. [Edd.]
Page 130
†1 Wittgenstein had sent Russell a copy of the Tractatus by the hand of Keynes, and the following letter is a
reply to Russell's queries about the book. [Edd.]
Page 131
Tractatus Logico-Philosophicus
Logisch-philosophische Abhandlung
By Ludwig Wittgenstein
First published by Kegan Paul (London), 1922.
SIDE-BY-SIDE-BY-SIDE EDITION, VERSION 0.25 (NOVEMBER 8, 2011),
containing the original German, alongside both the Ogden/Ramsey, and Pears/McGuinness English translations.
1
Introduction
By Bertrand Russell, F. R. S.
MR. WITTGENSTEIN’S Tractatus Logico-Philosophicus, whether or not
it prove to give the ultimate truth on the matters with which it deals,
certainly deserves, by its breadth and scope and profundity, to be considered
an important event in the philosophical world. Starting from
the principles of Symbolism and the relations which are necessary
between words and things in any language, it applies the result of this
inquiry to various departments of traditional philosophy, showing in
each case how traditional philosophy and traditional solutions arise
out of ignorance of the principles of Symbolism and out of misuse of
language.
The logical structure of propositions and the nature of logical inference
are first dealt with. Thence we pass successively to Theory of
Knowledge, Principles of Physics, Ethics, and finally the Mystical (das
Mystische).
In order to understand Mr. Wittgenstein’s book, it is necessary to
realize what is the problem with which he is concerned. In the part
of his theory which deals with Symbolism he is concerned with the
conditions which would have to be fulfilled by a logically perfect language.
There are various problems as regards language. First, there is
the problem what actually occurs in our minds when we use language
with the intention of meaning something by it; this problem belongs to
psychology. Secondly, there is the problem as to what is the relation
subsisting between thoughts, words, or sentences, and that which they
refer to or mean; this problem belongs to epistemology. Thirdly, there
is the problem of using sentences so as to convey truth rather that
falsehood; this belongs to the special sciences dealing with the subjectmatter
of the sentences in question. Fourthly, there is the question:
what relation must one fact (such as a sentence) have to another in
order to be capable of being a symbol for that other? This last is a logical
question, and is the one with which Mr. Wittgenstein is concerned.
He is concerned with the conditions for accurate Symbolism, i.e. for
Symbolism in which a sentence “means” something quite definite. In
practice, language is always more or less vague, so that what we assert
is never quite precise. Thus, logic has two problems to deal with in
regard to Symbolism: (1) the conditions for sense rather than nonsense
in combinations of symbols; (2) the conditions for uniqueness of meaning
or reference in symbols or combinations of symbols. A logically
perfect language has rules of syntax which prevent nonsense, and has
single symbols which always have a definite and unique meaning. Mr.
Wittgenstein is concerned with the conditions for a logically perfect
language—not that any language is logically perfect, or that we believe
ourselves capable, here and now, of constructing a logically perfect
language, but that the whole function of language is to have meaning,
and it only fulfills this function in proportion as it approaches to the
ideal language which we postulate.
The essential business of language is to assert or deny facts. Given
the syntax of language, the meaning of a sentence is determined as
soon as the meaning of the component words is known. In order that a
certain sentence should assert a certain fact there must, however the
language may be constructed, be something in common between the
structure of the sentence and the structure of the fact. This is perhaps
the most fundamental thesis of Mr. Wittgenstein’s theory. That which
has to be in common between the sentence and the fact cannot, he
contends, be itself in turn said in language. It can, in his phraseology,
only be shown, not said, for whatever we may say will still need to
have the same structure.
The first requisite of an ideal language would be that there should
be one name for every simple, and never the same name for two different
simples. A name is a simple symbol in the sense that it has no
parts which are themselves symbols. In a logically perfect language
nothing that is not simple will have a simple symbol. The symbol for
the whole will be a “complex”, containing the symbols for the parts. (In
speaking of a “complex” we are, as will appear later, sinning against
the rules of philosophical grammar, but this is unavoidable at the
outset. “Most propositions and questions that have been written about
philosophical matters are not false but senseless. We cannot, therefore,
2
answer questions of this kind at all, but only state their senselessness.
Most questions and propositions of the philosophers result from the
fact that we do not understand the logic of our language. They are
of the same kind as the question whether the Good is more or less
identical than the Beautiful” (4.003).) What is complex in the world
is a fact. Facts which are not compounded of other facts are what Mr.
Wittgenstein calls Sachverhalte, whereas a fact which may consist of
two or more facts is a Tatsache: thus, for example “Socrates is wise” is
a Sachverhalt, as well as a Tatsache, whereas “Socrates is wise and
Plato is his pupil” is a Tatsache but not a Sachverhalt.
He compares linguistic expression to projection in geometry. A
geometrical figure may be projected in many ways: each of these ways
corresponds to a different language, but the projective properties of
the original figure remain unchanged whichever of these ways may
be adopted. These projective properties correspond to that which in
his theory the proposition and the fact must have in common, if the
proposition is to assert the fact.
In certain elementary ways this is, of course, obvious. It is impossible,
for example, to make a statement about two men (assuming for the
moment that the men may be treated as simples), without employing
two names, and if you are going to assert a relation between the two
men it will be necessary that the sentence in which you make the
assertion shall establish a relation between the two names. If we say
“Plato loves Socrates”, the word “loves” which occurs between the word
“Plato” and the word “Socrates” establishes a certain relation between
these two words, and it is owing to this fact that our sentence is able
to assert a relation between the persons named by the words “Plato”
and “Socrates”. “We must not say, the complex sign ‘aRb’ says that ‘a
stands in a certain relation R to b’; but we must say, that ‘a’ stands in
a certain relation to ‘b’ says that aRb” (3.1432).
Mr. Wittgenstein begins his theory of Symbolism with the statement
(2.1): “We make to ourselves pictures of facts.” A picture, he says,
is a model of the reality, and to the objects in the reality correspond the
elements of the picture: the picture itself is a fact. The fact that things
have a certain relation to each other is represented by the fact that in
the picture its elements have a certain relation to one another. “In the
picture and the pictured there must be something identical in order
that the one can be a picture of the other at all. What the picture must
have in common with reality in order to be able to represent it after
its manner—rightly or falsely—is its form of representation” (2.161,
2.17).
We speak of a logical picture of a reality when we wish to imply only
so much resemblance as is essential to its being a picture in any sense,
that is to say, when we wish to imply no more than identity of logical
form. The logical picture of a fact, he says, is a Gedanke. A picture can
correspond or not correspond with the fact and be accordingly true or
false, but in both cases it shares the logical form with the fact. The
sense in which he speaks of pictures is illustrated by his statement:
“The gramophone record, the musical thought, the score, the waves
of sound, all stand to one another in that pictorial internal relation
which holds between language and the world. To all of them the logical
structure is common. (Like the two youths, their two horses and their
lilies in the story. They are all in a certain sense one)” (4.014). The
possibility of a proposition representing a fact rests upon the fact that
in it objects are represented by signs. The so-called logical “constants”
are not represented by signs, but are themselves present in the proposition
as in the fact. The proposition and the fact must exhibit the same
logical “manifold”, and this cannot be itself represented since it has
to be in common between the fact and the picture. Mr. Wittgenstein
maintains that everything properly philosophical belongs to what can
only be shown, or to what is in common between a fact and its logical
picture. It results from this view that nothing correct can be said
in philosophy. Every philosophical proposition is bad grammar, and
the best that we can hope to achieve by philosophical discussion is to
lead people to see that philosophical discussion is a mistake. “Philosophy
is not one of the natural sciences. (The word ‘philosophy’ must
mean something which stands above or below, but not beside the natural
sciences.) The object of philosophy is the logical clarification of
thoughts. Philosophy is not a theory but an activity. A philosophical
work consists essentially of elucidations. The result of philosophy is
not a number of ‘philosophical propositions’, but to make propositions
clear. Philosophy should make clear and delimit sharply the thoughts
which otherwise are, as it were, opaque and blurred” (4.111 and 4.112).
In accordance with this principle the things that have to be said in
3
leading the reader to understand Mr. Wittgenstein’s theory are all of
them things which that theory itself condemns as meaningless. With
this proviso we will endeavour to convey the picture of the world which
seems to underlie his system.
The world consists of facts: facts cannot strictly speaking be defined,
but we can explain what we mean by saying that facts are what
makes propositions true, or false. Facts may contain parts which are
facts or may contain no such parts; for example: “Socrates was a wise
Athenian”, consists of the two facts, “Socrates was wise”, and “Socrates
was an Athenian.” A fact which has no parts that are facts is called by
Mr. Wittgenstein a Sachverhalt. This is the same thing that he calls
an atomic fact. An atomic fact, although it contains no parts that are
facts, nevertheless does contain parts. If we may regard “Socrates is
wise” as an atomic fact we perceive that it contains the constituents
“Socrates” and “wise”. If an atomic fact is analyzed as fully as possible
(theoretical, not practical possibility is meant) the constituents finally
reached may be called “simples” or “objects”. It is a logical necessity
demanded by theory, like an electron. His ground for maintaining
that there must be simples is that every complex presupposes a fact.
It is not necessarily assumed that the complexity of facts is finite;
even if every fact consisted of an infinite number of atomic facts and if
every atomic fact consisted of an infinite number of objects there would
still be objects and atomic facts (4.2211). The assertion that there is
a certain complex reduces to the assertion that its constituents are
related in a certain way, which is the assertion of a fact: thus if we
give a name to the complex the name only has meaning in virtue of
the truth of a certain proposition, namely the proposition asserting
the relatedness of the constituents of the complex. Thus the naming
of complexes presupposes propositions, while propositions presuppose
the naming of simples. In this way the naming of simples is shown to
be what is logically first in logic.
The world is fully described if all atomic facts are known, together
with the fact that these are all of them. The world is not described
by merely naming all the objects in it; it is necessary also to know
the atomic facts of which these objects are constituents. Given this
totality of atomic facts, every true proposition, however complex, can
theoretically be inferred. A proposition (true or false) asserting an
atomic fact is called an atomic proposition. All atomic propositions are
logically independent of each other. No atomic proposition implies any
other or is inconsistent with any other. Thus the whole business of
logical inference is concerned with propositions which are not atomic.
Such propositions may be called molecular.
Wittgenstein’s theory of molecular propositions turns upon his
theory of the construction of truth-functions.
A truth-function of a proposition p is a proposition containing p
and such that its truth or falsehood depends only upon the truth or
falsehood of p, and similarly a truth-function of several propositions
p, q, r, ... is one containing p, q, r, ... and such that its truth or falsehood
depends only upon the truth or falsehood of p, q, r, ... It might
seem at first sight as though there were other functions of propositions
besides truth-functions; such, for example, would be “A believes p”,
for in general A will believe some true propositions and some false
ones: unless he is an exceptionally gifted individual, we cannot infer
that p is true from the fact that he believes it or that p is false from
the fact that he does not believe it. Other apparent exceptions would
be such as “p is a very complex proposition” or “p is a proposition
about Socrates”. Mr. Wittgenstein maintains, however, for reasons
which will appear presently, that such exceptions are only apparent,
and that every function of a proposition is really a truth-function. It
follows that if we can define truth-functions generally, we can obtain
a general definition of all propositions in terms of the original set of
atomic propositions. This Wittgenstein proceeds to do.
It has been shown by Dr. Sheffer (Trans. Am. Math. Soc., Vol. XIV.
pp. 481–488) that all truth-functions of a given set of propositions can
be constructed out of either of the two functions “not-p or not-q” or
“not-p and not-q”. Wittgenstein makes use of the latter, assuming a
knowledge of Dr. Sheffer’s work. The manner in which other truthfunctions
are constructed out of “not-p and not-q” is easy to see. “Not-p
and not-p” is equivalent to “not-p”, hence we obtain a definition of
negation in terms of our primitive function: hence we can define “p or
q”, since this is the negation of “not-p and not-q”, i.e. of our primitive
function. The development of other truth-functions out of “not-p” and
“p or q” is given in detail at the beginning of Principia Mathematica.
This gives all that is wanted when the propositions which are argu-
4
ments to our truth-function are given by enumeration. Wittgenstein,
however, by a very interesting analysis succeeds in extending the process
to general propositions, i.e. to cases where the propositions which
are arguments to our truth-function are not given by enumeration but
are given as all those satisfying some condition. For example, let fx be
a propositional function (i.e. a function whose values are propositions),
such as “x is human”—then the various values of fx form a set of propositions.
We may extend the idea “not-p and not-q” so as to apply to the
simultaneous denial of all the propositions which are values of fx. In
this way we arrive at the proposition which is ordinarily represented
in mathematical logic by the words “fx is false for all values of x”. The
negation of this would be the proposition “there is at least one x for
which fx is true” which is represented by “(∃x). fx”. If we had started
with not-fx instead of fx we should have arrived at the proposition “fx
is true for all values of x” which is represented by “(x). fx”. Wittgenstein’s
method of dealing with general propositions [i.e. “(x). fx” and
“(∃x). fx”] differs from previous methods by the fact that the generality
comes only in specifying the set of propositions concerned, and when
this has been done the building up of truth-functions proceeds exactly
as it would in the case of a finite number of enumerated arguments
p, q, r, ...
Mr. Wittgenstein’s explanation of his symbolism at this point is not
quite fully given in the text. The symbol he uses is [p, ξ, N(ξ)]. The
following is the explanation of this symbol:
p stands for all atomic propositions.
ξ stands for any set of propositions.
N(ξ) stands for the negation of all the propositions making up ξ.
The whole symbol [p, ξ, N(ξ)] means whatever can be obtained by
taking any selection of atomic propositions, negating them all, then
taking any selection of the set of propositions now obtained, together
with any of the originals—and so on indefinitely. This is, he says, the
general truth-function and also the general form of proposition. What
is meant is somewhat less complicated than it sounds. The symbol is
intended to describe a process by the help of which, given the atomic
propositions, all others can be manufactured. The process depends
upon:
(a). Sheffer’s proof that all truth-functions can be obtained out of
simultaneous negation, i.e. out of “not-p and not-q”;
(b). Mr. Wittgenstein’s theory of the derivation of general propositions
from conjunctions and disjunctions;
(c). The assertion that a proposition can only occur in another
proposition as argument to a truth-function. Given these three foundations,
it follows that all propositions which are not atomic can be
derived from such as are, buy a uniform process, and it is this process
which is indicated by Mr. Wittgenstein’s symbol.
From this uniform method of construction we arrive at an amazing
simplification of the theory of inference, as well as a definition of the
sort of propositions that belong to logic. The method of generation
which has just been described, enables Wittgenstein to say that all
propositions can be constructed in the above manner from atomic
propositions, and in this way the totality of propositions is defined.
(The apparent exceptions which we mentioned above are dealt with in
a manner which we shall consider later.) Wittgenstein is enabled to
assert that propositions are all that follows from the totality of atomic
propositions (together with the fact that it is the totality of them); that
a proposition is always a truth-function of atomic propositions; and
that if p follows from q the meaning of p is contained in the meaning
of q, from which of course it results that nothing can be deduced from
an atomic proposition. All the propositions of logic, he maintains, are
tautologies, such, for example, as “p or not p”.
The fact that nothing can be deduced from an atomic proposition
has interesting applications, for example, to causality. There cannot, in
Wittgenstein’s logic, be any such thing as a causal nexus. “The events
of the future”, he says, “cannot be inferred from those of the present.
Superstition is the belief in the causal nexus.” That the sun will rise
to-morrow is a hypothesis. We do not in fact know whether it will rise,
since there is no compulsion according to which one thing must happen
because another happens.
Let us now take up another subject—that of names. In Wittgenstein’s
theoretical logical language, names are only given to simples.
We do not give two names to one thing, or one name to two things.
There is no way whatever, according to him, by which we can describe
the totality of things that can be named, in other words, the totality of
5
what there is in the world. In order to be able to do this we should have
to know of some property which must belong to every thing by a logical
necessity. It has been sought to find such a property in self-identity,
but the conception of identity is subjected by Wittgenstein to a destructive
criticism from which there seems no escape. The definition of
identity by means of the identity of indiscernibles is rejected, because
the identity of indiscernibles appears to be not a logically necessary
principle. According to this principle x is identical with y if every property
of x is a property of y, but it would, after all be logically possible
for two things to have exactly the same properties. If this does not
in fact happen that is an accidental characteristic of the world, not a
logically necessary characteristic, and accidental characteristics of the
world must, of course, not be admitted into the structure of logic. Mr.
Wittgenstein accordingly banishes identity and adopts the convention
that different letters are to mean different things. In practice, identity
is needed as between a name and a description or between two descriptions.
It is needed for such propositions as “Socrates is the philosopher
who drank the hemlock”, or “The even prime is the next number after
1.” For such uses of identity it is easy to provide on Wittgenstein’s
system.
The rejection of identity removes one method of speaking of the
totality of things, and it will be found that any other method that may
be suggested is equally fallacious: so, at least, Wittgenstein contends
and, I think, rightly. This amounts to saying that “object” is a pseudoconcept.
To say “x is an object” is to say nothing. It follows from this
that we cannot make such statements as “there are more than three
objects in the world”, or “there are an infinite number of objects in the
world”. Objects can only be mentioned in connexion with some definite
property. We can say “there are more than three objects which are
human”, or “there are more than three objects which are red”, for in
these statements the word object can be replaced by a variable in the
language of logic, the variable being one which satisfies in the first
case the function “x is human”; in the second the function “x is red”.
But when we attempt to say “there are more than three objects”, this
substitution of the variable for the word “object” becomes impossible,
and the proposition is therefore seen to be meaningless.
We here touch one instance of Wittgenstein’s fundamental thesis,
that it is impossible to say anything about the world as a whole, and
that whatever can be said has to be about bounded portions of the
world. This view may have been originally suggested by notation, and
if so, that is much in its favor, for a good notation has a subtlety and
suggestiveness which at times make it seem almost like a live teacher.
Notational irregularities are often the first sign of philosophical errors,
and a perfect notation would be a substitute for thought. But although
notation may have first suggested to Mr. Wittgenstein the limitation
of logic to things within the world as opposed to the world as a whole,
yet the view, once suggested, is seen to have much else to recommend
it. Whether it is ultimately true I do not, for my part, profess to know.
In this Introduction I am concerned to expound it, not to pronounce
upon it. According to this view we could only say things about the
world as a whole if we could get outside the world, if, that is to say, it
ceased to be for us the whole world. Our world may be bounded for
some superior being who can survey it from above, but for us, however
finite it may be, it cannot have a boundary, since it has nothing
outside it. Wittgenstein uses, as an analogy, the field of vision. Our
field of vision does not, for us, have a visual boundary, just because
there is nothing outside it, and in like manner our logical world has no
logical boundary because our logic knows of nothing outside it. These
considerations lead him to a somewhat curious discussion of Solipsism.
Logic, he says, fills the world. The boundaries of the world are also its
boundaries. In logic, therefore, we cannot say, there is this and this in
the world, but not that, for to say so would apparently presuppose that
we exclude certain possibilities, and this cannot be the case, since it
would require that logic should go beyond the boundaries of the world
as if it could contemplate these boundaries from the other side also.
What we cannot think we cannot think, therefore we also cannot say
what we cannot think.
This, he says, gives the key to solipsism. What Solipsism intends
is quite correct, but this cannot be said, it can only be shown. That
the world is my world appears in the fact that the boundaries of language
(the only language I understand) indicate the boundaries of my
world. The metaphysical subject does not belong to the world but is a
boundary of the world.
We must take up next the question of molecular propositions which
6
are at first sight not truth-functions, of the propositions that they
contain, such, for example, as “A believes p.”
Wittgenstein introduces this subject in the statement of his position,
namely, that all molecular functions are truth-functions. He
says (5.54): “In the general propositional form, propositions occur in a
proposition only as bases of truth-operations.” At first sight, he goes on
to explain, it seems as if a propositions could also occur in other ways,
e.g. “A believes p.” Here it seems superficially as if the proposition p
stood in a sort of relation to the object A. “But it is clear that ‘A believes
that p,’ ‘A thinks p,’ ‘A says p’ are of the form “‘p’ says p”; and here
we have no co-ordination of a fact and an object, but a co-ordination of
facts by means of a co-ordination of their objects” (5.542).
What Mr. Wittgenstein says here is said so shortly that its point is
not likely to be clear to those who have not in mind the controversies
with which he is concerned. The theory which which he is disagreeing
will be found in my articles on the nature of truth and falsehood in
Philosophical Essays and Proceedings of the Aristotelian Society, 1906–
7. The problem at issue is the problem of the logical form of belief, i.e.
what is the schema representing what occurs when a man believes.
Of course, the problem applies not only to belief, but also to a host of
other mental phenomena which may be called propositional attitudes:
doubting, considering, desiring, etc. In all these cases it seems natural
to express the phenomenon in the form “A doubts p”, “A considers p”,
“A desires p”, etc., which makes it appear as though we were dealing
with a relation between a person and a proposition. This cannot, of
course, be the ultimate analysis, since persons are fictions and so are
propositions, except in the sense in which they are facts on their own
account. A proposition, considered as a fact on its own account, may be
a set of words which a man says over to himself, or a complex image, or
train of images passing through his mind, or a set of incipient bodily
movements. It may be any one of innumerable different things. The
proposition as a fact on its own account, for example, the actual set of
words the man pronounces to himself, is not relevant to logic. What is
relevant to logic is that common element among all these facts, which
enables him, as we say, to mean the fact which the proposition asserts.
To psychology, of course, more is relevant; for a symbol does not mean
what it symbolizes in virtue of a logical relation alone, but in virtue
also of a psychological relation of intention, or association, or what-not.
The psychological part of meaning, however, does not concern the logician.
What does concern him in this problem of belief is the logical
schema. It is clear that, when a person believes a proposition, the person,
considered as a metaphysical subject, does not have to be assumed
in order to explain what is happening. What has to be explained is the
relation between the set of words which is the proposition considered
as a fact on its own account, and the “objective” fact which makes the
proposition true or false. This reduces ultimately to the question of
the meaning of propositions, that is to say, the meaning of propositions
is the only non-psychological portion of the problem involved in the
analysis of belief. This problem is simply one of a relation of two facts,
namely, the relation between the series of words used by the believer
and the fact which makes these words true or false. The series of
words is a fact just as much as what makes it true or false is a fact.
The relation between these two facts is not unanalyzable, since the
meaning of a proposition results from the meaning of its constituent
words. The meaning of the series of words which is a proposition is a
function of the meaning of the separate words. Accordingly, the proposition
as a whole does not really enter into what has to be explained
in explaining the meaning of a propositions. It would perhaps help to
suggest the point of view which I am trying to indicate, to say that in
the cases which have been considering the proposition occurs as a fact,
not as a proposition. Such a statement, however, must not be taken
too literally. The real point is that in believing, desiring, etc., what
is logically fundamental is the relation of a proposition considered as
a fact, to the fact which makes it true or false, and that this relation
of two facts is reducible to a relation of their constituents. Thus the
proposition does not occur at all in the same sense in which it occurs
in a truth-function.
There are some respects, in which, as it seems to me, Mr. Wittgenstein’s
theory stands in need of greater technical development. This
applies in particular to his theory of number (6.02ff.) which, as it
stands, is only capable of dealing with finite numbers. No logic can
be considered adequate until it has been shown to be capable of dealing
with transfinite numbers. I do not think there is anything in Mr.
Wittgenstein’s system to make it impossible for him to fill this lacuna.
7
More interesting than such questions of comparative detail is Mr.
Wittgenstein’s attitude towards the mystical. His attitude upon this
grows naturally out of his doctrine in pure logic, according to which
the logical proposition is a picture (true or false) of the fact, and has in
common with the fact a certain structure. It is this common structure
which makes it capable of being a picture of the fact, but the structure
cannot itself be put into words, since it is a structure of words, as
well as of the fact to which they refer. Everything, therefore, which
is involved in the very idea of the expressiveness of language must
remain incapable of being expressed in language, and is, therefore,
inexpressible in a perfectly precise sense. This inexpressible contains,
according to Mr. Wittgenstein, the whole of logic and philosophy. The
right method of teaching philosophy, he says, would be to confine oneself
to propositions of the sciences, stated with all possible clearness
and exactness, leaving philosophical assertions to the learner, and
proving to him, whenever he made them, that they are meaningless.
It is true that the fate of Socrates might befall a man who attempted
this method of teaching, but we are not to be deterred by that fear, if
it is the only right method. It is not this that causes some hesitation
in accepting Mr. Wittgenstein’s position, in spite of the very powerful
arguments which he brings to its support. What causes hesitation is
the fact that, after all, Mr. Wittgenstein manages to say a good deal
about what cannot be said, thus suggesting to the sceptical reader that
possibly there may be some loophole through a hierarchy of languages,
or by some other exit. The whole subject of ethics, for example, is
placed by Mr. Wittgenstein in the mystical, inexpressible region. Nevertheless
he is capable of conveying his ethical opinions. His defence
would be that what he calls the mystical can be shown, although it
cannot be said. It may be that this defence is adequate, but, for my
part, I confess that it leaves me with a certain sense of intellectual
discomfort.
There is one purely logical problem in regard to which these difficulties
are peculiarly acute. I mean the problem of generality. In the
theory of generality it is necessary to consider all propositions of the
form fx where fx is a given propositional function. This belongs to the
part of logic which can be expressed, according to Mr. Wittgenstein’s
system. But the totality of possible values of x which might seem to be
involved in the totality of propositions of the form fx is not admitted
by Mr. Wittgenstein among the things that can be spoken of, for this is
no other than the totality of things in the world, and thus involves the
attempt to conceive the world as a whole; “the feeling of the world as
a bounded whole is the mystical”; hence the totality of the values of
x is mystical (6.45). This is expressly argued when Mr. Wittgenstein
denies that we can make propositions as to how may things there are
in the world, as for example, that there are more than three.
These difficulties suggest to my mind some such possibility as this:
that every language has, as Mr. Wittgenstein says, a structure concerning
which in the language, nothing can be said, but that there may
be another language dealing with the structure of the first language,
and having itself a new structure, and that to this hierarchy of languages
there may be no limit. Mr. Wittgenstein would of course reply
that his whole theory is applicable unchanged to the totality of such
languages. The only retort would be to deny that there is any such
totality. The totalities concerning which Mr. Wittgenstein holds that it
is impossible to speak logically are nevertheless thought by him to exist,
and are the subject-matter of his mysticism. The totality resulting
from our hierarchy would be not merely logically inexpressible, but a
fiction, a mere delusion, and in this way the supposed sphere of the
mystical would be abolished. Such a hypothesis is very difficult, and
I can see objections to it which at the moment I do not know how to
answer. Yet I do not see how any easier hypothesis can escape from
Mr. Wittgenstein’s conclusions. Even if this very difficult hypothesis
should prove tenable, it would leave untouched a very large part of
Mr. Wittgenstein’s theory, though possibly not the part upon which he
himself would wish to lay most stress. As one with a long experience
of the difficulties of logic and of the deceptiveness of theories which
seem irrefutable, I find myself unable to be sure of the rightness of
a theory, merely on the ground that I cannot see any point on which
it is wrong. But to have constructed a theory of logic which is not at
any point obviously wrong is to have achieved a work of extraordinary
difficulty and importance. This merit, in my opinion, belongs to Mr.
Wittgenstein’s book, and makes it one which no serious philosopher
can afford to neglect.
BERTRAND RUSSELL.
May 1922.
Tractatus Logico-Philosophicus
DEDICATED
TO THE MEMORY OF MY FRIEND
DAVID H. PINSENT
M o t t o: . . . und alles, was man weiss, nicht bloss rauschen and brausen gehört hat, lässt sich in drei Worten sagen. –KÜRNBERGER.
9
Vorwort (Preface)
German Ogden Pears/McGuinness
Dieses Buch wird vielleicht nur der ver- This book will perhaps only be under- Perhaps this book will be understood only
stehen, der die Gedanken, die darin ausge- stood by those who have themselves already by someone who has himself already had the
drückt sind—oder doch ähnliche Gedanken— thought the thoughts which are expressed in thoughts that are expressed in it—or at least
schon selbst einmal gedacht hat.—Es ist also it—or similar thoughts. It is therefore not similar thoughts.—So it is not a textbook.—
kein Lehrbuch.—Sein Zweck wäre erreicht, a text-book. Its object would be attained if Its purpose would be achieved if it gave pleawenn
es einem, der es mit Verständnis liest, there were one person who read it with under- sure to one person who read and understood
Vergnügen bereitete. standing and to whom it afforded pleasure. it.
Das Buch behandelt die philosophischen The book deals with the problems of The book deals with the problems of phiProbleme
und zeigt—wie ich glaube—dass philosophy and shows, as I believe, that losophy, and shows, I believe, that the reason
die Fragestellung dieser Probleme auf dem the method of formulating these problems why these problems are posed is that the logic
Missverständnis der Logik unserer Sprache rests on the misunderstanding of the logic of of our language is misunderstood. The whole
beruht. Man könnte den ganzen Sinn des our language. Its whole meaning could be sense of the book might be summed up in
Buches etwa in die Worte fassen: Was sich summed up somewhat as follows: What can the following words: what can be said at all
überhaupt sagen lässt, lässt sich klar sagen; be said at all can be said clearly; and whereof can be said clearly, and what we cannot talk
und wovon man nicht reden kann, darüber one cannot speak thereof one must be silent. about we must pass over in silence.
muss man schweigen.
Das Buch will also dem Denken eine The book will, therefore, draw a limit to Thus the aim of the book is to draw a limit
Grenze ziehen, oder vielmehr—nicht dem thinking, or rather—not to thinking, but to to thought, or rather—not to thought, but to
Denken, sondern dem Ausdruck der Gedan- the expression of thoughts; for, in order to the expression of thoughts: for in order to be
ken: Denn um dem Denken eine Grenze zu draw a limit to thinking we should have to able to draw a limit to thought, we should
ziehen, müssten wir beide Seiten dieser Gren- be able to think both sides of this limit (we have to find both sides of the limit thinkable
ze denken können (wir müssten als denken should therefore have to be able to think (i.e. we should have to be able to think what
können, was sich nicht denken lässt). what cannot be thought). cannot be thought).
Die Grenze wird also nur in der Sprache The limit can, therefore, only be drawn in It will therefore only be in language that
gezogen werden können und was jenseits der language and what lies on the other side of the limit can be drawn, and what lies on the
Grenze liegt, wird einfach Unsinn sein. the limit will be simply nonsense. other side of the limit will simply be non-
sense.
Wieweit meine Bestrebungen mit denen How far my efforts agree with those of I do not wish to judge how far my efforts
anderer Philosophen zusammenfallen, will other philosophers I will not decide. Indeed coincide with those of other philosophers. Inich
nicht beurteilen. Ja, was ich hier geschrie- what I have here written makes no claim to deed, what I have written here makes no
ben habe, macht im Einzelnen überhaupt novelty in points of detail; and therefore I claim to novelty in detail, and the reason
nicht den Anspruch auf Neuheit; und dar- give no sources, because it is indifferent to why I give no sources is that it is a matter of
um gebe ich auch keine Quellen an, weil es me whether what I have thought has already indifference to me whether the thoughts that
10
mir gleichgültig ist, ob das was ich gedacht been thought before me by another. I have had have been anticipated by someone
habe, vor mir schon ein anderer gedacht hat. else.
Nur das will ich erwähnen, dass ich den I will only mention that to the great I will only mention that I am indebted to
großartigen Werken Freges und den Arbei- works of Frege and the writings of my friend Frege’s great works and of the writings of my
ten meines Freundes Herrn Bertrand Russell Bertrand Russell I owe in large measure the friend Mr. Bertrand Russell for much of the
einen großen Teil der Anregung zu meinen stimulation of my thoughts. stimulation of my thoughts.
Gedanken schulde.
Wenn diese Arbeit einen Wert hat, so be- If this work has a value it consists in If this work has any value, it consists in
steht er in Zweierlei. Erstens darin, dass in two things. First that in it thoughts are ex- two things: the first is that thoughts are exihr
Gedanken ausgedrückt sind, und dieser pressed, and this value will be the greater the pressed in it, and on this score the better the
Wert wird umso größer sein, je besser die Ge- better the thoughts are expressed. The more thoughts are expressed—the more the nail
danken ausgedrückt sind. Je mehr der Nagel the nail has been hit on the head.—Here I am has been hit on the head—the greater will
auf den Kopf getroffen ist.—Hier bin ich mir conscious that I have fallen far short of the be its value.—Here I am conscious of having
bewusst, weit hinter dem Möglichen zurück- possible. Simply because my powers are in- fallen a long way short of what is possible.
geblieben zu sein. Einfach darum, weil meine sufficient to cope with the task.—May others Simply because my powers are too slight for
Kraft zur Bewältigung der Aufgabe zu gering come and do it better. the accomplishment of the task.—May others
ist.—Mögen andere kommen und es besser come and do it better.
machen.
Dagegen scheint mir die W a h r h e i t On the other hand the truth of the On the other hand the truth of the
der hier mitgeteilten Gedanken unantastbar thoughts communicated here seems to me thoughts that are here communicated seems
und definitiv. Ich bin also der Meinung, die unassailable and definitive. I am, therefore, to me unassailable and definitive. I therefore
Probleme im Wesentlichen endgültig gelöst of the opinion that the problems have in es- believe myself to have found, on all essenzu
haben. Und wenn ich mich hierin nicht sentials been finally solved. And if I am not tial points, the final solution of the problems.
irre, so besteht nun der Wert dieser Arbeit mistaken in this, then the value of this work And if I am not mistaken in this belief, then
zweitens darin, dass sie zeigt, wie wenig da- secondly consists in the fact that it shows the second thing in which the of this work
mit getan ist, dass diese Probleme gelöst how little has been done when these prob- consists is that it shows how little is achieved
sind. lems have been solved. when these problems are solved.
L. W. L. W. L. W.
Wien, 1918 Vienna, 1918 Vienna, 1918
11
Tractatus Logico-Philosophicus
German Ogden Pears/McGuinness
1* Die Welt ist alles, was der Fall ist. The world is everything that is the The world is all that is the case.
case.
1.1 Die Welt ist die Gesamtheit der Tatsa- The world is the totality of facts, not The world is the totality of facts, not
chen, nicht der Dinge. of things. of things.
1.11 Die Welt ist durch die Tatsachen be- The world is determined by the facts, The world is determined by the facts,
stimmt und dadurch, dass es a l l e Tat- and by these being all the facts. and by their being all the facts.
sachen sind.
1.12 Denn, die Gesamtheit der Tatsachen For the totality of facts determines For the totality of facts determines
bestimmt, was der Fall ist und auch, was both what is the case, and also all that is what is the case, and also whatever is
alles nicht der Fall ist. not the case. not the case.
1.13 Die Tatsachen im logischen Raum The facts in logical space are the The facts in logical space are the
sind die Welt. world. world.
1.2 Die Welt zerfällt in Tatsachen. The world divides into facts. The world divides into facts.
1.21 Eines kann der Fall sein oder nicht Any one can either be the case or not Each item can be the case or not the
der Fall sein und alles übrige gleich blei- be the case, and everything else remain case while everything else remains the
ben. the same. same.
2 Was der Fall ist, die Tatsache, ist das What is the case, the fact, is the exis- What is the case—a fact—is the exisBestehen
von Sachverhalten. tence of atomic facts. tence of states of affairs.
2.01 Der Sachverhalt ist eine Verbindung An atomic fact is a combination of ob- A state of affairs (a state of things) is
von Gegenständen. (Sachen, Dingen.) jects (entities, things). a combination of objects (things).
2.011 Es ist dem Ding wesentlich, der Be- It is essential to a thing that it can be It is essential to things that they
standteil eines Sachverhaltes sein zu kön- a constituent part of an atomic fact. should be possible constituents of states
nen. of affairs.
2.012 In der Logik ist nichts zufällig: Wenn In logic nothing is accidental: if a In logic nothing is accidental: if a
das Ding im Sachverhalt vorkommen thing can occur in an atomic fact the pos- thing can occur in a state of affairs, the
k a n n, so muss die Möglichkeit des Sach- sibility of that atomic fact must already possibility of the state of affairs must be
verhaltes im Ding bereits präjudiziert be prejudged in the thing. written into the thing itself.
sein.
* [German] Die Decimalzahlen als Nummern der einzelnen Sätze deuten das logische Gewicht der Sätze an, den Nachdruck, der auf ihnen in meiner Darstellung liegt,
Die Sätze n.1, n.2, n.3, etc., sind Bemerkungen zum Sätze No. n; die Sätze n.m1, n.m2, etc. Bemerkungen zum Satze No. n.m; und so weiter. / [Ogden] The decimal figures
as numbers of the separate propositions indicate the logical importance of the propositions, the emphasis laid upon them in my exposition. The propositions n.1, n.2, n.3,
etc., are comments on proposition No. n; the propositions n.m1, n.m2, etc., are comments on the proposition No. n.m; and so on. / [Pears & McGuinness] The decimal
numbers assigned to the individual propositions indicate the logical importance of the propositions, the stress laid on them in my exposition. The propositions n.1, n.2, n.3,
etc. are comments on proposition no. n; the propositions n.m1, n.m2, etc. are comments on proposition no. n.m; and so on.
12
2.0121 Es erschiene gleichsam als Zufall, It would, so to speak, appear as an It would seem to be a sort of accident,
wenn dem Ding, das allein für sich beste- accident, when to a thing that could exist if it turned out that a situation would fit
hen könnte, nachträglich eine Sachlage alone on its own account, subsequently a a thing that could already exist entirely
passen würde. state of affairs could be made to fit. on its own.
Wenn die Dinge in Sachverhalten vor- If things can occur in atomic facts, this If things can occur in states of affairs,
kommen können, so muss dies schon in possibility must already lie in them. this possibility must be in them from the
ihnen liegen. beginning.
(Etwas Logisches kann nicht nur- (A logical entity cannot be merely pos- (Nothing in the province of logic can
möglich sein. Die Logik handelt von jeder sible. Logic treats of every possibility, and be merely possible. Logic deals with evMöglichkeit
und alle Möglichkeiten sind all possibilities are its facts.) ery possibility and all possibilities are its
ihre Tatsachen.) facts.)
Wie wir uns räumliche Gegenstände Just as we cannot think of spatial ob- Just as we are quite unable to imagine
überhaupt nicht außerhalb des Raumes, jects at all apart from space, or tempo- spatial objects outside space or temporal
zeitliche nicht außerhalb der Zeit denken ral objects apart from time, so we cannot objects outside time, so too there is no
können, so können wir uns k e i n e n Ge- think of any object apart from the possi- object that we can imagine excluded from
genstand außerhalb der Möglichkeit sei- bility of its connexion with other things. the possibility of combining with others.
ner Verbindung mit anderen denken.
Wenn ich mir den Gegenstand im Ver- If I can think of an object in the con- If I can imagine objects combined in
bande des Sachverhalts denken kann, so text of an atomic fact, I cannot think of it states of affairs, I cannot imagine them
kann ich ihn nicht außerhalb der M ö g - apart from the possibility of this context. excluded from the possibility of such coml
i c h k e i t dieses Verbandes denken. binations.
2.0122 Das Ding ist selbständig, insofern es The thing is independent, in so far as Things are independent in so far as
in allen m ö g l i c h e n Sachlagen vor- it can occur in all possible circumstances, they can occur in all possible situations,
kommen kann, aber diese Form der Selb- but this form of independence is a form of but this form of independence is a form
ständigkeit ist eine Form des Zusammen- connexion with the atomic fact, a form of of connexion with states of affairs, a form
hangs mit dem Sachverhalt, eine Form dependence. (It is impossible for words to of dependence. (It is impossible for words
der Unselbständigkeit. (Es ist unmöglich, occur in two different ways, alone and in to appear in two different roles: by themdass
Worte in zwei verschiedenen Weisen the proposition.) selves, and in propositions.)
auftreten, allein und im Satz.)
2.0123 Wenn ich den Gegenstand kenne, so If I know an object, then I also know If I know an object I also know all its
kenne ich auch sämtliche Möglichkeiten all the possibilities of its occurrence in possible occurrences in states of affairs.
seines Vorkommens in Sachverhalten. atomic facts.
(Jede solche Möglichkeit muss in der (Every such possibility must lie in the (Every one of these possibilities must
Natur des Gegenstandes liegen.) nature of the object.) be part of the nature of the object.)
Es kann nicht nachträglich eine neue A new possibility cannot subsequently A new possibility cannot be discovered
Möglichkeit gefunden werden. be found. later.
2.01231 Um einen Gegenstand zu kennen, In order to know an object, I must If I am to know an object, though I
13
muss ich zwar nicht seine externen—aber know not its external but all its internal need not know its external properties, I
ich muss alle seine internen Eigenschaf- qualities. must know all its internal properties.
ten kennen.
2.0124 Sind alle Gegenstände gegeben, so If all objects are given, then thereby If all objects are given, then at the
sind damit auch alle m ö g l i c h e n Sach- are all possible atomic facts also given. same time all possible states of affairs are
verhalte gegeben. also given.
2.013 Jedes Ding ist, gleichsam, in einem Every thing is, as it were, in a space Each thing is, as it were, in a space of
Raume möglicher Sachverhalte. Diesen of possible atomic facts. I can think of possible states of affairs. This space I can
Raum kann ich mir leer denken, nicht this space as empty, but not of the thing imagine empty, but I cannot imagine the
aber das Ding ohne den Raum. without the space. thing without the space.
2.0131 Der räumliche Gegenstand muss im A spatial object must lie in infinite A spatial object must be situated in
unendlichen Raume liegen. (Der Raum- space. (A point in space is an argument infinite space. (A spatial point is an
punkt ist eine Argumentstelle.) place.) argument-place.)
Der Fleck im Gesichtsfeld muss zwar A speck in a visual field need not be A speck in the visual field, thought it
nicht rot sein, aber eine Farbe muss er red, but it must have a colour; it has, so need not be red, must have some colour:
haben: er hat sozusagen den Farbenraum to speak, a colour space round it. A tone it is, so to speak, surrounded by colourum
sich. Der Ton muss e i n e Höhe ha- must have a pitch, the object of the sense space. Notes must have some pitch, obben,
der Gegenstand des Tastsinnes e i - of touch a hardness, etc. jects of the sense of touch some degree of
n e Härte, usw. hardness, and so on.
2.014 Die Gegenstände enthalten die Mög- Objects contain the possibility of all Objects contain the possibility of all
lichkeit aller Sachlagen. states of affairs. situations.
2.0141 Die Möglichkeit seines Vorkommens The possibility of its occurrence in The possibility of its occurring in
in Sachverhalten, ist die Form des Gegen- atomic facts is the form of the object. states of affairs is the form of an object.
standes.
2.02 Der Gegenstand ist einfach. The object is simple. Objects are simple.
2.0201 Jede Aussage über Komplexe lässt Every statement about complexes can Every statement about complexes can
sich in eine Aussage über deren Bestand- be analysed into a statement about their be resolved into a statement about their
teile und in diejenigen Sätze zerlegen, constituent parts, and into those proposi- constituents and into the propositions
welche die Komplexe vollständig beschrei- tions which completely describe the com- that describe the complexes completely.
ben. plexes.
2.021 Die Gegenstände bilden die Substanz Objects form the substance of the Objects make up the substance of the
der Welt. Darum können sie nicht zusam- world. Therefore they cannot be com- world. That is why they cannot be commengesetzt
sein. pound. posite.
2.0211 Hätte die Welt keine Substanz, so wür- If the world had no substance, then If the world had no substance, then
de, ob ein Satz Sinn hat, davon abhängen, whether a proposition had sense would whether a proposition had sense would
ob ein anderer Satz wahr ist. depend on whether another proposition depend on whether another proposition
was true. was true.
14
2.0212 Es wäre dann unmöglich, ein Bild der It would then be impossible to form a In that case we could not sketch any
Welt (wahr oder falsch) zu entwerfen. picture of the world (true or false). picture of the world (true or false).
2.022 Es ist offenbar, dass auch eine von der It is clear that however different from It is obvious that an imagined world,
wirklichen noch so verschieden gedachte the real one an imagined world may be, however different it may be from the real
Welt Etwas—eine Form—mit der wirkli- it must have something—a form—in com- one, must have something—a form—in
chen gemein haben muss. mon with the real world. common with it.
2.023 Diese feste Form besteht eben aus den This fixed form consists of the objects. Objects are just what constitute this
Gegenständen. unalterable form.
2.0231 Die Substanz der Welt k a n n nur The substance of the world can only The substance of the world can only
eine Form und keine materiellen Eigen- determine a form and not any material determine a form, and not any material
schaften bestimmen. Denn diese werden properties. For these are first presented properties. For it is only by means of
erst durch die Sätze dargestellt—erst by the propositions—first formed by the propositions that material properties are
durch die Konfiguration der Gegenstände configuration of the objects. represented—only by the configuration of
gebildet. objects that they are produced.
2.0232 Beiläufig gesprochen: Die Gegenstän- Roughly speaking: objects are colour- In a manner of speaking, objects are
de sind farblos. less. colourless.
2.0233 Zwei Gegenstände von der gleichen Two objects of the same logical If two objects have the same logical
logischen Form sind—abgesehen von ih- form are—apart from their external form, the only distinction between them,
ren externen Eigenschaften—von einan- properties—only differentiated from one apart from their external properties, is
der nur dadurch unterschieden, dass sie another in that they are different. that they are different.
verschieden sind.
2.02331 Entweder ein Ding hat Eigenschaften, Either a thing has properties which no Either a thing has properties that
die kein anderes hat, dann kann man other has, and then one can distinguish nothing else has, in which case we can
es ohneweiteres durch eine Beschreibung it straight away from the others by a de- immediately use a description to distinaus
den anderen herausheben, und dar- scription and refer to it; or, on the other guish it from the others and refer to it;
auf hinweisen; oder aber, es gibt mehrere hand, there are several things which have or, on the other hand, there are several
Dinge, die ihre sämtlichen Eigenschaften the totality of their properties in common, things that have the whole set of their
gemeinsam haben, dann ist es überhaupt and then it is quite impossible to point to properties in common, in which case it is
unmöglich auf eines von ihnen zu zeigen. any one of them. quite impossible to indicate one of them.
Denn, ist das Ding durch nichts her- For it a thing is not distinguished by For it there is nothing to distinguish
vorgehoben, so kann ich es nicht hervor- anything, I cannot distinguish it—for oth- a thing, I cannot distinguish it, since othheben,
denn sonst ist es eben hervorgeho- erwise it would be distinguished. erwise it would be distinguished after all.
ben.
2.024 Die Substanz ist das, was unabhängig Substance is what exists indepen- The substance is what subsists indevon
dem was der Fall ist, besteht. dently of what is the case. pendently of what is the case.
2.025 Sie ist Form und Inhalt. It is form and content. It is form and content.
2.0251 Raum, Zeit und Farbe (Färbigkeit) Space, time and colour (colouredness) Space, time, colour (being coloured)
15
sind Formen der Gegenstände. are forms of objects. are forms of objects.
2.026 Nur wenn es Gegenstände gibt, kann Only if there are objects can there be There must be objects, if the world is
es eine feste Form der Welt geben. a fixed form of the world. to have unalterable form.
2.027 Das Feste, das Bestehende und der The fixed, the existent and the object Objects, the unalterable, and the subGegenstand
sind Eins. are one. sistent are one and the same.
2.0271 Der Gegenstand ist das Feste, Beste- The object is the fixed, the existent; Objects are what is unalterable and
hende; die Konfiguration ist das Wech- the configuration is the changing, the subsistent; their configuration is what is
selnde, Unbeständige. variable. changing and unstable.
2.0272 Die Konfiguration der Gegenstände The configuration of the objects forms The configuration of objects produces
bildet den Sachverhalt. the atomic fact. states of affairs.
2.03 Im Sachverhalt hängen die Gegen- In the atomic fact objects hang one in In a state of affairs objects fit into one
stände ineinander, wie die Glieder einer another, like the links of a chain. another like the links of a chain.
Kette.
2.031 Im Sachverhalt verhalten sich die Ge- In the atomic fact the objects are com- In a state of affairs objects stand in a
genstände in bestimmter Art und Weise bined in a definite way. determinate relation to one another.
zueinander.
2.032 Die Art und Weise, wie die Gegenstän- The way in which objects hang to- The determinate way in which objects
de im Sachverhalt zusammenhängen, ist gether in the atomic fact is the structure are connected in a state of affairs is the
die Struktur des Sachverhaltes. of the atomic fact. structure of the state of affairs.
2.033 Die Form ist die Möglichkeit der The form is the possibility of the struc- Form is the possibility of structure.
Struktur. ture.
2.034 Die Struktur der Tatsache besteht aus The structure of the fact consists of The structure of a fact consists of the
den Strukturen der Sachverhalte. the structures of the atomic facts. structures of states of affairs.
2.04 Die Gesamtheit der bestehenden The totality of existent atomic facts is The totality of existing states of afSachverhalte
ist die Welt. the world. fairs is the world.
2.05 Die Gesamtheit der bestehenden The totality of existent atomic facts The totality of existing states of afSachverhalte
bestimmt auch, welche also determines which atomic facts do not fairs also determines which states of afSachverhalte
nicht bestehen. exist. fairs do not exist.
2.06 Das Bestehen und Nichtbestehen von The existence and non-existence of The existence and non-existence of
Sachverhalten ist die Wirklichkeit. atomic facts is the reality. states of affairs is reality.
(Das Bestehen von Sachverhalten nen- (The existence of atomic facts we also (We call the existence of states of
nen wir auch eine positive, das Nichtbe- call a positive fact, their non-existence a affairs a positive fact, and their nonstehen
eine negative Tatsache.) negative fact.) existence a negative fact.)
2.061 Die Sachverhalte sind von einander Atomic facts are independent of one States of affairs are independent of
unabhängig. another. one another.
2.062 Aus dem Bestehen oder Nichtbeste- From the existence of non-existence of From the existence or non-existence of
hen eines Sachverhaltes kann nicht auf an atomic fact we cannot infer the exis- one state of affairs it is impossible to infer
16
das Bestehen oder Nichtbestehen eines tence of non-existence of another. the existence or non-existence of another.
anderen geschlossen werden.
2.063 Die gesamte Wirklichkeit ist die Welt. The total reality is the world. The sum-total of reality is the world.
2.1 Wir machen uns Bilder der Tatsachen. We make to ourselves pictures of facts. We picture facts to ourselves.
2.11 Das Bild stellt die Sachlage im logi- The picture presents the facts in logi- A picture presents a situation in logischen
Raume, das Bestehen und Nichtbe- cal space, the existence and non-existence cal space, the existence and non-existence
stehen von Sachverhalten vor. of atomic facts. of states of affairs.
2.12 Das Bild ist ein Modell der Wirklich- The picture is a model of reality. A picture is a model of reality.
keit.
2.13 Den Gegenständen entsprechen im To the objects correspond in the pic- In a picture objects have the elements
Bilde die Elemente des Bildes. ture the elements of the picture. of the picture corresponding to them.
2.131 Die Elemente des Bildes vertreten im The elements of the picture stand, in In a picture the elements of the picBild
die Gegenstände. the picture, for the objects. ture are the representatives of objects.
2.14 Das Bild besteht darin, dass sich seine The picture consists in the fact that its What constitutes a picture is that its
Elemente in bestimmter Art und Weise elements are combined with one another elements are related to one another in a
zu einander verhalten. in a definite way. determinate way.
2.141 Das Bild ist eine Tatsache. The picture is a fact. A picture is a fact.
2.15 Dass sich die Elemente des Bildes in That the elements of the picture are The fact that the elements of a picbestimmter
Art und Weise zu einander combined with one another in a definite ture are related to one another in a deverhalten,
stellt vor, dass sich die Sachen way, represents that the things are so terminate way represents that things are
so zu einander verhalten. combined with one another. related to one another in the same way.
Dieser Zusammenhang der Elemente This connexion of the elements of the Let us call this connexion of its eledes
Bildes heiße seine Struktur und ihre picture is called its structure, and the pos- ments the structure of the picture, and
Möglichkeit seine Form der Abbildung. sibility of this structure is called the form let us call the possibility of this structure
of representation of the picture. the pictorial form of the picture.
2.151 Die Form der Abbildung ist die Mög- The form of representation is the pos- Pictorial form is the possibility that
lichkeit, dass sich die Dinge so zu ein- sibility that the things are combined with things are related to one another in the
ander verhalten, wie die Elemente des one another as are the elements of the same way as the elements of the picture.
Bildes. picture.
2.1511 Das Bild ist s o mit der Wirklichkeit Thus the picture is linked with reality; That is how a picture is attached to
verknüpft—es reicht bis zu ihr. it reaches up to it. reality; it reaches right out to it.
2.1512 Es ist wie ein Maßstab an die Wirk- It is like a scale applied to reality. It is laid against reality like a mealichkeit
angelegt. sure.
2.15121 Nur die äußersten Punkte der Teil- Only the outermost points of the divid- Only the end-points of the graduating
striche b e r ü h r e n den zu messenden ing lines touch the object to be measured. lines actually touch the object that is to
Gegenstand. be measured.
2.1513 Nach dieser Auffassung gehört also According to this view the represent- So a picture, conceived in this way,
17
zum Bilde auch noch die abbildende Be- ing relation which makes it a picture, also also includes the pictorial relationship,
ziehung, die es zum Bild macht. belongs to the picture. which makes it into a picture.
2.1514 Die abbildende Beziehung besteht aus The representing relation consists of The pictorial relationship consists of
den Zuordnungen der Elemente des Bil- the co-ordinations of the elements of the the correlations of the picture’s elements
des und der Sachen. picture and the things. with things.
2.1515 Diese Zuordnungen sind gleichsam These co-ordinations are as it were These correlations are, as it were, the
die Fühler der Bildelemente, mit denen the feelers of its elements with which the feelers of the picture’s elements, with
das Bild die Wirklichkeit berührt. picture touches reality. which the picture touches reality.
2.16 Die Tatsache muss, um Bild zu sein, In order to be a picture a fact must If a fact is to be a picture, it must have
etwas mit dem Abgebildeten gemeinsam have something in common with what it something in common with what it dehaben.
pictures. picts.
2.161 In Bild und Abgebildetem muss etwas In the picture and the pictured there There must be something identical in
identisch sein, damit das eine überhaupt must be something identical in order that a picture and what it depicts, to enable
ein Bild des anderen sein kann. the one can be a picture of the other at the one to be a picture of the other at all.
all.
2.17 Was das Bild mit der Wirklichkeit ge- What the picture must have in com- What a picture must have in common
mein haben muss, um sie auf seine Art mon with reality in order to be able to with reality, in order to be able to depict
und Weise—richtig oder falsch—abbilden represent it after its manner—rightly or it—correctly or incorrectly—in the way
zu können, ist seine Form der Abbildung. falsely—is its form of representation. that it does, is its pictorial form.
2.171 Das Bild kann jede Wirklichkeit abbil- The picture can represent every real- A picture can depict any reality whose
den, deren Form es hat. ity whose form it has. form it has.
Das räumliche Bild alles Räumliche, The spatial picture, everything spa- A spatial picture can depict anything
das farbige alles Farbige, etc. tial, the coloured, everything coloured, spatial, a coloured one anything coloured,
etc. etc.
2.172 Seine Form der Abbildung aber, kann The picture, however, cannot repre- A picture cannot, however, depict its
das Bild nicht abbilden; es weist sie auf. sent its form of representation; it shows pictorial form: it displays it.
it forth.
2.173 Das Bild stellt sein Objekt von außer- The picture represents its object from A picture represents its subject from
halb dar (sein Standpunkt ist seine Form without (its standpoint is its form of rep- a position outside it. (Its standpoint is
der Darstellung), darum stellt das Bild resentation), therefore the picture repre- its representational form.) That is why a
sein Objekt richtig oder falsch dar. sents its object rightly or falsely. picture represents its subject correctly or
incorrectly.
2.174 Das Bild kann sich aber nicht außer- But the picture cannot place itself out- A picture cannot, however, place itself
halb seiner Form der Darstellung stellen. side of its form of representation. outside its representational form.
2.18 Was jedes Bild, welcher Form immer, What every picture, of whatever form, What any picture, of whatever form,
mit der Wirklichkeit gemein haben muss, must have in common with reality in must have in common with reality, in orum
sie überhaupt—richtig oder falsch— order to be able to represent it at all— der to be able to depict it—correctly or
18
abbilden zu können, ist die logische Form, rightly or falsely—is the logical form, that incorrectly—in any way at all, is logical
das ist, die Form der Wirklichkeit. is, the form of reality. form, i.e. the form of reality.
2.181 Ist die Form der Abbildung die logi- If the form of representation is the A picture whose pictorial form is logische
Form, so heißt das Bild das logische logical form, then the picture is called a cal form is called a logical picture.
Bild. logical picture.
2.182 Jedes Bild ist a u c h ein logisches. Every picture is also a logical picture. Every picture is at the same time a
(Dagegen ist z. B. nicht jedes Bild ein (On the other hand, for example, not ev- logical one. (On the other hand, not every
räumliches.) ery picture is spatial.) picture is, for example, a spatial one.)
2.19 Das logische Bild kann die Welt abbil- The logical picture can depict the Logical pictures can depict the world.
den. world.
2.2 Das Bild hat mit dem Abgebildeten The picture has the logical form of rep- A picture has logico-pictorial form in
die logische Form der Abbildung gemein. resentation in common with what it pic- common with what it depicts.
tures.
2.201 Das Bild bildet die Wirklichkeit ab, The picture depicts reality by repre- A picture depicts reality by representindem
es eine Möglichkeit des Bestehens senting a possibility of the existence and ing a possibility of existence and nonund
Nichtbestehens von Sachverhalten non-existence of atomic facts. existence of states of affairs.
darstellt.
2.202 Das Bild stellt eine mögliche Sachlage The picture represents a possible A picture represents a possible situaim
logischen Raume dar. state of affairs in logical space. tion in logical space.
2.203 Das Bild enthält die Möglichkeit der The picture contains the possibility of A picture contains the possibility of
Sachlage, die es darstellt. the state of affairs which it represents. the situation that it represents.
2.21 Das Bild stimmt mit der Wirklichkeit The picture agrees with reality or not; A picture agrees with reality or fails
überein oder nicht; es ist richtig oder un- it is right or wrong, true or false. to agree; it is correct or incorrect, true or
richtig, wahr oder falsch. false.
2.22 Das Bild stellt dar, was es darstellt, The picture represents what it repre- What a picture represents it repreunabhängig
von seiner Wahr- oder Falsch- sents, independently of its truth or false- sents independently of its truth or falsity,
heit, durch die Form der Abbildung. hood, through the form of representation. by means of its pictorial form.
2.221 Was das Bild darstellt, ist sein Sinn. What the picture represents is its What a picture represents is its sense.
sense.
2.222 In der Übereinstimmung oder Nicht- In the agreement or disagreement of The agreement or disagreement of its
übereinstimmung seines Sinnes mit der its sense with reality, its truth or falsity sense with reality constitutes its truth or
Wirklichkeit, besteht seine Wahrheit oder consists. falsity.
Falschheit.
2.223 Um zu erkennen, ob das Bild wahr In order to discover whether the pic- In order to tell whether a picture is
oder falsch ist, müssen wir es mit der ture is true or false we must compare it true or false we must compare it with
Wirklichkeit vergleichen. with reality. reality.
2.224 Aus dem Bild allein ist nicht zu erken- It cannot be discovered from the pic- It is impossible to tell from the picture
19
nen, ob es wahr oder falsch ist. ture alone whether it is true or false. alone whether it is true or false.
2.225 Ein a priori wahres Bild gibt es nicht. There is no picture which is a priori There are no pictures that are true a
true. priori.
3 Das logische Bild der Tatsachen ist The logical picture of the facts is the A logical picture of facts is a thought.
der Gedanke. thought.
3.001 „Ein Sachverhalt ist denkbar“ heißt: “An atomic fact is thinkable”—means: ‘A state of affairs is thinkable’: what
Wir können uns ein Bild von ihm machen. we can imagine it. this means is that we can picture it to
ourselves.
3.01 Die Gesamtheit der wahren Gedan- The totality of true thoughts is a pic- The totality of true thoughts is a picken
sind ein Bild der Welt. ture of the world. ture of the world.
3.02 Der Gedanke enthält die Möglichkeit The thought contains the possibility of A thought contains the possibility of
der Sachlage, die er denkt. Was denkbar the state of affairs which it thinks. What the situation of which it is the thought.
ist, ist auch möglich. is thinkable is also possible. What is thinkable is possible too.
3.03 Wir können nichts Unlogisches den- We cannot think anything unlogical, Thought can never be of anything ilken,
weil wir sonst unlogisch denken müs- for otherwise we should have to think un- logical, since, if it were, we should have
sten. logically. to think illogically.
3.031 Man sagte einmal, dass Gott alles It used to be said that God could cre- It used to be said that God could creschaffen
könne, nur nichts, was den logi- ate everything, except what was contrary ate anything except what would be conschen
Gesetzen zuwider wäre.—Wir kön- to the laws of logic. The truth is, we could trary to the laws of logic. The truth is
nen nämlich von einer „unlogischen“ Welt not say of an “unlogical” world how it that we could not say what an ‘illogical’
nicht s a g e n, wie sie aussähe. would look. world would look like.
3.032 Etwas „der Logik widersprechendes“ To present in language anything It is as impossible to represent in lanin
der Sprache darstellen, kann man which “contradicts logic” is as impossi- guage anything that ‘contradicts logic’ as
ebensowenig, wie in der Geometrie eine ble as in geometry to present by its co- it is in geometry to represent by its coden
Gesetzen des Raumes widersprechen- ordinates a figure which contradicts the ordinates a figure that contradicts the
de Figur durch ihre Koordinaten darstel- laws of space; or to give the co-ordinates laws of space, or to give the co-ordinates
len; oder die Koordinaten eines Punktes of a point which does not exist. of a point that does not exist.
angeben, welcher nicht existiert.
3.0321 Wohl können wir einen Sachverhalt We could present spatially an atomic Though a state of affairs that would
räumlich darstellen, welcher den Geset- fact which contradicted the laws of contravene the laws of physics can be repzen
der Physik, aber keinen, der den Ge- physics, but not one which contradicted resented by us spatially, one that would
setzen der Geometrie zuwiderliefe. the laws of geometry. contravene the laws of geometry cannot.
3.04 Ein a priori richtiger Gedanke wä- An a priori true thought would be one If a thought were correct a priori, it
re ein solcher, dessen Möglichkeit seine whose possibility guaranteed its truth. would be a thought whose possibility enWahrheit
bedingte. sured its truth.
3.05 Nur so könnten wir a priori wissen, Only if we could know a priori that a A priori knowledge that a thought was
dass ein Gedanke wahr ist, wenn aus dem thought is true if its truth was to be rec- true would be possible only if its truth
20
Gedanken selbst (ohne Vergleichsobjekt) ognized from the thought itself (without were recognizable from the thought itself
seine Wahrheit zu erkennen wäre. an object of comparison). (without anything to compare it with).
3.1 Im Satz drückt sich der Gedanke sinn- In the proposition the thought is ex- In a proposition a thought finds an
lich wahrnehmbar aus. pressed perceptibly through the senses. expression that can be perceived by the
senses.
3.11 Wir benützen das sinnlich wahrnehm- We use the sensibly perceptible sign We use the perceptible sign of a propobare
Zeichen (Laut- oder Schriftzeichen (sound or written sign, etc.) of the propo- sition (spoken or written, etc.) as a projecetc.)
des Satzes als Projektion der mögli- sition as a projection of the possible state tion of a possible situation.
chen Sachlage. of affairs.
Die Projektionsmethode ist das Den- The method of projection is the think- The method of projection is to think of
ken des Satz-Sinnes. ing of the sense of the proposition. the sense of the proposition.
3.12 Das Zeichen, durch welches wir den The sign through which we express I call the sign with which we express
Gedanken ausdrücken, nenne ich das the though I call the propositional sign. a thought a propositional sign.—And a
Satzzeichen. Und der Satz ist das Satzzei- And the proposition is the propositional proposition is a propositional sign in its
chen in seiner projektiven Beziehung zur sign in its projective relation to the world. projective relation to the world.
Welt.
3.13 Zum Satz gehört alles, was zur Projek- To the proposition belongs everything A proposition includes all that the protion
gehört; aber nicht das Projizierte. which belongs to the projection; but not jection includes, but not what is projected.
what is projected.
Also die Möglichkeit des Projizierten, Therefore the possibility of what is Therefore, though what is projected is
aber nicht dieses selbst. projected but not this itself. not itself included, its possibility is.
Im Satz ist also sein Sinn noch nicht In the proposition, therefore, its sense A proposition, therefore, does not acenthalten,
wohl aber die Möglichkeit, ihn is not yet contained, but the possibility of tually contain its sense, but does contain
auszudücken. expressing it. the possibility of expressing it.
(„Der Inhalt des Satzes“ heißt der In- (“The content of the proposition” (‘The content of a proposition’ means
halt des sinnvollen Satzes.) means the content of the significant the content of a proposition that has
proposition.) sense.)
Im Satz ist die Form seines Sinnes In the proposition the form of its sense A proposition contains the form, but
enthalten, aber nicht dessen Inhalt. is contained, but not its content. not the content, of its sense.
3.14 Das Satzzeichen besteht darin, dass The propositional sign consists in the What constitutes a propositional sign
sich seine Elemente, die Wörter, in ihm fact that its elements, the words, are com- is that in its elements (the words) stand
auf bestimmte Art und Weise zu einander bined in it in a definite way. in a determinate relation to one another.
verhalten.
Das Satzzeichen ist eine Tatsache. The propositional sign is a fact. A propositional sign is a fact.
3.141 Der Satz ist kein Wörtergemisch.— The proposition is not a mixture of A proposition is not a blend of words.—
(Wie das musikalische Thema kein Ge- words (just as the musical theme is not a (Just as a theme in music is not a blend
misch von Tönen.) mixture of tones). of notes.)
21
Der Satz ist artikuliert. The proposition is articulate. A proposition is articulate.
3.142 Nur Tatsachen können einen Sinn Only facts can express a sense, a class Only facts can express a sense, a set
ausdrücken, eine Klasse von Namen kann of names cannot. of names cannot.
es nicht.
3.143 Dass das Satzzeichen eine Tatsache That the propositional sign is a fact is Although a propositional sign is a fact,
ist, wird durch die gewöhnliche Aus- concealed by the ordinary form of expres- this is obscured by the usual form of exdrucksform
der Schrift oder des Druckes sion, written or printed. pression in writing or print.
verschleiert.
Denn im gedruckten Satz z. B. sieht For in the printed proposition, for ex- For in a printed proposition, for examdas
Satzzeichen nicht wesentlich ver- ample, the sign of a proposition does not ple, no essential difference is apparent
schieden aus vom Wort. appear essentially different from a word. between a propositional sign and a word.
(So war es möglich, dass Frege den (Thus it was possible for Frege to call (That is what made it possible for
Satz einen zusammengesetzten Namen the proposition a compounded name.) Frege to call a proposition a composite
nannte.) name.)
3.1431 Sehr klar wird das Wesen des Satzzei- The essential nature of the proposi- The essence of a propositional sign
chens, wenn wir es uns, statt aus Schrift- tional sign becomes very clear when we is very clearly seen if we imagine one
zeichen, aus räumlichen Gegenständen imagine it made up of spatial objects composed of spatial objects (such as ta(etwa
Tischen, Stühlen, Büchern) zusam- (such as tables, chairs, books) instead of bles, chairs, and books) instead of written
mengesetzt denken. written signs. signs.
Die gegenseitige räumliche Lage die- The mutual spatial position of these Then the spatial arrangement of these
ser Dinge drückt dann den Sinn des Sat- things then expresses the sense of the things will express the sense of the propozes
aus. proposition. sition.
3.1432 Nicht: „Das komplexe Zeichen ‚aRb‘ We must not say, “The complex sign Instead of, ‘The complex sign “aRb”
sagt, dass a in der Beziehung R zu b ‘aRb’ says ‘a stands in relation R to b’”; says that a stands to b in the relation R’,
steht“, sondern: D a s s „a“ in einer gewis- but we must say, “That ‘a’ stands in a we ought to put, ‘That “a” stands to “b” in
sen Beziehung zu „b“ steht, sagt, d a s s certain relation to ‘b’ says that aRb”. a certain relation says that aRb.’
aRb.
3.144 Sachlagen kann man beschreiben, States of affairs can be described but Situations can be described but not
nicht b e n e n n e n. not named. given names.
(Namen gleichen Punkten, Sätze Pfei- (Names resemble points; propositions (Names are like points; propositions
len, sie haben Sinn.) resemble arrows, they have sense.) like arrows—they have sense.)
3.2 Im Satze kann der Gedanke so ausge- In propositions thoughts can be so ex- In a proposition a thought can be exdrückt
sein, dass den Gegenständen des pressed that to the objects of the thoughts pressed in such a way that elements of
Gedankens Elemente des Satzzeichens correspond the elements of the proposi- the propositional sign correspond to the
entsprechen. tional sign. objects of the thought.
3.201 Diese Elemente nenne ich „einfache These elements I call “simple signs” I call such elements ‘simple signs’, and
Zeichen“ und den Satz „vollständig analy- and the proposition “completely anal- such a proposition ‘complete analysed’.
22
siert“. ysed”.
3.202 Die im Satze angewandten einfachen The simple signs employed in proposi- The simple signs employed in proposiZeichen
heißen Namen. tions are called names. tions are called names.
3.203 Der Name bedeutet den Gegenstand. The name means the object. The ob- A name means an object. The object is
Der Gegenstand ist seine Bedeutung. („A“ ject is its meaning. (“A” is the same sign its meaning. (‘A’ is the same sign as ‘A’.)
ist dasselbe Zeichen wie „A“.) as “A”.)
3.21 Der Konfiguration der einfachen Zei- To the configuration of the simple The configuration of objects in a situchen
im Satzzeichen entspricht die Konfi- signs in the propositional sign corre- ation corresponds to the configuration of
guration der Gegenstände in der Sachla- sponds the configuration of the objects simple signs in the propositional sign.
ge. in the state of affairs.
3.22 Der Name vertritt im Satz den Gegen- In the proposition the name repre- In a proposition a name is the represtand.
sents the object. sentative of an object.
3.221 Die Gegenstände kann ich nur n e n - Objects I can only name. Signs rep- Objects can only be named. Signs are
n e n. Zeichen vertreten sie. Ich kann nur resent them. I can only speak of them. their representatives. I can only speak
v o n ihnen sprechen, s i e a u s s p r e - I cannot assert them. A proposition can about them: I cannot put them into words.
c h e n kann ich nicht. Ein Satz kann nur only say how a thing is, not what it is. Propositions can only say how things are,
sagen, w i e ein Ding ist, nicht w a s es not what they are.
ist.
3.23 Die Forderung der Möglichkeit der The postulate of the possibility of the The requirement that simple signs be
einfachen Zeichen ist die Forderung der simple signs is the postulate of the deter- possible is the requirement that sense be
Bestimmtheit des Sinnes. minateness of the sense. determinate.
3.24 Der Satz, welcher vom Komplex han- A proposition about a complex stands A proposition about a complex stands
delt, steht in interner Beziehung zum Sat- in internal relation to the proposition in an internal relation to a proposition
ze, der von dessen Bestandteil handelt. about its constituent part. about a constituent of the complex.
Der Komplex kann nur durch seine A complex can only be given by its A complex can be given only by its
Beschreibung gegeben sein, und diese description, and this will either be right description, which will be right or wrong.
wird stimmen oder nicht stimmen. Der or wrong. The proposition in which there A proposition that mentions a complex
Satz, in welchem von einem Komplex die is mention of a complex, if this does not will not be nonsensical, if the complex
Rede ist, wird, wenn dieser nicht existiert, exist, becomes not nonsense but simply does not exist, but simply false.
nicht unsinnig, sondern einfach falsch false.
sein.
Dass ein Satzelement einen Komplex That a propositional element signifies When a propositional element signibezeichnet,
kann man aus einer Unbe- a complex can be seen from an indetermi- fies a complex, this can be seen from an
stimmtheit in den Sätzen sehen, worin es nateness in the propositions in which it indeterminateness in the propositions in
vorkommt. Wir w i s s e n, durch diesen occurs. We know that everything is not which it occurs. In such cases we know
Satz ist noch nicht alles bestimmt. (Die yet determined by this proposition. (The that the proposition leaves something unAllgemeinheitsbezeichnung
e n t h ä l t notation for generality contains a proto- determined. (In fact the notation for gen-
23
ja ein Urbild.) type.) erality contains a prototype.)
Die Zusammenfassung des Symbols The combination of the symbols of a The contraction of a symbol for a comeines
Komplexes in ein einfaches Symbol complex in a simple symbol can be ex- plex into a simple symbol can be exkann
durch eine Definition ausgedrückt pressed by a definition. pressed in a definition.
werden.
3.25 Es gibt eine und nur eine vollständige There is one and only one complete A proposition has one and only one
Analyse des Satzes. analysis of the proposition. complete analysis.
3.251 Der Satz drückt auf bestimmte, klar The proposition expresses what it ex- What a proposition expresses it exangebbare
Weise aus, was er ausdrückt: presses in a definite and clearly specifi- presses in a determinate manner, which
Der Satz ist artikuliert. able way: the proposition is articulate. can be set out clearly: a proposition is
articulated.
3.26 Der Name ist durch keine Definition The name cannot be analysed further A name cannot be dissected any furweiter
zu zergliedern: er ist ein Urzei- by any definition. It is a primitive sign. ther by means of a definition: it is a primchen.
itive sign.
3.261 Jedes difinierte Zeichen bezeichnet Every defined sign signifies via those Every sign that has a definition signiü
b e r jene Zeichen, durch welche es de- signs by which it is defined, and the defi- fies via the signs that serve to define it;
finier wurde; und die Definitionen weisen nitions show the way. and the definitions point the way.
den Weg.
Zwei Zeichen, ein Urzeichen, und Two signs, one a primitive sign, and Two signs cannot signify in the same
ein durch Urzeichen definiertes, können one defined by primitive signs, cannot manner if one is primitive and the other
nicht auf dieselbe Art und Weise bezeich- signify in the same way. Names cannot is defined by means of primitive signs.
nen. Namen k a n n man nicht durch be taken to pieces by definition (nor any Names cannot be anatomized by means
Definitionen auseinanderlegen. (Kein Zei- sign which alone and independently has of definitions. (Nor can any sign that has
chen, welches allein, selbständig eine Be- a meaning). a meaning independently and on its own.)
deutung hat.)
3.262 Was in den Zeichen nicht zum Aus- What does not get expressed in the What signs fail to express, their applidruck
kommt, das zeigt ihre Anwendung. sign is shown by its application. What the cation shows. What signs slur over, their
Was die Zeichen verschlucken, das spricht signs conceal, their application declares. application says clearly.
ihre Anwendung aus.
3.263 Die Bedeutung von Urzeichen können The meanings of primitive signs can The meanings of primitive signs can
durch Erläuterungen erklärt werden. Er- be explained by elucidations. Elucida- be explained by means of elucidations.
läuterungen sind Sätze, welche die Ur- tions are propositions which contain the Elucidations are propositions that conzeichen
enthalten. Sie können also nur primitive signs. They can, therefore, tain the primitive signs. So they can only
verstanden werden, wenn die Bedeutun- only be understood when the meanings of be understood if the meanings of those
gen dieser Zeichen bereits bekannt sind. these signs are already known. signs are already known.
3.3 Nur der Satz hat Sinn; nur im Zusam- Only the proposition has sense; only Only propositions have sense; only in
menhang des Satzes hat ein Name Bedeu- in the context of a proposition has a name the nexus of a proposition does a name
24
tung. meaning. have meaning.
3.31 Jeden Teil des Satzes, der seinen Sinn Every part of a proposition which char- I call any part of a proposition that
charakterisiert, nenne ich einen Aus- acterizes its sense I call an expression (a characterizes its sense an expression (or
druck (ein Symbol). symbol). a symbol).
(Der Satz selbst ist ein Ausdruck.) (The proposition itself is an expres- (A proposition is itself an expression.)
sion.)
Ausdruck ist alles, für den Sinn des Expressions are everything—essen- Everything essential to their sense
Satzes wesentliche, was Sätze miteinan- tial for the sense of the proposition—that that propositions can have in common
der gemein haben können. propositions can have in common with with one another is an expression.
one another.
Der Ausdruck kennzeichnet eine An expression characterizes a form An expression is the mark of a form
Form und einen Inhalt. and a content. and a content.
3.311 Der Ausdruck setzt die Formen aller An expression presupposes the forms An expression presupposes the forms
Sätze voraus, in welchem er vorkommen of all propositions in which it can occur. of all the propositions in which it can ockann.
Er ist das gemeinsame charakteri- It is the common characteristic mark of a cur. It is the common characteristic mark
stische Merkmal einer Klasse von Sätzen. class of propositions. of a class of propositions.
3.312 Er wird also dargestellt durch die all- It is therefore represented by the gen- It is therefore presented by means of
gemeine Form der Sätze, die er charakte- eral form of the propositions which it char- the general form of the propositions that
risiert. acterizes. it characterizes.
Und zwar wird in dieser Form der Aus- And in this form the expression is con- In fact, in this form the expression will
druck k o n s t a n t und alles übrige v a - stant and everything else variable. be constant and everything else variable.
r i a b e l sein.
3.313 Der Ausdruck wird also durch eine Va- An expression is thus presented by a Thus an expression is presented by
riable dargestellt, deren Werte die Sätze variable, whose values are the proposi- means of a variable whose values are the
sind, die den Ausdruck enthalten. tions which contain the expression. propositions that contain the expression.
(Im Grenzfall wird die Variable zur (In the limiting case the variable be- (In the limiting case the variable beKonstanten,
der Ausdruck zum Satz.) comes constant, the expression a proposi- comes a constant, the expression becomes
tion.) a proposition.)
Ich nenne eine solche Variable „Satz- I call such a variable a “propositional I call such a variable a ‘propositional
variable“. variable”. variable’.
3.314 Der Ausdruck hat nur im Satz Bedeu- An expression has meaning only in a An expression has meaning only in
tung. Jede Variable lässt sich als Satzva- proposition. Every variable can be con- a proposition. All variables can be conriable
auffassen. ceived as a propositional variable. strued as propositional variables.
(Auch der variable Name.) (Including the variable name.) (Even variable names.)
3.315 Verwandeln wir einen Bestandteil ei- If we change a constituent part of a If we turn a constituent of a propones
Satzes in eine Variable, so gibt es eine proposition into a variable, there is a class sition into a variable, there is a class of
Klasse von Sätzen, welche sämtlich Wer- of propositions which are all the values of propositions all of which are values of the
25
te des so entstandenen variablen Satzes the resulting variable proposition. This resulting variable proposition. In general,
sind. Diese Klasse hängt im allgemeinen class in general still depends on what, by this class too will be dependent on the
noch davon ab, was wir, nach willkürli- arbitrary agreement, we mean by parts meaning that our arbitrary conventions
cher Übereinkunft, mit Teilen jenes Sat- of that proposition. But if we change all have given to parts of the original proposizes
meinen. Verwandeln wir aber alle je- those signs, whose meaning was arbitrar- tion. But if all the signs in it that have arne
Zeichen, deren Bedeutung willkürlich ily determined, into variables, there al- bitrarily determined meanings are turned
bestimmt wurde, in Variable, so gibt es ways remains such a class. But this is into variables, we shall still get a class of
nun noch immer eine solche Klasse. Diese now no longer dependent on any agree- this kind. This one, however, is not depenaber
ist nun von keiner Übereinkunft ab- ment; it depends only on the nature of dent on any convention, but solely on the
hängig, sondern nur noch von der Natur the proposition. It corresponds to a logi- nature of the proposition. It corresponds
des Satzes. Sie entspricht einer logischen cal form, to a logical prototype. to a logical form—a logical prototype.
Form—einem logischen Urbild.
3.316 Welche Werte die Satzvariable anneh- What values the propositional vari- What values a propositional variable
men darf, wird festgesetzt. able can assume is determined. may take is something that is stipulated.
Die Festsetzung der Werte i s t die The determination of the values is the The stipulation of values is the variVariable.
variable. able.
3.317 Die Festsetzung der Werte der Satzva- The determination of the values of the To stipulate values for a propositional
riablen ist die A n g a b e d e r S ä t z e, propositional variable is done by indicat- variable is to give the propositions whose
deren gemeinsames Merkmal die Varia- ing the propositions whose common mark common characteristic the variable is.
ble ist. the variable is.
Die Festsetzung ist eine Beschreibung The determination is a description of The stipulation is a description of
dieser Sätze. these propositions. those propositions.
Die Festsetzung wird also nur von The determination will therefore deal The stipulation will therefore be conSymbolen,
nicht von deren Bedeutung only with symbols not with their mean- cerned only with symbols, not with their
handeln. ing. meaning.
Und n u r dies ist der Festset- And only this is essential to the deter- And the only thing essential to the
zung wesentlich, d a s s s i e n u r e i - mination, that it is only a description of stipulation is that it is merely a descripn
e B e s c h r e i b u n g v o n S y m b o - symbols and asserts nothing about what tion of symbols and states nothing about
l e n i s t u n d n i c h t ü b e r d a s is symbolized. what is signified.
B e z e i c h n e t e a u s s a g t.
Wie die Beschreibung der Sätze ge- The way in which we describe the How the description of the proposischieht,
ist unwesentlich. propositions is not essential. tions is produced is not essential.
3.318 Den Satz fasse ich—wie Frege und I conceive the proposition—like Frege Like Frege and Russell I construe a
Russell—als Funktion der in ihm enthal- and Russell—as a function of the expres- proposition as a function of the exprestenen
Ausdrücke auf. sions contained in it. sions contained in it.
3.32 Das Zeichen ist das sinnlich Wahr- The sign is the part of the symbol per- A sign is what can be perceived of a
nehmbare am Symbol. ceptible by the senses. symbol.
26
3.321 Zwei verschiedene Symbole können al- Two different symbols can therefore So one and the same sign (written or
so das Zeichen (Schriftzeichen oder Laut- have the sign (the written sign or the spoken, etc.) can be common to two difzeichen
etc.) miteinander gemein haben— sound sign) in common—they then sig- ferent symbols—in which case they will
sie bezeichnen dann auf verschiedene Art nify in different ways. signify in different ways.
und Weise.
3.322 Es kann nie das gemeinsame Merk- It can never indicate the common char- Our use of the same sign to signify two
mal zweier Gegenstände anzeigen, dass acteristic of two objects that we symbolize different objects can never indicate a comwir
sie mit demselben Zeichen, aber them with the same signs but by differ- mon characteristic of the two, if we use it
durch zwei verschiedene B e z e i c h - ent methods of symbolizing. For the sign with two different modes of signification.
n u n g s w e i s e n bezeichnen. Denn das is arbitrary. We could therefore equally For the sign, of course, is arbitrary. So we
Zeichen ist ja willkürlich. Man könnte well choose two different signs and where could choose two different signs instead,
also auch zwei verschiedene Zeichen wäh- then would be what was common in the and then what would be left in common
len, und wo bliebe dann das Gemeinsame symbolization? on the signifying side?
in der Bezeichnung?
3.323 In der Umgangssprache kommt es un- In the language of everyday life it very In everyday language it very fregemein
häufig vor, dass dasselbe Wort auf often happens that the same word sig- quently happens that the same word has
verschiedene Art und Weise bezeichnet— nifies in two different ways—and there- different modes of signification—and so
also verschiedene Symbolen angehört—, fore belongs to two different symbols—or belongs to different symbols—or that two
oder, dass zwei Wörter, die auf verschiede- that two words, which signify in different words that have different modes of signine
Art und Weise bezeichnen, äußerlich ways, are apparently applied in the same fication are employed in propositions in
in der gleichen Weise im Satz angewandt way in the proposition. what is superficially the same way.
werden.
So erscheint das Wort „ist“ als Kopu- Thus the word “is” appears as the cop- Thus the word ‘is’ figures as the copla,
als Gleichheitszeichen und als Aus- ula, as the sign of equality, and as the ula, as a sign for identity, and as an exdruck
der Existenz; „existieren“ als in- expression of existence; “to exist” as an pression for existence; ‘exist’ figures as an
transitives Zeitwort wie „gehen“; „iden- intransitive verb like “to go”; “identical” intransitive verb like ‘go’, and ‘identical’
tisch“ als Eigenschaftswort; wir reden von as an adjective; we speak of something as an adjective; we speak of something,
E t w a s, aber auch davon, dass e t w a s but also of the fact of something happen- but also of something’s happening.
geschieht. ing.
(Im Satze „Grün ist grün“—wo das (In the proposition “Green is green”— (In the proposition, ‘Green is green’—
erste Wort ein Personenname, das letz- where the first word is a proper name as where the first word is the proper name
te ein Eigenschaftswort ist—haben diese the last an adjective—these words have of a person and the last an adjective—
Worte nicht einfach verschiedene Bedeu- not merely different meanings but they these words do not merely have different
tung, sondern es sind v e r s c h i e d e n e are different symbols.) meanings: they are different symbols.)
S y m b o l e.)
3.324 So entstehen leicht die fundamental- Thus there easily arise the most fun- In this way the most fundamental consten
Verwechselungen (deren die ganze damental confusions (of which the whole fusions are easily produced (the whole of
27
Philosophie voll ist). of philosophy is full). philosophy is full of them).
3.325 Um diesen Irrtümern zu entgehen, In order to avoid these errors, we must In order to avoid such errors we must
müssen wir eine Zeichensprache verwen- employ a symbolism which excludes them, make use of a sign-language that excludes
den, welche sie ausschließt, indem sie by not applying the same sign in different them by not using the same sign for difnicht
das gleiche Zeichen in verschied- symbols and by not applying signs in the ferent symbols and by not using in a sunen
Symbolen, und Zeichen, welche auf same way which signify in different ways. perficially similar way signs that have
verschiedene Art bezeichnen, nicht äu- A symbolism, that is to say, which obeys different modes of signification: that is to
ßerlich auf die gleiche Art verwendet. the rules of logical grammar—of logical say, a sign-language that is governed by
Eine Zeichensprache also, die der l o - syntax. logical grammar—by logical syntax.
g i s c h e n Grammatik—der logischen
Syntax—gehorcht.
(Die Begriffsschrift Freges und Rus- (The logical symbolism of Frege and (The conceptual notation of Frege and
sells ist eine solche Sprache, die aller- Russell is such a language, which, how- Russell is such a language, though, it is
dings noch nicht alle Fehler ausschließt.) ever, does still not exclude all errors.) true, it fails to exclude all mistakes.)
3.326 Um das Symbol am Zeichen zu erken- In order to recognize the symbol in the In order to recognize a symbol by its
nen, muss man auf den sinnvollen Ge- sign we must consider the significant use. sign we must observe how it is used with
brauch achten. a sense.
3.327 Das Zeichen bestimmt erst mit sei- The sign determines a logical form A sign does not determine a logical
ner logisch-syntaktischen Verwendung only together with its logical syntactic ap- form unless it is taken together with its
zusammen eine logische Form. plication. logico-syntactical employment.
3.328 Wird ein Zeichen n i c h t g e - If a sign is not necessary then it is If a sign is useless, it is meaningless.
b r a u c h t, so ist es bedeutungslos. Das meaningless. That is the meaning of Oc- That is the point of Occam’s maxim.
ist der Sinn der Devise Occams. cam’s razor.
(Wenn sich alles so verhält als hätte (If everything in the symbolism works (If everything behaves as if a sign had
ein Zeichen Bedeutung, dann hat es auch as though a sign had meaning, then it has meaning, then it does have meaning.)
Bedeutung.) meaning.)
3.33 In der logischen Syntax darf nie die In logical syntax the meaning of a sign In logical syntax the meaning of a sign
Bedeutung eines Zeichens eine Rolle spie- ought never to play a rôle; it must admit should never play a role. It must be poslen;
sie muss sich aufstellen lassen, ohne of being established without mention be- sible to establish logical syntax without
dass dabei von der B e d e u t u n g eines ing thereby made of the meaning of a sign; mentioning the meaning of a sign: only
Zeichens die Rede wäre, sie darf n u r die it ought to presuppose only the descrip- the description of expressions may be preBeschreibung
der Ausdrücke vorausset- tion of the expressions. supposed.
zen.
3.331 Von dieser Bemerkung sehen wir in From this observation we get a fur- From this observation we turn to RusRussells
„Theory of types“ hinüber: Der ther view—into Russell’s Theory of Types. sell’s ‘theory of types’. It can be seen that
Irrtum Russells zeigt sich darin, dass er Russell’s error is shown by the fact that Russell must be wrong, because he had
bei der Aufstellung der Zeichenregeln von in drawing up his symbolic rules he has to mention the meaning of signs when
28
der Bedeutung der Zeichen reden musste. to speak about the things his signs mean. establishing the rules for them.
3.332 Kein Satz kann etwas über sich selbst No proposition can say anything about No proposition can make a statement
aussagen, weil das Satzzeichen nicht in itself, because the propositional sign can- about itself, because a propositional sign
sich selbst enthalten sein kann (das ist not be contained in itself (that is the cannot be contained in itself (that is the
die ganze „Theory of types“). “whole theory of types”). whole of the ‘theory of types’).
3.333 Eine Funktion kann darum nicht ihr A function cannot be its own argu- The reason why a function cannot be
eigenes Argument sein, weil das Funkti- ment, because the functional sign already its own argument is that the sign for a
onszeichen bereits das Urbild seines Ar- contains the prototype of its own argu- function already contains the prototype
guments enthält und es sich nicht selbst ment and it cannot contain itself. of its argument, and it cannot contain
enthalten kann. itself.
Nehmen wir nämlich an, die Funktion If, for example, we suppose that the For let us suppose that the function
F(fx) könnte ihr eigenes Argument sein; function F(fx) could be its own argu- F(fx) could be its own argument: in
dann gäbe es also einen Satz: „F(F(fx))“ ment, then there would be a proposition that case there would be a proposition
und in diesem müssen die äußere Funkti- “F(F(fx))”, and in this the outer function ‘F(F(fx))’, in which the outer function F
on F und die innere Funtion F verschie- F and the inner function F must have and the inner function F must have differdene
Bedeutungen haben, denn die in- different meanings; for the inner has the ent meanings, since the inner one has the
nere hat die Form φ(fx), die äußere die form φ(fx), the outer the form ψ(φ(fx)). form φ(fx) and the outer one has the form
Form ψ(φ(fx)). Gemeinsam ist den beiden Common to both functions is only the let- ψ(φ(fx)). Only the letter ‘F’ is common to
Funktionen nur der Buchstabe „F“, der ter “F”, which by itself signifies nothing. the two functions, but the letter by itself
aber allein nichts bezeichnet. signifies nothing.
Dies wird sofort klar, wenn wir statt This is at once clear, if instead of This immediately becomes clear if in„F(Fu)“
schreiben „(∃φ):F(φu) . φu = Fu“. “F(Fu)” we write “(∃φ):F(φu) . φu = Fu”. stead of ‘F(Fu)’ we write ‘(∃φ):F(φu) .
φu = Fu’.
Hiermit erledigt sich Russells Para- Herewith Russell’s paradox vanishes. That disposes of Russell’s paradox.
dox.
3.334 Die Regeln der logischen Syntax müs- The rules of logical syntax must follow The rules of logical syntax must go
sen sich von selbst verstehen, wenn man of themselves, if we only know how every without saying, once we know how each
nur weiß, wie ein jedes Zeichen bezeich- single sign signifies. individual sign signifies.
net.
3.34 Der Satz besitzt wesentliche und zu- A proposition possesses essential and A proposition possesses essential and
fällige Züge. accidental features. accidental features.
Zufällig sind die Züge, die von der Accidental are the features which are Accidental features are those that rebesonderen
Art der Hervorbringung des due to a particular way of producing the sult from the particular way in which the
Satzzeichens herrühren. Wesentlich die- propositional sign. Essential are those propositional sign is produced. Essenjenigen,
welche allein den Satz befähigen, which alone enable the proposition to ex- tial features are those without which the
seinen Sinn auszudrücken. press its sense. proposition could not express its sense.
3.341 Das Wesentliche am Satz ist also das, The essential in a proposition is there- So what is essential in a proposition
29
was allen Sätzen, welche den gleichen fore that which is common to all proposi- is what all propositions that can express
Sinn ausdrücken können, gemeinsam ist. tions which can express the same sense. the same sense have in common.
Und ebenso ist allgemein das Wesent- And in the same way in general the And similarly, in general, what is esliche
am Symbol das, was alle Symbole, essential in a symbol is that which all sential in a symbol is what all symbols
die denselben Zweck erfüllen können, ge- symbols which can fulfill the same pur- that can serve the same purpose have in
meinsam haben. pose have in common. common.
3.3411 Man könnte also sagen: Der eigentli- One could therefore say the real name So one could say that the real name
che Name ist das, was alle Symbole, die is that which all symbols, which signify of an object was what all symbols that
den Gegenstand bezeichnen, gemeinsam an object, have in common. It would then signified it had in common. Thus, one by
haben. Es würde sich so successive erge- follow, step by step, that no sort of compo- one, all kinds of composition would prove
ben, dass keinerlei Zusammensetzung für sition was essential for a name. to be unessential to a name.
den Namen wesentlich ist.
3.342 An unseren Notationen ist zwar etwas In our notations there is indeed some- Although there is something arbiwillkürlich,
aber d a s ist nicht willkür- thing arbitrary, but this is not arbitrary, trary in our notations, this much is
lich: Dass, w e n n wir etwas willkürlich namely that if we have determined any- not arbitrary—that when we have deterbestimmt
haben, dann etwas anderes der thing arbitrarily, then something else mined one thing arbitrarily, something
Fall sein muss. (Dies hängt von dem W e - must be the case. (This results from the else is necessarily the case. (This derives
s e n der Notation ab.) essence of the notation.) from the essence of notation.)
3.3421 Eine besondere Bezeichnungsweise A particular method of symbolizing A particular mode of signifying may
mag unwichtig sein, aber wichtig ist es may be unimportant, but it is always im- be unimportant but it is always imporimmer,
dass diese eine m ö g l i c h e Be- portant that this is a possible method of tant that it is a possible mode of signifyzeichnungsweise
ist. Und so verhält es symbolizing. And this happens as a rule ing. And that is generally so in philososich
in der Philosophie überhaupt: Das in philosophy: The single thing proves phy: again and again the individual case
Einzelne erweist sich immer wieder als over and over again to be unimportant, turns out to be unimportant, but the posunwichtig,
aber die Möglichkeit jedes Ein- but the possibility of every single thing sibility of each individual case discloses
zelnen gibt uns einen Aufschluss über das reveals something about the nature of the something about the essence of the world.
Wesen der Welt. world.
3.343 Definitionen sind Regeln der Überset- Definitions are rules for the transla- Definitions are rules for translating
zung von einer Sprache in eine andere. tion of one language into another. Every from one language into another. Any corJede
richtige Zeichensprache muss sich correct symbolism must be translatable rect sign-language must be translatable
in jede andere nach solchen Regeln über- into every other according to such rules. into any other in accordance with such
setzen lassen: D i e s ist, was sie alle It is this which all have in common. rules: it is this that they all have in comgemeinsam
haben. mon.
3.344 Das, was am Symbol bezeichnet, ist What signifies in the symbol is what What signifies in a symbol is what is
das Gemeinsame aller jener Symbole, is common to all those symbols by which common to all the symbols that the rules
durch die das erste den Regeln der lo- it can be replaced according to the rules of logical syntax allow us to substitute for
gischen Syntax zufolge ersetzt werden of logical syntax. it.
30
kann.
3.3441 Man kann z. B. das Gemeinsame aller We can, for example, express what For instance, we can express what
Notationen für die Wahrheitsfunktionen is common to all notations for the truth- is common to all notations for truthso
ausdrücken: Es ist ihnen gemeinsam, functions as follows: It is common to them functions in the following way: they have
dass sich alle—z. B.—durch die Notation that they all, for example, can be replaced in common that, for example, the notation
von „∼p“ („nicht p“) und „p ∨ q“ („p oder by the notations of “∼p” (“not p”) and that uses ‘∼p’ (‘not p’) and ‘p∨q’ (‘p or q’)
q“) e r s e t z e n l a s s e n. “p ∨ q” (“p or q”). can be substituted for any of them.
(Hiermit ist die Art und Weise gekenn- (Herewith is indicated the way in (This serves to characterize the way in
zeichnet, wie eine spezielle mögliche No- which a special possible notation can give which something general can be disclosed
tation uns allgemeine Aufschlüsse geben us general information.) by the possibility of a specific notation.)
kann.)
3.3442 Das Zeichen des Komplexes löst sich The sign of the complex is not arbitrar- Nor does analysis resolve the sign for
auch bei der Analyse nicht willkürlich auf, ily resolved in the analysis, in such a way a complex in an arbitrary way, so that it
so dass etwa seine Auflösung in jedem that its resolution would be different in would have a different resolution every
Satzgefüge eine andere wäre. every propositional structure. time that it was incorporated in a different
proposition.
3.4 Der Satz bestimmt einen Ort im lo- The proposition determines a place in A proposition determines a place in
gischen Raum. Die Existenz dieses logi- logical space: the existence of this logical logical space. The existence of this logical
schen Ortes ist durch die Existenz der place is guaranteed by the existence of the place is guaranteed by the mere existence
Bestandteile allein verbürgt, durch die constituent parts alone, by the existence of the constituents—by the existence of
Existenz des sinnvollen Satzes. of the significant proposition. the proposition with a sense.
3.41 Das Satzzeichen und die logischen Ko- The propositional sign and the logical The propositional sign with logical coordinaten:
Das ist der logische Ort. co-ordinates: that is the logical place. ordinates—that is the logical place.
3.411 Der geometrische und der logische Ort The geometrical and the logical place In geometry and logic alike a place is
stimmen darin überein, dass beide die agree in that each is the possibility of an a possibility: something can exist in it.
Möglichkeit einer Existenz sind. existence.
3.42 Obwohl der Satz nur einen Ort des lo- Although a proposition may only de- A proposition can determine only one
gischen Raumes bestimmen darf, so muss termine one place in logical space, the place in logical space: nevertheless the
doch durch ihn schon der ganze logische whole logical space must already be given whole of logical space must already be
Raum gegeben sein. by it. given by it.
(Sonst würden durch die Verneinung, (Otherwise denial, the logical sum, (Otherwise negation, logical sum,
die logische Summe, das logische Pro- the logical product, etc., would always in- logical product, etc., would introduce
dukt, etc. immer neue Elemente—in troduce new elements—in co-ordination.) more and more new elements—in coKoordinaten—eingeführt.)
ordination.)
(Das logische Gerüst um das Bild her- (The logical scaffolding round the pic- (The logical scaffolding surrounding
um bestimmt den logischen Raum. Der ture determines the logical space. The a picture determines logical space. The
Satz durchgreift den ganzen logischen proposition reaches through the whole force of a proposition reaches through the
31
Raum.) logical space.) whole of logical space.)
3.5 Das angewandte, gedachte Satzeichen The applied, thought, propositional A propositional sign, applied and
ist der Gedanke. sign, is the thought. thought out, is a thought.
4 Der Gedanke ist der sinnvolle Satz. The thought is the significant proposi- A thought is a proposition with a
tion. sense.
4.001 Die Gesamtheit der Sätze ist die Spra- The totality of propositions is the lan- The totality of propositions is lanche.
guage. guage.
4.002 Der Mensch besitzt die Fähigkeit Man possesses the capacity of con- Man possesses the ability to construct
Sprachen zu bauen, womit sich jeder Sinn structing languages, in which every sense languages capable of expressing every
ausdrücken lässt, ohne eine Ahnung da- can be expressed, without having an idea sense, without having any idea how each
von zu haben, wie und was jedes Wort how and what each word means—just as word has meaning or what its meaning
bedeutet.—Wie man auch spricht, ohne one speaks without knowing how the sin- is—just as people speak without knowing
zu wissen, wie die einzelnen Laute her- gle sounds are produced. how the individual sounds are produced.
vorgebracht werden.
Die Umgangssprache ist ein Teil des Colloquial language is a part of the hu- Everyday language is a part of the humenschlichen
Organismus und nicht we- man organism and is not less complicated man organism and is no less complicated
niger kompliziert als dieser. than it. than it.
Es ist menschenunmöglich, From it it is humanly impossible to It is not humanly possible to gather
die Sprachlogik aus ihr unmittelbar zu gather immediately the logic of language. immediately from it what the logic of lanentnehmen.
guage is.
Die Sprache verkleidet den Gedanken. Language disguises the thought; so Language disguises thought. So much
Und zwar so, dass man nach der äußeren that from the external form of the clothes so, that from the outward form of the
Form des Kleides, nicht auf deie Form des one cannot infer the form of the thought clothing it is impossible to infer the form
bekleideten Gedankens schließen kann; they clothe, because the external form of the thought beneath it, because the outweil
die äußere Form des Kleides nach of the clothes is constructed with quite ward form of the clothing is not designed
ganz anderen Zwecken gebildet ist als da- another object than to let the form of the to reveal the form of the body, but for ennach,
die Form des Körpers erkennen zu body be recognized. tirely different purposes.
lassen.
Die stillschweigenden Abmachungen The silent adjustments to understand The tacit conventions on which the
zum Verständnis der Umgangssprache colloquial language are enormously com- understanding of everyday language desind
enorm kompliziert. plicated. pends are enormously complicated.
4.003 Die meisten Sätze und Fragen, wel- Most propositions and questions, that Most of the propositions and questions
che über philosophische Dinge geschrie- have been written about philosophical to be found in philosophical works are not
ben worden sind, sind nicht falsch, son- matters, are not false, but senseless. We false but nonsensical. Consequently we
dern unsinnig. Wir können daher Fragen cannot, therefore, answer questions of cannot give any answer to questions of
dieser Art überhaupt nicht beantworten, this kind at all, but only state their sense- this kind, but can only point out that they
sondern nur ihre Unsinnigkeit feststellen. lessness. Most questions and propositions are nonsensical. Most of the propositions
32
Die meisten Fragen und Sätze der Philo- of the philosophers result from the fact and questions of philosophers arise from
sophen beruhen darauf, dass wir unsere that we do not understand the logic of our our failure to understand the logic of our
Sprachlogik nicht verstehen. language. language.
(Sie sind von der Art der Frage, ob das (They are of the same kind as the ques- (They belong to the same class as the
Gute mehr oder weniger identisch sei als tion whether the Good is more or less question whether the good is more or less
das Schöne.) identical than the Beautiful.) identical than the beautiful.)
Und es ist nicht verwunderlich, dass And so it is not to be wondered at that And it is not surprising that the deepdie
tiefsten Probleme eigentlich k e i n e the deepest problems are really no prob- est problems are in fact not problems at
Probleme sind. lems. all.
4.0031 Alle Philosophie ist „Sprachkritik“. All philosophy is “Critique of lan- All philosophy is a ‘critique of lan(Allerdings
nicht im Sinne Mauthners.) guage” (but not at all in Mauthner’s guage’ (though not in Mauthner’s sense).
Russells Verdienst ist es, gezeigt zu ha- sense). Russell’s merit is to have shown It was Russell who performed the service
ben, dass die scheinbar logische Form des that the apparent logical form of the of showing that the apparent logical form
Satzes nicht seine wirkliche sein muss. proposition need not be its real form. of a proposition need not be its real one.
4.01 Der Satz ist ein Bild der Wirklichkeit. The proposition is a picture of reality. A proposition is a picture of reality.
Der Satz ist ein Modell der Wirklich- The proposition is a model of the real- A proposition is a model of reality as
keit, so wie wir sie uns denken. ity as we think it is. we imagine it.
4.011 Auf den ersten Blick scheint der Satz— At the first glance the proposition— At first sight a proposition—one set
wie er etwa auf dem Papier gedruckt say as it stands printed on paper—does out on the printed page, for example—
steht—kein Bild der Wirklichkeit zu sein, not seem to be a picture of the reality of does not seem to be a picture of the reality
von der er handelt. Aber auch die No- which it treats. But nor does the musical with which it is concerned. But neither
tenschrift scheint auf den ersten Blick score appear at first sight to be a picture do written notes seem at first sight to
kein Bild der Musik zu sein, und unse- of a musical piece; nor does our phonetic be a picture of a piece of music, nor our
re Lautzeichen-(Buchstaben-)Schrift kein spelling (letters) seem to be a picture of phonetic notation (the alphabet) to be a
Bild unserer Lautsprache. our spoken language. picture of our speech.
Und doch erweisen sich diese Zeichen- And yet these symbolisms prove to be And yet these sign-languages prove to
sprachen auch im gewöhnlichen Sinne als pictures—even in the ordinary sense of be pictures, even in the ordinary sense, of
Bilder dessen, was sie darstellen. the word—of what they represent. what they represent.
4.012 Offenbar ist, dass wir einen Satz von It is obvious that we perceive a propo- It is obvious that a proposition of the
der Form „aRb“ als Bild empfinden. Hier sition of the form aRb as a picture. Here form ‘aRb’ strikes us as a picture. In this
ist das Zeichen offenbar ein Gleichnis des the sign is obviously a likeness of the sig- case the sign is obviously a likeness of
Bezeichneten. nified. what is signified.
4.013 Und wenn wir in das Wesentliche die- And if we penetrate to the essence of And if we penetrate to the essence of
ser Bildhaftigkeit eindringen, so sehen this pictorial nature we see that this is this pictorial character, we see that it is
wir, dass dieselbe durch s c h e i n b a - not disturbed by apparent irregularities not impaired by apparent irregularities
r e U n r e g e l m ä ß i g k e i t e n (wie (like the use of and in the score). (such as the use of and in musical nodie
Verwendung von und in der No- tation).
33
tenschrift) n i c h t gestört wird.
Denn auch diese Unregelmäßigkeiten For these irregularities also picture For even these irregularities depict
bilden das ab, was sie ausdrücken sollen; what they are to express; only in another what they are intended to express; only
nur auf eine andere Art und Weise. way. they do it in a different way.
4.014 Die Grammophonplatte, der musika- The gramophone record, the musical A gramophone record, the musical
lische Gedanke, die Notenschrift, die thought, the score, the waves of sound, idea, the written notes, and the soundSchallwellen,
stehen alle in jener abbil- all stand to one another in that pictorial waves, all stand to one another in the
denden internen Beziehung zu einander, internal relation, which holds between same internal relation of depicting that
die zwischen Sprache und Welt besteht. language and the world. holds between language and the world.
Ihnen allen ist der logische Bau ge- To all of them the logical structure is They are all constructed according to
meinsam. common. a common logical pattern.
(Wie im Märchen die zwei Jünglinge, (Like the two youths, their two horses (Like the two youths in the fairy-tale,
ihre zwei Pferde und ihre Lilien. Sie sind and their lilies in the story. They are all their two horses, and their lilies. They
alle in gewissem Sinne Eins.) in a certain sense one.) are all in a certain sense one.)
4.0141 Dass es eine allgemeine Regel gibt, In the fact that there is a general rule There is a general rule by means of
durch die der Musiker aus der Partitur by which the musician is able to read which the musician can obtain the symdie
Symphonie entnehmen kann, durch the symphony out of the score, and that phony from the score, and which makes it
welche man aus der Linie auf der Gram- there is a rule by which one could recon- possible to derive the symphony from the
mophonplatte die Symphonie und nach struct the symphony from the line on a groove on the gramophone record, and, usder
ersten Regel wieder die Partitur ab- gramophone record and from this again— ing the first rule, to derive the score again.
leiten kann, darin besteht eben die inne- by means of the first rule—construct the That is what constitutes the inner simire
Ähnlichkeit dieser scheinbar so ganz score, herein lies the internal similarity larity between these things which seem to
verschiedenen Gebilde. Und jene Regel between these things which at first sight be constructed in such entirely different
ist das Gesetz der Projektion, welches die seem to be entirely different. And the rule ways. And that rule is the law of proSymphonie
in die Notensprache projiziert. is the law of projection which projects the jection which projects the symphony into
Sie ist die Regel der Übersetzung der No- symphony into the language of the mu- the language of musical notation. It is the
tensprache in die Sprache der Grammo- sical score. It is the rule of translation rule for translating this language into the
phonplatte. of this language into the language of the language of gramophone records.
gramophone record.
4.015 Die Möglichkeit aller Gleichnisse, der The possibility of all similes, of all the The possibility of all imagery, of all
ganzen Bildhaftigkeit unserer Ausdrucks- images of our language, rests on the logic our pictorial modes of expression, is conweise,
ruht in der Logik der Abbildung. of representation. tained in the logic of depiction.
4.016 Um das Wesen des Satzes zu verste- In order to understand the essence In order to understand the essential
hen, denken wir an die Hieroglyphen- of the proposition, consider hieroglyphic nature of a proposition, we should conschrift,
welche die Tatsachen die sie be- writing, which pictures the facts it de- sider hieroglyphic script, which depicts
schreibt abbildet. scribes. the facts that it describes.
Und aus ihr wurde die Buchstaben- And from it came the alphabet with- And alphabetic script developed out
34
schrift, ohne das Wesentliche der Abbil- out the essence of the representation be- of it without losing what was essential to
dung zu verlieren. ing lost. depiction.
4.02 Dies sehen wir daraus, dass wir den This we see from the fact that we un- We can see this from the fact that we
Sinn des Satzzeichens verstehen, ohne derstand the sense of the propositional understand the sense of a propositional
dass er uns erklärt wurde. sign, without having had it explained to sign without its having been explained to
us. us.
4.021 Der Satz ist ein Bild der Wirklichkeit: The proposition is a picture of reality, A proposition is a picture of reality:
Denn ich kenne die von ihm dargestel- for I know the state of affairs presented for if I understand a proposition, I know
le Sachlage, wenn ich den Satz verstehe. by it, if I understand the proposition. And the situation that it represents. And I unUnd
den Satz verstehe ich, ohne dass mir I understand the proposition, without its derstand the proposition without having
sein Sinn erklärt wurde. sense having been explained to me. had its sense explained to me.
4.022 Der Satz z e i g t seinen Sinn. The proposition shows its sense. A proposition shows its sense.
Der Satz z e i g t, wie es sich verhält, The proposition shows how things A proposition shows how things stand
w e n n er wahr ist. Und er s a g t, d a s s stand, if it is true. And it says, that they if it is true. And it says that they do so
es sich so verhält. do so stand. stand.
4.023 Die Wirklichkeit muss durch den Satz The proposition determines reality to A proposition must restrict reality to
auf ja oder nein fixiert sein. this extent, that one only needs to say two alternatives: yes or no.
“Yes” or “No” to it to make it agree with
reality.
Dazu muss sie durch ihn vollständig Reality must therefore be completely In order to do that, it must describe
beschrieben werden. described by the proposition. reality completely.
Der Satz ist die Beschreibung eines A proposition is the description of a A proposition is a description of a
Sachverhaltes. fact. state of affairs.
Wie die Beschreibung einen Gegen- As the description of an object de- Just as a description of an object destand
nach seinen externen Eigenschaf- scribes it by its external properties so scribes it by giving its external properties,
ten, so beschreibt der Satz die Wirklich- propositions describe reality by its inter- so a proposition describes reality by its inkeit
nach ihren internen Eigenschaften. nal properties. ternal properties.
Der Satz konstruiert eine Welt mit The proposition constructs a world A proposition constructs a world with
Hilfe eines logischen Gerüstes und dar- with the help of a logical scaffolding, and the help of a logical scaffolding, so that
um kann man am Satz auch sehen, wie therefore one can actually see in the one can actually see from the proposition
sich alles Logische verhält, w e n n er proposition all the logical features pos- how everything stands logically if it is
wahr ist. Man kann aus einem falschen sessed by reality if it is true. One can true. One can draw inferences from a
Satz S c h l ü s s e z i e h e n. draw conclusions from a false proposition. false proposition.
4.024 Einen Satz verstehen, heißt, wissen To understand a proposition means to To understand a proposition means to
was der Fall ist, wenn er wahr ist. know what is the case, if it is true. know what is the case if it is true.
(Man kann ihn also verstehen, ohne (One can therefore understand it with- (One can understand it, therefore,
zu wissen, ob er wahr ist.) out knowing whether it is true or not.) without knowing whether it is true.)
35
Man versteht ihn, wenn man seine One understands it if one understands It is understood by anyone who underBestandteile
versteht. it constituent parts. stands its constituents.
4.025 Die Übersetzung einer Sprache in ei- The translation of one language into When translating one language into
ne andere geht nicht so vor sich, dass man another is not a process of translating another, we do not proceed by translating
jeden S a t z der einen in einen S a t z each proposition of the one into a proposi- each proposition of the one into a proposider
anderen übersetzt, sondern nur die tion of the other, but only the constituent tion of the other, but merely by translatSatzbestandteile
werden übersetzt. parts of propositions are translated. ing the constituents of propositions.
(Und das Wörterbuch übersetzt nicht (And the dictionary does not only (And the dictionary translates not
nur Substantiva, sondern auch Zeit-, translate substantives but also adverbs only substantives, but also verbs, adjecEigenschafts-
und Bindewörter etc.; und and conjunctions, etc., and it treats them tives, and conjunctions, etc.; and it treats
es behandelt sie alle gleich.) all alike.) them all in the same way.)
4.026 Die Bedeutung der einfachen Zeichen The meanings of the simple signs (the The meanings of simple signs (words)
(der Wörter) müssen uns erklärt werden, words) must be explained to us, if we are must be explained to us if we are to undass
wir sie verstehen. to understand them. derstand them.
Mit den Sätzen aber verständigen wir By means of propositions we explain With propositions, however, we make
uns. ourselves. ourselves understood.
4.027 Es liegt im Wesen des Satzes, dass er It is essential to propositions, that It belongs to the essence of a proposiuns
einen n e u e n Sinn mitteilen kann. they can communicate a new sense to us. tion that it should be able to communicate
a new sense to us.
4.03 Ein Satz muss mit alten Ausdrücken A proposition must communicate a A proposition must use old expreseinen
neuen Sinn mitteilen. new sense with old words. sions to communicate a new sense.
Der Satz teilt uns eine Sachlage mit, The proposition communicates to us a A proposition communicates a situaalso
muss er w e s e n t l i c h mit der state of affairs, therefore it must be essen- tion to us, and so it must be essentially
Sachlage zusammenhängen. tially connected with the state of affairs. connected with the situation.
Und der Zusammenhang ist eben, And the connexion is, in fact, that it is And the connexion is precisely that it
dass er ihr logisches Bild ist. its logical picture. is its logical picture.
Der Satz sagt nur insoweit etwas aus, The proposition only asserts some- A proposition states something only
als er ein Bild ist. thing, in so far as it is a picture. in so far as it is a picture.
4.031 Im Satz wird gleichsam eine Sachlage In the proposition a state of affairs is, In a proposition a situation is, as it
probeweise zusammengestellt. as it were, put together for the sake of were, constructed by way of experiment.
experiment.
Man kann geradezu sagen: statt, die- One can say, instead of, This propo- Instead of, ‘This proposition has such
ser Satz hat diesen und diesen Sinn; die- sition has such and such a sense, This and such a sense’, we can simply say, ‘This
ser Satz stellt diese und diese Sachlage proposition represents such and such a proposition represents such and such a
dar. state of affairs. situation’.
4.0311 Ein Name steht für ein Ding, ein an- One name stands for one thing, and One name stands for one thing, anderer
für ein anderes Ding und unterein- another for another thing, and they are other for another thing, and they are
36
ander sind sie verbunden, so stellt das connected together. And so the whole, like combined with one another. In this way
Ganze—wie ein lebendes Bild—den Sach- a living picture, presents the atomic fact. the whole group—like a tableau vivant—
verhalt vor. presents a state of affairs.
4.0312 Die Möglichkeit des Satzes beruht auf The possibility of propositions is based The possibility of propositions is based
dem Prinzip der Vertretung von Gegen- upon the principle of the representation on the principle that objects have signs
ständen durch Zeichen. of objects by signs. as their representatives.
Mein Grundgedanke ist, dass die „logi- My fundamental thought is that the My fundamental idea is that the ‘logschen
Konstanten“ nicht vertreten. Dass “logical constants” do not represent. That ical constants’ are not representatives;
sich die L o g i k der Tatsachen nicht ver- the logic of the facts cannot be repre- that there can be no representatives of
treten lässt. sented. the logic of facts.
4.032 Nur insoweit ist der Satz ein Bild der The proposition is a picture of its state It is only in so far as a proposition is
Sachlage, als er logisch gegliedert ist. of affairs, only in so far as it is logically logically articulated that it is a picture of
articulated. a situation.
(Auch der Satz: „ambulo“, ist zusam- (Even the proposition “ambulo” is com- (Even the proposition, Ambulo, is commengesetzt,
denn sein Stamm ergibt mit posite, for its stem gives a different sense posite: for its stem with a different ending
einer anderen Endung, und seine Endung with another termination, or its termina- yields a different sense, and so does its
mit einem anderen Stamm, einen ande- tion with another stem.) ending with a different stem.)
ren Sinn.)
4.04 Am Satz muss gerade soviel zu unter- In the proposition there must be ex- In a proposition there must be exactly
scheiden sein, als an der Sachlage, die er actly as many thing distinguishable as as many distinguishable parts as in the
darstellt. there are in the state of affairs, which it situation that it represents.
represents.
Die beiden müssen die gleiche logische They must both possess the same The two must possess the same logical
(mathematische) Mannigfaltigkeit besit- logical (mathematical) multiplicity (cf. (mathematical) multiplicity. (Compare
zen. (Vergleiche Hertz’s „Mechanik“, über Hertz’s Mechanics, on Dynamic Models). Hertz’s Mechanics on dynamical models.)
dynamische Modelle.)
4.041 Diese mathematische Mannigfaltig- This mathematical multiplicity natu- This mathematical multiplicity, of
keit kann man natürlich nicht selbst wie- rally cannot in its turn be represented. course, cannot itself be the subject of deder
abbilden. Aus ihr kann man beim Ab- One cannot get outside it in the represen- piction. One cannot get away from it
bilden nicht heraus. tation. when depicting.
4.0411 Wollten wir z. B. das, was wir durch If we tried, for example, to express If, for example, we wanted to express
„(x). fx“ ausdrücken, durch Vorsetzen ei- what is expressed by “(x). fx” by putting what we now write as ‘(x). fx’ by putting
nes Indexes von „fx“ ausdrücken—etwa an index before fx, like: “Gen. fx”, it an affix in front of ‘fx’—for instance by
so: „Alg. fx“—es würde nicht genügen— would not do, we should not know what writing ‘Gen. fx’—it would not be adewir
wüssten nicht, was verallgemeinert was generalized. If we tried to show it quate: we should not know what was bewurde.
Wollten wir es durch einen Index by an index g, like: “f (xg)” it would not ing generalized. If we wanted to signalize
„a“ anzeigen—etwa so: „f (xa)“—es würde do—we should not know the scope of the it with an affix ‘g’—for instance by writ-
37
auch nicht genügen—wir wüssten nicht generalization. ing ‘f (xg)’—that would not be adequate
den Bereich der Allgemeinheitsbezeich- either: we should not know the scope of
nung. the generality-sign.
Wollten wir es durch Einführung If we were to try it by introducing If we were to try to do it by introduceiner
Marke in die Argumentstellen a mark in the argument places, like ing a mark into the argument-places—for
versuchen—etwa so: „(A, A).F(A, A)“—es “(G,G).F(G,G)”, it would not do—we instance by writing ‘(G,G).F(G,G)’ —it
würde nicht genügen—wir könnten die could not determine the identity of the would not be adequate: we should not be
Identität der Variablen nicht feststellen. variables, etc. able to establish the identity of the variU.s.w.
ables. And so on.
Alle diese Bezeichnungsweisen genü- All these ways of symbolizing are in- All these modes of signifying are inadgen
nicht, weil sie nicht die notwendige adequate because they have not the nec- equate because they lack the necessary
mathematische Mannigfaltigkeit haben. essary mathematical multiplicity. mathematical multiplicity.
4.0412 Aus demselben Grunde genügt die For the same reason the idealist expla- For the same reason the idealist’s apidealistische
Erklärung des Sehens der nation of the seeing of spatial relations peal to ‘spatial spectacles’ is inadequate
räumlichen Beziehung durch die „Raum- through “spatial spectacles” does not do, to explain the seeing of spatial relations,
brille“ nicht, weil sie nicht die Mannig- because it cannot explain the multiplicity because it cannot explain the multiplicity
faltigkeit dieser Beziehungen erklären of these relations. of these relations.
kann.
4.05 Die Wirklichkeit wird mit dem Satz Reality is compared with the proposi- Reality is compared with propositions.
verglichen. tion.
4.06 Nur dadurch kann der Satz wahr oder Propositions can be true or false only A proposition can be true or false only
falsch sein, indem er ein Bild der Wirk- by being pictures of the reality. in virtue of being a picture of reality.
lichkeit ist.
4.061 Beachtet man nicht, dass der Satz If one does not observe that proposi- It must not be overlooked that a propoeinen
von den Tatsachen unabhängigen tions have a sense independent of the sition has a sense that is independent of
Sinn hat, so kann man leicht glauben, facts, one can easily believe that true and the facts: otherwise one can easily supdass
wahr und falsch gleichberechtigte false are two relations between signs and pose that true and false are relations of
Beziehungen von Zeichen und Bezeichne- things signified with equal rights. equal status between signs and what they
tem sind. signify.
Man könnte dann z. B. sagen, dass „p“ One could, then, for example, say that In that case one could say, for example,
auch die wahre Art bezeichnet, was „∼p“ “p” signifies in the true way what “∼p” that ‘p’ signified in the true way what ‘∼p’
auf die falsche Art, etc. signifies in the false way, etc. signified in the false way, etc.
4.062 Kann man sich nicht mit falschen Can we not make ourselves under- Can we not make ourselves underSätzen,
wie bisher mit wahren, verstän- stood by means of false propositions as stood with false propositions just as we
digen? Solange man nur weiß, dass sie hitherto with true ones, so long as we have done up till now with true ones?—So
falsch gemeint sind. Nein! Denn, wahr ist know that they are meant to be false? No! long as it is known that they are meant
ein Satz, wenn es sich so verhält, wie wir For a proposition is true, if what we as- to be false.—No! For a proposition is true
38
es durch ihn sagen; und wenn wir mit „p“ sert by means of it is the case; and if by if we use it to say that things stand in a
∼p meinen, und es sich so verhält wie wir “p” we mean ∼p, and what we mean is certain way, and they do; and if by ‘p’ we
es meinen, so ist „p“ in der neuen Auffas- the case, then “p” in the new conception mean ∼p and things stand as we mean
sung wahr und nicht falsch. is true and not false. that they do, then, construed in the new
way, ‘p’ is true and not false.
4.0621 Dass aber die Zeichen „p“ und „∼p“ That, however, the signs “p” and “∼p” But it is important that the signs ‘p’
das gleiche sagen k ö n n e n, ist wichtig. can say the same thing is important, for and ‘∼p’ can say the same thing. For it
Denn es zeigt, dass dem Zeichen „∼“ in it shows that the sign “∼” corresponds to shows that nothing in reality corresponds
der Wirklichkeit nichts entspricht. nothing in reality. to the sign ‘∼’.
Dass in einem Satz die Verneinung That negation occurs in a proposition, The occurrence of negation in a propovorkommt,
ist noch kein Merkmal seines is no characteristic of its sense (∼∼p = p). sition is not enough to characterize its
Sinnes (∼∼p = p). sense (∼∼p = p).
Die Sätze „p“ und „∼p“ haben entge- The propositions “p” and “∼p” have The propositions ‘p’ and ‘∼p’ have opgengesetzten
Sinn, aber es entspricht ih- opposite senses, but to them corresponds posite sense, but there corresponds to
nen eine und dieselbe Wirklichkeit. one and the same reality. them one and the same reality.
4.063 Ein Bild zur Erklärung des Wahrheits- An illustration to explain the concept An analogy to illustrate the concept
begriffes: Schwarzer Fleck auf weißem of truth. A black spot on white paper; the of truth: imagine a black spot on white
Papier; die Form des Fleckes kann man form of the spot can be described by say- paper: you can describe the shape of the
beschreiben, indem man für jeden Punkt ing of each point of the plane whether it spot by saying, for each point on the sheet,
der Fläche angibt, ob er weiß oder is white or black. To the fact that a point whether it is black or white. To the fact
schwarz ist. Der Tatsache, dass ein Punkt is black corresponds a positive fact; to the that a point is black there corresponds a
schwarz ist, entspricht eine positive—der, fact that a point is white (not black), a positive fact, and to the fact that a point is
dass ein Punkt weiß (nicht schwarz) ist, negative fact. If I indicate a point of the white (not black), a negative fact. If I deseine
negative Tatsache. Bezeichne ich plane (a truth-value in Frege’s terminol- ignate a point on the sheet (a truth-value
einen Punkt der Fläche (einen Frege- ogy), this corresponds to the assumption according to Frege), then this corresponds
schen Wahrheitswert), so entspricht dies proposed for judgment, etc. etc. to the supposition that is put forward for
der Annahme, die zur Beurteilung aufge- judgement, etc. etc.
stellt wird, etc. etc.
Um aber sagen zu können, ein Punkt But to be able to say that a point is But in order to be able to say that a
sei schwarz oder weiß, muss ich vorerst black or white, I must first know under point is black or white, I must first know
wissen, wann man einen Punkt schwarz what conditions a point is called white when a point is called black, and when
und wann man ihn weiß nennt; um sa- or black; in order to be able to say “p” is white: in order to be able to say, ‘“p” is
gen zu können: „p“ ist wahr (oder falsch), true (or false) I must have determined true (or false)’, I must have determined
muss ich bestimmt haben, unter welchen under what conditions I call “p” true, in what circumstances I call ‘p’ true, and
Umständen ich „p“ wahr nenne, und da- and thereby I determine the sense of the in so doing I determine the sense of the
mit bestimme ich den Sinn des Satzes. proposition. proposition.
Der Punkt, an dem das Gleichnis hin- The point at which the simile breaks Now the point where the simile breaks
39
kt ist nun der: Wir können auf einen down is this: we can indicate a point on down is this: we can indicate a point on
Punkt des Papiers zeigen, auch ohne the paper, without knowing what white the paper even if we do not know what
zu wissen, was weiß und schwarz ist; and black are; but to a proposition with- black and white are, but if a proposieinem
Satz ohne Sinn aber entspricht out a sense corresponds nothing at all, for tion has no sense, nothing corresponds
gar nichts, denn er bezeichnet kein Ding it signifies no thing (truth-value) whose to it, since it does not designate a thing
(Wahrheitswert) dessen Eigenschaften et- properties are called “false” or “true”; the (a truth-value) which might have properwa
„falsch“ oder „wahr“ hießen; das Ver- verb of the proposition is not “is true” ties called ‘false’ or ‘true’. The verb of a
bum eines Satzes ist nicht „ist wahr“ oder or “is false”—as Frege thought—but that proposition is not ‘is true’ or ‘is false’, as
„ist falsch“—wie Frege glaubte—, sondern which “is true” must already contain the Frege thought: rather, that which ‘is true’
das, was „wahr ist“, muss das Verbum verb. must already contain the verb.
schon enthalten.
4.064 Jeder Satz muss s c h o n einen Sinn Every proposition must already have Every proposition must already have
haben; die Bejahung kann ihn ihm nicht a sense; assertion cannot give it a sense, a sense: it cannot be given a sense by afgeben,
denn sie bejaht ja gerade den Sinn. for what it asserts is the sense itself. And firmation. Indeed its sense is just what is
Und dasselbe gilt von der Verneinung, the same holds of denial, etc. affirmed. And the same applies to negaetc.
tion, etc.
4.0641 Man könnte sagen: Die Verneinung One could say, the denial is already One could say that negation must be
bezieht sich schon auf den logischen Ort, related to the logical place determined by related to the logical place determined by
den der verneinte Satz bestimmt. the proposition that is denied. the negated proposition.
Der verneinende Satz bestimmt einen The denying proposition determines a The negating proposition determines
a n d e r e n logischen Ort als der vernein- logical place other than does the proposi- a logical place different from that of the
te. tion denied. negated proposition.
Der verneinende Satz bestimmt einen The denying proposition determines a The negating proposition determines
logischen Ort mit Hilfe des logischen Or- logical place, with the help of the logical a logical place with the help of the logical
tes des verneinten Satzes, indem er jenen place of the proposition denied, by saying place of the negated proposition. For it
als außerhalb diesem liegend beschreibt. that it lies outside the latter place. describes it as lying outside the latter’s
logical place.
Dass man den verneinten Satz wie- That one can deny again the denied The negated proposition can be
der verneinen kann, zeigt schon, dass das, proposition, shows that what is denied is negated again, and this in itself shows
was verneint wird, schon ein Satz und already a proposition and not merely the that what is negated is already a proponicht
erst die Vorbereitung zu einem Sat- preliminary to a proposition. sition, and not merely something that is
ze ist. preliminary to a proposition.
4.1 Der Satz stellt das Bestehen und A proposition presents the existence Propositions represent the existence
Nichtbestehen der Sachverhalte dar. and non-existence of atomic facts. and non-existence of states of affairs.
4.11 Die Gesamtheit der wahren Sätze ist The totality of true propositions is the The totality of true propositions is the
die gesamte Naturwissenschaft (oder die total natural science (or the totality of the whole of natural science (or the whole corGesamtheit
der Naturwissenschaften). natural sciences). pus of the natural sciences).
40
4.111 Die Philosophie ist keine der Natur- Philosophy is not one of the natural Philosophy is not one of the natural
wissenschaften. sciences. sciences.
(Das Wort „Philosophie“ muss etwas (The word “philosophy” must mean (The word ‘philosophy’ must mean
bedeuten, was über oder unter, aber nicht something which stands above or below, something whose place is above or below
neben den Naturwissenschaften steht.) but not beside the natural sciences.) the natural sciences, not beside them.)
4.112 Der Zweck der Philosophie ist die logi- The object of philosophy is the logical Philosophy aims at the logical clarifische
Klärung der Gedanken. clarification of thoughts. cation of thoughts.
Die Philosophie ist keine Lehre, son- Philosophy is not a theory but an ac- Philosophy is not a body of doctrine
dern eine Tätigkeit. tivity. but an activity.
Ein philosophisches Werk besteht we- A philosophical work consists essen- A philosophical work consists essensentlich
aus Erläuterungen. tially of elucidations. tially of elucidations.
Das Resultat der Philosophie sind The result of philosophy is not a num- Philosophy does not result in ‘philonicht
„philosophische Sätze“, sondern das ber of “philosophical propositions”, but to sophical propositions’, but rather in the
Klarwerden von Sätzen. make propositions clear. clarification of propositions.
Die Philosophie soll die Gedanken, die Philosophy should make clear and de- Without philosophy thoughts are, as it
sonst, gleichsam, trübe und verschwom- limit sharply the thoughts which other- were, cloudy and indistinct: its task is to
men sind, klar machen und scharf abgren- wise are, as it were, opaque and blurred. make them clear and to give them sharp
zen. boundaries.
4.1121 Die Psychologie ist der Philosophie Psychology is no nearer related to phi- Psychology is no more closely related
nicht verwandter als irgend eine andere losophy, than is any other natural science. to philosophy than any other natural sciNaturwissenschaft.
ence.
Erkenntnistheorie ist die Philosophie The theory of knowledge is the philos- Theory of knowledge is the philosophy
der Psychologie. ophy of psychology. of psychology.
Entspricht nicht mein Studium der Does not my study of sign-language Does not my study of sign-language
Zeichensprache dem Studium der Denk- correspond to the study of thought pro- correspond to the study of thoughtprozesse,
welches die Philosophen für die cesses which philosophers held to be so processes, which philosophers used to conPhilosophie
der Logik für so wesentlich essential to the philosophy of logic? Only sider so essential to the philosophy of
hielten? Nur verwickelten sie sich mei- they got entangled for the most part in logic? Only in most cases they got entanstens
in unwesentliche psychologische unessential psychological investigations, gled in unessential psychological investiUntersuchungen
und eine analoge Gefahr and there is an analogous danger for my gations, and with my method too there is
gibt es auch bei meiner Methode. method. an analogous risk.
4.1122 Die Darwinsche Theorie hat mit der The Darwinian theory has no more to Darwin’s theory has no more to do
Philosophie nicht mehr zu schaffen als do with philosophy than has any other with philosophy than any other hypotheirgendeine
andere Hypothese der Natur- hypothesis of natural science. sis in natural science.
wissenschaft.
4.113 Die Philosophie begrenzt das bestreit- Philosophy limits the disputable Philosophy sets limits to the much disbare
Gebiet der Naturwissenschaft. sphere of natural science. puted sphere of natural science.
41
4.114 Sie soll das Denkbare abgrenzen und It should limit the thinkable and It must set limits to what can be
damit das Undenkbare. thereby the unthinkable. thought; and, in doing so, to what cannot
be thought.
Sie soll das Undenkbare von innen It should limit the unthinkable from It must set limits to what cannot be
durch das Denkbare begrenzen. within through the thinkable. thought by working outwards through
what can be thought.
4.115 Sie wird das Unsagbare bedeuten, in- It will mean the unspeakable by It will signify what cannot be said, by
dem sie das Sagbare klar darstellt. clearly displaying the speakable. presenting clearly what can be said.
4.116 Alles was überhaupt gedacht werden Everything that can be thought at all Everything that can be thought at all
kann, kann klar gedacht werden. Alles, can be thought clearly. Everything that can be thought clearly. Everything that
was sich aussprechen läßt, läßt sich klar can be said can be said clearly. can be put into words can be put clearly.
aussprechen.
4.12 Der Satz kann die gesamte Wirklich- Propositions can represent the whole Propositions can represent the whole
keit darstellen, aber er kann nicht das reality, but they cannot represent what of reality, but they cannot represent what
darstellen, was er mit der Wirklichkeit they must have in common with reality they must have in common with reality
gemein haben muss, um sie darstellen zu in order to be able to represent it—the in order to be able to represent it—logical
können—die logische Form. logical form. form.
Um die logische Form darstellen zu To be able to represent the logical In order to be able to represent logical
können, müssten wir uns mit dem Satze form, we should have to be able to put form, we should have to be able to station
außerhalb der Logik aufstellen können, ourselves with the propositions outside ourselves with propositions somewhere
das heißt außerhalb der Welt. logic, that is outside the world. outside logic, that is to say outside the
world.
4.121 Der Satz kann die logische Form nicht Propositions cannot represent the log- Propositions cannot represent logical
darstellen, sie spiegelt sich in ihm. ical form: this mirrors itself in the propo- form: it is mirrored in them.
sitions.
Was sich in der Sprache spiegelt, kann That which mirrors itself in language, What finds its reflection in language,
sie nicht darstellen. language cannot represent. language cannot represent.
Was s i c h in der Sprache ausdrückt, That which expresses itself in lan- What expresses itself in language, we
können w i r nicht durch sie ausdrücken. guage, we cannot express by language. cannot express by means of language.
Der Satz z e i g t die logische Form The propositions show the logical form Propositions show the logical form of
der Wirklichkeit. of reality. reality.
Er weist sie auf. They exhibit it. They display it.
4.1211 So zeigt ein Satz „fa“, dass in seinem Thus a proposition “fa” shows that in Thus one proposition ‘fa’ shows that
Sinn der Gegenstand a vorkommt, zwei its sense the object a occurs, two propo- the object a occurs in its sense, two propoSätze
„fa“ und „ga“, dass in ihnen beiden sitions “fa” and “ga” that they are both sitions ‘fa’ and ‘ga’ show that the same
von demselben Gegenstand die Rede ist. about the same object. object is mentioned in both of them.
Wenn zwei Sätze einander widerspre- If two propositions contradict one an- If two propositions contradict one an-
42
chen. So zeigt dies ihre Struktur; ebenso, other, this is shown by their structure; other, then their structure shows it; the
wenn einer aus dem anderen folgt. U.s.w. similarly if one follows from another, etc. same is true if one of them follows from
the other. And so on.
4.1212 Was gezeigt werden k a n n, k a n n What can be shown cannot be said. What can be shown, cannot be said.
nicht gesagt werden.
4.1213 Jetzt verstehen wir auch unser Ge- Now we understand our feeling that Now, too, we understand our feeling
fühl: dass wir im Besitze einer richti- we are in possession of the right logical that once we have a sign-language in
gen logischen Auffassung seien, wenn nur conception, if only all is right in our sym- which everything is all right, we already
einmal alles in unserer Zeichensprache bolism. have a correct logical point of view.
stimmt.
4.122 Wir können in gewissem Sinne von We can speak in a certain sense of In a certain sense we can talk about
formalen Eigenschaften der Gegenstände formal properties of objects and atomic formal properties of objects and states of
und Sachverhalte bezw. von Eigenschaf- facts, or of properties of the structure of affairs, or, in the case of facts, about structen
der Struktur der Tatsachen reden, facts, and in the same sense of formal tural properties: and in the same sense
und in demselben Sinne von formalen Re- relations and relations of structures. about formal relations and structural relationen
und Relationen von Strukturen. lations.
(Statt Eigenschaft der Struktur sage (Instead of property of the structure (Instead of ‘structural property’ I also
ich auch „interne Eigenschaft“; statt Re- I also say “internal property”; instead of say ‘internal property’; instead of ‘struclation
der Strukturen „interne Relation“. relation of structures “internal relation”. tural relation’, ‘internal relation’.
Ich führe diese Ausdrücke ein, um den I introduce these expressions in order I introduce these expressions in order
Grund der bei den Philosophen sehr ver- to show the reason for the confusion, very to indicate the source of the confusion
breiteten Verwechslung zwischen den in- widespread among philosophers, between between internal relations and relations
ternen Relationen und den eigentlichen internal relations and proper (external) proper (external relations), which is very
(externen) Relationen zu zeigen.) relations.) widespread among philosophers.)
Das Bestehen solcher interner Eigen- The holding of such internal proper- It is impossible, however, to assert by
schaften und Relationen kann aber nicht ties and relations cannot, however, be as- means of propositions that such internal
durch Sätze behauptet werden, sondern serted by propositions, but it shows it- properties and relations obtain: rather,
es zeigt sich in den Sätzen, welche jene self in the propositions, which present the this makes itself manifest in the proposiSachverhalte
darstellen und von jenen facts and treat of the objects in question. tions that represent the relevant states
Gegenständen handeln. of affairs and are concerned with the relevant
objects.
4.1221 Eine interne Eigenschaft einer Tatsa- An internal property of a fact we also An internal property of a fact can also
che können wir auch einen Zug dieser Tat- call a feature of this fact. (In the sense in be called a feature of that fact (in the
sache nennen. (In dem Sinn, in welchem which we speak of facial features.) sense in which we speak of facial features,
wir etwa von Gesichtszügen sprechen.) for example).
4.123 Eine Eigenschaft ist intern, wenn es A property is internal if it is unthink- A property is internal if it is unthinkundenkbar
ist, dass ihr Gegenstand sie able that its object does not possess it. able that its object should not possess it.
43
nicht besitzt.
(Diese blaue Farbe und jene stehen (This bright blue colour and that (This shade of blue and that one stand,
in der internen Relation von heller und stand in the internal relation of bright eo ipso, in the internal relation of lighter
dunkler eo ipso. Es ist undenkbar, dass and darker eo ipso. It is unthinkable that to darker. It is unthinkable that these two
d i e s e beiden Gegenstände nicht in die- these two objects should not stand in this objects should not stand in this relation.)
ser Relation stünden.) relation.)
(Hier entspricht dem schwankenden (Here to the shifting use of the words (Here the shifting use of the word ‘obGebrauch
der Worte „Eigenschaft“ und “property” and “relation” there corre- ject’ corresponds to the shifting use of the
„Relation“ der schwankende Gebrauch des sponds the shifting use of the word “ob- words ‘property’ and ‘relation’.)
Wortes „Gegenstand“.) ject”.)
4.124 Das Bestehen einer internen Eigen- The existence of an internal property The existence of an internal property
schaft einer möglichen Sachlage wird of a possible state of affairs is not ex- of a possible situation is not expressed
nicht durch einen Satz ausgedrückt, son- pressed by a proposition, but it expresses by means of a proposition: rather, it exdern
es drückt sich in dem sie darstellen- itself in the proposition which presents presses itself in the proposition representden
Satz durch eine interne Eigenschaft that state of affairs, by an internal prop- ing the situation, by means of an internal
dieses Satzes aus. erty of this proposition. property of that proposition.
Es wäre ebenso unsinnig, dem Satze It would be as senseless to ascribe It would be just as nonsensical to aseine
formale Eigenschaft zuzusprechen, a formal property to a proposition as to sert that a proposition had a formal propals
sie ihm abzusprechen. deny it the formal property. erty as to deny it.
4.1241 Formen kann man nicht dadurch von- One cannot distinguish forms from It is impossible to distinguish forms
einander unterscheiden, dass man sagt, one another by saying that one has this from one another by saying that one has
die eine habe diese, die andere aber jene property, the other that: for this assumes this property and another that property:
Eigenschaft; denn dies setzt voraus, dass that there is a sense in asserting either for this presupposes that it makes sense
es einen Sinn habe, beide Eigenschaften property of either form. to ascribe either property to either form.
von beiden Formen auszusagen.
4.125 Das Bestehen einer internen Relati- The existence of an internal relation The existence of an internal relation
on zwischen möglichen Sachlagen drückt between possible states of affairs ex- between possible situations expresses itsich
sprachlich durch eine interne Relati- presses itself in language by an internal self in language by means of an internal
on zwischen den sie darstellenden Sätzen relation between the propositions present- relation between the propositions repreaus.
ing them. senting them.
4.1251 Hier erledigt sich nun die Streitfra- Now this settles the disputed question Here we have the answer to the vexed
ge, „ob alle Relationen intern oder extern “whether all relations are internal or ex- question ‘whether all relations are interseien“.
ternal”. nal or external’.
4.1252 Reihen, welche durch i n t e r n e Re- Series which are ordered by internal I call a series that is ordered by an
lationen geordnet sind, nenne ich Formen- relations I call formal series. internal relation a series of forms.
reihen.
Die Zahlenreihe ist nicht nach einer The series of numbers is ordered not The order of the number-series is not
44
externen, sondern nach einer internen Re- by an external, but by an internal rela- governed by an external relation but by
lation geordnet. tion. an internal relation.
Ebenso die Reihe der Sätze „aRb“, Similarly the series of propositions The same is true of the series of propo“aRb”,
sitions ‘aRb’,
„(∃x):aRx . xRb“, “(∃x):aRx . xRb”, ‘(∃x):aRx . xRb’,
„(∃x, y):aRx . xR y . yRb“, u. s. f. “(∃x, y):aRx . xR y . yRb”, etc. ‘(∃x, y):aRx . xR y . yRb’, and so forth.
(Steht b in einer dieser Beziehungen (If b stands in one of these relations (If b stands in one of these relations
zu a, so nenne ich b einen Nachfolder von to a, I call b a successor of a.) to a, I call b a successor of a.)
a.)
4.126 In dem Sinne, in welchem wir von for- In the sense in which we speak of for- We can now talk about formal conmalen
Eigenschaften sprechen, können mal properties we can now speak also of cepts, in the same sense that we speak
wir nun auch von formalen Begriffen re- formal concepts. of formal properties.
den.
(Ich führe diesen Ausdruck ein, um (I introduce this expression in order (I introduce this expression in order
den Grund der Verwechslung der forma- to make clear the confusion of formal con- to exhibit the source of the confusion
len Begriffe mit den eigentlichen Begrif- cepts with proper concepts which runs between formal concepts and concepts
fen, welche die ganze alte Logik durch- through the whole of the old logic.) proper, which pervades the whole of trazieht,
klar zu machen.) ditional logic.)
Dass etwas unter einen formalen Be- That anything falls under a formal When something falls under a formal
griff als dessen Gegenstand fällt, kann concept as an object belonging to it, can- concept as one of its objects, this cannot
nicht durch einen Satz ausgedrückt wer- not be expressed by a proposition. But it be expressed by means of a proposition.
den. Sondern es zeigt sich an dem Zeichen is shown in the symbol for the object it- Instead it is shown in the very sign for
dieses Gegenstandes selbst. (Der Name self. (The name shows that it signifies an this object. (A name shows that it signizeigt,
dass er einen Gegenstand bezeich- object, the numerical sign that it signifies fies an object, a sign for a number that it
net, das Zahlenzeichen, dass es eine Zahl a number, etc.) signifies a number, etc.)
bezeichnet etc.)
Die formalen Begriffe können ja nicht, Formal concepts, cannot, like proper Formal concepts cannot, in fact, be
wie die eigentlichen Begriffe, durch eine concepts, be presented by a function. represented by means of a function, as
Funktion dargestellt werden. concepts proper can.
Denn ihre Merkmale, die formalen Ei- For their characteristics, the formal For their characteristics, formal propgenschaften,
werden nicht durch Funktio- properties, are not expressed by the func- erties, are not expressed by means of funcnen
ausgedrückt. tions. tions.
Der Ausdruck der formalen Eigen- The expression of a formal property is The expression for a formal property
schaft ist ein Zug gewisser Symbole. a feature of certain symbols. is a feature of certain symbols.
Das Zeichen der Merkmale eines for- The sign that signifies the character- So the sign for the characteristics of
malen Begriffes ist also ein charakteristi- istics of a formal concept is, therefore, a formal concept is a distinctive feature
scher Zug aller Symbole, deren Bedeutun- a characteristic feature of all symbols, of all symbols whose meanings fall under
45
gen unter den Begriff fallen. whose meanings fall under the concept. the concept.
Der Ausdruck des formalen Begriffes, The expression of the formal concept So the expression for a formal concept
also, eine Satzvariable, in welcher nur is therefore a propositional variable in is a propositional variable in which this
dieser charakteristische Zug konstant ist. which only this characteristic feature is distinctive feature alone is constant.
constant.
4.127 Die Satzvariable bezeichnet den for- The propositional variable signifies The propositional variable signifies
malen Begriff und ihre Werte die Gegen- the formal concept, and its values signify the formal concept, and its values signify
stände, welche unter diesen Begriff fal- the objects which fall under this concept. the objects that fall under the concept.
len.
4.1271 Jede Variable ist das Zeichen eines Every variable is the sign of a formal Every variable is the sign for a formal
formalen Begriffes. concept. concept.
Denn jede Variable stellt eine konstan- For every variable presents a constant For every variable represents a conte
Form dar, welche alle ihre Werte be- form, which all its values possess, and stant form that all its values possess, and
sitzen, und die als formale Eigenschaft which can be conceived as a formal prop- this can be regarded as a formal property
dieser Werte aufgefasst werden kann. erty of these values. of those values.
4.1272 So ist der variable Name „x“ das So the variable name “x” is the proper Thus the variable name ‘x’ is the
eigentliche Zeichen des Scheinbegriffes sign of the pseudo-concept object. proper sign for the pseudo-concept object.
G e g e n s t a n d.
Wo immer das Wort „Gegenstand“ Wherever the word “object” (“thing”, Wherever the word ‘object’ (‘thing’,
(„Ding“, „Sache“, etc.) richtig gebraucht “entity”, etc.) is rightly used, it is ex- etc.) is correctly used, it is expressed in
wird, wird es in der Begriffsschrift durch pressed in logical symbolism by the vari- conceptual notation by a variable name.
den variablen Namen ausgedrückt. able name.
Zum Beispiel in dem Satz „es For example in the proposition “there For example, in the proposition,
gibt 2 Gegenstände, welche . . . “ durch are two objects which . . . ”, by “(∃x, y)...”. ‘There are 2 objects which . . . ’, it is ex„(∃x,
y)...“. pressed by ‘(∃x, y)...’.
Wo immer es anders, also als eigentli- Wherever it is used otherwise, i.e. as Wherever it is used in a different way,
ches Begriffswort gebraucht wird, entste- a proper concept word, there arise sense- that is as a proper concept-word, nonsenhen
unsinnige Scheinsätze. less pseudo-propositions. sical pseudo-propositions are the result.
So kann man z. B. nicht sagen „Es So one cannot, e.g. say “There are ob- So one cannot say, for example, ‘There
gibt Gegenstände“, wie man etwa sagt: jects” as one says “There are books”. Nor are objects’, as one might say, ‘There are
„Es gibt Bücher“. Und ebenso wenig: „Es “There are 100 objects” or “There are ℵ0 books’. And it is just as impossible to say,
gibt 100 Gegenstände“, oder „Es gibt ℵ0 objects”. ‘There are 100 objects’, or, ‘There are ℵ0
Gegenstände“. objects’.
Und es ist unsinnig, von der A n - And it is senseless to speak of the And it is nonsensical to speak of the
z a h l a l l e r G e g e n s t ä n d e zu number of all objects. total number of objects.
sprechen.
Dasselbe gilt von den Worten „Kom- The same holds of the words “Com- The same applies to the words ‘com-
46
plex“, „Tatsache“, „Funktion“, „Zahl“, etc. plex”, “Fact”, “Function”, “Number”, etc. plex’, ‘fact’, ‘function’, ‘number’, etc.
Sie alle bezeichnen formale Begriffe They all signify formal concepts and They all signify formal concepts, and
und werden in der Begriffsschrift durch are presented in logical symbolism by are represented in conceptual notation by
Variable, nicht durch Funktionen oder variables, not by functions or classes (as variables, not by functions or classes (as
Klassen dargestellt. (Wie Frege und Rus- Frege and Russell thought). Frege and Russell believed).
sell glaubten.)
Ausdrücke wie „1 ist eine Zahl“, „Es Expressions like “1 is a number”, ‘1 is a number’, ‘There is only one zero’,
gibt nur Eine Null“ und alle ähnlichen “there is only one number nought”, and and all similar expressions are nonsensisind
unsinnig. all like them are senseless. cal.
(Es ist ebenso unsinnig zu sagen: „Es (It is as senseless to say, “there is only (It is just as nonsensical to say, ‘There
gibt nur Eine 1“, als es unsinnig wäre, zu one 1” as it would be to say: 2+2 is at 3 is only one 1’, as it would be to say, ‘2+2
sagen: „2+2 ist um 3 Uhr gleich 4“.) o’clock equal to 4.) at 3 o’clock equals 4’.)
4.12721 Der formale Begriff ist mit einem Ge- The formal concept is already given A formal concept is given immediately
genstand, der unter ihn fällt, bereits ge- with an object, which falls under it. One any object falling under it is given. It
geben. Man kann also nicht Gegenstän- cannot, therefore, introduce both, the ob- is not possible, therefore, to introduce as
de eines formalen Begriffes u n d den jects which fall under a formal concept primitive ideas objects belonging to a forformalen
Begriff selbst als Grundbegrif- and the formal concept itself, as prim- mal concept and the formal concept itself.
fe einführen. Man kann also z. B. nicht itive ideas. One cannot, therefore, e.g. So it is impossible, for example, to introden
Begriff der Funktion, und auch spezi- introduce (as Russell does) the concept duce as primitive ideas both the concept
elle Funktionen (wie Russell) als Grund- of function and also special functions as of a function and specific functions, as
begriffe einführen; oder den Begriff der primitive ideas; or the concept of number Russell does; or the concept of a number
Zahl und bestimmte Zahlen. and definite numbers. and particular numbers.
4.1273 Wollen wir den allgemeinen Satz: „b If we want to express in logical sym- If we want to express in conceptual
ist ein Nachfolger von a“ in der Begriffs- bolism the general proposition “b is a suc- notation the general proposition, ‘b is a
schrift ausdrücken, so brauchen wir hier- cessor of a” we need for this an expression successor of a’, then we require an expreszu
einen Ausdruck für das allgemeine for the general term of the formal series: sion for the general term of the series of
Glied der Formenreihe: forms
aRb, aRb, aRb,
(∃x):aRx . xRb, (∃x):aRx . xRb, (∃x):aRx . xRb,
(∃x, y):aRx . xR y . yRb, (∃x, y):aRx . xR y . yRb, (∃x, y):aRx . xR y . yRb,
. . . . . . . . . . . .
Das allgemeine Glied einer Formenreihe The general term of a formal series can In order to express the general term of a
kann man nur durch eine Variable aus- only be expressed by a variable, for the series of forms, we must use a variable,
drücken, denn der Begriff: Glied dieser concept symbolized by “term of this for- because the concept ‘term of that series
Formenreihe, ist ein f o r m a l e r Begriff. mal series” is a formal concept. (This of forms’ is a formal concept. (This is
(Dies haben Frege und Russell übersehen; Frege and Russell overlooked; the way in what Frege and Russell overlooked: con-
47
die Art und Weise, wie sie allgemeine Sät- which they express general propositions sequently the way in which they want to
ze wie den obigen ausdrücken wollen, ist like the above is, therefore, false; it con- express general propositions like the one
daher falsch; sie enthält einen circulus tains a vicious circle.) above is incorrect; it contains a vicious
vitiosus.) circle.)
Wir können das allgemeine Glied der We can determine the general term of We can determine the general term of
Formenreihe bestimmen, indem wir ihr the formal series by giving its first term a series of forms by giving its first term
erstes Glied angeben und die allgemeine and the general form of the operation, and the general form of the operation that
Form der Operation, welche das folgen- which generates the following term out produces the next term out of the proposide
Glied aus dem vorhergehenden Satz of the preceding proposition. tion that precedes it.
erzeugt.
4.1274 Die Frage nach der Existenz eines for- The question about the existence of a To ask whether a formal concept exmalen
Begriffes ist unsinnig. Denn kein formal concept is senseless. For no propo- ists is nonsensical. For no proposition can
Satz kann eine solche Frage beantworten. sition can answer such a question. be the answer to such a question.
(Man kann also z. B. nicht fra- (For example, one cannot ask: “Are (So, for example, the question, ‘Are
gen: „Gibt es unanalysierbare Subjekt- there unanalysable subject-predicate there unanalysable subject-predicate
Prädikatsätze¿‘) propositions?”) propositions?’ cannot be asked.)
4.128 Die logischen Formen sind zahl l o s. The logical forms are anumerical. Logical forms are without number.
Darum gibt es in der Logik keine aus- Therefore there are in logic no pre- Hence there are no pre-eminent numgezeichneten
Zahlen und darum gibt es eminent numbers, and therefore there is bers in logic, and hence there is no possikeinen
philosophischen Monismus oder no philosophical monism or dualism, etc. bility of philosophical monism or dualism,
Dualismus, etc. etc.
4.2 Der Sinn des Satzes ist seine Überein- The sense of a proposition is its agree- The sense of a proposition is its agreestimmung
und Nichtübereinstimmung ment and disagreement with the possibil- ment and disagreement with possibilities
mit den Möglichkeiten des Bestehens und ities of the existence and non-existence of of existence and non-existence of states of
Nichtbestehens der Sachverhalte. the atomic facts. affairs.
4.21 Der einfachste Satz, der Elementar- The simplest proposition, the elemen- The simplest kind of proposition, an
satz, behauptet das Bestehen eines Sach- tary proposition, asserts the existence of elementary proposition, asserts the exisverhaltes.
an atomic fact. tence of a state of affairs.
4.211 Ein Zeichen des Elementarsatzes ist It is a sign of an elementary proposi- It is a sign of a proposition’s being elees,
dass kein Elementarsatz mit ihm in tion, that no elementary proposition can mentary that there can be no elementary
Widerspruch stehen kann. contradict it. proposition contradicting it.
4.22 Der Elementarsatz besteht aus Na- The elementary proposition consists of An elementary proposition consists of
men. Er ist ein Zusammenhang, eine Ver- names. It is a connexion, a concatenation, names. It is a nexus, a concatenation, of
kettung, von Namen. of names. names.
4.221 Es ist offenbar, dass wir bei der Analy- It is obvious that in the analysis of It is obvious that the analysis of propose
der Sätze auf Elementarsätze kommen propositions we must come to elementary sitions must bring us to elementary propomüssen,
die aus Namen in unmittelbarer propositions, which consist of names in sitions which consist of names in immedi-
48
Verbindung bestehen. immediate combination. ate combination.
Es frägt sich hier, wie kommt der Satz- The question arises here, how the This raises the question how such comverband
zustande. propositional connexion comes to be. bination into propositions comes about.
4.2211 Auch wenn die Welt unendlich kom- Even if the world is infinitely complex, Even if the world is infinitely complex,
plex ist, so dass jede Tatsache aus unend- so that every fact consists of an infinite so that every fact consists of infinitely
lich vielen Sachverhalten besteht und je- number of atomic facts and every atomic many states of affairs and every state of
der Sachverhalt aus unendlich vielen Ge- fact is composed of an infinite number of affairs is composed of infinitely many obgenständen
zusammengesetzt ist, auch objects, even then there must be objects jects, there would still have to be objects
dann müsste es Gegenstände und Sach- and atomic facts. and states of affairs.
verhalte geben.
4.23 Der Name kommt im Satz nur im Zu- The name occurs in the proposition It is only in the nexus of an elemensammenhange
des Elementarsatzes vor. only in the context of the elementary tary proposition that a name occurs in a
proposition. proposition.
4.24 Die Namen sind die einfachen Symbo- The names are the simple symbols, I Names are the simple symbols: I indile,
ich deute sie durch einzelne Buchsta- indicate them by single letters (x, y, z). cate them by single letters (‘x’, ‘y’, ‘z’).
ben („x“, „y“, „z“) an.
Den Elementarsatz schreibe ich als The elementary proposition I write as I write elementary propositions as
Funktion der Namen in der Form: „fx“, function of the names, in the form “fx”, functions of names, so that they have the
„φ(x, y)“, etc. “φ(x, y)”, etc. form ‘fx’, ‘φ(x, y)’, etc.
Oder ich deute ihn durch die Buchsta- Or I indicate it by the letters p, q, r. Or I indicate them by the letters ‘p’,
ben p, q, r an. ‘q’, ‘r’.
4.241 Gebrauche ich zwei Zeichen in ein und If I use two signs with one and the When I use two signs with one and the
derselben Bedeutung, so drücke ich dies same meaning, I express this by putting same meaning, I express this by putting
aus, indem ich zwischen beide das Zei- between them the sign “=”. the sign ‘=’ between them.
chen „=“ setze.
„a = b“ heißt also: das Zeichen „a“ ist “a = b” means then, that the sign “a” So ‘a = b’ means that the sign ‘b’ can
durch das Zeichen „b“ ersetzbar. is replaceable by the sign “b”. be substituted for the sign ‘a’.
(Führe ich durch eine Gleichung ein (If I introduce by an equation a new (If I use an equation to introduce a
neues Zeichen „b“ ein, indem ich bestim- sign “b”, by determining that it shall re- new sign ‘b’, laying down that it shall
me, es solle ein bereits bekanntes Zei- place a previously known sign “a”, I write serve as a substitute for a sign ‘a’ that is
chen „a“ ersetzen, so schreibe ich die the equation—definition—(like Russell) already known, then, like Russell, I write
Gleichung—Definition—(wie Russell) in in the form “a = b Def.”. A definition is a the equation—definition—in the form ‘a =
der Form „a = b Def.“. Die Definition ist symbolic rule.) b Def.’ A definition is a rule dealing with
eine Zeichenregel.) signs.)
4.242 Ausdrücke von der Form „a = b“ sind Expressions of the form “a = b” are Expressions of the form ‘a = b’ are,
also nur Behelfe der Darstellung; sie sa- therefore only expedients in presentation: therefore, mere representational devices.
gen nichts über die Bedeutung der Zei- They assert nothing about the meaning They state nothing about the meaning of
49
chen „a“, „b“ aus. of the signs “a” and “b”. the signs ‘a’ and ‘b’.
4.243 Können wir zwei Namen verstehen, Can we understand two names with- Can we understand two names withohne
zu wissen, ob sie dasselbe Ding oder out knowing whether they signify the out knowing whether they signify the
zwei verschiedene Dinge bezeichnen?— same thing or two different things? Can same thing or two different things?—Can
Können wir einen Satz, worin zwei Na- we understand a proposition in which we understand a proposition in which two
men vorkommen, verstehen, ohne zu wis- two names occur, without knowing if they names occur without knowing whether
sen, ob sie Dasselbe oder Verschiedenes mean the same or different things? their meaning is the same or different?
bedeuten?
Kenne ich etwa die Bedeutung eines If I know the meaning of an English Suppose I know the meaning of an
englischen und eines gleichbedeutenden and a synonymous German word, it is English word and of a German word that
deutschen Wortes, so ist es unmöglich, impossible for me not to know that they means the same: then it is impossible for
dass ich nicht weiß, dass die beiden gleich- are synonymous, it is impossible for me me to be unaware that they do mean the
bedeutend sind; es ist unmöglich, dass ich not to be able to translate them into one same; I must be capable of translating
sie nicht ineinander übersetzen kann. another. each into the other.
Ausdrücke wie „a = a“, oder von die- Expressions like “a = a”, or expres- Expressions like ‘a = a’, and those desen
abgeleitete, sind weder Elementar- sions deduced from these are neither el- rived from them, are neither elementary
sätze, noch sonst sinnvolle Zeichen. (Dies ementary propositions nor otherwise sig- propositions nor is there any other way in
wird sich später zeigen.) nificant signs. (This will be shown later.) which they have sense. (This will become
evident later.)
4.25 Ist der Elementarsatz wahr, so be- If the elementary proposition is true, If an elementary proposition is true,
steht der Sachverhalt; ist der Elementar- the atomic fact exists; if it is false the the state of affairs exists: if an elemensatz
falsch, so besteht der Sachverhalt atomic fact does not exist. tary proposition is false, the state of afnicht.
fairs does not exist.
4.26 Die Angabe aller wahren Elementar- The specification of all true elemen- If all true elementary propositions are
sätze beschreibt die Welt vollständig. Die tary propositions describes the world com- given, the result is a complete description
Welt ist vollständig beschrieben durch die pletely. The world is completely described of the world. The world is completely deAngaben
aller Elementarsätze plus der by the specification of all elementary scribed by giving all elementary proposiAngabe,
welche von ihnen wahr und wel- propositions plus the specification, which tions, and adding which of them are true
che falsch sind. of them are true and which false. and which false.
4.27 Bezüglich des Bestehens und Nicht- With regard to the existence of n For n states of affairs, there are Kn =
bestehens von n Sachverhalten gibt es atomic facts there are Kn =
n
ν=0
n
ν
pos-
n
ν=0
n
ν
possibilities of existence and nonKn
=
n
ν=0
n
ν
Möglichkeiten. sibilities. existence.
Es können alle Kombinationen der It is possible for all combinations of Of these states of affairs any combiSachverhalte
bestehen, die andern nicht atomic facts to exist, and the others not nation can exist and the remainder not
50
bestehen. to exist. exist.
4.28 Diesen Kombinationen entsprechen To these combinations correspond the There correspond to these combinaebenso
viele Möglichkeiten der Wahr- same number of possibilities of the truth— tions the same number of possibilities
heit—und Falschheit—von n Elementar- and falsehood—of n elementary proposi- of truth—and falsity—for n elementary
sätzen. tions. propositions.
4.3 Die Wahrheitsmöglichkeiten der Ele- The truth-possibilities of the elemen- Truth-possibilities of elementary propmentarsätze
bedeuten die Möglichkeiten tary propositions mean the possibilities ositions mean possibilities of existence
des Bestehens und Nichtbestehens der of the existence and non-existence of the and non-existence of states of affairs.
Sachverhalte. atomic facts.
4.31 Die Wahrheitsmöglichkeiten können The truth-possibilities can be pre- We can represent truth-possibilities
wir durch Schemata folgender Art dar- sented by schemata of the following kind by schemata of the following kind (‘T’
stellen („W“ bedeutet „wahr“, „F“, „falsch“. (“T” means “true”, “F” “false”. The rows means ‘true’, ‘F’ means ‘false’; the rows
Die Reihen der „W“ und „F“ unter der Rei- of T’s and F’s under the row of the ele- of ‘T’s’ and ‘F’s’ under the row of elemenhe
der Elementarsätze bedeuten in leicht- mentary propositions mean their truth- tary propositions symbolize their truthverständlicher
Symbolik deren Wahr- possibilities in an easily intelligible sym- possibilities in a way that can easily be
heitsmöglichkeiten): bolism). understood):
p q r
W W W
F W W
W F W
W W F
F F W
F W F
W F F
F F F
p q
W W
F W
W F
F F
p
W
F
p q r
T T T
F T T
T F T
T T F
F F T
F T F
T F F
F F F
p q
T T
F T
T F
F F
p
T
F
p q r
T T T
F T T
T F T
T T F
F F T
F T F
T F F
F F F
p q
T T
F T
T F
F F
p
T
F
4.4 Der Satz ist der Ausdruck der A proposition is the expression of A proposition is an expression of
Übereinstimmung und Nichtübereinstim- agreement and disagreement with the agreement and disagreement with truthmung
mit den Wahrheitsmöglichkeiten truth-possibilities of the elementary possibilities of elementary propositions.
der Elementarsätze. propositions.
4.41 Die Wahrheitsmöglichkeiten der Ele- The truth-possibilities of the elemen- Truth-possibilities of elementary propmentarsätze
sind die Bedingungen der tary propositions are the conditions of the ositions are the conditions of the truth
Wahrheit und Falschheit der Sätze. truth and falsehood of the propositions. and falsity of propositions.
4.411 Es ist von vornherein wahrscheinlich, It seems probable even at first sight It immediately strikes one as probable
dass die Einführung der Elementarsätze that the introduction of the elementary that the introduction of elementary propofür
das Verständnis aller anderen Satzar- propositions is fundamental for the com- sitions provides the basis for understandten
grundlegend ist. Ja, das Verständnis prehension of the other kinds of proposi- ing all other kinds of proposition. Indeed
der allgemeinen Sätze hängt f ü h l b a r tions. Indeed the comprehension of the the understanding of general propositions
51
von dem der Elementarsätze ab. general propositions depends palpably on palpably depends on the understanding
that of the elementary propositions. of elementary propositions.
4.42 Bezüglich der Übereinstimmung und With regard to the agreement and dis- For n elementary propositions there
Nichtüberein stimmung eines Satzes mit agreement of a proposition with the truth- are
Kn
κ=0
Kn
κ
= Ln ways in which a propoden
Wahrheitsmöglichkeiten von n Ele- possibilities of n elementary propositions sition can agree and disagree with their
mentarsätzen gibt es
Kn
κ=0
Kn
κ
= Ln Mög- there are
Kn
κ=0
Kn
κ
= Ln possibilities. truth possibilities.
lichkeiten.
4.43 Die Übereinstimmung mit den Wahr- Agreement with the truth-possibili- We can express agreement with truthheitsmöglichkeiten
können wir dadurch ties can be expressed by co-ordinating possibilities by correlating the mark ‘T’
ausdrücken, indem wir ihnen im Schema with them in the schema the mark “T” (true) with them in the schema.
etwa das Abzeichen „W“ (wahr) zuordnen. (true).
Das Fehlen dieses Abzeichens bedeu- Absence of this mark means disagree- The absence of this mark means distet
die Nichtübereinstimmung. ment. agreement.
4.431 Der Ausdruck der Übereinstimmung The expression of the agreement and The expression of agreement and disund
Nichtübereinstimmung mit den disagreement with the truth-possibilities agreement with the truth possibilities
Wahrheitsmöglichkeiten der Elementar- of the elementary propositions expresses of elementary propositions expresses the
sätze drückt die Wahrheitsbedingungen the truth-conditions of the proposition. truth-conditions of a proposition.
des Satzes aus.
Der Satz ist der Ausdruck seiner The proposition is the expression of A proposition is the expression of its
Wahrheitsbedingungen. its truth-conditions. truth-conditions.
(Frege hat sie daher ganz richtig als (Frege has therefore quite rightly put (Thus Frege was quite right to use
Erklärung der Zeichen seiner Begriffs- them at the beginning, as explaining them as a starting point when he exschrift
vorausgeschickt. Nur ist die Er- the signs of his logical symbolism. Only plained the signs of his conceptual notaklärung
des Wahrheitsbegriffes bei Fre- Frege’s explanation of the truth-concept tion. But the explanation of the concept of
ge falsch: Wären „das Wahre“ und „das is false: if “the true” and “the false” were truth that Frege gives is mistaken: if ‘the
Falsche“ wirklich Gegenstände und die real objects and the arguments in ∼p, etc., true’ and ‘the false’ were really objects,
Argumente in ∼p etc. dann wäre nach then the sense of ∼p would by no means and were the arguments in ∼p etc., then
Freges Bestimmung der Sinn von „∼p“ be determined by Frege’s determination.) Frege’s method of determining the sense
keineswegs bestimmt.) of ‘∼p’ would leave it absolutely undeter-
mined.)
4.44 Das Zeichen, welches durch die Zuord- The sign which arises from the co- The sign that results from correlating
nung jener Abzeichen „W“ und der Wahr- ordination of that mark “T” with the the mark ‘T’ with truth-possibilities is a
heitsmöglichkeiten entsteht, ist ein Satz- truth-possibilities is a propositional sign. propositional sign.
zeichen.
52
4.441 Es ist klar, dass dem Komplex der Zei- It is clear that to the complex of the It is clear that a complex of the signs
chen „F“ und „W“ kein Gegenstand (oder signs “F” and “T” no object (or complex ‘F’ and ‘T’ has no object (or complex of obKomplex
von Gegenständen) entspricht; of objects) corresponds; any more than to jects) corresponding to it, just as there is
so wenig, wie den horizontalen und ver- horizontal and vertical lines or to brack- none corresponding to the horizontal and
tikalen Strichen oder den Klammern.— ets. There are no “logical objects”. vertical lines or to the brackets.—There
„Logische Gegenstände“ gibt es nicht. are no ‘logical objects’.
Analoges gilt natürlich für alle Zei- Something analogous holds of course Of course the same applies to all signs
chen, die dasselbe ausdrücken wie die for all signs, which express the same as that express what the schemata of ‘T’s’
Schemata der „W“ und „F“. the schemata of “T” and “F”. and ‘F’s’ express.
4.442 Es ist z. B.: Thus e.g. For example, the following is a propositional
sign:
p q “
W W W
F W W
W F
„ F F W
“ p q
T T T
F T T
T F
F F T ”
‘ p q ’
T T T
F T T
T F
F F T
ein Satzzeichen. is a propositional sign.
(Frege’s „Urtelistrich“ „ “ ist logisch (Frege’s assertion sign “ ” is logically (Frege’s ‘judgement stroke’ ‘ ’ is logiganz
bedeutunglos; er zeigt bei Frege altogether meaningless; in Frege (and cally quite meaningless: in the works of
(und Russell) nur an, dass diese Autoren Russell) it only shows that these authors Frege (and Russell) it simply indicates
die so bezeichneten Sätze für wahr halten. hold as true the propositions marked in that these authors hold the propositions
„ “ gehört daher ebenso wenig zum Satz- this way. “ ” belongs therefore to the marked with this sign to be true. Thus ‘ ’
gefüge, wie etwa die Nummer des Satzes. propositions no more than does the num- is no more a component part of a proposiEin
Satz kann unmöglich von sich selbst ber of the proposition. A proposition can- tion than is, for instance, the proposition’s
aussagen, dass er wahr ist.) not possibly assert of itself that it is true. number. It is quite impossible for a proposition
to state that it itself is true.)
Ist die Reihenfolge der Wahrheitsmög- If the sequence of the truth-possibil- If the order or the truth-possibilities
lichkeiten im Schema durch eine Kombi- ities in the schema is once for all deter- in a schema is fixed once and for all by a
nationsregel ein für allemal festgesetzt, mined by a rule of combination, then the combinatory rule, then the last column by
dann ist die letzte Kolonne allein schon last column is by itself an expression of itself will be an expression of the truthein
Ausdruck der Wahrheitsbedingungen. the truth-conditions. If we write this col- conditions. If we now write this column as
Schreiben wir diese Kolonne als Reihe umn as a row the propositional sign be- a row, the propositional sign will become
hin, so wird das Satzzeichen zu comes:
„(WW−W) (p, q)“ “(TT−T) (p, q)”, “(TT−T) (p, q)”,
oder deutlicher
„(WWFW) (p, q)“.
or more plainly:
“(TTFT) (p, q)”.
or more explicitly
“(TTFT) (p, q)”.
53
(Die Anzahl der Stellen in der linken (The number of places in the left-hand (The number of places in the left-hand
Klammer ist durch die Anzahl der Glieder bracket is determined by the number of pair of brackets is determined by the numin
der rechten bestimmt.) terms in the right-hand bracket.) ber of terms in the right-hand pair.)
4.45 Für n Elementarsätze gibt es Ln mög- For n elementary propositions there For n elementary propositions there
liche Gruppen von Wahrheitsbedingun- are Ln possible groups of truth-condi- are Ln possible groups of truth-condigen.
tions. tions.
Die Gruppen von Wahrheitsbedingun- The groups of truth-conditions which The groups of truth-conditions that
gen, welche zu den Wahrheitsmöglichkeit- belong to the truth-possibilities of a num- are obtainable from the truth-possibilities
en einer Anzahl von Elementarsätzen ge- ber of elementary propositions can be or- of a given number of elementary proposihören,
lassen sich in eine Reihe ordnen. dered in a series. tions can be arranged in a series.
4.46 Unter den möglichen Gruppen von Among the possible groups of truth- Among the possible groups of truthWahrheitsbedingungen
gibt es zwei ex- conditions there are two extreme cases. conditions there are two extreme cases.
treme Fälle.
In dem einen Fall ist der Satz für In the one case the proposition is true In one of these cases the proposition
sämtliche Wahrheitsmöglichkeiten der for all the truth-possibilities of the ele- is true for all the truth-possibilities of the
Elementarsätze wahr. Wir sagen, die mentary propositions. We say that the elementary propositions. We say that the
Wahrheitsbedingungen sind t a u t o l o - truth-conditions are tautological. truth-conditions are tautological.
g i s c h.
Im zweiten Fall ist der Satz für sämt- In the second case the proposition is In the second case the proposition is
liche Wahrheitsmöglichkeiten falsch: Die false for all the truth-possibilities. The false for all the truth-possibilities: the
Wahrheitsbedingungen sind k o n t r a - truth-conditions are self-contradictory. truth-conditions are contradictory.
d i k t o r i s c h.
Im ersten Fall nennen wir den Satz In the first case we call the proposition In the first case we call the proposieine
Tautologie, im zweiten Fall eine Kon- a tautology, in the second case a contra- tion a tautology; in the second, a contratradiktion.
diction. diction.
4.461 Der Satz zeigt was er sagt, die Tau- The proposition shows what it says, Propositions show what they say: tautologie
und die Kontradiktion, dass sie the tautology and the contradiction that tologies and contradictions show that
nichts sagen. they say nothing. they say nothing.
Die Tautologie hat keine Wahrheits- The tautology has no truth-conditions, A tautology has no truth-conditions,
bedingungen, denn sie ist bedingungslos for it is unconditionally true; and the con- since it is unconditionally true: and a conwahr;
und die Kontradiktion ist unter kei- tradiction is on no condition true. tradiction is true on no condition.
ner Bedingung wahr.
Tautologie und Kontradiktion sind Tautology and contradiction are with- Tautologies and contradictions lack
sinnlos. out sense. sense.
(Wie der Punkt, von dem zwei Pfeile (Like the point from which two arrows (Like a point from which two arrows
in entgegengesetzter Richtung auseinan- go out in opposite directions.) go out in opposite directions to one andergehen.)
other.)
54
(Ich weiß z. B. nichts über das Wetter, (I know, e.g. nothing about the (For example, I know nothing about
wenn ich weiß, dass es regnet oder nicht weather, when I know that it rains or does the weather when I know that it is either
regnet.) not rain.) raining or not raining.)
4.4611 Tautologie und Kontradiktion sind Tautology and contradiction are, how- Tautologies and contradictions are
aber nicht unsinnig; sie gehören zum ever, not senseless; they are part of the not, however, nonsensical. They are part
Symbolismus, und zwar ähnlich wie die symbolism, in the same way that “0” is of the symbolism, much as ‘0’ is part of
„0“ zum Symbolismus der Arithmetik. part of the symbolism of Arithmetic. the symbolism of arithmetic.
4.462 Tautologie und Kontradiktion sind Tautology and contradiction are not Tautologies and contradictions are not
nicht Bilder der Wirklichkeit. Sie stellen pictures of the reality. They present no pictures of reality. They do not represent
keine mögliche Sachlage dar. Denn jene possible state of affairs. For the one al- any possible situations. For the former
lässt j e d e mögliche Sachlage zu, diese lows every possible state of affairs, the admit all possible situations, and latter
k e i n e. other none. none.
In der Tautologie heben die Bedin- In the tautology the conditions of In a tautology the conditions of agreegungen
der Übereinstimmung mit der agreement with the world—the present- ment with the world—the representaWelt—die
darstellenden Beziehungen— ing relations—cancel one another, so that tional relations—cancel one another, so
einander auf, so dass sie in keiner darstel- it stands in no presenting relation to real- that it does not stand in any representalenden
Beziehung zur Wirklichkeit steht. ity. tional relation to reality.
4.463 Die Wahrheitsbedingungen bestim- The truth-conditions determine the The truth-conditions of a proposition
men den Spielraum, der den Tatsachen range, which is left to the facts by the determine the range that it leaves open
durch den Satz gelassen wird. proposition. to the facts.
(Der Satz, das Bild, das Modell, sind (The proposition, the picture, the (A proposition, a picture, or a model
im negativen Sinne wie ein fester Kör- model, are in a negative sense like a solid is, in the negative sense, like a solid body
per, der die Bewegungsfreiheit der ande- body, which restricts the free movement that restricts the freedom of movement of
ren beschränkt; im positiven Sinne, wie of another: in a positive sense, like the others, and, in the positive sense, like
der von fester Substanz begrenzte Raum, space limited by solid substance, in which a space bounded by solid substance in
worin ein Körper Platz hat.) a body may be placed.) which there is room for a body.)
Die Tautologie lässt der Wirklich- Tautology leaves to reality the whole A tautology leaves open to reality
keit den ganzen—unendlichen—logisch- infinite logical space; contradiction fills the whole—the infinite whole—of logical
en Raum; die Kontradiktion erfüllt den the whole logical space and leaves no space: a contradiction fills the whole of
ganzen logischen Raum und lässt der point to reality. Neither of them, there- logical space leaving no point of it for reWirklichkeit
keinen Punkt. Keine von bei- fore, can in any way determine reality. ality. Thus neither of them can determine
den kann daher die Wirklichkeit irgend- reality in any way.
wie bestimmen.
4.464 Die Wahrheit der Tautologie ist ge- The truth of tautology is certain, of A tautology’s truth is certain, a propowiss,
des Satzes möglich, der Kontradik- propositions possible, of contradiction im- sition’s possible, a contradiction’s impostion
unmöglich. possible. sible.
(Gewiss, möglich, unmöglich: Hier ha- (Certain, possible, impossible: here (Certain, possible, impossible: here
55
ben wir das Anzeichen jener Gradation, we have an indication of that gradation we have the first indication of the scale
die wir in der Wahrscheinlichkeitslehre which we need in the theory of probabil- that we need in the theory of probability.)
brauchen.) ity.)
4.465 Das logische Produkt einer Tautolo- The logical product of a tautology and The logical product of a tautology and
gie und eines Satzes sagt dasselbe, wie a proposition says the same as the propo- a proposition says the same thing as the
der Satz. Also ist jenes Produkt identisch sition. Therefore that product is identical proposition. This product, therefore, is
mit dem Satz. Denn man kann das We- with the proposition. For the essence of identical with the proposition. For it is
sentliche des Symbols nicht ändern, ohne the symbol cannot be altered without al- impossible to alter what is essential to a
seinen Sinn zu ändern. tering its sense. symbol without altering its sense.
4.466 Einer bestimmten logischen Verbin- To a definite logical combination of What corresponds to a determinate
dung von Zeichen entspricht eine be- signs corresponds a definite logical combi- logical combination of signs is a determistimmte
logische Verbindung ihrer Be- nation of their meanings; every arbitrary nate logical combination of their meandeutungen;
j e d e b e l i e b i g e Verbin- combination only corresponds to the un- ings. It is only to the uncombined signs
dung entspricht nur den unverbundenen connected signs. that absolutely any combination correZeichen.
sponds.
Das heißt, Sätze, die für jede Sachlage That is, propositions which are true In other words, propositions that are
wahr sind, können überhaupt keine Zei- for every state of affairs cannot be combi- true for every situation cannot be combichenverbindungen
sein, denn sonst könn- nations of signs at all, for otherwise there nations of signs at all, since, if they were,
ten ihnen nur bestimmte Verbindungen could only correspond to them definite only determinate combinations of objects
von Gegenständen entsprechen. combinations of objects. could correspond to them.
(Und keiner logischen Verbindung ent- (And to no logical combination corre- (And what is not a logical combination
spricht k e i n e Verbindung der Gegen- sponds no combination of the objects.) has no combination of objects correspondstände.)
ing to it.)
Tautologie und Kontradiktion sind die Tautology and contradiction are the Tautology and contradiction are the
Grenzfälle der Zeichenverbindung, näm- limiting cases of the combination of sym- limiting cases—indeed the disintegralich
ihre Auflösung. bols, namely their dissolution. tion—of the combination of signs.
4.4661 Freilich sind auch in der Tautologie Of course the signs are also combined Admittedly the signs are still comund
Kontradiktion die Zeichen noch mit with one another in the tautology and con- bined with one another even in tautoloeinander
verbunden, d. h. sie stehen tradiction, i.e. they stand in relations to gies and contradictions—i.e. they stand
in Beziehungen zu einander, aber die- one another, but these relations are mean- in certain relations to one another: but
se Beziehungen sind bedeutungslos, dem ingless, unessential to the symbol. these relations have no meaning, they are
S y m b o l unwesentlich. not essential to the symbol.
4.5 Nun scheint es möglich zu sein, die all- Now it appears to be possible to give It now seems possible to give the most
gemeinste Satzform anzugeben: das heißt, the most general form of proposition; i.e. general propositional form: that is, to
eine Beschreibung der Sätze i r g e n d to give a description of the propositions give a description of the propositions of
e i n e r Zeichensprache zu geben, so dass of some one sign language, so that every any sign-language whatsoever in such a
jeder mögliche Sinn durch ein Symbol, possible sense can be expressed by a sym- way that every possible sense can be ex-
56
auf welches die Beschreibung passt, aus- bol, which falls under the description, and pressed by a symbol satisfying the degedrückt
werden kann, und dass jedes so that every symbol which falls under scription, and every symbol satisfying the
Symbol, worauf die Beschreibung passt, the description can express a sense, if the description can express a sense, provided
einen Sinn ausdrücken kann, wenn die meanings of the names are chosen accord- that the meanings of the names are suitBedeutungen
der Namen entsprechend ingly. ably chosen.
gewählt werden.
Es ist klar, dass bei der Beschreibung It is clear that in the description of It is clear that only what is essential
der allgemeinsten Satzform n u r ihr the most general form of proposition only to the most general propositional form
Wesentliches beschrieben werden darf,— what is essential to it may be described— may be included in its description—for
sonst wäre sie nämlich nicht die allge- otherwise it would not be the most gen- otherwise it would not be the most genmeinste.
eral form. eral form.
Dass es eine allgemeine Satzform gibt, That there is a general form is proved The existence of a general proposiwird
dadurch bewiesen, dass es keinen by the fact that there cannot be a propo- tional form is proved by the fact that
Satz geben darf, dessen Form man nicht sition whose form could not have been there cannot be a proposition whose form
hätte voraussehen (d. h. konstruieren) foreseen (i.e. constructed). The general could not have been foreseen (i.e. conkönnen.
Die allgemeine Form des Satzes form of proposition is: Such and such is structed). The general form of a propoist:
Es verhält sich so und so. the case. sition is: This is how things stand.
4.51 Angenommen, mir wären a l l e Ele- Suppose all elementary propositions Suppose that I am given all elemenmentarsätze
gegeben: Dann lässt sich ein- were given me: then we can simply ask: tary propositions: then I can simply ask
fach fragen: Welche Sätze kann ich aus what propositions I can build out of them. what propositions I can construct out of
ihnen bilden? Und das sind a l l e Sätze And these are all propositions and so are them. And there I have all propositions,
und s o sind sie begrenzt. they limited. and that fixes their limits.
4.52 Die Sätze sind alles, was aus der Ge- The propositions are everything which Propositions comprise all that follows
samtheit aller Elementarsätze folgt (na- follows from the totality of all elementary from the totality of all elementary propotürlich
auch daraus, dass es die G e - propositions (of course also from the fact sitions (and, of course, from its being the
s a m t h e i t a l l e r ist). (So könnte that it is the totality of them all). (So, in totality of them all). (Thus, in a certain
man in gewissem Sinne sagen, dass a l l e some sense, one could say, that all propo- sense, it could be said that all proposiSätze
Verallgemeinerungen der Elemen- sitions are generalizations of the elemen- tions were generalizations of elementary
tarsätze sind.) tary propositions.) propositions.)
4.53 Die allgemeine Satzform ist eine Va- The general proposition form is a vari- The general propositional form is a
riable. able. variable.
5 Der Satz ist eine Wahrheitsfunktion Propositions are truth-functions of el- A proposition is a truth-function of elder
Elementarsätze. ementary propositions. ementary propositions.
(Der Elementarsatz ist eine Wahr- (An elementary proposition is a truth- (An elementary proposition is a truthheitsfunktion
seiner selbst.) function of itself.) function of itself.)
5.01 Die Elementarsätze sind die Wahr- The elementary propositions are the Elementary propositions are the
heitsargumente des Satzes. truth-arguments of propositions. truth-arguments of propositions.
57
5.02 Es liegt nahe, die Argumente von It is natural to confuse the arguments The arguments of functions are readFunktionen
mit den Indices von Namen of functions with the indices of names. ily confused with the affixes of names. For
zu verwechseln. Ich erkenne nämlich so- For I recognize the meaning of the sign both arguments and affixes enable me to
wohl am Argument wie am Index die Be- containing it from the argument just as recognize the meaning of the signs condeutung
des sie enthaltenden Zeichens. much as from the index. taining them.
In Russells „+c“ ist z.B. „c“ ein Index, In Russell’s “+c”, for example, “c” is For example, when Russell writes ‘+c’,
der darauf hinweist, dass das ganze Zei- an index which indicates that the whole the ‘c’ is an affix which indicates that the
chen das Additionszeichen für Kardinal- sign is the addition sign for cardinal num- sign as a whole is the addition-sign for carzahlen
ist. Aber diese Bezeichnung be- bers. But this way of symbolizing depends dinal numbers. But the use of this sign
ruht auf willkürlicher Übereinkunft und on arbitrary agreement, and one could is the result of arbitrary convention and
man könnte statt „+c“ auch ein einfaches choose a simple sign instead of “+c”: but it would be quite possible to choose a simZeichen
wählen; in „∼p“ aber ist „p“ kein in “∼p” “p” is not an index but an argu- ple sign instead of ‘+c’; in ‘∼p’, however,
Index, sondern ein Argument: der Sinn ment; the sense of “∼p” cannot be under- ‘p’ is not an affix but an argument: the
von „∼p“ k a n n n i c h t verstanden wer- stood, unless the sense of “p” has pre- sense of ‘∼p’ cannot be understood unless
den, ohne dass vorher der Sinn von „p“ viously been understood. (In the name the sense of ‘p’ has been understood alverstanden
worden wäre. (Im Namen Ju- Julius Cæsar, Julius is an index. The in- ready. (In the name Julius Caesar ‘Julius’
lius Cäsar ist „Julius“ ein Index. Der In- dex is always part of a description of the is an affix. An affix is always part of a
dex ist immer ein Teil einer Beschreibung object to whose name we attach it, e.g. description of the object to whose name
des Gegenstandes, dessen Namen wir ihn The Cæsar of the Julian gens.) we attach it: e.g. the Caesar of the Julian
anhängen. Z. B. d e r Cäsar aus dem Ge- gens.)
schlechte der Julier.)
Die Verwechslung von Argument und The confusion of argument and index If I am not mistaken, Frege’s theory
Index liegt, wenn ich mich nicht irre, der is, if I am not mistaken, at the root of about the meaning of propositions and
Theorie Freges von der Bedeutung der Frege’s theory of the meaning of proposi- functions is based on the confusion beSätze
und Funktionen zugrunde. Für Fre- tions and functions. For Frege the propo- tween an argument and an affix. Frege rege
waren die Sätze der Logik Namen, und sitions of logic were names and their ar- garded the propositions of logic as names,
deren Argumente die Indices dieser Na- guments the indices of these names. and their arguments as the affixes of
men. those names.
5.1 Die Wahrheitsfunktionen lassen sich The truth-functions can be ordered in Truth-functions can be arranged in
in Reihen ordnen. series. series.
Das ist die Grundlage der Wahrschein- That is the foundation of the theory of That is the foundation of the theory of
lichkeitslehre. probability. probability.
5.101 Die Wahrheitsfunktionen jeder An- The truth-functions of every number The truth-functions of a given number
zahl von Elementarsätzen lassen sich in of elementary propositions can be written of elementary propositions can always be
einem Schema folgender Art hinschrei- in a schema of the following kind: set out in a schema of the following kind:
ben:
58
(WWWW) (p, q) Tautologie (Wenn p, so p; und wenn q, so q.)
(p ⊃ p . q ⊃ q)
(F WWW) (p, q) in Worten: Nicht beides p und q. (∼(p . q))
(W F WW) (p, q) ” ” Wenn q, so p. (q ⊃ p)
(WW F W) (p, q) ” ” Wenn p, so q. (p ⊃ q)
(WWW F) (p, q) ” ” p oder q. (p ∨ q)
(F F WW) (p, q) ” ” Nicht q. (∼q)
(F W F W) (p, q) ” ” Nicht p. (∼p)
(F WW F) (p, q) ” ” p, oder q, aber nicht beide.
(p . ∼q :∨: q . ∼p)
(W F F W) (p, q) ” ” Wenn p, so q; und wenn q, so p.
(p ≡ q)
(W F W F) (p, q) ” ” p
(WW F F) (p, q) ” ” q
(F F F W) (p, q) ” ” Weder p noch q.
(∼p . ∼q) oder (p | q)
(F F W F) (p, q) ” ” p und nicht q. (p . ∼q)
(F W F F) (p, q) ” ” q und nicht p. (q . ∼p)
(W F F F) (p, q) ” ” q und p. (q . p)
(F F F F) (p, q) Kontradiktion (p und nicht p; und
q und nicht q.) (p . ∼p . q . ∼q)
(TTTT) (p, q) Tautology (if p then p; and if q then q)
[p ⊃ p . q ⊃ q]
(FTTT) (p, q) in words: Not both p and q. [∼(p . q)]
(TFTT) (p, q) ” ” If q then p. [q ⊃ p]
(TTFT) (p, q) ” ” If p then q. [p ⊃ q]
(TTTF) (p, q) ” ” p or q. [p ∨ q]
(FFTT) (p, q) ” ” Not q. [∼q]
(FTFT) (p, q) ” ” Not p. [∼p]
(FTTF) (p, q) ” ” p or q, but not both.
[p . ∼q :∨: q . ∼p]
(TFFT) (p, q) ” ” If p, then q; and if q, then p.
[p ≡ q]
(TFTF) (p, q) ” ” p
(TTFF) (p, q) ” ” q
(FFFT) (p, q) ” ” Neither p nor q.
[∼p . ∼q or p | q]
(FFTF) (p, q) ” ” p and not q. [p . ∼q]
(FTFF) (p, q) ” ” q and not p. [q . ∼p]
(TFFF) (p, q) ” ” p and q. [p . q]
(FFFF) (p, q) Contradiction (p and not p; and
q and not q.) [p . ∼p . q . ∼q]
(TTTT) (p, q) Tautology (If p then p; and if q then q.)
(p ⊃ p . q ⊃ q)
(FTTT) (p, q) In words: Not both p and q. (∼(p . q))
(TFTT) (p, q) ” ” : If q then p. (q ⊃ p)
(TTFT) (p, q) ” ” : If p then q. (p ⊃ q)
(TTTF) (p, q) ” ” : p or q. (p ∨ q)
(FFTT) (p, q) ” ” : Not q. (∼q)
(FTFT) (p, q) ” ” : Not p. (∼p)
(FTTF) (p, q) ” ” : p or q, but not both.
(p . ∼q :∨: q . ∼p)
(TFFT) (p, q) ” ” : If p then q, and if q then p.
(p ≡ q)
(TFTF) (p, q) ” ” : p
(TTFF) (p, q) ” ” q
(FFFT) (p, q) ” ” : Neither p nor q.
(∼p . ∼q or p | q)
(FFTF) (p, q) ” ” : p and not q. (p . ∼q)
(FTFF) (p, q) ” ” : q and not p. (q . ∼p)
(TFFF) (p, q) ” ” : q and p. (q . p)
(FFFF) (p, q) Contradiction (p and not p, and
q and not q.) (p . ∼p . q . ∼q)
Diejenigen Wahrheitsmöglichkeiten Those truth-possibilities of its truth- I will give the name truth-grounds of
seiner Wahrheitsargumente, welche den arguments, which verify the proposition, a proposition to those truth-possibilities
Satz bewahrheiten, will ich seine W a h r - I shall call its truth-grounds. of its truth-arguments that make it true.
h e i t s g r ü n d e nennen.
5.11 Sind die Wahrheitsgründe, die ei- If the truth-grounds which are com- If all the truth-grounds that are comner
Anzahl von Sätzen gemeinsam sind, mon to a number of propositions are all mon to a number of propositions are at
sämtlich auch Wahrheitsgründe eines be- also truth-grounds of some one proposi- the same time truth-grounds of a certain
stimmten Satzes, so sagen wir, die Wahr- tion, we say that the truth of this proposi- proposition, then we say that the truth of
heit dieses Satzes folge aus der Wahrheit tion follows from the truth of those propo- that proposition follows from the truth of
jener Sätze. sitions. the others.
5.12 Insbesondere folgt die Wahrheit eines In particular the truth of a proposition In particular, the truth of a proposiSatzes
„p“ aus der Wahrheit eines an- p follows from that of a proposition q, if tion ‘p’ follows from the truth of another
deren „q“, wenn alle Wahrheitsgründe all the truth-grounds of the second are proposition ‘q’ if all the truth-grounds of
des zweiten Wahrheitsgründe des ersten truth-grounds of the first. the latter are truth-grounds of the former.
sind.
5.121 Die Wahrheitsgründe des einen sind The truth-grounds of q are contained The truth-grounds of the one are conin
denen des anderen enthalten; p folgt in those of p; p follows from q. tained in those of the other: p follows
aus q. from q.
5.122 Folgt p aus q, so ist der Sinn von „p“ If p follows from q, the sense of “p” is If p follows from q, the sense of ‘p’ is
im Sinne von „q“ enthalten. contained in that of “q”. contained in the sense of ‘q’.
5.123 Wenn ein Gott eine Welt erschafft, If a god creates a world in which cer- If a god creates a world in which cerworin
gewisse Sätze wahr sind, so schafft tain propositions are true, he creates tain propositions are true, then by that
59
er damit auch schon eine Welt, in welcher thereby also a world in which all proposi- very act he also creates a world in which
alle ihre Folgesätze stimmen. Und ähn- tions consequent on them are true. And all the propositions that follow from them
lich könnte er keine Welt schaffen, worin similarly he could not create a world in come true. And similarly he could not creder
Satz „p“ wahr ist, ohne seine sämtli- which the proposition “p” is true without ate a world in which the proposition ‘p’
chen Gegenstände zu schaffen. creating all its objects. was true without creating all its objects.
5.124 Der Satz bejaht jeden Satz, der aus A proposition asserts every proposi- A proposition affirms every proposiihm
folgt. tion which follows from it. tion that follows from it.
5.1241 „p . q“ ist einer der Sätze, welche „p“ “p . q” is one of the propositions which ‘p . q’ is one of the propositions that
bejahen, und zugleich einer der Sätze, assert “p” and at the same time one of the affirm ‘p’ and at the same time one of the
welche „q“ bejahen. propositions which assert “q”. propositions that affirm ‘q’.
Zwei Sätze sind einander entgegen- Two propositions are opposed to one Two propositions are opposed to one
gesetzt, wenn es keinen sinnvollen Satz another if there is no significant proposi- another if there is no proposition with a
gibt, der sie beide bejaht. tion which asserts them both. sense, that affirms them both.
Jeder Satz der einem anderen wider- Every proposition which contradicts Every proposition that contradicts anspricht,
verneint ihn. another, denies it. other negates it.
5.13 Dass die Wahrheit eines Satzes aus That the truth of one proposition fol- When the truth of one proposition folder
Wahrheit anderer Sätze folgt, ersehen lows from the truth of other propositions, lows from the truth of others, we can see
wir aus der Struktur der Sätze. we perceive from the structure of the this from the structure of the proposipropositions.
tions.
5.131 Folgt die Wahrheit eines Satzes aus If the truth of one proposition follows If the truth of one proposition follows
der Wahrheit anderer, so drückt sich dies from the truth of others, this expresses from the truth of others, this finds expresdurch
Beziehungen aus, in welchen die itself in relations in which the forms of sion in relations in which the forms of the
Formen jener Sätze zu einander stehen; these propositions stand to one another, propositions stand to one another: nor is
und zwar brauchen wir sie nicht erst in and we do not need to put them in these it necessary for us to set up these relajene
Beziehungen zu setzen, indem wir relations first by connecting them with tions between them, by combining them
sie in einem Satz miteinander verbinden, one another in a proposition; for these re- with one another in a single proposition;
sondern diese Beziehungen sind intern lations are internal, and exist as soon as, on the contrary, the relations are interund
bestehen, sobald, und dadurch dass, and by the very fact that, the propositions nal, and their existence is an immediate
jene Sätze bestehen. exist. result of the existence of the propositions.
5.1311 Wenn wir von p ∨ q und ∼p auf q When we conclude from p ∨ q and ∼p When we infer q from p ∨ q and ∼p,
schließen, so ist hier durch die Bezeich- to q the relation between the forms of the relation between the propositional
nungsweise die Beziehung der Satzfor- the propositions “p ∨ q” and “∼p” is here forms of ‘p∨ q’ and ‘∼p’ is masked, in this
men von „p∨q“ und „∼p“ verhüllt. Schrei- concealed by the method of symbolizing. case, by our mode of signifying. But if
ben wir aber z. B. statt „p ∨ q“ „p | q .|. p | But if we write, e.g. instead of “p ∨ q” instead of ‘p ∨ q’ we write, for example,
q“ und statt „∼p“ „p | p“ (p | q = weder p, “p | q .|. p | q” and instead of “∼p” “p | p” ‘p | q .|. p | q’, and instead of ‘∼p’, ‘p | p’
noch q), so wird der innere Zusammen- (p | q = neither p nor q), then the inner (p | q = neither p nor q), then the inner
hang offenbar. connexion becomes obvious. connexion becomes obvious.
60
(Dass man aus (x). fx auf fa schließen (The fact that we can infer fa from (The possibility of inference from
kann, das zeigt, dass die Allgemeinheit (x). fx shows that generality is present (x). fx to fa shows that the symbol (x). fx
auch im Symbol „(x). fx“ vorhanden ist.) also in the symbol “(x). fx”. itself has generality in it.)
5.132 Folgt p aus q, so kann ich von q auf p If p follows from q, I can conclude If p follows from q, I can make an
schließen; p aus q folgern. from q to p; infer p from q. inference from q to p, deduce p from q.
Die Art des Schlusses ist allein aus The method of inference is to be un- The nature of the inference can be
den beiden Sätzen zu entnehmen. derstood from the two propositions alone. gathered only from the two propositions.
Nur sie selbst können den Schluss Only they themselves can justify the They themselves are the only possible
rechtfertigen. inference. justification of the inference.
„Schlussgesetze“, welche—wie bei Fre- Laws of inference, which—as in Frege ‘Laws of inference’, which are supge
und Russell—die Schlüsse rechtferti- and Russell—are to justify the conclu- posed to justify inferences, as in the
gen sollen, sind sinnlos, und wären über- sions, are senseless and would be super- works of Frege and Russell, have no sense,
flüssig. fluous. and would be superfluous.
5.133 Alles Folgern geschieht a priori. All inference takes place a priori. All deductions are made a priori.
5.134 Aus einem Elementarsatz lässt sich From an elementary proposition no One elementary proposition cannot be
kein anderer folgern. other can be inferred. deduced form another.
5.135 Auf keine Weise kann aus dem Beste- In no way can an inference be made There is no possible way of making an
hen irgend einer Sachlage auf das Beste- from the existence of one state of affairs to inference from the existence of one situahen
einer von ihr gänzlich verschiedenen the existence of another entirely different tion to the existence of another, entirely
Sachlage geschlossen werden. from it. different situation.
5.136 Einen Kausalnexus, der einen solchen There is no causal nexus which justi- There is no causal nexus to justify
Schluss rechtfertigte, gibt es nicht. fies such an inference. such an inference.
5.1361 Die Ereignisse der Zukunft k ö n n e n The events of the future cannot be in- We cannot infer the events of the fuwir
nicht aus den gegenwärtigen erschlie- ferred from those of the present. ture from those of the present.
ßen.
Der Glaube an den Kausalnexus ist Superstition is the belief in the causal Belief in the causal nexus is superstider
A b e r g l a u b e. nexus. tion.
5.1362 Die Willensfreiheit besteht darin, The freedom of the will consists in The freedom of the will consists in the
dass zukünftige Handlungen jetzt nicht the fact that future actions cannot be impossibility of knowing actions that still
gewusst werden können. Nur dann könn- known now. We could only know them lie in the future. We could know them
ten wir sie wissen, wenn die Kausalität if causality were an inner necessity, like only if causality were an inner necessity
eine i n n e r e Notwendigkeit wäre, wie that of logical deduction.—The connexion like that of logical inference.—The condie
des logischen Schlusses.—Der Zusam- of knowledge and what is known is that nexion between knowledge and what is
menhang von Wissen und Gewusstem ist of logical necessity. known is that of logical necessity.
der der logischen Notwendigkeit.
(„A weiß, dass p der Fall ist“ ist sinn- (“A knows that p is the case” is sense- (‘A knows that p is the case’, has no
los, wenn p eine Tautologie ist.) less if p is a tautology.) sense if p is a tautology.)
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5.1363 Wenn daraus, dass ein Satz uns ein- If from the fact that a proposition is If the truth of a proposition does not
leuchtet, nicht f o l g t, dass er wahr ist, obvious to us it does not follow that it is follow from the fact that it is self-evident
so ist das Einleuchten auch keine Recht- true, then obviousness is no justification to us, then its self-evidence in no way jusfertigung
für unseren Glauben an seine for our belief in its truth. tifies our belief in its truth.
Wahrheit.
5.14 Folgt ein Satz aus einem anderen, so If a proposition follows from another, If one proposition follows from ansagt
dieser mehr als jener, jener weniger then the latter says more than the former, other, then the latter says more than the
als dieser. the former less than the latter. former, and the former less than the lat-
ter.
5.141 Folgt p aus q und q aus p, so sind sie If p follows from q and q from p then If p follows from q and q from p, then
ein und derselbe Satz. they are one and the same proposition. they are one and the same proposition.
5.142 Die Tautologie folgt aus allen Sätzen: A tautology follows from all proposi- A tautology follows from all proposisie
sagt nichts. tions: it says nothing. tions: it says nothing.
5.143 Die Kontradiktion ist das Gemeinsa- Contradiction is something shared by Contradiction is that common factor
me der Sätze, was k e i n Satz mit ei- propositions, which no proposition has of propositions which no proposition has
nem anderen gemein hat. Die Tautologie in common with another. Tautology is in common with another. Tautology is
ist das Gemeinsame aller Sätze, welche that which is shared by all propositions, the common factor of all propositions that
nichts miteinander gemein haben. which have nothing in common with one have nothing in common with one ananother.
other.
Die Kontradiktion verschwindet so- Contradiction vanishes so to speak Contradiction, one might say, vanzusagen
außerhalb, die Tautologie inner- outside, tautology inside all propositions. ishes outside all propositions: tautology
halb aller Sätze. vanishes inside them.
Die Kontradiktion ist die äußere Gren- Contradiction is the external limit Contradiction is the outer limit of
ze der Sätze, die Tautologie ihr substanz- of the propositions, tautology their sub- propositions: tautology is the unsubstanloser
Mittelpunkt. stanceless centre. tial point at their centre.
5.15 Ist Wr die Anzahl der Wahrheitsgrün- If Tr is the number of the truth- If Tr is the number of the truthde
des Satzes „r“, Wrs die Anzahl derjeni- grounds of the proposition “r”, Trs the grounds of a proposition ‘r’, and if Trs
gen Wahrheitsgründe des Satzes „s“, die number of those truth-grounds of the is the number of the truth-grounds of a
zugleich Wahrheitsgründe von „r“ sind, proposition “s” which are at the same proposition ‘s’ that are at the same time
dann nennen wir das Verhältnis: Wrs : time truth-grounds of “r”, then we call truth-grounds of ‘r’, then we call the ratio
Wr das Maß der W a h r s c h e i n l i c h - the ratio Trs : Tr the measure of the prob- Trs : Tr the degree of probability that the
k e i t, welche der Satz „r“ dem Satz „s“ ability which the proposition “r” gives to proposition ‘r’ gives to the proposition ‘s’.
gibt. the proposition “s”.
5.151 Sei in einem Schema wie dem obigen Suppose in a schema like that above In a schema like the one above in
in No. 5.101 Wr die Anzahl der „W“ im in No. 5.101 Tr is the number of the “T” ’s 5.101, let Tr be the number of ‘T’s’ in the
Satze r; Wrs die Anzahl derjenigen „W“ in the proposition r, Trs the number of proposition r, and let Trs, be the number
im Satze s, die in gleichen Kolonnen mit those “T” ’s in the proposition s, which of ‘T’s’ in the proposition s that stand in
62
„W“ des Satzes r stehen. Der Satz r gibt stand in the same columns as “T” ’s of columns in which the proposition r has
dann dem Satze s die Wahrscheinlichkeit: the proposition r; then the proposition r ‘T’s’. Then the proposition r gives to the
Wrs : Wr. gives to the proposition s the probability proposition s the probability Trs : Tr.
Trs : Tr.
5.1511 Es gibt keinen besonderen Gegen- There is no special object peculiar to There is no special object peculiar to
stand, der den Wahrscheinlichkeitssätzen probability propositions. probability propositions.
eigen wäre.
5.152 Sätze, welche keine Wahrheitsargu- Propositions which have no truth- When propositions have no truthmente
mit einander gemein haben, nen- arguments in common with one another arguments in common with one another,
nen wir von einander unabhängig. we call independent. we call them independent of one another.
Zwei Elementarsätze geben einander Independent propositions (e.g. any two Two elementary propositions give one
die Wahrscheinlichkeit 1
2. elementary propositions) give to one an- another the probability 1
2.
other the probability 1
2.
Folgt p aus q, so gibt der Satz „q“ dem If p follows from q, the proposition q If p follows from q, then the propoSatz
„p“ die Wahrscheinlichkeit 1. Die gives to the proposition p the probability sition ‘q’ gives to the proposition ‘p’ the
Gewissheit des logischen Schlusses ist ein 1. The certainty of logical conclusion is a probability 1. The certainty of logical inGrenzfall
der Wahrscheinlichkeit. limiting case of probability. ference is a limiting case of probability.
(Anwendung auf Tautologie und Kon- (Application to tautology and contra- (Application of this to tautology and
tradiktion.) diction.) contradiction.)
5.153 Ein Satz ist an sich weder wahrschein- A proposition is in itself neither prob- In itself, a proposition is neither problich
noch unwahrscheinlich. Ein Ereignis able nor improbable. An event occurs or able nor improbable. Either an event octrifft
ein, oder es trifft nicht ein, ein Mit- does not occur, there is no middle course. curs or it does not: there is no middle
telding gibt es nicht. way.
5.154 In einer Urne seien gleichviel weiße In an urn there are equal numbers of Suppose that an urn contains black
und schwarze Kugeln (und keine ande- white and black balls (and no others). I and white balls in equal numbers (and
ren). Ich ziehe eine Kugel nach der ande- draw one ball after another and put them none of any other kind). I draw one ball
ren und lege sie wieder in die Urne zu- back in the urn. Then I can determine by after another, putting them back into the
rück. Dann kann ich durch den Versuch the experiment that the numbers of the urn. By this experiment I can establish
feststellen, dass sich die Zahlen der gezo- black and white balls which are drawn that the number of black balls drawn and
genen schwarzen und weißen Kugeln bei approximate as the drawing continues. the number of white balls drawn approxifortgesetztem
Ziehen einander nähern. mate to one another as the draw contin-
ues.
D a s ist also kein mathematisches So this is not a mathematical fact. So this is not a mathematical truth.
Faktum.
Wenn ich nun sage: Es ist gleich wahr- If then, I say, It is equally probable Now, if I say, ‘The probability of my
scheinlich, dass ich eine weiße Kugel wie that I should draw a white and a black drawing a white ball is equal to the probeine
schwarze ziehen werde, so heißt das: ball, this means, All the circumstances ability of my drawing a black one’, this
63
Alle mir bekannten Umstände (die hy- known to me (including the natural laws means that all the circumstances that I
pothetisch angenommenen Naturgesetze hypothetically assumed) give to the occur- know of (including the laws of nature asmitinbegriffen)
geben dem Eintreffen des rence of the one event no more probability sumed as hypotheses) give no more probeinen
Ereignisses nicht m e h r Wahr- than to the occurrence of the other. That ability to the occurrence of the one event
scheinlichkeit als dem Eintreffen des an- is they give—as can easily be understood than to that of the other. That is to say,
deren. Das heißt, sie geben—wie aus den from the above explanations—to each the they give each the probability 1
2, as can
obigen Erklärungen leicht zu entnehmen probability 1
2. easily be gathered from the above definiist—jedem
die Wahrscheinlichkeit 1
2. tions.
Was ich durch den Versuch bestätige What I can verify by the experiment What I confirm by the experiment is
ist, dass das Eintreffen der beiden Ereig- is that the occurrence of the two events that the occurrence of the two events is innisse
von den Umständen, die ich nicht is independent of the circumstances with dependent of the circumstances of which
näher kenne, unabhängig ist. which I have no closer acquaintance. I have no more detailed knowledge.
5.155 Die Einheit des Wahrscheinlichkeits- The unit of the probability proposition The minimal unit for a probability
satzes ist: Die Umstände—die ich sonst is: The circumstances—with which I am proposition is this: The circumstances—
nicht weiter kenne—geben dem Eintref- not further acquainted—give to the occur- of which I have no further knowledge—
fen eines bestimmten Ereignisses den rence of a definite event such and such a give such and such a degree of probability
und den Grad der Wahrscheinlichkeit. degree of probability. to the occurrence of a particular event.
5.156 So ist die Wahrscheinlichkeit eine Ver- Probability is a generalization. It is in this way that probability is a
allgemeinerung. generalization.
Sie involviert eine allgemeine Be- It involves a general description of a It involves a general description of a
schreibung einer Satzform. propositional form. propositional form.
Nur in Ermanglung der Gewissheit Only in default of certainty do we need We use probability only in default of
gebrauchen wir die Wahrscheinlichkeit.— probability. If we are not completely ac- certainty—if our knowledge of a fact is
Wenn wir zwar eine Tatsache nicht voll- quainted with a fact, but know something not indeed complete, but we do know
kommen kennen, wohl aber e t w a s über about its form. something about its form.
ihre Form wissen.
(Ein Satz kann zwar ein unvollstän- (A proposition can, indeed, be an in- (A proposition may well be an incomdiges
Bild einer gewissen Sachlage sein, complete picture of a certain state of af- plete picture of a certain situation, but
aber er ist immer e i n vollständiges fairs, but it is always a complete picture.) it is always a complete picture of someBild.)
thing.)
Der Wahrscheinlichkeitssatz ist The probability proposition is, as it A probability proposition is a sort of
gleichsam ein Auszug aus anderen Sätz- were, an extract from other propositions. excerpt from other propositions.
en.
5.2 Die Strukturen der Sätze stehen in The structures of propositions stand The structures of propositions stand
internen Beziehungen zu einander. to one another in internal relations. in internal relations to one another.
5.21 Wir können diese internen Beziehun- We can bring out these internal re- In order to give prominence to these
gen dadurch in unserer Ausdrucksweise lations in our manner of expression, by internal relations we can adopt the follow-
64
hervorheben, dass wir einen Satz als Re- presenting a proposition as the result of ing mode of expression: we can represent
sultat einer Operation darstellen, die ihn an operation which produces it from other a proposition as the result of an operation
aus anderen Sätzen (den Basen der Ope- propositions (the bases of the operation). that produces it out of other propositions
ration) hervorbringt. (which are the bases of the operation).
5.22 Die Operation ist der Ausdruck einer The operation is the expression of a re- An operation is the expression of a reBeziehung
zwischen den Strukturen ih- lation between the structures of its result lation between the structures of its result
res Resultats und ihrer Basen. and its bases. and of its bases.
5.23 Die Operation ist das, was mit dem The operation is that which must hap- The operation is what has to be done
einen Satz geschehen muss, um aus ihm pen to a proposition in order to make an- to the one proposition in order to make
den anderen zu machen. other out of it. the other out of it.
5.231 Und das wird natürlich von ihren for- And that will naturally depend on And that will, of course, depend on
malen Eigenschaften, von der internen their formal properties, on the internal their formal properties, on the internal
Ähnlichkeit ihrer Formen abhängen. similarity of their forms. similarity of their forms.
5.232 Die interne Relation, die eine Reihe The internal relation which orders a The internal relation by which a series
ordnet, ist äquivalent mit der Operation, series is equivalent to the operation by is ordered is equivalent to the operation
durch welche ein Glied aus dem anderen which one term arises from another. that produces one term from another.
entsteht.
5.233 Die Operation kann erst dort auftre- The first place in which an operation Operations cannot make their appearten,
wo ein Satz auf logisch bedeutungs- can occur is where a proposition arises ance before the point at which one propovolle
Weise aus einem anderen entsteht. from another in a logically significant sition is generated out of another in a
Also dort, wo die logische Konstruktion way; i.e. where the logical construction logically meaningful way; i.e. the point at
des Satzes anfängt. of the proposition begins. which the logical construction of propositions
begins.
5.234 Die Wahrheitsfunktionen der Elemen- The truth-functions of elementary Truth-functions of elementary propotarsätze
sind Resultate von Operationen, proposition, are results of operations sitions are results of operations with eldie
die Elementarsätze als Basen haben. which have the elementary propositions ementary propositions as bases. (These
(Ich nenne diese Operationen Wahrheits- as bases. (I call these operations, truth- operations I call truth-operations.)
operationen.) operations.)
5.2341 Der Sinn einer Wahrheitsfunktion von The sense of a truth-function of p is a The sense of a truth-function of p is a
p ist eine Funktion des Sinnes von p. function of the sense of p. function of the sense of p.
Verneinung, logische Addition, logi- Denial, logical addition, logical multi- Negation, logical addition, logical mulsche
Multiplikation, etc., etc. sind Ope- plication, etc., etc., are operations. tiplication, etc. etc. are operations.
rationen.
(Die Verneinung verkehrt den Sinn (Denial reverses the sense of a propo- (Negation reverses the sense of a
des Satzes.) sition.) proposition.)
5.24 Die Operation zeigt sich in einer Va- An operation shows itself in a vari- An operation manifests itself in a variriablen;
sie zeigt, wie man von einer Form able; it shows how we can proceed from able; it shows how we can get from one
65
von Sätzen zu einer anderen gelangen one form of proposition to another. form of proposition to another.
kann.
Sie bringt den Unterschied der For- It gives expression to the difference It gives expression to the difference
men zum Ausdruck. between the forms. between the forms.
(Und das Gemeinsame zwischen den (And that which is common the the (And what the bases of an operation
Basen und dem Resultat der Operation bases, and the result of an operation, is and its result have in common is just the
sind eben die Basen.) the bases themselves.) bases themselves.)
5.241 Die Operation kennzeichnet keine The operation does not characterize An operation is not the mark of a form,
Form, sondern nur den Unterschied der a form but only the difference between but only of a difference between forms.
Formen. forms.
5.242 Dieselbe Operation, die „q“ aus „p“ The same operation which makes “q” The operation that produces ‘q’ from
macht, macht aus „q“ „r“ u. s. f. Dies kann from “p”, makes “r” from “q”, and so on. ‘p’ also produces ‘r’ from ‘q’, and so on.
nur darin ausgedrückt sein, dass „p“, „q“, This can only be expressed by the fact There is only one way of expressing this:
„r“, etc. Variable sind, die gewisse forma- that “p”, “q”, “r”, etc., are variables which ‘p’, ‘q’, ‘r’, etc. have to be variables that
le Relationen allgemein zum Ausdruck give general expression to certain formal give expression in a general way to cerbringen.
relations. tain formal relations.
5.25 Das Vorkommen der Operation cha- The occurrence of an operation does The occurrence of an operation does
rakterisiert den Sinn des Satzes nicht. not characterize the sense of a proposi- not characterize the sense of a proposition.
tion.
Die Operation sagt ja nichts aus, nur For an operation does not assert any- Indeed, no statement is made by an
ihr Resultat, und dies hängt von den Ba- thing; only its result does, and this de- operation, but only by its result, and this
sen der Operation ab. pends on the bases of the operation. depends on the bases of the operation.
(Operation und Funktion dürfen nicht (Operation and function must not be (Operations and functions must not
miteinander verwechselt werden.) confused with one another.) be confused with each other.)
5.251 Eine Funktion kann nicht ihr eigenes A function cannot be its own argu- A function cannot be its own arguArgument
sein, wohl aber kann das Re- ment, but the result of an operation can ment, whereas an operation can take one
sultat einer Operation ihre eigene Basis be its own basis. of its own results as its base.
werden.
5.252 Nur so ist das Fortschreiten von Glied Only in this way is the progress from It is only in this way that the step
zu Glied in einer Formenreihe (von Ty- term to term in a formal series possible from one term of a series of forms to anpe
zu Type in den Hierarchien Russells (from type to type in the hierarchy of Rus- other is possible (from one type to another
und Whiteheads) möglich. (Russell und sell and Whitehead). (Russell and White- in the hierarchies of Russell and WhiteWhitehead
haben die Möglichkeit dieses head have not admitted the possibility of head). (Russell and Whitehead did not
Fortschreitens nicht zugegeben, aber im- this progress but have made use of it all admit the possibility of such steps, but
mer wieder von ihr Gebrauch gemacht.) the same.) repeatedly availed themselves of it.)
5.2521 Die fortgesetzte Anwendung einer The repeated application of an opera- If an operation is applied repeatedly
Operation auf ihr eigenes Resultat tion to its own result I call its successive to its own results, I speak of successive
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nenne ich ihre successive Anwendung application (“O’O’O’a” is the result of the applications of it. (‘O’O’O’a’ is the result
(„O’O’O’a“ ist das Resultat der dreima- threefold successive application of “O’ξ” of three successive applications of the opligen
successiven Anwendung von „O’ξ“ to “a”). eration ‘O’ξ’ to ‘a’.)
auf „a“).
In einem ähnlichen Sinne rede ich von In a similar sense I speak of the suc- In a similar sense I speak of succesder
successiven Anwendung m e h r e - cessive application of several operations sive applications of more than one operar
e r Operationen auf eine Anzahl von to a number of propositions. tion to a number of propositions.
Sätzen.
5.2522 Das allgemeine Glied einer Formen- The general term of the formal se- Accordingly I use the sign ‘[a, x, O’x]’
reihe a, O’a, O’O’a, ... schreibe ich da- ries a, O’a, O’O’a, .... I write thus: for the general term of the series of forms
her so: „[a, x, O’x]“. Dieser Klammeraus- “[a, x, O’x]”. This expression in brackets a, O’a, O’O’a, .... This bracketed expresdruck
ist eine Variable. Das erste Glied is a variable. The first term of the expres- sion is a variable: the first term of the
des Klammerausdruckes ist der Anfang sion is the beginning of the formal series, bracketed expression is the beginning of
der Formenreihe, das zweite die Form ei- the second the form of an arbitrary term the series of forms, the second is the form
nes beliebigen Gliedes x der Reihe und x of the series, and the third the form of of a term x arbitrarily selected from the
das dritte die Form desjenigen Gliedes that term of the series which immediately series, and the third is the form of the
der Reihe, welches auf x unmittelbar follows x. term that immediately follows x in the
folgt. series.
5.2523 Der Begriff der successiven Anwen- The concept of the successive applica- The concept of successive applications
dung der Operation ist äquivalent mit tion of an operation is equivalent to the of an operation is equivalent to the condem
Begriff „und so weiter“. concept “and so on”. cept ‘and so on’.
5.253 Eine Operation kann die Wirkung ei- One operation can reverse the effect One operation can counteract the efner
anderen rückgängig machen. Opera- of another. Operations can cancel one fect of another. Operations can cancel one
tionen können einander aufheben. another. another.
5.254 Die Operation kann verschwinden (z. Operations can vanish (e.g. denial in An operation can vanish (e.g. negation
B. die Verneinung in „∼∼p“: ∼∼p = p). “∼∼p”. ∼∼p = p). in ‘∼∼p’: ∼∼p = p).
5.3 Alle Sätze sind Resultate von Wahr- All propositions are results of truth- All propositions are results of truthheitsoperationen
mit den Elementarsät- operations on the elementary proposi- operations on elementary propositions.
zen. tions.
Die Wahrheitsoperation ist die Art The truth-operation is the way in A truth-operation is the way in which
und Weise, wie aus den Elementarsätzen which a truth-function arises from ele- a truth-function is produced out of eledie
Wahrheitsfunktion entsteht. mentary propositions. mentary propositions.
Nach dem Wesen der Wahrheitsope- According to the nature of truth- It is of the essence of truth-operations
ration wird auf die gleiche Weise, wie operations, in the same way as out of el- that, just as elementary propositions
aus den Elementarsätzen ihre Wahrheits- ementary propositions arise their truth- yield a truth-function of themselves, so
funktion, aus Wahrheitsfunktionen eine functions, from truth-functions arises a too in the same way truth-functions yield
neue. Jede Wahrheitsoperation erzeugt new one. Every truth-operation creates a further truth-function. When a truth-
67
aus Wahrheitsfunktionen von Elemen- from truth-functions of elementary propo- operation is applied to truth-functions of
tarsätzen wieder eine Wahrheitsfunkti- sitions, another truth-function of elemen- elementary propositions, it always generon
von Elementarsätzen, einen Satz. Das tary propositions i.e. a proposition. The ates another truth-function of elementary
Resultat jeder Wahrheitsoperation mit result of every truth-operation on the re- propositions, another proposition. When
den Resultaten von Wahrheitsoperatio- sults of truth-operations on elementary a truth-operation is applied to the results
nen mit Elementarsätzen ist wieder das propositions is also the result of one truth- of truth-operations on elementary propoResultat
E i n e r Wahrheitsoperation operation on elementary propositions. sitions, there is always a single operation
mit Elementarsätzen. on elementary propositions that has the
same result.
Jeder Satz ist das Resultat von Wahr- Every proposition is the result of Every proposition is the result of
heitsoperationen mit Elementarsätzen. truth-operations on elementary proposi- truth-operations on elementary propositions.
tions.
5.31 Die Schemata No. 4.31 haben auch The Schemata No. 4.31 are also signif- The schemata in 4.31 have a meaning
dann eine Bedeutung, wenn „p“, „q“, „r“, icant, if “p”, “q”, “r”, etc. are not elemen- even when ‘p’, ‘q’, ‘r’, etc. are not elemenetc.
nicht Elementarsätze sind. tary propositions. tary propositions.
Und es ist leicht zu sehen, dass das And it is easy to see that the propo- And it is easy to see that the propoSatzzeichen
in No. 4.442, auch wenn „p“ sitional sign in No. 4.442 expresses one sitional sign in 4.442 expresses a single
und „q“ Wahrheitsfunktionen von Ele- truth-function of elementary propositions truth-function of elementary propositions
mentarsätzen sind, Eine Wahrheitsfunk- even when “p” and “q” are truth-functions even when ‘p’ and ‘q’ are truth-functions
tion von Elementarsätzen ausdrückt. of elementary propositions. of elementary propositions.
5.32 Alle Wahrheitsfunktionen sind Resul- All truth-functions are results of the All truth-functions are results of suctate
der successiven Anwendung einer successive application of a finite number cessive applications to elementary propoendlichen
Anzahl von Wahrheitsoperatio- of truth-operations to elementary propo- sitions of a finite number of truthnen
auf die Elementarsätze. sitions. operations.
5.4 Hier zeigt es sich, dass es „logische Here it becomes clear that there are At this point it becomes manifest that
Gegenstände“, „logische Konstante“ (im no such things as “logical objects” or “logi- there are no ‘logical objects’ or ‘logical conSinne
Freges und Russells) nicht gibt. cal constants” (in the sense of Frege and stants’ (in Frege’s and Russell’s sense).
Russell).
5.41 Denn: Alle Resultate von Wahrheits- For all those results of truth-oper- The reason is that the results of truthoperationen
mit Wahrheitsfunktionen ations on truth-functions are identical, operations on truth-functions are always
sind identisch, welche eine und dieselbe which are one and the same truth-func- identical whenever they are one and the
Wahrheitsfunktion von Elementarsätzen tion of elementary propositions. same truth-function of elementary proposind.
sitions.
5.42 Dass ∨, ⊃, etc. nicht Beziehungen That ∨, ⊃, etc., are not relations in the It is self-evident that ∨, ⊃, etc. are not
im Sinne von rechts und links etc. sind, sense of right and left, etc., is obvious. relations in the sense in which right and
leuchtet ein. left etc. are relations.
Die Möglichkeit des kreuzweisen Defi- The possibility of crosswise definition The interdefinability of Frege’s and
68
nierens der logischen „Urzeichen“ Freges of the logical “primitive signs” of Frege Russell’s ‘primitive signs’ of logic is
und Russells zeigt schon, dass diese kei- and Russell shows by itself that these are enough to show that they are not primine
Urzeichen sind, und schon erst recht, not primitive signs and that they signify tive signs, still less signs for relations.
dass sie keine Relationen bezeichnen. no relations.
Und es ist offenbar, dass das „⊃“, wel- And it is obvious that the “⊃” which And it is obvious that the ‘⊃’ defined
ches wir durch „∼“ und „∨“ definieren, we define by means of “∼” and “∨” is iden- by means of ‘∼’ and ‘∨’ is identical with
identisch ist mit dem, durch welches wir tical with that by which we define “∨” the one that figures with ‘∼’ in the def„∨“
mit „∼“ definieren, und dass dieses „∨“ with the help of “∼”, and that this “∨” is inition of ‘∨’; and that the second ‘∨’ is
mit dem ersten identisch ist. U. s. w. the same as the first, and so on. identical with the first one; and so on.
5.43 Dass aus einer Tatsache p unendlich That from a fact p an infinite num- Even at first sight it seems scarcely
viele a n d e r e folgen sollten, nämlich ber of others should follow, namely, ∼∼p, credible that there should follow from
∼∼p, ∼∼∼∼p, etc., ist doch von vornher- ∼∼∼∼p, etc., is indeed hardly to be be- one fact p infinitely many others, namely
ein kaum zu glauben. Und nicht weniger lieved, and it is no less wonderful that the ∼∼p, ∼∼∼∼p, etc. And it is no less
merkwürdig ist, dass die unendliche An- infinite number of propositions of logic (of remarkable that the infinite number of
zahl der Sätze der Logik (der Mathema- mathematics) should follow from half a propositions of logic (mathematics) follow
tik) aus einem halben Dutzend „Grundge- dozen “primitive propositions”. from half a dozen ‘primitive propositions’.
setzen“ folgen.
Alle Sätze der Logik sagen aber das- But the propositions of logic say the But in fact all the propositions of logic
selbe. Nämlich nichts. same thing. That is, nothing. say the same thing, to wit nothing.
5.44 Die Wahrheitsfunktionen sind keine Truth-functions are not material func- Truth-functions are not material funcmateriellen
Funktionen. tions. tions.
Wenn man z. B. eine Bejahung durch If e.g. an affirmation can be produced For example, an affirmation can be
doppelte Verneinung erzeugen kann, ist by repeated denial, is the denial—in produced by double negation: in such a
dann die Verneinung—in irgend einem any sense—contained in the affirmation? case does it follow that in some sense
Sinn—in der Bejahung enthalten? Ver- Does “∼∼p” deny ∼p, or does it affirm p; negation is contained in affirmation?
neint „∼∼p“ ∼p, oder bejaht es p; oder or both? Does ‘∼∼p’ negate ∼p, or does it affirm
beides? p—or both?
Der Satz „∼∼p“ handelt nicht von der The proposition “∼∼p” does not treat The proposition ‘∼∼p’ is not about
Verneinung wie von einem Gegenstand; of denial as an object, but the possibility negation, as if negation were an object:
wohl aber ist die Möglichkeit der Ver- of denial is already prejudged in affirma- on the other hand, the possibility of neganeinung
in der Bejahung bereits präju- tion. tion is already written into affirmation.
diziert.
Und gäbe es einen Gegenstand, der And if there was an object called “∼”, And if there were an object called ‘∼’,
„∼“ hieße, so müsste „∼∼p“ etwas anderes then “∼∼p” would have to say something it would follow that ‘∼∼p’ said something
sagen als „p“. Denn der eine Satz wür- other than “p”. For the one proposition different from what ‘p’ said, just because
de dann eben von ∼ handeln, der andere would then treat of ∼, the other would the one proposition would then be about
nicht. not. ∼ and the other would not.
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5.441 Dieses Verschwinden der scheinbaren This disappearance of the appar- This vanishing of the apparent loglogischen
Konstanten tritt auch ein, wenn ent logical constants also occurs if ical constants also occurs in the case
„∼(∃x).∼fx“ dasselbe sagt wie „(x). fx“, “∼(∃x).∼fx” says the same as “(x). fx”, or of ‘∼(∃x).∼fx’, which says the same as
oder „(∃x). fx . x = a“ dasselbe wie „fa“. “(∃x). fx . x = a” the same as “fa”. ‘(x). fx’, and in the case of ‘(∃x). fx . x = a’,
which says the same as ‘fa’.
5.442 Wenn uns ein Satz gegeben ist, so sind If a proposition is given to us then the If we are given a proposition, then
m i t i h m auch schon die Resultate aller results of all truth-operations which have with it we are also given the results of
Wahrheitsoperationen, die ihn zur Basis it as their basis are given with it. all truth-operations that have it as their
haben, gegeben. base.
5.45 Gibt es logische Urzeichen, so muss If there are logical primitive signs a If there are primitive logical signs,
eine richtige Logik ihre Stellung zuein- correct logic must make clear their posi- then any logic that fails to show clearly
ander klar machen und ihr Dasein recht- tion relative to one another and justify how they are placed relatively to one anfertigen.
Der Bau der Logik a u s ihren their existence. The construction of logic other and to justify their existence will be
Urzeichen muss klar werden. out of its primitive signs must become incorrect. The construction of logic out of
clear. its primitive signs must be made clear.
5.451 Hat die Logik Grundbegriffe, so müs- If logic has primitive ideas these must If logic has primitive ideas, they must
sen sie von einander unabhängig sein. Ist be independent of one another. If a prim- be independent of one another. If a primein
Grundbegriff eingeführt, so muss er itive idea is introduced it must be intro- itive idea has been introduced, it must
in allen Verbindungen eingeführt sein, duced in all contexts in which it occurs have been introduced in all the combinaworin
er überhaupt vorkommt. Man kann at all. One cannot therefore introduce tions in which it ever occurs. It cannot,
ihn also nicht zuerst für e i n e Verbin- it for one context and then again for an- therefore, be introduced first for one comdung,
dann noch einmal für eine andere other. For example, if denial is introduced, bination and later reintroduced for aneinführen.
Z. B.: Ist die Verneinung ein- we must understand it in propositions of other. For example, once negation has
geführt, so müssen wir sie jetzt in Sätzen the form “∼p”, just as in propositions like been introduced, we must understand it
von der Form „∼p“ ebenso verstehen, wie “∼(p∨ q)”, “(∃x).∼fx” and others. We may both in propositions of the form ‘∼p’ and
in Sätzen wie „∼(p ∨ q)“, „(∃x).∼fx“ u. a. not first introduce it for one class of cases in propositions like ‘∼(p ∨ q)’, ‘(∃x).∼fx’,
Wir dürfen sie nicht erst für die eine Klas- and then for another, for it would then etc. We must not introduce it first for
se von Fällen, dann für die andere ein- remain doubtful whether its meaning in the one class of cases and then for the
führen, denn es bliebe dann zweifelhaft, the two cases was the same, and there other, since it would then be left in doubt
ob ihre Bedeutung in beiden Fällen die would be no reason to use the same way whether its meaning were the same in
gleiche wäre und es wäre kein Grund vor- of symbolizing in the two cases. both cases, and no reason would have
handen, in beiden Fällen dieselbe Art der been given for combining the signs in the
Zeichenverbindung zu benützen. same way in both cases.
(Kurz, für die Einführung der Ur- (In short, what Frege (“Grundgesetze (In short, Frege’s remarks about inzeichen
gilt, mutatis mutandis, dassel- der Arithmetik”) has said about the in- troducing signs by means of definitions
be, was Frege („Grundgesetze der Arith- troduction of signs by definitions holds, (in The Fundamental Laws of Arithmetic)
metik“) für die Einführung von Zeichen mutatis mutandis, for the introduction of also apply, mutatis mutandis, to the intro-
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durch Definitionen gesagt hat.) primitive signs also.) duction of primitive signs.)
5.452 Die Einführung eines neuen Be- The introduction of a new expedient in The introduction of any new device
helfes in den Symbolismus der Logik the symbolism of logic must always be an into the symbolism of logic is necessarily
muss immer ein folgenschweres Ereig- event full of consequences. No new sym- a momentous event. In logic a new device
nis sein. Kein neuer Behelf darf in die bol may be introduced in logic in brackets should not be introduced in brackets or
Logik—sozusagen, mit ganz unschuldi- or in the margin—with, so to speak, an in a footnote with what one might call a
ger Miene—in Klammern oder unter dem entirely innocent face. completely innocent air.
Striche eingeführt werden.
(So kommen in den „Principia Mathe- (Thus in the “Principia Mathematica” (Thus in Russell and Whitehead’s
matica“ von Russell und Whitehead De- of Russell and Whitehead there occur Principia Mathematica there occur deffinitionen
und Grundgesetze in Worten definitions and primitive propositions in initions and primitive propositions exvor.
Warum hier plötzlich Worte? Dies be- words. Why suddenly words here? This pressed in words. Why this sudden apdürfte
einer Rechtfertigung. Sie fehlt und would need a justification. There was pearance of words? It would require a
muss fehlen, da das Vorgehen tatsächlich none, and can be none for the process is justification, but none is given, or could
unerlaubt ist.) actually not allowed.) be given, since the procedure is in fact
illicit.)
Hat sich aber die Einführung eines But if the introduction of a new expe- But if the introduction of a new device
neuen Behelfes an einer Stelle als nötig dient has proved necessary in one place, has proved necessary at a certain point,
erwiesen, so muss man sich nun sofort we must immediately ask: Where is this we must immediately ask ourselves, ‘At
fragen: Wo muss dieser Behelf nun i m - expedient always to be used? Its position what points is the employment of this dem
e r angewandt werden? Seine Stellung in logic must be made clear. vice now unavoidable?’ and its place in
in der Logik muss nun erklärt werden. logic must be made clear.
5.453 Alle Zahlen der Logik müssen sich All numbers in logic must be capable All numbers in logic stand in need of
rechtfertigen lassen. of justification. justification.
Oder vielmehr: Es muss sich heraus- Or rather it must become plain that Or rather, it must become evident that
stellen, dass es in der Logik keine Zahlen there are no numbers in logic. there are no numbers in logic.
gibt.
Es gibt keine ausgezeichneten Zahlen. There are no pre-eminent numbers. There are no privileged numbers.
5.454 In der Logik gibt es kein Nebeneinan- In logic there is no side by side, there In logic there is no co-ordinate status,
der, kann es keine Klassifikation geben. can be no classification. and there can be no classification.
In der Logik kann es nicht Allgemei- In logic there cannot be a more gen- In logic there can be no distinction
neres und Spezielleres geben. eral and a more special. between the general and the specific.
5.4541 Die Lösungen der logischen Probleme The solution of logical problems must The solutions of the problems of logic
müssen einfach sein, denn sie setzen den be simple for they set the standard of sim- must be simple, since they set the stanStandard
der Einfachheit. plicity. dard of simplicity.
Die Menschen haben immer geahnt, Men have always thought that there Men have always had a presentiment
dass es ein Gebiet von Fragen ge- must be a sphere of questions whose that there must be a realm in which
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ben müsse, deren Antworten—a priori— answers—a priori—are symmetrical and the answers to questions are symmetrisymmetrisch,
und zu einem abgeschlosse- united into a closed regular structure. cally combined—a priori—to form a selfnen,
regelmäßigen Gebilde vereint liegen. contained system.
Ein Gebiet, in dem der Satz gilt: sim- A sphere in which the proposition, A realm subject to the law: Simplex
plex sigillum veri. simplex sigillum veri, is valid. sigillum veri.
5.46 Wenn man die logischen Zeichen rich- When we have rightly introduced the If we introduced logical signs properly,
tig einführte, so hätte man damit auch logical signs, the sense of all their combi- then we should also have introduced at
schon den Sinn aller ihrer Kombinatio- nations has been already introduced with the same time the sense of all combinanen
eingeführt; also nicht nur „p∨ q“ son- them: therefore not only “p ∨ q” but also tions of them; i.e. not only ‘p ∨ q’ but
dern auch schon „∼(p∨∼q)“ etc. etc. Man “∼(p ∨ ∼q)”, etc. etc. We should then al- ‘∼(p ∨ ∼q)’ as well, etc. etc. We should
hätte damit auch schon die Wirkung aller ready have introduced the effect of all also have introduced at the same time
nur möglichen Kombinationen von Klam- possible combinations of brackets; and it the effect of all possible combinations of
mern eingeführt. Und damit wäre es klar would then have become clear that the brackets. And thus it would have been
geworden, dass die eigentlichen allgemei- proper general primitive signs are not made clear that the real general priminen
Urzeichen nicht die „p∨ q“, „(∃x). fx“, “p ∨ q”, “(∃x). fx”, etc., but the most gen- tive signs are not ‘p∨ q’, ‘(∃x). fx’, etc. but
etc. sind, sondern die allgemeinste Form eral form of their combinations. the most general form of their combinaihrer
Kombinationen. tions.
5.461 Bedeutungsvoll ist die scheinbar un- The apparently unimportant fact that Though it seems unimportant, it is in
wichtige Tatsache, dass die logischen the apparent relations like ∨ and ⊃ fact significant that the pseudo-relations
Scheinbeziehungen, wie ∨ und ⊃, der need brackets—unlike real relations—is of logic, such as ∨ and ⊃, need brackets—
Klammern bedürfen—im Gegensatz zu of great importance. unlike real relations.
den wirklichen Beziehungen.
Die Benützung der Klammern mit The use of brackets with these appar- Indeed, the use of brackets with these
jenen scheinbaren Urzeichen deutet ja ent primitive signs shows that these are apparently primitive signs is itself an inschon
darauf hin, dass diese nicht die not the real primitive signs; and nobody dication that they are not primitive signs.
wirklichen Urzeichen sind. Und es wird of course would believe that the brackets And surely no one is going to believe
doch wohl niemand glauben, dass die have meaning by themselves. brackets have an independent meaning.
Klammern eine selbständige Bedeutung
haben.
5.4611 Die logischen Operationszeichen sind Logical operation signs are punctua- Signs for logical operations are
Interpunktionen. tions. punctuation-marks.
5.47 Es ist klar, dass alles, was sich über- It is clear that everything which can It is clear that whatever we can say in
haupt v o n v o r n h e r e i n über die be said beforehand about the form of all advance about the form of all propositions,
Form aller Sätze sagen lässt, sich a u f propositions at all can be said on one oc- we must be able to say all at once.
e i n m a l sagen lassen muss. casion.
Sind ja schon im Elementarsatze alle For all logical operations are already An elementary proposition really conlogischen
Operationen enthalten. Denn contained in the elementary proposition. tains all logical operations in itself. For
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„fa“ sagt dasselbe wie For “fa” says the same as ‘fa’ says the same thing as
„(∃x). fx . x = a“. “(∃x). fx . x = a”. ‘(∃x). fx . x = a’.
Wo Zusammengesetztheit ist, da ist Where there is composition, there is Wherever there is compositeness, arArgument
und Funktion, und wo diese argument and function, and where these gument and function are present, and
sind, sind bereits alle logischen Konstan- are, all logical constants already are. where these are present, we already have
ten. all the logical constants.
Man könnte sagen: Die Eine logische One could say: the one logical con- One could say that the sole logical conKonstante
ist das, was a l l e Sätze, ihrer stant is that which all propositions, ac- stant was what all propositions, by their
Natur nach, mit einander gemein haben. cording to their nature, have in common very nature, had in common with one anwith
one another. other.
Das aber ist die allgemeine Satzform. That however is the general form of But that is the general propositional
proposition. form.
5.471 Die allgemeine Satzform ist das We- The general form of proposition is the The general propositional form is the
sen des Satzes. essence of proposition. essence of a proposition.
5.4711 Das Wesen des Satzes angeben, heißt, To give the essence of proposition To give the essence of a proposition
das Wesen aller Beschreibung angeben, means to give the essence of all descrip- means to give the essence of all descripalso
das Wesen der Welt. tion, therefore the essence of the world. tion, and thus the essence of the world.
5.472 Die Beschreibung der allgemeinsten The description of the most general The description of the most general
Satzform ist die Beschreibung des einen propositional form is the description of propositional form is the description of
und einzigen allgemeinen Urzeichens der the one and only general primitive sign the one and only general primitive sign
Logik. in logic. in logic.
5.473 Die Logik muss für sich selber sorgen. Logic must take care of itself. Logic must look after itself.
Ein m ö g l i c h e s Zeichen muss A possible sign must also be able to If a sign is possible, then it is also caauch
bezeichnen können. Alles was in der signify. Everything which is possible in pable of signifying. Whatever is possible
Logik möglich ist, ist auch erlaubt. („So- logic is also permitted. (“Socrates is iden- in logic is also permitted. (The reason
krates ist identisch“ heißt darum nichts, tical” means nothing because there is no why ‘Socrates is identical’ means nothing
weil es keine Eigenschaft gibt, die „iden- property which is called “identical”. The is that there is no property called ‘identisch“
heißt. Der Satz ist unsinnig, weil proposition is senseless because we have tical’. The proposition is nonsensical bewir
eine willkürliche Bestimmung nicht not made some arbitrary determination, cause we have failed to make an arbitrary
getroffen haben, aber nicht darum, weil not because the symbol is in itself unper- determination, and not because the symdas
Symbol an und für sich unerlaubt wä- missible.) bol, in itself, would be illegitimate.)
re.)
Wir können uns, in gewissem Sinne, In a certain sense we cannot make In a certain sense, we cannot make
nicht in der Logik irren. mistakes in logic. mistakes in logic.
5.4731 Das Einleuchten, von dem Russell Self-evidence, of which Russell has Self-evidence, which Russell talked
so viel sprach, kann nur dadurch in said so much, can only be discard in logic about so much, can become dispensable
der Logik entbehrlich werden, dass die by language itself preventing every logi- in logic, only because language itself
73
Sprache selbst jeden logischen Fehler cal mistake. That logic is a priori consists prevents every logical mistake.—What
verhindert.—Dass die Logik a priori ist, in the fact that we cannot think illogically. makes logic a priori is the impossibility
besteht darin, dass nicht unlogisch ge- of illogical thought.
dacht werden k a n n.
5.4732 Wir können einem Zeichen nicht den We cannot give a sign the wrong We cannot give a sign the wrong
unrechten Sinn geben. sense. sense.
5.47321 Occams Devise ist natürlich keine Occam’s razor is, of course, not an arbi- Occam’s maxim is, of course, not an
willkürliche, oder durch ihren prakti- trary rule nor one justified by its practical arbitrary rule, nor one that is justified by
schen Erfolg gerechtfertigte Regel: Sie be- success. It simply says that unnecessary its success in practice: its point is that unsagt,
dass u n n ö t i g e Zeicheneinheiten elements in a symbolism mean nothing. necessary units in a sign-language mean
nichts bedeuten. nothing.
Zeichen, die E i n e n Zweck erfül- Signs which serve one purpose are log- Signs that serve one purpose are loglen,
sind logisch äquivalent, Zeichen, die ically equivalent, signs which serve no ically equivalent, and signs that serve
k e i n e n Zweck erfüllen, logisch bedeu- purpose are logically meaningless. none are logically meaningless.
tungslos.
5.4733 Frege sagt: Jeder rechtmäßig gebilde- Frege says: Every legitimately con- Frege says that any legitimately conte
Satz muss einen Sinn haben; und ich structed proposition must have a sense; structed proposition must have a sense.
sage: Jeder mögliche Satz ist rechtmäßig and I say: Every possible proposition is And I say that any possible proposition is
gebildet, und wenn er keinen Sinn hat, legitimately constructed, and if it has legitimately constructed, and, if it has no
so kann das nur daran liegen, dass wir no sense this can only be because we sense, that can only be because we have
einigen seiner Bestandteile keine B e - have given no meaning to some of its con- failed to give a meaning to some of its
d e u t u n g gegeben haben. stituent parts. constituents.
(Wenn wir auch glauben, es getan zu (Even if we believe that we have done (Even if we think that we have done
haben.) so.) so.)
So sagt „Sokrates ist identisch“ darum Thus “Socrates is identical” says noth- Thus the reason why ‘Socrates is idennichts,
weil wir dem Wort „identisch“ als ing, because we have given no mean- tical’ says nothing is that we have not
E i g e n s c h a f t s w o r t k e i n e Be- ing to the word “identical” as adjec- given any adjectival meaning to the word
deutung gegeben haben. Denn, wenn es tive. For when it occurs as the sign of ‘identical’. For when it appears as a sign
als Gleichheitszeichen auftritt, so sym- equality it symbolizes in an entirely dif- for identity, it symbolizes in an entirely
bolisiert es auf ganz andere Art und ferent way—the symbolizing relation is different way—the signifying relation is a
Weise—die bezeichnende Beziehung ist another—therefore the symbol is in the different one—therefore the symbols also
eine andere,—also ist auch das Symbol two cases entirely different; the two sym- are entirely different in the two cases: the
in beiden Fällen ganz verschieden; die bols have the sign in common with one two symbols have only the sign in combeiden
Symbole haben nur das Zeichen another only by accident. mon, and that is an accident.
zufällig miteinander gemein.
5.474 Die Anzahl der nötigen Grundopera- The number of necessary fundamental The number of fundamental operationen
hängt n u r von unserer Notation operations depends only on our notation. tions that are necessary depends solely
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ab. on our notation.
5.475 Es kommt nur darauf an, ein Zeichen- It is only a question of constructing All that is required is that we should
system von einer bestimmten Anzahl von a system of signs of a definite number of construct a system of signs with a particDimensionen—von
einer bestimmten ma- dimensions—of a definite mathematical ular number of dimensions—with a parthematischen
Mannigfaltigkeit—zu bil- multiplicity. ticular mathematical multiplicity.
den.
5.476 Es ist klar, dass es sich hier nicht um It is clear that we are not concerned It is clear that this is not a question of
eine A n z a h l v o n G r u n d b e g r i f - here with a number of primitive ideas a number of primitive ideas that have to
f e n handelt, die bezeichnet werden müs- which must be signified but with the ex- be signified, but rather of the expression
sen, sondern um den Ausdruck einer Re- pression of a rule. of a rule.
gel.
5.5 Jede Wahrheitsfunktion ist ein Resul- Every truth-function is a result of the Every truth-function is a result of
tat der successiven Anwendung der Ope- successive application of the operation successive applications to elementary
ration (−−−−−W)(ξ, . . . . .) auf Elemen- (−−−−−T)(ξ, . . . . .) to elementary propo- propositions of the operation ‘(−−−−−T)
tarsätze. sitions. (ξ, . . . . .)’.
Diese Operation verneint sämtliche This operation denies all the proposi- This operation negates all the propoSätze
in der rechten Klammer, und ich tions in the right-hand bracket and I call sitions in the right-hand pair of brackets,
nenne sie die Negation dieser Sätze. it the negation of these propositions. and I call it the negation of those proposi-
tions.
5.501 Einen Klammerausdruck, dessen Glie- An expression in brackets whose When a bracketed expression has
der Sätze sind, deute ich—wenn die Rei- terms are propositions I indicate—if the propositions as its terms—and the orhenfolge
der Glieder in der Klammer order of the terms in the bracket is der of the terms inside the brackets is
gleichgültig ist—durch ein Zeichen von indifferent—by a sign of the form “(ξ)”. “ξ” indifferent—then I indicate it by a sign of
der Form „(ξ)“ an. „ξ“ ist eine Variable, is a variable whose values are the terms the form ‘(ξ)’. ‘ξ’ is a variable whose valderen
Werte die Glieder des Klammer- of the expression in brackets, and the line ues are terms of the bracketed expression
ausdruckes sind; und der Strich über der over the variable indicates that it stands and the bar over the variable indicates
Variablen deutet an, dass sie ihre sämtli- for all its values in the bracket. that it is the representative of all its valchen
Werte in der Klammer vertritt. ues in the brackets.
(Hat also ξ etwa die 3 Werte P, Q, R, (Thus if ξ has the 3 values P, Q, R, (E.g. if ξ has the three values P, Q, R,
so ist (ξ) = (P, Q, R).) then (ξ) = (P, Q, R).) then (ξ) = (P, Q, R). )
Die Werte der Variablen werden fest- The values of the variables must be What the values of the variable are is
gesetzt. determined. something that is stipulated.
Die Festsetzung ist die Beschreibung The determination is the description The stipulation is a description of the
der Sätze, welche die Variable vertritt. of the propositions which the variable propositions that have the variable as
stands for. their representative.
Wie die Beschreibung der Glieder des How the description of the terms of How the description of the terms of
Klammerausdruckes geschieht, ist unwe- the expression in brackets takes place is the bracketed expression is produced is
75
sentlich. unessential. not essential.
Wir k ö n n e n drei Arten der Be- We may distinguish 3 kinds of descrip- We can distinguish three kinds of deschreibung
unterscheiden: 1. Die direk- tion: 1. Direct enumeration. In this case scription: 1. direct enumeration, in which
te Aufzählung. In diesem Fall können we can place simply its constant values in- case we can simply substitute for the variwir
statt der Variablen einfach ihre kon- stead of the variable. 2. Giving a function able the constants that are its values; 2.
stanten Werte setzen. 2. Die Angabe ei- fx, whose values for all values of x are giving a function fx whose values for all
ner Funktion fx, deren Werte für alle the propositions to be described. 3. Giv- values of x are the propositions to be deWerte
von x die zu beschreibenden Sätze ing a formal law, according to which those scribed; 3. giving a formal law that govsind.
3. Die Angabe eines formalen Ge- propositions are constructed. In this case erns the construction of the propositions,
setzes, nach welchem jene Sätze gebildet the terms of the expression in brackets in which case the bracketed expression
sind. In diesem Falle sind die Glieder des are all the terms of a formal series. has as its members all the terms of a seKlammerausdrucks
sämtliche Glieder ei- ries of forms.
ner Formenreihe.
5.502 Ich schreibe also statt „(−−−−−W) Therefore I write instead of So instead of ‘(−−−−−T) (ξ, . . . . .)’,
(ξ, . . . . .)“ „N(ξ)“. “(−−−−−T) (ξ, . . . . .)”, “N(ξ)”. I write ‘N(ξ)’.
N(ξ) ist die Negation sämtlicher Werte N(ξ) is the negation of all the values N(ξ) is the negation of all the values
der Satzvariablen ξ. of the propositional variable ξ. of the propositional variable ξ.
5.503 Da sich offenbar leicht ausdrücken As it is obviously easy to express how It is obvious that we can easily
läßt, wie mit dieser Operation Sätze gebil- propositions can be constructed by means express how propositions may be condet
werden können und wie Sätze mit ihr of this operation and how propositions are structed with this operation, and how
nicht zu bilden sind, so muss dies auch not to be constructed by means of it, this they may not be constructed with it; so it
einen exakten Ausdruck finden können. must be capable of exact expression. must be possible to find an exact expression
for this.
5.51 Hat ξ nur einen Wert, so ist N(ξ) = ∼p If ξ has only one value, then N(ξ) = ∼p If ξ has only one value, then N(ξ) = ∼p
(nicht p), hat es zwei Werte, so ist N(ξ) = (not p), if it has two values then N(ξ) = (not p); if it has two values, then N(ξ) =
∼p . ∼q (weder p noch q). ∼p . ∼q (neither p nor q). ∼p . ∼q (neither p nor q).
5.511 Wie kann die allumfassende, welt- How can the all-embracing logic which How can logic—all-embracing logic,
spiegelnde Logik so spezielle Haken und mirrors the world use such special which mirrors the world—use such peManipulationen
gebrauchen? Nur, indem catches and manipulations? Only because culiar crotchets and contrivances? Only
sich alle diese zu einem unendlich feinen all these are connected into an infinitely because they are all connected with one
Netzwerk, zu dem großen Spiegel, ver- fine network, to the great mirror. another in an infinitely fine network, the
knüpfen. great mirror.
5.512 „∼p“ ist wahr, wenn „p“ falsch ist. “∼p” is true if “p” is false. Therefore ‘∼p’ is true if ‘p’ is false. Therefore, in
Also in dem wahren Satz „∼p“ ist „p“ in the true proposition “∼p” “p” is a false the proposition ‘∼p’, when it is true, ‘p’
ein falscher Satz. Wie kann ihn nun der proposition. How then can the stroke “∼” is a false proposition. How then can the
Strich „∼“ mit der Wirklichkeit zum Stim- bring it into agreement with reality? stroke ‘∼’ make it agree with reality?
76
men bringen?
Das, was in „∼p“ verneint, ist aber That which denies in “∼p” is however But in ‘∼p’ it is not ‘∼’ that negates, it
nicht das „∼“, sondern dasjenige, was al- not “∼”, but that which all signs of this is rather what is common to all the signs
len Zeichen dieser Notation, welche p ver- notation, which deny p, have in common. of this notation that negate p.
neinen, gemeinsam ist.
Also die gemeinsame Regel, nach wel- Hence the common rule according to That is to say the common rule that
cher „∼p“, „∼∼∼p“, „∼p ∨∼p“, „∼p . ∼p“, which “∼p”, “∼∼∼p”, “∼p∨∼p”, “∼p . ∼p”, governs the construction of ‘∼p’, ‘∼∼∼p’,
etc. etc. (ad inf.) gebildet werden. Und etc. etc. (to infinity) are constructed. And ‘∼p ∨∼p’, ‘∼p . ∼p’, etc. etc. (ad inf.). And
dies Gemeinsame spiegelt die Verneinung this which is common to them all mirrors this common factor mirrors negation.
wieder. denial.
5.513 Man könnte sagen: Das Gemeinsame We could say: What is common to all We might say that what is common to
aller Symbole, die sowohl p als q bejahen, symbols, which assert both p and q, is all symbols that affirm both p and q is
ist der Satz „p . q“. Das Gemeinsame aller the proposition “p . q”. What is common the proposition ‘p . q’; and that what is
Symbole, die entweder p oder q bejahen, to all symbols, which asserts either p or common to all symbols that affirm either
ist der Satz „p ∨ q“. q, is the proposition “p ∨ q”. p or q is the proposition ‘p ∨ q’.
Und so kann man sagen: Zwei Sätze And similarly we can say: Two propo- And similarly we can say that two
sind einander entgegengesetzt, wenn sie sitions are opposed to one another when propositions are opposed to one another
nichts miteinander gemein haben, und: they have nothing in common with one if they have nothing in common with one
Jeder Satz hat nur ein Negativ, weil es another; and every proposition has only another, and that every proposition has
nur einen Satz gibt, der ganz außerhalb one negative, because there is only one only one negative, since there is only one
seiner liegt. proposition which lies altogether outside proposition that lies completely outside
it. it.
Es zeigt sich so auch in Russells No- Thus in Russell’s notation also it ap- Thus in Russell’s notation too it is
tation, dass „q : p ∨∼p“ dasselbe sagt wie pears evident that “q : p ∨ ∼p” says the manifest that ‘q : p ∨ ∼p’ says the same
„q“; dass „p ∨∼p“ nichts sagt. same thing as “q”; that “p∨∼p” says noth- thing as ‘q’, that ‘p ∨∼p’ says nothing.
ing.
5.514 Ist eine Notation festgelegt, so gibt es If a notation is fixed, there is in it a Once a notation has been established,
in ihr eine Regel, nach der alle p vernei- rule according to which all the proposi- there will be in it a rule governing the connenden
Sätze gebildet werden, eine Regel, tions denying p are constructed, a rule struction of all propositions that negate
nach der alle p bejahenden Sätze gebil- according to which all the propositions as- p, a rule governing the construction of
det werden, eine Regel, nach der alle p serting p are constructed, a rule accord- all propositions that affirm p, and a rule
oder q bejahenden Sätze gebildet werden, ing to which all the propositions asserting governing the construction of all proposiu.
s. f. Diese Regeln sind den Symbolen p or q are constructed, and so on. These tions that affirm p or q; and so on. These
äquivalent und in ihnen spiegelt sich ihr rules are equivalent to the symbols and rules are equivalent to the symbols; and
Sinn wieder. in them their sense is mirrored. in them their sense is mirrored.
5.515 Es muss sich an unseren Symbolen It must be recognized in our symbols It must be manifest in our symbols
zeigen, dass das, was durch „∨“, „.“, etc. that what is connected by “∨”, “.”, etc., that it can only be propositions that are
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miteinander verbunden ist, Sätze sein must be propositions. combined with one another by ‘∨’, ‘∼’, etc.
müssen.
Und dies ist auch der Fall, denn das And this is the case, for the symbols And this is indeed the case, since the
Symbol „p“ und „q“ setzt ja selbst das „∨“, “p” and “q” presuppose “∨”, “∼”, etc. If symbol in ‘p’ and ‘q’ itself presupposes ‘∨’,
„∼“, etc. voraus. Wenn das Zeichen „p“ in the sign “p” in “p ∨ q” does not stand for ‘∼’, etc. If the sign ‘p’ in ‘p ∨ q’ does not
„p ∨ q“ nicht für ein komplexes Zeichen a complex sign, then by itself it cannot stand for a complex sign, then it cannot
steht, dann kann es allein nicht Sinn ha- have sense; but then also the signs “p∨p”, have sense by itself: but in that case the
ben; dann können aber auch die mit „p“ “p . p”, etc. which have the same sense as signs ‘p ∨ p’, ‘p . p’, etc., which have the
gleichsinnigen Zeichen „p∨ p“, „p . p“, etc. “p” have no sense. If, however, “p∨ p” has same sense as p, must also lack sense.
keinen Sinn haben. Wenn aber „p∨p“ kei- no sense, then also “p ∨ q” can have no But if ‘p ∨ p’ has no sense, then ‘p ∨ q’
nen Sinn hat, dann kann auch „p∨ q“ kei- sense. cannot have a sense either.
nen Sinn haben.
5.5151 Muss das Zeichen des negativen Sat- Must the sign of the negative proposi- Must the sign of a negative proposizes
mit dem Zeichen des positiven gebil- tion be constructed by means of the sign tion be constructed with that of the posdet
werden? Warum sollte man den nega- of the positive? Why should one not be itive proposition? Why should it not be
tiven Satz nicht durch eine negative Tat- able to express the negative proposition possible to express a negative proposition
sache ausdrücken können. (Etwa: Wenn by means of a negative fact? (Like: if “a” by means of a negative fact? (E.g. sup„a“
nicht in einer bestimmten Beziehung does not stand in a certain relation to “b”, pose that ‘a’ does not stand in a certain
zu „b“ steht, könnte das ausdrücken, dass it could express that aRb is not the case.) relation to ‘b’; then this might be used to
aRb nicht der Fall ist.) say that aRb was not the case.)
Aber auch hier ist ja der negative Satz But here also the negative proposition But really even in this case the negaindirekt
durch den positiven gebildet. is indirectly constructed with the positive. tive proposition is constructed by an indirect
use of the positive.
Der positive S a t z muss die Existenz The positive proposition must presup- The positive proposition necessarily
des negativen S a t z e s voraussetzen pose the existence of the negative propo- presupposes the existence of the negative
und umgekehrt. sition and conversely. proposition and vice versa.
5.52 Sind die Werte von ξ sämtliche Werte If the values of ξ are the total values If ξ has as its values all the values
einer Funktion fx für alle Werte von x, so of a function fx for all values of x, then of a function fx for all values of x, then
wird N(ξ) = ∼(∃x). fx. N(ξ) = ∼(∃x). fx. N(ξ) = ∼(∃x). fx.
5.521 Ich trenne den Begriff A l l e von der I separate the concept all from the I dissociate the concept all from truthWahrheitsfunktion.
truth-function. functions.
Frege und Russell haben die Allge- Frege and Russell have introduced Frege and Russell introduced genermeinheit
in Verbindung mit dem logi- generality in connexion with the logical ality in association with logical product
schen Produkt oder der logischen Summe product or the logical sum. Then it would or logical sum. This made it difficult to
eingeführt. So wurde es schwer, die Sätze be difficult to understand the propositions understand the propositions ‘(∃x). fx’ and
„(∃x). fx“ und „(x). fx“, in welchen beide “(∃x). fx” and “(x). fx” in which both ideas ‘(x). fx’, in which both ideas are embedIdeen
beschlossen liegen, zu verstehen. lie concealed. ded.
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5.522 Das Eigentümliche der Allgemein- That which is peculiar to the “symbol- What is peculiar to the generality-sign
heitsbezeichnung ist erstens, dass sie auf ism of generality” is firstly, that it refers is first, that it indicates a logical protoein
logisches Urbild hinweist, und zwei- to a logical prototype, and secondly, that type, and secondly, that it gives promitens,
dass sie Konstante hervorhebt. it makes constants prominent. nence to constants.
5.523 Die Allgemeinheitsbezeichnung tritt The generality symbol occurs as an The generality-sign occurs as an arguals
Argument auf. argument. ment.
5.524 Wenn die Gegenstände gegeben sind, If the objects are given, therewith are If objects are given, then at the same
so sind uns damit auch schon a l l e Ge- all objects also given. time we are given all objects.
genstände gegeben.
Wenn die Elementarsätze gegeben If the elementary propositions are If elementary propositions are given,
sind, so sind damit auch a l l e Elemen- given, then therewith all elementary then at the same time all elementary
tarsätze gegeben. propositions are also given. propositions are given.
5.525 Es ist unrichtig, den Satz „(∃x). fx“— It is not correct to render the propo- It is incorrect to render the proposiwie
Russell dies tut—in Worten durch „fx sition “(∃x). fx”—as Russell does—in the tion ‘(∃x). fx’ in the words, ‘fx is possible’
ist m ö g l i c h“ wiederzugeben. words “fx is possible”. as Russell does.
Gewißheit, Möglichkeit oder Unmög- Certainty, possibility or impossibility The certainty, possibility, or impossilichkeit
einer Sachlage wird nicht durch of a state of affairs are not expressed by a bility of a situation is not expressed by a
einen Satz ausgedrückt, sondern dadurch, proposition but by the fact that an expres- proposition, but by an expression’s being
dass ein Ausdruck eine Tautologie, ein sion is a tautology, a significant proposi- a tautology, a proposition with a sense, or
sinnvoller Satz oder eine Kontradiktion tion or a contradiction. a contradiction.
ist.
Jener Präzedenzfall, auf den man sich That precedent to which one would The precedent to which we are conimmer
berufen möchte, muss schon im always appeal, must be present in the stantly inclined to appeal must reside in
Symbol selber liegen. symbol itself. the symbol itself.
5.526 Man kann die Welt vollständig durch One can describe the world completely We can describe the world completely
vollkommen verallgemeinerte Sätze be- by completely generalized propositions, by means of fully generalized proposischreiben,
das heißt also, ohne irgend- i.e. without from the outset co-ordinating tions, i.e. without first correlating any
einen Namen von vornherein einem be- any name with a definite object. name with a particular object.
stimmten Gegenstand zuzuordnen.
Um dann auf die gewöhnliche Aus- In order then to arrive at the custom- Then, in order to arrive at the customdrucksweise
zu kommen, muss man ein- ary way of expression we need simply say ary mode of expression, we simply need
fach nach einem Ausdruck: „Es gibt ein after an expression “there is only and only to add, after an expression like, ‘There
und nur ein x, welches . . . “ sagen: Und one x, which . . . ”: and this x is a. is one and only one x such that . . . ’, the
dies x ist a. words, ‘and that x is a’.
5.5261 Ein vollkommen verallgemeinerter A completely generalized proposition A fully generalized proposition, like
Satz ist, wie jeder andere Satz, zusam- is like every other proposition compos- every other proposition, is composite.
mengesetzt. (Dies zeigt sich daran, dass ite. (This is shown by the fact that in (This is shown by the fact that in
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wir in „(∃x,φ).φx“ „φ“ und „x“ getrennt er- “(∃x,φ).φx” we must mention “φ” and “x” ‘(∃x,φ).φx’ we have to mention ‘φ’ and
wähnen müssen. Beide stehen unabhän- separately. Both stand independently in ‘x’ separately. They both, independently,
gig in bezeichnenden Beziehungen zur signifying relations to the world as in the stand in signifying relations to the world,
Welt, wie im unverallgemeinerten Satz.) ungeneralized proposition.) just as is the case in ungeneralized propo-
sitions.)
Kennzeichen des zusammengesetzten A characteristic of a composite symbol: It is a mark of a composite symbol that
Symbols: Es hat etwas mit a n d e r e n it has something in common with other it has something in common with other
Symbolen gemeinsam. symbols. symbols.
5.5262 Es verändert ja die Wahr- oder Falsch- The truth or falsehood of every proposi- The truth or falsity of every proposiheit
j e d e s Satzes etwas am allgemei- tion alters something in the general struc- tion does make some alteration in the
nen Bau der Welt. Und der Spielraum, ture of the world. And the range which general construction of the world. And
welcher ihrem Bau durch die Gesamt- is allowed to its structure by the totality the range that the totality of elementary
heit der Elementarsätze gelassen wird, of elementary propositions is exactly that propositions leaves open for its construcist
eben derjenige, welchen die ganz all- which the completely general propositions tion is exactly the same as that which
gemeinen Sätze begrenzen. delimit. is delimited by entirely general proposi-
tions.
(Wenn ein Elementarsatz wahr ist, so (If an elementary proposition is true, (If an elementary proposition is true,
ist damit doch jedenfalls Ein Elementar- then, at any rate, there is one more ele- that means, at any rate, one more true
satz m e h r wahr.) mentary proposition true.) elementary proposition.)
5.53 Gleichheit des Gegenstandes drücke Identity of the object I express by iden- Identity of object I express by idenich
durch Gleichheit des Zeichens aus, tity of the sign and not by means of a sign tity of sign, and not by using a sign for
und nicht mit Hilfe eines Gleichheitszei- of identity. Difference of the objects by identity. Difference of objects I express by
chens. Verschiedenheit der Gegenstände difference of the signs. difference of signs.
durch Verschiedenheit der Zeichen.
5.5301 Dass die Identität keine Relation zwi- That identity is not a relation between It is self-evident that identity is not
schen Gegenständen ist, leuchtet ein. objects is obvious. This becomes very a relation between objects. This becomes
Dies wird sehr klar, wenn man z. B. den clear if, for example, one considers the very clear if one considers, for example,
Satz „(x): fx .⊃. x = a“ betrachtet. Was die- proposition “(x): fx .⊃. x = a”. What this the proposition ‘(x): fx .⊃. x = a’. What
ser Satz sagt, ist einfach, dass n u r a proposition says is simply that only a sat- this proposition says is simply that only a
der Funktion f genügt, und nicht, dass isfies the function f , and not that only satisfies the function f , and not that only
nur solche Dinge der Funktion f genü- such things satisfy the function f which things that have a certain relation to a
gen, welche eine gewisse Beziehung zu a have a certain relation to a. satisfy the function f .
haben.
Man könnte nun freilich sagen, dass One could of course say that in fact Of course, it might then be said that
eben n u r a diese Beziehung zu a habe, only a has this relation to a, but in order only a did have this relation to a; but in
aber, um dies auszudrücken, brauchten to express this we should need the sign of order to express that, we should need the
wir das Gleichheitszeichen selber. identity itself. identity-sign itself.
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5.5302 Russells Definition von „=“ genügt Russell’s definition of “=” won’t do; Russell’s definition of ‘=’ is inadequate,
nicht; weil man nach ihr nicht sagen because according to it one cannot say because according to it we cannot say
kann, dass zwei Gegenstände alle Eigen- that two objects have all their properties that two objects have all their properties
schaften gemeinsam haben. (Selbst wenn in common. (Even if this proposition is in common. (Even if this proposition is
dieser Satz nie richtig ist, hat er doch never true, it is nevertheless significant.) never correct, it still has sense.)
S i n n.)
5.5303 Beiläufig gesprochen: Von z w e i Din- Roughly speaking: to say of two things Roughly speaking, to say of two things
gen zu sagen, sie seien identisch, ist ein that they are identical is nonsense, and that they are identical is nonsense, and
Unsinn, und von E i n e m zu sagen, to say of one thing that it is identical with to say of one thing that it is identical with
es sei identisch mit sich selbst, sagt gar itself is to say nothing. itself is to say nothing at all.
nichts.
5.531 Ich schreibe also nicht „f (a,b) . a = I write therefore not “f (a,b) . a = Thus I do not write ‘f (a,b) . a = b’, but
b“, sondern „f (a,a)“ (oder „f (b,b)“). Und b” but “f (a,a)” (or “f (b,b)”). And not ‘f (a,a)’ (or ‘f (b,b)’); and not ‘f (a,b) . ∼a =
nicht „f (a,b) . ∼a = b“, sondern „f (a,b)“. “f (a,b) . ∼a = b”, but “f (a,b)”. b’, but ‘f (a,b)’.
5.532 Und analog: Nicht „(∃x, y). f (x, y) . And analogously: not “(∃x, y). f (x, y) . And analogously I do not write
x = y“, sondern „(∃x). f (x,x)“; und nicht x = y”, but “(∃x). f (x,x)”; and not “(∃x, y). ‘(∃x, y). f (x, y) . x = y’, but ‘(∃x). f (x,x)’;
„(∃x, y). f (x, y) . ∼x = y“, sondern „(∃x, y). f (x, y) . ∼x = y”, but “(∃x, y). f (x, y)”. and not ‘(∃x, y). f (x, y) . ∼x = y’, but
f (x, y)“. ‘(∃x, y). f (x, y)’.
(Also statt des Russellschen „(∃x, y). (Therefore instead of Russell’s “(∃x, y). (So Russell’s ‘(∃x, y). f (x, y)’ becomes
f (x, y)“: „(∃x, y). f (x, y) .∨. (∃x). f (x,x)“.) f (x, y)”: “(∃x, y). f (x, y) .∨. (∃x). f (x,x)”.) ‘(∃x, y). f (x, y) .∨. (∃x). f (x,x)’.)
5.5321 Statt „(x): fx ⊃ x = a“ schreiben wir al- Instead of “(x): fx ⊃ x = a” we there- Thus, for example, instead of ‘(x): fx
so z. B. „(∃x). fx .⊃. fa : ∼(∃x, y). fx . f y“. fore write e.g. “(∃x). fx .⊃. fa : ∼(∃x, y). fx . ⊃ x = a’ we write ‘(∃x). fx .⊃. fa : ∼(∃x, y).
f y”. fx . f y’.
Und der Satz: „n u r Ein x befriedigt And if the proposition “only one x And the proposition, ‘Only one x satf
( )“ lautet: „(∃x). fx .⊃. fa : ∼(∃x, y). fx . satisfies f ( )” reads: “(∃x). fx .⊃. fa : isfies f ( )’, will read ‘(∃x). fx .⊃. fa :
f y“. ∼(∃x, y). fx . f y”. ∼(∃x, y). fx . f y’.
5.533 Das Gleichheitszeichen ist also kein The identity sign is therefore not an The identity-sign, therefore, is not an
wesentlicher Bestandteil der Begriffs- essential constituent of logical notation. essential constituent of conceptual notaschrift.
tion.
5.534 Und nun sehen wir, dass Scheinsät- And we see that the apparent proposi- And now we see that in a correct conze
wie: „a = a“, „a = b . b = c .⊃ a = c“, tions like: “a = a”, “a = b . b = c .⊃ a = c”, ceptual notation pseudo-propositions like
„(x).x = x“, „(∃x).x = a“, etc. sich in ei- “(x).x = x”. “(∃x).x = a”, etc. cannot be ‘a = a’, ‘a = b . b = c .⊃ a = c’, ‘(x).x = x’,
ner richtigen Begriffsschrift gar nicht hin- written in a correct logical notation at ‘(∃x).x = a’, etc. cannot even be written
schreiben lassen. all. down.
5.535 Damit erledigen sich auch alle Proble- So all problems disappear which are This also disposes of all the problems
me, die an solche Scheinsätze geknüpft connected with such pseudo-propositions. that were connected with such pseudowaren.
propositions.
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Alle Probleme, die Russells „Axiom of This is the place to solve all the prob- All the problems that Russell’s ‘axiom
Infinity“ mit sich bringt, sind schon hier lems with arise through Russell’s “Axiom of infinity’ brings with it can be solved at
zu lösen. of Infinity”. this point.
Das, was das Axiom of Infinity sagen What the axiom of infinity is meant What the axiom of infinity is intended
soll, würde sich in der Sprache dadurch to say would be expressed in language by to say would express itself in language
ausdrücken, dass es unendlich viele Na- the fact that there is an infinite number through the existence of infinitely many
men mit verschiedener Bedeutung gäbe. of names with different meanings. names with different meanings.
5.5351 Es gibt gewisse Fälle, wo man in Ver- There are certain cases in which one There are certain cases in which one
suchung gerät, Ausdrücke von der Form is tempted to use expressions of the form is tempted to use expressions of the form
„a = a“ oder „p ⊃ p“ u. dgl. zu benützen. “a = a” or “p ⊃ p” and of that kind. And ‘a = a’ or ‘p ⊃ p’ and the like. In fact,
Und zwar geschieht dies, wenn man von indeed this takes place when one would this happens when one wants to talk
dem Urbild: Satz, Ding, etc. reden möchte. speak of the archetype Proposition, Thing, about prototypes, e.g. about proposition,
So hat Russell in den „Principles of Ma- etc. So Russell in the Principles of Math- thing, etc. Thus in Russell’s Principles of
thematics“ den Unsinn „p ist ein Satz“ in ematics has rendered the nonsense “p Mathematics ‘p is a proposition’—which
Symbolen durch „p ⊃ p“ wiedergegeben is a proposition” in symbols by “p ⊃ p” is nonsense—was given the symbolic renund
als Hypothese vor gewisse Sätze ge- and has put it as hypothesis before cer- dering ‘p ⊃ p’ and placed as an hypothesis
stellt, damit deren Argumentstellen nur tain propositions to show that their places in front of certain propositions in order
von Sätzen besetzt werden könnten. for arguments could only be occupied by to exclude from their argument-places evpropositions.
erything but propositions.
(Es ist schon darum Unsinn, die Hypo- (It is nonsense to place the hypoth- (It is nonsense to place the hypothesis
these p ⊃ p vor einen Satz zu stellen, um esis p ⊃ p before a proposition in order ‘p ⊃ p’ in front of a proposition, in order
ihm Argumente der richtigen Form zu si- to ensure that its arguments have the to ensure that its arguments shall have
chern, weil die Hypothese für einen Nicht- right form, because the hypotheses for a the right form, if only because with a nonSatz
als Argument nicht falsch, sondern non-proposition as argument becomes not proposition as argument the hypothesis
unsinnig wird, und weil der Satz selbst false but meaningless, and because the becomes not false but nonsensical, and bedurch
die unrichtige Gattung von Argu- proposition itself becomes senseless for cause arguments of the wrong kind make
menten unsinnig wird, also sich selbst arguments of the wrong kind, and there- the proposition itself nonsensical, so that
ebenso gut, oder so schlecht, vor den un- fore it survives the wrong arguments no it preserves itself from wrong arguments
rechten Argumenten bewahrt wie die zu better and no worse than the senseless just as well, or as badly, as the hypothesis
diesem Zweck angehängte sinnlose Hypo- hypothesis attached for this purpose.) without sense that was appended for that
these.) purpose.)
5.5352 Ebenso wollte man „Es gibt keine Similarly it was proposed to express In the same way people have wanted
D i n g e“ ausdrücken durch „∼(∃x).x = x“. “There are no things” by “∼(∃x).x = x”. to express, ‘There are no things’, by writAber
selbst wenn dies ein Satz wäre— But even if this were a proposition— ing ‘∼(∃x).x = x’. But even if this were a
wäre er nicht auch wahr, wenn es zwar would it not be true if indeed “There were proposition, would it not be equally true
„Dinge gäbe“, aber diese nicht mit sich things”, but these were not identical with if in fact ‘there were things’ but they were
selbst identisch wären? themselves? not identical with themselves?
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5.54 In der allgemeinen Satzform kommt In the general propositional form, In the general propositional form
der Satz im Satze nur als Basis der Wahr- propositions occur in a proposition only propositions occur in other propositions
heitsoperationen vor. as bases of the truth-operations. only as bases of truth-operations.
5.541 Auf den ersten Blick scheint es, als At first sight it appears as if there At first sight it looks as if it were also
könne ein Satz in einem anderen auch were also a different way in which one possible for one proposition to occur in
auf andere Weise vorkommen. proposition could occur in another. another in a different way.
Besonders in gewissen Satzformen Especially in certain propositional Particularly with certain forms of
der Psychologie, wie „A glaubt, dass p forms of psychology, like “A thinks, that p proposition in psychology, such as ‘A beder
Fall ist“, oder „A denkt p“, etc. is the case”, or “A thinks p”, etc. lieves that p is the case’ and A has the
thought p’, etc.
Hier scheint es nämlich oberflächlich, Here it appears superficially as if the For if these are considered superfials
stünde der Satz p zu einem Gegen- proposition p stood to the object A in a cially, it looks as if the proposition p stood
stand A in einer Art von Relation. kind of relation. in some kind of relation to an object A.
(Und in der modernen Erkenntnis- (And in modern epistemology (Russell, (And in modern theory of knowledge
theorie (Russell, Moore, etc.) sind jene Moore, etc.) those propositions have been (Russell, Moore, etc.) these propositions
Sätze auch so aufgefasst worden.) conceived in this way.) have actually been construed in this way.)
5.542 Es ist aber klar, dass „A glaubt, dass But it is clear that “A believes that It is clear, however, that ‘A believes
p“, „A denkt p“, „A sagt p“ von der Form p”, “A thinks p”, “A says p”, are of the that p’, ‘A has the thought p’, and ‘A says
„‚p‘ sagt p“ sind: Und hier handelt es sich form “‘p’ says p”: and here we have no p’ are of the form ‘“p” says p’: and this
nicht um eine Zuordnung von einer Tatsa- co-ordination of a fact and an object, but does not involve a correlation of a fact
che und einem Gegenstand, sondern um a co-ordination of facts by means of a co- with an object, but rather the correlation
die Zuordnung von Tatsachen durch Zu- ordination of their objects. of facts by means of the correlation of
ordnung ihrer Gegenstände. their objects.
5.5421 Dies zeigt auch, dass die Seele—das This shows that there is no such thing This shows too that there is no such
Subjekt etc.—wie sie in der heutigen ober- as the soul—the subject, etc.—as it is con- thing as the soul—the subject, etc.—as it
flächlichen Psychologie aufgefasst wird, ceived in superficial psychology. is conceived in the superficial psychology
ein Unding ist. of the present day.
Eine zusammengesetzte Seele wäre A composite soul would not be a soul Indeed a composite soul would no
nämlich keine Seele mehr. any longer. longer be a soul.
5.5422 Die richtige Erklärung der Form des The correct explanation of the form of The correct explanation of the form
Satzes „A urteilt p“ muss zeigen, dass the proposition “A judges p” must show of the proposition, ‘A makes the judgees
unmöglich ist, einen Unsinn zu urtei- that it is impossible to judge a nonsense. ment p’, must show that it is impossible
len. (Russells Theorie genügt dieser Be- (Russell’s theory does not satisfy this con- for a judgement to be a piece of nonsense.
dingung nicht.) dition.) (Russell’s theory does not satisfy this re-
quirement.)
5.5423 Einen Komplex wahrnehmen heißt To perceive a complex means to per- To perceive a complex means to perwahrnehmen,
dass sich seine Bestandtei- ceive that its constituents are combined ceive that its constituents are related to
83
le so und so zu einander verhalten. in such and such a way. one another in such and such a way.
Dies erklärt wohl auch, dass man die This perhaps explains that the figure This no doubt also explains why there
Figur are two possible ways of seeing the figure
a
a
a
a
b
b b
b
a
a
a
a
b
b b
b
a
a
a
a
b
b b
b
auf zweierlei Art als Würfel sehen kann; can be seen in two ways as a cube; and as a cube; and all similar phenomena. For
und alle ähnlichen Erscheinungen. Denn all similar phenomena. For we really see we really see two different facts.
wir sehen eben wirklich zwei verschiede- two different facts.
ne Tatsachen.
(Sehe ich erst auf die Ecken a und nur (If I fix my eyes first on the corners a (If I look in the first place at the corflüchtig
auf b, so erscheint a vorne; und and only glance at b, a appears in front ners marked a and only glance at the b’s,
umgekehrt.) and b behind, and vice versa.) then the a’s appear to be in front, and vice
versa).
5.55 Wir müssen nun die Frage nach allen We must now answer a priori the ques- We now have to answer a priori the
möglichen Formen der Elementarsätze a tion as to all possible forms of the elemen- question about all the possible forms of
priori beantworten. tary propositions. elementary propositions.
Der Elementarsatz besteht aus Na- The elementary proposition consists Elementary propositions consist of
men. Da wir aber die Anzahl der Namen of names. Since we cannot give the num- names. Since, however, we are unable to
von verschiedener Bedeutung nicht ange- ber of names with different meanings, we give the number of names with different
ben können, so können wir auch nicht die cannot give the composition of the elemen- meanings, we are also unable to give the
Zusammensetzung des Elementarsatzes tary proposition. composition of elementary propositions.
angeben.
5.551 Unser Grundsatz ist, dass jede Fra- Our fundamental principle is that ev- Our fundamental principle is that
ge, die sich überhaupt durch die Logik ery question which can be decided at all whenever a question can be decided by
entscheiden läßt, sich ohne weiteres ent- by logic can be decided without further logic at all it must be possible to decide it
scheiden lassen muss. trouble. without more ado.
(Und wenn wir in die Lage kommen, (And if we get into a situation where (And if we get into a position where
ein solches Problem durch Ansehen der we need to answer such a problem by look- we have to look at the world for an answer
Welt beantworten zu müssen, so zeigt ing at the world, this shows that we are to such a problem, that shows that we are
dies, dass wir auf grundfalscher Fährte on a fundamentally wrong track.) on a completely wrong track.)
sind.)
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5.552 Die „Erfahrung“, die wir zum Verste- The “experience” which we need to un- The ‘experience’ that we need in order
hen der Logik brauchen, ist nicht die, derstand logic is not that such and such is to understand logic is not that something
dass sich etwas so und so verhält, son- the case, but that something is; but that or other is the state of things, but that
dern, dass etwas i s t: aber das ist eben is no experience. something is: that, however, is not an
k e i n e Erfahrung. experience.
Die Logik ist v o r jeder Erfahrung— Logic precedes every experience—that Logic is prior to every experience—
dass etwas s o ist. something is so. that something is so.
Sie ist vor dem Wie, nicht vor dem It is before the How, not before the It is prior to the question ‘How?’, not
Was. What. prior to the question ‘What?’
5.5521 Und wenn dies nicht so wäre, wie And if this were not the case, how And if this were not so, how could we
könnten wir die Logik anwenden? Man could we apply logic? We could say: if apply logic? We might put it in this way:
könnte sagen: Wenn es eine Logik gäbe, there were a logic, even if there were no if there would be a logic even if there were
auch wenn es keine Welt gäbe, wie könnte world, how then could there be a logic, no world, how then could there be a logic
es dann eine Logik geben, da es eine Welt since there is a world? given that there is a world?
gibt?
5.553 Russell sagte, es gäbe einfache Rela- Russell said that there were sim- Russell said that there were simple
tionen zwischen verschiedenen Anzahlen ple relations between different numbers relations between different numbers of
von Dingen (Individuals). Aber zwischen of things (individuals). But between things (individuals). But between what
welchen Anzahlen? Und wie soll sich das what numbers? And how should this be numbers? And how is this supposed to be
entscheiden?—Durch die Erfahrung? decided—by experience? decided?—By experience?
(Eine ausgezeichnete Zahl gibt es (There is no pre-eminent number.) (There is no privileged number.)
nicht.)
5.554 Die Angabe jeder speziellen Form wä- The enumeration of any special forms It would be completely arbitrary to
re vollkommen willkürlich. would be entirely arbitrary. give any specific form.
5.5541 Es soll sich a priori angeben lassen, How could we decide a priori whether, It is supposed to be possible to answer
ob ich z. B. in die Lage kommen kann, for example, I can get into a situation in a priori the question whether I can get
etwas mit dem Zeichen einer 27-stelligen which I need to symbolize with a sign of a into a position in which I need the sign
Relation bezeichnen zu müssen. 27-termed relation? for a 27-termed relation in order to signify
something.
5.5542 Dürfen wir denn aber überhaupt so May we then ask this at all? Can we But is it really legitimate even to ask
fragen? Können wir eine Zeichenform auf- set out a sign form and not know whether such a question? Can we set up a form of
stellen und nicht wissen, ob ihr etwas ent- anything can correspond to it? sign without knowing whether anything
sprechen könne? can correspond to it?
Hat die Frage einen Sinn: Was muss Has the question sense: what must Does it make sense to ask what there
s e i n, damit etwas der-Fall-sein kann? there be in order that anything can be the must be in order that something can be
case? the case?
5.555 Es ist klar, wir haben vom Elementar- It is clear that we have a concept of Clearly we have some concept of el-
85
satz einen Begriff, abgesehen von seiner the elementary proposition apart from its ementary propositions quite apart from
besonderen logischen Form. special logical form. their particular logical forms.
Wo man aber Symbole nach einem Sy- Where, however, we can build symbols But when there is a system by which
stem bilden kann, dort ist dieses System according to a system, there this system we can create symbols, the system is what
das logisch wichtige und nicht die einzel- is the logically important thing and not is important for logic and not the individnen
Symbole. the single symbols. ual symbols.
Und wie wäre es auch möglich, dass And how would it be possible that I And anyway, is it really possible that
ich es in der Logik mit Formen zu tun should have to deal with forms in logic in logic I should have to deal with forms
hätte, die ich erfinden kann; sondern mit which I can invent: but I must have to that I can invent? What I have to deal
dem muss ich es zu tun haben, was es mir deal with that which makes it possible for with must be that which makes it possible
möglich macht, sie zu erfinden. me to invent them. for me to invent them.
5.556 Eine Hierarchie der Formen der Ele- There cannot be a hierarchy of the There cannot be a hierarchy of the
mentarsätze kann es nicht geben. Nur forms of the elementary propositions. forms of elementary propositions. We can
was wir selbst konstruieren, können wir Only that which we ourselves construct foresee only what we ourselves construct.
voraussehen. can we foresee.
5.5561 Die empirische Realität ist begrenzt Empirical reality is limited by the to- Empirical reality is limited by the todurch
die Gesamtheit der Gegenstände. tality of objects. The boundary appears tality of objects. The limit also makes itDie
Grenze zeigt sich wieder in der Ge- again in the totality of elementary propo- self manifest in the totality of elementary
samtheit der Elementarsätze. sitions. propositions.
Die Hierarchien sind, und müssen un- The hierarchies are and must be inde- Hierarchies are and must be indepenabhängig
von der Realität sein. pendent of reality. dent of reality.
5.5562 Wissen wir aus rein logischen Grün- If we know on purely logical grounds, If we know on purely logical grounds
den, dass es Elementarsätze geben muss, that there must be elementary proposi- that there must be elementary proposidann
muss es jeder wissen, der die Sätze tions, then this must be known by ev- tions, then everyone who understands
in ihrer unanalysierten Form versteht. eryone who understands propositions in propositions in their unanalyzed form
their unanalysed form. must know it.
5.5563 Alle Sätze unserer Umgangssprache All propositions of our colloquial lan- In fact, all the propositions of our evsind
tatsächlich, so wie sie sind, logisch guage are actually, just as they are, log- eryday language, just as they stand, are
vollkommen geordnet.—Jenes Einfachste, ically completely in order. That simple in perfect logical order.—That utterly simwas
wir hier angeben sollen, ist nicht ein thing which we ought to give here is not a ple thing, which we have to formulate
Gleichnis der Wahrheit, sondern die volle model of the truth but the complete truth here, is not an image of the truth, but
Wahrheit selbst. itself. the truth itself in its entirety.
(Unsere Probleme sind nicht abstrakt, (Our problems are not abstract but (Our problems are not abstract, but
sondern vielleicht die konkretesten, die perhaps the most concrete that there are.) perhaps the most concrete that there are.)
es gibt.)
5.557 Die A n w e n d u n g der Logik ent- The application of logic decides what The application of logic decides what
scheidet darüber, welche Elementarsätze elementary propositions there are. elementary propositions there are.
86
es gibt.
Was in der Anwendung liegt, kann die What lies in its application logic can- What belongs to its application, logic
Logik nicht vorausnehmen. not anticipate. cannot anticipate.
Das ist klar: Die Logik darf mit ihrer It is clear that logic may not conflict It is clear that logic must not clash
Anwendung nicht kollidieren. with its application. with its application.
Aber die Logik muss sich mit ihrer But logic must have contact with its But logic has to be in contact with its
Anwendung berühren. application. application.
Also dürfen die Logik und ihre Anwen- Therefore logic and its application Therefore logic and its application
dung einander nicht übergreifen. may not overlap one another. must not overlap.
5.5571 Wenn ich die Elementarsätze nicht a If I cannot give elementary proposi- If I cannot say a priori what elemenpriori
angeben kann, dann muss es zu tions a priori then it must lead to obvious tary propositions there are, then the atoffenbarem
Unsinn führen, sie angeben nonsense to try to give them. tempt to do so must lead to obvious nonzu
wollen. sense.
5.6 D i e G r e n z e n m e i n e r S p r a - The limits of my language mean the The limits of my language mean the
c h e bedeuten die Grenzen meiner Welt. limits of my world. limits of my world.
5.61 Die Logik erfüllt die Welt; die Grenzen Logic fills the world: the limits of the Logic pervades the world: the limits
der Welt sind auch ihre Grenzen. world are also its limits. of the world are also its limits.
Wir können also in der Logik nicht We cannot therefore say in logic: This So we cannot say in logic, ‘The world
sagen: Das und das gibt es in der Welt, and this there is in the world, that there has this in it, and this, but not that.’
jenes nicht. is not.
Das würde nämlich scheinbar voraus- For that would apparently presuppose For that would appear to presuppose
setzen, dass wir gewisse Möglichkeiten that we exclude certain possibilities, and that we were excluding certain possibiliausschließen,
und dies kann nicht der this cannot be the case since otherwise ties, and this cannot be the case, since it
Fall sein, da sonst die Logik über die logic must get outside the limits of the would require that logic should go beyond
Grenzen der Welt hinaus müsste; wenn world: that is, if it could consider these the limits of the world; for only in that
sie nämlich diese Grenzen auch von der limits from the other side also. way could it view those limits from the
anderen Seite betrachten könnte. other side as well.
Was wir nicht denken können, das What we cannot think, that we cannot We cannot think what we cannot
können wir nicht denken; wir können also think: we cannot therefore say what we think; so what we cannot think we cannot
auch nicht s a g e n, was wir nicht denken cannot think. say either.
können.
5.62 Diese Bemerkung gibt den Schlüssel This remark provides a key to the This remark provides the key to the
zur Entscheidung der Frage, inwieweit question, to what extent solipsism is a problem, how much truth there is in solipder
Solipsismus eine Wahrheit ist. truth. sism.
Was der Solipsismus nämlich In fact what solipsism means, is quite For what the solipsist means is quite
m e i n t, ist ganz richtig, nur lässt es sich correct, only it cannot be said, but it correct; only it cannot be said, but makes
nicht s a g e n, sondern es zeigt sich. shows itself. itself manifest.
87
Dass die Welt m e i n e Welt ist, das That the world is my world, shows it- The world is my world: this is manizeigt
sich darin, dass die Grenzen d e r self in the fact that the limits of the lan- fest in the fact that the limits of language
Sprache (der Sprache, die allein ich ver- guage (the language which only I under- (of that language which alone I understehe)
die Grenzen m e i n e r Welt be- stand) mean the limits of my world. stand) mean the limits of my world.
deuten.
5.621 Die Welt und das Leben sind Eins. The world and life are one. The world and life are one.
5.63 Ich bin meine Welt. (Der Mikrokos- I am my world. (The microcosm.) I am my world. (The microcosm.)
mos.)
5.631 Das denkende, vorstellende, Subjekt The thinking, presenting subject; There is no such thing as the subject
gibt es nicht. there is no such thing. that thinks or entertains ideas.
Wenn ich ein Buch schriebe „Die Welt, If I wrote a book “The world as I found If I wrote a book called The World as I
wie ich sie vorfand“, so wäre darin auch it”, I should also have therein to report found it, I should have to include a report
über meinen Leib zu berichten und zu on my body and say which members obey on my body, and should have to say which
sagen, welche Glieder meinem Willen un- my will and which do not, etc. This then parts were subordinate to my will, and
terstehen und welche nicht, etc., dies ist would be a method of isolating the subject which were not, etc., this being a method
nämlich eine Methode, das Subjekt zu iso- or rather of showing that in an important of isolating the subject, or rather of showlieren,
oder vielmehr zu zeigen, dass es in sense there is no subject: that is to say, of ing that in an important sense there is
einem wichtigen Sinne kein Subjekt gibt: it alone in this book mention could not be no subject; for it alone could not be menVon
ihm allein nämlich könnte in diesem made. tioned in that book.—
Buche n i c h t die Rede sein.—
5.632 Das Subjekt gehört nicht zur Welt, The subject does not belong to the The subject does not belong to the
sondern es ist eine Grenze der Welt. world but it is a limit of the world. world: rather, it is a limit of the world.
5.633 Wo i n der Welt ist ein metaphysi- Where in the world is a metaphysical Where in the world is a metaphysical
sches Subjekt zu merken? subject to be noted? subject to be found?
Du sagst, es verhält sich hier ganz You say that this case is altogether You will say that this is exactly like
wie mit Auge und Gesichtsfeld. Aber das like that of the eye and the field of sight. the case of the eye and the visual field.
Auge siehst du wirklich n i c h t. But you do not really see the eye. But really you do not see the eye.
Und nichts a m G e s i c h t s f e l d And from nothing in the field of sight And nothing in the visual field allows
lässt darauf schließen, dass es von einem can it be concluded that it is seen from an you to infer that it is seen by an eye.
Auge gesehen wird. eye.
5.6331 Das Gesichtsfeld hat nämlich nicht For the field of sight has not a form For the form of the visual field is
etwa eine solche Form: like this: surely not like this
Auge — Eye — Eye —
5.634 Das hängt damit zusammen, dass This is connected with the fact that no This is connected with the fact that no
88
kein Teil unserer Erfahrung auch a priori part of our experience is also a priori. part of our experience is at the same time
ist. a priori.
Alles, was wir sehen, könnte auch an- Everything we see could also be other- Whatever we see could be other than
ders sein. wise. it is.
Alles, was wir überhaupt beschreiben Everything we describe at all could Whatever we can describe at all could
können, könnte auch anders sein. also be otherwise. be other than it is.
Es gibt keine Ordnung der Dinge a There is no order of things a priori. There is no a priori order of things.
priori.
5.64 Hier sieht man, dass der Solipsismus, Here we see that solipsism strictly car- Here it can be seen that solipsism,
streng durchgeführt, mit dem reinen Rea- ried out coincides with pure realism. The when its implications are followed out
lismus zusammenfällt. Das Ich des Solip- I in solipsism shrinks to an extensionless strictly, coincides with pure realism. The
sismus schrumpft zum ausdehnungslosen point and there remains the reality co- self of solipsism shrinks to a point withPunkt
zusammen, und es bleibt die ihm ordinated with it. out extension, and there remains the rekoordinierte
Realität. ality co-ordinated with it.
5.641 Es gibt also wirklich einen Sinn, in There is therefore really a sense in Thus there really is a sense in which
welchem in der Philosophie nichtpsycho- which the philosophy we can talk of a non- philosophy can talk about the self in a
logisch vom Ich die Rede sein kann. psychological I. non-psychological way.
Das Ich tritt in die Philosophie da- The I occurs in philosophy through the What brings the self into philosophy
durch ein, dass „die Welt meine Welt ist“. fact that the “world is my world”. is the fact that ‘the world is my world’.
Das philosophische Ich ist nicht der The philosophical I is not the man, The philosophical self is not the huMensch,
nicht der menschliche Körper, not the human body or the human soul man being, not the human body, or the
oder die menschliche Seele, von der die of which psychology treats, but the meta- human soul, with which psychology deals,
Psychologie handelt, sondern das meta- physical subject, the limit—not a part of but rather the metaphysical subject, the
physische Subjekt, die Grenze—nicht ein the world. limit of the world—not a part of it.
Teil—der Welt.
6 Die allgemeine Form der Wahrheits- The general form of truth-function is: The general form of a truth-function
funktion ist: [p, ξ, N(ξ)]. [p, ξ, N(ξ)]. is [p, ξ, N(ξ)].
Dies ist die allgemeine Form des Sat- This is the general form of proposi- This is the general form of a proposizes.
tion. tion.
6.001 Dies sagt nichts anderes, als dass je- This says nothing else than that ev- What this says is just that every
der Satz ein Resultat der successiven An- ery proposition is the result of successive proposition is a result of successive apwendung
der Operation N’(ξ) auf die Ele- applications of the operation N’(ξ) to the plications to elementary propositions of
mentarsätze ist. elementary propositions. the operation N(ξ).
6.002 Ist die allgemeine Form gegeben, wie If we are given the general form of the If we are given the general form acein
Satz gebaut ist, so ist damit auch way in which a proposition is constructed, cording to which propositions are conschon
die allgemeine Form davon gege- then thereby we are also given the gen- structed, then with it we are also given
ben, wie aus einem Satz durch eine Ope- eral form of the way in which by an oper- the general form according to which one
89
ration ein anderer erzeugt werden kann. ation out of one proposition another can proposition can be generated out of anbe
created. other by means of an operation.
6.01 Die allgemeine Form der Operation The general form of the opera- Therefore the general form of an operΩ’(η)
ist also: [ξ, N(ξ)]’(η) (= [η, ξ, N(ξ)]). tion Ω’(η) is therefore: [ξ, N(ξ)]’(η) (= ation Ω’(η) is [ξ, N(ξ)]’(η) (= [η, ξ, N(ξ)]).
[η, ξ, N(ξ)]).
Das ist die allgemeinste Form des This is the most general form of tran- This is the most general form of tranÜberganges
von einem Satz zum ande- sition from one proposition to another. sition from one proposition to another.
ren.
6.02 Und s o kommen wir zu den Zahlen: And thus we come to numbers: I de- And this is how we arrive at numbers.
Ich definiere fine I give the following definitions
x = Ω0
’x Def. und
Ω’Ων
’x = Ων+1
’x Def.
x = Ω0
’x Def. and
Ω’Ων
’x = Ων+1
’x Def.
x = Ω0
’x Def.
Ω’Ων
’x = Ων+1
’x Def.
Nach diesen Zeichenregeln schreiben According, then, to these symbolic So, in accordance with these rules,
wir also die Reihe rules we write the series which deal with signs, we write the se-
ries
x, Ω’x, Ω’Ω’x, Ω’Ω’Ω’x, ... x, Ω’x, Ω’Ω’x, Ω’Ω’Ω’x, ... x, Ω’x, Ω’Ω’x, Ω’Ω’Ω’x, ...
so: as: in the following way
Ω0
’x, Ω0+1
’x, Ω0+1+1
’x, Ω0+1+1+1
’x, ... Ω0
’x, Ω0+1
’x, Ω0+1+1
’x, Ω0+1+1+1
’x, ... Ω0
’x, Ω0+1
’x, Ω0+1+1
’x, Ω0+1+1+1
’x, ...
Also schreibe ich statt „[x, ξ, Ω’ξ]“: Therefore I write in place of “[x, ξ, Therefore, instead of ‘[x, ξ, Ω’ξ]’,
Ω’ξ]”,
„[Ω0
’x, Ων
’x, Ων+1
’x]“. “[Ω0
’x, Ων
’x, Ων+1
’x]”. I write ‘[Ω0
’x, Ων
’x, Ων+1
’x]’.
Und definiere: And I define: And I give the following definitions
0+1 = 1 Def.
0+1+1 = 2 Def.
0+1+1+1 = 3 Def.
(u. s. f.)
0+1 = 1 Def.
0+1+1 = 2 Def.
0+1+1+1 = 3 Def.
and so on.
0+1 = 1 Def.,
0+1+1 = 2 Def.,
0+1+1+1 = 3 Def.,
(and so on).
6.021 Die Zahl ist der Exponent einer Ope- A number is the exponent of an opera- A number is the exponent of an operaration.
tion. tion.
6.022 Der Zahlbegriff ist nichts anderes als The concept number is nothing else The concept of number is simply what
das Gemeinsame aller Zahlen, die allge- than that which is common to all num- is common to all numbers, the general
meine Form der Zahl. bers, the general form of a number. form of a number.
Der Zahlbegriff ist die variable Zahl. The concept number is the variable The concept of number is the variable
number. number.
Und der Begriff der Zahlengleichheit And the concept of equality of num- And the concept of numerical equality
90
ist die allgemeine Form aller speziellen bers is the general form of all special is the general form of all particular cases
Zahlengleichheiten. equalities of numbers. of numerical equality.
6.03 Die allgemeine Form der ganzen Zahl The general form of the cardinal num- The general form of an integer is
ist: [0,ξ,ξ+1]. ber is: [0,ξ,ξ+1]. [0,ξ,ξ+1].
6.031 Die Theorie der Klassen ist in der Ma- The theory of classes is altogether su- The theory of classes is completely suthematik
ganz überflüssig. perfluous in mathematics. perfluous in mathematics.
Dies hängt damit zusammen, dass die This is connected with the fact that This is connected with the fact that
Allgemeinheit, welche wir in der Mathe- the generality which we need in mathe- the generality required in mathematics is
matik brauchen, nicht die z u f ä l l i g e matics is not the accidental one. not accidental generality.
ist.
6.1 Die Sätze der Logik sind Tautologien. The propositions of logic are tautolo- The propositions of logic are tautologies.
gies.
6.11 Die Sätze der Logik sagen also nichts. The propositions of logic therefore say Therefore the propositions of logic say
(Sie sind die analytischen Sätze.) nothing. (They are the analytical proposi- nothing. (They are the analytic propositions.)
tions.)
6.111 Theorien, die einen Satz der Logik ge- Theories which make a proposition All theories that make a proposition
haltvoll erscheinen lassen, sind immer of logic appear substantial are always of logic appear to have content are false.
falsch. Man könnte z. B. glauben, dass false. One could e.g. believe that the One might think, for example, that the
die Worte „wahr“ und „falsch“ zwei Ei- words “true” and “false” signify two prop- words ‘true’ and ‘false’ signified two propgenschaften
unter anderen Eigenschaf- erties among other properties, and then erties among other properties, and then it
ten bezeichnen, und da erschiene es als it would appear as a remarkable fact that would seem to be a remarkable fact that
eine merkwürdige Tatsache, dass jeder every proposition possesses one of these every proposition possessed one of these
Satz eine dieser Eigenschaften besitzt. properties. This now by no means ap- properties. On this theory it seems to be
Das scheint nun nichts weniger als selbst- pears self-evident, no more so than the anything but obvious, just as, for instance,
verständlich zu sein, ebensowenig selbst- proposition “All roses are either yellow or the proposition, ‘All roses are either yelverständlich,
wie etwa der Satz: „Alle Ro- red” would seem even if it were true. In- low or red’, would not sound obvious even
sen sind entweder gelb oder rot“ klänge, deed our proposition now gets quite the if it were true. Indeed, the logical propoauch
wenn er wahr wäre. Ja, jener Satz character of a proposition of natural sci- sition acquires all the characteristics of a
bekommt nun ganz den Charakter eines ence and this is a certain symptom of its proposition of natural science and this is
naturwissenschaftlichen Satzes, und dies being falsely understood. the sure sign that it has been construed
ist das sichere Anzeichen dafür, dass er wrongly.
falsch aufgefasst wurde.
6.112 Die richtige Erklärung der logischen The correct explanation of logical The correct explanation of the propoSätze
muss ihnen eine einzigartige Stel- propositions must give them a peculiar sitions of logic must assign to them a
lung unter allen Sätzen geben. position among all propositions. unique status among all propositions.
6.113 Es ist das besondere Merkmal der logi- It is the characteristic mark of logi- It is the peculiar mark of logical proposchen
Sätze, dass man am Symbol allein cal propositions that one can perceive in sitions that one can recognize that they
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erkennen kann, dass sie wahr sind, und the symbol alone that they are true; and are true from the symbol alone, and this
diese Tatsache schließt die ganze Philoso- this fact contains in itself the whole phi- fact contains in itself the whole philosophie
der Logik in sich. Und so ist es auch losophy of logic. And so also it is one of phy of logic. And so too it is a very imeine
derwichtigsten Tatsachen, dass sich the most important facts that the truth or portant fact that the truth or falsity of
die Wahrheit oder Falschheit der nicht- falsehood of non-logical propositions can non-logical propositions cannot be recoglogischen
Sätze n i c h t am Satz allein not be recognized from the propositions nized from the propositions alone.
erkennen lässt. alone.
6.12 Dass die Sätze der Logik Tautolo- The fact that the propositions of The fact that the propositions of
gien sind, das z e i g t die formalen— logic are tautologies shows the formal— logic are tautologies shows the formal—
logischen—Eigenschaften der Sprache, logical—properties of language, of the logical—properties of language and the
der Welt. world. world.
Dass ihre Bestandteile s o verknüpft That its constituent parts connected The fact that a tautology is yielded
eine Tautologie ergeben, das charakteri- together in this way give a tautology char- by this particular way of connecting its
siert die Logik ihrer Bestandteile. acterizes the logic of its constituent parts. constituents characterizes the logic of its
constituents.
Damit Sätze, auf bestimmte Art und In order that propositions connected If propositions are to yield a tautolWeise
verknüpft, eine Tautologie ergeben, together in a definite way may give a tau- ogy when they are connected in a certain
dazu müssen sie bestimmte Eigenschaf- tology they must have definite properties way, they must have certain structural
ten der Struktur haben. Dass sie s o ver- of structure. That they give a tautology properties. So their yielding a tautology
bunden eine Tautologie ergeben, zeigt al- when so connected shows therefore that when combined in this way shows that
so, dass sie diese Eigenschaften der Struk- they possess these properties of structure. they possess these structural properties.
tur besitzen.
6.1201 Dass z. B. die Sätze „p“ und „∼p“ in That e.g. the propositions “p” and “∼p” For example, the fact that the propoder
Verbindung „∼(p . ∼p)“ eine Tautolo- in the connexion “∼(p . ∼p)” give a tau- sitions ‘p’ and ‘∼p’ in the combination
gie ergeben, zeigt, dass sie einander wi- tology shows that they contradict one an- ‘∼(p . ∼p)’ yield a tautology shows that
dersprechen. Dass die Sätze „p ⊃ q“, „p“ other. That the propositions “p ⊃ q”, “p” they contradict one another. The fact
und „q“ in der Form „(p ⊃ q) . (p) :⊃: (q)“ and “q” connected together in the form that the propositions ‘p ⊃ q’, ‘p’, and ‘q’,
miteinander verbunden eine Tautologie “(p ⊃ q) . (p) :⊃: (q)” give a tautology shows combined with one another in the form
ergeben, zeigt, dass q aus p und p ⊃ q that q follows from p and p ⊃ q. That ‘(p ⊃ q) . (p) :⊃: (q)’, yield a tautology
folgt. Dass „(x). fx :⊃: fa“ eine Tautologie “(x). fx :⊃: fa” is a tautology shows that shows that q follows from p and p ⊃ q.
ist, dass fa aus (x). fx folgt. etc. etc. fa follows from (x). fx, etc. etc. The fact that ‘(x). fx :⊃: fa’ is a tautology
shows that fa follows from (x). fx. Etc.
etc.
6.1202 Es ist klar, dass man zu demselben It is clear that we could have used It is clear that one could achieve the
Zweck statt der Tautologien auch die Kon- for this purpose contradictions instead of same purpose by using contradictions intradiktionen
verwenden könnte. tautologies. stead of tautologies.
6.1203 Um eine Tautologie als solche zu er- In order to recognize a tautology as In order to recognize an expression as
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kennen, kann man sich, in den Fällen, such, we can, in cases in which no sign of a tautology, in cases where no generalityin
welchen in der Tautologie keine All- generality occurs in the tautology, make sign occurs in it, one can employ the folgemeinheitsbezeichnung
vorkommt, fol- use of the following intuitive method: lowing intuitive method: instead of ‘p’,
gender anschaulichen Methode bedienen: I write instead of “p”, “q”, “r”, etc., ‘q’, ‘r’, etc. I write ‘T pF’, ‘T qF’, ‘TrF’, etc.
Ich schreibe statt „p“, „q“, „r“ etc. „W pF“, “T pF”, “T qF”, “TrF”, etc. The truth- Truth-combinations I express by means
„W qF“, „WrF“ etc. Die Wahrheitskombi- combinations I express by brackets, e.g.: of brackets, e.g.
nationen drücke ich durch Klammern aus,
z. B.:
W p F W q F T p F T q F T p F T q F,
und die Zuordnung der Wahr- oder Falsch- and the co-ordination of the truth or and I use lines to express the correlation
heit des ganzen Satzes und der Wahr- falsity of the whole proposition with of the truth or falsity of the whole propoheitskombinationen
der Wahrheitsargu- the truth-combinations of the truth- sition with the truth-combinations of its
mente durch Striche auf folgende Weise: arguments by lines in the following way: truth-arguments, in the following way
W p F W q F
F
W
T p F T q F
F
T
T p F T q F.
F
T
Dies Zeichen würde also z. B. den Satz This sign, for example, would therefore So this sign, for instance, would represent
p ⊃ q darstellen. Nun will ich z. B. den present the proposition p ⊃ q. Now I will the proposition p ⊃ q. Now, by way of exSatz
∼(p . ∼p) (Gesetz des Widerspruchs) proceed to inquire whether such a propo- ample, I wish to examine the proposition
daraufhin untersuchen, ob er eine Tauto- sition as ∼(p . ∼p) (The Law of Contra- ∼(p . ∼p) (the law of contradiction) in orlogie
ist. Die Form „∼ξ“ wird in unserer diction) is a tautology. The form “∼ξ” is der to determine whether it is a tautology.
Notation written in our notation In our notation the form ‘∼ξ’ is written as
93
„W ξ F“
W
F
“T ξ F”
T
F
‘T ξ F’,
T
F
geschrieben; die Form „ξ . η“ so: the form “ξ . η” thus:— and the form ‘ξ . η’ as
W ξ F W η F
F
W
T ξ F T η F
F
T
T ξ F T η F.
F
T
Daher lautet der Satz ∼(p . ∼q) so: Hence the proposition ∼(p . ∼q) runs Hence the proposition ∼(p . ∼q) reads as
thus:— follows
W q F W p F
W
F
W
F
F
W
T q F T p F
T
F
T
F
F
T
T q F T p F.
T
F
T
F
F
T
Setzen wir statt „q“ „p“ ein und unter- If here we put “p” instead of “q” and ex- If we here substitute ‘p’ for ‘q’ and
suchen die Verbindung der äußersten W amine the combination of the outermost T examine how the outermost T and F
und F mit den innersten, so ergibt sich, and F with the innermost, it is seen that are connected with the innermost ones,
dass die Wahrheit des ganzen Satzes a l - the truth of the whole proposition is co- the result will be that the truth of the
l e n Wahrheitskombinationen seines Ar- ordinated with all the truth-combinations whole proposition is correlated with all
gumentes, seine Falschheit keiner der of its argument, its falsity with none of the truth-combinations of its argument,
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Wahrheitskombinationen zugeordnet ist. the truth-combinations. and its falsity with none of the truth-
combinations.
6.121 Die Sätze der Logik demonstrieren die The propositions of logic demonstrate The propositions of logic demonstrate
logischen Eigenschaften der Sätze, indem the logical properties of propositions, by the logical properties of propositions by
sie sie zu nichtssagenden Sätzen verbin- combining them into propositions which combining them so as to form propositions
den. say nothing. that say nothing.
Diese Methode könnte man auch eine This method could be called a zero- This method could also be called a
Nullmethode nennen. Im logischen Satz method. In a logical proposition proposi- zero-method. In a logical proposition,
werden Sätze miteinander ins Gleich- tions are brought into equilibrium with propositions are brought into equilibrium
gewicht gebracht und der Zustand des one another, and the state of equilibrium with one another, and the state of equiGleichgewichts
zeigt dann an, wie diese then shows how these propositions must librium then indicates what the logical
Sätze logisch beschaffen sein müssen. be logically constructed. constitution of these propositions must
be.
6.122 Daraus ergibt sich, dass wir auch oh- Whence it follows that we can get on It follows from this that we can actune
die logischen Sätze auskommen kön- without logical propositions, for we can ally do without logical propositions; for
nen, da wir ja in einer entsprechenden recognize in an adequate notation the in a suitable notation we can in fact recNotation
die formalen Eigenschaften der formal properties of the propositions by ognize the formal properties of proposiSätze
durch das bloße Ansehen dieser Sät- mere inspection. tions by mere inspection of the proposize
erkennen können. tions themselves.
6.1221 Ergeben z. B. zwei Sätze „p“ und „q“ If for example two propositions “p” If, for example, two propositions ‘p’
in der Verbindung „p ⊃ q“ eine Tautologie, and “q” give a tautology in the connex- and ‘q’ in the combination ‘p ⊃ q’ yield a
so ist klar, dass q aus p folgt. ion “p ⊃ q”, then it is clear that q follows tautology, then it is clear that q follows
from p. from p.
Dass z. B. „q“ aus „p ⊃ q . p“ folgt, er- E.g. that “q” follows from “p ⊃ q . p” For example, we see from the two
sehen wir aus diesen beiden Sätzen selbst, we see from these two propositions them- propositions themselves that ‘q’ follows
aber wir können es auch s o zeigen, in- selves, but we can also show it by com- from ‘p ⊃ q . p’, but it is also possible to
dem wir sie zu „p ⊃ q . p :⊃: q“ verbinden bining them to “p ⊃ q . p :⊃: q” and then show it in this way: we combine them to
und nun zeigen, dass dies eine Tautologie showing that this is a tautology. form ‘p ⊃ q . p :⊃: q’, and then show that
ist. this is a tautology.
6.1222 Dies wirft ein Licht auf die Frage, This throws light on the question why This throws some light on the queswarum
die logischen Sätze nicht durch logical propositions can no more be empir- tion why logical propositions cannot be
die Erfahrung bestätigt werden können, ically established than they can be empiri- confirmed by experience any more than
ebensowenig wie sie durch die Erfah- cally refuted. Not only must a proposition they can be refuted by it. Not only must
rung widerlegt werden können. Nicht nur of logic be incapable of being contradicted a proposition of logic be irrefutable by
muss ein Satz der Logik durch keine mög- by any possible experience, but it must any possible experience, but it must also
liche Erfahrung widerlegt werden kön- also be incapable of being established by be unconfirmable by any possible experinen,
sondern er darf auch nicht durch any such. ence.
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eine solche bestätigt werden können.
6.1223 Nun wird klar, warum man oft fühl- It now becomes clear why we often Now it becomes clear why people have
te, als wären die „logischen Wahrheiten“ feel as though “logical truths” must be often felt as if it were for us to ‘postulate’
von uns zu „f o r d e r n“: Wir können sie “postulated” by us. We can in fact postu- the ‘truths of logic’. The reason is that
nämlich insofern fordern, als wir eine ge- late them in so far as we can postulate an we can postulate them in so far as we can
nügende Notation fordern können. adequate notation. postulate an adequate notation.
6.1224 Es wird jetzt auch klar, warum die It also becomes clear why logic has It also becomes clear now why logic
Logik die Lehre von den Formen und vom been called the theory of forms and of was called the theory of forms and of inSchließen
genannt wurde. inference. ference.
6.123 Es ist klar: Die logischen Gesetze dür- It is clear that the laws of logic cannot Clearly the laws of logic cannot in
fen nicht selbst wieder logischen Geset- themselves obey further logical laws. their turn be subject to laws of logic.
zen unterstehen.
(Es gibt nicht, wie Russell meinte, für (There is not, as Russell supposed, for (There is not, as Russell thought, a
jede „Type“ ein eigenes Gesetz des Wider- every “type” a special law of contradiction; special law of contradiction for each ‘type’;
spruches, sondern Eines genügt, da es auf but one is sufficient, since it is not applied one law is enough, since it is not applied
sich selbst nicht angewendet wird.) to itself.) to itself.)
6.1231 Das Anzeichen des logischen Satzes The mark of logical propositions is not The mark of a logical proposition is
ist n i c h t die Allgemeingültigkeit. their general validity. not general validity.
Allgemein sein heißt ja nur: zufälli- To be general is only to be acciden- To be general means no more than to
gerweise für alle Dinge gelten. Ein unver- tally valid for all things. An ungeneral- be accidentally valid for all things. An
allgemeinerter Satz kann ja ebensowohl ized proposition can be tautologous just ungeneralized proposition can be tautotautologisch
sein als ein verallgemeiner- as well as a generalized one. logical just as well as a generalized one.
ter.
6.1232 Die logische Allgemeingültigkeit Logical general validity, we could call The general validity of logic might be
könnte man wesentlich nennen, im Ge- essential as opposed to accidental gen- called essential, in contrast with the acgensatz
zu jener zufälligen, etwa des Sat- eral validity, e.g. of the proposition “all cidental general validity of such proposizes:
„Alle Menschen sind sterblich“. Sätze men are mortal”. Propositions like Rus- tions as ‘All men are mortal’. Propositions
wie Russells „Axiom of Reducibility“ sind sell’s “axiom of reducibility” are not log- like Russell’s ‘axiom of reducibility’ are
nicht logische Sätze, und dies erklärt un- ical propositions, and this explains our not logical propositions, and this explains
ser Gefühl: Dass sie, wenn wahr, so doch feeling that, if true, they can only be true our feeling that, even if they were true,
nur durch einen günstigen Zufall wahr by a happy chance. their truth could only be the result of a
sein könnten. fortunate accident.
6.1233 Es lässt sich eine Welt denken, in der We can imagine a world in which the It is possible to imagine a world in
das Axiom of Reducibility nicht gilt. Es axiom of reducibility is not valid. But it which the axiom of reducibility is not
ist aber klar, dass die Logik nichts mit is clear that logic has nothing to do with valid. It is clear, however, that logic has
der Frage zu schaffen hat, ob unsere Welt the question whether our world is really nothing to do with the question whether
wirklich so ist oder nicht. of this kind or not. our world really is like that or not.
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6.124 Die logischen Sätze beschreiben das The logical propositions describe the The propositions of logic describe the
Gerüst der Welt, oder vielmehr, sie stellen scaffolding of the world, or rather they scaffolding of the world, or rather they
es dar. Sie „handeln“ von nichts. Sie set- present it. They “treat” of nothing. They represent it. They have no ‘subjectzen
voraus, dass Namen Bedeutung, und presuppose that names have meaning, matter’. They presuppose that names
Elementarsätze Sinn haben: Und dies ist and that elementary propositions have have meaning and elementary proposiihre
Verbindung mit der Welt. Es ist klar, sense. And this is their connexion with tions sense; and that is their connexion
dass es etwas über die Welt anzeigen the world. It is clear that it must show with the world. It is clear that somemuss,
dass gewisse Verbindungen von something about the world that certain thing about the world must be indicated
Symbolen—welche wesentlich einen be- combinations of symbols—which essen- by the fact that certain combinations of
stimmten Charakter haben—Tautologien tially have a definite character—are tau- symbols—whose essence involves the possind.
Hierin liegt das Entscheidende. Wir tologies. Herein lies the decisive point. session of a determinate character—are
sagten, manches an den Symbolen, die We said that in the symbols which we use tautologies. This contains the decisive
wir gebrauchen, wäre willkürlich, man- much is arbitrary, much not. In logic only point. We have said that some things are
ches nicht. In der Logik drückt nur dieses this expresses: but this means that in arbitrary in the symbols that we use and
aus: Das heißt aber, in der Logik drücken logic it is not we who express, by means that some things are not. In logic it is only
nicht w i r mit Hilfe der Zeichen aus, of signs, what we want, but in logic the the latter that express: but that means
was wir wollen, sondern in der Logik sagt nature of the essentially necessary signs that logic is not a field in which we exdie
Natur der naturnotwendigen Zeichen itself asserts. That is to say, if we know press what we wish with the help of signs,
selbst aus: Wenn wir die logische Syn- the logical syntax of any sign language, but rather one in which the nature of the
tax irgendeiner Zeichensprache kennen, then all the propositions of logic are al- natural and inevitable signs speaks for
dann sind bereits alle Sätze der Logik ge- ready given. itself. If we know the logical syntax of
geben. any sign-language, then we have already
been given all the propositions of logic.
6.125 Es ist möglich, und zwar auch nach It is possible, even in the old logic, to It is possible—indeed possible even acder
alten Auffassung der Logik, von vorn- give at the outset a description of all “true” cording to the old conception of logic—to
herein eine Beschreibung aller „wahren“ logical propositions. give in advance a description of all ‘true’
logischen Sätze zu geben. logical propositions.
6.1251 Darum kann es in der Logik auch n i e Hence there can never be surprises in Hence there can never be surprises in
Überraschungen geben. logic. logic.
6.126 Ob ein Satz der Logik angehört, kann Whether a proposition belongs to logic One can calculate whether a proposiman
berechnen, indem man die logischen can be calculated by calculating the logi- tion belongs to logic, by calculating the
Eigenschaften des S y m b o l s berech- cal properties of the symbol. logical properties of the symbol.
net.
Und dies tun wir, wenn wir einen lo- And this we do when we prove a log- And this is what we do when we ‘prove’
gischen Satz „beweisen“. Denn, ohne uns ical proposition. For without troubling a logical proposition. For, without botherum
einen Sinn und eine Bedeutung zu ourselves about a sense and a meaning, ing about sense or meaning, we construct
kümmern, bilden wir den logischen Satz we form the logical propositions out of the logical proposition out of others using
97
aus anderen nach bloßen Z e i c h e n r e - others by mere symbolic rules. only rules that deal with signs.
g e l n.
Der Beweis der logischen Sätze be- We prove a logical proposition by cre- The proof of logical propositions consteht
darin, dass wir sie aus anderen lo- ating it out of other logical propositions sists in the following process: we produce
gischen Sätzen durch successive Anwen- by applying in succession certain opera- them out of other logical propositions by
dung gewisser Operationen entstehen las- tions, which again generate tautologies successively applying certain operations
sen, die aus den ersten immer wieder Tau- out of the first. (And from a tautology that always generate further tautologies
tologien erzeugen. (Und zwar f o l g e n only tautologies follow.) out of the initial ones. (And in fact only
aus einer Tautologie nur Tautologien.) tautologies follow from a tautology.)
Natürlich ist diese Art zu zeigen, dass Naturally this way of showing that Of course this way of showing that the
ihre Sätze Tautologien sind, der Logik its propositions are tautologies is quite propositions of logic are tautologies is not
durchaus unwesentlich. Schon darum, unessential to logic. Because the proposi- at all essential to logic, if only because the
weil die Sätze, von welchen der Beweis tions, from which the proof starts, must propositions from which the proof starts
ausgeht, ja ohne Beweis zeigen müssen, show without proof that they are tautolo- must show without any proof that they
dass sie Tautologien sind. gies. are tautologies.
6.1261 In der Logik sind Prozess und Re- In logic process and result are equiva- In logic process and result are equivasultat
äquivalent. (Darum keine Überra- lent. (Therefore no surprises.) lent. (Hence the absence of surprise.)
schung.)
6.1262 Der Beweis in der Logik ist nur ein Proof in logic is only a mechanical ex- Proof in logic is merely a mechanical
mechanisches Hilfsmittel zum leichteren pedient to facilitate the recognition of tau- expedient to facilitate the recognition of
Erkennen der Tautologie, wo sie kompli- tology, where it is complicated. tautologies in complicated cases.
ziert ist.
6.1263 Es wäre ja auch zu merkwürdig, wenn It would be too remarkable, if one Indeed, it would be altogether too reman
einen sinnvollen Satz l o g i s c h could prove a significant proposition logi- markable if a proposition that had sense
aus anderen beweisen könnte, und einen cally from another, and a logical propo- could be proved logically from others, and
logischen Satz a u c h. Es ist von vornher- sition also. It is clear from the begin- so too could a logical proposition. It is
ein klar, dass der logische Beweis eines ning that the logical proof of a significant clear from the start that a logical proof of
sinnvollen Satzes und der Beweis i n der proposition and the proof in logic must be a proposition that has sense and a proof
Logik zwei ganz verschiedene Dinge sein two quite different things. in logic must be two entirely different
müssen. things.
6.1264 Der sinnvolle Satz sagt etwas aus, und The significant proposition asserts A proposition that has sense states
sein Beweis zeigt, dass es so ist; in der something, and its proof shows that it is something, which is shown by its proof
Logik ist jeder Satz die Form eines Bewei- so; in logic every proposition is the form to be so. In logic every proposition is the
ses. of a proof. form of a proof.
Jeder Satz der Logik ist ein in Zeichen Every proposition of logic is a modus Every proposition of logic is a modus
dargestellter modus ponens. (Und den mo- ponens presented in signs. (And the ponens represented in signs. (And one
dus ponens kann man nicht durch einen modus ponens can not be expressed by cannot express the modus ponens by
98
Satz ausdrücken.) a proposition.) means of a proposition.)
6.1265 Immer kann man die Logik so auffas- Logic can always be conceived to be It is always possible to construe logic
sen, dass jeder Satz sein eigener Beweis such that every proposition is its own in such a way that every proposition is its
ist. proof. own proof.
6.127 Alle Sätze der Logik sind gleichberech- All propositions of logic are of equal All the propositions of logic are of
tigt, es gibt unter ihnen nicht wesentlich rank; there are not some which are essen- equal status: it is not the case that some
Grundgesetze und abgeleitete Sätze. tially primitive and others deduced from of them are essentially derived proposithere.
tions.
Jede Tautologie zeigt selbst, dass sie Every tautology itself shows that it is Every tautology itself shows that it is
eine Tautologie ist. a tautology. a tautology.
6.1271 Es ist klar, dass die Anzahl der „lo- It is clear that the number of “prim- It is clear that the number of the ‘primgischen
Grundgesetze“ willkürlich ist, itive propositions of logic” is arbitrary, itive propositions of logic’ is arbitrary,
denn man könnte die Logik ja aus Ei- for we could deduce logic from one prim- since one could derive logic from a sinnem
Grundgesetz ableiten, indem man itive proposition by simply forming, for gle primitive proposition, e.g. by simply
einfach z. B. aus Freges Grundgesetzen example, the logical produce of Frege’s constructing the logical product of Frege’s
das logische Produkt bildet. (Frege würde primitive propositions. (Frege would per- primitive propositions. (Frege would pervielleicht
sagen, dass dieses Grundgesetz haps say that this would no longer be haps say that we should then no longer
nun nicht mehr unmittelbar einleuchte. immediately self-evident. But it is re- have an immediately self-evident primAber
es ist merkwürdig, dass ein so exak- markable that so exact a thinker as Frege itive proposition. But it is remarkable
ter Denker wie Frege sich auf den Grad should have appealed to the degree of self- that a thinker as rigorous as Frege apdes
Einleuchtens als Kriterium des logi- evidence as the criterion of a logical propo- pealed to the degree of self-evidence as
schen Satzes berufen hat.) sition.) the criterion of a logical proposition.)
6.13 Die Logik ist keine Lehre, sondern ein Logic is not a theory but a reflexion of Logic is not a body of doctrine, but a
Spiegelbild der Welt. the world. mirror-image of the world.
Die Logik ist transzendental. Logic is transcendental. Logic is transcendental.
6.2 Die Mathematik ist eine logische Me- Mathematics is a logical method. Mathematics is a logical method.
thode.
Die Sätze der Mathematik sind Glei- The propositions of mathematics The propositions of mathematics
chungen, also Scheinsätze. are equations, and therefore pseudo- are equations, and therefore pseudopropositions.
propositions.
6.21 Der Satz der Mathematik drückt kei- Mathematical propositions express no A proposition of mathematics does not
nen Gedanken aus. thoughts. express a thought.
6.211 Im Leben ist es ja nie der mathema- In life it is never a mathematical Indeed in real life a mathematitische
Satz, den wir brauchen, sondern proposition which we need, but we use cal proposition is never what we want.
wir benützen den mathematischen Satz mathematical propositions only in order Rather, we make use of mathematin
u r, um aus Sätzen, welche nicht der to infer from propositions which do not cal propositions only in inferences from
Mathematik angehören, auf andere zu belong to mathematics to others which propositions that do not belong to mathe-
99
schließen, welche gleichfalls nicht der Ma- equally do not belong to mathematics. matics to others that likewise do not bethematik
angehören. long to mathematics.
(In der Philosophie führt die Frage: (In philosophy the question “Why do (In philosophy the question, ‘What do
„Wozu gebrauchen wir eigentlich jenes we really use that word, that proposition?” we actually use this word or this propoWort,
jenen Satz“ immer wieder zu wert- constantly leads to valuable results.) sition for?’ repeatedly leads to valuable
vollen Einsichten.) insights.)
6.22 Die Logik der Welt, die die Sätze der The logic of the world which the propo- The logic of the world, which is shown
Logik in den Tautologien zeigen, zeigt die sitions of logic show in tautologies, math- in tautologies by the propositions of logic,
Mathematik in den Gleichungen. ematics shows in equations. is shown in equations by mathematics.
6.23 Wenn zwei Ausdrücke durch das If two expressions are connected by If two expressions are combined by
Gleichheitszeichen verbunden werden, so the sign of equality, this means that they means of the sign of equality, that means
heißt das, sie sind durch einander ersetz- can be substituted for one another. But that they can be substituted for one anbar.
Ob dies aber der Fall ist, muss sich whether this is the case must show itself other. But it must be manifest in the two
an den beiden Ausdrücken selbst zeigen. in the two expressions themselves. expressions themselves whether this is
the case or not.
Es charakterisiert die logische Form It characterizes the logical form of two When two expressions can be substizweier
Ausdrücke, dass sie durch einan- expressions, that they can be substituted tuted for one another, that characterizes
der ersetzbar sind. for one another. their logical form.
6.231 Es ist eine Eigenschaft der Bejahung, It is a property of affirmation that it It is a property of affirmation that it
dass man sie als doppelte Verneinung auf- can be conceived as double denial. can be construed as double negation.
fassen kann.
Es ist eine Eigenschaft von „1 + 1 + It is a property of “1+1+1+1” that it It is a property of ‘1+1+1+1’ that it
1 + 1“, dass man es als „(1 + 1) + (1 + 1)“ can be conceived as “(1+1)+(1+1)”. can be construed as ‘(1+1)+(1+1)’.
auffassen kann.
6.232 Frege sagt, die beiden Ausdrücke ha- Frege says that these expressions Frege says that the two expressions
ben dieselbe Bedeutung, aber verschiede- have the same meaning but different have the same meaning but different
nen Sinn. senses. senses.
Das Wesentliche an der Gleichung ist But what is essential about equation But the essential point about an equaaber,
dass sie nicht notwendig ist, um zu is that it is not necessary in order to tion is that it is not necessary in order to
zeigen, dass die beiden Ausdrücke, die show that both expressions, which are show that the two expressions connected
das Gleichheitszeichen verbindet, diesel- connected by the sign of equality, have the by the sign of equality have the same
be Bedeutung haben, da sich dies aus den same meaning: for this can be perceived meaning, since this can be seen from the
beiden Ausdrücken selbst ersehen lässt. from the two expressions themselves. two expressions themselves.
6.2321 Und, dass die Sätze der Mathematik And, that the propositions of mathe- And the possibility of proving the
bewiesen werden können, heißt ja nichts matics can be proved means nothing else propositions of mathematics means simanderes,
als dass ihre Richtigkeit einzu- than that their correctness can be seen ply that their correctness can be persehen
ist, ohne dass das, was sie aus- without our having to compare what they ceived without its being necessary that
100
drücken, selbst mit den Tatsachen auf express with the facts as regards correct- what they express should itself be comseine
Richtigkeit hin verglichen werden ness. pared with the facts in order to determine
muss. its correctness.
6.2322 Die Identität der Bedeutung zweier The identity of the meaning of two ex- It is impossible to assert the identity of
Ausdrücke lässt sich nicht b e h a u p - pressions cannot be asserted. For in order meaning of two expressions. For in order
t e n. Denn, um etwas von ihrer Bedeu- to be able to assert anything about their to be able to assert anything about their
tung behaupten zu können, muss ich ihre meaning, I must know their meaning, and meaning, I must know their meaning,
Bedeutung kennen: und indem ich ihre if I know their meaning, I know whether and I cannot know their meaning withBedeutung
kenne, weiß ich, ob sie dassel- they mean the same or something differ- out knowing whether what they mean is
be oder verschiedenes bedeuten. ent. the same or different.
6.2323 Die Gleichung kennzeichnet nur den The equation characterizes only the An equation merely marks the point
Standpunkt, von welchem ich die beiden standpoint from which I consider the two of view from which I consider the two exAusdrücke
betrachte, nämlich vom Stand- expressions, that is to say the standpoint pressions: it marks their equivalence in
punkte ihrer Bedeutungsgleichheit. of their equality of meaning. meaning.
6.233 Die Frage, ob man zur Lösung der ma- To the question whether we need in- The question whether intuition is
thematischen Probleme die Anschauung tuition for the solution of mathematical needed for the solution of mathematical
brauche, muss dahin beantwortet werden, problems it must be answered that lan- problems must be given the answer that
dass eben die Sprache hier die nötige An- guage itself here supplies the necessary in this case language itself provides the
schauung liefert. intuition. necessary intuition.
6.2331 Der Vorgang des R e c h n e n s ver- The process of calculation brings The process of calculating serves to
mittelt eben diese Anschauung. about just this intuition. bring about that intuition.
Die Rechnung ist kein Experiment. Calculation is not an experiment. Calculation is not an experiment.
6.234 Die Mathematik ist eine Methode der Mathematics is a method of logic. Mathematics is a method of logic.
Logik.
6.2341 Das Wesentliche der mathematischen The essential of mathematical method It is the essential characteristic of
Methode ist es, mit Gleichungen zu arbei- is working with equations. On this mathematical method that it employs
ten. Auf dieser Methode beruht es näm- method depends the fact that every equations. For it is because of this method
lich, dass jeder Satz der Mathematik sich proposition of mathematics must be self- that every proposition of mathematics
von selbst verstehen muss. intelligible. must go without saying.
6.24 Die Methode der Mathematik, zu ih- The method by which mathematics ar- The method by which mathematics arren
Gleichungen zu kommen, ist die Sub- rives at its equations is the method of rives at its equations is the method of
stitutionsmethode. substitution. substitution.
Denn die Gleichungen drücken die Er- For equations express the substi- For equations express the substisetzbarkeit
zweier Ausdrücke aus und tutability of two expressions, and we pro- tutability of two expressions and, starting
wir schreiten von einer Anzahl von Glei- ceed from a number of equations to new from a number of equations, we advance
chungen zu neuen Gleichungen vor, in- equations, replacing expressions by oth- to new equations by substituting differdem
wir, den Gleichungen entsprechend, ers in accordance with the equations. ent expressions in accordance with the
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Ausdrücke durch andere ersetzen. equations.
6.241 So lautet der Beweis des Satzes 2×2 = Thus the proof of the proposition 2× Thus the proof of the proposition 2×
4: 2 = 4 runs: 2 = 4 runs as follows:
(Ων
)µ
’x = Ων×µ
’x Def. (Ων
)µ
’x = Ων×µ
’x Def. (Ων
)µ
’x = Ων×µ
’x Def.
Ω2×2
’x = (Ω2
)2
’x = (Ω2
)1+1
’x = Ω2
’Ω2
’x Ω2×2
’x = (Ω2
)2
’x = (Ω2
)1+1
’x = Ω2
’Ω2
’x Ω2×2
’x = (Ω2
)2
’x = (Ω2
)1+1
’x = Ω2
’Ω2
’x
= Ω1+1
’Ω1+1
’x = (Ω’Ω)’(Ω’Ω)’x = Ω1+1
’Ω1+1
’x = (Ω’Ω)’(Ω’Ω)’x = Ω1+1
’Ω1+1
’x = (Ω’Ω)’(Ω’Ω)’x
= Ω’Ω’Ω’Ω’x = Ω1+1+1+1
’x = Ω4
’x. = Ω’Ω’Ω’Ω’x = Ω1+1+1+1
’x = Ω4
’x. = Ω’Ω’Ω’Ω’x = Ω1+1+1+1
’x = Ω4
’x.
6.3 Die Erforschung der Logik bedeutet Logical research means the investiga- The exploration of logic means the exdie
Erforschung a l l e r G e s e t z m ä - tion of all regularity. And outside logic all ploration of everything that is subject to
ß i g k e i t. Und außerhalb der Logik ist is accident. law. And outside logic everything is accialles
Zufall. dental.
6.31 Das sogenannte Gesetz der Indukti- The so-called law of induction cannot The so-called law of induction cannot
on kann jedenfalls kein logisches Gesetz in any case be a logical law, for it is ob- possibly be a law of logic, since it is obsein,
denn es ist offenbar ein sinnvoller viously a significant propositions.—And viously a proposition with sense.—Nor,
Satz.—Und darum kann es auch kein Ge- therefore it cannot be a law a priori either. therefore, can it be an a priori law.
setz a priori sein.
6.32 Das Kausalitätsgesetz ist kein Gesetz, The law of causality is not a law but The law of causality is not a law but
sondern die Form eines Gesetzes. the form of a law.* the form of a law.
6.321 „Kausalitätsgesetz“, das ist ein Gat- “Law of Causality” is a class name. ‘Law of causality’—that is a general
tungsname. Und wie es in der Mechanik, And as in mechanics there are, for in- name. And just as in mechanics, for examsagen
wir, Minimum-Gesetze gibt—etwa stance, minimum-laws, such as that of ple, there are ‘minimum-principles’, such
der kleinsten Wirkung—so gibt es in der least actions, so in physics there are as the law of least action, so too in physics
Physik Kausalitätsgesetze, Gesetze von causal laws, laws of the causality form. there are causal laws, laws of the causal
der Kausalitätsform. form.
6.3211 Man hat ja auch davon eine Ahnung Men had indeed an idea that there Indeed people even surmised that
gehabt, dass es e i n „Gesetz der klein- must be a “law of least action”, before there must be a ‘law of least action’ before
sten Wirkung“ geben müsse, ehe man ge- they knew exactly how it ran. (Here, as they knew exactly how it went. (Here, as
nau wusste, wie es lautete. (Hier, wie im- always, the a priori certain proves to be always, what is certain a priori proves to
mer, stellt sich das a priori Gewisse als something purely logical.) be something purely logical.)
etwas rein Logisches heraus.)
6.33 Wir g l a u b e n nicht a priori an ein We do not believe a priori in a law of We do not have an a priori belief in a
Erhaltungsgesetz, sondern wir w i s s e n conservation, but we know a priori the law of conservation, but rather a priori
a priori die Möglichkeit einer logischen possibility of a logical form. knowledge of the possibility of a logical
Form. form.
6.34 Alle jene Sätze, wie der Satz vom All propositions, such as the law of All such propositions, including the
* [Ogden only] I.e. not the form of one particular law, but of any law of a certain sort (B. R.).
102
Grunde, von der Kontinuität in der Natur, causation, the law of continuity in nature, principle of sufficient reason, the laws of
vom kleinsten Aufwande in der Natur etc. the law of least expenditure in nature, continuity in nature and of least effort
etc., alle diese sind Einsichten a priori etc. etc., all these are a priori intuitions in nature, etc. etc.—all these are a priüber
die mögliche Formgebung der Sätze of possible forms of the propositions of ori insights about the forms in which the
der Wissenschaft. science. propositions of science can be cast.
6.341 Die Newtonsche Mechanik z. B. bringt Newtonian mechanics, for example, Newtonian mechanics, for example,
die Weltbeschreibung auf eine einheitli- brings the description of the universe to a imposes a unified form on the description
che Form. Denken wir uns eine weiße unified form. Let us imagine a white sur- of the world. Let us imagine a white surFläche,
auf der unregelmäßige schwar- face with irregular black spots. We now face with irregular black spots on it. We
ze Flecken wären. Wir sagen nun: Was say: Whatever kind of picture these make then say that whatever kind of picture
für ein Bild immer hierdurch entsteht, I can always get as near as I like to its these make, I can always approximate as
immer kann ich seiner Beschreibung be- description, if I cover the surface with a closely as I wish to the description of it
liebig nahe kommen, indem ich die Flä- sufficiently fine square network and now by covering the surface with a sufficiently
che mit einem entsprechend feinen qua- say of every square that it is white or fine square mesh, and then saying of evdratischen
Netzwerk bedecke und nun black. In this way I shall have brought ery square whether it is black or white.
von jedem Quadrat sage, dass es weiß the description of the surface to a unified In this way I shall have imposed a unioder
schwarz ist. Ich werde auf diese Wei- form. This form is arbitrary, because I fied form on the description of the surface.
se die Beschreibung der Fläche auf eine could have applied with equal success a The form is optional, since I could have
einheitliche Form gebracht haben. Die- net with a triangular or hexagonal mesh. achieved the same result by using a net
se Form ist beliebig, denn ich hätte mit It can happen that the description would with a triangular or hexagonal mesh. Posdem
gleichen Erfolge ein Netz aus drei- have been simpler with the aid of a tri- sibly the use of a triangular mesh would
eckigen oder sechseckigen Maschen ver- angular mesh; that is to say we might have made the description simpler: that
wenden können. Es kann sein, dass die have described the surface more accu- is to say, it might be that we could deBeschreibung
mit Hilfe eines Dreiecks- rately with a triangular, and coarser, than scribe the surface more accurately with a
Netzes einfacher geworden wäre; das with the finer square mesh, or vice versa, coarse triangular mesh than with a fine
heißt, dass wir die Fläche mit einem grö- and so on. To the different networks cor- square mesh (or conversely), and so on.
beren Dreiecks-Netz genauer beschreiben respond different systems of describing The different nets correspond to different
könnten als mit einem feineren quadra- the world. Mechanics determine a form systems for describing the world. Mechantischen
(oder umgekehrt) usw. Den ver- of description by saying: All propositions ics determines one form of description
schiedenen Netzen entsprechen verschie- in the description of the world must be of the world by saying that all proposidene
Systeme der Weltbeschreibung. Die obtained in a given way from a number tions used in the description of the world
Mechanik bestimmt eine Form der Welt- of given propositions—the mechanical ax- must be obtained in a given way from a
beschreibung, indem sie sagt: Alle Sätze ioms. It thus provides the bricks for build- given set of propositions—the axioms of
der Weltbeschreibung müssen aus einer ing the edifice of science, and says: What- mechanics. It thus supplies the bricks
Anzahl gegebener Sätze—den mechani- ever building thou wouldst erect, thou for building the edifice of science, and it
schen Axiomen—auf eine gegebene Art shalt construct it in some manner with says, ‘Any building that you want to erect,
und Weise erhalten werden. Hierdurch these bricks and these alone. whatever it may be, must somehow be
103
liefert sie die Bausteine zum Bau des wis- constructed with these bricks, and with
senschaftlichen Gebäudes und sagt: Wel- these alone.’
ches Gebäude immer du aufführen willst,
jedes musst du irgendwie mit diesen und
nur diesen Bausteinen zusammenbrin-
gen.
(Wie man mit dem Zahlensystem jede (As with the system of numbers one (Just as with the number-system we
beliebige Anzahl, so muss man mit dem must be able to write down any arbitrary must be able to write down any number
System der Mechanik jeden beliebigen number, so with the system of mechan- we wish, so with the system of mechanSatz
der Physik hinschreiben können.) ics one must be able to write down any ics we must be able to write down any
arbitrary physical proposition.) proposition of physics that we wish.)
6.342 Und nun sehen wir die gegenseitige And now we see the relative posi- And now we can see the relative poStellung
von Logik und Mechanik. (Man tion of logic and mechanics. (We could sition of logic and mechanics. (The net
könnte das Netz auch aus verschieden- construct the network out of figures of might also consist of more than one kind
artigen Figuren etwa aus Dreiecken und different kinds, as out of triangles and of mesh: e.g. we could use both trianSechsecken
bestehen lassen.) Dass sich hexagons together.) That a picture like gles and hexagons.) The possibility of deein
Bild, wie das vorhin erwähnte, durch that instanced above can be described by scribing a picture like the one mentioned
ein Netz von gegebener Form beschreiben a network of a given form asserts noth- above with a net of a given form tells us
lässt, sagt über das Bild n i c h t s aus. ing about the picture. (For this holds of nothing about the picture. (For that is
(Denn dies gilt für jedes Bild dieser Art.) every picture of this kind.) But this does true of all such pictures.) But what does
Das aber charakterisiert das Bild, dass es characterize the picture, the fact, namely, characterize the picture is that it can be
sich durch ein bestimmtes Netz von b e - that it can be completely described by a described completely by a particular net
s t i m m t e r Feinheit v o l l s t ä n d i g definite net of definite fineness. with a particular size of mesh.
beschreiben lässt.
So auch sagt es nichts über die Welt So too the fact that it can be described Similarly the possibility of describing
aus, dass sie sich durch die Newtonsche by Newtonian mechanics asserts nothing the world by means of Newtonian meMechanik
beschreiben lässt; wohl aber, about the world; but this asserts some- chanics tells us nothing about the world:
dass sie sich s o durch jene beschrei- thing, namely, that it can be described in but what does tell us something about it
ben lässt, wie dies eben der Fall ist. Auch that particular way in which as a matter is the precise way in which it is possible
das sagt etwas über die Welt, dass sie of fact it is described. The fact, too, that it to describe it by these means. We are also
sich durch die eine Mechanik einfacher can be described more simply by one sys- told something about the world by the
beschreiben lässt als durch die andere. tem of mechanics than by another says fact that it can be described more simply
something about the world. with one system of mechanics than with
another.
6.343 Die Mechanik ist ein Versuch, alle Mechanics is an attempt to construct Mechanics is an attempt to construct
w a h r e n Sätze, die wir zur Weltbe- according to a single plan all true proposi- according to a single plan all the true
schreibung brauchen, nach Einem Plane tions which we need for the description of propositions that we need for the descrip-
104
zu konstruieren. the world. tion of the world.
6.3431 Durch den ganzen logischen Apparat Through the whole apparatus of logic The laws of physics, with all their loghindurch
sprechen die physikalischen Ge- the physical laws still speak of the objects ical apparatus, still speak, however indisetze
doch von den Gegenständen der of the world. rectly, about the objects of the world.
Welt.
6.3432 Wir dürfen nicht vergessen, dass die We must not forget that the descrip- We ought not to forget that any deWeltbeschreibung
durch die Mechanik im- tion of the world by mechanics is always scription of the world by means of memer
die ganz allgemeine ist. Es ist in ihr quite general. There is, for example, chanics will be of the completely general
z. B. nie von b e s t i m m t e n materiel- never any mention of particular material kind. For example, it will never mention
len Punkten die Rede, sondern immer nur points in it, but always only of some points particular point-masses: it will only talk
von i r g e n d w e l c h e n. or other. about any point-masses whatsoever.
6.35 Obwohl die Flecke in unserem Bild Although the spots in our picture are Although the spots in our picture are
geometrische Figuren sind, so kann doch geometrical figures, geometry can obvi- geometrical figures, nevertheless geomeselbstverständlich
die Geometrie gar ously say nothing about their actual form try can obviously say nothing at all about
nichts über ihre tatsächliche Form und and position. But the network is purely their actual form and position. The netLage
sagen. Das Netz aber ist r e i n geo- geometrical, and all its properties can be work, however, is purely geometrical; all
metrisch, alle seine Eigenschaften kön- given a priori. its properties can be given a priori.
nen a priori angegeben werden.
Gesetze wie der Satz vom Grunde, etc. Laws, like the law of causation, etc., Laws like the principle of sufficient
handeln vom Netz, nicht von dem, was treat of the network and not what the reason, etc. are about the net and not
das Netz beschreibt. network describes. about what the net describes.
6.36 Wenn es ein Kausalitätsgesetz gäbe, If there were a law of causality, it If there were a law of causality, it
so könnte es lauten: „Es gibt Naturgeset- might run: “There are natural laws”. might be put in the following way: There
ze“. are laws of nature.
Aber freilich kann man das nicht sa- But that can clearly not be said: it But of course that cannot be said: it
gen: es zeigt sich. shows itself. makes itself manifest.
6.361 In der Ausdrucksweise Hertz’s könn- In the terminology of Hertz we might One might say, using Hertz’s terminolte
man sagen: Nur g e s e t z m ä ß i g e say: Only uniform connections are think- ogy, that only connexions that are subject
Zusammenhänge sind d e n k b a r. able. to law are thinkable.
6.3611 Wir können keinen Vorgang mit dem We cannot compare any process with We cannot compare a process with ‘the
„Ablauf der Zeit“ vergleichen—diesen gibt the “passage of time”—there is no such passage of time’—there is no such thing—
es nicht—, sondern nur mit einem an- thing—but only with another process (say, but only with another process (such as
deren Vorgang (etwa mit dem Gang des with the movement of the chronometer). the working of a chronometer).
Chronometers).
Daher ist die Beschreibung des zeit- Hence the description of the temporal Hence we can describe the lapse of
lichen Verlaufs nur so möglich, dass wir sequence of events is only possible if we time only by relying on some other prouns
auf einen anderen Vorgang stützen. support ourselves on another process. cess.
105
Ganz Analoges gilt für den Raum. Wo It is exactly analogous for space. Something exactly analogous applies
man z. B. sagt, es könne keines von zwei When, for example, we say that neither of to space: e.g. when people say that neiEreignissen
(die sich gegenseitig aus- two events (which mutually exclude one ther of two events (which exclude one anschließen)
eintreten, weil k e i n e U r - another) can occur, because there is no other) can occur, because there is nothing
s a c h e vorhanden sei, warum das eine cause why the one should occur rather to cause the one to occur rather than the
eher als das andere eintreten solle, da than the other, it is really a matter of other, it is really a matter of our being
handelt es sich in Wirklichkeit darum, our being unable to describe one of the unable to describe one of the two events
dass man gar nicht e i n e s der beiden two events unless there is some sort of unless there is some sort of asymmetry
Ereignisse beschreiben kann, wenn nicht asymmetry. And if there is such an asym- to be found. And if such an asymmetry
irgend eine Asymmetrie vorhanden ist. metry, we can regard this as the cause is to be found, we can regard it as the
Und w e n n eine solche Asymmetrie vor- of the occurrence of the one and of the cause of the occurrence of the one and the
handen i s t, so können wir diese als U r - non-occurrence of the other. non-occurrence of the other.
s a c h e des Eintreffens des einen und
Nicht- Eintreffens des anderen auffassen.
6.36111 Das Kant’sche Problem von der rech- The Kantian problem of the right and Kant’s problem about the right hand
ten und linken Hand, die man nicht zur left hand which cannot be made to cover and the left hand, which cannot be made
Deckung bringen kann, besteht schon in one another already exists in the plane, to coincide, exists even in two dimensions.
der Ebene, ja im eindimensionalen Raum, and even in one-dimensional space; where Indeed, it exists in one-dimensional space
wo die beiden kongruenten Figuren a und the two congruent figures a and b canb
auch nicht zur Deckung gebracht wer- not be made to cover one another without
den können, ohne aus diesem Raum
a
× ×
b a
× ×
b a
× ×
b
herausbewegt zu werden. Rechte und lin- moving them out of this space. The right in which the two congruent figures, a and
ke Hand sind tatsächlich vollkommen and left hand are in fact completely con- b, cannot be made to coincide unless they
kongruent. Und dass man sie nicht zur gruent. And the fact that they cannot be are moved out of this space. The right
Deckung bringen kann, hat damit nichts made to cover one another has nothing to hand and the left hand are in fact comzu
tun. do with it. pletely congruent. It is quite irrelevant
that they cannot be made to coincide.
Den rechten Handschuh könnte man A right-hand glove could be put on a A right-hand glove could be put on the
an die linke Hand ziehen, wenn man ihn left hand if it could be turned round in left hand, if it could be turned round in
im vierdimensionalen Raum umdrehen four-dimensional space. four-dimensional space.
könnte.
6.362 Was sich beschreiben lässt, das kann What can be described can happen too, What can be described can happen too:
auch geschehen, und was das Kausalitäts- and what is excluded by the law of causal- and what the law of causality is meant to
gesetz ausschließen soll, das lässt sich ity cannot be described. exclude cannot even be described.
106
auch nicht beschreiben.
6.363 Der Vorgang der Induktion besteht The process of induction is the process The procedure of induction consists in
darin, dass wir das e i n f a c h s t e Ge- of assuming the simplest law that can be accepting as true the simplest law that
setz annehmen, das mit unseren Erfah- made to harmonize with our experience. can be reconciled with our experiences.
rungen in Einklang zu bringen ist.
6.3631 Dieser Vorgang hat aber keine logi- This process, however, has no logical This procedure, however, has no logische,
sondern nur eine psychologische Be- foundation but only a psychological one. cal justification but only a psychological
gründung. one.
Es ist klar, dass kein Grund vorhan- It is clear that there are no grounds It is clear that there are no grounds
den ist, zu glauben, es werde nun auch for believing that the simplest course of for believing that the simplest eventuality
wirklich der einfachste Fall eintreten. events will really happen. will in fact be realized.
6.36311 Dass die Sonne morgen aufgehen That the sun will rise to-morrow, is an It is an hypothesis that the sun will
wird, ist eine Hypothese; und das heißt: hypothesis; and that means that we do rise tomorrow: and this means that we do
wir w i s s e n nicht, ob sie aufgehen wird. not know whether it will rise. not know whether it will rise.
6.37 Einen Zwang, nach dem Eines gesche- A necessity for one thing to happen There is no compulsion making one
hen müsste, weil etwas anderes gesche- because another has happened does not thing happen because another has haphen
ist, gibt es nicht. Es gibt nur eine exist. There is only logical necessity. pened. The only necessity that exists is
l o g i s c h e Notwendigkeit. logical necessity
6.371 Der ganzen modernen Weltanschau- At the basis of the whole modern view The whole modern conception of the
ung liegt die Täuschung zugrunde, dass of the world lies the illusion that the so- world is founded on the illusion that the
die sogenannten Naturgesetze die Erklä- called laws of nature are the explanations so-called laws of nature are the explanarungen
der Naturerscheinungen seien. of natural phenomena. tions of natural phenomena.
6.372 So bleiben sie bei den Naturgesetzen So people stop short at natural laws Thus people today stop at the laws of
als bei etwas Unantastbarem stehen, wie as something unassailable, as did the an- nature, treating them as something inviodie
Älteren bei Gott und dem Schicksal. cients at God and Fate. lable, just as God and Fate were treated
in past ages.
Und sie haben ja beide Recht, und Un- And they are both right and wrong. And in fact both are right and both
recht. Die Alten sind allerdings insofern but the ancients were clearer, in so far wrong: though the view of the ancients
klarer, als sie einen klaren Abschluss an- as they recognized one clear conclusion, is clearer in so far as they have a clear
erkennen, während es bei dem neuen Sy- whereas the modern system makes it and acknowledged terminus, while the
stem scheinen soll, als sei a l l e s erklärt. appear as though everything were ex- modern system tries to make it look as if
plained. everything were explained.
6.373 Die Welt ist unabhängig von meinem The world is independent of my will. The world is independent of my will.
Willen.
6.374 Auch wenn alles, was wir wünschen, Even if everything we wished were to Even if all that we wish for were to
geschähe, so wäre dies doch nur, sozusa- happen, this would only be, so to speak, happen, still this would only be a favour
gen, eine Gnade des Schicksals, denn es a favour of fate, for there is no logical granted by fate, so to speak: for there is
107
ist kein l o g i s c h e r Zusammenhang connexion between will and world, which no logical connexion between the will and
zwischen Willen und Welt, der dies ver- would guarantee this, and the assumed the world, which would guarantee it, and
bürgte, und den angenommenen physika- physical connexion itself we could not the supposed physical connexion itself is
lischen Zusammenhang könnten wir doch again will. surely not something that we could will.
nicht selbst wieder wollen.
6.375 Wie es nur eine l o g i s c h e Notwen- As there is only a logical necessity, so Just as the only necessity that exists
digkeit gibt, so gibt es auch nur eine l o - there is only a logical impossibility. is logical necessity, so too the only imposg
i s c h e Unmöglichkeit. sibility that exists is logical impossibility.
6.3751 Dass z. B. zwei Farben zugleich an For two colours, e.g. to be at one place For example, the simultaneous preseinem
Ort des Gesichtsfeldes sind, ist in the visual field, is impossible, logically ence of two colours at the same place in
unmöglich, und zwar logisch unmöglich, impossible, for it is excluded by the logical the visual field is impossible, in fact logdenn
es ist durch die logische Struktur structure of colour. ically impossible, since it is ruled out by
der Farbe ausgeschlossen. the logical structure of colour.
Denken wir daran, wie sich dieser Wi- Let us consider how this contradic- Let us think how this contradiction apderspruch
in der Physik darstellt: Unge- tion presents itself in physics. Somewhat pears in physics: more or less as follows—
fähr so, dass ein Teilchen nicht zu glei- as follows: That a particle cannot at the a particle cannot have two velocities at
cher Zeit zwei Geschwindigkeiten haben same time have two velocities, i.e. that at the same time; that is to say, it cannot be
kann; das heißt, dass es nicht zu gleicher the same time it cannot be in two places, in two places at the same time; that is to
Zeit an zwei Orten sein kann; das heißt, i.e. that particles in different places at the say, particles that are in different places
dass Teilchen an verschiedenen Orten zu same time cannot be identical. at the same time cannot be identical.
Einer Zeit nicht identisch sein können.
(Es ist klar, dass das logische Produkt It is clear that the logical product of (It is clear that the logical product of
zweier Elementarsätze weder eine Tauto- two elementary propositions can neither two elementary propositions can neither
logie noch eine Kontradiktion sein kann. be a tautology nor a contradiction. The be a tautology nor a contradiction. The
Die Aussage, dass ein Punkt des Gesichts- assertion that a point in the visual field statement that a point in the visual field
feldes zu gleicher Zeit zwei verschiedene has two different colours at the same time, has two different colours at the same time
Farben hat, ist eine Kontradiktion.) is a contradiction. is a contradiction.)
6.4 Alle Sätze sind gleichwertig. All propositions are of equal value. All propositions are of equal value.
6.41 Der Sinn der Welt muss außerhalb ih- The sense of the world must lie out- The sense of the world must lie outrer
liegen. In der Welt ist alles, wie es ist, side the world. In the world everything is side the world. In the world everything
und geschieht alles, wie es geschieht; es as it is and happens as it does happen. In is as it is, and everything happens as it
gibt i n ihr keinen Wert—und wenn es it there is no value—and if there were, it does happen: in it no value exists—and if
ihn gäbe, so hätte er keinen Wert. would be of no value. it did exist, it would have no value.
Wenn es einen Wert gibt, der Wert hat, If there is a value which is of value, it If there is any value that does have
so muss er außerhalb alles Geschehens must lie outside all happening and being- value, it must lie outside the whole sphere
und So-Seins liegen. Denn alles Gesche- so. For all happening and being-so is acci- of what happens and is the case. For all
hen und So-Sein ist zufällig. dental. that happens and is the case is accidental.
108
Was es nichtzufällig macht, kann What makes it non-accidental cannot What makes it non-accidental cannot
nicht i n der Welt liegen, denn sonst wäre lie in the world, for otherwise this would lie within the world, since if it did it would
dies wieder zufällig. again be accidental. itself be accidental.
Es muss außerhalb der Welt liegen. It must lie outside the world. It must lie outside the world.
6.42 Darum kann es auch keine Sätze der Hence also there can be no ethical So too it is impossible for there to be
Ethik geben. propositions. propositions of ethics.
Sätze können nichts Höheres aus- Propositions cannot express anything Propositions can express nothing that
drücken. higher. is higher.
6.421 Es ist klar, dass sich die Ethik nicht It is clear that ethics cannot be ex- It is clear that ethics cannot be put
aussprechen lässt. pressed. into words.
Die Ethik ist transzendental. Ethics are transcendental. Ethics is transcendental.
(Ethik und Ästhetik sind Eins.) (Ethics and æsthetics are one.) (Ethics and aesthetics are one and the
same.)
6.422 Der erste Gedanke bei der Aufstellung The first thought in setting up an ethi- When an ethical law of the form, ‘Thou
eines ethischen Gesetzes von der Form cal law of the form “thou shalt . . . ” is: And shalt . . . ’ is laid down, one’s first thought
„Du sollst . . . “ ist: Und was dann, wenn what if I do not do it? But it is clear that is, ‘And what if I do not do it?’ It is
ich es nicht tue? Es ist aber klar, dass die ethics has nothing to do with punishment clear, however, that ethics has nothing
Ethik nichts mit Strafe und Lohn im ge- and reward in the ordinary sense. This to do with punishment and reward in the
wöhnlichen Sinne zu tun hat. Also muss question as to the consequences of an ac- usual sense of the terms. So our question
diese Frage nach den F o l g e n einer tion must therefore be irrelevant. At least about the consequences of an action must
Handlung belanglos sein.—Zum Minde- these consequences will not be events. For be unimportant.—At least those consesten
dürfen diese Folgen nicht Ereignisse there must be something right in that for- quences should not be events. For there
sein. Denn etwas muss doch an jener Fra- mulation of the question. There must be must be something right about the quesgestellung
richtig sein. Es muss zwar ei- some sort of ethical reward and ethical tion we posed. There must indeed be
ne Art von ethischem Lohn und ethischer punishment, but this must lie in the ac- some kind of ethical reward and ethical
Strafe geben, aber diese müssen in der tion itself. punishment, but they must reside in the
Handlung selbst liegen. action itself.
(Und das ist auch klar, dass der Lohn (And this is clear also that the reward (And it is also clear that the reward
etwas Angenehmes, die Strafe etwas Un- must be something acceptable, and the must be something pleasant and the punangenehmes
sein muss.) punishment something unacceptable.) ishment something unpleasant.)
6.423 Vom Willen als dem Träger des Ethi- Of the will as the subject of the ethical It is impossible to speak about the will
schen kann nicht gesprochen werden. we cannot speak. in so far as it is the subject of ethical at-
tributes.
Und der Wille als Phänomen interes- And the will as a phenomenon is only And the will as a phenomenon is of
siert nur die Psychologie. of interest to psychology. interest only to psychology.
6.43 Wenn das gute oder böse Wollen die If good or bad willing changes the If the good or bad exercise of the will
Welt ändert, so kann es nur die Gren- world, it can only change the limits of does alter the world, it can alter only the
109
zen der Welt ändern, nicht die Tatsachen; the world, not the facts; not the things limits of the world, not the facts—not
nicht das, was durch die Sprache ausge- that can be expressed in language. what can be expressed by means of landrückt
werden kann. guage.
Kurz, die Welt muss dann dadurch In brief, the world must thereby be- In short the effect must be that it beüberhaupt
eine andere werden. Sie muss come quite another, it must so to speak comes an altogether different world. It
sozusagen als Ganzes abnehmen oder zu- wax or wane as a whole. must, so to speak, wax and wane as a
nehmen. whole.
Die Welt des Glücklichen ist eine an- The world of the happy is quite an- The world of the happy man is a difdere
als die des Unglücklichen. other than that of the unhappy. ferent one from that of the unhappy man.
6.431 Wie auch beim Tod die Welt sich nicht As in death, too, the world does not So too at death the world does not aländert,
sondern aufhört. change, but ceases. ter, but comes to an end.
6.4311 Der Tod ist kein Ereignis des Lebens. Death is not an event of life. Death is Death is not an event in life: we do
Den Tod erlebt man nicht. not lived through. not live to experience death.
Wenn man unter Ewigkeit nicht un- If by eternity is understood not end- If we take eternity to mean not infiendliche
Zeitdauer, sondern Unzeitlich- less temporal duration but timelessness, nite temporal duration but timelessness,
keit versteht, dann lebt der ewig, der in then he lives eternally who lives in the then eternal life belongs to those who live
der Gegenwart lebt. present. in the present.
Unser Leben ist ebenso endlos, wie Our life is endless in the way that our Our life has no end in just the way in
unser Gesichtsfeld grenzenlos ist. visual field is without limit. which our visual field has no limits.
6.4312 Die zeitliche Unsterblichkeit der See- The temporal immortality of the hu- Not only is there no guarantee of the
le des Menschen, das heißt also ihr ewiges man soul, that is to say, its eternal sur- temporal immortality of the human soul,
Fortleben auch nach dem Tode, ist nicht vival also after death, is not only in no that is to say of its eternal survival afnur
auf keine Weise verbürgt, sondern way guaranteed, but this assumption in ter death; but, in any case, this assumpvor
allem leistet diese Annahme gar nicht the first place will not do for us what we tion completely fails to accomplish the
das, was man immer mit ihr erreichen always tried to make it do. Is a riddle purpose for which it has always been inwollte.
Wird denn dadurch ein Rätsel ge- solved by the fact that I survive for ever? tended. Or is some riddle solved by my
löst, dass ich ewig fortlebe? Ist denn die- Is this eternal life not as enigmatic as our surviving for ever? Is not this eternal life
ses ewige Leben dann nicht ebenso rätsel- present one? The solution of the riddle of itself as much of a riddle as our present
haft wie das gegenwärtige? Die Lösung life in space and time lies outside space life? The solution of the riddle of life
des Rätsels des Lebens in Raum und Zeit and time. in space and time lies outside space and
liegt a u ß e r h a l b von Raum und Zeit. time.
(Nicht Probleme der Naturwissen- (It is not problems of natural science (It is certainly not the solution of any
schaft sind ja zu lösen.) which have to be solved.) problems of natural science that is re-
quired.)
6.432 W i e die Welt ist, ist für das Höhere How the world is, is completely indif- How things are in the world is a matvollkommen
gleichgültig. Gott offenbart ferent for what is higher. God does not ter of complete indifference for what is
sich nicht i n der Welt. reveal himself in the world. higher. God does not reveal himself in the
110
world.
6.4321 Die Tatsachen gehören alle nur zur The facts all belong only to the task The facts all contribute only to setting
Aufgabe, nicht zur Lösung. and not to its performance. the problem, not to its solution.
6.44 Nicht w i e die Welt ist, ist das Mysti- Not how the world is, is the mystical, It is not how things are in the world
sche, sondern d a s s sie ist. but that it is. that is mystical, but that it exists.
6.45 Die Anschauung der Welt sub spe- The contemplation of the world sub To view the world sub specie aeterni
cie aeterni ist ihre Anschauung als— specie aeterni is its contemplation as a is to view it as a whole—a limited whole.
begrenztes—Ganzes. limited whole.
Das Gefühl der Welt als begrenztes The feeling that the world is a limited Feeling the world as a limited whole—
Ganzes ist das mystische. whole is the mystical feeling. it is this that is mystical.
6.5 Zu einer Antwort, die man nicht aus- For an answer which cannot be ex- When the answer cannot be put into
sprechen kann, kann man auch die Frage pressed the question too cannot be ex- words, neither can the question be put
nicht aussprechen. pressed. into words.
D a s R ä t s e l gibt es nicht. The riddle does not exist. The riddle does not exist.
Wenn sich eine Frage überhaupt stel- If a question can be put at all, then it If a question can be framed at all, it is
len lässt, so k a n n sie auch beantwortet can also be answered. also possible to answer it.
werden.
6.51 Skeptizismus ist n i c h t unwiderleg- Scepticism is not irrefutable, but pal- Scepticism is not irrefutable, but obvilich,
sondern offenbar unsinnig, wenn er pably senseless, if it would doubt where a ously nonsensical, when it tries to raise
bezweifeln will, wo nicht gefragt werden question cannot be asked. doubts where no questions can be asked.
kann.
Denn Zweifel kann nur bestehen, wo For doubt can only exist where there is For doubt can exist only where a queseine
Frage besteht; eine Frage nur, wo a question; a question only where there is tion exists, a question only where an aneine
Antwort besteht, und diese nur, wo an answer, and this only where something swer exists, and an answer only where
etwas g e s a g t werden k a n n. can be said. something can be said.
6.52 Wir fühlen, dass, selbst wenn alle We feel that even if all possible scien- We feel that even when all possible
m ö g l i c h e n wissenschaftlichen Fra- tific questions be answered, the problems scientific questions have been answered,
gen beantwortet sind, unsere Lebenspro- of life have still not been touched at all. the problems of life remain completely
bleme noch gar nicht berührt sind. Frei- Of course there is then no question left, untouched. Of course there are then no
lich bleibt dann eben keine Frage mehr; and just this is the answer. questions left, and this itself is the anund
eben dies ist die Antwort. swer.
6.521 Die Lösung des Problems des Lebens The solution of the problem of life is The solution of the problem of life is
merkt man am Verschwinden dieses Pro- seen in the vanishing of this problem. seen in the vanishing of the problem.
blems.
(Ist nicht dies der Grund, warum Men- (Is not this the reason why men to (Is not this the reason why those who
schen, denen der Sinn des Lebens nach whom after long doubting the sense of life have found after a long period of doubt
langen Zweifeln klar wurde, warum diese became clear, could not then say wherein that the sense of life became clear to them
111
dann nicht sagen konnten, worin dieser this sense consisted?) have then been unable to say what constiSinn
bestand?) tuted that sense?)
6.522 Es gibt allerdings Unaussprechliches. There is indeed the inexpressible. There are, indeed, things that cannot
Dies z e i g t sich, es ist das Mystische. This shows itself; it is the mystical. be put into words. They make themselves
manifest. They are what is mystical.
6.53 Die richtige Methode der Philosophie The right method of philosophy would The correct method in philosophy
wäre eigentlich die: Nichts zu sagen, be this: To say nothing except what can would really be the following: to say nothals
was sich sagen lässt, also Sätze der be said, i.e. the propositions of natural sci- ing except what can be said, i.e. proposiNaturwissenschaft—also
etwas, was mit ence, i.e. something that has nothing to do tions of natural science—i.e. something
Philosophie nichts zu tun hat—, und with philosophy: and then always, when that has nothing to do with philosophy—
dann immer, wenn ein anderer etwas someone else wished to say something and then, whenever someone else wanted
Metaphysisches sagen wollte, ihm nach- metaphysical, to demonstrate to him that to say something metaphysical, to demonzuweisen,
dass er gewissen Zeichen in he had given no meaning to certain signs strate to him that he had failed to give
seinen Sätzen keine Bedeutung gegeben in his propositions. This method would be a meaning to certain signs in his propohat.
Diese Methode wäre für den anderen unsatisfying to the other—he would not sitions. Although it would not be satisunbefriedigend—er
hätte nicht das Ge- have the feeling that we were teaching fying to the other person—he would not
fühl, dass wir ihn Philosophie lehrten— him philosophy—but it would be the only have the feeling that we were teaching
aber s i e wäre die einzig streng richtige. strictly correct method. him philosophy—this method would be
the only strictly correct one.
6.54 Meine Sätze erläutern dadurch, dass My propositions are elucidatory in My propositions serve as elucidations
sie der, welcher mich versteht, am Ende this way: he who understands me finally in the following way: anyone who underals
unsinnig erkennt, wenn er durch sie— recognizes them as senseless, when he stands me eventually recognizes them as
auf ihnen—über sie hinausgestiegen ist. has climbed out through them, on them, nonsensical, when he has used them—
(Er muss sozusagen die Leiter wegwerfen, over them. (He must so to speak throw as steps—to climb up beyond them. (He
nachdem er auf ihr hinaufgestiegen ist.) away the ladder, after he has climbed up must, so to speak, throw away the ladder
on it.) after he has climbed up it.)
Er muss diese Sätze überwinden, He must surmount these propositions; He must transcend these propositions,
dann sieht er die Welt richtig. then he sees the world rightly. and then he will see the world aright.
7 Wovon man nicht sprechen kann, dar- Whereof one cannot speak, thereof one What we cannot speak about we must
über muss man schweigen. must be silent. pass over in silence.
112
Index (Pears/McGuinness)
[Original note by Pears and McGuinness.]
The translators’ aim has been to include all
the more interesting words, and, in each case,
either to give all the occurrences of a word,
or else to omit only a few unimportant ones.
Paragraphs in the preface are referred to as
P1, P2, etc. Propositions are indicated by numbers
without points [—the points have been
restored for the side-by-side-by-side edition—];
more than two consecutive propositions, by
two numbers joined by an en-rule, as 202–
2021.
In the translation it has sometimes been
necessary to use different English expressions
for the same German expression or the same
English expression for different German expressions.
The index contains various devices
designed to make it an informative guide to
the German terminology and, in particular, to
draw attention to some important connexions
between ideas that are more difficult to bring
out in English than in German.
First, when a German expression is of any
interest in itself, it is given in brackets after
the English expression that translates it, e.g.
situation [Sachlage]; also, whenever an English
expression is used to translate more than
one German expression, each of the German
expressions is given separately in numbered
brackets, and is followed by the list of passages
in which it is translated by the English expression,
e.g. reality 1. [Realität], 55561, etc. 2.
[Wirklichkeit], 206, etc.
Secondly, the German expressions given in
this way sometimes have two or more English
translations in the text; and when this is so,
if the alternative English translations are of
interest, they follow the German expression inside
the brackets, e.g. proposition [Satz: law;
principle].
The alternative translations recorded by
these two devices are sometimes given in an
abbreviated way. For a German expression
need not actually be translated by the English
expressions that it follows or precedes, as it is
in the examples above. The relationship may
be more complicated. For instance, the German
expression may be only part of a phrase
that is translated by the English expression,
e.g. stand in a relation to one another; are
related [sich verhalten: stand, how things;
state of things].
Thirdly, cross-references have been used to
draw attention to other important connexions
between ideas, e.g. true, cf. correct; right: and
a priori, cf. advance, in.
In subordinate entries and cross-references
the catchword is indicated by ∼, unless the
catchword contains /, in which case the part
preceding / is so indicated, e.g. accident; ∼al
for accident; accidental, and state of /affairs;
∼ things for state of affairs; state of
things. Cross-references relate to the last preceding
entry or numbered bracket. When references
are given both for a word in its own right
and for a phrase containing it, occurrences of
the latter are generally not also counted as
occurrences of the former, so that both entries
should be consulted.
about [von etwas handeln: concerned with;
deal with; subject-matter], 3.24, 5.44, 6.35;
cf. mention; speak; talk.
abstract, 5.5563
accident; ∼al [Zufall], 2.012, 2.0121, 3.34,
5.4733, 6.031, 6.1231, 6.1232, 6.3, 6.41
action, 5.1362, 6.422
activity, 4.112
addition, cf. logical.
adjectiv/e; ∼al, 3.323, 5.4733
advance, in [von vornherein], 5.47, 6.125; cf. a
priori. aesthetics, 6.421
affirmation [Bejahung], 4.064, 5.124, 5.1241,
5.44, 5.513, 5.514, 6.231
affix, [Index], 4.0411, 5.02
agreement
1. [stimmmen: right; true], 5.512
2. [Übereinstimmmung], 2.21, 2.222, 4.2,
4.4, 4.42–4.431, 4.462
analysis [Analyse], 3.201, 3.25, 3.3442, 4.1274,
4.221, 5.5562; cf. anatomize; dissect;
resolve.
analytic, 6.11
anatomize [auseinanderlegen], 3.261 ; cf.
analysis.
answer, 4.003, 4.1274, 5.4541, 5.55, 5.551,
6.5–6.52
apparent, 4.0031, 5.441, 5.461; cf. pseudo–.
application [Anwendung: employment], 3.262,
3.5, 5.2521, 5.2523, 5.32, 5.5, 5.5521,
5.557, 6.001, 6.123, 6.126
a priori, 2.225, 3.04, 3.05, 5.133, 5.4541,
5.4731, 5.55, 5.5541, 5.5571, 5.634, 6.31,
6.3211, 6.33, 6.34, 6.35; cf. advance, in.
arbitrary, 3.315, 3.322, 3.342, 3.3442, 5.02,
113
5.473, 5.47321, 5.554, 6.124, 6.1271
argument, 3.333, 4.431, 5.02, 5.251, 5.47,
5.523, 5.5351; cf. truth-argument.
∼-place, 2.0131, 4.0411, 5.5351
arithmetic, 4.4611, 5.451
arrow, 3.144, 4.461
articulated [artikuliert] 3.141, 3.251; cf.
segmented.
ascribe [aussagen: speak; state; statement;
tell] 4.1241
assert
1. [behaupten], 4.122, 4.21, 6.2322
2. [zusprechen], 4.124
asymmetry, 6.3611
axiom, 6.341
∼ of infinity, 5.535
∼ of reducibility, 6.1232, 6.1233
bad, 6.43
basis, 5.21, 5.22, 5.234, 5.24, 5.25, 5.251,
5.442, 5.54
beautiful, 4.003
belief, 5.1361, 5.1363, 5.541, 5.542, 6.33,
6.3631
bound; ∼ary [Grenze: delimit; limit], 4.112,
4.463
brackets, 4.441, 5.46, 5.461
build [Bau: construction], 6.341
calculation, 6.126, 6.2331
cardinal, cf. number.
case, be the
1. [der Fall sein], 1, 1.12, 1.21, 2, 2.024,
3.342, 4.024, 5.1362, 5.5151, 5.541,
5.5542, 6.23
2. [So-Sein], 6.41
causality, 5.136–5.1362, 6.32, 6.321, 6.36,
6.3611, 6.362; cf. law.
certainty [Gewißheit], 4.464, 5.152, 5.156,
5.525, 6.3211
chain, 2.03; cf. concatenation.
clarification, 4.112
class [Klasse: set]. 3.311, 3.315, 4.1272, 6.031
clear, P2, 3.251, 4.112, 4.115, 4.116
make ∼ [erklären: definition; explanation],
5.452
colour, 2.0131, 2.0232, 2.0251, 2.171, 4.123,
6.3751
∼-space, 2.0131
combination
1. [Kombination], 4.27, 4.28, 5.46; cf. rule,
combinatory; truth-∼.
2. [Verbindung: connexion], 2.01, 2.0121,
4.0311, 4.221, 4.466, 4.4661, 5.131,
5.451, 5.515, 6.12, 6.1201, 6.121,
6.1221, 6.124, 6.23, 6.232; cf. sign.
common, 2.022, 2.16, 2.17, 2.18, 2.2, 3.31,
3.311, 3.317, 3.321, 3.322, 3.333, 3.341,
3.3411, 3.343–3.3441, 4.014, 4.12, 5.11,
5.143, 5.152, 5.24, 5.47, 5.4733, 5.512,
5.513, 5.5261, 6.022
comparison, 2.223, 3.05, 4.05, 6.2321, 6.3611
complete
1. [vollkommen: folly], 5.156
2. [vollstädig], 5.156;
analyse ∼ly, 3.201, 3.25;
describe ∼ly, 2.0201, 4.023, 4.26, 5.526,
6.342
complex, 2.0201, 3.1432, 3.24, 3.3442, 4.1272,
4.2211, 4.441, 5.515, 5.5423
composite [zummmengesetzt], 2.021, 3.143,
3.1431, 3.3411, 4.032, 4.2211, 5.47, 5.5261,
5.5421, 5.55
compulsion, 6.37
concatenation [Verkettung], 4.022; cf. chain.
concept [Begriff: primitive idea], 4.063,
4.126–4.1274, 4.431, 5.2523, 5.521, 5.555,
6.022; cf. formal ∼; pseudo-∼.
∼ual notation [Begriffsschrift], 3.325,
4.1272, 4.1273, 4.431, 5.533, 5.534
∼-word, 4.1272
concerned with [von etwas handeln: about;
deal with; subject-matter], 4.011, 4.122
concrete, 5.5563
condition, 4.41, 4.461, 4.462; cf. truth-∼.
configuration, 2.0231, 2.0271, 2.0272, 3.21
connexion
1. [Verbindung: combination], 6.124, 6.232
2. [Zusammenhang: nexus], 2.0122, 2.032,
2.15, 4.03, 5.1311, 5.1362, 6.361, 6.374
consequences, 6.422
conservation, cf. law.
constant, 3.312, 3.313, 4.126, 4.1271, 5.501,
5.522; cf. logical ∼.
constituent [Bestandteil], 2.011, 2.0201, 3.24,
3.315, 3.4, 4.024, 4.025, 5.4733, 5.533,
5.5423, 6.12
construct [bilden], 4.51, 5.4733, 5.475, 5.501,
5.503, 5.512, 5.514, 5.5151, 6.126, 6.1271
construction
1. [Bau: build], 4.002, 4.014, 5.45, 5.5262,
6.002
2. [Konstruktion], 4.023, 4.5, 5.233, 5.556,
6.343
contain [enthalten], 2.014, 2.203, 3.02, 3.13,
3.24, 3.332, 3.333, 5.121, 5.122, 5.44, 5.47
content
1. [Gehalt], 6.111
2. [Inhalt], 2.025, 3.13, 3.31
continuity, cf. law.
contradiction
1. [Kontradiktion], 4.46–4.4661 , 5.101,
114
5.143, 5.152, 5.525, 6.1202, 6.3751
2. [Widerspruch], 3.032, 4.1211, 4.211,
5.1241, 6.1201, 6.3751; cf. law of ∼.
convention
1. [Abmachung], 4.002
2. [Übereinkunft], 3.315, 5.02
co-ordinate, 3.032, 3.41, 3.42, 5.64
copula, 3.323
correct [richtig], 2.17, 2.173, 2.18, 2.21, 3.04,
5.5302, 5.62, 6.2321; cf. incorrect; true.
correlate [zuordnen], 2.1514, 2.1515, 4.43,
4.44, 5.526, 5.542, 6.1203
correspond [entsprechen], 2.13, 3.2, 3.21,
3.315, 4.0621, 4.063, 4.28, 4.441, 4.466,
5.5542
creation, 3.031, 5.123
critique of language, 4.0031
cube, 5.5423
Darwin, 4.1122
deal with [von etwas handeln: about;
concerned with; subject-matter], 2.0121
death, 6.431–6.4312
deduce [folgern], 5.132–5.134; cf. infer.
definition
1. [Definition], 3.24, 3.26–3.262, 3.343,
4.241, 5.42, 5.451, 5.452, 5.5302, 6.02
2. [Erklärung: clear, make; explanation],
5.154
delimit [begrenzen: bound; limit], 5.5262
depiction [Abbildung: form, logico-pictorial;
form, pictorial; pictorial], 2.16–2.172, 2.18,
2.19, 2.2, 2.201, 4.013, 4.014, 4.015, 4.016,
4.041
derive [ableiten], 4.0141, 4.243, 6.127, 6.1271;
cf. infer.
description [Beschreibung], 2.0201, 2.02331,
3.144, 3.24, 3.317, 3.33, 4.016, 4.023,
4.0641, 4.26, 4.5, 5.02
∼ of the world [Weltb.], 6.341, 6.343, 6.3432
designate [bezeichnen: sign; signify], 4.063
determin/ate [bestimmt], 2.031, 2.032, 2.14,
2.15, 3.14, 3.23, 3.251, 4.466, 6.124; cf.
indeterminateness; undetermined.
∼e, 1.11, 1.12, 2.0231, 2.05, 3.327, 3.4, 3.42,
4.063, 4.0641, 4.431, 4.463
difference [Verschiedenheit], 2.0233, 5.135,
5.53, 6.232, 6.3751
display [aufweisen], 2.172, 4.121; cf. show.
dissect [zerlegen], 3.26; cf. analysis.
doctrine [Lehre: theory], 4.112, 6.13
doubt, 6.51, 6.521
dualism, 4.128
duration, 6.4311
dynamical model, 4.04
effort, least, cf. law.
element, 2.13–2.14, 2.15, 2.151, 2.1514,
2.1515, 3.14, 3.2, 3.201, 3.24, 3.42
∼ary proposition [Elementarsatz],
4.21–4.221, 4.23, 4.24, 4.243–4.26,
4.28–4.42, 4.431, 4.45, 4.46, 4.51, 4.52,
5, 5.01, 5.101, 5.134, 5.152, 5.234,
5.3–5.32, 5.41, 5.47, 5.5, 5.524, 5.5262,
5.55, 5.555–5.5571, 6.001, 6.124,
6.3751
elucidation [Erläuterung], 3.263, 4.112, 6.54
empirical, 5.5561
employment
1. [Anwendung: application], 3.202, 3.323,
5.452
2. [Verwendung: use], 3.327
enumeration, 5.501
equal value, of [gleichwertig], 6.4
equality/, numerical [Zahlengleichheit], 6.022
sign of ∼ [Gleichheitszeichen: identity, sign
for], 6.23, 6.232
equation [Gleichung], 4.241, 6.2, 6.22, 6.232,
6.2323, 6.2341, 6.24
equivalent, cf. meaning, ∼ n. [äquivalent],
5.232, 5.2523, 5.47321, 5.514, 6.1261
essence [Wesen], 2.011, 3.143, 3.1431, 3.31,
3.317, 3.34–3.3421, 4.013, 4.016, 4.027,
4.03, 4.112, 4.1121, 4.465, 4.4661, 4.5, 5.3,
5.471, 5.4711, 5.501, 5.533, 6.1232, 6.124,
6.126, 6.127, 6.232, 6.2341
eternity, 6.4311, 6.4312; cf. sub specie aeterni.
ethics, 6.42–6.423
everyday language [Umgangssprache], 3.323,
4.002, 5.5563
existence
1. [Bestehen: hold; subsist], 2, 2.0121,
2.04–2.06, 2.062, 2.11, 2.201, 4.1, 4.122,
4.124, 4.125, 4.2, 4.21, 4.25, 4.27, 4.3,
5.131, 5.135
2. [Existenz], 3.032, 3.24, 3.323, 3.4, 3.411,
4.1274, 5.5151
experience [Erfahrung] 5.552, 5.553, 5.634,
6.1222, 6.363
explanation [Erklärung: clear, make;
definition], 3.263, 4.02, 4.021, 4.026, 4.431,
5.5422, 6.371, 6.372
exponent, 6.021
expression [Ausdruck: say], P3, 3.1, 3.12, 3.13,
3.142, 3.1431, 3.2, 3.24, 3.251, 3.262,
3.31–3.314, 3.318, 3.323, 3.33, 3.34, 3.341,
3.3441, 4.002, 4.013, 4.03, 4.0411, 4.121,
4.124, 4.125, 4.126, 4.1272, 4.1273, 4.241,
4.4, 4.43, 4.431, 4.441, 4.442, 4.5, 5.131,
5.22, 5.24, 5.242, 5.31, 5.476, 5.503, 5.5151,
5.525, 5.53, 5.5301, 5.535, 5.5352, 6.124,
6.1264, 6.21, 6.23, 6.232–6.2323, 6.24
mode of ∼ [Ausdrucksweise], 4.015, 5.21,
115
5.526
external, 2.01231, 2.0233, 4.023, 4.122, 4.1251
fact [Tatsache], 1.1–1.2, 2, 2.0121, 2.034, 2.06,
2.1, 2.141, 2.16, 3, 3.14, 3.142, 3.143, 4.016,
4.0312, 4.061, 4.063, 4.122, 4.1221, 4.1272,
4.2211, 4.463, 5.156, 5.43, 5.5151, 5.542,
5.5423, 6.2321, 6.43, 6.4321; cf. negative ∼.
fairy tale, 4.014
false [falsch: incorrect], 2.0212, 2.21, 2.22,
2.222–2.224, 3.24, 4.003, 4.023,
4.06–4.063, 4.25, 4.26, 4.28, 4.31, 4.41,
4.431, 4.46, 5.512, 5.5262, 5.5351, 6.111,
6.113, 6.1203; cf. wrong.
fate, 6.372, 6.374
feature [Zug], 3.34, 4.1221, 4.126
feeling, 4.122, 6.1232, 6.45
finite, 5.32
follow, 4.1211, 4.52, 5.11–5.132, 5.1363–5.142,
5.152, 5.43, 6.1201, 6.1221, 6.126
foresee, 4.5, 5.556
form [Form], 2.0122, 2.0141, 2.022–2.0231,
2.025–2.026, 2.033, 2.18, 3.13, 3.31, 3.312,
3.333, 4.002, 4.0031, 4.012, 4.063, 4.1241,
4.1271, 4.241, 4.242, 4.5, 5.131, 5.156,
5.231, 5.24, 5.241, 5.2522, 5.451, 5.46, 5.47,
5.501, 5.5351, 5.542, 5.5422, 5.55, 5.554,
5.5542, 5.555, 5.556, 5.6331, 6, 6.002, 6.01,
6.022, 6.03, 6.1201, 6.1203, 6.1224, 6.1264,
6.32, 6.34–6.342, 6.35, 6.422; cf. ∼al;
general ∼; propositional ∼; series of ∼s.
logical ∼, 2.0233, 2.18, 2.181, 2.2, 3.315,
3.327, 4.12, 4.121, 4.128, 5.555, 6.23,
6.33
logico-pictorial ∼ [logische Form der
Abbildung], 2.2
pictorial ∼ [Form der Abbildung: depiction;
pictorial], 2.15, 2.151, 2.17, 2.172,
2.181, 2.22
representational ∼ [Form der Darstellung:
present; represent], 2.173, 2.174
formal [formal], 4.122, 5.501
∼ concept, 4.126–4.1273
∼ property, 4.122, 4.124, 4.126, 4.1271,
5.231, 6.12, 6.122
∼ relation [Relation], 4.122, 5.242
formulate [angeben: give; say], 5.5563
free will, 5.1362
Frege, P6, 3.143, 3.318, 3.325, 4.063, 4.1272,
4.1273, 4.431, 4.442, 5.02, 5.132, 5.4, 5.42,
5.451, 5.4733, 5.521, 6.1271, 6.232
fully [vollkommen: complete], ∼ generalized,
5.526, 5.5261
function [Funktion], 3.318, 3.333, 4.126,
4.1272, 4.12721, 4.24, 5.02, 5.2341, 5.25,
5.251, 5.44, 5.47, 5.501, 5.52, 5.5301; cf.
truth-∼.
Fundamental Laws of Arithmetic
[Grundgesetze der Arithmetik], 5.451; cf.
primitive proposition.
future, 5.1361, 5.1362
general [allgemein], 3.3441, 4.0141, 4.1273,
4.411, 5.1311, 5.156, 5.242, 5.2522, 5.454,
5.46, 5.472, 5.521, 5.5262, 6.031, 6.1231,
6.3432
∼ form, 3.312, 4.1273, 4.5, 4.53, 5.46, 5.47,
5.471, 5.472, 5.54, 6, 6.002, 6.01, 6.022,
6.03
∼-ity-sign, 3.24, 4.0411, 5.522, 5.523, 6.1203
∼ validity, 6.1231, 6.1232
generalization [verallgemeinerung], 4.0411,
4.52, 5.156, 5.526, 5.5261, 6.1231 ; cf. fully.
geometry, 3.032, 3.0321, 3.411, 6.35
give [angeben: formulate; say], 3.317, 4.5,
5.4711, 5.55, 5.554, 6.35
given [gegeben], 2.0124, 3.42, 4.12721, 4.51,
5.442, 5.524, 6.002, 6.124
God, 3.031, 5.123, 6.372, 6.432
good, 4.003, 6.43
grammar, cf. logical
happy, 6.374
Hertz, 4.04, 6.361
hierarchy, 5.252, 5.556, 5.5561
hieroglyphic script, 4.016
higher, 6.42, 6.432
hold [bestehen: existence; subsist], 4.014
how [wie], 6.432, 6.44; cf. stand, ∼ things.
∼) (what, 3.221, 5.552
hypothesis, 4.1122, 5.5351, 6.36311
idea, cf. primitive ∼.
1. [Gedanke: thought], musical ∼, 4.014
2. [Vorstellung: present; represent], 5.631
idealist, 4.0412
identical [identisch], 3.323, 4.003, 4.0411,
5.473, 5.4733, 5.5303, 5.5352, 6.3751; cf.
difference.
identity [Gleichheit], 5.53
sign for ∼ [Gleichheitszeichen: equality, sign
of], 3.323, 5.4733, 5.53, 5.5301, 5.533;
cf. equation.
illogical [unlogisch], 3.03, 3.031, 5.4731
imagine [sich etwas denken: think], 2.0121,
2.022, 4.01, 6.1233
immortality, 6.4312
impossibility [Unmöglichkeit], 4.464, 5.525,
5.5422, 6.375, 6.3751
incorrect
1. [falsch: false], 2.17, 2.173, 2.18
2. [unrichtig], 2.21
independence [Selbständigkeit], 2.0122, 3.261
independent [unabhängig], 2.024, 2.061, 2.22,
116
4.061, 5.152, 5.154, 5.451, 5.5261, 5.5561,
6.373
indetermlnateness [Unbestimmtheit] 3.24
indicate
1. [anzeigen], 3.322, 6.121, 6.124
2. [auf etwas zeigen: manifest; show],
2.02331, 4.063
individuals, 5.553
induction, 6.31, 6.363
infer [schließen], 2.062, 4.023, 5.1311, 5.132,
5.135, 5.1361, 5.152, 5.633, 6.1224, 6.211;
cf. deduce; derive.
infinite, 2.0131, 4.2211, 4.463, 5.43, 5.535,
6.4311
infinity, cf. axiom.
inner, 4.0141, 5.1311, 5.1362
internal, 2.01231, 3.24, 4.014, 4.023,
4.122–4.1252, 5.131, 5.2, 5.21, 5.231, 5.232
intuition [Anschauung], 6.233, 6.2331
intuitive [anschaulich], 6.1203
judgement [Urteil], 4.063, 5.5422
∼-stroke [Urteilstrich], 4.442
Julius Caesar, 5.02
Kant, 6.36111
know
1. [kennen], 2.0123, 2.01231, 3.263, 4.021,
4.243, 6.2322; cf. theory of knowledge.
2. [wissen], 3.05, 3.24, 4.024, 4.461, 5.1362,
5.156, 5.5562 [—the previous entry
was mistakenly listed as 5.562 in Pears
and McGuinness’s original index—],
6.3211, 6.33, 6.36311
language [Sprache], P2, P4, 3.032, 3.343,
4.001–4.0031, 4.014, 4.0141, 4.025, 4.121,
4.125, 5.4731, 5.535, 5.6, 5.62, 6.12, 6.233,
6.43; cf. critique of ∼; everyday ∼; sign-∼.
law
1. [Gesetz: minimum-principle; primitive
proposition], 3.031, 3.032, 3.0321,
4.0141, 5.501, 6.123, 6.3–6.3211,
6.3431, 6.35, 6.361, 6.363, 6.422;
∼ of causality [Kausalitätsg.], 6.32,
6.321;
∼ of conservation [Erhaltungsg.], 6.33;
∼ of contradiction [G. des Widerspruchs],
6.1203, 6.123;
∼ of least action [G. der kleinsten
Wirkung], 6.321, 6.3211;
∼ of nature [Naturg.], 5.154, 6.34, 6.36,
6.371, 6.372
2. [Satz: principle of sufficient reason;
proposition], 6.34;
∼ of continuity [S. von der Kontinuität],
6.34;
∼ of least effort [S. von kleinsten
Aufwande], 6.34
life, 5.621, 6.4311, 6.4312, 6.52, 6.521
limit [Grenze: bound; delimit], P3, P4, 4.113,
4.114, 4.51, 5.143, 5.5561, 5.6–5.62, 5.632,
5.641, 6.4311, 6.45
logic; ∼al, 2.012, 2.0121, 3.031, 3.032, 3.315,
3.41, 3.42, 4.014, 4.015, 4.023, 4.0312,
4.032, 4.112, 4.1121, 4.1213, 4.126, 4.128,
4.466, 5.02, 5.1362, 5.152, 5.233, 5.42,
5.43, 5.45–5.47, 5.472–5.4731, 5.47321,
5.522, 5.551–5.5521, 5.555, 5.5562–5.557,
5.61, 6.1–6.12, 6.121, 6.122, 6.1222–6.2,
6.22, 6.234, 6.3, 6.31, 6.3211, 6.342,
6.3431, 6.3631, 6.37, 6.374–6.3751; cf.
form, ∼al; illogical.
∼al addition, 5.2341
∼al constant, 4.0312, 5.4, 5.441, 5.47
∼al grammar, 3.325
∼al multiplication, 5.2341
∼al object, 4.441, 5.4
∼al picture, 2.18–2.19, 3, 4.03
∼al place, 3.41–3.42, 4.0641
∼al product, 3.42, 4.465, 5.521, 6.1271,
6.3751
∼al space, 1.13, 2.11, 2.202, 3.4, 3.42, 4.463
∼al sum, 3.42, 5.521
∼al syntax, 3.325, 3.33, 3.334, 3.344, 6.124
∼o-pictorial, cf. form.
∼o-syntactical, 3.327
manifest [sich zeigen: indicate; show], 4.122,
5.24, 5.4, 5.513, 5.515, 5.5561, 5.62, 6.23,
6.36, 6.522
material, 2.0231, 5.44
mathematics, 4.04–4.0411, 5.154, 5.43, 5.475,
6.031, 6.2–6.22, 6.2321, 6.233, 6.234–6.24
Mauthner, 4.0031
mean [meinen], 3.315, 4.062, 5.62
meaning [Bedeutung: signify], 3.203, 3.261,
3.263, 3.3, 3.314, 3.315, 3.317, 3.323,
3.328–3.331, 3.333, 4.002, 4.026, 4.126,
4.241–4.243, 4.466, 4.5, 5.02, 5.31 , 5.451,
5.461, 5.47321, 5.4733, 5.535, 5.55, 5.6,
5.62, 6.124, 6.126, 6.232, 6.2322, 6.53
equivalent in ∼ [Bedeutungsgleichheit],
4.243, 6.2323
∼ful [bedeutungsvoll], 5.233
∼less [bedeutungslos], 3.328, 4.442, 4.4661,
5.47321
mechanics, 4.04, 6.321, 6.341–6.343, 6.3432
mention [von etwas reden: talk about], 3.24,
3.33, 4.1211, 5.631, 6.3432; cf. about.
metaphysical, 5.633, 5.641, 6.53
method, 3.11, 4.1121, 6.121, 6.2, 6.234–6.24,
6.53; cf. projection, ∼ of; zero-∼.
117
microcosm, 5.63
minimum-principle [Minimum-Gesetz: law],
6.321
mirror, 4.121, 5.511, 5.512, 5.514, 6.13
∼-image [Spiegelbild: picture], 6.13
misunderstanding, P2
mode, cf. expression; signification.
model, 2.12, 4.01, 4.463; cf. dynamical ∼.
modus ponens, 6.1264
monism, 4.128
Moore, 5.541
multiplicity, 4.04–4.0412, 5.475
music, 3.141, 4.011, 4.014, 4.0141
mystical, 6.44, 6.45, 6.522
name
1. [Name], 3.142, 3.143, 3.144, 3.202, 3.203,
3.22, 3.26, 3.261, 3.3, 3.314, 3.3411,
4.0311, 4.126, 4.1272, 4.22, 4.221, 4.23,
4.24, 4.243, 4.5, 5.02, 5.526, 5.535, 5.55,
6.124; cf. variable ∼.
general ∼ [Gattungsn.], 6.321
proper ∼ of a person [Personenn.], 3.323
2. [benennen; nennen], 3.144, 3.221
natur/e, 2.0123, 3.315, 5.47, 6.124; cf. law of
∼e.
∼al phenomena, 6.371
∼al science, 4.11, 4.111, 4.1121–4.113,
6.111, 6.4312, 6.53
necessary, 4.041, 5.452, 5.474, 6.124; cf.
unnecessary.
negation
1. [Negation], 5.5, 5.502
2. [Verneinung], 3.42, 4.0621, 4.064, 4.0641,
5.1241, 5.2341, 5.254, 5.44, 5.451, 5.5,
5.512, 5.514, 6.231
negative [negativ], 4.463, 5.513, 5.5151
∼ fact, 2.06, 4.063, 5.5151
network, 5.511, 6.341, 6.342, 6.35
Newton, 6.341, 6.342
nexus
1. [Nexus], 5.136, 5.1361
2. [Zusammenhang: connexion], 3.3, 4.22,
4.23
non-proposition, 5.5351
nonsense [Unsinn], P4, 3.24, 4.003, 4.124,
4.1272, 4.1274, 4.4611, 5.473, 5.5303,
5.5351, 5.5422, 5.5571, 6.51, 6.54; cf. sense,
have no.
notation, 3.342, 3.3441, 5.474, 5.512–5.514,
6.1203, 6.122, 6.1223; cf. conceptual ∼.
number
1. [Anzahl], 4.1272, 5.474–5.476, 5.55, 5.553,
6.1271
2. [Zahl: integer], 4.1252, 4.126, 4.1272,
4.12721, 4.128, 5.453, 5.553, 6.02,
6.022; cf. equality, numerical;
privileged ∼s; series of ∼s; variable ∼.
cardinal ∼, 5.02;
∼-system, 6.341
object [Gegenstand], 2.01, 2.0121,
2.0123–2.0124, 2.0131–2.02, 2.021,
2.023–2.0233, 2.0251–2.032, 2.13, 2.15121,
3.1431, 3.2, 3.203–3.221, 3.322, 3.3411,
4.023, 4.0312, 4.1211, 4.122, 4.123, 4.126,
4.127, 4.1272, 4.12721, 4.2211, 4.431,
4.441, 4.466, 5.02, 5.123, 5.1511, 5.4, 5.44,
5.524, 5.526, 5.53–5.5302, 5.541, 5.542,
5.5561, 6.3431 ; cf. thing.
obvious [sich von selbst verstehen: say;
understand], 6.111; cf. self-evidence.
Occam, 3.328, 5.47321
occur [vorkommen], 2.012–2.0123, 2.0141,
3.24, 3.311, 4.0621, 4.1211, 4.23, 4.243,
5.25, 5.451, 5.54, 5.541, 6.1203
operation, 4.1273, 5.21–5.254, 5.4611, 5.47,
5.5, 5.503, 6.001–6.01, 6.021, 6.126; cf.
sign for a logical ∼; truth-∼.
oppos/ed; ∼ite [entgegengesetzt], 4.0621, 4.461,
5.1241, 5.513
order, 4.1252, 5.5563, 5.634
paradox, Russell’s, 3.333
particle, 6.3751
perceive, 3.1, 3.11, 3.32, 5.5423
phenomenon, 6.423; cf. natural ∼.
philosophy, P2, P5, 3.324, 3.3421, 4.003,
4.0031, 4.111–4.115, 4.122, 4.128, 5.641,
6.113, 6.211, 6.53
physics, 3.0321, 6.321, 6.341, 6.3751
pictorial
1. [abbilden: depict; form, logico-∼], 2.15,
2.151, 2.1513, 2.1514, 2.17, 2.172,
2.181, 2.22; cf. form, ∼.
2. [bildhaftig], 4.013, 4.015
picture [Bild: mirror-image; tableau vivant],
2.0212, 2.1–2.1512, 2.1513–3.01, 3.42,
4.01–4.012, 4.021, 4.03, 4.032, 4.06, 4.462,
4.463, 5.156, 6.341, 6.342, 6.35; cf. logical
∼; prototype.
place [Ort], 3.411, 6.3751; cf. logical ∼.
point-mass [materieller Punkt], 6.3432
positive, 2.06, 4.063, 4.463, 5.5151
possible, 2.012, 2.0121, 2.0123–2.0141, 2.033,
2.15, 2.151, 2.201–2.203, 3.02, 3.04, 3.11,
3.13, 3.23, 3.3421, 3.3441, 3.411, 4.015,
4.0312, 4.124, 4.125, 4.2, 4.27–4.3, 4.42,
4.45, 4.46, 4.462, 4.464, 4.5, 5.252, 5.42,
5.44, 5.46, 5.473, 5.4733, 5.525, 5.55, 5.61,
6.1222, 6.33, 6.34, 6.52; cf. impossibility;
truth-possibility.
postulate [Forderung: requirement], 6.1223
predicate, cf. subject.
118
present
1. [darstellen: represent], 3.312, 3.313, 4.115
2. [vorstellen: Idea; represent], 2.11, 4.0311
presuppose [voraussetzen], 3.31, 3.33, 4.1241,
5.515, 5.5151, 5.61, 6.124
primitive idea [Grundbegriff] 4.12721, 5.451,
5.476
primitive proposition [Grundgesetz], 5.43,
5.452, 6.127, 6.1271; cf. Fundamental
Laws of Arithmetic; law.
primitive sign [Urzeichen], 3.26, 3.261, 3.263,
5.42, 5.45, 5.451, 5.46, 5.461, 5.472
Principia Mathematica, 5.452
principle of sufficient reason [Satz vom
Grunde: law; proposition], 6.34, 6.35
Principles of Mathematics, 5.5351
privileged [ausgezeichnet], ∼ numbers, 4.128,
5.453, 5.553
probability, 4.464, 5.15–5.156
problem
1. [Fragestellung: question], P2, 5.62
2. [Problem], P2, 4.003, 5.4541, 5.535, 5.551,
5.5563, 6.4312, 6.521
product, cf. logical.
project/ion; ∼ive, 3.11–3.13, 4.0141
method of ∼ion, 3.11
proof [Beweis], 6.126, 6.1262, 6.1263–6.1265,
6.2321, 6.241
proper, cf. name.
property [Eigenschaft], 2.01231, 2.0231,
2.0233, 2.02331, 4.023, 4.063,
4.122–4.1241, 5.473, 5.5302, 6.111, 6.12,
6.121, 6.126, 6.231, 6.35; cf. formal ∼.
proposition [Satz: law; principle], 2.0122,
2.0201, 2.0211, 2.0231, 3.1 (& passim
thereafter); cf. non-∼; primitive ∼;
pseudo-∼; variable, ∼al; variable ∼.
∼al form, 3.312, 4.0031, 4.012, 4.5, 4.53,
5.131, 5.1311, 5.156, 5.231, 5.24, 5.241,
5.451, 5.47, 5.471, 5.472, 5.54–5.542,
5.5422, 5.55, 5.554, 5.555, 5.556, 6,
6.002
∼al sign, 3.12, 3.14, 3.143, 3.1431, 3.2, 3.21,
3.332, 3.34, 3.41, 3.5, 4.02, 4.44, 4.442,
5.31
prototype [Urbild], 3.24, 3.315, 3.333, 5.522,
5.5351; cf. picture.
pseudo-, cf. apparent.
∼-concept, 4.1272
∼-proposition, 4.1272, 5.534, 5.535, 6.2
∼relation, 5.461
psychology, 4.1121, 5.541, 5.5421, 5.641,
6.3631, 6.423
punishment, 6.422
question [Frage: problem], 4.003, 4.1274,
5.4541, 5.55, 5.551, 5.5542, 6.5–6.52
range [Spielraum], 4.463, 5.5262; cf. space.
real [wirklich], 2.022, 4.0031, 5.461
realism, 5.64
reality
1. [Realität], 5.5561, 5.64
2. [Wirklichkeit], 2.06, 2.063, 2.12, 2.1511,
2.1512, 2.1515, 2.17, 2.171, 2.18, 2.201,
2.21, 2.222, 2.223, 4.01, 4.011, 4.021,
4.023, 4.05, 4.06, 4.0621, 4.12, 4.121,
4.462, 4.463, 5.512
reducibility, cf. axiom.
relation
1. [Beziehung], 2.1513, 2.1514, 3.12, 3.1432,
3.24, 4.0412, 4.061, 4.0641, 4.462
4.4661, 5.131, 5.1311, 5.2–5.22, 5.42,
5.461, 5.4733, 5.5151, 5.5261, 5.5301;
cf. pseudo-.
2. [Relation], 4.122, 4.123, 4.125, 4.1251,
5.232, 5.42, 5.5301, 5.541, 5.553,
5.5541 ; cf. formal.
3. stand in a ∼ to one another; are related
[sich verhalten: stand, how things;
state of things], 2.03, 2.14, 2.15, 2.151,
3.14, 5.5423
represent
1. [darstellen: present], 2.0231, 2.173, 2.174,
2.201–2.203, 2.22, 2.221, 3.032, 3.0321,
4.011, 4.021, 4.031, 4.04, 4.1, 4.12,
4.121, 4.122, 4.124, 4.125, 4.126,
4.1271, 4.1272, 4.24, 4.31, 4.462, 5.21,
6.1203, 6.124, 6.1264; cf. form,
∼ational.
2. [vorstellen: idea; present], 2.15
representative, be the ∼ of [vertreten], 2.131,
3.22, 3.221, 4.0312, 5.501
requirement [Forderung: postulate], 3.23
resolve, cf. analysis.
1. [auflösen], 3.3442
2. [zerlegen], 2.0201
reward, 6.422
riddle, 6.4312, 6.5
right [stimmen: agreement; true], 3.24
rule [Regel], 3.334, 3.343, 3.344, 4.0141,
5.47321, 5.476, 5.512, 5.514
combinatory ∼ [Kombinationsr.], 4.442
∼ dealing with signs [Zeichenr.] 3.331, 4.241,
6.02, 6.126
Russell, P6, 3.318, 3.325, 3.331, 3.333, 4.0031,
4.1272–4.1273, 4.241, 4.442, 5.02, 5.132,
5.252, 5.4, 5.42, 5.452, 5.4731, 5.513,
5.521, 5.525, 5.5302, 5.532, 5.535, 5.5351,
5.541, 5.5422, 5.553, 6.123, 6.1232
say
1. [angeben: give], 5.5571
119
2. [audrücksen: expression], 5.5151
3. [aussprechen: words, put into], ∼ clearly,
3.262
4. [sagen], can be said, P3, 3.031, 4.115,
4.1212, 5.61 5.62, 6.36, 6.51, 6.53;
said) (shown, 4.022, 4.1212, 5.535, 5.62,
6.36;
∼ nothing, 4.461, 5.142, 5.43, 5.4733,
5.513, 5.5303, 6.11, 6.121, 6.342,
6.35
5. [sich von selbst verstehen: obvious;
understand], ∼ing,
go without, 3.334, 6.2341
scaffolding, 3.42, 4.023, 6.124
scepticism, 6.51
schema, 4.31, 4.43, 4.441, 4.442, 5.101, 5.151,
5.31
science, 6.34, 6.341, 6.52; cf. natural ∼.
scope, 4.0411
segmented [gegliedert], 4.032; cf. articulated.
self, the [das Ich], 5.64, 5.641
self-evidence [Einleuchten], 5.1363, 5.42,
5.4731, 5.5301, 6.1271; cf. obvious.
sense [Sinn; sinnvoll], P2, 2.0211, 2.221,
2.222, 3.11, 3.13, 3.142, 3.1431, 3.144,
3.23, 3.3, 3.31, 3.326, 3.34, 3.341, 3.4,
4.002, 4.011, 4.014, 4.02–4.022,
4.027–4.031, 4.032, 4.061, 4.0621–4.064,
4.1211, 4.122, 4.1221, 4.1241, 4.126, 4.2,
4.243, 4.431, 4.465, 4.52, 5.02, 5.122,
5.1241, 5.2341, 5.25, 5.2521, 5.4, 5.42,
5.44, 5.46, 5.4732, 5.4733, 5.514, 5.515,
5.5302, 5.5542, 5.631, 5.641, 6.124, 6.126,
6.232, 6.41, 6.422, 6.521
have the same ∼ [gleichsinnig], 5.515
have no ∼; lack ∼; without ∼ [sinnlos], 4.461,
5.132, 5.1362, 5.5351; cf. nonsense.
∼ of touch [Tastsinn], 2.0131
series [Reihe], 4.1252, 4.45, 5.1, 5.232, 6.02
∼ of forms [Formenr.], 4.1252, 4.1273, 5.252,
5.2522, 5.501
∼ of numbers [Zahlenr.], 4.1252
set [Klasse: class], 3.142
show [zeigen: indicate; manifest], 3.262, 4.022,
4.0621, 4.0641, 4.121–4.1212, 4.126, 4.461,
5.1311, 5.24, 5.42, 5.5261, 5.5421, 5.5422,
5.631, 6.12, 6.1201, 6.1221, 6.126, 6.127,
6.22, 6.232; cf. display; say.
sign [Zeichen], 3.11, 3.12, 3.1432, 3.201–3.203,
3.21, 3.221, 3.23, 3.261–3.263, 3.315,
3.32–3.322, 3.325–3.334, 3.3442, 4.012,
4.026, 4.0312, 4.061, 4.0621, 4.126, 4.1271,
4.1272, 4.241–4.243, 4.431–4.441, 4.466,
4.4661, 5.02, 5.451, 5.46, 5.473,
5.4732–5.4733, 5.475, 5.501, 5.512, 5.515,
5.5151, 5.53, 5.5541, 5.5542, 6.02, 6.1203,
6.124, 6.126, 6.1264, 6.53; cf. primitive ∼;
propositional ∼; rule dealing with ∼s;
simple ∼.
be a ∼ for [bezeichnen: designate; signify],
5.42
combination of ∼s [Zeichenverbindung],
4.466, 5.451
∼ for a logical operation [logisches
Operationsz.], 5.4611
∼-language [Zeichensprache], 3.325, 3.343,
4.011, 4.1121, 4.1213, 4.5, 6.124
signif/y
1. [bedeuten: meaning], 4.115
2. [bezeichnen: designate: sign], 3.24, 3.261,
3.317, 3.321, 3.322, 3.333, 3.334,
3.3411, 3.344, 4.012, 4.061, 4.126,
4.127, 4.1272, 4.243, 5.473, 5.4733,
5.476, 5.5261, 5.5541, 6.111;
mode of ∼ication [Bezeichnungsweise],
3.322, 3.323, 3.325, 3.3421, 4.0411,
5.1311
similarity, 4.0141, 5.231
simple, 2.02, 3.24, 4.21, 4.24, 4.51, 5.02,
5.4541, 5.553, 5.5563, 6.341, 6.342, 6.363,
6.3631;
∼ sign, 3.201, 3.202, 3.21, 3.23, 4.026
simplex sigillum veri, 5.4541
situation [Sachlage], 2.0121, 2.014, 2.11,
2.202, 2.203, 3.02, 3.11, 3.144, 3.21, 4.021,
4.03, 4.031, 4.032 4.04, 4.124, 4.125, 4.462,
4.466, 5.135, 5.156, 5.525
Socrates, 5.473, 5.4733
solipsism, 5.62, 5.64
solution, P8, 5.4541, 5.535, 6.4312, 6.4321,
6.521
soul, 5.5421, 5.641, 6.4312
space [Raum], 2.0121, 2.013, 2.0131, 2.0251,
2.11, 2.171, 2.182, 2.202, 3.032–3.0321,
3.1431, 4.0412, 4.463, 6.3611, 6.36111,
6.4312; cf. colour-∼; logical ∼; range.
speak/ about [von etwas sprechen], 3.221,
6.3431, 6.423, 7; cf. about.
∼ for itself [aussagen: ascribe; state;
statement; tell], 6.124
stand/, how things [sich verhalten: relation;
state of things], 4.022, 4.023, 4.062, 4.5
∼ for [für etwas stehen], 4.0311, 5.515
state [aussagen: ascribe; speak; statement;
tell], 3.317, 4.03, 4.242, 4.442, 6.1264
statement [Aussage], 2.0201, 6.3751
make a ∼ [aussagen: ascribe; speak; state;
tell], 3.332, 5.25
state of/ affairs [Sachverhalt: ∼ things],
2–2.013, 2.014, 2.0272- 2.062, 2.11, 2.201,
3.001, 3.0321, 4.023, 4.0311, 4.1, 4.122,
120
4.2, 4.21, 4.2211, 4.25, 4.27, 4.3
∼ things
1. [Sachverhalt: ∼ affairs], 2.01
2. [sich verhalten: relation; stand, how
things], 5.552
stipulate [festsetzen], 3.316, 3.317, 5.501
structure [Struktur], 2.032–2.034, 2.15,
4.1211, 4.122, 5.13, 5.2, 5.22, 6.12, 6.3751
subject
1. [Subjekt], 5.5421, 5.631–5.633, 5.641;
∼-predicate propositions, 4.1274
2. [Träger], 6.423
3. ∼-matter [von etwas handeln: about;
concerned with; deal with], 6.124
subsistent [bestehen: existence; hold], 2.024,
2.027, 2.0271
sub specie aeterni, 6.45; cf. eternity.
substance [Substanz], 2.021, 2.0211, 2.0231,
2.04
substitut/e, 3.344, 3.3441, 4.241, 6.23, 6.24
∼ion, method of, 6.24
successor [Nachfolger], 4.1252, 4.1273
sum, cf. logical.
sum-total [gesamt: totality; whole], 2.063
superstition, 5.1361
supposition [Annahme], 4.063
survival [Fortleben], 6.4312
symbol [Symbol], 3.24, 3.31, 3.317, 3.32, 3.321,
3.323, 3.325, 3.326, 3.341, 3.3411, 3.344,
4.126, 4.24, 4.31, 4.465, 4.4661, 4.5,
5.1311, 5.473, 5.4733, 5.513–5.515, 5.525,
5.5351, 5.555, 6.113, 6.124, 6.126
∼ism [Symbolismus], 4.461, 5.451
syntax, cf. logical.
system, 5.475, 5.555, 6.341, 6.372; cf.
number-∼.
tableau vivant [lebendes Bild: picture], 4.0311
talk about [von etwas reden: mention], P2,
5.641, 6.3432; cf. about.
tautology, 4.46–4.4661, 5.101, 5.1362, 5.142,
5.143, 5.152, 5.525, 6.1, 6.12–6.1203,
6.1221, 6.1231, 6.124, 6.126, 6.1262, 6.127,
6.22, 6.3751
tell [aussagen: ascribe; speak; state;
statement], 6.342
term [Glied], 4.1273, 4.442, 5.232, 5.252,
5.2522, 5.501
theory
1. [Lehre: doctrine], 6.1224;
∼ of probability, 4.464
2. [Theorie], 4.1122, 5.5422, 6.111;
∼ of classes, 6.031;
∼ of knowledge, 4.1121, 5.541;
∼ of types, 3.331, 3.332
thing, cf. object; state of affairs; state of ∼s.
1. [Ding], 1.1, 2.01–2.0122, 2.013, 2.02331,
2.151, 3.1431, 4.0311, 4.063, 4.1272,
4.243, 5.5301, 5.5303, 5.5351, 5.5352,
5.553, 5.634, 6.1231
2. [Sache], 2.01, 2.15, 2.1514, 4.1272
think [denken: imagine], P3, 3.02, 3.03, 3.11,
3.5, 4.114, 4.116, 5.4731, 5.541, 5.542,
5.61, 5.631
∼able [denkbar] P3, 3 3.001, 3.02, 6.361; cf.
unthinkable.
thought [Gedanke: idea], P3, 3, 3.01, 3.02,
3.04–3.1, 3.12, 3.2, 3.5, 4, 4.002, 4.112,
6.21
∼-process [Denkprozeß], 4.1121
time, 2.0121, 2.0251, 6.3611, 6.3751, 6.4311,
6.4312
totality [Gesamtheit: sum-total; whole], 1.1,
1.12, 2.04, 2.05, 3.01, 4.001, 4.11, 4.52,
5.5262, 5.5561
transcendental, 6.13, 6.421
translation, 3.343, 4.0141, 4.025, 4.243
tru/e
1. [Faktum], 5.154
2. [wahr], 2.0211, 2.0212, 2.21, 2.22,
2.222–2.225, 3.01, 3.04, 3.05,
4.022–4.024, 4.06–4.063, 4.11, 4.25,
4.26, 4.28, 4.31, 4.41, 4.43, 4.431, 4.442,
4.46, 4.461, 4.464, 4.466, 5.11, 5.12,
5.123, 5.13, 5.131, 5.1363, 5.512,
5.5262, 5.5352, 5.5563, 5.62, 6.111,
6.113, 6.1203, 6.1223, 6.1232, 6.125,
6.343; cf. correct; right.
come ∼e [stimmmen: agreement; right],
5.123
∼th-argument, 5.01, 5.101, 5.152, 6.1203
∼th-combination, 6.1203
∼th-condition, 4.431, 4.442, 4.45–4.461,
4.463
∼th-function, 3.3441, 5, 5.1, 5.101, 5.234,
5.2341, 5.3, 5.31, 5.41, 5.44, 5.5, 5.521,
6
∼th-ground, 5.101–5.121, 5.15
∼th-operation, 5.234, 5.3, 5.32, 5.41, 5.442,
5.54
∼th-possibility, 4.3–4.44, 4.442, 4.45, 4.46,
5.101
∼th-value, 4.063
type, 3.331, 3.332, 5.252, 6.123; cf. prototype.
unalterable [fest], 2.023, 2.026–2.0271
understand [verstehen: obvious; say], 3.263,
4.002, 4.003, 4.02, 4.021, 4.024, 4.026,
4.243, 4.411, 5.02, 5.451, 5.521, 5.552,
5.5562, 5.62; cf. misunderstanding.
make oneself understood [sich verständigen],
4.026, 4.062
undetermined [nicht bestimmt], 3.24, 4.431
121
unit, 5.155, 5.47321
unnecessary, 5.47321
unthinkable, 4.123
use
1. [Gebrauch], 3.326, 4.123, 4.1272, 4.241,
6.211;
∼less [nicht gebraucht], 3.328
2. [Verwendung: employment], 3.325, 4.013,
6.1202
validity, 6.1233; cf. general ∼.
value [Wert], 6.4, 6.41 ; cf. truth-∼.
∼ of a variable, 3.313, 3.315–3.317, 4.127,
4.1271, 5.501, 5.51, 5.52
variable, 3.312–3.317, 4.0411, 4.1271, 4.1272,
4.1273, 4.53, 5.24, 5.242, 5.2522, 5.501,
6.022
propositional ∼ [Satzvariable], 3.313, 3.317,
4.126, 4.127, 5.502
∼ name, 3.314, 4.1272
∼ number, 6.022
∼ proposition [variabler Satz], 3.315
visual field, 2.0131, 5.633, 5.6331, 6.3751,
6.4311
Whitehead, 5.252, 5.452
whole [gesamt: sum-total; totality], 4.11, 4.12
will [Wille; wollen] 5.1362, 5.631, 6.373, 6.374,
6.423, 6.43
wish [wünschen], 6.374
word [Wort], 2.0122, 3.14, 3.143, 3.323, 4.002,
4.026, 4.243, 6.211; cf. concept-∼.
put into ∼s [aussprechen; unausprechlich:
say], 3.221, 4.116, 6.421, 6.5, 6.522
world, 1–1.11, 1.13, 1.2, 2.021–2.022, 2.0231,
2.026, 2.063, 3.01, 3.12, 3.3421, 4.014,
4.023, 4.12, 4.2211, 4.26, 4.462, 5.123,
5.4711, 5.511, 5.526–5.5262, 5.551, 5.5521,
5.6–5.633, 5.641, 6.12, 6.1233, 6.124, 6.22,
6.342, 6.3431, 6.371, 6.373, 6.374, 6.41,
6.43, 6.431, 6.432, 6.44, 6.45, 6.54; cf.
description of the ∼.
wrong [nicht stimmen: agreement; true], 3.24;
cf. false.
zero-method, 6.121
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†1 In Russell's symbolism, the cardinal number of the universal class, i.e. of all objects. [Edd.]
Page 131
†2 Russell's Introduction to the Tractatus. [Edd.]
Page 133
†* See Preface.
PHILOSOPHICAL REMARKS
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Ludwig Wittgenstein from Blackwell Publishers
The Wittgenstein Reader
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Zettel
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Remarks on Colour
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Edited by G.E.M. Anscombe and Heikki Nyman
Last Writings on the Philosophy of Psychology, Vol. I
Edited by G.H. von Wright and Heikki Nyman
Last Writings on the Philosophy of Psychology, Vol. II
Edited by G.H. von Wright and Heikki Nyman
Philosophical Investigations (German-English edition)
Translated by G.E.M. Anscombe
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Titlepage
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LUDWIG WITTGENSTEIN
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PHILOSOPHICAL REMARKS
Edited from his posthumous
writings by Rush Rhees
and translated into English
by Raymond Hargreaves and
Roger White
BASIL BLACKWELL • OXFORD
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Copyright page
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Copyright © in this translation
Basil Blackwell 1975
Reprinted 1990, 1998
First German edition 1964
All Rights Reserved. No part of this publication
may be reproduced, stored in a retrieval system,
or transmitted, in any form or by any means, electronic,
mechanical, photocopying, recording or otherwise, without the prior
permission of Blackwell Publishers Limited.
The editor was enabled to prepare this volume for
publication through the generosity of the Nuffield
Foundation. He owes it particular gratitude. R. R.
ISBN 0 631 19130 5
Printed in Great Britain by Athenaeum Press Ltd, Gateshead,
Tyne & Wear.
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Et multi ante nos vitam istam agentes, praestruxerant aerumnosas vias, per quas transire cogebamur multiplicato labore
et dolore filiis Adam.
Augustine
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FOREWORD
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This book is written for such men as are in sympathy with its spirit. This spirit is different from the one which
informs the vast stream of European and American civilization in which all of us stand. That spirit expresses
itself in an onwards movement, in building ever larger and more complicated structures; the other in striving
after clarity and perspicuity in no matter what structure. The first tries to grasp the world by way of its
periphery--in its variety; the second at its centre--in its essence. And so the first adds one construction to
another, moving on and up, as it were, from one stage to the next, while the other remains where it is and what it
tries to grasp is always the same.
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I would like to say 'This book is written to the glory of God', but nowadays that would be chicanery, that
is, it would not be rightly understood. It means the book is written in good will, and in so far as it is not so
written, but out of vanity, etc., the author would wish to see it condemned. He cannot free it of these impurities
further than he himself is free of them.
November 1930 L. W.
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ANALYTICAL TABLE OF CONTENTS
I
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1 A proposition is completely logically analysed if its grammar is made clear--in no matter what idiom. All that is
possible and necessary is to separate what is essential from what is inessential in our language--which amounts to
the construction of a phenomenological language. Phenomenology as the grammar of those facts on which physics
builds its theories. 51
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2 The complexity of philosophy is not in its matter, but in our tangled understanding. 52
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3 How strange if logic were concerned with an 'ideal' language, and not with ours! 52
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4 If I could describe the point of grammatical conventions by saying they are made necessary by certain properties
of the colours (say)--then what would make the conventions superfluous, since in that case I would be able to say
precisely that which the conventions exclude my saying. 53
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5 Might we say: A child must of course learn to speak a particular language, but not to think?
53
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6 In a certain sense, the use of language is something that cannot be taught. 54
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7 Grammatical conventions cannot be justified by describing what is represented: any such description already
presupposes the grammatical rules. 54
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8 The kind of co-ordination on the basis of which a heard or seen language functions would be, say: 'If you hear a
shot or see me wave, run.' 55
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9 Have philosophers hitherto always spoken nonsense? 55
II
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10 Thinking of propositions as instructions for making models. For it to be possible for an expression to guide my
hand, it must have the same multiplicity as the action desired. And this must also explain the nature of negative
propositions. 57
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11 How do I know that I can recognize red when I see it? How do I know it is the colour that I meant?
57
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12 If the image of the colour is not identical with the colour that is really seen, how can a comparison be made?
58
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13 Language must have the same multiplicity as a control panel that sets off the actions corresponding to its
propositions. 58
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14 Only the application makes a rod into a lever.--Every instruction can be construed as a description, every
description as an instruction. 59
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15 What does it mean, to understand a proposition as a member of a system of propositions? Its complexity is only
to be explained by the use for which it is intended. 59
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16 How do I know that that was what I expected? How do I know that the colour which I now call 'white' is the
same as the one I saw here yesterday? By recognizing it again. 59
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17 Should Logic bother itself with the question whether the
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proposition was merely automatic or thoroughly thought? It is interested in the proposition as part of a language
system. 60
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18 I do not believe that logic can talk about sentences [propositions] in any other than the normal sense in which we
say, 'There's a sentence written here'. 61
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19 Agreement of a proposition with reality. We can look at recognition, like memory, in two different ways: as a
source of the concepts of the past and of identity, or as a way of checking what happened in the past, and on
identity. 61
III
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20 If you exclude the element of intention from language, its whole function then collapses.
63
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21 The essential difference between the picture conception (of intention) and Russell's conception is that the former
regards recognition as seeing an internal relation. The causal connection between speech and action is an external
relation. 63
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22 I believe Russell's theory amounts to the following: if I give someone an order and I am happy with what he then
does, then he has carried out my order. 64
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23 If, when learning a language, speech, as it were, is connected up to action, can these connections possibly break
down? If so, what means have I for comparing the original arrangement with the subsequent action?
64
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24 The intention is already expressed in the way I now compare the picture with reality.
65
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25 Expecting that p will be the case must be the same as expecting the fulfilment of this expectation.
65
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26 If there were only an external connection, no connection could be described at all, since we only describe the
external connection by means of the internal one. 66
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27 The meaning of a question is the method of answering it. Tell me how you are searching, and I will tell you what
you are searching for. 66
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28 Expecting is connected with looking for. I know what I am looking for, without what I am looking for having to
exist. The event that replaces an expectation is the reply to it. That of course implies that the expectation must be in
the same space as what is expected. 67
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29 Expectation is not given an external description by citing what is expected; describing it by means of what is
expected is giving an internal description. 68
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30 If I say 'This is the same event as I expected' and 'This is the same event as also happened on that occasion', then
the word 'same' has a different meaning in each case. 68
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31 Language and intention. If you say, 'That's a brake lever, but it doesn't work', you are speaking of intention.
69
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32 I only use the terms expectation, thought, wish, etc. of something which is articulated.
69
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33 How you search in one way or another expresses what you expect. Expectation prepares a yardstick for
measuring the event. If there were no connection between expectation and reality, you could expect a nonsense
. 70
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34 If I say that the representation must treat of my world, then I cannot say 'since otherwise I could not verify it', but
'since otherwise it wouldn't even begin to make sense to me'. 71
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35 The strange thing about expectation is that we know it is an
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expectation. And that is what shows expectation is immediately connected with reality. We have to be able to give a
description comparing expectation with the present. 72
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36 What I once called 'objects' were simply that which we can speak about no matter what may be the case. 'I
expect three knocks on the door.' What if I replied, 'How do you know three knocks exist?'
72
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37 Is a man who cannot see any red around him at present in the same position as someone incapable of seeing red?
If one of them imagines red, that is not a red he sees. 73
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38 The memory and the reality must be in one space. Also: the image and the reality are in one space.
73
IV
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39 If I can only see something black and say it isn't red, how do I know that I am not talking nonsense--i.e. that it
could be red, that there is red--if red weren't just another graduation mark on the same scale as black?
75
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40 If there is a valid comparison with a ruler, the word 'blue' must give the direction in which I gofrom black to blue.
But how do these different directions find expression in grammar? 75
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41 A man with red/green colour blindness has a different colour system from a normal man. Is the question then
'Can someone who doesn't know what red and green are like really see what we call "blue" and "yellow"?'
76
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42 Grey must already be conceived as being in lighter/darker space. The yardstick must already be applied: I cannot
choose between inner hearing and inner deafness. 76
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43 For any question there is always a corresponding method of finding. You cannot compare a picture with reality
unless you can set it against it as a yardstick. 77
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44 How is a 'formally certified proposition' possible? The application of a yardstick doesn't presuppose any particular
length for the object to be measured. That is why I can learn to measure in general.
78
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45 But are the words in the same space as the object whose length is described? The unit length is part of the
symbolism, and it is what contains the specifically spatial element. 78
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46 A language using a co-ordinate system. The written sign without the co-ordinate system is senseless.
79
V
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47 It doesn't strike us at all when we look round us, move about in space, feel our own bodies, etc., etc., because
there is nothing that contrasts with the form of our world. The self-evidence of the world expresses itself in the very
fact that language can and does only refer to it. 80
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48 The stream of life, or the stream of the world, flows on and our propositions are so to speak verified only at
instants. Then they are commensurable with the present. 80
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49 Perhaps the difficulty derives from taking the time concept from time in physics and applying it to the course of
immediate experience. We don't speak of present, past and future images. 81
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50 'I do not see the past, only a picture of the past.' But how do I know it's a picture of the past?
82
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51 On the film strip there is a present picture and past and future pictures: but on the screen there is only the present.
83
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52 We cannot say 'time flows' if by time we mean the possibility of change.--It also appears to us as though memory
were a faint picture of what we originally had before us in full clarity. And in the language of physical objects that is
so. 83
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53 But it can also be put differently; and that is important. The phrase 'optical illusion', for example, gives the idea of
a mistake, even when there is none. One could imagine an absolutely impartial language.
84
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54 Language can only say those things that we can also imagine otherwise. That everything flows must be expressed
in the application of language. And if someone says only the present experience has reality, then the word 'present'
must be redundant here. 84
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55 Certain important propositions describing an experience which might have been otherwise: such as the
proposition that my visual field is almost incessantly in a state of flux. 86
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56 If I make a proposition such as 'Julius Caesar crossed the Alps', do I merely describe my present mental
state?--The proposition states what I believe. If I wish to know what that is, the best thing to do is to ask why I
believe it. 86
VI
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57 One misleading representational technique in our language is the use of the word "I", particularly when it is used
in representing immediate experience. How would it be if such experience were represented without using the
personal pronoun? 88
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58 Like this, say: If I, L. W., have toothache, that is expressed as 'There is toothache'. In other cases: 'A is behaving
as L. W. does when there is toothache'. Language can have anyone as its centre. That it has me as its centre lies in
the application. This privileged status cannot be expressed. Whether I say that what is represented is not one thing
among others; or that I cannot express the advantage of my language--both approaches lead to the same result. 88
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59 It isn't possible to believe something for which you cannot find some kind of verification. In a case where I
believe someone is sad I can do this. But I cannot believe that I am sad. 89
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60 Does it make sense to say two people have the same body? 90
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61 What distinguishes his toothache from mine? 90
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62 'When I say he has toothache, I mean he now has what I once had.' But is this a relation toothache once had to
me and now has to him? 91
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63 I could speak of toothache (datum of feeling) in someone else's tooth in the sense that it would be possible to feel
pain in a tooth in someone else's mouth. 92
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64 If I say 'A has toothache', I use the image of feeling pain in the same way as, say, the concept of flowing when I
talk of an electric current flowing.--The hypotheses that (1) other people have toothache and that (2) they behave
just as I do but don't have toothache--possibly have identical senses. 93
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65 Our language employs the phrases 'my pain' and 'his pain' and also 'I have (or feel) a pain', but 'I feel my pain' or 'I
feel his pain' is
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nonsense. 94
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66 What would it be like if I had two bodies, i.e. my body were composed of two separate organisms?--Philosophers
who believe you can, in a manner of speaking, extend experience by thinking, ought to remember you can transmit
speech over the telephone, but not measles. 95
VII
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67 Suppose I had such a good memory that I could remember all my sense impressions. I could then describe them,
e.g. by representing the visual images plastically, only finishing them so far as I had actually seen them and moving
them with a mechanism. 97
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68 If I describe a language, I am describing something that belongs to physics. But how can a physical language
describe the phenomenal? 97
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69 A phenomenon (specious present) contains time, but isn't in time. Whereas language unwinds in time.
98
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70 We need a way of speaking with which we can represent the phenomena of visual space, isolated as such.
98
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71 Visual space is called subjective only in the language of physical space. The essential thing is that the
representation of visual space is the representation of an object and contains no suggestion of a subject.
99
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72 How can I tell that I see the world through the pupil of my eyeball? Surely not in an essentially different way
from that of my seeing it through the window. 100
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73 In visual space there isn't an eye belonging to me and eyes
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belonging to others. Only the space itself is asymmetrical. 102
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74 The exceptional position of my body in visual space derives from other feelings, and not from something purely
visual. 102
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75 Is the time of isolated 'visual' phenomena the time of our ordinary idioms of physics? I imagine the changes in
my visual space are discontinuous and in time with the beats of a metronome. I can then describe them and compare
the description with what actually happens. A delusion of memory? No, a delusion that, ex hypothesi, cannot be
unmasked isn't a delusion. And here the time of my memory is precisely the time I'm describing.
103
VIII
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76 Incompatible for red and green to be in one place at the same time. What would a mixed colour of red and green
be? And different degrees of red are also incompatible with one another.--And yet I can say: 'There's an even redder
blue than the redder of these two'. That is, from the given I can construct what is not given.--Is a construction
possible within the elementary proposition which doesn't work by means of truth functions and also has an effect on
one proposition's following logically from another? In that case, two elementary propositions can contradict one
another. 105
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77 This is connected with the idea of a complete description. 106
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78 That r and g completely occupy the f--that doesn't show itself in our signs. But it must show itself if we look, not
at the sign, but at the symbol. For since this includes the form of the objects, then the impossibility of 'f(r)•f(g)' must
show itself there in this form. 106
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79 That would imply I can write down two particular propositions, but not their logical product? We can say that the
'•' has a different meaning here. 107
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80 A mixed or intermediate colour of blue and red is such in virtue of an internal relation to the structures of red and
blue. But this internal relation is elementary. That is, it doesn't consist in the proposition 'a is blue-red' representing a
logical product of 'a is blue' and 'a is red'. 107
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81 As with colours, so with sounds or electrical charges. It's always a question of the complete description of a
certain state at one point or at the same time. But how can I express the fact that e.g. a colour is definitively
described? How can I bring it about that a second proposition of the same form contradicts the first?--Two
elementary propositions can't contradict one another. 108
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82 There are rules for the truth functions which also deal with the elementary part of the proposition. In which case
propositions become even more like yardsticks. The fact that one measurement is right automatically excludes all
others. It isn't a proposition that I put against reality as a yardstick, it's a system of propositions. Equally in the case
of negative description: I can't be given the zero point without the yardstick.
109
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83 The concept of the independent co-ordinates of description. The propositions joined e.g. by 'and' are not
independent of one another, they form one picture and can be tested for their compatibility or incompatibility.
111
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84 In that case every assertion would consist in setting a number of scales (yardsticks), and it's impossible to set one
scale simultaneously at two graduation marks. 112
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85 That all propositions contain time appears to be accidental when compared with the fact that the truth functions
can be applied to any proposition. 112
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86 Syntax prohibits a construction such as 'a is green and a is red', but for 'a is green' the proposition 'a is red' is not,
so to speak, another proposition, but another form of the same proposition. In this way syntax draws together the
propositions that make one determination. 113
IX
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87 The general proposition 'I see a circle on a red background'--a proposition which leaves possibilities open. What
would this generality have to do with a totality of objects? Generality in this sense, therefore, enters into the theory
of elementary propositions. 115
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88 If I describe only a part of my visual field, my description must necessarily include the whole visual space. The
form (the logical form) of the patch in fact presupposes the whole space. 115
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89 Can I leave some determination in a proposition open, without at the same time specifying precisely what
possibilities are left open? 'A red circle is situated in the square.' How do I know such a proposition? Can I ever
know it as an endless disjunction? 116
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90 Generality and negation. 'There is a red circle that is not in the square.' I cannot express the proposition 'This
circle is not in the square' by placing the 'not' at the front of the proposition. That is connected with the fact that it's
nonsense to give a circle a name. 116
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91 'All circles are in the square' can mean either 'A certain number of circles are in the square' or: 'There is no circle
outside it'. But the last proposition is again the negation of a generalization and not the generalization of a negation.
117
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92 The part of speech is only determined by all the grammatical
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rules which hold for a word, and seen from this point of view our language contains countless different parts of
speech. 118
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93 The subject-predicate form does not in itself amount to a logical form. The forms of the propositions: 'The plate is
round', 'The man is tall', 'The patch is red', have nothing in common.-- Concept and object: but that is subject and
predicate. 118
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94 Once you have started doing arithmetic, you don't bother about functions and objects.--The description of an
object may not express what would be essential for the existence of the object. 119
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95 If I give names to three visual circles of equal size--I always name (directly or indirectly) a location. What
characterizes propositions of the form 'This is...' is only the fact that the reality outside the so-called system of signs
somehow enters into the symbol. 119
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96 What remains in this case, if form and colour alter? For position is part of the form. It is clear that the phrase
'bearer of a property' conveys a completely wrong--an impossible--picture. 120
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97 Roughly speaking, the equation of a circle is the sign for the concept 'circle'. So it is as if what corresponds with
the objects falling under the concept were here the co-ordinates of the centres. In fact, the number pair that gives the
co-ordinates of the centre is not just anything, but characterizes just what in the symbol constitutes the 'difference' of
the circles. 121
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98 The specification of the 'here' must not prejudge what is here. F(x) must be an external description of x.--but if I
now say 'Here is a circle' and on another occasion 'Here is a sphere', are the two 'here's' of the same kind?
122
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X
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99 Number and concept. Does it make sense to ascribe a number to objects that haven't been brought under a
concept? But I can e.g. form the concept, 'link between a and b'. 123
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100 Numbers are pictures of the extensions of concepts. We could regard the extension of a concept as an object
whose name has sense only in the context of a proposition. In the symbolism there is an actual correlation, whereas
at the level of meaning only the possibility of correlation is at issue. 124
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101 I can surely always distinguish 3 and 4 in the sign 1 + 1 + 1 + 1 + 1 + 1 + 1. 124
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102 Numbers can only be defined from propositional forms, independently of the question which propositions are
true or false. The possibility of grouping these 4 apples into 2 and 2 refers to the sense, not the truth of a proposition.
125
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103 Can a proposition (A) in PM notation give the sense of 5 + 7 = 12? But how have I obtained the numerical sign
in the right-hand bracket if I don't know that it is the result of adding the two left-hand signs?
125
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104 What tells us that the 5 strokes and the 7 combine precisely to make 12 is always only insight into the internal
relations of the structures--not some logical consideration. 127
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105 An extension is a characteristic of the sense of a proposition. 127
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106 What A contains apart from the arithmetical schema can only be what is necessary in order to apply it. But
nothing at all is necessary. 128
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107 No investigation of concepts can tell us that 3 + 2 = 5;
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equally it is not an examination of concepts which tells us that A is a tautology. Numbers must be of a kind with
what we use to represent them. 129
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108 Arithmetic is the grammar of numbers. 129
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109 Every mathematical calculation is an application of itself and only as such does it have a sense. That is why it
isn't necessary to speak about the general form of logical operation here.--Arithmetic is a more general kind of
geometry. 130
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110 It's as if we're surprised that the numerals cut adrift from their definitions function so unerringly; which is
connected with the internal consistency of geometry. The general form of the application of arithmetic seems to be
represented by the fact that nothing is said about it. 131
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111 Arithmetical constructions are autonomous, like geometrical ones, and hence they guarantee their own
applicability. 132
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112 If 3 strokes on paper are the sign for the number 3, then you can say the number 3 is to be applied in the way in
which 3 strokes can be applied. (Cf. §107) 133
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113 A statement of number about the extension of a concept is a proposition, but a statement of number about the
range of a variable is not, since it can be derived from the variable itself. 133
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114 Do I know there are 6 permutations of 3 elements in the same way in which I know there are 6 people in this
room? No. Therefore the first proposition is of a different kind from the second. 134
XI
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115 A statement of number doesn't always contain something general or indefinite. For instance, 'I see 3 equal circles
equidistant from one another'. Something indefinite would be, say: I know that three things have the property φ, but
I don't know which.
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Here it would be nonsense to say I don't know which circles they are. 136
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116 There is no such concept as 'pure colour'. Similarly with permutations. If we say that A B admits of two
permutations, it sounds as though we had made a general assertion. But 'Two permutations are possible' cannot say
any less--i.e. something more general--than the schema A B, B A. They are not the extension of a concept: they are
the concept. 137
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117 There is a mathematical question: 'How many permutations of 4 elements are there?' which is the same kind as
'What is 25 × 18?' For in both cases there is a general method of solution. 139
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118 In Russell's theory only an actual correlation can show the "similarity" of two classes. Not the possibility of
correlation, for this consists precisely in the numerical equality. 140
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119 What sort of an impossibility is the impossibility of a 1-1 correlation between 3 circles and 2 crosses?--It is
nonsense to say of an extension that it has such and such a number, since the number is an internal property of the
extension. 140
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120 Ramsey explains the sign '=' like this: x = x is taut.; x = y is cont. What then is the relation of ' ' to '='?--You
may compare mathematical equations only with significant propositions, not with tautologies. 141
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121 An equation is a rule of syntax. You may construe sign-rules as propositions, but you don't have to construe
them so. The 'heterological' paradox. 143
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122 The generality of a mathematical assertion is different from the generality of the proposition proved. A
mathematical proposition is an allusion to a proof. A generalization only makes sense if it--i.e. all values of its
variables--is completely determined. 143
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XII
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123 I grasp an infinite stretch in a different way from an endless one. A proposition about it can't be verified by a
putative endless striding, but only in one stride. 146
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124 It isn't just impossible 'for us men' to run through the natural numbers one by one; it's impossible, it means
nothing. The totality is only given as a concept. 146
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125 That, in the case of the logical concept (1, ξ, ξ + 1), the existence of its objects is already given with the concept,
of itself shows that it determines them. What is fundamental is simply the repetition of an operation. The operation +
1 three times yields and is the number 3. 147
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126 It looks now as if the quantifiers make no sense for numbers. 148
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127 If no finite product makes a proposition true, that means no product makes it true. And so it isn't a logical
product. 148
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128 Can I know that a number satisfies the equation without a finite section of the infinite series being marked out as
one within which it occurs? 149
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129 A proposition about all propositions, or all functions, is impossible. Generality in arithmetic is indicated by an
induction. 150
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130 The defect (circle) in Dedekind's explanation of the concept of infinity lies in its application of the concept 'all' in
the formal implication. What really corresponds to what we mean isn't a proposition at all, it's the inference from φx
to ψx, if this inference is permitted--but the inference isn't expressed by a proposition.
151
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131 Generality in Euclidean geometry. Strange that what holds for one triangle should therefore hold for every other.
But once more the construction of a proof is not an experiment; no, a description of the construction must
suffice.--What is demonstrated can't be expressed by a proposition. 152
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132 'The world will eventually come to an end' means nothing at all, for it's compatible with this statement that the
world should still exist on any day you care to mention. 'How many 9s immediately succeed one another after
3.1415 in the development of π?' If this question is meant to refer to the extension, then it doesn't have the sense of
the question which interests us. ('I grasp an infinite stretch in a different way from an endless one.')
153
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133 The difficulty in applying the simple basic principles shakes our confidence in the principles themselves.
153
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134 'I saw the ruler move from t1 to t2, therefore I must have seen it at t.' If in such a case I appear to infer a
particular case from a general proposition, then the general proposition is never derived from experience, and the
proposition isn't a real proposition. 154
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135 'We only know the infinite by description.' Well then, there's just the description and nothing else.
155
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136 Does a notation for the infinite presuppose infinite space or infinite time? Then the possibility of such a
hypothesis must surely be prefigured somewhere. The problem of the smallest visible distinction.
155
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137 If I cannot visibly bisect the strip any further, I can't even try to, and so can't see the failure of such an attempt.
Continuity in our visual field consists in our not seeing discontinuity. 156
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138 Experience as experience of the facts gives me the finite; the objects contain the infinite. Of course, not as
something rivalling finite experience, but in intension. (Infinite possibility is not a quantity.) Space has no extension,
only spatial objects are extended, but infinity is a property of space. 157
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139 Infinite divisibility: we can conceive of any finite number of parts but not of an infinite number; but that is
precisely what constitutes infinite divisibility.--That a patch in visual space can be divided into three parts means that
a proposition describing a patch divided in this way makes sense. Whereas infinite divisibility doesn't mean there's a
proposition describing a line divided into infinitely many parts. Therefore this possibility is not brought out by any
reality of the signs, but by a possibility of a different kind in the signs themselves.
158
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140 Time contains the possibility of all the future now. The space of human movement is infinite in the same way as
time. 160
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141 The rules for a number system--say, the decimal system--contain everything that is infinite about the
numbers.--It all hangs on the syntax of reality and possibility. m = 2n contains the possibility of correlating any
number with another, but doesn't correlate all numbers with others. 161
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142 The propositions 'Three things can lie in this direction' and 'Infinitely many things can lie in this direction' are
only apparently formed in the same way, but are in fact different in structure: the 'infinitely many' of the second
structure doesn't play the same role as the 'three' of the first. 162
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143 Empty infinite time is only the possibility of facts which alone are the realities.--If there is an infinite reality, then
there is also contingency in the infinite. And so, for instance, also an
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infinite decimal that isn't given by a law.--Infinity lies in the nature of time, it isn't the extension it happens to have.
163
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144 The infinite number series is only the infinite possibility of finite series of numbers. The signs themselves only
contain the possibility and not the reality of their repetition. Mathematics can't even try to speak about their
possibility. If it tries to express their possibility, i.e. when it confuses this with their reality, we ought to cut it down
to size. 164
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145 An infinite decimal not given by a rule. 'The number that is the result when a man endlessly throws a die',
appears to be nonsense.--An infinite row of trees. If there is a law governing the way the trees' heights vary, then the
series is defined and can be imagined by means of this law. If I now assume there could be a random series, then
that is a series about which, by its very nature, nothing can be known apart from the fact that I can't know it.
165
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146 The multiplicative axiom. In the case of a finite class of classes we can in fact make a selection. But in the case
of infinitely many sub-classes I can only know the law for making a selection. Here the infinity is only in the rule.
167
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147 What makes us think that perhaps there are infinitely many things is only our confusing the things of physics
with the elements of knowledge. 'The patch lies somewhere between b and c': the infinite possibility of positions isn't
expressed in the analysis of this.--The illusion of an infinite hypothesis in which the parcels of matter are confused
with the simple objects. What we can imagine multiplied to infinity are the combinations of the things in accordance
with their infinite possibilities, never the things themselves. 168
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148 While we've as yet no idea how a certain proposition is to be proved, we still ask 'Can it be proved or not?' You
cannot have a logical plan of search for a sense you don't know. Every proposition teaches us through its sense how
we are to convince ourselves whether it is true or false. 170
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149 A proof of relevance would be a proof which, while yet not proving the proposition, showed the form of a
method for testing the proposition. 170
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150 I can assert the general (algebraic) proposition just as much or as little as the equation 3 × 3 = 9 or 3 × 3 = 11.
The general method of solution is in itself a clarification of the nature of the equation. Even in a particular case I see
only the rule. 'The equation yields a' means: if I transform the equation in accordance with certain rules I get a. But
these rules must be given to me before the word 'yields' has a meaning and before the question has a sense. 172
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151 We may only put a question in mathematics where the answer runs: 'I must work it out'. The question 'How
many solutions are there to this equation?' is the holding in readiness of the general method for solving it. And that,
in general, is what a question is in mathematics: the holding in readiness of a general method.
175
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152 I can't ask whether an angle can be trisected until I can see the system 'Ruler and Compasses' embedded in a
larger one, where this question has a sense.--The system of rules determining a calculus thereby determines the
'meaning' of its signs too. If I change the rules, then I change the form, the meaning.--In mathematics, we cannot talk
of systems in general, but only within systems. 177
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153 A mathematical proof is an analysis of a mathematical
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proposition. It isn't enough to say that p is provable, we have to say: provable according to a particular system.
Understanding p means understanding its system. 179
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154 I can ask 'What is the solution of this equation?', but not 'Has it a solution?'--It's impossible for us to discover
rules of a new type that hold for a form with which we are familiar.--The proposition: 'It's possible--though not
necessary--that p should hold for all numbers' is nonsense. For 'necessary' and 'all' belong together in mathematics.
181
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155 Finding a new system (Sheffer's discovery, for instance). You can't say: I already had all these results, now all
I've done is find a better way that leads to all of them. The new way amounts to a new system.
182
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156 Unravelling knots in mathematics. We may only speak of a genuine attempt at a solution to the extent that the
structure of the knot is clearly seen. 184
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157 You can't write mathematics, you can only do it.--Suppose I hit upon the right way of constructing a regular
pentagon by accident. If I don't understand this construction, as far as I'm concerned it doesn't even begin to be the
construction of a pentagon. The way I have arrived at it vanishes in what I understand.
186
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158 Where a connection is now known to exist which was previously unknown, there wasn't a gap before,
something incomplete which has now been filled in.--Induction: if I know the law of a spiral, that's in many respects
analogous with the case in which I know all the whorls. Yet not completely analogous--and that's all we can say. 187
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159 But doesn't it still count as a question, whether there is a finite number of primes or not? Once I can write down
the general form of primes, e.g. 'dividing... by smaller numbers leaves a
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remainder'--there is no longer a question of 'how many' primes there are. But since it was possible for us to have the
phrase 'prime number' before we had the strict expression, it was also possible for people to have wrongly formed
the question. Only in our verbal language are there in mathematics 'as yet unsolved problems'.
188
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160 A consistency proof can't be essential for the application of the axioms. For these are propositions of syntax.
189
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161 A polar expedition and a mathematical one. How can there be conjectures in mathematics? Can I make a
hypothesis about the distribution of primes? What kind of verification do I then count as valid? I can't conjecture the
proof. And if I've got the proof it doesn't prove what was conjectured. 189
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162 Sheffer's discovery. The systems are certainly not in one space, so that I could say: there are systems with 3 and
2 logical constants, and now I'm trying to reduce the number of constants in the same way.--A mathematical
proposition is only the immediately visible surface of a whole body of proof and this surface is the boundary facing
us. 191
XIV
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163 A proof for the associative law? As a basic rule of the system it cannot be proved. The usual mistake lies in
confusing the extension of its application with what the proof genuinely contains.--Can one prove that by addition
of forms ((1 + 1) + 1) etc. numbers of this form would always result? The proof lies in the rule, i.e. in the definition
and in nothing else. 193
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164 A recursive proof is only a general guide to arbitrary special proofs: the general form of continuing along this
series. Its
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generality is not the one we desire but consists in the fact that we can repeat the proof. What we gather from the
proof we cannot represent in a proposition at all. 196
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165 The correct expression for the associative law is not a proposition, but precisely its 'proof', which admittedly
doesn't state the law. I know the specific equation is correct just as well as if I had given a complete derivation of it.
That means it really is proved. The one whorl, in conjunction with the numerical forms of the given equation, is
enough. 198
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166 One says an induction is a sign that such and such holds for all numbers. But an induction isn't a sign for
anything but itself.--Compare the generality of genuine propositions with generality in arithmetic. It is differently
verified and so is of a different kind. 200
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167 An induction doesn't prove an algebraic equation, but it justifies the setting up of algebraic equations from the
standpoint of their application to arithmetic. That is, it is only through the induction that they gain their sense, not
their truth. An induction is related to an algebraic proposition not as proof is to what is proved, but as what is
designated to a sign. 201
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168 If we ask 'Does a + (b + c) = (a + b) + c?', what could we be after?--An algebraic proposition doesn't express a
generality; this is shown, rather, in the formal relation to the substitution, which proves to be a term of the inductive
series. 202
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169 One can prove any arithmetical equation of the form a × b = c or prove its opposite. A proof of this provability
would be the exhibition of an induction from which it could be seen what sort of propositions the ladder leads to.
204
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XV
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170 The theory of aggregates says that you can't grasp the actual infinite by means of arithmetical symbolism at all, it
can therefore only be described and not represented. So one could talk about a logical structure without reproducing
it in the proposition itself. A method of wrapping a concept up in such a way that its form disappears. 206
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171 Any proof of the continuity of a function must relate to a number system. The numerical scale, which comes to
light when calculating a function, should not be allowed to disappear in the general treatment.--Can the continuum
be described? A form cannot be described: it can only be presented. 207
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172 'The highest point of a curve' doesn't mean 'the highest point among all the points of the curve'. In the same way,
the maximum of a function isn't the largest value among all its values. No, the highest point is something I
construct, i.e. derive from a law. 208
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173 The expression '(n)...' has a sense if nothing more than the unlimited possibility of going on is
presupposed.--Brouwer--. The explanation of the Dedekind cut as if it were clear what was meant by: either R has a
last member and L a first, or, etc. In truth none of these cases can be conceived.
209
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174 Set theory builds on a fictitious symbolism, therefore on nonsense. As if there were something in Logic that
could be known, but not by us. If someone says (as Brouwer does) that for (x) • f1x = f2x there is, as well as yes and
no, also the case of undecidability, this implies that '(x)...' is meant extensionally and that we may talk of all x
happening to have a property. 211
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175 If one regards the expression 'the root of the equation φx = 0' as a Russellian description, then a proposition
about the
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root of the equation x + 2 = 6 must have a different sense from one saying the same about 4.
213
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176 How can a purely internal generality be refuted by the occurrence of a single case (and so by something
extensional)? But the particular case refutes the general proposition from within--it attacks the internal proof.--The
difference between the two equations x2 = x • x and x2 = 2x isn't one consisting in the extensions of their validity.
214
XVI
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177 That a point in the plane is represented by a number-pair, and in three-dimensional space by a number-triple, is
enough to show that the object represented isn't the point at all but the point-network.
216
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178 Geometry as the syntax of the propositions dealing with objects in space. Whatever is arranged in visual space
stands in this sort of order a priori, i.e. in virtue of its logical nature, and geometry here is simply grammar. What
the physicist sets into relation with one another in the geometry of physical space are instrument readings, which do
not differ in their internal nature whether we live in a linear space or a spherical one.
216
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179 I can approach any point of an interval indefinitely by always carrying out the bisection prescribed by tossing a
coin. Can I divide the rationals into two classes in a similar way, by putting either 0 or 1 in an infinite binary
expansion according to the way the coin falls (heads or tails)? No law of succession is described by the instruction to
toss a coin; and infinite indefiniteness does not define a number. 218
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180 Is it possible within the law to abstract from the law and see the extension presented as what is essential?--If I
cut at a place
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where there is no rational number, then there must be approximations to this cut. But closer to what? For the time
being I have nothing in the domain of number which I can approach.--All the points of a line can actually be
represented by arithmetical rules. In the case of approximation by repeated bisection we approach every point via
rational numbers. 221
XVII
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181 What criterion is there for the irrational numbers being complete? Every irrational number runs through a series
of rational approximations, and never leaves this series behind. If I have the totality of all irrational numbers except
π, and now insert π, I cannot cite a point at which π is really needed; at every point it has a companion agreeing with
it. This shows clearly that an irrational number isn't the extension of an infinite decimal fraction, it's a law. If π were
an extension, we would never feel the lack of it--it would be impossible for us to detect a gap.
223
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182 ' : a rule with an exception.--There must first be the rules for the digits, and then--e.g.--a root is expressed in
them. But this expression in a sequence of digits only has significance through being the expression for a real
number. If someone subsequently alters it, he has only succeeded in distorting the expression, but not in obtaining a
new number. 224
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183 If ' is anything at all, then it is the same as , only another expression for it; the expression in another
system. It doesn't measure until it is in a system. You would no more say of ' that it is a limit towards which the
sums of a series are tending than you would of the instruction to throw dice. 225
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184 That we can apply the law holds also for the law to throw digits like dice. And what distinguishes π' from this
can only consist in our knowing that there must be a law governing the occurrences
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of the digit 7 in π, even if we don't yet know what the law is. π' alludes to a law which is as yet unknown.
227
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185 Only a law approaches a value. 228
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186 The letter π stands for a law which has its position in arithmetical space. Whereas π' doesn't use the idioms of
arithmetic and so doesn't assign the law a place in this space. For substituting 3 for 7 surely adds absolutely nothing
to the law and in this system isn't an arithmetical operation at all. 228
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187 To determine a real number a rule must be completely intelligible in itself. That is to say, it must not be
essentially undecided whether a part of it could be dispensed with. If the extensions of two laws coincide as far as
we've gone, and I cannot compare the laws as such, then the numbers defined cannot be compared.
230
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188 The expansion of π is simultaneously an expression of the nature of π and of the nature of the decimal system.
Arithmetical operations only use the decimal system as a means to an end. They can be translated into the language
of any other number system, and do not have any of them as their subject matter.--A general rule of operation gets
its generality from the generality of the change it effects in the numbers. π' makes the decimal system into its subject
matter, and for that reason it is no longer sufficient that we can use the rule to form the extension.231
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189 A law where p runs through the series of whole numbers except for those for which Fermat's last theorem
doesn't hold. Would this law define a real number? The number F wants to use the spiral... and choose sections of
this spiral according to a principle. But this principle doesn't belong to the spiral. There is admittedly a law there, but
it doesn't refer directly to the number. The number is a sort of lawless by-product of the law.
232
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190 In this context we keep coming up against something that could be called an 'arithmetical experiment'. Thus the
primes come out from the method for looking for them, as the results of an experiment. I can certainly see a law in
the rule, but not in the numbers that result. 235
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191 A number must measure in and of itself. If it doesn't do that but leaves it to the rationals, we have no need of
it.--The true expansion is the one which evokes from the law a comparison with a rational number.
235
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192 A real number can be compared with the fiction of an infinite spiral, whereas structures like F, P or π' only with
finite sections of a spiral. 237
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193 To compare rational numbers with , I have to square them.--They then assume the form , where
is now an arithmetical operation. Written out in this system, they can be compared with , and it is for me as if
the spiral of the irrational number had shrunk to a point.
237
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194 Is an arithmetical experiment still possible when a recursive definition has been set up? No, because with the
recursion each stage becomes arithmetically comprehensible. 238
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195 Is it possible to prove a greater than b, without being able to prove at which place the difference will come to
light? 1.4--Is that the square root of 2? No, it's the root of 1.96. That is, I can immediately write it down as an
approximation to . 238
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196 If the real number is a rational number a, a comparison of its law with a must show this. That means the law
must be so formed as to 'click into' the rational number when it comes to the
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appropriate place. It wouldn't do, e.g., if we couldn't be sure whether really breaks off at 5.
240
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197 Can I call a spiral a number if it is one which, for all I know, comes to a stop at a rational point? There is a lack
of a method for comparing with the rationals. Expanding indefinitely isn't a method, even when it leads to a result of
the comparison. 241
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198 If the question how F compares with a rational number has no sense, since all expansion still hasn't given us an
answer, then this question also had no sense before we tried to settle the matter at random by means of an
extension. 242
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199 It isn't only necessary to be able to say whether a given rational number is the real number: we must also be able
to say how close it can possibly come to it. An order of magnitude for the distance apart. Decimal expansion doesn't
give me this, since I cannot know e.g. how many 9s will follow a place that has been reached in the expansion.--'e
isn't this number' means nothing; we have to say 'It is at least this interval away from it'.
243
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Appendix: From F. Waismann's shorthand notes of a conversation on 30 December 1930
245
XIX
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200 It appears to me that negation in arithmetic is interesting only in conjunction with a certain
generality.--Indivisibility and inequality.--I don't write '~(5 × 5 = 30)', I write 5 × 5 ≠ 30, since I'm not negating
anything but want to establish a relation between 5 × 5 and 30 (and hence something positive). Similarly, when I
exclude divisibility, this is equivalent to establishing indivisibility. 247
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Page 39
201 There is something recalcitrant to the application of the law of the excluded middle in mathematics--Looking for
a law for the distribution of primes. We want to replace the negative criterion for a prime number by a positive
one--but this negation isn't what it is in logic, but an indefiniteness.--The negation of an equation is as like and as
unlike the denial of a proposition as the affirmation of an equation is as like or unlike the affirmation of a
proposition. 248
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202 Where negation essentially--on logical grounds--corresponds to a disjunction or to the exclusion of one part of a
logical series in favour of another--then here it must be one and the same as those logical forms and therefore only
apparently a negation. 249
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203 Yet what is expressed by inequalities is essentially different from what is expressed by equations. And so you
can't immediately compare a law yielding places of a decimal expansion which works with inequalities, with one that
works with equations. Here we have completely different methods and consequently different kinds of arithmetical
structure. 250
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204 Can you use the prime numbers to define an irrational number? As far as you can foresee the primes, and no
further. 251
XX
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205 Can we say a patch is simpler than a larger one?--It seems as if it is impossible to see a uniformly coloured patch
as composite.--The larger geometrical structure isn't composed of smaller geometrical structures. The 'pure
geometrical figures' are of course only logical possibilities. 252
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206 Whether it makes sense to say 'This part of a red patch is red' depends on whether there is absolute position. It's
possible to
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establish the identity of a position in the visual field, since we would otherwise be unable to distinguish whether a
patch always stays in the same place. In visual space there is absolute position, absolute direction, and hence
absolute motion. If this were not so, there would be no sense in speaking in this context of the same or different
places. This shows the structure of our visual field: for the criterion for its structure is what propositions make sense
for it. 253
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207 Can I say: 'The top half of my visual field is red'?--There isn't a relation of 'being situated' which would hold
between a colour and a position. 257
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208 It seems to me that the concept of distance is given immediately in the structure of visual space. Measuring in
visual space. Equal in length, unequal in parts. Can I be sure that what I count is really the number I see?
257
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209 But if I can't say there is a definite number of parts in a and b, how in that case am I to describe the visual
image?--'Blurred' and 'unclear' are relative expressions.--If we were really to see 24 and 25 parts in a and b, we
couldn't then see a and b as equal. The word 'equal' has a meaning even for visual space which stamps this as a
contradiction. 260
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210 The question is, how to explain certain contradictions that arise when we apply the methods of inference used in
Euclidean space to visual space. This happens because we can only see the construction piecemeal and not as a
whole: because there's no visual construction that could be composed of these individual visual pieces.
261
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211 The moment we try to apply exact concepts of measurement to immediate experience, we come up against a
peculiar vagueness in this experience.--The words 'rough', 'approximate', etc. have
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only a relative sense, but they are still needed and they characterize the nature of our experience.--Problem of the
heap of sand.--What corresponds in Euclidean geometry to the visual circle isn't a circle, but a class of figures.--Here
it seems as though an exact demarcation of the inexactitude is impossible. We border off a swamp with a wall, and
the wall is not the boundary of the swamp. 263
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212 The correlation between visual space and Euclidean space. If a circle is at all the sort of thing we see, then we
must be able to see it and not merely something like it. If I cannot see an exact circle then in this sense neither can I
see approximations to one. 265
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213 We need new concepts and we continually resort to those of the language of physical objects. For instance
'precision'. If it is right to say 'I do not see a sharp line', then a sharp line is conceivable. If it makes sense to say 'I
never see an exact circle', then this implies: an exact circle is conceivable in visual space.--The word 'equal' used with
quite different meanings.--Description of colour patches close to the boundary of the visual field. Clear that the lack
of clarity is an internal property of visual space. 266
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214 What distinctions are there in visual space? The fact that you see a physical hundred-sided polygon as a circle
implies nothing as to the possibility of seeing a hundred-sided polygon. Is there a sense in speaking of a visual
hundred-sided polygon? 268
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215 Couldn't I say, 'Perhaps I see a perfect circle, but can never know it'? Only if it is established in what cases one
calls one measurement more precise than another. It means nothing to say the circle is only an ideal to which reality
can only approximate. But it may also be that we call an infinite possibility itself a circle. As with an irrational
number.--Now, is the imprecision of measurement the same concept as the imprecision of visual images? Certainly
not.--'Seems' and 'appears' ambiguous: in one
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case it is the result of measurement, in another a further appearance. 269
Page 42
216 'Sense datum' contains the idea: if we talk about 'the appearance of a tree' we are either taking for a tree
something which is one, or something which is not. But this connection isn't there.270
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217 Can you try to give 'the right model for visual space'? You cannot translate the blurredness of phenomena into
an imprecision in the drawing. That visual space isn't Euclidean is already shown by the occurrence of two different
kinds of lines and points. 271
XXI
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218 Simple colours--simple as psychological phenomena. I need a purely phenomenological colour theory in which
mention is only made of what is actually perceptible and no hypothetical objects--waves, rods, cones and all
that--occur. Can I find a metric for colours? Is there a sense in saying, e.g., that with respect to the amount of red in
it, one colour is halfway between two other colours? 273
Page 42
219 Orange is a mixture of red and yellow in a sense in which yellow isn't a mixture of red and green although
yellow comes between red and green in the colour circle.--If I imagine mixing a blue-green with a yellow-green I see
straightaway that it can't happen, that a component part would first have to be 'killed'.
273
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220 I must know what in general is meant by the expression 'mixture of colours A and B'. If someone says to me that
the colour of a patch lies between violet and red, I understand this and can imagine a redder violet than the one
given. But: 'The colour lies
Page Break 43
between this violet and an orange'? The way in which the mixed colour lies between the others is no different here
from the way red comes between blue and yellow.--'Red and yellow make orange' doesn't speak of a quantity of
components. It means nothing to say this orange and this violet contain the same amount of red.--False comparison
between the colour series and a system of two weights on a balance. 274
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221 The position here is just as it is with the geometry of visual space as compared with Euclidean geometry. There
are here quantities of a different sort from that represented by our rational numbers.--If the expression 'lie between'
on one occasion designates a mixture of two simple colours, and on another a simple component common to two
mixed colours, the multiplicity of its application is different in the two cases.--You can also arrange all the shades
along a straight line. But then you have to introduce rules to exclude certain transitions, and in the end the
representation on the lines has to be given the same kind of topological structure as the octahedron has. Completely
analogous to the relation of ordinary language to a 'logically purified' mode of expression.
276
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222 We can't say red has an orange tinge in the same sense as orange has a reddish tinge. 'x is composed of y and z'
and 'x is the common component of y and z' are not interchangeable here. 278
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223 When we see dots of one colour intermingled with dots of another we seem to have a different sort of colour
transition from that on the colour-circle. Not that we establish experimentally that certain colours arise in this way
from others. For whether or not such a transition is possible (or conceivable) is an internal property of the colours.
279
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224 The danger of seeing things as simpler than they really are. Understanding a Gregorian mode means hearing
something new; analogous with suddenly seeing 10 strokes, which I had hitherto only been able to see as twice five
strokes, as a characteristic whole. 281
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225 A proposition, an hypothesis, is coupled with reality--with varying degrees of freedom. All that matters is that
the signs in the end still refer to immediate experience and not to an intermediary (a thing in itself). A proposition
construed in such a way that it can be uncheckably true or false is completely detached from reality and no longer
functions as a proposition. 282
Page 44
226 An hypothesis is a symbol for which certain rules of representation hold. The choice of representation is a
process based on so-called induction (not mathematical induction). 283
Page 44
227 We only give up an hypothesis for an even higher gain. The question, how simple a representation is yielded by
assuming a particular hypothesis, is connected with the question of probability. 284
Page 44
228 What is essential to an hypothesis is that it arouses an expectation, i.e., its confirmation is never completed. It
has a different formal relation to reality from that of verification.--Belief in the uniformity of events. An hypothesis is
a law for forming propositions. 285
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229 The probability of an hypothesis has its measure in how much evidence is needed to make it profitable to throw
it out. If I say: I assume the sun will rise again tomorrow, because the opposite is so unlikely, I here mean by 'likely'
and 'unlikely' something completely different from 'It's equally likely that I'll throw heads or tails'.--The expectation
must make sense now; i.e. I must be able to compare it with how things stand at present.
286
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230 Describing phenomena by means of the hypothesis of a
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world of material things compared with a phenomenological description.--Thus the theory of Relativity doesn't
represent the logical multiplicity of the phenomena themselves, but that of the regularities observed. This multiplicity
corresponds not to one verification, but to a law by verifications. 286
Page 45
231 Hypothesis and postulate. No conceivable experience can refute a postulate, even though it may be extremely
inconvenient to hang on to it. Corresponding to the greater or slighter convenience, there is a greater or slighter
probability of the postulate. It is senseless to talk of a measure for this probability.
288
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232 If I say 'That will probably occur', this proposition is neither verified by the occurrence nor falsified by its
non-occurrence. If we argue about whether it is probable or not, we shall always adduce arguments from the past
only.--It's always as if the same state of affairs could be corroborated by experience, whose existence was evident a
priori. But that's nonsense. If the experience agrees with the computation, that means my computation is justified
by the experience--not its a priori element, but its bases, which are a posteriori: certain natural laws. In the case of
throwing a die the natural law takes the form that it is equally likely for any of the six sides to be the side uppermost.
It's this law that we test. 289
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233 Certain possible events must contradict the law if it is to be one at all; and should these occur, they must be
explained by a different law.--The prediction that there will be an equal distribution contains an assumption about
those natural laws that I don't know precisely. 290
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234 A man throwing dice every day for a week throws nothing but ones--and not because of any defect in the die.
Has he grounds
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for thinking that there's a natural law at work here which makes him throw nothing but ones?--When an insurance
company is guided by probability, it isn't guided by the probability calculus but by a frequency actually observed.
291
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235 'Straight line with deviations' is only one form of description. If I state 'That's the rule', that only has a sense as
long as I have determined the maximum number of exceptions I'll allow before knocking down the rule.
292
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236 It only makes sense to say of the stretch you actually see that it gives the general impression of a straight line,
and not of an hypothetical one you assume. An experiment with dice can only give grounds for expecting things to
go in the same way. 293
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237 Any 'reasonable' expectation is an expectation that a rule we have observed up to now will continue to hold. But
the rule must have been observed and can't, for its part too, be merely expected.--Probability is concerned with the
form and a standard of expectation. 294
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238 A ray of light strikes two different surfaces. The centre of each stretch seems to divide it into equally probable
possibilities. This yields apparently incompatible probabilities. But the assumption of the probability of a certain
event is verified by a frequency experiment; and, if confirmed, shows itself to be an hypothesis belonging to
physics. The geometrical construction merely shows that the equal lengths of the sections was no ground for
assuming equal likelihood. I can arbitrarily lay down a law, e.g. that if the lengths of the parts are equal, they are
equally likely; but any other law is just as permissible. Similarly with further examples. It is from experience that we
determine these possibilities as equally likely. But logic gives this stipulation no precedence.
295
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First Appendix
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Complex and Fact 301
The Concept of Infinity in Mathematics 304
Second Appendix (from F. Waismann's shorthand notes on Wittgenstein's talks and conversation between
December 1929 and September 1931)
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Yardstick and Propositional System 317
Consistency 318
Editor's Note 347
Translators' Note 352
Corrigenda for the German text 355
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PHILOSOPHICAL REMARKS
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I
Page 51
1 A proposition is completely logically analysed if its grammar is made completely clear: no matter what idiom it
may be written or expressed in.
Page 51
I do not now have phenomenological language, or 'primary language' as I used to call it, in mind as my goal. I
no longer hold it to be necessary. All that is possible and necessary is to separate what is essential from what is
inessential in our language.
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That is, if we so to speak describe the class of languages which serve their purpose, then in so doing we have
shown what is essential to them and given an immediate representation of immediate experience.
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Each time I say that, instead of such and such a representation, you could also use this other one, we take a
further step towards the goal of grasping the essence of what is represented.
Page 51
A recognition of what is essential and what inessential in our language if it is to represent, a recognition of
which parts of our language are wheels turning idly, amounts to the construction of a phenomenological language.
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Physics differs from phenomenology in that it is concerned to establish laws. Phenomenology only
establishes the possibilities. Thus, phenomenology would be the grammar of the description of those facts on which
physics builds its theories.
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To explain is more than to describe; but every explanation contains a description.
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An octahedron with the pure colours at the corner-points e.g. provides a rough representation of
colour-space, and this is a grammatical representation, not a psychological one. On the other hand, to say that in
such and such circumstances you can see a red
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after-image (say) is a matter of psychology. (This may, or may not, be the case--the other is a priori; we can
establish the one by experiment but not the other.)
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Using the octahedron as a representation gives us a bird's-eye view of the grammatical rules [†1].
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The chief trouble with our grammar is that we don't have a bird's-eye view of it [†2].
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What Mach calls a thought experiment is of course not an experiment at all [†3]. At bottom it is a
grammatical investigation.
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2 Why is philosophy so complicated? It ought, after all, to be completely simple. Philosophy unties the knots in our
thinking, which we have tangled up in an absurd way; but to do that, it must make movements which are just as
complicated as the knots. Although the result of philosophy is simple, its methods for arriving there cannot be so.
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The complexity of philosophy is not in its matter, but in our tangled understanding.
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3 How strange if logic were concerned with an 'ideal' language and not with ours. For what would this ideal language
express? Presumably, what we now express in our ordinary language; in that case, this is the language logic must
investigate. Or something else: but in that case how would I have any idea what that would be?--Logical analysis is
the analysis of something we have, not of something we don't have. Therefore it is the analysis of propositions as
they stand. (It would be odd if the human race had been speaking all this time without ever putting together a
genuine proposition.)
Page 52
When a child learns 'Blue is a colour, red is a colour, green, yellow--all are colours', it learns nothing new
about the colours,
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but the meaning of a variable in such propositions as: 'There are beautiful colours in that picture' etc. The first
proposition tells him the values of a variable.
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The words 'Colour', 'Sound', 'Number' etc. could appear in the chapter headings of our grammar. They need
not occur within the chapters but that is where their structure is given.
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4 Isn't the theory of harmony at least in part phenomenology and therefore grammar?
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The theory of harmony isn't a matter of taste.
Page 53
If I could describe the point of grammatical conventions by saying they are made necessary by certain
properties of the colours (say), then that would make the conventions superfluous, since in that case I would be able
to say precisely that which the conventions exclude my saying. Conversely, if the conventions were necessary, i.e. if
certain combinations of words had to be excluded as nonsensical, then for that very reason I cannot cite a property
of colours that makes the conventions necessary, since it would then be conceivable that the colours should not have
this property, and I could only express that by violating the conventions.
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It cannot be proved that it is nonsense to say of a colour that it is a semitone higher than another. I can only
say 'If anyone uses words with the meanings that I do, then he can connect no sense with this combination. If it
makes sense to him, he must understand something different by these words from what I do.'
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5 The arbitrariness of linguistic expressions: might we say: A child must of course learn to speak a particular
language, but doesn't have to learn to think, i.e. it would think spontaneously, even without learning any language?
Page 53
But in my view, if it thinks, then it forms for itself pictures and in a certain sense these are arbitrary, that is to
say, in so far as other
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pictures could have played the same role. On the other hand, language has certainly also come about naturally, i.e.
there must presumably have been a first man who for the first time expressed a definite thought in spoken words.
And besides, the whole question is a matter of indifference because a child learning a language only learns it by
beginning to think in it. Suddenly beginning; I mean: there is no preliminary stage in which a child already uses a
language, so to speak uses it for communication, but does not yet think in it.
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Of course, the thought processes of an ordinary man consist of a medley of symbols, of which the strictly
linguistic perhaps form only a small part.
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6 If I explain the meaning of a word 'A' to someone by pointing to something and saying 'This is A', then this
expression may be meant in two different ways. Either it is itself a proposition already, in which case it can only be
understood once the meaning of 'A' is known, i.e. I must now leave it to chance whether he takes it as I meant it or
not. Or the sentence is a definition. Suppose I have said to someone 'A is ill', but he doesn't know who I mean by 'A',
and I now point at a man, saying 'This is A'. Here the expression is a definition, but this can only be understood if he
has already gathered what kind of object it is through his understanding of the grammar of the proposition 'A is ill'.
But this means that any kind of explanation of a language presupposes a language already. And in a certain sense,
the use of language is something that cannot be taught, i.e. I cannot use language to teach it in the way in which
language could be used to teach someone to play the piano.--And that of course is just another way of saying: I
cannot use language to get outside language.
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7 Grammar is a 'theory of logical types'.
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I do not call a rule of representation a convention if it can be justified in propositions: propositions describing
what is represented and showing that the representation is adequate. Grammatical conventions cannot be justified by
describing what is represented. Any such description already presupposes the grammatical rules. That is to say, if
anything is to count as nonsense in the grammar which is to be justified, then it cannot at the same time pass for
sense in the grammar of the propositions that justify it (etc.).
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You cannot use language to go beyond the possibility of evidence.
Page 55
The possibility of explaining these things always depends on someone else using language in the same way
as I do. If he states that a certain string of words makes sense to him, and it makes none to me, I can only suppose
that in this context he is using words with a different meaning from the one I give them, or else is speaking without
thinking.
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8 Can anyone believe it makes sense to say 'That's not a noise, it's a colour'?
Page 55
On the other hand, you can of course say 'It's not the noise but the colour that makes me nervous', and here it
might look as if a variable assumed a colour and a noise as values. ('Sounds and colours can be used as vehicles of
communication.') It is clear that this proposition is of the same kind as 'If you hear a shot or see me wave, run.' For
this is the kind of co-ordination on the basis of which a heard or seen language functions.
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9 Asked whether philosophers have hitherto spoken nonsense, you could reply: no, they have only failed to notice
that they are using a word in quite different senses. In this sense, if we say it's nonsense to say that one thing is as
identical as another, this needs qualification, since if anyone says this with conviction, then at that
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moment he means something by the word 'identical' (perhaps 'large'), but isn't aware that he is using the word here
with a different meaning from that in 2 + 2 = 4.
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II
Page 57
10 If you think of propositions as instructions for making models, their pictorial nature becomes even clearer.
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Since, for it to be possible for an expression to guide my hand, it must have the same multiplicity as the
action desired.
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And this must also explain the nature of negative propositions. Thus, for example, someone might show his
understanding of the proposition 'The book is not red' by throwing away the red when preparing a model.
Page 57
This and the like would also show in what way the negative proposition has the multiplicity of the
proposition it denies and not of those propositions which could perhaps be true in its stead.
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11 What does it mean to say 'Admittedly I can't see any red, but if you give me a paint-box, I can point it out to
you'? How can you know that you will be able to point it out if...; and so, that you will be able to recognize it when
you see it?
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This might mean two different kinds of things: it might express the expectation that I shall recognize it if I am
shown it, in the same sense that I expect a headache if I'm hit on the head; then it is, so to speak, an expectation that
belongs to physics, with the same sort of grounds as any other expectation relating to the occurrence of a physical
event.--Or else it has nothing to do with expecting a physical event, and for that reason neither would my
proposition be falsified if such an event should fail to occur. Instead, it's as if the proposition is saying that I possess
a paradigm that I could at any time compare the colour with. (And the 'could' here is logical possibility.)
Page 57
Taking the first interpretation: if, on looking at a certain colour, I in fact do give a sign of recognition, how do
I know it is the colour that I meant?
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Page 58
The propositions of our grammar are always of the same sort as propositions of physics and not of the same
sort as the 'primary' propositions which treat of what is immediate.
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12 The idea that you 'imagine' the meaning of a word when you hear or read it, is a naive conception of the meaning
of a word. And in fact such imagining gives rise to the same question as a word meaning something. For if, e.g., you
imagine sky-blue and are to use this image as a basis for recognizing or looking for the colour, we are still forced to
say that the image of the colour isn't the same as the colour that is really seen; and in that case, how can one
compare these two?
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Yet the naive theory of forming-an-image can't be utterly wrong.
Page 58
If we say 'A word only has meaning in the context of a proposition', then that means that it's only in a
proposition that it functions as a word, and this is no more something that can be said than that an armchair only
serves its purpose when it is in space. Or perhaps better: that a cogwheel only functions as such when engaged with
other cogs.
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13 Language must have the same multiplicity as a control panel that sets off the actions corresponding to its
propositions.
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Strangely enough, the problem of understanding language is connected with the problem of the Will.
Page 58
Understanding a command before you obey it has an affinity with willing an action before you perform it.
Page 58
Just as the handles in a control room are used to do a wide variety of things, so are the words of language
that correspond to the handles. One is the handle of a crank and can be adjusted continuously; one belongs to a
switch and is always either on or
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off; a third to a switch which permits three or more positions; a fourth is the handle of a pump and only works when
it is being moved up and down, etc.; but all are handles, are worked by hand.
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14 A word only has meaning in the context of a proposition: that is like saying only in use is a rod a lever. Only the
application makes it into a lever.
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Every instruction can be construed as a description, every description as an instruction.
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15 What does it mean, to understand a proposition as a member of a system of propositions? (It's as if I were to say:
the use of a word isn't over in an instant, any more than that of a lever).
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Imagine a gearbox whose lever can take four positions. Now of course it can only take these positions in
succession, and that takes time; and suppose it happened that it only ever occupied one of these positions, since the
gearbox was then destroyed. Wasn't it still a gearbox with four positions? Weren't the four possible?
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Anyone who saw it would have seen its complexity, and its complexity is only to be explained by the use for
which it was intended, to which in fact it was not put. Similarly I would like to say in the case of language: What's
the point of all these preparations; they only have any meaning if they find a use.
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You might say: The sense of a proposition is its purpose. (Or, of a word 'Its meaning is its purpose'.)
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But the natural history of the use of a word can't be any concern of logic.
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16 If I expect an event and that which fulfils my expectation occurs, does it then make sense to ask whether that
really is the
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event I expected? i.e. how would a proposition that asserted this be verified? It is clear that the only source of
knowledge I have here is a comparison of the expression of my expectation with the event that has occurred.
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How do I know that the colour of this paper, which I call 'white', is the same as the one I saw here yesterday?
By recognizing it again; and recognizing it again is my only source of knowledge here. In that case, 'That it is the
same' means that I recognize it again.
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Then of course you also can't ask whether it really is the same and whether I might not perhaps be mistaken;
(whether it is the same and doesn't just seem to be.)
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Of course, it would also be possible to say that the colour is the same because chemical investigations do not
disclose any change. So that if it doesn't look the same to me then I am mistaken. But even then there must still be
something that is immediately recognized.
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And the 'colour' I can recognize immediately and the one I establish by chemical investigation are two
different things.
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One source only yields one thing.
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17 Is it an objection to my view that we often speak half or even entirely automatically? If someone asks me 'Is the
curtain in this room green?' and I look and say, 'No, red', I certainly don't have to hallucinate green and compare it
with the curtain. No, just looking at the curtain can automatically produce the answer, and yet this answer is of
interest to Logic, whereas a whistle, say, that I make automatically on seeing red is not. Isn't the point that Logic is
only interested in this answer as a part of a language system? The system our books are written in. Could we say
Logic considers language in extenso? And so, in the same way as grammar.
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Could you say that Logic has nothing to do with that utterance if it was merely automatic? For should Logic
bother itself with the question whether the proposition was also really thoroughly
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thought? And what would the criterion for that be? Surely not the lively play of images accompanying its
expression I It is plain that here we have got into a region that is absolutely no concern of ours and from which we
should retire with the utmost alacrity.
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18 Here we come to the apparently trivial question, what does Logic understand by a word--is it an ink-mark, a
sequence of sounds, is it necessary that someone should associate a sense with it, or should have associated one,
etc., etc.?--And here, the crudest conception must obviously be the only correct one.
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And so I will again talk about 'books'; here we have words; if a mark should happen to occur that looks like a
word, I say: that's not a word, it only looks like one, it's obviously unintentional. This can only be dealt with from the
standpoint of normal common sense. (It's extraordinary that that in itself constitutes a change in perspective.)
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I do not believe that Logic can talk about sentences [†1] in any other than the normal sense in which we say,
'There's a sentence written here' or 'No, that only looks like a sentence, but isn't', etc., etc.
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The question 'What is a word?' is completely analogous with the question 'What is a chessman?'
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19 Isn't it agreement and disagreement that is primary, just as recognition is what is primary and identity what is
secondary? If we see a proposition verified, what higher court is there to which we could yet appeal in order to tell
whether it really is true?
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The agreement of a proposition with reality only resembles the agreement of a picture with what it depicts to
the same extent as the agreement of a memory image with the present object.
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But we can look at recognition, like memory, in two different ways: as a source of the concepts of the past
and of identity, or as a way of checking what happened in the past, and on identity.
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If I can see two patches of colour alongside one another and say that they have the same colour, and if I say
that this patch has the same colour as one I saw earlier, the identity assertion means something different in the two
cases, since it is differently verified.
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To know that it was the same colour is something different from knowing that it is the same colour.
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III
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20 You can draw a plan from a description. You can translate a description into a plan.
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The rules of translation here are not essentially different from the rules for translating from one verbal
language into another.
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A wrong conception of the way language functions destroys, of course, the whole of logic and everything
that goes with it, and doesn't just create some merely local disturbance.
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If you exclude the element of intention from language, its whole function then collapses.
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21 What is essential to intention is the picture: the picture of what is intended.
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It may look as if, in introducing intention, we were introducing an uncheckable, a so-to-speak metaphysical
element into our discussion. But the essential difference between the picture conception and the conception of
Russell, Ogden and Richards, is that it regards recognition as seeing an internal relation, whereas in their view this is
an external relation.
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That is to say, for me, there are only two things involved in the fact that a thought is true, i.e. the thought and
the fact; whereas for Russell, there are three, i.e. thought, fact and a third event which, if it occurs, is just recognition.
This third event, a sort of satisfaction of hunger (the other two being hunger and eating a particular kind of food),
could, for example, be a feeling of pleasure. It's a matter of complete indifference here how we describe this third
event; that is irrelevant to the essence of the theory.
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The causal connection between speech and action is an external relation, whereas we need an internal one.
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22 I believe Russell's theory amounts to the following: if I give someone an order and I am happy with what he then
does, then he has carried out my order.
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(If I wanted to eat an apple, and someone punched me in the stomach, taking away my appetite, then it was
this punch that I originally wanted.)
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The difficulty here in giving an account of what's going on is that if someone makes false assumptions about
the way language works and tries to give an account of something with language conceived as functioning in this
way, the result is not something false but nonsense.
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Thus in terms of Russell's theory I could not express things by saying that the order is carried out if I am
made happy by what happens, because I have also to recognise my being made happy, and this requires that
something else should happen which I cannot describe in advance.
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23 Suppose you were now to say: pictures do occur, but they are not what is regular; but how strange then, if they
happen to be there and a conflict were now to arise between the two criteria of truth and falsity. How should it be
adjudicated?
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In that case, there would, of course, be no distinction between a command and its countermand, since both
could be obeyed in the same way.
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If when a language is first learnt, speech, as it were, is connected up to action--i.e. the levers to the
machine--then the question arises, can these connections possibly break down? If they can't, then I have to accept
any action as the right one; on the other hand if they can, what criterion have I for their having broken down? For
what means have I for comparing the original arrangement with the subsequent action?
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It is such comparison which is left out in Russell's theory. And comparison doesn't consist in confronting the
representation with what it represents and through this confrontation experiencing a phenomenon, which, as I have
said, itself could not be described in advance.
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(Experience decides whether a proposition is true or false, but not its sense.)
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24 How is a picture meant? The intention never resides in the picture itself, since, no matter how the picture is
formed, it can always be meant in different ways. But that doesn't mean that the way the picture is meant only
emerges when it elicits a certain reaction, for the intention is already expressed in the way I now compare the picture
with reality.
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In philosophy we are always in danger of giving a mythology of the symbolism, or of psychology: instead of
simply saying what everyone knows and must admit.
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What if someone played chess and, when he was mated, said, 'Look, I've won, for that is the goal I was
aiming at'? We would say that such a man simply wasn't trying to play chess, but another game; whereas Russell
would have to say that if anyone plays with the pieces and is satisfied with the outcome, then he has won at chess.
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I expect that the rod will be 2 m high in the same sense in which it is now 1 m 99 cm high.
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25 The fulfilment of an expectation doesn't consist in a third thing happening which you could also describe in
another way than just as 'the fulfilment of the expectation', thus for example as a feeling of satisfaction or pleasure or
whatever.
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For expecting that p will be the case must be the same as expecting that this expectation will be fulfilled;
whereas, if I am wrong,
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expecting p would be different from expecting that this expectation will be fulfilled.
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Isn't it like this: My theory is completely expressed in the fact that the state of affairs satisfying the
expectation of p is represented by the proposition p? And so, not by the description of a totally different event.
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26 I should like to say, if there were only an external connection no connection could be described at all, since we
only describe the external connection by means of the internal one. If this is lacking, we lose the footing we need for
describing anything at all--just as we can't shift anything with our hands unless our feet are planted firmly.
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Causality rests on an observed uniformity. Now, that doesn't mean that a uniformity we have observed until
now will go on for ever, but it must be an established fact that events have been uniform until now; that cannot in
turn be the insecure result of a series of observations which again is itself not a datum, but depends on another
equally insecure series, etc. ad inf.
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If I wish that p were the case, then of course p is not the case and there must be a surrogate for p in the state
of wishing, just as, of course, in the expression of the wish.
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There's nothing left for me, in answer to the question, 'What does p instruct you to do?', but to say it, i.e. to
give another sign.
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But can't you give someone an instruction by showing him how to do something? Certainly: and then you
have to tell him 'Now copy that'. Perhaps you have already had examples of this before but now you have to say to
him that what happened then should happen now. That still means: sooner or later there is a leap from the sign to
what is signified.
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27 The meaning of a question is the method of answering it: then what is the meaning of 'Do two men really mean
the same by the word "white"?'
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Tell me how you are searching, and I will tell you what you are searching for.
Page 67
If I understand an order but do not carry it out, then understanding it can only consist in a process which is a
surrogate for its execution, and so in a different process from its execution.
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I should like to say, assuming the surrogate process to be a picture doesn't get me anywhere, since even that
does not do away with the transition from the picture to what is depicted.
Page 67
If you were to ask: 'Do I expect the future itself, or only something similar to the future?', that would be
nonsense. Or, if you said, 'We can never be certain that that was what we really expected.'
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Co-ordinating signals always contains something general, otherwise the coordination is unnecessary. It is a
co-ordination which has to be understood in the particular case.
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If I say to someone that it will be fine tomorrow, he gives evidence of his having understood by not trying to
verify the proposition now.
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28 Expecting is connected with looking for: looking for something presupposes that I know what I am looking for,
without what I am looking for having to exist.
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Earlier I would have put this by saying that searching presupposes the elements of the complex, but not the
combination that I was looking for.
Page 67
And that isn't a bad image: for, in the case of language, that would be expressed by saying that the sense of a
proposition only presupposes the grammatically correct use of certain words.
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How do I know that I have found that which I was looking for? (That what I expected has occurred, etc.)
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I cannot confront the previous expectation with what happens.
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The event that replaces the expectation, is a reply to it.
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But for that to be so, necessarily some event must take its place, and that of course implies that the
expectation must be in the same space as what is expected.
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In this context I am talking about an expectation only as something that is necessarily either fulfilled or
disappointed: therefore not of an expectation in the void.
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29 The event which takes the place of an expectation, answers it: i.e. the replacement constitutes the answer, so that
no question can arise whether it really is the answer. Such a question would mean putting the sense of a proposition
in question. 'I expect to see a red patch' describes, let's say, my present mental state. 'I see a red patch' describes
what I expect: a completely different event from the first. Couldn't you now ask whether the word 'red' has a
different meaning in the two cases? Doesn't it look as if the first proposition uses an alien and inessential event to
describe my mental state? Perhaps like this: I now find myself in a state of expectation which I characterize by
saying that it is satisfied by the event of my seeing a red patch. That is, as though I were to say 'I am hungry and
know from experience that eating a particular kind of food will or would satisfy my hunger.' But expectation isn't
like that! Expectation is not given an external description by citing what is expected, as is hunger by citing what food
satisfies it--in the last resort the appropriate food of course can still only be a matter of conjecture. No, describing an
expectation by means of what is expected is giving an internal description.
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The way the word 'red' is used is such that it has a use in all these propositions: 'I expect to see a red patch', 'I
remember a red patch', 'I am afraid of a red patch', etc.
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30 If I say 'This is the same event as I expected', and 'This is the same event as also happened on that occasion', then
the word
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'same' has two different meanings. (And you wouldn't normally say 'This is the same as I expected' but 'This is what
I expected'.)
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Could we imagine any language at all in which expecting p was described without using 'p'?
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Isn't that just as impossible as a language in which ~p would be expressed without using 'p'?
Page 69
Isn't this simply because expectation uses the same symbol as the thought of its fulfilment?
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For if we think in signs, then we also expect and wish in signs.
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(And you could almost say that someone could hope in German and fear in English, or vice versa.)
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31 Another mental process belonging to this group, and which ties in with all these things, is intention. You could
say that language is like a control room operated with a particular intention or built for a particular purpose.
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If a mechanism is meant to act as a brake, but for some reason accelerates a machine then the purpose of the
mechanism cannot be found out from it alone.
Page 69
If you were then to say 'That's a brake lever but it doesn't work', you would be talking about intention. It is
just the same as when we still call a broken clock a clock.
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(Psychological--trivial--discussions of expectation, association, etc. always leave out what is really
remarkable, and you notice that they talk all around, without touching on the vital point.)
Page 69
32 I only use the terms the expectation, thought, wish, etc., that p will be the case, for processes having the
multiplicity that finds
Page Break 70
expression in p, and thus only if they are articulated. But in that case they are what I call the interpretation of signs.
Page 70
I only call an articulated process a thought: you could therefore say 'only what has an articulated
expression'.
Page 70
(Salivation--no matter how precisely measured--is not what I call expecting.)
Page 70
Perhaps we have to say that the phrase 'interpretation of signs' is misleading and instead we ought to say 'the
use of signs'. For 'interpretation' makes it sound as if one were now to correlate the word 'red' with the colour (when
it isn't even there), etc. And now the question again arises: what is the connection between sign and world? Could I
look for something unless the space were there to look for it in?
Page 70
Where does the sign link up with the world?
Page 70
33 To look for something is, surely, an expression of expectation. In other words: How you search in one way or
another expresses what you expect.
Page 70
Thus the idea would be: what expectation has in common with reality is that it refers to another point in the
same space. ('Space' in a completely general sense.)
Page 70
I see a patch getting nearer and nearer to the place where I expect it.
Page 70
If I say I remember a colour--say, the colour of a certain book--you could take as evidence for this the fact
that I was in a position to mix this colour or recognize it again, or say of other colours that they are more like or less
like the colour I remember.
Page 70
Expectation, so to speak, prepares a yardstick for measuring the event when it comes and what's more, in
such a way that it will necessarily be possible to measure the one with the other, whether the event coincides with
the expected graduation mark or not.
Page 70
It is, say, as if I guess a man's height by looking at him, saying
Page Break 71
'I believe he's 5 ft 8 in' and then set about measuring him with a tape measure. Even if I don't know how tall he is, I
still know that his height is measured with a tape measure and not a weighing machine.
Page 71
If I expect to see red, then I prepare myself for red.
Page 71
I can prepare a box for a piece of wood to fit in, just because the wood, whatever it's like, must have a
volume.
Page 71
If there were no connection between the act of expectation and reality, you could expect a nonsense.
Page 71
34 The expectation of p and the occurrence of p correspond perhaps to the hollow shape of a body and the solid
shape. Here p corresponds to the shape of the volume, and the different ways in which this shape is given
correspond to the distinction between expectation and occurrence.
Page 71
If I say 'I can make you a sketch of that any time you like', then that presupposes that I am in the same space
as the business involved.
Page 71
Our expectation anticipates the event. In this sense, it makes a model of the event. But we can only make a
model of a fact in the world we live in, i.e. the model must be essentially related to the world we live in and what's
more, independently of whether it's true or false.
Page 71
If I say that the representation must treat of my world, then you cannot say 'since otherwise I could not
verify it', but 'since otherwise it wouldn't even begin to make sense to me'.
Page 71
35 In expecting, the part corresponding to searching in a space is the directing of one's attention.
Page Break 72
Page 72
Surely the strange thing about expectation is that we know that it is an expectation. For we couldn't, e.g.,
imagine the following situation: I have some image or other before me and say: 'Now, I don't know whether it's an
expectation or a memory, or an image without any relation to reality.'
Page 72
And that is what shows that expectation is immediately connected with reality.
Page 72
For of course you couldn't say that the future the expectation speaks of--I mean the concept of the
future--was also only a surrogate for the real future.
Page 72
For I await in just as real a sense as I wait.
Page 72
Could you also say: You cannot describe an expectation unless you can describe the present reality; or, you
cannot describe an expectation unless you can give a description comparing the expectation with the present, of the
form: Now I see a red circle here, and expect a blue square there later on.
Page 72
That is to say the yardstick of language must be applied at the point which is present and then points out
beyond it--roughly speaking, in the direction of the expectation.
Page 72
36 It only makes sense to give the length of an object if I have a method for finding the object--since otherwise I
cannot apply a yardstick to it.
Page 72
What I once called 'objects', simples, were simply what I could refer to without running the risk of their
possible non-existence; i.e. that for which there is neither existence nor non-existence, and that means: what we can
speak about no matter what may be the case.
Page 72
The visual table is not composed of electrons.
Page 72
What if someone said to me 'I expect three knocks on the door' and I replied 'How do you know three
knocks exist?'--Wouldn't that be just like the question 'How do you know six feet exist?' after someone has said 'I
believe A is 6 feet high'?
Page Break 73
Page 73
37 Can absolute silence be confused with inner deafness, meaning having no acquaintance with the concept of
sound? If that were so, you couldn't distinguish lacking the sense of hearing from lacking any other sense.
Page 73
But isn't this exactly the same question as: 'Is a man who cannot see any red around him at present, in the
same position as someone incapable of seeing red?'
Page 73
You could of course say: The one can still imagine red, but the red we imagine is not the same as the red we
see.
Page 73
38 Our ordinary language has no means for describing a particular shade of a colour, such as the brown of my table.
Thus it is incapable of producing a picture of this colour.
Page 73
If I want to tell someone what colour some material is to be, I send him a sample, and obviously this sample
belongs to language; and equally the memory or image of a colour that I conjure by a word, belongs to language.
Page 73
The memory and the reality must be in one space.
Page 73
I could also say: the image and the reality are in one space.
Page 73
If I compare two colour samples in front of me with one another, and if I compare a colour sample with my
image of a sample, that is similar to comparing, on the one hand, the lengths of two rods standing up against each
other and on the other of two that are apart. In that case, I can say perhaps, they are the same height, if, turning my
gaze horizontally, I can glance from the tip of the one to the tip of the other.
Page 73
As a matter of fact I have never seen a black patch become gradually lighter and lighter until it is white and
then redden until it is red; but I know that this would be possible because I can imagine it; i.e. I operate with my
images in colour space and do with them what would be possible with the colours. And my words
Page Break 74
take their sense from the fact that they more or less completely reflect the operations of the images perhaps in the
way in which a score can be used to describe a piece of music that has been played, but for example, does not
reproduce the emphasis on each individual note.
Page 74
Grammar gives language the necessary degrees of freedom.
Page Break 75
IV
Page 75
39 The colour octahedron is grammar, since it says that you can speak of a reddish blue but not of a reddish green,
etc.
Page 75
If I can only see something black and say it isn't red, how do I know that I am not talking nonsense, i.e. that it
could be red, that there is red? Unless red is just another graduation mark on the same scale as black. What is the
difference between 'That is not red' and 'That is not abracadabra'? Obviously I need to know that 'black', which
describes the actual state of affairs (or is used in describing it), is that in whose place 'red' stands in the description.
Page 75
But what does that mean? How do I know it isn't 'soft' in whose place 'red' stands? Can you say red is less
different from black than from soft? That would of course be nonsense.
Page 75
40 How far can you compare the colours with points on a scale?
Page 75
Can you say that the direction leading from black to red is a different one from the one you must take from
black to blue?
Page 75
For, if there is black in front of me and I am expecting red, that's different from having black in front of me
and expecting blue. And if there is a valid comparison with a ruler, the word 'blue' must so to speak give me the
direction in which I go from black to blue; so to speak the method by which I reach blue.
Page 75
Couldn't we also say: 'The proposition must give a construction for the position of blue, the point the fact
must reach if such and such is to be blue'?
Page 75
The fact that I can say that one colour comes closer to what I expected than another belongs here.
Page Break 76
Page 76
But how do these different directions find expression in grammar? Isn't it the same case as my seeing a grey
and saying 'I expect this grey to go darker'? How does grammar deal with the distinction between 'lighter' and
'darker'? Or, how can the ruler going from white to black be applied to grey in a particular direction?
Page 76
It's still as if grey were only one point; and how can I see the two directions in that? And yet I should be able
to do so somehow or other if it is to be possible for me to get to a particular place in these directions [†1].
Page 76
41 The feeling is as if, for it to negate p, ~p has in a certain sense first to make it true. One asks 'What isn't the case?'
This must be represented but cannot be represented in such a way that p is actually made true.
Page 76
A man with red/green colour blindness has a different colour system from a normal man. He will be like a
man whose head was fixed in one position and so had a different kind of space, since for him there would only be
visual space and therefore, e.g., no 'behind'. That wouldn't of course mean that Euclidean space was bounded for
him. But that--at least as far as seeing things was concerned--he wouldn't acquire the concept of Euclidean space. Is
the question then: can someone who doesn't know what red and green are like, really see what we (or I) call 'blue'
and 'yellow'?
Page 76
This question must, of course, be just as nonsensical as the question whether someone else with normal
vision really sees the same as I do.
Page 76
42 Grey must already be conceived as being in lighter/darker space if I want to talk of its being possible for it to get
darker or lighter.
Page 76
So you might perhaps also say: the yardstick must already be applied, I cannot apply it how I like; I can only
pick out a point on it.
Page 76
This amounts to saying: if I am surrounded by absolute silence,
Page Break 77
I cannot join (construct) or not join auditory space on to this silence as I like, i.e. either it is for me 'silence' as
opposed to a sound, or the word 'silence' has no meaning for me, i.e. I cannot choose between inner hearing and
inner deafness.
Page 77
And in just the same way, I cannot while I am seeing greyness choose between normal inner vision and
partial or complete colour-blindness.
Page 77
Suppose we had a device for completely cutting out our visual activity so that we could lose our sense of
sight; and suppose I had so cut it out: could I say in such circumstances 'I can see a yellow patch on a red
background'? Could this way of talking make sense to me?
Page 77
43 I should like to say: for any question there is always a corresponding method of finding.
Page 77
Or you might say, a question denotes a method of searching.
Page 77
You can only search in a space. For only in space do you stand in a relation to where you are not.
Page 77
To understand the sense of a proposition means to know how the issue of its truth or falsity is to be decided.
Page 77
The essence of what we call the will is immediately connected with the continuity of the given.
Page 77
You must find the way from where you are to where the issue is decided.
Page 77
You cannot search wrongly; you cannot look for a visual impression with your sense of touch.
Page 77
You cannot compare a picture with reality, unless you can set it against it as a yardstick.
Page 77
You must be able to fit the proposition on to reality.
Page Break 78
Page 78
The reality that is perceived takes the place of the picture.
Page 78
If I am to settle whether two points are a certain distance apart, I must look at the distance that does separate
them.
Page 78
44 How is a 'formally certified proposition' possible? It would be a proposition that you could tell was true or false
by looking at it. But how can you discover by inspecting the proposition or thought that it is true? The thought is
surely something quite different from the state of affairs asserted by the proposition.
Page 78
The method of taking measurements, e.g. spatial measurements, is related to a particular measurement in
precisely the same way as the sense of a proposition is to its truth or falsity.
Page 78
The use, the application, of a yardstick doesn't presuppose any particular length for the object to be
measured.
Page 78
That is why I can learn how to measure in general, without measuring every measurable object. (This isn't
simply an analogy, but is in fact an example.)
Page 78
All that I need is: I must be able to be certain I can apply my yardstick.
Page 78
Thus if I say 'Three more steps and I'll see red', that presupposes that at any rate I can apply the yardsticks of
length and colour.
Page 78
Someone may object that a scale with a particular height marked on it can say that something has that height,
but not what has it.
Page 78
I would then perhaps reply that all I can do is say that something 3 m away from me in a certain direction is 2
m high.
Page 78
45 I will count any fact whose obtaining is a presupposition of a proposition's making sense, as belonging to
language.
Page 78
It's easy to understand that a ruler is and must be in the same space as the object measured by it. But in what
sense are words in
Page Break 79
the same space as an object whose length is described in words, or, in the same space as a colour, etc.? It sounds
absurd.
Page 79
A black colour can become lighter but not louder. That means that it is in light/dark space but not loud/soft
space.--But surely the object just stops being black when it becomes lighter. But in that case it was black and just as
I can see movement (in the ordinary sense), I can see a colour movement.
Page 79
The unit length is part of the symbolism. It belongs to the method of projection. Its length is arbitrary, but it
is what contains the specifically spatial element.
Page 79
And so if I call a length '3', the 3 signifies via the unit length presupposed in the symbolism.
Page 79
You can also apply these remarks to time.
Page 79
46 When I built language up by using a coordinate system for representing a state of affairs in space, I introduced
into language an element which it doesn't normally use. This device is surely permissible. And it shows the
connection between language and reality. The written sign without the coordinate system is senseless. Mustn't we
then use something similar for representing colours?
Page 79
If I say something is three feet long, then that presupposes that somehow or other I am given the foot length.
In fact it is given by a description: in such and such a place there is a rod one foot long. The 'such and such a place'
indirectly describes a method for getting there; otherwise the specification is senseless. The place name 'London'
only has a sense if it is possible to try to find London.
Page 79
A command is only then complete, when it makes sense no matter what may be the case. We might also say:
That is when it is completely analysed.
Page Break 80
V
Page 80
47 That it doesn't strike us at all when we look around us, move about in space, feel our own bodies, etc., etc., shows
how natural these things are to us. We do not notice that we see space perspectively or that our visual field is in
some sense blurred towards the edges. It doesn't strike us and never can strike us because it is the way we perceive.
We never give it a thought and it's impossible we should, since there is nothing that contrasts with the form of our
world.
Page 80
What I wanted to say is it's strange that those who ascribe reality only to things and not to our ideas [†1]
move about so unquestioningly in the world as idea [†1] and never long to escape from it.
Page 80
In other words, how much of a matter of course the given is. It would be the very devil if this were a tiny
picture taken from an oblique, distorting angle.
Page 80
This which we take as a matter of course, life, is supposed to be something accidental, subordinate; while
something that normally never comes into my head, reality!
Page 80
That is, what we neither can nor want to go beyond would not be the world.
Page 80
Time and again the attempt is made to use language to limit the world and set it in relief--but it can't be done.
The self-evidence of the world expresses itself in the very fact that language can and does only refer to it.
Page 80
For since language only derives the way in which it means from its meaning, from the world, no language is
conceivable which does not represent this world.
Page 80
48 If the world of data is timeless, how can we speak of it at all?
Page Break 81
Page 81
The stream of life, or the stream of the world, flows on and our propositions are so to speak verified only at
instants.
Page 81
Our propositions are only verified by the present.
Page 81
So they must be so constructed that they can be verified by it. And so in some way they must be
commensurable with the present; and they cannot be so in spite of their spatio-temporal nature; on the contrary this
must be related to their commensurability as the corporeality of a ruler is to its being extended--which is what
enables it to measure. In this case, too, you cannot say: 'A ruler does measure in spite of its corporeality; of course a
ruler which only has length would be the Ideal, you might say the pure ruler'. No, if a body has length, there can be
no length without a body and although I realize that in a certain sense only the ruler's length measures, what I put in
my pocket still remains the ruler, the body, and isn't the length.
Page 81
49 Perhaps this whole difficulty stems from taking the time concept from time in physics and applying it to the
course of immediate experience. It's a confusion of the time of the film strip with the time of the picture it projects.
For 'time' has one meaning when we regard memory as the source of time, and another when we regard it as a
picture preserved from a past event.
Page 81
If we take memory as a picture, then it's a picture of a physical event. The picture fades, and I notice how it
has faded when I compare it with other evidence of what happened. In this case, memory is not the source of time,
but a more or less reliable custodian of what 'actually' happened; and this is something we can know about in other
ways, a physical event.--It's quite different if we now take memory to be the source of time. Here it isn't a picture,
and cannot fade either--not in the sense in which a picture fades, becoming an ever less faithful representation of its
object. Both ways of talking are in order, and are equally legitimate, but cannot be mixed together. It's clear of
course that
Page Break 82
speaking of memory as a picture is only a metaphor; just as the way of speaking of images as 'pictures of objects in
our minds' (or some such phrase) is a metaphor. We know what a picture is, but images are surely no kind of picture
at all. For, in the first case I can see the picture and the object of which it is a picture. But in the other, things are
obviously quite different. We have just used a metaphor and now the metaphor tyrannizes us. While in the language
of the metaphor, I am unable to move outside of the metaphor. It must lead to nonsense if you try to use the
language of this metaphor to talk about memory as the source of our knowledge, the verification of our propositions.
We can speak of present, past and future events in the physical world, but not of present, past and future images, if
what we are calling an image is not to be yet another kind of physical object (say, a physical picture which takes the
place of the body), but precisely that which is present. Thus we cannot use the concept of time, i.e. the syntactical
rules that hold for the names of physical objects, in the world of the image [†1], that is, not where we adopt a
radically different way of speaking.
Page 82
50 If memory is no kind of seeing into the past, how do we know at all that it is to be taken as referring to the past?
We could then remember some incident and be in doubt whether in our memory image we have a picture of the past
or of the future.
Page 82
We can of course say: I do not see the past, only a picture of the past. But how do I know it's a picture of the
past unless this belongs to the essence of a memory-image? Have we, say, learnt from experience to interpret these
pictures as pictures of the past? But in this context what meaning would 'past' have at all?
Page 82
Yet it contradicts every concept of physical time that I should have perception into the past, and that again
seems to mean nothing
Page Break 83
else than that the concept of time in the first system must be radically different from that in physics.
Page 83
Can I conceive the time in which the experiences of visual space occur without experiences of sound? It
appears so. And yet how strange that something should be able to have a form, which would also be conceivable
without this content. Or does a man who has been given hearing also learn a new time along with it?
Page 83
The traditional questions are not suited to a logical investigation of phenomena. These generate their own
questions, or rather, give their own answers.
Page 83
51 If I compare the facts of immediate experience with the pictures on the screen and the facts of physics with
pictures in the film strip, on the film strip there is a present picture and past and future pictures. But on the screen,
there is only the present.
Page 83
What is characteristic about this image is that in using it I regard the future as preformed.
Page 83
There's a point in saying future events are pre-formed if it belongs to the essence of time that it does not
break off. For then we can say: something will happen, it's only that I don't know what. And in the world of physics
we can say that.
Page 83
52 It's strange that in ordinary life we are not troubled by the feeling that the phenomenon is slipping away from us,
the constant flux of appearance, but only when we philosophize. This indicates that what is in question here is an
idea suggested by a misapplication of our language.
Page 83
The feeling we have is that the present disappears into the past without our being able to prevent it. And here
we are obviously using the picture of a film strip remorselessly moving past us, that we are unable to stop. But it is
of course just as clear that the picture is misapplied: that we cannot say 'Time flows' if by time we mean the
possibility of change. What we are looking at here is really the possibility of motion: and so the logical form of
motion.
Page Break 84
Page 84
In this connection it appears to us as if memory were a somewhat secondary sort of experience, when
compared with experience of the present. We say 'We can only remember that'. As though in a primary sense
memory were a somewhat faint and uncertain picture of what we originally had before us in full clarity.
Page 84
In the language of physical objects, that's so: I say: 'I only have a vague memory of this house.'
Page 84
53 And why not let matters rest there? For this way of talking surely says everything we want to say, and everything
that can be said. But we wish to say that it can also be put differently; and that is important.
Page 84
It is as if the emphasis is placed elsewhere in this other way of speaking: for the words 'seem', 'error', etc.,
have a certain emotional overtone which doesn't belong to the essence of the phenomena. In a way it's connected
with the will and not merely with cognition.
Page 84
We talk for instance of an optical illusion and associate this expression with the idea of a mistake, although
of course it isn't essential that there should be any mistake; and if appearance were normally more important in our
lives than the results of measurement, then language would also show a different attitude to this phenomenon.
Page 84
There is not--as I used to believe--a primary language as opposed to our ordinary language, the 'secondary'
one. But one could speak of a primary language as opposed to ours in so far as the former would not permit any
way of expressing a preference for certain phenomena over others; it would have to be, so to speak, absolutely
impartial.
Page 84
54 What belongs to the essence of the world cannot be expressed by language.
Page 84
For this reason, it cannot say that everything flows. Language can only say those things that we can also
imagine otherwise.
Page Break 85
Page 85
That everything flows must be expressed in the application of language, and in fact not in one kind of
application as opposed to another but in the application. In anything we would ever call the application of language.
Page 85
By application I understand what makes the combination of sounds or marks into a language at all. In the
sense that it is the application which makes the rod with marks on it into a measuring rod [†1]: putting language up
against reality.
Page 85
We are tempted to say: only the experience of the present moment has reality. And then the first reply must
be: As opposed to what?
Page 85
Does it imply I didn't get up this morning? (For if so, it would be dubious.) But that is not what we mean.
Does it mean that an event that I'm not remembering at this instant didn't occur? Not that either.
Page 85
The proposition that only the present experience has reality appears to contain the last consequence of
solipsism. And in a sense that is so; only what it is able to say amounts to just as little as can be said by
solipsism.--For what belongs to the essence of the world simply cannot be said. And philosophy, if it were to say
anything, would have to describe the essence of the world.
Page 85
But the essence of language is a picture of the essence of the world; and philosophy as custodian of grammar
can in fact grasp the essence of the world, only not in the propositions of language, but in rules for this language
which exclude nonsensical combinations of signs.
Page 85
If someone says, only the present experience has reality, then the word 'present' must be redundant here, as
the word 'I' is in other contexts. For it cannot mean present as opposed to past and future.--Something else must be
meant by the word, something that isn't in a space, but is itself a space. That is to say, not something bordering on
something else (from which it could therefore be limited off). And so, something language cannot legitimately set in
relief.
Page Break 86
Page 86
The present we are talking about here is not the frame in the film reel that is in front of the projector's lens at
precisely this moment, as opposed to the frames before and after it, which have already been there or are yet to
come; but the picture on the screen which would illegitimately be called present, since 'present' would not be used
here to distinguish it from past and future. And so it is a meaningless epithet.
Page 86
55 There are, admittedly, very interesting, completely general propositions of great importance, therefore
propositions describing an actual experience which might have been otherwise, but just is as it is. For instance, that
I have only one body. That my sensations never reach out beyond this body (except in cases where someone has
had a limb, e.g. an arm, amputated, and yet feels pain in his fingers). These are remarkable and interesting facts.
Page 86
But it does not belong in this category, if someone says I cannot remember the future. For that means
nothing, and, like its opposite, is something inconceivable.
Page 86
That I always see with my eyes when I am awake is on the other hand a remarkable and interesting fact.
Equally, it is important that my visual field is almost incessantly in a state of flux.
Page 86
'I' clearly refers to my body, for I am in this room; and 'I' is essentially something that is in a place, and in a
place belonging to the same space as the one the other bodies are in too.
Page 86
From the very outset 'Realism', 'Idealism', etc., are names which belong to metaphysics. That is, they indicate
that their adherents believe they can say something specific about the essence of the world.
Page 86
56 Anyone wishing to contest the proposition that only the present experience is real (which is just as wrong as to
maintain it) will perhaps ask whether then a proposition like 'Julius Caesar crossed the Alps' merely describes my
present mental state which is occupied with the matter. And of course the answer is: no, it
Page Break 87
describes an event which we believe happened ca. 2,000 years ago. That is, if the word 'describes' is construed in the
same way as in the sentence 'The proposition "I am writing" describes what I am at present doing'. The name Julius
Caesar designates a person. But what does all that amount to? I seem to be fighting shy of the genuinely
philosophical answer! Propositions dealing with people, i.e. containing proper names, can be verified in very
different ways.--We still might find Caesar's corpse: that this is thinkable is directly connected with the sense of the
proposition about Caesar. But also that a manuscript might be found from which it emerged that such a man never
lived and that the accounts of his existence were concocted for particular purposes. Propositions about Julius Caesar
must, therefore, have a sense of a sort that covers this possibility. If I utter the proposition: I can see a red patch
crossing a green one, the possibilities provided for in 'Julius Caesar crossed the Alps' are not present here, and to that
extent I can say that the proposition about Caesar has its sense in a more indirect way than this one.
Page 87
Everything which, if it occurred, would legitimately confirm a belief, determines logically the nature of this
belief. That is, it shows something about the logical nature of the belief.
Page 87
The proposition about Julius Caesar is simply a framework (like that about any other person) that admits of
widely differing verifications, although not all those it would allow in speaking of other people--of living people, for
instance.
Page 87
Isn't all that I mean: between the proposition and its verification there is no go-between negotiating this
verification?
Page 87
Even our ordinary language has of course to provide for all cases of uncertainty, and if we have any
philosophical objection to it, this can only be because in certain cases it gives rise to misinterpretations.
Page Break 88
VI
Page 88
57 One of the most misleading representational techniques in our language is the use of the word 'I', particularly
when it is used in representing immediate experience, as in 'I can see a red patch'.
Page 88
It would be instructive to replace this way of speaking by another in which immediate experience would be
represented without using the personal pronoun; for then we'd be able to see that the previous representation wasn't
essential to the facts. Not that the representation would be in any sense more correct than the old one, but it would
serve to show clearly what was logically essential in the representation.
Page 88
The worst philosophical errors always arise when we try to apply our ordinary--physical--language in the
area of the immediately given.
Page 88
If, for instance, you ask, 'Does the box still exist when I'm not looking at it?', the only right answer would be
'Of course, unless someone has taken it away or destroyed it'. Naturally, a philosopher would be dissatisfied with
this answer, but it would quite rightly reduce his way of formulating the question ad absurdum.
Page 88
All our forms of speech are taken from ordinary, physical language and cannot be used in epistemology or
phenomenology without casting a distorting light on their objects.
Page 88
The very expression 'I can perceive x' is itself taken from the idioms of physics, and x ought to be a physical
object--e.g. a body--here. Things have already gone wrong if this expression is used in phenomenology, where x
must refer to a datum. For then 'I' and 'perceive' also cannot have their previous senses.
Page 88
58 We could adopt the following way of representing matters: if I, L. W., have toothache, then that is expressed by
means of the proposition 'There is toothache'. But if that is so, what we now
Page Break 89
express by the proposition 'A has toothache', is put as follows: 'A is behaving as L. W. does when there is
toothache'. Similarly we shall say 'It is thinking' [†1] and 'A is behaving as L. W. does when it is thinking'. (You
could imagine a despotic oriental state where the language is formed with the despot as its centre and his name
instead of L. W.) It's evident that this way of speaking is equivalent to ours when it comes to questions of
intelligibility and freedom from ambiguity. But it's equally clear that this language could have anyone at all as its
centre.
Page 89
Now, among all the languages with different people as their centres, each of which I can understand, the one
with me as its centre has a privileged status. This language is particularly adequate. How am I to express that? That
is, how can I rightly represent its special advantage in words? This can't be done. For, if I do it in the language with
me as its centre, then the exceptional status of the description of this language in its own terms is nothing very
remarkable, and in the terms of another language my language occupies no privileged status whatever.--The
privileged status lies in the application, and if I describe this application, the privileged status again doesn't find
expression, since the description depends on the language in which it's couched. And now, which description gives
just that which I have in mind depends again on the application.
Page 89
Only their application really differentiates languages; but if we disregard this, all languages are equivalent. All
these languages only describe one single, incomparable thing and cannot represent anything else. (Both these
approaches must lead to the same result: first, that what is represented is not one thing among others, that it is not
capable of being contrasted with anything; second, that I cannot express the advantage of my language.)
Page 89
59 It isn't possible to believe something for which you cannot imagine some kind of verification.
Page 89
If I say I believe that someone is sad, it's as though I am seeing his behaviour through the medium of
sadness, from the viewpoint
Page Break 90
of sadness. But could you say: 'It looks to me as if I'm sad, my head is drooping so'?
Page 90
60 Not only does epistemology pay no attention to the truth or falsity of genuine propositions, it's even a
philosophical method of focusing on precisely those propositions whose content seems to us as physically
impossible as can be imagined (e.g. that someone has an ache in someone else's tooth). In this way, epistemology
highlights the fact that its domain includes everything that can possibly be thought.
Page 90
Does it make sense to say that two people have the same body? That is an uncommonly important and
interesting question. If it makes no sense, then that means--I believe--that only our bodies are the principle of
individuation. It is clearly imaginable that I should feel a pain in the hand of a different body from the one called my
own. But suppose now that my old body were to become completely insensible and inert and from then on I only
felt my pains in the other body?
Page 90
You could say: Philosophy is constantly gathering a store of propositions without worrying about their truth
or falsity; only in the cases of logic and mathematics does it have to do exclusively with the 'true' propositions.
Page 90
61 In the sense of the phrase 'sense data' in which it is inconceivable that someone else should have them, it cannot,
for this very reason, be said that someone else does not have them. And by the same token, it's senseless to say that
I, as opposed to someone else, have them.
Page 90
We say, 'I cannot feel your toothache'; when we say this, do we only mean that so far we have never as a
matter of fact felt someone else's toothache? Isn't it, rather, that it's logically impossible?
Page Break 91
Page 91
What distinguishes his toothache from mine? If the word 'toothache' means the same in 'I have toothache'
and 'He has toothache', what does it then mean to say he can't have the same toothache as I do? How are toothaches
to be distinguished from one another? By intensity and similar characteristics, and by location. But suppose these
are the same in the two cases? But if it is objected that the distinction is simply that in the one case I have it, in the
other he; then the owner is a defining mark of the toothache itself; but then what does the proposition 'I have
toothache' (or someone else does) assert? Nothing at all.
Page 91
If the word 'toothache' has the same meaning in both cases, then we must be able to compare the toothaches
of the two people; and if their intensities, etc. coincide, they're the same. Just as two suits have the same colour, if
they match one another in brightness, saturation, etc.
Page 91
Equally, it's nonsense to say two people can't have the same sense datum, if by 'sense datum' what is
primary is really intended.
Page 91
62 In explaining the proposition 'He has toothache', we even say something like: 'Quite simple, I know what it means
for me to have toothache, and when I say he has toothache, I mean he now has what I once had.' But what does 'he'
mean and what does 'have toothache' mean? Is this a relation toothache once had to me and now has to him? So in
that case I would also be conscious of toothache now and of his having it now, just as I can now see a wallet in his
hand that I saw earlier in mine.
Page 91
Is there a sense in saying 'I have a pain, only I don't notice it'? For I could certainly substitute 'he has' for 'I
have' in this proposition. And conversely, if the propositions 'He has a pain' and 'I have a pain' are logically on a par,
I must be able to substitute 'I have' for 'he has' in the proposition 'He has a pain that I can't feel'.--I might also put it
like this: only in so far as I can have a pain I don't
Page Break 92
feel can he have a pain I don't feel. Then it might still be the case that in fact I always feel the pain I have, but it must
make sense to deny that I do.
Page 92
'I have no pain' means: if I compare the proposition 'I have a pain' with reality, it turns out false--so I must be
in a position to compare the proposition with what is in fact the case. And the possibility of such a
comparison--even though the result may be negative--is what we mean when we say: what is the case must happen
in the same space as what is denied; only it must be otherwise.
Page 92
63 Admittedly the concept of toothache as a datum of feeling can be applied to someone else's tooth just as readily
as it can to mine, but only in the sense that it might well be perfectly possible to feel pain in a tooth in someone else's
mouth. According to our present way of speaking we wouldn't, however, express this fact in the words 'I feel his
toothache' but by saying 'I've got a pain in his tooth'. Now we may say: Of course you haven't got his toothache, for
it is now more than possible that he will say, 'I don't feel anything in this tooth'. And in such a situation, am I
supposed to say 'You're lying, I can feel how your tooth is aching'?
Page 92
When I feel sorry for someone with toothache, I put myself in his place. But I put myself in his place.
Page 92
The question is, whether it makes sense to say: 'Only A can verify the proposition "A is in pain", I can't'. But
what would it be like if this were false, and I could verify it: can that mean anything other than that I'd have to feel
pain? But would that be a verification? Let's not forget: it's nonsense to say I must feel my or his pain.
Page Break 93
Page 93
We might also put the question like this: What in my experience justifies the 'my' in 'I feel my pain'? Where is
the multiplicity in the feeling that justifies this word? And it can only be justified if we could also replace it by
another word.
Page 93
64 'I have a pain' is a sign of a completely different kind when I am using the proposition, from what it is to me on
the lips of another; the reason being that it is senseless, as far as I'm concerned, on the lips of another until I know
through which mouth it was expressed. The propositional sign in this case doesn't consist in the sound alone, but in
the fact that the sound came out of this mouth. Whereas in the case in which I say or think it, the sign is the sound
itself.
Page 93
Suppose I had stabbing pains in my right knee and my right leg jerked with every pang. At the same time I
see someone else whose leg is jerking like mine and he complains of stabbing pains; and while this is going on my
left leg begins jerking like the right although I can't feel any pain in my left knee. Now I say: the other fellow
obviously has the same pains in his knee as I've got in my right knee. But what about my left knee, isn't it precisely
the same case here as that of the other's knee?
Page 93
If I say 'A has toothache', I use the image of feeling pain in the same way as, say, the concept of flowing
when I talk of an electric current flowing.
Page 93
The two hypotheses that other people have toothache and that they behave just as I do but don't have
toothache, possibly have identical senses. That is, if I had, for example, learnt the second form of expression, I
would talk in a pitying tone of voice about people who don't have toothache, but are behaving as I do when I have.
Page Break 94
Page 94
Can I imagine pains in the tips of my nails, or in my hair? Isn't that just as possible or impossible as it is to
imagine a pain in any part of the body whatever in which I have none at the moment, and cannot remember having
had any?
Page 94
65 The logic of our language is so difficult to grasp at this point: our language employs the phrases 'my pain' and 'his
pain', and also the expressions 'I have (or feel) a pain' and 'He has (or feels) a pain'. An expression 'I feel my pain' or
'I feel his pain' is nonsense. And it seems to me that, at bottom, the entire controversy over behaviourism turns on
this.
Page 94
The experience of feeling pain is not that a person 'I' has something.
Page 94
I distinguish an intensity, a location, etc. in the pain, but not an owner.
Page 94
What sort of a thing would a pain be that no one has? Pain belonging to no one at all?
Page 94
Pain is represented as something we can perceive in the sense in which we perceive a matchbox. What is
unpleasant is then naturally not the pain, only perceiving it.
Page 94
When I am sorry for someone else because he's in pain, I do of course imagine the pain, but I imagine that I
have it.
Page 94
Is it also to be possible for me to imagine the pain of a tooth lying on the table, or a teapot's pain? Are we
perhaps to say: it merely isn't true that the teapot is in pain, but I can imagine it being so?!
Page 94
The two hypotheses, that others have pain, and that they don't and merely behave as I do when I have, must
have identical senses if every possible experience confirming the one confirms the other
Page Break 95
as well. In other words, if a decision between them on the basis of experience is inconceivable.
Page 95
To say that others have no pain, presupposes that it makes sense to say they do have pains.
Page 95
I believe it's clear we say other people have pains in the same sense as we say a chair has none.
Page 95
66 What would it be like if I had two bodies, i.e. my body were composed of two separate organisms?
Page 95
Here again, I think, we see the way in which the self is not on a par with others, for if everyone else had two
bodies, I wouldn't be able to tell that this was so.
Page 95
Can I imagine experience with two bodies? Certainly not visual experience.
Page 95
The phenomenon of feeling toothache I am familiar with is represented in the idioms of ordinary language by
'I have a pain in such-and-such a tooth'. Not by an expression of the kind 'In this place there is a feeling of pain'. The
whole field of this experience is described in this language by expressions of the form 'I have...'. Propositions of the
form 'N has toothache' are reserved for a totally different field. So we shouldn't be surprised when for propositions
of the form 'N has toothache', there is nothing left that links with experience in the same way as in the first case.
Page 95
Philosophers who believe you can, in a manner of speaking, extend experience by thinking, ought to
remember that you can transmit speech over the telephone, but not measles.
Page 95
Similarly I cannot at will experience time as bounded, or the visual field as homogeneous, etc.
Page Break 96
Page 96
Visual space and retina. It's as if you were to project a sphere orthogonally on to a plane, for instance in the
way in which you represent the two hemispheres of the globe in an atlas, and now someone might believe that
what's on the page surrounding the two projections of the sphere somehow still corresponds to a possible extension
of what is to be found on the sphere. The point is that here a complete space is projected onto a part of another
space; and it is like this with the limits of language in a dictionary.
Page 96
If someone believes he can imagine four-dimensional space, then why not also four-dimensional
colours--colours which in addition to the degree of saturation, hue and intensity of light, are susceptible of being
determined in yet a fourth way.
Page Break 97
VII
Page 97
67 Suppose I had such a good memory that I could remember all my sense impressions. In that case, there would,
prima facie, be nothing to prevent me from describing them. This would be a biography. And why shouldn't I be
able to leave everything hypothetical out of this description?
Page 97
I could, e.g., represent the visual images plastically, perhaps with plaster-cast figures on a reduced scale
which I would only finish as far as I had actually seen them, designating the rest as inessential by shading or some
other means.
Page 97
So far everything would be fine. But what about the time I take to make this representation? I'm assuming I'd
be able to keep pace with my memory in 'writing' this language--producing this representation. But if we suppose I
then read the description through, isn't it now hypothetical after all?
Page 97
Let's imagine a representation such as this: the bodies I seem to see are moved by a mechanism in such a
way that they would give the visual images to be represented to two eyes fixed at a particular place in the model. The
visual image described is then determined from the position of the eyes in the model and from the position and
motion of the bodies.
Page 97
We could imagine that the mechanism could be driven by turning a crank and in that way the description
'read off'.
Page 97
68 Isn't it clear that this would be the most immediate description we can possibly imagine? That is to say, that
anything which tried to be more immediate still would inevitably cease to be a description.
Page 97
Instead of a description, what would then come out would be
Page Break 98
that inarticulate sound with which many writers would like to begin philosophy. ('I have, knowing of my knowledge,
consciousness of something.')
Page 98
You simply can't begin before the beginning.
Page 98
Language itself belongs to the second system. If I describe a language, I am essentially describing something
that belongs to physics. But how can a physical language describe the phenomenal?
Page 98
69 Isn't it like this: a phenomenon (specious present) contains time, but isn't in time?
Page 98
Its form is time, but it has no place in time.
Page 98
Whereas language unwinds in time.
Page 98
What we understand by the word 'language' unwinds in physical time. (As is made perfectly clear by the
comparison with a mechanism.)
Page 98
Only what corresponds to this mechanism in the primary world could be the primary language.
Page 98
I mean: what I call a sign must be what is called a sign in grammar; something on the film, not on the screen.
Page 98
'I cannot tell whether...' only makes sense if I can know, not when it's inconceivable.
Page 98
70 With our language we find ourselves, so to speak, in the domain of the film, not of the projected picture. And if I
want to make music to accompany what is happening on the screen, whatever produces the music must again
happen in the sphere of the film.
Page 98
On the other hand, it's clear we need a way of speaking with which we can represent the phenomena of
visual space, isolated as such.
Page 98
'I can see a lamp standing on the table', says, in the way in which it has to be understood in our ordinary
language, more than a
Page Break 99
description of visual space. 'It seems to me as if I were seeing a lamp standing on a table' would certainly be a correct
description: but this form of words is misleading since it makes it look as though nothing actual were being
described, but only something whose nature was unclear.
Page 99
Whereas 'it seems' is only meant to say that something is being described as a special case of a general rule,
and all that is uncertain is whether further events will be capable of being described as special cases of the same rule.
Page 99
It seems as if there is a sine curve on the film, of which we can see particular parts.
Page 99
That is to say, what we see can be described by means of a sine curve on the film, when the light projecting it
has been interrupted at particular points.
Page 99
A concentric circle seems to have been drawn round the circle K and a, b, c, d, e, f to have been drawn as
tangents to it.
Page 99
71 It could, e.g., be practical under certain circumstances to give proper names to my hands and to those of other
people, so that you
Page Break 100
wouldn't have to mention their relation to somebody when talking about them, since that relation isn't essential to
the hands themselves; and the usual way of speaking could create the impression that its relation to its owner was
something belonging to the essence of the hand itself.
Page 100
Visual space has essentially no owner.
Page 100
Let's assume that, with all the others, I can always see one particular object in visual space--viz my nose--.
Someone else naturally doesn't see this object in the same way. Doesn't that mean, then, that the visual space I'm
talking about belongs to me? And so is subjective? No. It has only been construed subjectively here, and an
objective space opposed to it, which is, however, only a construction with visual space as its basis. In
the--secondary--language of 'objective'--physical--space, visual space is called subjective, or rather, whatever in this
language corresponds directly with visual space is called subjective. In the same way that one might say that in the
language of real numbers whatever in their domain corresponds directly with the cardinal numbers is called the
'positive integers'.
Page 100
In the model described above neither the two eyes that see the objects, nor their position need be included.
That's only one technique of representation. It does just as well if e.g. the part of the objects that is 'seen', is indicated
by shading it in. Of course you can always work out the position of two eyes from the boundaries of this shaded
area; but that only corresponds to translation from one way of speaking into another.
Page 100
The essential thing is that the representation of visual space is the representation of an object and contains no
suggestion of a subject.
Page 100
72 Suppose all the parts of my body could be removed until only one eyeball was left; and this were to be firmly
fixed in a certain position, retaining its power of sight. How would the world appear to me? I wouldn't be able to
perceive any part of myself,
Page Break 101
and supposing my eyeball to be transparent for me, I wouldn't be able to see myself in the mirror either. One
question arising at this point is: would I be able to locate myself by means of my visual field? 'Locate myself', of
course here only means to establish a particular structure for the visual space.
Page 101
Does anything now force me into interpreting the tree I see through my window as larger than the window?
If I have a sense for the distance of objects from my eye, this is a justified interpretation. But even then it's a
representation in a different space from visual space, for what corresponds to the tree in visual space is, surely,
obviously smaller than what corresponds to the window.
Page 101
Or ought I to say: Well, that all depends on how you're using the words 'larger' and 'smaller'?
Page 101
And that's right: in visual space I can use the words 'larger' and 'smaller' both ways. And in one sense the
visual mountain is smaller, and in the other larger, than the visual window.
Page 101
Suppose my eyeball were fixed behind the window, so that I would see most things through it. In that case
this window could assume the role of a part of my body. What's near the window is near me. (I'm assuming I can
see three-dimensionally with one eye.) In addition, I assume that I'm in a position to see my eyeball in the mirror,
and perceive similar eyeballs on the trees outside, say.
Page 101
How can I in this case tell, or arrive at the assumption, that I see the world through the pupil of my eyeball?
Surely not in an essentially different way from that of my seeing it through the window, or, say, through a hole in a
board that my eye is directly behind.
Page 101
In fact, if my eye were in the open stuck on the end of a branch, my position could be made perfectly clear to
me by someone bringing a ring closer and closer until in the end I could see
Page Break 102
everything through it. They could even bring up the old surroundings of my eye: cheek bones, nose, etc., and I
would know where it all fits in.
Page 102
73 Does all this mean then that a visual image does essentially contain or presuppose a subject after all?
Page 102
Or isn't it rather that these experiments give me nothing but purely geometrical information?
Page 102
That is to say information that constantly only concerns the object.
Page 102
Objective information about reality.
Page 102
There isn't an eye belonging to me and eyes belonging to others in visual space. Only the space itself is
asymmetrical, the objects in it are on a par. In the space of physics however this presents itself in such a way that
one of the eyes which are on a par is singled out and called my eye.
Page 102
I want to know what's going on behind me and turn round. If I were prevented from doing this, wouldn't the
idea that space stretches out around me remain? And that I could manage to see the objects now behind me by
turning around. Therefore it's the possibility of turning around that leads me to this idea of space. The resulting
space around me is thus a mixture of visual space and the space of muscular sensation.
Page 102
Without the feeling of the ability 'to turn around', my idea of space would be essentially different.
Page 102
Thus the detached, immovable eye wouldn't have the idea of a space all around it.
Page 102
74 Immediate experience cannot contain any contradiction. If it is beyond all speaking and contradicting, then the
demand for an explanation cannot arise either: the feeling that there must be an explanation of what is happening,
since otherwise something would be amiss.
Page Break 103
Page 103
What about when we close our eyes: we don't stop seeing. But what we see in this case surely can't have any
relation to an eye. And it's the same with a dream image. But even in the case of normal seeing, it's clear that the
exceptional position of my body in visual space only derives from other feelings that are located in my body, and
not from something purely visual.
Page 103
Even the word 'visual space' is unsuitable for our purpose, since it contains an allusion to a sense organ
which is as inessential to the space as it is to a book that it belongs to a particular person; and it could be very
misleading if our language were constructed in such a way that we couldn't use it to designate a book without
relating it to an owner. It might lead to the idea that a book can only exist in relation to a person.
Page 103
75 If, now, phenomenological language isolates visual space and what goes on in it from everything else, how does it
treat time? Is the time of 'visual' phenomena the time of our ordinary idioms of physics?
Page 103
It's clear we're able to recognize that two time intervals are equal. I could, e.g., imagine what happens in
visual space being accompanied by the ticking of a metronome or a light flashing at regular intervals.
Page 103
To simplify matters, I'm imagining the changes in my visual space to be discontinuous, and, say, in time with
the beats of the metronome. Then I can give a description of these changes (in which I use numbers to designate the
beats).
Page 103
Suppose this description to be a prediction, which is now to be verified. Perhaps I know it by heart and now
compare it with what actually happens. Everything hypothetical is avoided here, apart from what is contained in the
presupposition that the description is given to me independently of the question of which elements in it are before
me at precisely this moment.
Page Break 104
Page 104
The whole is a talking film, and the spoken word that goes with the events on the screen is just as fleeting as
those events and not the same as the sound track. The sound track doesn't accompany the scenes on the screen.
Page 104
Does it now make any sense to say I could have been deceived by a demon and what I took for a description
wasn't one at all, but a memory delusion? No, that can have no sense. A delusion that, ex hypothesi, cannot be
unmasked isn't a delusion.
Page 104
And this means no more and no less than that the time of my memory is, in this instance, precisely the time
which I'm describing.
Page 104
This isn't the same as time as it's usually understood: for that, there are any number of possible sources, such
as the accounts other people give, etc. But here it is once again a matter of isolating the one time.
Page 104
If there are three pipes in which a black liquid, a yellow liquid and a red liquid are flowing respectively, and
these combine at some point to make a brown, then the resulting liquid has its own way of flowing too; but all I
want to say is that each of the liquids with a simple colour also has a way of flowing, and I wish to examine this at a
point before the three have run into one another.
Page 104
Of course the word 'present' is also out of place here. For to what extent can we say of reality that it is
present? Surely only if we embed it once more in a time that is foreign to it. In itself it isn't present. Rather, on the
contrary, it contains a time.
Page Break 105
VIII
Page 105
76 One's first thought is that it's incompatible for two colours to be in one place at the same time. The next is that
two colours in one place simply combine to make another. But third comes the objection: how about the
complementary colours? What do red and green make? Black perhaps? But do I then see green in the black
colour?--But even apart from that: how about the mixed colours, e.g. mixtures of red and blue? These contain a
greater or lesser element of red: what does that mean? It's clear what it means to say that something is red: but that it
contains more or less red?--And different degrees of red are incompatible with one another. Someone might
perhaps imagine this being explained by supposing that certain small quantities of red added together would yield a
specified degree of red. But in that case what does it mean if we say, for example, that five of these quantities of red
are present? It cannot, of course, be a logical product of quantity no. 1 being present, and quantity no. 2 etc., up to 5;
for how would these be distinguished from one another? Thus the proposition that 5 degrees of red are present can't
be analysed like this. Neither can I have a concluding proposition that this is all the red that is present in this colour:
for there is no sense in saying that no more red is needed, since I can't add quantities of red with the 'and' of logic.
Page 105
Neither does it mean anything to say that a rod which is 3 yards long is 2 yards long, because it is 2 + 1 yards
long, since we can't say it is 2 yards long and that it is 1 yard long. The length of 3 yards is something new.
Page 105
And yet I can say, when I see two different red-blues: there's an even redder blue than the redder of these
two. That is to say, from the given I can construct what is not given.
Page 105
You could say that the colours have an elementary affinity with one another.
Page Break 106
Page 106
That makes it look as if a construction might be possible within the elementary proposition. That is to say, as
if there were a construction in logic which didn't work by means of truth functions.
Page 106
What's more, it also seems that these constructions have an effect on one proposition's following logically
from another.
Page 106
For, if different degrees exclude one another it follows from the presence of one that the other is not present.
In that case, two elementary propositions can contradict one another.
Page 106
77 How is it possible for f(a) and f(b) to contradict one another, as certainly seems to be the case? For instance, if I
say 'There is red here now' and 'There is green here now'?
Page 106
This is connected with the idea of a complete description: 'The patch is green' describes the patch
completely, and there's no room left for another colour.
Page 106
It's no help either that red and green can, in a manner of speaking, pass one another by in the dimension of
time: for, suppose I say that throughout a certain period of time a patch was red and that it was green?
Page 106
If I say for example that a patch is simultaneously light red and dark red, I imagine as I say it that the one
shade covers the other. But then is there still a sense in saying the patch has the shade that is invisible and covered
over?
Page 106
Does it make any sense at all to say that a perfectly black surface is white, only we don't see the white
because it is covered by the black? And why does the black cover the white and not vice versa?
Page 106
If a patch has a visible and an invisible colour, then at any rate it has these colours in quite different senses.
Page 106
78 If f(r) and f(g) contradict one another, it is because r and g completely occupy the f and cannot both be in it. But
that doesn't show itself in our signs. But it must show itself if we look, not at
Page Break 107
the sign, but at the symbol. For since this includes the form of the objects, then the impossibility of 'f(r) &unk; f(g)'
must show itself there, in this form.
Page 107
It must be possible for the contradiction to show itself entirely in the symbolism, for if I say of a patch that it
is red and green, it is certainly at most only one of these two, and the contradiction must be contained in the sense of
the two propositions.
Page 107
That two colours won't fit at the same time in the same place must be contained in their form and the form of
space.
Page 107
But the symbols do contain the form of colour and of space, and if, say, a letter designates now a colour,
now a sound, it's a different symbol on the two occasions; and this shows in the fact that different syntactical rules
hold for it.
Page 107
Of course, this doesn't mean that inference could now be not only formal, but also material.--Sense follows
from sense and so form from form.
Page 107
'Red and green won't both fit into the same place' doesn't mean that they are as a matter of fact never
together, but that you can't even say they are together, or, consequently, that they are never together.
Page 107
79 But that would imply that I can write down two particular propositions, but not their logical product.
Page 107
The two propositions collide in the object.
Page 107
The proposition f(g) &unk; f(r) isn't nonsense, since not all truth possibilities disappear, even if they are all
rejected. We can, however, say that the '•' has a different meaning here, since 'x • y' usually means (TFFF); here, on
the other hand, it means (FFF). And something analogous holds for 'x v y', etc.
Page 107
80 A yellow tinge is not the colour yellow.
Page Break 108
Page 108
Strictly, I cannot mix yellow and red, i.e. not strictly see them at the same time, since if I want to see yellow
in this place, the red must leave it and vice versa.
Page 108
It's clear, as I've said, that the proposition that a colour contains five tints of yellow cannot say it contains tint
no. 1 and it contains tint no. 2 etc. On the contrary the addition of the tints must occur within the elementary
proposition. But what if these tints are objects lined up like links in a chain in a certain way; and now in one
proposition we are speaking of five such links, and in another proposition of three. All right, but these propositions
must exclude one another, while yet not being analysable.--But then do F5 and F6 have to exclude each other? Can't
I say, Fn doesn't mean that the colour contains only n tints, but that it contains at least n tints? It contains only n
tints would be expressed by the proposition F(n)•~F(n + 1). But even then the elementary propositions aren't
independent of one another, since F(n - 1) at any rate still follows from F(n), and F(5) contradicts ~F(4).
Page 108
The proposition asserting a certain degree of a property contradicts on the one interpretation the specification
of any other degree and on the other interpretation it follows from the specification of any higher degree.
Page 108
A conception which makes use of a product aRx&unk;xRy• yRb is inadequate too, since I must be able to
distinguish the things x, y, etc., if they are to yield a distance.
Page 108
A mixed colour, or better, a colour intermediate between blue and red is such in virtue of an internal relation
to the structures of blue and red. But this internal relation is elementary. That is, it doesn't consist in the proposition
'a is blue-red' representing a logical product of 'a is blue' and 'a is red'.
Page 108
To say that a particular colour is now in a place is to describe that place completely.
Page 108
81 Besides, the position is no different for colours than for sounds or electrical charges.
Page Break 109
Page 109
In every case it's a question of the complete description of a certain state at one point or at the same time.
Page 109
Wouldn't the following schema be possible: the colour at a point isn't described by allocating one number to
a point, but by allocating several numbers. Only a mixture of such numbers makes the colour; and to describe the
colour in full I need the proposition that this mixture is the complete mixture, i.e. that nothing more can be added.
That would be like describing the taste of a dish by listing its ingredients; then I must add at the end that these are all
the ingredients.
Page 109
In this way we could say the colour too is definitely described when all its ingredients have been specified, of
course with the addition that these are all there are.
Page 109
But how is such an addition to be made? If in the form of a proposition, then the incomplete description
would already have to be one as well. And if not in the form of a proposition, but by some sort of indication in the
first proposition, how can I then bring it about that a second proposition of the same form contradicts the first?
Page 109
Two elementary propositions can't contradict one another.
Page 109
What about all assertions which appear to be similar, such as: a point mass can only have one velocity at a
time, there can only be one charge at a point of an electrical field, at one point of a warm surface only one
temperature at one time, at one point in a boiler only one pressure etc.? No one can doubt that these are all
self-evident and that their denials are contradictions.
Page 109
82 This is how it is, what I said in the Tractatus doesn't exhaust the grammatical rules for 'and', 'not', 'or' etc.; there
are rules for the truth functions which also deal with the elementary part of the proposition.
Page Break 110
Page 110
In which case, propositions turn out to be even more like yardsticks than I previously believed.--The fact that
one measurement is right automatically excludes all others. I say automatically: just as all the graduation marks are
on one rod, the propositions corresponding to the graduation marks similarly belong together, and we can't measure
with one of them without simultaneously measuring with all the others.--It isn't a proposition which I put against
reality as a yardstick, it's a system of propositions [†1].
Page 110
We could now lay down the rule that the same yardstick may only be applied once in one proposition. Or
that the parts corresponding to different applications of one yardstick should be collated.
Page 110
'I haven't got stomach-ache' may be compared to the proposition 'These apples cost nothing'. The point is
that they don't cost any money, not that they don't cost any snow or any trouble. The zero is the zero point of one
scale. And since I can't be given any point on the yardstick without being given the yardstick, I can't be given its zero
point either. 'I haven't got a pain' doesn't refer to a condition in which there can be no talk of pain, on the contrary
we're talking about pain. The proposition presupposes the capacity for feeling pain, and this can't be a 'physiological
capacity'--for otherwise how would we know what it was a capacity for--it's a logical possibility.--I describe my
present state by alluding to something that isn't the case. If this allusion is needed for the description (and isn't
merely an ornament), there must be something in my present state making it necessary to mention (allude to) this. I
compare this state with another, it must therefore be comparable with it. It too must be located in pain-space, even if
at a different point.--Otherwise my proposition would mean something like: my present state has nothing to do with
a painful one; rather in the way I might say the colour of this rose has nothing to do with Caesar's conquest
Page Break 111
of Gaul. That is, there's no connection between them. But I mean precisely that there is a connection between my
present state and a painful one.
Page 111
I don't describe a state of affairs by mentioning something that has nothing to do with it and stating it has
nothing to do with it. That wouldn't be a negative description.
Page 111
'The sense consists in the possibility of recognition', but this is a logical possibility. I must be in the space in
which what is to be expected is located.
Page 111
83 The concept of an 'elementary proposition' now loses all of its earlier significance.
Page 111
The rules for 'and', 'or', 'not' etc., which I represented by means of the T-F notation, are a part of the grammar
of these words, but not the whole.
Page 111
The concept of independent co-ordinates of description: the propositions joined, e.g., by 'and' are not
independent of one another, they form one picture and can be tested for their compatibility or incompatibility.
Page 111
In my old conception of an elementary proposition there was no determination of the value of a co-ordinate;
although my remark that a coloured body is in a colour-space, etc., should have put me straight on to this.
Page 111
A co-ordinate of reality may only be determined once.
Page 111
If I wanted to represent the general standpoint I would say: 'You should not say now one thing and now
another about the same matter.' Where the matter in question would be the coordinate to which I can give one value
and no more.
Page Break 112
Page 112
84 The situation is misrepresented if we say we may not ascribe to an object two incompatible attributes. For seen
like that, it looks as if in every case we must first investigate whether two determinations are incompatible or not.
The truth is, two determinations of the same kind (co-ordinate) are impossible.
Page 112
What we have recognized is simply that we are dealing with yardsticks, and not in some fashion with isolated
graduation marks.
Page 112
In that case every assertion would consist, as it were, in setting a number of scales (yardsticks) and it's
impossible to set one scale simultaneously at two graduation marks.
Page 112
For instance, that would be the claim that a coloured circle, of colour... and radius... was located at.... We
might think of signals on a ship: 'Stop', 'Full Speed Ahead', etc.
Page 112
Incidentally, they don't have to be yardsticks. For you can't call a dial with two signals a yardstick.
Page 112
85 That every proposition contains time in some way or other appears to us to be accidental, when compared with
the fact that the truth-functions can be applied to any proposition.
Page Break 113
Page 113
The latter seems to be connected with their nature as propositions, the former with the nature of the reality
we encounter.
Page 113
True-false, and the truth functions, go with the representation of reality by propositions. If someone said:
Very well, how do you know that the whole of reality can be represented by propositions?, the reply is: I only know
that it can be represented by propositions in so far as it can be represented by propositions, and to draw a line
between a part which can and a part which can't be so represented is something I can't do in language. Language
means the totality of propositions.
Page 113
We could say: a proposition is that to which the truth functions may be applied.--The truth functions are
essential to language.
Page 113
86 Syntax prohibits a construction such as 'A is green and A is red' (one's first feeling is that it's almost as if this
proposition had been done an injustice; as though it had been cheated of its rights as a proposition), but for 'A is
green', the proposition 'A is red' is not, so to speak, another proposition--and that strictly is what the syntax
fixes--but another form of the same proposition.
Page 113
In this way syntax draws together the propositions that make one determination.
Page 113
If I say I did not dream last night, I must still know where I would have to look for a dream (i.e. the
proposition 'I dreamt', applied to this situation can at most be false, it cannot be nonsense).
Page 113
I express the present situation by a setting--the negative one of the signal dial 'dreams--no dreams'. But in
spite of its negative setting I must be able to distinguish it from other signal dials. I must know that this is the signal
dial I have in my hand.
Page 113
Someone might now ask: Does that imply you have, after all, felt something, so to speak, the hint of a dream,
which makes you conscious of the place where a dream would have been? Or if I say 'I haven't got a pain in my
arm', does that mean I have a sort of shadowy feeling, indicating the place where the pain would be? No, obviously
not.
Page Break 114
Page 114
In what sense does the present, painless, state contain the possibility of pain?
Page 114
If someone says 'For the word pain to have a meaning, it's necessary that pain should be recognized as such
when it occurs', we may reply 'It's no more necessary than that the absence of pain should be recognized as such'.
Page 114
'Pain' means, so to speak, the whole yardstick and not one of its graduation marks. That it is set at one
particular graduation mark can only be expressed by means of a proposition.
Page Break 115
IX
Page 115
87 The general proposition 'I see a circle on a red background' appears simply to be a proposition which leaves
possibilities open.
Page 115
A sort of incomplete picture. A portrait in which, e.g. the eyes have not been painted in.
Page 115
But what would this generality have to do with a totality of objects?
Page 115
There must be incomplete elementary propositions from whose application the concept of generality derives.
Page 115
This incomplete picture is, if we compare it with reality, right or wrong: depending on whether or not reality
agrees with what can be read off from the picture.
Page 115
(The theory of probability is connected with the fact that the more general, i.e. the more incomplete,
description is more likely to fit the facts than the more complete one.)
Page 115
Generality in this sense, therefore, enters into the theory of elementary propositions, and not into the theory
of truth functions.
Page 115
88 If I do not completely describe my visual field, but only a part of it, it is obvious that there is, as it were, a gap in
the fact. There is obviously something left out.
Page 115
If I were to paint a picture of this visual image, I would let the canvas show through at certain places. But of
course the canvas also has a colour and occupies space. I could not leave nothing in the place where something was
missing.
Page 115
My description must therefore necessarily include the whole visual space--and its being coloured, even if it
does not specify what the colour is at every place.
Page 115
That is, it must still say that there is a colour at every place.
Page 115
Does that mean that the description, in so far as it does not exhaust the space with constants, must exhaust it
with variables?
Page Break 116
Page 116
To this one might object that you cannot describe a part of the visual field separated from the whole at all,
since it is not even conceivable on its own.
Page 116
But the form (the logical form) of the patch in fact presupposes the whole space. And if you can only
describe the whole visual field, then why not only the whole flux of visual experience; for a visual image can only
exist in time.
Page 116
89 The question is this: can I leave some determination in a proposition open, without at the same time specifying
precisely what the possibilities left open are?
Page 116
Is the case of the general proposition 'A red circle is situated in the square' essentially different from a general
assertion of numerical equality, such as 'I have as many jackets as trousers'? and is not this proposition for its part
completely on all fours with 'There is a number of chairs in this room'? Of course, in everyday life you would not
need to develop the disjunction of numbers very far. But however far you go, you must stop somewhere. The
question here is always: How do I know such a proposition? Can I ever know it as an endless disjunction?
Page 116
Even if the first case is construed in such a way that we can establish the position and size of the circle by
taking measurements, even then the general proposition cannot be construed as a disjunction (or if so, then just as a
finite one). For what then is the criterion for the general proposition, for the circle's being in the square? Either,
nothing that has anything to do with a set of positions (or sizes), or else something that deals with a finite number of
such positions.
Page 116
90 Suppose this is my incomplete picture: a red circle stands on a differently coloured background with colour x. It
is clear that this picture can be used as a proposition in a positive sense, but also in a negative one. In the negative
sense it says what Russell expresses as ~(∃x) φx.
Page 116
Now is there also in my account an analogue to Russell's
Page Break 117
(∃x) ~φx? That would mean: there is an x of which it is not true that a red circle stands on a background with this
colour. Or in other words: there is a colour of the background on which there does not stand a red circle. And in this
context that is nonsense!
Page 117
But how about the proposition 'There is a red ball which is not in the box'? Or 'There is a red circle not in the
square'? That is once more a general description of a visual image. Here negation seems to be used in a different
way. For it certainly seems as if I could express the proposition 'This circle is not in the square' so that the 'not' is
placed at the front of the proposition.--But that seems to be an illusion. If you mean by the words 'this circle', 'the
circle that I am pointing at', then this case of course falls into line, for then it says 'It is not the case that I am pointing
at a circle in the square', but it does not say that I am pointing at a circle outside the square.
Page 117
This is connected with the fact that it's nonsense to give a circle a name. That is to say, I cannot say 'The
circle A is not in the square'. For that would only make sense if it made sense to say 'The circle A is in the square'
even when it wasn't.
Page 117
91 If generality no longer combines with truth functions into a homogeneous whole, then a negation cannot occur
within the scope of a quantifier.
Page 117
Of course I could say: 'There is a red circle outside the square' means 'It is not the case that all red circles are
in the square'. But what does 'all' refer to here?
Page 117
'All circles are in the square' can only mean, either 'A certain number of circles are in the square', or 'There is
no circle outside'. But the proposition 'There is no circle outside' is once again the negation of a generalization and
not the generalization of a negation.
Page Break 118
Page 118
92 If someone confronts us with the fact that language can express everything by means of nouns, adjectives and
verbs, we can only say that then it is at any rate necessary to distinguish between entirely different kinds of nouns
etc., since different grammatical rules hold for them. This is shown by the fact that it is not permissible to substitute
them for one another. This shows that their being nouns is only an external characteristic and that we are in fact
dealing with quite different parts of speech. The part of speech is only determined by all the grammatical rules which
hold for a word, and seen from this point of view our language contains countless different parts of speech.
Page 118
If you give a body a name then you cannot in the same sense give names to its colour, its shape, its position,
its surface. And vice versa.
Page 118
'A' is the name of a shape, not of a cluster of graphite particles.
Page 118
The different ways names are used correspond exactly to the different uses of the demonstrative pronoun. If
I say: 'That is a chair', 'That is the place where it stood', 'That is the colour it had', the word 'that' is used in that many
different ways. (I cannot in the same sense point at a place, a colour, etc.)
Page 118
93 Imagine two planes, with figures on plane I that we wish to map on to plane II by some method of projection. It
is then open to us to fix on a method of projection (such as orthogonal projection) and then to interpret the images
on plane II according to this method of mapping. But we could also adopt a quite different procedure: we might for
some reason lay down that the images on plane II should all be circles no matter what the figures on plane I may be.
That is, different figures on I are mapped on to II by different methods of projection. In order in this case to
construe the circles in II as images, I shall have to say for each circle what
Page Break 119
method of projection belongs to it. But the mere fact that a figure is represented on II as a circle will say nothing.--It
is like this with reality if we map it onto subject-predicate propositions. The fact that we use subject-predicate
propositions is only a matter of our notation. The subject-predicate form does not in itself amount to a logical form
and is the way of expressing countless fundamentally different logical forms, like the circles on the second plane.
The forms of the propositions: 'The plate is round', 'The man is tall', 'The patch is red', have nothing in common.
Page 119
One difficulty in the Fregean theory is the generality of the words 'concept' and 'object'. For even if you can
count tables and tones and vibrations and thoughts, it is difficult to bracket them all together.
Page 119
Concept and object: but that is subject and predicate. And we have just said that there is not just one logical
form which is the subject-predicate form.
Page 119
94 That is to say, it is clear that once you have started doing arithmetic, you don't bother about functions and
objects. Indeed, even if you decide only to deal with extensions, strangely enough you still ignore the form of the
objects completely.
Page 119
There is a sense in which an object may not be described.
Page 119
That is, the description may ascribe to it no property whose absence would reduce the existence of the object
itself to nothing, i.e. the description may not express what would be essential for the existence of the object.
Page 119
95 I see three circles in certain positions; I close my eyes, open them again and see three circles of the same size in
different positions. Does it make sense to ask whether these are the same circles and which is which? Surely not.
However, while I can see
Page Break 120
them, I can identify them (even if they move before my eyes, I can identify the circles in the new places with those
that were in the earlier ones). If I give them names, close my eyes, open them again and see that the circles are in the
same places, I can give to each its name once more. (I could still do this even if they had moved so as to exchange
places.) In any case, I always name (directly or indirectly) a location.
Page 120
Would it be possible to discover a new colour? (For the man who is colour-blind is of course in the same
position as ourselves, his colours form just as complete a system as ours; he does not see gaps where the remaining
colours fit in.) (Comparison with mathematics.)
Page 120
If someone says that substance is indestructible, then what he is really after is that it is senseless in any
context to speak of 'the destruction of a substance'--either to affirm or deny it.
Page 120
What characterizes propositions of the form 'This is...' is only the fact that the reality outside the so-called
system of signs somehow enters into the symbol.
Page 120
96 Russell and Frege construe a concept as a sort of property of a thing. But it is very unnatural to construe the
words 'man', 'tree', 'treatise', 'circle' as properties of a substratum.
Page 120
If a table is painted brown then it's easy to think of the wood as bearer of the property brown, and you can
imagine what remains when the colour changes. Even in the case of one particular circle which appears now red,
now blue. It is thus easy to imagine what is red, but difficult to imagine what is circular. What remains in this case, if
form and colour alter? For position is part of the form, and it is arbitrary for me to lay down that the centre should
stay fixed and the only changes in form be changes in the radius.
Page Break 121
Page 121
We must once more adhere to ordinary language and say that a patch is circular.
Page 121
It is clear that the phrase 'bearer of a property' in this context conveys a completely wrong--an
impossible--picture. If I have a lump of clay, I can consider it as the bearer of a form, and that, roughly, is where this
picture comes from.
Page 121
'The patch changes its form' and 'The lump of clay changes its form' are fundamentally different forms of
proposition.
Page 121
You can say 'Measure whether that is a circle' or 'See whether that over there is a hat'. You can also say
'Measure whether that is a circle or an ellipse', but not '... whether that is a circle or a hat'; not 'See whether that is a
hat or red'.
Page 121
If I point to a curve and say 'That is a circle', then someone can object that if it were not a circle, it would no
longer be that. That is to say, what I mean by the word 'that' must be independent of what I assert about it.
Page 121
('Was that thunder or gunfire?' Here you could not ask 'Was that a noise?')
Page 121
97 Roughly speaking, the equation of a circle is the sign for the concept 'circle', if it does not have definite values
substituted for the co-ordinates of its centre and for the radius, or even, if these are only given as lying within a
certain range. The object falling under the concept is then a circle whose position and size have been fixed.
Page 121
How are two red circles of the same size distinguished? This question makes it sound as if they were pretty
nearly one circle, and only distinguished by a nicety.
Page 121
In the technique of representation by equations, what is common is expressed by the form of the equation
and the difference by the difference in the co-ordinates of the centres.
Page 121
So it is as if what corresponds with the objects falling under the concept were here the co-ordinates of the
centres.
Page 121
Couldn't you then say, instead of 'This is a circle', 'This point
Page Break 122
is the centre of a circle'? For, to be the centre of a circle is an external property of the point.
Page 122
For the number pair that gives the co-ordinates of the centre is in fact not just anything, any more than the
centre is: the number pair characterizes just what in the symbol constitutes the 'difference' of the circles.
Page 122
98 What is necessary to a description that--say--a book is in a certain position? The internal description of the book,
i.e. of the concept, and a description of its place which it would be possible to give by giving the co-ordinates of
three points. The proposition, 'Such a book is here', would then mean that it had these three triples of co-ordinates.
For the specification of the 'here' must not pre-judge what is here.
Page 122
But doesn't it come to the same thing whether I say 'This is a book' or 'Here is a book'? The proposition
would then amount to saying 'Those are three particular corners of such a book'.
Page 122
Similarly you can also say 'This circle is the projection of a sphere' or 'This is a man's appearance'.
Page 122
All that I am saying comes back to this: F(x) must be an external description of x.
Page 122
If in this sense I now say in three-dimensional space 'Here is a circle' and on another occasion 'Here is a
sphere', are the two 'here's of the same type? Couldn't both refer to the three co-ordinates of the relevant
centre-point? But, the position of the circle in three-dimensional space is not fixed by the coordinates of its centre.
Page 122
Suppose my visual field consisted of two red circles of the same size on a blue background: what occurs
twice here and what once? And what does this question mean in any case?
Page 122
Here we have one colour, but two positions.
Page Break 123
X
Page 123
99 We can ask whether numbers are essentially concerned with concepts. I believe this amounts to asking whether it
makes sense to ascribe a number to objects that haven't been brought under a concept. For instance, does it mean
anything to say 'a and b and c are three objects'? I think obviously not. Admittedly we have a feeling: Why talk
about concepts; the number, of course, depends only on the extension of the concept, and once that has been
determined, the concept may drop out of the picture. The concept is only a method for determining an extension,
but the extension is autonomous and, in its essence, independent of the concept; for it's quite immaterial which
concept we have used to determine the extension. That is the argument for the extensional viewpoint. The
immediate objection to it is: if a concept is really only an expedient for arriving at an extension, then there is no place
for concepts in arithmetic; in that case we must simply divorce a class completely from the concept which happens
to be associated with it; but if it isn't like that, then an extension independent of a concept is just a chimera, and in
that case it's better not to speak of it at all, but only of the concept.
Page 123
How about the proposition '(∃x, y, z) • aRx•xRy
•yRz• zRb.∨.aRy •yRx• xRz • zRb. ∨. etc.' (all combinations)? Can't I write this in the perfectly intelligible form: '(∃3)x
• aRxRb'--say 'Three links are inserted between a and b'? Here we've formed the concept 'link between a and b'.
Page 123
(Things between these walls.)
Page 123
If I have two objects, then I can of course, at least hypothetically, bring them under one umbrella, but what
characterizes the extension is still the class, and the concept encompassing it still only a makeshift, a pretext.
Page Break 124
Page 124
100 Numbers are pictures of the extensions of concepts.
Page 124
Now we could regard the extension of a concept as an object, whose name, like any other, has sense only in
the context of a proposition. Admittedly, 'a and b and c' has no sense, it isn't a proposition. But then neither is 'a' a
proposition.
(E1)xφx • (E1)xψx • (x) • ~(φx • ψx). ⊃. (E2)xφx. ∨. ψx
Page 124
If φ and ψ here are of the form x = a. ∨. x = b, etc., then the whole proposition has become a contrivance for
ensuring that we add correctly.
Page 124
In the symbolism there is an actual correlation, whereas at the level of meaning only the possibility of
correlation is at issue.
Page 124
The problem is: How can we make preparations for the reception of something that may happen to exist?
Page 124
The axiom of infinity is nonsense if only because the possibility of expressing it would presuppose infinitely
many things--i.e. what it is trying to assert. You can say of logical concepts such as that of infinity that their essence
implies their existence.
Page 124
101. (3)xφx • (4)xψx • ~(∃x)φx. ψx. . (3 + 4)xφx. ∨. ψx[†1]
Page 124
This expression isn't equivalent to the substitution rule 3 + 4 = 7.
Page 124
We might also ask: Suppose I have four objects satisfying a function, does it always make sense to say that
these 4 objects are 2 + 2 objects? I certainly don't know whether there are functions grouping them into 2 and 2.
Does it make sense to say of 4 objects taken at random that they are composed of 2 objects and 2 objects?
Page Break 125
Page 125
The notation I used above '(3 + 4)x' etc. already contains the assumption that it always makes sense to
construe 7 as 3 + 4, since on the right hand side of the ' ' ψ I have so to speak already forgotten where the 3 and 4
have come from. On the other hand: I can surely always distinguish 3 and 4 in the sign 1 + 1 + 1 + 1 + 1 + 1 + 1.
Page 125
Perhaps this provides the answer? What would it be like for me to have a sign for 7 in which I couldn't
separate 3 and 4? Is such a sign conceivable?
Page 125
Does it make sense to say a relation holds between 2 objects, although for the rest there is no concept under
which they both fall?
Page 125
102 I want to say numbers can only be defined from propositional forms, independently of the question which
propositions are true or false.
Page 125
Only 3 of the objects a, b, c, d have the property φ. That can be expressed through a disjunction. Obviously
another case where a numerical assertion doesn't refer to a concept (although you could make it look as though it did
by using an '='.)
Page 125
If I say: If there are 4 apples on the table, then there are 2 + 2 on it, that only means that the 4 apples already
contain the possibility of being grouped into two and two, and I needn't wait for them actually to be grouped by a
concept. This 'possibility' refers to the sense, not the truth of a proposition. 2 + 2 = 4 may mean 'whenever I have
four objects, there is the possibility of grouping them into 2 and 2'.
Page 125
103 How am I to know that |||||||| and |||||||| are the same sign? It isn't enough that they look alike. For having roughly
the same Gestalt can't be what is to constitute the identity of the signs, but just their being the same in number.
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Page 126
If you write (E |||||) etc. • (E |||||||) etc. • ⊃. (E ||||||||||||)--A [†1] you may be in doubt as to how I obtained the
numerical sign in the right-hand bracket if I don't know that it is the result of adding the two left-hand signs. I believe
that makes it clear that this expression is only an application of 5 + 7 = 12 but doesn't represent this equation itself.
Page 126
If we ask: But what then does '5 + 7 = 12' mean--what kind of significance or point is left for this
expression--the answer is, this equation is a rule for signs which specifies which sign is the result of applying a
particular operation (addition) to two other particular signs. The content of 5 + 7 = 12 (supposing someone didn't
know it) is precisely what children find difficult when they are learning this proposition in arithmetic lessons.
Page 126
We can completely disregard the special structure of the proposition A and pay attention solely to the
relation, the connection, between the numerical signs in it. This shows that the relation holds independently of the
proposition--i.e. of the other features of its structure which make it a tautology.
Page 126
For if I look at it as a tautology I merely perceive features of its structure and can now perceive the addition
theorem in them, while disregarding other characteristics that are essential to it as a proposition.
Page 126
The addition theorem is in this way to be recognized in it (among other places), not by means of it.
Page 126
This thought would of course be nonsense if it were a question here of the sense of a proposition, and not of
the way the structure of a tautology functions.
Page Break 127
Page 127
104 You could reply: what I perceive in the sign A and call the relation between the numerical signs is once more
only the bringing together of extensions of concepts: I combine the first five strokes of the right-hand bracket, which
stand in 1-1 correspondence with the five in one of the left-hand brackets, with the remaining 7 strokes, which stand
in 1-1 correspondence with the seven in the other left-hand bracket, to make 12 strokes which do one or the other.
But even if I followed this train of thought, the fundamental insight would still remain, that the 5 strokes and the 7
combine precisely to make 12 (and so for example to make the same structure as do 4 and 4 and 4).--It is always
only insight into the internal relations of the structures and not some proposition or other or some logical
consideration which tells us this. And, as far as this insight is concerned, everything in the tautology apart from the
numerical structures is mere decoration; they are all that matters for the arithmetical proposition. (Everything else
belongs to the application of the arithmetical proposition.)
Page 127
Thus what I want to say is: it isn't what occasions our combining 5 and 7 that belongs to arithmetic, but the
process of doing so and its outcome.
Page 127
Suppose I wrote out the proposition A but put the wrong number of strokes in the right-hand bracket, then
you would and could only come upon this mistake by comparing the structures, not by applying theorems of logic.
Page 127
If asked how do you know that this number of strokes in the right-hand bracket is correct, I can only justify it
by a comparison of the structures.
Page 127
In this way it would turn out that what Frege called the 'gingersnap standpoint' in arithmetic could yet have
some justification.
Page 127
105 And now--I believe--the relation between the extensional conception of classes and the concept of a number as
a feature of a
Page Break 128
logical structure is clear: an extension is a characteristic of the sense of a proposition.
Page 128
106 Now if the transition in A were the only application of this arithmetical schema, wouldn't it then be possible or
necessary to replace it or define it by the tautology?
Page 128
That is to say, what would it be like for A to be the most general form of the application of the arithmetical
schema?
Page 128
If A were the only--and therefore essentially the only--application of the schema, then in the very nature of
the case the schema couldn't mean anything other than just the tautology.
Page 128
Or: the schema itself must then be the tautology and the tautology nothing other than the schema.
Page 128
In that case, you also could no longer say A was an application of the schema--A would be the schema, only
not as it were the implement on its own, but the implement with its handle, without which it is after all useless.
Page 128
What A contains apart from the schema can then only be what is necessary in order to apply it.
Page 128
But nothing at all is necessary, since we understand and apply the propositions of arithmetic perfectly well
without adding anything whatever to them.
Page 128
But forming a tautology is especially out of place here, as we can see perfectly well from the tautology itself,
since otherwise in order to recognise it as a tautology we should have to recognise yet another one as a tautology
and so on [†1].
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Page 129
107 Arithmetical propositions, like the multiplication table and things of that kind, or again like definitions which do
not have whole propositions standing on both sides, are used in application to propositions. And anyhow I certainly
can't apply them to anything else. (Therefore I don't first need some description of their application.)
Page 129
No investigation of concepts, only direct insight can tell us [†1] that 3 + 2 = 5.
Page 129
That is what makes us rebel against the idea that A could be the proposition 3 + 2 = 5. For what enables us to
tell that this expression is a tautology cannot itself be the result of an examination of concepts but must be
immediately visible.
Page 129
And if we say numbers are structures we mean that they must always be of a kind with what we use to
represent them.
Page 129
I mean: numbers are what I represent in my language by number schemata.
Page 129
That is to say, I take (so to speak) the number schemata of the language as what I know, and say numbers are
what these represent [†2].
Page 129
This is what I once meant when I said, it is with the calculus [system of calculation] that numbers enter into
logic.
Page 129
108 What I said earlier about the nature of arithmetical equations and about an equation's not being replaceable by a
tautology explains--I believe--what Kant means when he insists that 7 + 5 = 12 is not an analytic proposition, but
synthetic a priori.
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Page 130
Am I using the same numbers when I count the horses in a stall and when I count the different species of
animal in the stall? When I count the strokes in a line and the kinds of group (as defined by the different number of
strokes)?
Page 130
Whether they are cardinal numbers in the same sense depends on whether the same syntactical rules hold for
them.
Page 130
(It is conceivable that there should be no man in a room, but not that there should be a man of no race in it.)
Page 130
Arithmetic is the grammar of numbers. Kinds of number can only be distinguished by the arithmetical rules
relating to them.
Page 130
109 One always has an aversion to giving arithmetic a foundation by saying something about its application. It
appears firmly enough grounded in itself. And that of course derives from the fact that arithmetic is its own
application.
Page 130
Arithmetic doesn't talk about numbers, it works with numbers.
Page 130
The calculus presupposes the calculus.
Page 130
Aren't the numbers a logical peculiarity of space and time?
Page 130
The calculus itself exists only in space and time.
Page 130
Every mathematical calculation is an application of itself and only as such does it have a sense. That is why it
isn't necessary to speak about the general form of logical operation when giving a foundation to arithmetic.
Page 130
A cardinal number is applicable to the subject-predicate form, but not to every variety of this form. And the
extent to which it is applicable simply characterizes the subject-predicate form.
Page 130
On the one hand it seems to me that you can develop arithmetic completely autonomously and its
application takes care of itself,
Page Break 131
since wherever it's applicable we may also apply it. On the other hand a nebulous introduction of the concept of
number by means of the general form of operation--such as I gave--can't be what's needed.
Page 131
You could say arithmetic is a kind of geometry; i.e. what in geometry are constructions on paper, in
arithmetic are calculations (on paper).--You could say it is a more general kind of geometry.
Page 131
And can't I say that in this sense chess (or any other game) is also a kind of geometry.
Page 131
But in that case it must be possible to work out an application of chess that is completely analogous to that
of arithmetic.
Page 131
You could say: Why bother to limit the application of arithmetic, that takes care of itself. (I can make a knife
without bothering which sorts of material it will cut: that will show soon enough.)
Page 131
What speaks against our demarcating a region of application is the feeling that we can understand arithmetic
without having any such region in mind. Or put it like this: instinct rebels against anything that isn't restricted to an
analysis of the thoughts already before us.
Page 131
110
Page 131
'Look, it always turns out the same.' Seen like that, we have performed an experiment. We have applied the
rules of one-and-one and from those you can't tell straight off that they lead to the same result in the three cases.
Page 131
It's as if we're surprised that the numerals cut adrift from their definitions function so unerringly. Or rather:
that the rules for the numerals work so unerringly (when they are not under the supervision of the definitions).
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Page 132
This is connected (oddly enough) with the internal consistency of geometry.
Page 132
For, you can say the rules for the numerals always presuppose the definitions. But in what sense? What does
it mean to say one sign presupposes another that strictly speaking isn't there at all? It presupposes its possibility; its
possibility in sign-space (in grammatical space).
Page 132
It's always a question of whether and how it's possible to represent the most general form of the application
of arithmetic. And here the strange thing is that in a certain sense it doesn't seem to be needed. And if in fact it isn't
needed then it's also impossible.
Page 132
The general form of its application seems to be represented by the fact that nothing is said about it. (And if
that's a possible representation, then it is also the right one.)
Page 132
What is characteristic of a statement of number is that you may replace one number by any other and the
proposition must always still be significant; and so the infinite formal series of propositions.
Page 132
111 The point of the remark that arithmetic is a kind of geometry is simply that arithmetical constructions are
autonomous like geometrical ones, and hence so to speak themselves guarantee their applicability.
Page 132
For it must be possible to say of geometry, too, that it is its own application.
Page 132
That is an arithmetical construction, and in a somewhat extended sense also a
geometrical one.
Page 132
Suppose I wish to use this calculation to solve the following problem: if I have 11 apples and want to share
them among some people in such a way that each is given 3 apples, how many people
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can there be? The calculation supplies me with the answer 3. Now, suppose I were to go through the whole process
of sharing and at the end 4 people each had 3 apples in their hands. Would I then say that the computation gave a
wrong result? Of course not. And that of course means only that the computation was not an experiment.
Page 133
It might look as though the mathematical computation entitled us to make a prediction--say, that I could give
3 people their share and there will be two apples left over. But that isn't so. What justifies us in making this
prediction is an hypothesis of physics, which lies outside the calculation. The calculation is only a study of logical
forms, of structures, and of itself can't yield anything new.
Page 133
112 Different as strokes and court cases are, you can still use strokes to represent court cases on a calendar. And you
can count the former instead of the latter.
Page 133
This isn't so, if, say, I want to count hat-sizes. It would be unnatural to represent three hat-sizes by three
strokes. Just as if I were to represent a measurement, 3 ft, by 3 strokes. You can certainly do so, but then '|||'
represents in a different way.
Page 133
If 3 strokes on paper are the sign for the number 3, then you can say the number 3 is to be applied in the way
in which the 3 strokes can be applied.
Page 133
Of what 3 strokes are a picture, of that they can be used as a picture.
Page 133
113 The natural numbers are a form given in reality through things, as the rational numbers are through extensions
etc. I mean, by actual forms. In the same way, the complex numbers are given by actual manifolds. (The symbols are
actual.)
Page 133
What distinguishes a statement of number about the extension of a concept from one about the range of a
variable? The first is a proposition, the second not. For the statement of number about
Page Break 134
a variable can be derived from the variable itself. (It must show itself.)
Page 134
But can't I specify a variable by saying that its values are to be all objects satisfying a certain material
function? In that case the variable is not a form! And then the sense of one proposition depends on whether another
is true or false.
Page 134
A statement of number about a variable consists in a transformation of the variable rendering the number of
its values visible.
Page 134
114 What kind of proposition is 'There is a prime number between 5 and 8'? I would say 'That shows itself'. And
that's correct, but can't you draw attention to this internal state of affairs? You could surely say: Search the interval
between 10 and 20 for prime numbers. How many are there? Wouldn't that be a straightforward problem? And how
would its result be correctly expressed or represented? What does the proposition 'There are 4 primes between 10
and 20' mean?
Page 134
This proposition seems to draw our attention to a particular aspect of the matter.
Page 134
If I ask someone, 'How many primes are there between 10 and 20?', he may reply, 'I don't know straight off,
but I can work it out any time you like.' For it's as if there were somewhere where it was already written out.
Page 134
If you want to know what a proposition means, you can always ask 'How do I know that?' Do I know that
there are 6 permutations of 3 elements in the same way in which I know there are 6 people in this room? No.
Therefore the first proposition is of a different kind from the second.
Page 134
Another equally useful question is 'How would this proposition actually be used in practice?'; and there the
proposition from the theory of combinations is of course used as a law of inference in the transition from one
proposition to another, each of which describes a reality, not a possibility.
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Page 135
You can, I think, say in general that the use of apparent propositions about possibilities--and
impossibilities--is always in the passage from one actual proposition to another.
Page 135
Thus I can, e.g., infer from the proposition 'I label 7 boxes with permutations of a, b, c' that at least one of the
labels is repeated.--And from the proposition 'I distribute 5 spoons among 4 cups' it follows that one cup gets 2
spoons, etc.
Page 135
If someone disagrees with us about the number of men in this room, saying there are 7, while we can only
see 6, we can understand him even though we disagree with him. But if he says that for him there are 5 pure colours,
in that case we don't understand him, or must suppose we completely misunderstand one another. This number is
demarcated in dictionaries and grammars and not within language.
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XI
Page 136
115 A statement of number doesn't always contain something general or indefinite: 'The interval AB is divided into
two (3, 4 etc.) equal parts.'
Page 136
There doesn't even need to be a certain element of generality in a statement of number.' Suppose, e.g., I say 'I
see three equal circles equidistant from one another'.
Page 136
If I give a correct description of a visual field in which three red circles stand on a blue ground, it surely won't
take the form of saying '(∃x, y, z): x is circular and red and y is circular and red, etc. etc.'
Page 136
You might of course write it like this: there are 3 circles with the property of being red. But at this point the
difference emerges between improper objects--colour patches in a visual field, sounds, etc. etc.--and the elements of
knowledge, the genuine objects.
Page 136
It is plain that the proposition about the three circles isn't general or indefinite in the way a proposition of the
form (∃x, y, z) • φx • φy • φz is. That is, in such a case, you may say: Certainly I know that three things have the
property φ, but I don't know which; and you can't say this in the case of three circles.
Page 136
'There are now 3 red circles of such and such a size and in such and such a place in my visual field'
determines the facts completely and it would be nonsense to say I don't know which circles they are.
Page 136
Think of such 'objects' as: a flash of lightning, the simultaneous occurrence of two events, the point at which
a line cuts a circle, etc.; the three circles in the visual field are an example for all these cases.
Page 136
You can of course treat the subject-predicate form (or, what comes to the same thing, the argument-function
form) as a norm
Page Break 137
of representation, and then it is admittedly important and characteristic that whenever we use numbers, the number
may be represented as the property of a predicate. Only we must be clear about the fact that now we are not dealing
with objects and concepts as the results of an analysis, but with moulds into which we have squeezed the
proposition. And of course it's significant that it can be fitted into this mould. But squeezing something into a mould
is the opposite of analysis. (If you want to study the natural growth of an apple tree, you don't look at an espalier
tree--except to see how this tree reacts to this pressure.)
Page 137
That implies the Fregean theory of number would be applicable provided we were not intending to give an
analysis of propositions. This theory explains the concept of number for the idioms of everyday speech. Of course,
Frege would have said (I remember a conversation we had) that the simultaneous occurrence of an eclipse of the
moon and a court case was an object. And what's wrong with that? Only that we in that case use the word 'object'
ambiguously, and so throw the results of the analysis into disarray.
Page 137
If I say, 'There are 4 men in this room', then at any rate a disjunction seems to be involved, since it isn't said
which men. But this is quite inessential. We could imagine all men to be indistinguishable from one another apart
from their location (so that it would be a question of humanity at a particular place), and in that case all
indefiniteness would vanish.
Page 137
116 If I am right, there is no such concept as 'pure colour'; the proposition 'A has a pure colour' simply means 'A is
red, or yellow, or green, or blue'. 'This hat belongs to either A or B or C' isn't the same proposition as 'This hat
belongs to someone in this room', even if as a matter of fact only A, B, and C are in the room, since that needs
saying.--'There are two pure colours on this surface', means 'On this surface there is red and yellow, or red and blue,
or red and green etc.'
Page 137
Even if I may not say 'There are four pure colours', still the pure colours and the number 4 are somehow
connected, and that must
Page Break 138
come out in some way or other, e.g. if I say, 'I can see 4 colours on this surface: yellow, blue, red and green.'
Page 138
The situation must be exactly similar for permutations. The permutations (without repetition) of AB are AB,
BA. They are not the extension of a concept: they alone are the concept. But in that case you cannot say of these that
they are two. And yet apparently we do just that in the theory of combinations. It strikes me as a question of a
correlation similar to the one between algebra and inductions in arithmetic. Or is the connection the same as that
between geometry and arithmetic? The proposition that there are two permutations of AB is in fact completely on all
fours with the proposition that a line meets a circle in 2 points. Or, that a quadratic equation has two roots.
Page 138
If we say that AB admits of two permutations, it sounds as though we had made a general assertion,
analogous with 'There are two men in the room', in which nothing further is said or need be known about the men.
But this isn't so in the AB case. I cannot give a more general description of AB, BA, and so the proposition that two
permutations are possible cannot say any less than that the permutations AB, BA are possible. To say that 6
permutations of 3 elements are possible cannot say less, i.e. anything more general, than is shown by the schema:
ABC
ACB
BAC
BCA
CAB
CBA
For it's impossible to know the number of possible permutations without knowing which they are. And if this weren't so,
the theory of combinations wouldn't be capable of arriving at its general formulae. The law which we see in the
formulation of the permutations is represented by the equation p = n! In the same sense, I believe, as that in which a
circle is given by its equation.
--Of course, I can correlate the number 2 with the permutations AB, BA, just as I can 6 with the complete set of
permutations of A, B, C, but this doesn't give me the theorem of combination
Page Break 139
theory--What I see in AB, BA is an internal relation which therefore cannot be described.--That is, what cannot be
described is that which makes this class of permutations complete.--I can only count what is actually there, not
possibilities. But I can, e.g., work out how many rows a man must write if in each row he puts a permutation of 3
elements and goes on until he can't go any further without repetition. And this means he needs 6 rows to write down
the permutations ABC, ACB, etc. since these just are 'the permutations of A, B, C'. But it makes no sense to say that
these are all the permutations of A, B, C.
Page 139
117 We could imagine a combination computer exactly like the Russian abacus.
Page 139
It is clear that there is a mathematical question: 'How many permutations of--say--4 elements are there?', a
question of precisely the same kind as 'What is 25 × 18?'. For in both cases there is a general method of solution.
Page 139
But still it is only with respect to this method that this question exists.
Page 139
The proposition that there are 6 permutations of 3 elements is identical with the permutation schema, and
thus there isn't here a proposition, 'There are 7 permutations of 3 elements', for no such schema corresponds to it.
Page 139
You may also say that the proposition 'There are 6 permutations of 3 elements' is related to the proposition
'There are 6 people in this room' in precisely the same way as is '3 + 3 = 6', which you could also cast in the form
'There are 6 units in 3 + 3'. And just as in the one case I can count the rows in the permutation schema, so in the
other I can count the strokes in .
Page 139
Just as I can prove that 4 × 3 = 12 by means of the schema:
I can also prove 3! = 6 by means of the permutation schema.
Page Break 140
Page 140
118 What is meant by saying I have as many spoons as can be put in 1-1 correspondence with a dozen bowls?
Page 140
Either this proposition assumes I have 12 spoons, in which case I can't say that they can be correlated with
the 12 bowls, since the opposite would be impossible; or else it doesn't assume I have 12 spoons, in which case it
says I can have 12 spoons, and that's self-evident and once more cannot be said.
Page 140
You could also ask: does this proposition say any less than that I have 12 spoons? Does it say something
which only together with another proposition implies that I have 12 spoons? If p follows q alone, then q already says
p. An apparent process of thought, making the transition, doesn't come in.
Page 140
The symbol for a class is a list.
Page 140
Can I know there are as many apples as pears on this plate, without knowing how many? And what is meant
by not knowing how many? And how can I find out how many? Surely by counting. It is obvious that you can
discover that there are the same number by correlation, without counting the classes.
In Russell's theory only an actual correlation can show the 'similarity' of two classes. Not the possibility of
correlation, for this consists precisely in the numerical equality. Indeed, the possibility must be an internal relation
between the extensions of the concepts, but this internal relation is only given through the equality of the 2 numbers.
Page 140
A cardinal number is an internal property of a list.
Page 140
119 We divide the evidence for the occurrence of a physical event according to the various kinds of such evidence,
into the heard, seen, measured etc., and see that in each of these taken singly there is a formal element of order,
which we can call space.
Page Break 141
Page 141
What sort of an impossibility is the impossibility, e.g., of a 1-1 correlation between 3 circles and 2 crosses? We
could also ask--and it would obviously be a question of the same sort--what sort of an impossibility is the
impossibility of making a correlation by drawing parallel lines, if the arrangement is the given one?
Page 141
That a 1-1 correlation is possible is shown in that a significant proposition--true or false--asserts that it
obtains. And that the correlation discussed above is not possible is shown by the fact that we cannot describe it.
Page 141
We can say that there are 2 circles in this square, even if in reality there are 3, and this proposition is simply
false. But I cannot say that this group of circles is comprised of 2 circles, and just as little that it's comprised of 3
circles, since I should then be ascribing an internal property.
Page 141
It is nonsense to say of an extension that it has such and such a number, since the number is an internal
property of the extension. But you can ascribe a number to the concept that collects the extension (just as you can
say that this extension satisfies the concept).
Page 141
120 It is remarkable that in the case of a tautology or contradiction you actually could speak of sense and reference
in Frege's sense.
Page 141
If we call its property of being a tautology the reference of a tautology, then we may call the way in which
the tautology comes about here the sense of the tautology. And so for contradiction.
Page 141
If, as Ramsey proposed, the sign '=' were explained by saying that x = x is a tautology, and x = y a
contradiction, then we may say that the tautology and the contradiction have no 'sense' here.
Page Break 142
Page 142
So, if a tautology shows something through the fact that just this sense gives this reference, then a tautology
à la Ramsey shows nothing, since it is a tautology by definition.
Page 142
What then is the relation between the sign ' ' and the equals sign explained by means of tautology and
contradiction?
Page 142
Is 'p • q = ~ (~ p ∨ ~ q)' a tautology for the latter? You could say 'p • q = p • q' is taut., and since according to
the definition you may substitute '~(~p ∨ ~q)' for one of the signs 'p • q', the previous expression is also a tautology.
Page 142
Hence you should not write the explanation of the equals sign thus:
x = x is taut.
x = y is contra.
but must say: if, and only if, 'x' and 'y' have the same reference according to the sign-rules, then 'x = y' is taut: if 'x' and
'y' do not have the same reference according to the sign-rules, then 'x = y' is contra. In that case it would be to the
point to write the equals sign thus defined differently, to distinguish it from 'x = y', which represents a rule for signs
and says that we may substitute y for x. That is just what I cannot gather from the sign as explained above, but only
from the fact that it is a tautology, but I don't know that either unless I already know the rules of substitution.
Page 142
It seems to me that you may compare mathematical equations only with significant propositions, not with
tautologies. For an equation contains precisely this assertoric element--the equals sign--which is not designed for
showing something. Since whatever shows itself, shows itself without the equals sign. The equals sign doesn't
correspond to the '.⊃.' in 'p • (p ⊃ q).⊃.q' since the '.⊃.' is only one element among others which go to make up the
tautology. It doesn't drop out of its context, but belongs to the proposition, in the same way that the '•' or '⊃' do. But
the '=' is a copula, which alone makes the equation into something
Page Break 143
propositional. A tautology shows something, an equation shows nothing: rather, it indicates that its sides show
something.
Page 143
121 An equation is a rule of syntax.
Page 143
Doesn't that explain why we cannot have questions in mathematics that are in principle unanswerable? For if
the rules of syntax cannot be grasped, they're of no use at all. And equally, it explains why an infinity that
transcends our powers of comprehension cannot enter into these rules. And it also makes intelligible the attempts of
the formalists to see mathematics as a game with signs.
Page 143
You may certainly construe sign-rules, for example definitions, as propositions about signs, but you don't
have to treat them as propositions at all. They belong to the devices of language. Devices of a different kind from the
propositions of language.
Page 143
Ramsey's theory of identity makes the mistake that would be made by someone who said that you could use
a painting as a mirror as well, even if only for a single posture. If we say this, we overlook that what is essential to a
mirror is precisely that you can infer from it the posture of a body in front of it, whereas in the case of the painting
you have to know that the postures tally before you can construe the picture as a mirror image.
Page 143
Weyl's [†1] 'heterological' paradox:
~ φ ('φ') 'φ' is heterological F ('φ')
F('F') = ~F('F') = ~[~(~ (' '))('~ (' ')')]
Page 143
122 We may imagine a mathematical proposition as a creature which itself knows whether it is true or false. (In
contrast with genuine propositions.)
Page 143
A mathematical proposition itself knows that it is true or that it
Page Break 144
is false. If it is about all the numbers, it must also survey all the numbers.
Page 144
Its truth or falsity must be contained in it, as is its sense.
Page 144
It's as though the generality of such a proposition as '(n) ~ Chr n' were only a pointer to the genuine, actual,
mathematical generality of a proposition [†1]. As though it were only a description of the generality, not the
generality itself. As if the proposition formed a sign only in a purely external way and you still needed to give the
sign a sense from within.
Page 144
We feel the generality possessed by the mathematical assertion to be different from the generality of the
proposition proved.
Page Break 145
Page 145
How is a mathematical problem related to its solution?
Page 145
We could say: a mathematical proposition is an allusion to a proof.
Page 145
A generalization cannot be both empirical and provable.
Page 145
If a proposition is to have a definite sense (and it's nonsense otherwise), it must comprehend--survey--its
sense completely; a generalization only makes sense if it--i.e. all values of its variables--is completely determined.
Page Break 146
XII
Page 146
123 If, by making a series of tests, I advance along an endless stretch, why should it be any different in the case of an
infinite one? And in that case of course, I can never get to the end.
Page 146
But if I only advance along the infinite stretch step by step, then I can't grasp the infinite stretch at all.
Page 146
So I grasp it in a different way; and if I have grasped it, then a proposition about it can only be verified in the
way in which the proposition has taken it.
Page 146
So now it can't be verified by putative endless striding, since even such striding wouldn't reach a goal, since
of course the proposition can outstrip our stride just as endlessly as before. No: it can only be verified by one stride,
just as we can only grasp the totality of numbers at one stroke.
Page 146
We may also say: there is no path to infinity, not even an endless one.
Page 146
The situation would be something like this: We have an infinitely long row of trees, and so as to inspect
them, I make a path beside them. All right, the path must be endless. But if it is endless, then that means precisely
that you can't walk to the end of it. That is, it does not put me in a position to survey the row. (Ex hypothesi not.)
Page 146
That is to say, the endless path doesn't have an end 'infinitely far away', it has no end.
Page 146
124 It isn't just impossible 'for us men' to run through the natural numbers one by one; it's impossible, it means
nothing.
Page 146
Nor can you say, 'A proposition cannot deal with all the numbers
Page Break 147
one by one, so it has to deal with them by means of the concept of number', as if this were a pis aller: 'Because we
can't do it like this, we have to do it another way.' But it's not like that: of course it's possible to deal with the
numbers one by one, but that doesn't lead to the totality. For the totality is only given as a concept.
Page 147
If it's objected that 'if I run through the number series I either eventually come to the number with the
required property, or I never do', we need only reply that it makes no sense to say that you eventually come to the
number and just as little that you never do. Certainly it's correct to say 101 is or is not the number in question. But
you can't talk about all numbers, because there's no such thing as all numbers.
Page 147
125 Can you say you couldn't foresee that 6 - 4 would be precisely 2, but only see it when you get there?
Page 147
That, in the case of the logical concept (1, ξ, ξ + 1), the existence of its objects is already given with the
concept, of itself shows that it determines them.
Page 147
Besides, it's quite clear that every number has its own irreducible individuality. And if I want to prove that a
number has a certain property, in one way or another I must always bring in the number itself.
Page 147
In this sense you might say that the properties of a particular number cannot be foreseen. You can only see
them when you've got there.
Page 147
Someone may say: Can't I prove something about the number 310, even though I can't write it down? Well,
310 already is the number, only written in a different way.
Page 147
What is fundamental is simply the repetition of an operation. Each stage of the repetition has its own
individuality.
Page 147
But it isn't as if I use the operation to move from one individual to another so that the operation would be the
means for getting from one to the other--like a vehicle stopping at every number
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which we can then study: no, applying the operation + 1 three times yields and is the number 3.
Page 148
An 'infinitely complicated law' means no law at all. How are you to know it's infinitely complicated? Only by
there being as it were infinitely many approximations to the law. But doesn't that imply that they in fact approach a
limit? Or could the infinitely many descriptions of intervals of the prime number series be called such
approximations to a law? No, since no description of a finite interval takes us any nearer the goal of a complete
description.
Page 148
Then how would an infinitely complicated law in this sense differ from no law at all?
Page 148
In that case, the law would, at best, run 'Everything is as it is'.
Page 148
126 Yet it still looks now as if the quantifiers make no sense for numbers. I mean: you can't say '(n)•φn', precisely
because 'all natural numbers' isn't a bounded concept. But then neither should one say a general proposition follows
from a proposition about the nature of number.
Page 148
But in that case it seems to me that we can't use generality--all, etc.--in mathematics at all. There's no such
thing as 'all numbers', simply because there are infinitely many. And because it isn't a question here of the
amorphous 'all', such as occurs in 'All the apples are ripe', where the set is given by an external description: it's a
question of a collection of structures, which must be given precisely as such.
Page 148
It's, so to speak, no business of logic how many apples there are when we talk of all the apples. Whereas it's
different in the case of the numbers: there, it has an individual responsibility for each one of them.
Page 148
127 What is the meaning of such a mathematical proposition as '(∃n) • 4 + n = 7'? It might be a disjunction - 4 + 0 =
7 • ∨ • 4 + 1 = 7 • ∨ • 4 + 2 = 7 • ∨ • etc. ad inf. But what does that mean? I can
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understand a proposition with a beginning and an end. But can one also understand a proposition with no end? [†1]
Page 149
I also find it intelligible that one can give an infinite rule by means of which you may form infinitely many
finite propositions. But what does an endless proposition mean?
Page 149
If no finite product makes a proposition true, that means no product makes it true. And so it isn't a logical
product.
Page 149
128 But then can't I say of an equation 'I know it doesn't hold for some substitution--I've forgotten now which; but
whether it doesn't hold in general, I don't know'? Doesn't that make good sense, and isn't it compatible with the
generality of the inequality?
Page 149
Is the reply: 'If you know that the inequality holds for some substitution, that can never mean "for some
(arbitrary) member of the infinite number series", but I always know too that this number lies between 1 and 107, or
within some such limits'?
Page 149
Can I know that a number satisfies the equation without a finite section of the infinite series being marked
out as one within which it occurs? No.
Page 149
'Can God know all the places of the expansion of π?' would have been a good question for the schoolmen to
ask. In all such cases the answer runs, 'The question is senseless.'
Page Break 150
Page 150
'No degrees of brightness below this one hurt my eyes': that means, I have observed that my previous
experiences correspond to a formal law.
Page 150
129 A proposition about all propositions, or all functions, is a priori an impossibility: what such a proposition is
intended to express would have to be shown by an induction. (For instance, that all propositions ~ 2np say the
same.) [†1]
Page 150
This induction isn't itself a proposition, and that excludes there being any vicious circle.
Page 150
What do we want to differentiate propositions from when we form the concept 'proposition'?
Page 150
Isn't it the case that we can only give an external description of propositions in general?
Page 150
Equally, if we ask: Is there a general form of law? As opposed to what? Laws must of course fill the whole of
logical space, and so I can no longer mark them off.
Generality in arithmetic is indicated by an induction.
An induction is the expression for arithmetical generality.
Page 150
Suppose one of the rules of a game ran 'Write down a fraction between 0 and 1'. Wouldn't we understand it?
Do we need any limits here? And what about the rule 'Write down a number greater than 100'? Both seem
thoroughly intelligible.
Page 150
I have always said you can't speak of all numbers, because there's no such thing as 'all numbers'. But that's
only the expression of a feeling. Strictly, one should say,... 'In arithmetic we never are talking about all numbers,
and if someone nevertheless does speak in that way, then he so to speak invents something--nonsensical
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--to supplement the arithmetical facts.' (Anything invented as a supplement to logic must of course be nonsense.)
Page 151
130 It is difficult to extricate yourself completely from the extensional viewpoint: You keep thinking 'Yes, but there
must still be an internal relation between x3 + y3 and z3, since the extension, if only I knew it, would have to show
the result of such a relation.' Or perhaps: 'It must surely be either essential to all n to have the property or not, even
if I can't know it.'
Page 151
If I write '(∃x)•x2 = 2x', and don't construe the '(∃x)' extensionally, it can only mean: 'If I apply the rules for
solving such an equation, I arrive at a particular number, in contrast with the cases in which I arrive at an identity or a
prohibited equation.'
Page 151
The defect (circle) in Dedekind's explanation of the concept of infinity lies in its application of the concept
'all' in the formal implication that holds independently if one may put it like this--of the question whether a finite or
an infinite number of objects falls under its concepts. The explanation simply says: if the one holds of an object, so
does the other. It does not consider the totality of objects at all, it only says something about the object at the
moment in front of it, and its application is finite or infinite as the case may be.
Page 151
But how are we to know such a proposition?--How is it verified? What really corresponds to what we mean
isn't a proposition at all, it's the inference from φx to ψx, if this inference is permitted--but the inference isn't
expressed by a proposition.
Page 151
What does it mean to say a line can be extended indefinitely? Isn't this a case of an 'and so on ad inf.' that is
quite different from that in mathematical induction? According to what's gone before, the expression for the
possibility of extending it further would exist in the sense of a description of the extended line or of the act
Page Break 152
of extending it. Now at first this doesn't seem to be connected with numbers at all. I can imagine the pencil drawing
the line going on moving and keeping on for ever. But is it also conceivable that there should be no possibility of
accompanying this process with a countable process? I think not.
Page 152
131 The generality of a Euclidean proof. We say, the demonstration is carried out for one triangle, but the proof
holds for all triangles--or for an arbitrary triangle. First, it's strange that what holds for one triangle should therefore
hold for every other. It wouldn't be possible for a doctor to examine one man and then conclude that what he had
found in his case must also be true of every other. And if I now measure the angles of a triangle and add them, I
can't in fact conclude that the sum of the angles in every other triangle will be the same. It is clear that the Euclidean
proof can say nothing about a totality of triangles. A proof can't go beyond itself.
Page 152
But once more the construction of the proof is not an experiment, and were it so, its outcome couldn't prove
anything for other cases. That is why it isn't even necessary for the construction actually to be carried out with pencil
and paper, but a description of the construction must be sufficient to show all that is essential. (The description of an
experiment isn't enough to give us the result of the experiment: it must actually be performed.) The construction in a
Euclidean proof is precisely analogous to the proof that 2 + 2 = 4 by means of the Russian abacus.
Page 152
And isn't this the kind of generality the tautologies of logic have, which are demonstrated for p, q, r, etc.?
Page 152
The essential point in all these cases is that what is demonstrated can't be expressed by a proposition.
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Page 153
132 If I say 'The world will eventually come to an end' then that means nothing at all if the date is left indefinitely
open. For it's compatible with this statement that the world should still exist on any day you care to mention.--What
is infinite is the possibility of numbers in propositions of the form 'In n days the world will come to an end'.
Page 153
To understand the sense of a question, consider what an answer to it would look like.
Page 153
To the question 'Is A my ancestor?' the only answers I can imagine are 'A is to be found in my ancestral
gallery' or 'A is not to be found in my ancestral gallery' (Where by my ancestral gallery I understand the sum total of
all kinds of information about my predecessors). But in that case the question can't mean anything more than 'Is A
to be found in my ancestral gallery?' (An ancestral gallery has an end: that is a proposition of syntax.) If a god were
to reveal to me that A was my ancestor, but not at what remove, even this revelation could only mean to me that I
shall find A among my ancestors if I search long enough; but since I shall search through N ancestors, the revelation
must mean that A is one of those N.
Page 153
If I ask how many 9s immediately succeed one another after 3.1415 in the development of π, meaning my
question to refer to the extension, the answer runs either, that in the development of the extension up to the place
last developed (the Nth) we have gone beyond the series of 9s, or, that 9s succeed one another up to the Nth place.
But in this case the question cannot have a different sense from 'Are the first N - 5 places of π all 9s or not?'--But of
course, that isn't the question which interests us.
Page 153
133 In philosophy it's always a matter of the application of a series of utterly simple basic principles that any child
knows, and the--enormous--difficulty is only one of applying these in the confusion our language creates. It's never
a question of the latest
Page Break 154
results of experiments with exotic fish or the most recent developments in mathematics. But the difficulty in
applying the simple basic principles shakes our confidence in the principles themselves.
Page 154
134 What sort of proposition is: 'On this strip you may see all shades of grey between black and white'? Here it looks
at first glance as if we're talking about infinitely many shades.
Page 154
Indeed we are apparently confronted here by the paradox that we can, of course, only distinguish a finite
number of shades, and naturally the distinction between them isn't infinitely slight, and yet we see a continuous
transition.
Page 154
It is just as impossible to conceive of a particular grey as being one of the infinitely many greys between
black and white as it is to conceive of a tangent t as being one of the infinitely many transitional stages in going from
t1 to t2. If I see a ruler roll from t1 to t2, I see--if its motion is continuous--none of the individual intermediate
positions in the sense in which I see t when the tangent is at rest; or else I see only a finite number of such positions.
Page 154
But if in such a case I appear to infer a particular case from a general proposition, then the general
proposition is never derived from experience, and the proposition isn't a real proposition.
Page 154
If, e.g., I say 'I saw the ruler move from t1 to t2, therefore I must have seen it at t', this doesn't give us a valid
logical inference. That is, if what I mean is that the ruler must have appeared to me at t--and so, if I'm talking about
the position in visual space--then it doesn't in the least follow from the premiss. But if I'm talking about the physical
ruler, then of course it's possible for the ruler to have skipped over
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position t, and yet for the phenomena in visual space to have remained continuous.
Page 155
135 Ramsey proposed to express the proposition that infinitely many objects satisfied a function by denying all
propositions of the form:
~(∃x) • φx
(∃x) • φx • ~(∃x, y) • φx • φy
(∃x, y) • φx • φy • ~(∃x, y, z) • φx • φy • φz, etc.
But let's suppose there are only three objects, i.e., there are only three names with a meaning. Then we can no longer
write down the fourth proposition of the series, since it makes no sense to write: (∃x, y, z, u) • φx • φy • φz • φu. So I
don't arrive at the infinite by denying all the propositions in this series.
Page 155
'We only know the infinite by description.' Well then, there's just the description and nothing else.
Page 155
136 To what extent does a notation for the infinite presuppose infinite space or infinite time?
Page 155
Of course it doesn't presuppose an infinitely large sheet of paper. But how about the possibility of one?
Page 155
We can surely imagine a notation which extends through time, not space. Such as speech. Here too we
clearly find it possible to imagine a representation of infinity, yet in doing so we certainly don't make any hypothesis
about time. Time appears to us to be essentially an infinite possibility.
Page 155
Indeed, obviously infinite from what we know of its structure.
Page 155
Surely it's impossible that mathematics should depend on an hypothesis concerning physical space. And
surely in this sense visual space isn't infinite.
Page 155
And if it's a matter, not of the reality, but of the possibility of the hypothesis of infinite space, then this
possibility must surely be prefigured somewhere.
Page Break 156
Page 156
Here we run into the problem that also arises for the extension of visual space that of the smallest visible
distinction. The existence of a smallest visible distinction conflicts with continuity, and yet the two must be
reconcilable with one another.
Page 156
137 If I have a series of alternately black and white patches, as shown in the diagram
then by continual bisection, I will soon arrive at a limit where I'm no longer able to distinguish the black and white
patches, that is, where I have the impression of a grey strip.
Page 156
But doesn't that imply that a strip in my visual field cannot be bisected indefinitely often? And yet I don't see
a discontinuity and of course I wouldn't, since I could only see a discontinuity if I hadn't yet reached the limit of
divisibility.
Page 156
This seems very paradoxical.
Page 156
But what about the continuity between the individual rows? Obviously we have a last but one row of
distinguishable patches and then a last row of uniform grey; but could you tell from this last row that it was in fact
obtained by bisecting the last but one? Obviously not. On the other hand, could you tell from the so-called last but
one row that it can no longer be visibly bisected? It seems to me, just as little. In that case, there would be no last
visibly bisected row!
Page 156
If I cannot visibly bisect the strip any further, I can't even try to, and so can't see the failure of such an
attempt. (This is like the case of the limitlessness of visual space.)
Page 156
Obviously, the same would hold for distinctions between colours.
Page Break 157
Page 157
Continuity in our visual field consists in our not seeing discontinuity.
Page 157
138 But if I can always see only a finite number of things, divisions, colours, etc., than there just isn't an infinity at
all; in no sense whatever. The feeling here is: if I am always able to see only so few, then there aren't any more. As if
it were a case like this: if I can only see 4, then there aren't 100. But infinity doesn't have the role of a number here.
It's perfectly true; if I can only see 4, there aren't 100, there aren't even 5. But there is the infinite possibility that isn't
exhausted by a small number any more than by a large: and in fact just because it isn't itself a quantity.
Page 157
We all of course know what it means to say there is an infinite possibility and a finite reality, since we say
space and time are infinite but we can always only see or live through finite bits of them. But from where, then, do I
derive any knowledge of the infinite at all? In some sense or other, I must have two kinds of experience: one which
is of the finite, and which cannot transcend the finite (the idea of such a transcendence is nonsense even on its own
terms), and one of the infinite. And that's how it is. Experience as experience of the facts gives me the finite; the
objects contain the infinite. Of course not as something rivalling finite experience, but in intension. Not as though I
could see space as practically empty, with just a very small finite experience in it. But, I can see in space the
possibility of any finite experience. That is, no experience could be too large for it or exhaust it: not of course
because we are acquainted with the dimensions of every experience and know space to be larger, but because we
understand this as belonging to the essence of space.--We recognize this essential infinity of space in its smallest
part.
Page 157
Where the nonsense starts is with our habit of thinking of a large number as closer to infinity than a small
one.
Page 157
As I've said, the infinite doesn't rival the finite. The infinite is that whose essence is to exclude nothing finite.
Page Break 158
Page 158
The word 'nothing' occurs in this proposition and, once more, this should not be interpreted as the expression
for an infinite disjunction, on the contrary, 'essentially' and 'nothing' belong together. It's no wonder that time and
again I can only explain infinity in terms of itself, i.e. cannot explain it.
Page 158
Space has no extension, only spatial objects are extended, but infinity is a property of space.
Page 158
(This of itself shows it isn't an infinite extent.)
Page 158
And the same goes for time.
Page 158
139 How about infinite divisibility? Let's remember that there's a point to saying we can conceive of any finite
number of parts but not of an infinite number; but that this is precisely what constitutes infinite divisibility.
Page 158
Now, 'any' doesn't mean here that we can conceive of the sum total of all divisions (which we can't, for
there's no such thing). But that there is the variable 'divisibility' (i.e. the concept of divisibility) which sets no limit to
actual divisibility; and that constitutes its infinity.
Page 158
But how do we construct an infinite hypothesis, such as that there are infinitely many fixed stars (it's clear
that in the end only a finite reality can correspond to it)? Once more it can only be given through a law. Let's think
of an infinite series of red spheres.--Let's think of an infinite film strip. (It would give the possibility of everything
finite that happens on the screen.) This is a typical case of an hypothesis reaching out to infinity. It's clear to us that
no experience corresponds with it. It only exists in 'the second system', that is, in language; but how is it expressed
there? (If a man can imagine an infinite strip, then as far as he's concerned there is an infinite reality, and also the
'actual infinite' of mathematics.) It is expressed by a proposition of the form '(n): (∃nx). φx'. Everything relating to the
infinite possibility (every infinite assertion about the
Page Break 159
film), is reproduced in the expression in the first bracket, and the reality corresponding to it in the second.
Page 159
But what then has divisibility to do with actual division, if something can be divisible that never is divided?
Page 159
Indeed, what does divisibility mean at all in the case of that which is given as primary? How can you
distinguish between reality and possibility here?
Page 159
It must be wrong to speak as I do of restricting infinite possibility to what is finite.
Page 159
For it makes it look as if an infinite reality were conceivable even if there isn't one and so once more as
though it were a question of a possible infinite extension and an actual finite one: as though infinite possibility were
the possibility of an infinite number.
Page 159
And that again shows we are dealing with two different meanings of the word 'possible' when we say 'The
line can be divided into 3 parts' and when we say 'The line can be divided infinitely often'. (This is also indicated by
the proposition above, which questions whether there are actual and possible in visual space.)
Page 159
What does it mean to say a patch in visual space can be divided into three parts? Surely it can mean only that
a proposition describing a patch divided in this way makes sense. (Provided it isn't a question of a confusion
between the divisibility of physical objects and that of a visual patch.)
Page 159
Whereas infinite--or better unlimited--divisibility doesn't mean there's a proposition describing a line divided
into infinitely many parts, since there isn't such a proposition. Therefore this possibility is not brought out by any
reality of the signs, but by a possibility of a different kind in the signs themselves.
Page 159
If you say space is infinitely divisible, then strictly speaking that means: space isn't made up of individual
things (parts).
Page 159
In a certain sense, infinite divisibility means that space is indivisible, that it is not affected by any division.
That it is above such
Page Break 160
things: it doesn't consist of parts. Much as if it were saying to reality: you may do what you like in me (you can be
divided as often as you like in me.)
Page 160
Space gives to reality an infinite opportunity for division.
Page 160
And that is why there is only one letter in the first bracket. Obviously only an opportunity, nothing more.
Page 160
140 Is primary time infinite? That is, is it an infinite possibility? Even if it is only filled out as far as memory extends,
that in no way implies that it is finite. It is infinite in the same sense as the three-dimensional space of sight and
movement is infinite, even if in fact I can only see as far as the walls of my room. For what I see presupposes the
possibility of seeing further. That is to say, I could correctly represent what I see only by an infinite form.
Page 160
Is it possible to imagine time with an end, or with two ends?
Page 160
What can happen now, could also have happened earlier, and could always happen in the future if time
remains as it is. But that doesn't depend on a future experience. Time contains the possibility of all the future now.
Page 160
But all that of itself implies that time isn't infinite in the sense of the primitive conception of an infinite set.
Page 160
And so for space. If I say that I can imagine a cylinder extended to infinity, that is already contained in its
nature. So again, contained in the nature of the homogeneity of the cylinder and of the space in which it is--and the
one of course presupposes the other and this homogeneity is in the finite bit that I see.
Page 160
The space of human movement is infinite in the same way as time.
Page 160
141 The rules for a number-system--say, the decimal system--
Page Break 161
contain everything that is infinite about the numbers. That, e.g. these rules set no limits on the left or right hand to
the numerals; this is what contains the expression of infinity.
Page 161
Someone might perhaps say: True, but the numerals are still limited by their use and by writing materials and
other factors. That is so, but that isn't expressed in the rules for their use, and it is only in these that their real essence
is expressed.
Page 161
Does the relation m = 2n correlate the class of all numbers with one of its subclasses? No. It correlates any
arbitrary number with another, and in that way we arrive at infinitely many pairs of classes, of which one is
correlated with the other, but which are never related as class and subclass. Neither is this infinite process itself in
some sense or other such a pair of classes.
Page 161
In the superstition that m = 2n correlates a class with its subclass, we merely have yet another case of
ambiguous grammar.
Page 161
What's more, it all hangs on the syntax of reality and possibility. m = 2n contains the possibility of
correlating any number with another, but doesn't correlate all numbers with others.
Page 161
The word 'possibility' is of course misleading, since someone will say, let what is possible now become
actual. And in thinking this, we always think of a temporal process and infer from the fact that mathematics has
nothing to do with time, that in its case possibility is (already) actuality.
Page 161
(But in truth the opposite is the case, and what is called possibility in mathematics is precisely the same as it
is in the case of time.)
Page 161
m = 2n points along the number series, and if we add 'to infinity', that simply means that it doesn't point at an
object a definite distance away.
Page Break 162
Page 162
142 The infinite number series is itself only such a possibility as emerges clearly from the single symbol for it '(1, x, x
+ 1)'[†1]. This symbol is itself an arrow with the first '1' as the tail of the arrow and 'x + 1' as its tip and what is
characteristic is that just as length is inessential in an arrow--the variable x shows here that it is immaterial how far
the tip is from the tail.
Page 162
It is possible to speak of things which lie in the direction of the arrow but nonsense to speak of all possible
positions for things lying in the direction of the arrow as an equivalent for this direction itself.
Page 162
A searchlight sends out light into infinite space and so illuminates everything in its direction, but you can't
say it illuminates infinity.
Page 162
You could also put it like this: it makes sense to say there can be infinitely many objects in a direction, but no
sense to say there are infinitely many. And this conflicts with the way the word 'can' is normally used. For, if it
makes sense to say a book can lie on this table, it also makes sense to say it is lying there. But here we are led astray
by language. The 'infinitely many' is so to speak used adverbially and is to be understood accordingly.
Page 162
That is to say, the propositions 'Three things can lie in this direction' and 'Infinitely many things can lie in this
direction' are only apparently formed in the same way, but are in fact different in structure: the 'infinitely many' of
the second proposition doesn't play the same role as the 'three' of the first.
Page 162
It is, again, only the ambiguity of our language that makes it appear as if numerals and the word 'infinite' are
both given as answers to the same question. Whereas the questions which have these words as an answer are in
reality fundamentally different.
Page Break 163
Page 163
(The usual conception really amounts to the idea that the absence of a limit is itself a limit. Even if it isn't put
as baldly as that.)
Page 163
If two arrows point in the same direction, isn't it in such a case absurd to call these directions equally long
because whatever lies in the direction of the one arrow, also lies in that of the other?
Page 163
Generality in mathematics is a direction, an arrow pointing along the series generated by an operation. And
you can even say that the arrow points to infinity; but does that mean that there is something--infinity--at which it
points, as at a thing? Construed in that way, it must of course lead to endless nonsense.
Page 163
It's as though the arrow designates the possibility of a position in its direction.
Page 163
143 In what sense is endless time a possibility and not a reality? For someone might object to what I am saying by
arguing that time must be just as much a reality as, say, colour.
Page 163
But isn't colour taken by itself also only a possibility, until it is in a particular time and place? Empty infinite
time is only the possibility of facts which alone are the realities.
Page 163
But isn't the infinite past to be thought of as filled out, and doesn't that yield an infinite reality?
Page 163
And if there is an infinite reality, then there is also contingency in the infinite. And so, for instance, also an
infinite decimal that isn't given by a law. Everything in Ramsey's conception stands or falls with that.
Page 163
That we don't think of time as an infinite reality, but as infinite in intension, is shown in the fact that on the
one hand we can't imagine an infinite time interval, and yet see that no day can be the last, and so that time cannot
have an end.
Page Break 164
Page 164
We might also say: infinity lies in the nature of time, it isn't the extension it happens to have.
Page 164
We are of course only familiar with time--as it were--from the bit of time before our eyes. It would be
extraordinary if we could grasp its infinite extent in this way (in the sense, that is to say, in which we could grasp it if
we ourselves were its contemporaries for an infinite time.)
Page 164
We are in fact in the same position with time as with space. The actual time we are acquainted with is limited
(finite). Infinity is an internal quality of the form of time.
Page 164
144 The infinite number series is only the infinite possibility of finite series of numbers. It is senseless to speak of the
whole infinite number series, as if it, too, were an extension.
Page 164
Infinite possibility is represented by infinite possibility. The signs themselves only contain the possibility and
not the reality of their repetition [†1].
Page 164
Doesn't it come to this: the facts are finite, the infinite possibility of facts lies in the objects. That is why it is
shown, not described.
Page 164
Corresponding to this is the fact that numbers--which of course are used to describe the facts--are finite,
whereas their possibility, which corresponds with the possibility of facts, is infinite. It finds expression, as I've said,
in the possibilities of the symbolism.
Page 164
The feeling is that there can't be possibility and actuality in mathematics. It's all on one level. And is, in a
certain sense, actual.
Page Break 165
Page 165
And that is correct. For what mathematics expresses with its signs is all on one level; i.e. it doesn't speak
sometimes about their possibility, and sometimes about their actuality. No, it can't even try to speak about their
possibility. On the other hand, there is a possibility in its signs, i.e. the possibility found in genuine propositions, in
which mathematics is applied.
Page 165
And when (as in set theory) it tries to express their possibility, i.e. when it confuses them with their reality,
we ought to cut it down to size.
Page 165
We reflect far too little on the fact that a sign really cannot mean more than it is.
Page 165
The infinite possibility in the symbol relates--i.e. refers--only to the essence of a finite extension, and this is
its way of leaving its size open.
Page 165
If I were to say, 'If we were acquainted with an infinite extension, then it would be all right to talk of an actual
infinite', that would really be like saying, 'If there were a sense of abracadabra then it would be all right to talk about
abracadabraic sense-perception'.
Page 165
We see a continuous colour transition and a continuous movement, but in that case we just see no parts, no
leaps (not infinitely many).
Page 165
145 What is an infinite decimal not given by a rule? Can you give an infinite sequence of digits by a
non-mathematical--and so external--description, instead of a law? (Very strange that there should be two modes of
comprehension.)
Page 165
'The number that is the result when a man endlessly throws a die', appears to be nonsense because no infinite
number results from it.
Page 165
But why is it easier to imagine life without end than an endless series in space? Somehow, it's because we
simply take the endless
Page Break 166
life as never complete, whereas the infinite series in space ought, we feel, already to exist as a whole.
Page 166
Let's imagine a man whose life goes back for an infinite time and who says to us: 'I'm just writing down the
last digit of π, and it's a 2'. Every day of his life he has written down a digit, without ever having begun; he has just
finished.
Page 166
This seems utter nonsense, and a reductio ad absurdum of the concept of an infinite totality.
Page 166
Suppose we travel out along a straight line in Euclidean Space and say that at 10 m intervals we encounter an
iron sphere of a certain diameter, ad infinitum; is this a construction? It seems so. What is strange is that you can
construe such an infinite complex of spheres as the endless repetition of the same sphere in accordance with a
certain law--and yet the moment you think of these spheres as each having its own individual characteristics, their
infinite number seems to become nonsense.
Page 166
Let's imagine an infinite row of trees, all of different heights between 3 and 4 yards. If there is a law
governing the way the heights vary, then the series is defined and can be imagined by means of this law (this is
assuming that the trees are indistinguishable save in their height). But what if the heights vary at random?
Then--we're forced to say--there's only an infinitely long, an endless description. But surely that's not a description! I
can suppose there to be infinitely many descriptions of the infinitely many finite stretches of the infinite row of trees,
but in that case I have to know these infinitely many descriptions by means of a law governing their sequence. Or, if
there's no such law, I once more require an infinite description of these descriptions. And that would again get me
nowhere.
Page 166
Now I could of course say that I'm aware of the law that each tree must differ in height from all its
predecessors. It's true that this
Page Break 167
is a law, but still not such as to define the series. If I now assume there could be a random series, then that is a series
about which, by its very nature, nothing can be known apart from the fact that I can't know it. Or better, that it can't
be known. For is this a situation in which 'the human intellect is inadequate, but a higher might succeed'? How then
does it come about that human understanding frames this question at all, sets out on this path whose end is out of its
reach?
Page 167
What is infinite about endlessness is only the endlessness itself.
Page 167
146 What gives the multiplicative axiom its plausibility? Surely that in the case of a finite class of classes we can in
fact make a selection [choice]. But what about the case of infinitely many subclasses? It's obvious that in such a case
I can only know the law for making a selection.
Page 167
Now I can make something like a random selection from a finite class of classes. But is that conceivable in
the case of an infinite class of classes? It seems to me to be nonsense.
Page 167
Let's imagine someone living an endless life and making successive choices of an arbitrary fraction from the
fractions between 1 and 2, 2 and 3, etc. ad. inf. Does that yield us a selection from all those intervals? No, since he
does not finish. But can't I say nonetheless that all those intervals must turn up, since I can't cite any which he
wouldn't eventually arrive at? But from the fact that given any interval, he will eventually arrive at it, it doesn't follow
that he will eventually have arrived at them all.
Page 167
But doesn't this still give the description of a process that generates selections without end, and doesn't that
mean precisely that an endless selection is formed? But here the infinity is only in the rule.
Page 167
Imagine the following hypothesis: there is in space an infinite series of red spheres, each one metre behind its
predecessor. What
Page Break 168
conceivable experience could correspond to this hypothesis? I think for instance of my travelling along this series
and every day passing a certain number, n, of red spheres. In that case, my experience ought to consist in the fact
that on every possible day in the future I see n more spheres. But when shall I have had this experience? Never!
Page 168
147 You can only answer the objection 'But if nevertheless there were infinitely many things?' by saying 'But there
aren't'. And what makes us think that perhaps there are is only our confusing the things of physics with the elements
of knowledge [†1].
Page 168
For this reason we also can't suppose an hypothetical infinite visual space in which an infinite series of red
patches is visible.
Page 168
What we imagine in physical space is not that which is primary, which we can know only to a greater or
lesser extent; rather, what we can know of physical space shows us how far what is primary reaches and how we
have to interpret physical space.
Page 168
But how is a proposition of the form 'The red patch a lies somewhere between b and c' to be analysed? This
doesn't mean: 'The
patch a corresponds to one of the infinitely many numbers lying between the numbers of b and c' (it isn't a question
of a disjunction). It's clear that the infinite possibility of positions of a between b and c isn't expressed in the
proposition. Just as, in the case of 'I have locked him in the room', the infinitely many possible positions
Page Break 169
of the man shut in the room play no role whatever.
Page 169
'Each thing has one and only one predecessor; a has no successor; everything except for a has one and only
one successor.' These propositions appear to describe an infinite series (and also to say that there are infinitely many
things. But this would be a presupposition of the propositions' making sense). They appear to describe a structure
amorphously. We can sketch out a structure in accordance with these propositions, which they describe
unambiguously. But where can we discover this structure in them?
Page 169
But couldn't we regard this set of propositions simply as propositions belonging to physics, setting out a
scientific hypothesis? In that case they would have to be unassailable. But what would it be like if a biologist
discovered a species of animal, in which every individual appeared to be the offspring of one earlier one, and put this
forward as an hypothesis?
Page 169
Are we misled in such a case by the illusion that parcels of matter--i.e. in this case the members of a species
of animal--were the simple objects?
Page 169
That is to say, isn't that which we can imagine multiplied to infinity never the things themselves, but
combinations of the things in accordance with their infinite possibilities?
Page 169
The things themselves are perhaps the four basic colours, space, time and other data of the same sort.
Page 169
In that case, how about a series of fixed stars, in which each has a predecessor (in a particular spacial
direction)? And this hypothesis would come out in the same way as that of an endless life. This seems to me to
make sense, precisely because it doesn't conflict with the insight that we cannot make any hypothesis concerning the
number of objects (elements of facts). Its analysis only presupposes the infinite possibility of space and time and a
finite number of elements of experience.
Page Break 170
XIII
Page 170
148 We might also ask: What is it that goes on when, while we've as yet no idea how a certain proposition is to be
proved, we still ask 'Can it be proved or not?' and proceed to look for a proof? If we 'try to prove it', what do we do?
Is this a search which is essentially unsystematic, and therefore strictly speaking not a search at all, or can there be
some plan involved? How we answer this question is a pointer as to whether the as yet unproved--or as yet
unprovable--proposition is senseless or not. For, in a very important sense, every significant proposition must teach
us through its sense how we are to convince ourselves whether it is true or false. 'Every proposition says what is the
case if it is true.' And with a mathematical proposition this 'what is the case' must refer to the way in which it is to be
proved. Whereas--and this is the point--you cannot have a logical plan of search for a sense you don't know. The
sense would have to be, so to speak, revealed to us: revealed from without--since it can't be obtained from the
propositional sign alone--in contrast with its truth, where the proposition itself tells us how to look for its truth and
compare the truth with it.
Page 170
This amounts to asking: Does a mathematical proposition tie something down to a Yes or No answer? (i.e.
precisely a sense.)
Page 170
My explanation mustn't wipe out the existence of mathematical problems. That is to say, it isn't as if it were
only certain that a mathematical proposition made sense when it (or its opposite) had been proved. (This would
mean that its opposite would never have a sense (Weyl).) On the other hand, it could be that certain apparent
problems lose their character as problems--the question as to Yes or No.
Page 170
149 Is it like this: I need a new insight at each step in a proof? This is connected with the question of the individuality
of each
Page Break 171
number. Something of the following sort: Supposing there to be a certain general rule (therefore one containing a
variable), I must recognize each time afresh that this rule may be applied here. No act of foresight can absolve me
from this act of insight [†1]. Since the form to which the rule is applied is in fact different at every step.
Page 171
A proof of relevance would be a proof which, while yet not proving the proposition [would show the form of
a method to be followed in order to test such a proposition] [†2]. And just that could make such a proof possible. It
wouldn't get to the top of the ladder since that requires that you pass every rung; but only show that the ladder leads
in this direction. That is, there's no substitute for stepping on every rung, and whatever is equivalent to doing so
must in its turn possess the same multiplicity as doing so. (There are no surrogates in logic.) Neither is an arrow a
surrogate for going through all the stages towards a particular goal. This is also connected with the impossibility of
an hierarchy of proofs.
Page 171
Wouldn't the idea of an hierarchy mean that merely posing a problem would have to be already preceded by
a proof, i.e. a proof that the question makes sense? But then I would say that a proof of sense must be a radically
different sort of proof from one of a proposition's truth, or else it would in its turn presuppose yet another and we'd
get into an infinite regress.
Page 171
Does the question of relevance make sense? If it does, it must always be possible to say whether the axioms
are relevant to this proposition or not, and in that case this question must always be decidable, and so a question of
the first type has already been decided. And if it can't be decided, it's completely senseless.
Page Break 172
Page 172
What, apart from Fermat's alleged proof, drives us to concern ourselves with the formula xn + yn = zn... (F), is
the fact that we never happen upon cardinal numbers that satisfy the equation; but that doesn't give the slightest
support (probability) to the general theorem and so doesn't give us any good reason for concerning ourselves with
the formula. Rather, we may look on it simply as a notation for a particular general form and ask ourselves whether
syntax is in any way at all concerned with this form.
Page 172
I said: Where you can't look for an answer, you can't ask either, and that means: Where there's no logical
method for finding a solution, the question doesn't make sense either.
Page 172
Only where there's a method of solution is there a problem (of course that doesn't mean 'Only when the
solution has been found is there a problem').
Page 172
That is, where we can only expect the solution from some sort of revelation, there isn't even a problem. A
revelation doesn't correspond to any question.
Page 172
It would be like wanting to ask about experiences belonging to a sense organ we don't yet possess. Our being
given a new sense, I would call revelation.
Page 172
Neither can we look for a new sense (sense-perception).
Page 172
150 We come back to the question: In what sense can we assert a mathematical proposition? That is: what would
mean nothing would be to say that I can only assert it if it's correct.--No, to be able to make an assertion, I must do
so with reference to its sense, not its truth. As I've already said, it seems clear to me that I can assert a general
proposition as much or as little as the equation 3 × 3 = 9 or 3 × 3 = 11.
Page 172
It's almost unbelievable, the way in which a problem gets completely barricaded in by the wrong expressions
which generation
Page Break 173
upon generation throw up for miles around it, so that it's virtually impossible to get at it.
Page 173
What makes understanding difficult is the misconception of the general method of solution as only
an--incidental--expedient for deriving numbers satisfying the equation. Whereas it is in itself a clarification of the
essence (nature) of the equation. Again, it isn't an incidental device for discovering an extension, it's an end in itself.
Page 173
What questions can be raised concerning a form, e.g. fx = gx?--Is fx = gx or not (x as a general constant)? Do
the rules yield a solution of this equation or not (x as unknown)? Do the rules prohibit the form fx = gx or not (x
construed as a stop-gap)?
Page 173
None of these cases admits of being tested empirically, and so extensionally.
Page 173
Not even the last two, since I see that e.g., 'x2 = 4' is permitted no less from 72 = 4 than from 22 = 4, and that
x2 = -4 is prohibited isn't shown me by 22 ≠ -4 in any different way from that by which 82 ≠ -4 shows it me. That is,
in the particular case, I again see here only the rule.
Page 173
The question 'Do any numbers satisfy the equation?' makes no sense, no more than does 'It is satisfied by
numbers', or, of course, than 'It is satisfied by all (no) numbers'.
Page 173
The important point is that, even in the case where I am given that 32 + 42 = 52, I ought not to say '(∃x, y, z,
n) • xn + yn = zn', since taken extensionally that's meaningless, and taken intensionally this doesn't provide a proof of
it. No, in this case I ought to express only the first equation.
Page 173
It's clear that I may only write down the general proposition (using general constants), when it is analogous
to the proposition 25 × 25 = 625, and that will be when I know the rules for calculating with a and b just as I know
those for 6, 2 and 5. This illustrates
Page Break 174
precisely what it means to call a and b constants here--i.e. constant forms [†1].
Page 174
Is it like this: I can't use the word 'yields' while I am unaware of any method of solution, since yields refers to
a structure that I cannot designate unless I am aware of it. Because the structure must be represented.
Page 174
Every proposition is the signpost for a verification.
Page 174
If I construe the word 'yields' in an essentially intensional way, then the proposition 'The equation E yields
the solution a' means nothing until the word 'yields' stands for a definite method. For it is precisely this which I wish
to refer to.
Page Break 175
Page 175
I have here nothing other than the old case that I cannot say two complexes stand in a relation without using
a logical mapping of the relation.
Page 175
'The equation yields a' means: If I transform the equation in accordance with certain rules, I get a, just as the
equation 25 × 25 = 620 says that I get 620 if I apply the rules for multiplication to 25 × 25. But these rules must
already be given to me before the word 'yields' has a meaning and before the question whether the equation yields a
has a sense.
Page 175
Thus Fermat's proposition makes no sense until I can search for a solution to the equation in cardinal
numbers.
Page 175
And 'search' must always mean: search systematically. Meandering about in infinite space on the look-out for
a gold ring is no kind of search.
Page 175
You can only search within a system: And so there is necessarily something you can't search for [†1].
Page 175
151 We may only put a question in mathematics (or make a conjecture), where the answer runs: 'I must work it out'.
Page 175
But can't I also say this is the case of 1/3 = , where the result isn't an extension, but the formation of the
inductive relationship?
Page 175
But even so, we must also have a clear idea of this inductive relationship if we are to expect it.
Page 175
That is: even here we still can't conjecture or expect in a void.
Page 175
What 'mathematical questions' share with genuine questions is simply that they can be answered.
Page Break 176
Page 176
If the in 1/3 = refers to a definite method, then 0.110 as connected with F means nothing, since a
method is lacking here [†1].
Page 176
A law I'm unaware of isn't a law.
Page 176
A mathematical question must be as exact as a mathematical proposition.
Page 176
The question 'How many solutions are there to this equation?' is the holding in readiness of the general
method for solving it. And that, in general, is what a question is in mathematics: the holding in readiness of a general
method.
Page 176
I need hardly say that where the law of the excluded middle doesn't apply, no other law of logic applies
either, because in that case we aren't dealing with propositions of mathematics. (Against Weyl and Brouwer.)
Page 176
Wouldn't all this lead to the paradox that there are no difficult problems in mathematics, since, if anything is
difficult, it isn't a problem?
Page 176
But it isn't like that: The difficult mathematical problems are those for whose solution we don't yet possess a
written system. The mathematician who is looking for a solution then has a system in some sort of psychic
symbolism, in images, 'in his head', and endeavours to get it down on paper. Once that's done, the rest is easy. But if
he has no kind of system, either in written or unwritten
Page Break 177
symbols, then he can't search for a solution either, but at best can only grope around.--Now, of course you may find
something even by random groping. But in that case you haven't searched for it, and, from a logical point of view,
the process was synthetic; whereas searching is a process of analysis.
Page 177
Whatever one can tackle is a problem [†1].
Page 177
Only where there can be a problem can something be asserted.
Page 177
If I know the rules of elementary trigonometry, I can check the proposition sin 2x = 2 sin x • cos x, but not
the proposition sin x = x - + .... But that means that the sine function of elementary trigonometry and that of
higher trigonometry are different concepts. If we give them the same name, we admittedly do so with good reason,
since the second concept includes within itself the multiplicity of the first; but as far as the system of elementary
trigonometry is concerned, the second proposition makes no sense, and it is of course senseless in this context to
ask whether sin x = x--etc.
Page 177
152 Is it a genuine question if we ask whether it's possible to trisect an angle? And of what sort is the proposition and
its proof that it's impossible with ruler and compasses alone?
Page 177
We might say, since it's impossible, people could never even have tried to look for a construction.
Page 177
Until I can see the larger system encompassing them both, I can't try to solve the higher problem.
Page 177
I can't ask whether an angle can be trisected with ruler and compasses, until I can see the system 'Ruler and
Compasses' as embedded in a larger one, where the problem is soluble; or better, where the problem is a problem,
where this question has a sense.
Page Break 178
Page 178
This is also shown by the fact that you must step outside the Euclidean system for a proof of the
impossibility.
Page 178
A system is, so to speak, a world.
Page 178
Therefore we can't search for a system: What we can search for is the expression for a system that is given
me in unwritten symbols.
Page 178
A schoolboy, equipped with the armoury of elementary trigonometry and asked to test the equation sin x =
x--etc., simply wouldn't find what he needs to tackle the problem. If the teacher nevertheless expects a solution from
him, he's assuming that the multiplicity of the syntax which such a solution presupposes is in some way or other
present in a different form in the schoolboy's head--present in such a way that the schoolboy sees the symbolism of
elementary trigonometry as a part of this unwritten symbolism and now translates the rest from an unwritten into a
written form.
Page 178
The system of rules determining a calculus thereby determines the 'meaning' of its signs too. Put more
strictly: The form and the rules of syntax are equivalent. So if I change the rules--seemingly supplement them,
say--then I change the form, the meaning.
Page 178
I cannot draw the limits of my world, but I can draw limits within my world. I cannot ask whether the
proposition p belongs to the system S, but I can ask whether it belongs to the part s of S. So that I can locate the
problem of the trisection of an angle within the larger system, but can't ask, within the Euclidean system, whether it's
soluble. In what language should I ask this? in the Euclidean? But neither can I ask in Euclidean language about the
possibility of bisecting an angle within the Euclidean system. For
Page Break 179
in this language that would boil down to a question about absolute possibility, and such a question is always
nonsense.
Page 179
But there's nothing to be found here which we could call a hierarchy of types.
Page 179
In mathematics, we cannot talk of systems in general, but only within systems. They are just what we can't
talk about. And so, too, what we can't search for.
Page 179
The schoolboy that lacked the equipment for answering the second question, couldn't merely not answer it,
he couldn't even understand it. (It would be like the task the prince set the smith in the folk tale: Fetch me a
'Fiddle-de-dee'.)
Page 179
Every legitimate mathematical proposition must put a ladder up against the problem it poses, in the way that
12 × 13 = 137 does--which I can then climb if I choose.
Page 179
This holds for propositions of any degree of generality. (N.B. there is no ladder with 'infinitely many' rungs.)
Page 179
Now suppose I have two systems: I can't enquire after one system encompassing them both, since not only
am I unable now to search for this system, but even in the event of one turning up that encompasses two systems
analogous to the original ones, I see that I could never have looked for it.
Page 179
153 Proofs proving the same thing may be translated into one another, and to that extent are the same proof. The
only proofs for which this doesn't hold are like: 'From two things, I infer that he's at home: first his jacket's in the
hall, and also I can hear him whistling'. Here we have two independent ways of knowing. This proof requires
grounds that come from outside, whereas a mathematical proof is an analysis of the mathematical proposition.
Page Break 180
Page 180
What is a proof of provability? It's different from the proof of proposition.
Page 180
And is a proof of provability perhaps the proof that a proposition makes sense? But then, such a proof would
have to rest on entirely different principles from those on which the proof of the proposition rests. There cannot be
an hierarchy of proofs!
Page 180
On the other hand there can't in any fundamental sense be such a thing as metamathematics. Everything
must be of one type (or, what comes to the same thing, not of a type).
Page 180
Now, is it possible to show that the axioms are relevant to a proposition (i.e. prove it or its opposite), without
actually bringing them to bear directly on it? That is, do we only know whether they are relevant when we've got
there, or is it possible to know this at an earlier stage? And is the possibility of checking 36 × 47 = 128 a proof of
this? It obviously makes sense to say 'I know how you check that', even before you've done so.
Page 180
Thus, it isn't enough to say that p is provable, what we must say is: provable according to a particular system.
Page 180
Further, the proposition doesn't assert that p is provable in the system S, but in its own system, the system of
p. That p belongs to the system S cannot be asserted, but must show itself.
Page 180
You can't say p belongs to the system S; you can't ask which system p belongs to; you can't search for the
system of p. Understanding p means understanding its system. If p appears to go over from one system into another,
then p has, in reality, changed its sense.
Page 180
Ramsey believed that what I call recognizing the system was nothing more than the application--perhaps
unconscious--of a general mathematical proposition. So that if I know that the question of the correctness of sin 3a
= 5 cos a is decidable, I
Page Break 181
merely deduce that from the laws for sin (a + b) etc. But this isn't right: on the contrary, I derive it from the fact that
there is such a law, not from how it runs.
Page 181
154 I could gather together numerical equations and equations using variables as follows: Transforming the left-hand
side in accordance with certain rules either does or does not yield the right-hand side.
Page 181
But for this to be so, the two sides of the equation (N.B. of the general equation) must, so to speak, be
commensurable.
Page 181
The classifications made by philosophers and psychologists are like those that someone would give who
tried to classify clouds by their shapes.
Page 181
What a mathematical proposition says is always what its proof proves. That is to say, it never says more than
its proof proves.
Page 181
If I had a method for distinguishing equations with a solution from those without, then in terms of this
method the expression '(∃x)•x2 = 2x' would make sense.
Page 181
I can ask 'What is the solution of the equation x2 = 2x?', but not 'Has it a solution?' For what would it look
like for it to have none? A proposition has a sense only if I know what is the case if it is false.--But suppose the
alternative case were something like the equation '(∃x) • x2 - 2x - x(x - 2) = 0'? In that case at any rate, the proposition
(∃x) • x2 = 2x would make sense and it would be proved by the fact that the rules don't allow us to reduce the two
sides to one another. In reply to the question 'Is there a solution of the equation xn + axn - 1 + ... + z = 0?', we may
always ask 'As opposed to what?'
Page 181
25 × 25 = 625. What constitutes the system which shows me the commensurability in this case?
Page 181
Surely that multiplication of two numbers written in this form always gives me a number of the same form,
and a rule for two
Page Break 182
numerical signs of this form decides whether they designate the same or different numbers.
Page 182
We could also characterize this idea as follows: 'It's impossible for us to discover rules of a new type that
hold for a form with which we are familiar. If they are rules which are new to us, then it isn't the old form. The
edifice of rules must be complete, if we are to work with a concept at all--we cannot make any discoveries in
syntax.--For, only the group of rules defines the sense of our signs, and any alteration (e.g. supplementation) of the
rules means an alteration of the sense.
Page 182
Just as we can't alter the marks of a concept without altering the concept itself. (Frege)
Page 182
A system is a formal series, and it is precisely in the rules of the series that the iterations yielding its
successive members are described.
Page 182
The negation of 'It is necessary that p holds for all numbers' is of course 'It is not necessary that...', not 'It is
necessary that not...'. But now we think: if it isn't necessary that p holds for all numbers, it's surely still possible. But
this is where the fallacy lies, since we don't see that we've slipped into the extensional way of looking at things: the
proposition 'It's possible--though not necessary--that p should hold for all numbers' is nonsense. For in mathematics
'necessary' and 'all' go together. (Unless we replace these idioms throughout by ones which are less misleading.)
Page 182
155 What kind of discovery was it, e.g., that Sheffer made when he showed that we can reduce all the
truth-functions to p|q? Or the discovery of the method for extracting the cube root? What's going on when we have
to resort to a dodge in Mathematics? (As when solving an equation or integrating.) Here it's like unravelling a knot. I
can try out one way or another at random and
Page Break 183
the knot may get even more tangled or come undone. (Whatever happens, each operation is a permissible one and
leads somewhere.)
Page 183
I want to say that finding a system for solving problems which previously could only be solved one by one
by separate methods isn't merely discovering a more convenient vehicle, but is something completely new which
previously we didn't have at all. The uniform method precisely isn't just the method for constructing an object,
which is the same no matter how it was constructed. The method isn't a vehicle taking us to a place which is our real
goal no matter how we arrive at it.
Page 183
That is to say: In my opinion, no way can be found in mathematics which isn't also a goal. You can't say: I
already had all these results, now all I've done is find an even better way that leads to all of them. No: this way is a
new place that we previously lacked. The new way amounts to a new system.
Page 183
Wouldn't this imply that we can't learn anything new about an object in mathematics, since, if we do, it is a
new object?
Page 183
This boils down to saying: If I hear a proposition of, say, number theory, but don't know how to prove it,
then I don't understand the proposition either. This sounds extremely paradoxical. It means, that is to say, that I
don't understand the proposition that there are infinitely many primes, unless I know its so-called proof: when I learn
the proof, I learn something completely new, and not just the way leading to a goal with which I'm already familiar.
Page 183
But in that case it's unintelligible that I should admit, when I've got the proof, that it's a proof of precisely this
proposition, or of the induction meant by this proposition.
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Page 184
I want to say, it isn't the prose which is the mathematical proposition, it's the exact expression [†1].
Page 184
There can't be two independent proofs of one mathematical proposition.
Page 184
156 Unravelling knots in mathematics: Can someone try to unravel a knot which is subsequently shown to be
impossible to untie? People succeeded in resolving the cubic equation, they could not have succeeded in trisecting
an angle with ruler and compasses; people had been grappling with both these problems long before they knew how
to solve the one and that the other was insoluble.
Page 184
Let's consider something that appears to be a knot, but is in reality made up of many loops of thread and
perhaps also a few threads with loose ends. I now set someone the task of unravelling the knot. If he can see the
arrangement of the threads clearly, he'll
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say 'That's not a knot and so there's no such thing as unravelling it'. If he can only see a jumble of threads, then he
may try to untie it, pulling various ends at random, or actually making a few transformations that are the result of
having a clear picture of some parts of the knot, even though he hasn't seen its structure as a whole.
Page 185
I would say here we may only speak of a genuine attempt at a solution to the extent that the structure of the
knot is seen clearly. To the extent that it isn't, everything is groping in the dark, since it's certainly possible that
something which looks like a knot to me isn't one at all; (the best proof that I in fact had no method for searching for
a solution). We can't compare what goes on here with what happens when I make a methodical search of a room for
something, and so discover that it isn't there. For in this case I am searching for a possible state of affairs, not an
impossible one.
Page 185
But now I want to say that the analogy with a knot breaks down, since I can have a knot and get to know it
better and better, but in the case of mathematics I want to say it isn't possible for me to learn more and more about
something which is already given me in my signs, it's always a matter of learning and designating something new.
Page 185
I don't see how the signs, which we ourselves have made for expressing a certain thing, are supposed to
create problems for us.
Page 185
It's more like a situation in which we are gradually shown more and more of a knot or tangle, and we make
ourselves a series of pictures of as much of it as we can see. We have no idea what the part of the knot not yet
revealed to us is like and can't in any way make conjectures about this (say, by examining the pictures of the part we
know already).
Page Break 186
Page 186
157 What did people discover when they found there was an infinity of primes? What did people discover when
they realized there was an infinity of cardinals?--Isn't it exactly similar to the recognition--if that is what it is--that
Euclidean Space is infinite, long after we have been forming propositions about the objects in this space.
Page 186
What is meant by an investigation of space?--For every mathematical investigation is a sort of investigation
of space [†1]. It's clear that we can investigate the things in space--but space itself! (Geometry and Grammar always
correspond with one another.)
Page 186
Let's remember that in mathematics, the signs themselves do mathematics, they don't describe it. The
mathematical signs are like the beads of an abacus. And the beads are in space, and an investigation of the abacus is
an investigation of space.
Page 186
What wasn't foreseen, wasn't foreseeable; for people lacked the system within which it could have been
foreseen. (And would have been.)
Page 186
You can't write mathematics [†2], you can only do it. (And for that very reason, you can't 'fiddle' the signs in
mathematics.)
Page 186
Suppose I wanted to construct a regular pentagon but didn't know how, and were now to make experiments
at random, finally coming upon the right construction by accident: Haven't we here an actual case of a knot which is
untied by trial and error? No, since if I don't understand this construction, as far as I'm concerned it doesn't even
begin to be the construction of a pentagon.
Page 186
Of course I can write down the solution of a quadratic equation by accident, but I can't understand it by
accident. The way I have
Page Break 187
arrived at it vanishes in what I understand. I then understand what I understand. That is, the accident can only refer
to something purely external, as when I say 'I found that out after drinking strong coffee.' The coffee has no place in
what I discovered.
Page 187
158 The discovery of the connection between two systems wasn't in the same space as those two systems, and if it
had been in the same space, it wouldn't have been a discovery (but just a piece of homework).
Page 187
Where a connection is now known to exist which was previously unknown, there wasn't a gap before,
something incomplete which has now been filled in!--(At the time, we weren't in a position to say 'I know this much
about the matter, from here on it's unknown to me.')
Page 187
That is why I have said there are no gaps in mathematics. This contradicts the usual view.
Page 187
In mathematics there is no 'not yet' and no 'until further notice' (except in the trivial sense that we haven't yet
multiplied two 1,000-digit numbers together).
Page 187
An induction has a great deal in common with the multiplicity of a class (a finite class, of course). But all the
same it isn't one, and now it is called an infinite class.--
Page 187
If, e.g., I say, 'If I know one whorl, I know the whole spiral', that strictly means: if I know the law of a spiral,
that's in many respects analogous with the case in which I know a totality of whorls.--But naturally a 'finite' totality,
since that's the only kind there is.--We can't now say: I agree it's in many ways analogous with a finite totality, but
on the other hand it's completely analogous with an infinite one; that an induction isn't completely analogous with a
totality is simply all we can say.
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Page 188
Mathematics cannot be incomplete; any more than a sense can be incomplete. Whatever I can understand, I
must completely understand. This ties up with the fact that my language is in order just as it stands, and that logical
analysis does not have to add anything to the sense present in my propositions in order to arrive at complete clarity.
So that even the most unclear seeming proposition retains its previous content intact after the analysis and all that
happens is that its grammar is made clear.
Page 188
159 But doesn't it still have to count as a question, whether there is a finite number of primes or not?--Once you
have acquired this concept at all. For it certainly seems that the moment I'm introduced to the concept of a prime
number, I can ask 'How many are there?' Just as I can ask 'How many are there?' straight off when I am given the
concept 'man in this room'.
Page 188
If I am misled by this analogy, it can only be because the concept 'prime number' is given me in a completely
different way from a genuine concept. For, what is the strict expression of the proposition '7 is a prime number'?
Obviously it is only that dividing 7 by a smaller number always leaves a remainder. There cannot be a different
expression for that, since we can't describe mathematics, we can only do it. (And that of itself abolishes every 'set
theory'.)
Page 188
Therefore once I can write down the general form of prime number, i.e. an expression in which anything
analogous to 'the number of prime numbers' is contained at all, then there is no longer a question of 'how many'
primes there are, and until I can do this, I also can't put the question. For, I can't ask 'Does the series of primes
eventually come to an end?' nor, 'Does another prime ever come after 7?'
Page 188
For since it was possible for us to have the phrase 'prime number' in ordinary language, even before there
was the strict expression which so to speak admitted of having a number assigned
Page Break 189
to it, it was also possible for people to have wrongly formed the question how many primes there were. This is what
creates the impression that previously there was a problem which is now solved. Verbal language seemed to permit
this question both before and after, and so created the illusion that there had been a genuine problem which was
succeeded by a genuine solution. Whereas in exact language people originally had nothing of which they could ask
how many, and later an expression from which one could immediately read off its multiplicity.
Page 189
Thus I want to say: only in our verbal language (which in this case leads to a misunderstanding of logical
form) are there in mathematics 'as yet unsolved problems' or the problem of the finite 'solubility of every
mathematical problem'.
Page 189
160 It seems to me that the idea of the consistency of the axioms of mathematics, by which mathematicians are so
haunted these days, rests on a misunderstanding.
Page 189
This is tied up with the fact that the axioms of mathematics are not seen for what they are, namely,
propositions of syntax.
Page 189
There is no question as to provability, and in that sense no proof of provability either. The so-called proof of
provability is an induction, to recognize which is to recognize a new system.
Page 189
A consistency proof can't be essential for the application of the axioms.
Page 189
A postulate is only the postulation of a form of expression. The 'axioms' are postulates of the form of
expression.
Page 189
161 The comparison between a mathematical expedition and a polar expedition. There is a point in drawing this
comparison, and it is a very useful one.
Page Break 190
Page 190
How strange it would be if a geographical expedition were uncertain whether it had a goal, and so whether it
had any route whatsoever. We can't imagine such a thing, it's nonsense. But this is precisely what it is like in a
mathematical expedition. And so perhaps it's a good idea to drop the comparison altogether.
Page 190
It would be an expedition, which was uncertain of the space it was in!
Page 190
How can there be conjectures in mathematics? Or better: what sort of thing is it that looks like a conjecture in
mathematics? Such as making a conjecture about the distribution of primes.
Page 190
I might, e.g., imagine that someone is writing the primes in series in front of me without my knowing they
are the primes--I might for instance believe he is writing down numbers just as they occur to him--and I now try to
detect a law in them. I might now actually form an hypothesis about this number sequence, just as I could about any
other sequence yielded by an experiment in physics.
Page 190
Now in what sense have I, by so doing, made an hypothesis about the distribution of primes?
Page 190
You might say that an hypothesis in mathematics has the value that it trains your thoughts on a particular
object--I mean a particular region--and we might say 'we shall surely discover something interesting about these
things'.
Page 190
The trouble is that our language uses each of the words 'question', 'problem', 'investigation', 'discovery' to
refer to such basically different things. It's the same with 'inference', 'proposition', 'proof'.
Page 190
The question again arises, what kind of verification do I count as valid for my hypothesis? Or, can I--faute de
mieux--allow an empirical one to hold for the time being, until I have a 'strict
Page Break 191
proof'? No. Until there is such a proof, there's no connection at all between hypothesis and the 'concept' of a prime
number.
Page 191
The concept of a prime number is the general law by means of which I test whether a number is a prime
number or not.
Page 191
Only the so-called proof establishes any connection between my hypothesis and the primes as such. And
that is shown by the fact that--as I've said--until then, the hypothesis can be construed as one belonging purely to
physics.--On the other hand, when we have supplied a proof, it doesn't prove what was conjectured at all, since I
can't conjecture to infinity. I can only conjecture what can be confirmed, but experience can only confirm a finite
number of conjectures and you can't conjecture the proof until you've got it--and not then either.
Page 191
The concept 'prime number' is the general form of investigation of a number for the relevant property; the
concept 'composite' the general form of investigation for divisibility etc.
Page 191
162 What kind of discovery did Sheffer make when he found that p ∨ q and ~ p can be expressed by means of p|q?
People had no method for looking for p|q, and if someone were to find one today, it wouldn't make any difference.
Page 191
What was it we didn't know before the discovery? It wasn't anything that we didn't know, it was something
with which we weren't acquainted.
Page 191
You can see this very clearly if you imagine someone objecting that p|p isn't at all the same as is said by ~p.
The reply, of course, is that it's only a question of the system p|q etc. having the necessary multiplicity. Thus Sheffer
found a symbolic system with the necessary multiplicity.
Page 191
Does it count as looking for something, if I am unaware of Sheffer's system and say I would like to construct
a system with only one logical constant? No!
Page 191
Systems are certainly not all in one space, so that I could say:
Page Break 192
there are systems with 3 and with 2 logical constants, and now I am trying to reduce the number of constants in the
same way. There is no 'same way' here.
Page 192
We might also put it like this: the completely analysed mathematical proposition is its own proof.
Page 192
Or like this: a mathematical proposition is only the immediately visible surface of a whole body of proof and
this surface is the boundary facing us.
Page 192
A mathematical proposition--unlike a genuine proposition--is essentially the last link in a demonstration that
renders it visibly right or wrong.
Page 192
We can imagine a notation in which every proposition is represented as the result of performing certain
operations--transitions--on particular 'axioms' treated as bases. (Roughly in the same way as chemical compounds
are represented by means of such names as 'trimethylamide' etc.)
Page 192
With a few modifications, such a notation could be constructed out of the instructions with which Russell
and Whitehead preface the propositions of Principia Mathematica.
Page 192
A mathematical proposition is related to its proof as the outer surface of a body is to the body itself. We
might talk of the body of proof belonging to the proposition.
Page 192
Only on the assumption that there's a body behind the surface, has the proposition any significance for us.
Page 192
We also say: a mathematical proposition is the last link in a chain of proof.
Page Break 193
XIV
Page 193
163 'a + (b + c) = (a + b) + c'... A(c) can be construed as a basic rule of a system. As such, it can only be laid down,
but not asserted, or denied (hence no law of the excluded middle). But I can apparently also regard the proposition
as the result of a proof. Is this proof the answer to a question and, if so, which? Has it shown an assertion to be true
and so its negation to be false?
Page 193
But now it looks as though I cannot prove the proposition at all in the sense in which it is a basic rule of a
system. Rather, I prove something about it.
Page 193
This is tied up with the question whether you can deny 2 = 2, as you can 2 × 35 = 70, and why you cannot
deny a definition.
Page 193
In school, children admittedly learn that 2 × 2 = 4, but not that 2 = 2.
Page 193
If we want to see what has been proved, we ought to look at nothing but the proof.
Page 193
We ought not to confuse the infinite possibility of its application with what is actually proved. The infinite
possibility of application is not proved!
Page 193
The most striking thing about a recursive proof is that what it alleges to prove is not what comes out of it.
Page 193
The proof shows that the form 'a + (b + (c + 1)) = (a + b) + (c + 1)'... 'A(c + 1)' follows from the form 1) 'A(c)'
in accordance with the rule 2) 'a + (b + 1) (a + b) + 1'... 'A(1)'. Or, what comes to the same thing, by means of
the rules 1) and 2) the form 'a + (b + (c + 1))' can be transformed into '(a + b) + (c + 1)'. This is the sum total of what
is actually in the proof. Everything else, and the whole of the usual interpretation, lies in the possibility of its
application. And the usual mistake, in
Page Break 194
confusing the extension of its application with what it genuinely contains.
Page 194
Of course, a definition is not something that I can deny. So it does not have a sense, either. It is a rule by
which I can proceed (or have to proceed).
Page 194
I cannot negate the basic rules of a system.
Page 194
In Skolem's proof the 'c' doesn't have any meaning during the proof, it stands for 1 or for what may perhaps
come out of the proof, and after the proof we are justified in regarding it as some number or other. But it must
surely have already meant something in the proof. If 1, why then don't we write '1' instead of 'c'? And if something
else, what? [†1]
Page Break 195
Page 195
If we now suppose that I wish to apply the theorem to 5, 6, 7, then the proof tells me I am certainly entitled
to do so. That is to say, if I write these numbers in the form ((1 + 1) + 1) etc., then I can recognize that the
proposition is a member of the series of propositions that the final proposition of Skolem's chain presents me with.
Once more, this recognition is not provable, but intuitive.
Page Break 196
Page 196
'Every symbol is what it is and not another symbol.'
Page 196
Can there be no proof which merely shows that every multiplication in the decimal system in accordance
with the rules must yield a number of the decimal system? (So that recognizing the same system would thus after all
depend on recognizing the truth of a mathematical proposition.)
Page 196
It would have to be analogous to a proof that by addition of forms ((1 + 1) + 1) etc. numbers of this form
would always result. Now, can that be proved? The proof obviously lies in the rule for addition of such expressions,
i.e. in the definition and in nothing else.
Page 196
Indeed, we might also reply to the question for which this proof is to provide the answer: Well, what else do
you expect the addition to yield?
Page 196
164 A recursive proof is only a general guide to an arbitrary special proof. A signpost that shows every proposition
of a particular form a particular way home. It says to the proposition 2 + (3 + 4) = (2 + 3) + 4: 'Go in this direction
(run through this spiral), and you will arrive home.'
Page 196
To what extent, now, can we call such a guide to proofs the proof of a general proposition? (Isn't that like
wanting to ask: 'To what extent can we call a signpost a route?')
Page 196
Yet it surely does justify the application of A(c) to numbers. And so mustn't there, after all, be a legitimate
transition from the proof schema to this expression?
Page 196
I know a proof with endless possibility, which, e.g., begins with 'A(1)' and continues through 'A(2)' etc. etc.
The recursive proof is the general form of continuing along this series. But it itself must still prove something since it
in fact spares me the trouble of proving each proposition of the form 'A(7)'. But how can it
Page Break 197
prove this proposition? It obviously points along the series of proofs
Page 197
That is a stretch of the spiral taken out of the middle.
Page 197
ξ is a stop-gap for what only emerges in the course of the development.
Page 197
If I look at this series, it may strike me that it is akin to the definition A(1); that if I substitute '1' for 'c' and '1'
for 'd', the two systems are the same.
Page 197
In the proof, at any rate, what is to be proved is not the end of the chain of equations.
Page 197
The proof shows the spiral form of the law.
Page 197
But not in such a way that it comes out as the conclusion of the chain of inferences.
Page 197
It is also very easy to imagine a popular introduction of the proof using 1, perhaps followed by dots to
indicate what we are to look out for. It wouldn't in essence be any less strict. (For the peculiarity of the proof stands
out even more clearly in this case.)
Page 197
Let's imagine it like this. How does it then justify the proposition A(c)?
Page 197
If one regards the proof as being of the same sort as the derivation of (x + y)2 = x2 + 2xy + y2, then it proves
the proposition 'A(c + 1)' (on the hypothesis 'A(c)', and so of the proposition I really want to prove). And justifies--on
this assumption special cases such as 3 + (5 + (4 + 1)) = (3 + 5) + (4 + 1). It also has a generality, but not the one we
desire. This generality does not lie
Page Break 198
in the letters, but just as much in the particular numbers and consists in the fact that we can repeat the proof.
Page 198
But how can I use the sign 'f(a)' to indicate what I see in the passage from f(1) to f(2)? (i.e. the possibility of
repetition.)
Neither can I prove that a + (b + 1) = (a + b) + 1
is a special case of a + (b + c) = (a + b) + c:
I must see it.
(No rule can help me here either, since I would still have to know what would be a special case of this general rule.)
Page 198
This is the unbridgeable gulf between rule and application, or law and special case.
Page 198
A(c) [†1] is a definition, a rule for algebraic calculation. It is chosen in such a way that this calculation agrees
with arithmetical calculation. It permits the same transition in algebraic calculation that holds for cardinal numbers,
as may be seen in the recursive proof. Thus A(c) is not the result of a proof, it so to speak runs parallel with it.
Page 198
What we gather from the proof, we cannot represent in a proposition at all and of course for the same reason
we can't deny it either.
Page 198
But what about a definition like A(1) [†2]? This is not meant as a rule for algebraic calculation, but as a device
for explaining arithmetical expressions. It represents an operation, which I can apply to an arbitrary pair of
numbers.
Page 198
165 The correct expression for the associative law is not a proposition, but precisely its 'proof', which admittedly
doesn't state the law, but shows it. And at this point it becomes clear that we cannot
Page Break 199
deny the law, since it doesn't figure in the form of a proposition at all. We could of course deny the individual
equations of the law, but would not thereby deny the law: that eludes affirmation and denial.
Page 199
To know that you can prove something is to have proved it.
Page 199
7 + (8 + 9) = (7 + 8) + 9. How do I know that this is so, without having to give a particular proof of it? And
do I know just as well as if I had given a complete derivation of it? Yes!--Then that means it really is proved. What's
more, in that case it cannot have a still better proof; say, by my carrying out the derivation as far as this proposition
itself. So it must be possible for me to say after running through one turn of the spiral 'Stop! I don't need any more, I
can already see how it goes on', and then every higher step must be purely superfluous and doesn't make the matter
clearer. If I draw all the whorls of the spiral as far as my point, I cannot see that the spiral leads to it any better than if
I draw only one. It is only that each shows the same thing in a different form. I can so to speak blindly follow the
spiral that has been completely drawn in and arrive at my point, whereas the one whorl which has been drawn has to
be interpreted in a particular way for me to perceive that, if continued, it leads to the point A.
Page 199
That is to say: from the proof for 6 + (7 + 8) = (6 + 7) + 8 which has been completely worked out, I can
extract the same as from the one which only describes one 'whorl', but in a different way. And
Page Break 200
at any rate the one whorl in conjunction with the numerical forms of the given equation is a complete proof of this
equation. It's as though I were to say: 'You want to get to the point A? Well, you can get there along this spiral.'
Page 200
When we teach someone how to take his first step, we thereby enable him to go any distance.
Page 200
166 As the immediate datum is to a proposition which it verifies, so is the arithmetical relation we see in the structure
to the equation which it verifies.
Page 200
It is the real thing, not an expression for something else for which another expression could also be
substituted. That is, it is not a symptom of something else, but the thing itself.
Page 200
For that is how it is usually construed (i.e. in the wrong way). One says an induction is a sign that such and
such holds for all numbers. But the induction isn't a sign for anything but itself. If there were yet something besides
the induction, for which it was a sign, this something would have to have its specific expression which would be
nothing but the complete expression for this something.
Page 200
And this conception is then developed into the idea that the algebraic equation tells us what we see in the
arithmetical induction. For that, it would have to have the same multiplicity as what it describes.
Page 200
How a proposition is verified is what it says. Compare the generality of genuine propositions with generality
in arithmetic. It is differently verified and so is of a different kind.
Page 200
The verification is not one token of the truth, it is the sense of the proposition. (Einstein: How a magnitude is
measured is what it is.)
Page 200
Indeed Russell has really already shown by his theory of descriptions, that you can't get a knowledge of
things by sneaking up on
Page Break 201
them from behind and it can only look as if we knew more about things than they have shown us openly and
honestly. But he has obscured everything again by using the phrase 'indirect knowledge'.
Page 201
167 An algebraic schema derives its sense from the way in which it is applied. So this must always be behind it. But
then so must the inductive proof, since that justifies the application.
Page 201
An algebraic proposition is just as much an equation as 2 × 2 = 4, it is only applied differently. Its relation to
arithmetic is different. It deals with the substitutability of other parts of speech.
Page 201
That is to say, an algebraic equation as an equation between real numbers is, to be sure, an arithmetical
equation, since something arithmetical lies behind it. Only it lies behind the algebraic equation in a different way
from the way it lies behind 1 + 1 = 2.
Page 201
An induction doesn't prove the algebraic proposition, since only an equation can prove an equation. But it
justifies the setting up of algebraic equations from the standpoint of their application to arithmetic.
Page 201
That is, it is only through the induction that they gain their sense, not their truth.
Page 201
For this reason, what can no longer be reduced to other equations, and can only be justified by induction, is a
stipulation.
Page 201
Which is connected with the fact that in the application of this algebraic proposition I cannot appeal to it, but
once again only to the induction.
Page 201
Hence these last equations can't be denied--i.e. there is no arithmetical content corresponding to their
negation.
Page 201
Through them alone the algebraic system becomes applicable to numbers.
Page 201
And so in a particular sense they are certainly the expression of
Page Break 202
something arithmetical, but as it were the expression of an arithmetical existence.
Page 202
They alone make algebra into clothing for arithmetic--and are therefore arbitrary to the extent that no one
compels us to use algebra in this way. They fit algebra on to arithmetic.
Page 202
And while it is wearing these clothes, it can move about in them.
Page 202
They are not the expression of something computable and to that extent are stipulations.
Page 202
Can someone who sees these stipulations learn something from them, in arithmetic? And, if so, what?--Can I
learn an arithmetical state of affairs, and, if so, which?
Page 202
A stipulation is more like a name than like a proposition.
Page 202
You can only prove those propositions whose truth you can enquire into. 'Is it so or not?' 'I will prove to you
it is so.'
Page 202
An induction is related to an algebraic proposition not as a proof is to what is proved, but as what is
designated to a sign.
Page 202
The system of algebraic propositions corresponds to a system of inductions.
Page 202
168 A proof by induction, if it were a proof, would be a proof of generality, not a proof of a certain property of all
numbers.
Page 202
We can only ask from a standpoint from which a question is possible. From which a doubt is possible.
Page 202
If one wanted to ask about A(c), it wouldn't strictly speaking be the induction that answered us, but the
humiliating feeling that we
Page Break 203
could only arrive at the thought of this equation by way of the induction.
Page 203
If we ask 'Does a + (b + c) = (a + b) + c?', what could we be after? Taken purely algebraically, the question
means nothing, since the answer would be: 'Just as you like, as you decide.' Neither can the question mean 'Does
this hold for all numbers?', it can only ask what the induction says: it, however, says nothing at all to us.
Page 203
We cannot ask about that which alone makes questions possible at all.
Page 203
Not about what first gives the system a foundation.
Page 203
That some such thing must be present is clear.
Page 203
And it is also clear that in algebra this first thing must present itself as a rule of calculation, which we can
then use to test the other propositions.
Page 203
An algebraic proposition always gains only arithmetical significance if you replace the letters in it by
numerals, and then always only particular arithmetical significance.
Page 203
Its generality doesn't lie in itself, but in the possibility of its correct application. And for that it has to keep on
having recourse to the induction.
Page 203
That is, it does not assert its generality, it does not express it; the generality is, rather, shown in the formal
relation to the substitution, which proves to be a term of the inductive series.
Page 203
(∃24x) • φx • (∃18x) • ψx • Ind: ⊃:(∃24 + 18x) • φx ∨ ψx. How do I know that this is so, unless I have
introduced the concept of addition for the context of this application? I can only arrive at this proposition by
induction. That is to say, corresponding to the general proposition--or rather, the tautology--(∃nx) • φx • (∃mx) • ψx •
Ind: ⊃:(∃n + mx) • ψx ∨ ψx, there is an induction, and this induction is the proof of the above proposition (∃24x) •
etc.', even before we have actually worked out 24 + 18 and have tested whether it is a tautology.
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Page 204
To believe Goldbach's Conjecture, means to believe you have a proof of it, since I can't, as it were, believe it
in extenso, because that doesn't mean anything, and you cannot imagine an induction corresponding to it until you
have one.
Page 204
If the proof that every equation has a root is a recursive proof, then that means the Fundamental Theorem of
Algebra isn't a genuine mathematical proposition [†1].
Page 204
169 If I want to know what '1/3 = ' means, it's a relevant question to ask 'How can I know that?' For the proof
comes as a reply to this 'how?', and more than is shown by this proof I certainly do not know.
Page 204
It is clear that every multiplication in the decimal system has a solution and therefore that one can prove any
arithmetical equation of the form a × b = c or prove its opposite. What would a proof of this provability look like? It
is obviously nothing further than a clarification of the symbolism and an exhibition of an induction from which it
can be seen what sort of propositions the ladder leads to.
Page 204
I cannot deny the generality of a general arithmetical proposition.
Page 204
Isn't it only this generality that I cannot reflect in the algebraic proposition?
Page 204
An equation can only be proved by reducing it to equations.
Page 204
The last equations in this process are definitions.
Page 204
If an equation can't be reduced to other equations, then it is a definition.
Page 204
An induction cannot justify an equation.
Page 204
Therefore, e.g. the introduction of the notation cannot refer to the induction whose sign it appears to be.
The relation must be similar to that of 'A(c)' to its proof by induction.
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Page 205
Or better, it certainly refers to the bare facts of the induction, but not to the generality, which is its true sense.
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XV
Page 206
170 The theory of aggregates attempts to grasp the infinite at a more general level than a theory of rules. It says that
you can't grasp the actual infinite by means of arithmetical symbolism at all and that therefore it can only be
described and not represented. The description would encompass it in something like the way in which you carry a
number of things that you can't hold in your hands by packing them in a box. They are then invisible but we still
know we are carrying them (so to speak, indirectly). The theory of aggregates buys a pig in a poke. Let the infinite
accommodate itself in this box as best it can.
Page 206
With this there goes too the idea that we can use language to describe logical forms. In a description of this
sort the structures and e.g. correlations etc., are presented in a package, and so it does indeed look as if one could
talk about a structure without reproducing it in the proposition itself. Concepts which are packed up like this and so
whose structures are not recognisable may, to be sure, be used, but they always derive their meaning from
definitions which package the concepts in this way; and if we trace our way back through these definitions the
concepts are then unpacked again and so are present in their structure.
Page 206
This is what Russell does with R*; he wraps the concept up in such a way that its form disappears.
Page 206
The point of this method is to make everything amorphous and treat it accordingly.
Page 206
If in logic a question can be answered (1) generally and (2) in particular, the particular answer must always
show itself to be a special case of the general answer; put differently: the general case must always include the
particular as a possibility.
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Page 207
A case of this is the calculation of a limit with δ and ν, which must include the number system of the
particular computation.
Page 207
There must be a determinate way for translating the general form and the particular form into one another.
Page 207
171 Any proof of the continuity of a function must relate to a numerical scale--a number system.
Page 207
For if I say, 'Given any ν there is a δ for which the function is less than ν', I am ipso facto referring to a
general arithmetical criterion that indicates when φ(δ) is less than ν.
Page 207
It is impossible that what in the nature of this case comes to light when calculating the function--namely, the
numerical scale should be allowed to disappear in the general treatment.
Page 207
If the numerical system belongs to the essence of number, then the general treatment cannot do away with it.
Page 207
And if, therefore, the notation of the numerical system reflects the essence of number, then this essential
element must also find its way into the general notation. In this way the general notation acquires the structure of the
numbers.
Page 207
If, in the nature of the case, I cannot write down a number independently of a number system, that must also
be reflected in the general treatment of number.
Page 207
A number system is not something inferior--like a Russian abacus--that is only of interest to elementary
schools, while the higher, general discussion can afford to disregard it.
Page 207
The Cretan liar paradox could also be set up with someone writing the proposition: 'This proposition is false'.
The demonstrative takes over the role of the 'I' in 'I'm lying'. The basic mistake
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consists, as in the previous philosophy of logic, in assuming that a word can make a sort of allusion to its object
(point at it from a distance) without necessarily going proxy for it.
Page 208
So the question would really be: Can the continuum be described? As Cantor and others tried to do.
Page 208
A form cannot be described: it can only be presented.
Page 208
Dedekind's definition of an infinite set is another example of an attempt to describe the infinite, without
presenting it.
Page 208
It would be like describing an illness by the external symptoms we know always accompany it. Only in this
case there is a connection which isn't formal in nature.
Page 208
172 'The highest point of a curve' doesn't mean 'the highest point among all the points of the curve'--after all, we
don't see these--it is a specific point yielded by the curve. In the same way the maximum of a function isn't the
largest value among all its values (that's nonsense, save in the case of finitely many, discrete points), it's a point
yielded by a law and a condition; which, to be sure, is higher than any other point taken at random (possibility, not
reality). And so also the point of intersection of two lines isn't the common member of two classes of points, it's the
meeting of two laws. As indeed is perfectly clear in analytic geometry.
Page 208
The maximum of a function is susceptible of an intensional explanation. The highest point of a curve is
admittedly higher than any other point taken at random, but I don't find it by sifting through the points of the curve
one by one, looking for a yet higher one.
Page 208
Here again it is grammar which, as always in the sphere of the infinite, is playing tricks on us.
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Page 209
We say 'the highest point of the curve'. But that can't mean 'the highest point of all the points in the curve' in
the sense in which we talk of the largest of these three apples, since we don't have all the points of the curve before
us--in fact that's a nonsense expression.
Page 209
It's the same defect in our syntax which presents the proposition 'The apple may be divided into two parts' as
of the same form as 'a length may be divided without limit', with the result that we can apparently say in both cases,
'Let's suppose the possible division to have been carried out'.
Page 209
But in truth the expressions 'divisible into two parts' and 'divisible without limit' have completely different
forms.
Page 209
This is, of course, the same case as the one in which someone operates with the word 'infinite' as if it were a
number word; because, in everyday speech, both are given as answers to the question 'How many?'
Page 209
The curve exists, independently of its individual points. This also finds expression in the fact that I construct
its highest point: that is, derive it from a law and not by examining individual points.
Page 209
We don't say 'among all its points, there is only one at which it intersects the straight line'; no, we only talk
about one point.
Page 209
So to speak, about one that runs along the straight line, but not about one among all the points of the line.
Page 209
The straight line isn't composed of points.
Page 209
173 But then what would a correct, as opposed to an amorphous explanation of R* be like? Here I do need '(n)...'. In
this case, this expression seems to be admissible.
Page 209
But, to be sure, '(∃x) • φx' also says 'There is a number of x satisfying φx', and yet the expression '(∃x) • φx'
can't be taken to presuppose the totality of all numbers.
Page 209
Ramsey's explanation of infinity also is nonsense for the same reason, since '(n):(∃nx) • φx' would
presuppose that we were given
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the actual infinite and not merely the unlimited possibility of going on [†1].
Page 210
But is it inconceivable that I should know someone to be my ancestor without having any idea at what
remove, so that no limits would be set to the number of people in between [†2].
Page 210
But how would we put the proposition: 'φ is satisfied by the same number of objects as ψ'? One would
suppose: '(∃n):(∃nx) • φx • (∃nx) • ψx'.
Page 210
Brouwer is right when he says that the properties of his pendulum number are incompatible with the law of
the excluded middle. But, saying this doesn't reveal a peculiarity of propositions about infinite aggregates. Rather, it
is based on the fact that logic presupposes that it cannot be a priori--i.e. logically--impossible to tell whether a
proposition is true or false. For, if the question of the truth or falsity of a proposition is a priori undecidable, the
consequence is that the proposition loses its sense and the consequence of this is precisely that the propositions of
logic lose their validity for it.
Page 210
Just as in general the whole approach that if a proposition is valid for one region of mathematics it need not
necessarily be valid for a second region as well, is quite out of place in mathematics, completely contrary to its
essence. Although these authors hold
Page Break 211
just this approach to be particularly subtle, and to combat prejudice.
Page 211
Mathematics is ridden through and through with the pernicious idioms of set theory. One example of this is
the way people speak of a line as composed of points. A line is a law and isn't composed of anything at all. A line as
a coloured length in visual space can be composed of shorter coloured lengths (but, of course, not of points). And
then we are surprised to find, e.g., that 'between the everywhere dense rational points' there is still room for the
irrationals! What does a construction like that for show? Does it show how there is yet room for this point in
between all the rational points? It merely shows that the point yielded by the construction is not rational.
Page 211
And what corresponds to this construction and to this point in arithmetic? A sort of number which manages
after all to squeeze in between the rational numbers? A law that is not a law of the nature of a rational number.
Page 211
The Dedekind cut proceeds as if it were clear what was meant when one says: There are only three cases:
either R has a last member and L a first, or, etc. In truth none of these cases can be conceived (or imagined).
Page 211
174 Set theory is wrong because it apparently presupposes a symbolism which doesn't exist instead of one that does
exist (is alone possible). It builds on a fictitious symbolism, therefore on nonsense.
Page 211
There is no such thing as an hypothesis in logic.
Page 211
When people say 'The set of all transcendental numbers is greater that[[sic]] that of algebraic numbers', that's
nonsense. The set is of a different kind. It isn't 'no longer' denumerable, it's simply not denumerable!
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Page 212
The distribution of primes would then for once provide us with something in logic that a god could know and
we couldn't. That is to say, there would be something in logic that could be known, but not by us.
Page 212
That there is a process of solution is something that cannot be asserted. For, were there none, the equation,
as a general proposition, would be nonsensical.
Page 212
We can assert anything which can be checked in practice.
Page 212
It's a question of the possibility of checking.
Page 212
If someone says (as Brouwer does) that for (x) • f1x = f2x, there is, as well as yes and no, also the case of
undecidability, this implies that '(x)...' is meant extensionally and we may talk of the case in which all x happen to
have a property. In truth, however, it's impossible to talk of such a case at all and the '(x)...' in arithmetic cannot be
taken extensionally.
Page 212
We might say, 'A mathematical proposition is a pointer to an insight'. The assumption that no insight
corresponded to it would reduce it to utter nonsense.
Page 212
We cannot understand the equation unless we recognize the connection between its two sides.
Page 212
Undecidability presupposes that there is, so to speak, a subterranean connection between the two sides; that
the bridge cannot be made with symbols.
Page 212
A connection between symbols which exists but cannot be represented by symbolic transformations is a
thought that cannot be thought. If the connection is there, then it must be possible to see it.
Page 212
For it exists in the same way as the connection between parts of visual space. It isn't a causal connection.
The transition isn't
Page Break 213
produced by means of some dark speculation different in kind from what it connects. (Like a dark passage between
two sunlit places.)
Page 213
Of course, if mathematics were the natural science of infinite extensions of which we can never have
exhaustive knowledge, then a question that was in principle undecidable would certainly be conceivable.
Page 213
175 Is there a sense in saying: 'I have as many shoes as the value of a root of the equation x3 + 2x - 3 = 0'? Even if
solving it were to yield a positive integer?
Page 213
For, on my view, we would have here a notation in which we cannot immediately tell sense from nonsense.
Page 213
If one regards the expression 'the root of the equation φx = 0' as a Russellian description, then a proposition
about the root of the equation x + 2 = 6 must have a different sense from one saying the same about 4.
Page 213
I cannot use a proposition before knowing whether it makes sense, whether it is a proposition. And this I
don't know in the above case of an unsolved equation, since I don't know whether cardinal numbers correspond to
the roots in the prescribed manner. It is clear that the proposition in the given case becomes nonsense and not false
(not even a contradiction), since 'I have n shoes and n2 = 2' is obviously tantamount to 'I have shoes'.
Page 213
But I can establish this--or at any rate it can be established--if we only look at the signs. But I mustn't chance
my luck and incorporate the equation in the proposition, I may only incorporate it if I know that it determines a
cardinal number, for in that case it is simply a different notation for the cardinal number. Otherwise, it's just like
throwing the signs down like so many dice and leaving it to chance whether they yield a sense or not.
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Page 214
(x + y)2 = x2 + y2 + 2xy is correct in the same sense as 2 × 2 = 4.
Page 214
And 2 + n = 1 (where n is a cardinal number) is just as wrong as 2 + 3 = 1 and 2 + n ≠ 1 as correct as the
above.
Page 214
176 What makes one dubious about a purely internal generality is the fact that it can be refuted by the occurrence of
a single case (and so by something extensional).
Page 214
But what sort of collision is there here between the general and the particular proposition? The particular case
refutes the general proposition from within, not in an external way.
Page 214
It attacks the internal proof of the proposition and refutes it thus--not in the way that the existence of a
one-eyed man refutes the proposition 'Every man has two eyes'.
Page 214
'(x) • x2 = x + x' seems to be false because working out the equation gives x = and not that the two sides
cancel out. Trying the substitution x = 3, say, also yields the general result--(∃x) • x2 ≠ 2x--and so must, to the extent
that its result tallies with that of the general solution, itself tally with the general method.
Page 214
If the equation x2 + 2x + 2 = 0 yields, by applying the algebraic rules, x = - 1 ± , that is quite in order
so long as we don't require the rules for x to accord with the rules for real numbers. In that case the outcome of the
algebraic calculation means that the equation has no solution.
Page 214
My difficulty is: When I solve equations in the real, rational or whole number domain by the appropriate
rules, in certain cases I arrive at apparent nonsense. Now, when that happens: am I to say this proves that the original
equation was nonsense? Which would mean I could only see whether it was sense or nonsense after I had finished
applying the rules?! Isn't what we have to say, rather: The result of the apparently nonsensical equation does
Page Break 215
after all show something about the general form, and does indeed establish a connection between the forbidden
equation and equations which have a normal solution. After all, the solution always does show the distance of the
abnormal solution from a normal one. If, e.g., is the result, I at least know that would be a
normal root. The continuity, the connection with the normal solution, has not been disrupted. Would this imply that
in the concept of the real numbers as we represent it in our symbolism and its rules, the concept of the imaginary
numbers is already prefigured?
Page 215
That would amount to roughly the same thing as saying of a straight line g that it is a distance a away from
cutting the circle, instead of simply saying it doesn't cut it.
Page 215
One might say 'It fails to intersect it by a certain amount' and this would represent the continuity with normal
intersection. 'It misses it by a certain amount.'
Page 215
The difference between the two equations x2 = x • x and x2 = 2x isn't one consisting in the extensions of their
validity.
Page 215
I can, it is true, define m > n as (∃x) • n + x = m, but I only know whether x = m - n yields a number if I know
the rule for subtraction, and in this context this does duty for the rule for determining greater and less. Thus
formulated, this rule runs: m is greater than n, if, according to the rule for subtraction, m - n yields a number.
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XVI
Page 216
177 What makes it apparent that space is not a collection of points, but the realization of a law?
Page 216
It seems as though we should have first to erect the entire structure of space without using propositions; and
then one may form all the correct propositions within it.
Page 216
In order to represent space we need--so it appears to me--something like an expansible sign.
Page 216
A sign that makes allowance for an interpolation, similar to the decimal system.
Page 216
The sign must have the multiplicity and properties of space.
Page 216
The axioms of a geometry are not to include any truths.
Page 216
We can be sure of getting the correct multiplicity for the designations if we use analytic geometry.
Page 216
That a point in the plane is represented by a number-pair, and in three-dimensional space by a number-triple,
is enough to show that the object represented isn't the point at all but the point-network.
Page 216
178 The geometry of visual space is the syntax of the propositions about objects in visual space.
Page 216
The axioms--e.g.--of Euclidean geometry are the disguised rules of a syntax. This becomes very clear if you
look at what corresponds to them in analytic geometry.
Page 216
You could imagine the constructions of Euclidean geometry actually being carried out, say by using the
edges of bodies as straight lines and the surfaces as planes. The axiom--for instance--that a
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straight line can be drawn through any two points has here the clear sense that, although no straight line is drawn
between any 2 arbitrary points, it is possible to draw one, and that only means that the proposition 'A straight line
passes through these points' makes sense. That is to say, Euclidean geometry is the syntax of assertions about
objects in Euclidean space. And these objects are not lines, planes and points, but bodies.
Page 217
How are the equations of analysis connected with the results of spatial measurements? I believe, in such a
way that they (the equations) fix what is to count as an accurate measurement and what as an error.
Page 217
One could almost speak of an external and an internal geometry. Whatever is arranged in visual space stands
in this sort of order a priori, i.e. in virtue of its logical nature, and geometry here is simply grammar. What the
physicist sets into relation with one another in the geometry of physical space are instrument readings, which do not
differ in their internal nature whether we live in a linear space or a spherical one. That is to say, it isn't an
investigation of the logical properties of these readings that leads the physicist to make an assumption about the
nature of physical space, but the facts that are derived from them.
Page 217
The geometry of physics in this sense doesn't have to do with possibility, but with the facts. It is
corroborated by facts: in the sense, that is, in which a part of an hypothesis is corroborated.
Page 217
Comparison between working with an adding machine and measuring geometrical structures: In such
measurement, do we perform an experiment, or is the situation here the same as in the case of the adding machine,
where we only establish internal relations and the physical result of our operations proves nothing?
Page 217
In visual space, of course, there is no such thing as a geometrical experiment.
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Page 218
I believe this is where the main source of misunderstanding about the a priori and the a posteriori in
geometry lies.
Page 218
The question is, in what sense can the results of measurement tell us something concerning that which we
also see.
Page 218
What about the proposition 'The sum of the angles of a triangle is 180°'? At any rate, it doesn't give the
appearance of a proposition of syntax.
Page 218
The proposition 'Vertically opposite angles are equal' means that if they turn out to be different when they
are measured, I shall declare the measurement to be in error; and 'The sum of the angles of a triangle is 180°' means
that, if the sum doesn't turn out to be 180° when measured, I shall assume an error in the measurement. In this way,
the proposition is a postulation concerning the way to describe the facts: and so a proposition of syntax.
Page 218
There is obviously a method for constructing a straight ruler. This method involves an ideal: I mean, an
approximation procedure with unlimited possibility, for it is simply this procedure which is the ideal.
Page 218
Or better: Only if it is a procedure with unlimited possibility, can (not must) the geometry of this procedure
be Euclidean.
Page 218
Euclidean geometry doesn't presuppose any method for measuring angles and lengths, it no more says when
two angles are to be counted as equal, than probability theory says when two cases are to be counted as equally
likely. If a particular method of measurement is now adopted--say one with metal rulers--the question then arises
whether the results of measurements carried out in this way yield Euclidean results.
Page 218
179 Imagine we are throwing a two-sided die, such as a coin. I now want to determine a point of the interval A B by
continually tossing the coin, and always bisecting the side prescribed by the
Page Break 219
throw: say: heads means I bisect the right-hand interval, tails the left-hand one
Page 219
Now, does it describe the position of a point of the interval if I say 'It is the point which is approached
indefinitely by bisection as prescribed by the endless tossing of the coin'?
Page 219
In this way I can approach without limit every point of the interval by continual bisection, and with infinitely
fine eyesight and instruments every stage of the bisection would be determined. (The infinitely fine eyesight doesn't
introduce a vicious circle.)
Page 219
Now, could we call an infinite series of digits which was thus determined an infinite decimal? That is, does
this geometrical process define a number?
Page 219
The geometrical process does not involve a vicious circle, since only an infinite possibility is presupposed by
it, not an infinite reality. (Lines and points being given by the boundaries of coloured surfaces.)
Page 219
To what extent can you say that I have in this way really divided the rationals into two classes? In fact, of
course, this division is never accomplished. But I have a process by means of which I approach such a division
indefinitely? I have an unlimited process, whose results as such don't lead me to the goal, but whose unlimited
possibility is itself the goal. But what does this limitlessness consist in--don't we here have once more merely an
operation and the ad infinitum? Certainly. But the operation is not an arithmetical one.
Page 219
(And the point which I call to my aid in my endless construction can't be given arithmetically at all.)
Page 219
At this point many people would say: it doesn't matter that the
Page Break 220
method was geometrical, it is only the resulting extension itself that is our goal. But, then, do I have this?
Page 220
What is the analogue in arithmetic to the geometrical process of bisection? It must be the converse process:
that of determining a point by a law. (Instead of the law by a point.)
Page 220
It would in fact correspond to the endless process of choosing between 0 and 1 in an infinite binary
expansion ... ad inf. The law here would run 'You must put a 1 or a 0 in succession ad inf., each yields a
law, each a different one.'
Page 220
But that doesn't imply that a law would be given in that I say 'In every case throw either heads or tails.' Of
course, in this way I would necessarily obtain a special case of the general law mentioned, but wouldn't know from
the outset which. No law of succession is described by the instruction to toss a coin.
Page 220
What is arithmetical about the process of tossing the coin isn't the actual result, it is its infinite indefiniteness.
But that simply does not define a number [†1].
Page 220
If I adumbrate a law thus: '0.001001001... ad inf.', what I want to show is not the finite series as a specimen of
a section of an infinite series, but rather the kind of regularity to be perceived in it. But in ... I don't perceive
any law--on the contrary, precisely that a law is absent. Save, say, the law that the results of the specific laws are
written with '0' and '1' and no other signs.
Page 220
The rules of combination for 0 and 1 yield the totality of all finite
Page Break 221
fractions. This would be an infinite extension, in which the infinite extension of fractions 0.1, 0.101, 0.10101 etc. ad
inf. would also have to occur and, generally, all the irrational numbers.
Page 221
What is it like if someone so to speak checks the various laws by means of the set of finite combinations?
Page 221
The results of a law run through the finite combinations, and hence the laws are complete as far as their
extensions are concerned, once all the finite combinations have been gone through.
Page 221
Neither may we say: Two laws are identical in the case where they yield the same result at every stage. No,
they are identical if it is of their essence to yield the same result, i.e. if they are identical.
Page 221
180 If an amorphous theory of infinite aggregates is possible, it can describe and represent only what is amorphous
about these aggregates.
Page 221
It would then really have to construe the laws as merely inessential devices for representing an aggregate.
And abstract from this inessential feature and attend only to what is essential. But to what?
Page 221
Is it possible within the law to abstract from the law and see the extension presented as what is essential?
Page 221
This much at least is clear: that there isn't a dualism: the law and the infinite series obeying it; that is to say,
there isn't something in logic like description and reality.
Page 221
Suppose I cut at a place where there is no rational number. Then there must surely be approximations to this
cut. But what does 'closer to' mean here? Closer to what? For the time being I have nothing in the domain of number
which I can approach. But I do in the case of a geometrical stretch. Here it is clear I can come arbitrarily close to any
non-rational cut. And it is also clear that this is a process without an end and I am shown unambiguously how to go
on by the spatial fact.
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Page 222
Once more it is only infinite possibility, but now the law is given in a different way.
Page 222
But can I be in doubt whether all the points of a line can actually be represented by arithmetical rules [†1].
Can I then ever find a point for which I can show that this is not the case? If it is given by means of a construction,
then I can translate this into an arithmetical rule, and if it is given by chance, then there is, no matter how far I
continue the approximation, an arithmetically defined decimal expansion which is concomitant with it.
Page 222
It is clear that a point corresponds to a rule.
Page 222
What is the situation with regard to types [†2] of rules, and does it make sense to talk of all rules, and so, of
all points?
Page 222
In some sense or other there can't be irrational numbers of different types.
Page 222
My feeling here is the following: no matter how the rule is formulated, in every case I still arrive at nothing
else but an endless series of rational numbers. We may also put it like this: no matter how the rule is formulated,
when I translate it into geometrical notation, everything is of the same type.
Page 222
In the case of approximation by repeated bisection, we approach every point via rational numbers. There is
no point which we could only approach with irrational numbers of a specific type.
Page 222
It is of course possible that in determining a maximum I may stumble upon a new rule, but this has nothing
essential to do with determining the maximum; it doesn't refer explicitly to a totality of real numbers.
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XVII
Page 223
181 The question would be: what criterion is there for the irrational numbers being complete?
Page 223
Let us look at an irrational number: it runs through a series of rational approximations. When does it leave
this series behind? Never. But then, the series also never comes to an end.
Page 223
Suppose we had the totality of all irrational numbers with one single exception. How would we feel the lack
of this one? And--if it were to be added--how would it fill the gap?--Suppose that it's π. If an irrational number is
given through the totality of its approximations, then up to any point taken at random there is a series coinciding
with that of π. Admittedly, for each such series there is a point where they diverge. But this point can lie arbitrarily
far 'out'. So that for any series agreeing with π, I can find one agreeing with it still further. And so if I have the totality
of all irrational numbers except π, and now insert π, I cannot cite a point at which π is now really needed. At every
point it has a companion agreeing with it from the beginning on.
Page 223
This shows clearly that an irrational number isn't the extension of an infinite decimal fraction, that it's a law.
Page 223
Somehow it seems to follow from this--and this is something I find very compelling--that infinitely long isn't
a measure of distance.
Page 223
Our answer to the question above must be 'If π were an extension, we would never feel the lack of it', i.e. it
would be impossible for us to observe a gap. If someone were to ask us, 'But have you then an infinite decimal
expansion with m in the r-th place and n in the s-th place, etc.?', we could always oblige him.
Page Break 224
Page 224
Now let's assume we have been given all the irrational numbers that can be represented by laws, but that
there are yet other irrationals, and I am given a cut representing a number not belonging to the first class: How am I
to tell that this is so? This is impossible, since no matter how far I go with my approximations, there will always also
be a corresponding fraction.
Page 224
And so we cannot say that the decimal fractions developed in accordance with a law still need supplementing
by an infinite set of irregular infinite decimal fractions that would be 'brushed under the carpet' if we were to restrict
ourselves to those generated by a law. Where is there such an infinite decimal that is generated by no law? And how
would we notice that it was missing? Where is the gap it is needed to fill?
Page 224
If from the very outset only laws reach to infinity, the question whether the totality of laws exhausts the
totality of infinite decimal fractions can make no sense at all.
Page 224
The usual conception is something like this: it is true that the real numbers have a different multiplicity from
the rationals, but you can still write the two series down alongside one another to begin with, and sooner or later the
series of real numbers leaves the other behind and goes infinitely further on.
Page 224
But my conception is: you can only put finite series alongside one another and in that way compare them;
there's no point in putting dots after these finite stretches (as signs that the series goes on to infinity). Furthermore,
you can compare a law with a law, but not a law with no law.
Page 224
182 [†1]: I'm tempted to say, the individual digits are always only the results, the bark of the fully grown tree.
What
Page Break 225
counts, or what something new can still grow from, is the inside of the trunk, where the tree's vital energy is. Altering
the surface doesn't change the tree at all. To change it, you have to penetrate the trunk which is still living.
Page 225
Thus it's as though the digits were dead excretions of the living essence of the root. Just as when in the
course of its vital processes a snail discharges chalk, so building on to its shell.
Page 225
There must first be the rules for the digits, and then--e.g.--a root is expressed in them. But this expression in a
sequence of digits only has significance through being the expression for a real number. If someone subsequently
alters it, he has only succeeded in distorting the expression, but not in obtaining a new number.
Page 225
The rules for the digits belong at the beginning, as a preparation for the expression.
Page 225
For the construction of the system in which the law lives out its life.
Page 225
183 And so I would say: If ' is anything at all, then it is the same as , only another expression for it; the
expression in another system.
Page 225
In that case we might also put this quite naively as follows: I understand what ' means, but not ' ,
since has no places at all, and I can't substitute others for none.
Page 225
How about 1/7 (5→3)? Of course isn't an infinite extension but once again an infinite rule, with
which an extension
Page Break 226
can be formed. But it is such a rule as can so to speak digest the (5→3).
Page 226
The suffix (1→5) so to speak strikes at the heart of the law 0.1010010001.... The law talks of a 1 and a 5 is to
be substituted for this 1.
Page 226
Could we perhaps put it by saying isn't a measure until it is in a system?
Page 226
It's as if a man were needed before the rule could be carried out. Almost: if it's to be of any concern to
arithmetic, the rule itself must understand itself. The rule doesn't do this, it is made up out of two
heterogeneous parts. The man applying it puts these parts together [†1].
Page 226
Does this mean that the rule lacks something, viz. the connection between the system of the root and
the system of the sequence of digits?
Page 226
You would no more say of that it is a limit towards which the values of a series were tending, than
you would of the instruction to throw dice.
Page 226
How far does have to be expanded for us to have some acquaintance with it? This of course means
nothing. So we are already acquainted with it without its having been expanded at all. But in that case doesn't
mean a thing.
Page Break 227
Page 227
The idea behind is this: we look for a rational number which, multiplied by itself, yields 2. There isn't
one. But there are those which in this way come close to 2 and there are always some which approach 2 more
closely still. There is a procedure permitting me to approach 2 indefinitely closely. This procedure is itself something.
And I call it a real number.
Page 227
It finds expression in the fact that it yields places of a decimal fraction lying ever further to the right.
Page 227
Only what can be foreseen about a sequence of digits is essential to the real number.
Page 227
184 That we can apply the law, holds also for the law to throw the digits like dice.
Page 227
And what distinguishes π' from this can only be its arithmetical definiteness. But doesn't that consist in our
knowing that there must be a law governing the occurrences of the digit 7 in π; even if we don't yet know what this
law is?
Page 227
And so we could also say: π' alludes to a law which is as yet unknown (unlike 1/7').
Page 227
But mightn't we now say: π contains the description of a law 'the law in accordance with which 7 occurs in
the expansion of π'? Or would this allusion only make sense if we knew how to derive this law? (Solution of a
mathematical problem.)
Page 227
In that case I confessedly cannot simply read this law off from the prescription, and so the law in it is
contained in a language I can't read. And so in this sense too I don't understand π'.
Page 227
But then how about the solubility of the problem of finding this law? Isn't it only a problem in so far as we
know the method for solving it?
Page 227
And if it is known, precisely that gives π' its sense, and if
Page Break 228
unknown, we can't speak about the law which we don't yet know, and π' is bereft of all sense. For if no law presents
itself, π' becomes analogous to the instruction to follow the throws of dice.
Page 228
185 A real number lives in the substratum of the operations out of which it is born.
Page 228
We could also say: " " means the method whereby x2 approximates to 2.
Page 228
Only a path approaches a goal, positions do not. And only a law approaches a value.
Page 228
x2 approaching 2, we call x approaching .
Page 228
186 The letter π stands for a law. The sign π' (or: ) means nothing, if there isn't any talk of a 7 in the law for π,
which we could then replace by a 3. Similarly for . (Whereas might mean ).
Page 228
A real number yields extensions, it is not an extension.
Page 228
A real number is: an arithmetical law which endlessly yields the places of a decimal fraction.
Page 228
This law has its position in arithmetical space. Or you might also say: in algebraic space.
Page 228
Whereas π' doesn't use the idioms of arithmetic and so doesn't assign the law a place in this space.
Page 228
It's as though what is lacking is the arithmetical creature which produces these excretions.
Page 228
The impossibility of comparing the sizes of π and π' ties in with this homelessness of π'.
Page 228
You cannot say: two real numbers are identical, if all their places coincide. You cannot say: they are different,
if they disagree in one
Page Break 229
of the places of their expansion. No more can you say the one is greater than the other if the first place at which they
disagree is greater for the first than the second.
Page 229
Suppose someone were to invent a new arithmetical operation, which was normal multiplication, but
modified so that every 7 in the product was replaced by a 3. Then the operation ×' would have something about it we
didn't understand so long as we lacked a law through which we could understand the occurrence of 7 in the product
in general.
Page 229
We would then have here the extraordinary fact that my symbolism would express something I don't
understand. (But that is absurd.)
Page 229
It is clear that were I able to apply ×' all doubts about its legitimacy would be dispelled. For the possibility of
application is the real criterion for arithmetical reality.
Page 229
Even if I wasn't already familiar with the rule for forming , and I took to be the original
prescription, I would still ask: what's the idea of this peculiar ceremony of replacing 7 by 3? Is it perhaps that 7 is
tabu, so that we are forbidden to write it down? For substituting 3 for 7 surely adds absolutely nothing to the law,
and in this system it isn't an arithmetical operation at all.
Page 229
Put geometrically: it's not enough that someone should--supposedly--determine a point ever more closely by
narrowing down its whereabouts; we must be able to construct it.
Page 229
To be sure, continual throwing of a die indefinitely restricts the possible whereabouts of a point, but it doesn't
determine a point.
Page 229
After every throw the point is still infinitely indeterminate.
Page 229
Admittedly, even in the course of extracting a root in the usual way, we constantly have to apply the rules of
multiplication
Page Break 230
appropriate at that point, and their application has also not been anticipated. But neither is there any mention of
them and their application in the principle of .
Page 230
A number must measure. And this doesn't mean merely: values in its expansion must measure. For we can't
talk of all values, and that rational numbers (which I have formed in accordance with some rule) measure goes
without saying.
Page 230
What I mean might be put like this: for a real number, a construction and not merely a process of
approximation must be conceivable.--The construction corresponds to the unity of the law.
Page 230
187 . That we come full circle is what I actually see and express by means of doesn't mean
'nothing but 3s come', but 'a 3 must recur again and again'.
Page 230
Understanding the rule and how to carry it out in practice always only helps us over finite stretches. To
determine a real number it must be completely intelligible in itself. That is to say, it must not be essentially
undecided whether a part of it could be dispensed with.
Page 230
For in that case it simply isn't clearly given, for there is no extension which would be equivalent to it, and in
itself it is indeterminate. π' in that case sets out to seek its fortune in infinite space.
Page 230
Of course, if a and b do not agree for the first time at the fourth place, we can say that therefore they are
unequal. This fourth place clearly does belong to both numbers; but not the indefinite nth place in the infinite
progression.
Page 230
Thus we can certainly tell that π and e are different from the difference in their first place. But we can't say
that they would be equal, if all their places were equal.
Page Break 231
Page 231
If the extensions of two laws coincide as far as we've gone, and I cannot compare the laws as such, the
numbers defined, if I have any right to talk of such numbers, cannot be compared, and the question which is greater
or whether they are equal is nonsense. Indeed, an equation between them must be nonsense. And that gives one
pause for thought. And it's true: we cannot mean anything by equating them, if there is no inner connection between
them; if they belong to different systems. (And the extension is of no use to us.)
Page 231
But then, are what can't be compared with one another really numbers?
Page 231
Doesn't that contradict the simple image of the number line?
Page 231
188 There is no number outside a system.
Page 231
The expansion of π is simultaneously an expression of the nature of π and of the nature of the decimal
system.
Page 231
Arithmetical operations only use the decimal system as a means to an end; that is, the rules for the operations
are of such a kind that they can be translated into the language of any other number system, and do not have any of
them as their subject matter.
Page 231
The expansion of π is admittedly an expression both of the nature of π and of the decimal notation, but our
interest is usually restricted exclusively to what is essential to π, and we don't bother about the latter. That is a
servant which we regard merely as a tool and not as an individual in its own right. But if we now regard it as a
member of society, then that alters society.
Page 231
A general rule of operation gets its generality from the generality of the alteration it effects in the numbers.
That is why won't do as a general rule of operation, since the result of a b doesn't
Page Break 232
depend solely on the nature of the numbers a and b; the decimal system also comes in. Now, that wouldn't matter,
of course, if this system underlay the operation as a further constant , and we can certainly find an
operation that corresponds to ×': which then has not only a and b but also the decimal system as its subject-matter.
This operation will be written in a number system which withdraws itself as a servant into the background, and of
which there is no mention in the operation.
Page 232
In just this way makes the decimal system into its subject matter (or would have to do so, if it were
genuine), and for that reason it is no longer sufficient that we can use the rule to form the extension. For this
application has now ceased to be the criterion for the rule's being in order, since it is not the expression of the
arithmetical law at all, but only makes a superficial alteration to the language.
Page 232
And so, if the decimal system is to stop being a servant, it must join the others at table, observing all the
required forms, and leave off serving, since it can't do both at once.
Page 232
This is how it is: the number π is expressed in the decimal system. You can't achieve a modification of this
law by fixing on the specific expression in the decimal system. What you thereby influence isn't the law, it's its
accidental expression. The influence does not penetrate as far as the law at all. Indeed it stands separated from it on
the other side. It's like trying to influence a creature by working on a secretion that has already been discharged.
Page 232
189 What about a law , (where p runs through the series of primes)? Or where p runs through the
series of whole numbers except for those for which the equation of Fermat's Last Theorem doesn't hold.--Do these
laws define real numbers?
Page 232
I say: the so-called 'Fermat's Last Theorem' isn't a proposition [†1]. (Not even in the sense of a proposition of
arithmetic.) Rather, it
Page Break 233
corresponds to a proof by induction. Now, if there is a number F = 0.110000 etc., and that proof succeeds, that
would surely prove that F = 0.11--and isn't that a proposition?! Or: it is a proposition, if the law F is a number.
Page 233
A proof proves what it proves and no more.
Page 233
The number F wants to use the spiral and choose sections of this spiral according to a
principle. But this principle doesn't belong to the spiral.
Page 233
If I imagine the windings of the spiral
1/100, 1/100 + 1/101, 1/100 + 1/101 + 1/102, etc.
to have been written out, then the number F makes a comment on each winding, it puts a tick or a cross against each
one; and, what's more, making its choices according to a law we don't know.
Page 233
And this is how we arrive at the paradox that it's nonsense to ask whether F = 0.11. For, accepting F depends
upon accepting the assumption of a law, an infinite law, which governs the behaviour of the numbers in the Fermat
formula.--But what indicates to us the infinity of the law? Only the induction. And where is that to be found here? In
the infinite possibility of the exponent n in xn + yn = zn, and so in the infinite possibility of making tests. But that
doesn't have any different value for us from that of the infinite possibility of throwing dice, since we don't know a
law to which the results of such tests would conform.
Page 233
There is admittedly a law there (and so also an arithmetical interest), but it doesn't refer directly to the
number. The number is a sort of lawless by-product of the law. As though someone went along the street at a
regulation pace, throwing a die at every pace, and constantly either putting a peg in the ground or not, as directed by
the fall of the die; in this case these pegs wouldn't be spaced out in accordance with a law.
Page Break 234
Page 234
Or rather, the law spacing them would only be that governing the strides and no other.
Page 234
Does it then make no sense to say, even after Fermat's Last Theorem has been proved, that F = 0.110? (If,
say, I were to read about it in the papers.)
Page 234
The true nature of real numbers must be the induction. What I must look at in the real number, its sign, is the
induction.--The 'So' of which we may say 'and so on'. If the law, the winding of the spiral, is a number, then it must
be comparable with all the others through its position (on the number scale).
Page 234
I certainly do not define that position by means of anything but the law.
Page 234
Only what I see, is a law; not what I describe.
Page 234
I believe that is the only thing standing in the way of my expressing more by my signs than I can understand.
Page Break 235
XVIII
Page 235
190 In this context we keep coming up against something that could be called an 'arithmetical experiment'.
Admittedly the data determine the result, but I can't see in what way they determine it. (cf. e.g. the occurrences of 7
in π.) The primes likewise come out from the method for looking for them, as the results of an experiment. To be
sure, I can convince myself that 7 is a prime, but I can't see the connection between it and the condition it satisfies.--I
have only found the number, not generated it.
Page 235
I look for it, but I don't generate it. I can certainly see a law in the rule which tells me how to find the primes,
but not in the numbers that result. And so it is unlike the case + 1/1!, - 1/3!, + 1/5!, etc., where I can see a law in the
numbers.
Page 235
I must be able to write down a part of the series, in such a way that you can recognize the law.
Page 235
That is to say, no description is to occur in what is written down, everything must be represented.
Page 235
The approximations must themselves form what is manifestly a series.
Page 235
That is, the approximations themselves must obey a law.
Page 235
Can we say that unless I knew the geometrical representation of π and , I would only have an
approximate knowledge of these numbers? Surely not.
Page 235
191 A number must measure in and of itself.
Page 235
It seems to me as though that's its job.
Page 235
If it doesn't do that but leaves it to the rationals, we have no need of it.
Page Break 236
Page 236
It seems to be a good rule that what I will call a number is that which can be compared with any rational
number taken at random. That is to say, that for which it can be established whether it is greater than, less than, or
equal to a rational number.
Page 236
That is to say, it makes sense to call a structure a number by analogy, if it is related to the rationals in ways
which are analogous to (of the same multiplicity as) greater, less and equal to.
Page 236
A real number is what can be compared with the rationals.
Page 236
When I say I call irrational numbers only what can be compared with the rationals, I am not seeking to place
too much weight on the mere stipulation of a name. I want to say that this is precisely what has been meant or
looked for under the name 'irrational number'.
Page 236
Indeed, the way the irrationals are introduced in text books always makes it sound as if what is being said is:
Look, that isn't a rational number, but still there is a number there. But why then do we still call what is there 'a
number'? And the answer must be: because there is a definite way for comparing it with the rational numbers.
Page 236
'The process would only define a number when it has come to an end, but since it goes on to infinity and is
never complete, it doesn't define a number.'
Page 236
The process must have infinite foresight, or else it won't define a number. There must be no 'I don't as yet
know', since there's no 'as yet' in the infinite.
Page 236
Every rational number must stand in a visible relation to the law which is a number.
Page 236
The true expansion is just the method of comparison with the rationals.
Page Break 237
Page 237
The true expansion of a number is the one which permits a direct comparison with the rationals.
Page 237
If we bring a rational number into the neighbourhood of the law, the law must give a definite reaction to it.
Page 237
It must reply to the question 'Is it this one?'
Page 237
I should like to say: The true expansion is that which evokes from the law a comparison with a rational
number.
Page 237
Narrowing down the interval of course contributes to the comparison through the fact that every number
thereby comes to lie to the left or right of it.
Page 237
This only holds when comparing the law with a given rational number forces the law to declare itself in
comparison with this number.
Page 237
192 A real number can be compared with the fiction of an infinite spiral, whereas structures like F, P[†1], or π' only
with finite sections of a spiral.
Page 237
For my inability to establish on which side it passes by a point, simply means that it is absurd to compare it
with a complete (whole) spiral, for with that I would see how it goes past the point.
Page 237
You see, at the back of our minds here, we always have the idea that while I don't know the spiral as a whole,
and so don't know what its path is at this point, what I don't know is still in fact thus or so.
Page 237
193 If I say ( )2 approaches 2 and so eventually reaches the numbers 1.9, 1.99, 1.999, that is nonsense unless I
can state within how many stages these values will be reached, for 'eventually' means nothing.
Page 237
To compare rational numbers with , I have to square them.--They then assume the form , where
is now an arithmetical operation.
Page 237
Written out in this system, they can be compared with , and it is for me as if the spiral of the irrational
number had shrunk to a point.
Page Break 238
Page 238
We don't understand why there is a 4 at the third decimal place of , but then we don't need to
understand it.--For this lack of understanding is swallowed up in the wider (consistent) use of the decimal system.
Page 238
In fact, in the end the decimal system as a whole withdraws into the background, and then only what is
essential to remains in the calculation.
Page 238
194 Is an arithmetical experiment still possible when a recursive definition has been set up? I believe, obviously not;
because via the recursion each stage becomes arithmetically comprehensible.
Page 238
And in recursion we do not start from another generality, but from a particular arithmetical case.
Page 238
A recursive definition conveys understanding by building on one particular case that presupposes no
generality.
Page 238
To be sure, I can give a recursive explanation of the rule for investigating numbers in the cases)([†1], F, and
P, but not of the outcome of such investigation. I cannot construct the result.
Page 238
Let →4 mean: 'The fourth prime number'. Can → 4 be construed as an arithmetical operation applied to the
basis 4? So that →4 = 5 is an equation of arithmetic, like 42 = 16?
Page 238
Or is it that →4 can 'only be sought, but not constructed'?
Page 238
195 Is it possible to prove a greater than b, without being able to prove at which place the difference will come to
light? I think not.
Page Break 239
Page 239
How many noughts can occur in succession in e? If the nth decimal place is followed by a 0, and if it is fixed
after n + r stages of the summation [†1] then the 0 must be fixed at the same time as the nth place, for another
number can only give way to an 0, if the place preceding it also still changes. Thus the number of noughts is limited.
Page 239
We can and must show the decimal places to be fixed after a definite number of stages.
Page 239
If I don't know how many 9s may follow after 3.1415, it follows that I can't specify an interval smaller than
the difference between π and 3.1416; and that implies, in my opinion, that π doesn't correspond to a point on the
number line, since, if it does correspond to a point, it must be possible to cite an interval which is smaller than the
interval from this point to 3.1416.
Page 239
If the rational number with which I want to compare my real number is given in decimal notation, then if I
am to carry out the comparison I must be given a relation between the law of the real number and the decimal
notation.
Page 239
1.4--Is that the square root of 2? No, it's the root of 1.96. That is, I can immediately write it in the form of an
approximation to ; and of course, see whether it is an approximation above or below [†2].
Page 239
What is an approximation? (Surely all rational numbers lie either above or below the irrational number.) An
approximation is a rational number written in such a form that it can be compared with the irrational number.
Page 239
In that case, decimal expansion is a method of comparison with
Page Break 240
the rationals, if it is determined in advance how many places I must expand to in order to settle the issue.
Page 240
196 A number as the result of an arithmetical experiment, and so the experiment as the description of a number, is
an absurdity.
Page 240
The experiment would be the description, not the representation of a number.
Page 240
I cannot compare F with 11/100, and so it isn't a number.
Page 240
If the real number is a rational number a, comparison of its law with a must show this. That means the law
must be so formed as to 'click into' the rational number when it comes to the appropriate place (this number).
Page 240
It wouldn't do, e.g., if we couldn't be sure whether really breaks off at 5.
Page 240
We might also put it like this: the law must be such that any rational number can be inserted and tried out.
Page 240
But in that case what about a number like P = 0.11101010001 etc.? Suppose someone claimed it was a
recurring decimal, and also at some stage or other it looked as if it were, then I would have to be able to try out the
supposed number in the law, just as I can see directly by multiplication whether 1.414 is . But that isn't possible.
Page 240
The mark of an arithmetical experiment is that there is something opaque about it.
Page Break 241
Page 241
197 A subsequent proof of convergence cannot justify construing a series as a number.
Page 241
Where a convergence reveals itself, that is where the number must be sought.
Page 241
A proof showing something has the properties necessary to a number, must indicate the number. That is, the
proof is simply what points the number out.
Page 241
Isn't F also an unlimited contraction of an interval? [†1]
Page 241
How can I know that or whether the spiral will not focus on this point? In the case of I do know.
Page 241
Now, can I also call such a spiral a number? A spiral which can, for all I know, come to a stop at a rational
point.
Page 241
But that isn't possible either: there is a lack of a method for comparing with the rationals.
Page 241
For developing the extension isn't such a method, since I can never know if or when it will lead to a decision.
Page 241
Expanding indefinitely isn't a method, even when it leads to a
Page Break 242
result of the comparison.
Page 242
Whereas squaring a and seeing whether the result is greater or less than 2 is a method.
Page 242
Could we say: what is a real number is the general method of comparison with the rationals?
Page 242
It must make sense to ask: 'Can this number be π?'
Page 242
198 F is not the interval 0-0.1; for I can also make a certain decision within this interval, but it isn't a number inside
this interval, since we can't force the issues that would be necessary for that.
Page 242
Might we then say: F is certainly an arithmetical structure, only not a number (nor an interval)?
Page 242
That is, I can compare F neither with a point nor with an interval. Is there a geometrical structure to which
that corresponds?
Page 242
The law, i.e. the method of comparison, only says that it will yield either the answers 'greater, less or
equal'--or--'greater' (but not equal). As though I go into a dark room and say: I can only find out whether it is lower
than I am or the same height--or--higher. And here we might say: and so you can't find out a height; and so what is it
you can find out? The comparison only goes lame because I can surely establish the height if I bump my head,
whereas, in the case of F, I cannot, in principle, ask: 'Is it this point?'
Page 242
I don't know a method for determining whether it is this point, and so it isn't a point.
Page 242
If the question how F compares with a rational number has no sense, since all expansion still hasn't given us
an answer, then this question also had no sense before we tried to settle the question at random by means of an
extension.
Page 242
If it makes no sense now to ask 'Does F = 0.11?', then it had no
Page Break 243
sense even before people examined 100 places of the extension, and so even before they had examined only one.
Page 243
But then it would make no sense at all to ask in this case whether the number is equal to any rational number
whatsoever. As long, that is, as we don't possess a method which necessarily settles the question.
Page 243
This is as much as I know to date about the 'number'.
Page 243
The rational number given is either equal to, less than or greater than the interval that has so far been worked
out. In the first case the point forms the lower end point of the interval, in the second it lies below, in the third, above
the interval. In none of these cases are we talking about comparing the positions of two points.
Page 243
199 is not a result of 1/3 in the same sense as, say, 0.25 is of 1/4; it points to a different arithmetical fact.
Page 243
Suppose the division continually yielded the same digit, 3, but without our seeing in it the necessity for this,
would it then make sense to form the conjecture that the result will be ?
Page 243
That is, doesn't simply designate an induction we have seen and not--an extension?
Page 243
We must always be able to determine the order of magnitude. Suppose (in my notation) nothing speaks
against 100 threes following in succession in e at a particular point, then something
Page Break 244
speaks against 10100 threes doing so. (A great deal must be left open in the decimal system, that is definite in the
binary system.)
Page 244
It isn't only necessary to be able to say whether a given rational number is the real number: we must also be
able to say how close it can possibly come to it. That is, it isn't enough to be able to say that the spiral doesn't go
through this point but beneath it: we must also know limits within which the distance from the point lies. We must
know an order of magnitude for the distance apart.
Page 244
Decimal expansion doesn't give me this, since I cannot know, e.g. how many 9s will follow a place that has
been reached in the expansion.
Page 244
The question 'Is e ?' is meaningless, since it doesn't ask about an extension, but about a law, i.e. an
induction of which we have, however, no conception here. We can put this question in the case of division only
because we know the form of the induction we call .
Page 244
The question 'Are the decimal places of e eventually fixed?' and the answer 'They are eventually fixed', are
both nonsense. The question runs: After how many stages do the places have to remain fixed?
Page 244
May we say: 'e isn't this number' means nothing--we have to say 'It is at least this interval away from it'?
Page 244
I believe that's how it is. But that implies it couldn't be answered at all unless we were simultaneously given a
concept of the distance apart.
Page Break 245
Page 245
From F. Waismann's shorthand notes of a conversation on 30 December 1930
A proof for all real numbers is not analogous to a proof for all rational numbers in such a way that you could
say: What we have proved for all rational numbers--i.e. by induction--we can prove for all real numbers in the same
way by an extension of the method of proof. You can only speak about a real number when you've got it.
Page 245
Let us assume e.g., I have proved the formula am • an = am + n for rational values of m and n (by induction),
and would now like to prove it for real number values. What do I do in this case? Obviously it's no longer possible to
carry through the proof by induction.
Page 245
Therefore such a formula doesn't mean: for all real numbers such and such holds, but: given a real number, I
interpret the formula so as to mean: such and such holds for the limit and I prove this by the rules of calculation laid
down for the real numbers. I prove the formula for rational numbers by induction, and then show it carries over to
the real numbers, in fact simply by means of the rules of calculation I have laid down for the real numbers. But I
don't prove the formula holds for 'all real' numbers, precisely because the rules of calculation for real numbers do not
have the form of an induction.
Page 245
This, then, is how things are: I treat as given a particular real number and keep this constant throughout the
proof. That's something totally different from what happens with the rationals; for there it was precisely a question
of whether the formula remained correct when the rational numbers were varied, and that is why the proof had the
form of an induction. But here the question simply doesn't arise whether the formula holds for 'all real'
numbers--just because we don't allow the real number to vary at all.
Page 245
We don't let the variable real number run through all values--
Page Break 246
i.e. all laws. We simply rely on the rules of calculation and nothing else.
Page 246
You might now think: strictly the proposition ought to be proved for all real numbers, and what we give is no
more than a pointer. That's wrong. A formula that is proved for real numbers doesn't say: For all real numbers...
holds; what it says is: Given a real number, then... holds. And in fact not on the basis of a proof, but of an
interpretation.
Page Break 247
XIX
Page 247
200 Our interest in negation in arithmetic appears to be limited in a peculiar way. In fact it appears to me as if a
certain generality is needed to make the negation interesting.
Page 247
But you can represent indivisibility perspicuously (e.g. in Eratothenes' sieve). You can see how all the
composite numbers lie above or below the number under consideration.
Page 247
Here, arithmetical negation is represented by spatial negation, the 'somewhere else'.
Page 247
What do I know, if I know a mathematical inequality? Is it possible to know just an inequality, without
knowing something positive?
Page 247
It seems clear that negation means something different in arithmetic from what it means in the rest of
language. If I say 7 is not divisible by 3, then I can't even make a picture of this, I can't imagine how it would be if 7
were divisible by 3. All this follows naturally from the fact that mathematical equations aren't a kind of proposition.
Page 247
It is very odd that for a presentation of mathematics we should be obliged to use false equations as well. For
that is what all this is about. If negation or disjunction is necessary in arithmetic, then false equations are an essential
element in its presentation.
Page Break 248
Page 248
What does '~(5 × 5 = 30)' mean? It seems to me as if you oughtn't to write it like that, but '5 × 5 ≠ 30'; the
reason being that I am not negating anything, but want to establish a relation, even if an indefinite one, between 5 ×
5 and 30 (and so, something positive).--Admittedly you could say: 'True, but this relation is at any rate incompatible
with 5 × 5 = 30'.--And so is the relation of indivisibility with the relation of divisibility! It's quite clear that when I
exclude divisibility, in this logical system this is equivalent to establishing indivisibility.--And isn't this the same case
as that of a number which is less than 5, if it is not greater than or equal to it?
Page 248
201 Now, there is something recalcitrant to the application of the law of the excluded middle in mathematics.
Page 248
(Of course even the name of this law is misleading: it always sounds as though this were the same sort of
case as 'A frog is either green or brown, there isn't a third type'.)
Page 248
You can show by induction that when you successively subtract 3 from a number until it won't go any
further, the remainder can only be either 0 or 1 or 2. You call the cases of the first class those in which the division
goes out.
Page 248
Looking for a law for the distribution of primes is simply an attempt to replace the negative criterion for a
prime number by a positive one. Or more correctly, the indefinite one by a definite one.
Page 248
I believe that negation here is not what it is in logic but an indefiniteness. For how do I recognize--verify--a
negative? By something indefinite or positive.
Page 248
An inequality, like an equation, must be either the result of a calculation or a stipulation.
Page Break 249
Page 249
Just as equations can be construed, not as propositions, but as rules for signs, so it must be possible to treat
inequalities in the same way.
Page 249
How, then, can you use an inequality? That leads to the thought that in logic there is also the internal relation
of not following, and it can be important to recognize that one proposition does not follow from another.
Page 249
The denial of an equation is as like and as unlike the denial of a proposition as the affirmation of an equation
is like and unlike the affirmation of a proposition.
Page 249
202 It is quite clear that negation in arithmetic is completely different from the genuine negation of a proposition.
Page 249
And it is of course clear that where it essentially--on logical grounds--corresponds to a disjunction or to the
exclusion of one part of a logical series in favour of another, it must have a completely different meaning.
Page 249
It must in fact be one and the same as those logical forms and therefore be only apparently a negation.
Page 249
If 'not equal to' means greater or less than, then that cannot be, as it were, an accident which befalls the 'not'.
Page 249
A mathematical proposition can only be either a stipulation, or a result worked out from stipulations in
accordance with a definite method. And this must hold for '9 is divisible by 3' or '9 is not divisible by 3'.
Page 249
How do you work out 2 × 2 ≠ 5? Differently from 2 × 2 = 4? If at all, then by 2 × 2 = 4 and 4 ≠ 5.
Page 249
And how do you work out '9 is divisible by 3'? You could treat it as a disjunction and first work out 9 ÷ 3 = 3,
and then, instead of this definite proposition, use a rule of inference to derive the disjunction.
Page Break 250
Page 250
Aren't we helped here by the remark that negation in arithmetic is important only in the context of
generality? But the generality is expressed by an induction.
Page 250
And that is what makes it possible for negation or disjunction, which appear as superfluously indefinite in
the particular case, yet to be essential to arithmetic in the general 'proposition', i.e. in the induction.
Page 250
203 It is clear to me that arithmetic doesn't need false equations for its construction, but it seems to me that you may
well say 'There is a prime number between 11 and 17', without ipso facto referring to false equations.
Page 250
Isn't an inequality a perfectly intelligible rule for signs, just as an equation is? The one permits a substitution,
the other forbids a substitution.
Page 250
Perhaps all that's essential is that you should see that what is expressed by inequalities is essentially different
from what is expressed by equations. And so you certainly can't immediately compare a law yielding places of a
decimal expansion which works with inequalities, with one that works with equations. Here we have before us
completely different methods and consequently different kinds of arithmetical structure.
Page 250
In other words, in arithmetic you cannot just put equations and something else (such as inequalities) on one
level, as though they were different species of animals. On the contrary, the two methods will be categorically
different, and determine (define) structures not comparable with one another.
Page 250
Negation in arithmetic cannot be the same as the negation of a proposition, since otherwise, in 2 × 2 ≠ 5, I
should have to make myself a picture of how it would be for 2 × 2 to be 5.
Page Break 251
Page 251
204 You could call '= 5', 'divisible by 5', 'not divisible by 5', 'prime', arithmetical predicates and say: arithmetical
predicates always correspond to the application of a definite, generally defined, method. You might also define a
predicate in this way: (ξ × 3 = 25) F(ξ.
Page 251
Arithmetical predicates, which in the particular case are unimportant because the definite form makes the
indefinite superfluous--become significant in the general law, i.e. in induction. Since here they are not--so to
speak--superseded by a definite form. Or better: In the general law, they are in no way indefinite.
Page 251
Could the results of an engineer's computations be such that, let us say, it is essential for certain machine
parts to have lengths corresponding to the prime number series? No.
Page 251
Can you use the prime numbers to construct an irrational number? The answer is always: as far as you can
foresee the primes and no further.
Page 251
If you can foresee that a prime number must occur in this interval then this interval is what can be foreseen
and constructed, and so, I believe, it can play a role in the construction of an irrational number.
Page Break 252
XX
Page 252
205 Can we say a patch is simpler than a larger one?
Page 252
Let's suppose they are uniformly coloured circles: what is supposed to constitute the greater simplicity of the
smaller circle?
Page 252
Someone might reply that the larger one can be made up out of the smaller one and a further part, but not
vice versa. But why shouldn't I represent the smaller one as the difference between the larger one and the ring?
Page 252
And so it seems to me that the smaller patch is not simpler than the larger one.
Page 252
It seems as if it is impossible to see a uniformly coloured patch as composite, unless you imagine it as not
uniformly coloured. The image of a dissecting line gives the patch more than one colour, since the dissecting line
must have a different colour from the rest of the patch.
Page 252
May we say: If we see a figure in our visual field--a red triangle say--we cannot then describe it by e.g.
describing one half of the triangle in one proposition, and the other half in another. That is, we may say that there is
a sense in which there is no such thing as a half of this triangle. We can only speak of the triangle at all if its
boundary lines are the boundaries between two colours.
Page 252
This is how it is with the composition of spatial structures out of their smaller spatial components: the larger
geometrical structure isn't composed of smaller geometrical structures, any more than you can say that 5 is
composed of 3 and 2, or for that matter 2 of 5 and--3. For here the larger determines the smaller quite as much as the
smaller the larger. The rectangle isn't composed of the rectangles ; instead the first geometrical
figure determines the other two and conversely. Here, then, Nicod would be right
Page Break 253
when he says [†1] that the larger figure doesn't contain the smaller ones as components. But it is different in actual
space: the figure is actually composed of the components , even though the
purely geometrical figure of the large rectangle is not composed of the figures of the two squares.
Page 253
These 'purely geometrical figures' are of course only logical possibilities.--Now, you can in fact see an actual
chess board as a unity not as composed of its squares--by seeing it as a large rectangle and disregarding its
squares.--But if you don't disregard the squares, then it is a complex and the squares are its component parts--they
are, in Nicod's phrase, what constitute it.
Page 253
(Incidentally, I am unable to understand what is supposed to be meant by saying that something is
'determined' by certain objects but not 'constituted' by them. If these two expressions make sense at all, it's the same
sense.)
Page 253
An intellect which takes in the component parts and their relations, but not the whole, is a nonsense notion.
Page 253
206 Whether it makes sense to say 'This part of a red patch (which isn't demarcated by any visible boundary) is red'
depends on whether there is absolute position. For if we can speak of an absolute location in visual space, I can then
also ascribe a colour to this absolute location even if its surroundings are the same colour.
Page 253
I see, say, a uniformly yellow visual field and say: 'The centre of my visual field is yellow.' But then can I
describe a shape in this way?
Page 253
An apparent remedy would appear to be to say that red and circular are (external) properties of two objects,
which one might call patches and that in addition these patches are spatially related to each other in a certain way;
but this is nonsense.
Page 253
It's obviously possible to establish the identity of a position in the visual field, since we would otherwise be
unable to distinguish
Page Break 254
whether a patch always stays in the same place or whether it changes its place. Let's imagine a patch which vanishes
and then reappears, we can surely say whether it reappears in the same place or at another.
Page 254
So we can really speak of certain positions in the visual field, and in fact with the same justification as in
speaking of different positions on the retina.
Page 254
Would it be appropriate to compare such a space with a surface that has a different curvature at each of its
points, so that each point is marked out as distinct?
Page 254
If every point in visual space is marked out as distinct, then there is certainly a sense in speaking of here and
there in visual space, and that now seems to me to simplify the presentation of visual states of affairs. But is this
property of having points marked out as distinct really essential to visual space; I mean, couldn't we imagine a visual
space in which we would only perceive certain spatial relations but no absolute position? That is, could we picture an
experience so? In something like the sense in which we can imagine the experiences of a one-eyed man?--I don't
believe we could. For instance, one wouldn't be able to perceive the whole visual field turning, or rather this would
be inconceivable. How would the hand of a clock look, say, when it moved around the edge of the dial? (I am
imagining the sort of dial you find on many large clocks, that only has points on it, and not digits.) We would then
indeed be able to perceive the movement from one point to another--if it didn't just jump from one position to the
other--but once the hand had arrived at a point, we wouldn't be able to distinguish its position from the one it was in
at the last point. I believe it speaks for itself that we can't visualise this.
Page 254
In visual space there is absolute position and hence also absolute motion. Think of the image of two stars in a
pitch-black night, in which I can see nothing but these stars and they orbit around one another.
Page Break 255
Page 255
We can also say visual space is an oriented space, a space in which there is an above and below and a right
and left.
Page 255
And this above and below, right and left have nothing to do with gravity or right and left hands. It would,
e.g., still retain its sense even if we spent our whole lives gazing at the stars through a telescope.
Page 255
Suppose we are looking at the night sky through a telescope, then our visual field would be completely dark
with a brighter circle and there would be points of light in this circle. Let us suppose further that we have never seen
our bodies, always only this image, so that we couldn't compare the position of a star with that of our head or our
feet. What would then show me that my space has an above and below etc., or simply that it is oriented? I can at any
rate perceive that the whole constellation turns in the bright circle and that implies I can perceive different
orientations of the constellation. If I hold a book the wrong way, I can't read the print at all, or only with difficulty.
Page 255
It's no sort of explanation of this situation to say: it's just that the retina has an above and below etc., and this
makes it easy to understand that there should exist the analogue in the visual field. Rather, that is just a
representation of the situation by the roundabout route of the relations on the retina.
Page 255
We might also say, the situation in our visual field is always such as would arise if we could see, along with
everything else, a coordinate frame of reference, in accordance with which we can establish any direction.--But even
that isn't an accurate representation, since if we really saw such a set of axes of co-ordinates (say, with arrows), we
would in fact be in a position, not only to establish the orientations of objects relative to these axes, but also the
position of the cross itself in the space, as though in relation to an unseen coordinate system contained in the
essence of this space.
Page 255
What would our visual field have to be like, if this were not so? Then of course I could see relative positions
and motions, but not
Page Break 256
absolute ones. But that would mean, e.g., that there would be no sense in speaking of a rotation of the whole visual
field. Thus far it is perhaps comprehensible. But now let's assume that, say, we saw with our telescope only one star
at a certain distance from the black edge: that this star vanishes and reappears at the same distance from the edge. In
that case we couldn't know whether it reappears at the same place or in another. Or if two stars were to come and go
at the same distance from the edge, we couldn't say whether--or that--they were the same or different stars.
Page 256
Not only: 'we couldn't know whether', but: there would be no sense in speaking in this context of the same or
different places. And since in reality it has a sense, this isn't the structure of our visual field. The genuine criterion for
the structure is precisely which propositions make sense for it--not, which are true. To look for these is the method
of philosophy.
Page 256
We might also put it like this: let's suppose that a set of coordinate axes once flared up in our visual field for a
few moments and disappeared again, provided our memory were good enough, we could then establish the
orientation of every subsequent image by reference to our memory of the axes. If there were no absolute direction,
this would be logically impossible.
Page 256
But that means we have the possibility of describing a possible location and so a position--in the visual field,
without referring to anything that happens to be there at the time. Thus we can for instance say that something can
be at the top on the right, etc.
Page 256
(The analogy with a curved surface would be to say something like: a patch on an egg can be located near the
broad end.)
Page 256
I can obviously see the sign V at one time as a v, at another as an A, as a 'greater than' or 'less than' sign, even
if I were to see it
Page Break 257
through a telescope and cannot compare its position with the position of my body.
Page 257
Perhaps someone might reply that I feel the position of my body without seeing it. But position in feeling
space (as I'd like to put it for once) has nothing to do with position in visual space, the two are independent of each
other, and unless there were absolute direction in visual space, you couldn't correlate direction in feeling space with
it at all.
Page 257
207 Now, can I say something like: The top half of my visual field is red? And what does that mean? Can I say that
an object (the top half) has the property of being red?
Page 257
In this connection, it should be remembered that every part of visual space must have a colour, and that
every colour must occupy a part of visual space. The forms colour and visual space permeate one another.
Page 257
It is clear that there isn't a relation of 'being situated' which would hold between a colour and a position, in
which it 'was situated'. There is no intermediary between colour and space.
Page 257
Colour and space saturate one another.
Page 257
And the way in which they permeate one another makes up the visual field.
Page 257
208 It seems to me that the concept of distance is given immediately in the structure of visual space. Were it not so,
and the concept of distance only associated with visual space by means of a correlation between a visual space
without distance and another structure that contained distance, then the case would be conceivable in which,
through an alteration in this association, the length a, for example, will appear greater than the length b, even though
we still observe the point B as always between A and C.
Page Break 258
Page 258
How about when we measure an object in visual space with a yardstick for a time? Is it also measured when
the yardstick isn't there?
Page 258
Yes, if any sense can be made at all of establishing the identity between what was measured and what is not.
Page 258
If I can say: 'I have measured this length and it was three times as long as that', then it makes sense and it is
correct to say that the lengths are still in the same relationship to one another now.
Page 258
'CC between AA' follows from 'CC between BB', but only if the partition is really given, through colour
boundaries.
Page 258
It is obviously possible for the intervals a and b to appear to me to be the same in length and for the
segments c and the segments d also to appear to me to be the same in length but for there still to be 25 cs and 24 ds
when I count them. And the question arises: how can that be possible? Is it correct to say here: but it is so, and all
we see is that visual space does not obey the rules of--say--Euclidean space? This would imply that the question
'How can that be possible?' was nonsense and so unjustified. And so there wouldn't be anything paradoxical in this
at all, we would simply have to accept it. But is it conceivable that a should appear equal to b and the cs to the ds,
and a visibly different number of cs and ds be present?
Page 258
Or should I now say that even in visual space something can after all appear different from what it is?
Certainly not! Or that n times an interval and n + 1 times the same interval can yield
Page Break 259
precisely the same result in visual space? That is just as unacceptable. Except if there is no sense at all in saying of
intervals in visual space that they are equal. If, that is, in visual space it only made sense to talk of a 'seeming' and
this expression weren't only appropriate for the relationship between two independent experiences. And so if there
were an absolute seeming.
Page 259
And so perhaps also an absolute vagueness or an absolute unclarity. (Whereas on my view, something can
only be vague or unclear with reference to something we have posited as the standard of clarity: therefore relatively.)
Page 259
Can't I then--in the first case--if I can't take in the number at a glance, make a mistake in determining this
number? Or: are a and b composed of a number of parts at all--in the ordinary sense--if I can't see this number in a
and b? For it certainly seems I've no right at all to conclude that the same number of cs and of ds must be present.
And this still holds, even if counting really does yield the same number! I mean: even if counting were never to yield
different results where a and b are equal etc.
Page 259
(This shows, incidentally, how difficult it is to describe what it is that we really see.)
Page 259
But suppose we have the right to talk of a number of parts--N.B. remaining throughout at the purely visual
level--even when we don't see the number immediately; then the question would arise: can I in that case be sure that
what I count is really the number I see or rather whose visual result I see? Could I be sure that the number of parts
doesn't instantaneously change from 24 to 25 without my noticing it?
Page 259
If I see a = b and c = d and someone else counts the parts and finds the numbers equal, I shall at least feel
that that doesn't contradict what I see. But I am also aware that I can see the same when there are 25 cs in a and 24
ds in b. From this I can conclude that I don't notice when there's one part more or one less, and
Page Break 260
therefore also cannot notice if the number of parts in b changes between 24 and 25.
Page 260
209 But if you can't say that there is a definite number of parts in a and b, how am I in that case to describe the
visual image? Here it emerges--I believe--that the visual image is much more complicated than it seems to be at first
glance. What makes it so much more complicated is e.g. the factor introduced by eye movements.
Page 260
If--say--I were to describe what is seen at a glance by painting a picture instead of using the language of
words, then I ought not really to paint all the parts c and d. In many places I should have to paint something 'blurred'
instead, i.e. a grey section.
Page 260
'Blurred' [†1] and 'unclear' are relative expressions. If this often doesn't seem to be so, that results from the
fact that we still know too little of the real nature of the given phenomena, that we imagine them as more primitive
than they are. Thus it is, e.g., possible that no coloured picture of any kind whatever is able to represent the
impression of 'blurredness' correctly. But it doesn't follow from that that the visual image is blurred in itself and so
can't be represented by any kind of definite picture whatever. No, this would only indicate that a factor plays a part
in the visual image--say through eye movements--which admittedly can't be reproduced by a painted picture, but
which is in itself as 'definite' as any other. You might in that case say, what is really given is still always indefinite or
blurred relative to the painted picture, but merely because we have in that case made the painted picture arbitrarily
into a standard for the given, when this has a greater multiplicity than that of a painted representation.
Page 260
It[[sic]] we were really to see 24 and 25 parts in a and b, we couldn't then see a and b as equal.
Page 260
If this is wrong, the following must be possible: it ought to be possible to distinguish immediately between
the cases where a
Page Break 261
and b both equal 24 and the case where a is 24 and b 25, but it would only be possible to distinguish between the
number of parts, and not between the lengths of a and b that result.
Page 261
We might also put this more simply thus: it would then have to be possible to see immediately that one
interval is made up of 24 parts, the other of 25, without it being possible to distinguish the resulting lengths.--I
believe that the word 'equal' has a meaning even for visual space which stamps this as a contradiction.
Page 261
Do I tell that two intervals of visual space are equal by not telling that they are unequal? This is a question
with far-reaching implications.
Page 261
Couldn't I have two impressions in succession: in one, an interval visibly divided into 5 parts, in the other one
visibly divided in the same way into 6 parts, while I yet couldn't say that I had seen the parts or the entire intervals as
having different lengths?
Page 261
Asked 'Were the intervals different in length or the same?' I couldn't answer 'I saw them as different in
length', since no difference in length has so to speak 'struck' me. And yet I couldn't --I believe--say that I saw them as
equal in length. On the other hand, I couldn't say either: 'I don't know whether they were the same or different in
length' (unless my memory had deserted me), for that means nothing so long as I go on talking only of the
immediately given.
Page 261
210 The question is, how to explain certain contradictions that arise when we apply the methods of inference used in
Euclidean space to visual space.
Page 261
I mean: it is possible to carry through a construction (i.e. a chain of inference) in visual space in which we
appreciate every step (transition), but whose result contradicts our geometrical concepts.
Page Break 262
Page 262
Now I believe this happens because we can only see the construction piecemeal and not as one whole. The
explanation would then consist in saying that there isn't a visual construction at all that is composed of these
individual visual pieces. This would be something like what happens when I show someone a small section of a large
spherical surface and ask him whether he accepts the great circle which is visible on it as a straight line; and if he did
so, I would then rotate the sphere and show him that it came back to the same place on the circle. But I surely
haven't proved to him in this way that a straight line in visual space returns to meet itself.
Page 262
This explanation would be: these are visual pieces which do not, however, add up to a visual whole, or at any
rate not to the whole the final result of which I believe I can see at the end.
Page 262
The simplest construction of this sort would, indeed, be the one above of the two equally long intervals into
one of which a piece will go n times and into the other n + 1 times. The steps of the construction would lie in
proceeding from one component part to another and discovering the equality of these parts.
Page 262
Here you could explain that, in making this progression, I am not really investigating the original visual image
of the equally long intervals. But that something else obtrudes into the investigation, which then leads to the startling
result.
Page 262
But there's an objection to this explanation. Someone might say: we didn't hide any part of the construction
from you while you were examining the individual parts, did we? So you ought to have been able to see whether
anything about the rest changed, shifted in the meantime. If that didn't happen, then you really ought to be able to
see, oughtn't you, that everything was above board?
Page 262
To speak of divisibility in visual space has a sense, since in a description it must be possible to substitute a
divided stretch for
Page Break 263
an undivided one. And then it is clear what, in the light of what I elaborated earlier, the infinite divisibility of this
space means.
Page 263
211 The moment we try to apply exact concepts of measurement to immediate experience, we come up against a
peculiar vagueness in this experience. But that only means a vagueness relative to these concepts of measurement.
And, now, it seems to me that this vagueness isn't something provisional, to be eliminated later on by more precise
knowledge, but that this is a characteristic logical peculiarity. If, e.g., I say: 'I can now see a red circle on a blue
ground and remember seeing one a few minutes ago that was the same size or perhaps a little smaller and a little
lighter,' then this experience cannot be described more precisely.
Page 263
Admittedly the words 'rough', 'approximate' etc. have only a relative sense, but they are still needed and they
characterise the nature of our experience; not as rough or vague in itself, but still as rough and vague in relation to
our techniques of representation.
Page 263
This is all connected with the problem 'How many grains of sand make a heap?'
Page 263
You might say: any group with more than a hundred grains is a heap and less than ten grains do not make a
heap: but this has to be taken in such a way that ten and a hundred are not regarded as limits which could be
essential to the concept 'heap'.
Page 263
And this is the same problem as the one specifying which of the vertical strokes we first notice to have a
different length from the first.
Page 263
What corresponds in Euclidean geometry to the visual circle isn't a circle, but a class of figures, including the
circle, but also, e.g., the hundred-sided regular polygon. The defining characteristic of this class could be something
like their all being figures
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contained in a band which arises through the vibration of a circle.--But even that is wrong: for why should I take
precisely the band which arises from vibrating a circle and not that produced by vibrating the hundred-sided
polygon?
Page 264
And here I come up against the cardinal difficulty, since it seems as though an exact demarcation of the
inexactitude is impossible. For the demarcation is arbitrary, since how is what corresponds to the vibrating circle
distinguished from what corresponds to the vibrating hundred-sided polygon?
Page 264
There is something attractive about the following explanation: everything that is within a a appears as the
visual circle C, everything that is outside b b does not appear as C. We would then have the case of the word 'heap'.
There would be an indeterminate zone left open, and the boundaries a and b are not essential to the concept
defined.--The boundaries a and b are still only like the walls of the forecourts. They are drawn arbitrarily at a point
where we can still draw something firm.--Just as if we were to border off a swamp with a wall, where the wall is not
the boundary of the swamp, it only stands around it on firm ground. It is a sign which shows there is a swamp inside
it, but not, that the swamp is exactly the same size as that of the surface bounded by it.
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Page 265
212 Now isn't the correlation between visual space and Euclidean space as follows: no matter what Euclidean figure I
show to the observer, he must be able to distinguish whether or not it is the visual circle C. That is to say, by
constantly reducing the interval between the figures shown I shall be able to reduce the indeterminate interval
indefinitely, be able 'to approach indefinitely close to a limit between what I see as C and what I see as not C'.
Page 265
But on the other hand, I shall never be able to draw such a limit as a curve in Euclidean space, for if I could, it
would itself then have to belong to one of the two classes and be the last member of this class, in which case I would
have to be able to see a Euclidean curve after all.
Page 265
If someone says e.g. that we never see a real circle but always only approximations to one, this has a sound,
unobjectionable sense, if it means that, given a body which looks circular, we can still always discover inaccuracies
by precise measurement or by looking through a magnifying glass. But we lose this sense the moment we substitute
the immediately given, the patch or whatever we choose to call it, for the circular body.
Page 265
If a circle is at all the sort of thing that we see--see in the same sense as that in which we see a blue
patch--then we must be able to see it and not merely something like it.
Page 265
If I cannot see an exact circle then in this sense neither can I see approximations to one.--But then the
Euclidean circle--and the Euclidean approximation to one--is in this sense not an object of my perception at all, but,
say, only a different logical construction which could be obtained from the objects of a quite different space from
the space of immediate vision.
Page 265
But even this way of talking is misleading, and we must rather say that we see the Euclidean circle in a
different sense.
Page 265
And so that a different sort of projection exists between the
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Euclidean circle and the circle perceived than one would naïvely suppose.
Page 266
If I say you can't distinguish between a chiliagon and a circle, then the chiliagon must here be given through
its construction, its origin. For how else would I know that it is 'in fact' a chiliagon and not a circle?
Page 266
In visual space there is no measurement.
Page 266
We could, e.g., perfectly well give the following definitions for visual space: 'A straight line is one that isn't
curved' and 'A circle is a curve with constant curvature'.
Page 266
213 We need new concepts and we continually resort to those of the language of physical objects. The word
'precision' is one of these dubious expressions. In ordinary language it refers to a comparison and then it is quite
intelligible. Where a certain degree of imprecision is present, perfect precision is also possible. But what is it
supposed to mean when I say I can never see a precise circle, and am now using this word not relatively, but
absolutely?
Page 266
The words 'I see' in 'I see a patch' and 'I see a line' therefore have different meanings.
Page 266
Suppose I have to say 'I never see a perfectly sharp line'. I have then to ask 'Is a sharp line conceivable?' If it
is right to say 'I do not see a sharp line', than a sharp line is conceivable. If it makes sense to say 'I never see an exact
circle', then this implies: an exact circle is conceivable in visual space.
Page 266
If an exact circle in a visual field is inconceivable, the proposition 'I never see an exact circle in my visual
field' must be the same sort of proposition as 'I never see a high C in my visual field'.
Page 266
If I say 'The upper interval is as long as the lower' and mean by this what is usually said by the proposition
'the upper interval
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appears to me as long as the lower', then the word 'equal' [†1] means something quite different in this proposition
from what it means in the proposition expressed in the same words but which is verified by comparing lengths with
dividers. For this reason I can, e.g., speak in the latter case of improving the techniques of comparison, but not in the
former. The use of the same word 'equal' with quite different meanings is very confusing. This is the typical case of
words and phrases which originally referred to the 'things' of the idioms for talking about physical objects, the
'bodies in space', being applied to the elements of our visual field; in the course of this they inevitably change their
meanings utterly and statements which previously had had a sense now lose it and others which had had no sense in
the first way of speaking now acquire one. Even though a certain analogy does persist--just the one which tricks us
into using the same expression.
Page 267
It is, e.g., important that the word 'close' means something different in the proposition 'There is a red patch
close to the boundary of the visual field' and in such a proposition as 'The red patch in the visual field is close to the
brown one'. Furthermore the word 'boundary' in the first proposition also has a different meaning--and is a different
sort of word from 'boundary' in the proposition: 'the boundary between red and blue in the visual field is a circle'.
Page 267
What sense does it make to say: our visual field is less clear at the edges than towards the middle? That is, if
we aren't here talking about the fact that we see physical objects more clearly in the middle of the visual field?
Page 267
One of the clearest examples of the confusion between physical and phenomenological language is the
picture Mach [†2] made of his visual field, in which the so-called blurredness of the figures near the edge of the
visual field was reproduced by a blurredness (in a quite different sense) in the drawing. No, you can't make a visual
picture of our visual image.
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Page 268
Can I therefore say that colour patches near the edge of the visual field no longer have sharp contours: are
contours then conceivable there? I believe it is clear that this lack of clarity is an internal property of visual space.
Has, e.g., the word 'colour' a different meaning when it refers to figures near to the edge?
Page 268
Without this 'blurredness' the limitlessness of visual space isn't conceivable.
Page 268
214 The question arises what distinctions are there in visual space. Can we learn anything about this from the
co-ordination, e.g., of tactile space with visual space? Say, by specifying which changes in one space do not
correspond to a change in the other?
Page 268
The fact that you see a physical hundred-sided polygon as a circle--cannot distinguish it from a physical
circle--implies nothing here as to the possibility of seeing a hundred-sided polygon.
Page 268
That it proves impossible for me to find physical bodies which give the visual image of a hundred-sided
visual polygon is of no significance for logic. The question is: is there a sense in speaking of a hundred-sided
polygon? Or: Does it make sense to talk of thirty strokes in a row taken in at one look? I believe there is none.
Page 268
The process isn't at all one of seeing first a triangle, then a square, pentagon etc. up to e.g. a fifty-sided
polygon and then the circle coming; no[[sic]] we see a triangle, a square etc., up to, maybe, an octagon, then we see
only polygons with sides of varying length. The sides get shorter, then a fluctuation towards the circle begins, and
then comes the circle.
Page 268
Neither does the fact that a physical straight line drawn as a tangent to a circle gives the visual image of a
straight line which for a stretch merges with the curve prove that our visual space isn't Euclidean, for a different
physical configuration could perfectly well produce the image corresponding to the Euclidean tangent. But in fact
such an image is inconceivable.
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Page 269
215 What is meant by the proposition 'We never see a precise circle'? What is the criterion of precision? Couldn't I
also perfectly well say: 'Perhaps I see a precise circle, but can never know it'? All this only makes sense, once it has
been established in what cases one calls one measurement more precise than another. Now the concept of a circle
presupposes--I believe--a concept of 'greater precision', which contains an infinite possibility of being increased. And
we may say the concept of a circle is the concept of the infinite possibility of greater precision. This infinite
possibility of increase would be a postulate of this idiom. Of course, it must then be clear in every case what I would
regard as an increase in precision.
Page 269
It obviously means nothing to say the circle is only an ideal to which reality can only approximate. This is a
misleading metaphor. For you can only approximate to something that is there, and if the circle is given us in any
form that makes it possible for us to approximate, then precisely that form would be the important thing for us, and
approximation to another form in itself of secondary importance. But it may also be that we call an infinite
possibility itself, a circle. So that the circle would then be in the same position as an irrational number.
Page 269
It seems essential to the application of Euclidean geometry that we talk of an imprecise circle, an imprecise
sphere etc. And also that this imprecision must be logically susceptible of an unlimited reduction. And so, if
someone is to understand the application of Euclidean geometry, he has to know what the word 'imprecise' means.
For nothing is given us over and above the result of our measuring and the concept of imprecision. These two
together must correspond to Euclidean geometry.
Page 269
Now, is the imprecision of measurement the same concept as the imprecision of visual images? I believe:
Certainly not.
Page 269
If the assertion that we never see a precise circle is supposed to mean, e.g., that we never see a straight line
touch a circle at one point (i.e. that nothing in our visual space has the multiplicity of a
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line touching a circle), then for this imprecision, an indefinitely high degree of precision is not conceivable.
Page 270
The word 'equality' has a different meaning when applied to intervals in visual space and when applied in
physical space. Equality in visual space has a different multiplicity from equality in physical space, so that in visual
space g1 and g2 can be straight
lines (visually straight) and the lengths a1 = a2, a2 = a3 etc., but not a1 = a5. Equally, a circle and a straight line in
visual space have a different multiplicity from a circle and a straight line in physical space, for a short stretch of a
seen circle can be straight; 'circle' and 'straight line' simply used in the sense of visual geometry.
Page 270
Here ordinary language resorts to the words 'seems' or 'appears'. It says a1 seems to be equal to a2, whereas
this appearance has ceased to exist in the case of a1 and a5. But it uses the word 'seems' ambiguously. For its
meaning depends on what is opposed to this appearance as reality. In one case it is the result of measurement, in
another a further appearance. And so the meaning of the word 'seem' is different in these two cases.
Page 270
216 The time has now come to subject the phrase 'sense-datum' to criticism. A sense-datum is the appearance of this
tree, whether 'there really is a tree standing there' or a dummy, a mirror image, an hallucination, etc. A sense-datum
is the appearance of the tree, and what we want to say is that its representation in language is only one description,
but not the essential one. Just as you can say of the expression 'my visual image' that it is only one form of
description, but by no means the only possible and correct one. For the form of words 'the appearance of this tree'
incorporates
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the idea of a necessary connection between what we are calling the appearance and 'the existence of a tree', in fact
whether it be veridical perception or a mistake. That is to say, if we talk about 'the appearance of a tree', we are either
taking for a tree something which is one, or something which is not one. But this connection isn't there.
Page 271
Idealists would like to reproach language with presenting what is secondary as primary and what is primary
as secondary. But that is only the case with these inessential valuations which are independent of cognition ('only' an
appearance). Apart from that, ordinary language makes no decision as to what is primary or secondary. We have no
reason to accept that the expression 'the appearance of a tree' represents something which is secondary in relation to
the expression 'tree'. The expression 'only an image' goes back to the idea that we can't eat the image of an apple.
Page 271
217 We might think that the right model for visual space would be a Euclidean drawing-board with its ideally fine
constructions which we make vibrate so that all the constructions are to a certain extent blurred (further, the surface
vibrates equally in all the directions lying in it). We could in fact say: it is to be vibrated precisely as far as it can
without its yet being noticeable, and then its physical geometry will be a picture of our phenomenological geometry.
Page 271
But the big question is: Can you translate the 'blurredness' of phenomena into an imprecision in the drawing?
It seems to me that you can't.
Page 271
It is, for instance, impossible to represent the imprecision of what is immediately seen by thick strokes and
dots in the drawing.
Page 271
Just as we cannot represent the memory of a picture by this picture painted in faint colours. The faintness of
memory is something quite other than the faintness of a hue we see; and the unclarity of vision different in kind
from the blurredness of an
Page Break 272
imprecise drawing. (Indeed, an imprecise drawing is seen with precisely the unclarity we are trying to represent by
its imprecision.)
Page 272
(In films, when a memory or dream is to be represented, the pictures are given a bluish tint. But memory
images have no bluish tint, and so the bluish projections are not visually accurate pictures of the dream, but pictures
in a sense which is not immediately visual.)
Page 272
A line in the visual field need not be either straight or curved. Of course the third possibility should not be
called 'doubtful' (that is nonsense); we ought to use another word for it, or rather replace the whole way of speaking
by a different one.
Page 272
That visual space isn't Euclidean is already shown by the occurrence of two different kinds of lines and
points: we see the fixed stars as points: that is, we can't see the contours of a fixed star, and in a different sense two
colour boundaries also intersect in a point; similarly for lines. I can see a luminous line without thickness, since
otherwise I should be able to see a cross-section of it as a rectangle or at least the four points of intersection of its
contours.
Page 272
A visual circle and a visual straight line can have a stretch in common.
Page 272
If I look at a drawn circle with a tangent, it wouldn't be remarkable that I never see a perfect circle and a
perfect straight line touch one another, it only becomes interesting when I see this happen, and the straight line and
the circle then coincide for a stretch.
Page 272
For only that would imply that the visual circle and the visual line are essentially different from the circle and
line of Euclidean geometry; but the first case, that we have never seen a perfect circle and a perfect line touch one
another, would not.
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XXI
Page 273
218 There appear to be simple colours. Simple as psychological phenomena. What I need is a psychological or rather
phenomenological colour theory, not a physical and equally not a physiological one.
Page 273
Furthermore, it must be a theory in pure phenomenology in which mention is only made of what is actually
perceptible and in which no hypothetical objects--waves, rods, cones and all that--occur.
Page 273
Now, we can recognize colours as mixtures of red, green, blue, yellow, white and black immediately. Where
this is still always the colour itself, and not pigment, light, process on or in the retina, etc.
Page 273
We can also see that one colour is redder than another or whiter, etc. But can I find a metric for colours? Is
there a sense in saying, for instance, that with respect to the amount of red in it one colour is halfway between two
other colours?
Page 273
At least there seems to be a sense in saying that one colour is closer in this respect to a second than it is to a
third.
Page 273
219 You might say, violet and orange partially obliterate one another when mixed, but not red and yellow.
Page 273
At any rate orange is a mixture of red and yellow in a sense in which yellow isn't a mixture of red and green,
although yellow comes between red and green in the colour circle.
Page 273
And if that happens to be manifest nonsense, the question arises, at what point does it begin to make sense;
that is, if I now move on the circle from red and from green towards yellow and call yellow a mixture of the two
colours I have now reached.
Page 273
That is, in yellow I recognize an affinity with red and with green, viz. the possibility of reddish yellow--and
yet I still don't
Page Break 274
recognize green and red as component parts of yellow in the sense in which I recognize red and yellow as
component parts of orange.
Page 274
I want to say that red is between violet and orange only in the sense in which white is between pink and
greenish-white. But, in this sense, isn't any colour between any two other colours, or at least between any two which
may be reached from the first by independent routes?
Page 274
Can one say that in this sense a colour only lies between two others with reference to a specified continuous
transition? And so, say, blue between red and black?
Page 274
Is this then how it is: to say the colour of a patch is a mixture of orange and violet is to ascribe to it a different
colour from that ascribed by saying that the patch has the colour common to violet and orange?--But that won't
work either; for, in the sense in which orange is a mixture of red and yellow, there isn't a mixture of orange and
violet at all. If I imagine mixing a blue-green with a yellow-green, I see straightaway that it can't happen, that, on the
contrary, a component part would first have to be 'killed' before the union could occur. This isn't the case with red
and yellow. And in this I don't have an image of a continuous transition (via green), only the discrete hues play a
part here.
Page 274
220 I must know what in general is meant by the expression 'mixture of colours A and B', since its application isn't
limited to a finite number of pairs. Thus if, e.g., someone shows me any shade of orange and a white and says the
colour of a patch is a mixture of these two, then I must understand this, and I can understand it.
Page 274
If someone says to me that the colour of a patch lies between violet and red, I understand this and can
imagine a redder violet than the one given. If, now, someone says to me the colour lies between this violet and an
orange--where I don't have a specific continuous transition before me in the shape of a painted colour circle--then I
can at best think that here, too, a redder violet is meant, but a redder orange might also be what is meant, since,
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leaving on one side a given colour circle, there is no colour lying halfway between the two colours, and for just this
reason neither can I say at what point the orange forming one limit is already too close to the yellow for it still to be
mixed with the violet; the point is that I can't tell which orange lies at a distance of 90° from violet on the colour
circle. The way in which the mixed colour lies between the others is no different here from the way red comes
between blue and yellow.
Page 275
If I say in the ordinary sense that red and yellow make orange, I am not talking here about a quantity of the
components. And so, given an orange, I can't say that yet more red would have made it a redder orange (I'm not of
course speaking about pigments), even though there is of course a sense in speaking of a redder orange. But there is,
e.g., no sense in saying this orange and this violet contain the same amount of red. And how much red would red
contain?
Page 275
The comparison we are seduced into making is one between the colour series and a system of two weights
on the beam of a balance, where I can move the centre of gravity of the system just as I choose, by increasing or
moving the weights.
Page 275
Now it's nonsense to believe that if I held the scale A at violet and moved the scale B into the red-yellow
region, C will then move towards red.
Page 275
And what about the weights I put on the scales: Does it mean anything to say, 'more of this red'? when I'm
not talking about pigments. That can only mean something if I understand by pure red a number of units, where the
number is stipulated at the outset. But then the complete number of units means nothing but that the
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scale is standing at red. And so the relative numbers again only specify a point on the balance, and not a point and a
weight.
Page 276
Now so long as I keep my two end colours in say, the blue-red region, and move the redder colour, I can say
that the resultant also moves towards red. But if I move one end-colour beyond red, and move it towards yellow, the
resultant doesn't now become redder! Mixing a yellowish red with a violet doesn't make the violet redder than
mixing pure red with the violet. That the one red has now become yellower even takes away something of the red
and doesn't add red.
Page 276
We could also describe this as follows: if I have a paint pot of violet pigment and another of orange, and now
increase the amount of orange added to the mixture, the colour of the mixture will gradually move away from violet
towards orange, but not via pure red.
Page 276
I can say of two different shades of orange that I have no grounds for saying of either that it is closer to red
than yellow.--There simply isn't a 'midpoint' here.--On the other hand, I can't see two different reds and be in doubt
whether one of them, and if so which, is pure red. That is because pure red is a point, but the midway between red
and yellow isn't.
Page 276
221 Admittedly it's true that we can say of an orange that it's almost yellow, and so that it is 'closer to yellow than to
red' and analogously for an almost red orange. But it doesn't follow from this that there must also be a midpoint
between red and yellow. Here the position is just as it is with the geometry of visual space as compared with
Euclidean geometry. There are here quantities of a different sort from that represented by our rational numbers. The
concepts 'closer to' and 'further from' are simply of no use at all or are misleading when we apply these phrases.
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Page 277
Or again: to say of a colour that it lies between red and blue doesn't define it sharply (unambiguously). But
the pure colours must be defined unambiguously when it is stated that they lie between certain mixed colours. And
so the phrase 'lie between' means something different from what it meant in the first case. That is to say, if the
expression 'lie between' on one occasion designates a mixture of two simple colours, and on another a simple
component common to two mixed colours, the multiplicity of its application is different in the two cases. And this is
not a difference in degree, it's an expression of the fact that we are dealing with two entirely different categories.
Page 277
We say a colour can't be between green-yellow and blue-red in the same sense as between red and yellow,
but we can only say this because in the latter case we can distinguish the angle of 90°; because we see yellow and
red as points. But there simply is no such distinguishing in the other case where the mixed colours are regarded as
primary. And so in this case we can, so to speak, never be certain whether the mixture is still possible or not. To be
sure, I could choose mixed colours at random and stipulate that they include an angle of 90°; but this would be
completely arbitrary, whereas it isn't arbitrary when we say that in the first sense there is no mixture of blue-red and
green-yellow.
Page 277
So in the one case grammar yields the 'angle of 90°', and now we are misled into thinking: we only need to
bisect it and the adjacent segment too, to arrive at another 90° segment. But here the metaphor of an angle collapses.
Page 277
Of course you can also arrange all the shades in a straight line, say with black and white as endpoints, as has
been done, but then you have to introduce rules to exclude certain transitions, and in the end the representation on
the line must be given the same kind of topological structure as the octahedron has. In this, it's completely analogous
to the relation of ordinary language to a 'logically purified' mode of expression. The two are completely equivalent;
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it's just that one of them already wears the rules of grammar on its face.
Page 278
To what extent can you say that grey is a mixture of black and white in the same sense as orange is a mixture
of red and yellow? And doesn't lie between black and white in the sense in which red lies between blue-red and
orange?
Page 278
If we represent the colours by means of a double-cone, instead of an octahedron, there is only one between
on the colour circle, and red appears on it between blue-red and orange in the same sense as that in which bluered
lies between blue and red. And if in fact that is all there is to be said, then a representation by means of a
double-cone is adequate, or at least one using a double eight-sided pyramid is.
Page 278
222 Now strangely enough, it seems clear from the outset that we can't say red has an orange tinge in the same sense
as orange
Page Break 279
has a reddish tinge. That is to say, it seems to be clear that the phrases 'x is composed of (is a mixture of) y and z' and
'x is the common component of y and z' are not interchangeable here: Were they so, the relation between would be
all we needed for a representation.
Page 279
In general, the phrases 'common component of' and 'mixture of' have different meanings only if one can be
used in a context where the other can't.
Page 279
Now, it's irrelevant to our investigation that when I mix blue and green pigments I get a blue-green, but when
I mix blue-green and blue-red the result isn't a blue.
Page 279
If I am right in my way of thinking, then 'Red is a pure colour' isn't a proposition, and what it is meant to
show is not susceptible to experimental testing. So that it is inconceivable that at one time red and at another
blue-red should appear to us to be pure.
Page 279
223 Besides the transition from colour to colour on the colour-circle, there seems to be another specific transition
that we have before us when we see dots of one colour intermingled with dots of another. Of course I mean here a
seen transition.
Page 279
And this sort of transition gives a new meaning to the word 'mixture', which doesn't coincide with the
relation 'between' on the colour-circle.
Page 279
You might describe it like this: I can imagine an orange-coloured patch as having arisen from intermingling
red and yellow dots, whereas I can't imagine a red patch as having arisen from intermingling violet and orange
dots.--In this sense, grey is a mixture of black and white, but white isn't a mixture of pink and a whitish green.
Page 279
But I don't mean that it is established by experimental mixing that certain colours arise in this way from
others. I could, say,
Page Break 280
perform the experiment with a rotating coloured disc. Then it might work or might not, but that will only show
whether or not the visual process in question can be produced by these physical means--it doesn't show whether the
process is possible. Just as physical dissection of a surface can neither prove nor refute visual divisibility. For
suppose I can no longer see a physical dissection as a visual dissection, but when drunk see the undivided surface as
divided, then wasn't the visual surface divisible?
Page 280
If I am given two shades of red, say, which are close to one another, it's impossible to be in doubt whether
both lie between red and blue, or both between red and yellow, or one between red and blue and the other between
red and yellow. And in deciding this, we have also decided whether both will mix with blue or with yellow, or one
with blue and one with yellow, and this holds no matter how close the shades are brought together so long as we are
still capable of distinguishing the pigments by colour at all.
Page 280
If we ask whether the musical scale carries with it an infinite possibility of being continued, then it's no
answer to say that we can no longer perceive vibrations of the air that exceed a certain rate of vibration as notes,
since it might be possible to bring about sensations of higher notes in another way. Rather, the finitude of the
musical scale can only derive from its internal properties. For instance, from our being able to tell from a note itself
that it is the final one, and so that this last note, or the last notes, exhibit inner properties which the notes in between
don't have.
Page 280
Just as thin lines in our visual field exhibit internal properties not possessed by the thicker ones, so that there
is a line in our visual field, which isn't a colour boundary but is itself coloured, and yet in a specific sense has no
breadth, so that when it intersects another such line we do not see four points A, B, C, D.
Page Break 281
Page 281
224 Nowadays the danger that lies in trying to see things as simpler than they really are is often greatly exaggerated.
But this danger does actually exist to the highest degree in the phenomenological investigation of sense impressions.
These are always taken to be much simpler than they are.
Page 281
If I see that a figure possesses an organization which previously I hadn't noticed, I now see a different figure.
Thus I can see |||||| as a special case of || || || or of ||| ||| or of | |||| | etc. This merely shows that that which we see isn't as
simple as it appears.
Page 281
Understanding a Gregorian mode doesn't mean getting used to the sequence of notes in the sense in which I
can get used to a smell and after a while cease to find it unpleasant. No, it means hearing something new, which I
haven't heard before, much in the same way--in fact it's a complete analogy--as it would be if I were suddenly able to
see 10 strokes ||||||||||, which I had hitherto only been able to see as twice five strokes, as a characteristic whole. Or
suddenly seeing the picture of a cube as 3-dimensional when I had previously only been able to see it as a flat
pattern.
Page 281
The limitlessness of visual space stands out most clearly, when we can see nothing, in pitch-darkness.
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XXII
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225 A proposition, an hypothesis, is coupled with reality--with varying degrees of freedom. In the limit case there's
no longer any connection, reality can do anything it likes without coming into conflict with the proposition: in which
case the proposition (hypothesis) is senseless!
Page 282
All that matters is that the signs, in no matter how complicated a way, still in the end refer to immediate
experience and not to an intermediary (a thing in itself).
Page 282
All that's required for our propositions (about reality) to have a sense, is that our experience in some sense or
other either tends to agree with them or tends not to agree with them. That is, immediate experience need confirm
only something about them, some facet of them. And in fact this image is taken straight from reality, since we say
'There's a chair here', when we only see one side of it.
Page 282
According to my principle, two assumptions must be identical in sense if every possible experience that
confirms the one confirms the other too. Thus, if no empirical way of deciding between them is conceivable.
Page 282
A proposition construed in such a way that it can be uncheckably true or false is completely detached from
reality and no longer functions as a proposition.
Page 282
The views of modern physicists (Eddington) tally with mine completely, when they say that the signs in their
equations no longer have 'meanings', and that physics cannot attain to such meanings but must stay put at the signs.
But they don't see that these signs have meaning in as much as--and only in as much
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as--immediately observable phenomena (such as points of light) do or do not correspond to them.
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A phenomenon isn't a symptom of something else: it is the reality.
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A phenomenon isn't a symptom of something else which alone makes the proposition true or false: it itself is
what verifies the proposition.
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226 An hypothesis is a logical structure. That is, a symbol for which certain rules of representation hold.
Page 283
The point of talking of sense-data and immediate experience is that we're after a description that has nothing
hypothetical in it. If an hypothesis can't be definitively verified, it can't be verified at all, and there's no truth or falsity
for it [†1].
Page 283
My experience speaks in favour of the idea that this hypothesis will be able to represent it and future
experience simply. If it turns out that another hypothesis represents the material of experience more simply, then I
choose the simpler method. The choice of representation is a process based on socalled induction (not mathematical
induction).
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Page 284
This is how someone might try to represent the course of an experience which presents itself as the
development of a curve by means of various curves, each of which is based on how much of the actual course is
known to us.
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The curve ----- is the actual course, so far as it is to be observed at all. The curves ---, -•-•-•-, -••-••-, show
different attempts to represent it that are based on a greater or lesser part of the whole material of observation.
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227 We only give up an hypothesis for an even higher gain.
Page 284
Induction is a process based on a principle of economy.
Page 284
Any hypothesis has a connection with reality which is, as it were, looser than that of verification.
Page 284
The question, how simple a representation is yielded by assuming a particular hypothesis, is directly
connected, I believe, with the question of probability.
Page 284
You could obviously explain an hypothesis by means of pictures. I mean, you could, e.g., explain the
hypothesis, 'There is a book lying here', with pictures showing the book in plan, elevation and various cross-sections.
Page 284
Such a representation gives a law. Just as the equation of a curve gives a law, by means of which you may
discover the ordinates, if you cut at different abscissae.
Page 284
In which case the verifications of particular cases correspond to cuts that have actually been made.
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Page 285
If our experiences yield points lying on a straight line, the proposition that these experiences are various
views of a straight line is an hypothesis.
Page 285
The hypothesis is a way of representing this reality, for a new experience may tally with it or not, or possibly
make it necessary to modify the hypothesis.
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228 What is essential to an hypothesis is, I believe, that it arouses an expectation by admitting of future
confirmation. That is, it is of the essence of an hypothesis that its confirmation is never completed.
Page 285
When I say an hypothesis isn't definitively verifiable, that doesn't mean that there is a verification of it which
we may approach ever more nearly, without ever reaching it. That is nonsense--of a kind into which we frequently
lapse. No, an hypothesis simply has a different formal relation to reality from that of verification. (Hence, of course,
the words 'true' and 'false' are also inapplicable here, or else have a different meaning.)
Page 285
The nature of the belief in the uniformity of events is perhaps clearest in a case where we are afraid of what
we expect to happen. Nothing could persuade me to put my hand in the fire, even though it's only in the past that
I've burnt myself.
Page 285
If physics describes a body of a particular shape in physical space, it must assume, even if tacitly, the
possibility of verification. The points at which the hypothesis is connected with immediate experience must be
anticipated.
Page 285
An hypothesis is a law for forming propositions.
Page 285
You could also say: An hypothesis is a law for forming expectations.
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Page 286
A proposition is, so to speak, a particular cross-section of an hypothesis.
Page 286
229 The probability of an hypothesis has its measure in how much evidence is needed to make it profitable to throw
it out.
Page 286
It's only in this sense that we can say that repeated uniform experience in the past renders the continuation of
this uniformity in the future probable.
Page 286
If, in this sense, I now say: I assume the sun will rise again tomorrow, because the opposite is so unlikely, I
here mean by 'likely' and 'unlikely' something completely different from what I mean by these words in the
proposition 'It's equally likely that I'll throw heads or tails'. The two meanings of the word 'likely' are, to be sure,
connected in certain ways, but they aren't identical.
Page 286
What's essential is that I must be able to compare my expectation not only with what is to be regarded as its
definitive answer (its verification or falsification), but also with how things stand at present. This alone makes the
expectation into a picture.
Page 286
That is to say: it must make sense now.
Page 286
If I say I can see, e.g., a sphere, that means nothing other than that I am seeing a view such as a sphere
affords, but this only means I can construct views in accordance with a certain law--that of the sphere--and that this
is such a view.
Page 286
230 Describing phenomena by means of the hypothesis of a world of material objects is unavoidable in view of its
simplicity when compared with the unmanageably complicated phenomenological description. If I can see different
discrete parts of a circle, it's perhaps impossible to give precise direct description of them, but the statement that
they're parts of a circle, which, for reasons which haven't been gone into any further, I don't see as a whole--is
simple.
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Page 287
Description of this kind always introduces some sort of parameter, which for our purposes we don't need to
investigate.
Page 287
What is the difference between the logical multiplicity of an explanation of appearances by the natural
sciences and the logical multiplicity of a description?
Page 287
If e.g. a regular ticking sound were to be represented in physics, the multiplicity of the picture
would suffice, but here it's not a question of the logical multiplicity of the sound,
but of that of the regularity of the phenomenon observed. And just so, the theory of Relativity doesn't represent the
logical multiplicity of the phenomena themselves, but that of the regularities observed.
Page 287
If, for instance, we use a system of co-ordinates and the equation for a sphere to express the proposition that
a sphere is located at a certain distance from our eyes, this description has a greater multiplicity than that of a
verification by eye. The first multiplicity corresponds not to one verification, but to a law obeyed by verifications.
Page 287
As long as someone imagines the soul as a thing, a body in our heads, there's no harm in the hypothesis. The
harm doesn't lie in the imperfection and crudity of our models, but in their lack of clarity (vagueness).
Page 287
The trouble starts when we notice that the old model is inadequate, but then, instead of altering it, only as it
were sublimate it. While I say thoughts are in my head, everything's all right; it becomes harmful when we say
thoughts aren't in my head, they're in my mind.
Page 287
Whatever someone can mean by a proposition, he also may mean by it. When people say, by the
proposition 'There's a chair here', I don't merely mean what is shown me by immediate experience,
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but something over and above that, you can only reply: whatever you can mean must connect with some sort of
experience, and whatever you can mean is unassailable.
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231 We may compare a part of an hypothesis with the movement of a part of a gear, a movement that can be
stipulated without prejudicing the intended motion. But then of course you have to make appropriate adjustments to
the rest of the gear if it is to produce the desired motion. I'm thinking of a differential gear.--Once I've decided that
there is to be no deviation from a certain part of my hypothesis no matter what the experience to be described may
be, I have stipulated a mode of representation and this part of my hypothesis is now a postulate. A postulate must be
such that no
conceivable experience can refute it, even though it may be extremely inconvenient to cling to the postulate. To the
extent to which we can talk here of greater or slighter convenience, there is a greater or slighter probability of the
postulate.
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Page 289
It's senseless to talk of a measure for this probability at this juncture. The situation here is like that in the case
of two kinds of numbers where we can with a certain justice say that the one is more like the other (is closer to it)
than a third, but there isn't any numerical measure of the similarity. Of course you could imagine a measure being
constructed in such cases, too, say by counting the postulates or axioms common to the two systems, etc., etc.
Page 289
232 We may apply our old principle to propositions expressing a probability and say, we shall discover their sense
by considering what verifies them.
Page 289
If I say 'That will probably occur', is this proposition verified by the occurrence or falsified by its
non-occurrence? In my opinion, obviously not. In that case it doesn't say anything about either. For if a dispute were
to arise as to whether it is probable or not, it would always only be arguments from the past that would be adduced.
And this would be so even when what actually happened was already known.
Page 289
If you look at ideas about probability and its application, it's always as though a priori and a posteriori were
jumbled together, as if the same state of affairs could be discovered or corroborated by experience, whose existence
was evident a priori. This of course shows that something's amiss in our ideas, and in fact we always seem to
confuse the natural law we are assuming with what we experience.
Page 289
That is, it always looks as if our experience (say in the case of card shuffling) agreed with the probability
calculated a priori. But that is nonsense. If the experience agrees with the computation, that means my computation
is justified by the experience, and of course it isn't its a priori element which is justified, but its bases, which are a
posteriori. But those must be certain natural laws which I take as the basis for my calculation, and it is these that are
confirmed, not the calculation of the probability.
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Page 290
The calculation of the probability can only cast the natural law in a different form. It transforms the natural
law. It is the medium through which we view and apply the natural law.
Page 290
If for instance I throw a die, I can apparently predict a priori that the 1 will occur on average once every 6
throws, and that can then be confirmed empirically. But it isn't the calculation I confirm by the experiment, but the
natural law which the probability calculation can present to me in different forms. Through the medium of the
probability calculation I check the natural law lying behind the calculation.
Page 290
In our case the natural law takes the form that it is equally likely for any of the six sides to be the side
uppermost. It's this law that we test.
Page 290
233 Of course this is only a natural law if it can be confirmed by a particular experiment, and also refuted by a
particular experiment. This isn't the case on the usual view, for if any event can be justified throughout an arbitrary
interval of time, then any experience whatever can be reconciled with the law. But that means the law is idling; it's
senseless.
Page 290
Certain possible events must contradict the law if it is to be one at all; and should these occur, they must be
explained by a different law.
Page 290
When we wager on a possibility, it's always on the assumption of the uniformity of nature.
Page 290
If we say the molecules of a gas move in accordance with the laws of probability, it creates the impression
that they move in accordance with some a priori laws or other. Naturally, that's nonsense. The laws of probability,
i.e. those on which the calculation is based,
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are hypothetical assumptions, which are then rehashed by the calculation and then in another form empirically
confirmed--or refuted.
Page 291
If you look at what is called the a priori probability and then at its confirmation by the relative frequency of
events, the chief thing that strikes you is that the a priori probability, which is, as it were, something smooth, is
supposed to govern the relative frequency, which is something irregular. If both bundles of hay are the same size
and the same distance away, that would explain why the donkey stands still between them, but it's no explanation of
its eating roughly as often from the one as from the other. That requires different laws of nature to explain it.--The
facts that the die is homogeneous and that each side is exactly the same, and that the natural laws with which I'm
familiar tell me nothing about the result of a throw, don't give me adequate grounds for inferring that the numbers 1
to 6 will be distributed roughly equally among the numbers thrown. Rather, the prediction that such a distribution
will be the case contains an assumption about those natural laws that I don't know precisely: the assumption that
they will produce such a distribution.
Page 291
234 Isn't the following fact inconsistent with my conception of probability: it's obviously conceivable that a man
throwing dice every day for a week--let's say--throws nothing but ones, and not because of any defect in the die, but
simply because the movement of his hand, the position of the die in the box, the friction of the table top always
conspire to produce the same result. The man has inspected the die, and also found that when others throw it the
normal results are produced. Has he grounds, now, for thinking that there's a natural law at work here which makes
him throw nothing but ones? Has he grounds for believing that it's sure now to go on like this, or has he grounds for
believing that this regularity can't last much longer? That is, has he grounds for abandoning the game since it has
become clear that he can only
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throw ones, or for playing on since it is in these circumstances all the more probable that he will now throw a higher
number? In actual fact, he will refuse to accept it as a natural law that he can throw nothing but ones. At least, it will
have to go on for a long time before he will entertain this possibility. But why? I believe, because so much of his
previous experience in life speaks against there being a law of nature of such a sort, and we have--so to speak--to
surmount all that experience, before embracing a totally new way of looking at things.
Page 292
If we infer from the relative frequency of an event its relative frequency in the future, we can of course only
do that from the frequency which has in fact been so far observed. And not from one we have derived from
observation by some process or other for calculating probabilities. For the probability we calculate is compatible
with any frequency whatever that we actually observe, since it leaves the time open.
Page 292
When a gambler or insurance company is guided by probability, they aren't guided by the probability
calculus, since one can't be guided by this on its own, because anything that happens can be reconciled with it: no,
the insurance company is guided by a frequency actually observed. And that, of course, is an absolute frequency.
Page 292
235 We can represent the equation of this curve:
as the equation of a straight line with a variable parameter, whose course expresses the deviations from a straight
line. It isn't essential that these deviations should be 'slight'. They can be so large that the curve doesn't look like a
straight line at all. 'Straight line with deviations' is only one form of description. It makes it possible for me to neglect
a particular component of the description--if I so wish. (The form: 'rule with exceptions'.)
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Page 293
A Galtonian photograph is the picture of a probability [†1].
Page 293
A probability law is the natural law you see when you screw up your eyes.
Page 293
To say that the points yielded by this experiment distribute themselves around this curve--e.g. a straight
line--means something like: seen from a certain distance, they appear to lie on a straight line.
Page 293
If I state 'That's the rule', that only has a sense as long as I have determined the maximum number of
exceptions I'll allow before knocking down the rule.
Page 293
236 I can say of a curve that the general impression is one of a straight line, but not of
the curve even though it might be possible to see this stretch in the course of a long stretch of
curve in which its divergence from a straight line would be swallowed up.
Page 293
I mean: it only makes sense to say of the stretch you actually see (and not of an hypothetical one you
assume) that it gives the general impression of a straight line.
Page 293
What is meant in a statistical experiment by 'in the long run'? An experiment must have a beginning and an
end.
Page 293
An experiment with dice lasts a certain time, and our expectations for the future can only be based on
tendencies we observe in what happens during this experiment. That is to say, the experiment can only give grounds
for expecting that things will go on in the way shown by the experiment; but we can't expect that the experiment,
Page Break 294
if continued, will now yield results that tally better with a preconceived idea of its course than did those of the
experiment we have actually performed.
Page 294
So if, for instance, I toss a coin and find no tendency in the results of the experiment itself for the numbers of
heads and of tails to approximate to each other more closely, then the experiment gives me no reason to suppose
that if it were continued such an approximation would emerge. Indeed, the expectation of such an approximation
must itself refer to a definite point in time, since I can't expect something to happen eventually, without setting any
finite limit whatever to the time when.
Page 294
I can't say: 'The curve looks straight, since it could be part of a line which taken as a
whole gives me the impression of a straight line.'
Page 294
237 Any 'reasonable' expectation is an expectation that a rule we have observed up to now will continue to hold.
Page 294
But the rule must have been observed and can't, for its part too, be merely expected.
Page 294
Probability Theory is only concerned with the state of expectation in the sense in which logic is with
thinking.
Page 294
Rather, probability is concerned with the form and a standard of expectation.
Page 294
It's a question of expecting that future experience will obey a law which has been obeyed by previous
experience.
Page 294
'It's likely that an event will occur' means: something speaks in favour of its occurring.
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238
Page 295
A ray is emitted from the light source S striking the surface AB to form a point of light there, and then
striking the surface AB' to form one there. We have no reason to suppose that the point on AB lies to the left or to
the right of M, and equally none for supposing that the point on AB' lies to the right or to the left of m; this yields
apparently incompatible probabilities.
Page 295
But suppose I have made an assumption about the probability of the point on AB lying in AM, how is this
assumption verified? Surely by a frequency experiment. Supposing this confirms the one view, then that is
recognized as the right one and so shows itself to be an hypothesis belonging to physics. The geometrical
construction merely shows that the fact that AM = MB was no ground for assuming equal likelihood.
Page 295
I give someone the following piece of information, and no more: at such and such a time you will see a point
of light appear in the interval AB.
Page 295
Does the question now make sense, 'Is it more likely that this point will appear in the interval AC than in
CB?'? I believe,
Page Break 296
obviously not.--I can of course decide that the probability of the event's happening in CB is to be in the ratio CB/AC
to the probability of its happening in AC; however, that's a decision I can have empirical grounds for making, but
about which there is nothing to be said a priori. It is possible for the observed distribution of events not to lead to
this assumption [†1]. The probability, where infinitely many possibilities come into consideration, must of course be
treated as a limit. That is, if I divide the stretch AB into arbitrarily many parts of arbitrary lengths and regard it as
equally likely that the event should occur in any one of these parts, we immediately have the simple case of dice
before us. And now I can--arbitrarily--lay down a law for constructing parts of equal likelihood. For instance, the law
that, if the lengths of the parts are equal, they are equally likely. But any other law is just as permissible.
Page 296
Couldn't I, in the case of dice too, take, say, five faces together as one possibility, and oppose them to the
sixth as the second possibility? And what, apart from experience, is there to prevent me from regarding these two
possibilities as equally likely?
Page 296
Let's imagine throwing, say, a red ball with just one very small green patch on it. Isn't it much more likely in
this case for the red area to strike the ground than for the green?--But how would we support this proposition?
Presumably by showing that when we throw the ball, the red strikes the ground much more often than the green.
But that's got nothing to do with logic.--We may always project the red and green surfaces and what befalls them
onto a surface in such a way that the projection of the green surface is greater than or equal to the red; so that the
events, as seen in this projection, appear to have a quite different probability ratio from the one they had on the
original surface.
Page 296
If, e.g., I reflect the events in a suitably curved mirror and now imagine what I would have held to be the
more probable event if I had only seen the image in the mirror.
Page 296
The one thing the mirror can't alter is the number of clearly
Page Break 297
demarcated possibilities. So that if I have n coloured patches on my ball, the mirror will also show n, and if I have
decided that these are to be regarded as equally likely, then I can stick to this decision for the mirror image too.
Page 297
To make myself even clearer: if I carry out the experiment with a concave mirror, i.e., make the observations
in a concave mirror, it will perhaps then look as if the ball falls more often on the small surface than on the much
larger one; and it's clear that neither experiment--in the mirror or outside it has a claim to precedence.
Page 297
What now, does it really mean to decide that two possibilities are equally likely?
Page 297
Doesn't it mean that, first, the natural laws known to us give no preference to either of the possibilities, and
second, that under certain conditions the relative frequencies of the two events approach one another?
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APPENDIX I
Page 299
Written in 1931
Page Break 300
Page Break 301
Complex and Fact (June 1931)
Page 301
The use of the words 'fact' and 'act'.--'That was a noble act.'--'But, that never happened.'--
Page 301
It is natural to want to use the word 'act' so that it only corresponds to a true proposition. So that we then
don't talk of an act which was never performed. But the proposition 'That was a noble act' must still have a sense
even if I am mistaken in thinking that what I call an act occurred. And that of itself contains all that matters, and I
can only make the stipulation that I will only use the words 'fact', 'act' (perhaps also 'event') in a proposition which,
when complete, asserts that this fact obtains.
Page 301
It would be better to drop the restriction on the use of these words, since it only leads to confusion, and say
quite happily: 'This act was never performed', 'This fact does not obtain', 'This event did not occur'.
Page 301
Complex is not like fact. For I can, e.g., say of a complex that it moves from one place to another, but not of
a fact.
Page 301
But that this complex is now situated here is a fact.
Page 301
'This complex of buildings is coming down' is tantamount to: 'The buildings thus grouped together are
coming down'.
Page 301
I call a flower, a house, a constellation, complexes: moreover, complexes of petals, bricks, stars, etc.
Page 301
That this constellation is located here, can of course be described by a proposition in which only its stars are
mentioned and neither the word 'constellation' nor its name occurs.
Page 301
But that is all there is to say about the relation between complex and fact. And a complex is a spatial object,
composed of spatial
Page Break 302
objects. (The concept 'spatial' admitting of a certain extension.)
Page 302
A complex is composed of its parts, the things of a kind which go to make it up. (This is of course a
grammatical proposition concerning the words 'complex', 'part' and 'compose'.)
Page 302
To say that a red circle is composed of redness and circularity, or is a complex with these component parts, is
a misuse of these words and is misleading. (Frege was aware of this and told me.)
Page 302
It is just as misleading to say the fact that this circle is red (that I am tired) is a complex whose component
parts are a circle and redness (myself and tiredness).
Page 302
Neither is a house a complex of bricks and their spatial relations; i.e. that too goes against the correct use of
the word.
Page 302
Now, you can of course point at a constellation and say: this constellation is composed entirely of objects
with which I am already acquainted; but you can't 'point at a fact' and say this.
Page 302
'To describe a fact', or 'the description of a fact', is also a misleading expression for the assertion stating that
the fact obtains, since it sounds like: 'describing the animal that I saw'.
Page 302
Of course we also say: 'to point out a fact', but that always means; 'to point out the fact that...'. Whereas 'to
point at (or point out) a flower' doesn't mean to point out that this blossom is on this stalk; for we needn't be talking
about this blossom and this stalk at all.
Page 302
It's just as impossible for it to mean: to point out the fact that this flower is situated there.
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Page 303
To point out a fact means to assert something, to state something. 'To point out a flower' doesn't mean this.
Page 303
A chain, too, is composed of its links, not of these and their spatial relations.
Page 303
The fact that these links are so concatenated, isn't 'composed' of anything at all.
Page 303
The root of this muddle is the confusing use of the word 'object'.
Page 303
The part is smaller than the whole: applied to fact and component part (constituent), that would yield an
absurdity.†*
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The Concept of Infinity in Mathematics
(The end of 1931)
Infinitely Long
Page 304
If you speak of the concept 'infinity', you must remember that this word has many different meanings and
bear in mind which one we are going to speak of at this particular moment. Whether, e.g., of the infinity of a number
series and of the cardinals in particular. If, for example, I say 'infinite' is a characteristic of a rule, I am referring to
one particular meaning of the word. But we might perfectly well say a continuous transition of colour was a
transition 'through infinitely many stages', provided we don't forget that here we are defining the meaning of the
phrase 'infinitely many stages' anew by means of the experience of a colour transition. (Even if by analogy with
other ways of using the word 'infinite'.)
Page 304
(If we say that this area of our subject is extraordinarily difficult, that isn't true in the sense that we are, say,
talking about things which are extraordinarily complicated or difficult to imagine, but only in the sense that it is
extraordinarily difficult to negotiate the countless pitfalls language puts in our path here.)
Page 304
'I once said there was no extensional infinity. Ramsey replied: "Can't we imagine a man living for ever, that is
simply, never dying, and isn't that extensional infinity?" I can surely imagine a wheel spinning and never coming to
rest' [†1]. What a peculiar
Page Break 305
argument: 'I can imagine...'! Let's consider what experience we would regard as a confirmation or proof of the fact
that the wheel will never stop spinning. And compare this experience with that which would tell us that the wheel
spins for a day, for a year, for ten years, and we shall find it easy to see the difference in the grammar of the
assertions '... never comes to rest' and '... comes to rest in 100 years'. Let's think of the kind of evidence we might
adduce for the claim that two heavenly bodies will orbit around one another, without end. Or of the Law of Inertia
and how it is confirmed.
Page 305
'Suppose we travel out along a straight line into Euclidean space and that at 10m. intervals we encounter an
iron sphere, ad. inf.' [†1]. Again: What sort of experience would I regard as confirmation for this, and what on the
other hand for there being 10,000 spheres in a row?--A confirmation of the first sort would be something like the
following: I observe the pendulum movement of a body. Experiments have shown me that this body is attracted by
iron spheres in accordance with a certain law; supposing there to be 100 such spheres in a series at a particular
position with respect to the body under test would explain, on the assumption of this law of attraction, an
approximation to the observed (or supposed) behaviour; but the more spheres we suppose there to be in the series,
the more closely does the result calculated agree with the one observed. In that case, it makes sense to say
experience confirms the supposition of an infinite series of spheres. But the difference between the sense of the
statement of number and that of 'an infinite number' is as great as the difference between this experience and that of
seeing a number of spheres.
Page 305
'The merely negative description of not stopping cannot yield a positive infinity' [†2]. With the phrase 'a
positive infinity' I
Page Break 306
thought of course of a countable (= finite) set of things (chairs in this room) and wanted to say that the presence of a
colossal number of such things can't be inferred from whatever it is that indicates to us that they don't stop. And so
here in the form of my assertion I make the strange mistake of denying a fact, instead of denying that a particular
proposition makes sense, or more strictly, of showing that two similar sounding remarks have different grammars.
Page 306
What an odd question: 'Can we imagine an endless row of trees?'! If we speak of an 'endless row of trees', we
will surely still link what we mean with the experiences we call 'seeing a row of trees', 'counting a row of trees',
'measuring a row of trees' etc. 'Can we imagine an endless row of trees?'! Certainly, once we have laid down what we
are to understand by this; that is once we have brought this concept into relation with all these things, with the
experiences which define for us the concept of a row of trees.
Page 306
What in experience is the criterion for the row of trees being infinite? For that will show me how this
assertion is to be understood. Or, if you give me no such criterion, what am I then supposed to do with the concept
'infinite row of trees'? What on earth has this concept to do with what I ordinarily call a row of trees? Or didn't you
in the end only mean: an enormously long row of trees?
Page 306
'But we are surely familiar with an experience, when we walk along a row of trees, which we can call the row
coming to an end. Well, an endless row of trees is one such that we never have this experience.'--But what does
'never' mean here? I am familiar with an experience I describe by the words 'He never coughed during the whole
hour', or 'He never laughed in his whole life'. We cannot speak of an analogous experience where the 'never' doesn't
refer to a time interval. And so once again analogy leaves us in the lurch here and I must try to find out ab initio
how the word
Page Break 307
'never' can be used so as to make sense in this case.--Admittedly such uses can be found, but their rules are to be
examined in their own right. For example, the proposition that a row of trees is infinitely long (or that we shall never
come to its end), could be a natural law of the same sort as the Law of Inertia, which certainly says that under certain
conditions a body moves in a straight line with constant velocity; and here it could indeed be said that under those
conditions the movement will never end. But if we ask about the verification of such a proposition, the main thing to
be said is that it is falsified if the movement (row of trees) comes to an end. There can be no talk of a verification
here, and that means we are dealing with a fundamentally different kind of proposition (or with a proposition, in a
different sense of that word). Naturally I don't want to say that this is the only significant use of the expression
'infinite row of trees' or of the word 'never' (in all eternity). But each such use must be described in its own right and
has its own laws. It's no help to us that we find a way of speaking ready-made in our ordinary language, since this
language uses each of its words with the most varied meanings, and understanding the use of the word in one
context does not relieve us from investigating its grammar in another. Thus we think something like 'It is still surely
possible to imagine an infinitely long life, for a man lives for an infinite length of time, if he simply never dies.' But
the use of the word 'never' just isn't that simple.
Page 307
Let's now discuss an endless life in the sense of an hypothesis (cf. the Law of Inertia), and the man living it
choosing in succession an arbitrary fraction from the fractions between 1 and 2, 2 and 3, 3 and 4, etc. ad inf. and
writing it down. Does this give us a 'selection from all those intervals'? No,... /see above, §146, p. 167.
Page 307
'But now let's imagine a man who becomes more and more adept at choosing from intervals, so that he
would take an hour for the first choice, half an hour for the second, a quarter for the third, etc. ad inf. In that case he
would have done the whole job
Page Break 308
in two hours!' [†1] Let's now imagine the process. The choice would consist, say, in his writing down a fraction, and
so in moving his hand. This movement would get quicker and quicker; but however quick it becomes, there will
always be a last interval that has been dealt with in a particular time. The consideration behind our objection depends
on forming the sum 1 + ½ + ¼ +..., but that is of course a limit of a series of sums, and not a sum in the sense in
which, e.g., 1 + ½ + ¼ is a sum. If I were to say 'He needs an hour for the first choice, half an hour for the second, a
quarter for the third, etc. ad. inf.' then this remark only makes sense so long as I don't ask about the velocity of the
choice at the time instant t = 2, since our reckoning gives no value for this (for there's no value c = infinity here as far
as we're concerned, since we haven't correlated any experience with it). My law gives me a velocity for any instant
before t = 2, and so is to that extent applicable and in order. Thus the fallacy lies only in the sentence 'In that case he
would have done the whole job in two hours'. (If we can call that a fallacy, since the sentence is in fact senseless in
this context.)
Page 308
Let's now consider the hypothesis that under certain conditions someone will throw the digits of π (say,
expressed in a system to the base 6). This hypothesis is, then, a law according to which I can work out for any throw
the number of spots showing. But what if we modified the hypothesis into one that under certain conditions
someone would not throw the digits of π! Shouldn't that make sense too? But how could we ever know that this
hypothesis was correct, since up to any given time he may have thrown in agreement with π, and yet this would not
refute the hypothesis. But that surely just means that we have to do with a different kind of hypothesis; with a kind
of proposition for which no provision is made in its grammar for a falsification. And it is open
Page Break 309
to me to call that a 'proposition' or 'hypothesis' or something quite different, as I like. (π is not a decimal fraction, but
a law according to which decimal fractions are formed.)
Page 309
The infinity of time is not a duration.
Page 309
If we ask 'What constitutes the infinity of time?' the reply will be 'That no day is the last, that each day is
followed by another'. But here we are misled again into seeing the situation in the light of a false analogy. For we are
comparing the succession of days with the succession of events, such as the strokes of a clock. In such a case we
sometimes experience a fifth stroke following four strokes. Now, does it also make sense to talk of the experience of
a fifth day following four days? And could someone say 'See, I told you so: I said there would be another after the
fourth'? (You might just as well say it's an experience that the fourth is followed by the fifth and no other.) But we
aren't talking here about the prediction that the sun will continue to move after the fourth day as before, that's a
genuine prediction. No, in our case it's not a question of a prediction, no event is prophesied; what we're saying is
something like this: that it makes sense, in respect of any sunrise or sunset, to talk of a next. For what is meant by
designation of a period of time is of course bound up with something happening: the movement of the hand of a
clock, of the earth, etc., etc.; but when we say 'each hour is succeeded by a next', having defined an hour by means
of the revolution of a particular pointer (as a paradigm), we are still not using that assertion in order to prophesy that
this pointer will go on in the same way for all eternity:--but we want to say: that it 'can go on in the same way for
ever'; and that is simply an assertion concerning the grammar of our determinations of time.
Page 309
Let's compare the propositions 'I'm making my plans on the assumption that this situation will last for two
years' and 'I'm
Page Break 310
making them on the assumption that this situation will last for ever'.--Does the proposition make sense 'I believe (or
expect, or hope), that it will stay like that throughout infinite time'?
Page 310
We may say: 'I am making arrangements for the next three days', or 10 years, etc., and also 'I'm making
arrangements for an indefinite period';--but also 'for an infinite time'? If I 'make arrangements for an indefinite
period', it is surely possible to mention a time for which at any rate I am no longer making arrangements. That is, the
proposition 'I am making arrangements for an indefinite period' does not imply every arbitrary proposition of the
form 'I am making arrangements for n years'.
Page 310
Just think of the proposition: 'I suspect this situation will continue like this without end'!
Page 310
Or how comical this rebuttal sounds: 'You said this clockwork would run for ever--well, it's already stopped
now'. We feel that surely every finite prediction of too long a run would also be refuted by the fact, and so in some
sense or other the refutation is incommensurable with the claim.--For it is nonsense to say 'The clockwork didn't go
on running for an infinite time, but stopped after 10 years' (or, even more comically: '... but stopped after only 10
years').
Page 310
How odd, if someone were to say: 'You have to be very bold to predict something for 100 years;--but how
bold you must be to predict something for infinite time, as Newton did with his Law of Inertia!'
Page 310
'I believe it will go on like that for ever.'--'Isn't it enough (for all practical purposes) to say you believe it will
go on like that for 10,000 years?'--That is to say, we must ask: Can there be grounds for this belief? What are they?
What are the grounds for assuming that it will go on for 1,000 more years; what for assuming it will go on for 10,000
more years;--and, now, what are the grounds for the infinite assumption?!--That's what makes the sentence 'I
suspect it will go on without end' so comic; we want to ask, why do you suspect that? For we want to say it's
senseless to say you suspect that: because it's senseless to talk of grounds for such a suspicion.
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Page 311
Let's consider the proposition 'This comet will move in a parabola with equation...'. How is this proposition
used? It cannot be verified; that is to say: we have made no provision for a verification in its grammar (that doesn't of
course mean we can't say it's true; for 'p is true' says the same as 'p'). The proposition can bring us to make certain
observations. But for those a finite prediction would always have done equally well. It will also determine certain
actions. For instance, it might dissuade us from looking for the comet in such and such a place. But for that too, a
finite claim would have been enough. The infinity of the hypothesis doesn't consist of its largeness, but in its
open-endedness.
Page 311
'Eventually, the world will come to an end': an infinite hypothesis.
Page 311
The proposition that eventually--in the infinite future--an event (e.g. the end of the world) will occur, has a
certain formal similarity with what we call a tautology.
Infinite Possibility
Page 311
Different use of the word 'can' in the propositions 'Three objects can lie in this direction' and 'Infinitely many objects
can lie in this direction' [†1]. What sense, that is to say, what grammar could such a way of talking possess? We
might for example say: 'In the natural number series 1, 2, 3, 4.... infinitely many numerals can follow the "1"'; that is
tantamount to: 'The operation + 1 may be applied ever again (or: without end)'. And so if, for example, someone
writes the numeral 100 + 1 after the numeral 100, that rule gives him the right to do so. On the other hand, there is no
Page Break 312
sense here in saying: 'If it's permissible to write down infinitely many numerals, let's write infinitely many numerals
(or try to)!'--
Page 312
Analogously, if I say a division yields an infinite decimal fraction, then there isn't one result of the division
called 'an infinite decimal', in the sense in which the number 0.142 is a result of 1 ÷ 7. The division doesn't yield as its
final result one decimal number, or a number of decimal numbers but rather we can't talk of 'its final result': it
endlessly yields decimal fractions; not 'an endless decimal fraction'. 'Endlessly' not 'endless'.
Page 312
Let's now imagine the following case: I have constructed a particular kind of die, and am now going to
predict: 'I shall throw the places of π with this die'. This claim is of a different form from the apparently similar 'I
shall throw the first ten places of π with this die'. For in the second case there is a proposition 'I shall have thrown the
first ten places of π within the hour', but this proposition becomes nonsense (not false), if I substitute 'the places' for
'the first ten places'. In the sentence 'Any arbitrary number of throws is possible', 'possible' may be equivalent to
'logically possible' ('conceivable'), and then it is a rule, not an empirical proposition, and is of a similar sort to that of
the rule 'numerals can follow 1 without end'. But we might also construe it as a kind of empirical proposition, a kind
of hypothesis: but then it would be the kind of hypothesis which has no verification provided for it, only a
falsification, and so it would be a different kind of proposition (a 'proposition' in a different sense) from the empirical
proposition: 'Three throws are possible with this die'. This--unlike the rule 'Three throws are conceivable'--would
mean something like: 'The die will still be usable after three throws'; the hypothesis 'Infinitely many throws are
possible with this die' would mean 'However often you throw it, this die will not wear out'. It's very clear that these
propositions are different in kind, if you think of the nonsensical order 'Throw it infinitely often' or 'Throw ad
infinitum', as opposed to the significant 'Throw it three times'. For it is essential to a command that we can check
whether it has been carried out.
Page Break 313
Page 313
If we wish to say infinity is an attribute of possibility, not of reality, or: the word 'infinite' always goes with
the word 'possible', and the like,--then this amounts to saying: the word 'infinite' is always a part of a rule.
Page 313
Let's suppose we told someone: 'I bought a ruler yesterday with infinite curvature'. Here, however, the word
'infinite' surely occurs in the description of a reality.--But still, I can never have the experience which would justify
me in saying that the ruler actually had an infinite radius of curvature, since a radius of 100100 km would surely do
just as well.--Granted, but in that case I can't have the experience which would justify me in saying the ruler is
straight either. And the words 'straight' (or in another context 'parallel') and 'infinite' are in the same boat. I mean: If
the word 'straight', ('parallel', 'equally long', etc., etc.) may occur in a description of reality, then so may the word
'infinite' [†1].
Page 313
'All that's infinite is the possibility' means '"infinite" qualifies "etc.".' And so far as this is what it does, it
belongs in a rule, a law. It's out of place in describing experience only when we mean by 'experience that
corresponds to a law' an endless series of experiences.--The slogan 'all that's infinite is the possibility, not the reality'
is misleading. We may say: 'In this case all that's infinite is the possibility'.--And we justifiably ask: what is it that is
infinite about this hypothesis (e.g. about the path of a comet)? Is there something huge about this assumption, this
thought?
Page 313
If we say 'The possibility of forming decimal places in the division 1 ÷ 3 is infinite', we don't pin down a fact
of nature, but give a rule of the system of calculation. But if I say: 'I give you infinite freedom to develop as many
places as you like, I won't stop you', then this isn't enunciating the rule of the system of calculation, it's saying
something about the future. 'Yes, but still only as the description of a possibility.'--No, of a reality! But of
Page Break 314
course not that of 'infinitely many places'; to say that would be to fall into the very grammatical trap we must avoid.
Page 314
The fact that it permits the endless formation of numerals doesn't make grammar infinitely complicated.
Page 314
To explain the infinite possibility, it must be sufficient to point out the features of the sign which lead us to
assume this infinite possibility, or better: from which we read off this infinite possibility. That is, what is actually
present in the sign must be sufficient, and the possibilities of the sign, which once more could only emerge from a
description of signs, do not come into the discussion [†1]. And so everything must be already contained in the sign
'/1, x, x + 1/'--the expression for the rule of formation. In introducing infinite possibility, I mustn't reintroduce a
mythical element into grammar. If we describe the process of division 1.0 ÷ 3 = 0.3, which leads to the quotient 0.3
and remainder 1, the infinite possibility of going on with always the same result must be contained in this
description, since we certainly aren't given anything else, when we see 'that it must always go on in the same way'.
Page 314
And when we 'see the infinite possibility of going on', we still can't see anything that isn't described when we
simply describe the sign we see.
Page Break 315
APPENDIX 2
Page 315
From F. Waismann's shorthand transcript of
Wittgenstein's talks and conversation between December
1929 and September 1931.
Page Break 316
Page Break 317
Yardstick and System of Propositions
Page 317
(From F. Waismann's notes for 25 December 1929.)
Page 317
I once wrote: 'A proposition is laid like a yardstick against reality. Only the outermost tips of the graduation
marks touch the object to be measured.' I should now prefer to say: a system of propositions is laid like a yardstick
against reality. What I mean by this is: when I lay a yardstick against a spatial object, I apply all the graduation
marks simultaneously. It's not the individual graduation marks that are applied, it's the whole scale. If I know that the
object reaches up to the tenth graduation mark, I also know immediately that it doesn't reach the eleventh, twelfth,
etc. The assertions telling me the length of an object form a system, a system of propositions. It's such a whole
system which is compared with reality, not a single proposition. If, for instance, I say such and such a point in the
visual field is blue, I not only know that, I also know that the point isn't green, isn't red, isn't yellow etc. I have
simultaneously applied the whole colour scale. This is also the reason why a point can't have different colours
simultaneously; why there is a syntactical rule against fx being true for more than one value of x. For if I apply a
system of propositions to reality, that of itself already implies--as in the spatial case--that in every case only one state
of affairs can obtain, never several.
Page 317
When I was working on my book I was still unaware of all this and thought then that every inference
depended on the form of a tautology. I hadn't seen then that an inference can also be of the form: A man is 6 ft tall,
therefore he isn't 7 ft. This is bound up with my then believing that elementary propositions had to be independent
of one another: from the fact that one state of affairs obtained you couldn't infer another did not. But if my present
conception of a system of propositions is right, then it's even the rule that from the fact that one state of affairs
obtains we can infer that all the others described by the system of propositions do not.
Page Break 318
Consistency
Page 318
(Waismann's notes, Wednesday, 17 December, 1930, Neuwaldegg. Transcript of Wittgenstein's words,
unless otherwise indicated.)
Page 318
I've been reading a work by Hilbert on consistency. It strikes me that this whole question has been put
wrongly. I should like to ask: Can mathematics be inconsistent at all? I should like to ask these people: Look, what
are you really up to? Do you really believe there are contradictions hidden in mathematics?
Page 318
Axioms have a twofold significance, as Frege saw.
Page 318
1) The rules, according to which you play.
Page 318
2) The opening positions of the game.
Page 318
If you take the axioms in the second way, I can attach no sense to the claim that they are inconsistent. It
would be very queer to say: this configuration (e.g. in the Hilbertian formula game, 0 ≠ 0) is a contradiction. And if I
do call some configuration or other a contradiction, that has no essential significance, at least for the game qua
game. If I arrange the rules so that this configuration can't arise, all I've done is made up a different game. But the
game's a game, and I can't begin to understand why anyone should attach such great importance to the occurrence
of this configuration: they behave as though this particular one were tabu. I then ask: and what is there to get excited
about if this configuration does arise?
Page 318
The situation is completely different if the axioms are taken as the rules according to which the game is
played. The rules are--in a certain sense--statements. They say: you may do this or this, but not that. Two rules can
be inconsistent. Suppose, e.g., that in chess one rule ran: under such and such circumstances the piece concerned
must be taken. But another rule said: a knight may never be taken. If now the piece concerned happens to be a
knight, the
Page Break 319
rules contradict one another: I don't know what I'm supposed to do. What do we do in such a case? Nothing easier:
we introduce a new rule, and the conflict is resolved.
Page 319
My point, then, is: if inconsistencies were to arise between the rules of the game of mathematics, it would be
the easiest thing in the world to remedy. All we have to do is to make a new stipulation to cover the case in which
the rules conflict, and the matter's resolved.
Page 319
But here I must make an important point. A contradiction is only a contradiction when it arises. People have
the idea that there might at the outset be a contradiction hidden away in the axioms which no-one has seen, like
tuberculosis: a man doesn't suspect anything and then one day he's dead. That's how people think of this case too:
one day the hidden contradiction might break out, and then the catastrophe would be upon us.
Page 319
What I'm saying is: to ask whether the derivations might not eventually lead to a contradiction makes no
sense at all as long as I'm given no method for discovering it.
Page 319
While I can play, I can play, and everything's all right.
Page 319
The truth of the matter is that the calculus qua calculus is all right. It doesn't make any sense whatever to talk
about contradiction. What we call a contradiction arises when we step outside the calculus and say in prose:
'Therefore all numbers have this property, but the number 17 doesn't have it.'
Page 319
In the calculus the contradiction can't be expressed at all.
Page 319
I can play with the chessmen according to certain rules. But I can also invent a game in which I play with the
rules themselves. The pieces in my game are now the rules of chess, and the rules of the game are, say, the laws of
logic. In that case I have yet another game and not a metagame. What Hilbert does is mathematics and not
metamathematics. It's another calculus, just like any other.
Page 319
(Sunday, 28 December 1930, at Schlick's home.)
The problem of the consistency of mathematics stems from two
Page Break 320
sources: (1) From the idea of non-Euclidean geometry where it was a matter of proving the axiom of parallels by
means of the given paradigm of a reductio ad absurdum. (2) From the Burali-Forti and Russellian antinomies.
Page 320
The impetus behind the present preoccupation with consistency came mainly from the antinomies. Now, it
has to be said that these antinomies haven't got anything to do with consistency in mathematics, that there's no
connection here at all. For the antinomies didn't in fact arise in the calculus, but in ordinary everyday speech,
precisely because we use words ambiguously. So that resolving the antinomies consists in replacing the vague idiom
by a precise one (by reflecting on the strict meaning of the words). And so the antinomies vanish by means of an
analysis, not of a proof.
Page 320
If the contradictions in mathematics arise through an unclarity, I can never dispel this unclarity by a proof.
The proof only proves what it proves. But it can't lift the fog.
Page 320
This of itself shows that there can be no such thing as a consistency proof (if we are thinking of the
inconsistencies of mathematics as being of the same sort as the inconsistencies of set theory), that the proof can't
begin to offer what we want of it. If I'm unclear about the nature of mathematics, no proof can help me. And if I'm
clear about the nature of mathematics, the question of consistency can't arise at all.
Page 320
Russell had the idea that his 5 'primitive propositions' were to be both the basic configurations and the rules
for going on. But he was under an illusion here, and this came out in the fact that he himself had to add further rules
(in words!).
Page 320
So we must distinguish: the basic configurations of the calculus (the opening positions of the game) and the
rules telling us how to get from one configuration to another. This was already made clear by Frege in his critique of
the theories of Heine and Thomae: 'How surprising. What would someone say if he asked what the rules of chess
were and instead of an answer was shown a group of chessmen on the chessboard? Presumably that he couldn't find
Page Break 321
a rule in this, since he didn't attach any sense at all to these pieces and their layout' (Grundgesetze, II, p. 113).
Page 321
Now if I take the calculus as a calculus, the positions in the game can't represent contradictions (unless I
arbitrarily call one of the positions that arise in the game a 'contradiction' and exclude it; all I'm doing in that case is
declaring that I'm playing a different game).
Page 321
The idea of inconsistency --this is what I'm insisting on--is contradiction [†1], and this can only arise in the
true/false game, i.e. only where we are making assertions.
Page 321
That is to say: A contradiction can only occur among the rules of the game. I can for instance have one rule
saying the white piece must jump over the black one.
Page 321
Now if the black is at the edge of the board, the rule breaks down. So the situation can arise where I don't
know what I'm meant to do. The rule doesn't tell me anything any more. What would I do in such a case? There's
nothing simpler than removing the inconsistency: I must make a decision, i.e. introduce another rule.
Page 321
By permission and prohibition, I can always only define a game, never the game. What Hilbert is trying to
show by his proof is that the axioms of arithmetic have the properties of the game, and that's impossible. It's as if
Hilbert would like to prove that a contradiction is inadmissible.
Page 321
Incidentally, suppose two of the rules were to contradict one another. I have such a bad memory that I never
notice this, but always forget one of the two rules or alternately follow one and then the other. Even in this case I
would say everything's in order.
Page Break 322
The rules are instructions how to play, and as long as I can play they must be all right. They only cease to be all right
the moment I notice that they are inconsistent, and the only sign for that is that I can't apply them any more. For the
logical product of the two rules is a contradiction, and the contradiction no longer tells me what to do. And so the
conflict only arises when I notice it. While I could play there was no problem.
Page 322
In arithmetic, too, we arrive at 'the edge of the chessboard', e.g. with the problem %. (Were I to say % = 1, I
could prove 3 = 5 and thus would come into conflict with the other rules of the game.)
Page 322
We see then that as long as we take the calculus as a calculus the question of consistency cannot arise as a
serious question at all. And so is consistency perhaps connected with the application of the calculus? With this in
mind, we must ask ourselves:
Page 322
What does it mean, to apply a calculus?
Page 322
It can mean two different things.
Page 322
1) We apply the calculus in such a way as to provide the grammar of a language. For, what is permitted or
forbidden by the rules then corresponds in the grammar to the words 'sense' and 'senseless'. For example: Euclidean
geometry construed as a system of syntactical rules according to which we describe spatial objects. 'Through any 2
points, a straight line can be drawn' means: a claim mentioning the line determined by these 2 points makes sense
whether it happens to be true or false.
Page 322
A rule of syntax corresponds to the position in the game. (Can the rules of syntax contradict one another?)
Syntax cannot be justified.
Page 322
2) A calculus can be applied so that true and false propositions correspond to the configurations of the
calculus. Then the calculus yields a theory, describing something.
Page 322
Newton's three laws have a completely different significance from those of geometry. There is a verification
for them--by experiments in physics. But there is no such thing as a justification for a game. That is highly
important. You can construe geometry in this way too, by taking it as the description of actual measurements. Now
we have claims before us, and claims can indeed be inconsistent.
Page Break 323
Page 323
Whether the theory can describe something depends on whether the logical product of the axioms is a
contradiction. Either I see straight off that they form a contradiction, in which case the situation's clear; or I don't
what then? Then there's a hidden contradiction present. For instance: Euclid's axioms together with the axiom 'The
sum of the angles of a triangle is 181°'. Here I can't see the contradiction straight off, since I can't see straight off that
a sum of 180° follows from the axioms.
Page 323
As long as we stay within the calculus, we don't have any contradiction. For s = 180°, s = 181° don't
contradict one another at all. We can simply make two different stipulations. All we can say is: the calculus is
applicable to everything to which it is applicable. Indeed, even here an application might still be conceivable, e.g. in
such a way that, measured by one method, the sum of the angles of a triangle comes to 180°, and by another to
181°. It's only a matter of finding a domain whose description requires the multiplicity possessed by the axioms.
Page 323
Now if a contradiction occurs at this point in a theory, that would mean that the propositions of the theory
couldn't be translated into statements about how a galvanometer needle is deflected, etc., any more. It might for
instance come out that the needle stays still or is deflected, and so this theory couldn't be verified.
Page 323
Unlike geometrical equations, Maxwell's equations don't represent a calculus, they are a fragment, a part of a
calculus.
Page 323
What does it mean, mathematics must be 'made secure' [†1]? What would happen if mathematics weren't
secure? For is it any kind of claim at all, to say that the axioms are consistent?
Page 323
Can one look for a contradiction? Only if there is a method for looking. There can be no such question as
whether we will ever come upon a contradiction by going on in accordance with the rules. I believe that's the crucial
point, on which everything
Page Break 324
depends in the question of consistency.
Page 324
WAISMANN ASKS: But doesn't it make sense to ask oneself questions about an axiom system? Let's
consider, for instance, the propositional calculus which Russell derives from 5 axioms. Bernays has shown one of
these axioms is redundant, and that just 4 will do. He has gone on to show that these axioms form a 'complete
system', i.e. that adding another axiom which can't be derived from these 4 makes it possible to derive any
proposition you write down whatever. This comes down to the same thing as saying that every proposition follows
from a contradiction. Now, isn't that a material insight into the Russellian calculus? Or, to take another case, I
choose 3 axioms. I can't derive the same set of propositions from these as I can from all 5. Isn't that a material
insight? And so can't you look upon a consistency proof as the recognition of something substantial?
Page 324
WITTGENSTEIN: If I first take 3 propositions and then 5 propositions, I can't compare the classes of
consequences at all unless I form a new system in which both groups occur.
Page 324
And so it isn't as if I have both systems--the one with 3 axioms and the one with 5 axioms--in front of me
and now compare them from outside. I can't do that any more than I can compare, say, the integers and the rational
numbers until I've brought them into one system. And I don't gain a material insight either; what I do is once more
to construct a calculus. And in this calculus the proposition 'The one class includes more than the other' doesn't
occur at all: that is the prose accompanying the calculus.
Page 324
Can one ask: When have I applied the calculus? Is it possible for me not to know whether I have applied the
calculus, and to have to wait until I have a consistency proof?
Page Break 325
Page 325
(Tuesday, 30 December, at Schlick's house)
WAISMANN reads out §117 and §118 of Frege's Grundgesetze.
Page 325
WITTGENSTEIN comments:
If one looks at this naïvely, the chief thing that strikes one is that mathematicians are always afraid of only
one thing, which is a sort of nightmare to them: contradiction. They're not at all worried, e.g. by the possibility of a
proposition's being a tautology, even though a contradiction is surely no worse than a tautology. In logic,
contradiction has precisely the same significance as tautology, and I could study logic just as well by means of
contradictions. A contradiction and a tautology of course say nothing, they are only a method for demonstrating the
logical interrelations between propositions.
Page 325
It's always: 'the law [Satz] of contradiction'. I believe in fact that the fear of contradiction is bound up with its
being construed as a proposition [Satz]: ~(p •~p). There's no difficulty in construing the law of contradiction as a
rule: I forbid the formation of the logical product p •~p. But the contradiction ~(p•~p) doesn't begin to express this
prohibition. How could it? The contradiction doesn't say anything at all, but the rule says something.
Page 325
WAISMANN REPEATS HIS QUESTION: You said that by prohibitions and permissions I can always only
determine a game, but never the game. But is that right? Imagine for instance the case where I permit any move in
chess and forbid nothing--would that still be a game? Mustn't the rules of a game then still have certain properties for
them to define a game at all? Couldn't we then interpret the demand for consistency as one to exclude the
'tautological' game--the game in which anything is permissible? That is to say, if the formula '0 ≠ 0' can be derived by
a legitimate proof and if in addition with Hilbert we add the axiom '0 ≠ 0 → ', where stands for an arbitrary
formula, then we can derive the formula from the inference pattern
Page Break 326
and write it down, too. But that means that in this case any formula can be derived, and so the game loses its
character and its interest.
Page 326
WITTGENSTEIN: Not at all! There's a mistake here, i.e. a confusion between 'a rule of the game' and
'position in the game'. This is how it is: the game is tautological if the rules of the game are tautological (i.e. if they no
longer forbid or permit anything); but that isn't the case here. This game, too, has its own particular rules: it's one
game among many, and that the configuration '0 ≠ 0' arises in it is neither here nor there. It's just a configuration
which arises in this game, and if I exclude it, then I have a different game before me. It simply isn't the case that in
the first instance I don't have a game before me but in the second I do. One class of rules and prohibitions borders
on another class of rules and prohibitions, but a game doesn't border on a non-game. The 'tautological' game must
arise as the limit case of games, as their natural limit. The system of games must be delimited from within, and this
limit consists simply in the fact that the rules of the game vanish. I can't get this limit case by myself setting up
particular rules and prohibitions; for that just gives me once more one game among many. So if I say: The
configuration '0 ≠ 0' is to be permitted, I am once more stating a rule, defining a game; merely a different one from
the one where I exclude this configuration.
Page 326
That is to say: by rules I can never define the game, always only a game.
Page 326
WAISMANN ASKS: There's a theory of chess, isn't there? So we can surely use this theory to help us obtain
information about the possibilities of the game--e.g. whether in a particular position I can force mate in 8, and the
like. If, now, there is a theory of the game of chess, then I don't see why there shouldn't also be a theory of the game
of arithmetic and why we shouldn't apply the propositions of this theory to obtain material information about the
possibilities of this game. This theory is Hilbert's metamathematics.
Page Break 327
Page 327
WITTGENSTEIN: What is known as the 'theory of chess' isn't a theory describing something, it's a kind of
geometry. It is of course in its turn a calculus and not a theory.
Page 327
To make this clear I ask you whether, in your opinion, there is a difference between the following two
propositions: 'I can force mate in 8' and 'By the theory I have proved I can force mate in 8'? No! For if in the theory I
use a symbolism instead of a chessboard and chess set, the demonstration that I can mate in 8 consists in my
actually doing it in the symbolism, and so, by now doing with signs what I do with pieces on the board. When I
make the moves and when I prove their possibility--then I have surely done the same thing over again in the proof.
I've made the moves with symbols, that's all. The only thing missing is in fact the actual movement; and of course
we agree that moving the little piece of wood across the board is inessential.
Page 327
I am doing in the proof what I do in the game, precisely as if I were to say: You, Herr Waismann, are to do a
sum, but I am going to predict what digits will be your result: I then simply do the sum for my own part, only
perhaps with different signs (or even with the same, which I construe differently). I can now work out the result of
the sum all over again; I can't come to the same result by a totally different route. It isn't as if you are the calculator
and I recognize the result of your calculation on the strength of a theory. And precisely the same situation obtains in
the case of the 'theory of chess'.
Page 327
And so if I establish in the 'theory' that such and such possibilities are present, I am again moving about
within the game, not within a metagame. Every step in the calculus corresponds to a move in the game, and the
whole difference consists only in the physical movement of a piece of wood.
Page 327
Besides, it is highly important that I can't tell from looking at the pieces of wood whether they are pawns,
bishops, rooks, etc.
Page Break 328
I can't say: that is a pawn and such and such rules hold for this piece. No, it is the rules alone which define this
piece: a pawn is the sum of the rules for its moves (a square is a piece too), just as in the case of language the rules
define the logic of a word.
Page 328
WAISMANN RAISES THE OBJECTION: Good, I can see all that. But so far we have only been dealing
with the case that the theory says such and such a position is possible. But what if the theory proves the
impossibility of a certain position--e.g. the 4 rooks in a straight line next to one another? And just this sort of case is
dealt with by Hilbert. In this case the theory simply cannot reproduce the game. The steps in the calculus no longer
correspond to moves in the game.
Page 328
WITTGENSTEIN: Certainly they don't. But even in this case it must come out that the theory is a calculus,
just a different one from the game. Here we have a new calculus before us, a calculus with a different multiplicity.
Page 328
In the first instance: If I prove that I cannot do such and such, I don't prove a proposition, I give an
induction.
Page 328
I can see the induction on the chessboard, too. I'll explain in a moment what I mean by that. What I prove is
that no matter how long I play I can't reach a particular position. A proof like that can only be made by induction. It
is now essential for us to be quite clear about the nature of proof by induction.
Page 328
In mathematics there are two sorts of proof:
Page 328
1) A proof proving a particular formula. This formula appears in the proof itself as its last step.
Page 328
2) Proof by induction. The most striking fact here is that the proposition to be proved doesn't itself occur in
the proof at all. That is to say, induction isn't a procedure leading to a proposition. Instead, induction shows us an
infinite possibility, and the essence of proof by induction consists in this alone.
Page 328
We subsequently express what the proof by induction showed us as a proposition and in doing so use the
word 'all'. But this proposition adds something to the proof, or better: the proposition stands
Page Break 329
to the proof as a sign does to what is designated. The proposition is a name for the induction. It goes proxy for it, it
doesn't follow from it.
Page 329
You can also render the induction visible on the chessboard itself, e.g., by my saying I can go there and back,
there and back, etc. But the induction no longer corresponds to a move in the game.
Page 329
So if I prove in the 'theory' that such and such a position cannot occur, I have given an induction which
shows something but expresses nothing. And so there is also no proposition at all in the 'theory' which says 'such
and such is impossible'.
Page 329
Now it will be said that there must still be some connection between the actual game and the induction. And
there is indeed such a connection--it consists in the fact that once I have been given the proof by induction I will no
longer try to set up this position in the game. Before I might perhaps have tried and then finally given up. Now I
don't try any more. It's exactly the same here as when I prove by induction that there are infinitely many primes or
that is irrational. The effect of these proofs on the actual practice of arithmetic is simply that people stop
looking for a 'greatest prime number', or for a fraction equal to . But here we must be even more precise than
this. Could people previously search at all? What they did had, to be sure, a certain likeness to searching, but was
something of an entirely different sort; they did something, expecting that as a result something else would occur.
But that wasn't searching at all, any more than I can search for a way of waggling my ears. All I can do is move my
eyebrows, my forehead and such parts of my body in the hope that my ears will move as well. But I can't know
whether they will, and thus I can't search for a way of making them.
Page 329
Within the system in which I tell that a number is prime, I can't even ask what the number of primes is. The
question only arises when you apply the substantival form to it. And if you have discovered the induction, that's also
something different from computing a number.
Page Break 330
Page 330
The inductions correspond to the formulae of algebra (calculation with letters)--for the reason that the
internal relations between the inductions are the same as the internal relations between the formulae. The system of
calculating with letters is a new calculus; but it isn't related to ordinary numerical calculation as a metacalculus to a
calculus. Calculation with letters isn't a theory. That's the crucial point. The 'theory' of chess--when it investigates
the impossibility of certain positions--is like algebra in relation to numerical calculation. In the same way, Hilbert's
'metamathematics' must reveal itself to be mathematics in disguise.
Page 330
Hilbert's proof: ('Neubegründung der Mathematik', 1922)
Page 330
'But if our formalism is to provide a complete substitute for the earlier theory which actually consists of
inferences and statements, then material contradiction must find its formal equivalent' [†1] a = b and a ≠ b are never
to be simultaneously provable formulae.
Page 330
The demonstration of consistency on Hilbert's simple model does in fact turn out to be inductive: the proof
shows us by induction the possibility that signs must continue to occur on and on →.
Page 330
The proof lets us see something. But what it shows can't be expressed by a proposition. And so we also can't
say: 'The axioms are consistent'. (Any more than you can say there are infinitely many primes. That is prose.)
Page 330
I believe giving a proof of consistency can mean only one thing: looking through the rules. There's nothing
else for me to do. Imagine I give someone a long list of errands to run in the town. The list is so long that perhaps I
have forgotten one errand and given another instead, or I have put together different people's errands. What am I to
do to convince myself that the errands can be
Page Break 331
carried out? I have to run through the list. But I can't 'prove' anything. (We mustn't forget that we are only dealing
here with the rules of the game and not with its configurations. In the case of geometry, it's perfectly conceivable
that when I go through the axioms I don't notice the contradiction.) If I say: I want to check whether the logical
product is a contradiction, this comes to the same thing. Writing it out in the form of a contradiction only makes the
matter easier. If you now choose to call that a 'proof', you are welcome to do so: even then it is just a method of
making it easier to check. But it must be said: in and of itself even such a proof cannot guarantee that I don't
overlook something.
Page 331
No calculation can do what checking does.
Page 331
But suppose I search systematically through the rules of the game? The moment I work within a system, I
have a calculus once more; but then the question of consistency arises anew. And so in fact there is nothing for me
to do but inspect one rule after another.
Page 331
What would it mean if a calculus were to yield '0 ≠ 0'? It's clear we wouldn't then be dealing with a sort of
modified arithmetic, but with a totally different kind of arithmetic, one without the slightest similarity to cardinal
arithmetic. We couldn't then say: in such and such respects it still accords with our arithmetic (as non-Euclidean
geometry does with Euclidean--here the modification of an axiom doesn't have such radical implications); no, there
wouldn't be the slightest trace of similarity left at all. Whether I can apply such a calculus is another matter.
Page 331
Besides, there are various difficulties here. In the first instance, one thing isn't clear to me: a = b surely only
expresses the substitutability of b for a. And so the equation is a rule for signs, a rule of the game. Then how can it
be an axiom, i.e. a position in the game? From this angle a formula such as 0 ≠ 0 is utterly unintelligible.
Page Break 332
For it would in fact mean: you can't substitute 0 for 0--am I then supposed to inspect whether perhaps the one 0 has
a flourish which the other doesn't? What on earth does such a prohibition mean? It's the same story as if I say a = a.
That too is rubbish, no matter how often it gets written down. Schoolmasters are perfectly right to teach their
children that 2 + 2 = 4, but not that 2 = 2. The way children learn to do sums in school is already in perfect order and
there's no reason to wish it any stricter. In fact it's also evident that a = a means nothing from the fact that this
formula is never used.
Page 332
If, when I was working with a calculus, I arrive at the formula 0 ≠ 0, do you think that as a result the calculus
would lose all interest?
Page 332
SCHLICK: Yes, a mathematician would say such a thing was of no interest to him.
Page 332
WITTGENSTEIN: But excuse me! It would be enormously interesting that precisely this came out! In the
calculus, we are always interested in the result. How strange! This comes out here--and that there! Who would have
thought it? Then how interesting if it were a contradiction which came out! Indeed, even at this stage I predict a time
when there will be mathematical investigations of calculi containing contradictions, and people will actually be proud
of having emancipated themselves even from consistency.
Page 332
But suppose I want to apply such a calculus? Would I apply it with an uneasy conscience if I hadn't already
proved there was no contradiction? But how can I ask such a question? If I can apply a calculus, I have simply
applied it; there's no subsequent correction. What I can do, I can do. I can't undo the application by saying: strictly
speaking that wasn't an application.
Page 332
Need I first wait for a consistency proof before applying a calculus? Have all the calculations people have
done so far been really--sub specie aeterni--a hostage against fortune? And is it conceivable that a time will come
when it will all be shown to be illegitimate? Don't I know what I'm doing? It in fact amounts to a wish to prove that
certain propositions are not nonsense.
Page Break 333
Page 333
So the question is this: I have a series of propositions, e.g. p, q, r... and a series of rules governing operations,
e.g. '•', '∨', '~', and someone asks: if you apply these rules concerning the operations to the given propositions, can
you ever arrive at nonsense? The question would be justified if by 'nonsense' I mean contradiction or tautology. I
must then make the rules for forming statements in such a way that these forms do not occur.
Page 333
(1 January 1931, at Schlick's house)
Is one justified in raising the question of consistency? The strange thing here is that we are looking for
something and have no idea what it really is we are looking for. How, for example, can I ask whether Euclidean
geometry is consistent when I can't begin to imagine what it would be for it to contain a contradiction? What would
it be like for there to be a contradiction in it? This question has first to be answered before we investigate such
questions.
Page 333
One thing is clear: I can only understand a contradiction [Widerspruch], if it is a contradiction
[Kontradiktion] [†1]. So I put forward the case in which I have a series of propositions, let's say p, q, r..., and form
their logical product. I can now check whether this logical product is a contradiction. Is that all the question of
consistency amounts to? That would take five minutes to settle. Surely in this sense, no one can doubt whether the
Euclidean axioms are consistent.
Page 333
But what else could the question mean? Perhaps: if we go on drawing inferences sooner or later we will
arrive at a contradiction? To that we must say: Have we a method for discovering the contradiction? If not, then
there isn't a question here at all. For we cannot search to infinity.
Page 333
WAISMANN: Perhaps we can still imagine something--namely the schema for indirect proof. By analogy,
we transfer that to an axiom system. We must distinguish two things: a problem which can be formulated within
mathematics, which therefore also already possesses a method of solution; and the idea which
Page Break 334
precedes and guides the construction of mathematics. Mathematicians do in fact have such guiding ideas even in the
case of Fermat's Last Theorem. I'm inclined to think that the question of consistency belongs to this complex of
pre-mathematical questioning.
Page 334
WITTGENSTEIN: What's meant by analogy? E.g. analogy with indirect proof? Here it's like the trisection of
an angle. I can't look for a way to trisect an angle. What really happens when a mathematician concerns himself with
this question? Two things are possible: (1) He imagines the angle divided into 3 parts (a drawing); (2) He thinks of
the construction for dividing an angle into 2 parts, into 4 parts. And this is where the mistake occurs: people think,
since we can talk of dividing into 2, into 4 parts, we can also talk of dividing into 3 parts, just as we can count 2, 3
and 4 apples. But trisection--if there were such a thing--would in fact belong to a completely different category, a
completely different system, from bisection, quadrisection. In the system in which I talk of dividing into 2 and 4
parts I can't talk of dividing into 3 parts. These are completely different logical structures. I can't group dividing into
2, 3, 4 parts together since they are completely different forms. You can't count forms as though they were actual
things. You can't bring them under one concept.
Page 334
It's like waggling your ears. The mathematician naturally lets himself be led by associations, by certain
analogies with the previous system. I'm certainly not saying: if anyone concerns himself with Fermat's Last
Theorem, that's wrong or illegitimate. Not at all! If, for instance, I have a method for looking for whole numbers
satisfying the equation x2 + y2 = z2, the formula xn + yn = zn can intrigue me. I can allow myself to be intrigued by a
formula. And so I shall say: there's a fascination here but not a question. Mathematical 'problems' always fascinate
like this. This kind of fascination is in no way the preparation of a calculus.
Page 334
WAISMANN: But what then is the significance of the proof of the consistency of non-Euclidean geometry?
Let's think of the simplest case, that we give a model, say, of two-dimensional Riemannian
Page Break 335
geometry on a sphere. We then have a translation: every concept (theorem) of the one geometry corresponds to a
concept (theorem) of the other. If now the theorems in the one case were to include a contradiction, this
contradiction would also have to be detectable in the other geometry. And so we may say: the Riemannian axiom
system is consistent provided the Euclidean one is. We have then demonstrated consistency relative to Euclidean
geometry.
Page 335
WITTGENSTEIN: Consistency 'relative to Euclidean geometry' is altogether nonsense. What happens here is
this: a rule corresponds to another rule (a configuration in the game to a configuration in the game). We have a
mapping. Full stop! Whatever else is said is prose. One says: Therefore the system is consistent. But there is no
'therefore', any more than in the case of induction. This is once more connected with the proof being misconstrued,
with something being read into it which it doesn't contain. The proof is the proof. The internal relations in which
the rules (configurations) of one group stand to one another are similar to those in which the rules
(configurations) of the other group stand. That's what the proof shows and no more.
Page 335
On Independence: Let's suppose we have 5 axioms. We now make the discovery that one of these axioms
can be derived from the other four, and so was redundant. I now ask: What's the significance of such a discovery? I
believe that the situation here is exactly as it is in the case of Sheffer's discovery that we can get by with one logical
constant.
Page 335
Above all, let's be clear: the axioms define--when taken with the rules for development of the calculus--a
group of propositions. This domain of propositions isn't also given to us in some other way, but only by the 5
axioms. So we can't ask: Is the same domain perhaps also defined by 4 axioms alone? For the domain isn't
detachable from the 5 axioms. These 5 axioms and whatever derives from them are--so to speak--my whole world. I
can't get outside of this world.
Page 335
Now how about the question: are these 5 axioms independent?
Page Break 336
I would reply: is there a method for settling this question? Here various cases can arise.
Page 336
1) There is no such method. Then the situation is as I've described it: All that I have are the 5 axioms and the
rules of development. In that case I can't seek to find out whether one of these axioms will perhaps ever emerge as a
consequence of the others. And so I can't raise the question of independence at all.
Page 336
2) But suppose now that one of the axioms does come out as the result of a proof, then we haven't at all
proved that only 4 axioms are sufficient, that one is redundant. No, I can't come to this insight through a logical
inference, I must see it, just as Sheffer saw that he could get by with one logical constant. I must see the new system
in the system within which I am working and in which I perform the proof.
Page 336
It's a matter of seeing, not of proving. No proposition corresponds to what I see--to the possibility of the
system. Nothing is claimed, and so neither is there anything I can prove. So if in this case I give 5 axioms where 4
would do, I have simply been guilty of an oversight. For I should certainly have been in a position to know at the
outset that one of these 5 axioms was redundant, and if I have written it down all the same, that was simply a
mistake. Granted, it isn't enough in this case simply to set up the axioms, we must also prove that they really are
independent.
Page 336
Now Hilbert appears to adopt this last course in the case of geometry. However, one important point remains
unclear here: is the use of models a method? Can I look for a model systematically, or do I have to depend on a
happy accident? What if I can't find a model which fits?
Page 336
To summarize: The question whether an axiom system is independent only makes sense if there is a method
for deciding the question. Otherwise we can't raise this question at all. And if,
Page Break 337
e.g., someone discovers one axiom to be redundant, then he hasn't proved a proposition, he has read a new system
into the old one.
Page 337
And the same goes for consistency.
Page 337
Hilbert's axioms:
I.1 and I.2: 'Any two distinct points A, B define a straight line a'
'Any two distinct points on a straight line define that line.'
Page 337
I already don't know how these axioms are to be construed, what their logical form is.
Page 337
WAISMANN: You can of course write them as truth functions by saying, e.g.: 'for all x, if x is a point,
then...', I believe, however, that we miss the real sense of the axioms in this way. We mustn't introduce the points
one after another. It seems much more correct to me to introduce the points, lines, planes by means of co-ordinates
as it were at one fell swoop.
Page 337
WITTGENSTEIN: That's my view too. But one thing I don't understand: What would it mean to say that
these axioms form a contradiction? The position is that as they stand they can't yield a contradiction, unless I
determine by a rule that their logical product is a contradiction. That is, the situation with contradiction here is just as
it is with the incompatibility of the propositions: 'This patch is green' and 'This patch is red'. As they stand, these
propositions don't contradict one another at all. They only contradict one another once we introduce a further rule of
syntax forbidding treating both propositions as true. Only then does a contradiction arise. (Cf. above p. 323: the
example with 'the sum of the angles of a triangle = 180°'.)
Page 337
(September 1931)
WAISMANN ASKS: (You said earlier) there was no contradiction at all in the calculus. I now don't
understand how that fits in with the nature of indirect proof, for this proof of course depends precisely on producing
a contradiction in the calculus.
Page Break 338
Page 338
WITTGENSTEIN: What I mean has nothing at all to do with indirect proof. There's a confusion here. Of
course there are contradictions in the calculus; all I mean is this: it makes no sense to talk of a hidden contradiction.
For what would a hidden contradiction be? I can, e.g., say: the divisibility of 357, 567 by 7 is hidden, that is, as long
as I have not applied the criterion--say the rule for division. In order to turn the divisibility from a hidden one into an
open one, I need only apply the criterion. Now, is it like that with contradiction? Obviously not. For I can't bring the
contradiction into the light of day by applying a criterion. Now I say: then all this talk of a hidden contradiction
makes no sense, and the danger mathematicians talk about is pure imagination.
Page 338
You might now ask: But what if one day a method were discovered for establishing the presence or absence
of a contradiction? This proposition is very queer. It makes it look as if you could look at mathematics on an
assumption, namely the assumption that a method is found. Now I can, e.g., ask whether a red-haired man has been
found in this room, and this question makes perfect sense, for I can describe the man even if he isn't there. Whereas
I can't ask after a method for establishing a contradiction, for I can only describe it (the method) once it's there. If it
hasn't yet been discovered, I'm in no position to describe it, and what I say are empty words. And so I can't even
begin to ask the question what would happen if a method were discovered.
Page 338
The situation with the method for demonstrating a contradiction is precisely like that with Goldbach's
conjecture: what is going on is a random attempt at constructing a calculus. If the attempt succeeds I have a calculus
before me again, only a different one from that which I have used till now. But I still haven't proved the calculus is a
calculus, nor can that be proved at all.
Page 338
If someone were to describe the introduction of irrational
Page Break 339
numbers by saying he had discovered that between the rational points on a line there were yet more points, we
would reply: 'Of course you haven't discovered new points between the old ones: you have constructed new points.
So you have a new calculus before you.' That's what we must say to Hilbert when he believes it to be a discovery
that mathematics is consistent. In reality the situation is that Hilbert doesn't establish something, he lays it down.
When Hilbert says: 0 ≠ 0 is not to occur as a provable formula, he defines a calculus by permission and prohibition.
Page 339
WAISMANN ASKS: But still you said a contradiction can't occur in the calculus itself, only in the rules. The
configurations can't represent a contradiction. Is that still your view now?
Page 339
WITTGENSTEIN: I would say that the rules, too, form a calculus, but a different one. The crucial point is for
us to come to an understanding on what we mean by the term contradiction. For if you mean one thing by it and I
mean another, we can't reach any agreement.
Page 339
The word 'contradiction' is taken in the first instance from where we all use it, namely from the truth
functions, and mean, say, p • ~p. So in the first instance we can only talk of a contradiction where it's a matter of
assertions. Since the formulae of a calculus are not assertions, there can't be contradictions in the calculus either. But
of course you can stipulate that a particular configuration of the calculus, e.g. 0 ≠ 0, is to be called a contradiction.
Only then there is always the danger of your thinking of contradiction in logic and so confounding 'contradictory'
and 'forbidden'. For if I call a particular configuration of signs in the calculus a contradiction, then that only means
that the formation of this configuration is forbidden: if in a proof you stumble upon such a formula, something must
be done about it, e.g. the opening formula must be struck out.
Page 339
In order to avoid this confusion, I should like to propose that in place of the word 'contradiction' we use a
completely new sign which has no associations for us except what we have explicitly
Page Break 340
laid down: let's say the sign S. In the calculus, don't take anything for granted. If the formula S occurs, as yet that
has no significance at all. We have first to make further stipulations.
Page 340
WAISMANN ASKS: An equation in arithmetic has a twofold significance: it is a configuration and it is a
substitution rule. Now what would happen if in arithmetic or analysis a proof of the formula 0 ≠ 0 were to be found?
Then arithmetic would have to be given a completely different meaning, since we wouldn't any longer be entitled to
interpret an equation as a substitution rule. 'You cannot substitute 0 for 0' certainly doesn't mean anything. A disciple
of Hilbert could now say: there you see what the consistency proof really achieves. Namely this proof is intended to
show us that we are entitled to interpret an equation as a substitution rule.
Page 340
WITTGENSTEIN: That, of course, is something it can't mean. Firstly: how does it come about that we may
interpret an equation as a substitution rule? Well, simply, because the grammar of the word 'substitute' is the same as
the grammar of an equation. That is why there is from the outset a parallelism between substitution rules and
equations. (Both, e.g., are transitive.) Imagine I were to say to you 'You cannot substitute a for a'. What would you
do?
Page 340
WAISMANN: I wouldn't know what to think, since this claim is incompatible with the grammar of the word
'substitute'.
Page 340
WITTGENSTEIN: Good, you wouldn't know what to think, and quite right too, for in fact you have a new
calculus in front of you, one with which you're unfamiliar. If I now explain the calculus to you by giving the
grammatical rules and the application, then you will also understand the claim 'You cannot substitute a for a'. You
can't understand this claim while you remain at the standpoint of the old calculus. Now, if say the formula 0 ≠ 0
could be proved, that would only mean we have two different calculi before us: one calculus which is the grammar
of the verb 'substitute', and another in which the formula 0 ≠ 0 can be proved. These two calculi would then exist
alongside one another.
Page 340
If someone now wanted to ask whether it wouldn't be possible
Page Break 341
to prove that the grammar of the word 'substitute' is the same as the grammar of an equation, i.e. that an equation
can be interpreted as a substitution rule, we should have to reply: there can be no question of a proof here. For how
is the claim supposed to run which is to be proved? That I apply the calculus, after all, only means that I set up rules
telling me what to do when this or that comes out in the calculus. Am I then supposed to prove that I have set up
rules? For surely that's the only sense the question whether I have applied the calculus can have. I once wrote: the
calculus is not a concept of mathematics.
Page 341
WAISMANN: You said on one occasion that no contradictions can occur in the calculus. If, e.g., we take the
axioms of Euclidean geometry and add in the further axiom: 'The sum of the angles of a triangle is 181°', even this
would not give rise to a contradiction. For it could of course be that the sum of the angles has two values, just as
does . Now if you put it like that, I no longer understand what an indirect proof achieves. For indirect proof
rests precisely on the fact that a contradiction is derived in the calculus. Now what happens if I lay down as an
axiom an assumption which has been refuted by an indirect proof? Doesn't the system of axioms thus extended
present a contradiction? For instance: in Euclidean geometry it is proved that you can only drop one perpendicular
from a point onto a straight line, what's more by an indirect proof. For suppose there are two perpendiculars, then
these would form a triangle with two right angles, the sum of whose angles would be greater than 180°, contradicting
the well known theorem about the sum of the angles. If I now lay down the proposition 'there are two
perpendiculars' as an axiom and add the rest of the axioms of Euclidean geometry--don't I now obtain a
contradiction?
Page 341
WITTGENSTEIN: Not at all. What is indirect proof? Manipulating signs. But that surely isn't everything. A
further rule now comes in, too, telling me what to do when an indirect proof is performed. (The rule, e.g., might run:
if an indirect proof has been performed,
Page Break 342
all the assumptions from which the proof proceeds may not be struck out.) Nothing is to be taken for granted here.
Everything must be explicitly spelt out. That this is so easily left undone is bound up with the fact that we can't
break away from what the words 'contradiction' etc. mean in ordinary speech.
Page 342
If I now lay down the axiom 'Two perpendiculars can be dropped from one point onto a straight line', the
sign picture of an indirect proof is certainly contained in this calculus. But we don't use it as such.
Page 342
What then would happen if we laid down such an axiom? 'I'd reach a point where I wouldn't know how to go
on.' Quite right, you don't know how to go on because you have a new calculus before you with which you are
unfamiliar. A further stipulation needs making.
Page 342
WAISMANN: But you could always do that when an indirect proof is given in a normal calculus. You could
retain the refuted proposition by altering the stipulation governing the use of indirect proof, and then the proposition
simply wouldn't be refuted any more.
Page 342
WITTGENSTEIN: Of course we could do that. We have then simply destroyed the character of indirect
proof, and what is left of the indirect proof is the mere sign picture.
Page 342
(December 1931)
WAISMANN formulates the problem of consistency:
Page 342
The significance of the problem of consistency is as follows: How do I know that a proposition I have proved
by transfinite methods cannot be refuted by a finite numerical calculation? If, e.g., a mathematician finds a proof of
Fermat's Last Theorem which uses essentially transfinite methods--say the Axiom of Choice, or the Law of the
Excluded Middle in the form: either Goldbach's conjecture holds for all numbers, or there is a number for which it
doesn't hold--how do I know that such a theorem can't be refuted by a counter-example? That is not in the least
self-evident. And yet it is remarkable that mathematicians place so
Page Break 343
much trust in the transfinite modes of inference that, once such a proof is known, no one would any longer try to
discover a counter example. The question now arises: Is this trust justifiable? That is, are we sure that a proposition
which has been proved by transfinite methods can never be refuted by a concrete numerical calculation? That's the
mathematical problem of consistency.
Page 343
I will at the same time show how the matter seems to stand to me by raising the analogous question for
ordinary algebra. How do I know, when I have proved a theorem by calculating with letters, that it can't be refuted
by a numerical example? Suppose, e.g., I've proved that 1 + 2 + 3 +... + n = [n(n + 1)]/2--how do I know this formula
can survive the test of numerical calculation? Here we have precisely the same situation. I believe we have to say the
following: the reason why a calculation made with letters and a numerical calculation lead to the same result, i.e., the
reason why calculating with letters can be applied to concrete numbers, lies in the fact that the axioms of the letter
calculus--the commutative, associative laws of addition etc.--are chosen from the outset so as to permit such an
application. This is connected with the fact that we choose axioms in accordance with a definite recipe. That is, an
axiom corresponds to an induction, and this correspondence is possible because the formulae possess the same
multiplicity as the induction, so that we can project the system of induction onto the system of formulae. And so
there is no problem in this case, and we can't raise the question at all whether a calculation with letters can even
come into conflict with a numerical calculation. But what about analysis? Here there really seems at first sight to be a
problem.
Page 343
WITTGENSTEIN: First of all: what are we really talking about? If by 'trust' is meant a disposition, I would
say: that is of no interest to me. That has to do with the psychology of mathematicians. And so presumably
something else is meant by 'trust'. Then it can only be something which can be written down in symbols. What we
Page Break 344
have to ask about seems to be the reason why two calculations tally. Let's take a quite simple example:
2 + (3 + 4) = 2 + 7 = 9
(2 + 3) + 4 = 5 + 4 = 9
I have here performed two independent calculations and arrived on both occasions at the same result. 'Independent'
here means: the one calculation isn't a copy of the other. I have two different processes.
Page 344
And what if they didn't agree? Then there's simply nothing I can do about it. The symbols would then just
have a different grammar. The associative and commutative laws of addition hold on the basis of grammar. But in
group theory it's no longer the case that AB = BA; and so, e.g., we couldn't calculate in two ways, and yet we still
have a calculus.
Page 344
This is how things are: I must have previously laid down when a calculation is to be correct. That is, I must
state in what circumstances I will say a formula is proved. Now if the case arose that a formula counted as having
been proved on the basis of one method, but as refuted on the basis of another, then that wouldn't in the least imply
we now have a contradiction and are hopelessly lost; on the contrary we can say: the formula simply means different
things. It belongs to two different calculi. In the one calculus it's proved, in the other refuted. And so we really have
two different formulae in front of us which by mere accident have their signs in common.
Page 344
A whole series of confusions has arisen around the question of consistency.
Page 344
Firstly, we have to ask where the contradiction is supposed to arise: in the rules or in the configurations of
the game.
Page 344
What is a rule? If, e.g., I say 'Do this and don't do this', the other doesn't know what he is meant to do; that is,
we don't allow a contradiction to count as a rule. We just don't call a contradiction a rule--or more simply the
grammar of the word 'rule' is such that a contradiction isn't designated as a rule. Now if a contradiction
Page Break 345
occurs among my rules, I could say: these aren't rules in the sense I normally speak of rules. What do we do in such
a case? Nothing could be more simple: we give a new rule and the matter's resolved.
Page 345
An example here would be a board game. Suppose there is one rule here saying 'black must jump over
white'. Now if the white piece is at the edge of the board, the rule can't be applied any more. We then simply make a
new stipulation to cover this case, and with that the difficulty is wiped off the face of the earth.
Page 345
But here we must be even more precise. We have here a contradiction (namely between the rules 'white must
jump over black' and 'you may not jump over the edge of the board'). I now ask: Did we possess a method for
discovering the contradiction at the outset? There are two possibilities here:
Page 345
1) In the board game this possibility was undoubtedly present. For the rule runs: 'In general...' If this means
'in this position and in this position...' then I obviously had the possibility right at the beginning of discovering the
contradiction--and if I haven't seen it, that's my fault. Perhaps I've been too lazy to run through all the cases, or I
have forgotten one. There's no serious problem present in this case at all. Once the contradiction arises, I simply
make a new stipulation and so remove it. We can always wipe the contradiction off the face of the earth.
Page 345
But whether there is a contradiction can always be settled by inspecting the list of rules. That is, e.g., in the
case of Euclidean geometry a matter of five minutes. The rules of Euclidean geometry don't contradict one another,
i.e., no rule occurs which cancels out an earlier one (p and ~p), and with that I'm satisfied.
Page 345
2) But now let's take the second case, that we have no such method. My list of rules is therefore in order. I
can't see any contradiction. Now I ask: Is there now still any danger? It's out of the question. What are we supposed
to be afraid of? A contradiction? But a contradiction is only given me with the method for discovering it. As long as
the contradiction hasn't arisen, it's no concern of mine. So I can quite happily go on calculating. Would
Page Break 346
the calculations mathematicians have made through the centuries suddenly come to an end because a contradiction
had been found in mathematics? Would we say they weren't calculations? Certainly not. If a contradiction does
arise, we will simply deal with it. But we don't need to worry our heads about it now.
Page 346
What people are really after is something quite different. A certain paradigm hovers before their mind's eye,
and they want to bring the calculus into line with this paradigm.
Page Break 347
EDITOR'S NOTE
Page 347
Our text is a typescript that G. E. Moore gave us soon after Wittgenstein's death: evidently the one which
Wittgenstein left with Bertrand Russell in May, 1930, and which Russell sent to the Council of Trinity College,
Cambridge, with his report in favour of a renewal of Wittgenstein's research grant.[†1] All the passages in it were
written in manuscript volumes between February 2nd, 1929, and the last week of April, 1930. The latest manuscript
entry typed and included here is dated April 24th, 1930. On May 5th, 1930, Russell wrote to Moore that he had 'had
a second visit from Wittgenstein.... He left me a large quantity of typescript which I am to forward to Littlewood as
soon as I have read it.' [J. E. Littlewood, F.R.S., Fellow of Trinity College.] In his report to the Council, dated May
8th, Russell says he spent five days in discussion with Wittgenstein, 'while he explained his ideas, and he left with
me a bulky typescript, Philosophische Bemerkungen, of which I have read about a third. The typescript, which
consists merely of rough notes, would have been very difficult to understand without the help of the conversations'.
The Council had authorized Moore to ask Russell for a report, and I suppose they asked Moore to return the
typescript to Wittgenstein. Wittgenstein asked Moore to take care of it. Apparently he did not keep a copy himself.
Page 347
Wittgenstein had made an earlier visit to Russell, in March, 1930, and Russell wrote to Moore then that 'he
intends while in Austria to make a synopsis of his work which would make it much easier for me to report
adequately..... He intends to visit me again in Cornwall just before the beginning of the May term, with his synopsis'.
Wittgenstein can hardly have called the Bemerkungen a synopsis when he gave it to Russell; nor did Russell when
he wrote to Moore and then to the Council about it. On the other hand it
Page Break 348
is a selection and a rearrangement of passages from what he had in manuscript and in typescript at the time. It was
made from slips cut from a typescript and pasted into a blank ledger in a new order. The 'continuous' typescript was
made from Wittgenstein's manuscript volumes I, II, III and the first half of IV. In the typing Wittgenstein left out
many of the manuscript passages--more from I and II than from the others. In this typing he changed the order of
some large blocks of material. This was nothing like his rearrangement of the slips he cut from it for the
Philosophische Bemerkungen as Moore gave it to us.
Page 348
It was not 'merely rough notes'. But it was not easy to read, and Wittgenstein would not have published it
without polishing. No spacing showed where a group of remarks hang closely together and where a new topic
begins. Paragraphs were not numbered. And it was hard to see the arrangement and unity of the work until one had
read it a number of times. (Russell did not have time to read it through once.) If he had thought of making this
typescript more übersichtlich Wittgenstein might have introduced numbers for paragraphs, as he did in the Brown
Book and in the Investigations. He might have divided into chapters, as he did with the typescript of 1932/33 (No.
213)--although he never did this again. (See Part II of Philosphische Grammatik.) The 1932/33 typescript was the
only one for which he made a table of contents. We cannot guess whether he would have tried to do this for the
Philosophische Bemerkungen. For the Blue Book he gave neither numbers nor chapter divisions. If he did at first
think of the Bemerkungen which he left with Russell as a 'synopsis' of his writing over the past 18 months, he might
have thought of some system of numbering (I am thinking of the Tractatus) which would show the arrangement.
But I have found no reference or note in this sense.
Page 348
He went on writing directly after he had seen Russell--from the entries in the manuscript volume you would
think there had not been a break. He had kept a copy of the typescript he had cut into slips to form the
Philosophische Bemerkungen. (The top copy, in fact--and the least legible, for the first 60 pages of it: his typewriter
ribbon was exhausted and hardly left ink marks. Later he used to dictate to a professional typist.) In a number of
places he
Page Break 349
added to what he had typed or revised it--writing in pencil or ink on the back of the typed sheets or between the lines
or in the margins. And in certain places he took pages from this copy and cut sections from it--pasting them into
manuscript books or clipping them together with slips from other typescripts to form the 1932/33 typescript. In one
or two places I have quoted a pencilled revision in a footnote here. But I do not think generally that the revisions he
made there can be counted as revisions of the typescript (with its title page and motto) which he had left with
Russell. They belonged to a far-reaching change and development in Wittgenstein's thinking, and a different book.
Page 349
He had arrived in Cambridge sometime in January, 1929, and began his first manuscript volume on February
2nd. (Manuscript 'volume', as distinct from 'notebook': it was generally a hard covered notebook, often a large ledger.
Such a volume is never rough notes, as the paper-covered and pocket notebooks often are.) Volumes I and II cannot
be separated, for he wrote at first on the right-hand pages only; when he reached the end of volume II in this way he
continued what he was writing on the left-hand pages of volume II, and when he came to the end of the volume on
this side he continued on the left-hand pages of volume I--and on to the end of I. Volume III begins where this
leaves off. He started III at the end of August or the beginning of September, 1929. (After the first week he did not
date his entries in I and II. The first date in III comes on page 87 of the manuscript. It is: 11.9.29.)
Page 349
He wrote Some Remarks on Logical Form to read to the Aristotelian Society and Mind Association Joint
Session in July, 1929--but he disliked the paper and when the time came he ignored it and talked about infinity in
mathematics. In volumes I and II there are discussions of suggestions he makes in the Logical Form paper. The
Philosophische Bemerkungen has a few of these; but many--those central to the paper, I think--are left out. Some
remarks about 'phenomenological language' may refer to the earlier view in that paper. And the first sentence in §46
may refer to the example using coordinates there.
Page 349
In Appendix I: Wittgenstein's typescript of 'Complex and Fact'
Page Break 350
(Komplex und Tatsache) is taken from manuscript volume VI, in an entry dated 30.6.31.--more than a year after the
last passage in the Bemerkungen. It is the first of three sections of what he had put together as a single typescript
(now printed together in Philosophische Grammatik). These are numbered consecutively as forming a single essay,
although they do not come close to one another in the manuscripts. 'Infinitely Long' and 'Infinite Possibility' are
numbered consecutively, as though forming one essay, as well. Both groups seem to have been typed along with the
1932/33 typescript and Wittgenstein kept them together with it, but they do not form part of it.
Page 350
Of the two sections that follow 'Complex and Fact' in the typescript, about two-thirds of the remarks are here
in Philosophische Bemerkungen, especially §93 to the end of §98. In the manuscript (volume VI) the discussion of
'complex ≠ fact' grows out of 'I describe the fact which would be the fulfilment' of an order or of an expectation--and
the apparent difficulty that this description is 'general in character', whereas the 'fulfilment' when it arrives is
something you can point to. As though I had painted the apple that was going to grow on this branch, and now the
apple itself is there: as though the grown apple was the event I was expecting. But the apple is not an event at all.
'The fact (or event) is described in general terms.' But then the fact is treated as though it were on all fours with a
house or some other sort of complex. I expected the man, and my expectation is fulfilled. But the man I expected is
not the fulfilment; but that he has come. This leads directly to 'complex ≠ fact'. And this was important for the
understanding of 'describing'.
Page 350
Expectation and fulfilment have a different role in mathematics--not like their role in experience. And in
mathematics there is not this distinction.
'In mathematics description and object are equivalent. "The fifth number in this series of numbers has these properties"
says the same as "5 has these properties". The properties of a house do not follow from its position in a row of houses;
but the properties of a number are the properties of a position.'
Page 350
You cannot describe what you expect to happen in the development of a decimal fraction, except by writing
out the calculation
Page Break 351
in which it comes. Wittgenstein used to put this by saying 'the description of a calculation accomplishes the same as
the calculation; the description of a language accomplishes the same as the language'.
Page 351
The passage I have quoted comes just before the discussion of 'Infinitely Long' in the manuscript (volume
IX). Revising his account of 'description' brought with it a revision of the account of the application of mathematics,
and especially of the use of expressions like 'infinity' in the mathematical sense for the description of physical
happenings.
RUSH RHEES
Page Break 352
TRANSLATORS' NOTE
Page 352
This note, like the translation itself, runs the risk of overlying and obscuring the original. Yet we very much
wish to express our thanks for help received, and we also feel we should mention one or two significant areas where
we are conscious of having only half-resolved the difficulties of translation.
Page 352
Wittgenstein's style often depends on repeating certain words or groups of words in order to create a series of
sometimes unexpected interrelations and back-references within the discussion. Some of these terms, e.g. Raum
[space], have a straightforward English equivalent, whereas others prove less tractable. In such cases to use the same
word in English throughout would unduly strain against the natural bent of English usage, whilst varying the English
equivalents blunts the point of Wittgenstein's remarks. One term important enough to single out is Bild: 'picture'
covers much the same area as this term in Wittgenstein's usage, but various phrases, e.g. Erinnerungsbild [memory
image] are beyond its natural scope. For Maßstab we have usually said 'yardstick', since like Maßstab it can be used
to convey a general notion of measure or standard whilst retaining, to some degree, its characteristics as a measure
of length. At times these characteristics come into their own, for instance when Wittgenstein says someone shows
his understanding of the notion of height in that he knows it is measured by a Maßstab and not by a weighing
machine. At another point he reflects that the yardstick [Maßstab] needn't of course be a yardstick, it could equally
well be a dial or scale with a needle. And, of course, it isn't strictly correct to say that the application is what makes a
stick into a yardstick, it makes a stick (of any length) into a measuring stick, or a rod into a measuring rod
[Maßstab]. Occasionally, we felt that 'ruler' or 'measure' would be more natural--neither of us thought people would
keep yardsticks in their pockets--and that such mild variation in equivalents might possibly emphasize the unity of
the underlying principle: they are all standards of measurement. A word curiously
Page Break 353
difficult to translate is suchen: 'seek', 'search (for)' and 'look (for)' all create difficulties, the first because it has the
wrong kind of resonance, the others because of the occurrence of such sentences as 'Tell me how you suchen, and I
will tell you what you suchen'. 'Tell me how you are searching and I will tell you what you are searching for' throws
unwanted emphasis on 'for', and it is obvious that 'looking' and 'looking for' don't fit in with this pattern of syntax.
We have generally said 'proposition' for Satz, but some contexts ask for 'sentence' in English and others for
'theorem'. In such cases important links are lost, and wherever it seemed appropriate we have indicated this fact. The
word Übersichtlichkeit, which occurs at the beginning of the book, must have given trouble to all translators, who
have variously rendered it as 'perspicuity', 'surveyability' and 'synoptic view', whereas we have rendered it freely as
'bird's-eye view'. The idea at stake here is that a range of phenomena is made übersichtlich or übersehbar when
presented in such a way that we can simultaneously grasp the phenomena individually and as a whole, i.e. if their
interrelations can be seen or surveyed in their entirety (you could say in RFM that a proof is übersehbar if it may be
taken in as a single proof). Although differing from the German in that it suggests the elimination of the less
important elements, 'bird's-eye view' seems to work well enough in our context. However, in view of the use
Wittgenstein later makes of this concept, we felt we should signal its presence in the German text and express the
hope that 'bird's-eye view' has something of its immediacy and simplicity.
Page 353
In preparing this translation we have received help and encouragement from a number of people, especially
members of the Philosophy Department in the University of Leeds, and also members of the German Department.
We are indebted to Mrs Inge Hudson for some suggestions about German usage and particularly to Peter Long for
suggesting apt and pleasing ways of catching the force of the original at a number of points. Finally, we owe a
special debt of gratitude to Rush Rhees for allowing us a completely free hand with the translation and for making
considered and valuable suggestions on virtually every paragraph. We hope that these remarks indicate in some
degree the extent to which we have benefited from his generosity.
Page Break 354
Page 354
This is a translation of the German edition of Philosophische Bemerkungen, edited by Rush Rhees,
published Oxford 1964. The only divergences are a few additional footnotes requested by Rush Rhees. The
following list of Corrigenda to the German edition was prepared by Rush Rhees and ourselves.
Page 354
As far as practicable the pagination of the translation follows that of the German edition.
FOOTNOTES
Page 52
†1] German: Die Oktaeder-Darstellung ist eine übersichtliche Darstellung der grammatischen Regeln.
Page 52
†2] German: Unserer Grammatik fehlt es vor allem an Übersichtlichkeit.
Page 52
†3] E. Mach, Erkenntnis und Irrtum, 2nd ed., Leipzig 1906. p. 186, 191.
Page 61
†1] The German word here is 'Satz', which spans both 'proposition' and 'sentence': in English the translation
'sentence' would frequently become intolerably strained and we have therefore normally rendered 'Satz' by
'proposition': in this context 'sentence' is clearly required and yet this passage should link with those other passages
in the book where we have adopted 'proposition' as our translation. [trans.]
Page 76
†1] Cf. diagram on p. 278.
Page 80
†1] ideas = Vorstellungen; the world as idea--Vontellungswelt. [trans.]
Page 82
†1] world of the image = Welt der Vorstellung. [trans.]
Page 85
†1] Maßstab (usually rendered 'yardstick'). [trans.]
Page 89
†1] To be construed by analogy with 'It is snowing'. [trans.]
Page 110
†1] cf. Appendix, p. 317.
Page 124
†1] In the manuscript W. precedes this formula by the following definitions:
'(∃x)φx (∃1)xφ(x)
(∃x, y)φx.φy (∃1 + 1)x)φ(x)
(∃x, y, z)φx • φy • φz (∃1 + 1 + 1)xφ(x)
etc.
further:
(∃n)xφ(x) • ~ (∃n + 1)xφ(x) (n)xφx'
and then leads into the formula in the text with the words 'Then you can e.g. write:...' ['Dann kann man z. B.
schreiben:...'].
Page 126
†1] In the manuscript Wittgenstein first wrote 'the proposition A' as follows:
Page 126
'(∃2x) φx • (∃2x)x • Ind.. ⊃. (∃4x) φxv ψx--A. This proposition doesn't--of course--say that 2 + 2 = 4 but that
the expression is a tautology shows it. φ and ψ must be disjoint variables.'
Page 126
He also writes in this connection: 'For isn't '(∃2x) φx • (∃2x)ψx • Ind. (∃4x)φx ∨ ψx' an application of 2 + 2 =
4, just as much as '(E2x)φx etc., etc.'?
Page 126
By this stage he is using '(E2x)φx' in contrast with '(∃2x)φx' to mean 'There are exactly two φs', as opposed to
'There are at least two φs'.
Page 128
†1] (Later marginal note): That we can make the calculus with strokes and without tautologies shows we do
not need tautologies for it. Everything that does not belong to the number calculus is mere decoration.
Page 129
†1] (Later correction):... only direct insight into the number calculus can tell us... (with a mark of
dissatisfaction under the words 'direct insight').
Page 129
†2] (Later marginal note): Instead of a question of the definition of number, it's only a question of the
grammar of numerals.
Page 143
†1] Grelling's.
Page 144
†1] 'Chromatic number': a fictitious designation of a number that is perhaps not given by any law or
calculation. In the manuscript, W. wrote, among other things:
Page 144
You could ask: What does (x)2x = x + x say? It says that all equations of the form 2x = x + x are correct. But
does that mean anything? May one say: Yes, I see that all equations of this form are correct, so now I may write
'(x)2x = x + x'?
Page 144
Its meaning must derive from its proof. What the proof proves-that is the meaning of the proposition (neither
more nor less)...
Page 144
An algebraic proof is the general form of a proof which can be applied to any number. If, referring to this
proof, I say I have demonstrated that there is no chromatic number, then this proposition obviously says something
other than '~(∃n)•Chr n.'
Page 144
And in that case what does the proposition 'There is a chromatic number' say? It ought, surely, to say the
opposite of what was demonstrated by the proof. But then, it doesn't say '(∃n)•Chr n.'
Page 144
If you make the wrong transition from the variable proposition to the general proposition (in the way Russell
and Whitehead said was permissible), then the proof looks as if it's only a source for knowledge of the general
proposition, instead of an analysis of its actual sense.
Footnote Page Break 145
Page 145
Then you could also say: perhaps the proposition is correct, even though it can't be proved.
Page 145
If this proof yields the proposition Fa≠fa, what in this case is its opposite? (Surely--in our sense--not Fa =
fa).
Page 145
Indeed, I have here only a form that I have proved possesses certain properties. On the strength of these
properties, I can use it now in certain ways, viz. to show in any particular case that the number in question isn't
chromatic. I can, of course, negate that form, but that doesn't give me the sense I want, and all I can do now is to
deny the proof. But what does that mean? It doesn't of course, mean that it was wrongly--fallaciously--executed, but
that it can't be carried out. That then means: the inequality doesn't derive from the forms in question; the forms do
not exclude the inequality. But then, what else does? Does the development depend then on something further? That
is, can the case arise in which the equation doesn't hold, and the form doesn't exclude it?
Page 149
†1] (Later marginal note): Is '(∃n) 4 + n = 7'... α a disjunction? No, since a disjunction wouldn't have the sign
'etc. ad inf.' at the end but a term of the form 4 + x.
Page 149
α is neither the same sort of proposition as 'There are men in this house', nor as 'There's a colour that goes
well with this', nor as 'There are problems I find too difficult for me', nor as 'There's a time of day when I like to go
for a walk'.
Page 150
†1] In the manuscript: 'that all propositions, ~ p, ~~~p, ~~~~~p, etc. say the same.'
Page 162
†1] In the manuscript he writes ξ instead of x, cf. above, §125.
Page 164
†1] W. had just written in the manuscript: One is constantly confused by the thought 'but can there be a
possibility without a corresponding reality?'
Page 168
†1] The only reason why you can't say there are infinitely many things is that there aren't. If there were, you
could also express the fact I [Manuscript]
Page 171
†1] (Later marginal note): Act of decision, not insight.
Page 171
†2] This sentence is incomplete in the manuscript. [ed.]
Page 174
†1] In this connection, W. had written in the manuscript:
Page 174
'I still haven't stressed sufficiently that 25 × 25 = 625 is on precisely the same level as and of precisely the
same kind as x2 + y2 + 2xy = (x + y)2.............
Page 174
In the case of general propositions, what corresponds to the proposition 252 ≠ 620? It would be ~(x, y)(x +
y)2 = x2 + y2, or in words, it is not a rule that '(x + y)2 = x2 + y2', or more precisely, the two sides of this
equation--taken in the way the '(x)' indicates--are not equal to one another.'
Page 174
But this sign '(x)' says exactly the opposite of what it says in non-mathematical cases... i.e. precisely that we
should treat the variables in the proposition as constants. You could paraphrase the above proposition by saying 'It
isn't correct--if we treat x and y as constants--that (x + y)2 = x2 + y2'.....
(a + b)2 = a2 + xab + b2 yields x = 2}
(x + b)2 = x2 + 2xb + b2 yields x = x} and refer to the same state of affairs which is also affirmed by (a + b)2 = a2 +
2ab + b2. (But is that true?)'
Page 174
And after the remark given in the text: 'But what is meant by knowing the rules for calculating?'
Page 175
†1] Cf. below p. 178.
Page 176
†1] By this point in his manuscript Wittgenstein had already written the paragraph in §189: 'I say that the
so-called "Fermat's last Theorem"†1 is no kind of proposition†1 (not even in the sense of a proposition of
arithmetic). It would, rather, correspond to a proof by induction. But if, now, there is a number F = 0.110000 etc. and
the proof succeeds, then it would surely give us a proof that F = 0.11 and surely that's a proposition! Or: it's a
proposition in the case of the law F being a number.'
Page 357
†1 Satz.
Page 177
†1] Manuscript (a bit earlier): 'Whatever one can tackle is a problem. (So mathematics is all right.)' [In
English]
Page 184
†1] Earlier in the manuscript W. had distinguished 'the real mathematical proposition' (i.e. the proof) from
'the so-called mathematical proposition' (existing without its proof), e.g. in the following context:
Page 184
'What sort of a proposition is "there is a prime number between 5 and 8"? I would say: "That shows itself".
And that is correct; but can't we draw attention to this internal state of affairs?...
... I can, e.g., write the number 5 in such a way that you can clearly see that it's only divisible by 1 and itself:...
Perhaps this comes down to the same thing as what I meant earlier when I said that the real mathematical
proposition is a proof of a so-called mathematical proposition. The real mathematical proposition is the proof: that is
to say, the thing which shows how matters stand.'
Page 186
†1] See above p. 132. [ed.]
Page 186
†2] In the sense in which you write history. [ed.]
Page 194
†1] This remark refers to the first section of Th. Skolem's Begründung der Elementaren Arithmetik durch
die Rekurrierende Denkweise ohne Anwendung Scheinbarer Veränderlichen mit Unendlichem
Ausdehnungsbereich (Videnskapsselskapets Skrifter. 1 Math.-Naturv. Klasse 1923, No. 6.). [Translated in van
Heijenhoort: From Frege to Gödel, Harvard University Press, 1967, pp. 302-333.]
Page 194
The text runs:
§1
Addition
Page 194
I will introduce a descriptive function of two variables a and b, which I will designate by means of a + b and
call the sum of a and b, in that, for b = 1, it is to mean simply the successor of a, a + 1. And so this function is to be
regarded as already defined for b = 1 and arbitrary a. In order to define it in general, I in that case only need to define
it for b + 1 and arbitrary a, on the assumption that it is already defined for b and arbitrary a. This is done by means
of the following definition:
Page 194
Def. 1. a + (b + 1) = (a + b) + 1
Page 194
In this manner, the sum of a and b + 1 is equated with the successor of a + b. And so if addition is already
defined for arbitrary values of a for a certain number b, then by Def. 1 addition is explained for b + 1 for arbitrary a,
and
Footnote Page Break 195
thereby is defined in general. This is a typical example of a recursive definition.
Page 195
Theorem 1. The associative law: a + (b + c) = (a + b) + c.
Page 195
Proof: The theorem holds for c = 1 in virtue of Def. 1. Assume that it is valid for a certain c for arbitrary a and
b.
Page 195
Then we must have, for arbitrary values of a and b
(α) a + (b + (c + 1)) = a + ((b + c) + 1)
since, that is to say, by Def. 1 b + (c + 1) = (b + c) + 1. But also by Def. 1
(β) a +((b + c) + 1) = (a + (b + c)) + 1.
Now, by hypothesis, a + (b + c) = (a + b) + c, whence
(γ) (a + (b + c)) + 1 = ((a + b) + c) + 1.
Finally, by Def. 1 we also have
(δ) ((a + b) + c) + 1 = (a + b) + (c + 1).
From (α), (β), (γ), and (δ) there follows
a + (b + (c + 1)) = (a + b) + (c + 1),
whence the theorem is proved for c + 1 with a and b left undetermined. Thus the theorem holds generally. This is a
typical example of a recursive proof (proof by complete induction).
Page 194
In the margin of his copy, Wittgenstein drew arrows pointing to 'Theorem 1' and to (α) and wrote: 'the c is a
different kind of variable in these two cases'. He puts a question mark against the equals sign in (γ); and another
question mark and 'Transition?' against the words 'whence the theorem is proved for c + 1 with a and b left
undetermined'. Cf. above footnote on p. 144: 'If one makes the wrong transition from a variable proposition to a
general proposition....'
Page 198
†1] a + (b + c) = (a + b) + c.
Page 198
†2] a + (b + 1) (a + b) + 1.
Page 204
†1] Fundamental Theorem = Hauptsatz, proposition = Satz. [trans.]
Page 210
†1] In the notebook W. then wrote: 'Strangely enough the general concept of the ancestral relation also seems
to me to be nonsense now. It seems to me that the variable n must always be confined as lying within two limits.'
Page 210
†2] 'If it were admissible to write '(∃nx) • φx' instead of '(∃x) • φx', then it would be admissible to write '(nx) •
φx' instead of '~(∃x) ~ • φx'--and so of '(x) • φx'--and that presupposes that there are infinitely many objects.
Page 210
And so the (∃nx)..., if it has any justification at all, here may not mean 'there is one among all the numbers...'
[Manuscript]
Page 220
†1] (Later gloss): This isn't true! The process of measuring! Nothing determines what the outcome of the
process of bisection is going to be.
Page 222
†1] In this, and the following paragraphs: rule = Vorschrift (prescription). [trans.]
Page 222
†2] In Russell's sense of type. [trans.]
Page 224
†1] This sign is to express the prescription: write out the digits of : but whenever a 5 occurs in ,
substitute a 3 for it. Wittgenstein also writes this prescription in the form means: expanded to 4
places in a given
Footnote Page Break 225
system--say the decimal system. Sometimes he also writes , or π', and that means: write out the decimal
expansion of π, but whenever a 7 occurs, substitute a 3 for it. " " would mean: π expanded to 4 places (e.g. in the
decimal system). Thus would be 3, , 3.1 in the decimal system.
Page 226
†1] Cf. for example what A. Fränkel says about Cantor's "diagonal number": "... for whilst in general, for
every k, δk is to be set equal to 1, this rule is subject to an exception solely for those values of k for which the digit 1
is already in the relevant diagonal place of Φ, i.e. at the position of the kth digit of the kth decimal fraction; for such,
and only for such, values of k, δk is to be set equal to 2." Einleitung in die Mengenlehre, 3rd ed. 1928, p. 47. [ed.]
Page 232
†1] Theorem, proposition = Satz. [trans.]
Page 237
†1] Cf. The opening sentence of §189 (p. 232 above), and below p. 240.
Page 238
†1] (Manuscript): What about an operation such as x)(y; we form the product of x and y; if it is greater than
100, the result is the greater of the two numbers, otherwise it is 0?
Page 238
12)(10 = 12. The operation isn't arithmetically comprehensible.
Page 239
†1] The words 'of the summation' have been added in the translation to make clearer Wittgenstein's meaning
in this highly compressed paragraph: we take the 'stages' referred to in the original as referring to the stages in some
method of computing e, and for what Wittgenstein says here to be true, he must have in mind a method such as the
summing of the usual series for e, which produces a monotone increasing series of approximations. [trans.]
Page 239
†2] Similar to the above: 'Is 3.14 the circumference of a unit circle? No, it's the perimeter of a ----gon.'
[Manuscript]
Page 241
†1] (Manuscript): I once asked: 'How can I call a law which represents the endless nesting of intervals
anything but a number?'--But why do I term such a nesting a number?--Because we can significantly say that every
number lies to the left or right of this nesting.
Page 253
†1] Nicod, The Foundations of Geometry and Induction.
Page 260
†1] Throughout this section we have translated 'verschwommen' by either 'vague' or 'blurred' for purely
stylistic reasons. [trans.]
Page 267
†1] As Wittgenstein has written these two propositions, the word 'equal' doesn't occur; but this hardly
matters.
Page 267
†2] In the first chapter of Analysis of Sensations, p. 19.
Page 283
†1] Cf. below, p. 285. [ed.]
Page 293
†1] Francis Galton, Inquiries into Human Faculty, London, 1883, Ch. 1 and Appendix A on 'Composite
portraiture'.
Page 296
†1] In the manuscript, Wittgenstein writes this sentence in the form, 'Die beobachtete Verteilung von
Ereignissen kann mich zu dieser Annahme führen.' ['The observed distribution of events can lead me to this
assumption.'] In the typescript the 'mich' is changed to 'nicht'; this is the reading given in
Footnote Page Break 297
the German text and which we have followed. It is possible that the typescript reading is a misprint, but it does seem
to make neatest sense in the context, provided one allows the slightly strained way of taking 'kann nicht', ['The
observed distribution of events cannot lead to this assumption' appears not to fit the context at all.] [trans.]
Page 303
†* The following translations have been used throughout this discussion: Tat, act; Tatsache, fact; Ereignis,
event; hinweisen auf, point out; zeigen auf, point at. The one nuance it seems impossible to pick up in English is the
link between 'Diese Tatsache besteht' = 'this fact obtains' and 'Dieser Komplex besteht aus...' = 'this complex is
composed of'; we have rendered the noun from bestehen, 'Bestandteil', 'component part'; in translations of the
Tractatus and other related discussions these last two are usually rendered 'consists of', 'constituent' (as Wittgenstein
indicates in the last paragraph). [trans.]
Page 304
†1] Cf. §145, p. 165, 166.
Page 305
†1] See p. 166.
Page 305
†2] In the Manuscript, this sentence immediately follows the paragraph about the red spheres.
Page 308
†1] An apparent reference to a remark of H. Hahn.
Page 311
†1] Cf. §142, p. 162.
Page 313
†1] Cf. §147, p. 169; §212, p. 265.
Page 314
†1] Cf. §139, p. 159; §144, p. 164.
Page 321
†1] 'Die Idee des Widerspruchs ist die Kontradiktion'. The use of the ordinary German word for contradiction
and the (practically synonymous) loan-word gives to this sentence overtones difficult to reproduce in English. The
main point of the remark seems to be 'Look, by a contradiction we must mean a contradiction' (Everything's what it
is and not another thing), where he goes on to spell out what is involved in the notion of contradiction, and what can
be overlooked in talking of contradictions within calculi. But there are additional overtones introduced by the fact
that the word Kontradiktion is the word used in the Tractatus as the converse of Tautologie. It would produce
artificialities to translate Widerspruch and Kontradiktion by different words: in most contexts Wittgenstein uses
them virtually interchangeably. Where this has seemed the most natural rendering (as in 'consistency proof') we have
translated 'Widerspruch' by 'inconsistency'. We think that the force of Wittgenstein's remarks comes over clearly
enough to render clumsy and unnecessary the use of different standard translations for the two words. [trans.]
Page 323
†1] Cf. D. Hilbert Neubegründung der Mathematik, Gesammelte Abhandlungen III, p. 74.
Page 330
†1] D. Hilbert: Gesammelte Abhandlungen III, p. 170.
Page 333
†1] Cf. Footnote on p. 321 above.
Page 347
†1] The Autobiography of Bertrand Russell, Volume II, pp. 196-200. London, 1968.
PHILOSOPHICAL GRAMMAR
Page 1
Page Break 2
Page Break 3
Titlepage
Page 3
LUDWIG WITTGENSTEIN:
PHILOSOPHICAL GRAMMAR
PART I: The Proposition and its Sense
PART II: On Logic and Mathematics
Edited by
RUSH RHEES
Translated by
ANTHONY KENNY
BASIL BLACKWELL
Page Break 4
Copyright page
Page 4
Copyright © Basil Blackwell Ltd 1974
First published 1974
Paperback edition 1980
Reprinted 1988, 1990, 1993
Blackwell Publishers
108 Cowley Road, Oxford OX4 1JF, UK
All rights reserved. Except for the quotation of short passages for the purposes of criticism and review, no part of
this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher.
Except in the United States of America, this book is sold subject to the condition that it shall not, by way of trade or
otherwise, be lent, resold, hired out, or otherwise circulated without the publisher's prior consent in any form of
binding or cover other than that in which it is published and without a similar condition including this condition
being imposed on the subsequent purchaser.
ISBN 0 631 11891 8
Printed in Great Britain by Athenæum Press, Newcastle-upon-Tyne
This book is printed on acid-free paper
Page Break 5
CONTENTS
Part I: The Proposition and its Sense
Page 5
I
Page 5
1 How can one talk about "understanding" and "not understanding" a proposition?
Surely it's not a proposition until it's understood? 39
Page 5
2 Understanding and signs. Frege against the formalists. Understanding like seeing a picture that makes all the rules
clear; in that case the picture is itself a sign, a calculus.
"To understand a language"--to take in a symbolism as a whole.
Language must speak for itself. 39
Page 5
3 One can say that meaning drops out of language.
In contrast: "Did you mean that seriously or as a joke?"
When we mean (and don't just say) words it seems to us as if there were something coupled to the words.
41
Page 5
4 Comparison with understanding a piece of music: for explanation I can only translate the musical picture into a
picture in another medium--and why just that picture? Comparison with understanding a picture. Perhaps we see
only patches and lines--"we do not understand the picture". Seeing a genre-picture in different ways.
41
Page 5
5 "I understand that gesture"--it says something.
In a sentence a word can be felt as belonging first with one word and then with another.
A 'proposition' may be what is conceived in different ways or the way of conceiving itself.
Page Break 6
A sentence from the middle of a story I have not read.
The concept of understanding is a fluid one. 42
Page 6
6 A sentence in a code: at what moment of translating does understanding begin?
The words of a sentence are arbitrary; so I replace them with letters. But now I cannot immediately think the
sense of the sentence in the new expression.
The notion that we can only imperfectly exhibit our understanding: the expression of understanding has
something missing that is essentially inexpressible. But in that case it makes no sense to speak of a more complete
expression. 43
Page 6
7 What is the criterion for an expression's being meant thus? A question about the relationship between two
linguistic expressions. Sometimes a translation into another mode of representation. 45
Page 6
8 Must I understand a sentence to be able to act on it? If "to understand a sentence" means somehow or other to act
on it, then understanding cannot be a precondition for our acting on it.--What goes on when I suddenly understand
someone else? There are many possibilities here. 45
Page 6
9 Isn't there a gap between an order and its execution? "I understand it, but only because I add something to it,
namely the interpretation."--But if one were to say "any sentence still stands in need of an interpretation", that would
mean: no sentence can be understood without a rider. 46
Page 6
10 "Understanding a word"--being able to apply it.--"When I said 'I can play chess' I really could." How did I know
that I could? My answer will show in what way I use the word "can".
Being able is called a state. "To describe a state" can mean various things. "After all I can't have the whole
mode of application of a word in my head all at once." 47
Page Break 7
Page 7
11 It is not a question of an instantaneous grasping.--
When a man who knows the game watches a game of chess, the experience he has when a move is made
usually differs from that of someone else watching without understanding the game. But this experience is not the
knowledge of the rules.--The understanding of language seems like a background; like the ability to multiply.
Page 7
12 When do we understand a sentence?--When we've uttered the whole of it? Or while uttering it?
50
Page 7
13 When someone interprets, or understands, a sign in one sense or another, what he is doing is taking a step in a
calculus.--"Thought" sometimes means a process which may accompany the utterance of a sentence and sometimes
the sentence itself in the system of language. 50
Page 7
II
14 Grammar as (e.g.) the geometry of negation. We would like to say: "Negation has the property that when it is
doubled it yields an affirmation". But the rule doesn't give a further description of negation, it constitutes negation.
52
Page 7
15 Geometry no more speaks about cubes than logic does about negation.
It looks as if one could infer from the meaning of negation that "~~p" means p. 52
Page 7
16 What does it mean to say that the "is" in "The rose is red" has a different meaning from the "is" in "twice two is
four"? Here we have one word but as it were different meaning-bodies with a single end surface: different
possibilities of constructing sentences. The comparison of the glass cubes. The rule for the arrangement of the red
sides contains the possibilities, i.e. the geometry of the cube. The cube can also serve as a notation for the rule if it
belongs to a system of propositions. 53
Page Break 8
Page 8
17 "The grammatical possibilities of the negation-sign". The T-F notation can illustrate the meaning of "not". The
written symbol becomes a sign for negation only by the way it works--the way it is used in the game.
55
Page 8
18 If we derive geometrical propositions from a drawing or a model, then the model has the role of a sign in a game.
We use the drawing of a cube again and again in different contexts. It is this sign that we take to be the cube in
which the geometrical laws are already laid up. 55
Page 8
19 My earlier concept of meaning originates in a primitive philosophy of language.--Augustine on the learning of
language. He describes a calculus of our language, only not everything that we call language is this calculus.
56
Page 8
20 As if words didn't also have functions quite different from the naming of tables, chairs, etc.--Here is the origin of
the bad expression: a fact is a complex of objects. 57
Page 8
21 In a familiar language we experience different parts of speech as different. It is only in a foreign language that we
see clearly the uniformity of words. 58
Page 8
22 If I decide to use a new word instead of "red", how would it come out that it took the place of the word "red"?
59
Page 8
23 The meaning of a word: what the explanation of its meaning explains. (If, on the other hand by "meaning" we
mean a characteristic sensation, then the explanation of meaning would be a cause.) 59
Page Break 9
Page 9
24 Explanation can clear up misunderstandings. In that case understanding is a correlate of
explanation.--Definitions.
It seems as if the other grammatical rules for a word had to follow from its ostensive definition. But is this
definition really unambiguous? One must understand a great deal of a language in order to understand the definition.
60
Page 9
25 The words "shape", "colour" in the definitions determine the kind of use of the word. The ostensive definition has
a different role in the grammar of each part of speech. 61
Page 9
26 So how does it come about that on the strength of this definition we understand the word?
What's the sign of someone's understanding a game? Can't he learn a game simply by watching it being
played? Learning and speaking without explicit rules. We are always comparing language with a game according to
rules. 61
Page 9
27 The names I give to bodies, shapes, colours, lengths have different grammars in each case. The meaning of a
name is not the thing we point to when we give an ostensive definition of the name. 63
Page 9
28 What constitutes the meaning of a word like "perhaps"?
I know how it is used. The case is similar when someone is explaining to me a calculation "that I don't quite
understand". "Now I know how to go on." How do I know that I know how to go on?
64
Page 9
29 Is the meaning really only the use of the word? Isn't it the way this use meshes with our life?
65
Page 9
30 The words "fine", "oh", "perhaps"... can each be the expression of a feeling. But I don't call that feeling the
meaning of the word.
Page Break 10
I can replace the sensations by intonation and gestures.
I could also treat the word (e.g. "oh") itself as a gesture. 66
Page 10
31 A language spoken in a uniform metre.
Relationships between tools in a toolbox.
"The meaning of a word: its role in the calculus of language." Imagine how we calculate with "red". And
then: the word "oh"--what corresponds now to the calculus? 67
Page 10
32 Describing ball-games. Perhaps one will be unwilling to call some of them ball-games; but it is clear where the
boundary is to be drawn here?
We consider language from one point of view only.
The explanation of the purpose or the effect of a word is not what we call the explanation of its meaning. It
may be that if it is to achieve its effect a particular word cannot be replaced by any other, just as it may be that a
gesture cannot be replaced by any other.--We only bother about what's called the explanation of meaning and not
about meaning in any other sense. 68
Page 10
33 Aren't our sentences parts of a mechanism? As in a pianola? But suppose it is in bad condition? So it is not the
effect but the purpose that is the sense of the signs (the holes in the pianola roll). Their purpose within the
mechanism.
We need an explanation that is part of the calculus.
"A symbol is something that produces this effect."--How do I know that it is the one I meant?"
We could use a colour-chart: and then our calculus would have to get along with the visible colour-sample.
69
Page 10
34 "We could understand a penholder too, if we had given it a meaning." Does the understanding contain the whole
system of its application?
Page Break 11
When I read a sentence with understanding something happens: perhaps a picture comes into my mind. But
before we call "understanding" is related to countless things that happen before and after the reading of this
sentence.
When I don't understand a sentence--that can be different things in different cases.
"Understanding a word"--that is infinitely various. 71
Page 11
35 "Understanding" is not the name of a single process but of more or less interrelated processes against a
background of the actual use of a learnt language.--We think that if I use the word "understanding" in all these cases
there must be some one thing that happens in all of them. Well, the concept-word certainly does show a kinship but
this need not be the sharing of a common property or constituent.--The concept-word "game". "By 'knowledge' we
mean these processes, and these, and similar ones." 74
Page 11
III
36 If for our purposes we wish to regulate the use of a word by definite rules, then alongside its fluctuating use we
set a different use. But this isn't like the way physics gives a simplified description of a natural phenomenon. It is not
as if we were saying something that would hold only of an ideal language. 77
Page 11
37 We understand a genre-picture if we recognize what the people in it are doing. If this recognition does not come
easily, there is a period of doubt followed by a familiar process of recognition. If on the other hand we take it in at
first glance it is difficult to say what the understanding--the recognition say--consists of. There is no one thing that
happens that could be called recognition.
If I want to say "I understand it like that" then the "like that" stands for a translation into a different
expression. Or is it a sort of intransitive understanding? 77
Page Break 12
Page 12
38 Forgetting the meaning of a word. Different cases. The man feels, as he looks at blue objects, that the connection
between the word "blue" and the colour has been broken off. We might restore the connection in various ways. The
connection is not made by a single phenomenon, but can manifest itself in very various processes. Do I mean then
that there is no such thing as understanding but only manifestations of understanding?--a senseless question. 79
Page 12
39 How does an ostensive definition work? Is it put to work again every time the word is used? Definition as a part
of the calculus acts only by being applied. 80
Page 12
40 In what cases shall we say that the man understands the word "blue"? In what circumstances will he be able to
say it? or to say that he understood it in the past?
If he says "I picked the ball out by guesswork, I didn't understand the word", ought we to believe him? "He
can't be wrong if he says that he didn't understand the word": a remark on the grammar of the statement "I didn't
understand the word". 81
Page 12
41 We call understanding a mental state, and characterize it as a hypothetical process. Comparison between the
grammar of mental processes and the grammar of brain processes.
In certain circumstances both our picking out a red object from others on demand and our being able to give
the ostensive definition of the word "red" are regarded as signs of understanding.
We aren't interested here in the difference between thinking out loud (or in writing) and thinking in the
imagination.
What we call "understanding" is not the behaviour that shows us the understanding, but a state of which this
behaviour is a sign. 82
Page Break 13
Page 13
42 We might call the recital of the rules on its own a criterion of understanding, or alternatively tests of use on their
own. Or we may regard the recital of the rules as a symptom of the man's being able to do something other than
recite the rules.
To understand = to let a proposition work on one.
When one remembers the meaning of a word, the remembering is not the mental process that one imagines
at first sight.
The psychological process of understanding is in the same case as the arithmetical object Three.
84
Page 13
43 An explanation, a chart, is first used by being looked up, then by being looked up in the head, and finally as if it
had never existed.
A rule as the cause or history behind our present behaviour is of no interest to us. But a rule can be a
hypothesis, or can itself enter into the conduct of the game. If a disposition is hypothesized in the player to give the
list of rules on request, it is a disposition analogous to a physiological one. In our study of symbolism there is no
foreground and background. 85
Page 13
44 What interests us in the sign is what is embodied in the grammar of the sign. 87
Page 13
IV
45 The ostensive definition of signs is not an application of language, but part of the grammar: something like a rule
for translation from a gesture language into a word-language.--What belongs to grammar are all the conditions
necessary for comparing the proposition with reality--all the conditions necessary for its sense.
88
Page 13
46 Does our language consist of primary signs (gestures) and secondary signs (words)?
Obviously we would not be able to replace an ordinary sentence by gestures.
Page Break 14
Page 14
"Is it an accident that in order to define the signs I have to go outside the written and spoken signs?" In that
case isn't it strange that I can do anything at all with the written signs? 88
Page 14
47 We say that a red label is the primary sign for the colour red, and the word a secondary sign.--But must a
Frenchman have a red image present to his mind when he understands my explanation "red = rouge"?
89
Page 14
48 Are the primary signs incapable of being misinterpreted? Can one say they don't any longer need to be
understood? 90
Page 14
49 A colour chart might be arranged differently or used differently, and yet the words mean the same colours as with
us.
Can a green label be a sample of red?
Can it be said that when someone is painting a certain shade of green he is copying the red of a label?
A sample is not used like a name. 90
Page 14
50 "Copy" can mean various things. Various methods of comparison.
We do not understand what is meant by "this shade of colour is a copy of this note on the violin." It makes
no sense to speak of a projection-method for association. 91
Page 14
51 We can say that we communicate by signs whether we use words or samples, but the game of acting in
accordance with words is different from the game of acting in accordance with samples.
92
Page 14
52 "There must be some sort of law for reading the chart.--Otherwise how would one know how the table was to be
used?" It is part of human nature to understand pointing with the finger in the way we do.
The chart does not compel me to use it always in the same way. 93
Page Break 15
Page 15
53 Is the word "red" enough to enable one to look for something red? Does one need a memory image to do so?
An order. Is the real order "Do now what you remember doing then?"
If the colour sample appears darker than I remember it being yesterday, I need not agree with my memory.
94
Page 15
54 "Paint from memory the colour of the door of your room" is no more unambiguous than "paint the green you see
on this chart."
I see the colour of the flower and recognize it.
Even if I say "no, this colour is brighter than the one I saw there," there is no process of comparing two
simultaneously given shades of colour.
Think of reading aloud from a written test (or writing to dictation). 95
Page 15
55 "Why do you choose this colour when given this order?"--"Because this colour is opposite to the word 'red' in
my chart." In that case there is no sense in this question: "Why do you call 'red' the colour in the chart opposite the
word 'red'?"
The connection between "language and reality" is made by definitions of words--which belong to grammar.
96
Page 15
56 A gesture language used to communicate with people who have no word-language in common with us. Do we
feel there too the need to go outside language to explain its signs?
The correlation between objects and names is a part of the symbolism. It gives the wrong idea if you say that
the connection is a psychological one. 97
Page 15
57 Someone copies a figure on the scale of 1 to 10. Is the understanding of the general rule of such mapping
contained in the process of copying?
Page Break 16
Or was the process merely in agreement with that rule, but also in agreement with other rules?
97
Page 16
58 Even if my pencil doesn't always do justice to the model, my intention always does. 98
Page 16
59 For our studies it can never be essential that a symbolic phenomenon occurs in the mind and not on paper.
An explanation of a sign can replace the sign itself--this contrasts with causal explanation. 99
Page 16
60 Reading.--Deriving a translation from the original may also be a visible process.
Always what represents is the system in which a sign is used.
If 'mental' processes can be true and false, their descriptions must be able to as well. 99
Page 16
61 Every case of deriving an action from a command is the same kind of thing as the written derivation of a result.
"I write the number '16' here because it says 'x²' there."
It might appear that some causality was operating here, but that would be a confusion between 'reason' and
'cause'. 101
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V
62 "That's him"--that contains the whole problem of representation.
I make a plan: I see myself acting thus and so. "How do I know that it's myself?" Or "How do I know that the
word 'I' stands for me?"
The delusion that in thought the objects do what the proposition states about them.
"I meant the victor of Austerlitz"--the past tense, which looks as if it was giving a description, is deceptive.
102
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63 "How does one think a proposition? How does thought use its expression?"
Let's compare belief with the utterance of a sentence: the processes in the larynx etc. accompany the spoken
sentence which alone interests us--not as part of a mechanism, but as part of a calculus.
We think we can't describe thought after the event because the delicate processes have been lost sight of.
What is the function of thought? Its effect does not interest us. 103
Page 17
64 But if thinking consists only in writing or speaking, why shouldn't a machine do it?
Could a machine be in pain?
It is a travesty of the truth to say: thinking is an activity of our mind, as writing is an activity of the hand.
105
Page 17
65 'Thinking' 'Language' are fluid concepts.
The expression "mental process" is meant to distinguish 'experience' from 'physical processes'; or else we talk
of 'unconscious thoughts'--of processes in a mind-model; or else the word "thought" is taken as synonymous with
"sense of a sentence". 106
Page 17
66 The idea that one language in contrast to others can have an order of words which corresponds to the order of
thinking.
Is it, as it were, a contamination of the sense that we express it in a particular language? Does it impair the
rigour and purity of the proposition 25 × 25 = 625 that it is written down in a particular number system?
Thought can only be something common-or-garden. But we are affected by this concept as we are by that of
the number one. 107
Page 17
67 What does man think for? There is no such thing as a "thought-experiment".
Page Break 18
I believe that more boilers would explode if people did not calculate when making boilers. Does it follow that there
will in fact be fewer? The belief that fire will burn me is of the same nature as the fear that it will burn me.
109
Page 18
68 My assumption that this house won't collapse may be the utterance of a sentence which is part of a calculation. I
do have reasons for it. What counts as a reason for an assumption determines a calculus.--So is the calculus
something we adopt arbitrarily? No more so than the fear of fire.
As long as we remain in the province of true-false games a change of grammar can only lead us from one
game to another, and never from something true to something false. 110
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VI
69 What is a proposition?--Do we have a single general concept of proposition? 112
Page 18
70 "What happens when a new proposition is taken into the language: what is the criterion for its being a
proposition?"
In this respect the concept of number is like the concept of proposition. On the other hand the concept of
cardinal number can be called a rigorously circumscribed concept, that's to say it's a concept in a different sense of
the word. 113
Page 18
71 I possess the concept 'language' from the languages I have learnt. "But language can expand": if "expand" makes
sense here, I must now be able to specify how I imagine such an expansion.
No sign leads us beyond itself.
Does every newly constructed language broaden the concept of language?--Comparison with the concept of
number. 114
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72 The indeterminacy of generality is not a logical indeterminacy.
The task of philosophy is not to create an ideal language, but to clarify the use of existing language.
I'm allowed to use the word "rule" without first tabulating the rules for the word.--If philosophy was
concerned with the concept of the calculus of all calculi, there would be such a thing as metaphilosophy. But there is
not. 115
Page 19
73 It isn't on the strength of a particular property, the property of being a rule, that we speak of the rules of a
game.--We use the word "rule" in contrast to "word", "projection" and some other words.
116
Page 19
74 We learnt the meaning of the word "plant" by examples. And if we disregard hypothetical dispositions, these
examples stand only for themselves.--
The grammatical pace of the word "game" "rule" etc is given by examples in rather the way in which the
place of a meeting is specified by saying that it will take place beside such and such a tree.117
Page 19
75 Meaning as something which comes before our minds when we hear a word.
"Show the children a game".
The sentence "The Assyrians knew various games" would strike us as curious since we wouldn't be certain
that we could give an example. 118
Page 19
76 Examples of the use of the word "wish". Our aim is not to give a theory of wishing, which would have to explain
every case of wishing.
The use of the words "proposition", "language", etc. has the haziness of the normal use of concept-words in
our language. 119
Page 19
77 The philosophy of logic speaks of sentences and words in the sense in which we speak of them in ordinary life.
Page Break 20
(We are not justified in having any more scruples about our language than the chess player has about chess,
namely none.) 121
Page 20
78 Sounding like a sentence. We don't call everything 'that sounds like a sentence' a sentence.--If we disregard
sounding like a sentence do we still have a general concept of proposition?
The example of a language in which the order of the words in a sentence is the reverse of the present one.
122
Page 20
79 The definition "A proposition is whatever can be true or false".--The words "true" and "false" are items in a
particular notation for the truth-functions.
Does "'p' is true" state anything about the sign 'p'? 123
Page 20
80 In the schema "This is how things stand" the "how things stand" is a handle for the truth-functions.
A general propositional form determines a proposition as part of a calculus. 124
Page 20
81 The rules that say that such and such a combination of words yields no sense.
"How do I know that red can't be cut into bits?" is not a question. I must begin with the distinction between
sense and nonsense. I can't give it a foundation. 125
Page 20
82 "How must we make the grammatical rules for words if they are to give a sentence sense?"--
A proposition shows the possibility of the state of affairs it describes. "Possible" here means the same as
"conceivable"; representable in a particular system of propositions.
The proposition "I can imagine such and such a colour transition connects the linguistic representation with
another form of representation; it is a proposition of grammar. 127
Page 20
83 It looks as if we could say: Word-language allows of senseless combinations of words, but the language of
imagining does not allow us to imagine anything senseless.
Page Break 21
"Can you imagine it's being otherwise?"--How strange that one should be able to say that such and such a
state of affairs is inconceivable! 128
Page 21
84 The role of a proposition in the calculus is its sense.
It is only in language that something is a proposition. To understand a proposition is to understand a
language. 130
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VII
85 Symbols appear to be of their nature unsatisfied.
A proposition seems to demand that reality be compared with it.
"A proposition like a ruler laid against reality." 132
Page 21
86 If you see the expression of an expectation you see what is being expected.
It looks as if the ultimate thing sought by an order had to remain unexpressed.--As if the sign was trying to
communicate with us.
A sign does its job only in a grammatical system. 132
Page 21
87 It seems as if the expectation and the fact satisfying the expectation fitted together somehow. Solids and
hollows.--Expectation is not related to its satisfaction in the same way as hunger is related to its satisfaction.
133
Page 21
88 The strange thing that the event I expected isn't distinct from the one I expected.--"The report was not so loud as I
had expected."
"How can you say that the red which you see in front of you is the same as the red you imagined?"--One
takes the meaning of the word "red" as being the sense of a proposition saying that something is red.
134
Page 21
89 A red patch looks different from one that is not red. But it would be odd to say "a red patch looks different when
it is there from when it isn't there". Or: "How do you know that you are expecting a red patch?"
135
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Page 22
90 How can I expect the event, when it isn't yet there at all?--I can imagine a stag that is not there, in this meadow,
but not kill one that is not there.--It is not the expected thing that is the fulfilment, but rather its coming about. It is
difficult for us to shake off this comparison: a man makes his appearance--an event makes its appearance. 136
Page 22
91 A search for a particular thing (e.g. my stick) is a particular kind of search, and differs from a search for
something else because of what one does (says, thinks) while searching, not because of what one finds.--Contrast
looking for the trisection of the angle. 138
Page 22
92 The symptoms of expectation are not the expression of expectation.
In the sentence "I expect that he is coming" is one using the words "he is coming" in a different sense from
the one they have in the assertion "he is coming"?
What makes it the expectation precisely of him?
Various definitions of "expecting a person X".
It isn't a later experience that decides what we are expecting.
"Let us put the expression of expectation in place of the expectation." 138
Page 22
93 Expectation as preparatory behaviour.
"Expectation is a thought"
If hunger is called a wish it is a hypothesis that just that will satisfy the wish.
In "I have been expecting him all day" "expect" does not mean a persistent condition. 140
Page 22
94 When I expect someone,--what happens?
What does the process of wanting to eat an apple consist in? 141
Page Break 23
Page 23
95 Intention and intentionality.--
"The thought that p is the case doesn't presuppose that it is the case; yet I can't think that something is red if
the colour red does not exist." Here we mean the existence of a red sample as part of our language.
142
Page 23
96 It's beginning to look somehow as if intention could never be recognized as intention from the outside. But the
point is that one has to read off from a thought that it is the thought that such and such is the case.
143
Page 23
97 This is connected with the question whether a machine could think. This is like when we say: "The will can't be a
phenomenon, for whatever phenomenon you take is something that simply happens, not something we do." But
there's no doubt that you also have experiences when you move your arm voluntarily, although the phenomena of
doing are indeed different from the phenomena of observing. But there are very different cases here.
144
Page 23
98 The intention seems to interpret, to give the final interpretation.
Imagine an 'abstract' sign-language translated into an unambiguous picture-language. Here there seem to be
no further possibilities of interpretation.--We might say we didn't enter into the sign-language but did enter into the
painted picture. Examples: picture, cinema, dream. 145
Page 23
99 What happens is not that this symbol cannot be further interpreted, but: I do no interpreting.
I imagine N. No interpretation accompanies this image; what gives the image its interpretation is the path on
which it lies. 147
Page 23
100 We want to say: "Meaning is essentially a mental process, not a process in dead matter."--What we are
dissatisfied with here is the grammar of process, not the specific kind of process. 148
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Page 24
101 Doesn't the system of language provide me with a medium in which the proposition is no longer dead?--"Even if
the expression of the wish is the wish, still the whole language isn't present during this expression." But that is not
necessary. 149
Page 24
102 In the gesture we don't see the real shadow of the fulfilment, the unambiguous shadow that admits of no further
interpretation. 149
Page 24
103 It's only considering the linguistic manifestation of a wish that makes it appear that my wish prefigures the
fulfilment.--Because it's the wish that just that were the case.--It is in language that wish and fulfilment meet.
150
Page 24
104 "A proposition isn't a mere series of sounds, it is something more." Don't I see a sentence as part of a system of
consequences? 152
Page 24
105 "This queer thing, thought."--It strikes us as queer when we say that it connects objects in the mind.--We're all
ready to pass from it to the reality.--"How was it possible for thought to deal with the very person himself?" Here I
am being astonished by my own linguistic expression and momentarily misunderstanding it.
154
Page 24
106 "When I think of what will happen tomorrow I am mentally already in the future."--Similarly people think that
the endless series of cardinal numbers is somehow before our mind's eye, whenever we can use that expression
significantly.
A thought experiment is like a drawing of an experiment that is not carried out. 155
Page 24
107 We said "one cannot recognize intention as intention from the outside"--i.e. that it is not something that
happens, or happens to us, but something we do. It is almost as if we said: we cannot see ourselves going to a place,
because it is we who are doing the going. One does have a particular experience if one is doing the going oneself.
156
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Page 25
108 Fulfilment of expectation doesn't consist in some third thing's happening, such as a feeling of satisfaction.
157
Page 25
VIII
109 A description of language must achieve the same result as language itself.
Suppose someone says that one can infer from a propsotion [[sic]] the fact that verifies it. What can one infer
from a proposition apart from itself?
The shadowy anticipation of a fact consists in our being able already to think that that very thing will happen
which hasn't yet happened. 159
Page 25
110 However many steps I insert between the thought and its application, each intermediate step always follows the
previous one without any intermediate link, and so too the application follows the last intermediate step.--We can't
cross the bridge to the execution (of an order) until we are there. 160
Page 25
111 It is the calculus of thought that connects with extra-mental reality. From expectation to fulfilment is a step in a
calculation. 160
Page 25
112 We are--as it were--surprised, not at anyone's knowing the future, but at his being able to prophesy at all (right
or wrong). 161
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IX
113 Is the pictorial character of thought an agreement with reality? In what sense can I say that a proposition is a
picture? 163
Page 25
114 The sense of a proposition and the sense of a picture. The different grammar of the expressions:
Page Break 26
"This picture shows people at a village inn."
"This picture shows the coronation of Napoleon." 164
Page 26
115 A picture's telling me something will consist in my recognizing in it objects in some sort of characteristic
arrangement.--
What does "this object is familiar to me" mean? 165
Page 26
116 "I see what I see." I say that because I don't want to give a name to what I see.--I want to exclude from my
consideration of familiarity everything that is 'historical'.--The multiplicity of familiarity is that of feeling at home in
what I see. 165
Page 26
117 Understanding a genre picture: don't we recognize the painted people as people and the painted trees as trees,
etc.?
A picture of a human face is a no less familiar object than the human face itself. But there is no question of
recognition here. 166
Page 26
118 The false concept that recognizing always consists in comparing two impressions with one another.--
"We couldn't use words at all if we didn't recognize them and the objects they denote." Have we any sort of
check on this recognition? 167
Page 26
119 This shape I see is not simply a shape, but is one of the shapes I know.--But it is not as if I were comparing the
object with a picture set beside it, but as if the object coincided with the picture. I see only one thing, not two. 168
Page 26
120 "This face has a quite particular expression." We perhaps look for words and feel that everyday language is
here too crude. 169
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Page 27
121 That a picture tells me something consists in its own form and colours. Or it narrates something to me: it uses
words so to speak, and I am comparing the picture with a combination of linguistic forms.--That a series of signs
tells me something isn't constituted by its now making this impression on me. "It's only in a language that something
is a proposition." 169
Page 27
122 'Language' is languages.--Languages are systems.
It is units of languages that I call "propositions". 170
Page 27
123 Certainly, I read a story and don't give a hang about any system of language, any more than if it was a story in
pictures. Suppose we were to say at this point "something is a picture only in a picture-language."?
171
Page 27
124 We might imagine a language in whose use the impression made on us by the signs played no part.
What I call a "proposition" is a position in the game of language.
Thinking is an activity, like calculating. 171
Page 27
125 A puzzle picture. What does it amount to to say that after the solution the picture means something to us,
whereas it meant nothing before? 172
Page 27
126 The impression is one thing, and the impression's being determinate is another thing. The impression of
familiarity is perhaps the characteristics of the determinacy that every strong impression has.
174
Page 27
127 Can I think away the impression of individual familiarity where it exists; and think it into a situation where it
does not? The difficulty is not a psychological one. We have not determined what that is to mean.
Can I look at a printed English word and see it as if I hadn't learnt to read?
I can ascribe meaning to a meaningless shape. 175
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Page 28
128 We can read courage into a face and say "now once more courage fits this face". This is related to "an attributive
adjective agrees with the subject".
What do I do if I take a smile now as a kind one, now as malicious? This is connected with the contrast
between saying and meaning. 176
Page 28
129 A friendly mouth, friendly eyes, the wagging of a dog's tail are primary symbols of friendliness: they are parts of
the phenomena that are called friendliness. If we want to imagine further appearances as expressions of friendliness,
we read these symbols into them. It is not that I can imagine that this man's face might change so that it looked
courageous, but that there is a quite definite way in which it can change into a courageous face.
Think of the multifariousness of what we call "language": word-language, picture-language, gesture-language,
sound-language. 178
Page 28
130 "'This object is familiar to me' is like saying 'this object is portrayed in my catalogue'." We are making the
assumption that the picture in our catalogue is itself familiar.
The sheath in my mind as a "form of imagining".--The pattern is no longer presented as an object, which
means that it didn't make sense to talk of a pattern at all.
"Familiarity: an object's fitting into a sheath"--that's not quite the same as our comparing what is seen with a
copy.
The question is "What do I recognize as what?" For "to recognize a thing as itself" is meaningless.
179
Page 28
131 The comparison between memory and a notebook.
How did I read off from the memory image that I stood thus at the window yesterday? What made you so
certain when you spoke those words? Nothing; I was certain.
How do I react to a memory? 181
Page 28
132 Operating with written signs and operating with "imagination pictures".
Page Break 29
An attitude to a picture (to a thought) is what connects it with reality. 182
Page 29
X
133 Grammatical rules determine a meaning and are not answerable to any meaning that they could contradict.
Why don't I call cookery rules arbitrary, and why am I tempted to call the rules of grammar arbitrary?
I don't call an argument good just because it has the consequences I want.
The rules of grammar are arbitrary in the same sense as the choice of a unit of measurement.
184
Page 29
134 Doesn't grammar put the primary colours together because there is a kind of similarity between them? Or
colours, anyway, in contrast to shapes or notes?
The rules of grammar cannot be justified by shewing that their application makes a representation agree with
reality.
The analogy between grammar and games. 185
Page 29
135 Langauge considered as a part of a psychological mechanism.
I do not use "this is the sign for sugar" in the same way as the sentence "if I press this button, I get a piece of
sugar". 187
Page 29
136 Suppose we compare grammar to a keyboard which I can use to direct a man by pressing different
combinations of keys. What corresponds in this case to the grammar of language?
If the utterance of a 'nonsensical' combination of words has the effect that the other person stares at me, I
don't on that account call it the order to stare. 188
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Page 30
137 Language is not defined for us as an arrangement fulfilling a definite purpose. 189
Page 30
138 Grammar consists of conventions--say in a chart. This might be a part of a mechanism. But it is the connection
and not the effect which determines the meaning.
Can one speak of a grammar in the case where a language is taught to a person by a mere drill?
190
Page 30
139 I do not scruple to invent causal connections in the mechanism of language.
To invent a keyboard might mean to invent something that had the desired effect; or else to devise new
forms which were similar to the old ones in various ways.
"It is always for living beings that signs exist." 191
Page 30
140 Inventing a language--inventing an instrument--inventing a game.
If we imagine a goal for chess--say entertainment--then the rules are not arbitrary. So too for the choice of a
unit of measurement.
We can't say "without language we couldn't communicate with one another". The concept of language is
contained in the concept of communication. 192
Page 30
141 Philosophy is philosophical problems. Their common element extends as far as the common element in
different regions of our language.
Something that at first sight looks like a proposition and is not one. Something that looks like a design for a
steamroller and is not one. 193
Page 30
142 Are we willing to call a series of independent signals "a language"?
Imagine a diary kept with signals. Are explanations given so that the signals are connected to another
language?
Page Break 31
A language consisting of commands. We wouldn't say that a series of such signals alone would enable me to
derive a picture of the movement of a man obeying them unless in addition to the signal there is something that
might be called a general rule for translating into drawing.
The grammar explains the meaning of the signs and thus makes the language pictorial. 194
Page 31
Appendix
1 Complex and Fact. 199
Page 31
2 Concept and Object, Property and Substrate. 202
Page 31
3 Objects. 208
Page 31
4 Elementary propositions. 210
Page 31
5 Is time essential to propositions? Comparison between time and truth-functions. 215
Page 31
6 The nature of hypotheses. 219
Page 31
7 Probability. 224
Page 31
8 The concept "about". The problem of the heap. 236
Part II: On Logic and Mathematics
I Logical Inference
Page 31
1 Is it because we understand the propositions that we know that q entails p? Does a sense give rise to the
entailment? 243
Page Break 32
Page 32
2 "If p follows from q, then thinking that q must involve thinking that p." 247
Page 32
3 The case of infinitely many propositions following from a single one. 250
Page 32
4 Can an experience show that one proposition follows from another? 255
II Generality
Page 32
5 The proposition "The circle is in the square" is in a certain sense independent of the assignment of a particular
position. (In a certain sense it is totally unconnected.) 257
Page 32
6 The proposition "The circle is in the square" is not a disjunction of cases. 261
Page 32
7 The inadequacy of the Frege-Russell notation for generality. 265
Page 32
8 Criticism of my former view of generality. 268
Page 32
9 The explanation of generality by examples. 270
Page 32
10 The law of a series. "And so on". 280
III The Foundations of Mathematics
11 The comparison between Mathematics and a game. 289
Page 32
12 There is no metamathematics. 296
Page 32
13 Proofs of relevance. 299
Page 32
14 Consistency proofs 303
Page Break 33
Page 33
15 Justifying arithmetic and preparing it for its application (Russell, Ramsey). 306
Page 33
16 Ramsey's theory of identity. 315
Page 33
17 The concept of the application of arithmetic (mathematics) 319
Page 33
IV On Cardinal Numbers
18 Kinds of cardinal number 321
Page 33
19 2 + 2 + 4. 332
Page 33
20 Statements of number within mathematics. 348
Page 33
21 Sameness of number and sameness of length. 351
Page 33
V Mathematical Proof
22 In other cases, if I am looking for something, then even before it is found I can describe what finding is; not so, if
I am looking for the solution of a mathematical problem.
Mathematical expeditions and Polar expeditions. 359
Page 33
23 Proof and the truth and falsehood of mathematical propositions. 366
Page 33
24 If you want to know what is proved, look at the proof. 369
Page 33
25 Mathematical problems. Kinds of problems. Search. "Projects" in mathematics.377
Page 33
26 Euler's Proof. 383
Page Break 34
Page 34
27 The trisection of an angle etc. 387
Page 34
28 Searching and trying. 393
Page 34
VI Inductive Proofs and Periodicity
29 How far is a proof by induction a proof of a proposition? 395
Page 34
30 Recursive proof and the concept of proposition. Is the proof a proof that a proposition is true and its
contradictory false? 397
Page 34
31 Induction, (x).φx and (∃x).φx. Does the induction prove the general proposition true and an existential
proposition false? 400
Page 34
32 Is there a further step from writing the recursive proof to the generalization? Doesn't the recursion schema
already say all that is to be said? 405
Page 34
33 How far does a recursive proof deserve the name of "proof". How far is a step in accordance with the paradigm A
justified by the proof of B? 408
Page 34
34 The recursive proof does not reduce the number of fundamental laws. 425
Page 34
35 Recurring decimals 1:3 = 0 • 3. 427
Page 34
36 The recursive proof as a series of proofs. 430
Page 34
37 Seeing or viewing a sign in a particular manner. Discovering an aspect of a mathematical expression. "Seeing an
expression in a particular way." Marks of emphasis. 437
Page 34
38 Proof by induction, arithmetic and algebra. 448
Page Break 35
Page 35
VII Infinity in Mathematics
39 Generality in arithmetic. 451
Page 35
40 On set theory. 460
Page 35
41 The extensional conception of the real numbers. 471
Page 35
42 Kinds of irrational number (π' P, F). 475
Page 35
43 Irregular infinite decimals. 483
Page 35
Note in Editing. 487
Page 35
Translator's note. 491
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Part I:
The Proposition and its Sense
Page 37
Page Break 38
Page Break 39
I
Page 39
1 How can one talk about 'understanding' and 'not understanding' a proposition? Surely it is not a proposition until
it's understood?
Page 39
Does it make sense to point to a clump of trees and ask "Do you understand what this clump of trees says?"
In normal circumstances, no; but couldn't one express a sense by an arrangement of trees? Couldn't it be a code?
Page 39
One would call 'propositions' clumps of trees one understood; others, too, that one didn't understand,
provided one supposed the man who planted them had understood them.
Page 39
"Doesn't understanding only start with a proposition, with a whole proposition? Can you understand half a
proposition?"--Half a proposition is not a whole proposition.--But what the question means can perhaps be
understood as follows. Suppose a knight's move in chess was always carried out by two movements of the piece,
one straight and one oblique; then it could be said "In chess there are no half knight's moves" meaning: the
relationship of half a knight's move to a whole knight's move is not the same as that of half a bread roll to a whole
bread roll. We want to say that it is not a difference of degree.
Page 39
It is strange that science and mathematics make use of propositions, but have nothing to say about
understanding those propositions.
Page 39
2 We regard understanding as the essential thing, and signs as something inessential.--But in that case, why have the
signs at all? If you think that it is only so as to make ourselves understood by others, then you are very likely looking
on the signs as a drug which is to produce in other people the same condition as my own.
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Page 40
Suppose that the question is "what do you mean by that gesture?" and the answer is "I mean you must
leave". The answer would not have been more correctly phrased: "I mean what I mean by the sentence 'you must
leave'."
Page 40
In attacking the formalist conception of arithmetic, Frege says more or less this: these petty explanations of
the signs are idle once we understand the signs. Understanding would be something like seeing a picture from which
all the rules followed, or a picture that makes them all clear. But Frege does not seem to see that such a picture
would itself be another sign, or a calculus to explain the written one to us.
Page 40
What we call "understanding a language" is often like the understanding we get of a calculus when we learn
its history or its practical application. And there too we meet an easily surveyable symbolism instead of one that is
strange to us.--Imagine that someone had originally learnt chess as a writing game, and was later shown the
'interpretation' of chess as a board game.
Page 40
In this case "to understand" means something like "to take in as a whole".
Page 40
If I give anyone an order I feel it to be quite enough to give him signs. And if I am given an order, I should
never say: "this is only words, and I have got to get behind the words". And when I have asked someone something
and he gives me an answer I am content--that was just what I expected--and I don't raise the objection: "but that's a
mere answer."
Page 40
But if you say: "How am I to know what he means, when I see nothing but the signs he gives?" then I say:
"How is he to know what he means, when he has nothing but the signs either?"
Page 40
What is spoken can only be explained in language, and so in this sense language itself cannot be explained.
Page 40
Language must speak for itself.
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Page 41
3 One can say that meaning drops out of language; because what a proposition means is told by yet another
proposition.
Page 41
"What did you mean by those words?" "Did you mean those words." The first question is not a more precise
specification of the second. The first is answered by a proposition replacing the proposition which wasn't
understood. The second question is like: "Did you mean that seriously or as a joke?"
Page 41
Compare: "Did you mean anything by that gesture--if so what?"
Page 41
In certain of their applications the words "understand", "mean" refer to a psychological reaction while
hearing, reading, uttering etc. a sentence. In that case understanding is the phenomenon that occurs when I hear a
sentence in a familiar language and not when I hear a sentence in a strange language.
Page 41
Learning a language brings about the understanding of it. But that belongs to the past history of the
reaction.--The understanding of a sentence is as much something that happens to me as is the hearing of a sentence;
it accompanies the hearing.
Page 41
I can speak of 'experiencing' a sentence. "I am not merely saying this, I mean something by it." When we
consider what is going on in us when we mean (and don't just say) words, it seems to us as if there were something
coupled to the words, which otherwise would run idle. As if they connected with something in us.
Page 41
4 Understanding a sentence is more akin to understanding a piece of music than one might think. Why must these
bars be played just so? Why do I want to produce just this pattern of variation in loudness and tempo? I would like
to say "Because I know what it's all about." But what is it all about? I should not be able to say. For explanation I
can only translate the musical picture into a picture in another medium and let the one picture throw light on the
other.
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Page 42
The understanding of a sentence can also be compared with what we call understanding a picture. Think of a
still-life picture, and imagine that we were unable to see it as a spatial representation and saw only patches and lines
on the canvas. We could say in that case "we do not understand the picture". But we say the same thing in a
different sense when although we see the picture spatially we do not recognize the spatial objects as familiar things
like books, animals and bottles.
Page 42
Suppose the picture is a genre-picture and the people in it are about an inch long. If I had ever seen real
people of that size, I would be able to recognize them in the picture and regard it as a life-size representation of them.
In that case my visual experience of the picture would not be the same as the one I have when I see the picture in the
normal way as a representation in miniature, although the illusion of spatial vision is the same in each
case.--However, acquaintance with real inch-high people is put forward here only as one possible cause of the visual
experience; except for that the experience is independent. Similarly, it may be that only someone who has already
seen many real cubes can see a drawn cube spatially; but the description of the spatial visual presentation contains
nothing to differentiate a real cube from a painted one.
Page 42
The different experiences I have when I see a picture first one way and then another are comparable to the
experience I have when I read a sentence with understanding and without understanding.
Page 42
(Recall what it is like when someone reads a sentence with a mistaken intonation which prevents him from
understanding it--and then realizes how it is to be read.)
Page 42
(To see a watch as a watch, i.e. as a dial with hands, is like seeing Orion as a man striding across the sky.)
Page 42
5 How curious: we should like to explain the understanding of a gesture as a translation into words, and the
understanding of words as a translation into gestures.
Page 42
And indeed we really do explain words by a gesture, and a gesture by words.
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Page 43
On the other hand we say "I understand that gesture" in the same sense as "I understand this theme", "it says
something" and what that means is that I have a particular experience as I follow it.
Page 43
Consider the difference it makes to the understanding of a sentence when a word in it is felt as belonging first
with one word and then with another. I might have said: the word is conceived, understood, seen, pronounced as
belonging first with one word and then with another.
Page 43
We can call a 'proposition' that which is conceived first in one way and then in another; we can also mean the
various ways of conceiving it. This is a source of confusions.
Page 43
I read a sentence from the middle of a story: "After he had said this, he left her as he did the day before." Do
I understand the sentence?--It's not altogether easy to give an answer. It is an English sentence, and to that extent I
understand it. I should know how the sentence might be used, I could invent a context for it. And yet I do not
understand it in the sense in which I should understand it if I had read the story. (Compare various language-games:
describing a state of affairs, making up a story, etc. What counts as a significant sentence in the several cases?)
Page 43
Do we understand Christian Morgenstern's poems, or Lewis Carroll's poem "Jabberwocky"? In these cases
it's very clear that the concept of understanding is a fluid one.
Page 43
6 A sentence is given me in unfamiliar code together with the key for deciphering it. Then, in a certain sense,
everything required for the understanding of the sentence has been given me. And yet if I were asked whether I
understood the sentence I should reply "I must first decode it" and only when I had it in front of me decoded as an
English sentence, would I say "now I understand it".
Page 43
If we now raise the question "At what moment of translating into English does understanding begin?" we get
a glimpse into the
Page Break 44
nature of what is called "understanding".
Page 44
I say the sentence "I see a black patch there"; but the words are after all arbitrary: so I will replace them one
after the other by the first six letters of the alphabet. Now it goes "a b c d e f". But now it is dear that--as one would
like to say--I cannot think the sense of the above sentence straight away in the new expression. I might also put it
like this: I am not used to saying "a" instead of "I", "b" instead of "see", "c" instead of "a" and so on. I don't mean
that I am not used to making an immediate association between the word "I" and the sign "a"; but that I am not used
to using "a" in the place of "I".
Page 44
"To understand a sentence" can mean "to know what the sentence signifies"; i.e. to be able to answer the
question "what does this sentence say?"
Page 44
It is a prevalent notion that we can only imperfectly exhibit our understanding; that we can only point to it
from afar or come close to it, but never lay our hands on it, and that the ultimate thing can never be said. We say:
"Understanding is something different from the expression of understanding. Understanding cannot be exhibited; it
is something inward and spiritual."--Or "Whatever I do to exhibit understanding, whether I repeat an explanation of
a word, or carry out an order to show that I have understood it, these bits of behaviour do not have to be taken as
proofs of understanding." Similarly, people also say "I cannot show anyone else my toothache; I cannot prove to
anyone else that I have toothache." But the impossibility spoken of here is supposed to be a logical one. "Isn't it the
case that the expression of understanding is always an incomplete expression?" That means, I suppose, an
expression with something missing--but the something missing is essentially inexpressible, because otherwise I
Page Break 45
might find a better expression for it. And "essentially inexpressible" means that it makes no sense to talk of a more
complete expression.
Page 45
The psychological processes which are found by experience to accompany sentences are of no interest to us.
What does interest us is the understanding that is embodied in an explanation of the sense of the sentence.
Page 45
7 To understand the grammar of the word "to mean" we must ask ourselves what is the criterion for an expression's
being meant thus. What should be regarded as a criterion of the meaning?
Page 45
An answer to the question "How is that meant?" exhibits the relationship between two linguistic expressions.
So the question too is a question about that relationship.
Page 45
The process we call the understanding of a sentence or of a description is sometimes a process of translation
from one symbolism into another; tracing a picture, copying something, or translating into another mode of
representation.
Page 45
In that case understanding a description means making oneself a picture of what is described. And the
process is more or less like making a drawing to match a description.
Page 45
We also say: "I understand the picture exactly, I could model it in day".
Page 45
8 We speak of the understanding of a sentence as a condition of being able to apply it. We say "I cannot obey an
order if I do not understand it" or "I cannot obey it before I understand it".
Page 45
"Must I really understand a sentence to be able to act on it?--Certainly, otherwise you wouldn't know what
you had to do."--But how does this knowing help me? Isn't there in turn a jump from knowing to doing?
Page 45
"But all the same I must understand an order to be able to act according to it"--here the "must" is fishy. If it is
a logical must, then the sentence is a grammatical remark.
Page Break 46
Page 46
Here it could be asked: How long before obeying it must you understand the order?--But of course the
proposition "I must understand the order before I can act on it" makes good sense: but not a metalogical sense.--And
'understanding' and 'meaning' are not metalogical concepts.
Page 46
If "to understand a sentence" means somehow or other to act on it, then understanding cannot be a
precondition for our acting on it. But of course experience may show that the specific behaviour of understanding is
a precondition for obedience to an order.
Page 46
"I cannot carry out the order because I don't understand what you mean.--Yes, I understand you
now."--What went on when I suddenly understood him? Here there are many possibilities. For example: the order
may have been given in a familiar language but with a wrong emphasis, and the right emphasis suddenly occurred to
me. In that case perhaps I should say to a third party: "Now I understand him: he means..." and should repeat the
order with the right emphasis. And when I grasped the familiar sentence I'd have understood the order,--I mean, I
should not first have had to grasp an abstract sense.--Alternatively: I understood the order in that sense, so it was a
correct English sentence, but it seemed preposterous. In such a case I would say: "I do not understand you: because
you can't mean that." But then a more comprehensible interpretation occurred to me. Before I understand several
interpretations, several explanations, may pass through my mind, and then I decide on one of them.
Page 46
(Understanding, when an absent-minded man at the order "Right turn!" turns left, and then, clutching his
forehead, says "Oh! right turn" and does a right turn.)
Page 46
9 Suppose the order to square a series of numbers is written in the form of a table, thus:
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Page 47
--It seems to us as if by understanding the order we add something to it, something that fills the gap between
command and execution. So that if someone said "You understand it, don't you, so it is not incomplete" we could
reply "Yes, I understand it, but only because I add something to it, namely the interpretation."--But what makes you
give just this interpretation? Is it the order? In that case it was already unambiguous, since it demanded this
interpretation. Or did you attach the interpretation arbitrarily? In that case what you understood was not the
command, but only what you made of it.
Page 47
(While thinking philosophically we see problems in places where there are none. It is for philosophy to show
that there are no problems.)
Page 47
But an interpretation is something that is given in signs. It is this interpretation as opposed to a different one
(running differently). So if one were to say "Any sentence still stands in need of an interpretation" that would mean:
no sentence can be understood without a rider.
Page 47
Of course sometimes I do interpret signs, give signs an interpretation; but that does not happen every time I
understand a sign. (If someone asks me "What time is it?" there is no inner process of laborious interpretation; I
simply react to what I see and hear. If someone whips out a knife at me, I do not say "I interpret that as a threat".)
Page 47
10 "Understanding a word" may mean: knowing how it is used; being able to apply it.
Page 47
"Can you lift this ball?"--"Yes". Then I try and fail. Then perhaps I say "I was wrong, I cannot". Or perhaps
"I can't now, because I am too tired; but when I said I could, I really could." Similarly "I thought I could play chess,
but now I have forgotten
Page Break 48
how", but on the other hand "When I said 'I can play chess' I really could, but now I've lost it."--But what is the
criterion for my being able at that particular time? How did I know that I could? To that question I would answer
"I've always been able to lift that sort of weight", "I lifted it just a moment before", "I've played chess quite recently
and my memory is good", "I'd just recited the rules" and so on. What I regard as an answer to that question will
show me in what way I use the word "can".
Page 48
Knowing, being able to do something, a capacity is what we would call a state. Let us compare with each
other propositions which all in various senses describe states.
Page 48
"I have had toothache since yesterday."
"I have been longing for him since yesterday."
"I have been expecting him since yesterday."
"I have known since yesterday that he is coming."
"Since yesterday I can play chess."
Page 48
Can one say: "I have known continuously since yesterday that he is coming?" In which of the above
sentences can one sensibly insert the word "continuously"?
Page 48
If knowledge is called a "state" it must be in the sense in which we speak of the state of a body or of a
physical model. So it must be in a physiological sense or in the sense used in a psychology that talks about
unconscious states of a mind-model. Certainly no one would object to that; but in that case one still has to be clear
that we have moved from the grammatical realm of 'conscious states' into a different grammatical realm. I can no
doubt speak of unconscious toothache, if the sentence "I have unconscious toothache" means something like "I have
a bad tooth that doesn't ache". But the expression "conscious state" (in its old sense) doesn't have the same
grammatical relationship to the expression
Page Break 49
"unconscious state" as the expression "a chair which I see" has to "a chair which I don't see because it's behind me".
Page 49
Instead of "to know something" we might say "to keep a piece of paper on which it is written".
Page 49
If "to understand the meaning of a word" means to know the grammatically possible ways of applying it,
then I can ask "How can I know what I mean by a word at the moment I utter it? After all, I can't have the whole
mode of application of a word in my head all at once".
Page 49
I can have the possible ways of applying a word in my head in the same sense as the chess player has all the
rules of chess in his head, and the alphabet and the multiplication table. Knowledge is the hypothesized reservoir out
of which the visible water flows.
Page 49
11 So we mustn't think that when we understand or mean a word what happens is an act of instantaneous, as it were
non-discursive, grasp of grammar. As if it could all be swallowed down in a single gulp.
Page 49
It is as if I get tools in the toolbox of language ready for future use.
Page 49
"I can use the word 'yellow'" is like "I know how to move the king in chess".
Page 49
In this example of chess we can again observe the ambiguity of the word "understand". When a man who
knows the game watches a game of chess, the experience he has when a move is made usually differs from that of
someone else watching without understanding the game. (It differs too from that of a man who doesn't even know
that it's a game.) We can also say that it's the knowledge of the rules of chess which makes the difference between
the two spectators, and so too that it's the knowledge of the
Page Break 50
rules which makes the first spectator have the particular experience he has. But this experience is not the knowledge
of the rules. Yet we are inclined to call them both "understanding".
Page 50
The understanding of language, as of a game, seems like a background against which a particular sentence
acquires meaning.--But this understanding, the knowledge of the language, isn't a conscious state that accompanies
the sentences of the language. Not even if one of its consequences is such a state. It's much more like the
understanding or mastery of a calculus, something like the ability to multiply.
Page 50
12 Suppose it were asked: "When do you know how to play chess? All the time? Or just while you say that you
can? Or just during a move in the game?"--How queer that knowing how to play chess should take such a short
time, and a game of chess so much longer!
Page 50
(Augustine: "When do I measure a period of time?")
Page 50
It can seem as if the rules of grammar are in a certain sense an unpacking of something we experience all at
once when we use a word.
Page 50
In order to get clearer about the grammar of the word "understand", let's ask: when do we understand a
sentence?--When we've uttered the whole of it? Or while uttering it?--Is understanding, like the uttering of a
sentence, an articulated process and does its articulation correspond exactly to that of the sentence? Or is it
non-articulate, something accompanying the sentence in the way a pedal note accompanies a melody?
Page 50
How long does it take to understand a sentence?
Page 50
And if we understand a sentence for a whole hour, are we always starting afresh?
Page 50
13 Chess is characterized by its rules (by the list of rules). If I define the game (distinguish it from draughts) by its
rules, then
Page Break 51
these rules belong to the grammar of the word "chess". Does that mean that if someone uses the word "chess"
intelligently he must have a definition of the word in mind? Certainly not.--He will only give one if he's asked what
he means by "chess".
Page 51
Suppose I now ask: "When you uttered the word, what did you mean by it?"--If he answered "I meant the
game we've played so often, etc. etc." I would know that this explanation hadn't been in his mind at all when he used
the word, and that he wasn't giving an answer to my question in the sense of telling me what "went on inside him"
while he was uttering the word.
Page 51
When someone interprets, or understands, a sign in one sense or another, what he is doing is taking a step in
a calculus (like a calculation). What he does is roughly what he does if he gives expression to his interpretation.
Page 51
"Thought" sometimes means a particular mental process which may accompany the utterance of a sentence
and sometimes the sentence itself in the system of language.
Page 51
"He said those words, but he didn't think any thoughts with them."--"Yes, I did think a thought while I said
them". "What thought?" "Just what I said."
Page 51
On hearing the assertion "This sentence makes sense" you cannot really ask "what sense?" Just as on hearing
the assertion "this combination of words is a sentence" you cannot ask "what sentence?"
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II
Page 52
14 Can what the rules of grammar say about a word be described in another way by describing the process which
takes place when understanding occurs?
Page 52
Suppose the grammar is the geometry of negation for example, can I replace it by the description of what
"lies behind" the word "not" when it is applied?
Page 52
We say: "Anyone who understands negation knows that two negations yield an affirmation."
Page 52
That sounds like "Carbon and oxygen yield carbonic acid". But in reality a doubled negation does not yield
anything, it is something.
Page 52
Something here gives us the illusion of a fact of physics. As if we saw the result of a logical process. Whereas
the only result is the result of the physical process.
Page 52
We would like to say: "Negation has the property that when it is doubled it yields an affirmation," But the
rule doesn't give a further description of negation, it constitutes negation.
Page 52
Negation has the property that it denies truly such and such a sentence.
Page 52
Similarly, a circle--say one painted on a flat surface--has the property of being in such and such a position, of
having the colour it has, of being bisected by a certain line (a boundary between two colours) and so on; but it
doesn't have the properties that geometry seems to ascribe to it (i.e. the ability to have the other properties).
Page 52
Likewise one doesn't have the property that when it's added to itself it makes two.
Page 52
15 Geometry no more speaks about cubes than logic does about negation.
Page 52
Geometry defines the form of a cube but does not describe it. If the description of a cube says that it is red
and hard, then 'a
Page Break 53
description of the form of a cube' is a sentence like "This box has the form of a cube".
Page 53
But if I describe how to make a cubical box, doesn't this contain a description of the form of a cube? A
description only insofar as this thing is said to be cubical, and for the rest an analysis of the concept of cube.
Page 53
"This paper is not black, and two such negations yield an affirmation".
Page 53
The second clause is reminiscent of "and two such horses can pull the cart". But it contains no assertion
about negation; it is a rule for the replacement of one sign by another.
Page 53
"That two negations yield an affirmation must already be contained in the negation that I am using now."
Here I am on the verge of inventing a mythology of symbolism.
Page 53
It looks as if one could infer from the meaning of negation that "~~p" means p. As if the rules for the
negation sign follow from the nature of negation. So that in a certain sense there is first of all negation, and then the
rules of grammar.
Page 53
It also looks as if the essence of negation had a double expression in language: the one whose meaning I
grasp when I understand the expression of negation in a sentence, and the consequences of this meaning in the
grammar.
Page 53
16 What does it mean to say that the "is" in "The rose is red" has a different meaning from the "is" in "Twice two is
four"? If it is answered that it means that different rules are valid for these two words, we can say that we have only
one word here.--And if all I am attending to is grammatical rules, these do allow the use of the word "is" in both
connections.--But the rule which shews that the word "is" has different meanings in the two sentences is the one
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allowing us to replace the word "is" in the second sentence by "equals" and forbidding this substitution in the first
sentence.
Page 54
"Is this rule then only the consequence of the first rule, that the word 'is' has different meanings in the two
sentences? Or is it rather that this very rule is the expression of the word's having a different meaning in the two
contexts?"
Page 54
It looks as if a sentence with e.g. the word "ball" in it already contained the shadow of other uses of this
word. That is to say, the possibility of forming those other sentences. To whom does it look like that? And under
what circumstances?
Page 54
The comparison suggests itself that the word "is" in different cases has different meaning-bodies behind it;
that it is perhaps each time a square surface, but in one case it is the end surface of a prism and in the other the end
surface of a pyramid.
Page 54
Imagine the following case. Suppose we have some completely transparent glass cubes which have one face
painted red. If we arrange these cubes together in space, only certain arrangements of red squares will be permitted
by the shape of the glass bodies. I might then express the rule for the possible arrangements of the red squares
without mentioning the cubes; but the rule would none the less contain the essence of the form of cube--Not, of
course, the fact that there are glass cubes behind the red squares, but the geometry of the cube.
Page 54
But suppose we see such a cube: are we immediately presented with the rules for the possible combinations,
i.e. the geometry of the cube? Can I read off the geometry of the cube from a cube?
Page 54
Thus the cube is a notation for the rule. And if we had discovered such a rule, we really wouldn't be able to
find anything better than the drawing of a cube to use as a notation for it. (And it is significant that here a drawing of
a cube will do instead of a cube.)
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Page 55
But how can the cube (or the drawing) serve as a notation for a geometrical rule? Only if it belongs, as a
proposition or part of a proposition, to a system of propositions.
Page 55
17 "Of course the grammatical possibilities of the negation sign reveal themselves bit by bit in the use of the signs,
but I think negation all at once. The sign 'not' is only a pointer to the thought 'not'; it is only a stimulus to produce
the right thought, only a signal."
Page 55
(If I were asked what I mean by the word "and" in the sentence "pass me the bread and butter" I would
answer by a gesture of gathering together; and that gesture would illustrate what I mean, in the same way as a green
pattern illustrates the meaning of "green" and the T-F notation illustrates the meaning of "not", "and" etc.)
Page 55
For instance, this sign for negation: is worth no more and no less than any other negation sign;
it is a complex of lines just like the expression "not-p" and it is only made into a sign for negation by the way it
works--I mean, the way it is used in the game.
Page 55
(The same goes for the T-F schemata for tautology and contradiction.)
Page 55
What I want to say is that to be a sign a thing must be dynamic, not static.
Page 55
18 Here it can easily seem as if the sign contained the whole of the grammar; as if the grammar were contained in the
sign like a string of pearls in a box and he had only to pull it out. (But this kind of picture is just what is misleading
us). As if understanding were an instantaneous grasping of something from which later we only draw consequences
which already exist in an ideal sense before they are drawn. As if the cube already contained the geometry of the
cube, and I had only to unpack it. But which cube? Or is there
Page Break 56
an ideal geometrical cube?--Often we have in mind the process of deriving geometrical propositions from a drawing,
a representation (or a model). But what is the role of the model in such a case? It has the role of a sign, a sign
employed in a particular game.--And it is an interesting and remarkable thing how this sign is employed, how we
perhaps use the drawing of a cube again and again in different contexts.--And it is this sign, (which has the identity
proper to a sign) that we take to be the cube in which the geometrical laws are already laid up. (They are no more
laid up there than the disposition to be used in a certain way is laid up in the chessman which is the king).
Page 56
In philosophy one is constantly tempted to invent a mythology of symbolism or of psychology, instead of
simply saying what we know.
Page 56
19 The concept of meaning I adopted in my philosophical discussions originates in a primitive philosophy of
language.
Page 56
The German word for "meaning" is derived from the German word for "pointing".
Page 56
When Augustine talks about the learning of language he talks about how we attach names to things, or
understand the names of things. Naming here appears as the foundation, the be all and end all of language.
Page 56
Augustine does not speak of there being any difference between parts of speech and means by "names"
apparently words like "tree", "table", "bread" and of course, the proper names of people; also no doubt "eat", "go",
"here", "there"--all words, in fact. Certainly he's thinking first and foremost of nouns, and of the remaining words as
something that will take care of itself. (Plato too says that a sentence consists of nouns and verbs.)†1
Page Break 57
Page 57
They describe the game as simpler than it is.
Page 57
But the game Augustine describes is certainly a part of language. Imagine I want to put up a building using
building stones someone else is to pass me; we might first make a convention by my pointing to a building stone
and saying "that is a pillar", and to another and saying "that is called 'a block' ", "that is called 'a slab'" and so on.
And then I call out the words "pillar", "slab", etc. in the order in which I need the stones.
Page 57
Augustine does describe a calculus of our language, only not everything that we call language is this calculus.
(And one has to say this in many cases where we are faced with the question "Is this an appropriate description or
not?" The answer is: "Yes, it is appropriate, but only here, and not for the whole region that you were claiming to
describe.") So it could be said that Augustine represents the matter too simply; but also that he represents something
simpler.
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It is as if someone were to say "a game consists in moving objects about on a surface according to certain
rules..." and we replied: You must be thinking of board games, and your description is indeed applicable to them.
But they are not the only games. So you can make your definitions correct by expressly restricting it to those games.
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20 The way Augustine describes the learning of language can show us the way of looking at language from which
the concept of the meaning of words derives.
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The case of our language could be compared with a script in which the letters were used to stand for sounds,
and also as signs of emphasis and perhaps as marks of punctuation. If one conceives this script as a language for
describing sound-patterns, one can imagine someone misinterpreting the script as if there were simply a
correspondence of letters to sounds and as if the letters had not also completely different functions.
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Just as the handles in the cabin of a locomotive have different kinds of job, so do the words of language,
which in one way are like handles. One is the handle of a crank, it can be moved continuously since it operates a
valve; another works a switch, which has two positions; a third is the handle of a pump and only works when it is
being moved up and down etc. But they all look alike, since they are all worked by hand.
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A connected point: it is possible to speak perfectly intelligibly of combinations of colours and shapes (e.g.
of the colours red and blue and the shapes square and circle) just as we speak of combinations of different shapes or
spatial objects. And this is the origin of the bad expression: a fact is a complex of objects. Here the fact that a man is
sick is compared with a combination of two things, one of them the man and the other the sickness.
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21 A man who reads a sentence in a familiar language experiences the different parts of speech in quite different
ways. (Think of the comparison with meaning-bodies.) We quite forget that the written and spoken words "not",
"table" and "green" are similar to each other. It is only in a foreign language that we see clearly the uniformity of
words. (Compare William James on the feelings that correspond to words like "not", "but" and so on.)
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("Not" makes a gesture of rejection.
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No, it is a gesture of rejection. To grasp negation is to understand a gesture of rejection.)
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Compare the different parts of speech in a sentence with lines on a map with different functions (frontiers,
roads, meridians, contours.) An uninstructed person sees a mass of lines and does not know the variety of their
meanings.
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Think of a line on a map crossing a sign out to show that it is void
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The difference between parts of speech is comparable to the differences between chessmen, but also to the
even greater difference between a chessman and the chess board.
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22 We say: the essential thing in a word is its meaning. We can replace the word by another with the same meaning.
That fixes a place for the word, and we can substitute one word for another provided we put it in the same place.
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If I decide to say a new word instead of "red" (perhaps only in thought), how would it come out that it took
the place of the word "red"?
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Suppose it was agreed to say "non" in English instead of "not", and "not" instead of "red". In that case the
word "not" would remain in the language, and one could say that "non" was now used in the way in which "not"
used to be, and that "not" now had a different use.
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Would it not be similar if I decided to alter the shape of the chess pieces, or to use a knight-shaped piece as
the king? How would it be clear that the knight is the king? In this case can't I very well talk about a change of
meaning?
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23 I want to say the place of a word in grammar is its meaning.
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But I might also say: the meaning of a word is what the explanation of its meaning explains.
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"What 1 c.c. of water weighs is called '1 gram'--Well, what does it weigh?"
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The explanation of the meaning explains the use of the word.
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The use of a word in the language is its meaning.
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Grammar describes the use of words in the language.
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So it has somewhat the same relation to the language as the description of a game, the rules of a game, have
to the game.
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Meaning, in our sense, is embodied in the explanation of meaning. If, on the other hand, by the word
"meaning" we mean a characteristic sensation connected with the use of a word, then the relation between the
explanation of a word and its meaning is rather that of cause to effect.
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24 An explanation of meaning can remove every disagreement with regard to a meaning. It can clear up
misunderstandings.
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The understanding here spoken of is a correlate of explanation.
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By "explanation of the meaning of a sign" we mean rules for use but above all definitions. The distinction
between verbal definitions and ostensive definitions gives a rough division of these types of explanation.
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In order to understand the role of a definition in the calculus we must investigate the particular case.
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It may seem to us as if the other grammatical rules for a word had to follow from its ostensive definition;
since after all an ostensive definition, e.g. "that is called 'red'" determines the meaning of the word "red".
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But this definition is only those words plus pointing to a red object, e.g. a red piece of paper. And is this
definition really unambiguous? Couldn't I have used the very same one to give the word "red" the meaning of the
word "paper", or "square", or "shiny", or "light", or "thin" etc. etc.?
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However, suppose that instead of saying "that is called 'red'"
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I had phrased my definition "that colour is called 'red'". That certainly is unambiguous, but only because the
expression "colour" settles the grammar of the word "red" up to this last point. (But here questions could arise like
"do you call just this shade of colour red, or also other similar shades?"). Definitions might be given like this: the
colour of this patch is called "red", its shape "ellipse".
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I might say: one must already understand a great deal of a language in order to understand that definition.
Someone who understands that definition must already know where the words ("red", "ellipse") are being put, where
they belong in language.
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25 The words "shape" and "colour" in the definitions determine the kind of use of the word, and therefore what one
may call the part of speech. And in ordinary grammar one might well distinguish "shape words", "colour words",
"sound words", "substance words" and so on as different parts of speech. (There wouldn't be the same reason for
distinguishing "metal words", "poison words", "predator words". It makes sense to say "iron is a metal",
"phosphorus is a poison", etc. but not "red is a colour", "a circle is a shape" and so on.)
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I can ostensively define a word for a colour or a shape or a number, etc. etc. (children are given ostensive
explanations of numerals and they do perfectly well); negation, too, disjunction and so on. The same ostension
might define a numeral, or the name of a shape or the name of a colour. But in the grammar of each different part of
speech the ostensive definition has a different role; and in each case it is only one rule.
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(Consider also the grammar of definitions like: "today is called Monday", "I will call this day of the year 'the
day of atonement'").
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26 But when we learn the meaning of a word, we are very often given only the single rule, the ostensive definition.
So how does it
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come about that on the strength of this definition we understand the word? Do we guess the rest of the rules?
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Think also of teaching a child to understand words by showing it objects and uttering words. The child is
given ostensive definitions and then it understands the words.--But what is the criterion of understanding here?
Surely, that the child applies the words correctly. Does it guess rules?--Indeed we must ask ourselves whether we are
to call these signs and utterances of words "definitions" at all. The language game is still very simple and the
ostensive definition has not the same role in this language-game as in more developed ones. (For instance, the child
cannot yet ask "What is that called?") But there is no sharp boundary between primitive forms and more
complicated ones. I wouldn't know what I can and what I can't still call "definition". I can only describe language
games or calculi; whether we still want to call them calculi or not doesn't matter as long as we don't let the use of the
general term divert us from examining each particular case we wish to decide.
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I might also say of a little child "he can use the word, he knows how it is applied." But I only see what that
means if I ask "what is the criterion for this knowledge?" In this case it isn't the ability to state rules.
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What's the sign of someone's understanding a game? Must he be able to recite the rules? Isn't it also a
criterion that he can play the game, i.e. that he does in fact play it, even if he's baffled when asked for the rules? Is it
only by being told the rules that the game is learnt and not also simply by watching it being played? Of course a man
will often say to himself while watching "oh, so that's the rule"; and he might perhaps write down the rules as he
observes them; but there's certainly such a thing as learning the game without explicit rules.
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The grammar of a language isn't recorded and doesn't come into existence until the language has already
been spoken by human
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beings for a long time. Similarly, primitive games are played without their rules being codified, and even without a
single rule being formulated.
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But we look at games and language under the guise of a game played according to rules. That is, we are
always comparing language with a procedure of that kind.
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27 The names I give to bodies, shapes, colours, lengths have different grammars in each case. ("A" in "A is yellow"
has one grammar if A is a body and another if A is the surface of a body; for instance it makes sense to say that the
body is yellow all through, but not to say that the surface is.) And one points in different sense to a body, and to its
length or its colour; for example, a possible definition would be: "to point to a colour" means, to point to the body
which has the colour. Just as a man who marries money doesn't marry it in the same sense as he marries the woman
who owns the money.
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Money, and what one buys with it. Sometimes a material object, sometimes the right to a seat in the theatre,
or a title, or fast travel, or life, etc.
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A name has meaning, a proposition has sense in the calculus to which it belongs. The calculus is as it were
autonomous.--Language must speak for itself.
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I might say: the only thing that is of interest to me is the content of a proposition and the content of a
proposition is something internal to it. A proposition has its content as part of a calculus.
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The meaning is the role of the word in the calculus.
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The meaning of a name is not the thing we point to when we give an ostensive definition of the name; that is,
it is not the bearer of
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the name.--The expression "the bearer of the name 'N'" is synonymous with the name "N". The expression can be
used in place of the name. "The bearer of the name 'N' is sick" means "N is sick". We don't say: The meaning of "N"
is sick.
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The name doesn't lose its meaning if its bearer ceases to exist (if he dies, say).
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But doesn't "Two names have a single bearer" mean the same as "two names have the same meaning?"
Certainly, instead of "A = B" one can write "the bearer of the name 'A' = the bearer of the name 'B'".
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28 What does "to understand a word" mean?
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We say to a child "No, no more sugar" and take it away from him. Thus he learns the meaning of the word
"no". If, while saying the same words, we had given him a piece of sugar he would have learnt to understand the
word differently. (In this way he has learnt to use the word, but also to associate a particular feeling with it, to
experience it in a particular way.)
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What constitutes the meaning of a word like "perhaps"? How does a child learn the use of the word
"perhaps"? It may repeat a sentence it has heard from an adult like "perhaps she will come"; it may do so in the
same tone of voice as the adult. (That is a kind of a game). In such a case the question is sometimes asked: Does it
already understand the word "perhaps" or is it only repeating it?--What shows that it really understands the
word?--Well, that it uses it in particular circumstances in a particular manner--in certain contexts and with a
particular intonation.
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What does it mean "to understand the word 'perhaps'"?--Do I understand the word "perhaps"?--And how do
I judge whether I do? Well, something like this: I know how it's used, I can explain its use to somebody, say by
describing it in made-up cases. I can describe the occasions of its use, its position in sentences, the intonation it has
in speech.--Of course this only means that "I
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understand the word 'perhaps'" comes to the same as: "I know how it is used etc."; not that I try to call to mind its
entire application in order to answer the question whether I understand the word. More likely I would react to this
question immediately with the answer "yes", perhaps after having said the word to myself once again, and as it were
convinced myself that it's familiar, or else I might think of a single application and pronounce the word with the
correct intonation and a gesture of uncertainty. And so on.
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This is like the case in which someone is explaining to me a calculation "that I don't quite understand", and
when he has reached a particular point of his explanation, I say: "ah, now I understand; now I know how to go on".
How do I know that I know how to go on? Have I run through the rest of the calculation at that moment? Of course
not. Perhaps a bit of it flashed before my mind; perhaps a particular application or a diagram. If I were asked: how
do you know that you can use the word "perhaps" I would perhaps simply answer "I have used it a hundred times".
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29 But it might be asked: Do I understand the word just be describing its application? Do I understand its point?
Haven't I deluded myself about something important?
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At present, say, I know only how men use this word. But it might be a game, or a form of etiquette. I do not
know why they behave in this way, how language meshes with their life.
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Is meaning then really only the use of a word? Isn't it the way this use meshes with our life?
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But isn't its use a part of our life?
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Do I understand the word "fine" when I know how and on what occasions people use it? Is that enough to
enable me to use it myself? I mean, so to say, use it with conviction.
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Wouldn't it be possible for me to know the use of the word and yet follow it without understanding? (As, in a
sense, we follow the
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singing of birds). So isn't it something else that constitutes understanding--the feeling "in one's own breast", the
living experience of the expressions?--They must mesh with my own life.
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Well, language does connect up with my own life. And what is called "language" is something made up of
heterogeneous elements and the way it meshes with life is infinitely various.
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30 We may say that the words "fine", "oh", and also "perhaps" are expressions of sensation, of feeling. But I don't
call the feeling the meaning of the word. We are not interested in the relation of the words to the senesation [[sic]],
whatever it may be, whether they are evoked by it, or are regularly accompanied by it, or give it an outlet. We are not
interested in any empirical facts about language, considered as empirical facts. We are only concerned with the
description of what happens and it is not the truth but the form of the description that interests us. What happens
considered as a game.
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I am only describing language, not explaining anything.
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For my purposes I could replace the sensation the word is said to express by the intonation and gestures with
which the word is used.
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I might say: in many cases understanding a word involves being able to use it on certain occasions in a
special tone of voice.
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You might say that certain words are only pegs to hang intonations on.
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But instead of the intonation and the accompanying gestures, I might for my own purposes treat the word
itself as a gesture. (Can't I say that the sound "ha ha" is a laugh and the sound "oh!" is a sigh?)
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31 I could imagine a language that was spoken in a uniform metre, with quasi-words intercalated between the words
of the sentences to maintain the metre. Suppose we talked about the meaning of these quasi-words. (The smith
putting in extra taps between the real strokes in order to maintain a rhythm in striking).
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Language is like a collection of very various tools. In the tool box there is a hammer, a saw, a rule, a lead, a
glue pot and glue. Many of the tools are akin to each other in form and use, and the tools can be roughly divided into
groups according to their relationships; but the boundaries between these groups will often be more or less arbitrary
and there are various types of relationship that cut across one another.
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I said that the meaning of a word is its role in the calculus of language. (I compared it to a piece in chess).
Now let us think how we calculate with a word, for instance with the word "red". We are told where the colour is
situated; we are told the shape and size of the coloured patch or the coloured object; we are told whether the colour
is pure or mixed, light or dark, whether it remains constant or changes, etc. etc. Conclusions are drawn from the
propositions, they are translated into diagrams and into behaviour, there is drawing, measurement and calculation.
But think of the meaning of the word "oh!" If we were asked about it, we would say "'oh'! is a sigh; we say, for
instance, things like 'Oh, it is raining again already'". And that would describe the use of the word. But what
corresponds now to the calculus, the complicated game that we play with other words? In the use of the words
"oh!", or "hurrah", or "hm", there is nothing comparable.
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Moreover, we mustn't confuse signs with symptoms here. The sound "hm" may be called an expression of
dubiousness and also, for other people, a symptom of dubiousness, in the way that
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clouds are a symptom of rain. But "hm" is not the name of dubiousness.
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32 Suppose we want to describe ball-games. There are some games like football, cricket and tennis, which have a
well-developed and complicated system of rules; there is a game consisting simply of everyone's throwing a ball as
high as he can; and there is the game little children play of throwing a ball in any direction and then retrieving it. Or
again someone throws a ball high into the air for the fun of it and catches it again without any element of
competition. Perhaps one will be unwilling to call some of these ball games at all; but is it clear where the boundary
is to be drawn here?
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We are interested in language as a procedure according to explicit rules, because philosophical problems are
misunderstandings which must be removed by clarification of the rules according to which we are inclined to use
words.
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We consider language from one point of view only.
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We said that when we understood the use we didn't yet understand the purpose of the word "perhaps". And
by "purpose" in this case we meant the role in human life. (This role can be called the "meaning" of the word in the
sense in which one speaks of the 'meaning of an event for our life'.)
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But we said that by "meaning" we meant what an explanation of meaning explains. And an explanation of
meaning is not an empirical proposition and not a causal explanation, but a rule, a convention.
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It might be said that the purpose of the word "hey!" in our language is to alarm the person spoken to. But
what does its having this purpose amount to? What is the criterion for it? The word "purpose" like all the words of
our language is used in various more or less related ways. I will mention two characteristic games. We might say that
the purpose of doing something is what the person doing it would say if asked what its purpose was. On the
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other hand if we say that the hen clucks in order to call her chicks together we infer this purpose from the effect of
the clucking. We wouldn't call the gathering of the chicks the purpose of the clucking if the clucking didn't have this
result always, or at least commonly or in specifiable circumstances.--One may now say that the purpose, the effect
of the word "hey" is the important thing about the word; but explaining the purpose or the effect is not what we call
explaining the meaning.
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It may be that if it is to achieve its effect a particular word cannot be replaced by any other; just as it may be
that a gesture cannot be replaced by any other. (The word has a soul and not just a meaning.) No one would believe
that a poem remained essentially unaltered if its words were replaced by others in accordance with an appropriate
convention.
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Our proposition "meaning is what an explanation of meaning explains" could also be interpreted in the
following way: let's only bother about what's called the explanation of meaning, and let's not bother about meaning
in any other sense.
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33 But one might say something like this. The sentences that we utter have a particular purpose, they are to produce
certain effects. They are parts of a mechanism, perhaps a psychological mechanism, and the words of the sentences
are also parts of the mechanism (levers, cogwheels and so on). The example that seems to illustrate what we're
thinking of here is an automatic music player, a pianola. It contains a roll, rollers, etc., on which the piece of music is
written in some kind of notation (the position of holes, pegs and so on). It's as if these written signs gave orders
which are carried out by the keys and hammers. And so shouldn't we say that the sense of the sign is its effect?--But
suppose the pianola is in bad condition and the signs on the roll produce hisses and bangs instead of the
notes.--Perhaps you will say that the sense of the signs is their effect on a mechanism in good condition, and
correspondingly
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that the sense of an order is its effect on an obedient man. But what is regarded as a criterion of obedience here?
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You might then say that the sense of the signs is not their effect, but their purpose. But consider too, that
we're tempted to think that this purpose is only a part of the larger purpose served by the pianola.--This purpose,
say, is to entertain people. But it's clear that when we spoke of "the sense of the signs" we didn't mean any part of
that purpose. We were thinking rather of the purpose of the signs within the mechanism of the pianola.--And so
you can say that the purpose of an order is its sense, only so far as the purpose can be expressed by a rule of
language. "I am saying 'go away' because I want you to leave me alone", "I am saying 'perhaps' because I am not
quite sure."
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An explanation of the operation of language as a psychophysical mechanism is of no interest to us. Such an
explanation itself uses language to describe phenomena (association, memory etc); it is itself a linguistic act and
stands outside the calculus; but we need an explanation which is part of the calculus.
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"How is he to know what colour he is to pick out when he hears the word 'red'?--Very simple: he is to take
the colour whose image occurs to him when he hears the word"--But how will he know what that means, and which
colour it is "which occurs to him when he hears the word"?
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Certainly there is such a procedure as choosing the colour which occurs to you when you hear that word.
And the sentence "red is the colour that occurs to you when you hear the word 'red'" is a definition.
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If I say, "a symbol is something which produces this effect" the question remains: how can I speak of "this
effect"? And if it occurs, how do I know that it's the one I meant?" "Very simple", we may say "we compare it with
our memory image." But this explanation does not get to the root of our dissatisfaction. For
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how are we given the method we're to use in making the comparison--i.e. how do we know what we're to do when
we're told to compare?
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In our language one of the functions of the word "red" is to call that particular colour to mind; and indeed it
might be discovered that this word did so better than others, even that it alone served that purpose. But instead of
the mechanism of association we might also have used a colour chart or some such apparatus; and then our calculus
would have to get along with the associated, or visible, colour sample. The psychological effectiveness of a sign does
not concern us. I wouldn't even scruple to invent that kind of mechanism.
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Investigating whether the meaning of a word is its effect or its purpose, etc. is a grammatical investigation.
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34 Why can one understand a word and not a penholder? Is it the difference between their shapes? But you will say
that we could understand a penholder too, if we had given it a meaning. But then how is giving it a meaning
done?--How was meaning given to the word "red"? Well, you point at something, and you say "I call that 'red'". Is
that a kind of consecration of mystical formula? How does this pointing and uttering words work? It works only as
part of a system containing other bits of linguistic behaviour. And so now one can understand a penholder too; but
does this understanding contain the whole system of its application? Impossible. We say that we understand its
meaning when we know its use, but we've also said that the word "know" doesn't denote a state of consciousness.
That is: the grammar of the word "know" isn't the grammar of a "state of consciousness", but something different.
And there is only one way to learn it: to watch how the word is used in practice.
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A truthful answer to the question "Did you understand the
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sentence (that you have just read)" is sometimes "yes" and sometimes "no". "So something different must take place
when I understand it and when I don't understand it."
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Right. So when I understand a sentence something happens like being able to follow a melody as a melody,
unlike the case when it's so long or so developed that I have to say "I can't follow this bit". And the same thing might
happen with a picture, and here I mean an ornament. First of all I see only a maze of lines; then they group
themselves for me into well-known and accustomed forms and I see a plan, a familiar system. If the ornamentation
contains representations of well-known objects the recognition of these will indicate a further stage of
understanding. (Think in this connection of the solution of a puzzle picture.) I then say "Yes, now I see the picture
rightly".
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Asked "what happened when you read that sentence with understanding" I would have to say "I read it as a
group of English words linked in a familiar way". I might also say that a picture came into my mind when I heard it.
But then I am asked: "Is that all? After all, the understanding couldn't consist in that and nothing else!" Well, that or
something like it is all that happened while I read the sentence and immediately afterwards; but what we call
"understanding" is related to countless things that happen before and after the reading of this sentence.
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What of when I don't understand a sentence? Well, it might be a sentence in a foreign language and all I see
is a row of unknown words. Or what I read seemed to be an English sentence, but it contained an unfamiliar phrase
and when I tried to grasp it (and that again can mean various things) I didn't succeed. (Think of what goes on when
we try to understand the sense of a poem in our native language which makes use of constructions we don't yet
understand.)
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But I can say that I understand a sentence in a foreign language
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say a Latin one that I can only decipher by a painful effort to construe--even if I have only turned it into English bit
by bit and have never succeeded in grasping the overall phrasing of the sentence.
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But all the same, in order to understand a sentence I have to understand the words in it! And when I read, I
understand some words and not others.
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I hear a word and someone asks me "did you understand it?" and I reply truly "yes". What happened when I
understood? How was the understanding different from what happens when I don't understand a word?--Suppose
the word was "tree". If I am to say truly that I understood it, must the image of a tree have come before my mind?
No; nor must any other image. All I can say is that when I was asked "do you understand the word tree?" I'd have
answered "yes" unthinkingly and without lying.--If the other person had asked me further "and what is a tree?" I
would have described one for him, or shown him one, or drawn one; or perhaps I would have answered "I know,
but I don't want to explain." And it may be that when I gave my reply the image of a tree came before my mind, or
perhaps I looked for something which had some similarity with a tree, or perhaps other words came into my head,
etc. etc.
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Let's just look how we actually use the word "understand".
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The word might also have been one of which I would say "I used to know what it meant, and it will come
back to me", and then later on say "now it's come back to me". What happened then?--Perhaps there came into my
mind the situation in which the word was first explained to me: I saw myself in a room with others, etc. etc. (But if
now I read and understand the word in a sentence that picture wouldn't have to come before my mind; perhaps no
picture at all comes to mind.)
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Or it was a word in a foreign language; and I had already often heard it, but never understood it. Perhaps I
said to myself "what can it mean?" and tried to give it a meaning which fitted the context
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(again various possibilities). Perhaps now this situation comes to mind and I say "I don't understand the word". But I
might also react immediately to the foreign word with the answer "I don't understand it", just as I reacted to the word
"tree" with the opposite answer.
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Suppose it is the word "red" and I say automatically that I understood it; then he asks again "do you really
understand it?" Then I summon up a red image in my mind as a kind of check. But how do I know that it's the right
colour that appears to me? And yet I say now with full conviction that I understand it.--But I might also look at a
colour chart with the word "red" written beneath the colour.--I could carry on for ever describing such processes.
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35 The problem that concerns us could be summed up roughly thus: "Must one see an image of the colour blue in
one's mind whenever one reads the word 'blue' with understanding?" People have often asked this question and have
commonly answered no; they have concluded from this answer that the characteristic process of understanding is
just a different process which we've not yet grasped.--Suppose then by "understanding" we mean what makes the
difference between reading with understanding and reading without understanding; what does happen when we
understand? Well, "Understanding" is not the name of a single process accompanying reading or hearing, but of
more or less interrelated processes against a background, or in a context, of facts of a particular kind, viz. the actual
use of a learnt language or languages.--We say that understanding is a "psychological process", and this label is
misleading, in this as in countless other cases. It compares understanding to a particular process like translation from
one language into another, and it suggests the same conception of thinking, knowing, wishing, intending, etc. That is
to say, in all these cases we see that what we would perhaps naively suggest as the hallmark of such a process is not
present in
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every case or even in the majority of cases. And our next step is to conclude that the essence of the process is
something difficult to grasp that still awaits discovery. For we say: since I use the word "understand" in all these
cases, there must be some one thing which happens in every case and which is the essence of understanding
(expecting, wishing etc.). Otherwise, why should I call them by all the same name?
Page 75
This argument is based on the notion that what is needed to justify characterizing a number of processes or
objects by a general concept-word is something common to them all.
Page 75
This notion is, in a way, too primitive. What a concept-word indicates is certainly a kinship between objects,
but this kinship need not be the sharing of a common property or a constituent. It may connect the objects like the
links of a chain, so that one is linked to another by intermediary links. Two neighbouring members may have
common features and be similar to each other, while distant ones belong to the same family without any longer
having anything in common. Indeed even if a feature is common to all members of the family it need not be that
feature that defines the concept.
Page 75
The relationship between the members of a concept may be set up by the sharing of features which show up
in the family of the concept, crossing and overlapping in very complicated ways.
Page 75
Thus there is probably no single characteristic which is common to all the things we call games. But it can't
be said either that "game" just has several independent meanings (rather like the word "bank"). What we call
"games" are procedures interrelated in various ways with many different transitions between one and another.
Page 75
It might be said that the use of the concept-word or common noun is justified in this case because there are
transitional steps
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between the members.--Then it might be objected that a transition can be made from anything to anything, and so
the concept isn't bounded. To this I have to say that for the most part it isn't in fact bounded and the way to specify
it is perhaps: "by 'knowledge' we mean these processes, and these, and similar ones". And instead of "and similar
ones" I might have said "and others akin to these in many ways".
Page 76
But if we wish to draw boundaries in the use of a word, in order to clear up philosophical paradoxes, then
alongside the actual picture of the use (in which as it were the different colours flow into one another without sharp
boundaries) we may put another picture which is in certain ways like the first but is built up of colours with clear
boundaries between them.
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III
Page 77
36 If we look at the actual use of a word, what we see is something constantly fluctuating.
Page 77
In our investigations we set over against this fluctuation something more fixed, just as one paints a stationary
picture of the constantly altering face of the landscape.
Page 77
When we study language we envisage it as a game with fixed rules. We compare it with, and measure it
against, a game of that kind.
Page 77
If for our purposes we wish to regulate the use of a word by definite rules, then alongside its fluctuating use
we set up a different use by codifying one of its characteristic aspects.
Page 77
Thus it could be said that the use of the word "good" (in an ethical sense) is a combination of a very large
number of interrelated games, each of them as it were a facet of the use. What makes a single concept here is
precisely the connection, the relationship, between these facets.
Page 77
But this isn't like the way physics gives a simplified description of a natural phenomenon, abstracting from
secondary factors. It can't be said that logic depicts an idealised reality, or that it holds strictly only for an ideal
language and so on. For where do we get the concept of this ideal? The most that could be said is that we are
constructing an ideal language which contrasts with ordinary language; but it can't be said that we are saying
something that would hold only of an ideal language.
Page 77
37 There is something else I would like to say about the understanding of a picture. Take a genre-picture: we say we
understand
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it if we recognise what is happening in it, what the people in it are doing. Here the criterion for this recognition is
perhaps that if asked what they are doing we explain it in words or represent it in mime etc. It's possible that this
recognition doesn't come easily, perhaps because we don't immediately see the figures in the picture as figures (as in
puzzle pictures), perhaps because we can't make out what they are doing together, etc. In these cases there may be a
period of doubt followed by a familiar process of recognition. On the other hand, it may be the kind of picture we'd
say we took in at first glance, and in that case we find it difficult to say what the understanding really consists of. In
the first place what happened was not that we took the painted objects for real ones. And again "I understand it" in
this case doesn't mean that finally, after an effort, I understand that it is this picture. And nothing takes place like
recognizing an old acquaintance in the street, no saying "oh, there's... ". If you insist on saying there is a recognition,
what does this recognition consist of? Perhaps I recognize a certain part of the picture as a human face. Do I have to
look at a real face, or call before my mind's eye the memory of a face I've seen before? Is what happens that I
rummage in the cupboard of my memory until I find something which resembles the picture? Is the recognition just
this finding? In our case there is no one thing that happens that could be called recognition, and yet if the person
who sees the picture is asked "do you recognize what it is?" he may truly answer "yes", or perhaps reply "it is a
face". It can indeed be said that when he sees the complex of signs as a face he sees something different from when
he doesn't do so. In that case I'd like to say that I see something familiar†1 in front of me. But what constitutes the
familiarity is not the historical fact that I've often seen objects like that etc; because the history behind the experience
is certainly not present in the experience itself. Rather, the familiarity
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lies in the fact that I immediately grasp a particular ryhthm of the picture and stay with it, fell at home with it, so to
speak. For the rest it is a different experience that constitutes the familiarity in each particular case; a picture of a
table carries one experience with it and a picture of a bed another.
Page 79
If I say: "I understand this picture" the question arises: do I mean "I understand it like that"? With the "like
that" standing for a translation of what I understand into a different expression? Or is it a sort of intransitive
understanding? When I'm understanding one thing do I as it were think of another thing? Does understanding, that
is, consist of thinking of something else? And if that isn't what I mean, then what's understood is as it were
autonomous, and the understanding of it is comparable to the understanding of a melody.
Page 79
(It is interesting to observe that the pictures which come before our minds when we read an isolated word
and try to understand it correctly just like that are commonly altogether absent when we read a sentence; the picture
that comes before our minds when we read a sentence with understanding is often something like a resultant of the
whole sentence).
Page 79
38 It is possible for a person to forget the meaning of a word (e.g. "blue"). What has he forgotten?--How is that
manifested?
Page 79
He may point, for instance, to a chart of different colours and say "I don't know any longer which of these is
called 'blue'". Or again, he may not any longer know at all what the word means (what purpose it serves); he may
know only that is it an English word.
Page 79
We might say: if someone has forgotten the meaning of the word "blue" and is asked to choose a blue object
from among others he feels as he looks at the objects that the connection between the word "blue" and the colour no
longer holds but has been broken off. The connection will be reestablished, it might be said, if we repeat the
definition of the word for him. But we might reestablish the connection in various ways: we might point to a blue
object and say "that is blue", or say "remember your blue
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patch" or we perhaps utter the German word "blau", etc. etc. And if I now say there are these different ways in
which we can establish the connection this suggests there's a single particular phenomenon I call the connection
between word and colour, or the understanding of the word, a phenomenon I've produced in all these different
ways, just as I can use objects of different shapes and materials as conductors to connect the ends of two wires. But
there is no need for such a phenomenon of connection, no need, say, that when I hear the word a picture of the
colour should occur before my inner eye. For if what is reestablished in his understanding of the word, this can
manifest itself in very various processes. There isn't a further process hidden behind, which is the real understanding,
accompanying and causing these manifestations in the way that toothache causes one to groan, hold one's cheek,
pull faces, etc. If I am now asked if I think that there's no such thing as understanding but only manifestations of
understanding, I must answer that this question is as senseless as the question whether there is a number three. I can
only describe piecemeal the grammar of the word "understand" and point out that it differs from what one is inclined
to portray without looking closely. We are like the little painter Klecksel who drew two eyes in a man's profile, since
he knew that human beings have two eyes.
Page 80
39 The effect of an explanation of the meaning of a word is like 'knowing how to go on', when you recite the
beginning of a poem to someone until he says "now I know how to go on". (Tell yourself the various psychological
forms this knowing how to go on may take.)
Page 80
The way in which language was learnt is not contained in its use. (Any more than the cause is contained in
the effect.)
Page 80
How does an ostensive definition work? Is it put to work again
Page Break 81
every time the word is used, or is it like a vaccination which changes us once and for all?
Page 81
A definition as a part of the calculus cannot act at a distance. It acts only by being applied.
Page 81
40 Once more: in what cases shall we say that the man understands the word "blue"? Well, if he picks out a blue
object from others on demand; or if he credibly says that he could now pick out the blue object but doesn't want to
(perhaps we notice that while he says this he glances involuntarily at the blue object; perhaps we believe him simply
on account of his previous behaviour). And how does he know that he understands the word? i.e. in what
circumstances will he be able to say it? Sometimes after some kind of test, but sometimes also without. But in that
case won't he sometimes have to say later "I was wrong, I did not understand it after all" if it turns out that he can't
apply it? Can he justify himself in such cases by saying that he did indeed understand the word when he said he did,
but that the meaning later slipped his memory? Well, what can he offer as a criterion (proof) that he did understand
the word at the previous time?--Perhaps he says "At that time I saw the colour in my mind's eye, but now I can't
remember it." Well, if that implies that he understood it, he did understand it then.--Or he says: "I can only say I've
used the word a hundred times before", or "I'd used it just before, and while I was saying I understood it I was
thinking of that occasion." It is what is regarded as the justification of an assertion that constitutes the sense of the
assertion.
Page 81
Suppose we say "he understands the word 'blue', he picked the blue ball out from the others right away" and
then he says "I just picked it out by guesswork, I didn't understand the word". What sort of criterion did he have for
not having understood the word? Ought we to believe him? If one asks oneself "How do I know that
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I don't understand this word" it produces a very strange thought sensation. One wants to say "I don't connect
anything with it", "it says nothing to me", "It's a mere noise", and in order to understand these utterances one has to
call to mind what it's like "when one connects something with a word", when a definition has made the sound into a
meaningful word, when one can do something with the word.
Page 82
You will say: "But he certainly can't be wrong when he says that he didn't understand the word." And that is
an observation about the grammar of the statement "I didn't understand the word". It is also an observation about
grammar when we say, "Whether he understood, is something he knows which we cannot know but only guess".
Moreover the statement "I didn't understand the word" doesn't describe a state at the time of hearing the word; there
are many different ways in which the processes characteristic of not understanding may have taken place later.
Page 82
41 We speak of understanding (a process of understanding, and also a state of understanding) and also of certain
processes which are criteria for this understanding.
Page 82
We are inclined to call understanding a mental process or a state of mind. This characterizes it as a
hypothetical process etc., or rather as a process (or state) in the sense of a hypothesis. That is, we banish the word
"understanding" to a particular region of grammar.
Page 82
The grammar of a mental state or process is indeed in many respects similar to that of e.g. a brain-process.
The principal difference is perhaps that in the case of a brain-process a direct check is admitted to be possible; the
process in question may perhaps be seen by opening the skull. But there is no room for a similar "immediate
perception" in the grammar of mental process. (There is no such move in this game.)
Page Break 83
Page 83
What is the criterion for our understanding the word "red"? That we pick out a red object from others on
demand, or that we can give the ostensive definition of the word "red"?
Page 83
We regard both things as signs of understanding. If we hear someone use the word "red" and are in doubt
whether he understands it, we can check by asking: "which colour do you call red?" On the other hand, if we'd given
someone the ostensive definition of the word and then wanted to see whether he'd understood it rightly, we wouldn't
ask him to repeat it, but we would set him a task like picking out the red objects from a row.
Page 83
Here it can be asked: "are we talking about my understanding or other people's understanding?"
Page 83
"Only I can know whether I understand, others can only guess." "'He understands' is a hypothesis; 'I
understand' is not."
Page 83
If that's what we say, then we're conceiving "understanding" as an experience, analogous e.g. to a pain.
Page 83
People say: "You cannot know whether I understand (whether I am glad), etc.; you can't look inside me."
"You can't know what I think." Yes, but that's so only as long as you don't think aloud; and we aren't interested here
in the difference between thinking out loud (or in writing) and thinking in the imagination.
Page 83
Here you may object that thinking is after all private even if it is only the visual experience of writing, and
that though another person can see what my physical hand is writing he cannot have my visual experience. These
questions must occupy us in another place.
Page 83
But for our present purpose can't we say "he is writing" and "I am writing" instead of "he understands" and "I
understand?"
Page 83
Then we leave the question of experience completely out of the
Page Break 84
game. Also, for instance, the question of private understanding. For then it appears unimportant here.
Page 84
What we call "understanding" is not the behaviour--whatever it may be--that shows us the understanding,
but a state of which this behaviour is a sign. And that is a statement about the grammar of denoting such a state.
Page 84
42 We might call the recital of the rules on its own a criterion for understanding, or alternatively tests of use on their
own. Then in the one case "he understands" would mean: "if you ask him for for [[sic]] the rules, he will tell you
them"; in the other case "if you require him to apply the rule, he will carry out your order".
Page 84
Or we may regard the recital of the rules as a symptom of the man's being able to do something other than
recite the rules. As when I hold a watch to my ear, hear it ticking and say: it is going. In that case I don't just expect it
to go on ticking, but also to show the time.
Page 84
One might say: "The recital of the rules is a criterion of understanding, if the man recites them with
understanding and not purely mechanically." But here once again an intelligent intonation during the recitation can
count as understanding; and so why not the recitation itself?
Page 84
To understand is to grasp, to receive a particular impression from an object, to let it work on one. To let a
proposition work on one; to consider consequences of the proposition, to imagine them, etc.
Page 84
What we call "understanding" is a psychological phenomenon that has a special connection with the
phenomena of learning and using our human language.
Page 84
What happens when I remember the meaning of a word? I see before me an object of a certain colour and I
say "this book is
Page Break 85
brown and I have always called this colour 'brown'". What sort of act of remembering must take place for me to be
able to say that? This question could be put in a much more general form. For instance, if someone asked me "have
you ever before seen the table at which you are now sitting?" I would answer "yes, I have seen it countless times".
And if I were pressed I would say "I have sat at it every day for months".--What act or acts of remembering occur in
such a case? After all I don't see myself in my mind's eye "sitting at this table every day for months". And yet I say
that I remember that I've done so, and I can later corroborate it in various ways. Last summer too, for example, I was
living in this room. But how do I know that? Do I see it in my mind's eye? No. In this case what does the
remembering consist of? If I as it were hunt for the basis of the memory, isolated pictures of my earlier sojourn
surface in my mind; but even so they don't have, say, a date written into them. And even before they've surfaced and
before I've called any particular evidence into my mind, I can say truly that I remember that I lived here for months
and saw this table. Remembering, then, isn't at all the mental process that one imagines at first sight. If I say, rightly,
"I remember it" the most varied things may happen; perhaps even just that I say it. And when I here say "rightly" of
course I'm not laying down what the right and wrong use of the expression is; on the contrary I'm just describing the
actual use.
Page 85
The psychological process of understanding is in the same case as the arithmetical object Three. The word
"process" in the one case, and the word "object" in the other produce a false grammatical attitude to the word.
Page 85
43 Isn't it like this? First of all, people use an explanation, a chart, by looking it up; later they as it were look it up in
the head (by calling it before the inner eye, or the like) and finally they work
Page Break 86
without the chart, as if it had never existed. In this last case they are playing a different game. For it isn't as if the
chart is still in the background, to fall back on; it is excluded from our game, and if I "fall back on it" I am like a
blinded man falling back on the sense of touch. An explanation provides a chart and when I no longer use the chart
it becomes mere history.
Page 86
I must distinguish between the case in which I follow the table, and the case in which I behave in accordance
with the table without making use of it.--The rule we learnt which makes us now behave in such and such a way is of
no interest to us considered as the cause or history behind our present behaviour.--But as a general description of
our manner of behaving it is a hypothesis. It is the hypothesis that the two people who sit at the chess board will
behave (move) in such and such a manner. (Here even a breach of the rules falls under the hypothesis, since it says
something about the behaviour of the players when they become aware of the breach). But the players might also
use the rules by looking up in each particular case what is to be done; here the rule would enter into the conduct of
the game itself and would not be related to it as hypothesis to confirmation. But there is a difficulty here. For a
player who plays without using the list of rules, and indeed has never seen one, might nevertheless if asked give the
rules of his game--not by ascertaining through repeated observation what he does in such and such a position in the
game, but by superintending a move and saying "in such a case this is how one moves".--But, if that is so, that just
shows that in certain circumstances he can enunciate the rules, not that he makes explicit use of them while playing.
Page 86
It is a hypothesis that he will if asked recite a list of rules; if a disposition or capacity for this is postulated in
him, it is a psychological disposition analogous to a physiological one. If it is said
Page Break 87
that this disposition characterizes the process of playing, it characterizes it as the psychological or physiological one
it really is. (In our study of symbolism there is no foreground and background; it isn't a matter of a tangible sign with
an accompanying intangible power or understanding.)
Page 87
44 What interests us in the sign, the meaning which matters for us is what is embodied in the grammar of the sign.
Page 87
We ask "How do you use the word, what do you do with it"--that will tell us how you understand it.
Page 87
Grammar is the account books of language. They must show the actual transactions of language, everything
that is not a matter of accompanying sensations.
Page 87
In a certain sense one might say that we are not concerned with nuances.
Page 87
(I could imagine a philosopher who thought that he must have a proposition about the essence of knowing
printed in red, otherwise it would not really express what it was meant to express.)
Page Break 88
IV
Page 88
45 The interpretation of written and spoken signs by ostensive definitions is not an application of language, but part
of the grammar. The interpretation remains at the level of generality preparatory to any application.
Page 88
The ostensive definition may be regarded as a rule for translating from a gesture language into a word
language. If I say "the colour of this object is called 'violet'", I must already have denoted the colour, already
presented it for christening, with the words "the colour of that object" if the naming is to be able to take place. For I
might also say "the name of this colour is for you to decide" and the man who gives the name would in that case
already have to know what he is to name (where in the language he is stationing the name).
Page 88
That one empirical proposition is true and another false is no part of grammar. What belongs to grammar are
all the conditions (the method) necessary for comparing the proposition with reality. That is, all the conditions
necessary for the understanding (of the sense).
Page 88
In so far as the meaning of words becomes clear in the fulfilment of an expectation, in the satisfaction of a
wish, in the carrying out of an order etc., it already shows itself when we put the expectation into language. It is
therefore completely determined in the grammar, in what could be foreseen and spoken of already before the
occurrence of the event.
Page 88
46 Does our language consist of primary signs (ostensive gestures) and secondary signs (words)? One is inclined to
ask, whether it isn't the case that our language has to have primary signs while it could get by without the secondary
ones.
Page 88
The false note in this question is that it expects an explanation of
Page Break 89
existing language instead of a mere description.
Page 89
It sounds like a ridiculous truism to say that a man who thinks that gestures are the primitive signs underlying
all others would not be able to replace an ordinary sentence by gestures.
Page 89
One is inclined to make a distinction between rules of grammar that set up "a connection between language
and reality" and those that do not. A rule of the first kind is "this colour is called 'red'",--a rule of the second kind is
"~~p = p". With regard to this distinction there is a common error; language is not something that is first given a
structure and then fitted on to reality.
Page 89
One might wish to ask: So is it an accident that in order to define signs and complete the sign-system I have
to go outside the written and spoken signs? When I do that don't I go right into the realm where what is to be
described occurs?--But in that case isn't it strange that I can do anything at all with the written signs?--We say
perhaps that the written signs are mere representatives of the things the ostensive definition points to.--But how then
is this representing possible? I can't after all make just anything stand in for anything else.--It is indeed important
that such representing is possible; for the representative must, in certain cases at least, do the job as well as the
principal.
Page 89
47 We say that something like a red label is the primary sign for the colour red, and the word is a secondary sign,
because the meaning of the word "red" is explained if I point, etc. to a red label, but not if I say "red" means the
same as "rouge". But don't I explain the meaning of the word "red" to a Frenchman in just this way? "Yes, but only
because he has learnt the meaning of 'rouge' by ostensive definition". But if he understands my explanation "red =
Page Break 90
rouge" does he have to have this definition--or a red image--present to his mind? If not, it is mere history. Must he
have such a picture present whenever we would say he was using the word "rouge" with understanding? (Think of
the order: "Imagine a round red patch.")
Page 90
48 Are the signs one wants to call 'primary' incapable of being misinterpreted?
Page 90
Can one perhaps say, they don't really any longer need to be understood?--If that means that they don't have
to be further interpreted, that goes for words too; if it means, they cannot be further interpreted, then it's false.
(Think of the explanation of gestures by words and vice versa).
Page 90
Is it correct, and if so in what sense, to say that the ostensive definition is like the verbal definition in
replacing one sign by another--the pointing by the word?
Page 90
49 Suppose I lay down a method of designation. Suppose, for example, I want to give names to shades of colours
for my private use. I may do so by means of a chart; and of course I won't write a name beside a wrong colour
(beside a colour I don't want to give that name to). But why not? Why shouldn't "red" go beside the green label and
"green" beside the red, etc.? If the ostensive definition merely replaces one sign by another, that shouldn't make any
difference.--Here there are at any rate two different possibilities. It may be that the table with green beside "red" is
used in such a way that a man who 'looks it up' goes diagonally from the word "red" to the red label, and from the
word "green" to the green one and so on. We would then say that though the table was arranged differently (had a
different spatial scheme) it connected the signs in the same way as the usual one.--But it might also be that the
person using the table looks from one side horizontally to another, and in some sentences uses a green label instead
of the word "red", and yet obeys an order like "give me a
Page Break 91
red book" not by bringing a green book, but perfectly correctly by bringing a red one (i.e. one that we too would call
"red"). Such a man would have used the table in a different way from the first, but still in such a way that the word
"red" means the same colour for him as it does for us.
Page 91
Now it is the second case which interests us, and the question is can a green label be a sample of red?--
Page 91
I can imagine an arrangement according to which a man to whom I show a green label with the words "paint
me this colour" is to paint me red and if I show him blue with the same words, is to paint me yellow (always the
complementary colour). And someone might interpret my order in that way even without such a convention. The
convention might also have been "if I say, 'paint this colour', then always paint one slightly darker", and again we
could imagine the order being thus interpreted even without this prearrangement.--But can it be said that when
someone is painting a certain shade of green he is copying the red of the label--as he may copy a geometrical figure
according to various methods of projection, copying it in different but equally exact ways?--Can I compare colours
with shapes? Can a green label be used both as the name of a particular shade of red, and also as a sample of it just
as a circle can serve as the name of a particular elliptical shape, and also as a sample for it?
Page 91
It is clear that a sample is not used like a word (a name). And an ostensive definition, a table, which leads us
from words to samples, is used differently from a table which replaces one name by another.
Page 91
50 However, the word "copy" has different meanings in different cases and what I mean by "pattern" changes
correspondingly. What does "to copy a figure exactly" mean? Does it mean copy it exactly with the unaided eye? Or
with measuring instruments, and if so which? What shall we be willing to call the same colour as that of the pattern?
Think of various methods of comparison. How far is the rule to copy darker comparable to a rule to copy a figure
Page Break 92
on a larger, or small scale?
Page 92
Imagine a man who claimed to be able to copy shades of red into green, who fixed his eye on a red sample
and with every outward sign of exact copying mixed a shade of green. For us he would be on a par with someone
who listened carefully and mixed colours in accordance with notes on a violin. In such a case we'd say "I don't know
how he does it"; not because we didn't understand the processes in his brain or in his muscles, but because we don't
understand what is meant by "this shade of colour is a copy of this note on the violin". Unless that means that as a
matter of experience a man associates a particular shade of colour with a particular note (sees it in his mind's eye,
paints it etc). The difference between the meanings of "associate" and "copy" shows itself in the fact that it doesn't
make sense to speak of a projection-method (rule of translation) for association. We say: "you haven't copied
correctly", but not "you haven't associated correctly".
Page 92
On the other hand it is certainly conceivable that human beings might agree so exactly with each other in
associating colours with violin notes that one might say to another: "No, you haven't represented that violin note
correctly, it was yellower than you painted it" and the other would answer something like "you're right, the same
thought occurred to me".
Page 92
51 If the table connects the word with a sample, then it isn't indifferent which label the word is linked with when the
table is consulted--"So then there are signs that are arbitrary and signs that are not!" Compare the giving of
information by maps and drawings, with the giving of information by sentences. The sentences are no more arbitrary
than the drawings are; only the words are arbitrary. On the other hand the projection of the maps is arbitrary; and
how would you decide which of the two is the more arbitrary?
Page Break 93
Page 93
Certainly I can compare deciding on the meanings of words with deciding on a method of projection, such as
that for the representation of spatial forms ("the proposition is a picture"). That is a good comparison, but it doesn't
exempt us from investigating the way words signify, which has its own rules. We can of course say--that is, it
accords with usage--that we communicate by signs whether we use words or patterns, but the game of acting in
accordance with words is not the same game as acting in accordance with patterns. (Words are not essential to what
we call "language", and neither are samples). Word-language is only one of many possible kinds of language, and
there are transitions between one kind and another. (Think of two ways of writing the proposition "I see a red circle":
it might be done by writing a circle and giving it the appropriate colour (red), or by writing a circle with a red patch
beside it. Consider what corresponds in a map to the form of expression of a word-language.)
Page 93
52 "I won't insist that the red pattern in the explanatory chart must be horizontally opposite the word 'red', but there
must be some sort of law for reading the table or it will lose its sense." But is there no law if the chart is read in the
way indicated by the arrows of the following schema?
"But in that case mustn't this schema of arrows be given in advance?"--Well, must you give this schema before we
follow the normal use?
Page Break 94
Page 94
"But in that case mustn't there at least be a regularity through time in the use of the table? Would it work if
we were to use the table in accordance with different schemata at different times? How would one know in that case
how the table was to be used?" Well, how does one know anyway? Explanations of signs come to an end
somewhere.
Page 94
Of course if I showed someone the way by pointing my finger not in the direction in which he was to go, but
in the opposite direction, in the absence of a special arrangement I should cause a misunderstanding. It is part of
human nature to understand pointing with the finger in the way we do. (As it is also part of human nature to play
board games and to use sign languages that consist of written signs on a flat surface.)
Page 94
The chart doesn't guarantee that I shall pass from one part of it to another in a uniform manner. It doesn't
compel me to use it always in the same way. It's there, like a field, with paths leading through it: but I can also cut
across.--Each time I apply the chart I make a fresh transition. The transitions aren't made as it were once for all in the
chart (the chart merely suggests to me that I make them).
Page 94
(What kind of propositions are these?--They are like the observation that explanations of signs come to an
end somewhere. And that is rather like saying "How does it help you to postulate a creator, it only pushes back the
problem of the beginning of the world." This observation brings out an aspect of my explanation that I perhaps
hadn't noticed. One might also say: "Look at your explanation in this way--now are you still satisfied with it?")
Page 94
53 Is the word "red" enough to enable one to look for something red? Does one need a memory image to do so?
Page 94
Can one say that the word "red" needs a supplement in memory in order to be a usable sign?
Page 94
If I use the words "there is a red book in front of me" to describe an experience, is the justification of the
choice of these words,
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apart from the experience described, the fact that I remember that I've always used the word "red" for this colour?
Does that have to be the justification?
Page 95
In order to be ableto [[sic]] obey a spoken order do we need something like a memory picture of what we did
when we last obeyed it?
Page 95
So is the real order "Do now what you remember doing then"? This order too might be given. But does that
mean that in order to obey it, I need a memory image of searching my memory?
Page 95
The order "do now what you remember doing then" tells me that I am to look in a particular place for a
picture that will tell me what I am to do. So the order is very similar to "Do what is written on the piece of paper in
this drawer". If there is nothing on the piece of paper then the order lacks sense.
Page 95
If the use of the word "red" depends on the picture that my memory automatically reproduces at the sound
of this word, then I am as much at mercy of this reproduction as if I had decided to settle the meaning by looking up
a chart in such a way that I would surrender unconditionally to whatever I found there.
Page 95
If the sample I am to work with appears darker than I remember it being yesterday, I need not agree with the
memory and in fact I do not always do so. And I might very well speak of a darkening of my memory.
Page 95
54 If I tell someone "paint from memory the colour of the door of your room" that doesn't settle what he is to do any
more unambiguously than the order "paint the green you see on this chart". Here too it is imaginable that the first of
the sentences might be understood in the way one would normally understand a sentence
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Page 96
like "paint a colour somewhat lighter than the one you remember seeing there". On the other hand the man
ordered to paint the shade of colour in accordance with the sample will usually be in no doubt about the method of
projection.
Page 96
If I'm told: "look for a red flower in this meadow and bring it to me" and then I find one--do I compare it with
my memory picture of the colour red?--And must I consult yet another picture to see whether the first is still
correct?--In that case why should I need the first one?--I see the colour of the flower and recognize it. (It would
naturally be conceivable that someone should hallucinate a colour sample and compare it, like a real sample, with
the object he was looking for.)
Page 96
But if I say "no, this colour isn't the right one, it's brighter than the colour I saw there" that doesn't mean that
I see the colour in my mind's eye and go through a process of comparing two simultaneously given shades of
colour. Again, it isn't as if when the right colour is found a bell rings somewhere in my mind and I carry round a
picture of this ringing, so as to be able to judge when it rings.
Page 96
Searching with a sample which one places beside objects to test whether the colours match is one game;
acting in accordance with the words of a word-language without a sample is another. Think of reading aloud from a
written text (or writing to dictation). We might of course imagine a kind of table that might guide us in this; but in
fact there isn't one, there's no act of memory, or anything else, which acts as an intermediary between the written
sign and the sound.
Page 96
55 Suppose I am now asked "why do you choose this colour when given this order; how do you justify the choice?"
In the one case I can answer "because this colour is opposite the word 'red' in my chart." In the other case there is no
answer to the question and the question makes no sense. But in the first game there is no sense in this question:
"why do you call 'red' the colour in the chart
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opposite the word 'red'"? A reason can only be given within a game. The links of the chain of reasons come to an
end, at the boundary of the game. (Reason and cause.)
Page 97
If one calls to mind "that the chart does not compel us" to use it in a particular way, or even always to use it
in the same way, it becomes clear to everyone that our use of the words "rule" and "game" is a fluctuating one
(blurred at the edges).
Page 97
The connection between "language and reality" is made by definitions of words, and these belong to
grammar, so that language remains self-contained and autonomous.
Page 97
56 Imagine a gesture language used to communicate with people who have no word-language in common with us.
Do we feel there too the need to go outside the language to explain its signs?
Page 97
"The connection between words and things is set up by the teaching of language." What kind or sort of
connection is this? A mechanical, electrical, psychological connection is something which may or may not function.
Mechanism and Calculus.
Page 97
The correlation between objects and names is simply the one set up by a chart, by ostensive gestures and
simultaneous uttering of the name etc. It is a part of the symbolism. Giving an object a name is essentially the same
kind of thing as hanging a label on it.
Page 97
It gives the wrong idea if you say that the connection between name and object is a psychological one.
Page 97
57 Imagine someone copying a figure on the scale of 1 to 10. Is the understanding of the general rule of such
mapping contained in the process of copying?--The pencil in my hand was free from
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presuppositions, so to speak, and was guided (influenced) only by the length of the lines in the pattern.--I would say
that if the pattern had been longer, I should have drawn my pencil further and if it had been shorter, not so far. But is
the mind which thus expresses itself already contained in the copying of the line?
Page 98
Suppose I want to meet someone on the street. I can decide "I will go on until I find N"--and then go along
the street and stop when I meet him at a particular point. Did the process of walking, or some other simultaneous
process, include acting in accordance with the general rule I intended? Or was what I did only in agreement with
that rule, but also in agreement with other rules?
Page 98
I give someone the order to draw from A a line parallel to a. He tries (intends) to do it, but with the result that
the line is parallel to b. Was what happened when he copied the same as if he had intended to draw a line parallel to
b and carried out his intention?
Page 98
If I succeed in reproducing a paradigm in accordance with a prescribed rule, is it possible to use a different
general rule to describe the process of copying, the way it took place? Or can I reject such a description with the
words "No, I was guided by this rule, and not by the other, though admittedly in this case the other would have
given the same result"?
Page 98
58 One is inclined to say: If I intentionally copy a shape, then the process of copying has the shape in common with
the pattern. The form is a facet of the process of copying; a facet which fits the copied object and coincides with it
there.
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Page 99
Even if my pencil doesn't do justice to the model, my intention always does.
Page 99
If I intend to play the piano from written music, it is experience that will show which notes I actually play
and the description of what is played need not have anything in common with the written notes. But if I want to
describe my intention, the description must be that I wanted to reproduce these written notes in sounds.--That alone
can be the expression of the fact that intention reaches up to the paradigm and contains a general rule.
Page 99
An expression of intention describes the model to be copied; describing the copy does not.
Page 99
59 For the purposes of our studies it can never be essential that a symbolic phenomenon occurs in the mind and not
on paper so that others can see it. One is constantly tempted to explain a symbolic process by a special
psychological process; as if the mind "could do much more in these matters" than signs can.
Page 99
We are misled by the idea of a mechanism that works in special media and so can explain special
movements. As when we say: this movement can't be explained by any arrangement of levers.
Page 99
A description of what is psychological must be something which can itself be used as a symbol.
Page 99
A connected point is that an explanation of a sign can replace the sign itself. This gives an important insight
into the nature of the explanation of signs, and brings out a contrast between the idea of this sort of explanation and
that of causal explanation.
Page 99
60 It could be said that it can't be decided by outward observation whether I am reading or merely producing
sounds while a text runs before my eyes. But what is of interest to us in reading can't be essentially something
internal. Deriving a translation from the
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original may also be a visible process. For instance, it must be possible to regard as a derivation what takes place on
paper when the terms of the series 100, 121, 144, 169 are derived from the terms of the series 10, 11, 12, 13 by the
following calculations
Page 100
(The distinction between "inner" and "outer" does not interest us.)
Page 100
Every such more or less behaviourist account leaves one with the feeling that it is crude and heavy handed;
but this is misleading we are tempted to look for a "better" account, but there isn't one. One is as good as the other
and in each case what represents is the system in which a sign is used.--("Representation is dynamic, not static.").
Page 100
(Even a psychological process cannot "leave anything open" in any way essentially different from the way in
which an empty bracket in the symbolism leaves open an argument place.)
Page 100
One may not ask "What sort of thing are mental processes, since they can be true and false, and non-mental
ones cannot?" For, if the 'mental' ones can, then the others must be able to do as well and vice versa.--For, if the
mental processes can, their descriptions must be able to as well. For how this is possible must show itself in their
descriptions.
Page 100
If one says that thought is a mental activity, or an activity of the mind, one thinks of the mind as a cloudy
gaseous medium in which many things can happen which cannot occur in a different sphere, and from which many
things can be expected that are otherwise not possible.
Page 100
(The process of thinking in the human mind, and the process of digestion).
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Page 101
61 Every case of copying (acting in obedience to, not just in accordance with, particular rules), every case of
deriving an action from a command or justifying an action by a command, is the same kind of thing as writing
down the steps that lead to the answer of a sum, or pointing to signs standing beside each other in a table.
Page 101
"I write the number '16' here because it says 'x²' there, and '64' here because it says x3 there." That is what
every justification looks like. In a certain sense it takes us no further. But indeed it can't take us further i.e. into the
realm of metalogic.
Page 101
(The difficulty here is: in not trying to justify what admits of no justification.)
Page 101
Suppose, though, I said "I write a '+' here because it says 'x²' there? You would ask "Do you always write a '+'
where it says "x²"?--that is, you would look for a general rule; otherwise the "because" in my sentence makes no
sense. Or you might ask "So how do you know that that is why you wrote it?"
Page 101
In that case you've taken the "because" as introducing a statement of the cause, instead of the reason.
Page 101
If I write "16" under "4" in accordance with the rule, it might appear that some causality was operating that
was not a matter of hypothesis, but something immediately perceived (experienced).
Page 101
(Confusion between 'reason' and 'cause'.)
Page 101
What connection do I mean in the sentence "I am going out, because he's telling me to"? And how is this
sentence related to "I am going out, although he told me to". (Or "I am going out, but not because he told me to" "I
am going out, because he told me not to".)
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V
Page 102
62 "That's him" (this picture represents him)--that contains the whole problem of representation.
Page 102
What is the criterion, how is it to be verified, that this picture is the portrait of that object, i.e. that it is meant
to represent it? It is not similarity that makes the picture a portrait (it might be a striking resemblance of one person,
and yet be a portrait of someone else it resembles less).
Page 102
How can I know that someone means the picture as a portrait of N?--Well, perhaps because he says so, or
writes it underneath.
Page 102
What is the connection between the portrait of N and N himself? Perhaps, that the name written underneath
is the name used to address him.
Page 102
When I remember my friend and see him "in my mind's eye", what is the connection between the memory
image and its subject? The likeness between them?
Page 102
Well, the image, qua picture, can't do more than resemble him.
Page 102
The image of him is an unpainted portrait.
Page 102
In the case of the image too, I have to write his name under the picture to make it the image of him.
Page 102
I have the intention of carrying out a particular task and I make a plan. The plan in my mind is supposed to
consist in my seeing myself acting thus and so. But how do I know, that it is myself that I'm seeing? Well, it isn't
myself, but a kind of a picture. But why do I call it the picture of me?
Page 102
"How do I know that it's myself?": the question makes sense if it means, for example, "how do I know that
I'm the one I see there". And the answer mentions characteristics by which I can be recognized.
Page 102
But it is my own decision that makes my image represent myself. And I might as well ask "how do I know
that the word 'I' stands for myself?" For my shape in the picture was only another word "I".
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Page 103
"I can imagine your being about to go out of the door." We suffer from a strange delusion that in the
proposition, the thought, the objects do what the proposition states about them. It's as though the command
contained a shadow of the execution. But a shadow of just this execution. It is you in the command who go to such
and such a place.--Otherwise it would be just a different command.
Page 103
This identity is indeed the identity contrasted with the diversity of two different commands.
Page 103
"I thought Napoleon was crowned in the year 1805."--What has your thought got to do with
Napoleon?--What connection is there between your thought and Napoleon?--It may be, for example, that the word
"Napoleon" occurs in the expression of my thought, plus the connection that word had with its bearer; e.g. that was
the way he signed his name, that was how he was spoken to and so on.
Page 103
"But when you utter the word 'Napoleon' you designate that man and no other"--"How then does this act of
designating work, in your view? Is it instantaneous? Or does it take time?"--"But after all if someone asks you 'did
you mean the very man who won the battle of Austerlitz' you will say 'yes'. So you meant that man when you
uttered the sentence."--Yes, but only in the kind of way that I then knew also that 6 × 6 = 36.
Page 103
The answer "I meant the victor of Austerlitz" is a new step in our calculus. The past tense is deceptive,
because it looks as if it was giving a description of what went on "inside me" while I was uttering the sentence.
Page 103
("But I meant him". A strange process, this meaning! Can you mean in Europe someone who's in America?
Even if he no longer exists?)
Page 103
63 Misled by our grammar, we are tempted to ask "How does one think a proposition, how does one expect such
and such to happen? (how does one do that?)"
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Page 104
"How does thought work, how does it use its expression?"--This question looks like "How does a Jacquard
loom work, how does it use the cards".
Page 104
In the proposition "I believe that p is the case" we feel that the essential thing, the real process of belief, isn't
expressed but only hinted at; we feel it must be possible to replace this hint by a description of the mechanism of
belief, a description in which the series of words "p" would occur as the cards occur in the description of the loom.
This description, we feel, would be at last the full expression of the thought.
Page 104
Let's compare belief with the utterance of a sentence; there too very complicated processes take place in the
larynx, the speech muscles, the nerves, etc. These are accompaniments of the spoken sentence. And the sentence
itself remains the only thing that interests us--not as part of a mechanism, but as part of a calculus.
Page 104
"How does thought manage to represent?"--the answer might be "Don't you really know? You certainly see it
when you think." For nothing is concealed.
Page 104
How does a sentence do it? Nothing is hidden.
Page 104
But given this answer "But you know how sentences do it, for nothing is concealed" one would like to say
"yes, but it all goes by so quick, and I should like to see it as it were laid open to view".
Page 104
We feel that thoughts are like a landscape that we have seen and are supposed to describe, but don't
remember exactly enough to describe how all the parts fitted together. Similarly, we think, we can't describe thought
after the event because then the many delicate processes have been lost sight of. We would like as it were to see
these intricacies under the magnifying glass. (Think of the proposition "Everything is in flux".)
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Page 105
We ask: "What is a thought? What kind of thing must something be to perform the function of thought?"
This question is like: "What is a sewing machine, how does it work?--And the answer which would be like ours
would be "Look at the stitch it is meant to sew; you can see from that what is essential in the machine, everything
else is optional."
Page 105
So what is the function, that makes thought what it is?--
Page 105
If it is its effect, then we are not interested in it.
Page 105
We are not in the realm of causal explanations, and every such explanation sounds trivial for our purposes.
Page 105
64 If one thinks of thought as something specifically human and organic, one is inclined to ask "could there be a
prosthetic apparatus for thinking, an inorganic substitute for thought?" But if thinking consists only in writing or
speaking, why shouldn't a machine do it? "Yes, but the machine doesn't know anything." Certainly it is senseless to
talk of a prosthetic substitute for seeing and hearing. We do talk of artificial feet, but not of artificial pains in the foot.
Page 105
"But could a machine think?"--Could it be in pain?--Here the important thing is what one means by
something being in pain. I can look on another person--another person's body--as a machine which is in pain. And
so, of course, I can in the case of my own body. On the other hand, the phenomenon of pain which I describe when
I say something like "I have toothache" doesn't presuppose a physical body. (I can have toothache without teeth.)
And in this case there is no room for the machine.--It is clear that the machine can only replace a physical body. And
in the sense in which we can say of such a body that it is in pain, we can say it of a machine as well. Or again, what
we can compare with machines and call machines is the bodies we say are in pain.
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Page 106
In the consideration of our problems one of the most dangerous ideas is the idea that we think with, or in,
our heads.
Page 106
The idea of a process in the head, in a completely enclosed space, makes thinking something occult.†1
Page 106
"Thinking takes place in the head" really means only "the head is connected with thinking".--Of course one
says also "I think with my pen" and this localisation is at least as good.
Page 106
It is a travesty of the truth to say "Thinking is an activity of our mind, as writing is an activity of the hand".
(Love in the heart. The head and the heart as loci of the soul).
Page 106
65 We may say "Thinking is operating with symbols". But 'thinking' is a fluid concept, and what 'operating with
symbols' is must be looked at separately in each individual case.
Page 106
I might also say "Thinking is operating with language" but 'language' is a fluid concept.
Page 106
It is correct to say "Thinking is a mental process" only if we also call seeing a written sentence or hearing a
spoken one a mental process. In the sense, that is, in which pain is called a mental state. In that case the expression
"mental process" is intended to distinguish 'experience' from 'physical processes'.--On the other hand, of course, the
expression "mental process" suggests that we are concerned with imperfectly understood processes in an
inaccessible sphere.
Page 106
Psychology too talks of 'unconscious thought' and here "thought" means a process in a mind-model. ('Model'
in the sense in which one speaks of a mechanical model of electrical processes).
Page 106
By contrast, when Frege speaks of the thought a sentence expresses the word "thought" is more or less
equivalent to the expression "sense of the sentence".
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Page 107
It might be said: in every case what is meant by "thought" is the living element in the sentence, without
which it is dead, a mere succession of sounds or series of written shapes.
Page 107
But if I talked in the same way about a something that gives meaning to an arrangement of chessmen,
something that makes it different from an arbitrary collection of bits of wood, I might mean almost anything! I might
mean the rules that make the arrangement of chessmen a position in a game, or the special experiences we connect
with positions in the game or the use of the game.
Page 107
It is the same if we speak of a something that makes the difference between paper money and mere printed
bits of paper, something that gives it its meaning, its life.
Page 107
Though we speak of a thought and its expression, the thought is not a kind of condition that the sentence
produces as a potion might. And communication by language is not a process by which I use a drug to produce in
others the same pains as I have myself.
Page 107
(What sort of process might be called "thought-transference" or "thought-reading"?)
Page 107
66 A French politician once said it was a special characteristic of the French language that in French sentences words
occurred in the sequence in which one thinks them.
Page 107
The idea that one language in contrast to others has a word order which corresponds to the order of thinking
arises from the notion that thought is an essentially different process going on independently of the expression of
the thoughts.
Page 107
(No one would ask whether the written multiplication of two numbers in the decimal system runs parallel
with the thought of the multiplication.)
Page 107
"I meant something definite by it, when I said..."
Page 107
--"Did you mean something different when you said each word, or did you mean the same thing throughout
the whole sentence?"
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Page 108
It is strange, though: you can mean something by each word and the combination of them can still be
nonsense!
Page 108
"At the time when you said the sentence, did you think of the fact that..."
Page 108
"I thought only what I said."
Page 108
(It perplexes us that there is no moment at which the thought of a sentence is completely present. Here we
see that we are comparing the thought with a thing that we manufacture and possess as a whole; but in fact as soon
as one part comes into being another disappears. This leaves us in some way unsatisfied, since we are misled by a
plausible simile into expecting something different.)
Page 108
Does the child learn only to talk, or also to think? Does it learn the sense of multiplication before or after it
learns multiplication?
Page 108
Is it, as it were, a contamination of the sense that we express it in a particular language which has accidental
features, and not as it were bodiless and pure?
Page 108
Do I really not play chess itself because the chessmen might have had a different shape?
Page 108
(Is a mathematical proof in the general theory of irrational numbers less general or rigorous because we go
through it using the decimal notation for those numbers? Does it impair the rigour and purity of the proposition 25 ×
25 = 625 that it is written down in a particular number system?)
Page 108
Thought can only be something common-or-garden and ordinary. (We are accustomed to thinking of it as
something ethereal and unexplored, as if we were dealing with something whose exterior alone is known to us, and
whose interior is yet unknown like our brain.) One is inclined to say: "Thought, what a strange thing!" But when I
say that thought is something quite common-or-garden, I mean that we are affected by this concept as we are by a
concept like that of the number one. There seems to be something mysterious about it, because we misunderstand
its grammar and feel the lack of a tangible substance
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to correspond to the substantive. (It is almost like hearing a human voice coming from in front of us, and seeing
nobody there.)
Page 109
67 What does man think for? What use is it? Why does he calculate the thickness of the walls of a boiler and not
leave it to chance or whim to decide? After all it is a mere fact of experience that boilers do not explode so often if
made according to calculations. But just as having once been burnt he would do anything rather than put his hand
into the fire, so he would do anything rather than not calculate for a boiler.--Since we are not interested in causes, we
might say: human beings do in fact think: this, for instance, is how they proceed when they make a boiler.--Now,
can't a boiler produced in this way explode? Certainly it can.
Page 109
We think over our actions before we do them. We make pictures of them--but why? After all, there is no
such thing as a "thought-experiment".
Page 109
We expect something, and act in accordance with the expectation; must the expectation come true? No.
Then why do we act in accordance with the expectation? Because we are impelled to, as we are impelled to get out
of the way of a car, to sit down when we are tired, to jump up if we have sat on a thorn.
Page 109
What the thought of the uniformity of nature amounts to can perhaps be seen most clearly when we fear the
event we expect. Nothing could induce me to put my hand into a flame--although after all it is only in the past that I
have burnt myself.
Page 109
The belief that fire will burn me is of the same nature as the fear that it will burn me.
Page 109
Here I see also what "it is certain" means.
Page 109
If someone pushed me into the fire, I would struggle and go on
Page Break 110
resisting; and similarly I would cry out "it will burn me!" and not "perhaps it will be quite agreeable".
Page 110
"But after all you do believe that more boilers would explode if people did not calculate when making
boilers!" Yes, I believe it;--but what does that mean? Does it follow that there will in fact be fewer explosions?--Then
what is the foundation of this belief?
Page 110
68 I assume that his house in which I am writing won't collapse during the next half hour.--When do I assume this?
The whole time? And what sort of an activity is this assuming?
Page 110
Perhaps what is meant is a psychological disposition; or perhaps the thinking and expressing of particular
thoughts. In the second case perhaps I utter a sentence which is part of a train of thought (a calculation). Now
someone says: you must surely have a reason to assume that, otherwise the assumption is unsupported and
worthless.--(Remember that we stand on the earth, but the earth doesn't stand on anything else; children think it'll
have to fall if it's not supported). Well, I do have reasons for my assumption. Perhaps that the house has already
stood for years, but not so long that it may already be rickety, etc. etc.--What counts as a reason for an assumption
can be given a priori and determines a calculus, a system of transitions. But if we are asked now for a reason for the
calculus itself, we see that there is none.
Page 110
So is the calculus something we adopt arbitrarily? No more so than the fear of fire, or the fear of a raging
man coming at us.
Page 110
"Surely the rules of grammar by which we act and operate are not arbitrary!" Very well; why then does a
man think in the way he does, why does he go through these activities of thought? (This question of course asks for
reasons, not for causes.) Well, reasons can be given within the calculus, and at the very end one is tempted to say "it
just is very probable, that things will behave in this case as they always have"--or something similar. A turn of
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phrase which masks the beginning of the chain of reasons. (The creator as the explanation at the beginning of the
world).†1
Page 111
The thing that's so difficult to understand can be expressed like this. As long as we remain in the province of
the true-false games a change in the grammar can only lead us from one such game to another, and never from
something true to something false. On the other hand if we go outside the province of these games, we don't any
longer call it 'language' and 'grammar', and once again we don't come into contradiction with reality.
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VI
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69 What is a proposition?--What am I distinguishing a proposition from? What do I want to distinguish it from?
From things which are only parts of propositions in the same grammatical system (like the parts of an equation)? Or
from everything we don't call propositions, including this chair and my watch, etc. etc?
Page 112
The question "how is the general concept of proposition bounded?" must be countered with another: "Well,
do we have a single concept of proposition?"
Page 112
"But surely I have a definite concept of what I mean by 'proposition'." Well, and how would I explain it to
another or to myself? This explanation will make clear what my concept is (I am not concerned with a feeling
accompanying the word 'proposition'). I would explain the concept by means of examples.--So my concept goes as
far as the examples.--But after all they're only examples, and their range is capable of extension.--All right, but in
that case you must tell me what "capable of extension" means here. The grammar of this word must have definite
boundaries.
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"But I know a proposition when I see one, so I must also be able to draw the boundaries of the concept
precisely." But is it really the case that no doubt is possible?--Imagine a language in which all sentences are
commands to go in a particular direction. (This language might be used by a primitive kind of human beings
exclusively in war. Remember how restricted the use of written language once was.) Well, we would still call the
commands "come here", "go there", "sentences".†1 But suppose now the language consisted only in pointing the
finger in one direction or the other.--Would this sign still be a proposition?--And what about a language like the early
speech of children whose signs expressed only desire for particular objects, a language which consisted simply of
signs for these objects (of nouns, as it were)? Or consider
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a system consisting of two signs, one expressing acceptance and the other rejection of proferred objects. Is this a
language, does it consist of propositions?
Page 113
And on the other hand: does everything that sounds like a sentence in English fall under our concept of
proposition? "I am tired", "2 × 2 = 4", "time passes", "there is only one zero"?
Page 113
The word "proposition" does not signify a sharply bounded concept. If we want to put a concept with sharp
boundaries beside our use of this word, we are free to define it, just as we are free to narrow down the meaning of
the primitive measure of length "a pace" to 75 cm.
Page 113
70 "What happens when a new proposition is taken into the language: what is the criterion for its being a
proposition?" Let us imagine such a case. We become aquainted with a new experience, say the tingling of an
electric shock, and we say it's unpleasant. What right have I to call this newly formed statement a "proposition"?
Well, what right did I have to speak of a new "experience", or a new "muscular sensation"? Surely I did so by
analogy with my earlier use of these words. But, on the other hand, did I have to use the word "experience" and the
word "proposition" in the new case? Do I already assert something about the sensation of the electric shock when I
call it an experience? And what difference would it make if I excluded the statement "the tingling is unpleasant"
from the concept of proposition, because I had already drawn its boundaries once and for all?
Page 113
Compare the concept of proposition with the concept 'number' and then with the concept of cardinal
number. We count as numbers cardinal numbers, rational numbers, irrational numbers, complex numbers; whether
we call other constructions numbers because of their similarities with these, or draw a definitive boundary here or
elsewhere, depends on us. In this respect the concept of number is like the concept of proposition. On the
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other hand the concept of cardinal number [1, ξ, ξ + 1] can be called a rigorously circumscribed concept, that's to
say it's a concept in a different sense of the word.
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71 How did I come by the concept 'proposition' or the concept 'language'? Only through the languages I've
learnt.--But in a certain sense they seem to have led me beyond themselves, since I'm now able to construct a new
language, for instance to invent words.--So this construction too belongs to the concept of language. But only if I so
stipulate. The sense of my "etc." is constantly given limits by its grammar.
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That's also what I meant when I said "there are surprises in reality but not in grammar."
Page 114
"But language can expand"--Certainly; but if this word "expand" has a sense here, then I know already what
I mean by it. I must be able to specify how I imagine such an expansion. And what I can't think, I can't now express
or even hint at. And in this case the word "now" means: "in this calculus" or "if the words are used according to
these grammatical rules".
Page 114
But here we also have this nagging problem: how is it possible even to think of the existence of things when
we always see only images, copies of them?--We ask: "Then how did I come by this concept at all?" It would be
quite correct to add in thought the rider: "It is not as if I was able to transcend my own thought", "It is not as if I
could sensibly transcend what has sense for me." We feel that there is no way of smuggling in by the back door a
thought I am barred from thinking directly.
Page 114
No sign leads us beyond itself, and no argument either.
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Page 115
What does a man do when he constructs (invents) a new language; on what principle does he operate? For
this principle is the concept of 'language'. Does every newly constructed language broaden (alter) the concept of
language?--Consider its relationship to the earlier concept: that depends on how the earlier concept was
established.--Think of the relation of complex numbers to the earlier concept of number; and again of the relation of
a new multiplication to the general concept of the multiplication of cardinal numbers, when two particular (perhaps
very large) cardinal numbers are written down for the first time and multiplied together.
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72 In logic one cannot employ generality in a void. If I determine the grammar of my generality, then there are no
more surprises in logic. And if I do not determine it, then I am no longer in the realm of an exact grammar.
Page 115
That's to say, the indeterminacy of generality is not a logical indeterminacy. Generality is a freedom of
movement, not an indeterminacy of geometry.
Page 115
But if the general concept of language dissolves in this way, doesn't philosophy dissolve as well? No, for the
task of philosophy is not to create a new, ideal language, but to clarify the use of our language, the existing language.
Its aim is to remove particular misunderstandings; not to produce a real understanding for the first time.
Page 115
If a man points out that a word is used with several different meanings, or that a certain misleading picture
comes to mind when we use a certain expression, if he sets out (tabulates) rules according to which certain words
are used, he hasn't committed himself to giving an explanation (definition) of the words "rule", "proposition",
"word", etc.
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I'm allowed to use the word "rule" without first tabulating the rules for the use of the word. And those rules
are not super-rules.
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Page 116
Philosophy is concerned with calculi in the same sense as it is concerned with thoughts, sentences and
languages. But if it was really concerned with the concept of calculus, and thus with the concept of the calculus of all
calculi, there would be such a thing as metaphilosophy. (But there is not. We might so present all that we have to say
that this would appear as a leading principle.)
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73 How do we use the word "rule", say when we are talking of games? In contrast to what?--We say for instance
"that follows from this rule", but in that case we could cite the rule in question and thus avoid the word "rule". Or we
speak of "all the rules of a game" and in that case either we've listed them (in which case we have a repetition of the
first case) or we're speaking of the rules as a group of expressions produced in a certain way from given basic rules,
and then the word "rule" stands for the expression of those basic rules and operations. Or we say "this is a rule and
that isn't"--if the second, say, is only an individual word or a sentence which is incomplete by the standards of
English grammar, or the illustration of a position of pieces in a game. (Or "No, according to the new convention that
too is a rule"). If we had to write down the list of rules of the game, something like that might be said and then it
would mean "this belongs in it and that doesn't". But this isn't on the strength of a particular property, the property
of being a rule, like the case when one wants to pack only apples in a box and says, "no, that shouldn't go in there,
that's a pear".
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Yes, but there are many things we call games and many we don't, many things we call rules and many we
don't!--But it's never a question of drawing a boundary between everything we call games and everything else. For
us games are the games of which we have heard, the games we can list, and perhaps some others newly devised by
analogy; and if someone wrote a book on games, he wouldn't really need to use the word "game" in the title of the
book, he could use as a title a list of the names of the individual games.
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Page 117
If he's asked "but what's common to all these things that makes you collect them together?" he might say: I
can't give it straight off--but surely you may see many analogies. Anyway the question seems to me idle, because
proceeding by analogy, I can also come by imperceptible steps to things that no one in ordinary life would any
longer call "games". Hence I call games things on this list, and whatever is similar to these games up to a certain
point that I don't further specify. Moreover, I reserve the right to decide in every new case whether I will count
something as a game or not.
Page 117
The case is the same with the concepts 'rule', 'proposition', 'language', etc. It is only in special cases (i.e. not
every time we use the word "rule") that it is a question of drawing a boundary between rules and what are not rules,
and in all cases it is easy to give the distinguishing mark. We use the word "rule" in contrast to "word", "projection"
and some other words and these demarcations can be clearly drawn. On the other hand we commonly do not draw
boundaries where we do not need them. (Just as in certain games a single line is drawn in the middle of the field to
separate the sides, but the field is not otherwise bounded since it is unnecessary.)
Page 117
We are able to use the word "plant" in a way that gives rise to no misunderstanding, yet countless borderline
cases can be constructed in which no one has yet decided whether something still falls under the concept 'plant'.
Does this mean that the meaning of the word "plant" in all other cases is infected by uncertainty, so that it might be
said we use the word without understanding it? Would a definition which bounded this concept on several sides
make the meaning of the word clearer to us in all sentences? Would we understand better all the sentences in which
it occurs?
Page 117
74 How did we learn to understand the word "plant", then? Perhaps we learnt a definition of the concept, say in
botany, but I leave out that of account since it only has a role in botany. Apart
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from that, it is clear that we learnt the meaning of the word by example; and if we disregard hypothetical
dispositions, these examples stand only for themselves. Hypotheses about learning and using language and causal
connections don't interest us. So we don't assume that the examples produce something in the learner, that they set
before his mind an essence, the meaning of the concept-word, the concept 'plant'. If the examples should have an
effect, say they produce a particular visual picture in the learner, the causal connection between the examples and
this picture does not concern us, and for us they are merely coincidental. So we can perhaps disregard the examples
altogether and look on the picture alone as a symbol of the concept; or the picture and the examples together.
Page 118
If someone says "we understand the word 'chair', since we know what is common to all chairs"--what does it
mean to say we know that? That we are ready to say it (like "we know that 6 × 6 is 36")? What is it that is common,
then? Isn't it only because we can apply the word "chair" that we say here we know what is common? Suppose I
explained the word "red" by pointing to a red wall, a red book, and a red cloth and in accordance with this
explanation someone produced a sample of the colour red by exhibiting a red label. One might say in this case that
he had shown that he had grasped the common element in all the examples I gave him. Isn't it an analogy like this
that misleads us in the case of "chair"?
Page 118
The grammatical place of the words "game", "rule", etc. is given by examples in rather the way in which the
place of a meeting is specified by saying that it will take place beside such and such a tree.
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75 One imagines the meaning as something which comes before our minds when we hear a word.
Page 118
What comes before our minds when we hear a word is certainly something characteristic of the meaning. But
what comes before
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my mind is an example, an application of the word. And this coming to mind doesn't really consist in a particular
image's being present whenever I utter or hear the word, but in fact that when I'm asked the meaning of the word,
applications of the word occur to me.
Page 119
Someone says to me: "Shew the children a game" I teach them gaming with a dice, and the other says "I
didn't mean that sort of game". Must the exclusion of the game with dice have come before his mind when he gave
me the order?
Page 119
Suppose someone said: "No. I didn't mean that sort of game; I used 'game' in the narrower sense." How does
it come out, that he used the word in the narrower sense?
Page 119
But can't one also use the word "game" in its broadest sense? But which is that? No boundaries have been
drawn unless we fix some on purpose.
Page 119
If we found a sentence like "The Assyrians knew various games" in a history book without further
qualifications, it would strike us as very curious; for we wouldn't be certain that we could give an example that even
roughly corresponded to the meaning of the word "game" in this case.
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Someone wants to include in the list of rules of a game the proposition that the game was invented in such
and such a year. I say "No, that doesn't belong to the list of rules, that's not a rule." So I'm excluding historical
propositions from the rules. And similarly I would exclude from the rules, as an empirical proposition, a proposition
like "this game can only be learnt by long practice". But it could easily be misleading to say boundaries had thereby
been drawn around the area of rules.
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76 If I try to make clear to someone by characteristic examples
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the use of a word like "wish", it is quite likely that the other will adduce as an objection to the examples I offered
another one that suggests a different type of use. My answer then is that the new example may be useful in
discussion, but isn't an objection to my examples. For I didn't want to say that those examples gave the essence of
what one calls "wishing". At most they present different essences which are all signified by this word because of
certain inter-relationships. The error is to suppose that we wanted the examples to illustrate the essence of wishing,
and that the counter examples showed that this essence hadn't yet been correctly grasped. That is, as if our aim were
to give a theory of wishing, which would have to explain every single case of wishing.
Page 120
But for this reason, the examples given are only useful if they are clearly worked out and not just vaguely
hinted at.
Page 120
The use of the words "proposition", "language", etc. has the haziness of the normal use of concept-words in
our language. To think this makes them unusable, or ill-adapted to their purpose, would be like wanting to say "the
warmth this stove gives is no use, because you can't feel where it begins and where it ends".
Page 120
If I wish to draw sharp boundaries to clear up or avoid misunderstandings in the area of a particular use of
language, these will be related to the fluctuating boundaries of the natural use of language in the same way as sharp
contours in a pen-and-ink sketch are related to the gradual transitions between patches of colour in the reality
depicted.
Page 120
Socrates pulls up the pupil who when asked what knowledge is enumerates cases of knowledge. And
Socrates doesn't regard that as even a preliminary step to answering the question.
Page 120
But our answer consists in giving such an emuneration and a
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few analogies. (In a certain sense we are always making things easier and easier for ourselves in philosophy.)
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77 The philosophy of logic speaks of sentences and words in exactly the sense in which we speak of them in
ordinary life when we say "Here is a Chinese sentence", or "No, that only looks like writing; it is actually just an
ornament" and so on.
Page 121
We are talking about the spatial and temporal phenomenon of language, not about some non-spatial,
non-temporal phantasm. But we talk about it as we do about the pieces in chess when we are stating the rules of the
game, not describing their physical properties.
Page 121
The question "what is a word?" is analogous to "What is a piece in chess (say the king)?"
Page 121
In reflecting on language and meaning we can easily get into a position where we think that in philosophy we
are not talking of words and sentences in a quite common-or-garden sense, but in a sublimated and abstract
sense.--As if a particular proposition wasn't really the thing that some person utters, but an ideal entity (the "class of
all synonymous sentences" or the like). But is the chess king that the rules of chess deal with such an ideal and
abstract entity too?
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(We are not justified in having any more scruples about our language than the chess player has about chess,
namely none.)
Page 121
Again, we cannot achieve any greater generality in philosophy than in what we say in life and in science.
Here too (as in mathematics) we leave everything as it is.
Page 121
When I talk about language (words, sentences, etc.) I must speak the language of every day. Is this language
somehow too coarse and material for what we want to say? Then how is another one to be constructed?--And how
strange that we should be able to do anything at all with the one we have!
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Page 122
In giving philosophical explanations about language I already have to use language full-blown (not some sort
of preparatory, provisional one); this by itself shows that I can adduce only exterior facts about language.
Page 122
"Yes, but then how can these explanations satisfy us?"--Well, your very questions were framed in this
language!--And your scruples are misunderstandings. Your questions refer to words, so I have to talk about words.
Page 122
You say: the point isn't the word, but its meaning, and you think of the meaning as a thing of the same kind
as the word, though also different from the word. Here the word, there the meaning. The money, and the cow that
you can buy with it. (But contrast: money, and its use).
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78 If we ask about the general form of proposition--bear in mind that in normal language sentences have a particular
rhythm and sound but we don't call everything 'that sounds like a sentence' a sentence.--Hence we speak also of
significant and non-significant "sentences".
Page 122
On the other hand, sounding like a sentence in this way isn't essential to what we call a proposition in logic.
The expression "sugar good" doesn't sound like an English sentence, but it may very well replace the proposition
"sugar tastes good". And not e.g. in such a way that we should have to add in thought something that is missing.
(Rather, all that matters is the system of expressions to which the expression "sugar good" belongs.)
Page 122
So the question arises whether if we disregard this misleading business of sounding like a sentence we still
have a general concept of proposition.
Page 122
Imagine the English language altered in such a way that the order of the words in a sentence is the reverse of
the present one. The result would be the series of words which we get if we read sentences of an English book from
right to left. It's clear that the multiplicity of possible ways of expression in this language must be exactly the same
as in English; but if a longish sentence were read thus we could understand it only with great difficulty and we'd
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perhaps never learn "to think in this language". (The example of such a language can make clear a lot about the
nature of what we call "thought".)
Page 123
79 The definition "A proposition is whatever can be true or false" fixes the concept of proposition in a particular
language system as what in that system can be an argument of a truth-function.
Page 123
And if we speak of what makes a proposition a proposition, we are inclined to mean the truth-functions.
Page 123
"A proposition is whatever can be true or false" means the same as "a proposition is whatever can be
denied".
"p" is true = p
"p" is false = ~p
What he says is true = Things are as he says.
Page 123
One might say: the words "true" and "false" are only items in a particular notation for truth-functions.
Page 123
So is it correct to write "'p' is true", "'p' is false"; mustn't it be "p is true" (or false)? The ink mark is after all
not true; in the way in which it's black and curved.
Page 123
Does "'p' is true" state anything about the sign "p" then? "Yes, it says that 'p' agrees with reality." Instead of a
sentence of our word language consider a drawing that can be compared with reality according to exact
projection-rules. This surely must show as clearly as possible what "'p' is true" states about the picture "p". The
proposition "'p' is true" can thus be compared with the proposition "this object is as long as this metre rule" and "p"
to the proposition "this object is one metre long". But the comparison is incorrect, because "this metre rule" is a
description, whereas "metre rule" is the determination of a concept. On the other hand in "'p' is true" the ruler enters
immediately into the proposition. "p" represents here simply the length and not the metre rule. For the representing
drawing is also not 'true' except in accordance
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with a particular method of projection which makes the ruler a purely geometrical appendage of the measured line.
Page 124
It can also be put thus: The proposition "'p' is true" can only be understood if one understands the grammar
of the sign "p" as a propositional sign; not if "p" is simply the name of the shape of a particular ink mark. In the end
one can say that the quotation marks in the sentence "'p' is true" are simply superfluous.
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If one explains: "(x).fx" is true, if "f( )" gives true sentences for all substitutions--we must reflect that the
sentence "(x).fx" follows from the proposition 'f( )' gives true sentences for all substitutions", and vice versa. So the
two propositions say the same.
Page 124
So that explanation does not assemble the mechanism of generality from its parts.
Page 124
One can't of course say that a proposition is whatever one can predicate "true" or "false" of, as if one could
put symbols together with the words "true" and "false" by way of experiment to see whether the result makes sense.
For something could only be decided by this experiment if "true" and "false" already have definite meanings, and
they can only have that if the contexts in which they can occur are already settled.--(Think also of identifying parts
of speech by questions. "Who or what...?")
Page 124
80 In the schema "This is how things stand" the "how things stand" is really a handle for the truth-functions.
Page 124
"Things stand", then, is an expression from a notation for truth-functions. An expression which shows us
what part of grammar comes into play here.
Page 124
If I let "that is how things stand" count as the general form of proposition, then I must count "2 + 2 = 4" as a
proposition. Further rules are needed if we are to exclude the propositions of arithmetic.
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Page 125
Can one give the general form of a proposition?--Why not? In the same way as one might give the general
form of a number, for example by the sign "|0, ξ, ξ + 1|". I am free to restrict the name "number" to that, and in the
same way I can give an analogous formula for the construction of propositions or laws and use the word
"proposition" or "law" as equivalent to that formula.--If someone objects and says that this will only demarcate
certain laws from others, I reply: of course you can't draw a boundary if you've decided in advance not to recognize
one. But of course the question remains: how do you use the word "proposition"? In contrast to what?
Page 125
("Can a proposition treat of all propositions, or all propositional functions?" What is meant by that? Are you
thinking of a proposition of logic?--What does the proof of such a proposition look like?)
Page 125
A general propositional form determines a proposition as part of a calculus.
Page 125
81 Are the rules that say that such and such a combination of words yields no sense comparable to the stipulations
in chess that the game does not allow two pieces to stand on the same square, for instance, or a piece to stand on a
line between two squares? Those propositions in their turn are like certain actions; like e.g. cutting a chess board out
of a larger sheet of squared paper. They draw a boundary.
Page 125
So what does it mean to say "this combination of words has no sense"? One can say of a name (of a
succession of sounds): "I haven't given anyone this name"; and name-giving is a definite action (attaching a label).
Think of the representation of an explorer's route by a line drawn in each of the two hemispheres projected on the
page: we may say that a bit of line going outside the circles on the page makes no sense in this projection. We might
also express it thus: no stipulation has been made about it.
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"How do I manage always to use a word significantly? Do I always look up the grammar? No, the fact that I
mean something,--what I mean prevents me from talking nonsense."--But what do I mean?--I would like to say: I
speak of bits of an apple, but not bits of the colour red, because in connection with the words "bits of an apple",
unlike the expression "bits of the colour red", I can imagine something, picture something, want something. It would
be more correct to say that I do imagine, picture, or want something in connection with the words "bits of an apple"
but not in connecttion [[sic]] with the expression "bits of the colour red".
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But the expression "I'm cutting red into bits" can have a sense (e.g. the sense of the proposition "I'm cutting
something red into bits").--Suppose I asked: which word is it, which mistake, that makes the expression senseless?
This shows that this expression, in spite of its senselessness, makes us think of a quite definite grammatical system.
That's why we also say "red can't be cut into bits" and so give an answer; whereas one wouldn't make any answer to
a combination of words like "is has good". But if one is thinking of a particular system, a language game plus its
application, then what is meant by "'I'm cutting red into bits' is senseless" is first and foremost that this expression
doesn't belong to the particular game its appearance makes it seem to belong to.
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If we do give a sense to the set of words "I'm cutting red into bits" how do we do it?--We can indeed turn it
into quite different things; an empirical proposition, a proposition of arithmetic (like 2 + 2 = 4), an unproved theorem
of mathematics (like Goldbach's conjecture), an exclamation, and other things. So I've a free choice: how is it
bounded? That's hard to say--by various types of utility, and by the expression's formal similarity to certain primitive
forms of proposition; and all these boundaries are blurred.
Page 126
"How do I know that the colour red can't be cut into bits?" That isn't a question either.
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I would like to say: "I must begin with the distinction between
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sense and nonsense. Nothing is possible prior to that. I can't give it a foundation."
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82 Can one ask: "How must we make the grammatical rules for words if they are to give a sentence sense"?
Page 127
I say, for instance: There isn't a book here, but there could be one; on the other hand it's nonsensical to say
that the colours green and red could be in a single place at the same time. But if what gives a proposition sense is its
agreement with grammatical rules then let's make just this rule, to permit the sentence "red and green are both at this
point at the same time". Very well; but that doesn't fix the grammar of the expression. Further stipulations have yet
to be made about how such a sentence is to be used; e.g. how it is to be verified.
Page 127
If a proposition is conceived as a picture of the state of affairs it describes and a proposition is said to show
just how things stand if it's true, and thus to show the possibility of the asserted state of affairs, still the most that the
proposition can do is what a painting or relief does: and so it can at any rate not set forth what is just not the case. So
it depends wholly on our grammar what will be called possible and what not, i.e. what that grammar permits. But
surely that is arbitrary! Certainly; but the grammatical constructions we call empirical propositions (e.g. ones which
describe a visible distribution of objects in space and could be replaced by a representational drawing) have a
particular application, a particular use. And a construction may have a superficial resemblance to such an empirical
proposition and play a somewhat similar role in a calculus without having an analogous application; and if it hasn't
we won't be inclined to call it a proposition.
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"Possible" here means the same as "conceivable"; but "conceivable" may mean "capable of being painted",
"capable of being modelled", "capable of being imagined"; i.e. representable
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in a particular system of propositions. What matters is the system.--For example someone asks: "is it conceivable
that a row of trees might go on forever in the same direction without coming to an end?" Why shouldn't it be
'conceivable'? After all it's expressible in a grammatical system. But if so what's the application of the proposition?
How is it verified? What is the relation between its verification and the verification of a proposition like "this row of
trees ends at the hundredth tree"? That will tell us how much this conceivability is worth, so to speak.
Page 128
Chemically possible †1
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"I haven't ever in fact seen a black line gradually getting lighter until it was white, and then more reddish until
it was red: but I know that it is possible, because I can imagine it." The form of expression "I know that it is possible,
because..." is taken from cases like "I know that it is possible to unlock the door with this key, because I once did
so". So am I making that sort of conjecture: that the colour transition will be possible since I can imagine it?--Isn't
this rather the way it is: here "the colour transition is possible" has the same meaning as "I can imagine it?" What
about this: "The alphabet can be said aloud, because I can recite it in my mind"?
Page 128
"I can imagine the colour transition" isn't an assertion here about a particular power of my own imagination,
in the way that "I can lift this stone" is about the power of my own muscles. The sentence "I can imagine the
transition", like "this state of affairs can be drawn", connects the linguistic representation with another form of
representation; it is to be understood as a proposition of grammar.
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83 It looks as if we could say: Word-language allows of senseless combination of words, but the language of
imagining does not
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allow us to imagine anything senseless. Hence too the language of drawing doesn't allow of senseless drawings.--But
that isn't how it is: for a drawing can be senseless in the same way as a proposition. Think of a blueprint from which
a turner is to work; here it is very easy to represent an exact analogy with a senseless pseudo-proposition.
Remember too the example of drawing a route on a projection of the globe.
Page 129
When one wants to show the senselessness of metaphysical turns of phrase, one often says "I couldn't
imagine the opposite of that", or "What would it be like if it were otherwise?" (When, for instance, someone has said
that my images are private, that only I alone can know if I am feeling pain, etc.) Well, if I can't imagine how it might
be otherwise, I equally can't imagine that it is so. For here "I can't imagine" doesn't indicate a lack of imaginative
power. I can't even try to imagine it; it makes no sense to say "I imagine it". And that means, no connection has been
made between this sentence and the method of representation by imagination (or by drawing).
Page 129
But why does one say "I can't imagine how it could be otherwise" and not "I can't imagine the thing itself"?
One regards the senseless sentence (e.g. "this rod has a length") as a tautology as opposed to a contradiction. One
says as it were: "Yes, it has a length; but how could it be otherwise; and why say so?" To the proposition "This rod
has a length" we respond not "Nonsense!" but "Of course!" We might also put it thus: when we hear the two
propositions, "This rod has a length" and its negation "This rod has no length", we take sides and favour the first
sentence, instead of declaring them both nonsense. But this partiality is based on a confusion: we regard the first
proposition as verified (and the second as falsified) by the fact "that the rod has a length of 4 metres". "After all, 4
metres is a length"--but one forgets that this is a grammatical proposition.
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Page 130
It is often possible to show that a proposition is meant metaphysically by asking "Is what you affirm meant
to be an empirical proposition? Can you conceive (imagine) its being otherwise?"--Do you mean that substance has
never yet been destroyed, or that it is inconceivable that it should be destroyed? Do you mean that experience
shows that human beings always prefer the pleasant to the unpleasant?
Page 130
How strange that one should be able to say that such and such a state of affairs is inconceivable! If we regard
thought as essentially an accompaniment going with an expression, the words in the statement that specify the
inconceivable state of affairs must be unaccompanied. So what sort of sense is it to have? Unless it says these words
are senseless. But it isn't as it were their sense that is senseless; they are excluded from our language like some
arbitrary noise, and the reason for their explicit exclusion can only be that we are tempted to confuse them with a
sentence of our language.
Page 130
84 The role of a sentence in the calculus is its sense.
Page 130
A method of measurement--of length, for example--has exactly the same relation to the correctness of a
statement of length as the sense of a sentence has to its truth or falsehood.
Page 130
What does "discovering that an assertion doesn't make sense" mean?--and what does it mean to say: "If I
mean something by it, surely it must make sense to say it"? "If I mean something by it"--if I mean what by it?--
Page 130
One wants to say: a significant sentence is one which one can not merely say, but also think. But that would
be like saying: a significant picture is one that can not merely be drawn but also represented plastically [[sic]]. And
saying this would make sense. But the thinking of a sentence is not an activity which one does from the words (like
singing from a score). The following example shews this.
Page 130
Does it make sense to say "The number of my friends is equal to a root of the equation x² + 2x - 3 = 0?"
Here, one might think,
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we have a notation whose grammar doesn't settle by itself whether a sentence makes sense or not, so that it wasn't
determined in advance. That is a fine example of what is meant by understanding a proposition.
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If the expression "the root of the equation..." were a Russellian description, then the proposition "I have n
apples and 2 + n = 6" would have a different sense from the proposition "I have 4 apples".
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The sense of a proposition (or athought) isn't anything spiritual; it's what is given as an answer to a request
for an explanation of the sense. Or: one sense differs from another in the same way as the explanation of the one
differs from the explanation of the other. So also: the sense of one proposition differs from the sense of another in
the same way as the one proposition differs from the other.
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The sense of a proposition is not a soul.
Page 131
It is only in a language that something is a proposition. To understand a proposition is to understand a
language.
Page 131
A proposition is a sign in a system of signs. It is one combination of signs among a number of possible ones,
and as opposed to other possible ones. As it were one position of an indicator as opposed to other possible ones.
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"Go in the direction the arrow points."
"Go a hundred times as far as the arrow is long."
"Go as many paces as I draw arrows."
"Draw a copy of this arrow."
"Come at the time shown by this arrow considered as the hour hand of a clock."
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For all of these commands the same arrow might do.
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in contrast to is a different sign from in contrast to .
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VII
Page 132
85 Symbols appear to be of their nature unsatisfied.
Page 132
Wishes, conjectures, beliefs, commands appear to be something unsatisfied, something in need of
completion. Thus I would like to characterize my feeling of grasping a command as a feeling of an innervation. But
the innervation in itself isn't anything unsatisfied, it doesn't leave anything open, or stand in need of completion.
Page 132
And I want to say: "A wish is unsatisfied because it's a wish for something; opinion is unsatisfied, because
it's the opinion that something is the case, something real, something outside the process of opining."
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I would like to say: "my expectation is so made that whatever happens has to accord with it or not".
Page 132
The proposition seems set over us as a judge and we feel answerable to it.--It seems to demand that reality be
compared with it.
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I said that a proposition was laid against reality like a ruler. And a ruler--like all logical comparisons for a
proposition--is itself in a particular case a propositional sign. Now one would like to say: "Put the ruler against a
body: it does not say that the body is of such-and-such a length. Rather it is in itself dead and achieves nothing of
what thought achieves." It is as if we had imagined that the essential thing about a living being was the outward
form. Then we made a lump of wood in that form, and were abashed to see the stupid block, which hasn't even any
similarity to life.
Page 132
86 I want to say: "if someone could see the process of expectation, he would necessarily be seeing what was
expected."--But that is the case: if you see the expression of an expectation you see what is being expected. And in
what other way, in what other sense would it be possible to see it?
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When we give an order, it can look as if the ultimate thing sought by the order had to remain unexpressed, as
there is always a gulf between an order and its execution. Say I want someone to make a particular movement, say
to raise his arm. To make it quite clear, I do the movement. This picture seems unambiguous till we ask: how does he
know that he is to make that movement?--How does he know at all what use he is to make of the signs I give him,
whatever they are? Perhaps I shall now try to supplement the order by means of further signs, by pointing from
myself to him, making encouraging gestures etc. Here it looks as if the order were beginning to stammer.
Page 133
Suppose I wanted to tell someone to square the number 4, and did so by means of the schema:
Page 133
Now I'm tempted to say that the question mark only hints at something it doesn't express.
Page 133
As if the sign were precariously trying to produce understanding in us. But if we now understand it, by what
token do we understand?
Page 133
The appearance of the awkwardness of the sign in getting its meaning across, like a dumb person who uses
all sorts of suggestive gestures--this disappears when we remember that the sign does its job only in a grammatical
system.
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(In logic what is unnecessary is also useless.)
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87 In what sense can one call wishes as such, beliefs, expectations etc. 'unsatisfied'? What is the prototype of
nonsatisfaction from which we take our concept? Is it a hollow space? And would one call that unsatisfied?
Wouldn't this be a metaphor too? Isn't what we call nonsatisfaction a feeling--say hunger?
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Page 134
In a particular system of expressions we can describe an object by means of the words "satisfied" and
"unsatisfied". For example, if we lay it down that we call a hollow cylinder an "unsatisfied cylinder" and the solid
cylinder that fills it its "satisfaction".
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It seems as if the expectation and the fact satisfying the expectation fitted together somehow. Now one
would like to describe an expectation and a fact which fit together, so as to see what this agreement consists in. Here
one thinks at once of the fitting of a solid into a corresponding hollow. But when one wants to describe these two
one sees that, to the extent that they fit, a single description holds for both. (On the other hand compare the meaning
of: "These trousers don't go with this jacket"!)
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Expectation is not related to its satisfaction in the same way as hunger is related to its satisfaction. I can
describe the hunger, and describe what takes it away, and say that it takes it away. And it isn't like this either: I have
a wish for an apple, and so I will call 'an apple' whatever takes away the wish.
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88 The strange thing is expressed in the fact that if this is the event I expected, it isn't distinct from the one I
expected.
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I say: "that's just how I imagined it"; and someone says something like "That's impossible, because the one
was an image and the other isn't. Did you take your image for reality?"
Page 134
I see someone pointing a gun and say "I expect a report". The shot is fired.--Well, that was what you
expected, so did that bang somehow already exist in your expectation? Or is it just that there is some other kind of
agreement between your expectation and what occurred; that that noise was not contained in your expectation and
merely accidentally supervened when the expectation was being fulfilled? But no, if the noise had not occurred, my
expectation
Page Break 135
would not have been fulfilled; the noise fulfilled it; it was not an accompaniment of the fulfilment like a second guest
accompanying the one I expected.--Was the thing about the event that was not in the expectation too an accident, an
extra provided by fate?--But then what was not an extra? Did something of the shot already occur in my
expectation?--Then what was extra? for wasn't I expecting the whole shot?
Page 135
"The report was not so loud as I had expected." "Then was there a louder bang in your expectation?"
Page 135
"The red which you imagine is surely not the same (the same thing) as the red which you see in front of you;
so how can you say that it is what you imagined?"--But haven't we an analogous case with the propositions "Here is
a red patch" and "Here there isn't a red patch"? The word "red" occurs in both; so this word cannot indicate the
presence of something red. The word "red" does its job only in the propositional context. Doesn't the
misunderstanding consist in taking the meaning of the word "red" as being the sense of a sentence saying that
something is red?
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The possibility of this misunderstanding is also contained in the ambiguity of expressions like "the colour red
as the common element of two states of affairs"--This may mean that in each something is red, has the colour red; or
else that both propositions are about the colour red.
Page 135
What is common in the latter case is the harmony between reality and thought to which indeed a form of our
language corresponds.
Page 135
89 If we say to someone "imagine the colour red" he is to imagine a patch or something that is red, not one that is
green, since that is not red.
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(Could one define the word "red" by pointing to something that was not red? That would be as if one were
supposed to explain
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the word "modest" to someone whose English was weak, and one pointed to an arrogant man and said "That man is
not modest". That it is ambiguous is no argument against such a method of definition. Any definition can be
misunderstood. But it might well be asked: are we still to call this "definition"?--For, of course, even if it has the
same practical consequences, the same effect on the learner, it plays a different part in the calculus from what we
ordinarily call "ostensive definition" of the word "red".)
Page 136
It would be odd to say: "A process looks different when it happens from when it doesn't happen." Or "a red
patch looks different when it is there from when it isn't there; but language abstracts from this difference, for it
speaks of a red patch whether it is there or not."
Page 136
Reality is not a property still missing in what is expected and which accedes to it when one's expectation is
fulfilled.--Nor is reality like the daylight that things need to acquire colour, when they are already there, as it were
colourless, in the dark.
Page 136
"How do you know that you are expecting a red patch; that is, how do you know that a red patch is the
fulfilment of your expectation?" But I might just as well ask: "how do you know that that is a red patch?"
Page 136
How do you know that what you did really was to recite the alphabet in your head?--But how do you know
that what you are reciting aloud really is the alphabet?
Page 136
Of course that is the same question as "How do you know that what you call 'red' is really the same as what
another calls 'red'?" And in its metaphysical use the one question makes no more sense than the other.
Page 136
In these examples, I would like to say, you see how the words are really used.
Page 136
90 One might think: What a remarkable process willing must be,
Page Break 137
if I can now will the very thing I won't be doing until five minutes hence!
Page 137
How can I expect the event, when it isn't yet there at all?
Page 137
"Socrates: so if someone has an idea of what is not, he has an idea of nothing?--Theaetetus: It seems so.
Socrates: But surely if he has an idea of nothing, then he hasn't any idea at all?--Theaetetus: That seems plain."†1
Page 137
If we put the word "kill", say, in place of "have an idea of" in this argument, then there is a rule for the use of
this word: it makes no sense to say "I am killing something that does not exist". I can imagine a stag that is not there,
in this meadow, but not kill one that is not there. And "to imagine a stag in this meadow" means to imagine that a
stag is there. But to kill a stag does not mean to kill that... But if someone says "in order for me to be able to imagine
a stag it must after all exist in some sense", the answer is: no, it does not have to exist in any sense. And if it should
be replied: "But the colour brown at any rate must exist, for me to be able to have an idea of it"--then we can say 'the
colour brown exists' means nothing at all; except that it exists here or there as the colouring of an object, and that is
not necessary in order for me to be able to imagine a brown stag.
Page 137
We say that the expression of expectation 'describes' the expected fact and think of an object or complex
which makes its appearance as fulfilment of the expectation.--But it is not the expected thing that is the fulfilment,
but rather: its coming about.
Page 137
The mistake is deeply rooted in our language: we say "I expect him" and "I expect his arrival".
Page 137
It is difficult for us to shake off this comparison: a man makes his appearance--an event makes its
appearance. As if an event even now stood in readiness before the door of reality and were then to make its
appearance in reality--like coming into a room.
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Page 138
91 I can look for him when he is not there, but not hang him when he is not there.
Page 138
One might want to say: "But he must be somewhere there if I am looking for him."--Then he must be
somewhere there too if I don't find him and even if he doesn't exist at all.
Page 138
A search for a particular thing (e.g. my stick) is a particular kind of search, and differs from a search for
something else because of what one does (says, thinks) while searching, not because of what one finds.
Page 138
Suppose while I am searching I carry with me a picture or an image--very well. If I say that the picture is a
picture of what I am looking for, that merely tells the place of the picture in the process of searching. And if I find it
and say "There it is! That's what I was looking for" those words aren't a kind of definition of the name of the object
of the search (e.g. of the words "my stick"), a definition that couldn't have been given until the object had been
found.
Page 138
"You were looking for him? You can't even have known if he was there!" (Contrast looking for the trisection
of the angle.)
Page 138
One may say of the bearer of a name that he does not exist; and of course that is not an activity, although
one may compare it with one and say: he must be there all the same, if he does not exist. (And this has certainly
already been written some time by a philosopher.)
Page 138
The idea that it takes finding to show what we were looking for, and fulfilment of a wish to show what we
wanted, means one is judging the process like the symptoms of expectation or search in someone else. I see him
uneasily pacing up and down his room; then someone comes in at the door and he relaxes and gives signs of
satisfaction. And I say "obviously he was expecting this person."
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The symptoms of expectation are not the expression of expectation.
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Page 139
One may have the feeling that in the sentence "I expect he is coming" one is using the words "he is coming"
in a different sense from the one they have in the assertion "he is coming". But if it were so, how could I say that my
expectation had been fulfilled? And the words "he is coming" mean the same in the expression of expectation as in
the description of its fulfilment, because if I wanted to explain the words "he" and "is coming", say by means of
ostensive definitions, the same definitions of these words would go for both sentences.
Page 139
But it might now be asked: what's it like for him to come?--The door opens, someone walks in, and so
on.--What's it like for me to expect him to come?--I walk up and down the room, look at the clock now and then,
and so on. But the one set of events has not the smallest similarity to the other! So how can one use the same words
in describing them? What has become now of the hollow space and the corresponding solid?
Page 139
But perhaps I say as I walk up and down: "I expect he'll come in." Now there is a similarity somewhere. But
of what kind?!
Page 139
But of course I might walk up and down in my room and look at the clock and so on without expecting him
to come. I wouldn't describe doing that by saying "I expect he is coming". So what made it e.g. the expectation
precisely of him?
Page 139
I may indeed say: to walk restlessly up and down in my room, to look at the door, to listen for a noise is: to
expect N.--That is simply a definition of the expression "to expect N". Of course it isn't a definition of the word
"expect", because it doesn't explain what e.g. "to expect M" means. Well, we can take care of that; we say something
like: to expect X means to act as described and to utter the name "X" while doing so. On this definition the person
expected is the person whose name is uttered. Or I may give as a definition: to expect a person X is to do what I
described in the second example, and to make a drawing of a person. In that case, the person expected is the bearer
of the name X, the person who corresponds to the drawing.--That of course wouldn't explain
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what "to expect N to go" means, and I would have to give either an independent definition of that, or a general
definition including going and coming. And even that wouldn't explain say what "to expect a storm" means; etc. etc.
Page 140
What characterizes all these cases is, that the definition can be used to read off the object of the expectation
from the expectant behaviour. It isn't a later experience that decides what we are expecting.
Page 140
And I may say: it is in language that expectation and its fulfilment make contact.
Page 140
So in this case the behaviour of the expectant person is behaviour which can be translated in accordance
with given rules into the proposition "He is expecting it to happen that p". And so the simplest typical example to
illustrate this use of the word "expect" is that the expectation of its happening that p should consist in the expectant
person saying "I expect it to happen that p". Hence in so many cases it clarifies the grammatical situation to say: let
us put the expression of expectation in place of the expectation, the expression of the thought in place of the
thought.
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93 One can conceive expectation as expectant, preparatory behaviour. Expectation is like a player in a ball game
holding his hands in the right position to catch the ball. The expectation of the player might consist in his holding
out his hands in a particular way and looking at the ball.
Page 140
Some will perhaps want to say: "An expectation is a thought". Obviously, that corresponds to one use of the
word "expect". And we need to remember that the process of thinking may be very various.†1
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And if expectation is the thought "I am expecting it to happen that p" it is senseless to say that I won't
perhaps know until later what I expected.
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Something analogous might be said of wishing, fear and hope. (Plato called hope "a speech"†1).
Page 141
But it is different if hunger is called "a wish", say the body's wish for food to satisfy it. For it is a hypothesis
that just that will satisfy the wish; there's room for conjecture and doubt on the topic.
Page 141
Similarly if what I call "expectation" is a feeling, say a feeling of disquiet or dissatisfaction. But of course
these feelings are not thoughts in an amorphous form.
Page 141
The idea of thought as an unexplained process in the human mind makes it possible to imagine it turned into
a persistent amorphous condition.
Page 141
If I say "I have been expecting him all day", "expect" here doesn't mean a persistent condition including as
ingredients the person expected and his arrival, in the way that a dough may contain flour, sugar and eggs mixed
into a paste. What constitutes expectation is a series of actions, thoughts and feelings.
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94 When I expect someone,--what happens? I perhaps look at my calendar and see his name against today's date
and the note "5 p.m." I say to someone else "I can't come to see you today, because I'm expecting N". I make
preparations to receive a guest. I wonder "Does N smoke?", I remember having seen him smoke and put out
cigarettes. Towards 5 p.m. I say to myself "Now he'll come soon", and as I do so I imagine a man looking like N;
then I imagine him coming into the room and my greeting him and calling him by his name. This and many other
more or less similar trains of events are called "expecting N to come".
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Page 142
But perhaps I'm also prepared to say "I have been expecting N" in a case where the only thing that connects
him with my expectant activity is for instance that on a particular day I prepare a meal for myself and one other
person, and that N. has announced his intention of taking that meal with me.
Page 142
What does the process or state of wanting an apple consist in? Perhaps I experience hunger or thirst or both,
and meanwhile imagine an apple, or remember that I enjoyed one yesterday; perhaps I say "I would like to eat an
apple"; perhaps I go and look in a cupboard where apples are normally kept. Perhaps all these states and activities
are combined among themselves and with others.
Page 142
95 The same sort of thing must be said of intention. If a mechanism is meant to act as a brake, but for some reason
does not slow down the motion of the machine, then the purpose of the mechanism cannot be found out
immediately from it and from its effect. If you were to say "that is a brake, but it doesn't work" you would be talking
about intention. But now suppose that whenever the mechanism didn't work as a brake a particular person became
angry. Wouldn't the intention of the mechanism now be expressed in its effect? No, for now it could be said that the
lever sometimes triggers the brake and sometimes triggers the anger. For how does it come out that the man is angry
because the lever doesn't operate the brake? "Being annoyed that the apparatus does not function" is itself
something like "wishing that it did function in that way".--Here we have the old problem, which we would like to
express in the following way: "the thought that p is the case doesn't presuppose that it is the case; yet on the other
hand there must be something in the fact that is a presupposition even of having the thought (I can't think that
something is red, if the colour red does not exist)". It is the problem of the harmony between world and thought.--To
this it may be replied that thoughts are in the same
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space as the things that admit of doubt; they are laid against them in the same way as a ruler is laid against what is to
be measured.
Page 143
What I really want to say is this: the wish that he should come is the wish that really he should really come. If
a further explanation of this assurance is wanted, I would go on to say "and by 'he' I mean that man there, and by
'come' I mean doing this..." But these are just grammatical explanations, explanations which create language.
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It is in language that it's all done.
Page 143
"I couldn't think that something is red if red didn't exist." What that proposition really means is the image of
something red, or the existence of a red sample as part of our language. But of course one can't say that our
language has to contain such a sample; if it didn't contain it, it would just be another, a different language. But one
can say, and emphasize, that it does contain it.
Page 143
96 It's beginning to look somehow as if intention could never be recognized as intention from outside; as if one
must be doing the meaning of it oneself in order to understand it as meaning. That would amount to considering it
not as a phenomenon or fact but as something intentional which has a direction given to it. What this direction is, we
do not know; it is something which is absent from the phenomenon as such.
Page 143
Here, of course, our earlier problem returns, because the point is that one has to read off from a thought that
it is the thought that such and such is the case. If one can't read it off (as one can't read off the cause of a
stomach-ache) then it is of no logical interest.
Page 143
My idea seems nonsensical if it is expressed like this. It's supposed to be possible to see what someone is
thinking of by opening up his head. But how is that possible? The objects he's thinking about are certainly not in his
head--any more than in his thoughts!
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Page 144
If we consider them 'from outside' we have to understand thoughts as thoughts, intentions as intentions and
so on, without getting any information about something's meaning. For it is with the phenomenon of thinking that
meaning belongs.
Page 144
If a thought is observed there can be no further question of an understanding; for if the thought is seen it
must be recognized as a thought with a certain content; it doesn't need to be interpreted!--That really is how it is;
when we are thinking, there isn't any interpretation going on.
Page 144
97 If I said "but that would mean considering intention as something other than a phenomenon" that would make
intention reminiscent of the will as conceived by Schopenhauer. Every phenomenon seems dead in comparison with
the living thought.
Page 144
"Intention seen from outside" is connected with the question whether a machine could think. "Whatever
phenomenon we saw, it couldn't ever be intention; for that has to contain the very thing that is intended, and any
phenomenon would be something complete in itself and unconcerned with anything outside itself, something
merely dead if considered by itself."
Page 144
This is like when we say: "The will can't be a phenomenon, for whatever phenomenon you take is something
that simply happens, something we undergo, not something we do. The will isn't something I see happen, it's more
like my being involved in my actions, my being my actions." Look at your arm and move it and you will experience
this very vividly: "You aren't observing it moving itself, you aren't having an experience--not just an experience,
anyway--you're doing something." You may tell yourself that you could also imagine exactly the same thing
happening to your hand, but merely observed and not willed by you. But shut your eyes, and move your arm so that
you have, among other things, a certain experience: now ask yourself whether you still can imagine that you were
having the same experience but without willing it.
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Page 145
If someone wants to express the distinction between voluntary and involuntary movements by saying that
voluntary movements of the arm, for example, are differentiated from involuntary ones by a feeling of innervation,
you feel an urge to say "But I don't undergo this experience, I do it --" But can one speak of a distinction between
undergoing and doing in the case of an experience of innervation? I would like to say: "If I will, then there isn't
anything that happens to me, neither the movement nor a feeling; I am the agent." Very well; but there's no doubt
that you also have experiences when you voluntarily move your arm; because you see (and feel) it moving whether
or not you take up the attitude of an observer. So just for once try to distinguish between all the experiences of
acting plus the doing (which is not an experience) and all those experiences without the element of doing. Think
over whether you still need this element, or whether it is beginning to appear redundant.--Of course you can say
correctly that when you do something, there isn't anything happening to you, because the phenomena of doing are
different from the phenomena of observing something like a reflex movement. But this doesn't become clear until
one considers the very different sorts of things that people call voluntary activities and that people call unintentional
or involuntary processes in our life. (More about this in another place.)
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98 By "intention" I mean here what uses a sign in a thought. The intention seems to interpret, to give the final
interpretation; which is not a further sign or picture, but something else, the thing that cannot be further interpreted.
But what we have reached is a psychological, not a logical terminus.
Page 145
Think of a sign language, an 'abstract' one, I mean one that is strange to us, in which we do not feel at home,
in which, as we should say, we do not think (we used a similar example once before), and let us imagine this
language interpreted by a translation into--as we should like to say--an unambiguous picture-language, a language
consisting of pictures painted in perspective. It is quite
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clear that it is much easier to imagine different interpretations of the written language than of a picture painted in
the usual way depicting say a room with normal furniture. Here we shall also be inclined to think that there is no
further possibility of interpretation.
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Here we might also say we didn't enter into the sign-language, but did enter into the painted picture.
Page 146
(This is connected with the fact that what we call a 'picture by similarity' is not a picture in accordance with
some established method of projection. In this case the "likeness" between two objects means something like the
possibility of mistaking one for the other.)
Page 146
"Only the intended picture reaches up to reality like a yardstick. Looked at from outside, there it is, lifeless
and isolated."--It is as if at first we looked at a picture so as to enter into it and the objects in it surrounded us like
real ones; and then we stepped back, and were now outside it; we saw the frame, and the picture was a painted
surface. In this way, when we intend, we are surrounded by our intention's pictures and we are inside them. But
when we step outside intention, they are mere patches on a canvas, without life and of no interest to us. When we
intend, we exist among the pictures (shadows) of intention, as well as with real things. Let us imagine we are sitting
in a darkened cinema and entering into the happenings in the film. Now the lights are turned on, though the film
continues on the screen. But suddenly we see it "from outside" as movements of light and dark patches on a screen.
Page 146
(In dreams it sometimes happens that we first read a story and then are ourselves participants in it. And after
waking up after a dream it is sometimes as if we had stepped back out of the dream and now see it before us as an
alien picture.) And it also means something to speak of "living in the pages of a book". That is
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connected with the fact that our body is not at all essential for the occurrence of our experience. (Cf. eye and visual
field.)
Page 147
(Compare also the remark: if we understand a sentence, it has a certain depth for us.)
Page 147
99 What happens is not that this symbol cannot be further interpreted, but: I do no interpreting. I do not interpret
because I feel natural in the present picture. When I interpret, I step from one level of my thought to another.
Page 147
If I see the thought symbol "from outside", I become conscious that it could be interpreted thus or thus; if it
is a step in the course of my thoughts, then it is a stopping-place that is natural to me, and its further interpretability
does not occupy (or trouble) me. As I have a railway time-table and use it without being concerned with the fact that
a table can be interpreted in various ways.
Page 147
When I said that my image wouldn't be a portrait unless it bore the name of its subject, I didn't mean that I
have to imagine it and his name at the same time. Suppose I say something like: "What I see in my mind isn't just a
picture which is like N (and perhaps like others too). No, I know that it is him, that he is the person it portrays." I
might then ask: when do I know that and what does knowing it amount to? There's no need for anything to take
place during the imagining that could be called "knowing" in this way. Something of that sort may happen after the
imagining; I may go on from the picture to the name, or perhaps say that I imagined N, even though at the time of
the imagining there wasn't anything, except a kind of similarity, to characterize the image as N's. Or again there
might be something preceding the image that made the connection with N. And so the interpretation isn't something
that accompanies the image; what gives the image its interpretation is the path on which it lies.
Page 147
That all becomes clearer if one imagines images replaced by
Page Break 148
drawings, if one imagines people who go in for drawing instead of imagining.
Page 148
100 If I try to describe the process of intention, I feel first and foremost that it can do what it is supposed to only by
containing an extremely faithful picture of what it intends. But further, that that too does not go far enough, because
a picture, whatever it may be, can be variously interpreted; hence this picture too in its turn stands isolated. When
one has the picture in view by itself it is suddenly dead, and it is as if something had been taken away from it, which
had given it life before. It is not a thought, not an intention; whatever accompaniments we imagine for it, articulate or
inarticulate processes, or any feeling whatsoever, it remains isolated, it does not point outside itself to a reality
beyond.
Page 148
Now one says: "Of course it is not the picture that intends, but we who use it to intend." But if this intending,
this meaning, is something that is done with the picture, then I cannot see why that has to involve a human being.
The process of digestion can also be studied as a chemical process, independently of whether it takes place in a
living being. We want to say "Meaning is surely essentially a mental process, a process of consciousness and life,
not of dead matter." But what will give such a thing the specific character of what goes on?--so long as we speak of it
as a process. And now it seems to us as if intending could not be any process at all, of any kind whatever.--For what
we are dissatisfied with here is the grammar of process, not the specific kind of process.--It could be said: we should
call any process 'dead' in this sense.
Page 148
Let's say the wish for this table to be a little higher is the act of my holding my hand above the table at the
height I wish it to be. Now comes the objection: "The hand above the table can't be the wish: it doesn't express that
the table is to be higher; it is
Page Break 149
where it is and the table is where it is. And whatever other gesture I made it wouldn't make any difference."
Page 149
(It might almost be said: "Meaning moves, whereas a process stands still.")
Page 149
101 However, if I imagine the expression of a wish as the act of wishing, the problem appears solved, because the
system of language seems to provide me with a medium in which the proposition is no longer dead.
Page 149
If we imagine the expression of a wish as the wish, it is rather as if we were led by a train of thought to
imagine something like a network of lines spread over the earth, and living beings who moved only along the lines.
Page 149
But now someone will say: even if the expression of the wish is the wish, still the whole language isn't
present during this expression, yet surely the wish is!
Page 149
So how does language help? Well, it just isn't necessary that anything should be present except the
expression.
Page 149
102 You might as it were locate (look up) all of the connections in the grammar of the language. There you can see
the whole network to which the sentence belongs.
Page 149
Suppose we're asked "When we're thinking, meaning and so on why don't we come upon the bare picture?"
We must tell ourselves that when we're thinking we don't wonder whether the picture is the thought or the meaning,
we simply use pictures, sentences and so on and discard them one after the other.
Page 149
But of course if you call the picture the wish (e.g. that this table were higher) then what you're doing is
comparing the picture with an expression of our language, and certainly it doesn't correspond to such an expression
unless it's part of a system translatable into our language.
Page 149
One says: how can this way of holding the hand, this picture
Page Break 150
be the wish that such and such were the case? It is nothing more than a hand over a table, and there it is, alone and
without a sense. Like a single bit of scenery from the production of a play which has been left by itself. It had life
only in the play.
Page 150
In the gesture we don't see the real shadow of the fulfilment, the unambiguous shadow that admits of no
further interpretation.
Page 150
We ask: "does the hand above a table wish?" Does anything, spiritual or material, that we might add, wish? Is
there any such situation or process that really contains what is wished?--And what is our paradigm of such
containing? Isn't it our language? Where are we to find what makes the wish this wish, even though it's only a wish?
Nowhere but in the expressed wish.
Page 150
"After all, the wish must show what is wished, it must prefigure in the realm of wishes that which is wished."
But what actual process do you have in mind here as the prefiguring? (What is the mirror in which you think you
saw what was wished?)
Page 150
"The gesture tries to prefigure" one wants to say "but it can't".
Page 150
103 Can one say that while I'm wishing my wish seems to prefigure the fulfilment? While I'm wishing it doesn't
seem to do anything; I notice nothing odd about it. It's only considering the linguistic manifestation of the wish that
produces this appearance.
Page 150
We are considering an event that we might call an instance of the wish that this table were higher. But this
event doesn't even seem to contain the fulfilment. Now someone says: "But this event does have to be a shadow of
the very state of affairs that is wished, and
Page Break 151
these actions aren't that." But why do you say that's what a wish has to be? "Well, because it's the wish that just that
were the case". Precisely: that's the only answer you can give to the question. So after all that event is the shadow,
insofar as it corresponds within a system to the expression of the wish in the word-language. (It is in language that
wish and fulfilment meet.) Remember that the expression of a wish can be the wish, and that the expression doesn't
derive its sense from the presence of some extraordinary spirit.
Page 151
Think also of a case very similar to the present one: "This table isn't 80 cm high". Must the fact that it is 90
cm, and so not 80 cm, high contain the shadow of the fact of its being 80 cm high? What gives this impression?
When I see a table which is 90 cm high does it give a shadowy impression of having a height it doesn't have?
Page 151
This is rather as if we misunderstood the assertion " ~ p" in such a way as to think that it contained the
assertion " p", rather as " p.q" contains in its sense " p".
Page 151
Someone describes to me what went on when he, as he says, had the wish that the table were 10 cm higher.
He says that he held his hand 10 cm above the table. I reply "But how do you know that you weren't just wishing
that the table were higher, since in that case too you would have held your hand at some height above the table." He
says "After all, I must know what I wished" I reply "Very well, but I want to know by what token you remember
when you remember your wish. What happened when you wished, and what makes you say you wished just that?"
He says "I know that I intentionally held my hand just 10 cm above." I say "But what constituted just that
intention?"--I might also ask "Is it certain that when you were wishing you were using the scale 1:1? How do you
know that?"
Page 151
If he had described the process of wishing by saying "I said 'I would like to have the table 10 cm higher'",
then the question
Page Break 152
how he could know what he wished wouldn't have arisen. (Unless someone had gone on to ask: "And did you mean
those words in the way they are usually meant?".)
Page 152
What it always comes to in the end is that without any further meaning he calls what happened the wish that
that should happen. [Manifestation, not description.]
Page 152
"How do I know it's him I'm remembering, if the remembering is a picture?" To what extent do I know it?
("When two men look perfectly alike, how can I remember one of them in particular?")
Page 152
104 We say "A proposition isn't just a series of sounds, it is something more". We think of the way a Chinese
sentence is a mere series of sounds for us, which just means that we don't understand it, and we say this is because
we don't have any thoughts in connection with the Chinese sentence (e.g. the Chinese word for "red" doesn't call up
any image in us). "So what distinguishes a significant sentence from mere sounds is the thoughts it evokes." The
sentence is like a key-bit whose indentations are constructed to move levers in the soul in a particular way. The
sentence, as it were, plays a melody (the thought) on the instrument of the soul. But why should I now hypothesize,
in addition to the orderly series of words, another series of mental elements running parallel? That simply duplicates
language with something else of the same kind.
Page 152
Suppose the sentence is: "This afternoon N went into the Senate House." The sentence isn't a mere noise for
me, it evokes an image of a man in the vicinity of the Senate House, or something similar. But the sentence and the
image aren't just a noise plus a faint image; calling up the image, and having certain other consequences, is
something as it were internal to the sentence; that is what its sense is. The image seems only a faint copy of the
sense, or shall we say, only a single view of the sense.--But what do I mean by this? Don't I just see the sentence as
part of a system of consequences?
Page Break 153
Page 153
Let us suppose that proposition evoked in me a very clear picture of N on the way to the Senate House and
that in the picture there could also be seen the setting sun ("evening") and a calendar with today's date. Suppose that
instead of letting the sentence call up this picture, I actually painted it and showed it to someone else as a means of
communication in place of the sentence. He might say of this too that it expressed a thought but needed to be
understood; what he would think of as an act of understanding would probably be a translation into word languages.
Page 153
"I arrive in Vienna on the 24th of December." They aren't mere words! Of course not: when I read them
various things happen inside me in addition to the perception of the words: maybe I feel joy, I have images, and so
on.--But I don't just mean that various more or less inessential concomitant phenomena occur in conjunction with
the sentence; I mean that the sentence has a definite sense and I perceive it. But then what is this definite sense?
Well, that this particular person, whom I know, arrives at such and such a place etc. Precisely: when you are giving
the sense, you are moving around in the grammatical background of the sentence. You're looking at the various
transformations and consequences of the sentence as laid out in advance; and so they are, in so far as they are
embodied in a grammar. (You are simply looking at the sentence as a move in a given game.)
Page 153
I said that it is the system of language that makes the sentence a thought and makes it a thought for us.
Page 153
That doesn't mean that it is while we are using a sentence that the system of language makes it into a thought
for us, because the system isn't present then and there isn't any need for anything to make the sentence alive for us,
since the question of being alive doesn't arise. But if we ask: "why doesn't a sentence strike us as isolated and dead
when we are reflecting on its essence, its sense, the thought etc." it can be said that we are continuing to move in the
system of language.
Page Break 154
Page 154
To match the words "I grasp the sense" or "I am thinking the thought of this sentence" you hypothesize a
process which unlike the bare propositional sign contains these consequences.
Page 154
105 "This queer thing, thought": but it does not strike us as queer while we are thinking it. It strikes us as queer
when we tell ourselves that it connects objects in the mind, because it is the very thought that this person is doing
that; or that it isn't a sign or a picture, because I would still have to know how they were meant in their turn; or that
thought isn't something dead, because for me what I think really happens.
Page 154
What is the source of this odd way of looking at things?
Page 154
What makes us think that a thought, or a proposition we think, contains the reality? It's that we're all ready to
pass from it to the reality, and we feel this transition as something already potentially contained in it (when, that is,
we reflect on it), because we say "that word meant him". We feel this transition as something just as legitimate as a
permitted move in a game.
Page 154
Thought does not strike us as mysterious while we are thinking, but only when we say, as it were
retrospectively: "How was that possible?" How was it possible for thought to deal with the very person himself? But
here I am merely being astonished by my own linguistic expression and momentarily misunderstanding it.
Page 154
Thought strikes us as mysterious. But not while we think. And we don't mean that it's psychologically
remarkable. It isn't only that we see it as an extraordinary way of producing pictures and signs, we actually feel as if
by means of it we had caught reality in our net.
Page 154
It isn't while we're looking at it that it seems a strange process; but when we let ourselves be guided by
language, when we look at what we say about it.
Page 154
We mistakenly locate this mystery in the nature of the process.
Page Break 155
(We interpret the enigma created by our misunderstanding as the enigma of an incomprehensible process.)
Page 155
106 "Thought is a remarkable process, because when I think of what will happen tomorrow, I am mentally already in
the future." If one doesn't understand the grammar of the proposition "I am mentally in the future" one will believe
that here the future is in some strange way caught in the sense of a sentence, in the meaning of words. Similarly
people think that the endless series of cardinal numbers is somehow before our mind's eye, whenever we can use
that expression significantly.
Page 155
"For me this portrait is him"? What does that mean? I have the same attitude to the portrait, as to the man
himself. For I do of course distinguish between him and his picture.
Page 155
A thought experiment comes to much the same as an experiment which is sketched or painted or described
instead of being carried out. And so the result of a thought experiment is the fictitious result of a fictitious
experiment.
Page 155
"The sense of this proposition was present to me." What was it that happened?
Page 155
"Only someone who is convinced can say that".--How does the conviction help him when he says it?--Is it
somewhere at hand by the side of the spoken expression? (Or is it masked by it, as a soft sound by a loud one, so
that it can, as it were, no longer be heard when one expresses it out loud?) What if someone were to say "In order to
be able to sing a tune from memory one has to hear it in one's mind and sing from that"?
Page 155
Try the following experiment: Say a sentence, perhaps "The weather is very fine today"; right, and now think
the thought of the sentence, but unadulterated, without the sentence.
Page Break 156
Page 156
107 "It looks as if intention could never be recognized as intention "from outside", as if one must be doing the
meaning of it oneself in order to understand it as meaning."†1
Page 156
Can one recognize stomach-ache as such "from outside"? What are stomach-aches "from outside"? Here
there is no outside or inside! Of course, in so far as meaning is a specific experience, one wouldn't call any other
experience "meaning". Only it isn't any remarkable feature of the sensation which explains the directionality of
meaning. And if we say "from outside intention cannot be recognised as intention etc." we don't want to say that
meaning is a special experience, but that it isn't anything which happens, or happens to us, but something that we
do, otherwise it would be just dead. (The subject--we want to say--does not here drop out of the experience but is so
much involved in it that the experience cannot be described.)
Page 156
It is almost as if one said: we can't see ourselves going hither and thither, because it is we who are doing the
going (and so we can't stand still and watch). But here, as so very often, we are suffering from an inadequate form of
expression, which we are using at the very time we want to shake it off. We clothe the protest against our form of
expression in an apparently factual proposition expressed in that very form. For if we say "we see ourselves going
thither" we mean simply that we see what someone sees when he is going himself and not what he sees if someone
else is going. And one does indeed have a particular visual experience if one is doing the going oneself.
Page 156
That is to say, what we are speaking of is a case in which contrary to experience the subject is linked like an
element in a chemical compound. But where do we get this idea from? The concept of living activity in contrast with
dead phenomena.
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Page 157
Imagine someone now saying: "going somewhere oneself isn't an experience".
Page 157
We want to say: "When we mean something, there isn't a dead picture (of any kind); it's as if we went up to
someone. We go up to what we mean."
Page 157
But here we're constructing a false contrast between experience and something else, as if experience
consisted of sitting still and letting pictures pass in front of one.
Page 157
"When one means, it is oneself doing the meaning"; similarly, it is oneself that does the moving. One rushes
forward oneself, and one can't simultaneously observe the rushing. Of course not.
Page 157
Yes, meaning something is like going up to someone.
Page 157
108 Fulfilment of expectation doesn't consist in this: a third thing happens which can be described otherwise than as
"the fulfilment of this expectation", i.e. as a feeling of satisfaction or joy or whatever it may be. The expectation that
something will be the case is the same as the expectation of the fulfilment of that expectation.
Page 157
Could the justification of an action as fulfilment of an order run like this: "You said 'bring me a yellow
flower', upon which this one gave me a feeling of satisfaction; that is why I have brought it"? Wouldn't one have to
reply: "But I didn't set you to bring me the flower which should give you that sort of feeling after what I said!"
Page 157
(I go to look for the yellow flower. Suppose that while I am looking a picture comes before my mind,--even
so, do I need it when I see the yellow flower--or another flower? If I say: "as soon as I see a yellow flower,
something as it were clicks into place in my memory"--rather like a lever into a cog in the striking mechanism of a
clock--can I foresee, or expect, this clicking into place any better than the yellow flower? Even if in a particular case
Page Break 158
it really is true that what I'm expecting isn't what I am looking for, but some other (indirect) criterion, that certainly
isn't an explanation of expectation.)
Page 158
But isn't the occurrence of what is expected always accompanied by a phenomenon of agreement (or
satisfaction?). Is this phenomenon something different from the occurrence of what is expected? If so, then I don't
know whether fulfilment is always accompanied by such a phenomenon.
Page 158
If I say: the person whose expectation is fulfilled doesn't have to shout out "yes, that is it" or the like--I may
be told: "Certainly, but he must know that the expectation is fulfilled." Yes, if the knowledge is part of its being
fulfilled. "Yes, but when someone has his expectation fulfilled, there's always a relaxation of tension!"--How do you
know that?
Page Break 159
VIII
Page 159
109 A description of language must achieve the same result as language itself. "For in that case I really can learn
from the proposition, from the description of reality, how things are in reality."--Of course it's only this that is called
description, or "learning how things are". And that is all that is ever said when we say that we learn from the
description how things are in reality.
Page 159
"From the order you get the knowledge of what you have to do. And yet the order only gives you itself, and
its effect is neither here nor there." But here we are simply misled by the form of expression of our language, when it
says "the knowledge of what you have to do" or "the knowledge of the action". For then it looks as if this
something, the action, is a thing which is to come into existence when the order is carried out, and as if the order
made us acquainted with this very thing by showing it us in such a way that it already in a certain sense brought it
into existence. (How can a command--an expectation--show us a man before he has come into the room?)
Page 159
Suppose someone says that one can infer from an order the action that obeys it, and from a proposition the
fact which verifies it. What on earth can one infer from a proposition apart from itself? How can one pull the action
out of the order before it takes place? Unless what is meant is a different form of description of the action, such as
say making a drawing, in accordance with the order, of what I'm to do. But even this further description isn't there
until I have drawn it; it doesn't have a shadowy existence in the order itself.
Page 159
Being able to do something seems like a shadow of the actual doing, just as the sense of a sentence seems
like the shadow of a fact, and the understanding of an order the shadow of its execution.
Page Break 160
In the order the fact as it were "casts its shadow before it"! But this shadow, whatever it may be, is not the event.
Page 160
The shadowy anticipation of the fact consists in our being able already to think that that very thing will
happen, which hasn't yet happened. Or, as it is misleadingly put, in our being now able to think of (or about) what
hasn't yet happened.
Page 160
110 Thinking plus its application proceeds step by step like a calculus.--However many intermediate steps I insert
between the thought and its application, each intermediate step always follows the previous one without any
intermediate link, and so too the application follows the last intermediate step. It is the same as when we want to
insert intermediate links between decision and action.
Page 160
The ambiguity of our ways of expressing ourselves: If an order were given us in code with the key for
translating in into English, we might call the procedure of constructing the English form of the order "derivation of
what we have to do from the code" or "derivation of what executing the order is". If on the other hand we act
according to the order, obey it, here too in certain cases one may speak of a derivation of the execution.
Page 160
We can't cross the bridge to the execution until we are there.
Page 160
111 It is as a calculus that thinking has an interest for us; not as an activity of the human imagination.
Page 160
It is the calculus of thought that connects with extra-mental reality.
Page 160
From expectation to fulfilment is a step in a calculation. Indeed, the relation between the calculation
and its result 625 is exactly the same as that between expectation
Page Break 161
and its fulfilment. Expectation is a picture of its fulfilment to exactly the same degree as this calculation is a picture
of its result, and the fulfilment is determined by the expectation to exactly the same degree as the result is
determined by the calculation.
Page 161
112 When I think in language, there aren't meanings going through my mind in addition to the verbal expressions;
the language is itself the vehicle of thought.
Page 161
In what sense does an order anticipate its execution? By ordering just that which later on is carried out? But
one would have to say "which later on is carried out, or again is not carried out". And that is to say nothing.
Page 161
"But even if my wish does not determine what is going to be the case, still it does so to speak determine the
theme of a fact, whether the fact fulfils the wish or not." We are--as it were--surprised, not at anyone's knowing the
future, but at his being able to prophesy at all (right or wrong).
Page 161
As if the mere prophecy, no matter whether true or false, foreshadowed the future; whereas it knows nothing
of the future and cannot know less than nothing.
Page 161
Suppose you now ask: then are facts defined one way or another by an expectation--that is, is it defined for
whatever event may occur whether it fulfils the expectation or not? The answer has to be: Yes, unless the expression
of the expectation is indefinite, e.g. by containing a disjunction of different possibilities.
Page 161
"The proposition determines in advance what will make it true." Certainly, the proposition "p" determines
that p must be the case in order to make it true; and that means:
Page 161
(the proposition p) = (the proposition that the fact p makes true). And the statement that the wish for it to be
the case that p is satisfied by the event p, merely enunciates a rule for signs:
Page Break 162
(the wish for it to be the case that p) = (the wish that is satisfied by the event p).
Page 162
Like everything metaphysical the harmony between thought and reality is to be found in the grammar of the
language.
Page Break 163
IX
Page 163
113 Here instead of harmony or agreement of thought and reality one might say: the pictorial character of thought.
But is this pictorial character an agreement? In the Tractatus I had said something like: it is an agreement of form.
But that is misleading.
Page 163
Anything can be a picture of anything, if we extend the concept of picture sufficiently. If not, we have to
explain what we call a picture of something, and what we want to call the agreement of the pictorial character, the
agreement of the forms.
Page 163
For what I said really boils down to this: that every projection must have something in common with what is
projected no matter what is the method of projection. But that only means that I am here extending the concept of
'having in common' and am making it equivalent to the general concept of projection. So I am only drawing
attention to a possibility of generalization (which of course can be very important).
Page 163
The agreement of thought and reality consists in this: if I say falsely that something is red, then, for all that, it
isn't red. And when I want to explain the word "red" to someone, in the sentence "That is not red", I do it by
pointing to something red.
Page 163
In what sense can I say that a proposition is a picture? When I think about it, I want to say: it must be a
picture if it is to show me what I am to do, if I am to be able to act in accordance with it. But in that case all you want
to say is that you act in accordance with a proposition in the same sense as you act in accordance with a picture.
Page 163
To say that a proposition is a picture gives prominence to certain features of the grammar of the word
"proposition".
Page 163
Thinking is quite comparable to the drawing of pictures.
Page Break 164
Page 164
But one can also say that what looks like an analogue of a proposition is actually a particular case of our
general concept. When I compared the proposition with a ruler, strictly speaking what I did was to take the use of a
ruler in making a statement of length as an example for all propositions.
Page 164
114 The sense of a proposition and the sense of a picture. If we compare a proposition with a picture, we must think
whether we are comparing it to a portrait (a historical representation) or to a genre-picture. And both comparisons
have point.
Page 164
Sentences in fiction correspond to genre-pictures.
Page 164
"When I look at a genre-picture, it 'tells' me something even though I don't believe (imagine) for a moment
that the people I see in it really exist, or that there have really been people in that situation."
Page 164
Think of the quite different grammar of the expressions:
Page 164
"This picture shows people in a village inn."
Page 164
"This picture shows the coronation of Napoleon."
Page 164
(Socrates: "And if you have an idea must it not be an idea of something?"--Theaetetus:
"Necessarily."--Socrates: "And if you have an idea of something, mustn't it be of something real?"--Theaetetus: "It
seems so").†1
Page 164
Does the picture tell me, for instance, "two people are sitting in an inn drinking wine?" Only if this
proposition somehow enters into the process of understanding outside the picture, say if I say which I look at the
picture "here two people are sitting etc." If the picture tells me something in this sense, it tells me words. But how far
does it declare itself in these words? After all, if reality is declaring itself via language, it is taking a long way round.
Page 164
So for the picture to tell me something it isn't essential that words should occur to me while I look at it;
because the picture was supposed to be the more direct language.
Page 164
Here it is important to realise that instead of a picture one might have considered a slice of material reality.
For although our
Page Break 165
attitude to a painted table derives historically from our attitude to real tables, the latter is not a part of the former.
Page 165
115 So what the picture tells me is itself.
Page 165
Its telling me something will consist in my recognizing in it objects in some sort of characteristic
arrangement. (If I say: "I see a table in this picture" then what I say characterizes the picture--as I said--in a manner
which has nothing to do with the existence of a 'real' table. "The picture shows me a cube" can e.g. mean: It contains
the form .)
Page 165
Asked "Did you recognize your desk when you entered your room this morning?"--I should no doubt say
"Certainly!" and yet it would be misleading to call what took place "a recognition". Certainly the desk was not
strange to me; I was not surprised to see it, as I should have been if another one had been standing there, or some
unfamiliar kind of object.
Page 165
"Something is familiar if I know what it is."
Page 165
"What does it mean: 'this object is familiar to me'?"--"Well, I know that it's a table." But that can mean any
number of things, such as "I know how it's used", "I know it looks like a table when it's opened out", "I know that it's
what people call 'a table'."
Page 165
What kind of thing is "familiarity"? What constitutes a view's being familiar to me? (The question itself is
peculiar; it does not sound like a grammatical question.)
Page 165
I would like to say: "I see what I see". And the familiarity can only consist in my being at home in what I see.
Page 165
116 "I see what I see": I say that because I don't want to give a name to what I see. I don't want to say "I see a
flower" because that
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presupposes a linguistic convention, and I want a form of expression that makes no reference to the history of the
impression.
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The familiarity consists in my recognizing that what I see is a flower. I may say: the utterance of the words
"that is a flower" is a recognition reaction; but the criterion for recognition isn't that I name the object correctly, but
that when I look at it I utter a series of sounds and have a certain experience. For that the sounds are the correct
English word, or that they are a word at all in any existent language, isn't part of my experience during the utterance.
Page 166
I want to exclude from my consideration of familiarity everything that is 'historical'. When that's been done
what remains is impressions (experiences, reactions). Even where language does enter into our experience, we don't
consider it as an existing institution.
Page 166
So the multiplicity of familiarity, as I understand it, is that of feeling at home in what I see. It might consist in
such facts as these: my glance doesn't move restlessly (inquiringly) around the object. I don't keep changing the way
I look at it, but immediately fix on one and hold it steady.
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I see the picture of a heavy coat and have a feeling of warmth and cosiness; I see the picture of a winter
landscape and shiver. These reactions, it might be said, are justified by earlier experience. But we aren't concerned
now about the history of our experiences or about any such justification.
Page 166
No one will say that every time I enter a room, my long familiar surroundings, there is enacted a recognition
of all that I see and have seen hundreds of times before.
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117 If we think of our understanding of a picture, of a genre
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picture say, we are perhaps inclined to assume that there is a particular phenomenon of recognition and that we
recognize the painted people as people, the painted trees as trees and so on.
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But when I look at a genre picture do I compare the painted people with real people etc.?
Page 167
So should I say that I recognize the painted people as painted people? And similarly real people as real
people?
Page 167
Of course there is a phenomenon of recognition in a case where it takes some sort of investigation to
recognize a drawing as a representation of a human being; but when I see a drawing immediately as the
representation of a human being, nothing of that kind happens.
Page 167
A picture of a human face is a no less familiar object than the human face itself. But there is no question of
recognition here.
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118 It is easy to have a false concept of the processes called "recognizing"; as if recognizing always consisted in
comparing two impressions with one another. It is as if I carried a picture of an object with me and used it to
perform an identification of an object as the one represented by the picture. Our memory seems to us to be the agent
of such a comparison, by preserving a picture of what has been seen before, or by allowing us to look into the past
(as if down a spy-glass).
Page 167
In most cases of recognition no such process takes place.
Page 167
Someone meets me in the street and my eyes are drawn to his face; perhaps I ask myself "who is that?";
suddenly the face begins to look different in a particular way, "it becomes familiar to me"; I smile, go up to him and
greet him by name; then we talk of the past and while we do so perhaps a memory image of him comes before my
mind, and I see him in a particular situation.
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Page 168
Perhaps someone will say: if I hadn't kept his image in my memory, I couldn't have recognized him. But here
he is either using a metaphor, or expressing a hypothesis.
Page 168
One might say: "What I saw was memory-laden."
Page 168
We say: "we couldn't use words at all, if we didn't recognize them and the objects they denote." If (because
of a faulty memory) we didn't recognize the colour green for what it is then we couldn't use the word "green". But
have we any sort of check on this recognition, so that we know that it is really a recognition? If we speak of
recognition, we mean that we recognize something as what, in accordance with other criteria, it is. "To recognize"
means "to recognize what is".
Page 168
119 Familiarity gives confirmation to what we see, but not by comparing it with anything else. It gives it a stamp, as
it were.
Page 168
On the other hand I would like to say: "what I see here in front of me is not any old shape seen in a particular
manner: what I see is my shoes, which I know, and not anything else". But here it is just that two forms of
expression fight against each other.
Page 168
This shape that I see--I want to say--is not simply a shape; it is one of the shapes I know; it is a shape marked
out in advance. It is one of those shapes of which I already had a pattern in me; and only because it corresponds to
such a pattern is it this familiar shape. (I as it were carry a catalogue of such shapes around with me, and the objects
portrayed in it are the familiar ones.)
Page 168
But my already carrying the pattern round with me would be only a causal explanation of the present
impression. It is like saying: this movement is made as easily as if it had been practised.
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Page 169
And it is not so much as if I were comparing the object with a picture set beside it, but as if the object
coincided with the picture. So I see only one thing, not two.
Page 169
120 We say: "This face has a quite particular expression", and look perhaps for words to characterise it.
Page 169
Here it is easy to get into that dead-end in philosophy, where one believes that the difficulty of the task
consists in this: our having to describe phenomena that are hard to get hold of, the present experience that slips
quickly by, or something of the kind. Where we find ordinary language too crude, and it looks as if we were having
to do, not with the phenomena of every-day, but with ones that "easily elude us, and in their coming to be and
passing away, produce those others as an average effect".
Page 169
And here one must remember that all the phenomena that now strike us as so remarkable are the very
familiar phenomena that don't surprise us in the least when they happen. They don't strike us as remarkable until we
put them in a strange light by philosophizing.
Page 169
121 "What the picture tells me is itself" is what I want to say. That is, its telling me something consists in its own
structure, in its own forms and colours.
Page 169
It is as if, e.g. "it tells me something" or "it is a picture" meant: it shows a certain combination of cubes and
cylinders.
Page 169
"It tells me something" can mean: it narrates something to me, it is a story.
Page 169
It tells me itself, just as a proposition, a story tells me itself.
Page 169
The concept of a narrative picture is surely like that of a genre picture (or a battle scene). If I wanted to
explain what a battle
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scene is, I wouldn't need to refer to any reality outside the picture, I would only have to talk about painted men,
painted horses, painted cannon and so on.
Page 170
"The picture tells me something": it uses words, so to speak: here are eyes, mouth, nose, hands etc. I am
comparing the picture to a combination of linguistic forms.
Page 170
The system of language, however, is not in the category of experience. The experiences characteristic of
using the system are not the system. (Compare: the meaning of the word "or" and the or-feeling).
Page 170
"Now, that series of signs tells me something; earlier, before I learnt the language, it said nothing to me". Let
us suppose what we mean by that is that the sentence is now read with a particular experience. Certainly, before I
learnt the language, that series of signs used not to make the same impression on me. Of course, if we disregard the
causal element, the impression is quite independent of the system of language.--And there is something in me that is
reluctant to say: the sentence's telling me something is constituted by its making this impression on me.
Page 170
"It's only in a language that something is a proposition" is what I want to say.
Page 170
122 'Language' is only languages, plus things I invent by analogy with existing languages. Languages are systems.
Page 170
"A proposition belongs to a language." But that just means: it is units of languages that I call "propositions".
Page 170
But we must pay attention to the use of the expression "English language", otherwise we shall ask questions
like "What is the language? Is it all the sentences which have so far been spoken? Or the set of rules and words? etc.
etc." What is the system? Where is it? What is chess? All the games that have been played? The list of rules?
Page 170
"A proposition is a unit of language." "After all, what constitutes
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propositions is the combination of words which might be otherwise combined." But that means: what constitutes
propositions for me. That is the way I regard language.
Page 171
What we want to attend to is the system of language.
Page 171
123 Certainly I read a story and don't give a hang about any system of language. I simply read, have impressions, see
pictures in my mind's eye, etc. I make the story pass before me like pictures, like a cartoon story. (Of course I do not
mean by this that every sentence summons up one or more visual images, and that that is, say, the purpose of a
sentence.)
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Let us imagine a picture story in schematic pictures, and thus more like the narrative in a language than a
series of realistic pictures. Using such a picture-language we might in particular e.g. keep our hold on the course of
battles. (Language-game.) And a sentence of our word-language approximates to a picture in this picture language
much more closely than we think.
Page 171
Let us remember too that we don't have to translate such pictures into realistic ones in
order to 'understand' them, any more than we ever translate photographs or film pictures into coloured pictures,
although black-and-white men or plants in reality would strike us as unspeakably strange and frightful.
Page 171
Suppose we were to say at this point: "Something is a picture only in a picture-language"?
Page 171
A sentence in a story gives us the same satisfaction as a picture.
Page 171
124 We can on the other hand imagine a language in whose use the impression made on us by the signs played no
part; in which there was no question of an understanding, in the sense of such an impression. The signs are e.g.
written and transmitted to us and we are able to take notice of them. (That is to say, the only impression that comes
in here is the pattern of the sign.) If the sign is an order,
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we translate it into action by means of rules, tables. It does not get as far as an impression like that of a picture; nor
are stories written in this language. But there is perhaps a kind of reading for entertainment which consists in certain
series of signs being translated into bodily movements to make a kind of dance. (Compare the remark about
translation and code.)
Page 172
In this case one really might say "the series of signs is dead without the system".
Page 172
We could of course also imagine that we had to use rules and translate a verbal sentence into a drawing in
order to get an impression from it. (That only the picture had a soul.)
Page 172
(I might say to my pupils: When you have been through these exercises you will think differently.)
Page 172
But even in our normal speech we may often quite disregard the impression made by a sentence so that all
that is important is how we operate with the sentence (Frege's conception of logic).
Page 172
"There is no such thing as an isolated proposition." For what I call a "proposition" is a position in the game of
language.
Page 172
Isn't what misleads us the fact that I can look ever so closely at a position in a game without discovering that
it is a position in a game? What misleads us here is something in the grammar of the expression "position in a
game".
Page 172
Thinking is an activity, like calculating. No one would call calculating, or playing chess, a state.
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125 Let us imagine a kind of puzzle picture: there is not one particular object to find; at first glance it appears to us as
a jumble
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of meaningless lines, and only after some effort do we see it as, say, a picture of a landscape.--What makes the
difference between the look of the picture before and after the solution? It is clear that we see it differently the two
times. But what does it amount to to say that after the solution the picture means something to us, whereas it meant
nothing before?
Page 173
We can also put this question like this: What is the general mark of the solution's having been found?
Page 173
I will assume that as soon as it is solved I make the solution obvious by strongly tracing certain lines in the
puzzle picture and perhaps putting in some shadows. Why do you call the picture you have sketched in a solution?
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a) Because it is the clear representation of a group of spatial objects.
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b) Because it is the representation of a regular solid.
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c) Because it is a symmetrical figure.
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d) Because it is a shape that makes an ornamental impression on me.
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e) Because it is the representation of a body I am familiar with.
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f) Because there is a list of solutions and this shape (this body) is on the list.
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g) Because it represents a kind of object that I am very familiar with; for it gives me an instantaneous
impression of familiarity, I instantly have all sorts of associations in connexion with it; I know what it is called; I
know I have often seen it; I know what it is used for etc.
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h) Because it represents a face which strikes me as familiar.
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i) Because it represents a face which I recognize: α) it is the face of my friend so and so; β) it is a face which I
have often seen pictures of.
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k) Because it represents an object which I remember having seen at some time.
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l) Because it is an ornament that I know well (though I don't remember where I have seen it).
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Page 174
m) Because it is an ornament that I know well; I know its name, I know where I have seen it.
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n) Because it represents part of the furniture of my room.
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o) Because I instinctively traced out those lines and now feel easy.
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p) Because I remember that this object has been described.
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q) Because I seem to be familiar with the object, a word occurs to me at once as its name (although the word
does not belong to any existent language); I tell myself "of course, that is an α such as I have often seen in β; one γs
δ's with it until they ε." Something of the kind occurs e.g. in dreams.
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And so on.
Page 174
(Anyone who does not understand why we talk about these things must feel what we say to be mere trifling.)
Page 174
126 The impression is one thing, and the impression's being determinate is another thing. What I call the impression
of familiarity is as multifarious as being determinate is.
Page 174
When we look into a human face that we know very well, we need not have any impression, our wits may be
completely dull, so to speak; and between that case and a strong impression there are any number of stages.
Page 174
Suppose the sight of a face has a strong effect on us, inspiring us say with fear. Am I to say: first of all there
must occur an impression of familiarity, the form of the human face as such must make an impression of familiarity
on me, and only then is the impression of fear added to that impression?--Isn't it like this, that what I call the
impression of specific familiarity is a characteristic of every strong impression that a face makes on me?--The
characteristic, say of determinacy. I did indeed say that the impression of familiarity consists in things like our
feeling at home in what we see, in our not changing our way of looking and the like.
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Page 175
127 Can I think away the impression of individual familiarity where it exists; and think it into a situation where it
does not? And what does that mean? I see e.g. the face of a friend and ask myself: what does this face look like if I
see it as an unfamiliar face (as if I were seeing it now for the first time)? What remains, as it were, of the look of this
face, if I think away, subtract, the impression of familiarity from it? Here I am inclined to say: "It is very difficult to
separate the familiarity from the impression of the face." But I also feel that this is a misleading way of putting
things. For I have no notion how I should so much as try to separate these two things. The expression "to separate
them" does not have any clear sense for me.
Page 175
I know what this means: "Imagine this table black instead of brown"; it means something like: "paint a
picture of this table, but black instead of brown"; or similarly: "draw this man but with longer legs than he has".
Page 175
Suppose someone were to say "Imagine this butterfly exactly as it is, but ugly instead of beautiful"?!
Page 175
"It is very difficult to think away ... ": here it looks as if it was a matter of a psychological difficulty, a
difficulty of introspection or the like. (That is true of a large range of philosophical problems: think of the problem of
the exact reproduction or description of what is seen in the visual field; of the description of the perpetual flux of
phenomena; also of "how many raindrops do you see, if you look at the rain?")
Page 175
Compare: "It is difficult to will that table to move from a distance."
Page 175
In this case we have not determined what thinking the familiarity away is to mean.
Page 175
It might mean, say, to recall the impression which I had when I saw the face for the first time. And here again
one must know what it means to "try" to remember the impression. For that has several meanings. Let us ask
ourselves what activities we call
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"trying to remember something". What do we do if we want to remember what we had for lunch yesterday? Is that
method available for the early memories of an adult? Can one try to remember one's own birth?
Page 176
I tell myself: I want to try to look at a printed English word and see it as if I hadn't learnt to read, as if the
black shapes on the paper were strange drawings whose purpose I couldn't imagine or guess. And then what
happens is that I can't look at the printed word without the sound of the word or of the letter I'm actually looking at
coming before my mind.
Page 176
For someone who has no knowledge of such things a diagram representing the inside of a radio receiver will
be a jumble of meaningless lines. But if he is acquainted with the apparatus and its function, that drawing will be a
significant picture for him.
Page 176
Given some solid figure (say in a picture) that means nothing to me at present--can I at will imagine it as
meaningful? That's as if I were asked: Can I imagine a body of any old shape as an appliance? But for what sort of
use?
Page 176
Well, at any rate one class of corporeal shapes can readily be imagined as dwellings for beasts or men.
Another class as weapons. Another as models of landscapes. Etc. etc. So here I know how I can ascribe meaning to
a meaningless shape.
Page 176
128 If I say that this face has an expression of gentleness, or kindness, or cowardice, I don't seem just to mean that
we associate such and such feelings with the look of the face, I'm tempted to say that the face is itself one aspect of
the cowardice, kindness, etc. (Compare e.g. Weininger). It is possible to say: I see cowardice in this face (and might
see it in another too) but at all events it doesn't seem to be merely associated, outwardly connected, with the face;
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the fear has the multiplicity of the facial features. And if, for example, the features change slightly, we can speak of a
corresponding change in the fear. If we were asked "Can you think of this face as an expression of courage
too?"--we should, as it were, not know how to lodge courage in these features. Then perhaps I say "I don't know
what it would mean if this is a courageous face." [This sentence cannot be corrected by saying "for this to be a
courageous face" instead of "if this is a courageous face"].†1 But what would an answer to such a question be like?
Perhaps one says: "Yes, now I understand: the face as it were shews indifference to the outer world." So we have
somehow read courage into the face. Now once more, one might say, courage fits this face. But what fits what here?
Page 177
There is a related case (though perhaps it will not seem so) when for example we (Germans) are surprised
that the French do not simply say "the man is good" but put an attributive adjective where there should be a
predicative one; and when we solve the problem for ourselves by saying: they mean: "the man is a good one".
Page 177
Couldn't different interpretations of a facial expression consist in my imagining each time a different kind of
sequel? Certainly that's often how it is. I see a picture which represents a smiling face. What do I do if I take the
smile now as a kind one, now as malicious? Don't I imagine it with a spatial and temporal context which I call kind
or malicious? Thus I might supply the picture with the fancy that the smiler was smiling down at a child at play, or
again on the suffering of an enemy.
Page 177
This is in no way altered by the fact that I can also take the at first sight gracious situation and interpret it
differently by putting it into a wider context.--If no special circumstances reverse my interpretation I shall conceive a
particular smile as kind, call it a "kind" one, react correspondingly.
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Page 178
That is connected with the contrast between saying and meaning.
Page 178
"Any expression can lie": but you must think what you mean by "lie". How do you imagine a lie? Aren't you
contrasting one expression with another? At any rate, you are contrasting with the expression some other process
which might very well be an expression.
Page 178
129 What does it mean: "to read kindness into the smile"? Perhaps it means: I make a face which is coordinated
with the smiling face in a particular way. I coordinate my face to the other one in some such way as to exaggerate
one or other of its features.
Page 178
A friendly mouth, friendly eyes. How would one think of a friendly hand? Probably open and not as a
fist.--And could one think of the colour of a man's hair as an expression of friendliness or the opposite? Put like that
the question seems to ask whether we can manage to. The question ought to run: Do we want to call anything a
friendly or unfriendly hair-colour? If we wanted to give such words a sense, we should perhaps imagine a man
whose hair darkened when he got angry. The reading of an angry expression into dark hair, however, would work
via a previously existent conception.
Page 178
It may be said: the friendly eyes, the friendly mouth, the wagging of a dog's tail, are among the primary and
mutually independent symbols of friendliness; I mean: They are parts of the phenomena that are called friendliness.
If one wants to imagine further appearances as expressions of friendliness, one reads these symbols into them. We
say: "He has a black look", perhaps because the eyes are more strongly shadowed by the eyebrows; and now we
transfer the idea of darkness to the colour of the hair. He has glowering hair. If I were asked whether I could imagine
a chair with a friendly expression, it would be above all a friendly facial expression I would want to imagine it with;
I would want to read a friendly face into it.
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Page 179
I say "I can think of this face (which at first gives an impression of timidity) as courageous too." We do not
mean by this that I can imagine someone with this face perhaps saving someone's life (that, of course, is imaginable
in connexion with any face). I am speaking rather of an aspect of the face itself. Nor do I mean that I can imagine
that this man's face might change so that, in the ordinary sense, it looked courageous; though I may very well mean
that there is quite a definite way in which it can change into a courageous face. The reinterpretation of a facial
expression can be compared with the reinterpretation of a chord in music, when we hear it as a modulation first into
this, then into that key. (Compare also the distinction between mixed colours and intermediary colours).
Page 179
Suppose we ask ourselves "what proper name would suit the character of this man"--and portray it in sound?
The method of projection we use for the portrayal is something which as it were stands firm. (A writer might ask
himself what name he wants to give to a person.) But sometimes we project the character into the name that has
been given. Thus it appears to us that the great masters have names which uniquely fit the character of their works.
Page 179
Experience of the real size. Suppose we saw a picture showing a chair-shape; we are told it represents a
construction the size of a house. Now we see it differently.
Page 179
What happens when we learn to feel the ending of a church mode as an ending?
Page 179
Think of the multifariousness of what we call "language". Word-language, picture-language,
gesture-language, sound-language.
Page 179
130 "'This object is familiar to me' is like saying 'this object is portrayed in my catalogue'." In that case it would
consist in the fact that it was a picture filed with others in a particular folder, in this drawer. But if that really is what I
imagine--if I think I simply
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compare the seen object with pictures in my catalogue and find it to agree with one of them--it is something quite
unlike the phenomenon of familiarity. That is, we are making the assumption that the picture in our catalogue is itself
familiar. If it were something strange, then the fact that it was in this folder, in this drawer, would mean nothing to
us.
Page 180
When I speak of a pattern in my mental catalogue, or of a sheath into which an object fits if it is familiar,
what I would like to say is that the sheath in my mind is, as it were, the "form of imagining", so that it isn't possible
for me to say of a pattern that it is in my mind unless it really is there.--The pattern as it were retires into my mind, so
that it is no longer presented to it as an object. But that only means: it didn't make sense to talk of a pattern at all.
(The spatial spectacles we can't take off.)
Page 180
If we represent familiarity as an object's fitting into a sheath, that's not quite the same as our comparing what
is seen with a copy. What we really have in mind is the feeling when the object slips smoothly into the contour of
the sheath. But that is a feeling we might have even if there were no such perfectly fitting sheath there at all.
Page 180
We might also imagine that every object had an invisible sheath; that alters nothing in our experience, it is an
empty form of representation.
Page 180
It shouldn't really be "Yes, I recognize it, it's a face" but "I recognize it, I see a face". (Here the word face
might mean for me the mere ornament and have no reference to the human face; it might be on a level with
any other familiar figure, e.g. a swastika.) For the question is: "What do I recognize as what?" For "to recognize a
thing as itself" is meaningless.
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Page 181
131 The comparison between memory and a notebook. On the one hand this comparison serves as a picture of the
conscious phenomena, and on the other hand it provides a psychological model. (And the word "conscious" is a
reference to a chapter of the grammar and is not one side of the psychological contrast between "conscious" and
"unconscious".)
Page 181
Many very different things happen when we remember.
Page 181
"Have you been in your room?" "Yes". "Are you sure?" "I would know if I hadn't been here yesterday!" For
this I don't need to see myself, even for a moment, in memory in my room. But let's assume that when I said that I
saw myself standing at the window in my room; how does the picture show me that it was yesterday? Of course, the
picture could show that as well, if for instance I saw in it a wall-calendar with yesterday's date. But if that wasn't the
case, how did I read off from the memory image, or from the memory, that I stood thus at the window yesterday?
How do I translate the experience of remembering into words?--But did I translate an experience into words? Didn't
I just utter the words in a particular tone of voice with other experiences of certainty? But wasn't that the experience
of remembering? (The experience of translating is the same kind of thing as the experience of the tone of voice.) But
what made you so certain when you spoke those words? Nothing made me certain; I was certain.
Page 181
Of course, I have other ways of checking--as one might say--what I then uttered. That is: I can now try to
remember particular things that happened yesterday and to call up pictures before my mind's eye etc. But certainly
that didn't have to have happened before I answered.
Page 181
When we narrate a set of events from memory we do sometimes see memory pictures in our mind; but
commonly they are only scattered through the memory like illustrations in a story book.
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Someone says to me "Imagine a patch of the colour called 'red' on this white wall". I do so--shall I now say
that I remembered
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which colour is called 'red'? When I talk about this table, do I remember that this object is called a 'table'?
Page 182
Mightn't someone object: "So if a man has not learned a language is he unable to have certain memories?" Of
course--he cannot have verbal memories, verbal wishes and so on. And memories etc. in language are not mere
threadbare representations of the real experiences; for is what is linguistic not an experience? (Words are deeds.)
Page 182
Some men recall a musical theme by having an image of the score rise before them, and reading it off.
Page 182
It could be imagined that what we call "memory" in some man consisted in his seeing himself looking things
up in a notebook in spirit, and that what he read in that book was what he remembered. (How do I react to a
memory?)
Page 182
Incidentally, when I treat the objects around me as familiar, do I think of that comparison? Of course not. I
only do so when I look at the act of recognition (individual recognition) after the event; and not so much when I
look at it to see what actually happened, as when I look at it through a preconceived schema. (The flux of time.)†1
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132 If one takes it as obvious that a man takes pleasure in his own fantasies, let it be remembered that fantasy does
not correspond to a painted picture, to a sculpture or a film, but to a complicated formation out of heterogeneous
components--words, pictures, etc. Then one will not contrast operating with written and spoken
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signs with operating with "imagination-pictures" of events.
Page 183
(The ugliness of a human being can repel in a picture, in a painting, as in reality, but so it can too in a
description, in words.)
Page 183
Attitude to a picture (to a thought). The way we experience a picture makes it real for us, that is, connects it
with reality; it establishes a continuity with reality.
Page 183
(Fear connects a picture with the terrors of reality.)
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X
Page 184
133 Can an ostensive definition come into collision with the other rules for the use of a word?--It might appear so;
but rules can't collide, unless they contradict each other. That aside, it is they that determine a meaning; there isn't a
meaning that they are answerable to and could contradict.
Page 184
Grammar is not accountable to any reality. It is grammatical rules that determine meaning (constitute it) and
so they themselves are not answerable to any meaning and to that extent are arbitrary.
Page 184
There cannot be a question whether these or other rules are the correct ones for the use of "not" (that is,
whether they accord with its meaning). For without these rules the word has as yet no meaning; and if we change
the rules, it now has another meaning (or none), and in that case we may just as well change the word too.
Page 184
"The only correlate in language to an intrinsic necessity is an arbitrary rule. It is the only thing which one can
milk out of this intrinsic necessity into a proposition."†1
Page 184
Why don't I call cookery rules arbitrary, and why am I tempted to call the rules of grammar arbitrary?
Because I think of the concept "cookery" as defined by the end of cookery, and I don't think of the concept
"language" as defined by the end of language. You cook badly if you are guided in your cooking by rules other than
the right ones; but if you follow other rules than those of chess you are playing another game; and if you follow
grammatical
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rules other than such and such ones, that does not mean you say something wrong, no, you are speaking of
something else.
Page 185
If I want to carve a block of wood into a particular shape any cut that gives it the right shape is a good one.
But I don't call an argument a good argument just because it has the consequences I want (Pragmatism). I may call a
calculation wrong even if the actions based on its result have led to the desired end. (Compare the joke "I've hit the
jackpot and he wants to give me lessons!"†1) That shows that the justifications in the two cases are different, and
also that "justification" means something different in each case. In the one case one can say "Just wait, you will soon
see that it will come out right (i.e. as desired)". In the other case that is no justification.
Page 185
The connection between the rules of cookery and the grammar of the word "cook" is not the same as that
between the rules of chess and the expression "play chess" or that between the rules of multiplication and the
grammar of the word "multiply".
Page 185
The rules of grammar are arbitrary in the same sense as the choice of a unit of measurement. But that means
no more than that the choice is independent of the length of the objects to be measured and that the choice of one
unit is not 'true' and of another 'false' in the way that a statement of length is true or false. Of course that is only a
remark on the grammar of the word "unit of length".
Page 185
134 One is tempted to justify rules of grammar by sentences like "But there are really four primary colours". And if
we say that the
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rules of grammar are arbitrary, that is directed against the possibility of this justification. Yet can't it after all be said
that the grammar of colour words characterizes the world as it actually is? One would like to say: May I not really
look in vain for a fifth primary colour? (And if looking is possible, then finding is conceivable.) Doesn't grammar put
the primary colours together because there is a kind of similarity between them? Or colours, anway, in contrast to
shapes or notes? Or, when I set this up as the right way of dividing up the world, have I a preconceived idea in my
head as a paradigm? Of which in that case I can say: "Yes, that is the way we look at things" or "We just do want to
form this sort of picture." For if I say "there is a particular similarity among the primary colours"--whence do I derive
the idea of this similarity? Just as the idea 'primary colour' is nothing else but 'blue or red or green or yellow' is not
the idea of that similarity too given simply by the four colours? Indeed, aren't these concepts the same? (For here it
can be said: "what would it be like if these colours did not have this similarity?") (Think of a group containing the
four primary colours plus black and white, or the visible colours plus ultraviolet and infrared.)
Page 186
I do not call rules of representation conventions if they can be justified by the fact that a representation made
in accordance with them will agree with reality. For instance the rule "paint the sky brighter than anything that
receives its light from it" is not a convention.
Page 186
The rules of grammar cannot be justified by shewing that their application makes a representation agree with
reality. For this justification would itself have to describe what is represented. And if something can be said in the
justification and is permitted by its grammar--why shouldn't it also be permitted by the grammar that I am trying to
justify? Why shouldn't both forms of expression
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have the same freedom? And how could what the one says restrict what the other can say?
Page 187
But can't the justification simply point to reality?
Page 187
How far is such pointing a justification? Does it have the multiplicity of a justification? Of course it may be
the cause of our saying one sentence rather than another. But does it give a reason for it? Is that what we call a
justification?
Page 187
No one will deny that studying the nature of the rules of games must be useful for the study of grammatical
rules, since it is beyond doubt there is some sort of similarity between them.--The right thing is to let the certain
instinct that there is a kinship lead one to look at the rules of games without any preconceived judgement or
prejudice about the analogy between games and grammar. And here again one should simply report what one sees
and not be afraid that one is undermining a significant and correct intuition, or, on the other hand, wasting one's time
with something superfluous.
Page 187
135 One can of course consider language as part of a psychological mechanism. The simplest case is if one uses a
restricted concept of language in which language consists only of commands.
Page 187
One can then consider how a foreman directs the work of a group of people by shouting.
Page 187
One can imagine a man inventing language, imagine him discovering how to train other human beings to
work in his place, training them through reward and punishment to perform certain tasks when he shouts. This
discovery would be like the invention of a machine.
Page 187
Can one say that grammar describes language? If we consider language as part of the psycho-physical
mechanism which we use
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when we utter words--like pressing keys on a keyboard--to make a human machine work for us, then we can say
that grammar describes that part of the machine. In that case a correct language would be one which would stimulate
the desired activities.
Page 188
Clearly I can establish by experience that a human being (or animal) reacts to one sign as I want him to, and
to another not. That e.g. a human being goes to the right at the sign "→" and goes to the left at the sign "←"; but that
he does not react to the sign " " as to "←".
Page 188
I do not even need to fabricate a case, I have only to consider what is in fact the case; namely, that I can
direct a man who has learned only German, only by using the German language. (For here I am looking at learning
German as adjusting (conditioning) a mechanism to respond to a certain kind of influence; and it may be all one to
us whether someone else has learned the language, or was perhaps from birth constituted to react to sentences in
German like a normal person who has learned it.)
Page 188
Suppose I now made the discovery that someone would bring me sugar at a sign plus the cry "Su", and
would bring me milk at a sign and the cry "Mi", and would not do so in response to other words. Should I say that
this shows that "Su" is the correct (the only correct) sign for sugar, "Mi" the correct sign for milk?
Page 188
Well, if I say that, I am not using the expression "sign for sugar" in the way it is ordinarily used or in the way
I intended to use it.
Page 188
I do not use "that is the sign for sugar" in the same way as the sentence "if I press this button, I get a piece of
sugar".
Page 188
136 All the same, let us compare grammar with a system of buttons, a keyboard which I can use to direct a man or a
machine by pressing
Page Break 189
different combinations of keys. What corresponds in this case to the grammar of language?
Page 189
It is easy to construct such a keyboard, for giving different "commands" to the machine. Let's look at a very
simple one: it consists of two keys, the one marked "go" and the other "come". Now one might think it must
obviously be a rule of the grammar that the two keys shouldn't be depressed simultaneously (that would give rise to
a contradiction). But what does happen if we press them both at the same time? Am I assuming that this has an
effect? Or that it has no effect? In each case I can designate the effect, or the absence of an effect, as the point and
sense of the simultaneous depression of both keys.
Page 189
Or: When I say that the orders "Bring me sugar" and "Bring me milk" make sense, but not the combination
"Milk me sugar", that does not mean that the utterance of this combination of words has no effect. And if its effect is
that the other person stares at me and gapes, I don't on that account call it the order to stare and gape, even if that
was precisely the effect that I wanted to produce.
Page 189
"This combination of words makes no sense" does not mean it has no effect.
Page 189
Not even "it does not have the desired effect".
Page 189
137 To say "This combination of words makes no sense" excludes it from the sphere of language and thereby
bounds the domain of language. But when one draws a boundary it may be for various kinds of reasons. If I
surround an area with a fence or a line or otherwise, the purpose may be to prevent someone from getting in or out;
but it may also be part of a game, and the players be supposed, say, to jump over the boundary; or it may show
where A's property ends and B's begins; and so on. So if I draw a boundary line that is not yet to say what I am
drawing it for.
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Page 190
Language is not defined for us as an arrangement fulfilling a definite purpose. Rather "language" is for me a
name for a collection and I understand it as including German, English, and so on, and further various systems of
signs which have more or less affinity with these languages.
Page 190
Language is of interest to me as a phenomenon and not as a means to a particular end.
Page 190
138 Grammar consists of conventions. An example of such conventions be one saying "the word 'red' means this
colour". Such a convention may be included say in a chart.--Well, now, how could a convention find a place in a
mechanism (like the works of a pianola?) Well, it is quite possible that there is a part of the mechanism which
resembles a chart, and is inserted between the language-like part of the mechanism and the rest of it.
Page 190
Of course an ostensive definition of a word sets up a connection between a word and 'a thing', and the
purpose of this connection may be that the mechanism of which our language is a part should function in a certain
way. So the definition can make it work properly, like the connection between the keys and the hammers in a piano;
but the connection doesn't consist in the hearing of the words now having this effect, since the effect may actually
be caused by the making of the convention. And it is the connection and not the effect which determines the
meaning.
Page 190
When someone is taught language, does he learn at the same time what is sense and nonsense? When he
uses language to what extent does he employ grammar, and in particular the distinction between sense and
nonsense?
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Page 191
When someone learns musical notation, he is supplied with a kind of grammar. This is to say: this note
corresponds to this key on the piano, the sign sharpens a note, the sign cancels the etc. etc. If the
pupil asked whether there was a distinction between and or what the sign
meant, we would tell him that the distance between the top of the note and the stave didn't
mean anything, and so on. One can view this instruction as part of the preparation that makes the pupil into a
playing-machine.
Page 191
So he can speak of a grammar in the case where a language is taught to a person by a mere drill? It is clear
that if I want to use the word "grammar" here I can do so only in a "degenerate" sense, because it is only in a
degenerate sense that I can speak of "explanation", or of "convention".
Page 191
And a trained child or animal is not acquainted with any problems of philosophy.
Page 191
139 When I said that for us a language was not something that achieved a particular end, but a concept defined by
certain systems we call "languages" and such systems as are constructed by analogy with them--I could also have
expressed the same thing in the following way: causal connections in the mechanism of language are things that I
don't scruple to invent.
Page 191
Imagine that someone were to explain "Language is whatever one can use to communicate". What
constitutes communication? To complete the explantation we should have to describe what happens when one
communicates; and in the process certain causal connections and empirical regularities would come out. But these
are just the things that wouldn't interest me; they are the kinds of
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connection I wouldn't hesitate to make up. I wouldn't call just anything that opened the door a "key-bit", but only
something with a particular form and structure.
Page 192
"Language" is a word like "keyboard". There are machines which have keyboards. For some reason or other I
might be interested in forms of keyboard (both ones in actual use and others merely devised by myself). And to
invent a keyboard might mean to invent something that had the desired effect; or else to devise new forms which
were similar to the old ones in various ways.
Page 192
"It is always for living beings that signs exist, so that must be something essential to a sign." Yes, but how is
a "living" being defined? It appears that here I am prepared to use its capacity to use a sign-language as a defining
mark of a living being.
Page 192
And the concept of a living being really has an indeterminacy very similar to that of the concept "language".
Page 192
140 To invent a language could mean to invent an instrument for a particular purpose on the basis of the laws of
nature (or consistently with them); but it also has the other sense, analogous to that in which we speak of the
invention of a game.
Page 192
Here I am stating something about the grammar of the word "language" by connecting it with the grammar
of the word "invent".
Page 192
Are the rules of chess arbitrary? Imagine that it turned out that only chess entertained and satisfied people.
Then the rules aren't arbitrary if the purpose of the game is to be achieved.
Page 192
"The rules of a game are arbitrary" means: the concept 'game' is not defined by the effect the game is
supposed to have on us.
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Page 193
There is an analogous sense in which it is arbitrary which unit of measurement we use to express a length,
and another sense in which the choice of units is limited or determined.
Page 193
For us language is a calculus; it is characterized by linguistic activities.
Page 193
Where does language get its significance? Can we say "Without language we couldn't communicate with one
another"? No. It's not like "without the telephone we couldn't speak from Europe to America". We can indeed say
"without a mouth human beings couldn't communicate with each other". But the concept of language is contained
in the concept of communication.
Page 193
141 Is philosophy a creation of word-language? Is word-language a necessary condition for the existence of
philosophy? It would be more proper to ask: is there anything like philosophy outside the region of our
word-languages? For philosophy isn't anything except philosophical problems, the particular individual worries that
we call "philosophical problems". Their common element extends as far as the common element in different regions
of our language.
Page 193
Let us consider a particular philosophical problem, such as "How is it possible to measure a period of time,
since the past and the future aren't present and the present is only a point?" The characteristic feature of this is that a
confusion is expressed in the form of a question that doesn't acknowledge the confusion, and that what releases the
questioner from his problem is a particular alteration of his method of expression.
Page 193
I could imagine an organ whose stops were to be operated by keys distributed among the keys of the manual
which looked exactly like them. There might then arise a philosophical problem: "How are silent notes possible?"
And the problem would be
Page Break 194
solved by someone having the idea of replacing the stop-keys by stops which had no similarity with the note-keys.
Page 194
A problem or worry like a philosophical one might arise because someone played on all the keys of the
manual, and the result didn't sound like music, and yet he was tempted to think that it must be music etc.
Page 194
(Something that at first sight looks like a sentence and is not one.)
Page 194
The following design for the construction of a steam roller was shown to me and seems to be of
philosophical interest. The inventor's mistake is akin to a philosophical mistake. The invention consists of a motor
inside a hollow roller. The crank-shaft runs through the middle of the roller and is connected at both ends by spokes
with the wall of the roller. The cylinder of the petrol-engine is fixed onto the inside of the roller. At first glance this
construction looks like a machine. But it is a rigid system and the piston cannot move to and fro in the cylinder.
Unwittingly we have deprived it of all possibility of movement.
Page 194
142 "Could a language consist simply of independent signals?" Instead of this we might ask: Are we willing to call a
series of independent signals "a language"? To the question "can such a language achieve the same as one which
consists of sentences, or combinations of signs?" one would have to answer: it is experience
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that will show us whether e.g. these signals have the same effect on human beings as sentences. But the effect is of
no interest to us; we are looking at the phenomenon, the calculus of language.
Page 195
Imagine something like a diary kept with signals. One side is divided into sections for the hours of the day,
like a timetable. The sign "A" means: I am sleeping; "B" means "I am working"; "C" I am eating, etc. etc. But now
the question is: are explanations like this given, so that the signals are connected to another language? Is the
signal-language supplemented with ostensive definitions of the signals? Or is the language really only to consist of
the signs A, B, C etc.?
Page 195
Suppose someone asked: "how do you know, that you are now doing the same as you were an hour ago?",
and I answered: "I wrote it down, yes, here there's a 'C'"--Can one ask whether the sign "A" always means the same?
In what circumstances can this question be answered one way or the other? (One can imagine a language in which
the words, the names of the colours, say, changed their meanings with the day of the week; this colour is called
"red" on Monday, "blue" on Tuesday. "A = A" might say that in the language to which this rule applies there is no
change in the meaning of the sign "A".)
Page 195
Imagine again a language consisting of commands. It is to be used to direct the movements of a human
being; a command specifies the distance, and adds one of the words "forwards", "backwards", "right" and "left" and
one of the words "fast" and "slowly". Now of course all the commands which will actually be used to direct the
movements of a human being; a command such signals in the first place as abbreviations of the sentences of the first
language, perhaps translating them back into it before obeying them, and then later on act immediately in response
to the signals.--In that case we might speak of two languages and say the first was more pictorial than the second.
That is, we wouldn't
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say that a series of such signals by itself would enable me to derive a picture of the movement of a man obeying
them unless in addition to the signal there is something that might be called a general rule for translating into
drawing. We wouldn't say: from the sign a b b c d you can derive the figure
but we would say that you can derive it from a b b c d plus the table
Page 196
We can say: the grammar explains the meaning of the signs and thus makes the language pictorial.
Page 196
I can justify the choice of a word by a grammar. But that doesn't mean that I do, or have to, use definitions to
justify the words I use in a description or something similar.
Page 196
A comparable case is when ordinary grammar completes an elliptical sentence, and so takes a particular
construction as an abbreviated sentence.
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Appendix
Page 197
Page Break 198
Page Break 199
1:
Complex and Fact
Page 199
The use of the words 'fact' and 'act'--'That was a noble act.'--'But, that never happened.'--
Page 199
It is natural to want to use the word 'act' so that it only corresponds to a true proposition. So that we then
don't talk of an act which was never performed. But the proposition 'That was a noble act' must still have a sense
even if I am mistaken in thinking that what I call an act occurred. And that of itself contains all that matters, and I
can only make the stipulation that I will only use the words 'fact', 'act' (perhaps also 'event') in a proposition which,
when complete, asserts that this fact obtains.
Page 199
It would be better to drop the restriction on the use of these words, since it only leads to confusion, and say
quite happily: 'This act was never performed', 'This fact does not obtain', 'This event did not occur'.
Page 199
Complex is not like fact. For I can, e.g., say of a complex that it moves from one place to another, but not of
a fact.
Page 199
But that this complex is now situated here is a fact.
Page 199
'This complex of buildings is coming down' is tantamount to: 'The buildings thus grouped together are
coming down'.
Page 199
I call a flower, a house, a constellation, complexes: moreover, complexes of petals, bricks, stars etc.
Page 199
That this constellation is located here, can of course be described by a proposition in which only its stars are
mentioned and neither the word 'constellation' nor its name occurs.
Page 199
But that is all there is to say about the relation between complex
Page Break 200
and fact. And a complex is a spatial object, composed of spatial objects. (The concept 'spatial' admitting of a certain
extension.)
Page 200
A complex is composed of its parts, the things of a kind which go to make it up. (This is of course a
grammatical proposition concerning the words 'complex', 'part' and 'compose'.)
Page 200
To say that a red circle is composed of redness and circularity, or is a complex with these component parts, is
a misuse of these words and is misleading. (Frege was aware of this and told me.)
Page 200
It is just as misleading to say the fact that this circle is red (that I am tired) is a complex whose component
parts are a circle and redness (myself and tiredness).
Page 200
Neither is a house a complex of bricks and their spatial relations. i.e. that too goes against the correct use of
the word.
Page 200
Now, you can of course point at a constellation and say: this constellation is composed entirely of objects
with which I am already acquainted; but you can't 'point at a fact' and say this.
Page 200
'To describe a fact', or 'the description of a fact', is also a misleading expression for the assertion stating that
the fact obtains, since it sounds like: 'describing the animal that I saw'.
Page 200
Of course we also say: 'to point out a fact', but that always means; 'to point out the fact that...'. Whereas 'to
point at (or point out) a flower' doesn't mean to point out that this blossom is on this stalk; for we needn't be talking
about this blossom and this stalk at all.
Page 200
It's just as impossible for it to mean: to point out the fact that this flower is situated there.
Page 200
To point out a fact means to assert something, to state something. 'To point out a flower' doesn't mean this.
Page Break 201
Page 201
A chain, too, is composed of its links, not of these and their spatial relations.
Page 201
The fact that these links are so concatenated, isn't 'composed' of anything at all.
Page 201
The root of this muddle is the confusing use of the word 'object'.
Page 201
The part is smaller than the whole: applied to fact and component part (constituent), that would yield an
absurdity.
Page 201
The schema: thing-property. We say that actions have properties, like swiftness, or goodness.
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2:
Concept and Object, Property and Substrate
Page 202
When Frege and Russell talk of concept and object they really mean property and thing; and here I'm
thinking in particular of a spatial body and its colour. Or one can say: concept and object are the same as predicate
and subject. The subject-predicate form is one of the forms of expression that occur in human languages. It is the
form "x is y" ("x ∈ y"): "My brother is tall", "The storm is nearby", "This circle is red", "Augustus is strong", "2 is a
number", "This thing is a piece of coal".
Page 202
The concept of a material point in physics is an abstraction from the material objects of experience; in the
same way the subject-predicate form of logic is an abstraction from the subject-predicate form of our languages. The
pure subject-predicate form is supposed to be a ∈ f(x), where "a" is the name of an object. Now let's look for an
application of this schema. The first things that come to mind as "names of objects" are the names of persons and of
other spatial objects (the Koh-i-Noor). Such names are given by ostensive definitions ("that is called 'N'").
Such a definition might be conceived as a rule substituting the word "N" for a gesture pointing to the object, with the
proviso that the gesture can always be used in place of the name. Thus, I may have explained "this man is called 'N'",
and I go on to say "'N' is a mathematician", "N is lazy", and in each of these sentences I might have said "this man"
(with the ostensive gesture) instead of "N". (In that case, incidentally it would have been better to phrase the
ostensive definition "this man is called 'N'"†1 or "I want to call this man 'N'", because the version above is also the
proposition that this man bears this name).
Page 202
However, this isn't the normal way of using a name; it is an essential feature of the normal use that I can't fall
back on to a sign of the gesture language in place of the name. That's to say, in the way in which we use the name
"N", if N goes out of the room and later a man comes into the room it makes sense to ask whether
Page Break 203
this man is N, whether he is the same man as the one who left the room earlier. And the sentence "N has come back
into the room" only makes sense if I can decide the question. And its sense will vary with the criterion for this being
the object that I earlier called 'N'. Different kinds of criteria will make different rules hold for the sign 'N', will make it
a 'name' in a different sense of the word. Thus the word 'name' and the corresponding word 'object' are each
headings to countless different lists of rules.
Page 203
If we give names to spatial objects, our use of such names depends on a criterion of identity which
presupposes the impenetrability of bodies and the continuity of their movement. So if I could treat two bodies A and
B as I can treat their shadows on the wall, making two into one and one into two again, it would be senseless to ask
which of the two after the division is A and which is B, unless I go on to introduce a totally new criterion of identity
e.g. the direction of their movements. (There is a rule for the name of a river arising from the confluence of two
rivers, thus:
The resulting river takes the name of that source in whose approximate direction it flows onward.)
Page 203
Think of the possible criteria of identity for things like colour patches in my visual field (or figures on a
cinema screen) and of the different kinds of use of names given to such patches or figures.
Page 203
If we turn to the form of expression "(∃x).fx" it's clear that this is a sublimation of the form of expression in
our language: "There are human beings on this island" "There are stars that we do not see". To every proposition of
the form "(∃x).fx" there is supposed to correspond a proposition "fa", and "a" is supposed to be a name. So one
must be able to say "(∃x).fx, namely a and b",
Page Break 204
("There are some values of x, which satisfy fx, namely a and b"), or "(∃x).fx, e.g. a", etc. And this is indeed possible
in a case like "There are human beings on this island, namely Messrs A, B, C, D." But then is it essential to the sense
of the sentence "There are men on this island" that we should be able to name them, and fix a particular criterion for
their identification? That is only so in the case where the proposition "(∃x).fx" is defined as a disjunction of
propositions of the form "f(x)", if e.g. it is laid down that "There are men on this island" means "Either Mr. A or Mr.
B or Mr. C or Mr. D. or Mr. E is on this island"--if, that is, one determines the concept "man" extensionally (which
of course is quite contrary to the normal use of this word.) (On the other hand the concept "primary colour" really is
determined extensionally.)
Page 204
So it doesn't always make sense when presented with a proposition "(∃x).fx" to ask "Which xs satisfy f?"
"Which red circle a centimetre across is in the middle of this square"?--One mustn't confuse the question "which
object satisfies f?" with the question "what sort of object... etc.?" The first question would have to be answered by a
name, and so the answer would have to be able to take the form "f(a)"; the question "what sort of...?" is answered
by "(∃x).fx.φx". So it may be senseless to ask "which red spot do you see?" and yet make sense to ask "what kind of
a red spot do you see (a round one, a square one, etc.)?"
Page 204
I would like to say: the old logic contains more convention and physics than has been realised. If a noun is
the name of a body, a verb is to denote a movement, and an adjective to denote a property of a body, it is easy to see
how much that logic presupposes; and it is reasonable to conjecture that those original presuppositions go still
deeper into the application of the words, and the logic of propositions.
Page 204
(Suppose we were set the task of projecting figures of various shapes on a given plane I into a plane II. We
could then fix a
Page Break 205
method of projection (say orthogonal projection) and carry out the mapping in accordance with it. We could also
easily make inferences from the representations on plane II about the figures on plane I. But we could also adopt
another procedure: we might decide that the representations in the second plane should all be circles, no matter what
the copied figures in the first plane might be. (Perhaps this is the most convenient form of representation for us.)
That is, different figures on I are mapped onto II by different methods of projection. In order in this case to construe
the circles in II as representations of the figures in I, I shall have to give the method of projection for each circle; the
mere fact that a figure in I is represented as a circle in II†1 by itself tells us nothing about the shape of the figure
copied. That an image in II is a circle is just the established norm of our mapping.--Well, the same thing happens
when we depict reality in our language in accordance with the subject-predicate form. The subject-predicate form
serves as a projection of countless different logical forms.
Page 205
Frege's "Concept and Object" is the same as subject and predicate.
Page 205
If a table is painted brown, then it's easy to think of the wood as bearer of the property brown and you can
imagine what remains the same when the colour changes. Even in the case of one particular circle which appears
now red, now blue. It is thus easy to imagine what is red, but difficult to imagine what is circular. What remains in
this case if form and colour alter? For position is part of the form and it is arbitrary for me to lay down that the
centre should stay fixed and the only changes in form be changes in the radius.
Page 205
We must once more adhere to ordinary language and say that a patch is circular.
Page 205
It is clear that here the phrase "bearer of a property" in this context conveys a completely wrong--an
impossible--picture. If I have a lump of clay, I can consider it as the bearer of a form, and that, roughly, is where this
picture comes from.
Page Break 206
Page 206
"The patch changes its form" and "the lump of clay changes its form" are different forms of propositions.
Page 206
You can say "Measure whether that is a circle" or "See whether that over there is a hat". You can also say
"Measure whether that is a circle or an ellipse", but not "... whether that is a circle or a hat"; nor "See whether that is
a hat or red".
Page 206
If I point to a curve and say "That is a circle" then someone can object that if it were not a circle it would no
longer be that. That is to say, what I mean by the word "that" must be independent of what I assert about it.
Page 206
("Was that thunder, or gunfire?" Here you could not ask "Was that a noise?")
Page 206
How are two circles of the same size distinguished? This question makes it sound as if they were pretty
nearly one circle and only distinguished by a nicety.
Page 206
In the technique of representation by equations what is common is expressed by the form of the equation,
and the difference by the difference in the coordinates of the centres.
Page 206
So it is as if what corresponds with the objects falling under the concept were here the coordinates of the
centres.
Page 206
Couldn't you then say, instead of "This is a circle", "This point is the centre of a circle"? For to be the centre
of a circle is an external property of the point.
Page 206
What is necessary to a description that--say--a book is in a certain position? The internal description of the
book, i.e. of the concept, and a description of its place which it would be possible to give by giving the co-ordinates
of three points. The proposition "Such a book is here" would mean that it had these three co-ordinates. For the
specification of the "here" must not prejudge what is here.
Page 206
But doesn't it come to the same thing whether I say "This is a book" or "Here is a book"? The proposition
would then amount
Page Break 207
to saying, "These are three corners of such a book".
Page 207
Similarly you can also say "This circle is the projection of a sphere" or "This is a man's appearance".
Page 207
All that I am saying comes to this, that Φ(x) must be an external description of x.
Page 207
If in this sense I now say in three-dimensional space "Here is a circle" and on another occasion "Here is a
sphere" are the two "here's" of the same type? I want to ask: can one significantly say of the same 'object': it is a
circle, and: it is a sphere? Is the subject of each of these predicates of the same type? Both could be the three
coordinates of the relevant centre-point. But the position of the circle in three-dimensional space is not fixed by the
coordinates of its centre.
Page 207
On the other hand you can of course say "It's not the noise, but the colour that makes me nervous" and here
it might look as if a variable assumed a colour and a noise as values. ("Sounds and colours can be used as vehicles of
communication".) It is clear that this proposition is of the same kind as "if you hear a shot, or see me wave, run". For
this is the kind of co-ordination on the basis of which a heard or seen language functions.
Page 207
"Is it conceivable that two things have all their properties in common?"--If it isn't conceivable, then neither is
its opposite.
Page 207
We do indeed talk about a circle, its diameter, etc. etc., as if we were describing a concept in complete
abstraction from the objects falling under it.--But in that case 'circle' is not a predicate in the original sense. And in
general geometry is the place where concepts from the most different regions get mixed up together.
Page Break 208
3:
Objects
Page 208
"In a certain sense, an object cannot be described." (So too Plato: "You can't give an account of one but only
name it.") Here "object" means "reference of a not further definable word", and "description" or "explanation" really
means: "definition". For of course it isn't denied that the object can be "described from outside", that properties can
be ascribed to it and so on.
Page 208
So when we use the proposition above we are thinking of a calculus with signs or names that are
indefinable--or, more accurately, undefined--and we are saying that no account can be given of them.
Page 208
"What a word means a proposition cannot tell."
Page 208
What is the distinction, then, between blue and red?
Page 208
We aren't of the opinion that one colour has one property and the other another. In any case, the properties
of blue and red are that this body (or place) is blue, and that other is red.
Page 208
When asked "what is the distinction between blue and red?" we feel like answering: one is blue and the other
red. But of course that means nothing and in reality what we're thinking of is the distinction between the surfaces or
places that have these colours. For otherwise the question makes no sense at all.
Page 208
Compare the different question: "What is the distinction between orange and pink?" One is a mixture of
yellow and red, the other a mixture of white and red. And we may say accordingly: blue comes from purple when it
gets more bluish, and red comes from purple when that gets more and more reddish.
Page Break 209
Page 209
So what I am saying means: red can't be described. But can't we represent it in painting by painting
something red?
Page 209
No, that isn't a representation in painting of the meaning of the word 'red' (there's no such thing).
Page 209
The portrait of red.
Page 209
Still, it's no accident that in order to define the meaning of the word "red" the natural thing is to point to a red
object.
Page 209
(What is natural about it is portrayed in that sentence by the double occurrence of the word 'red').
Page 209
To say that blue is on the bluish side of blue-red and red on the reddish side is a grammatical sentence and
therefore akin to a definition. And indeed one can also say: more bluish = more like blue.
Page 209
"If you call the colour green an object, you must be saying that it is an object that occurs in the symbolism.
Otherwise the sense of the symbolism, and thus its very existence as a symbolism, would not be guaranteed."
Page 209
But what does that assert about green, or the word "green"? ((That sentence is connected with a particular
conception of the meaning-relation and a particular formulation of the problem the relation raises)).
Page Break 210
4:
Elementary Propositions
A†1
Page 210
Can a logical product be hidden in a proposition? And if so, how does one tell, and what methods do we
have of bringing the hidden element of a proposition to light? If we haven't yet got a method, then we can't speak of
something being hidden or possibly hidden. And if we do have a method of discovery then the only way in which
something like a logical product can be hidden in a proposition is the way in which a quotient like 753 /3 is hidden
until the division has been carried out.
Page 210
The question whether a logical product is hidden in a sentence is a mathematical problem.
Page 210
So an elementary proposition is a proposition which, in the calculus as I am now using it, is not represented
as a truth-function of other sentences.
Page 210
The idea of constructing elementary propositions (as e.g. Carnap has tried to do) rests on a false notion of
logical analysis. It is not the task of that analysis to discover a theory of elementary propositions, like discovering
principles of mechanics.
Page 210
My notion in the Tractatus Logico-Philosophicus was wrong: 1) because I wasn't clear about the sense of
the words "a logical product is hidden in a sentence" (and suchlike), 2) because I too thought that logical analysis
had to bring to light what was hidden (as chemical and physical analysis does).
Page 210
The proposition "this place is now red" (or "this circle is now red") can be called an elementary proposition if
this means that it is
Page Break 211
neither a truth-function of other propositions nor defined as such. (Here I am disregarding combinations such as p.:
q ∨ ~ q and the like.)
Page 211
But from "a is now red" there follows "a is now not green" and so elementary propositions in this sense aren't
independent of each other like the elementary propositions in the calculus I once described--a calculus to which,
misled as I was by a false notion of reduction, I thought that the whole use of propositions must be reducible.
B†1
Page 211
If you want to use the appellation "elementary proposition" as I did in the Tractatus Logico-Philosophicus,
and as Russell used "atomic proposition", you may call the sentence "Here there is a red rose" an elementary
proposition. That is to say, it doesn't contain a truth-function and it isn't defined by an expression which contains
one. But if we're to say that a proposition isn't an elementary proposition unless its complete logical analysis shows
that it isn't built out of other propositions by truth-functions, we are presupposing that we have an idea of what such
an 'analysis' would be. Formerly, I myself spoke of a 'complete analysis', and I used to believe that philosophy had
to give a definitive dissection of propositions so as to set out clearly all their connections and remove all possibilities
of misunderstanding. I spoke as if there was a calculus in which such a dissection would be possible. I vaguely had
in mind something like the definition that Russell had given for the definite article, and I used to think that in a
similar way one would be able to use visual impressions etc. to define the concept say of a sphere, and thus exhibit
once for all the connections between the concepts and lay bare the source of all misunderstandings, etc. At the root
of all this there was a false and idealized picture of the use of language. Of course, in particular cases one can
Page Break 212
clarify by definitions the connections between the different types of use of expressions. Such a definition may be
useful in the case of the connection between 'visual impression' and 'sphere'. But for this purpose it is not a definition
of the concept of a physical sphere that we need; instead we must describe a language game related to our own, or
rather a whole series of related language games, and it will be in these that such definitions may occur. Such a
contrast destroys grammatical prejudices and makes it possible for us to see the use of a word as it really is, instead
of inventing the use for the word.
Page 212
There could perhaps be a calculus for dissecting propositions; it isn't hard to imagine one. Then it becomes a
problem of calculation to discover whether a proposition is or is not an elementary proposition.
Page 212
The question whether e.g. a logical product is hidden in a sentence is a mathematical problem.--What
"hidden" means here is defined by the method of discovery (or, as it might be, by the lack of a method).
Page 212
. . . . . . . . . . .
What gives us the idea that there is a kind of agreement between thought and reality?--Instead of
"agreement" here one might say with a clear conscience "pictorial character".†1
Page 212
But is this pictorial character an agreement? In the Tractatus Logico-Philosophicus I said something like: it
is an agreement of form. But that is an error.
Page 212
First of all, "picture" here is ambiguous. One wants to say that an order is the picture of the action which was
carried out on the order; but also, a picture of the action which is to be carried out as an order.
Page Break 213
Page 213
We may say: a blueprint serves as a picture of the object which the workman is to make from it.
Page 213
And here we might call the way in which the workman turns such a drawing into an artefact "the method of
projection". We might now express ourselves thus: the method of projection mediates between the drawing and the
object, it reaches from the drawing to the artefact. Here we are comparing the method of projection with projection
lines which go from one figure to another.--But if the method of projection is a bridge, it is a bridge which isn't built
until the application is made.--This comparison conceals the fact that the picture plus the projection lines leaves open
various methods of application; it makes it look as if what is depicted, even if it does not exist in fact, is determined
by the picture and the projection lines in an ethereal manner; every bit as determined, that is to say, as if it did exist.
(It is 'determined give or take a yes or no.') In that case what we may call 'picture' is the blueprint plus the method of
its application. And we now imagine the method as something which is attached to the blueprint whether or not it is
used. (One can "describe" an application even if it doesn't exist).†1
Page 213
Now I would like to ask "How can the blueprint be used as a representation, unless there is already an
agreement with what is to be made?"--But what does that mean? Well, perhaps this: how could I play the notes in
the score on the piano if they didn't already have a relationship to particular types of movement of the hand? Of
course such a relationship sometimes consists in a certain agreement, but sometimes not in any agreement, but
merely in our having learnt to apply the signs in a particular way. What the comparison between the method of
projection and the projection lines connecting the picture with the object does is to make all these cases
alike--because that is what attracts us. You may say: I count the projection lines as part of the picture--but not the
method of projection.
Page Break 214
Page 214
You may of course also say: I count a description of a method of projection as part of the picture.
Page 214
So I am imagining that the difference between proposition and reality is ironed out by the lines of projection
belonging to the picture, the thought, and that no further room is left for a method of application, but only for
agreement and disagreement.
Page Break 215
5:
Is time essential to propositions?
Comparison between time and truth-functions
Page 215
If we had grammar set out in the form of a book, it wouldn't be a series of chapters side by side, it would
have quite a different structure. And it is here, if I am right, that we would have to see the distinction between
phenomenological and non-phenomenological. There would be, say, a chapter about colours, setting out the rules
for the use of colour-words; but there would be nothing comparable in what the grammar had to say about the
words "not", "or", etc. (the "logical constants").
Page 215
It would, for instance, be a consequence of the rules, that these latter words unlike the colour words were
usable in every proposition; and the generality belonging to this "every" would not be the kind that is discovered by
experience, but the generality of a supreme rule of the game admitting of no appeal.
Page 215
How does the temporal character of facts manifest itself? How does it express itself, if not by certain
expressions having to occur in our sentences? That means: how does the temporal character of facts express itself, if
not grammatically? "Temporal character"--that doesn't mean that I come at 5 o'clock, but that I come at some time or
other, i.e. that my proposition has the structure it has.
Page 215
We are inclined to say that negation and disjunction are connected with the nature of the proposition, but
that time is connected with its content rather than with its nature.
Page 215
But if two things are equally universal, how can it show itself in grammar that one of them is connected with
the nature of the proposition and the other is not?
Page 215
Or should I have said that time is not equally universal since mathematical propositions can be negated and
occur in disjunctions, without being temporal? There is indeed a connection here, though this form of portraying the
matter is misleading.
Page Break 216
Page 216
But that shows what I mean by "proposition." or "nature of the proposition".
Page 216
Why--I want to ask--is the temporal character of propositions so universal?
Page 216
Might one also put the question thus: "How does it happen that every fact of experience can be brought into
a relationship with what is shown by a clock?"
Page 216
Having two kinds of generality in the way I spoke of would be as strange as if there were two equally
exceptionless rules of a game and one of them were pronounced to be more fundamental. As if one could ask
whether in chess the king or the chess board was more important; which of the two was more essential, and which
more accidental.
Page 216
There's at least one question that seems in order: suppose I had written up the grammar, and the different
chapters on the colour words, etc. etc. were there one after the other, like rules for each of the chess pieces, how
would I know that those were all the chapters? If there turns out to be a common property in all the chapters so far
in existence, we seem to have encountered a logical generality that is not an essential, i.e. a priori generality. But we
can't say that the fact that chess is played with 16 pieces is any less essential to it than its being played on a
chessboard.
Page 216
Since time and the truth-functions taste so different, and since they manifest their nature only and wholly in
grammar, it is grammar that must explain the different taste.
Page 216
One tastes like content, the other like form of representation.
Page 216
They taste as different as a plan and a line through a plan.
Page Break 217
Page 217
It appears to me that the present, as it occurs in the proposition "the sky is blue" (if this proposition isn't
meant as a hypothesis), is not a form of time, so that the present in this sense is atemporal.
Page 217
Does time enter into a landscape picture? or into a still life?
Page 217
Literature consisting of descriptions of landscapes.
Page 217
It is noteworthy that the time of which I am here speaking is not time in a physical sense. We are not
concerned with measuring time. It is fishy that something which is unconnected with measurement is supposed to
have a role in propositions like that of physical time in the hypotheses of physics.
Page 217
Discuss:
Page 217
The distinction between the logic of the content and the logic of the propositional form in general. The
former seems, so to speak, brightly coloured, and the latter plain; the former seems to be concerned with what the
picture represents, the latter to be a characteristic of the pictorial form like a frame.
Page 217
By comparison with the way in which the truth-functions are applicable to all propositions, it seems to us
accidental that all propositions contain time in some way or other.
Page 217
The former seems to be connected with their nature as propositions, the latter with the nature of the reality
we encounter.
((Added later in the margins))
A sentence can contain time in very different senses.
You are hurting me.
The weather is marvellous outside.
The Inn flows into the Danube.
Water freezes at 0°.
I often make slips of the pen
Some time ago...
I hope he will come.
At 5 o'clock.
Page Break 218
This kind of steel is excellent.
The earth was once a ball of gas.
Page Break 219
6:
The Nature of Hypotheses
Page 219
You could obviously explain an hypothesis by means of pictures. I mean, you could e.g. explain the
hypothesis "there is a book lying here" with pictures showing the book in plan, elevation and various cross-sections.
Page 219
Such a representation gives a law. Just as the equation of a curve gives a law, by means of which you may
discover the ordinates, if you cut at different abscissae.
Page 219
In which case the verifications of particular cases correspond to cuts that have actually been made.
Page 219
If our experiences yield points lying on a straight line, the proposition that these experiences are various
views of a straight line is an hypothesis.
Page 219
The hypothesis is a way of representing this reality, for a new experience may tally with it or not, or possibly
make it necessary to modify the hypothesis.
Page 219
If for instance we use a system of coordinates and the equation for a sphere to express the proposition that a
sphere is located at a certain distance from our eyes, this description has a greater multiplicity than that of a
verification by eye. The first multiplicity corresponds not to one verification but to a law obeyed by verifications.
Page 219
An hypothesis is a law for forming propositions.
Page 219
You could also say: an hypothesis is a law for forming expectations.
Page 219
A proposition is, so to speak, a particular cross-section of an hypothesis.
Page 219
According to my principle two suppositions must have the same sense if every possible experience that
confirms the one also
Page Break 220
confirms the other, if, that is, no decision between the two is conceivable on the basis of experience.
Page 220
The representation of a curve as a straight line with deviations. The equation of the curve includes a
parameter whose course expresses the deviations from a straight line. It isn't essential that these deviations should be
"slight". They can be so large that the curve doesn't look like a straight line at all. "Straight line with deviations" is
only one form of description. It makes it easier for me to eliminate, or neglect, a particular component of the
description if I so wish. (The form "rule with exceptions").
Page 220
What does it mean, to be certain that one has toothache? (If one can't be certain, then grammar doesn't allow
the use of the word "certain" in this connection.)
Page 220
The grammar of the expression "to be certain".
Page 220
We say "If I say that I see a chair there, I am saying more than I know for certain". And commonly that
means "But all the same, there's one thing that I do know for certain." But if we now try to say what it is, we find
ourselves in a certain embarrassment.
Page 220
"I see something brown--that is certain." That's meant to say that the brown colour is seen and not perhaps
merely conjectured from other symptoms. And we do indeed say quite simply: "I see something brown."
Page 220
If someone tells me "Look into this telescope, and make me a sketch of what you see", the sketch I make is
the expression of a proposition, not of a hypothesis.
Page 220
If I say "Here there is a chair", I mean more--people say--than the mere description of what I perceive. This
can only mean that that proposition doesn't have to be true, even though the description fits what is seen. Well, in
what circumstances would I say that that proposition wasn't true? Apparently, if certain other
Page Break 221
propositions aren't true that were implicit in the first. But it isn't as if the first turns out to have been a logical product
all along.
Page 221
The best comparison for every hypothesis,--something that is itself an example of an hypothesis--is a body
in relation to a systematic series of views of it from different angles.
Page 221
Making a discovery in a scientific investigation (say in experimental physics) is of course not the same thing
as making a discovery in ordinary life outside the laboratory; but the two are similar and a comparison with the
former can throw light on the latter.
Page 221
There is an essential distinction between propositions like "That is a lion", "The sun is larger than the earth",
and propositions like "Men have two hands". Propositions like the first pair contain a "this", "now", "here" and thus
connect immediately with reality. But if there happened to be no men around, how would I go about checking the
third proposition?
Page 221
It is always single faces of hypotheses that are verified.
Page 221
Perhaps this is how it is: what an hypothesis explains is itself only expressible by an hypothesis. Of course,
this amounts to asking whether there are any primary propositions that are definitively verifiable and not merely
facets of an hypothesis. (That is rather like asking: are there surfaces that aren't surfaces of bodies?)
Page 221
At all events, there can't be any distinction between an hypothesis used as an expression of an immediate
experience and a proposition in the stricter sense.
Page 221
There is a distinction between a proposition like "Here there is a sphere in front of me" and "It looks as if
there is a sphere in front of me". The same thing shows itself also thus: one can say "There seems to be a sphere in
front of me", but it is senseless to say "It looks as if there seems to be a sphere here". So too one can say
Page Break 222
"Here there is probably a sphere", but not "Here there probably appears to be a sphere". In such a case people would
say "After all, you must know whether there appears to be".
Page 222
There is nothing hypothetical in what connects the proposition with the given fact.
Page 222
It's clear that reality--I mean immediate experience--will sometimes give an hypothesis the answer yes, and
sometimes the answer no (here of course the "yes" and "no" express only confirmation and lack of confirmation);
and it's clear that these affirmations and denials can be given expression.
Page 222
The hypothesis, if that face of it is laid against reality, becomes a proposition.
Page 222
It may be doubtful whether the body I see is a sphere, but it can't be doubtful that from here it looks to be
something like a sphere.--The mechanism of hypothesis would not function if appearance too were doubtful so that
one couldn't verify beyond doubt even a facet of the hypothesis. If there were a doubt here, what could take the
doubt away? If this connection too were loose, there would be no such thing as confirming an hypothesis and it
would hang entirely in the air, quite pointless (and therefore senseless).
Page 222
If I say "I saw a chair", that (in one sense) isn't contradicted by the proposition "there wasn't one there". For I
could use the first proposition in the description of a dream and then nobody would use the second to contradict me.
But the description of the dream throws a light on the sense of the words "I saw".
Page 222
Again, in the proposition "there wasn't one there", the word "there" may have more than one meaning.
Page Break 223
Page 223
I am in agreement with the opinions of contemporary physicists when they say that the signs in their
equations no longer have any "meanings" and that physics cannot attain to any such meanings, but must stay put at
the signs. But they don't see that the signs have meaning in as much as--and only in as much as--observable
phenomena do or do not correspond to them, in however circuitous a manner.
Page 223
Let us imagine that chess had been invented not as a board game, but as a game to be played with numbers
and letters on paper, so that no one had ever imagined a board with 64 squares in connection with it. And now
suppose someone made the discovery that the game corresponded exactly to a game which could be played on a
board in such and such a way. This discovery would have been a great simplification of the game (people who
would earlier have found it too difficult could now play it). But it is clear that this new illustration of the rules of the
game would be nothing more than a new, more easily surveyable symbolism, which in other respects would be on
the same level as the written game. Compare with this the talk about physics nowadays not working with mechanical
models but "only with symbols".
Page Break 224
7:
Probability
Page 224
The probability of an hypothesis has its measure in how much evidence is needed to make it profitable to
throw it out.
Page 224
It's only in this sense that we can say that repeated uniform experience in the past renders the continuation of
this uniformity in the future probable.
Page 224
If, in this sense, I now say: I assume the sun will rise again tomorrow, because the opposite is so unlikely, I
here mean by "likely" and "unlikely" something completely different from what I mean by these words in the
proposition "It's equally likely that I'll throw heads or tails". The two meanings of the word "likely" are, to be sure,
connected in certain ways, but they aren't identical.
Page 224
We only give up an hypothesis for an ever higher gain.
Page 224
Induction is a process based on a principle of economy.
Page 224
The question how simple a representation is yielded by assuming a particular hypothesis is directly
connected, I believe, with the question of probability.
Page 224
We may compare a part of an hypothesis with the movement of a part of a gear, a movement that can be
stipulated without prejudicing the intended motion. But then of course you have to make appropriate adjustments to
the rest of the gear if it is to produce the desired motion. I'm thinking of a differential gear.--Once I've decided that
there is to be no deviation from a certain part of my hypothesis no matter what the experience to be described may
be, I have stipulated a mode of representation and this part of my hypothesis is now a postulate.
Page Break 225
Page 225
A postulate must be such that no conceivable experience can refute it, even though it may be extremely
inconvenient to cling to the postulate. To the extent to which we can talk here of greater or slighter convenience,
there is a greater or slighter probability of the postulate.
Page 225
It's senseless to talk of a measure for this probability at this juncture. The situation here is like that in the case
of two kinds of numbers where we can with a certain justice say that the one is more like the other (is closer to it)
than a third, but there isn't any numerical measure of the similarity. Of course you could imagine a measure being
constructed in such cases, too, say by counting the postulates or axioms common to the two systems, etc. etc.
Page 225
I give someone the following piece of information, and no more: at such and such a time you will see a point
of light appear in the interval AB.
Does the question now makes sense "Is it more likely that this point will appear in the interval AC than in CB"? I
believe, obviously not.--I can of course decide that the probability of the event's happening in CB is to be in the ratio
CB/AC to the
Page Break 226
probability of its happening in AC; however, that's a decision I can have empirical grounds for making, but about
which there is nothing to be said a priori. It is possible for the observed distribution of events not to lead to this
assumption. The probability, where infinitely many possibilities come into consideration, must of course be treated
as a limit. That is, if I divide the stretch AB into arbitrarily many parts of arbitrary lengths and regard it as equally
likely that the event should occur in any one of these parts, we immediately have the simple case of dice before us.
And now I can--arbitrarily--lay down a law for constructing parts of equal likelihood. For instance, the law that, if
the lengths of the parts are equal, they are equally likely. But any other law is just as permissible.
Page 226
Couldn't I, in the case of dice too, take, say, five faces together as one possibility, and oppose them to the
sixth as the second possibility? And what, apart from experience, is there to prevent me from regarding these two
possibilities as equally likely?
Page 226
Let's imagine throwing, say, a red ball with just one very small green patch on it. Isn't it much more likely in
this case for the red area to strike the ground than for the green?--But how would we support this proposition?
Presumably by showing that when we throw the ball, the red strikes the ground much more often than the green.
But that's got nothing to do with logic.--We may always project the red and green surfaces and what befalls them
onto a surface in such a way that the projection of the green surface is greater than or equal to the red; so that the
events, as seen in this projection, appear to have a quite different probability ratio from the one they had on the
original surface. If, e.g. I reflect the events
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in a suitably curved mirror and now imagine what I would have held to be the more probable event if I had only seen
the image in the mirror.
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The one thing the mirror can't alter is the number of clearly demarcated possibilities. So that if I have n
coloured patches on my ball, the mirror will also show n, and if I have decided that these are to be regarded as
equally likely, then I can stick to this decision for the mirror image too.
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To make myself even clearer: if I carry out the experiment with a concave mirror, i.e. make the observations
in a concave mirror, it will perhaps then look as if the ball falls more often on the small surface than on the much
larger one; and it's clear that neither experiment--in the mirror or outside it--has a claim to precedence.
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We may apply our old principle to propositions expressing a probability and say, we shall discover their
sense by considering what verifies them.
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If I say "That will probably occur", is this proposition verified by the occurrence or falsified by its
non-occurrence? In my opinion, obviously not. In that case it doesn't say anything about either. For if a dispute were
to arise as to whether it is probable or not, it would always be arguments from the past that would be adduced. And
this would be so even when what actually happened was already known.
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Causality depends on an observed uniformity. This does not mean that a uniformity so far observed will
always continue, but what cannot be altered is that the events so far have been uniform; that can't be the uncertain
result of an empirical series which in its turn isn't something given but something dependent on another uncertain
one and so on ad infinitum.
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When people say that the proposition "it is probable that p will occur" says something about the event p,
they forget that the probability remains even when the event p does not occur.
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The proposition "p will probably occur" does indeed say something about the future, but not something
"about the event p", as the grammatical form of the statement makes us believe.
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If I ask for the grounds of an assertion, the answer to the question holds not only for this person and for this
action (assertion), but quite generally.
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If I say "the weather looks like rain" do I say anything about future weather? No; I say something about the
present weather, by means of a law connecting weather at any given time with weather at an earlier time. This law
must already be in existence, and we are using it to construct certain statements about our experience.--
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We might say the same of historical statements too. But I was too quick to say that the proposition "the
weather looks like rain" says nothing about future weather. It all depends what is meant by "saying something about
something". The sentence says just what it says.
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The sentence "p will probably occur" says something about the future only in a sense in which its truth and
falsehood are completely independent of what will happen in the future.
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If we say: "the gun is now aiming at the point p" we aren't saying anything about where the shot will hit.
Giving the point at which it is aiming is a geometrical means of assigning its direction. That this is the means we use
is certainly connected with certain observations (projectile parabolas, etc.) but these observations don't enter into our
present description of the direction.
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A Galtonian photograph is the picture of a probability.
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The law of probability is the natural law you see when you screw up your eyes.
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"On average, the points yielded by the experiment lie on a straight line". "If I throw with a good die, then on
average I throw a one every six throws". What does that mean? Is the proposition compatible with any experience I
may have? If so, it says nothing. Have I decided in advance which experiences are incompatible with it and what is
the limit beyond which exceptions may not go without upsetting the rule? No. But couldn't I have set such a limit?
Of course.--Suppose that the limit had been set thus: if 4 out of 6 successive throws turn out the same, then it's a bad
die. Now someone says: "But if that happens only very seldom, mayn't it be a good one after all?"--To that the
answer is as follows. If I permit the turning up of 4 similar throws among 6 successive ones to occur within a certain
number of throws, then I am replacing the first limit with a different one. But if I say "any number of similar
successive throws is allowed, as long as it happens sufficiently rarely", then strictly speaking I've defined the
goodness of the die in a way that makes it independent of the result of the throws; unless by the goodness of a die I
do not mean a property of the die, but a property of a particular game played with it. In that case I can certainly say:
in any game I call the die good provided that among the N throws of the game there occur not more than log N
similar successive throws. However, that doesn't give a test for the checking of dice, but a criterion for judging a
particular game.
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We say that if the die is quite regular and isn't interfered with then the distribution of the numbers 1, 2, 3, 4,
5, 6 among the throws must be uniform, since there is no reason why one number should occur more often than
another.
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But now let's represent the throws by the values of the function (x - 3)² for the arguments 1 to 6, i.e. by the
numbers 0, 1, 4, 9 instead of by the numbers 1 to 6. Is there a reason why one of these numbers should turn up in
the new results more often than another? This shows us that the a priori law of probability, like the
minimum-principles of mechanics etc., is a form that laws may take. If it had been discovered by experiment that the
distribution of the throws 1 to 6 with a regular die was such that the distribution of the values of (x -3)² was uniform,
it would have been this regularity that was defined as the a priori regularity.
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We do the same thing in the kinetic theory of gases: we represent the distribution of molecular movements in
the form of some sort of uniform distribution; but we make the choice of what is uniformly distributed--and in the
other case of what is reduced to a minimum--in such a way that our theory agrees with experience.
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"The molecules move purely according to the laws of probability" is supposed to mean: physics gets out of
the way, and now the molecules move as it were purely according to laws of logic. This idea is similar to the idea
that the law of inertia is an a priori proposition: there too one speaks of what a body does when it isn't interfered
with. But what is the criterion for its not being interfered with? Is it ultimately that it moves uniformly in a straight
line? Or is it something different? If the latter, then it's a matter of experience whether the law of inertia holds; if the
former, then it wasn't a law at all but a definition. So too with the proposition, "if the particles aren't interfered with,
then the distribution of their motions is such and such". What is the criterion for their not being interfered with? etc.
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To say that the points yielded in this experiment lie roughly on this line, e.g. a straight line, means something
like: "seen for this distance, they seem to lie on a straight line".
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Page 231
I may say that a stretch gives the general impression of a straight line; but I cannot say: "This bit of line looks
straight, for it could be a bit of a line that as a whole gives me the impression of being straight." (Mountains on the
earth and moon. The earth a ball.)
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An experiment with dice lasts a certain time, and our expectations about future throws can only be based on
tendencies we observe in what happens during this experiment. That is to say, the experiment can only give grounds
for expecting that things will go in in the way shown by the experiment; but we can't expect that the experiment, if
continued, will now yield results that tally better with a preconceived idea of its course than did those of the
experiment we have actually performed. So if, for instance, I toss a coin and find no tendency in the results of the
experiment itself for the number of heads and tails to approximate to each other more closely, then the experiment
gives me no reason to suppose that if it were continued such an approximation would emerge. Indeed, the
expectation of such an approximation must itself refer to a definite point in time, since we can't say we're expecting
something to happen eventually, in the infinite future.
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Any "reasonable expectation" is an expectation that a rule we have observed up to now will continue to hold.
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(But the rule must have been observed and can't, for its part too, be merely expected.)
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The logic of probability is only concerned with the state of expectation in the sense in which logic in general
is concerned with thinking.
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A ray is emitted from the light source S striking the surface AB to form a point of light there, and then
striking the surface AB'. We have no reason to suppose that the point on AB lies to the left or to the right of M, and
equally none for supposing that the
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point on AB' lies on one side or the other of m. This yields therefore incompatible probabilities. But if I make an
assumption about the probability of the point on AB lying in AM, how is this assumption verified? Surely, we think,
by a frequency experiment. Supposing this confirms the view that the probabilities of AM and BM are equal (and so
the probabilities of Am and B'm differ), then it is recognized as the right one and thus shows itself to be an
hypothesis belonging to physics. The geometrical construction merely shows that the fact that AM = MB was no
ground for assuming equal likelihood.
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Suppose that measurement shows the die to be accurate and regular, that the numbers on its sides don't
influence the throws, and that it is thrown by a hand whose movements follow no definite rules: does it follow that
the distribution among the throws of each of the throws from 1 to 6 will be uniform on average? Where is the
uniform distribution supposed to come from? The accuracy and regularity of the die can't establish that the
distribution of throws will be uniform on average. (It would be, as it were, a monochrome premise with a mottle
conclusion.) And we
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haven't made any suppositions about the movements while throwing. (Making the bundles of hay equal gives reason
to believe that the donkey will starve to death between them; it doesn't give reason to believe that he will eat from
each with roughly equal frequency.)--It is perfectly compatible with our assumptions for one hundred ones to be
thrown in succession, if friction, hand-movements and air-resistance coincide appropriately. The experimental fact
that this never happens is a fact about those factors, and the hypothesis that the throws will be uniformly distributed
is an hypothesis about the operation of those factors.
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Suppose someone says that a lever with arms of equal length must remain at rest under the influence of
equal and opposite forces, since there is no cause to make it move to one side rather than to the other. That only
means that if the lever moves to one side after we have ascertained the equality of the arms and the equal and
opposite nature of the forces, then we can't explain this on the basis of the preconditions we know or have assumed.
(The form that we call "explanation" must be asymmetrical: like the operation which makes "2a + 3b" out of "a +
b"). But on the basis of our presuppositions we can indeed explain the lever's continuance at rest.--Could we also
explain a swing to left and right with roughly equal frequency? No, because once again the swing involves
asymmetry; we would only explain the symmetry in this asymmetry. If the lever had rotated to the right with a
uniform motion, one could similarly have said: given the symmetry of the conditions I can explain the uniformity of
the motion, but not its direction.
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A lack of uniformity in the distribution of the throws is not to be explained by the symmetry of the die. It is
only to this extent that the symmetry explains the uniformity of the distribution.--For one can of course say: if the
numbers on the sides of the die have no effect, then the difference between them cannot explain an irregularity in the
distribution; and of course similar circumstances can't explain differences; and so to that extent one might infer a
regularity. But in that case why is there any difference at
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all between different throws? Whatever explains that must also explain their approximate regularity. It's just that the
regularity of the die doesn't interfere with that regularity.
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Suppose that a man throwing dice every day threw nothing but ones for a week, using dice that proved good
by every other method of testing and that gave the usual results when thrown by others. Has he grounds, now, for
supposing that there is a law of nature that he will always throw ones? Has he grounds for believing that it will go on
like this, or has he grounds for believing that this regularity can't last much longer? Has he reason to abandon the
game since it has become clear that he can only throw ones, or reason to play on since in these circumstances it is all
the more probable that he will throw a higher number at the next throw? In actual fact, he will refuse to accept the
regularity as a natural law: at least, it will have to go on for a long time before he will entertain the possibility. But
why? I believe it is because so much of his previous experience in life speaks against there being a law of nature of
such a sort, and we have--so to speak--to surmount all that experience, before embracing a totally new way of
looking at things.
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If we infer from the relative frequency of an event its relative frequency in the future, we can of course only
do that from the frequency which has in fact been so far observed. And not from one we have derived from
observation by some process or other for calculating probabilities. For the probability we calculate is compatible
with any frequency whatever that we actually observe, since it leaves the time open.
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When a gambler or insurance company is guided by probability, they aren't guided by the probability
calculus, since one can't be guided by this on its own, because anything that happens can be reconciled with it: no,
the insurance company is guided by a
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frequency actually observed. And that, of course, is an absolute frequency.
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8:
The concept "about"
Problem of the "heap"
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"He came from about there →."
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"About there is the brightest point of the horizon".
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"Make the plank about 2 m long".
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In order to say this, must I know of limits which determine the margin of tolerance of this length? Obviously
not. Isn't it enough e.g. to say "A margin of ± 1 cm is perfectly permissible; 2 would be too much"?--Indeed it's an
essential part of the sense of my proposition that I'm not in a position to give "precise" bounds to the margin. Isn't
that obviously because the space in which I am working here doesn't have the same metric as the Euclidean one?
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Suppose one wanted to fix the margin of tolerance exactly by experiment, by altering the length,
approaching the limits of the margin and asking in each case whether such a length would do or not. After a few
shortenings one would get contradictory results: at one time a point would be described as being within the limits,
and at another time a point closer in would be described as impermissible, each time perhaps with the remark that
the answers were no longer quite certain.
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It is the same sort of uncertainty as occurs in giving the highest point of a curve. We just aren't in Euclidean
space and here there isn't a highest point in the Euclidean sense. The answer will mean "The highest point is about
there" and the grammar of the word "about"--in this context--is part of the geometry of our space.
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Surely it is like the way the butcher weighs things only to the nearest ounce, though that is arbitrary and
depends on what are the customary counterweights. Here it is enough to know: it doesn't weigh more than P1 and it
doesn't weigh less than P2. One might say: in principle giving the weight thus isn't giving a
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number, but an interval, and the intervals make up a discontinuous series.
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Yet one might say: "at all events keep within ± 1 cm", thus setting an arbitrary limit.--If someone now said
"Right, but that isn't the real limit of the permissible tolerance; so what is?" the answer would be e.g. "I don't know
of any; I only know that ± 2 is too much".
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Imagine the following psychological experiment.
The subject is shown curves g1 g2 with a straight line A drawn across them. I will call the section of this line between
g1 and g2 a. Parallel to a we now draw b at an arbitrary distance and ask the subject whether he sees the section b as
bigger than a, or cannot any longer distinguish between the two lengths. He replies that b seems bigger than a. Next
we move closer to a, measuring half the distance from a to b and drawing c. "Do you see c as bigger than a?"
"Yes."--We halve the distance c-a and draw d. "Do you see d as bigger than a?" "Yes." We halve a-d. "Do you see e
as bigger than a?"--"No."--So we halve e-d. "Do you see f as bigger than e?"--"Yes."--So we halve e-f and draw h.
We might approach the line a from the left hand side as well and then say that what corresponds in Euclidean space
to a seen length a is not a single length but an interval of lengths, and in a similar way what corresponds to a single
seen position of a line (say the pointer of an instrument) is an interval of positions in Euclidean space; but this
interval has no precise limits. That means: it is bounded not by
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points, but by converging intervals which do not converge upon a point. (Like the series of binary fractions that we
get by throwing heads and tails.) The special thing about two intervals which are bounded in this blurred way
instead of by points is that in certain cases the answer to the question whether they overlap or are quite distinct is
"undecided"; and the question whether they touch, whether they have an end-point in common, is always a
senseless one since they don't have end-points at all. But one might say "they have de facto end-points", in the sense
in which the development of π has a de facto end. There is of course nothing mysterious about this property of
"blurred" intervals; the somewhat paradoxical character is explained by the double use of the word "interval".
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The case is the same as that of the double use of the word "chess" to mean at one time the totality of the
currently valid chess rules, and at another time the game invented in Persia by N. N. which developed in such and
such a way. In one case it is nonsensical to talk of a development of the rules of chess and in another not. What we
mean by "the length of a measured section" may be either what results from a particular measurement which I carry
out today at 5 o'clock--in that case there is no "± etc." for this assignment of length--or, something to which
measurements approximate, etc.; in the two cases the word "length" is used with quite different grammars. So too
the word "interval" if what I mean by an interval is at one time something fixed and at another time something in
flux.
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But we must not be surprised that an interval should have such a strange property; for we're now just using
the word "interval" in a sense different from the usual one. And we can't say that we have discovered new properties
of certain intervals, any more than we would discover new properties of the king in chess if we altered the rules of
the game while keeping the designation "chess" and "king". (On the other hand cf. Brouwer on the law of excluded
middle.)
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Page 239
I) the intervals are separate
II) they are searate with a de facto contact
III) undecided
IV) undecided
V) undecided
VI) they overlap
VII) they overlap
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So basically that experiment gives what we have called a "blurred" interval; on the other hand of course we
could conceive experiments which would give a sharp interval instead. Suppose we moved a straight-edge from the
starting position b, in the direction of a, keeping it parallel to b, until our subject began to display a particular
reaction; in that case we could call the point at which the reaction first occurs the limit of our strip. Likewise we
might of course call the result of a weighing "the weight of a body" and in that sense there would be an absolutely
exact weighing, that is, one whose result did not have the form "W ± w". We would thus have altered the form of
our expression, and we would have to say that the weight of bodies varied according to a law that was unknown to
us. (The distinction between "absolutely exact" weighing and "essentially inexact weighing" is a grammatical
distinction connected with two different meanings of the expression "result of weighing").
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Page 240
The indeterminacy of the word "heap". I could give as a definition: a body of a certain form and consistency
etc. is a heap, if it has a volume of K cubic metres, or more; anything less than that I will call a heaplet. In that case
there is no largest heaplet; that means, it is senseless to speak of a largest heaplet. Conversely, I could decide:
whatever is bigger than K cubic metres is to be a heap, and in that case the expression "the smallest heap" has no
meaning. But isn't this distinction an idle one? Certainly--if by the volume we mean a result of measurement in the
normal sense; for such a result has the form "V ± v". But otherwise the distinction would be no more idle than the
distinction between threescore apples and 61 apples.
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About the problem of the "heap": Here, as in similar cases, one might think that there is an official concept
like the official length of a pace; say "A heap is anything that is bigger than half a cubic metre". But this would still
not be the concept we normally use. For that there exists no delimitation (and if we fix one, we are altering the
concept); it is just that there are cases that we count as within the extension of the concept, and cases that we no
longer count as within the extension of the concept.
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"Make me a heap of sand here."--"Fine, that is certainly something he would call a heap." I was able to obey
the command, so it was in order. But what about this command "Make me the smallest heap you would still call a
heap"? I would say: that is nonsense; I can only determine a de facto upper and lower limit.
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Part II:
On Logic and Mathematics
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I LOGICAL INFERENCE
Is it because we understand the propositions that we know that q entails p?
Does a sense give rise to the entailment?
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p.q. = .p means "q follows from p".
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(∃x).fx ∨ fa. = .(∃x).fx, (∃x).fx.fa. = .fa. How do I know that? (Because for the equation above I gave a kind
of proof). One might say something like: "I just understand '(∃x).fx'". (An excellent example of what "understand"
means).
Page 243
But I might equally ask "How do I know that (∃x).fx follows from fa?" and answer "because I understand
'(∃x).fx'."
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But really how do I know that it follows?--Because that is the way I calculate.
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How do I know that (∃3x).fx follows from fa? Is it that I as it were see behind the sign "(∃x).fx", that I see
the sense lying behind it and see from that that it follows from fa? Is that what understanding is?
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No, what that equation expresses is a part of the understanding (that is thus unpacked before my eyes).
Page 243
Compare the idea that understanding is first of all grasping in a flash something which then has to be
unpacked like that.
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If I say "I know that (∃x).fx follows, because I understand it" that would mean, that when I understand it, I
see something different from the sign I'm given, a kind of definition of the sign which gives rise to the entailment.
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Page 244
Isn't it rather that the connection is set up and prescribed by the equations? For there is no such thing as a
hidden connection.
Page 244
But, I used to think, mustn't (∃x).fx be a truth function of fa for that to be possible, for that connection to be
possible?
Page 244
For doesn't (∃x).Fx ∨ Fa = (∃x).fx simply say that fa is already contained in (∃x).fx? Doesn't it show the
connection between the fa and the (∃x).fx? Not unless (∃x).fx is defined as a logical sum (with fa as one of the terms
of the sum).--If that is the case, then (∃x).fx is merely an abbreviation.
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In logic there is no such thing as a hidden connection.
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You can't get behind the rules, because there isn't any behind.
Page 244
fE.fa. = fa. Can one say: that is only possible if fE follows from fa? Or must one say: that settles that fE is to
follow from fa?
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If the former, it must be the structure that makes it follow, say because fE is so defined as to have the
appropriate structure. But can the entailment really be a kind of result of the visible structure of the signs, in the way
that a physical reaction is the result of a physical property? Doesn't it rather always depend on stipulations like the
equation fE.fa. = .fa? Can it be read off from p ∨ q that
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it follows from p, or only from the rules Russell gives for the truth-functions?
Page 245
And why should the rule fE.fa. = .fa be an effect of another rule rather than being itself the primary rule?
Page 245
For what is "fE must somehow contain fa" supposed to mean? It doesn't contain it, in so far as we can work
with fE without mentioning fa; but it does in so far as the rule fE.fa = .fa holds.
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But the idea is that fE.fa. = fa can only hold in virtue of a definition of fE.
Page 245
That is, I think, because otherwise it looks, wrongly, as if a further stipulation had been made about fE after it
had already been introduced into the language. But in fact there isn't any stipulation left for future experience to
make.
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And the definition of fE in terms of "all particular cases" is no less impossible than the enumeration of all
rules of the form fE.fx. = fx.
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Indeed the individual equations fE.fx. = fx are just precisely an expression of this impossibility.
Page 245
If we are asked: but is it now really certain that it isn't a different calculus being used, we can only say: if that
means "don't we use other calculi too in our real language?" I can only answer "I don't know any others at present".
(Similarly, if someone asked "are these all the calculi of contemporary mathematics?" I might say "I don't remember
any others, but I can read it up and find out more exactly"). But the question cannot mean "can no other calculus be
used?" For how is the answer to that question to be discovered?
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A calculus exists when one describes it.
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Page 246
Can one say 'calculus' is not a mathematical concept?
Page 246
If I were to say "whether p follows from q must result from p and q alone": it would have to mean this: that p
follows from q is a stipulation that determines the sense of p and q, not some extra truth that can be asserted about
the sense of both of them. Hence one can indeed give rules of inference, but in doing so one is giving for the use of
the written signs rules which determine their as yet undetermined sense; and that means simply that the rules must
be laid down arbitrarily, i.e. are not to be read off from reality like a description. For when I say that the rules are
arbitrary, I mean that they are not determined by reality in the way the description of reality is. And that means: it is
nonsense to say that they agree with reality, e.g. that the rules for the words "blue" and "red" agree with the facts
about those colours etc.†1
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What the equation p.q = p really shows is the connection between entailment and the truth-functions.
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2
"If p follows from q, then thinking that q must involve thinking that p."
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Remember that a general proposition might entail a logical sum of a hundred or so terms, which we certainly didn't
think of when we uttered the general proposition. Yet can't we say that it follows from it?
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"What follows from a thought must be involved in thinking it. For there is nothing in a thought that we aren't
aware of while we are thinking it. It isn't a machine which might be explored with unexpected results, a machine
which might achieve something that couldn't be read off from it. That is, the way it works is logical, it's quite
different from the way a machine works. Qua thought, it contains nothing more than was put into it. As a machine
functioning causally, it might be believed capable of anything; but in logic we get out of it only what we meant by
it."
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If I say that the square is entirely white, I don't think of ten smaller rectangles contained in it which are white,
and I can't think of "all" rectangles or patches contained in it. Similarly in the proposition "he is in the room" I don't
think of a hundred possible positions he might be in and certainly not of all possible positions.
Page 247
"Wherever you hit the target you've won. You've hit it in the upper right hand section, so... "
Page 247
At first sight there seem to be two kinds of deduction: in one of them the premise mentions everything the
conclusion does and in the other not. An instance of the first kind is the inference from p.q to q; an instance of the
second is the inference; the whole stick is white, so the middle third of it is white too. This conclusion mentions
boundaries that are not mentioned in the first proposition. (That is dubious.) Again, if I say "If you hit the target
anywhere in this circle you will win the prize..." and then "You have hit it here, so..." the place mentioned in the
second proposition was not prescribed in the first. The target after the shot stands in a certain internal relation to the
target as I saw it before, and that
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relation consists in the shot's falling within the bounds of the general possibility that we foresaw. But the shot was
not in itself foreseen and did not occur, or at least need not have occurred, in the first picture. For even supposing
that at the time I thought of a thousand definite possibilities, it was at least possible for the one that was later realised
to have been omitted. And if the foreseeing of that possibility really had been essential, the overlooking of this single
case would have given the premise the wrong sense and the conclusion wouldn't any longer follow from it.
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On the other hand you don't add anything to the proposition "Wherever you hit this circle..." by saying
"Wherever you hit this circle, and in particular if you hit the black dot..." If the black dot was already there when the
first proposition was uttered, then of course it was meant too; and if it wasn't there, then the actual sense of the
proposition has been altered by it.
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But what is it supposed to mean to say "If one proposition follows from another, thinking the second must
involve thinking the first", since in the proposition "I am 170 cm tall" it isn't necessary to think of even a single one
of the negative statements of height that follow from it?
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"The cross is situated thus on the straight line: --
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"So it is between the strokes".
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"It is 16½° here"--"So it is certainly more than 15°"
Page 248
Incidentally, if you are surprised that one proposition can follow from another even though one doesn't think
of the former while thinking of the latter, you should consider that p ∨ q follows from p, and I certainly don't think
all propositions of the form p ∨ ξ while I am thinking p.
Page 248
The whole idea that a proposition has to be thought along with any proposition that entails it rests on a false,
psychologising notion. We must concern ourselves only with what is contained in the signs and the rules.
Page Break 249
Page 249
If the criterion for p's following from q consists in "thinking of p being involved in thinking of q" then while
thinking of the proposition "in this box there are 105 grains of sand", you are thinking also of the 105 sentences "In
this box there is one grain of sand" "... 2 grains of sand", etc. etc. What's the criterion here for the thought of one
proposition's being involved in the thought of another?
Page 249
And what about a proposition like "There is a patch (F) between the limits AA"?
Page 249
Doesn't it follow from that that F is also between BB and CC and so on? Don't infinitely many propositions
follow from a single one? Does that make it infinitely significant?--From the proposition "There is a patch between
the limits AA" there follow as many propositions of the type "there is a patch between the limits BB" as I write
out--and no more than I write out. Similarly, from p there follow as many propositions of the form p ∨ ξ as I write
out (or utter etc.).
Page 249
(A proof by induction proves as many propositions of the form... as I write out.)
Page Break 250
3
The case of infinitely many propositions following from a single one
Page 250
Is it impossible that infinitely many propositions should follow from a single one--in the sense, that is, that
we might go on ad infinitum constructing new propositions from a single one according to a rule?
Page 250
Suppose that we wrote the first thousand propositions of the series in conjunction. Wouldn't the sense of this
product necessarily approximate more closely to the sense of our first proposition than the product of the first
hundred propositions? Wouldn't we obtain an ever closer approximation to the first proposition the further we
extended the product? And wouldn't that show that it can't be the case that from one proposition infinitely many
others follow, since I can't understand even the product with 1010 terms and yet I understood the proposition to
which the product with 10100 terms is a closer approximation than the one with 1010 terms?
Page 250
We imagine, perhaps, that the general proposition is an abbreviated expression of the product. But what is
there in the product to abbreviate? It doesn't contain anything superfluous.
Page 250
If we need an example of infinitely many propositions following from a single one, perhaps the simplest is
the way in which "a is red" entails the negation of all propositions that ascribe a different colour to a. The negative
propositions are certainly not contained in the thought of the single positive one. Of course we might say that we
don't distinguish infinitely many shades of colour; but the question is whether the number of shades of colour we
distinguish has anything at all to do with the complexity of the first sentence: is it more or less complex the more or
fewer colours we distinguish?
Page 250
Wouldn't this be what we'd have to say: it's only when a proposition exists that it follows from it. It's only
when we have constructed ten propositions following from the first one that ten propositions do follow from it.
Page Break 251
Page 251
I want to say that one proposition doesn't follow from another until it is confronted with it. The "etc ad
infinitum" indicates only the possiblity [[sic]] of constructing propositions following from the first; it doesn't yield a
definite number of such propositions.
Page 251
So mightn't I simply say: it is because it is impossible to write out infinitely many propositions (i.e. to say
that is a piece of nonsense) that infinitely many propositions don't follow from a single proposition.
Page 251
What about the proposition "the surface is white from A to B"? It does follow from it that the surface is
white from A' to B'. It needn't be a seen patch of white that is in question; and certainly the inference from the first
proposition to the second is often drawn. Someone says to me "I have painted the patch white from A to B" and
then I say "so it's certainly painted white from A' to B'".
Page 251
It must be possible to say a priori that F(A'B') would follow from F(AB).
Page 251
If the lines A' and B' exist, then the second proposition certainly does follow from the first (in that case the
compositeness is already there in the first proposition); but in that case it is only as many propositions as correspond
to its compositeness that follow from the first proposition (and so never infinitely many).
Page 251
"The whole is white, therefore a part bounded by such and such a line is white." "The whole was white, so
that part of it also was white even if I didn't then perceive it bounded within it."
Page 251
"A surface seen as undivided has no parts."
Page 251
But let's imagine a ruler laid against the surface, so that the
Page Break 252
appearance we are presented with is first and then and then
. It doesn't at all follow from the first strip's being entirely white that in the second and the
third everything except the graduating lines is white.
Page 252
"If you hit the target anywhere within the circle, you have won."
Page 252
"I think you will hit the target somewhere within the circle."
Page 252
Someone might ask about the first proposition: how do you know? Have you tried all possible places? And
the answer would have to be: that isn't a proposition at all, it is a general stipulation.
Page 252
The inference doesn't go like this: "If the shot hits the target anywhere, you have won. You have hit the target
there, so you have won". For where is this there? Is it marked out in any way other than by the shot--say by a
circle? And was that already there on the target beforehand? If not, then the target has changed; if so, it must have
been foreseen as a possible place to hit. We should rather say: "You have hit the target, so..."
Page 252
The place on the target does not necessarily have to be given by a mark on the target, like a circle. For there
are always descriptions like "nearer the centre", "nearer the edge", "on the right side at the top", etc. Wherever the
target is hit such descriptions must always be possible. (But there are not "infinitely many" such descriptions.)
Page 252
Does it make sense to say: "But if you hit the target, you must hit it somewhere" or "Wherever he hits the
surface it won't be a surprise, we won't have to say 'I didn't expect that. I didn't know there was such a place'?" What
that means is that it can't be a geometrical surprise.
Page Break 253
Page 253
What sort of proposition is: "On this strip you may see all shades of grey between black and white"? Here it
looks at first glance as if we're talking about infinitely many shades.
Page 253
Indeed, we are apparently confronted here by the paradox that we can, of course, only distinguish a finite
number of shades, and naturally the distinction between them isn't infinitely slight, and yet we see a continuous
transition.
Page 253
It is just as impossible to conceive of a particular grey as being one of the infinitely many greys between
black and white as it is to conceive of a tangent t as being one of the infinitely many transitional stages in going from
t1 to t2. If I see a ruler roll around the circle from t1 to t2 I see--if its motion is continuous--none of the intermediate
positions in the sense in which I see t when the tangent is at rest; or else I see only a finite number of such positions.
But if in such a case I appear to infer a particular case from a general proposition, then the general proposition is
never derived from experience, and the proposition isn't a real proposition.
Page 253
If, e.g., I say "I saw the ruler move from t1 to t2 therefore I must have seen it at t" this doen't [[sic]] give us a
valid logical inference. That is, if what I mean is that the ruler must have appeared to me at t and so, if I'm talking
about the position in visual space, then it doesn't
Page Break 254
in the least follow from the premise. But if I'm talking about the physical ruler, then of course it's possible for the
ruler to have skipped over position t and yet for the phenomenon in visual space to have remained continuous.
Page Break 255
4
Can an experience show that one proposition follows from another?
Page 255
The only essential point is that we cannot say that it was through experience we were made aware of an extra
application of grammar. For in making that statement we would have to describe the application, and even if this is
the first time I have realised that the description is true I must have been able to understand it even before the
experience.
Page 255
It is the old question: how far can one now speak of an experience that one is not now having?
Page 255
What I cannot foresee I can not foresee.
Page 255
And what I can now speak of, I can now speak of independently of what I can't now speak of.
Page 255
Logic just is always complex.
Page 255
"How can I know everything that's going to follow?" What I can know then, I can also know now.
Page 255
But are there general rules of grammar, or only rules for general signs?
Page 255
What kind of thing in chess (or some other game) would count as a general rule or a particular rule? Every
rule is general.
Page 255
Still, there is one kind of generality in the rule that p ∨ q follows from p and a different kind in the rule that
every proposition of the form p, ~~p, ~~~~p... follows from p.q. But isn't the generality of the rule for the knight's
move different from the generality of the rule for the beginning of a game?
Page 255
Is the word "rule" altogether ambiguous? So should we talk only about particular cases of rules, and stop
talking about rules in general, and indeed about languages in general?
Page 255
"If F1(a) [= a has the colour F1] entails ~F2 (a) then the possibility of the second proposition must have been
provided for in the
Page Break 256
grammar of the first (otherwise how could we call F1 and F2 colours?)
Page 256
"If the second proposition as it were turned up without being expected by the first it couldn't possibly follow
from it."
Page 256
"The first proposition must acknowledge the second as its consequence. Or rather they must be united in a
single grammar which remains the same before and after the inference."
Page 256
(Here it is very difficult not to tell fairy tales about symbolic processes, just as elsewhere it is hard not to tell
fairy tales about psychological processes. But everything is simple and familiar (there is nothing new to be
discovered). That is the terrible thing about logic, that its extraordinary difficulty lies in the fact that nothing must be
constructed, and everything is already present familiar.)
Page 256
"No proposition is a consequence of p unless p acknowledges it as its consequence."
Page 256
Whether a proposition entails another proposition must be clear from the grammar of the proposition and
from that alone. It cannot be the result of any insight into a new sense: only of an insight into the old sense. It is not
possible to construct a new proposition that follows from the old one which could not have been constructed
(perhaps without knowing whether it was true or false) when the old one was constructed. If a new sense were
discovered and followed from the first proposition, wouldn't that mean that that proposition had altered its sense?
Page Break 257
II GENERALITY
The proposition "The circle is in the square" is in a certain sense independent of the assignment of a particular
position. (In a certain sense it is totally unconnected.)
Page 257
I would like to say: a general picture like |0| does not have the same metric as a particular one.
Page 257
In the general sign "|0|" the distances play no greater part than they do in the sign "aRb".
Page 257
The drawing |0| can be looked on as a representation of the "general case". It is as if it were not in a
measurable space: the distances between the circle and the lines are of no consequence. The picture, taken thus, is
not seen as occurring in the same system as when one sees it as the representation of a particular position of the
circle between the lines. Or rather, taken thus, it is a part of a different calculus. The rules that govern variables are
not the same as those that govern their particular values.
Page 257
"How do you know he is in the room?" "Because I put him in and there is no way he can get out." Then your
knowledge of the general fact that he is somewhere in the room has the same multiplicity as that reason.
Page 257
Let us take the particular case of the general state of affairs of the cross being between the end-lines.
Page 257
Each of these cases, for instance, has its own individuality. Is there any way in which this individuality enters
into the sense of the general sentence? Obviously not.
Page 257
'Being between the lines, or the walls' seems something simple and the particular positions (both the visual
appearances and the
Page Break 258
positions established by measurement) seem quite independent of it.
Page 258
That is, when we talk about the individual (seen) positions we appear to be talking about something quite
different from the topic of the general proposition.
Page 258
There is one calculus containing our general characterization and another containing the disjunction. If we
say that the cross is between the lines we don't have any disjunction ready to take the place of the general
proposition.
Page 258
If we consider a general proposition like "the circle is in the square" it appears time and again that the
assignment of a position in the square is not (at least so far as visual space is concerned) a more precise specification
of the statement that the circle is in the square any more than a statement of the colour of a material is a more
precise specification of a statement of its hardness.--Rather, "in the square" appears a complete specification which
in itself does not admit of any more precise description. Now of course the statements about the circle are not related
to each other like the statements about colour and hardness, and yet that feeling is not baseless.
Page 258
The grammatical rules for the terms of the general proposition must contain the multiplicity of possible
particular cases provided for by the proposition. What isn't contained in the rules isn't provided for.
Page 258
All these patterns might be the same state of affairs distorted. (Imagine the two white strips and the middle
black strip as elastic.)
Page 258
Does fa's following from (x). fx mean that a is mentioned in
Page Break 259
(x). fx? Yes, if the general proposition is meant in such a way that its verification consists in an enumeration.
Page 259
If I say "there is a black circle in the square", it always seems to me that here again I have something simple
in mind, and don't have to think of different possible positions or sizes of the circle. And yet one may say: if there is
a circle in the square, it must be somewhere and have some size. But in any case there cannot be any question of my
thinking in advance of all the possible positions and sizes.--It is rather that in the first proposition I seem to put them
through a kind of sieve so that "circle in a square" corresponds to a single impression, which doesn't take any
account of the where etc., as if it were (against all appearance) something only physically, and not logically,
connected with the first state of affairs.
Page 259
The point of the expression "sieve" is this. If I look at a landscape or something similar through a glass which
transmits only the distinction between brightness and darkness and not the distinctions between colours, such a
glass can be called a sieve; and if one thinks of the square as being looked at through a glass which transmits only
the distinction "circle in the square or not in the square" and no distinction between positions or sizes of the circle,
here too we might speak of a sieve.
Page 259
I would like to say that in the proposition "there is a circle in the square" the particular positions are not
mentioned at all. In the picture I don't see the position, I disregard it, as if the distances from the sides of the square
were elastic and their lengths of no account.
Page 259
Indeed, can't the patch actually be moving in the square? Isn't that just a special case of being in the square?
So in that case it wouldn't be true that the patch has to be in a particular position in the square if it is there at all.
Page 259
I want to say that the patch seems to have a relation to the edge that is independent of its distance.--Almost
as if I were using a
Page Break 260
geometry in which there is no such thing as distance, but only inside and outside. Looked at in this way, there is no
doubt that the two pictures and are the same.
Page 260
By itself the proposition "The patch is in the square" does no more than hold the patch in the square, as it
were; it is only in this way that it limits the patch's freedom; within the square it allows it complete freedom. The
proposition constructs a frame that limits the freedom of the patch but within the frame it leaves it free, that is, it has
nothing to do with its position. For that to be so the proposition must have the logical nature of the frame (like a box
enclosing the patch). And so it has, because I could explain the proposition to someone and set out the possibilities,
quite independently of whether such a proposition is true or not, independently of a fact.
Page 260
"Wherever the patch is in the square..." means "as long as it is in the square..." and here all that is meant is
the freedom (lack of restraint) in the square, not a set of positions.
Page 260
Of course between this freedom and the totality of possibilities, there is a logical similarity (formal analogy),
and that is why the same words are often used in the two cases ("all", "every", etc.).
Page 260
"No degrees of brightness below this one hurt my eyes." Test the type of generality.
Page 260
"All points on this surface are white." How do you verify that?--then I will know what it means.
Page Break 261
6
The proposition "The circle is in the square" is not a disjunction of cases.
Page 261
If I say the patch is in the square, I know--and must know--that it may have various possible positions. I know too
that I couldn't give a definite number of all such positions. I do not know in advance how many positions "I could
distinguish".--And trying it out won't tell me what I want to know here either.
Page 261
The darkness veiling the possible positions etc. is the current logical situation, just as dim lighting is a
particular sort of lighting.
Page 261
Here it always seems as if we can't quite get an overall view of a logical form because we don't know how
many or what possible positions there are for the patch in the square. But on the other hand we do know, because
we aren't surprised by any of them when they turn up.
Page 261
Of course "position of the circle in this square" isn't a concept which particular positions fall under as objects.
You couldn't discover objects and ascertain that they were positions of the circle in the square which you didn't
know about beforehand.
Page 261
Incidentally, the centre and other special positions in the circle are quite analogous to the primary colours on
the colour scale. (This comparison might be pursued with profit.)
Page 261
Space is as it were a single possibility; it doesn't consist of several possibilities.
Page 261
So if I hear that the book is somewhere on the table, and then find it in a particular position, it isn't possible
for me to be surprised and say "oh, I didn't know that there was this position"; and yet I hadn't foreseen this
particular position i.e. envisaged it in
Page Break 262
advance as a particular possibility. It is physical, not logical possibilities that take me by surprise!
Page 262
But what is the difference between "the book is somewhere on the table" and "the event will occur sometime
in the future?" Obviously the difference is that in the one case we have a sure method of verifying whether the book
is on the table, while in the other case there is no such method. If a particular event were supposed to occur at one of
the infinitely many bisections of a line, or better, if it were supposed to occur when we cut the line at a single point,
not further specified, and then waited a minute at that point, that statement would be as senseless as the one about
the infinite future.
Page 262
Suppose I stated a disjunction of so many positions that it was impossible for me to see a single position as
distinct from all those given; would that disjunction be the general proposition (∃x).fx? Wouldn't it be a kind of
pedantry to continue to refuse to recognize the disjunction as the general proposition? Or is there an essential
distinction, and is the disjunction totally unlike the general proposition?
Page 262
What so strikes us is that the one proposition is so complicated and the other so simple. Or is the simple one
only an abbreviation for the more complicated one?
Page 262
What then is the criterion for the general proposition, for the circle's being in the square? Either, nothing that
has anything to do with a set of positions (or sizes) or something that deals with a finite number of such positions.
Page 262
If one says that the patch A is somewhere between the limits B and C, isn't it obviously possible to describe
or portray a number of positions of A between B and C in such a way that I see the succession of all the positions as
a continuous transition? And in
Page Break 263
that case isn't the disjunction of all those N positions the very proposition that A is somewhere between B and C?
Page 263
But what are these N pictures really like? It is clear that a picture must not be visually discernible from its
immediate successor, or the transition will be discontinuous.
Page 263
The positions whose succession I see as a continuous transition are positions which are not in visual space.
Page 263
How is the extension of the concept "lying between" determined? Because it has to be laid down in advance
what possibilities belong to this concept. As I say, it cannot be a surprise that I call that too "lying between". Or:
how can the rules for the expression "lie between" be given when I can't enumerate the cases of lying between? Of
course that itself must be a characteristic of the meaning of the expression.
Page 263
Indeed if we wanted to explain the word to someone we wouldn't try to do so by indicating all particular
instances, but by showing him one or two such instances and intimating in some way that it wasn't a question of the
particular case.
Page 263
It is not only that the enumeration of positions is unnecessary: in the nature of things there can be no
question of such an enumeration here.
Page 263
Saying "The circle is either between the two lines or here" (where "here" is a place between the lines)
obviously means no more than "The circle is between the two lines", and the rider "or here" is superfluous. You will
say: the "here" is already included in the "somewhere". But that is strange, since it isn't mentioned in it.
Page 263
There is a particular difficulty when the signs don't appear to say what the thought grasps, or the words don't
say what the thought appears to grasp.
Page Break 264
Page 264
As when we say "this theorem holds of all numbers" and think that in our thought we have comprehended
all numbers like apples in a box.
Page 264
But now it might be asked: how can I know in advance which propositions entail this general proposition, if I
can't specify the propositions?
Page 264
But can one say "We can't say which propositions entail this proposition"? That sounds like: we don't know.
But of course that isn't how it is. I can indeed say, and say in advance, propositions that entail it. "Only not all of
them." But that just has no meaning.
Page 264
There is just the general proposition and particular propositions (not the particular propositions). But the
general proposition does not enumerate particular propositions. In that case what characterizes it as general, and
what shows that it doesn't simply comprise the particular propositions we are speaking of in this particular case?
Page 264
It cannot be characterized by its instantiations, because however many we enumerate, it could still be
mistaken for the product of the cited cases. Its generality, therefore, lies in a property (a grammatical property) of the
variables.
Page Break 265
7
The inadequacy of the Frege-Russell notation for generality
Page 265
The real difficulty lies in the concept of "(∃n)" and in general of "(∃x)". The original source of this notation is
the expression of our word-language: "There is a... with such and such properties". And here what replaces the dots
is something like "book from my library" or "thing (body) in this room", "word in this letter", etc. We think of
objects that we can go through one after the other. As so often happens a process of sublimation turned this form
into "there is an object such that..." and here too people imagined originally the objects of the world as like 'objects'
in the room (the tables, chairs, books, etc.), although it is clear that in many cases the grammar of this "(∃x), etc." is
not at all the same as the grammar of the primitive case which serves as a paradigm. The discrepancy between the
original picture and the one to which the notation is now applied becomes particularly palpable when a proposition
like "there are two circles in this square" is rendered as "there is no object that has the property of being a circle in
this square without being the circle a or the circle b" or "there are not three objects that have the property of being a
circle in this square". The proposition "there are only two things that are circles in this square" (construed on the
model of the proposition "there are only two men who have climbed this mountain") sounds crazy, with good
reason. That is to say, nothing is gained by forcing the proposition "there are two circles in this square" into that
form; it only helps to conceal that we haven't cleared up the grammar of the proposition. But at the same time the
Russellian notation here gives an appearance of exactitude which makes people believe the problems are solved by
putting the proposition into the Russellian form. (This is no less dangerous than using the word "probably" without
further investigation into the use of the word in this particular case. For understandable reasons the word
Page Break 266
"probably", too, is connected with an idea of exactitude.)
Page 266
"One of the four legs of this table doesn't hold", "There are Englishmen with black hair", "There is a speck on
this wall" "The two pots have the same weight", "There are the same number of words on each of the two pages". In
all these cases in the Russellian notation the "(∃...)..." is used, and each time with a different grammar. The point I
want to make is that nothing much is gained by translating such a sentence from word-language into Russellian
notation.
Page 266
It makes sense to say "write down any cardinal number" but not "write down all cardinal numbers". "There is
a circle in the square" [(∃x).fx)] makes sense, but not ~∃x.~fx: "all circles are in the square." "There is a red circle on
a background of a different colour" makes sense, but not "there isn't a background-colour other than red that doesn't
have a red circle on it."
Page 266
"In this square there is a black circle". If this proposition has the form "(∃x).x is a black circle in a square"
what sort of thing is it that has the property of being a black circle (and so can also have the property of not being a
black circle)? Is it a place in the square? But then there is no proposition "(x).x is a black..." On the other hand the
proposition could mean "There is a speck in the square that is a black circle". How is that proposition verified? Well,
we take the different specks in the square in turn and investigate whether they are quite black and circular. But what
kind of proposition is "There isn't a speck in the square"? For if in the former case the 'x' in '(∃x)' meant 'speck in the
square', then though "(∃x).fx" is a possible proposition both "(∃x)" and "~(∃x)" are not. Or again, I might ask: what
sort of thing is it that has (or does not have) the property of being a speck in the square?
Page 266
And if we can say "There is a speck in the square" does it then also make sense to say "All specks are in the
square"? All which?
Page Break 267
Page 267
Ordinary language says "In this square there is a red circle"; the Russellian notation says "There is an object
which is a red circle in this square". That form of expression is obviously modelled on "There is a substance which
shines in the dark" "There is a circle in this square which is red".--Perhaps even the expression "there is" is
misleading. "There is" really means the same as "Among these circles there is one..." or "... there exists one...".
Page 267
So if we go as far as we can in the direction of the Russellian mode of expression and say "In this square
there is a place where there is a red circle", that really means, among these places there is one where... etc.
Page 267
(In logic the most difficult standpoint is that of sound common sense. For in order to justify its view it
demands the whole truth; it will not help by the slightest concession or construction.)
Page 267
The correct expression of this sort of generality is therefore the expression of ordinary language "There is a
circle in the square", which simply leaves the position of the circle open (leaves it undecided). ("Undecided" is a
correct expression, since there just has not been any decision.)
Page Break 268
8
Criticism of my former view of generality
Page 268
My view about general propositions was that (∃x).φx is a logical sum and that though its terms aren't
enumerated here, they are capable of being enumerated (from the dictionary and the grammar of language).
Page 268
For if they can't be enumerated we don't have a logical sum. (A rule, perhaps, for the construction of logical
sums).
Page 268
Of course, the explanation of (∃x).φx as a logical sum and of (x).φx as a logical product is indefensible. It
went with an incorrect notion of logical analysis in that I thought that some day the logical product for a particular
(x).φx would be found.--Of course it is correct that (∃x).φx behaves in some ways like a logical sum and (x).φx like a
product; indeed for one use of words "all" and "some" my old explanation is correct,--for instance for "all the
primary colours occur in this picture" or "all the notes of the C major scale occur in this theme". But for cases like
"all men die before they are 200 years old" my explanation is not correct. The way in which (∃x).φx behaves like a
logical sum is expressed by its following from φa and from φa ∨ φb, i.e. in the rules
(∃x).φx:φa. = .φa and
(∃x).φx:φa ∨ φb. = .φa ∨ φb
Page 268
From these rules Russell's fundamental laws follow as tautologies:
φx.⊃.(∃z).φz
φx ∨ φy.⊃.(∃z).φz
Page 268
For (∃x).φx we need also the rules:
(∃x).φx ∨ ψx. = .(∃x).φx.∨.(∃x).ψx
(∃x,y)φx.ψy.∨.(∃x).φx.ψx. = .(∃x).φx:(∃x).ψx.
Page 268
Every such rule is an expression of the analogy between (∃x).φx and a logical sum.
Page Break 269
Page 269
Incidentally, we really could introduce a notation for (∃x).φx in which it was replaced by a sign "φr ∨ φs ∨
φt..." which could then be used in calculation like a logical sum; but we would have to provide rules for reconverting
this notation at any time into the "(∃x).φx" notation and thus distinguishing the sign "φa ∨ φb ∨ φc..." from the sign
for a logical sum. The point of this notation could simply be to enable us to calculate more easily with (∃x).φx in
certain cases.
Page 269
If I am right, there is no concept "pure colour"; the proposition "A's colour is a pure colour" simply means
"A is red, or yellow, or blue, or green". "This hat belongs either to A or B or C" is not the same proposition as "This
hat belongs to a person in this room" even when in fact only A, B and C are in the room, for that itself is something
that has to be added.--"On this surface there are two pure colours" means: on this surface there is red and yellow, or
red and green, or... etc.
Page 269
If this means I can't say "there are 4 pure colours", still the pure colours and the number 4 are somehow
connected with each other and that must express itself in some way.--For instance, I may say "on this surface I see 4
colours: yellow, blue, red, green".
Page 269
The generality notation of our ordinary language grasps the logical form even more superficially than I earlier
believed. In this respect it is comparable with the subject-predicate form.
Page 269
Generality is as ambiguous as the subject-predicate form.
Page 269
There are as many different "alls" as there are different "ones".
Page 269
So it is no use using the word "all" for clarification unless we know its grammar in this particular case.
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9
The explanation of generality by examples
Page 270
Let us think how we explain the concept plant. We show someone several objects and say they are plants;
then he points to another object and asks "is that a plant too?" and we reply "yes, that too" etc. I would once have
said that he has now seen in what he has been shown the concept 'plant'--the common element--and that he does not
see the examples used in the explanation in the same way when he sees the concept in them as when he views them
just as representatives of a particular shape and colour or the like. (Just as I also used to say that when he
understands variables as variables he sees something in them which he doesn't see in the sign for the particular case).
But the notion of "seeing in" is taken from the case in which I see a figure like |||| differently "phrased". In that case, I
really do see different figures, but in a different sense; and what these have in common, apart from their similarity, is
their being caused by the same physical pattern.
Page 270
But this explanation cannot be applied without further ado to the case of the understanding of a variable or of
the examples illustrating the concept "plant". For suppose we really had seen something in them that we don't see in
plants that are shown only for their own sake, the question remains whether this, or any other, picture can entitle us
to apply them as variables. I might have shown someone the plants by way of explanation and given him in addition
a drug causing him to see the examples in the special way. (Just as it would be possible that a drunken man might
always see a group like |||| as ||| |). And this would give the explanation of the concept in an unambiguous manner,
and the specimens exhibited and the accompanying gestures would communicate to anyone who understood just
this picture. But that is not the way it is.--It may well be true that someone who sees a sign like |||||| as a numeral for
6 sees it differently (sees something different in it) from someone who views it only as a
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sign for "some", since he fixes his attention on something different; but what matters is the system of rules
governing the signs, and it isn't seeing the signs in a particular manner that is the essence of understanding.
Page 271
It would be possible to say "now I don't see it as a rose, but as a plant".
Page 271
Or "now I see it only as a rose, and no longer as this rose".
Page 271
"I see the patch merely in the square and no longer in a specific position."
Page 271
The mental process of understanding is of no interest to us (any more than the mental process of an
intuition).
Page 271
"Still, there's no doubt that someone who understands the examples as arbitrary cases chosen to illustrate the
concept doesn't understand the same as a man who regards them as a definitely bounded enumeration." Quite right,
but what does the first man understand that the second doesn't? Well, in the things he is shown he sees only
examples to illustrate certain features; he doesn't think that I am showing him the things for their own sake as well.--
Page 271
I would like to call the one class "logically bounded" and the other "logically unbounded".
Page 271
Yes, but is it really true that he sees only these features in the things? In a leaf, say, does he see only what is
common to all leaves? That would be as if he saw everything else blank like an uncompleted form with the essential
features ready printed. (But the function "f(...)" is just such a form.)
Page 271
But what sort of a process is it when someone shows me several different things as examples of a concept to
get me to see what is common to them, and when I look for it and then actually see it? He may draw my attention to
what is common.--But by doing this
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does he make me see the object differently? Perhaps so; for surely I may take a special look at one of the parts,
when otherwise I would have seen the whole with equal clarity. But this seeing is not the understanding of the
concept. For what we see isn't something with an empty argument place.
Page 272
One might also ask: Does a man who regards the sign "|||..." as a sign for the concept of number (in contrast
with "|||" to denote 3) see the first group of lines differently from the second? Even if he does see it differently
(perhaps, as it were, more blurred) does he see there anything like the essence of the concept of number? Wouldn't
that mean that he would actually have to be unable to distinguish "|||..." and "||||..." from each other? (As indeed he
would, if I had given him some drug that made him see the concept.)
Page 272
For if I say: by giving us a few examples he makes us see the common element in them and disregard the
rest, that really means that the rest falls into the background, as it were becomes paler (or altogether disappears--why
not?) and "the common element", say the oval shape, remains alone in the foreground.
Page 272
But that isn't the way it is. Apart from anything else, the multiplicity of examples would be no more than a
mechanical device, and once I had seen what I was supposed to, I could see it in a single example too. (As indeed
'(∃x).fx' itself contains only one example.)
Page 272
So it is the rules governing the example that make it an example.
Page 272
But by now at any rate, if someone says to me something like "make an egg shape" the bare concept word
without any illustration suffices to make itself understood (and the past history of this understanding is of no interest
to us): and I do not want to say that when I understand the command (and the word "egg") I see
Page Break 273
the concept of an egg before my mind's eye.
Page 273
When I make an application of the concept "egg" or "plant" there certainly isn't some general picture in front
of my mind before I do so, and when I hear the word "plant" it isn't that there comes before my mind a picture of a
certain object which I then describe as a plant. No, I make the application as it were spontaneously. Still, in the case
of certain applications I might say "No, I didn't mean that by 'plant'", or, "Yes, I meant that too". But does that mean
that these pictures came before my mind and that mentally I expressly rejected and admitted them?--And yet that is
what it looks like, when I say: "Yes, I meant all those things, but not that." But one might then ask: "But did you
foresee all those cases?" and then the answer might be "yes" or "no, but I imagined there must be something
between this form and that one" or the like. But commonly at that moment I did not draw any bounds, and they can
only be produced in a roundabout way after reflection. For instance, I say "Bring me a flower about so big"; he
brings one and I say: Yes, that is the size I meant. Perhaps I do remember a picture which came before my mind, but
it isn't that that makes the flower that has been brought acceptable. What I am doing is making an application of the
picture, and the application was not anticipated.
Page 273
The only thing of interest to us is the exact relationship between the example and the behaviour that accords
with it.
Page 273
The example is the point of departure for further calculation.
Page 273
Examples are decent signs, not rubbish or hocus-pocus.
Page 273
The only thing that interests us is the geometry of the mechanism. (That means, the grammar of its
description.)
Page 273
But how does it come out in our rules, that the instances of fx
Page Break 274
we are dealing with are not essentially closed classes?--Only indeed in the generality of the general rule.--How does
it come out that they don't have the same significance for the calculus as a closed group of primitive signs (like the
names of the 6 basic colours)? How else could it come out except in the rules given for them?--Suppose that in some
game I am allowed to help myself to as many pieces as I like of a certain kind, while only a limited number of
another kind is available; or suppose a game is unbounded in time but spatially bounded, or something similar. The
case is exactly the same. The distinction between the two different types of piece in the game must be laid down in
the rules; they will say about the one type that you can take as many pieces as you want of that kind. And I mustn't
look for another more restrictive expression of that rule.
Page 274
That means that the expression for the unboundedness of the particular instances in question will be a
general expression; there cannot be some other expression in which the other unconsidered instances appear in
some shadowy way.
Page 274
It is clear that I do not recognize any logical sum as a definition of the proposition "the cross is between the
lines". And that says everything that is to be said.
Page 274
There is one thing I always want to say to clarify the distinction between instances that are offered as
examples for a concept and instances that make up a definite closed group in the grammar. Suppose, after explaining
"a, b, c, d are books", someone says "Now bring me a book". If the person brings a book which isn't one of the ones
shown him he can still be said to have acted correctly in accordance with the rule given. But if what had been said
was "a, b, c, d, are my books.--Bring me one of my books", it would have been incorrect to bring a different one and
he would have been told "I told you that a, b, c, d are my books". In the first case it isn't against the rule to bring an
object other than those named,
Page Break 275
in the second case it is. But if in the order you named only a, b, c, and d, and yet you regarded the behaviour f(e) as
obeying the order, doesn't that mean that by F(a, b, c, d,...) you meant F(a, b, c, d, e) after all? Again, how are these
orders distinct from each other if the same thing obeys both of them?--But f(g) too would have been in accordance
with the order and not only f(e). Right, then your first order must have meant F(a, b, c, d, e, g) etc. Whatever you
bring me is something I could have included in a disjunction. So if we construct the disjunction of all the cases we
actually use, how would it differ syntactically from the general proposition? For we can't say: by the fact that the
general proposition is also made true by r (which doesn't occur in the disjunction), because that doesn't distinguish
the general proposition from a disjunction which contains r. (And every other similar answer too is impossible.) But
it will make sense to say: F(a, b, c, d, e) is the disjunction of all the cases we have actually used, but there are also
other cases (we won't of course, mention any) that make true the general proposition "F(a, b, c, d,...)". And here of
course we can't put the general proposition in place of F(a, b, c, d, e,).
Page 275
It is, by the way, a very important fact that the parenthesis in the previous paragraph "and every other similar
answer too is impossible" is senseless, because though you can give as instances of a generalization different
particular cases, you can't give different variables because the variables r, s, t don't differ in their meaning.
Page 275
Of course one couldn't say that when we do f(d) we don't obey f(∃) in the same way as we obey a
disjunction containing f(d), because f(∃) = f(∃) ∨ f(d). If you give someone the order "bring me some plant or other,
or this one" (giving him a picture of it), he will simply discard the picture and say to himself "since any one will do,
the picture doesn't matter". By contrast, we won't simply
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discard the picture if we are given it plus five others and the order to bring one of these six plants. (So what matters
is which disjunction contains the particular command.) And you wouldn't be guided in the same way by the order
"f(a) ∨ f(b) ∨ f(c)" as by the order "f(∃)" (= f(∃) ∨ f(c)), even if in each case you do f(c).--The picture f(c) sinks into
f(∃). (It is no good sitting in a boat, if you and it are under water and sinking). Someone may be inclined to say:
"Suppose you do f(c) on the command f(∃); in that case f(c) might have been expressly permitted and then how
would the general command have differed from a disjunction?"--But if the permission had occurred in a disjunction
with the general sentence, you couldn't have appealed to it.
Page 276
So is this how it is: "bring me a flower" can never be replaced by an order of the form "bring me a or b or c",
but must always be "bring me a or b or c or some other flower"?
Page 276
But why does the general sentence behave so indeterminately when every case which actually occurs is
something I could have described in advance?
Page 276
But even that seems to me not to get to the heart of the matter; because what matters, I believe, isn't really
the infinity of the possibilities, but a kind of indeterminacy. Indeed, if I were asked how many possibilities a circle in
the visual field has of being within a particular square, I could neither name a finite number, nor say that there were
infinitely many (as in a Euclidean plane). Here, although we don't ever come to an end, the series isn't endless in the
way in which |1, ξ, ξ + 1| is.
Page 276
Rather, no end to which we come is really the end; that is, I could always say: I don't understand why these
should be all the possibilities.--And doesn't that just mean that it is senseless to speak of "all the possibilities"? So
enumeration doesn't touch the concepts "plant" and "egg" at all.
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Page 277
And although we say that we could always have forseen f(a) as a possible particular execution of the order,
still we didn't in fact ever do so.--But even if I do foresee the possibility f(a) and expressly include it in my order, it
gets lost beside the general proposition, because I can see from the general proposition itself that this particular case
is permitted; it isn't just from its being expressly permitted in the order that I see this. If the general proposition is
there, the addition of the particular case isn't any extra use to me (that is, it doesn't make the command more
explicit). Indeed it was only the general proposition that gave me the justification for placing this particular case
beside it. What my whole argument is aiming at, is that someone might believe that the addition of the particular
case supersedes the--as it were blurred--generality of the proposition, that you could say "we don't need it any more,
now we have the particular case." Yes, but say I admit that the reason I put in the particular case is that it agrees with
the general proposition! Or suppose I admit that I recognize that f(a) is a particular case of f(∃)! For I can't say: that
just means that f(∃) is a disjunction with f(a) as one of its terms; for if that is so, the disjunction must be capable of
being stated, and f(∃) must be defined as a disjunction. There would be no difficulty in giving such a definition, but
it wouldn't correspond to the use of f(∃) that we have in mind. It isn't that the disjunction always leaves something
over; it is that it just doesn't touch the essential thing in generality, and even if it is added to it it depends on the
general proposition for its justification.
Page 277
First I command f(∃); he obeys the order and does f(a). Then I think that I could just as well have given him
the command "f(∃) ∨ f(a)". (For I knew in advance that f(a) obeyed the order f(∃) and to command him f(∃) ∨ f(a)
would come to the same.) In that case when he obeyed the order he would have been acting on the disjunction "do
something or f(a)". And if he obeys the order
Page Break 278
by doing f(a) isn't it immaterial what else is disjoined with f(a)? If he does f(a) in any case, the order is obeyed
whatever the alternative is.
Page 278
I would also like to say: in grammar nothing is supplementary, no stipulations come after others, everything
is there simultaneously.
Page 278
Thus I can't even say that I first gave the command f(∃) and only later realised that f(a) was a case of f(∃); at
all events my order was and remained f(∃) and I added f(a) to it in the knowledge that f(a) was in accordance with
f(∃). And the stipulation that f(a) is in accordance with f(∃) presupposes the sense that belongs to the proposition
f(∃) if it is taken as an independent unit and not defined as replaceable by a disjunction. And my proposition "at all
events my order was and remained f(∃) etc." only means that I didn't replace the general order by a disjunction.
Page 278
Suppose I give the order p ∨ f(a), and the addressee doesn't clearly understand the first part of the order but
does understand that the order goes "... ∨ f(a)". He might then do f(a) and say "I know for certain that I've obeyed
the command, even though I didn't understand the first part". And that too is how I imagine it when I say that the
other alternative doesn't matter. But in that case he didn't obey the order that was given, but simply treated it as
"f(a)!" One might ask: if someone does f(a) at the command "f(∃) ∨ f(a)" is he obeying the order because (i.e. in so
far as) the order is of the form ξ ∨ f(a), or because f(∃) ∨ f(a) = f(∃)? If you understand f(∃) and therefore know that
f(∃) ∨ f(a) = f(∃), then by doing f(a) you are obeying f(∃) even if I write it "f(∃) ∨ f(a)" because you can see none the
less that f(a) is a case of f(∃). And now someone might object: if you see that Fa is a case of F(∃) that just means
that f(a) is contained disjunctively in f(∃), and therefore that f(∃) is defined by means of f(a). The remaining parts of
the disjunction--he will have to say--don't concern me because the terms I see are the only ones I now need.--By
explaining 'that f(a) is an instance of f(∃)' you have said no more than that f(a) occurs in f(∃) alongside certain other
terms."--But that is precisely what we don't mean. It isn't as if our stipulation was an incomplete definition of f(∃);
Page Break 279
for that would mean that a complete definition was possible. That would be the disjunction which would make the
addition "∨ f(∃)" as it were ridiculous, since it would only be the enumerated instances which concerned us. But
according to our idea of f(∃), the stipulation that f(a) is a case of f(∃) is not an incomplete definition of f(∃); it is not a
definition of f(∃) at all. That means that I don't approximate to the sense of f(∃) by multiplying the number of cases
in the disjunction; though the disjunction of the cases ∨ f(∃) is equivalent to f(∃), it is never equivalent to the
disjunction of the cases alone; it is a totally different proposition.
Page 279
What is said about an enumeration of individual cases cannot ever be a roundabout explanation of generality.
Page 279
But can I give the rules of entailment that hold in this case? How do I know that (∃x).fx does follow from fa?
After all I can't give all the propositions from which it follows.--But that isn't necessary; if (∃x).fx follows from fa,
that at any rate was something that could be known in advance of any particular experience, and stated in the
grammar.
Page 279
I said "in advance of any experience it was possible to know and to state in the grammar that (∃x).fx follows
from fa". But it should have been: '(∃x).fx follows from fa' is not a proposition (empirical proposition) of the
language to which '(∃x).fx' and 'fa' belong; it is a rule laid down in their grammar.
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10
The law of a series
"And so on"
Page 280
We can of course set up a rule for the use of the variables, and the fact that in order to do so we need the same kind
of variable does not make it pleonastic. For if we didn't use it, then the variable would be defined by the rules, and
we don't assume that it can be defined, or that it must be defined (for sooner or later definitions come to an end).
Page 280
This means only that--e.g.--the variable "x²" is not an abbreviation (say for a logical sum), and that in our
thought too there is only a sign for this multiplicity.
Page 280
For suppose I had enumerated 7 particular instances and said "but their logical sum isn't the general
proposition" that still wouldn't be enough; and I want to say further that no other number of instances yields the
general proposition either. But in this rider once again I seem to go through an enumeration, in a kind of shadowy
manner if not in actuality. But that is not the way it is, because the words that occur in the rider are quite different
from the numerals.
Page 280
"But how can I forbid a particular numeral to be inserted in such and such a place? I surely can't foresee what
number someone will want to insert, so that I can forbid it". You can forbid it when it comes.--But here we are
already speaking of the general concept of number!
Page 280
But what makes a sign an expression of infinity? What gives the peculiar character that belongs to what we
call infinite? I believe that it is like the case of a sign for an enormous number. For the characteristic of the infinite,
conceived in this way, is its enormous size.
Page 280
But there isn't anything that is an enumeration and yet not an enumeration; a generality that enumerates in a
cloudy kind of
Page Break 281
way without really enumerating or enumerating to a determined limit.
Page 281
The dots in "1 + 1 + 1 + 1..." are just the four dots: a sign, for which it must be possible to give certain rules.
(The same rules, in fact, as for the sign "and so on ad inf.".) This sign does in a manner ape enumeration, but it isn't
an enumeration. And that means that the rules governing it don't totally agree with those which govern an
enumeration; they agree only up to a point.
Page 281
There is no third thing between the particular enumeration and the general sign.
Page 281
Of course the natural numbers have only been written down up to a certain highest point, let's say 1010. Now
what constitutes the possibility of writing down numbers that have not yet been written down? How odd is this
feeling that they are all somewhere already in existence! (Frege said that before it was drawn a construction line was
in a certain sense already there.)
Page 281
The difficulty here is to fight off the thought that possibility is a kind of shadowy reality.
Page 281
In the rules for the variable a a variable b may occur and so may particular numerals; but not any totality of
numbers.
Page 281
But now it seems as if this involved denying the existence of something in logic: perhaps generality itself, or
what the dots indicate; whatever is incomplete (loose, capable of further extension) in the number series. And of
course we may not and cannot deny the existence of anything. So how does this indeterminacy find expression?
Roughly thus: if we introduce numbers substitutible for the variable a, we don't say of any of them that it is the last,
or the highest.
Page Break 282
Page 282
But suppose someone asked us after the explanation of a form of calculation "and is 103 the last sign I can
use?" What are we to answer? "No, it isn't the last" or "there isn't a last?" Mustn't I ask him in turn "If it isn't the last,
what would come next?" And if he then says "104" I should say "Quite right, you can continue the series yourself".
Page 282
Of an end to the possibility, I cannot speak at all.
Page 282
(In philosophy the one thing we must guard against is waffle. A rule that can be applied in practice is always
in order.)
Page 282
It is clear that we can follow a rule like |a, ξ, ξ + 1|. I mean by really following the rule for constructing it
without previously being able to write down the series. In that case it's the same as if I were to begin a series with a
number like 1 and then say "now add 7, multiply by 5, take the square root of the result, and always apply this
complex operation once again to the result". (That would be the rule |1, ξ, |.)
Page 282
The expression "and so on" is nothing but the expression "and so on" (nothing, that is, but a sign in a
calculus which can't do more than have meaning via the rules that hold of it; which can't say more than it shows).
Page 282
That is, the expression "and so on" does not harbour a secret power by which the series is continued without
being continued.
Page 282
Of course it doesn't contain that, you'll say, but still it contains the meaning of infinite continuation.
Page 282
But we might ask: how does it happen that someone who now applies the general rule to a further number is
still following this rule? How does it happen that no further rule was necessary
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to allow him to apply the general rule to this case in spite of the fact that this case was not mentioned in the general
rule?
Page 283
And so we are puzzled that we can't bridge over this abyss between the individual numbers and the general
proposition.
Page 283
"Can one imagine an empty space?" (Surprisingly, this is where this question belongs.)
Page 283
It is one of the most deep rooted mistakes of philosophy to see possibility as a shadow of reality.
Page 283
But on the other hand it can't be an error; not even if one calls the proposition such a shadow.
Page 283
Here again, of course, there is a danger of falling into a positivism, of a kind which deserves a special name,
and hence of course must be an error. For we must avoid accepting party lines or particular views of things; we must
not disown anything that anyone has ever said on the topic, except where he himself had a particular view or theory.
Page 283
For the sign "and so on", or some sign corresponding to it, is essential if we are to indicate
endlessness--through the rules, of course, that govern such a sign. That is to say, we can distinguish the limited
series "1, 1 + 1, 1 + 1 + 1" from the series "1, 1 + 1, 1 + 1 + 1 and so on". And this last sign and its use is no less
essential for the calculus than any other.
Page 283
What troubles me is that the "and so on" apparently has to occur also in the rules for the sign "and so on".
For instance, 1, 1 + 1 and so on. =. 1, 1 + 1, 1 + 1 + 1 and so on, and so on.
Page 283
But then isn't this simply the old point that we can describe language only from the outside? So that we can't
expect by describing language to penetrate to depths deeper than language
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itself reveals: for it is by means of language that we describe language.
Page 284
We might say: there's no occasion to be afraid of our using the expression "and so on" in a way that
transcends the finite.
Page 284
Moreover, the distinctive part of the grammar of "and so on" can't consist in rules connecting "and so on"
with particular numerals (not "the particular numerals")--for these rules in turn mention some bit of a series--but in
rules connecting "and so on" with "and so on".
Page 284
The possibility of introducing further numbers. The difficulty seems to be that the numbers I've in fact
introduced aren't a group that is essential and yet there is nothing to indicate that they are an arbitrary collection:
Out of all numbers just those numbers that happen to have been written down.
Page 284
(As if I had all the pieces of a game in a box and a chance selection from the box on the table beside it.
Page 284
Or, as if one lot of numerals was traced in ink, while all of them are as it were drawn faintly in advance.)
Page 284
But apart from the ones we happen to have used we have only the general form.
Page 284
Isn't it here, by the way,--odd as it may sound--that the distinction between numerals and numbers comes?
Page 284
Suppose, for example, I say "By 'cardinal number' I mean whatever results from 1 by continued addition of
1". The word "continued" doesn't represent a nebulous continuation of 1, 1 + 1, 1 + 1 + 1; on the contrary the sign
"1, 1 + 1, 1 + 1 + 1..." is to be taken as perfectly exact; governed by definite rules which are different from those for
"1, 1 + 1, 1 + 1 + 1", and not a substitute for a series "which cannot be written down".
Page Break 285
Page 285
In other words: we calculate with the sign "1, 1 + 1, 1 + 1 + 1 ..." just as with the numerals, but in accordance
with different rules.
Page 285
But what is it then that we imagine? What is the mistake we make? What kind of thing do we take the sign
"1, 1 + 1..." to be? That is: where does what we think we see in this sign really occur? Something like when I say "he
counted 1, 2, 3, 4 and so on up to 1000", where it would also be possible really to write down all the numbers.
Page 285
What do we see "1, 1 + 1, 1 + 1 + 1 ...." as?
Page 285
As an inexact form of expression. The dots are like extra numerals indistinctly visible. It is as if we stopped
writing numerals, because after all we can't write them all down, but as if they are there all right in a kind of box.
Again, it is something like when I sing only the first notes of a melody distinctly, and then merely hint at the rest and
let it taper off into nothing. (Or when in writing one writes only a few letters of a word distinctly and ends with an
unarticulated line.) In all such cases the 'indistinctly' has a 'distinctly' corresponding to it.
Page 285
I once said that there couldn't be both numbers and and the concept of number. And that is quite correct, if it
means that a variable doesn't have the same relation to a number as the concept apple has to an apple (or the concept
sword to Nothung [[sic]]).
Page 285
On the other hand, a number-variable is not a numeral.
Page 285
But I also wanted to say that the concept of number couldn't be given independently of the numbers, and
that isn't true. A number-variable is independent of particular numbers in the sense that there does exist a calculus
with a class of our numerals and without the general number-variable. In that calculus, of course, not all the rules
which hold of our numerals will be valid, but those numerals will correspond to ours in the way that the
draughtsmen in draughts correspond to those in losing draughts.
Page 285
What I am opposing is the view that the infinite number series is
Page Break 286
something given concerning which there are both particular number theorems and also general theorems about all
numbers of the series; so that the arithmetical calculus wouldn't be complete if it didn't contain the general theorems
about cardinal numbers, i.e. general equations of the form a + (b + c) = (a + b) + c. Whereas even 1/3 =
belongs to a different calculus from 1/3 = 0 • 3. And similarly a general sign-rule (e.g. a recursive definition) that
holds for 1, (1) + 1, ((1) + 1) + 1, (((1) + 1) + 1) +1, and so on is something different from a particular definition. The
general rule adds to the number calculus something extra, without which it would have been no less complete than
the arithmetic of the number series 1, 2, 3, 4, 5.
Page 286
The question also arises: where is the concept of number (or of cardinal number) indispensable? Number, in
contrast to what? |/ 1, ξ, ξ + 1|, perhaps, in contrast to |5, | etc.--For if I really do introduce such a sign (like
|1, ξ, ξ + 1|) and don't just take it along as a luxury, then I must do something with it, i.e. use it in a calculus, and
then it loses its solitary splendour and occurs in a system of signs coordinated with it.
Page 286
You will perhaps say: but surely "cardinal number" is contrasted with "rational number", "real number", etc.
But this distinction is a distinction between the rules (the rules of the appropriate game)--not a distinction between
positions on the chessboard--not a distinction demanding different coordinated words in the same calculus.
Page 286
We say "this theorem is proved for all cardinal numbers". But let us just see how the concept of cardinal
numbers enters into the proof. Only because 1 and the operation ξ + 1 are spoken of in the proof--not in contrast to
anything the rational numbers have. So if we use the concept-word "cardinal number" to describe the proof in prose,
we see--don't we?--that no concept corresponds to that word.
Page Break 287
Page 287
The expressions "the cardinal numbers", "the real numbers", are extraordinarily misleading except where
they are used to help specify particular numbers, as in "the cardinal numbers from 1 to 100", etc. There is no such
thing as "the cardinal numbers", but only "cardinal numbers" and the concept, the form "cardinal number". Now we
say "the number of the cardinal numbers is smaller than the number of the real numbers" and we imagine that we
could perhaps write the two series side by side (if only we weren't weak humans) and then the one series would end
in endlessness, whereas the other would go on beyond it into the actual infinite. But this is all nonsense. If we can
talk of a relationship which can be called by analogy "greater" and "smaller", it can only be a relationship between
the forms "cardinal number" and "real number". I learn what a series is by having it explained to me and only to the
extent that it is explained to me. A finite series is explained to me by examples of the type 1, 2, 3, 4, and infinite one
by signs of the type "1, 2, 3, 4, and so on" or "1, 2, 3, 4..."
Page 287
It is important that I can understand (see) the rule of projection without having it in front of me in a general
notation. I can detect a general rule in the series 1/1, 2/4, 3/9, 4/16--of course I can detect any number of others too,
but still I can detect a particular one, and that means that this series was somehow for me the expression of that
one rule.
Page 287
If you have "intuitively" understood the law of a series, e.g. the series m, so that you are able to construct an
arbitrary term m(n), then you've completely understood the law, just as well as anything like an algebraic
formulation could convey it. That is, no such formulation can now make you understand it better, and therefore to
that extent no such formulation is any more rigorous, although it may of course be easier to take in.
Page 287
We are inclined to believe that the notation that gives a series by writing down a few terms plus the sign "and
so on" is essentially inexact, by contrast with the specification of the general term.
Page Break 288
Page 288
Here we forget that the general term is specified by reference to a basic series which cannot in turn be
described by a general term. Thus 2n + 1 is the general term of the odd numbers, if n ranges over the cardinal
numbers, but it would be nonsense to say that n was the general term of the series of cardinal numbers. If you want
to define that series, you can't do it by specifying "the general term n", but of course only by a definition like "1, 1 +
1, 1 + 1 + 1 and so on". And of course there is no essential difference between that series and "1, 1 + 1 + 1, 1 + 1 + 1
+ 1 + 1 and so on", which I could just as well have taken as the basic series (so that then the general term of the
cardinal number series would have been ½(n--1).)
(∃x).φx:~(∃x, y).φx.φy
(∃x, y).φx.φy:~(∃x, y, z).φx.φy.φz
(∃x, y, z).φx.φy.φz: ~(∃x, y, z, u).φx.φy.φz.φu
"How would we now go about writing the general form of such propositions? The question manifestly has a good
sense. For if I write down only a few such propositions as examples, you understand what the essential element in
these propositions is meant to be."
Page 288
Well, in that case the row of examples is already a notation: for understanding the series consists in our
applying the symbol, and distinguishing it from others in the same system, e.g. from
(∃x).φx
(∃x, y, z).φx.φy.φz
(∃x, y, z, u, v).φx.φy.φz.φu.φv
Page 288
But why shouldn't we write the general term of the first series thus:
?†1
Page 288
Is this notation inexact? It isn't supposed by itself to make anything graphic; all that matters are the rules for
its use, the system in which it is used. The scruples attaching to it date from a train of thought which was concerned
with the number of primitive signs in the calculus of Principia Mathematica.
Page Break 289
III FOUNDATIONS OF MATHEMATICS
11
The comparison between mathematics and a game
Page 289
What are we taking away from mathematics when we say it is only a game (or: it is a game)?
Page 289
A game, in contrast to what?--What are we awarding to mathematics if we say it isn't a game, its propositions
have a sense?
Page 289
The sense outside the proposition.
Page 289
What concern is it of ours? Where does it manifest itself and what can we do with it? (To the question "what
is the sense of this proposition?" the answer is a proposition.)
Page 289
("But a mathematical proposition does express a thought."--What thought?--.)
Page 289
Can it be expressed by another proposition? Or only by this proposition?--Or not at all? In that case it is no
concern of ours.
Page 289
Do you simply want to distinguish mathematical propositions from other constructions, such as hypotheses?
You are right to do so: there is no doubt that there is a distinction.
Page 289
If you want to say that mathematics is played like chess or patience, and the point of it is like winning or
coming out, that is manifestly incorrect.
Page 289
If you say that the mental processes accompanying the use of mathematical symbols are different from those
accompanying chess, I wouldn't know what to say about that.
Page 289
In chess there are some positions that are impossible although each individual piece is in a permissible
position. (E.g. if all the pawns are still in their initial position, but a bishop is already in
Page Break 290
play.) But one could imagine a game in which a record was kept of the number of moves from the beginning of the
game and then there would be certain positions which could not occur after n moves and yet one could not read off
from a position by itself whether or not it was a possible nth position.
Page 290
What we do in games must correspond to what we do in calculating. (I mean: it's there that the
correspondence must be, or again, that's the way that the two must be correlated with each other.)
Page 290
Is mathematics about signs on paper? No more than chess is about wooden pieces.
Page 290
When we talk about the sense of mathematical propositions, or what they are about, we are using a false
picture. Here too, I mean, it looks as if there are inessential, arbitrary signs which have an essential element in
common, namely the sense.
Page 290
Since mathematics is a calculus and hence isn't really about anything, there isn't any metamathematics.
Page 290
What is the relation between a chess problem and a game of chess?--It is clear that chess problems
correspond to arithmetical problems, indeed that they are arithmetical problems.
Page 290
The following would be an example of an arithmetical game: We write down a four-figure number at
random, e.g. 7368; we are to get as near to this number as possible by multiplying the numbers 7, 3, 6, 8 with each
other in any order. The players calculate with pencil and paper, and the person who comes nearest to the number
7368 in the smallest number of steps wins. (Many mathematical puzzles, incidentally, can be turned into games of
this kind.)
Page 290
Suppose a human being had been taught arithmetic only for use in an arithmetical game: would he have
learnt something different from a person who learns arithmetic for its ordinary use? If he multiplies 21 by 8 in the
game and gets 168, does he do something
Page Break 291
different from a person who wanted to find out how many 21 × 8 is?
Page 291
It will be said: the one wanted to find out a truth, but the other did not want to do anything of the sort.
Page 291
Well, we might want to compare this with a game like tennis. In tennis the player makes a particular
movement which causes the ball to travel in a particular way, and we can view his hitting the ball either as an
experiment, leading to the discovery of a particular truth, or else as a stroke with the sole purpose of winning the
game.
Page 291
But this comparison wouldn't fit, because we don't regard a move in chess as an experiment (though that too
we might do); we regard it as a step in a calculation.
Page 291
Someone might perhaps say: In the arithmetical game we do indeed do the multiplication (21 × 8)/168, but
the equation 21 × 8 = 168 doesn't occur in the game. But isn't that a superficial distinction? And why shouldn't we
multiply (and of course divide) in such a way that the equations were written down as equations?
Page 291
So one can only object that in the game the equation is not a proposition. But what does that mean? How
does it become a proposition? What must be added to it to make it a proposition?--Isn't it a matter of the use of the
equation (or of the multiplication)?--And it is certainly a piece of mathematics when it is used in the transition from
one proposition to another. And thus the specific difference between mathematics and a game gets linked up with
the concept of proposition (not 'mathematical proposition') and thereby loses its actuality for us.
Page 291
But one could say that the real distinction lay in the fact that in the game there is no room for affirmation and
negation. For
Page Break 292
instance, there is multiplication and 21 × 8 = 148 would be a false move, but "(21 × 8 = 148)", which is a correct
arithmetical proposition, would have no business in our game.
Page 292
(Here we may remind ourselves that in elementary schools they never work with inequations. The children
are only asked to carry out multiplications correctly and never--or hardly ever--asked to prove an inequation.)
Page 292
When I work out 21 × 8 in our game the steps in the calculation, at least, are the same as when I do it in order
to solve a practical problem (and we could make room in a game for inequations also). But my attitude to the sum in
other respects differs in the two cases.
Page 292
Now the question is: can we say of someone playing the game who reaches the position "21 × 8 = 168" that
he has found out that 21 × 8 is 168? What does he lack? I think the only thing missing is an application for the sum.
Page 292
Calling arithmetic a game is no more and no less wrong than calling moving chessmen according to
chess-rules a game; for that might be a calculation too.
Page 292
So we should say: No, the word "arithmetic" is not the name of a game. (That too of course is trivial)--But
the meaning of the word "arithmetic" can be clarified by bringing out the relationship between arithmetic and an
arithmetical game, or between a chess problem and the game of chess.
Page 292
But in doing so it is essential to recognize that the relationship is not the same as that between a tennis
problem and the game of tennis.
Page 292
By "tennis problem" I mean something like the problem of returning a ball in a particular direction in given
circumstances. (A billiard problem would perhaps be a clearer case.) A billiard problem isn't a mathematical problem
(although its solution may be an application of mathematics). A billiard problem is a physical
Page Break 293
problem and therefore a "problem" in the sense of physics; a chess problem is a mathematical problem and so a
"problem" in a different sense, a mathematical sense.
Page 293
In the debate between "formalism" and "contentful mathematics" what does each side assert? This dispute is
so like the one between realism and idealism in that it will soon have become obsolete, for example, and in that both
parties make unjust assertions at variance with their day-to-day practice.
Page 293
Arithmetic isn't a game, it wouldn't occur to anyone to include arithmetic in a list of games played by human
beings.
Page 293
What constitutes winning and losing in a game (or success in patience)? It isn't of course, just the winning
position. A special rule is needed to lay down who is the winner. ("Draughts" and "losing draughts" differ only in
this rule.)
Page 293
Now is the rule which says "The one who first has his pieces in the other one's half is the winner" a
statement? How would it be verified? How do I know if someone has won? Because he is pleased, or something of
the kind? Really what the rule says is: you must try to get your pieces as soon as possible, etc.
Page 293
In this form the rule connects the game with life. And we could imagine that in an elementary school in
which one of the subjects taught was chess the teacher would react to a pupil's bad moves in exactly the same way
as to a sum worked out wrongly.
Page 293
I would almost like to say: It is true that in the game there isn't any "true" and "false" but then in arithmetic
there isn't any "winning" and "losing".
Page Break 294
Page 294
I once said that is was imaginable that wars might be fought on a kind of huge chessboard according to the
rules of chess. But if everything really went simply according to the rules of chess, then you wouldn't need a
battlefield for the war, it could be played on an ordinary board; and then it wouldn't be a war in the ordinary sense.
But you really could imagine a battle conducted in accordance with the rules of chess--if, say, the "bishop" could
fight with the "queen" only when his position in relation to her was such that he would be allowed to "take" her in
chess.
Page 294
Could we imagine a game of chess being played (i.e. a complete set of chess moves being carried out) in
such different surroundings that what happened wasn't something we could call the playing of a game?
Page 294
Certainly, it might be a case of the two participants collaborating to solve a problem. (And we could easily
construct a case on these lines in which such a task would have a utility).
Page 294
The rule about winning and losing really just makes a distinction between two poles. It is not concerned with
what later happens to the winner (or loser)--whether, for instance, the loser has to pay anything.
Page 294
(And similarly, the thought occurs, with "right" and "wrong" in sums.)
Page 294
In logic the same thing keeps happening as happened in the dispute about the nature of definition. If
someone says that a definition is concerned only with signs and does no more than substitute one sign for another,
people resist and say that that isn't all a definition does, or that there are different kinds of definition and the
interesting and important ones aren't the mere "verbal definitions".
Page 294
They think, that is, that if you make definition out to be a mere substitution rule for signs you take away its
significance and importance. But the significance of a definition lies in its application, in its importance for life. The
same thing is happening today in the
Page Break 295
dispute between formalism and intuitionism, etc. People cannot separate the importance, the consequences, the
application of a fact from the fact itself; they can't separate the description of a thing from the description of its
importance.
Page 295
We are always being told that a mathematician works by instinct (or that he doesn't proceed mechanically
like a chessplayer or the like), but we aren't told what that's supposed to have to do with the nature of mathematics.
If such a psychological phenomenon does play a part in mathematics we need to know how far we can speak about
mathematics with complete exactitude, and how far we can only speak with the indeterminacy we must use in
speaking of instincts etc.
Page 295
Time and again I would like to say: What I check is the account books of mathematicians; their mental
processes, joys, depressions and instincts as they go about their business may be important in other connections,
but they are no concern of mine.
Page Break 296
12
There is no metamathematics.
Page 296
No calculus can decide a philosophical problem.
Page 296
A calculus cannot give us information about the foundations of mathematics.
Page 296
So there can't be any "leading problems" of mathematical logic, if those are supposed to be problems whose
solution would at long last give us the right to do arithmetic as we do.
Page 296
We can't wait for the lucky chance of the solution of a mathematical problem.
Page 296
I said earlier "calculus is not a mathematical concept"; in other words, the word "calculus" is not a chesspiece
that belongs to mathematics.
Page 296
There is no need for it to occur in mathematics.--If it is used in a calculus nonetheless, that doesn't make the
calculus into a metacalculus; in such a case the word is just a chessman like all the others.
Page 296
Logic isn't metamathematics either; that is, work within the logical calculus can't bring to light essential truths
about mathematics. Cf. here the "decision problem" and similar topics in modern mathematical logic.
Page 296
(Through Russell and Whitehead, especially Whitehead, there entered philosophy a false exactitude that is
the worst enemy of real exactitude. At the bottom of this there lies the erroneous opinion that a calculus could be the
mathematical foundation of mathematics.)
Page 296
Number is not at all a "fundamental mathematical concept"†1.
Page Break 297
There are so many calculations in which numbers aren't mentioned.
Page 297
So far as concerns arithmetic, what we are willing to call numbers is more or less arbitrary. For the rest, what
we have to do is to describe the calculus--say of cardinal numbers--that is, we must give its rules and by doing so we
lay the foundations of arithmetic.
Page 297
Teach it to us, and then you have laid its foundations.
Page 297
(Hilbert sets up rules of a particular calculus as rules of metamathematics.)
Page 297
A system's being based on first principles is not the same as its being developed from them. It makes a
difference whether it is like a house resting on its lowest walls or like a celestial body floating free in space which we
have begun to build beneath although we might have built anywhere else.
Page 297
Logic and mathematics are not based on axioms, any more than a group is based on the elements and
operations that define it. The idea that they are involves the error of treating the intuitiveness, the self-evidence, of
the fundamental propositions as a criterion for correctness in logic.
Page 297
A foundation that stands on nothing is a bad foundation.
Page 297
(p.q) ∨ (p.~q) ∨ (~p.q) ∨ (~p.~q.): That is my tautology, and then I go on to say that every "proposition of
logic" can be brought into this form in accordance with specified rules. But that means the same as: can be derived
from it. This would take us as far as the Russellian method of demonstration and all we add to it
Page Break 298
is that this initial form is not itself an independent proposition, and that like all other "laws of logic" it has the
property that p.Log = p, p ∨ Log = Log.
Page 298
It is indeed the essence of a "logical law" that when it is conjoined with any proposition it yields that
proposition. We might even begin Russell's calculus with definitions like
p ⊃ p:q. = .q
p:p ∨ q. = .p, etc.
Page Break 299
13
Proofs of Relevance
Page 299
If we prove that a problem can be solved, the concept "solution" must occur somewhere in the proof. (There
must be something in the mechanism corresponding to the concept.) But the concept cannot have an external
description as its proxy; it must be genuinely spelt out.
Page 299
The only proof of the provability of a proposition is a proof of the proposition itself. But there is something
we might call a proof of relevance: an example would be a proof convincing me that I can verify the equation 17 ×
38 = 456 before I have actually done so. Well, how is it that I know that I can check 17 × 38 = 456, whereas I perhaps
wouldn't know, merely by looking, whether I could check an expression in the integral calculus? Obviously, it is
because I know that the equation is constructed in accordance with a definite rule and because I know the kind of
connection between the rule for the solution of the sum and the way in which the proposition is put together. In that
case a proof of relevance would be something like a formulation of the general method of doing things like
multiplication sums, enabling us to recognize the general form of the propositions it makes it possible to check. In
that case I can say I recognise that this method will verify the equation without having actually carried out the
verification.
Page 299
When we speak of proofs of relevance (and other similar mathematical entities) it always looks as if in
addition to the particular series of operations called proofs of relevance, we had a quite definite inclusive concept of
such proofs or of mathematical proof in general; but in fact the word is applied with many different, more or less
related, meanings. (Like words such as "people", "king", "religion", etc.; cf Spengler.) Just think of the role of
examples in the explanation of such words. If I want to explain what I mean by "proof", I will have to point to
examples of proofs,
Page Break 300
just as when explaining the word "apple" I point to apples. The definition of the word "proof" is in the same case as
the definition of the word "number". I can define the expression "cardinal number" by pointing to examples of
cardinal numbers; indeed instead of the expression I can actually use the sign "1, 2, 3, 4, and so on ad inf". I can
define the word "number" too by pointing to various kinds of number; but when I do so I am not circumscribing the
concept "number" as definitely as I previously circumscribed the concept cardinal number, unless I want to say that
it is only the things at present called numbers that constitute the concept "number", in which case we can't say of
any new construction that it constructs a kind of number. But the way we want to use the word "proof" in is one in
which it isn't simply defined by a disjunction of proofs currently in use; we want to use it in cases of which at
present we "can't have any idea". To the extent that the concept of proof is sharply circumscribed, it is only through
particular proofs, or through series of proofs (like the number series), and we must keep that in mind if we want to
speak absolutely precisely about proofs of relevance, of consistency etc.
Page 300
We can say: A proof of relevance alters the calculus containing the proposition to which it refers. It cannot
justify a calculus containing the proposition, in the sense in which carrying out the multiplication 17 × 23 justifies
the writing down of the equation 17 × 23 = 391. Not, that is, unless we expressly give the word "justify" that
meaning. But in that case we mustn't believe that if mathematics lacks this justification, it is in some more general
and widely established sense illegitimate or suspicious. (That would be like someone wanting to say: "the use of the
expression 'pile of stones' is fundamentally illegitimate, until we have laid down officially how many stones make a
pile." Such a stipulation would modify the use of the word "pile" but it wouldn't "justify" it in any generally
recognized sense; and if such an official definition were given, it wouldn't mean that the use earlier made of the word
would be stigmatized as incorrect.)
Page Break 301
Page 301
The proof of the verifiability of 17 × 23 = 391 is not a "proof" in the same sense of the word as the proof of
the equation itself. (A cobbler heels, a doctor heals: both...) We grasp the verifiability of the equation from its proof
somewhat as we grasp the verifiability of the proposition "the points A and B are not separated by a turn of the
spiral" from the figure. And we see that the proposition stating verifiability isn't a "proposition" in the same sense as
the one whose verifiability is asserted. Here again, one can only say: look at the proof, and you will see what is
proved here, what gets called "the proposition proved".
Page 301
Can one say that at each step of a proof we need a new insight? (The individuality of numbers.) Something
of the following sort: if I am given a general (variable) rule, I must recognize each time afresh that this rule may be
applied here too (that it holds for this case too). No act of foresight can absolve me from this act of insight. Since
the form in which the rule is applied is in fact a new one at every step. But it is not a matter of an act of insight, but
of an act of decision.
Page 301
What I called a proof of relevance does not climb the ladder to its proposition--since that requires that you
pass every rung--but only shows that the ladder leads in the direction of that proposition.
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(There are no surrogates in logic). Neither is an arrow that points the direction a surrogate for going through all the
stages towards a particular goal.
Page Break 303
14
Consistency proofs
Page 303
Something tells me that a contradiction in the axioms of a system can't really do any harm until it is revealed.
We think of a hidden contradiction as like a hidden illness which does harm even though (and perhaps precisely
because) it doesn't show itself in an obvious way. But two rules in a game which in a particular instance contradict
each other are perfectly in order until the case turns up, and it's only then that it becomes necessary to make a
decision between them by a further rule.
Page 303
Mathematicians nowadays make so much fuss about proofs of the consistency of axioms. I have the feeling
that if there were a contradiction in the axioms of a system it wouldn't be such a great misfortune. Nothing easier
than to remove it.
Page 303
"We may not use a system of axioms before its consistency has been proved."
Page 303
"In the rules of the game no contradictions may occur."
Page 303
Why not? "Because then one wouldn't know how to play."
Page 303
But how does it happen that our reaction to a contradiction is a doubt?
Page 303
We don't have any reaction to a contradiction. We can only say: if it's really meant like that (if the
contradiction is supposed to be there) I don't understand it. Or: it isn't something I've learnt. I don't understand the
sign. I haven't learnt what I am to do with it, whether it is a command, etc.
Page 303
Suppose someone wanted to add to the usual axioms of arithmetic the equation 2 × 2 = 5. Of course that
would mean that the sign of equality had changed its meaning, i.e. that there would now be different rules for the
equals-sign.
Page Break 304
Page 304
If I inferred "I cannot use it as a substitution sign" that would mean that its grammar no longer fitted the
grammar of the word "substitute" ("substitution sign", etc.). For the word "can" in that proposition doesn't indicate a
physical (physiological, psychological) possibility.
Page 304
"The rules many not contradict each other" is like "negation, when doubled, may not yield a negation". That
is, it is part of the grammar of the word "rule" that if "p" is a rule, "p.~p" is not a rule.
Page 304
That means we could also say: the rules may contradict each other, if the rules for the use of the word "rule"
are different--if the word "rule" has a different meaning.
Page 304
Here too we cannot give any foundation (except a biological or historical one or something of the kind); all
we can do is to establish the agreement, or disagreement between the rules for certain words, and say that these
words are used with these rules.
Page 304
It cannot be shown, proved, that these rules can be used as the rules of this activity.
Page 304
Except by showing that the grammar of the description of the activity fits the rules.
Page 304
"In the rules there mustn't be a contradiction" looks like an instruction: "In a clock the hand mustn't be loose
on the shaft." We expect a reason: because otherwise... But in the first case the reason would have to be: because
otherwise it wouldn't be a set of rules. Once again we have a grammatical structure that cannot be given a logical
foundation.
Page 304
In the indirect proof that a straight line can have only one continuation through a certain point we make the
supposition that a straight line could have two continuations.--If we make that
Page Break 305
supposition, then the supposition must make sense.--But what does it mean to make that supposition? It isn't
making a supposition that goes against natural history, like the supposition that a lion has two tails.--It isn't making a
supposition that goes against an ascertained fact. What it means is supposing a rule; and there's nothing against that
except that it contradicts another rule, and for that reason I drop it.
Page 305
Suppose that in the proof there occurs the following drawing to represent a
straight line bifurcating. There is nothing absurd (contradictory) in that unless we have made some stipulation that it
contradicts.
Page 305
If a contradiction is found later on, that means that hitherto the rules have not been clear and unambiguous.
So the contradiction doesn't matter, because we can now get rid of it by enunciating a rule.
Page 305
In a system with a clearly set out grammar there are no hidden contradictions, because such a system must
include the rule which makes the contradiction is discernible. A contradiction can only be hidden in the sense that it
is in the higgledy-piggledy zone of the rules, in the unorganized part of the grammar; and there it doesn't matter
since it can be removed by organizing the grammar.
Page 305
Why may not the rules contradict one another? Because otherwise they wouldn't be rules.
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15
Justifying arithmetic and preparing it for its applications
(Russell, Ramsey)
Page 306
One always has an aversion to giving arithmetic a foundation by saying something about its application. It
appears firmly enough grounded in itself. And that of course derives from the fact that arithmetic is its own
application.
Page 306
You could say: why bother to limit the application of arithmetic, that takes care of itself. (I can make a knife
without bothering about what kinds of materials I will have cut with it; that will show soon enough.)
Page 306
What speaks against our demarcating a region of application is the feeling that we can understand arithmetic
without having any such region in mind. Or put it like this: our instinct rebels against anything that isn't restricted to
an analysis of the thoughts already before us.
Page 306
You could say arithmetic is a kind of geometry; i.e. what in geometry are constructions on paper in
arithmetic are calculations (on paper). You could say, it is a more general kind of geometry.
Page 306
It is always a question of whether and how far it's possible to represent the most general form of the
application of arithmetic. And here the strange thing is that in a certain sense it doesn't seem to be needed. And if in
fact it isn't needed, then it's also impossible.
Page 306
The general form of its application seems to be represented by the fact that nothing is said about it. (And if
that's a possible representation, then it is also the right one.)
Page 306
The point of the remark that arithmetic is a kind of geometry is simply that arithmetical constructions are
autonomous like
Page Break 307
geometrical ones and hence so to speak themselves guarantee their applicability.
Page 307
For it must be possible to say of geometry too that it is its own application.
Page 307
(In the sense in which we can speak of lines which are possible and lines which are actually drawn we can
also speak of possible and actually represented numbers.)
Page 307
That is an arithmetical construction, and in a somewhat extended sense also a geometrical one.
Page 307
Suppose I wish to use this calculation to solve the following problem: if I have 11 apples and want to share
them among some people in such a way that each is given 3 apples how many people can there be? The calculation
supplies me with the answer 3. Now suppose I were to go through the whole process of sharing and at the end 4
people each had 3 apples in their hands. Would I then say that the computation gave a wrong result? Of course not.
And that of course means only that the computation was not an experiment.
Page 307
It might look as though the mathematical computation entitled us to make a prediction, say, that I could give
three people their share and there will be two apples left over. But that isn't so. What justifies us in making this
prediction is an hypothesis of physics, which lies outside the calculation. The calculation is only a study of logical
forms, of structures, and of itself can't yield anything new.
Page 307
If 3 strokes on the paper are the sign for the number 3, then you can say the number 3 is to be applied in our
language in the way in which the 3 strokes can be applied.
Page 307
I said "One difficulty in the Fregean theory is the generality of the words 'Concept' and 'Object'. For, even if
you can count
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tables, tones, vibrations and thoughts, it is difficult to bracket them all together."†1 But what does "you can count
them" mean? What it means is that it makes sense to apply the cardinal numbers to them. But if we know that, if we
know these grammatical rules, why do we need to rack our brains about the other grammatical rules when we are
only concerned to justify the application of cardinal arithmetic? It isn't difficult "to bracket them all together"; so far
as is necessary for the present purpose they are already bracketed together.
Page 308
But (as we all know well) arithmetic isn't at all concerned about this application. Its applicability takes care of
itself.
Page 308
Hence so far as the foundations of arithmetic are concerned all the anxious searching for distinctions
between subject-predicate forms, and constructing functions 'in extension' (Ramsey) is a waste of time.
Page 308
The equation 4 apples + 4 apples = 8 apples is a substitution rule which I use if instead of substituting the
sign "8" for the sign "4 + 4", I substitute the sign "8 apples" for the sign "4 + 4 apples."
Page 308
But we must beware of thinking that "4 apples + 4 apples = 8 apples" is the concrete equation and 4 + 4 = 8
the abstract proposition of which the former is only a special case, so that the arithmetic of apples, though much less
general than the truly general arithmetic, is valid in its own restricted domain (for apples). There isn't any "arithmetic
of apples", because the equation 4 apples + 4 apples = 8 apples is not a proposition about apples. We may say that in
this equation the word "apples" has no reference. (And we can always say this about a sign in a rule which helps to
determine its meaning.)
Page Break 309
Page 309
How can we make preparations for the reception of something that may happen to exist--in the sense in
which Russell and Ramsey always wanted to do this? We get logic ready for the existence of many-placed relations,
or for the existence of an infinite number of objects, or the like.
Page 309
Well, we can make preparations for the existence of a thing: e.g. I may make a casket for jewellery which
may be made some time or other--But in this case I can say what the situation must be--what the situation is--for
which I am preparing. It is no more difficult to describe the situation now than after it has already occurred; even, if
it never occurs at all. (Solution of mathematical problems). But what Russell and Ramsey are making preparations
for is a possible grammar.
Page 309
On the one hand we think that the nature of the functions and of the arguments that are counted in
mathematics is part of its business. But we don't want to let ourselves be tied down to the functions now known to
us, and we don't know whether people will ever discover a function with 100 argument places; and so we have to
make preparations and construct a function to get everything ready for a 100-place relation in case one turns
up.--But what does "a 100-place relation turns up (or exists)" mean at all? What concept do we have of one? Or of a
2-place relation for that matter?--As an example of a 2-place relation we give something like the relation between
father and son. But what is the significance of this example for the further logical treatment of 2-place relations?
Instead of every "aRb" are we now to imagine "a is the father of b"?--If not, is this example or any example
essential? Doesn't this example have the same role as an example in arithmetic, when I use 3 rows of 6 apples to
explain 3 × 6 = 18 to somebody?
Page 309
Here it is a matter of our concept of application.--We have an image of an engine which first runs idle, and
then works a machine.
Page 309
But what does the application add to the calculation? Does it
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introduce a new calculus? In that case it isn't any longer the same calculation. Or does it give it substance in some
sense which is essential to mathematics (logic)? If so, how can we abstract from the application at all, even only
temporarily?
Page 310
No, calculation with apples is essentially the same as calculation with lines or numbers. A machine is an
extension of an engine, an application is not in the same sense an extension of a calculation.
Page 310
Suppose that, in order to give an example, I say "love is a 2-place relation"--am I saying anything about love?
Of course not. I am giving a rule for the use of the word "love" and I mean perhaps that we use this word in such
and such a way.
Page 310
Yet we do have the feeling that when we allude to the 2-place relation 'love' we put meaning into the husk of
the calculus of relations.--Imagine a geometrical demonstration carried out using the cylinder of a lamp instead of a
drawing or analytical symbols. How far is this an application of geometry? Does the use of the glass cylinder in the
lamp enter into the geometrical thought? And does the use of word "love" in a declaration of love enter into my
discussions of 2-place relations?
Page 310
We are concerned with different uses or meanings of the word "application". "Division is an application of
multiplication"; "the lamp is an application of the glass cylinder"; "the calculation is applied to these apples".
Page 310
At this point we can say: arithmetic is its own application. The calculus is its own application.
Page 310
In arithmetic we cannot make preparations for a grammatical application. For if arithmetic is only a game, its
application too is only a game, and either the same game (in which case it takes us no further) or a different
game--and in that case we could play it in pure arithmetic also.
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Page 311
So if the logician says that he has made preparations in arithmetic for the possible existence of 6-place
relations, we may ask him: when what you have prepared finds its application, what will be added to it? A new
calculus?--but that's something you haven't provided. Or something which doesn't affect the calculus?--then it
doesn't interest us, and the calculus you have shown us is application enough.
Page 311
What is incorrect is the idea that the application of a calculus in the grammar of real language correlates it to a
reality or gives it a reality that it did not have before.
Page 311
Here as so often in this area the mistake lies not in believing something false, but in looking in the direction
of a misleading analogy.
Page 311
So what happens when the 6-place relation is found? Is it like the discovery of a metal that has the desired
(and previously described) properties (the right specific weight, strength, etc.)? No; what is discovered is a word that
we in fact use in our language as we used, say, the letter R. "Yes, but this word has meaning, and 'R' has none. So
now we see that something can correspond to 'R'." But the meaning of the word does not consist in something's
corresponding to it, except in a case like that of a name and what it names; but in our case the bearer of the name is
merely an extension of the calculus, of the language. And it is not like saying "this story really happened, it was not
pure fiction".
Page 311
This is all connected with the false concept of logical analysis that Russell, Ramsey and I used to have,
according to which we are writing for an ultimate logical analysis of facts, like a chemical analysis of compounds--an
analysis which will enable us really
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to discover a 7-place relation, like an element that really has the specific weight 7.
Page 312
Grammar is for us a pure calculus (not the application of a calculus to reality).
Page 312
"How can we make preparations for something which may or may not exist?" means: how can we hope to
make an a priori construction to cope with all possible results while basing arithmetic upon a logic in which we are
still waiting for the results of an analysis of our propositions in particular cases?--One wants to say: "we don't know
whether it may not turn out that there are no functions with 4 argument places, or that there are only 100 arguments
that can significantly be inserted into functions of one variable. Suppose, for example (the supposition does appear
possible) that there is only one four-place function F and 4 arguments a, b, c, d; does it make sense in that case to
say '2 + 2 = 4' since there aren't any functions to accomplish the division into 2 and 2?" So now, one says to oneself,
we will make provision for all possible cases. But of course that has no meaning. On the one hand the calculus
doesn't make provision for possible existence; it constructs for itself all the existence that it needs. On the other hand
what look like hypothetical assumptions about the logical elements (the logical structure) of the world are merely
specifications of elements in a calculus; and of course you can make these in such a way that the calculus does not
contain any 2 + 2.
Page 312
Suppose we make preparations for the existence of 100 objects by introducing 100 names and a calculus to
go with them. Then let us suppose 100 objects are really discovered. What happens now that the names have objects
correlated with them which weren't correlated with them before? Does the calculus change?--What has the
correlation got to do with it at all? Does it make it acquire more reality? Or did the calculus previously belong only to
mathematics, and now to logic as well?--What sort of questions are "are there 3-place relations", "are there 1000
objects"? How is
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it to be decided?--But surely it is a fact that we can specify a 2-place relation, say love, and a 3-place one, say
jealousy, but perhaps not a 27-place one!--But what does "to specify a 2-place relation" mean? It sounds as if we
could point to a thing and say "you see, that is the kind of thing" (the kind of thing we described earlier). But nothing
of that kind takes place (the comparison with pointing is altogether wrong). "The relation of jealousy cannot be
reduced to 2-place relationships" sounds like "alcohol cannot be decomposed into water plus a solid substance". Is
that something that is part of the nature of jealousy? (Let's not forget: the proposition "A is jealous of B because of
C" is no more and no less reducible than the proposition "A is not jealous of B because of C".) What is pointed to is,
say, the group of people A, B and C.--"But suppose that living beings at first knew only plane surfaces, but none the
less developed a 3-dimensional geometry, and that they suddenly became acquainted with 3-dimensional space!"
Would this alter their geometry, would it become richer in content?--"Isn't this the way it is? Suppose at some time I
had made arbitrary rules for myself prohibiting me from moving in my room in certain directions where there were
no physical hindrances to get in my way; and then suppose the physical conditions changed, say furniture was put
in the room, in such a way as to force me to move in accordance with the rules which I had originally imposed on
myself arbitrarily. Thus, while the 3-dimensional calculus was only a game, there weren't yet three dimensions in
reality because the x, y, z belonged to the rules only because I had so decided; but now that we have linked them up
to the real 3 dimensions, no other movements are possible for them." But that is pure fiction. There isn't any
question here of a connection with reality which keeps grammar on the rails. The "connection of language with
reality", by means of ostensive definitions and the like, doesn't make the grammar inevitable or provide a
justification for the grammar. The grammar remains a free-floating calculus which can only be extended and never
supported. The "connection with
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reality" merely extends language, it doesn't force anything on it. We speak of discovering a 27-place relation but on
the one hand no discovery can force me to use the sign or the calculus for a 27-place relation, and on the other hand
I can describe the operation of the calculus itself simply by using this notation.
Page 314
When it looks in logic as if we are discussing several different universes (as with Ramsey), in reality we are
considering different games. The definition of a "universe" in a case like Ramsey's would simply be a definition like
(∃x).φx φa ∨ φb ∨ φc ∨ φd.
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16
Ramsey's theory of identity
Page 315
Ramsey's theory of identity makes the mistake that would be made by someone who said that you could use
a painting as a mirror as well, even if only for a single posture. If we say this we overlook that what is essential to a
mirror is precisely that you can infer from it the posture of a body in front of it, whereas in the case of the painting
you have to know that the postures tally before you can construe the picture as a mirror image.
Page 315
If Dirichlet's conception of function has a strict sense, it must be expressed in a definition that uses the table
to define the function-signs as equivalent.
Page 315
Ramsey defines†1 x = y as
(φe)•φex ∃ φey
Page 315
But according to the explanations he gives of his function-sign "φe"
(φe)•φex ∃ φex is the statement: "every sentence is equivalent to itself."
(φe)•φex ∃ φey is the statement: "every sentence is equivalent to every sentence."
So all he has achieved by his definition is what is laid down by the two definitions
x = x. . Tautology
x = y. . Contradiction
(Here the word "tautology" can be replaced by any arbitrary tautology, and similarly with "contradiction"). So far all
that has happened is that definitions have been given of the two distinct signs x = x and x = y. These definitions
could of course be replaced by two sets of definitions, e.g.
Page Break 316
Page 316
But then Ramsey writes:
"(∃x,y).x ≠ y", i.e. "(∃x, y).~(x = y)"--
but he has no right to: for what does the "x = y" mean in this expression? It is neither the sign "x = y" used in the
definition above, nor of course the "x = x" in the preceding definition. So it is a sign that is still unexplained.
Moreover to see the futility of these definitions, you should read them (as an unbiased person would) as follows: I
permit the sign "Taut", whose use we know, to be replaced by the sign "a = a" or "b = b", etc.; and the sign "Contr."
("~Taut.") to be replaced by the sign "a = b" or "a = c", etc. From which, incidentally, it follows that (a = b) = (c = d)
= (a ≠ a) = etc.!
Page 316
It goes without saying that an identity sign defined like that has no resemblance to the one we use to express
a substitution rule.
Page 316
Of course I can go on to define "(∃x, y). x ≠ y", say as a ≠ a. ∨. a ≠ b. ∨. b ≠ c. ∨. a ≠ c; but this definition is
pure humbug and I should have written straightaway (∃x, y).x ≠ y. . Taut. (That is, I would be given the sign on
the left side as a new unnecessary--sign for "Taut.") For we mustn't forget that according to the definitions "a = a", "a
= b", etc. are independent signs, no more connected with each other than the signs "Taut." and "Contr." themselves.
Page 316
What is in question here is whether functions in extension are any use; because Ramsey's explanation of the
identity sign is just such a specification by extension. Now what exactly is the specification of a function by its
extension? Obviously, it is a group of definitions, e.g.
fa = p Def.
fb = q Def.
fc = r Def.
Page 316
These definitions permit us to substitute for the known propositions "p", "q", "r" the signs "fa" "fb" "fc". To
say that these three definitions determine the function f(ξ) is either to say nothing, or to say the same as the three
definitions say.
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Page 317
For the signs "fa" "fb" "fc" are no more function and argument than the words "Co(rn)", "Co(al)" and
"Co(lt)" are. (Here it makes no difference whether or not the "arguments" "rn", "al", "lt" are used elsewhere as
words).
Page 317
(So it is hard to see what purpose the definitions can have except to mislead us.)
Page 317
To begin with, the sign "(∃x).fx" has no meaning; because here the rules for functions in the old sense of the
word don't hold at all. According to them a definition like fa =... would be nonsense. If no explicit definition is given
for it, the sign "(∃x).fx" can only be understood as a rebus in which the signs have some kind of spurious meaning.
Page 317
Each of the signs "a = a", "a = c", etc. in the definitions (a = a). .Taut. etc. is a word.
Page 317
Moreover, the purpose of the introduction of functions in extension was to analyse propositions about
infinite extensions, and it fails of this purpose when a function in extension is introduced by a list of definitions.
Page 317
There is a temptation to regard the form of an equation as the form of tautologies and contradictions, because
it looks as if one can say that x = x is self-evidently true and x = y self-evidently false. The comparison between x =
x and a tautology is of course better than that between x = y and a contradiction, because all correct (and
"significant") equations of mathematics are actually of the form x = y. We might call x = x a degenerate equation
(Ramsey quite correctly called tautologies and contradictions degenerate propositions) and indeed a correct
degenerate equation (the limiting case of an equation). For we use expressions of the form x = x like correct
equations, and when we do so we are fully conscious that we are dealing with degenerate equations. In geometrical
proofs there are propositions in the same case, such as "the angle α is equal to the angle β, the angle γ is equal to
itself..."
Page 317
At this point the objection might be made that correct equations of the form x = y must be tautologies, and
incorrect ones contradictions, because it must be possible to prove a correct equation
Page Break 318
by transforming each side of it until an identity of the form x = x is reached. But although the original equation is
shown to be correct by this process, and to that extent the identity x = x is the goal of the transformation, it is not its
goal in the sense that the purpose of the transformation is to give the equation its correct form--like bending a
crooked object straight; it is not that the equation at long last achieves its perfect form in the identity. So we can't
say: a correct equation is really an identity. It just isn't an identity.
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17
The concept of the application of arithmetic†1 (mathematics)
Page 319
If we say "it must be essential to mathematics that it can be applied" we mean that its applicability isn't the
kind of thing I mean of a piece of wood when I say "I will be able to find many applications for it".
Page 319
Geometry isn't the science (natural science) of geometric planes, lines and points, as opposed to some other
science of gross physical lines, stripes and surfaces and their properties. The relation between geometry and
propositions of practical life, about stripes, colour boundaries, edges and corners, etc. isn't that the things geometry
speaks of, though ideal edges and corners, resemble those spoken of in practical propositions; it is the relation
between those propositions and their grammar. Applied geometry is the grammar of statements about spatial
objects. The relation between what is called a geometrical line and a boundary between two colours isn't like the
relation between something fine and something coarse, but like the relation between possibility and actuality. (Think
of the notion of possibility as a shadow of actuality.)
Page 319
You can describe a circular surface divided diametrically into 8 congruent parts, but it is senseless to give
such a description of an elliptical surface. And that contains all that geometry says in this connexion about circular
and elliptical surfaces.
Page 319
(A proposition based on a wrong calculation (such as "he cut a 3-metre board into 4 one metre parts") is
nonsensical, and that throws light on what is meant by "making sense" and "meaning something by a proposition".)
Page Break 320
Page 320
What about the proposition "the sum of the angles of a triangle is 180 degrees"? At all events you can't tell by
looking at it that it is a proposition of syntax.
Page 320
The proposition "corresponding angles are equal" means that if they don't appear equal when they are
measured I will treat the measurement as incorrect; and "the sum of the angles of a triangle is 180 degrees" means
that if it doesn't appear to be 180 degrees when they are measured I will assume there has been a mistake in the
measurement. So the proposition is a postulate about the method of describing facts, and therefore a proposition of
syntax.
Page Break 321
IV ON CARDINAL NUMBERS
18
Kinds of cardinal number
Page 321
What are numbers?--What numerals signify; an investigation of what they signify is an investigation of the
grammar of numerals.
Page 321
What we are looking for is not a definition of the concept of number, but an exposition of the grammar of
the word "number" and of the numerals.
Page 321
The reason why there are infinitely many cardinal numbers, is that we construct this infinite system and call it
the system of cardinal numbers. There is also a number system "1, 2, 3, 4, 5, many" and even a system "1, 2, 3, 4, 5".
Why shouldn't I call that too a system of cardinal numbers (a finite one)?
Page 321
It is clear that the axiom of infinity is not what Russell took it for; it is neither a proposition of logic, nor--as it
stands--a proposition of physics. Perhaps the calculus to which it belongs, transplanted into quite different
surroundings (with a quite different "interpretation"), might somewhere find a practical application; I do not know.
Page 321
One might say of logical concepts (e.g. of the, or a, concept of infinity) that their essence proves their
existence.
Page 321
(Frege would still have said: "perhaps there are people who have not got beyond the first five in their
acquaintance with the series of cardinal numbers (and see the rest of the series only in an indeterminate form or
something of the kind), but this series exists independently of us". Does chess exist independently of us, or not?--)
Page 321
Here is a very interesting question about the position of the concept of number in logic: what happens to the
concept of
Page Break 322
number if a society has no numerals, but for counting, calculating, etc. uses exclusively an abacus like a Russian
abacus?
Page 322
(Nothing would be more interesting than to investigate the arithmetic of such people; it would make one
really understand that here there is no distinction between 20 and 21.)
Page 322
Could we also imagine, in contrast with the cardinal numbers, a kind of number consisting of a series like the
cardinal numbers without the 5? Certainly; but this kind of number couldn't be used for any of the things for which
we use the cardinal numbers. The way in which these numbers are missing a five is not like the way in which an
apple may have been taken out of a box of apples and can be put back again; it is of their essence to lack a 5; they do
not know the 5 (in the way that the cardinal numbers do not know the number ½). So these numbers (if you want to
call them that) would be used in cases where the cardinal numbers (with the 5) couldn't meaningfully be used.
Page 322
(Doesn't the nonsensicality of the talk of the "basic intuition" show itself here?)
Page 322
When the intuitionists speak of the "basic intuition"--is this a psychological process? If so, how does it come
into mathematics? Isn't what they mean only a primitive sign (in Frege's sense); an element of a calculus?
Page 322
Strange as it sounds, it is possible to know the prime numbers--let's say--only up to 7 and thus to have a
finite system of prime numbers. And what we call the discovery that there are infinitely many primes is in truth the
discovery of a new system with no greater rights than the other.
Page 322
If you close your eyes and see countless glimmering spots of light coming and going, as we might say, it
doesn't make sense to speak of a 'number' of simultaneously seen dots. And you can't say "there is always a definite
number of spots of light there, we
Page Break 323
just don't know what it is"; that would correspond to a rule applied in a case where you can speak of checking the
number.
Page 323
(It makes sense to say: I divide many among many. But the proposition "I couldn't divide the many nuts
among the many people" can't mean that it was logically impossible. Also you can't say "in some cases it is possible
to divide many among many and in others not"; for in that case I ask: in which cases is this possible and in which
impossible? And to that no further answer can be given in the many-system.)
Page 323
To say of a part of my visual field that it has no colour is nonsense; and of course it is equally nonsense to
say that it has colour (or a colour). On the other hand it makes sense to say it has only one colour (is monochrome,
or uniform in colour) or that it has at least two colours, only two colours, etc.
Page 323
So in the sentence "this square in my visual field has at least two colours" I cannot substitute "one" for
"two". Or again: "the square has only one colour" does not mean--on the analogy of (∃x).φx.~(∃x, y).φx.φy--"the
square has one colour but not two colours".
Page 323
I am speaking here of the case in which it is senseless to say "that part of space has no colour". If I am
counting the uniformly coloured (monochrome) patches in the square, it does incidentally make sense to say that
there aren't any there at all, if the colour of the square is continually changing. In that case of course it also makes
sense to say that there are one or more uniformly coloured patches in the square and also that the square has one
colour and not two.--But for the moment I am disregarding that use of the sentence "the square has no colour" and
am speaking of a system in which it would be called a matter of course that an area of a surface had a colour, a
system, therefore, in which strictly speaking there is no such proposition. If you call the proposition self-evident you
really mean something that is expressed by a grammatical rule giving the form of propositions about visual space,
for
Page Break 324
instance. If you now begin the series of statements giving the number of colours in the square with the proposition
"there is one colour in the square", then of course that mustn't be the proposition of grammar about the
"colouredness" of space.
Page 324
What do you mean if you say "space is coloured"? (And, a very interesting question: what kind of question
is this?) Well, perhaps you look around for confirmation and look at the different colours around you and feel the
inclination to say: "wherever I look there is a colour", or "it's all coloured, all as it were painted." Here you are
imagining colours in contrast to a kind of colourlessness, which on closer inspection turns into a colour itself.
Incidentally, when you look around for confirmation you look first and foremost at static and monochromatic parts
of space, rather than at unstable unclearly coloured parts (flowing water, shadows, etc.) If you then have to admit
that you call just everything that you see colour, what you want to say is that being coloured is a property of space
in itself, not of the parts of space. But that comes to the same as saying of chess that it is chess; and at best it can't
amount to more than a description of the game. So what we must do is describe spatial propositions; but we can't
justify them, as if we had to bring them into agreement with an independent reality.
Page 324
In order to confirm the proposition "the visual field is coloured" one looks round and says "that there is
black, and black is a colour; that is white, and white is a colour", etc. And one regards "black is a colour" as like "iron
is a metal" (or perhaps better, "gypsum is a sulphur compound").
Page 324
If I make it senseless to say that a part of the visual field has a colour, then asking for the analysis of a
statement assigning the number of colours in a part of the visual space becomes very like asking for the analysis of a
statement of the number of parts of a rectangle that I divide up into parts by lines.
Page 324
Here too I can regard it as senseless to say that the rectangle "consists of no parts". Hence, one cannot say
that it consists of one or more parts, or that it has at least one part. Imagine the special case of a rectangle divided by
parallel lines. It doesn't matter that this is a very special case, since we don't regard a game as less remarkable just
because it has only a very limited application.
Page Break 325
Here I can if I want count the parts in the usual manner, and then it is meaningless to say there are 0 parts. But I
could also imagine a way of counting which so to say regards the first part as a matter of course and doesn't count it
or counts it as 0, and counts only the parts which are added to this by division. Again, one could imagine a custom
according to which, say, soldiers in rank and file were always counted by giving the number of soldiers in a line over
and above the first soldier (perhaps because we wanted the number of possible combinations of the fugleman with
another soldier of the rank.). But a custom might also exist of always giving the number of soldiers as 1 greater than
the real one. Perhaps this happened originally in order to deceive a particular officer about the real number, and later
came into general use as a way of counting soldiers. (The academic quarter).†1 The number of different colours on a
surface might also be given by the number of their possible combinations in pairs and in that case the only numbers
that would count would be numbers of the form n/2(n - 1); it would be as senseless then to talk of the 2 or 4 colours
of a surface as it now is to talk of the or i colours. I want to say that it is not the case that the cardinal numbers
are essentially primary and what we might call the combination numbers--1, 2, 6, 10 etc.--are secondary. We might
construct an arithmetic of the combination numbers and it would be as self-contained as the arithmetic of the
cardinal numbers. But equally of course there might be an arithmetic of the even numbers or of the numbers 1, 3, 4,
5, 6, 7... Of course the decimal system is ill-adapted for the writing of these kinds of number.
Page 325
Imagine a calculating machine that calculates not with beads but with colours on a strip of paper. Just as we
now use our fingers, or the beads on an abacus, to count the colours on a strip
Page Break 326
so then we would use the colours on a strip to count the beads on a bar or the fingers on our hand. But how would
this colour-calculating machine have to be made in order to work? We would need a sign for there being no bead on
the bar. We must imagine the abacus as a practical tool and as an instrument in language. Just as we can now
represent a number like 5 by the five fingers of a hand (imagine a gesture language) so we would then represent it by
a strip with five colours. But I need a sign for the o, otherwise I do not have the necessary multiplicity. Well, I can
either stipulate that a black surface is to denote the o (this is of course arbitrary and a monochromatic red surface
would do just as well): or that any one-coloured surface is to denote zero, a two-coloured surface 1, etc. It is
immaterial which method of denotation I choose. Here we see how the multiplicity of the beads is projected onto the
multiplicity of the colours on a surface.
Page 326
It makes no sense to speak of a black two-sided figure in a white circle; this is analogous to its being
senseless to say that the rectangle consists of o parts (no part). Here we have something like a lower limit of counting
before we reach the number one.
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Page 327
Is counting parts in I the same as counting points in IV? What makes the difference? We may regard
counting the parts in I as counting rectangles; but in that case one can also say: "in this row there is no rectangle";
and then one isn't counting parts. We are disturbed both by the analogy between counting the points and counting
the parts, and by the breakdown of the analogy.
Page 327
There is something odd in counting the undivided surface as "one"; on the other hand we find no difficulty in
seeing the surface after a single division as a picture of 2. Here we would much prefer to count "0, 2, 3", etc. And this
corresponds to the series of propositions "the rectangle is undivided", "the rectangle is divided into 2 parts", etc.
Page 327
If it's a question of different colours, you can imagine a way of thinking in which you don't say that here we
have two colours, but that here we have a distinction between colours; a style of thought which does not see 3 at all
in red, green and yellow; which does indeed recognise as a series a series like: red; blue, green; yellow, black, white;
etc., but doesn't connect it with the series |; ||; |||; etc., or not in such a way as to correlate | with the term red.
Page 327
From the point of view from which it is 'odd' to count the undivided surface as one, it is also natural to count
the singly divided one as two. That is what one does if one regards it as two rectangles, and that would mean looking
at it from the standpoint from which the undivided one might well be counted as one rectangle. But if one regards
the first rectangle in I as the undivided surface, then the second appears as a whole with one division (one
distinction) and division here does not necessarily mean dividing line. What I am paying attention to is the
distinctions, and here there is a series of an increasing number of distinctions. In that case I will count the rectangles
in I "0, 1, 2, etc."
Page 327
This is all right where the colours on a strip border on each other, as in the schema
Page Break 328
Page 328
But it is different if the arrangement is or . Of course
I might also correlate each of these two schemata with the schema and correlate schemata like
with the schema , etc. And that way of thinking though
certainly unnatural is perfectly correct.
Page 328
The most natural thing is to conceive the series of schemata as etc. And here we may
denote the first schema by '0', the second by '1', but the third say with '3', if we think of all possible distinctions, and
the fourth by '6' Or we may call the third schema '2' (if we are concerned simply with an arrangement) and the fourth
'3'.
Page 328
We can describe the way a rectangle is divided by saying: it is divided into five parts, or: 4 parts have been
cut off it, or: its division-schema is ABCDE, or: you can reach every part by crossing four boundaries or: the
rectangle is divided (i.e. into 2 parts), one part is divided again, and both parts of this part divided, etc. I want to
show that there isn't only one method of describing the way it is divided.
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Page 329
But perhaps we might refrain altogether from using a number to denote the distinction and keep solely to the
schemata A, AB, ABC, etc.; or we might describe it like this: 1, 12, 123 etc., or, what comes to the same, 0, 01, 012
etc.
Page 329
We may very well call these too numerals.
Page 329
The schemata A, AB, ABC etc., |, ||, |||, etc.; etc.; 0, 1, 2, 3, etc.; 1, 2, 3, etc.; 1,
12, 121323, etc., etc., are all equally fundamental.
Page 329
We are surprised that the number-schema by which we count soldiers in a barracks isn't supposed also to
hold for the parts of a rectangle. But the schema for the soldiers in the barracks is etc., the
one for the parts of the rectangle is , etc. Neither is primary in comparison with the
other.
Page 329
I can compare the series of division-schemata with the series 1, 2, 3, etc. as well as with the series 0, 1, 2, 3,
etc.
Page 329
If I count the parts, then there is no 0 in my number series because the series
etc. begins with one letter whereas the series , etc. does not begin with one dot.
On the other hand, I can represent any fact about the division by this series too, only in that case "I'm not counting
the parts".
Page 329
A way of expressing the problem which, though incorrect, is natural is: why can one say "there are 2 colours
on this surface" but not "there is one colour on this surface?" Or: how must I express the grammatical rule so that it
is obvious and so that I'm not any longer tempted to talk nonsense? Where is the false thought, the
Page Break 330
false analogy by which I am misled into misusing language? How must I set out the grammar so that this temptation
ceases? I think that setting it out by means of the series
removes the unclarity.
Page 330
What matters is whether in order to count 1 use a number series that begins with 0 or one that begins with 1.
Page 330
It is the same if I am counting the lengths of sticks or the size of hats.
Page 330
If I counted with strokes, I might write them thus , in order to show that what matters
is the distinction between the directions and that a simple stroke corresponds to 0 (i.e. is the beginning).
Page 330
Here incidentally there is a certain difficulty about the numerals (1), ((1) + 1), etc.: beyond a certain length we
cannot distinguish them any further without counting the strokes, and so without translating the signs into different
ones. "||||||||||" and "|||||||||||" cannot be distinguished in the same sense as 10 and 11, and so they aren't in the same
sense distinct signs. The same thing could also happen incidentally in the decimal system (think of the numbers
1111111111 and 11111111111), and that is not without significance.
Page 330
Imagine someone giving us a sum to do in a stroke-notation, say |||||||||| + |||||||||||, and, while we are calculating,
amusing himself by removing and adding strokes without our noticing. He would keep on saying: "but the sum isn't
right", and we would keep going through it again, fooled every time.--Indeed, strictly speaking, we wouldn't have
any concept of a criterion for the correctness of the calculation.
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Page 331
Here one might raise questions like: is it only very probable that 464 + 272 = 736? And in that case isn't 2 + 3
= 5 also only very probable? And where is the objective truth which this probability approaches? That is, how do we
get a concept of 2 + 3's really being a certain number, apart from what it seems to us to be?
Page 331
For if it were asked: what is the criterion in the stroke-notation for our having the same numeral in front of us
twice?--the answer might be: "if it looks the same both times" or "if it contains the same number of lines both times".
Or should it be: if a one-one correlation etc. is possible?
Page 331
How can I know that |||||||||| and |||||||||| are the same sign? After all it is not enough that they look alike. For
having roughly the same gestalt can't be what is to constitute the identity of the signs, but just their being the same in
number.
Page 331
(The problem of the distinction between 1 + 1 + 1 + 1 + 1 + 1 + 1 and 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 is much
more fundamental than appears at first sight. It is a matter of the distinction between physical and visual number.)
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19
2 + 2 = 4
Page 332
A cardinal number is an internal property of a list. Are numbers essentially concerned with concepts? I
believe this amounts to asking whether it makes sense to ascribe a number to objects that haven't been brought
under a concept. Does it, for example, make sense to say "a, b and c are three objects"?--Admittedly we have a
feeling: why talk about concepts, the number of course depends only on the extension of the concept, and once that
has been determined the concept may drop out of the picture. The concept is only a method for determining an
extension, but the extension is autonomous and, in its essence, independent of the concept; for it's quite immaterial
which concept we have used to determine the extension. That is the argument for the extensional viewpoint. The
immediate objection to it is: if a concept is really only an expedient for aiming at an extension, then there is no place
for concepts in arithmetic; in that case we must simply divorce a class completely from the concept which happens
to be associated with it. But if it isn't like that, then an extension independent of the concept is just a chimaera, and
in that case it's better not to speak of it at all, but only of the concept.
Page 332
The sign for the extension of a concept is a list. We might say, as an approximation, that a number is an
external property of a concept and an internal property of its extension (the list of objects that fall under it). A
number is a schema for the extension of a concept. That is, as Frege said, a statement of number is a statement about
a concept (a predicate). It's not about the extension of a concept, i.e. a list that may be something like the extension
of a concept. But a number-statement about a concept has a similarity to a proposition saying that a determinate list
is the extension of the concept. I use such a list when I say "a, b, c, d, fall under the concept F(x)": "a, b, c, d," is the
list. Of course this proposition
Page Break 333
says the same as Fa.Fb.Fc.Fd; but the use of the list in writing the proposition shows its relationship to "(∃x, y, z, u).
Fx.Fy.Fz.Fu" which we can abbreviate as "(∃||||x).F(x)."
Page 333
What arithmetic is concerned with is the schema ||||.--But does arithmetic talk about the lines that I draw with
pencil on paper?--Arithmetic doesn't talk about the lines, it operates with them.
Page 333
A statement of number doesn't always contain a generalization or indeterminacy: "The line AB is divided into
2 (3, 4, etc.) equal parts."
Page 333
If you want to know what 2 + 2 = 4 means, you have to ask how we work it out. That means that we consider
the process of calculation as the essential thing; and that's how we look at the matter in ordinary life, at least as far as
concerns the numbers that we have to work out. We mustn't feel ashamed of regarding numbers and sums in the
same way as the everyday arithmetic of every trader. In everyday life we don't work out 2 + 2 = 4 or any of the rules
of the multiplication table; we take them for granted like axioms and use them to calculate. But of course we could
work out 2 + 2 = 4 and children in fact do so by counting off. Given the sequence of numbers 1 2 3 4 5 the
calculation is
Page 333
Abbreviative Definitions:
(∃x).φx:~(∃x, y).φx.φy .(εx).φx
(∃x, y).φx.φy:~(∃x, y, z).φx.φy.φz. .(εx, y).φx.φy, etc.
(εx).φx. .(ε|x).φx
(εx, y).φx.φy. .(ε||x).φx. .(ε2x).φx, etc.
Page 333
It can be shewn that
(ε||x).φx.(ε|||x).(ψx). .⊃ .(ε|||||x).φx ∨ ψx
is a tautology.
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Page 334
Does that prove the arithmetical proposition 2 + 3 = 5? Of course not. It does not even show that
(ε||x).φx.(ε|||x).ψx.Ind..⊃.(ε|| + |||x).φx ∨ ψx is tautologous, because nothing was said in our definitions about a sum
(|| + |||). (I will write the tautology in the abbreviated form "ε||. ε|||. ⊃. ε|||||".) Suppose the question is, given a left
hand side, to find what number of lines to the right of "⊃" makes the whole a tautology. We can find the number, we
can indeed discover that in the case above it is || + |||; but we can equally well discover that it is | + |||| or | + ||| + |, for
it is all of these. We can also find an inductive proof that the algebraic expression
εn.εm. ⊃.ε n + m
is tautologous. Then I have a right to regard a proposition like
ε17.ε28. ⊃ .ε(17 + 28)
as a tautology. But does that give us the equation 17 + 28 = 45? Certainly not. I still have to work it out. In
accordance with this general rule, it also makes sense to write ε2. ε3. ⊃. ε5 as a tautology if, as it were, I don't yet
know what 2 + 3 yields; for 2 + 3 only has sense in so far as it has still to be worked out.
Page 334
Hence the equation || + ||| = ||||| only has a point if the sign "|||||" can be recognised in the same way as the sign
"5", that is, independently of the equation.
Page 334
The difference between my point of view and that of contemporary writers on the foundations of arithmetic
is that I am not obliged to despise particular calculi like the decimal system. For me one calculus is as good as
another. To look down on a particular calculus is like wanting to play chess without real pieces, because playing with
pieces is too particularized and not abstract enough. If the pieces really don't matter then one lot is just as good as
another. And if the games are really distinct from each other, then one game is as good, i.e. as interesting, as the
other. None of them is more sublime than any other.
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Page 335
Which proof of ε||. ε|||. ⊃. ε||||| expresses our knowledge that this is a correct logical proposition?
Page 335
Obviously, one that makes use of the fact that one can treat (∃x)... as a logical sum. We may translate from a
symbolism like ("if there is a star in each square, then there are two in the whole rectangle") into the
Russellian one. And it isn't as if the tautologies in that notation expressed an idea that is confirmed by the proof after
first of all appearing merely plausible; what appears plausible to us is that this expression is a tautology (a law of
logic).
Page 335
The series of propositions
(∃x): aRx.xRb
(∃x, y): aRx.xRy.yRb
(∃x, y, z): aRx.xRy.yRz.zRb, etc.
may perfectly well be expressed as follows:
Page 335
"There is one term between a and b".
Page 335
"There are two terms between a and b", etc.,
and may be written in some such way as:
(∃1x).aRxRb, (∃2x).aRxRb, etc.
Page 335
But it is clear that in order to understand this expression we need the explanation above, because otherwise
by analogy with (∃2x).φx. = .(∃x, y).φx.φy you might believe that (∃2x).aRxRb was equivalent to the expression
(∃x, y).aRxRb.aRyRb.
Page 335
Of course I might also write "(∃2x, y).F(x, y)" instead of "(∃x, y).F(x, y)". But then the question would be:
what am I to take "(∃3x, y).F(x, y)" as meaning? But here a rule can be given; and indeed we need one that takes us
further in the number series as far as we want to go. E.g.:
(∃3x, y).F(x, y). =.(∃x, y, z):F(x, y).F(x, z).F(y, z)
(∃4x, y).F(x, y). = .(∃x, y, z, u):F(x, y).F(x, z)....
followed by the combinations of two elements, and so on. But we might also give the following definition:
(∃3x, y).F(x, y). =.(∃x, y, z):F(x, y).F(y, x).F(x, z).
F(z, x).F(y, z).F(x, y), and so on.
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Page 336
"(∃3 x, y).F(x, y)" would perhaps correspond to the proposition in word-language "F(x, y) is satisfied by 3
things"; and that proposition too would need an explanation if it was not to be ambiguous.
Page 336
Am I now to say that in these different cases the sign "3" has different meanings? Isn't it rather that the sign
"3" expresses what is common to the different interpretations? Why else would I have chosen it? Certainly, in each
of these contexts, the same rules hold for the sign "3". It is replaceable by 2 + 1 as usual and so on. But at all events a
proposition on the pattern of ε||. ε|||. ⊃. ε||||| is no longer a tautology. Two men who live at peace with each other and
three other men who live at peace with each other do not make five men who live at peace with each other. But that
does not mean that 2 + 3 are no longer 5; it is just that addition cannot be applied in that way. For one might say: 2
men who... and 3 men who..., each of whom lives at peace with each of the first group, = 5 men who...
Page 336
In other words, the signs of the form (∃1x, y).F(x, y), (∃2x, y). F(x, y) etc. have the same multiplicity as the
cardinal numbers, like the signs (∃1x).φx, (∃2x).φx, etc. and also like the signs (ε1x).φx, (ε2x).φx, etc.
Page 336
"There are only 4 red things, but they don't consist of 2 and 2, as there is no function under which they fall in
pairs". That would mean regarding the proposition 2 + 2 = 4 thus: if you can see 4 circles on a surface, every two of
them always have a particular property in common; say a sign inside the circle. (In that case of course every three of
the circles too will have to have a sign in common etc.) If I am to make any assumption at all about reality, why not
that? The 'axiom of reducibility' is essentially the same kind of thing. In this sense one might say that 2 and 2 do
always make 4, but 4 doesn't always consist of 2 and 2. (It is only because of the utter vagueness and generality of
the axiom of reducibility that we are seduced
Page Break 337
into believing that--if it is a significant sentence at all--it is more than an arbitrary assumption for which there is no
ground. For this reason, in this and all similar cases, it is very illuminating to drop this generality, which doesn't
make the matter any more mathematical, and in its place to make very specific assumptions.)
Page 337
We feel like saying: 4 does not always have to consist of 2 and 2, but if it does consist of groups it can
consist of 2 and 2, or of 3 and 1 etc.; but not of 2 and 1 or 3 and 2, etc. In that way we get everything prepared in
case 4 is actually divisible into groups. But in that case arithmetic doesn't have anything to do with the actual
division, but only with the possibility of division. The assertion might just as well be the assertion that any two of a
group of 4 dots on paper are always joined by a line.
Page 337
Or that around every 2 such groups of 2 dots in the real world there is always a circle drawn.
Page 337
Add to this that a statement like "you can see two black circles in a white rectangle" doesn't have the form
"(∃x, y), etc.". For, if I give the circles names, the names refer to the precise location of the circles and I can't say of
them that they are either in this rectangle or in the other. I can indeed say "there are 4 circles in both rectangles taken
together" but that doesn't mean that I can say of each individual circle that it is in one rectangle or the other. For in
the case supposed the sentence "this circle is in this rectangle" is senseless.
Page 337
But what does the proposition "there are 4 circles in the 2 rectangles taken together" mean? How do I
establish that? By adding the numbers in each? In that case the number of the circles in the two rectangles means the
result of the addition of the two numbers.--Or is it something like the result of taking a count through both
rectangles? Or the number of lines I get if I correlate a line to a circle no matter whether it is in this rectangle or in the
Page Break 338
other? If "this circle" is individuated by its position, we can say "every line is correlated either to a circle in this
rectangle or to a circle in the other rectangle" but not "this circle is either in this
rectangle or in the other". This can only be here if "this" and "here" do not mean the same. By contrast this line can
be correlated to a circle in this rectangle because it remains this line, even if it is correlated to a circle in the other
rectangle.
Page 338
In these two circles together are there 9 dots or 7? As one normally understands the question, 7. But must I
understand it so? Why shouldn't I count twice the points that are common to both circles?
Page 338
It is a different matter if we ask "how many dots are within the black lines?" For here I can say: in the sense
in which there are 5 and 4 in the circles, there are 7.
Page Break 339
Page 339
Now we might say: by the sum of 4 and 5 I mean the number of the objects which fall under the concept φx
∨ ψx, if it is the case that (E 4x)†1.φx.(E 5x).ψx.Ind. That doesn't mean that the sum of 4 and 5 may only be used in
the context of propositions like (∃ 4x).φx; it means: if you want to construct the sum of n and m, insert the numbers
on the left hand side of "⊃" in the form (∃nx).φx.(∃mx).ψx, etc., and the sum of m and n will be the number which
has to go on the right hand side in order to make the whole proposition a tautology. So that is a method of
addition--a very long-winded one.
Page 339
Compare: "Hydrogen and oxygen yield water", "2 dots and 3 dots yield 5 dots".
Page 339
So do e.g. 2 dots in my visual field, that I "see as 4" and not "as 2 and 2", consist of 2 and 2? Well, what does
that mean? Is it asking whether in some way they are divided into groups of 2 dots each? Of course not (for in that
case they would presumably have had to be divided in all other conceivable ways as well). Does it mean that they
can be divided into groups of 2 and 2, i.e. that it makes sense to speak of such groups in the four?--At any rate it
does correspond to the sentence 2 + 2 = 4 that I can't say that the group of 4 dots I saw consisted of separate groups
of 2 and 3. Everyone will say: that's impossible, because 3 + 2 = 5. (And "impossible" here means "nonsensical".)
Page 339
"Do 4 dots consist of 2 and 2?" may be a question about a physical or visual fact; for it isn't the question in
arithmetic. The
Page Break 340
arithmetical question, however, certainly could be put in the form: "Can a group of 4 dots consist of separate groups
of 2?"
Page 340
"Suppose that I used to believe that there wasn't anything at all except one function and the 4 objects that
satisfy it. Later I realise that it is satisfied by a fifth thing too: does this make the sign '4' become senseless?"--Well, if
there is no 4 in the calculus then '4' is senseless.
Page 340
If you say it would be possible when adding to make use of the tautology (E 2x).φx.(E 3x).ψx.Ind. ⊃.(E
5x).φx ∨ ψx... A) this is how it would have to be understood: first it is possible to establish according to certain rules
that (E x).φx. (E x).ψx.Ind..⊃.(E x, y):φx ∨ ψx.φy ∨ φy. is tautological. (E x).φx is an abbreviation for (∃x).φx.~(∃x,
y).φx.φy. I will abbreviate further tautologies like A thus: (E).(E).⊃.(E)
Page 340
Therefore
(E x).(E x). ⊃.(E x, y).(E x, y).(E x). ⊃.(E x, y, z).
and other tautologies follow from the rules. I write "and other tautologies" and not "and so on ad inf." since one
doesn't yet have to use that concept.
Page 340
†1 When the numbers were written out in the decimal system there were rules, namely the addition rules for
every pair of numbers from 0 to 9, and, used appropriately, these sufficed for the addition of all numbers. Now
which rule corresponds to these elementary
Page Break 341
rules? It is obvious that in a calculation like σ we don't have to keep as many rules in mind as in 17 + 28. Indeed we
need only one general rule. We don't need any rules like 3 + 2; on the contrary, we now seem to be able to deduce,
or work out, how many 3 + 2 makes.
Page 341
We are given the sum 2 + 3 =? and we write
1, 2, 3, 4, 5, 6, 7
1, 2; 1, 2, 3
Page 341
That is in fact how children calculate when they "count off". (And that calculus must be as good as any
other.)
Page 341
It is clear incidentally that the problem whether 5 + (4 + 3) = (5 + 4) + 3 can be solved in this way:
for this construction has precisely the same multiplicity as every other proof of that proposition.
A B C D E F G H I J K L MNO
A B C D E, A B C D
A I, A B C
A A B C D, A B C
A A G
A E, A G
A L
Page 341
If I name each number after its last letter, that is a proof that (E + D) + C = E + (D + C) = L
Page 341
This is a good form of proof, because it shows clearly that the result is really worked out and because from it
you can read off the general proof as well.
Page 341
It may sound odd, but it is good advice at this point: don't do philosophy here, do mathematics.
Page Break 342
Page 342
Our calculus doesn't at all need to be acquainted with the construction of a series '(E x)', '(E x, y)', '(E x, y, z)'
etc.; we can simply introduce two or three such signs without the "etc.". We can then introduce a calculus with a
finite series of signs by laying down a sequence of certain signs, say the letters of the alphabet, and writing:
(E a).(E a). ⊃.(E a, b)
(E a, b).(E a). ⊃.(E a, b, c)
(E a, b).(E a, b). ⊃.(E a, b, c, d)
etc. up to z.
Page 342
The right hand side (the side to the right of "⊃") can then be found from the left hand side by a calculus like:
Page 342
This calculus could be derived from the rules for the construction of tautologies as a simplification.--If I
presuppose this law for constructing a fragment of the series out of two others, I can then introduce as a designation
of that fragment the expression "sum of the two others", and thus give the definition:
a + a ab
a + ab abc
and so on up to z.
Page 342
If the rules for the calculus B had been explained by examples, we could regard those definitions too as
particular cases of a general rule and then set problems like "abc + ab =?". It is now tempting to confuse the
tautology
α) (E a, b). (E a, b). ⊃. (E a, b, c, d)
with the equation
β) ab + ab = abcd
But the latter is a replacement rule, the former isn't a rule but just a tautology. The sign "⊃" in α in no way
corresponds to the "=" in β.
Page 342
We forget that the sign "⊃" in α doesn't say that the two signs to the left and right of it yield a tautology.
Page Break 343
Page 343
On the other hand we might construct a calculus in which the equation ξ + η = η was obtained as a
transformation of the equation
γ) (E ξ).(E η). ⊃.(E ζ) = Taut.
Page 343
So that I as it were get ζ = ξ + η if I work out ζ from the equation γ.
Page 343
In these discussions, how does the concept of sum make its entry?--There is no mention of summation in the
original calculus that lays down that the form
δ) (Eξ).(Eη). ⊃ (Eζ)
is tautologous where ξ = xy, y = x and ζ = xyz.--Later we introduce into the calculus a number system (say the
system a, b, c, d,... z), and finally we define the sum of two numbers as the number that solves the equation γ.
Page 343
If we wrote "(E x).(E x). ⊃.(E x, y)" instead of "(E x). (E x). ⊃. (E x + x)" it would make no sense; unless the
notation already went, not
Page 343
ι) "(E x), etc.", "(E x, y) etc.", "(E x, y, z), etc."
but
Page 343
κ) "(E x), etc.", (E x + x) etc.", "(E x + x + x), etc."
For why should we suddenly write
"(E x, y).(E x). ⊃.(E xy + x)" instead of "(E x, y).(E x). ⊃.(E x, y, z)"?
That would just confuse the notation.--Then we say: it will greatly simplify the writing of the tautologies if we can
write in the right bracket simply the expressions in the two left brackets. But so far that notation hasn't been
explained: I don't know what (E xy + x) means, or that (E xy + x) = (E x, y, z).
Page 343
But if the notation already went "(E x)", "(E x + x)", "(E x + x + x)" that would only give a sense to the
expression "(E x + x + x + x + x) and not to (E (x + x) + (x + x)).
Page 343
The notation κ is in the same case as ι. A quick way of calculating whether you get a tautology of the form δ
is to draw connecting lines, thus
Page Break 344
and analogously
Page 344
The connecting lines only correspond to the rule which we have to give in any case for checking the
tautology. There is still no mention of addition; that doesn't come in until I decide--e.g.--to write "xy + yx" instead of
"x, y, z, u" and adjoin a calculus with rules that allow the derivation of the replacement rule "xy + yx = xyzu".
Again, addition doesn't come in when I write in the notation κ "(E x).(E x). ⊃.(E x + x)"; it only comes in when I
distinguish between "x + x" and "(x) + (x)" and write (x) + (x) = (x + x)
Page 344
I can define "the sum of ξ and η" "(ξ + η)" as the number (or "the expression" if we are afraid to use the
word "number")--I can define "ξ + η" as the number ζ that makes the expression δ tautologous; but we can also
define "ξ + η" (independently of the calculus of tautologies) by the calculus B and then derive the equation (E ξ).(E
η). ⊃. (E ξ + η) = Taut.
Page 344
A question that suggests itself is this: must we introduce the cardinal numbers in connection with the
notation (∃x, y,...).φx.φy...? Is the calculus of the cardinal numbers somehow bound up with the calculus of the
signs "(∃x, y...).φx.φy...? Is that kind of calculus perhaps in the nature of things the only application of the cardinal
numbers? So far as concerns the "application of the cardinal numbers in the grammar", we can refer to what we said
about the concept of the application of a calculus. We might put our question in this way too: in the propositions of
our language--if we imagine them translated into Russell's notation--do the cardinal numbers always occur after the
sign "∃"? This question
Page Break 345
is closely connected with another: Is a numeral always used in language as a characterization of a concept--a
function? The answer to that is that our language does always use the numerals as attributes of concept-words--but
that these concept-words belong to different grammatical systems that are so totally distinct from each other (as you
see from the fact that some of them have meaning in contexts in which others are senseless), that a norm making
them all concept-words is an uninteresting one. But the notation "(∃x, y...) etc." is just such a norm. It is a straight
translation of a norm of our word-language, the expression "there is...", which is a form of expression into which
countless grammatical forms are squeezed.
Page 345
Moreover there is another sense of numeral in which numerals are not connected with "∃": that is, in so far as
"(∃3)x..." is not contained in "(∃2 + 3)x...".
Page 345
If we disregard functions containing "=" (x = a. ∨ .x = b, etc.), then on Russell's theory 5 = 1 if there are no
functions that are satisfied by only one argument, or by only 5 arguments. Of course at first this proposition seems
nonsensical; for in that case how can one sensibly say that there are no such functions? Russell would have to say
that the statement that there are five-functions and the statement that these are one-functions can only be separated
if we have in our symbolism a five-class and a one-class. Perhaps he could say that his view is correct because
without the paradigm of the class 5 in the symbolism, I can't say at all that a function is satisfied by five arguments.
That is to say, from the existence of the sentence "(∃φ:(E1 x).φx" its truth already follows.--So you seem to be able
to say: look at this sentence, and you will see that it is true. And in a sense irrelevant for our purposes that is indeed
possible: think of the wall of a room on which is written in red "in this room there is something red".--
Page Break 346
Page 346
This problem is connected with the fact that in an ostensive definition I do not state anything about the
paradigm (sample); I only use it to make a statement. It belongs to the symbolism and is not one of the objects to
which I apply the symbolism.
Page 346
For instance, suppose that "1 foot" were defined as the length of a particular rod in my room, so that instead
of saying "this door is 6 ft high" I would say "this door is six times as high as this length" (pointing to the unit rod).
In that case we couldn't say things like "the proposition 'there is an object whose length is 1 ft' proves itself, because
I couldn't express the proposition at all if there were no object of that length". (That is, if I introduced the sign "this
length" instead of "1 foot", then the statement that the unit rod is 1 foot long would mean "this rod has this length"
(where I point both times to the same rod).) Similarly one cannot say of a group of strokes serving as a paradigm of
3, that it consists of 3 strokes.
Page 346
"If the proposition isn't true, then the proposition doesn't exist" means: "if the proposition doesn't exist, then
it doesn't exist". And one proposition can never describe the paradigm in another, unless it ceases to be a paradigm.
If the length of the unit rod can be described by assigning it the length "1 foot", then it isn't the paradigm of the unit
of length; if it were, every statement of length would have to be made by means of it.
Page 346
If we can give any sense at all to a proposition of the form "~(∃φ):(E x).φx" it must be a proposition like:
"there is no circle on this surface containing only one black speck" (I mean: it must have that sort of determined
sense, and not remain vague as it did in Russellian logic and in my logic in the Tractatus).
Page 346
If it follows from the propositions
ρ) ~(∃φ):(E x).φx
and σ ~(∃φ):(E x, y).φx.φy
that 1 = 2, then here "1" and "2" don't mean what we commonly
Page Break 347
mean by them, because in word-language the propositions ρ and σ would be "there is no function that is satisfied by
only one thing" and "there is no function that is satisfied by only two things." And according to the rules of our
language these are propositions with different senses.
Page 347
One is tempted to say: "In order to express '(∃x, y).φx.φy' we need 2 signs 'x' and 'y'." But that has no
meaning. What we need for it, is, perhaps, pen and paper; and the proposition means no more than "to express 'p'
we need 'p'."
Page 347
If we ask: but what then does "5 + 7 = 12" mean--what kind of significance or point is left for this expression
after the elimination of the tautologies, etc. from the arithmetical calculus?--the answer is: this equation is a
replacement rule which is based on certain general replacement rules, the rules of addition. The content of 5 + 7 = 12
(supposing someone didn't know it) is precisely what children find difficult when they are learning this proposition
in arithmetic lessons.
Page 347
No investigation of concepts, only insight into the number-calculus can tell us that 3 + 2 = 5. That is what
makes us rebel against the idea that
"(E 3 x).φx.(E 2 x).ψx.Ind.: ⊃.(E 5 x).φx ∨ ψx"†1
could be the proposition 3 + 2 = 5. For what enables us to tell that this expression is a tautology cannot itself be the
result of an examination of concepts, but must be recognizable from the calculus. For the grammar is a calculus.
That is, nothing of what the tautology calculus contains apart from the number calculus serves to justify it and if it is
number we are interested in the rest is mere decoration.
Page 347
Children learn in school that 2 × 2 = 4, but not that 2 = 2.
Page Break 348
20
Statements of number within mathematics
Page 348
What distinguishes a statement of number about a concept from one about a variable? The first is a proposition
about the concept, the second a grammatical rule concerning the variable.
Page 348
But can't I specify a variable by saying that its values are to be all objects satisfying a certain function? In that
way I do not indeed specify the variable unless I know which objects satisfy the function, that is, if these objects are
given me in another way (say by a list); and then giving the function becomes superfluous. If we do not know
whether an object satisfies the function, then we do not know whether it is to be a value of the variable, and the
grammar of the variable is in that case simply not expressed in this respect.
Page 348
Statements of number in mathematics (e.g. "The equation x2 = 1 has 2 roots") are therefore quite a different
kind of thing from statements of number outside mathematics ("There are 2 apples on the table").
Page 348
If we say that A B admits of 2 permutations, it sounds as if we had made a general assertion, analogous to
"There are 2 men in the room" in which nothing further is said or need be known about the men. But this isn't so in
the A B case. I cannot give a more general description of A B, B A and so the proposition that no permutations are
possible cannot say less than that the permutations A B, B A are possible. To say that 6 permutations of 3 elements
are possible cannot say less, i.e. anything more general, than is shown by the schema:
A B C
A C B
B A C
B C A
C A B
C B A
Page Break 349
Page 349
For it's impossible to know the number of possible permutations without knowing which they are. And if
this weren't so, the theory of combinations wouldn't be capable of arriving at its general formulae. The law which we
see in the formulation of the permutations is represented by the equation p = n! In the same sense, I believe, as that
in which a circle is given by its equation.--Of course I can correlate the number 2 with the permutations A B, B A
just as I can 6 with the complete set of permutations of A, B, C, but that does not give me the theorem of
combination theory.--What I see in A B, B A is an internal relation which therefore cannot be described. That is,
what cannot be described is that which makes this class of permutations complete.--I can only count what is actually
there, not possibilities. But I can e.g. work out how many rows a man must write if in each row he puts a
permutation of 3 elements and goes on until he cannot go any further without repetition. And this means, he needs 6
rows to write down the permutations A B C, A C B, etc., since these just are "the permutations of A, B, C". But it
makes no sense to say that these are all permutations of A B C.
Page 349
We could imagine a combination computer exactly like the Russian abacus.
Page 349
It is clear that there is a mathematical question: "How many permutations of--say--4 elements are there?", a
question of precisely the same kind as "What is 25 × 18?". For in both cases there is a general method of solution.
Page 349
But still it is only with respect to this method that this question exists.
Page 349
The proposition that there are 6 permutations of 3 elements is identical with the permutation schema and
thus there isn't here a proposition "There are 7 permutations of 3 elements", for no such schema corresponds to it.
Page Break 350
Page 350
You could also conceive the number 6 in this case as another kind of number, the permutation-number of A,
B, C. Permutation as another kind of counting.
Page 350
If you want to know what a proposition means, you can always ask "How do I know that?" Do I know that
there are 6 permutations of 3 elements in the same way in which I know that there are 6 people in this room? No.
Therefore the first proposition is of a different kind from the second.
Page 350
You may also say that the proposition "There are 6 permutations of 3 elements" is related to the proposition
"There are 6 people in this room" in precisely the same way as is "3 + 3 = 6", which you could also cast in the form
"There are 6 units in 3 + 3". And just as in the one case I can count the rows in the permutation schema, so in the
other I can count the strokes in
|||
|||
Just as I can prove that 4 × 3 = 12 by means of the schema
I can also prove 3! = 6 by means of the permutation schema.
Page 350
The proposition "the relation R links two objects", if it is to mean the same as "R is a two-place relation", is a
proposition of grammar.
Page Break 351
21
Sameness of number and sameness of length
Page 351
How should we regard the propositions "these hats are of the same size", or "these rods have the same
length" or "these patches have the same colour"? Should we write them in the form "(∃L).La. Lb"? But if that is
intended in the usual way, and so is used with the usual rules, it would mean that it made sense to write "(∃L).La",
i.e. "the patch has a colour", "the rod has a length". Of course I can write "(∃L).La.Lb" for "a and b have the same
length" provided that I know and bear in mind that "(∃L).La" is senseless; but then the notation becomes misleading
and confusing ("to have a length", "to have a father").--What we have here is something that we often express in
ordinary language as follows: "If a has the length L, so does b"; but here the sentence "a has the length L" has no
sense, or at least not as a statement about a; the proposition should be reworded "if we call the length of a 'L', then
the length of b is L" and 'L' here is essentially a variable. The proposition incidentally has the form of an example, of
a proposition that could serve as an example for the general sentence; we might go on: "for example, if the length of
a is 5 metres, then the length of b is 5 metres, etc."--Saying "the rods a and b have the same length" says nothing
about the length of each rod; for it doesn't even say "that each of the two has a length". So it is quite unlike "A and B
have the same father" and "the name of the father of A and B is 'N' ", where I simply substitute the proper name for
the general description. It is not that there is a certain length of which we are at first only told that a and b both
possess it, and of which '5m' is the name. If the lengths are lengths in the visual field we can say the two lengths are
the same, without in general being able to "name" them with a number.--The written form of the proposition "if L is
the length of a, the length of b too is L" is derived from the form of an example. And we might express the general
proposition by actually enumerating
Page Break 352
examples and adding "etc.". And If I say, "a and b are the same length; if the length of a is L, then the length of b is
L; if a is 5 m long then b is 5 m long, if a is 7 m long, then b is 7 m long, etc.", I am repeating the same proposition.
The third formulation shows that the "and" in the proposition doesn't stand between two forms, as it does in
"(∃x).φx.ψx", where one can also write "(∃x).φx" and "(∃x).ψx".
Page 352
Let us take as an example the proposition "there are the same number of apples in each of the two boxes". If
we write this proposition in the form "there is a number that is the number of the apples in each of the boxes" here
too we cannot construct the form "there is a number that is the number of apples in this box" or "the apples in this
box have a number". If I write: (∃x).φx.~(∃x, y).φx.φy. =.(∃n1x).φx. = φ1, etc. then we might write the proposition
"the number of apples in both boxes is the same" as "(∃x).φn.ψn". But "(∃n).φn" would not be a proposition.
Page 352
If you want to write the proposition "the same number of objects fall under φ and ψ" in a perspicuous
notation, the first temptation is to write it in the form "φn.ψn". And that doesn't feel as if it were a logical product of
φn and ψn, which would mean that it made sense to write φn.ψ5; it is essential that the same letter should follow ψ
as follows φ, and φn.ψn is an abstraction from the logical products φ4.ψ4, φ5.ψ5, etc., rather than itself a logical
product. (So φn doesn't follow from φn.ψn. The relation of φn • ψn to a logical product is more like that of a
differential quotient to a quotient.) It is no more a logical product than the photograph of a family group is a group
of photographs. Therefore the form "φn.ψn" can be misleading and perhaps we should prefer a notation of the form
" "; or even "(∃n).φn.ψn", provided that the grammar of this sign is fixed. We can then stipulate (∃n).φn =
Taut., which is the same as (∃n).φn.p. = .p.
Page Break 353
Therefore (∃n).φn ∨ ψn. = .Taut., (∃n).φn. ⊃ ψn. = .Taut., (∃n).φn|ψn. = Cont., etc.
φ1.ψ1.(∃n).φn.ψn = .φ1.(∃n).φn.ψn
φ2.ψ2.(∃n).φn.ψn = .φ2.(∃n).φn.ψn
etc. ad inf.
Page 353
And in general the calculation rules for (∃n)φn.ψn can be derived from the fact that we can write
(∃n).φn.ψn =.φ0.ψ0. ∨ .φ1.ψ1. ∨ .φ2.ψ2. ∨ .φ3.φ3
and so on ad inf.
Page 353
It is clear that this is not a logical sum, because "and so on ad inf." is not a sentence. The notation (∃n).φn.ψn
however is not proof against misunderstanding, because you might wonder why you shouldn't be able to put Φn
instead of φn.ψn though if you did (∃n).Φn should of course be meaningless. Of course we can clear that up by
going back to the notation ~(∃x).φx for φ0, (∃x)φx.~(∃x,y).φx.φx for φ1, etc., i.e. to (∃n0x). φx for φ0, (∃n1x).φx for
φ1 respectively, and so on. For then we can distinguish between
(∃n1x).φx(∃n1x).ψx and (∃n1x).φx.ψx
Page 353
And if we go back to (∃n).φn.ψn, that means (∃n):(∃nnx).φx.(∃nnx).ψx (which is not nonsensical) and not
(∃n):(∃nnx).φx.ψx, which is nonsensical.
Page 353
The expressions "same number", "same length", "same colour", etc. have grammars which are similar but not
the same. In each case it is tempting to regard the proposition as an endless logical sum whose terms have the form
φn.ψn. Moreover, each of these words has several different meanings, i.e. can itself be replaced by several words
with different grammars. For "same number" does not mean the same when applied to lines simultaneously present
in the visual field as in connection with the apples in two boxes; and "same length" applied in visual space is
different from "same length" in Euclidean space; and the meaning of "same colour" depends on the criterion we
adopt for sameness of colour.
Page 353
If we are talking about patches in the visual field seen simultaneously, the expression "same length" varies in
meaning
Page Break 354
depending on whether the lines are immediately adjacent or at a distance from each other. In word-language we
often get out of the difficulty by using the expression "it looks".
Page 354
Sameness of number, when it is a matter of a number of lines "that one can take in at a glance", is a different
sameness from that which can only be established by counting the lines.
Page 354
Different criteria for sameness of number: in I and II the number that one immediately recognizes; in III the
criterion of correlation; in IV we have to count both groups; in V we recognize the same pattern. (Of course these
are not the only cases.)
Page 354
We want to say that equality of length in Euclidean space consists in both lines measuring the same number
of cm, both 5 cm, both 10 cm etc.; but where it is a case of two lines in visual space being equally long, there is no
length L that both lines have.
Page Break 355
Page 355
One wants to say: two rods must always have either the same length or different lengths. But what does that
mean? What it is, of course, is a rule about modes of expression. "There must either be the same number or a
different number of apples in the two boxes." The method whereby I discover whether two lines are of the same
length is supposed to be the laying of a ruler against each line: but do they have the same length when the rulers are
not applied? In that case we would say we don't know whether during that time the two lines have the same or
different lengths. But we might also say that during that time they have no length, or perhaps no numerical length.
Page 355
Something similar, if not exactly the same, holds of sameness between numbers.
Page 355
When we cannot immediately see the number of dots in a group, we can sometimes keep the group in view
as a whole while we count, so that it makes sense to say it hasn't altered during the counting. It is different when we
have a group of bodies or patches that we cannot keep in a single view while we count them, so that we don't have
the same criterion for the group's not changing while it is counted.
Page 355
Russell's definition of sameness of number is unsatisfactory for various reasons. The truth is that in
mathematics we don't need any such definition of sameness of number. He puts the cart before the horse.
Page 355
What seduces us into accepting the Russellian or Fregean explanation is the thought that two classes of
objects (apples in two boxes) have the same number if they can be correlated 1 to 1. We imagine correlation as a
check of sameness of number. And here we do distinguish in thought between being correlated and being connected
by a relation; and correlation becomes something that is related to connection as the "geometrical straight line" is
Page Break 356
related to a real line, namely a kind of ideal connection that is as it were sketched in advance by Logic so that reality
only has to trace it. It is possibility conceived as a shadowy actuality. This in turn is connected with the idea of
("∃x).φx" as an expression of the possibility of φx.
Page 356
"φ and ψ have the same number" (I will write this "S(φ, ψ)" or simply "S") is supposed to follow from
"φ5.ψ5"; but it doesn't follow from φ5.ψ5 that φ and ψ are connected by a 1-1 relation R (this I will write Π(φ, ψ)" or
"Π"). We get out of the difficulty by saying that in that case there is a relation like
"x = a.y = b. ∨.x = c.y = d.∨., etc."
Page 356
But if so, then in the first place why don't we define S without more ado as the holding of such a relation?
And if you reply that this definition wouldn't include sameness of number in the case of infinite numbers, we shall
have to say that this only boils down to a question of "elegance", because for finite numbers in the end I have to take
refuge in "extensional" relations. But these too get us nowhere; because saying that between φ and ψ there holds a
relation e.g. of the form x = a.y = b. ∨ .x = c.y = d says only that
(∃x, y).φx.ψy.~(∃x, y, z).φx.φy.φz:(∃x, y)ψx.ψy.~(∃x, y, z).ψx.ψy.ψz.
(Which I write in the form
(∃n2x).φx.(∃n2x).ψx.)
Page 356
And saying that between φ and ψ there holds one of the relations x = a.y = b; x = a.y = b. ∨ .x = c.y = d; etc.
etc. means only that there obtains one of the facts φ1.ψ1; φ2.ψ2 etc. etc. Then we retreat into greater generality,
saying that between φ and ψ there holds some 1-1 relation, forgetting that in order to specify this generality we have
to make the rule that "some relation" includes also relations of the form x = a.y = b, etc. By saying more one does
Page Break 357
not avoid saying the less that is supposed to be contained in the more. Logic cannot be duped.
Page 357
So in the sense of S in which S follows from φ5.ψ5, it is not defined by Russell's definition. Instead, what we
need is a series of definitions.
Page 357
On the other hand Π is used as a criterion of sameness of number and of course in another sense of S it can
also be equated with S. (And then we can only say: if in a given notation S = Π, then S means the same as Π.)
Page 357
Though Π does not follow from φ5.ψ5, φ5.ψ5 does from Π.φ5.
Π.φ5 = Π.φ5.ψ5 = Π.ψ5
etc.
Page 357
We can therefore write:
Π.φ0 = Π.φ0.ψ0 = Π.ψ0.S
Π.φ1 = Π.φ1.ψ1 = Π.ψ1.S ...β
Π.φ2 = Π.φ2.ψ2 = Π.ψ2.S
and so on ad inf.
And we can express this by saying that the sameness of number follows from Π. And we can also give the rule Π.S
= Π; it accords with the rules, or the rule, β and the rule α.
Page 357
We could perfectly well drop the rule "S follows from Π", that is, Π.S = Π; the rule β does the same job.
Page 357
If we write S in the form
φ0.ψ0. ∨.φ1.ψ1. ∨.φ2ψ2. ∨... ad inf.
we can easily derive Π.S = Π by grammatical rules that correspond to ordinary language. For
(φ0.ψ0. ∨.φ1.ψ1. ∨ etc. ad inf.).Π = φ0.ψ0.Π. ∨.φ1.ψ1 Π. ∨ etc. ad inf. = φ0.Π. ∨ φ1.Π. ∨ φ2.Π. ∨ . etc. ad inf. =
Π.(φ0 ∨ φ1 ∨ φ2 ∨ etc. ad inf.) = Π
The proposition "φ0 ∨ φ1 ∨ φ2 ∨. etc. ad inf." must be treated as a tautology.
Page Break 358
We can regard the concept of sameness of number in such a way that it makes no sense to attribute sameness of
number or its opposite to two groups of points except in the case of two series of which one is correlated 1-1 to at
least a part of the other. Between such series all we can talk about is unilateral or mutual inclusion.
Page 358
This has really no more connection with particular numbers than equality or inequality of length in the visual
field has with numerical measurement. We can, but need not, connect it with numbers. If we connect it with the
number series, then the relation of mutual inclusion or equality of length between the rows becomes a relation of
sameness of number. But then it isn't only that ψ5 follows from Π.φ5. We also have Π following from φ5.ψ5. That
means that here S = Π.
Page Break 359
V MATHEMATICAL PROOF
22
In other cases, if I am looking for something, then even before it is found I can describe what finding it is; not so,
if I am looking for the solution of a mathematical problem.
Mathematical Expeditions and Polar Expeditions
Page 359
How can there be conjectures in Mathematics? Or better, what sort of thing is it that looks like a conjecture in
mathematics? Such as making a conjecture about the distribution of the primes.
Page 359
I might e.g. imagine that someone is writing primes in series in front of me without my knowing they are the
primes--I might for instance believe he is writing numbers just as they occur to him--and I now try to detect a law in
them. I might now actually form an hypothesis about this number sequence, just as I could about any sequence
yielded by an experiment in physics.
Page 359
Now in what sense have I, by so doing, made an hypothesis about the distribution of the primes?
Page 359
You might say that an hypothesis in mathematics has the value that it trains your thoughts on a particular
object--I mean a particular region--and we might say "we shall surely discover something interesting about these
things".
Page 359
The trouble is that our language uses each of the words "question", "problem", "investigation", "discovery",
to refer to such basically different things. It's the same with "inference", "proposition", "proof".
Page 359
The question again arises, what kind of verification do I count as valid for my hypothesis? Or can I faute de
mieux allow an empirical one to hold for the time being until I have a "strict proof"? No. Until there is such a proof,
there is no connection at all between my hypothesis and the "concept" of a prime number.
Page Break 360
Page 360
Only the so-called proof establishes any connection between the hypothesis and the primes as such. And
that is shown by the fact that--as I've said--until then the hypothesis can be construed as one belonging purely to
physics.--On the other hand when we have supplied a proof, it doesn't prove what was conjectured at all, since I
can't conjecture to infinity. I can only conjecture what can be confirmed, but experience can only confirm a finite
number of conjectures, and you can't conjecture the proof until you've got it, and not then either.
Page 360
Suppose that someone, without having proved Pythagoras' theorem, has been led by measuring the sides and
hypoteneuses of right angled triangles to "conjecture" it. And suppose he later discovered the proof, and said that he
had then proved what he had earlier conjectured. At least one remarkable question arises: at what point of the proof
does what he had earlier confirmed by individual trials emerge? For the proof is essentially different from the earlier
method.--Where do these methods make contact, if the proof and the tests are only different aspects of the same
thing (the same generalisation) if, as alleged, there is some sense in which they give the same result?
Page 360
I have said: "from a single source only one stream flows", and one might say that it would be odd if the same
thing were to come from such different sources. The thought that the same thing can come from different sources is
familiar from physics, i.e. from hypotheses. In that area we are always concluding from symptoms to illnesses and
we know that the most different symptoms can be symptoms of the same thing.
Page 360
How could one guess from statistics the very thing the proof later showed?
Page Break 361
Page 361
How can the proof produce the same generalisation as the earlier trials made probable?
Page 361
I am assuming that I conjectured the generalisation without conjecturing the proof. Does the proof now
prove exactly the generalisation that I conjectured?!
Page 361
Suppose someone was investigating even numbers to see if they confirmed Goldbach's conjecture. Suppose
he expressed the conjecture--and it can be expressed--that if he continued with this investigation, he would never
meet a counterexample as long as he lived. If a proof of the theorem is then discovered, will it also be a proof of the
man's conjecture? How is that possible?
Page 361
Nothing is more fatal to philosophical understanding than the notion of proof and experience as two different
but comparable methods of verification.
Page 361
What kind of discovery did Sheffer make when he found that p ∨ q and ~p can be expressed by p|q? People
had no method of looking for p|q, and if someone were to find one today, it wouldn't make any difference.
Page 361
What was it we didn't know before the discovery? (It wasn't anything that we didn't know, it was something
with which we weren't acquainted.)
Page 361
You can see this very clearly if you imagine someone objecting that p|p isn't at all the same as is said by ~p.
The reply of course is that it's only a question of the system p|q, etc. having the necessary multiplicity. Thus Sheffer
found a symbolic system with the necessary multiplicity.
Page 361
Does it count as looking for something, if I am unaware of Sheffer's system and say I would like to construct
a system with only one logical constant? No!
Page 361
Systems are certainly not all in one space, so that I could say: there are systems with 3 and with 2 logical
constants and now I am trying to reduce the number of constants in the same way. There is no "same way" here.
Page Break 362
Page 362
Suppose prizes are offered for the solution--say--of Fermat's problem. Someone might object to me: How
can you say that this problem doesn't exist? If prizes are offered for the solution, then surely the problem must exist.
I would have to say: Certainly, but the people who talk about it don't understand the grammar of the expression
"mathematical problem" or of the word "solution". The prize is really offered for the solution of a scientific problem;
for the exterior of the solution (hence also for instance we talk about a Riemannian hypothesis). The conditions of
the problem are external conditions; and when the problem is solved, what happens corresponds to the setting of the
problem in the way in which solutions correspond to problems in physics.
Page 362
If we set as a problem to find a construction for a regular pentagon, the way the construction is specified in
the setting of the problem is by the physical attribute that it is to yield a pentagon that is shown by measurement to
be regular. For we don't get the concept of constructive division into five (or of a constructive pentagon) until we
get it from the construction.
Page 362
Similarly in Fermat's theorem we have an empirical structure that we interpret as a hypothesis, and not--of
course--as the product of a construction. So in a certain sense what the problem asks for is not what the solution
gives.
Page 362
Of course a proof of the contradictory of Fermat's theorem (for instance) stands in the same relation to the
problem as a proof of the proposition itself. (Proof of the impossibility of a construction.)
Page 362
We can represent the impossibility of the trisection of an angle as a physical impossibility, by saying things
like "don't try to divide the angle into 3 equal parts, it is hopeless!" But in so far as we can do that, it is not this that
the "proof of impossibility" proves. That it is hopeless to attempt the trisection is something connected with physical
facts.
Page Break 363
Page 363
Imagine someone set himself the following problem. He is to discover a game played on a chessboard, in
which each player is to have 8 pieces; the two white ones which are in the outermost files at the beginning of the
game (the "consuls") are to be given some special status by the rules so that they have a greater freedom of
movement than the other pieces; one of the black pieces (the "general") is to have a special status; a white piece
takes a black one by being put in its place (and vice versa); the whole game is to have a certain analogy with the
Punic wars. Those are the conditions that the game is to satisfy.--There is no doubt that that is a problem, a problem
not at all like the problem of finding out how under certain conditions white can win in chess.--But now imagine the
problem: "How can white win in 20 moves in the war-game whose rules we don't yet know precisely?"--That
problem would be quite analogous to the problems of mathematics (other than problems of calculation).
Page 363
What is hidden must be capable of being found. (Hidden contradictions.)
Page 363
Also, what is hidden must be completely describable before it is found, no less than if it had already been
found.
Page 363
It makes good sense to say that an object is so well hidden that it is impossible to find it; but of course the
impossiblity [[sic]] here is not a logical one; i.e. it makes sense to speak of finding an object to describe the finding;
we are merely denying that it will happen.
Page 363
[We might put it like this: If I am looking for something,--I mean, the North Pole, or a house in London--I
can completely describe what I am looking for before I have found it (or have found that it isn't there) and either way
this description will be logically acceptable. But when I'm "looking for" something in mathematics, unless I am
doing so within a system, what I am looking for cannot be described, or can only apparently be described; for if I
could describe it in every particular, I would already
Page Break 364
actually have it; and before it is completely described I can't be sure whether what I am looking for is logically
acceptable, and therefore describable at all. That is to say, the incomplete description leaves out just what is
necessary for something to be capable of being looked for at all. So it is only an apparent description of what is
being "looked for."]†1
Page 364
Here we are easily misled by the legitimacy of an incomplete description when we are looking for a real
object, and here again there is an unclarity about the concepts "description" and "object". If someone says, I am
going to the North Pole and I expect to find a flag there, that would mean, on Russell's account, I expect to find
something (an x) that is a flag--say of such and such a colour and size. In that case too it looks as if the expectation
(the search) concerns only an indirect knowledge and not the object itself; as if that is something that I don't really
know (knowledge by acquaintance) until I have it in front of me (having previously been only indirectly acquainted
with it). But that is nonsense. There whatever I can perceive--to the extent that it is a fulfilment of my expectation--I
can also describe in advance. And here "describe" means not saying something or other about it, but rather
expressing it. That is, if I am looking for something I must be able to describe it completely.
Page 364
The question is: can one say that at present mathematics is as it were jagged--or frayed--and for that reason
we shall be able to round it off? I think you can't say that, any more than you can say that reality is untidy, because
there are 4 primary colours, seven notes in an octave, three dimensions in visual space, etc.
Page 364
You can't round off mathematics any more than you can say "let's round off the four primary colours to
eight or ten" or "let's round off the eight tones in an octave to ten".
Page Break 365
Page 365
The comparison between a mathematical expedition and a polar expedition. There is a point in drawing this
comparison and it is a very useful one.
Page 365
How strange it would be if a geographical expedition were uncertain whether it had a goal, and so whether it
had any route whatsoever. We can't imagine such a thing, it's nonsense. But this is precisely what it is like in a
mathematical expedition. And so perhaps it is a good idea to drop the comparison altogether.
Page 365
Could one say that arithmetical or geometrical problems can always look, or can falsely be conceived, as if
they referred to objects in space whereas they refer to space itself?
Page 365
By "space" I mean what one can be certain of while searching.
Page Break 366
23
Proof and the truth and falsehood of mathematical propositions
Page 366
A mathematical proposition that has been proved has a bias towards truth in its grammar. In order to
understand the sense of 25 × 25 = 625 I may ask: how is this proposition proved? But I can't ask how its
contradictory is or would be proved, because it makes no sense to speak of a proof of the contradictory of 25 × 25 =
625. So if I want to raise a question which won't depend on the truth of the proposition, I have to speak of checking
its truth, not of proving or disproving it. The method of checking corresponds to what one may call the sense of the
mathematical proposition. The description of this method is a general one and brings in a system of propositions, for
instance of propositions of the form a × b = c.
Page 366
We can't say "I will work out that it is so", we have to say "whether it is so", i.e., whether it is so or
otherwise.
Page 366
The method of checking the truth corresponds to the sense of a mathematical proposition. If it's impossible
to speak of such a check, then the analogy between "mathematical proposition" and the other things we call
propositions collapses. Thus there is a check for propositions of the form " ..." and " ..." which
bring in intervals.
Page 366
Now consider the question "does the equation x2 + ax + b = 0" have a solution in the real numbers?". Here
again there is a check and the check decides between (∃...), etc. and ~(∃...), etc. But can I in the same sense also ask
and check "whether the equation has a solution"? Not unless I include this case too in a system with others.
Page 366
(In reality the "proof of the fundamental theorem of algebra..." constructs a new kind of number.)
Page Break 367
Page 367
Equations are a kind of number. (That is, they can be treated similarly to the numbers.)
Page 367
A "proposition of mathematics" that is proved by an induction is not a "proposition" in the same sense as the
answer to a mathematical question unless one can look for the induction in a system of checks.
Page 367
"Every equation G has a root." And suppose it has no root? Could we describe that case as we can describe
its not having a rational solution? What is the criterion for an equation not having a solution? For this criterion must
be given if the mathematical question is to have a sense and if the apparent existence proposition is to be a
"proposition" in the sense of an answer to a question.
Page 367
(What does the description of the contradictory consist of? What supports it? What are the examples that
support it, and how are they related to particular cases of the proved contradictory? These questions are not
side-issues, but absolutely essential.)
Page 367
(The philosophy of mathematics consists in an exact scrutiny of mathematical proofs--not in surrounding
mathematics with a vapour.)
Page 367
In discussions of the provability of mathematical propositions it is sometimes said that there are substantial
propositions of mathematics whose truth or falsehood must remain undecided. What the people who say that don't
realize is that such propositions, if we can use them and want to call them "propositions", are not at all the same as
what are called "propositions" in other cases; because a proof alters the grammar of a proposition. You can certainly
use one and the same piece of wood first as a weathervane and then as a signpost; but you can't use it fixed as a
weathervane and moving as a signpost. If some one wanted to say "There are also moving signposts" I would
answer "You really mean 'There are also moving pieces of wood'. I don't say that a moving piece of wood can't
possibly be used at all, but only that it can't be used as a signpost".
Page Break 368
Page 368
The word "proposition", if it is to have any meaning at all here, is equivalent to a calculus: to a calculus in
which p ∨ ~p is a tautology (in which the "law of the excluded middle" holds). When it is supposed not to hold, we
have altered the concept of proposition. But that does not mean we have made a discovery (found something that is
a proposition and yet doesn't obey such and such a law); it means we have made a new stipulation, or set up a new
game.
Page Break 369
24
If you want to know what is proved, look at the proof
Page 369
Mathematicians only go astray, when they want to talk about calculi in general; they do so because they
forget the particular stipulations that are the foundations of each particular calculus.
Page 369
The reason why all philosophers of mathematics miss their way is that in logic, unlike natural history, one
cannot justify generalizations by examples. Each particular case has maximum significance, but once you have it the
story is complete, and you can't draw from it any general conclusion (or any conclusion at all).
Page 369
There is no such thing as a logical fiction and hence you can't work with logical fictions; you have to work
out each example fully.
Page 369
In mathematics there can only be mathematical troubles, there can't be philosophical ones.
Page 369
The philosopher only marks what the mathematician casually throws off about his activities.
Page 369
The philosopher easily gets into the position of a ham-fisted director, who, instead of doing his own work
and merely supervising his employees to see they do their work well, takes over their jobs until one day he finds
himself overburdened with other people's work while his employees watch and criticize him. He is particularly
inclined to saddle himself with the work of the mathematician.
Page 369
If you want to know what the expression "continuity of a function" means, look at the proof of continuity;
that will show what it proves. Don't look at the result as it is expressed in prose, or in the Russellian notation, which
is simply a translation of the prose expression; but fix your attention on the calculation actually going
Page Break 370
on in the proof. The verbal expression of the allegedly proved proposition is in most cases misleading, because it
conceals the real purport of the proof, which can be seen with full clarity in the proof itself.
Page 370
"Is the equation satisfied by any numbers?"; "It is satisfied by numbers"; "It is satisfied by all (no) numbers."
Does your calculus have proofs? And what proofs? It is only from them that we will be able to gather the sense of
these proportions and questions.
Page 370
Tell me how you seek and I will tell you what you are seeking.
Page 370
We must first ask ourselves: is the mathematical proposition proved? If so, how? For the proof is part of the
grammar of the proposition!--The fact that this is so often not understood arises from our thinking once again along
the lines of a misleading analogy. As usual in these cases, it is an analogy from our thinking in natural sciences. We
say, for example, "this man died two hours ago" and if someone asks us "how can you tell that?" we can give a
series of indications (symptoms). But we also leave open the possibility that medicine may discover hitherto
unknown methods of ascertaining the time of death. That means that we can already describe such possible
methods; it isn't their description that is discovered. What is ascertained experimentally is whether the description
corresponds to the facts. For example, I may say: one method consists in discovering the quantity of haemoglobin in
the blood, because this diminishes according to such and such a law in proportion to the time after death. Of course
that isn't correct, but if it were correct, nothing in my imaginary description would change. If you call the medical
discovery "the discovery of a proof that the man died two hours ago" you must go on to say that this discovery does
not change anything in the grammar of the proposition "the man died two hours ago". The discovery is the
discovery that a particular hypothesis is true (or: agrees with
Page Break 371
the facts). We are so accustomed to these ways of thinking, that we take the discovery of a proof in mathematics,
sight unseen, as being the same or similar. We are wrong to do so because, to put it concisely, the mathematical
proof couldn't be described before it is discovered.
Page 371
The "medical proof" didn't incorporate the hypothesis it proved into any new calculus, so it didn't give it any
new sense; a mathematical proof incorporates the mathematical proposition into a new calculus, and alters its
position in mathematics. The proposition with its proof doesn't belong to the same category as the proposition
without the proof. (Unproved mathematical propositions--signposts for mathematical investigation, stimuli to
mathematical constructions.)
Page 371
Are all the variables in the following equations variables of the same kind?
x2 + y2 + 2xy = (x + y)2
x2 + 3x + 2 = 0
x2 + ax + b = 0
x2 + xy + z = 0?
Page 371
That depends on the use of the equations.--But the distinction between no. 1 and no. 2 (as they are ordinarily
used) is not a matter of the extension of the values satisfying them. How do you prove the proposition "No. 1 holds
for all values of x and y" and how do you prove the proposition "there are values of x that satisfy No. 2?" There is
no more and no less similarity between the senses of the two propositions than there is between the proofs.
Page 371
But can't I say of an equation "I know it doesn't hold for some substitutions--I've forgotten now which; but
whether it doesn't hold in general, I don't know?" But what do you mean when you say you know that? How do you
know? Behind the words "I know..." there isn't a certain state of mind to be the sense of those words. What can you
do with that knowledge? That's what
Page Break 372
will show what the knowledge consists in. Do you know a method for ascertaining that the equation doesn't hold in
general? Do you remember that the equation doesn't hold for some values of x between 0 and 1000? Or did
someone just show you the equation and say he had found values of x that didn't satisfy the equation, so that
perhaps you don't yourself know how to establish it for a given value? etc. etc.
Page 372
"I have worked out that there is no number that..."--In what system of calculation does that calculation
occur?--That will show us to which proposition-system the worked-out proposition belongs. (One also asks: "how
does one work out something like that?")
Page 372
"I have discovered that there is such a number."
Page 372
"I have worked out that there is no such number."
Page 372
In the first sentence I cannot substitute "no such" for "such a". What if in the second I put "such a" for "no
such"? Let's suppose the result of a calculation isn't the proposition "~(∃n)" but "(∃n) etc." Does it then make sense
to say something like "Cheer up! Sooner or later you must come to such a number, if only you try long enough"?
That would only make sense if the result of the proof had not been "(∃n) etc." but something that sets limits to
testing, and therefore a quite different result. That is, the contradictory of what we call an existence theorem, a
theorem that tells us to look for a number, is not the proposition "(n) etc." but a proposition that says in such and
such an interval there is no number which... What is the contradictory of what is proved?--For that you must look at
the proof. We can say that the contradictory of a proved proposition is what would have been proved instead of it if
a particular miscalculation had been made in the proof. If now, for instance, the proof that ~(∃n) etc is the case is an
induction that shows that however far I go such a number cannot occur, the contradictory of this proof (using this
expression for the sake of argument) is not an existence proof in our sense. This case isn't like a proof that one or
none of the numbers a, b, c, d has the
Page Break 373
property ε; and that is the case that one always has before one's mind as a paradigm. In that case I could make a
mistake by believing that c had the property and after I had seen the error I would know that none of the numbers
had the property. But at this point the analogy just collapses.
Page 373
(This is connected with the fact that I can't eo ipso use the negations of equations in every calculus in which I
use equations. For 2 × 3 ≠ 7 doesn't mean that the equation 2 × 3 = 7 isn't to occur, like the equation 2 × 3 = sine; the
negation is an exclusion within a predetermined system. I can't negate a definition as I can negate an equation
derived by rules.)
Page 373
If you say that in an existence proof the interval isn't essential, because another interval might have done as
well, of course that doesn't mean that not specifying an interval would have done as well.--The relation of a proof of
non-existence to a proof of existence is not the same as that of a proof of p to a proof of its contradictory.
Page 373
One should suppose that in a proof of the contradictory of "(∃n)" it must be possible for a negation to creep
in which would enable "~(∃n)" to be proved erroneously. Let's for once start at the other end with the proofs, and
suppose we were shown them first and then asked: what do these calculations prove? Look at the proofs and then
decide what they prove.
Page 373
I don't need to assert that it must be possible to construct the n roots of equations of the n-th degree; I
merely say that the proposition "this equation has n roots" hasn't the same meaning if I've proved it by enumerating
the constructed roots as if I've proved it in a different way. If I find a formula for the roots of an equation, I've
constructed a new calculus; I haven't filled in a gap in an old one.
Page 373
Hence it is nonsense to say that the proposition isn't proved until such a construction is produced. For when
we do that we construct
Page Break 374
something new, and what we now mean by the fundamental theorem of algebra is what the present 'proof' shows us.
Page 374
"Every existence proof must contain a construction of what it proves the existence of." You can only say "I
won't call anything an 'existence proof' unless it contains such a construction". The mistake lies in pretending to
possess a clear concept of existence.
Page 374
We think we can prove a something, existence, in such a way that we are then convinced of it independently
of the proof. (The idea of proofs independent of each other--and so presumably independent of what is proved.)
Really, existence is what is proved by the procedures we call "existence proofs". When the intuitionists and others
talk about this they say: "This state of affairs, existence, can be proved only thus and not thus." And they don't see
that by saying that they have simply defined what they call existence. For it isn't at all like saying "that a man is in
the room can only be proved by looking inside, not by listening at the door".
Page 374
We have no concept of existence independent of our concept of an existence proof.
Page 374
Why do I say that we don't discover a proposition like the fundamental theorem of algebra, and that we
merely construct it?--Because in proving it we give it a new sense that it didn't have before. Before the so-called
proof there was only a rough pattern of that sense in the word-language.
Page 374
Suppose someone were to say: chess only had to be discovered, it was always there! Or: the pure game of
chess was always there; we only made the material game alloyed with matter.
Page 374
If a calculus in mathematics is altered by discoveries, can't we preserve the old calculus? (That is, do we have
to throw it away?)
Page Break 375
That is a very interesting way of looking at the matter. After the discovery of the North Pole we don't have two
earths, one with and one without the North pole. But after the discovery of the law of the distribution of the primes,
we do have two kinds of primes.
Page 375
A mathematical question must be no less exact than a mathematical proposition. You can see the misleading
way in which the mode of expression of word-language represents the sense of mathematical propositions if you call
to mind the multiplicity of a mathematical proof and consider that the proof belongs to the sense of the proved
proposition, i.e. determines that sense. It isn't something that brings it about that we believe a particular proposition,
but something that shows us what we believe--if we can talk of believing here at all. In mathematics there are
concept words: cardinal number, prime number, etc. That is why it seems to make sense straight off if we ask "how
many prime numbers are there?" (Human beings believe, if only they hear words...) In reality this combination of
words is so far nonsense; until it's given a special syntax. Look at the proof "that there are infinitely many primes,"
and then at the question that it appears to answer. The result of an intricate proof can have a simple verbal
expression only if the system of expressions to which this expression belongs has a multiplicity corresponding to a
system of such proofs. Confusions in these matters are entirely the result of treating mathematics as a kind of natural
science. And this is connected with the fact that mathematics has detached itself from natural science; for, as long as
it is done in immediate connection with physics, it is clear that it isn't a natural science. (Similarly, you can't mistake
a broom for part of the furnishing of a room as long as you use it to clean the furniture).
Page 375
The main danger is surely that the prose expression of the result of a mathematical operation may give the
illusion of a calculus that doesn't exist, by bearing the outward appearance of belonging to a system that isn't there at
all.
Page Break 376
Page 376
A proof is a proof of a particular proposition if it goes by a rule correlating the proposition to the proof. That
is, the proposition must belong to a system of propositions, and the proof to a system of proofs. And every
proposition in mathematics must belong to a calculus of mathematics. (It cannot sit in solitary glory and refuse to
mix with other propositions.)
Page 376
So even the proposition "every equation of nth degree has n roots" isn't a proposition of mathematics unless
it corresponds to a system of propositions and its proof corresponds to an appropriate system of proofs. For what
good reason have I to correlate that chain of equations etc. (that we call the proof) to this prose sentence? Must it
not be clear--according to a rule--from the proof itself which proposition it is a proof of?
Page 376
Now it is a part of the nature of what we call propositions that they must be capable of being negated. And
the negation of what is proved also must be connected with the proof; we must, that is, be able to show in what
different, contrasting, conditions it would have been the result.
Page Break 377
25
Mathematical problems,
Kinds of problem,
Search
"Projects" in mathematics
Page 377
Where you can ask you can look for an answer, and where you cannot look for an answer you cannot ask
either. Nor can you find an answer.
Page 377
Where there is no method of looking for an answer, there the question too cannot have any sense.--Only
where there is a method of solution is there a question (of course that doesn't mean: "only where the solution has
been found is there a question"). That is: where we can only expect the solution of the problem from some sort of
revelation, there isn't even a question. To a revelation no question corresponds.
Page 377
The supposition of undecidability presupposes that there is, so to speak, an underground connection
between the two sides of an equation; that though the bridge cannot be built in symbols, it does exist, because
otherwise the equation would lack sense.--But the connection only exists if we have made it by symbols; the
transition isn't produced by some dark speculation different in kind from what it connects (like a dark passage
between two sunlit places).
Page 377
I cannot use the expression "the equation E yields the solution S" unambiguously until I have a method of
solution; because "yields" refers to a structure that I cannot designate unless I am acquainted with it. For that would
mean using the word "yields" without knowing its grammar. But I might also say: When I use the word "yields" in
such a way as to bring in a method of solution, it doesn't have the same meaning as when this isn't the case. Here the
word "yields" is like the word "win" (or "lose") when at one time the criterion for "winning" is a particular set of
events in the game (in that case I must know the rules of the game in order to be able to say that someone has won)
and at another by "winning"
Page Break 378
I mean something that I could express roughly by "must pay".
Page 378
If we employ "yields" in the first meaning, then "the equation yield S" means: if I transform the equation in
accordance with certain rules, I get S. Just as the equation 25 × 25 = 620 says that I get 620 if I apply the rules for
multiplication to 25 × 25. But in this case these rules must already be given to me before the word "yields" has a
meaning, and before the question whether the equation yields S has a sense.
Page 378
It is not enough to say "p is provable"; we should say: provable according to a particular system.
Page 378
And indeed the proposition doesn't assert that p is provable according to the system S, but according to its
own system, the system that p belongs to. That p belongs to the system S cannot be asserted (that has to show
itself).--We can't say, p belongs to the system S; we can't ask, to which system does p belong; we cannot search for
p's system. "To understand p" means, to know its system. If p appears to cross over from one system to another, it
has in fact changed its sense.
Page 378
It is impossible to make discoveries of novel rules holding of a form already familiar to us (say the sine of an
angle). If they are new rules, then it is not the old form.
Page 378
If I know the rules of elementary trigonometry, I can check the proposition sin 2x = 2 sin x.cos x, but not the
proposition sin x = ... but that means that the sine function of elementary trigonometry and that of
higher trigonometry are different concepts.
Page 378
The two propositions stand as it were on two different planes. However far I travel on the first plane I will
never come to the proposition on the higher plane.
Page 378
A schoolboy, equipped with the armoury of elementary trigonometry and asked to test the equation sin x =
simply wouldn't find what he needs to tackle the problem. He not merely couldn't answer the question,
he couldn't even understand it.
Page Break 379
(It would be like the task the prince set the smith in the fairy tale: fetch me a 'Fiddle-de-dee'. Busch, Volksmärchen).
Page 379
We call it a problem, when we are asked "how many are 25 × 16", but also when we are asked: what is ™
sin2x dx. We regard the first as much easier than the second, but we don't see that they are "problems" in different
senses. Of course the distinction is not a psychological one; it isn't a question of whether the pupil can solve the
problem, but whether the calculus can solve it, or which calculus can solve it.
Page 379
The distinctions to which I can draw attention are ones that are familiar to every schoolboy. Later on we look
down on those distinctions, as we do on the Russian abacus (and geometrical proofs using diagrams); we regard
them as inessential, instead of seeing them as essential and fundamental.
Page 379
Whether a pupil knows a rule for ensuring a solution to ™ sin2x.dx is of no interest; what does interest us is
whether the calculus we have before us (and that he happens to be using) contains such a rule.
Page 379
What interests us is not whether the pupil can do it, but whether the calculus can do it, and how it does it.
Page 379
In the case of 25 × 16 = 370 the calculus we use prescribes every step for the checking of the equation.
Page 379
"I succeeded in proving this" is a remarkable expression. (That is something no one would say in the case of
25 × 16 = 400).
Page 379
One could lay down: "whatever one can tackle is a problem.--Only where there can be a problem, can
something be asserted."
Page Break 380
Page 380
Wouldn't all this lead to the paradox that there are no difficult problems in mathematics, since if anything is
difficult it isn't a problem? What follows is, that the "difficult mathematical problems", i.e. the problems for
mathematical research, aren't in the same relationship to the problem "25 × 25 = ?" as a feat of acrobatics is to a
simple somersault. They aren't related, that is, just as very easy to very difficult; they are 'problems' in different
meanings of the word.
Page 380
"You say 'where there is a question, there is also a way to answer it', but in mathematics there are questions
that we do not see any way to answer." Quite right, and all that follows from that is that in this case we are not using
the word 'question' in the same sense as above. And perhaps I should have said "here there are two different forms
and I want to use the word 'question' only for the first". But this latter point is a side-issue. What is important is that
we are here concerned with two different forms. (And if you want to say they are just two different kinds of
question you do not know your way about the grammar of the word "kind".)
Page 380
"I know that there is a solution for this problem, although I don't yet know what kind of solution"†1--In what
symbolism do you know it?
Page 380
"I know that here there must be a law." Is this knowledge an amorphous feeling accompanying the utterance
of the sentence?
Page Break 381
Page 381
That does not interest us. And if it is a symbolic process--well, then the problem is to represent it in a visible
symbolism.
Page 381
What does it mean to believe Goldbach's theorem? What does this belief consist in? In a feeling of certainty
as we state or hear the theorem? That does not interest us. I don't even know how far this feeling may be caused by
the proposition itself. How does the belief connect with this proposition? Let us look and see what are the
consequences of this belief, where it takes us. "It makes me search for a proof of the proposition."--Very well; and
now let us look and see what your searching really consists in. Then we shall know what belief in the proposition
amounts to.
Page 381
We may not overlook a difference between forms--as we may overlook a difference between suits, if it is
very slight.
Page 381
For us--that is, in grammar--there are in a certain sense no 'fine distinctions'. And altogether the word
distinction doesn't mean at all the same as it does when it is a question of a distinction between two things.
Page 381
A philosopher feels changes in the style of a derivation which a contemporary mathematician passes over
calmly with a blank face. What will distinguish the mathematicians of the future from those of today will really be a
greater sensitivity, and that will--as it were--prune mathematics; since people will then be more intent on absolute
clarity than on the discovery of new games.
Page 381
Philosophical clarity will have the same effect on the growth of mathematics as sunlight has on the growth of
potato shoots. (In a dark cellar they grow yards long.)
Page 381
A mathematician is bound to be horrified by my mathematical comments, since he has always been trained
to avoid indulging in
Page Break 382
thoughts and doubts of the kind I develop. He has learned to regard them as something contemptible and, to use an
analogy from psycho-analysis (this paragraph is reminiscent of Freud), he has acquired a revulsion from them as
infantile. That is to say, I trot out all the problems that a child learning arithmetic, etc., finds difficult, the problems
that education represses without solving. I say to those repressed doubts: you are quite correct, go on asking,
demand clarification!
Page Break 383
26
Euler's proof
Page 383
From the inequality
1 + 1/2 + 1/3 + 1/4 +... ≠ (1 + 1/2 + 1/22 + 1/23 +...).(1 + 1/3 + 1/32 +...)
can we derive a number which is still missing from the combinations on the right hand side? Euler's proof that there
are infinitely many prime numbers is meant to be an existence proof, but how is such a proof possible without a
construction?
~1 + 1/2 + 1/3 +... = (1 + 1/2 + 1/22 +...).(1 + 1/3 + 1/32 +...)
The argument goes like this: The product on the right is a series of fractions 1/n in whose denominators all multiples
of the form 2ν 3µ occur; if there were no numbers besides these, then this series would necessarily be the same as
the series 1 + ½ + 1/3 +... and in that case the sums also would necessarily be the same. But the left hand side is ∞
and the right hand side only a finite number 2/1. 3/2 = 3, so there are infinitely many fractions missing in the
right-hand series, that is, there are on the left hand side fractions that do not occur on the right.†1 And now the
question is: is this argument correct? If it were a question of finite series, everything would be perspicuous. For then
the method of summation would enable us to find out which terms occurring in the left hand series were missing
from the right hand series. Now we might ask: how does it come about that the left hand series gives ∞? What must
it contain in addition to the terms on the right to make it infinite? Indeed the question arises: does an equation, like 1
+ ½ + 1/3... = 3 above have any sense at all? I certainly can't find out from it which are the extra terms on the left.
How do we know that all the terms on the right hand side also occur on the left? In the case of finite series I can't say
that until I have ascertained it term by term;--and if I do so I see at the same time which are the extra ones.--Here
there is no connection between the result of the sum and the terms, and only such a connection could furnish a
proof. Everything becomes clearest if we imagine the business done with a finite equation:
Page Break 384
1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 ≠ (1 + 1/2).(1 + 1/3) = 1 + 1/2 + 1/3 + 1/6
Here again we have that remarkable phenomenon that we might call proof by circumstantial evidence in
mathematics--something that is absolutely never permitted. It might also be called a proof by symptoms. The result
of the summation is (or is regarded as) a symptom that there are terms on the left that are missing on the right. The
connection between the symptom and what we would like to have proved is a loose connection. That is, no bridge
has been built, but we rest content with seeing the other bank.
Page 384
All the terms on the right hand side occur on the left, but the sum on the left hand side is ∞ and the sum of
the right hand side is only a finite number, so there must... but in mathematics nothing must be except what is.
Page 384
The bridge has to be built.
Page 384
In mathematics there are no symptoms: it is only in a psychological sense that there can be symptoms for
mathematicians.
Page 384
We might also put it like this: in mathematics nothing can be inferred unless it can be seen.
Page 384
That reasoning with all its looseness no doubt rests on the confusion between a sum and the limiting value of
a sum.
Page 384
We do see dearly that however far we continue the right-hand series we can always continue the left hand
one far enough to contain all the terms of the right hand one. (And that leaves it open whether it then contains other
terms as well).
Page 384
We might also put the question thus: if you had only this proof, what would you bet on it? If we discovered
the primes up to N, could we later go on for ever looking for a further prime number--since the proof guarantees that
we will find one?--Surely that is nonsense. For "if we only search long enough" has no meaning. (That goes for
existence proofs in general).
Page 384
Could I add further prime numbers to the left hand side in this proof? Certainly not, because I don't know
how to discover any, and that means that I have no concept of prime number; the proof
Page Break 385
hasn't given me one. I could only add arbitrary numbers (or series).
Page 385
(Mathematics is dressed up in false interpretations).
Page 385
("Such a number has to turn up" has no meaning in mathematics. That is closely connected with the fact that
"in logic nothing is more general or more particular than anything else").
Page 385
If the numbers were all multiples of 2 and 3 then
would have to yield
but it does not... What follows from that? (The law excluded middle). Nothing follows, except that the limiting
values of the sums are different; that is, nothing. But now we might investigate how this comes about. And in so
doing we may hit on numbers that are not representable as 2ν • 3µ. Thus we shall hit on larger prime numbers, but
we will never see that no number of such original numbers will suffice for the formulation of all numbers.
1 + 1/2 + 1/3 +... ≠ 1 + 1/2 + 1/22 + 1/23
Page 385
However many terms of the form 1/2ν I take they never add up to more than 2, whereas the first four terms
of the left-hand series already add up to more than 2. (So this must already contain the proof.) This also gives us at
the same time the construction of a number that is not a power of 2, for the rule now says: find a segment of the
series that adds up to more than 2: this must contain a number that is not a power of 2.
(1 + 1/2 + 1/22 +...).(1 + 1/3 + 1/33 +...)...(1 + 1/n + 1/n2...) = n
Page 385
If I extend the sum 1 + ½ + 1/3 +... until it is greater than n, this part must contain a term that doesn't occur
in the right hand series, for if the right hand series contained all those terms it would yield a larger and not a smaller
sum.
Page Break 386
Page 386
The condition for a segment of the series 1 + ½ + 1/3 +... say 1/n + 1/(n + 1) + 1/(n + 2) +... 1/(n + ν) being
equal to or greater than 1 is as follows.
To make:
1/n + 1/(n + 1) + 1/(n + 2) +... 1/(n + ν) 1.
transform the left hand side into:
Page Break 387
27
The trisection of an angle, etc.
Page 387
We might say: in Euclidean plane geometry we can't look for the trisection of an angle, because there is no
such thing, and we can't look for the bisection of an angle, because there is such a thing.
Page 387
In the world of Euclid's Elements I can no more ask for the trisection of an angle than I can search for it. It
just isn't mentioned.
Page 387
(I can locate the problem of the trisection of an angle within a larger system but can't ask within the system
of Euclidean geometry whether it's soluble. In what language should I ask this? In the Euclidean?--But neither can I
ask in Euclidean language about the possibility of bisecting an angle within the Euclidean system. For in that
language that would boil down to a question about absolute possibility, which is always nonsense.)
Page 387
Incidentally, here we must make a distinction between different sorts of question, a distinction which will
show once again that what we call a "question" in mathematics is not the same as what we call by that name in
everyday life. We must distinguish between the question "how does one divide an angle into two equal parts?" and
the question "is this construction the bisection of an angle?" A question makes sense only in a calculus which gives
us a method for its solution; and a calculus may well give us a method for answering the one question without giving
us a method for answering the other. For instance, Euclid doesn't shew us how to look for the solutions to his
problems; he gives them to us and then proves that they are solutions. And this isn't a psychological or pedagogical
matter, but a mathematical one. That is, the calculus (the one he gives us) doesn't enable us to look for the
construction. A calculus which does enable us to do that is a different one.
Page Break 388
(Compare methods of integration with methods of differentiation, etc.)
Page 388
In mathematics there are very different things that all get called proofs, and the differences between them are
logical differences. The things called 'proofs' have no more internal connection with each other than the things called
'numbers'.
Page 388
What kind of proposition is "It is impossible to trisect an angle with ruler and compass"? The same kind, no
doubt, as "There is no F(3) in the series of angle-divisions F(n), just as there is no 4 in the series of
combination-numbers [n.(n - 1)]/2". But what kind of proposition is that? The same kind as "there is no ½ in the
series of cardinal numbers". That is obviously a (superfluous) rule of the game, something like: in draughts there is
no piece that is called "the queen". The question whether trisection is possible is then the question whether there is
such a thing in the game as trisection, whether there is a piece in draughts called "the queen" that has some kind of a
role like that of the queen in chess. Of course this question could be answered simply by a stipulation; but it
wouldn't set any problem or task of calculation, and so it wouldn't have the same sense as a question whose answer
was: I will work out whether there is such a thing. (Something like: I will work out whether any of the numbers 5, 7,
18, 25 is divisible by 3). Now is the question about the possibility of trisecting an angle that sort of question? It is if
you have a general system in the calculus for calculating the possibility of division into n equal parts.
Page 388
Now why does one call this proof the proof of this proposition? A proposition isn't a name; as a proposition
it belongs to a system of language. If I can say "there is no such thing as trisection" then it makes sense to say "there
is no such thing as quadrisection", etc., etc. And if this is a proof of the first proposition (a part of its syntax), then
there must be corresponding proofs (or disproofs) for the other propositions of the proposition-system, otherwise
they don't belong to the same system.
Page 388
I can't ask whether 4 occurs among the combination-numbers
Page Break 389
if that is my number-system. And I can't ask whether ½ occurs in the cardinal numbers, or show that it isn't one of
them, unless by "cardinal numbers" I mean part of a system that contains ½ as well. (Equally I can't either say or
prove that 3 is one of the cardinal numbers.) The question really means something like this: "If you divide 1/2 do
you get whole numbers?", and that can only be asked in a system in which divisibility and indivisibility is familiar.
(The working out must make sense.)
Page 389
If we don't mean by "cardinal numbers" a subset of the rational numbers, then we can't work out whether
81/3 is a cardinal number, but only whether the division 81/3 comes out or not.
Page 389
Instead of the problem of trisecting an angle with straightedge and compass we might investigate a parallel,
and much more perspicuous problem. There is nothing to prevent us restricting the possibilities of construction with
straightedge and compass still further. We might for instance lay down the condition that the angle of the compass
may not be changed. And we might lay down that the only construction we know--or better: that our calculus
knows--is the one used to bisect a line AB, namely
Page 389
(That might actually be the primitive geometry of a tribe. I said above that the number series "1, 2, 3, 4, 5,
many" has equal rights with the series of cardinal numbers†1 and that would go for this geometry too. In general it is
a good dodge in our investigations to imagine the arithmetic or geometry of a primitive people.)
Page Break 390
Page 390
I will call this geometry the system α and ask: "in the system α is it possible to trisect a line?"
Page 390
What kind of trisection is meant in this question?--that's obviously what the sense of the question depends
on. For instance, is what is meant physical trisection--trisection, that is, by trial and error and measurement? In that
case the answer is perhaps yes. Or optical trisection--trisection, that is, which yields three parts which look the same
length? It is quite easily imaginable that the parts a, b, and c might look the same length if, for instance, we were
looking through some distorting medium.
Page 390
We might represent the results of division in the system α by the numbers 2, 22, 23, etc. in accordance with
the number of the segments produced; and the question whether trisection is possible might mean: does any of the
numbers in this series = 3? Of course that question can only be asked if 2, 22, 23, etc. are imbedded in another
system (say the cardinal number system); it can't be asked if these numbers are themselves our number system for
in that case we, or our system, are not acquainted with the number 3. But if our question is: is one of the numbers 2,
22, etc. equal to 3, then here nothing is really said about a trisection of the line. None the less, we might look in this
manner at the question about the possibility of trisection.--We get a different view, if we adjoin to the system α a
system in which lines are divided in the manner of this figure:
Page Break 391
Page 391
It can then be asked: is a division into 180 sections a division of type α? And this question might again boil
down to: is 108 a power of 2? But it might also indicate a different decision procedure (have a different sense) if we
connected the systems α and β to a system of geometrical constructions in such a way that it could be proved in the
system that the two constructions "must yield" the same division points B, C, D.
Page 391
Suppose that someone, having divided a line AB into 8 sections in the system α, groups these into the lines
a, b, c, and asks: is that a trisection into 3 equal sections? (We could make the case more easily imaginable if we
took a larger number of original sections,
Page Break 392
which would make it possible to form groups of sections which looked the same length). The answer to that
question would be a proof that 23 is not divisible by 3; or an indication that the sections are in the ratio 1:3:4. And
now you might ask: but surely I do have a concept of trisection in the system, a concept of a division which yields
the parts a, b, c, in the ratio 1:1:1? Certainly, I have now introduced a new concept of 'trisection of a line'; we might
well say that by dividing the line AB into eight parts we have divided the line CB into 3 equal parts, if that is just to
mean we have produced a line that consists of 3 equal parts.
Page 392
The perplexity in which we found ourselves in relation to the problem of trisection was roughly this: if the
trisection of an angle is impossible--logically impossible--how can we ask questions about it at all? How can we
describe what is logically impossible and significantly raise the question of its possibility? That is, how can one put
together logically ill-assorted concepts (in violation of grammar, and therefore nonsensically) and significantly ask
about the possibility of the combination?--But the same paradox would arise if we asked "is 25 × 25 = 620?"; for
after all it's logically impossible that that equation should be correct; I certainly can't describe what it would be like
if...--Well, a doubt whether 25 × 25 = 620 (or whether it = 625) has no more and no less sense than the method of
checking gives it. It is quite correct that we don't here imagine, or describe, what it is like for 25 × 25 to be 620; what
that means is that we are dealing with a type of question that is (logically) different from "is this street 620 or 625
metres long"?
Page 392
(We talk about a "division of a circle into 7 segments" and also of a division of a cake into 7 segments).
Page Break 393
28
Searching and trying
Page 393
If you say to someone who has never tried "try to move your ears", he will first move some part of his body
near his ears that he has moved before, and either his ears will move at once or they won't. You might say of this
process: he is trying to move his ears. But if it can be called trying, it isn't trying in at all the same sense as trying to
move your ears (or your hands) in a case where you already "know how to do it" but someone is holding them so
that you can move them only with difficulty or not at all. It is the first sense of trying that corresponds to trying "to
solve a mathematical problem" when there is no method for its solution. One can always ponder on the apparent
problem. If someone says to me "try by sheer will power to move that jug at the other end of the room" I will look at
it and perhaps make some strange movements with my face muscles; so that even in that case there seems to be
such a thing as trying.
Page 393
Think of what it means to search for something in one's memory.
Here there is certainly something like a search in the strict sense.
Page 393
But trying to produce a phenomenon is not the same as searching for it.
Page 393
Suppose I am feeling for a painful place with my hand. I am searching in touch-space not in pain-space. That
means: what I find, if I find it, is really a place and not the pain. That means that even if experience shows that
pressing produces a pain, pressing isn't searching for a pain, any more than turning the handle of a generator is
searching for a spark.
Page Break 394
Page 394
Can one try to beat the wrong time to a melody? How does such an attempt compare with trying to lift a
weight that is too heavy?
Page 394
It is highly significant that one can see the group ||||| in different ways (in different groupings); but what is still
more noteworthy is that one can do it at will. That is, that there is a quite definite process of producing a particular
"view" at will; and correspondingly a quite definite process of unsuccessfully attempting to do so. Similarly, you can
to order see the figure below in such a way that first one and then the other vertical line is the nose, and first one and
then the other line becomes the mouth; in certain circumstances you can try in vain to do the one or the other.
Page 394
The essential thing here is that this attempt is the same kind of thing as trying to lift a weight with the hand; is
isn't like the sort of trying where one does different things, tries out different means, in order (e.g.) to lift a weight. In
the two cases the word "attempt" has quite different meanings. (An extremely significant grammatical fact.)
Page Break 395
VI INDUCTIVE PROOFS AND PERIODICITY
29
How far is a proof by induction a proof of a proposition?
Page 395
If a proof by induction is a proof of a + (b + c) = (a + b) + c, we must be able to say: the calculation gives
the result that a + (b + c) = (a + b) + c (and no other result).
Page 395
In that case the general method of calculating it must already be known, and we must be able to work out a +
(b + c) straight off in the way we can work out 25 × 16. So first there is a general rule taught for working out all such
problems, and later the particular cases are worked out.--But what is the general method of working out here? It
must be based on general rules for signs (--say, the associative law--).
Page 395
If I negate a + (b + c) = (a + b) + c it only makes sense if I mean to say something like: a + (b + c) isn't (a + b)
+ c, but (a + 2b) + c. For the question is: In what space do I negate the proposition? If I mark it off and exclude it,
what do I exclude it from?
Page 395
To check 25 × 25 = 625 I work out 25 × 25 until I get the right hand side;--can I work out a + (b + c) = (a + b)
+ c, and get the result (a + b) + c? Whether it is provable or not depends on whether we treat it as calculable or not.
For if the proposition is a rule, a paradigm, which every calculation has to follow, then it makes no more sense to
talk of working out the equation, than to talk of working out a definition.
Page 395
What makes the calculation possible is the system to which the proposition belongs; and that also determines
what miscalculations can be made in the working out. E.g. (a + b)2 is a2 + 2ab + b2 and not a2 + ab + b2; but (a + b)2
= -4 is not a possible miscalculation in this system.
Page Break 396
Page 396
I might also say very roughly (see other remarks): "25 × 64 = 160, 64 × 25 = 160; that proves that a × b = b ×
a" (this way of speaking need not be absurd or incorrect; you only have to interpret it correctly). The conclusion can
be correctly drawn from that; so in one sense "a.b = b.a" can be proved.
Page 396
And I want to say: It is only in the sense in which you can call working out such an example a proof of the
algebraic proposition that the proof by induction is a proof of the proposition. Only to that extent is it a check of the
algebraic proposition. (It is a check of its structure, not its generality).
Page 396
(Philosophy does not examine the calculi of mathematics, but only what mathematicians say about these
calculi.)
Page Break 397
30
Recursive proof and the concept of proposition. Is the proof a proof that a proposition is true and its
contradictory false?
Page 397
Is the recursive proof of
a + (b + c) = (a + b) + c... A
an answer to a question? If so, what question? Is it a proof that an assertion is true and its contradictory false?
Page 397
What Skolem†1 calls a recursive proof of A can be written thus;
Page 397
In this proof the proposition proved obviously doesn't occur at all.--What we have to do is to make a general
stipulation permitting the step to it. This stipulation could be expressed thus
Page 397
If three equations of the from α, β, γ are proved, we say "the equation ∆ is proved for all cardinal numbers".
This is a definition of this latter form of expression in terms of the first. It shows that we aren't using the word
"prove" in the second case in the same way as in the first. In any case it is misleading to say that we have proved the
equation ∆ or A. Perhaps it is better to say that we have proved its generality, though that too is misleading in other
respects.
Page 397
Now has the proof B answered a question, or proved an assertion true? And which is the proof B? Is it the
group of three equations of the form α, β, γ or the class of proofs of these equations? These equations do assert
something (they don't prove anything in the sense in which they are proved). But the proofs
Page Break 398
of α, β, γ answer the question whether these three equations are correct and prove true the assertion that they are
correct. All I can do is to explain: the question whether A holds for all cardinal numbers is to mean: "for the
functions
φ(ξ) = a + (b + ξ), ψ(ξ) = (a + b) + ξ
are the equations α, β, γ valid?" And then that question is answered by the recursive proof of A, if what that means
is the proofs of α, β, γ (or the laying down of α and the use of it to prove β and γ).
Page 398
So I can say that the recursive proof shows that the equation A satisfies a certain condition; but it isn't the
kind of condition that the equation (a + b)2 = a2 + 2b + b2 has to fulfil in order to be called "correct". If I call A
"correct" because equations of the form α, β, γ can be proved for it, I am no longer using the word "correct" in the
same way as in the case of the equations α, β, γ or (a + b)2 = a2 + 2ab + b2.
Page 398
What does "1/3 = " mean? Does it mean the same as " "?--Or is that division the proof of
the first proposition? That is, does it have the same relationship to it as a calculation has to what is proved?
" " is not the same kind of thing as
"1/2 = 0 • 5";
what " " corresponds to is " " not
" "†1
Page 398
Instead of the notation "1/4 = 0 • 25" I will adopt for this occasion the following " " So, for
example, . Then I can say, what corresponds to this proposition is not
Page Break 399
, but e.g. " is not a result of division (quotient) in the same sense as 0 • 375. For we
were acquainted with the numeral "0 • 375" before the division 3/8; but what does " " mean when detached from
the periodic division?--The assertion that the division a:b gives 0. as quotient is the same as the assertion that the
first place of the quotient is c and the first remainder is the same as the dividend.
Page 399
The relation of B to the assertion that A holds for all cardinal numbers is the same as that of
Page 399
The contradictory of the assertion "A holds for all cardinal numbers" is: one of the equations α, β, γ is false.
And the corresponding question isn't asking for a decision between a (x).fx and a (∃x).~fx.
Page 399
The construction of the induction is not a proof, but a certain arrangement of proofs (a pattern in the sense of
an ornament). And one can't exactly say either: if I prove three equations, then I prove one. Just as the movements
of a suite don't amount to a single movement.
Page 399
We can also say: we have a rule for constructing, in a certain game, decimal fractions consisting only of 3's;
but if you regard this rule as a kind of number, it can't be the result of a division; the only result would be what we
may call periodic division which has the form .
Page Break 400
31
Induction. (x).φx and (∃x).φx. Does the induction prove the general proposition true and an existential
proposition false?
Page 400
3 × 2 = 5 + 1
3 × (a + 1) = 3 + (3 × a) = (5 + b) + 3 = 5 + (b + 3)
Page 400
Why do you call this induction the proof that (n):n > 2.⊃.3 × n ≠ 5?! Well, don't you see that if the
proposition holds for n = 2, it also holds for n = 3, and then also for n = 4, and that it goes on like that for ever?
(What am I explaining when I explain the way a proof by induction works?) So you call it a proof of f(2).f(3).f(4),
etc." but isn't it rather the form of the proofs of "f(2)" and "f(3)" and "f(4)", etc.? Or does that come to the same
thing? Well, if I call the induction the proof of one proposition, I can do so only if that is supposed to mean no more
than that it proves every proposition of a certain form. (And my expression relies on the analogy with the
relationship between the proposition "all acids turn litmus paper red" and the proposition "sulphuric acid turns
litmus paper red").
Page 400
Suppose someone says "let us check whether f(n) holds for all n" and begins to write the series
3 × 2 = 5 + 1
3 × (2 + 1) = (3 × 2) + 3 = (5 + 1) + 3 = 5 + (1 + 3)
3 × (2 + 2) = (3 × (2 + 1)) + 3 = (5 + (1 + 3)) + 3 = 5 + ((1 + 3) + 3)
and then he breaks off and says "I see it holds for all n"--So he has seen an induction! But was he looking for an
induction? He didn't have any method for looking for one. And if he hadn't discovered one, would he ipso facto
have found a number which does not satisfy the condition?--The rule for checking can't be: let's see whether there is
an induction or a case for which the law does not hold.--If the law of excluded middle doesn't hold, that can only
mean that our expression isn't comparable to a proposition.
Page 400
When we say that the induction proves the general proposition, we think: it proves that this proposition and
not its contradictory
Page Break 401
is true. But what would be the contradictory of the proposition proved? Well, that (∃n).~fn is the case. Here we
combine two concepts: one derived from my current concept of the proof of (n).fn, and another taken from the
analogy with (∃)x.φx. (Of course we have to remember that "(n). fn" isn't a proposition until I have a criterion for its
truth; and then it only has the sense that the criterion gives it. Although, before getting the criterion, I could look out
for something like an analogy to (x).fx†1). What is the opposite of what the induction proves? The proof of (a + b)2
= a2 + 2ab + b2 works out this equation in contrast to something like (a + b)2 = a2 + 3ab + b2. What does the
inductive proof work out? The equations: 3 × 2 = 5 + 1, 3 × (a + 1) = (3 × a) + 3, (5 + b) + 3 = 5 + (b + 3) as opposed
to things like 3 × 2 = 5 + 6, 3 × (a + 1) = (4 × a) + 2, etc. But this opposite does not correspond to the proposition
(∃x).φx--Further, what does conflict with the induction is every proposition of the form ~f(n), i.e. the propositions
"~f(2)", "~f(3)", etc.; that is to say, the induction is the common element in the working out of f(2), f(3), etc.; but it
isn't the working out of "all propositions of the form f(n)", since of course no class of propositions occurs in the
proof that I call "all propositions of the form f(n)". Each one of these calculations is a checking of a proposition of
the form f(n). I was able to investigate the correctness of this proposition and employ a method to check it; all the
induction did was to bring this into a simple form. But if I call the induction "the proof of a general proposition", I
can't ask whether that proposition is correct (any more than whether the form of the cardinal numbers is correct).
Because the things I call inductive proofs give me no method of checking whether the general proposition is correct
or incorrect; instead, the method has to show me how to work out (check) whether or not an induction can be
constructed for a particular case within a system of propositions. (If I may so put it, what is checked in this way is
whether all n have this or that property; not whether all of them have it, or whether there are some that don't have it.
For example, we work
Page Break 402
out that the equation x2 + 3x + 1 = 0 has no rational roots (that there is no rational number that...), and the equation
x2 + 2x + ½ = 0 has none, but the equation x2 + 2x + 1 = 0 does, etc.)
Page 402
Hence we find it odd if we are told that the induction is a proof of the general proposition; for we feel rightly
that in the language of the induction we couldn't have posed the general question at all. It wasn't that we began with
an alternative between which we had to decide. (We only seemed to, so long as we had in mind a calculus with finite
classes).
Page 402
Prior to the proof asking about the general proposition made no sense at all, and so wasn't even a question,
because the question would only have made sense if a general method of decision had been known before the
particular proof was discovered.
Page 402
The proof by induction isn't something that settles a disputed question.
Page 402
If you say: "the proposition '(n).fn' follows from the induction" only means that every proposition of the
form f(n) follows from the induction and "the proposition (∃n).~fn contradicts the induction" only means "every
proposition of the form ~f(n) is disproved by the induction", then we may agree; but we shall ask: what is the correct
way for us to use the expression "the proposition (n).f(n)"? What is its grammar? (For from the fact that I use it in
certain contexts it doesn't follow that I use it everywhere in the same way as the expression "the proposition
(x).φx.")
Page 402
Suppose that people argued whether the quotient of the division 1/3 must contain only threes, but had no
method of deciding it. Suppose one of them noticed the inductive property of = 0.3 and said: now I know
that there must be only threes in the quotient. The others had not thought of that kind of decision. I suppose that
they had vaguely imagined some kind of decision by checking each step, though of course they could never have
reached a decision in this way. If they hold on to their extensional
Page Break 403
viewpoint, the induction does produce a decision because in the case of each extension of the quotient it shows that
it consists of nothing but threes. But if they drop their extensional viewpoint the induction decides nothing, or
nothing that is not decided by working out = 0 • 3, namely that the remainder is the same as the dividend.
But nothing else. Certainly, there is a valid question that may arise, namely, is the remainder left after this division
the same as the dividend? This question now takes the place of the old extensional question, and of course I can
keep the old wording, but it is now extremely misleading since it always makes it look as if having the induction
were only a vehicle--a vehicle that can take us into infinity. (This is also connected with the fact that the sign "etc."
refers to an internal property of the bit of the series that precedes it, and not to its extension.)
Page 403
Of course the question "is there a rational number that is a root of x2 × 3x + 1 = 0?" is decided by an
induction; but in this case I have actually constructed a method of forming inductions; and the question is only so
phrased because it is a matter of constructing inductions. That is, a question is settled by an induction, if I can look
for the induction in advance; if everything in its sign is settled in advance bar my acceptance or rejection of it in such
a way that I can decide yes or no by calculating; as I can decide, for instance, whether in 5/7 the remainder is equal
to the dividend or not. (The employment in these cases of the expressions "all..." and "there is..." has a certain
similarity with the employment of the word "infinite" in the sentence "today I bought a straightedge with an infinite
radius of curvature").
Page 403
The periodicity of decides nothing that had been left open. Suppose someone had been looking in
vain, before the discovery of the periodicity, for a 4 in the development of 1/3, he
Page Break 404
still couldn't significantly have put the question "is there a 4 in the development of 1/3?". That is, independently of
the fact that he didn't actually discover any 4s, we can convince him that he doesn't have a method of deciding his
question. Or we might say: quite apart from the result of his activity we could instruct him about the grammar of his
question and the nature of his search (as we might instruct a contemporary mathematician about analogous
problems). "But as a result of discovering the periodicity he does stop looking for a 4! So it does convince him that
he will never find one."--No. The discovery of the periodicity will cure him of looking if he makes the appropriate
adjustment. We might ask him: "Well, how about it, do you still want to look for a 4?" (Or has the periodicity so to
say, changed your mind?)
Page 404
The discovery of the periodicity is really the construction of a new symbol and a new calculus. For it is
misleading to say that it consists in our having realised that the first remainder is the same as the dividend. For if we
had asked someone unacquainted with periodic division whether the first remainder in this division was the same as
the dividend, of course he would have answered "yes"; and so he did realise. But that doesn't mean he must have
realised the periodicity; that is, it wouldn't mean he had discovered the calculus with the sign .
Page 404
Isn't what I am saying what Kant meant, by saying that 5 + 7 = 12 is not analytic but synthetic a priori?
Page Break 405
32
Is there a further step from writing the recursive proof to the generalization? Doesn't the recursion schema
already say all that is to be said?
Page 405
We commonly say that the recursive proofs show that the algebraic equations hold for all cardinal numbers;
for the time being it doesn't matter whether this expression is well or ill chosen, the point is whether it has the same
clearly defined meaning in all cases.
Page 405
And isn't it clear that the recursive proofs in fact show the same for all "proved" equations?
Page 405
And doesn't that mean that between the recursive proof and the proposition it proves there is always the
same (internal) relation?
Page 405
Anyway it is quite clear that there must be a recursive, or better, iterative "proof" of this kind (A proof
conveying the insight that "that's the way it must be with all the numbers".)
Page 405
I.e. it seems clear to me; and it seems that by a process of iteration I could make the correctness of these
theorems for the cardinal numbers intelligible to someone else.
Page 405
But how do I know that 28 + (45 + 17) = (28 + 45) + 17 without having proved it? How can a general proof
give me a particular proof? I might after all go through the particular proof, and how would the two proofs meet in
it? What happens if they do not agree?
Page 405
In other words: suppose I wanted to show someone that the associative law is really part of the nature of
number, and isn't something that only accidentally holds in this particular case; wouldn't I use a process of iteration
to try to show that the law holds and must go on holding? Well--that shows us what we mean here by saying that a
law must hold for all numbers.
Page Break 406
Page 406
And what is to prevent us calling this process a proof of the law?
Page 406
This concept of "making something comprehensible" is a boon in a case like this.
Page 406
For we might say: the criterion of whether something is a proof of a proposition is whether it could be used
for making it comprehensible. (Of course here again all that is involved is an extension of our grammatical
investigation of the word "proof" and not any psychological interest in the process of making things
comprehensible.)
Page 406
"This proposition is proved for all numbers by the recursive procedure." That is the expression that is so very
misleading. It sounds as if here a proposition saying that such and such holds for all cardinal numbers is proved true
by a particular route, and as if this route was a route through a space of conceivable routes.
Page 406
But really the recursion shows nothing but itself, just as periodicity too shows nothing but itself.
Page 406
We are not saying that when f(1) holds and when f (c + 1) follows from f(c), the proposition f(x) is therefore
true of all cardinal numbers; but: "the proposition f(x) holds for all cardinal numbers" means "it holds for x = 1, and
f(c + 1) follows from f(c)".
Page 406
Here the connection with generality in finite domains is quite clear, for in a finite domain that would certainly
be a proof that f(x) holds for all values of x, and that is the reason why we say in the arithmetical case that f(x) holds
for all numbers.
Page 406
At least I have to say that any objection that holds against the proof B†1 holds also e.g. against the formula
(a + b)n = etc.
Page 406
Here too, I would have to say, I am merely assuming an algebraic rule that agrees with the inductions of
arithmetic.
Page Break 407
f(n) × (a + b) = f(n + 1)
f(1) = a + b
therefore f(1) × (a + b) = (a + b)2 = f(2)
therefore f(2) × (a + b) = (a + b)3 = f(3), etc.
Page 407
So far all is clear. But then: "therefore (a + b)n = f(n)"!
Page 407
Is a further inference drawn here? Is there still something to be established?
Page 407
But if someone shows me the formula (a + b)n = f(n) I could ask: how have we got there? And the answer
would be the group
f(n) × (a + b) = f(n + 1)
f(1) = a + b
So isn't it a proof of the algebraic proposition?--Or is it rather an answer to the question "what does the algebraic
proposition mean?"
Page 407
I want to say: once you've got the induction, it's all over.
Page 407
The proposition that A holds for all cardinal numbers is really the complex B plus its proof, the proof of β
and γ. But that shows that this proposition is not a proposition in the same sense as an equation, and this proof is not
in the same sense a proof of a proposition.
Page 407
Don't forget that it isn't that we first of all have the concept of proposition, and then come to know that
equations are mathematical propositions, and later realise that there are also other kinds of mathematical
propositions.
Page Break 408
33
How far does a recursive proof deserve the name of "proof"? How far is a step in accordance with the paradigm
A justified by the proof of B?
Page 408
(Editor's note: What follows between the square brackets we have taken from one of the manuscript books
that Wittgenstein used for this chapter; although it is not in the typescript--"A" and "B" are given above, on p. 397.)
Page 408
(A step by step investigation of this proof would be very instructive.) The first step in I, a + (b + (c + 1)) = a +
((b + c) + 1), if it is made in accordance with R, shows that the variables in R are not meant in the same way as those
in the equations of I; since R would otherwise allow only the replacement of a + (b + 1) by (a + b) + 1, and not the
replacement of b + (c + 1) by (b + c) + 1.†1
Page Break 409
The same appears in the other steps in the proof.
Page 409
If I said that the proof of the two lines of the proof justifies me in inferring the rule a + (b + c) = (a + b) + c,
that wouldn't mean anything, unless I had deduced that in accordance with a previously established rule. But this
rule could only be
Page 409
But this rule is vague in respect of F1, F2 and f.]
Page 409
We cannot appoint a calculation to be a proof of a proposition.
Page 409
I would like to say: Do we have to call the recursive calculation the proof of proposition I? That is, won't
another relationship do?
Page 409
(What is infinitely difficult is to "see all round" the calculus.)
Page 409
In the one case "The step is justified" means that it can be carried out in accordance with definite forms that
have been given. In the other case the justification might be that the step is taken in accordance with paradigms that
themselves satisfy a certain condition.
Page 409
Suppose that for a certain board game rules are given containing only words with no "r" in them, and that I
call a rule justified, if it contains no "r". Suppose someone then said, he had laid down only one rule for a certain
game, namely, that its moves must obey rules containing no "r"s. Is that a rule of the game (in the first sense)? Isn't
the game played in accordance with the class of rules all of which have only to satisfy the first rule?
Page 409
Someone shows me the construction of B and then says that A has been proved. I ask "How? All I see is that
you have used α[ρ] to build a construction around A". Then he says "But when that is
Page Break 410
possible, I say that A is proved". To that I answer: "That only shows me the new sense you attach to the word
'prove'."
Page 410
In one sense it means that you have used α[ρ] to construct the paradigm in such and such a way, in another,
it means as before that an equation is in accordance with the paradigm.
Page 410
If we ask "is that a proof or not?" we are keeping to the word-language.
Page 410
Of course there can be no objection if someone says: if the terms of a step in a construction are of such and
such a kind, I say that the legitimacy of the step is proved.
Page 410
What is it in me that resists the idea of B as a proof of A? In the first place I observe that in my calculation I
now here use the proposition about "all cardinal numbers". I used ρ to construct the complex B and then I took the
step to the equation A; in all that there was no mention of "all cardinal numbers". (This proposition is a bit of
word-language accompanying the calculation, and can only mislead me.) But it isn't only that this general
proposition completely drops out, it is that no other takes it place.
Page 410
So the proposition asserting the generalisation drops out; "nothing is proved", "nothing follows".
Page 410
"But the equation A follows, it is that that takes the place of the general proposition." Well, to what extent
does it follow? Obviously, I'm here using "follows" in a sense quite different from the normal one, because what A
follows from isn't a proposition. And that is why we feel that the word "follows" isn't being correctly applied.
Page 410
If you say "it follows from the complex B that a + (b + c) = (a + b) + c", we feel giddy. We feel that
somehow or other you've said something nonsensical although outwardly it sounds correct.
Page Break 411
Page 411
That an equation follows, already has a meaning (has its own definite grammar).
Page 411
If I am told "A follows from B", I want to ask: "what follows?" That a + (b + c) is equal to (a + b) + c, is
something postulated, if it doesn't follow in the normal way from an equation.
Page 411
We can't fit our concept of following from to A and B; it doesn't fit.
Page 411
"I will prove to you that a + (b + n) = (a + b) + n." No one then expects to see the complex B. You expect to
hear another rule for a, b, and n permitting the passage from one side to the other. If instead of that I am given B
with the schema ρ†1 I can't call it a proof, because I mean something else by "proof".
Page 411
I shall very likely say something like "oh, so that's what you call a 'proof', I had imagined..."
Page 411
The proof of 17 + (18 + 5) = (17 + 18) + 5 is certainly carried out in accordance with the schema B, and this
numerical proposition is of the form A. Or again: B is a proof of the numerical proposition: but for that very reason,
it isn't a proof of A.
Page 411
"I will derive AI, AII, AIII from a single proposition."†2--This
Page Break 412
of course makes one think of a derivation that makes use of these propositions--We think we shall be given smaller
links of some kind to replace all these large ones in the chain.
Page 412
Here we have a definite picture; and we are offered something quite different.
Page 412
The inductive proof puts the equation together as it were crossways instead of lengthways.
Page 412
If we work out the derivation, we finally come to the point at which the construction of B is completed. But
at this point we say "therefore this equation holds"! But these words now don't mean the same as they do when we
elsewhere deduce an equation from equations. The words "The equation follows from it" already have a meaning.
And although an equation is constructed here, it is by a different principle.
Page 412
If I say "the equation follows from the complex", then here an equation is 'following' from something that is
not an equation.
Page 412
We can't say: if the equation follows from B, then it does follow from a proposition, namely from α.β.γ; for
what matters is how I get A from that proposition; whether I do so in accordance with a rule of inference; and what
the relationship is between the equation and the proposition α.β.γ. (The rule leading to A in this case makes a kind of
cross-section through α.β.γ; it doesn't view the proposition in the same way as a rule of inference does.)
Page 412
If we have been promised a derivation of A from α and now see the step from B to A, we feel like saying
"oh, that isn't what was meant". It is as if someone had promised to give me something and then says: see, I'm giving
you my trust.
Page 412
The fact that the step from B to A is not an inference indicates also what I meant when I said that the logical
product α.β.γ does not express the generalization.
Page Break 413
Page 413
I say that AI, AII etc. are used in proving (a + b)2 = etc. because the steps from (a + b)2 to a2 + 2ab + b2 are all
of the form AI or AII, etc. In this sense the step in III from (b + 1) + a to (b + a) + 1 is also made in accordance with
AI, but the step from a + n to n + a isn't!
Page 413
The fact that we say "the correctness of the equation is proved" shows that not every construction of the
equation is a proof.
Page 413
Someone shows me the complexes B and I say "they are not proofs of the equations A". Then he says: "You
still haven't seen the system on which the complexes are constructed", and points it out to me. How could that make
the Bs into proofs?
Page 413
This insight makes me ascend to another, a higher, level; whereas a proof would have to be carried out on the
lower level.
Page 413
Nothing except a definite transition to an equation from other equations is a proof of that equation. Here
there is no such thing, and nothing else can do anything to make B into a proof of A.
Page 413
But can't I say that if I have proved this about A, I have thereby proved A? Wherever did I get the illusion
that by doing this I had proved it? There must surely be some deep reason for this.
Page 413
Well, if it is an illusion, at all events it arose from our expression in word-language "this proposition holds for
all numbers"; for on this view the algebraic proposition is only another way of writing the proposition of
word-language. And that form of expression caused us to confuse the case of all the numbers with the case of 'all
the people in this room'. (What we do to distinguish the cases is to ask: how does one verify the one and the other?)
Page Break 414
Page 414
If I suppose the functions φ, ψ, F exactly defined and then write the schema for the inductive proof:
Even then I can't say that the step from φr to ψr is taken on the basis of ρ (if the step in α, β, γ was made in
accordance with ρ--in particular cases ρ = α). It is still the equation A it is made in accordance with, and I can only
say that it corresponds to the complex B if I regard that as another sign in place of the equation A.
Page 414
For of course the schema for the step had to include α, β and γ.
Page 414
In fact R isn't the schema for the inductive proof BIII; that is much more complicated, since it has to include
the schema BI.
Page 414
The only time it is inadvisable to call something a 'proof' is when the ordinary grammar of the word 'proof'
doesn't accord with the grammar of the object under consideration.
Page 414
What causes the profound uneasiness is in the last analysis a tiny but obvious feature of the traditional
expression.
Page 414
What does it mean, that R justifies a step of the form A? No doubt it means that I have decided to allow in
my calculus only steps in accordance with a schema B in which the propositions α, β, γ are derivable in accordance
with ρ. (And of course that would only mean that I allowed only the steps AI, AII etc., and that those had schemata
B corresponding to them).
Page 414
It would be better to write "and those schemata had the form R corresponding to them". The sentence added
in brackets was intended to say that the appearance of generality--I mean the generality of the concept of the
inductive method--is unnecessary,
Page Break 415
for in the end it only amounts to the fact that the particular constructions BI, BII, etc. are constructed flanking the
equations AI, AII, etc. Or that in that case it is superfluous to pick out the common feature of the constructions; all
that is relevant are the constructions themselves, for there is nothing there except these proofs, and the concept
under which the proofs fall is superfluous, because we never made any use of it. Just as if I only want to
say--pointing to three objects--"put that and that and that in my room", the concept chair is superfluous even though
the three objects are chairs. (And if they aren't suitable furniture for sitting on, that won't be changed by someone's
drawing attention to a similarity between them.) But that only means, that the individual proof needs our acceptance
of it as such (if 'proof' is to mean what it means); and if it doesn't have it no discovery of an analogy with other such
constructions can give it to it. The reason why it looks like a proof is that α, β, γ and A are equations, and that a
general rule can be given, according to which we can construct (and in that sense derive) A from B.
Page 415
After the event we may become aware of this general rule. (But does that make us aware that the Bs are
really proofs of A?) What we become aware of is a rule we might have started with and which in conjunction with α
would have enabled us to construct AI, AII, etc. But no one would have called it a proof in this game.
Page 415
Whence this conflict: "That isn't a proof!" "That surely is a proof."?
Page 415
We might say that it is doubtless true, that in proving B by α I use α to trace the contours of the equation A,
but not in the way I call "proving A by α".
Page 415
The difficulty that needs to be overcome in these discussions is the difficulty of looking at the proof by
induction as something new, naively as it were.
Page Break 416
Page 416
So when we said above we could begin with R, this beginning with R is in a way a piece of humbug. It isn't
like beginning a calculation by working out 526 × 718. For in the latter case setting out the problem is the first step
on the journey to the solution. But in the former case I immediately drop the R and have to begin again somewhere
else. And when it turns out that I construct a complex of the form R, it is again immaterial whether I explicitly set it
out earlier, since setting it out hasn't helped me at all mathematically, i.e. in the calculus. So what is left is just the
fact that I now have a complex of the form R in front of me.
Page 416
We might imagine we were acquainted only with the proof BI and could then say: all we have is this
construction--no mention of an analogy between this and other constructions, or of a general principle in carrying
out the constructions.--If I then see B and A like this I'm bound to ask: but why do you call that a proof of A
precisely?--(I am not asking: why do you call it a proof of A)! What has this complex to do with AI? Any reply will
have to make me aware of the relation between A and B which is expressed in V.†1
Page 416
Someone shows us B, and explains to us the relationship with AI, that is, that the right side of A was
obtained in such and such a manner etc. etc. We understand him; and he asks us: is that a proof of A? We would
answer: certainly not!
Page Break 417
Page 417
Had we understood everything there was to understand about the proof? Had we seen the general form of
the connection between A and B? Yes!
Page 417
We might also infer from that that in this way we can construct a B from every A and therefore conversely
an A from every B as well.
Page 417
The proof is constructed on a definite plan (a plan used to construct other proofs as well). But this plan
cannot make the proof a proof. For all we have here is one of the embodiments of the plan, and we can altogether
disregard the plan as a general concept. The proof has to speak for itself and the plan is only embodied in it, it isn't
itself a constituent part of the proof. (That is what I've been wanting to say all the time). Hence it's no use to me if
someone draws my attention to the similarity between proofs in order to convince me that they are proofs.
Page 417
Isn't our principle: not to use a concept-word where one isn't necessary?--That means, in cases where the
concept word really stands for an enumeration, to say so.
Page 417
When I said earlier "that isn't a proof" I meant 'proof' in an already established sense according to which it
can be gathered from A and B by themselves. In this sense I can say: I understand perfectly well what B does and
what relationship it has to A; all further information is superfluous and what is there isn't a proof. In this sense I am
concerned only with A and B; I don't see anything beyond them, and nothing else concerns me.
Page 417
If I do this, I can see clearly enough the relationship in accordance with the rule V, but it doesn't enter my
head to use it as an expedient in construction. If someone told me while I was considering B and A that there is a
rule according to which we could have constructed B from A (or conversely), I could only say to him "don't bother
me with irrelevant trivialities." Because of course it's something that's obvious, and I see immediately that it doesn't
make B a proof of A. For the general rule couldn't shew that B is a proof of A and not of some other proposition,
unless it were
Page Break 418
a proof in the first place. That means, that the fact that the connection between B and A is in accordance with a rule
can't show that B is a proof of A. Any and every such connection could be used as a construction of B from A (and
conversely).
Page 418
So when I said "R certainly isn't used for the construction, so we have no concern with it" I should have said:
I am only concerned with A and B. It is enough if I confront A and B with each other and ask: "is B a proof of A?"
So I don't need to construct A from B according to a previously established rule; it is sufficient for me to place the
particular As--however many there are--in confrontation with particular Bs. I don't need a previously established
construction rule (a rule needed to obtain the As).
Page 418
What I mean is: in Skolem's calculus we don't need any such concept, the list is sufficient.
Page 418
Nothing is lost if instead of saying "we have proved the fundamental laws A in this fashion" we merely show
that we can coordinate with them constructions that resemble them in certain respects.
Page 418
The concept of generality (and of recursion) used in these proofs has no greater generality than can be read
immediately from the proofs.
Page 418
The bracket } in R, which unites α, β, and γ†1 can't mean any more than that we regard the step in A (or a
step of the form A) as justified
Page Break 419
if the terms (sides) of the steps are related to each other in the ways characterized by the schema B. B then takes the
place of A. And just as before we said: the step is permitted in my calculus if it corresponds to one of the As, so we
now say: it is permitted if it corresponds to one of the Bs.
Page 419
But that wouldn't mean we had gained any simplification or reduction.
Page 419
We are given the calculus of equations. In that calculus "proof" has a fixed meaning. If I now call the
inductive calculation a proof, it isn't a proof that saves me checking whether the steps in the chain of equations have
been taken in accordance with these particular rules (or paradigms). If they have been, I say that the last equation of
the chain is proved, or that the chain of equations is correct.
Page 419
Suppose that we were using the first method to check the calculation (a + b)3 = ... and at the first step
someone said: "yes, that step was certainly taken in accordance with a (b + c) = a.b + a.c, but is that right?" And
then we showed him the inductive derivation of that equation.--
Page 419
The question "Is the equation G right?†1" means in one meaning: can it be derived in accordance with the
paradigms?--In the other case it means: can the equations α, β, γ be derived in accordance with the paradigm (or the
paradigms?)--And here we have put the two meanings of the question (or of the word "proof") on the same level
(expressed them in a single system) and can now compare them (and see that they are not the same).
Page 419
And indeed the new proof doesn't give you what you might expect: it doesn't base the calculus on a smaller
foundation--as happens if we replace p ∨ q and ~p by p|q, or reduce the number of axioms, or something similar.
For if we now say that all the basic equations A have been derived from ρ alone, the word "derived" here means
something quite different. (After this promise we expect
Page Break 420
the big links in the chain to be replaced by smaller ones, not by two half links.†1) And in one sense these derivations
leave everything as it was. For in the new calculus the links of the old one essentially continue to exist as links. The
old structure is not taken to pieces. So that we have to say the proof goes on in the same way as before. And in the
old sense the irreducibility remains.
Page 420
So we can't say that Skolem has put the algebraic system on to a smaller foundation, for he hasn't 'given it
foundations' in the same sense as is used in algebra.
Page 420
In the inductive proof doesn't α show a connection between the As? And doesn't this show that we are here
concerned with proofs?--The connection shown is not the one that breaking up the A steps into ρ steps would
establish. And one connection between the As is already visible before any proof.
Page 420
I can write the rule R like this
or like this
a + (b + 1) = (a + b) + 1
if I take R or S as a definition or substitute for that form†2.
Page 420
Page Break 421
Page 421
If I then say that the steps in accordance with the rule R are justified thus:
you can reply: "If that's what you call a justification, then you have justified the steps. But you haven't told us any
more than if you had just drawn our attention to the rule R and its formal relationship to α (or to α, β, and γ)."
Page 421
So I might also have said: I take the rule R in such and such a way as a paradigm for my steps.
Page 421
Suppose now that Skolem, following his proof of the associative law, takes the step to:
If he says the first and third steps in the third line are justified according to the already proved associative law, that
tells us no more than if he said the steps were taken in accordance with the paradigm a + (b + c) = (a + b) + c (i.e.
they correspond to the paradigm) and a schema α, β, γ was derived by steps according to the paradigm α.--"But
does B justify these steps, or not?"--"What do you mean by the word 'justify'?--"Well, the step is
Page Break 422
justified if a theorem really has been proved that holds for all numbers"--But in what case would that have
happened? What do you call a proof that a theorem holds for all cardinal numbers? How do you know whether a
theorem is really valid for all cardinal numbers, since you can't test it? Your only criterion is the proof itself. So you
stipulate a form and call it the form of the proof that a proposition holds for all cardinal numbers. In that case we
really gain nothing by being first shown the general form of these proofs; for that doesn't show that the individual
proof really gives us what we want from it; because, I mean, it doesn't justify the proof or demonstrate that it is a
proof of a theorem for all cardinal numbers. Instead, the recursive proof has to be its own justification. If we really
want to justify our proof procedure as a proof of a generalisation of this kind, we do something different: we give a
series of examples and then we are satisfied by the examples and the law we recognize in them, and we say: yes our
proof really gives us what we want. But we must remember that by giving this series of examples we have only
translated the notations B and C into a different notation. (For the series of examples is not an incomplete
application of the general form, but another expression of the law.) An explanation in word-language of the proof
(of what it proves) only translates the proof into another form of expression: because of this we can drop the
explanation altogether. And if we do so, the mathematical relationships become much clearer, no longer obscured
by the equivocal expressions of word-language. For example, if I put B right beside A, without interposing any
expression of word-language like "for all cardinal numbers, etc." then the misleading appearance of a proof of A by
B cannot arise. We then see quite soberly how far the relationships between B and A and a + b = b + a extend and
where they stop. Only thus do we learn the real structure and important features of that relationship, and escape the
confusion caused by the form of word-language, which makes everything uniform.
Page 422
Here we see first and foremost that we are interested in the tree
Page Break 423
of the structures B, C, etc., and that in it is visible on all sides, like a particular kind of branching, the following form
φ(1) = ψ(1)
φ(n + 1) = F(φn)
ψ(n + 1) = F(ψn)
These forms turn up in different arrangements and combinations but they are not elements of the construction in the
same sense as the paradigms in the proof of (a + (b + (c + 1))) = (a + (b + c)) + 1 or (a + b)2 = a2 + 2ab + b2. The aim
of the "recursive proofs" is of course to connect the algebraic calculus with the calculus of numbers. And the tree of
the recursive proofs doesn't "justify" the algebraic calculus unless that is supposed to mean that it connects it with
the arithmetical one. It doesn't justify it in the sense in which the list of paradigms justifies the algebraic calculus, i.e.
the steps in it.
Page 423
So tabulating the paradigms for the steps makes sense in the cases where we are interested in showing that
such and such transformations are all made by means of those transition forms, arbitrarily chosen as they are. But it
doesn't make sense where the calculation is to be justified in another sense, where mere looking at the
calculation--independently of any comparison with a table of previously established norms--must shew us whether
we are to allow it or not. Skolem did not have to promise us any proof of the associative and commutative laws; he
could simply have said he would show us a connection between the paradigms of algebra and the calculation rules
of arithmetic. But isn't this hair-splitting? Hasn't he reduced the number of paradigms? Hasn't he, for instance,
replaced very pair of laws with a single one, namely, a + (b + 1) = (a + b) + 1? No. When we prove e.g. (a + b)4 = etc.
(k) we can while doing so make use of the previously proved proposition (a + b)2 = etc. (l). But in that case the steps
in k which are justified by 1 can also be justified by the rules used to prove l. And then the relation of 1 to those first
rules is the same as that of a sign introduced by definition to the primary signs used to define it: we can always
eliminate the definitions and go back to the primary signs. But when we take a step in C that is justified by B,
Page Break 424
we can't take the same step with a + (b + 1) = (a + b) + 1 alone. What is called proof here doesn't break a step in to
smaller steps but does something quite different.
Page Break 425
34
The recursive proof does not reduce the number of fundamental laws
Page 425
So here we don't have a case where a group of fundamental laws is proved by a smaller set while everything
else in the proofs remains the same. (Similarly in a system of fundamental concepts nothing is altered in the later
development if we use definitions to reduce the number of fundamental concepts.)
Page 425
(Incidentally, how very dubious is the analogy between "fundamental laws" and "fundamental concepts"!)
Page 425
It is something like this: all that the proof of a ci-devant fundamental proposition does is to continue the
system of proofs backwards. But the recursive proofs don't continue backwards the system of algebraic proofs (with
the old fundamental laws); they are a new system, that seems only to run parallel with the first one.
Page 425
It is a strange observation that in the inductive proofs the irreducibility (independence) of the fundamental
rules must show itself after the proof no less than before. Suppose we said the same thing about the case of normal
proofs (or definitions), where fundamental rules are further reduced, and a new relationship between them is
discovered (or constructed).
Page 425
If I am right that the independence remains intact after the recursive proof, that sums up everything I have to
say against the concept of recursive "proof".
Page 425
The inductive proof doesn't break up the step in A. Isn't it that that makes me baulk at calling it a proof? It's
that that tempts me to say that whatever it does--even if it is constructed by R and α--it can't do more than show
something about the step.
Page Break 426
Page 426
If we imagine a mechanism constructed from cogwheels made simply out of uniform wedges held together
by a ring, it is still the cogwheels that remain in a certain sense the units of the mechanism.
Page 426
It is like this: if the barrel is made of hoops and wattles, it is these, combined as they are (as a complex) that
hold the liquid and form new units as containers.
Page 426
Imagine a chain consisting of links which can each be replaced by two smaller ones. Anything which is
anchored by the chain can also be anchored entirely by the small links instead of by the large ones. But we might
also imagine every link in the chain being made of two parts, each perhaps shaped like half a ring, which together
formed a link, but could not individually be used as links.
Page 426
Then it wouldn't mean at all the same to say, on the one hand: the anchoring done by the large links can be
done entirely by small links--and on the other hand: the anchoring can be done entirely by half large links. What is
the difference?
Page 426
One proof replaces a chain with large links by a chain with small links, the other shows how one can put
together the old large links from several parts.
Page 426
The similarity as well as the difference between the two cases is obvious.
Page 426
Of course the comparison between the proof and the chain is a logical comparison and therefore a
completely exact expression of what it illustrates.
Page Break 427
35
Recurring decimals
Page 427
We regard the periodicity of a fraction, e.g. of 1/3 as consisting in the fact that something called the extension
of the infinite decimal contains only threes; we regard the fact that in this division the remainder is the same as the
dividend as a mere symptom of this property of the infinite extension. Or else we correct this view by saying that it
isn't an infinite extension that has this property, but an infinite series of finite extensions; and it is of this that the
property of the division is a symptom. We may then say: the extension taken to one term is 0•3, to two terms 0•33,
to three terms 0•333 and so on. That is a rule and the "and so on" refers to the regularity; the rule might also be
written "|0•3, 0•ξ, 0•ξ3|" But what is proved by the division is this regularity in contrast to another, not
regularity in contrast to irregularity. The periodic division (in contrast to ) proves a
periodicity in the quotient, that is it determines the rule (the repetend), it lays it down; it isn't a symptom that a
regularity is "already there". Where is it already? In things like the particular expansions that I have written on this
paper. But they aren't "the expansions". (Here we are misled by the idea of unwritten ideal extensions, which are a
phantasm like those ideal, undrawn, geometric straight lines of which the actual lines we draw are mere tracings.)
When I said "the 'and so on' refers to the regularity" I was distinguishing it from the 'and so on' in "he read all the
letters of the alphabet: a, b, c and so on". When I say "the extensions of 1/3 are 0•3, 0•33, 0•333 and so on" I give
three three
Page Break 428
extensions and--a rule. That is the only thing that is infinite, and only in the same way as the division
Page 428
One can say of the sign that it is not an abbreviation.
Page 428
And the sign "|0•3, 0•ξ, 0•ξ3|" isn't a substitute for an extension, but the undevalued sign itself; and " "
does just as well. It should give us food for thought, that a sign like " " is enough to do what we need. It isn't a
mere substitute in the calculus there are no substitutes.
Page 428
If you think that the peculiar property of the division is a symptom of the periodicity of the
infinite decimal fraction, or the decimal fractions of the expansion, it is indeed a sign that something is regular, but
what? The extensions that I have constructed? But there aren't any others. It would be a most absurd manner of
speaking to say: the property of the division is an indication that the result has the form "|0•a, 0•ξ, 0•ξa|"; that is like
wanting to say that a division was an indication that the result was a number. The sign " " does not express its
meaning from any greater distance than "0•333...", because this sign gives an extension of three terms and a rule; the
extension 0•333 is inessential for our purposes and so there remains only the rule, which is given just as well by
"|0•3, 0•ξ, 0•ξ3|". The proposition "After the first place the division is periodic" just means "The first remainder is the
same as the dividend". Or again: the proposition "After the first place the division will yield the same number to
infinity", means "The first remainder is the same as the dividend", just as the proposition "This straightedge has an
infinite radius" means it is straight.
Page 428
We might now say: the places of a quotient of 1/3 are necessarily
Page Break 429
all 3s, and all that could mean would be again that the first remainder is like the dividend and the first place of the
quotient is 3. The negation of the first proposition is therefore equivalent to the negation of the second. So the
opposite of "necessarily all" isn't what one might call "accidentally all"; "necessarily all" is as it were one word. I only
have to ask: what is the criterion of the necessary generalization, and what would be the criterion of the accidental
generalization (the criterion for all numbers accidentally having the property ε)?
Page Break 430
36
The recursive proof as a series of proofs
Page 430
A "recursive proof" is the general term of a series of proofs. So it is a law for the construction of proofs. To
the question how this general form can save me the proof of a particular proposition, e.g. 7 + (8 + 9) = (7 + 8) + 9,
the answer is that it merely gets everything ready for the proof of the proposition, it doesn't prove it (indeed the
proposition doesn't occur in it). The proof consists rather of the general form plus the proposition.
Page 430
Our normal mode of expression carries the seeds of confusion right into its foundations, because it uses the
word "series" both in the sense of "extension", and in the sense of "law". The relationship of the two can be
illustrated by a machine for making coiled springs, in which a wire is pushed through a helically shaped
passage to make as many coils as are desired. What is called an infinite helix need not be anything like a finite piece
of wire, or something that that approaches the longer it becomes; it is the law of the helix, as it is embodied in the
short passage. Hence the expression "infinite helix" or "infinite series" is misleading.
Page 430
So we can always write out the recursive proof as a limited series with "and so on" without its losing any of
its rigour. At the same time this notation shows more clearly its relation to the equation A. For then the recursive
proof no longer looks at all like a justification of A in the sense of an algebraic proof--like the proof
Page Break 431
of (a + b)2 = a2 + 2ab + b2. That proof with algebraic calculation rules is quite like calculation with numbers.
5 + (4 + 3) = 5 + (4 + (2 + 1)) = 5 + ((4 + 2) + 1) =
= (5 + (4 + 2)) + 1 = (5 + (4 + (1 + 1))) + 1 =
= (5 + ((4 + 1) + 1)) + 1 = ((5 + (4 + 1)) + 1) + 1 =
= (((5 + 4) + 1) + 1) + 1 = ((5 + 4) + 2) + 1 = (5 + 4) + 3)...(L)
Page 431
That is a proof of 5 + (4 + 3) = (5 + 4) + 3, but we can also let it count, i.e. use it, as a proof of 5 + (4 + 4) = (5
+ 4) + 4, etc.
Page 431
If I say that L is the proof of the proposition a + (b + c) = (a + b) + c, the oddness of the step from the proof
to the proposition becomes much more obvious.
Page 431
Definitions merely introduce practical abbreviations; we could get along without them. But is that true of
recursive definitions?
Page 431
Two different things might be called applications of the rule a + (b + 1) = (a + b) + 1: in one sense 4 + (2 + 1)
= (4 + 2) + 1 is an application, in another sense 4 + (2 + 1) = ((4 + 1) + 1) + 1 = (4 + 2) + 1 is.
Page 431
The recursive definition is a rule for constructing replacement rules, or else the general term of a series of
definitions. It is a signpost that shows the same way to all expressions of a certain form.
Page 431
As we said, we might write the inductive proof without using letters at all (with no loss of rigour). Then the
recursive definition a + (b + 1) = (a + b) + 1 would have to be written as a series of definitions. As things are, this
series is concealed in the explanation of its use. Of course we can keep the letters in the definition for the sake of
convenience, but in that case in the explanation we have to bring in a sign like "1, (1) + 1, ((1) + 1) + 1 and so on",
or, what boils down to the same thing, "|1, ξ, ξ + 1|". But here we mustn't believe that this sign should really be
"(ξ).|1, ξ, ξ + 1|"!
Page Break 432
Page 432
The point of our formulation is of course that the concept "all numbers" is given only by a structure like "|1,
ξ, ξ + 1|". The generality is set out in the symbolism by this structure and cannot be described by an (x).fx.
Page 432
Of course the so-called "recursive definition" isn't a definition in the customary sense of the word, because it
isn't an equation, since the equation "a + (b + 1) = (a + b) + 1" is only a part of it. Nor is it a logical product of
equations. Instead, it is a law for the construction of equations; just as |1, ξ, ξ + 1| isn't a number but a law etc. (The
bewildering thing about the proof of a + (b + c) = (a + b) + c is of course that it's supposed to come out of the
definition alone. But α isn't a definition, but a general rule for addition).
Page 432
On the other hand the generality of this rule is no different from that of the periodic division .
That means, there isn't anything that the rule leaves open or in need of completion or the like.
Page 432
Let us not forget: the sign "|1, ξ, ξ + 1|"... N interests us not as a striking expression for the general term of
the series of cardinal numbers, but only in so far as it is contrasted with signs of similar construction. N as opposed
to something like |2, ξ, ξ + 3|; in short, as a sign, or an instrument, in a calculus. And of course the same holds for
. (The only thing left open in the rule is its application.)
1 + (1 + 1) = (1 + 1) + 1, 2 + (1 + 1) = (2 + 1) + 1, 3 + (1 + 1) = (3 + 1) + 1... and so on
1 + (2 + 1) = (1 + 2) + 1, 2 + (2 + 1) = (2 + 2) + 1, 3 + (2 + 1) = (3 + 2) + 1... and so on
1 + (3 + 1) = (1 + 3) + 1, 2 + (3 + 1) = (2 + 3) + 1, 3 + (3 + 1) = (3 + 3) + 1... and so on
and so on.
Page Break 433
Page 433
We might write the rule "a + (b + 1) = (a + b) + 1", thus.†1
Page 433
In the application of the rule R (and the description of the application is of course an inherent part of the sign
for the rule), a ranges over the series |1, ξ, ξ + 1|; and of course that might be expressly stated by an additional sign,
say "a → N". (We might call the second and third lines of the rule R taken together the operation, like the second
and third term of the sign N.) Thus too the explanation of the use of the recursive definition "a + (b + 1) = (a + b) +
1" is a part of that rule itself; or if you like a repetition of the rule in another form; just as "1, 1 + 1, 1 + 1 + 1 and so
on" means the same as (i.e. is translatable into) "|1, ξ, ξ + 1|". The translation into word-language casts light on the
calculus with the new signs, because we have already mastered the calculus with the signs of word-language.
Page 433
The sign of a rule, like any other sign, is a sign belonging to a calculus; its job isn't to hypnotize people into
accepting an application, but to be used in the calculus in accordance with a system. Hence the exterior form is no
more essential than that of an arrow →; what is essential is the system in which the sign for the rule is employed.
The system of contraries--so to speak--from which the sign is distinguished etc.
Page 433
What I am here calling the description of the application is itself of course something that contains an "and
so on", and so it can itself be no more than a supplement to or substitute for the rule-sign.
Page 433
What is the contradictory of a general proposition like a + (b + (1 + 1)) = a + ((b + 1) + 1)? What is the
system of propositions within which this proposition is negated? Or again, how, and in what form, can this
proposition come into contradiction with others? What question does it answer? Certainly not the question
Page Break 434
whether (n).fn or (∃n).~fn is the case, because it is the rule R that contributes to the generality of the proposition.
The generality of a rule is eo ipso incapable of being brought into question.
Page 434
Now imagine the general rule written as a series
p11, p12, p13...
p21, p22, p23...
p31, p32, p33...
............
and then negated. If we regard it as (x).fx, then we are treating it as a logical product and its opposite is the logical
sum of the denials of p11, p12 etc. This disjunction can be combined with any random product p11 • p21 • p22 • • •
pmn. (Certainly if you compare the proposition with a logical product, it becomes infinitely significant and its
opposite void of significance). (But remember that the "and so on" in the proposition comes after a comma, not after
an "and" (".") The "and so on" is not a sign of incompleteness.)
Page 434
Is the rule R infinitely significant? Like an enormously long logical product?
Page 434
That one can run the number series though the rule is a form that is given; nothing is affirmed about it and
nothing can be denied about it.
Page 434
Running the stream of numbers through is not something which I can say I can prove. I can only prove
something about the form, or pattern, through which I run the numbers.
Page 434
But can't we say that the general number rule a + (b + c) = (a + b) + c... A) has the same generality as a + (1 +
1) = (a + 1) + 1 (in that the latter holds for every cardinal number and the former for every triple of cardinal
numbers) and that the inductive proof of A justifies the rule A? Can we say that we can give the rule A, since the
proof shows that it is always right? Does justify the rule
and so on?"...P)
A is a completely intelligible rule; just like the replacement rule P. But I can't give such a rule, for the reason that I
can already
Page Break 435
calculate the particular instances of A by another rule; just as I cannot give P as a rule if I have given a rule whereby I
can calculate etc.
Page 435
How would it be if someone wanted to lay down "25 × 25 = 625" as a rule in addition to the multiplication
rules. (I don't say "25 × 25 = 624"!)--25 × 25 = 625 only makes sense if the kind of calculation to which the equation
belongs is already known, and it only makes sense in connection with that calculation. A only makes sense in
connection with A's own kind of calculation. For the first question here would be: is that a stipulation, or a derived
proposition? If 25 × 25 = 625 is a stipulation, then the multiplication sign does not mean the same as it does, e.g. in
reality (that is, we are dealing with a different kind of calculation). And if A is a stipulation, it doesn't define addition
in the same way as if it is a derived proposition. For in that case the stipulation is of course a definition of the
addition sign, and the rules of calculation that allow A to be worked out are a different definition of the same sign.
Here I mustn't forget that α, β, γ isn't the proof of A, but only the form of the proof, or of what is proved; so α, β, γ
is a definition of A.
Page 435
Hence I can only say "25 × 25 = 625 is proved" if the method of proof is fixed independently of the specific
proof. For it is this method that settles the meaning of "ξ × η" and so settles what is proved. So to that extent the
form belongs to the method of proof that explains the sense of c. Whether I have calculated correctly is
another question. And similarly α, β, γ belong to the method of proof that defines the sense of the proposition A.
Page 435
Arithmetic is complete without a rule like A; without it it doesn't lack anything. The proposition A is
introduced into arithmetic with the discovery of a periodicity, with the construction of a new calculus. Before this
discovery or construction a question about the correctness of that proposition would have as little sense as a
question about the correctness of "1/3 = 0 • 3, 1/3 = 0• 33... ad inf."
Page Break 436
Page 436
The stipulation of P is not the same thing as the proposition "1/3 = 0 • 3" and in that sense "a + (b + ) = (a +
b) + ) is different from a rule (stipulation) such as A. The two belong to different calculi. The proof of α, β, γ is a
proof or justification of a rule like A only in so far as it is the general form of the proof of arithmetical propositions of
the form A.
Page 436
Periodicity is not a sign (symptom) of a decimal's recurring; the expression "it goes on like that for ever" is
only a translation of the sign for periodicity into another form of expression. (If there was something other than the
periodic sign of which periodicity was only a symptom, that something would have to have a specific expression,
which could be nothing less than the complete expression of that something.)
Page Break 437
37
Seeing or viewing a sign in a particular manner. Discovering an aspect of a mathematical expression. "Seeing an
expression in a particular way". Marks of emphasis.
Page 437
Earlier I spoke of the use of connection lines, underlining etc. to bring out the corresponding, homologous, parts of
the equations of a recursion proof. In the proof
the one marked α for example corresponds not to β but to c in the next equation; and β corresponds not to δ but to
ε; and γ not to δ but to c + δ, etc.
Or in
ι doesn't correspond to κ and ε doesn't correspond to λ; it is β that ι corresponds to; and β does not correspond to
ξ, but ξ corresponds to θ and α to δ and β to γ and γ to µ, not to θ, and so on.
Page 437
What about a calculation like
(5 + 3)2 = (5 + 3) • (5 + 3) = 5 • (5 + 3) + 3 • (5 + 3) = 5 • 5 + 5 • 3 + 3 • 5 + 3 •3 = 52 + 2 • 5 • 3 + 32...R)
from which we can also read a general rule for the squaring of a binomial?
Page 437
We can as it were look at this calculation arithmetically or algebraically.
Page Break 438
Page 438
This difference between the two ways of looking at it would have been brought out e.g. if the example had
been written
In the algebraic way of looking at it we would have to distinguish the 2 in the position marked α from the 2s in the
positions marked β but in the arithmetical one they would not need to be distinguished. We are--I believe--using a
different calculus in each case.
Page 438
According to one but not the other way of looking at it the calculation above, for instance, would be a proof
of (7 + 8)2 = 82 + 2 • 7 • 8 + 82.
Page 438
We might work out an example to make sure that (a + b)2 is equal to a2 + b2 + 2ab, not to a2 + b2 + 3ab--if
we had forgotten it for instance; but we couldn't check in that sense whether the formula holds generally. But of
course there is that sort of check too, and in the calculation
(5 + 3)2 = ... = 52 + 2 • 5 • 3 + 32
I might check whether the 2 in the second summand is a general feature of the equation or something that depends
on the particular numbers occurring in the example.
I turn (5 + 2)2 = 52 + 2 • 2 • 5 into another sign, if I write
and thus "indicate which features of the right hand side originate from the particular numbers on the left" etc.
Page 438
(Now I realize the importance of this process of coordination. It expresses a new way of looking at the
calculation and therefore a way of looking at a new calculation.)
Page 438
'In order to prove A'--we could say--I first of all have to draw attention to quite definite features of B. (As in
the division ).
Page Break 439
Page 439
(And α had no suspicion, so to speak, of what I see if I do.)
Page 439
Here the relationship between generality and proof of generality is like the relationship between existence
and proof of existence.
Page 439
When α, β, γ are proved, the general calculus has still to be discovered.
Page 439
Writing "a + (b + c) = (a + b) + c" in the induction series seems to us a matter of course, because we don't see
that by doing so we are starting a totally new calculus. (A child just learning to do sums would see dearer than we do
in this connection.)
Page 439
Certain features are brought out by the schema R; they could be specially marked thus: †1
Of course it would also have: been enough (i.e. it would have been a symbol of the same multiplicity) if we had
written B and added
f1ξ = a + (b + ξ), f2ξ = (a + b) + ξ
Page 439
(Here we must also remember that every symbol--however explicit--can be misunderstood.)
Page 439
The first person to draw attention to the fact that B can be seen in that way introduces a new sign whether or
not he goes on to attach special marks to B or to write the schema R beside it. In
Page Break 440
the latter case R itself is the new sign, or, if you prefer, B plus R. It is the way in which he draws attention to it that
produces the new sign.
Page 440
We might perhaps say that here the lower equation is used as a + b = b + a; or similarly that here B is used as
A, by being as it were read sideways. Or: B was used as A, but the new proposition was built up from α.β.γ, in such
a way that though A is now read out of B, α.β.γ don't appear in the sort of abbreviation in which the premisses turn
up in the conclusion.
Page 440
What does it mean to say: "I am drawing your attention to the fact that the same sign occurs here in both
function signs (perhaps you didn't notice it)"? Does that mean that he didn't understand the proposition?--After all,
what he didn't notice was something which belonged essentially to the proposition; it wasn't as if it was some
external property of the proposition he hadn't noticed (Here again we see what kind of thing is called "understanding
a proposition".)
Page 440
Of course the picture of reading a sign lengthways and sideways is once again a logical picture, and for that
reason it is a perfectly exact expression of a grammatical relation. We mustn't say of it "it's a mere metaphor, who
knows what the facts are really like?"
Page 440
When I said that the new sign with the marks of emphasis must have been derived from the old one without
the marks, that was meaningless, because of course I can consider the sign with the marks without regard to its
origin. In that case it presents itself to me as three equations [Frege]†1, that is as the shape of three equations with
certain underlinings, etc.
Page Break 441
Page 441
It is certainly significant that this shape is quite similar to the three equations without the underlinings; it is
also significant that the cardinal number 1 and the rational number 1 are governed by similar rules; but that does not
prevent what we have here from being a new sign. What I am now doing with this sign is something quite new.
Page 441
Isn't this like the supposition I once made that people might have operated the Frege-Russell calculus of
truth-functions with the signs "~" and "." combined into "~p.~q" without anyone noticing, and that Sheffer, instead
of giving a new definition, had merely drawn attention to a property of the signs already in use.
Page 441
We might have gone on dividing without ever becoming aware of recurring decimals. When we have seen
them, we have seen something new.
Page 441
But couldn't we extend that and say "I might have multiplied numbers together without ever noticing the
special case in which I multiply a number by itself; and that means x² is not simply x.x"? We might call the
invention of the sign 'x2' the expression of our having become aware of that special case. Or, we might have gone on
multiplying a by b and dividing it by c without noticing that we could write " " as "a.(b|c)" or that the latter is
similar to a.b. Or again, this is like a savage who doesn't yet see the analogy between ||||| and ||||||, or between || and
|||||.
[a + (b + 1) (a + b) + 1] & [a + (b + (c + 1)) (a + (b + c))+ 1] & [(a + b) + (c + 1) ((a + b) + c) + 1].
.a + (b + c).ℑ.(a + b) + c... U)
and in general:
[f1(1) f2(1)] & [f1(c + 1) f1(c) + 1] & [f2(c + 1) = f2(c) + 1]. .f1(c).ℑ.f2(c)... V)
Page Break 442
Page 442
You might see the definition U, without knowing why I use that abbreviation.
Page 442
You might see the definition without understanding its point.--But its point is something new, not something
already contained in it as a specific replacement rule.
Page 442
Of course, "ℑ" isn't an equals-sign in the same sense as the ones occurring in α, β, γ.
Page 442
But we can easily show that "ℑ" has certain formal properties in common with =.
Page 442
It would be incorrect--according to the postulated rules--to use the equals-sign like this:
∆... |(a + b)2 = a.(a + b) + b.(a + b) =... =
= a2 + 2ab + b2|. = .|(a + b)2 = a2 + 2ab + b2|
if that is supposed to mean that the left hand side is the proof of the right.
Page 442
But mightn't we imagine this equation regarded as a definition? For instance, if it had always been the
custom to write out the whole chain instead of the right hand side, and we introduced the abbreviation.
Page 442
Of course ∆ can be regarded as a definition! Because the sign on the left hand side is in fact used, and there's
no reason why we shouldn't abbreviate it according to this convention. Only in that case either the sign on the right
or the sign on the left is used in a way different from the one now usual.
Page 442
It can never be sufficiently emphasized that totally different kinds of sign-rules get written in the form of an
equation.
Page 442
The 'definition' x.x = x2 might be regarded as merely allowing us to replace the sign "x.x" by the sign "x2,"
like the definition "1 + 1 = 2"; but it can also be regarded (and in fact is regarded) as allowing us to put a2 instead of
a.a, and (a + b)2 instead of (a + b).(a + b) and in such a way that any arbitrary number can be substituted for the x.
Page Break 443
Page 443
A person who discovers that a proposition p follows from one of the form q ⊃ p.q constructs a new sign, the
sign for that rule. (I am assuming that a calculus with p, q, ⊃, has already been in use, and that this rule is now added
to make it a new calculus.)
Page 443
It is true that the notation "x2" takes away the possibility of replacing one of the factors x by another number.
Indeed, we could imagine two stages in the discovery (or construction) of x2. At first, people might have written
"x=" instead of "x2", before it occurred to them that there was a system x.x, x.x.x, etc.; later, they might have hit
upon that too. Similar things have occurred in mathematics countless times. (In Liebig's sign for an oxide oxygen
did not appear as an element in the same way as what was oxidized. Odd as it sounds, we might even today, with all
the data available to us, give oxygen a similarly privileged position--only, of course, in the form of
representation--by adopting an incredibly artificial interpretation, i.e. grammatical construction.)
Page 443
The definitions x.x = x2, x.x.x = x3 don't bring anything into the world except the signs "x2" and "x3" (and
thus so far it isn't necessary to write numbers as exponents).
|The process of generalization creates a new sign-system.|
Page 443
Of course Sheffer's discovery is not the discovery of the definition ~p.~q = p|q. Russell might well have
given that definition without being in possession of Sheffer's system, and on the other hand Sheffer might have built
up his system without the definition. His system is contained in the use of the signs "~p.~p" for "~p" and
"~(~p.~q).~(~p.~q)" for "p ∨ q" and all "p|q" does is to permit an abbreviation. Indeed, we can say that someone
could well have been acquainted with the use of the sign "~(~p.~q).~(~p.~q)" for "p ∨ q" without recognizing the
system p|q.|.p|q in it.
Page Break 444
Page 444
It makes matters clearer if we adopt Frege's two primitive signs "~" and ".". The discovery isn't lost if the
definitions are written ~p.~p = ~p and ~(~p.~p).~(~q.~q) = p.q. Here apparently nothing at all has been altered in
the original signs.
Page 444
But we might also imagine someone's having written the whole Fregean or Russellian logic in this system,
and yet, like Frege, calling "~" and "." his primitive signs, because he did not see the other system in his proposition.
Page 444
It is clear that the discovery of Sheffer's system in ~.p.~p = ~p and ~(~p.~p).~(~q.~q) = p.q corresponds to
the discovery that x2 + ax + a2/4 is a specific instance of a2 + 2ab + b2.
Page 444
We don't see that something can be looked at in a certain way until it is so looked at.
Page 444
We don't see that an aspect is possible until it is there.
Page 444
That sounds as if Sheffer's discovery wasn't capable of being represented in signs at all. (Periodic division.)
But that is because we can't smuggle the use of the sign into its introduction (the rule is and remains a sign, separated
from its application).
Page 444
Of course I can only apply the general rule for the induction proof when I discover the substitution that
makes it applicable. So it would be possible for someone to see the equations
(a + 1) + 1 = (a + 1) + 1
1 + (a + 1) = (1 + a) + 1
without hitting on the substitution
Page Break 445
Page 445
Moreover, if I say that I understand the equations as particular cases of the rule, my understanding has to be
the understanding that shows itself in the explanations of the relations between the rule and the equations, i.e. what
we express by the substitutions. If I don't regard that as an expression of what I understand, then nothing is an
expression of it; but in that case it makes no sense either to speak of understanding or to say that I understand
something definite. For it only makes sense to speak of understanding in cases where we understand one thing as
opposed to another. And it is this contrast that signs express.
Page 445
Indeed, seeing the internal relation must in its turn be seeing something that can be described, something of
which one can say: "I see that such and such is the case"; it has to be really something of the same kind as the
correlation-signs (like connecting lines, brackets, substitutions, etc.). Everything else has to be contained in the
application of the sign of the general rule in a particular case.
Page 445
It is as if we had a number of material objects and discovered they had surfaces which enabled them to be
placed in a continuous row. Or rather, as if we discovered that such and such surfaces, which we had seen before,
enabled them to be placed in a continuous row. That is the way many games and puzzles are solved.
Page 445
The person who discovers periodicity invents a new calculus. The question is, how does the calculus with
periodic division differ from the calculus in which periodicity is unknown?
Page 445
(We might have operated a calculus with cubes without having had the idea of putting them together to make
prisms.)
Page Break 446
Appendix†1
Page 446
(On: The process of generalization creates a new sign-system)
Page 446
It is a very important observation that the c in A is not the same variable as the c in β and γ. So the way I
wrote out the proof was not quite correct in a respect which is very important for us. In A we could substitute n for
c, whereas the cs in β and γ are identical.
Page 446
But another question arises: can I derive from A that i + (k + c) = (i + k) + c? If so, why can't I derive it in the
same way from B? Does that mean that a and b in A are not identical with a and b in α, β and γ?
Page 446
We see clearly that the variable c in B isn't identical with the c in A if we put a number instead of it. Then B is
something like
but that doesn't have corresponding to it an equation like AW: 4 + (5 + 6) = (4 + 5) + 6!
Page 446
What makes the induction proof different from a proof of A is expressed in the fact that the c in B is not
identical with the one in A, so that we could use different letters in the two places.
Page 446
All that is meant by what I've written above is that the reason it looks like an algebraic proof of A is that we
think we meet the same variables a, b, c in the equations A as in α, β, γ and so we
Page Break 447
regard A as the result of a transformation of those equations. (Whereas of course in reality I regard the signs α, β, γ
in quite a different way, which means that the c in β and γ isn't used as a variable in the same way as a and b. Hence
one can express this new view of B, by saying that the c does not occur in A.)
Page 447
What I said about the new way of regarding α, β, γ might be put like this: α is used to build up β and γ in
exactly the same way as the fundamental algebraic equations are used to build up an equation like (a + b)2 = a2 +
2ab + b2. But if that is the way they are derived, we are regarding the complex α β γ in a new way when we give the
variable c a function which differs from that of a and b (c becomes the hole through which the stream of numbers
has to flow).
Page Break 448
38
Proof by induction, arithmetic and algebra
Page 448
Why do we need the commutative law? Not so as to be able to write the equation 4 + 6 = 6 + 4, because that
equation is justified by its own particular proof. Certainly the proof of the commutative law can also be used to
prove it, but in that case it becomes just a particular arithmetical proof. So the reason I need the law, is to apply it
when using letters.
Page 448
And it is this justification that the inductive proof cannot give me.
Page 448
However, one thing is clear: if the recursive proof gives us the right to calculate algebraically, then so does
the arithmetical proof L†1.
Page 448
Again: the recursive proof is--of course--essentially concerned with numbers. But what use are numbers to
me when I want to operate purely algebraically? Or again, the recursion proof is only of use to me when I want to
use it to justify a step in a number-calculation.
Page 448
But someone might ask: do we need both the inductive proof and the associative law, since the latter cannot
provide a foundation for calculation with numbers, and the former cannot provide one for transformations in
algebra?
Page 448
Well, before Skolem's proof was the associative law, for example, just accepted without anyone's being able
to work out the corresponding step in a numerical calculation? That is, were we previously unable to work out 5 + (4
+ 3) = (5 + 4) + 3, and did we treat it as an axiom?
Page 448
If I say that the periodic calculation proves the proposition that justifies me in those steps, what would the
proposition have been
Page Break 449
like if it had been assumed as an axiom instead of being proved?
Page 449
What would a proposition be like that permitted one to put 5 + (7 + 9) = (5 + 7) + 9) without being able to
prove it? It is obvious that there never has been such a proposition.
Page 449
But couldn't we also say that the associative law isn't used at all in arithmetic and that we work only with
particular number calculations?
Page 449
Even when algebra uses arithmetical notation, it is a totally different calculus, and cannot be derived from
the arithmetical one.
Page 449
To the question "is 5 × 4 = 20"? one might answer: "let's check whether it is in accord with the basic rules of
arithmetic" and similarly I might say: let's check whether A is in accord with the basic rules. But with which rules?
Presumably with α.
Page 449
But before we can bring α and A together we need to stipulate what we want to call "agreement" here.
Page 449
That means that α and A are separated by the gulf between arithmetic and algebra,†1 and if B is to count as a
proof of A, this gulf has to be bridged over by a stipulation.
Page 449
It is quite clear that we do use an idea of this kind of agreement when, for instance, we quickly work out a
numerical example to check the correctness of an algebraic proposition.
Page 449
And in this sense I might e.g. calculate
Page Break 450
and say: "yes, it's right, a.b is equal to b.a"--if I imagine that I have forgotten.
Page 450
Considered as a rule for algebraic calculation, A cannot be proved recursively. We would see that especially
dearly if we wrote down the "recursive proof" as a series of arithmetical expressions. Imagine them written down
(i.e. a fragment of the series plus "and so on") without any intention of "proving" anything, and then suppose
someone asks: "does that prove a + (b + c) = (a + b) + c?". We would ask in astonishment "How can it prove
anything of the kind? The series contains only numbers, it doesn't contain any letters".--But no doubt we might say:
if I introduce A as a rule for calculation with letters, that brings this calculus in a certain sense into unison with the
calculus of the cardinal numbers, the calculus I established by the law for the rules of addition (the recursive
definition a + (b + 1) = (a + b) + 1).
Page Break 451
VII INFINITY IN MATHEMATICS:
THE EXTENSIONAL VIEWPOINT
39
Generality in arithmetic
Page 451
"What is the sense of such a proposition as '(∃n).3 + n = 7'?" Here we are in an odd difficulty: on the one
hand we feel it to be a problem that the proposition has the choice between infinitely many values of n, and on the
other hand the sense of the proposition seems guaranteed in itself and only needing further research on our part,
because after all we "know 'what (∃x) φx' means". If someone said he didn't know what was the sense of "(∃n).3 + n
= 7", he would be answered "but you do know what this proposition says: 3 + 0 = 7. ∨.3 + 1 = 7. ∨.3 + 2 = 7 and so
on!" But to that one can reply "Quite correct--so the proposition isn't a logical sum, because a logical sum doesn't
end with 'and so on'. What I am not clear about is this propositional form 'φ(0) ∨ φ(1) ∨ φ(2) ∨ and so on'--and all
you have done is to substitute a second unintelligible kind of proposition for the first one, while pretending to give
me something familiar, namely a disjunction."
Page 451
That is, if we believe that we do understand "(∃n) etc." in some absolute sense, we have in mind as a
justification other uses of the notation "(∃...)... ", or of the ordinary-language expression "There is..." But to that one
can only say: So you are comparing the proposition "(∃n)..." with the proposition "There is a house in this city
which..." or "There are two foreign words on this page". But the occurrence of the words "there is" in those
sentences doesn't suffice to determine the grammar of this generalization, all it does is to indicate a certain analogy in
the rules. And so we can still investigate the grammar of the generalisation"(∃n) etc." with an open mind, that is,
without letting the meaning of "(∃...)..." in other cases get in our way.
Page 451
"Perhaps all numbers have the property ε". Again the question is:
Page Break 452
what is the grammar of this general proposition? Our being acquainted with the use of the expression "all..." in other
grammatical systems is not enough. If we say "you do know what it means: it means ε(0).ε(1).ε(2) and so on", again
nothing is explained except that the proposition is not a logical product. In order to understand the grammar of the
proposition we ask: how is the proposition used? What is regarded as the criterion of its truth? What is its
verification?--If there is no method provided for deciding whether the proposition is true or false, then it is pointless,
and that means senseless. But then we delude ourselves that there is indeed a method of verification, a method
which cannot be employed, but only because of human weakness. This verification consists in checking all the
(infinitely many) terms of the product ε(0).ε(1).ε(2)... Here there is confusion between physical impossibility and
what is called 'logical impossibility". For we think we have given sense to the expression "checking of the infinite
product" because we take the expression "infinitely many" for the designation of an enormously large number. And
when we hear of "the impossibility of checking the infinite number of propositions" there comes before our mind
the impossibility of checking a very large number of propositions, say when we don't have sufficient time.
Page 452
Remember that in the sense in which it is impossible to check an infinite number of propositions it is also
impossible to try to do so.--If we are using the words "But you do know what 'all' means" to appeal to the cases in
which this mode of speech is used, we cannot regard it as a matter of indifference if we observe a distinction
between these cases and the case for which the use of the words is to be explained.--Of course we know what is
meant by "checking a number of propositions for correctness", and it is this understanding that we are appealing to
when we claim that one should understand also the expression "... infinitely many propositions". But doesn't the
sense of the first expression depend on the specific experiences that correspond to it? And these experiences are
lacking in the employment (the calculus) of the second expression; if any experiences at all are correlated to it they
are fundamentally different ones.
Page Break 453
Page 453
Ramsey once proposed to express the proposition that infinitely many objects satisfied a function f(ξ) by the
denial of all propositions like
~(∃x).fx
(∃x).fx.~(∃x, y).fx.fy
(∃x, y).fx.fy.~(∃x, y, z).fx.fy.fz
and so on.
But this denial would yield the series
(∃x).fx
(∃x, y).fx.fy
(∃x, y, z)..., etc., etc.
Page 453
But this series too is quite superfluous: for in the first place the last proposition at any point surely contains
all the previous ones, and secondly even it is of no use to us, because it isn't about an infinite number of objects. So
in reality the series boils down to the proposition:
"(∃x, y, z... ad inf.).fx.fy.fz... ad inf."
and we can't make anything of that sign unless we know its grammar. But one thing is clear: what we are dealing
with isn't a sign of the form "(∃x, y, z).fx.fy.fz" but a sign whose similarity to that looks purposely deceptive.
Page 453
I can certainly define "m > n" as (∃x):m - n = x, but by doing so I haven't in any way analysed it. You think,
that by using the symbolism "(∃...)..." you establish a connection between "m > n" and other propositions of the
form "there is... "; what you forget is that that can't do more than stress a certain analogy, because the sign "(∃...)..."
is used in countlessly many different 'games'. (Just as there is a 'king' in chess and draughts.) So we have to know the
rules governing its use here; and as soon as we do that it immediately becomes clear that these rules are connected
with the rules for subtraction. For if
Page Break 454
we ask the usual question "how do I know--i.e. where do I get it from--that there is a number x that satisfies the
condition m - n = x?" it is the rules for subtraction that provide the answer. And then we see that we haven't gained
very much by our definition. Indeed we might just as well have given as an explanation of 'm > n' the rules for
checking a proposition of that kind--e.g. '32 > 17'.
Page 454
If I say: "given any n there is a δ for which the function is less than n", I am ipso facto referring to a general
arithmetical criterion that indicates when F(δ) < n.
Page 454
If in the nature of the case I cannot write down a number independently of a number system, that must be
reflected in the general treatment of number. A number system is not something inferior--like a Russian abacus--that
is only of interest to elementary schools while a more lofty general discussion can afford to disregard it.
Page 454
Again, I don't lose anything of the generality of my account if I give the rules that determine the correctness
and incorrectness (and thus the sense) of 'm > n' for a particular system like the decimal system. After all I need a
system, and the generality is preserved by giving the rules according to which one system can be translated into
another.
Page 454
A proof in mathematics is general if it is generally applicable. You can't demand some other kind of
generality in the name of rigour. Every proof rests on particular signs, produced on a particular occasion. All that
can happen is that one type of generality may appear more elegant than another. ((Cf. the employment of the
decimal system in proofs concerning δ and η)).
Page 454
"Rigorous" means: clear.†1
Page Break 455
"We may imagine a mathematical proposition as a creature which itself knows whether it is true or false (in contrast
with propositions of experience).
Page 455
A mathematical proposition itself knows that it is true or that it is false. If it is about all numbers, it must also
survey all the numbers. "Its truth or falsity must be contained in it as is its sense."
Page 455
"It's as though the generality of a proposition like '(n).ε(n)' were only a pointer to the genuine, actual,
mathematical generality, and not the generality itself. As if the proposition formed a sign only in a purely external
way and you still needed to give the sign a sense from within."
Page 455
"We feel the generality possessed by the mathematical assertion to be different from the generality of the
proposition proved."
Page 455
"We could say: a mathematical proposition is an allusion to a proof."†1
Page 455
What would it be like if a proposition itself did not quite grasp its sense? As if it were, so to speak, too grand
for itself? That is really what logicians suppose.
Page 455
A proposition that deals with all numbers cannot be thought of as verified by an endless striding, for, if the
striding is endless, it does not lead to any goal.
Page 455
Imagine an infinitely long row of trees, and, so that we can inspect them, a path beside them. All right, the
path must be endless. But if it is endless, then that means precisely that you can't walk to the end of it. That is, it
does not put me in a position to survey the row. That is to say, the endless path does not have an end 'infinitely far
away', it has no end.
Page 455
Nor can you say: "A proposition cannot deal with all the numbers one by one, so it has to deal with them by
means of the concept of number" as if this were a pis aller: "Because we can't do it like this, we have to do it another
way." But it is indeed possible to deal with the numbers one by one, only that doesn't
Page Break 456
lead to the totality. That doesn't lie on the path on which we go step by step, not even at the infinitely distant end of
that path. (This all only means that "ε(0).ε(1).ε(2) and so on" is not the sign for a logical product.)
Page 456
"It cannot be a contingent matter that all numbers possess a property; if they do so it must be essential to
them."--The proposition "men who have red noses are good-natured" does not have the same sense as the
proposition "men who drink wine are goodnatured" even if the men who have red noses are the same as the men
who drink wine. On the other hand, if the numbers m, n, o are the extension of a mathematical concept, so that is the
case that fm.fn.fo, then the proposition that the numbers that satisfy f have the property ε has the same sense as
"ε(m).ε(n)ε.(o)". This is because the propositions "f(m).f(n).f(o)" and "ε(m).ε(n).ε(o)" can be transformed into each
other without leaving the realm of grammar.
Page 456
Now consider the proposition: "all the n numbers that satisfy the condition F(ξ) happen by chance to have
the property ε". Here what matters is whether the condition F(ξ) is a mathematical one. If it is, then I can indeed
derive ε(x) from F(x), if only via the disjunction of the n values of F(ξ). (For what we have in this case is in fact a
disjunction). So I won't call this chance.--On the other hand if the condition is a non-mathematical one, we can
speak of chance. For example, if I say: all the numbers I saw today on buses happened to prime numbers. (But, of
course, we can't say: the numbers 17, 3, 5, 31 happen to be prime numbers" any more than "the number 3 happens
to be a prime number"), "By chance" is indeed the opposite of "in accordance with a general rule", but however odd
it sounds one can say that the proposition "17, 3, 5, 31 are prime numbers" is derivable by a general rule just like the
proposition 2 + 3 = 5.
Page Break 457
Page 457
If we now return to the first proposition, we may ask again: How is the proposition "all numbers have the
property ε" supposed to be meant? How is one supposed to be able to know? For to settle its sense you must settle
that too! The expression "by chance" indicates a verification by successive tests, and that is contradicted by the fact
that we are not speaking of a finite series of numbers.
Page 457
In mathematics description and object are equivalent. "The fifth number of the number series has these
properties" says the same as "5 has these properties". The properties of a house do not follow from its position in a
row of houses; but the properties of a number are the properties of a position.
Page 457
You might say that the properties of a particular number cannot be foreseen. You can only see them when
you've got there.
Page 457
What is general is the repetition of an operation. Each stage of the repetition has its own individuality. But it
isn't as if I use the operation to move from one individual to another so that the operation would be the means for
getting from one to the other--like a vehicle stopping at every number which we can then study: no, applying the
operation + 1 three times yields and is the number 3.
Page 457
(In the calculus process and result are equivalent to each other.)
Page 457
But before deciding to speak of "all these individualities" or "the totality of these individualities" I had to
consider carefully what stipulations I wanted to make here for the use of the expressions "all" and "totality".
Page 457
It is difficult to extricate yourself completely from the extensional viewpoint: You keep thinking "Yes, but
there must still be an internal relation between x3 + y3 and z3 since at least extensions of these expressions if I only
knew them would have to show the result of such a relation". Or perhaps: "It must surely be either essential to all
numbers to have the property or not, even if I can't know it."
Page Break 458
Page 458
"If I run through the number series, I either eventually come to a number with the property ε or I never do."
The expression "to run through the number series" is nonsense; unless a sense is given to it which removes the
suggested analogy with "running through the numbers from 1 to 100".
Page 458
When Brouwer attacks the application of the law of excluded middle in mathematics, he is right in so far as
he is directing his attack against a process analogous to the proof of empirical propositions. In mathematics you can
never prove something like this: I saw two apples lying on the table, and now there is only one there, so A has eaten
an apple. That is, you can't by excluding certain possibilities prove a new one which isn't already contained in the
exclusion because of the rules we have laid down. To that extent there are no genuine alternatives in mathematics. If
mathematics was the investigation of empirically given aggregates, one could use the exclusion of a part to describe
what was not excluded, and in that case the non-excluded part would not be equivalent to the exclusion of the
others.
Page 458
The whole approach that if a proposition is valid for one region of mathematics it need not necessarily be
valid for a second region as well, is quite out of place in mathematics, completely contrary to its essence. Although
many authors hold just this approach to be particularly subtle and to combat prejudice.
Page 458
It is only if you investigate the relevant propositions and their proofs that you can recognize the nature of the
generality of the propositions of mathematics that treat not of "all cardinal numbers" but e.g. of "all real numbers".
Page 458
How a proposition is verified is what it says. Compare generality in arithmetic with the generality of
non-arithmetical propositions.
Page Break 459
It is differently verified and so is of a different kind. The verification is not a mere token of the truth, but determines
the sense of the proposition. (Einstein: how a magnitude is measured is what it is.)
Page Break 460
40
On set theory
Page 460
A misleading picture: "The rational points lie close together on the number-line."
Page 460
Is a space thinkable that contains all rational points, but not the irrational ones? Would this structure be too
coarse for our space, since it would mean that we could only reach the irrational points approximately? Would it
mean that our net was not fine enough? No. What we would lack would be the laws, not the extensions.
Page 460
Is a space thinkable that contains all rational points but not the irrational ones?
Page 460
That only means: don't the rational numbers set a precedent for the irrational numbers?
Page 460
No more than draughts sets a precedent for chess.
Page 460
There isn't any gap left open by the rational numbers that is filled up by the irrationals.
Page 460
We are surprised to find that "between the everywhere dense rational points", there is still room for the
irrationals. (What balderdash!) What does a construction like that for show? Does it show how there is yet
room for this point in between all the rational points? It shows that the point yielded by the construction, yielded by
this construction, is not rational.--And what corresponds to this construction in arithmetic? A sort of number which
manages after all to squeeze in between the rational numbers? A law that is not a law of the nature of a rational
number.
Page 460
The explanation of the Dedekind cut pretends to be clear when it says: there are 3 cases: either the class R
has a first member and L no last member, etc. In fact two of these 3 cases cannot be imagined, unless the words
"class", "first member", "last member", altogether change the everyday meanings thay [[sic]] are supposed to have
retained.
Page Break 461
Page 461
That is, if someone is dumbfounded by our talk of a class of points that lie to the right of a given point and
have no beginning, and says: give us an example of such a class--we trot out the class of rational numbers; but that
isn't a class of points in the original sense.
Page 461
The point of intersection of two curves isn't the common member of two classes of points, it's the meeting of
two laws. Unless, very misleadingly, we use the second form of expression to define the first.
Page 461
After all I have already said, it may sound trivial if I now say that the mistake in the set-theoretical approach
consists time and again in treating laws and enumerations (lists) as essentially the same kind of thing and arranging
them in parallel series so that one fills in gaps left by another.
Page 461
The symbol for a class is a list.
Page 461
Here again, the difficulty arises from the formation of mathematical pseudo-concepts. For instance, when we
say that we can arrange the cardinal numbers, but not the rational numbers, in a series according to their size, we are
unconsciously presupposing that the concept of an ordering by size does have a sense for rational numbers, and
that it turned out on investigation that the ordering was impossible (which presupposes that the attempt is
thinkable).--Thus one thinks that it is possible to attempt to arrange the real numbers (as if that were a concept of the
same kind as 'apple on this table') in a series, and now it turned out to be impracticable.
Page 461
For its form of expression the calculus of sets relies as far as possible on the form of expression of the
calculus of cardinal numbers. In some ways that is instructive, since it indicates certain formal similarities, but it is
also misleading, like calling something a knife that has neither blade nor handle. (Lichtenberg.)
Page Break 462
Page 462
(The only point there can be to elegance in a mathematical proof is to reveal certain analogies in a particularly
striking manner, when that is what is wanted; otherwise it is a product of stupidity and its only effect is to obscure
what ought to be clear and manifest. The stupid pursuit of elegance is a principal cause of the mathematicians' failure
to understand their own operations; or perhaps the lack of understanding and the pursuit of elegance have a
common origin.)
Page 462
Human beings are entangled all unknowing in the net of language.
Page 462
"There is a point where the two curves intersect." How do you know that? If you tell me, I will know what
sort of sense the proposition "there is..." has.
Page 462
If you want to know what the expression "the maximum of a curve" means, ask yourself: how does one find
it?--If something is found in a different way it is a different thing. We define the maximum as the point on the curve
higher than all the others, and from that we get the idea that it is only our human weakness that prevents us from
sifting through the points of the curve one by one and selecting the highest of them. And this leads to the idea that
the highest point among a finite number of points is essentially the same as the highest point of a curve, and that we
are simply finding out the same thing by two different methods, just as we find out in two different ways that there
is no one in the next room; one way if the door is shut and we aren't strong enough to open it, and another way if we
can get inside. But, as I said, it isn't human weakness that's in question where the alleged description of the action
"that we cannot perform" is senseless. Of course it does no harm, indeed it's very interesting, to see the analogy
between the maximum of a curve and the maximum (in another sense) of a class of points, provided that the
analogy doesn't instil the prejudice that in each case we have fundamentally the same thing.
Page Break 463
Page 463
It's the same defect in our syntax which presents the geometrical proposition "a length may be divided by a
point into two parts" as a proposition of the same form as "a length may be divided for ever"; so that it looks as if in
both cases we can say "Let's suppose the possible division to have been carried out". "Divisible into two parts" and
"infinitely divisible" have quite different grammars. We mistakenly treat the word "infinite" as if it were a number
word, because in everyday speech both are given as answers to the question "how many?"
Page 463
"But after all the maximum is higher than any other arbitrary points of the curve." But the curve is not
composed of points, it is a law that points obey, or again, a law according to which points can be constructed. If you
now ask: "which points?" I can only say, "well, for instance, the points P, Q, R, etc." On the one hand we can't give a
number of points and say that they are all the points that lie on the curve, and on the other hand we can't speak of a
totality of points as something describable which although we humans cannot count them might be called the
totality of all the points of the curve--a totality too big for us human beings. On the one hand there is a law, and on
the other points on the curve;--but not "all the points of the curve". The maximum is higher than any point of the
curve that happens to be constructed, but it isn't higher than a totality of points, unless the criterion for that, and thus
the sense of the assertion, is once again simply construction according to the law of the curve.
Page 463
Of course the web of errors in this region is a very complicated one. There is also e.g. the confusion between
two different meanings of the word "kind". We admit, that is, that the infinite numbers are a different kind of
number from the finite ones, but then we misunderstand what the difference between different kinds amounts to in
this case. We don't realise, that is, that it's not a matter of distinguishing between objects by their properties in the
way we distinguish between red and yellow apples, but a matter of different logical forms.--Thus Dedekind tried to
describe an infinite class
Page Break 464
by saying that it is a class which is similar to a proper subclass of itself. Here it looks as if he has given a property
that a class must have in order to fall under the concept "infinite class" (Frege).†1 Now let us consider how this
definition is applied. I am to investigate in a particular case whether a class is finite or not, whether a certain row of
trees, say, is finite or infinite. So, in accordance with the definition, I take a subclass of the row of trees and
investigate whether it is similar (i.e. can be co-ordinated one-to-one) to the whole class! (Here already the whole
thing has become laughable.) It hasn't any meaning; for, if I take a "finite class" as a sub-class, the attempt to
coordinate it one-to-one with the whole class must eo ipso fail: and if I make the attempt with an infinite class--but
already that is a piece of nonsense, for if it is infinite, I cannot make an attempt to co-ordinate it.--What we call
'correlation of all the members of a class with others' in the case of a finite class is something quite different from
what we, e.g., call a correlation of all cardinal numbers with all rational numbers. The two correlations, or what one
means by these words in the two cases, belong to different logical types. An infinite class is not a class which
contains more members than a finite one, in the ordinary sense of the word "more". If we say that an infinite number
is greater than a finite one, that doesn't make the two comparable, because in that statement the word "greater"
hasn't the same meaning as it has say in the proposition 5 > 4!
Page 464
That is to say, the definition pretends that whether a class is finite or infinite follows from the success or
failure of the attempt to correlate a proper subclass with the whole class; whereas there just isn't any such decision
procedure.--'Infinite class' and 'finite class' are different logical categories; what can be significantly
Page Break 465
asserted of the one category cannot be significantly asserted of the other.
Page 465
With regard to finite classes the proposition that a class is not similar to its sub-classes is not a truth but a
tautology. It is the grammatical rules for the generality of the general implication in the proposition "k is a subclass
of K" that contain what is said by the proposition that K is an infinite class.
Page 465
A proposition like "there is no last cardinal number" is offensive to naive--and correct--common sense. If I
ask "Who was the last person in the procession?" and am told "There wasn't a last person" I don't know what to
think; what does "There wasn't a last person" mean? Of course, if the question had been "Who was the standard
bearer?" I would have understood the answer "There wasn't a standard bearer"; and of course the bewildering
answer is modelled on an answer of that kind. That is, we feel, correctly, that where we can speak at all of a last one,
there can't be "No last one". But of course that means: The proposition "There isn't a last one" should rather be: it
makes no sense to speak of a "last cardinal number", that expression is ill-formed.
Page 465
"Does the procession have an end?" might also mean: is the procession a compact group? And now
someone might say: "There, you see, you can easily imagine a case of something not having an end; so why can't
there be other such cases?"--But the answer is: The "cases" in this sense of the word are grammatical cases, and it is
they that determine the sense of the question. The question "Why can't there be other such cases?" is modelled on:
"Why can't there be other minerals that shine in the dark"; but the latter is about cases where a statement is true, the
former about cases that determine the sense.
Page 465
The form of expression "m = 2n correlates a class with one of its proper subclasses" uses a misleading
analogy to clothe a trivial sense in a paradoxial form. (And instead of being ashamed of
Page Break 466
this paradoxical form as something ridiculous, people plume themselves on a victory over all prejudices of the
understanding). It is exactly as if one changed the rules of chess and said it had been shown that chess could also be
played quite differently. Thus we first mistake the word "number" for a concept word like "apple", then we talk of a
"number of numbers" and we don't see that in this expression we shouldn't use the same word "number" twice; and
finally we regard it as a discovery that the number of the even numbers is equal to the number of the odd and even
numbers.
Page 466
It is less misleading to say "m = 2n allows the possiblity of correlating every time with another" than to say
"m = 2n correlates all numbers with others". But here too the grammar of the meaning of the expression "possibility
of correlation" has to be learnt.
Page 466
(It's almost unbelievable, the way in which a problem gets completely barricaded in by the misleading
expressions which generation upon generation throw up for miles around it, so that it becomes virtually impossible
to get at it.)
Page 466
If two arrows point in the same direction, isn't it in such a case absurd to call these directions equally long,
because whatever lies in the direction of the one arrow, also lies in that of the other?--The generality of m = 2n is an
arrow that points along the series generated by the operation. And you can even say that the arrow points to infinity;
but does that mean that there is something--infinity--at which it points, as at a thing?--It's as though the arrow
designates the possibility of a position in its direction. But the word "possibility" is misleading, since someone will
say: let what is possible now become actual. And in thinking this we always think of a temporal process, and infer
from the fact that mathematics has nothing to do with time, that in its case possibility is already actuality.
Page 466
The "infinite series of cardinal numbers" or "the concept of cardinal number" is only such a possibility--as
emerges clearly
Page Break 467
from the symbol "|0, ξ, ξ + 1|". This symbol is itself an arrow with the "0" as its tail and the "ξ + 1" as its tip. It is
possible to speak of things which lie in the direction of the arrow, but misleading or absurd to speak of all possible
positions for things lying in the direction of the arrow as an equivalent for the arrow itself. If a searchlight sends out
light into infinite space it illuminates everything in its direction, but you can't say it illuminates infinity.
Page 467
It is always right to be extremely suspicious when proofs in mathematics are taken with greater generality
than is warranted by the known application of the proof. This is always a case of the mistake that sees general
concepts and particular cases in mathematics. In set theory we meet this suspect generality at every step.
Page 467
One always feels like saying "let's get down to brass tacks".
Page 467
These general considerations only make sense when we have a particular region of application in mind.
Page 467
In mathematics there isn't any such thing as a generalization whose application to particular cases is still
unforseeable. That's why the general discussions of set theory (if they aren't viewed as calculi) always sound like
empty chatter, and why we are always astounded when we are shown an application for them. We feel that what is
going on isn't properly connected with real things.
Page 467
The distinction between the general truth that one can know, and the particular that one doesn't know, or
between the known description of the object, and the object itself that one hasn't seen, is another example of
something that has been taken over into logic from the physical description of the world. And that too is where we
get the idea that our reason can recognize questions but not their answers.
Page Break 468
Page 468
Set theory attempts to grasp the infinite at a more general level than the investigation of the laws of the real
numbers. It says that you can't grasp the actual infinite by means of mathematical symbolism at all and therefore it
can only be described and not represented. The description would encompass it in something like the way in which
you carry a number of things that you can't hold in your hand by packing them in a box. They are then invisible but
we still know we are carrying them (so to speak, indirectly). One might say of this theory that it buys a pig in a poke.
Let the infinite accommodate itself in this box as best it can.
Page 468
With this there goes too the idea that we can use language to describe logical forms. In a description of this
sort the structures are presented in a package and so it does look as if one could speak of a structure without
reproducing it in the proposition itself. Concepts which are packed up like this may, to be sure, be used, but our
signs derive their meaning from definitions which package the concepts in this way; and if we follow up these
definitions, the structures are uncovered again. (Cf. Russell's definition of "R*".)
Page 468
When "all apples" are spoken of, it isn't, so to speak, any concern of logic how many apples there are. With
numbers it is different; logic is responsible for each and every one of them.
Page 468
Mathematics consists entirely of calculations.
Page 468
In mathematics everything is algorithm and nothing is meaning; even when it doesn't look like that because
we seem to be using words to talk about mathematical things. Even these words are used to construct an algorithm.
Page 468
In set theory what is calculus must be separated off from what attempts to be (and of course cannot be)
theory. The rules of the game have to be separated off from inessential statements about the chessmen.
Page 468
In Cantor's alleged definition of "greater", "smaller", "+", "-"
Page Break 469
Page 469
Frege replaced the signs with new words to show the definition wasn't really a definition.†1 Similarly in the
whole of mathematics one might replace the usual words, especially the word "infinite" and its cognates, with
entirely new and hitherto meaningless expressions so as to see what the calculus with these signs really achieves and
what it fails to achieve. If the idea was widespread that chess gave us information about kings and castles, I would
propose to give the pieces new shapes and different names, so as to demonstrate that everything belonging to chess
has to be contained in the rules.
Page 469
What a geometrical proposition means, what kind of generality it has, is something that must show itself
when we see how it is applied. For even if someone succeeded in meaning something intangible by it it wouldn't
help him, because he can only apply it in a way which is quite open and intelligible to every one.
Page 469
Similarly, if someone imagined the chess king as something mystical it wouldn't worry us since he can only
move him on the 8 × 8 squares of the chess board.
Page 469
We have a feeling "There can't be possibility and actuality in mathematics. It's all on one level. And is in a
certain sense, actual.--And that is correct. For mathematics is a calculus; and the calculus does not say of any sign
that it is merely possible, but is concerned only with the signs with which it actually operates. (Compare the
foundations of set theory with the assumption of a possible calculus with infinite signs).
Page 469
When set theory appeals to the human impossibility of a direct symbolisation of the infinite it brings in the
crudest imaginable misinterpretation of its own calculus. It is of course this very misinterpretation that is responsible
for the invention of the calculus. But of course that doesn't show the calculus in itself to
Page Break 470
be something incorrect (it would be at worst uninteresting) and it is odd to believe that this part of mathematics is
imperilled by any kind of philosophical (or mathematical) investigations. (As well say that chess might be imperrilled
by the dicovery that wars between two armies do not follow the same course as battles on the chessboard.) What set
theory has to lose is rather the atmosphere of clouds of thought surrounding the bare calculus, the suggestion of an
underlying imaginary symbolism, a symbolism which isn't employed in its calculus, the apparent description of
which is really nonsense. (In mathematics anything can be imagined, except for a part of our calculus.)
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41
The extensional conception of the real numbers
Page 471
Like the enigma of time for Augustine, the enigma of the continuum arises because language misleads us
into applying to it a picture that doesn't fit. Set theory preserves the inappropriate picture of something
discontinuous, but makes statements about it that contradict the picture, under the impression that it is breaking with
prejudices; whereas what should really have been done is to point out that the picture just doesn't fit, that it certainly
can't be stretched without being torn, and that instead of it one can use a new picture in certain respects similar to the
old one.
Page 471
The confusion in the concept of the "actual infinite" arises from the unclear concept of irrational number, that
is, from the fact that logically very different things are called "irrational numbers" without any clear limits being
given to the concept. The illusion that we have a firm concept rests on our belief that in signs of the the [sic]] form
"o.abcd... ad infinitum" we have a pattern to which they (the irrational numbers) have to conform whatever happens.
Page 471
"Suppose I cut a length at a place where there is no rational point (no rational number)." But can you do
that? What sort of a length are you speaking of? "But if my measuring instruments were fine enough, at least I could
approximate without limit to a certain point by continued bisection"!--No, for I could never tell whether my point
was a point of this kind. All I could tell would always be that I hadn't reached it. "But if I carry out the construction
of with absolutely exact drawing instruments, and then by bisection approximate to the point I get, I know that
this process will never reach the constructed point." But it would be odd if one construction could as it were
prescribe something to the others in this way! And indeed that isn't the way it is. It is very possible that
Page Break 472
the point I get by means of the 'exact' construction of is reached by the bisection after say 100 steps;--but in
that case we could say: our space is not Euclidean.
Page 472
The "cut at the irrational point" is a picture, and a misleading picture.
Page 472
A cut is a principle of division into greater and smaller.
Page 472
Does a cut through a length determine in advance the results of all bisections meant to approach the point of
the cut? No.
Page 472
In the previous example†1 in which I threw dice to guide me in the successive reduction of an interval by the
bisection of a length I might just as well have thrown dice to guide me in the writing of a decimal. Thus the
description "endless process of choosing between 1 and 0" does not determine a law in the writing of a decimal.
Perhaps you feel like saying: the prescription for the endless choice between 0 and 1 in this case could be
reproduced by a symbol like " ... ad. inf.". But if I adumbrate a law thus '0 • 001001001 ... ad inf.", what I
want to show is not the finite section of the series as a specimen of the infinite series, but rather the kind of regularity
to be perceived in it. But in " ... ad. inf." I don't perceive any law,--on the contrary, precisely that a law is
absent.
Page 472
(What criterion is there for the irrational numbers being complete? Let us look at an irrational number: it runs
through a series of rational approximations. When does it leave this series behind? Never. But then, the series also
never comes to an end.
Page 472
Suppose we had the totality of all irrational numbers with one single exception. How would we feel the lack
of this one? And--if
Page Break 473
it were to be added--how would it fill the gap? Suppose that it's π. If an irrational number is given through the totality
of its approximations, then up to any point taken at random there is a series coinciding with that of π. Admittedly for
each such series there is a point where they diverge. But this point can lie arbitrarily far 'out', so that for any series
agreeing with π I can find one agreeing with it still further. And so if I have the totality of all irrational numbers
except π, and now insert π I cannot cite a point at which π is now really needed. At every point it has a companion
agreeing with it from the beginning on.
Page 473
To the question "how would we feel the lack of π" our answer must be "if π were an extension, we would
never feel the lack of it". i.e. it would be impossible for us to observe a gap that it filled. But if someone asked us 'But
have you then an infinite decimal expansion with the figure m in the r-th place and n in the s-th place, etc?' we could
always oblige him.)
Page 473
"The decimal fractions developed in accordance with a law still need supplementing by an infinite set of
irregular infinite decimal fractions that would be 'brushed under the carpet' if we were to restrict ourselves to those
generated by a law." Where is there such an infinite decimal that is generated by no law? And how would we notice
that it was missing? Where is the gap it is needed to fill?
Page 473
What is it like if someone so to speak checks the various laws for the construction of binary fractions by
means of the set of finite combinations of the numerals 0 and 1?--The results of a law run through the finite
combinations and hence the laws are complete as far as their extensions are concerned, once all the finite
combinations have been gone through.
Page 473
If one says: two laws are identical in the case where they yield the same result at every stage, this looks like a
quite general rule. But in reality the proposition has different senses depending on
Page Break 474
what is the criterion for their yielding the same result at every stage. (For of course there's no such thing as the
supposed generally applicable method of infinite checking!) Thus under a mode of speaking derived from an
analogy we conceal the most various meanings, and then believe that we have united the most various cases into a
single system.
Page 474
(The laws corresponding to the irrational numbers all belong to the same type to the extent that they must all
ultimately be recipes for the successive construction of decimal fractions. In a certain sense the common decimal
notation gives rise to a common type.)
Page 474
We could also put it thus: every point in a length can be approximated to by rational numbers by repeated
bisection. There is no point that can only be approximated to by irrational steps of a specified type. Of course, that is
only a way of clothing in different words the explanation that by irrational numbers we mean endless decimal
fractions; and that explanation in turn is only a rough explanation of the decimal notation, plus perhaps an indication
that we distinguish between laws that yield recurring decimals and laws that don't.
Page 474
The incorrect idea of the word "infinite" and of the role of "infinite expansion" in the arithmetic of the real
numbers gives us the false notion that there is a uniform notation for irrational numbers (the notation of the infinite
extension, e.g. of infinite decimal fractions).
Page 474
The proof that for every pair of cardinal numbers x and y (x/y)2 ≠ 2 does not correlate with a single
type of number--called "the irrational numbers". It is not as if this type of number was constructed before I construct
it; in other words, I don't know any more about this new type of number than I tell myself.
Page Break 475
42
Kinds of irrational numbers
Page 475
(π' P, F)
π' is a rule for the formation of decimal fractions: the expansion of π' is the same as the expansion of π except
where the sequence 777 occurs in the expansion of π; in that case instead of the sequence 777 there occurs the
sequence 000. There is no method known to our calculus of discovering where we encounter such a sequence in the
expansion of π.
Page 475
P is a rule for the construction of binary fractions. At the nth place of the expansion there occurs a 1 or
a 0 according to whether n is prime or not.
Page 475
F is a rule for the construction of binary fractions. At the nth place there is a 0 unless a triple x, y, z
from the first 100 cardinal numbers satisfies the equation xn + yn = zn.
Page 475
I'm tempted to say, the individual digits of the expansion (of π for example) are always only the results, the
bark of the fully grown tree. What counts, or what something new can still grow from, is the inside of the trunk,
where the tree's vital energy is. Altering the surface doesn't change the tree at all. To change it, you have to penetrate
the trunk which is still living.
Page 475
I call "πn" the expansion of π up to the nth place. Then I can say: I understand what π'100 means, but not
what π' means, since π has no places, and I can't substitute others for none. It would be different if I e.g. defined the
division as a rule for the formation of decimals by division and the replacements of every 5 in the quotient
by a 3. In this case I am acquainted, for instance, with the number .--And if our calculus contains a method, a
law, to calculate the position of 777 in the expansion of π, then the law of π includes a mention of 777 and the law
can be altered by the substitution of 000 for 777. But in that case π' isn't the same as what
Page Break 476
I defined above; it has a different grammar from the one I supposed. In our calculus there is no question whether π
π' or not, no such equation or inequality. π' is not comparable with π. And one can't say "not yet comparable",
because if at some time I construct something similar to π' that is comparable to π, then for that very reason it will no
longer be π'. For π' like π is a way of denoting a game, and I cannot say that draughts is not yet played with as many
pieces as chess, on the grounds that it might develop into a game with 16 pieces. In that case it will no longer be
what we call "draughts" (unless by this word I mean not a game, but a characteristic of several games or something
similar; and this rider can be applied to π and π' too). But since being comparable with other numbers is a
fundamental characteristic of a number, the question arises whether one is to call π' a number, and a real number;
but whatever it is called the essential thing is that π' is not a number in the same sense as π. I can also call an interval
a point and on occasion it may even be practical to do so; but does it become more like a point if I forget that I have
used the word "point" with two different meanings?
Page 476
Here it is clear that the possibility of the decimal expansion does not make π' a number in the same sense as
π. Of course the rule for this expansion is unambiguous, as unambiguous as that for π or ; but that is no proof
that π' is a real number, if one takes comparability with rational numbers as an essential mark of real numbers. One
can indeed abstract from the distinction between the rational and irrational numbers, but that does not make the
distinction disappear. Of course, the fact that π' is an unambiguous rule for decimal fractions naturally signifies a
similarity between π' and π or ; but equally an interval has a similarity with a point etc. All the errors that have
been made in this chapter of the philosophy of mathematics are based on the confusion between internal properties
of a form (a rule as one among a list of rules) and what we call "properties" in everyday life (red as a property
Page Break 477
of this book). We might also say: the contradictions and unclarities are brought about by people using a single word,
e.g. "number", to mean at one time a definite set of rules, and at another time a variable set, like meaning by "chess"
on one occasion the definite game we play today, and on another occasion the substratum of a particular historical
development.
Page 477
"How far must I expand π in order to have some acquaintance with it?"--Of course that is nonsense. We are
already acquainted with it without expanding it at all. And in the same sense I might say that I am not acquainted
with π' at all. Here it is quite clear that π' belongs to a different system from π; that is something we recognize if we
keep our eyes on the nature of the laws instead of comparing "the expansions" of both.
Page 477
Two mathematical forms, of which one but not the other can be compared in my calculus with every rational
number, are not numbers in the same sense of the word. The comparison of a number to a point on the number-line
is valid only if we can say for every two numbers a and b whether a is to the right of b or b to the right of a.
Page 477
It is not enough that someone should--supposedly--determine a point ever more closely by narrowing down
its whereabouts. We must be able to construct it. To be sure, continued throwing of a die indefinitely restricts the
possible whereabouts of a point, but it doesn't determine a point. After every throw (or every choice) the point is still
infinitely indeterminate--or, more correctly, after every throw it is infinitely indeterminate. I think that we are here
misled by the absolute size of the objects in our visual field; and on the other hand, by the ambiguity of the
expression "to approach a point". We can say of a line in the visual field that by shrinking it is approximating more
and more to a point--that is, it is becoming more and more similar to a point. On the other hand when a Euclidean
line shrinks it does not become any more like a point; it always remains totally dissimilar, since its length, so to say,
Page Break 478
never gets anywhere near a point. If we say of a Euclidean line that it is approximating to a point by shrinking, that
only makes sense if there is an already designated point which its ends are approaching; it cannot mean that by
shrinking it produces a point. To approach a point has two meanings: in one case it means to come spatially nearer
to it, and in that case the point must already be there, because in this sense I cannot approach a man who doesn't
exist; in the other case, it means "to become more like a point", as we say for instance that the apes as they
developed approached the stage of being human, their development produced human beings.
Page 478
To say "two real numbers are identical if their expansions coincide in all places" only has sense in the case in
which, by producing a method of establishing the coincidence, I have given a sense to the expression "to coincide in
all places". And the same naturally holds for the proposition "they do not coincide if they disagree in any one place".
Page 478
But conversely couldn't one treat π' as the original, and therefore as the first assumed point, and then be in
doubt about the justification of π? As far as concerns their extension, they are naturally on the same level; but what
causes us to call π a point on the number-line is its comparability with the rational numbers.
Page 478
If I view π, or let's say , as a rule for the construction of decimals, I can naturally produce a modification
of this rule by saying that every 7 in the development of is to be replaced by a 5; but this modification is of
quite a different nature from one which is produced by an alteration of the radicant or the exponent of the radical
sign or the like. For instance, in the modified law I am including a reference to the number system of the expansion
which wasn't in the original rule for . The alternation of the law is of a much more fundamental kind than might
at first appear.
Page Break 479
Of course, if we have the incorrect picture of the infinite extension before our minds, it can appear as if appending
the substitution rule 7 → 5 to alters it much less than altering into , because the expansions of
are very similar to those of , whereas the expansion of deviates from that of from the
second place onwards.
Page 479
Suppose I give a rule ρ for the formation of extensions in such a way that my calculus knows no way of
predicting what is the maximum number of times an apparently recurring stretch of the extension can be repeated.
That differs from a real number because in certain cases I can't compare ρ - a with a rational number, so that the
expression ρ - a = b becomes nonsensical. If for instance the expansion of ρ so far known to me is 3 • 14 followed
by an open series of ones (3•1411 11...), it wouldn't be possible to say of the difference ρ - whether it was
greater or less than 0; so in this sense it can't be compared with 0 or with a point on the number axis and it and ρ
can't be called number in the same sense as one of these points.
Page 479
|The extension of a concept of number, or of the concept 'all', etc. seems quite harmless to us; but it stops
being harmless as soon as we forget that we have in fact changed our concept.|
Page 479
|So far as concerns the irrational numbers, my investigation says only that it is incorrect (or misleading) to
speak of irrational numbers in such a way as to contrast them with cardinal numbers and rational numbers as a
different kind of number; because what are called "irrational numbers" are species of number that are really
different--as different from each other as the rational numbers are different from each of them.|
Page 479
"Can God know all the places of the expansion of π?" would have been a good question for the schoolmen
to ask.
Page Break 480
Page 480
In these discussions we are always meeting something that could be called an "arithmetical experiment".
Admittedly the data determine the result, but I can't see in what way they determine it. That is how it is with the
occurrences of the 7s in the expansion of π; the primes likewise are yielded as the result of an experiment. I can
ascertain that 31 is a prime number, but I do not see the connection between it (its position in the series of cardinal
numbers) and the condition it satisfies.--But this perplexity is only the consequence of an incorrect expression. The
connection that I think I do not see does not exist. There is not an--as it were irregular--occurrence of 7s in the
expansion of π, because there isn't any series that is called the expansion of π. There are expansions of π, namely
those that have been worked out (perhaps 1000) and in those the 7s don't occur "irregularly" because their
occurrence can be described. (The same goes for the "distribution of the primes". If you give as a law for this
distribution, you give us a new number series, new numbers.) (A law of the calculus that I do not know is not a law).
(Only what I see is a law; not what I describe. That is the only thing standing in the way of my expressing more in
my signs that I can understand.)
Page 480
Does it make no sense to say, even after Fermat's last theorem has been proved, that F = 0 • 11? (If, say I
were to read about it in the papers.) I will indeed then say, "so now we can write 'F = 0 • 11'." That is, it is tempting to
adopt the sign "F" from the earlier calculus, in which it didn't denote a rational number, into the new one and now to
denote 0 • 11 with it.
Page 480
F was supposed to be a number of which we did not know whether it was rational or irrational. Imagine a
number, of which we do not know whether it is a cardinal number or a rational number. A description in the calculus
is worth just as much as this particular set of words and it has nothing to do with an object given by description
which may someday be found.
Page Break 481
Page 481
What I mean could also be expressed in the words: one cannot discover any connection between parts of
mathematics or logic that was already there without one knowing.
Page 481
In mathematics there is no "not yet" and no "until further notice" (except in the sense in which we can say
that we haven't yet multiplied two 1000 digit numbers together.)
Page 481
"Does the operation yield a rational number for instance?"--How can that be asked, if we have no method for
deciding the question? For it is only in an established calculus that the operation yields results. I mean: "yields" is
essentially timeless. It doesn't mean "yields, given time"--but: yields in accordance to the rules already known and
established.
Page 481
"The position of all primes must somehow be predetermined. We work them out only successively, but they
are all already determined. God, as it were, knows them all. And yet for all that it seems possible that they are not
determined by a law."--Always this picture of the meaning of a word as a full box which is given us with its contents
packed in it all ready for us to investigate.--What do we know about the prime numbers? How is the concept of them
given to us at all? Don't we ourselves make the decisions about them? And how odd that we assume that there must
have been decisions taken about them that we haven't taken ourselves! But the mistake is understandable. For we
use the expression "prime number" and it sounds similar to "cardinal number", "square number", "even number"
etc. So we think it will be used in the same way, and we forget that for the expression "prime number" we have
given quite different rules--rules different in kind--and we find ourselves at odds with ourselves in a strange
way.--But how is that possible? After all the prime numbers are familiar cardinal numbers--how can one say that the
concept of prime number is not a number concept in the same sense as the concept of cardinal number? But here
again we are tricked by the image of an "infinite extension" as an analogue to the familiar "finite "extension. Of
course the concept 'prime number'
Page Break 482
is defined by means of the concept 'cardinal number', but "the prime numbers" aren't defined by means of "the
cardinal numbers", and the way we derived the concept 'prime number' from the concept 'cardinal number' is
essentially different from that in which we derived, say, the concept 'square number'. (So we cannot be surprised if it
behaves differently.) One might well imagine an arithmetic which--as it were--didn't stop at the concept 'cardinal
number' but went straight on to that of square numbers. (Of course that arithmetic couldn't be applied in the same
way as ours.) But then the concept "square number" wouldn't have the characteristic it has in our arithmetic of being
essentially a part-concept, with the square numbers essentially a sub-class of the cardinal numbers; in that case the
square numbers would be a complete series with a complete arithmetic. And now imagine the same done with the
prime numbers! That will make it clear that they are not "numbers" in the same sense as e.g. the square numbers or
the cardinal numbers.
Page 482
Could the calculations of an engineer yield the result that the strength of a machine part in proportion to
regularly increasing loads must increase in accordance with the series of primes?
Page Break 483
43
Irregular infinite decimals
Page 483
"Irregular infinite decimals". We always have the idea that we only have to bring together the words of our everyday
language to give the combinations a sense, and all we then have to do is to inquire into it--supposing it's not quite
clear right away. It is as if words were ingredients of a chemical compound, and we shook them together to make
them combine with each other, and then had to investigate the properties of the compound, If someone said he
didn't understand the expression "irregular infinite decimals" he would be told "that's not true, you understand it
very well: don't you know what the words "irregular", "infinite", and "decimal" mean?--well, then, you understand
their combination as well." And what is meant by "understanding" here is that he knows how to apply these words
in certain cases, and say connects an image with them. In fact, someone who puts these words together and asks
"what does it mean" is behaving rather like small children who cover a paper with random scribblings, show it to
grown-ups, and ask "what is that?"
Page 483
"Infinitely complicated law", "infinitely complicated construction" ("Human beings believe, if only they hear
words, there must be something that can be thought with them").
Page 483
How does an infinitely complicated law differ from the lack of any law?
Page 483
(Let us not forget: mathematicians' discussions of the infinite are clearly finite discussions. By which I mean,
they come to an end.)
Page 483
"One can imagine an irregular infinite decimal being constructed by endless dicing, with the number of pips
in each case being a decimal place." But, if the dicing goes on for ever, no final result ever comes out.
Page Break 484
Page 484
"It is only the human intellect that is incapable of grasping it, a higher intellect could do so!" Fine, then
describe to me the grammar of the expression "higher intellect"; what can such an intellect grasp and what can't it
grasp and in what cases (in experience) do I say that an intellect grasps something? You will then see that describing
grasping is itself grasping. (Compare: the solution of a mathematical problem.)
Page 484
Suppose we throw a coin heads and tails and divide an interval AB in accordance with the following rule:
"Heads" means: take the left half and divide it in the way the next throw prescribes. "Tails" says "take the right half,
etc." By repeated throws I then
get dividing-points that move in an ever smaller interval. Does it amount to a description of the position of a point if
I say that it is the one infinitely approached by the cuts as prescribed by the repeated tossing of the coin? Here one
believes oneself to have determined a point corresponding to an irregular infinite decimal. But the description doesn't
determine any point explicitly; unless one says that the words "point on this line" also "determine a point"! Here we
are confusing the recipe for throwing with a mathematical rule like that for producing decimal places of . Those
mathematical rules are the points. That is, you can find relations between those rules that resemble in their grammar
the relations "larger" and "smaller" between two lengths, and that is why they are referred to by these words. The
rule for working out places of is itself the numeral for the irrational number; and the reason I here speak of a
"number" is that I can calculate with these signs (certain rules for the construction of rational numbers) just as I can
with rational numbers themselves. If I want to say similarly
Page Break 485
that the recipe for endless bisection according to heads and tails determines a point, that would have to mean that
this recipe could be used as a numeral, i.e. in the same way as other numerals. But of course that is not the case. If
the recipe were to correspond to a numeral at all, it would at best correspond to the indeterminate numeral "some",
for all it does is to leave a number open. In a word, it corresponds to nothing except the original interval.
Page Break 486
Page Break 487
Note in Editing
Page 487
In June 1931 Wittgenstein wrote a parenthesis in his manuscript book: "(My book might be called:
Philosophical Grammar. This title would no doubt have the smell of a textbook title but that doesn't matter, for
behind it there is the book.)" In the next four manuscript volumes after this he wrote nearly everything that is in the
present work. The second of these he called "Remarks towards Philosophical Grammar" and the last two
"Philosophical Grammar".
Page 487
The most important source for our text is a large typescript completed probably in 1933, perhaps some of it
1932. Our "Part II" makes up roughly the second half of this typescript. In most of the first half of it Wittgenstein
made repeated changes and revisions--between the lines and on the reverse sides of the typed sheets--and probably
in the summer of 1933 he began a "Revision" in a manuscript volume (X and going over into XI). This, with the
"Second Revision" (which I will explain), is the text of our Part I up to the Appendix.--Wittgenstein simply wrote
"Umarbeitung" (Revision) as a heading, without a date; but he clearly wrote it in 1933 and the early weeks of 1934.
He did not write the "second revision" in the manuscript volume but on large folio sheets. He He crossed out the text
that this was to replace, and showed in margins which parts went where. But it is a revision of only a part, towards
the beginning, of the first and principal "Revision". The passages from the second revision are, in our text, §§1-13
and §§ 23-43. The second revision is not dated either, but obviously it is later than the passages it replaces; probably
not later than 1934.
Page 487
So we may take it that he wrote part of this work somewhat earlier, and part at the same time as his dictation
of The Blue Book. Many things in the Blue Book are here (and they are better expressed). There are passages also
which are in the Philosophical Remarks and others later included in the Investigations. It would be easy to give the
reference and page number for each of these. We decided not to. This book should be compared with Wittgenstein's
earlier and later writings. But this means: the method and the development
Page Break 488
of his discussion here should be compared with the Philosophical Remarks and again with the Investigations. The
footnotes would be a hindrance and, as often as not, misleading. When Wittgenstein writes a paragraph here that is
also in the Remarks, this does not mean that he is just repeating what he said there. The paragraph may have a
different importance, it may belong to the discussion in a different way. (We know there is more to be said on this
question.)
Page 488
Wittgenstein refers to "my book" at various times in his manuscripts from the start of 1929 until the latest
passages of the Investigations. It is what his writing was to produce. The first attempt to form the material into a
book was the typescript volume he made in the summer of 1930--the Philosophical Remarks (published in German
in 1964). The large typescript of 1933--the one we mentioned as a source of this volume--looks like a book.
Everyone who sees it first thinks it is. But it is unfinished; in a great many ways. And Wittgenstein evidently looked
on it as one stage in the ordering of his material. (Cf. the simile of arranging books on the shelves of a library, in Blue
Book p. 44-45.)
Page 488
Most of the passages which make up the text of the 1933 typescript (called "213" in the catalogue) he had
written in manuscript volumes between July 1930 and July 1932; but not in the order they have in the typescript.
From the manuscript volumes he dictated two typescripts, one fairly short and the other much longer--about 850
pages together. There was already a typescript made from manuscripts written before July 1930--not the typescript
which was the Philosophical Remarks but a typescript which he cut into parts and sifted and put together in a
different way to make the Philosophical Remarks. He now used an intact copy of this typescript together with the
two later ones in the same way, cutting them into strips: small strips sometimes with just one paragraph or one
sentence, sometimes groups of paragraphs; and arranging them in the order he saw they ought to have. Groups of
slips in their order were clipped together to form 'chapters', and he gave each chapter a title. He then brought the
chapters together--in a definite order--to form 'sections'. He gave each section a title and arranged them also in a
definite order. In this order the whole was finally typed.--Later he made a table of contents out of the titles of
sections and chapter headings.
Page Break 489
Page 489
Certain chapters, especially, leave one feeling that he cannot have thought the typing of the consecutive copy
had finished the work barring clerical details. He now wrote, over and over again, between the lines of typescript or
in the margin: "Does not belong here", "Belongs on page... above", "Belongs to 'Meaning', § 9", "Goes with 'What is
an empirical proposition?'", "Belongs with § 14, p. 58 or § 89 p. 414", and so on. But more than this, about 350
pages--most of the first half of the typescript--are so written over with changes, additions, cancellations, questions
and new versions, that no one could ever find the 'correct' text here and copy it--saving the author himself should
write it over to include newer versions and make everything shorter.
Page 489
He now makes no division into chapters and sections. He has left out paragraph numbers and any suggestion
of a table of contents. We do not know why. (We do not find chapters or tables of contents anywhere else in
Wittgenstein's writings. He may have found disadvantages in the experiment he tried here.)--The extra spaces
between paragraphs and groups of paragraphs are his own; and he thought these important. He would have
numbered paragraphs, probably, as he did in the Investigations. But the numbers in Part I here are the editor's, not
Wittgenstein's. Neither is the division in chapters Wittgenstein's, nor the table of contents.--On the other hand, Part II
has kept the chapters and the table of contents which Wittgenstein gave this part of the typescript. Perhaps this
makes it look as though Part I and Part II were not one work. But we could not make them uniform in this (division
and arrangement of chapters) without moving away from Wittgenstein's way of presenting what he wrote. Anyone
who reads both parts will see connections.
Page 489
And the appendix may make it plainer. Appendices 5, 6, 7, 8 and the first half of 4 are chapters of 'typescript
213'. Appendix I, Fact and Complex, is also an appendix in Philosophical Remarks. But Wittgenstein had fastened it
together with appendices 2 and 3 and given them a consecutive paging as one essay; with what intention we do not
know. Each one of the eight appendices here discusses something connected with 'proposition' and with 'sense of a
proposition'. The whole standpoint is somewhat earlier (the manuscripts often bear earlier dates) than that of
Page Break 490
Part I here, but later than the Philosophical Remarks.--But the appendices also discuss questions directly connected
with the themes of 'generality' and 'logical inference' in Part II.
Page 490
Part I is concerned with the generality of certain expressions or concepts, such as 'language', 'proposition' and
'number'. For instance, § 70, page 113:
Page 490
"Compare the concept of proposition with the concept 'number' and then on the other hand with the concept
of cardinal number. We count as numbers cardinal numbers, rational numbers, irrational numbers, complex
numbers; whether we call other constructions numbers because of their similarities with these, or draw a definitive
boundary here or elsewhere, depends on us. In this respect the concept of number is like the concept of proposition.
On the other hand the concept of cardinal number |1, ξ, ξ + 1| can be called a rigorously circumscribed concept,
that's to say it's a concept in a different sense of the word."
Page 490
This discussion is closely related to the chapter on 'Kinds of Cardinal Numbers' and on '2 + 2 = 4' in Part II;
and with the section on Inductive Proof. These are the most important things in Part II.
London, 1969
Page 490
Page Break 491
Translator's Note
Page 491
Many passages in the Philosophical Grammar appear also in the Philosophical Remarks, the Philosophical
Investigations, and the Zettel. In these cases I have used the translations of Mr Roger White and Professor G. E. M.
Anscombe, so that variations between the styles of translators should not be mistaken for changes of mind on
Wittgenstein's part. Rare departures from this practice are marked in footnotes. Passages from the Philosophical
Grammar appear also in The Principles of Linguistic Philosophy of F. Waismann (Macmillan 1965): in these cases
I have not felt obliged to follow the English text verbatim, but I am indebted to Waismann's translator.
Page 491
Three words or groups of words constantly presented difficulties in translation.
Page 491
The German word "Satz" may be translated "proposition" or "sentence" or (in mathematical and logical
contexts) "theorem". I have tried to follow what appears to have been Wittgenstein's own practice when writing
English, by using the word "proposition" when the syntactical or semantic properties of sentences were in question,
and the word "sentence" when it was a matter of the physical properties of sounds or marks. But it would be idle to
pretend that this rule provides a clear decision in every case, and sometimes I have been obliged to draw attention in
footnotes to problems presented by the German word.
Page 491
From the Tractatus onward Wittgenstein frequently compared a proposition to a Maßstab. The German
word means a rule or measuring rod: when Wittgenstein used it is clear that he had in mind a rigid object with
calibrations. Finding the word "rule" too ambiguous, and the word "measuring-rod" too cumbersome, I have
followed the translators of the Tractatus in using the less accurate but more natural word "ruler".
Page 491
Translators of Wittgenstein have been criticised for failing to adopt a uniform translation of the word
"übersehen" and its derivatives, given the importance of the notion of "übersichtliche Darstellung" in Wittgenstein's
later conception of philosophy. I have
Page Break 492
been unable to find a natural word to meet the requirement of uniformity, and have translated the word and its
cognates as seemed natural in each context.
Page 492
Like other translators of Wittgenstein I have been forced to retain a rather Germanic style of punctuation to
avoid departing too far from the original. For instance, Wittgenstein often introduced oratio recta by a colon instead
of by inverted commas. This is not natural in English, but to change to inverted commas would involve making a
decision--often a disputable one--about where the quotation is intended to end.
Page 492
I have translated the text of the Suhrkamp-Blackwell edition of 1969 as it stands, with the exception of the
passages listed below in which I took the opportunity to correct in translation errors of transcription or printing
which had crept into the German text. The pagination of the translation, so far as practicable, matches that of the
original edition.
Page 492
I am greatly indebted to Professor Ernst Tugendhat, who assisted me in the first draft of my translation; and
to Mr John Thomas, Dr Peter Hacker, Mr Brian McGuinness, Professor G. E. M. Anscombe, Professor Norman
Malcolm, Professor G. H. von Wright, Mr Roger White, Dr Anselm Müller, Mr and Mrs J. Tiles and Mr R.
Heinaman who assisted me on particular points. My greatest debt is to Mr Rush Rhees, who went very carefully
through large sections of a draft version and saved me from many errors while improving the translation in many
ways. The responsibility for remaining errors is entirely mine.
Page 492
I am grateful to the British Academy for a Visiting Fellowship which supported me while writing the first
draft translation.
Oxford 1973
Page 492
Page Break 493
Corrections to the 1969 German Edition
Page 493
page 17 line 31 For "Gedanken" read "Gedanke".
Page 493
24 15 For "selten" read "seltsam".
Page 493
25 17 For "Vom Befehl" read "Von der Erwartung".
Page 493
43 23 For "Carroll's" read "Carroll's Gedicht".
Page 493
44 15 For "was besagt" read "was sagt".
Page 493
52 13 For "uns da" read "uns da etwas".
Page 493
52 27 For "geben" read "ergeben".
Page 493
61 22 For "kann" read "kann nun".
Page 493
72 4 There should be no space between the paragraphs.
Page 493
75 28 For "übergrenzt" read "überkreuzt".
Page 493
88 9 For "gezeichnet haben" read "bezeichnet haben, sie schon zur Taufe gehalten haben".
Page 493
92 27 For "nun" read "nun nicht".
Page 493
97 7 For "Jedem" read "jedem".
Page 493
107 5 For "sie" read "sie von".
Page 493
108 17 For "Körperlos" read "körperlos".
Page 493
119 8 For "nun" read "um".
Page 493
123 16 There should be no space between the paragraphs.
Page 493
147 21 For "daß" read "das".
Page 493
148 1 For "Zeichen" read "Zeichnen".
Page 493
151 21 For "schreibt" read "beschreibt".
Page 493
152 19 For "Auszahlungen" read "Auszahnungen".
Page 493
152 30 For "hervorgerufen" read "hervorzurufen".
Page 493
160 21 There should be a space between the paragraphs.
Page 493
160 15 For "Ciffre" read "Chiffre".
Page 493
163 30 There should be no space between the paragraphs.
Page 493
171 12 For "systematischen" read "schematischen".
Page 493
178 3 For "lügen:" read "lügen"".
Page 493
205 9 For "in der" read "der".
Page 493
215 3 For "Buches" read "Buches uns".
Page 493
221 8 For "Erkenntis" read "Erkenntnis".
Page Break 494
Page 494
page 244 line 20 For "folgen" read "Folgen".
Page 494
244 23 For "etwa" read "es dazu".
Page 494
244 24 For "den" read "dem".
Page 494
245 19 For "fy" read "fx".
Page 494
251 12 For "dem" read "den".
Page 494
251 13 For "mit" read "mir".
Page 494
253 The "t1" in the figure is misplaced.
Page 494
256 23 For "folge" read "folgte".
Page 494
260 23 For "weh" read "nicht weh".
Page 494
261 12 For "ist" read "ist da".
Page 494
268 1ff For 'f' read 'φ' passim.
Page 494
278 31 For "Fall von f(∃) ist," read "Fall von f(∃) ist. Und nun kann mans uns entgegenhalten:
Wenn er sieht, dass f(a) ein Fall von f(∃) ist,"
Page 494
286 6 For " " read " ".
Page 494
288 11 For "1 + 1 + 1 + 1" read "1 + 1 + 1 + 1+ 1".
Page 494
288 30 For "ψx" read "φx".
Page 494
292 5 For "wird" read "wird. Vom Kind nur die richtige Ausführung der Multiplikation verlangt
wird".
Page 494
294 13 For "uns" read "nun uns".
Page 494
296 12 For "Ich habe gesagt" read "Ich sagte oben".
Page 494
305 12 Insert new paragraph: "Wenn nachträglich ein Widerspruch gefunden wird, so waren
vorher die Regeln noch nicht klar und eindeutig. Der Widerspruch macht also nichts denn
er ist dann durch das Aussprechen einer Regel zu entfernen."
Page 494
313 8 For "Hiweisen" read "Hinweisen".
Page 494
316 21 For "unnötiges Zeichen für "Taut." geben" read "unnötiges--Zeichen für "Taut." gegeben".
Page 494
317 6 For "den" read "dem".
Page 494
317 15 For "Def." read " ".
Page 494
317 29 For "Sätze)." read "Sätze) und zwar eine richtige degenerierte Gleichung (den Grenzfall
einer Gleichung)."
Page 494
325 21 For "könne" read "könnte".
Page Break 495
Page 495
page 325 line 27 For "nenne" read "nennen".
Page 495
328 9 For "Schma" read "Schema".
Page 495
344 1 For "(x" read "(E x".
Page 495
344 9 For "(x)" read "(E x)".
Page 495
353 2 For "Cont." read "Kont."
Page 495
353 13 For "(∃)" read "(∃n)
Page 495
383 22 For "nur" read "nun".
Page 495
386 13 For "3n" read "3n2".
Page 495
388 11 For "the" read "der".
Page 495
393 13 For "keine" read "eine".
Page 495
393 27 For "Schreibmaschine" read "Schreibweise".
Page 495
398 25 For "1:2" read " ".
Page 495
405 1 Seventh to tenth words of title should be roman.
Page 495
405 5 For "die Kardinalzahlen" read "alle Kardinalzahlen".
Page 495
411 fn2 For "→" read "S.414".
Page 495
415 27 For "können." read "können. Niemand aber würde sie in diesem Spiel einen Beweis gennant
haben!"
Page 495
416 fn4 For "(b + 1" read "(b + 1)".
Page 495
431 20 For "((4 + 1) + 1" read "((4 + 1) + 1)".
Page 495
441 16 For "gesehn" read "gesehen".
Page 495
441 33 For "ρ||" read " ".
Page 495
451 4 For "3n + 7" read "3 + n = 7".
Page 495
453 32 For "(...)" read "(...)".
Page 495
456 35 For "Kann nicht" read "kann".
Page 495
460 7 For "könnten?" read "könnten? Unser Netz wäre also nicht fein genug?"
Page 495
461 13 For "listen" read "Listen".
Page 495
465 28 For "andere Mineralien" read "auch andere Fälle".
Page 495
471 29 For "erhöhten" read "erhaltenen".
Page 495
472 19 The symbol should read: " ".
Page 495
481 27 For "Quadratzahlen", gerade Zahlen" read ""Quadratzehlen", "gerade Zahlen".
Page 495
481 30 For "primzahl" read "Primzahl".
FOOTNOTES
Page 56
†1. Sophist, 261E, 262A. [I have replaced "kinds of word" which appears in the translation of the parallel
passages in Philosophical Investigations § 1 with "parts of speech", which appears to have been Wittgenstein's
preferred translation. I am indebted for this information to Mr. R. Rhees. Trs.]
Page 78
†1. Cf. p. 165f (Ed.)
Page 106
†1. The parallel passage in Zettel 606 is translated in a way that does not fit this context (Trs.)
Page 111
†1. Cf. p. 94
Page 112
†1. The same German word corresponds to "sentence" as to "proposition" (Tr.)
Page 128
†1. Cf. Philosophical Investigations § 521 (Ed.)
Page 137
†1. Plato: Theaetetus 189A. (I have translated Wittgenstein's German rather than the Greek original. Trs.)
Page 140
†1. A line has dropped out of the translation of the corresponding passage in Zettel (§63).
Page 141
†1. Philebus, 40A. (The Greek word in the context means rather "a word", "a proposition". Tr.)
Page 156
†1. p. 143 above.
Page 164
†1. Theaetetus, 189A (immediately before the passage quoted in §90).
Page 177
†1. Cf. Philosophical Investigations, I, §537 (Trs.)
Page 182
†1. [Earlier draft of the parenthesis]. (Something very similar to this is the problem of the nature and flow of
time).
Page 184
†1. In pencil in the MS: [Perhaps apropos of the paradox that mathematics consists of rules.]
Page 185
†1. A tells B that he has hit the jackpot in the lottery; he saw a box lying in the street with the numbers 5 and
7 on it. He worked out that 5 × 7 = 64--and took the number 64.
Page 185
B: But 5 × 7 isn't 64!
Page 185
A: I've hit the jackpot and he wants to give me lessons!
Page 202
†1. There appears to be something wrong with the German text here. Possibly Wittgenstein meant to write
"let this man be called 'N'" and inadvertently wrote a version which is the same as the one he is correcting. (Trs.)
Page 205
†1. I have here corrected an inadvertent transposition of "I" and "II" in Wittgenstein's German. (Trs.)
Page 210
†1. From the 1932(?) typescript where it appears as a chapter by itself.
Page 211
†1. From a later MS note book, probably written in summer 1936, some two years after the main text of this
volume.
Page 212
†1. Cf. p. 163.
Page 213
†1. Cf. Tractatus 2. 1513 (Editor).
Page 246
†1. Cf. Tractatus 5. 132 (Ed.).
Page 288
†1. Perhaps Wittgenstein inadvertently omitted a negation sign before the second quantifier. (Trs.)
Page 296
†1. According to Dr. C. Lewy Wittgenstein wrote in the margin of F. P. Ramsey's copy of the Tractatus at
6.02: "Number is the fundamental idea of calculus and must be introduced as such." This was, Lewy thinks, in the
year 1923. See Mind, July 1967, p. 422.
Page 308
†1. Philosophical Remarks, p. 119.
Page 315
†1. F. P. Ramsey, The Foundations of Mathematics, London 1931, p. 53.
Page 319
†1. The section does not mention arithmetic. It may be conjectured that it was never completed. (Ed.)
Page 325
†1. This is an allusion to the German academic custom of announcing a lecture for, say, 11.15 by scheduling
it "11.00 c.t." (Trs.)
Page 339
†1. For the explanation of this notation, see below, p. 343 f.
Page 340
†1. In the manuscript this paragraph is preceded by the remark: I can work out 17 + 28 according to the rules,
I don't need to give 17 + 28 = 45 (α) as a rule. So if in a proof there occurs the step from f(17 + 28) to f(45) I don't
need to say it took place according to (α); I can cite other rules of the addition table.
Page 340
But what is this like in the (((1) + 1) + 1) notation? Can I say I could work out e.g. 2 + 3 in it? And according
to which rules? It would go like this:
Page 340
{(1) + 1} + {((1) + 1) + 1} = (({(1) + 1} + 1) = {((((1) + 1) + 1) + 1}... σ
Page 347
†1. Thus according to the typescript. The manuscript reads "(∃3x).φx.(∃2x).ψx.Ind.: ⊃.(∃5x).φx ∨ ψx".
Page 364
†1. This paragraph is crossed out in the typescript.
Page 380
†1. Perhaps the problem is to find the number of ways in which we can trace the joins in this wall without
interruption, omission or repetition. Cf. Remarks on the Foundations of Mathematics p. 174.
Page 383
†1. Here and at one point further down, I have corrected a confusion in Wittgenstein's typescript between
"left" and "right" (Tr.).
Page 389
†1. p. 321.
Page 397
†1. Begründung der Elementaren Arithmetik von Th. Skolem, Skrifter utg. av. Vid.-Selsk i Kristiana 1923. I
Mat.-nat. K. No. 6. p. 5. Translated in van Heijenoort, From Frege to Gödel, Harvard University Press, 1967, pp.
302-333.
Page 398
†1. The dash underneath emphasizes that the remainder is equal to the dividend. So the expression becomes
the symbol for periodic division. (Ed.)
Page 401
†1. ? (x).φx. (Ed.)
Page 406
†1. Above, p. 397.
Page 408
†1. See the appendix on p. 446. Cf. also Philosophical Remarks, p. 194 n.
Page 411
†1. "The schema ρ"--or: the group of equations α, β and γ on p. 397. A little further on Wittgenstein refers to
the same group as "R", p. 414 below. Later on p. 433, he speaks again of "the rule R" as here on p. 408, where it is: a
+ (b + 1) = (a + b) + 1. (Ed.)
Page 411
†2. This, probably, refers to those equations on p. 408 to the right of the brackets, that is: a + (b + c) = (a + b)
+ c, a + 1 = 1 + a, a + b = b + a. BI, BII, BIII... will then be the complexes of equations left of the brackets. On the
meaning of the brackets, see below (Ed.)
Page 416
†1. "V" denotes a definition which will be given below, p. 441. In the manuscript that passage comes
somewhat earlier than the remark above. The passage runs: "And if we now settle by definition:
"ℑ" is mentioned in the context below. V here is a definition of ℑ). (Ed.)
Page 418
†1. The schema R above, p. 414. In the manuscript shortly after this schema there follows the remark:
"I have put a bracket } between α, β, γ and A, as if it was self-evident what this bracket meant.
One might conjecture that the bracket meant the same as an equals sign. Such a bracket, incidentally, might be put
between " " and ".
Page 419
†1. Earlier version:... the question "is that too right?"
Page 420
†1. See below, p. 426.
Page 420
†2. Compare the form of the rule R on p. 433 below. In the manuscript Wittgenstein introduced this
formulation thus: "Perhaps the matter will become clearer, if we give the following rule for addition instead of the
recursive rule 'a + (b + 1) = (a + b) + 1'
a + (1 + 1) = (a + 1) + 1
a + (1 + 1) + 1) = ((a + 1) + 1) + 1 [[sic, missing (]]
a + (((1 + 1) + 1) + 1) = (((a + 1) + 1) + 1) + 1
... etc....
Footnote Page Break 421
We write this rule in the form, |1, ξ, ξ + 1| thus
In the application of the rule R... a ranges over the series |1, ξ, ξ + 1|."
Page 421
He then says of this rule that it can be written also in the form S or in the form a + (b + 1) = (a + b) + 1.
Page 433
†1. Cf. footnote, p. 420.
Page 439
†1. The schema R as above on p. 414. Cf B on p. 397. (Ed.)
Page 440
†1. Cf. perhaps: Grundgesetze der Arithmetik, II, p. 114, 115 §§ 107, 108. Waismann cited excerpts from
these §§. (Wittgenstein und der Wiener Kreis, pp. 150-151). Cf here Wittgenstein's remarks on them (ibid. pp.
151-157). (Ed.)
Page 446
†1. Remarks taken from the Manuscript volume. We must not forget that Wittgenstein omitted them. Even in
the MS they are not set out together as they are here. (Ed.)
Page 448
†1. Above, p. 431.
Page 449
†1. To repeat, α is: a + (b + 1) = (a + b) + 1
A is: a + (b + c) = (a + b) + c. (Ed.)
Page 454
†1. (Remark in the margin in pencil.) A defence, against Hardy, of the decimal system in proofs, etc.
Page 455
†1. Philosophical Remarks, 122, pp. 143-145.
Page 464
†1. Cf. The Foundations of Arithmetic, §84. (Ed.)
Page 469
†1. Grundgesetze d. Arithmetik, II, § 83, pp. 93, 94.
Page 472
†1. See below, p. 484
The Big Typescript
TS 213
WBTA01 30/05/2005 16:12 Page i
The Big Typescript
TS 213
Ludwig Wittgenstein
Kritische zweisprachige Ausgabe
Deutsch–Englisch
herausgegeben und übersetzt
von
C. Grant Luckhardt und Maximilian A. E. Aue
WBTA01 30/05/2005 16:12 Page 2
The Big Typescript
TS 213
Ludwig Wittgenstein
German–English Scholars’ Edition
edited and translated
by
C. Grant Luckhardt and Maximilian A. E. Aue
WBTA01 30/05/2005 16:12 Page 3
© 2005 by the Trustees of the Wittgenstein Estate
BLACKWELL PUBLISHING
350 Main Street, Malden, MA 02148-5020, USA
9600 Garsington Road, Oxford OX4 2DQ , UK
550 Swanston Street, Carlton, Victoria 3053, Australia
The right of C. Grant Luckhardt and Maximilian A. E. Aue to be identified as the Authors of the Editorial
Material in this Work has been asserted in accordance with the UK Copyright, Designs, and Patents Act 1988.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted,
in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted
by the UK Copyright, Designs, and Patents Act 1988, without the prior permission of the publisher.
First published 2005 by Blackwell Publishing Ltd
1 2005
Library of Congress Cataloging-in-Publication Data
Wittgenstein, Ludwig, 1889–1951.
[Big Typescript, TS 213. English & German]
The Big Typescript, TS 213 / Ludwig Wittgenstein ; edited and translated by C. Grant Luckhardt and
Maximilian A. E. Aue.— German–English scholars’ ed.
p. cm.
English and German.
Includes bibliographical references and index.
ISBN-13: 978-1-4051-0699-3 (hardcover : alk. paper)
ISBN-10: 1-4051-0699-9 (hardcover : alk. paper)
1. Semantics (Philosophy) 2. Logic, Symbolic and mathematical. 3. Mathematics—Philosophy. I.
Luckhardt, C. Grant, 1943– II. Aue, Maximilian A. E. III. Title.
B3376.W563B4713 2005
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A catalogue record for this title is available from the British Library.
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WBTA01 30/05/2005 16:12 Page 4
Wittgensteins „Big Typescript“ war Herrn Professor G. H. von Wright ein
besonderes Anliegen. Er hat unser Vorhaben stets unterstützt. Dieser Band ist
seinem Andenken agewidmet.
To the memory of Professor G. H. von Wright, who supported our efforts and
who was eager for this book to appear.
WBTA01 30/05/2005 16:12 Page 5
Einleitung der Herausgeber
Der Text
Vom sogenannten „Big Typescript“1
gibt es drei Versionen. Die erste ist der unkorrigierte
Durchschlag des Typoskripts, das von einem „Typisten“,2
wahrscheinlich im Sommer 1933,
hergestellt wurde.3
Die zweite Version – die wir im folgenden zusammen mit einer englischen
Übersetzung im en face Format herausgeben – hat das von Wittgenstein mit vielen
handschriftlichen Zusätzen und Änderungen versehene Original zur Grundlage. Die Arbeit
an diesen Zusätzen und Änderungen erstreckte sich wohl von unmittelbar nach der
Fertigstellung des Typoskripts bis ins Jahr 1937. (Die dritte Version ist bis auf einige handschriftliche
Bemerkungen, die nach Wittgensteins Tod von G. H. von Wright und G. Kreisel
den Grundlagen der Mathematik, dem letzten Teil des „Big Typescripts“, hinzugefügt wurden,
mit der zweiten identisch.)
Für das Typoskript gibt es viele Quellen. Am 18. 1. 1929 kehrte Wittgenstein nach Cambridge
zurück, um als „fortgeschrittener Student“ sein Studium wiederaufzunehmen.4
Rund
zwei Wochen danach begann er, ein Kontobuch mit Notizen zu füllen, denen er die Überschrift
„I. Band, Philosophische Bemerkungen“ gab. Die erste Eintragung lautet: „Ist ein
Raum denkbar, der nur alle rationalen aber nicht die irrationalen Punkte enthält? Und das heißt
nur: Sind die irrationalen Zahlen nicht in den rationalen präjudiziert?“ Da diese Bemerkung
später unverändert in das „Big Typescript“ übernommen wurde, kann man sagen, daß die
Arbeit daran 1929, mit Wittgensteins Rückkehr zur Philosophie, einsetzt.
Zwischen diesem Eckdatum und 1933 wuchs das Corpus dieser Bemerkungen – verteilt
auf weitere 9 „Bände“ und 4 Taschennotizbücher – auf ca. 3292 Seiten an. Aus diesen 14
Manuskripten wurden zunächst durch Auswahl, Korrektur und Zusätze vier Typoskripte.5
1 Von wem dieser Name stammt, ist uns
unbekannt. Dazu befragt, haben die Professoren
Elizabeth Anscombe, Norman Malcolm und
Georg Henrik von Wright übereinstimmend
ausgesagt, sie wüßten es nicht.
2 Siehe unten, S. 204, Fußnote 143.
3 Die erste Version erschien als 11. Band der
Wiener Ausgabe von Wittgensteins Werken, (hg.
v. Michael Nedo, Wien: Springer-Verlag, 2000).
4 Man weiß das deswegen so genau, weil John
Maynard Keynes in einem mit diesem Datum
datierten Brief an seine Schwester schrieb:
„Well, God has arrived. I met him on the
5.15 train.“ (Zit. nach: Ray Monk, Ludwig
Wittgenstein: The Duty of Genius, New York:
Penguin Books, 1990, S. 255).
5 Im von von Wright erstellten Werkverzeichnis
sind diese Bände folgendermaßen numeriert:
die zehn von Wittgenstein mit „I“ bis „X“
bezeichneten Manuskriptbände tragen die
Nummern 105–14, die vier Taschennotizbücher
die Nummern 153a, 153b, 154 und 155, die vier
Typoskripte (TS) die Nummern 208, 209, 210
und 211. Die Manuskriptbände I–IV wurden
zuerst geschrieben – zwischen 1929 und 1930. Die
Bände I–III und die erste Hälfte von Band IV
bilden die Grundlage des 1930 entstandenen
TS 208. 1930 entstand aus Band III und Teilen
von Band IV das TS 209, das Wittgenstein mit
der Überschrift „Philosophische Bemerkungen“
versah. Gleichfalls in diesem Jahr entstand
auch das TS 210, zusammengestellt aus dem
vi
WBTA01 30/05/2005 16:12 Page 6
Editors’ and Translators’ Introduction
The Text
“The Big Typescript”, as it has come to be called,1
exists in three versions. The first is the
unmarked carbon-copy of the text, as it was produced by a typist,2
probably in the summer
of 1933.3
The second and third versions consist of the top copy of the typed text, on which
Wittgenstein made numerous handwritten additions and changes, probably beginning
almost immediately after it was typed and continuing into 1937. It is the second version that
we are publishing here, together with an en face English translation. (The third version differs
from the second in containing some handwritten annotations in the mathematics section entered
after Wittgenstein’s death by G. H. von Wright and G. Kreisel.)
The sources for the typescript are numerous. Wittgenstein returned to Cambridge on 18
January 1929, to resume his studies, studying for a PhD as an “Advanced Student”.4
Some
two weeks later, on 2 February, he began writing in a blank accounts book, which he titled
“Volume I, Philosophical Remarks”. The first entry reads: “Is a space conceivable that contains
only all rational points but not the irrational ones? And that only means: Aren’t the
irrational numbers prejudged in the rational ones?” In so far as this remark is later incorporated
without change into “The Big Typescript”, it can be said that this work “begins”
at the time of his return to philosophy in 1929.
In the intervening years Wittgenstein produced nine more so-called “Volumes” and four
sets of pocket notebooks, totalling some 3,292 pages of handwritten remarks. He made
selections from all fourteen of these manuscripts, editing and adding to them as he went
along, to produce four typescripts.5
These typescripts were then cut into bits, edited, added
1 We do not know who gave it this name. Professors
Elizabeth Anscombe, Norman Malcolm
and Georg Henrik von Wright have all said (in
private conversations) that they did not know.
2 We infer that this was a man, from Wittgenstein’s
use of the masculine noun in footnote 114,
p. 204e.
3 This version has been edited by Michael Nedo
and published as Volume 11 of the Wiener
Ausgabe (Vienna: Springer-Verlag, 2000).
4 This date is based on a letter of the same date from
John Maynard Keynes to his sister, in which he
remarks, “Well, God has arrived. I met him on
the 5.15 train” (quoted in Ray Monk’s Ludwig
Wittgenstein: The Duty of Genius, New York:
Penguin Books, 1990, p. 255).
5 The numbers for these volumes in the von
Wright catalogue are as follows: the ten manuscript
volumes (numbered by Wittgenstein I–X)
are numbers 105–14, the four pocket notebooks
are 153a, 153b, 154 and 155, and the typescripts
are numbers 208, 209, 210, and 211. Manuscript
volumes I–IV (105–8) were the first to be
written, between 1929 and 1930. In 1930 Wittgenstein
based TS 208 on the first 3 of these
volumes plus the first half of the fourth. TS 209,
to which he gave the title “Philosophical
Remarks”, was based on the third and part of the
fourth and was typed in 1930. TS 210 was
created out of the second half of Volume IV, also
in 1930. Volumes V to the first part of Volume
X were written between 1930 and 1932, and
TS 211, produced in 1931/2, was based on
those volumes. Volumes VI–IX were in turn
based on the four pocket notebooks, which were
written in 1931/2.
vie
WBTA01 30/05/2005 16:12 Page 7
Diese Typoskripte wurden dann durch Zerschneiden und Neuordnen, durch Umformulierungen
und Zusätze zu einem neuen Typoskript, dem TS 212. Die Bemerkungen darin
sind Kapiteln mit schlagzeilenartigen Kapitelüberschriften zugeordnet, und diese Kapitel sind
ihrerseits wieder zu größeren, gleichfalls mit Überschriften versehenen Abschnitten zusammengefaßt.
Das „Big Typescript“ entstand dann, als Wittgenstein das TS 212 1933 unter
Einbeziehung weiterer Änderungen und Zusätze und unter Hinzufügung eines Inhaltsverzeichnisses
von einem Typisten kopieren ließ. Angesichts dieser Entstehungsgeschichte
verwundert es nicht, daß die Abfolge der Bemerkungen im „Big Typescript“ eine gänzlich
andere ist als in den ihm zugrundeliegenden Manuskripten. Beispielsweise findet sich die
erste Bemerkung von Manuskriptband I erst ganz am Ende des „Big Typescripts“, auf
S. 738 (Siehe S. 489 dieser Ausgabe).
Bald, vielleicht unmittelbar nach Fertigstellung des „Big Typescripts“, begann Wittgenstein
mit der Revision. Manche der zwischen doppelten Schrägstrichen plazierten alternativen Worte
und Satzteile strich er aus, andere blieben unberührt; er strich auch getippte und handgeschriebene
Worte, Wortgruppen, Sätze, Absätze und Bemerkungen aus, setzte handschriftlich
neue Alternativen ein, korrigierte Tippfehler, trennte, bzw. verband Buchstaben, Absätze
und Bemerkungen, zeigte mit Hilfe von Pfeilen und anderen Verweiszeichen an, wohin er
einzelne Bemerkungen verlegt haben wollte, entwarf sowohl auf den Vorder- als auch auf
den Rückseiten der getippten Blätter neue Bemerkungen, notierte sich Fragen und setzte
Randbemerkungen und Randzeichen (Schrägstriche, Haken, Fragezeichen und – als Zeichen
für „schlecht“ – langgestreckte „s“–Zeichen) ein.
Es hat den Anschein, daß die Arbeit an diesen Zusätzen den größeren Teil des Jahres
1937 in Anspruch nahm. In einer chiffrierten Tagebucheintragung vom 23. 10. dieses Jahres
(MS. 119, S. 79r) schreibt Wittgenstein: „Fing an meine alte Maschinschrift anzusehen und
den Weizen vom Spreu zu sondern“. Drei Tage danach heißt es:
Schreibe jetzt nicht mehr, sondern lese nur den ganzen Tag meine Maschinschrift und mache
Zeichen zu jedem Absatz. Es ist viel denken hinter diesen Bemerkungen. Aber brauchbar
für ein Buch sind doch nur wenige ohne Umarbeitung, aus verschiedenen Gründen. Ich
habe jetzt beinahe ein Viertel des Ganzen durchgesehen. Wenn es also glatt geht, könnte
ich in ca. 6 Tagen damit fertig sein. Aber was dann? Nun, versuchen das brauchbare zu
sammeln. – Freilich, das ist sehr schwer! und ich dachte heute manchmal, es werde vielleicht
für mich bedeuten, von hier wegzugehen, etwa zu Drury, so Gott will. Denn ich weiß
nicht ob ich diese Arbeit in dieser Einsamkeit machen kann. Aber es wird sich alles zeigen.
Wahrscheinlich fand diese Sonderung des Spreus vom Weizen Ende 1937 statt und kam
im ersten Teil von Manuskriptband XII (MS 116) zum Ausdruck.6
In die ersten 135 Seiten
dieses Manuskripts ist viel von den ersten 196 Seiten des „Big Typescripts“ eingegangen.
Manuskriptband XII setzt mit der ersten Bemerkung des „Big Typescripts“ ein, und es
folgen dann – mehr oder weniger in der Reihenfolge des „Big Typescripts“ – viele weitere,
oft wörtlich übernommene Bemerkungen. Einige wurden freilich geändert und es kam auch
neues Material hinzu.
vii EINLEITUNG DER HERAUSGEBER
Material des zweiten Teils von Band IV. Das
1931/32 entstandene TS 211 hat die Manuskriptbände
V–IX und den ersten Teil von
Band X – geschrieben zwischen 1930 und 1932
– zur Grundlage. Die Manuskriptbände VI
bis einschließlich IX beruhen ihrerseits auf
den vier Taschennotizbüchern, die 1931/32
entstanden.
6 Belege für diese Annahme finden sich bei G. P.
Baker und P. M. S. Hacker in Wittgenstein:
Understanding and Meaning, Volume 1 of An
Analytical Commentary on the Philosophical Investigations,
Part II – Exegesis §§1–184, Second
Edition, G. P. Baker and P. M. S. Hacker, Extensively
Revised by P. M. S. Hacker (Oxford:
Blackwell, 2004).
WBTA01 02/06/2005 14:32 Page 8
to and collected together. He grouped remarks together to form chapters, to which he gave
titles – often in the style of newspaper headlines – and groups of chapters were collected as
sections, which he also titled. It is from this collection of clippings – TS 212 – that Wittgenstein
produced “The Big Typescript” in 1933, making changes and additions as he read from the
clippings and adding a Table of Contents. It is important to note that owing to this method
of composition, the remarks in this work do not occur in anything that resembles the order
in which they were written in the predecessor manuscripts. The first entry in Volume I, for
example, does not appear in “The Big Typescript” until very near the end – page 489 of
this edition.
Soon – perhaps immediately – after the original version of “The Big Typescript”
emerged from the typewriter, Wittgenstein set about revising its pages – striking through
some alternative words and phrases he had had typed in between sets of double slashes,
leaving other alternatives unmarked, striking through and crossing out both typed and
handwritten words, sentences, phrases, paragraphs within remarks and remarks, adding new
handwritten alternatives, correcting typographical errors, inserting and deleting spaces
between letters, paragraphs and remarks, drawing arrows or adding symbols and other directions
to indicate where he wanted remarks moved, composing new remarks on both the front
and reverse sides of the typed pages, asking questions and making marginal comments, and
entering marginal marks such as slashes, ticks, question marks, and Schlecht marks (∫ – indicating
that he thought a passage bad).
It appears that some of these additions were being made well into 1937. In a coded diary
entry on 23 October of that year (MS 119, p. 79r) he says he “[b]egan to look at my old
typescript and to separate the wheat from the chaff”. Three days later he writes:
I’m not writing any more now, but spent the entire day reading my typescript and making
marks on each paragraph. There’s a great deal of thinking behind these remarks. But
without revision they’re not of much use for a book, for various reasons. So far I’ve looked
through almost a quarter of the whole thing. If things go this smoothly I could be finished
with it in about 6 days. But what then? Well, to try to gather together what is useful. –
Of course that’s very difficult! And today I thought perhaps it would mean leaving here,
perhaps to go to Drury’s, God willing. For I don’t know whether I can do this work in
solitude. But we’ll see.
It seems likely that this process of separating the wheat from the chaff occurred in late
1937 and resulted in what is contained in the first part of Volume XII (MS 116).6
In the
first 135 pages of that manuscript Wittgenstein includes a great deal of material from
the first 196 pages of “The Big Typescript”. Volume XII begins with the first remark in
“The Big Typescript” and continues with many verbatim remarks in more or less the same
order as they appear there. A few are changed, and new material is added.
EDITORS’ AND TRANSLATORS’ INTRODUCTION viie
6 This is well documented in G. P. Baker and
P. M. S. Hacker, Wittgenstein: Understanding and
Meaning, Volume 1 of An Analytical Commentary
on the Philosophical Investigations, Part II:
Exegesis §§1–184, Second Edition, G. P. Baker and
P. M. S. Hacker, Extensively Revised by P. M. S.
Hacker (Oxford: Blackwell Publishing, 2004).
WBTA01 02/06/2005 14:32 Page 9
Manuskriptband XII ist allerdings nur einer von drei Versuchen, in neuen Manuskripten
am Material aus dem „Big Typescript“ weiterzuarbeiten. Die anderen beiden Versuche, hier
als (1) und (2) gekennzeichnet, setzen gleichfalls mit der ersten Bemerkung des „Big
Typescripts“ ein, entwickeln sich dann aber in unterschiedliche Richtungen:
(1) ist der längere dieser Versuche. Er umfaßt den Manuskriptband X (MS 114) ab S. 31
und findet dann im Manuskriptband XI (MS 115) von S. 1 bis einschließlich. S. 117
seine Fortsetzung. Die erste Seite des Manuskriptbands XI ist auf den 14. 12. 1933
datiert. Diese Bände X und XI enthalten viel dem „Big Typescript“ entstammendes,
zum Teil revidiertes Material. Punkto Anordnung unterscheidet sich dieses Material
aber beträchtlich von dem im „Big Typescript“: Die übernommenen Bemerkungen
sind vielfach mit neuem Material untermischt. Mit Ausnahme einiger Seiten, die
Wittgenstein aus dem TS 211 ausgeschnitten und hier eingeklebt hatte, sind alle
Bemerkungen mit der Hand geschrieben.7
Nach Fertigstellung dieser Revision versah sie Wittgenstein (auf S. 1) mit der Überschrift
„Umarbeitung“. Darunter schrieb er: „Zweite Umarbeitung im großen Format“,
ein Hinweis auf den obenerwähnten zweiten Versuch, Material aus dem „Big
Typescript“ umzuarbeiten.
(2) Das Große Format (MS 140) ist also der zweite Ansatz zur Umarbeitung von Material,
das dem „Big Typescript“ entnommen wurde. Er erfolgte möglicherweise zur gleichen
Zeit wie der erste oder kurz danach. Dieses Manuskript ist kurz – 39 Seiten lang –
und befaßt sich vor allem mit den Themen Verstehen und Verwendung von Worten,
Sätzen und Sprache. Wieder sind die Bemerkungen, die dem „Big Typescript“ unter
Einhaltung ihrer Abfolge entnommen sind, mit neuem Material untermischt, darunter
auch mit solchem aus TS 212, das nicht ins „Big Typescript“ eingegangen ist. In
den beiden Versuchen zur Umarbeitung (1) und (2) geht aus Hinweisen hervor, daß
Wittgenstein Materialien aus dem einen in den jeweils anderen übertragen wollte.
Das „Big Typescript“ ist als Ursprung vieler Bemerkungen, die in späteren Manuskripten
einer strengen Auswahl unterzogen wurden, von großem Nutzen. Es ist eine Quelle
vieler Textstellen in den diversen Versionen der Philosophischen Untersuchungen und nimmt
thematisch viel von dem vorweg, was in späteren Schriften zum Ausdruck kommt.
Editorische Richtlinien
Wie bereits erwähnt, ist die zweite Version des „Big Typescripts“, die wir hiermit veröffentlichen,
mit unzähligen Änderungen versehen, und wir möchten im folgenden erklären,
wie wir sie dargestellt haben.
Unser übergreifendes Ziel war es, den Text mit seinen Varianten zugänglich zu machen,
ohne seine Lesbarkeit wesentlich zu beeinträchtigen. Infolgedessen haben wir im Fall von
Varianten jeweils nur eine in den Text aufgenommen und alle anderen – stehengelassene
und ausgestrichene – in die Fußnoten versetzt. Meist haben wir uns für die Variante
entschieden, die wir für die zuletzt eingetragene hielten. In einigen Fällen haben wir uns
aber von grammatischen Kriterien leiten lassen. D.h. dort, wo Wittgenstein die durch eine
spätere Variante bedingten grammatischen Unstellungen nicht durchführte, haben wir eine
viii EINLEITUNG DER HERAUSGEBER
7 Das eingeklebte Material befindet sich auf den
Seiten 95–100 des Manuskriptbands XI. Die
gleichen Seiten finden sich auch – mit neuer
Numerierung – im „Big Typescript“.
WBTA01 30/05/2005 16:12 Page 10
Volume XII is in fact only one of three attempts to rework material from “The Big
Typescript” in successor manuscripts. Like Volume XII, the other two also begin with
the first remark of “The Big Typescript” and then proceed in various directions. These
volumes are:
(1) Volume X (von Wright no. 114), beginning on page 31 and continuing for 197 pages
to the end of that volume, and resuming in Volume XI (no. 115) and continuing up
to page 117. The first page of Volume XI is dated 14 December 1933. Volumes X
and XI contain much material from “The Big Typescript”, some of it revised, but
the order in which it appears is quite different from its order in “The Big Typescript”.
The remarks are copiously interspersed with new material. All of the remarks are in
handwriting, except for several pages Wittgenstein pasted in from TS 211.7
Some time
after he had completed this revision he went back to the first page and wrote
“Reworking” as its title. Under that he wrote “Second reworking in the Großes Format”,
which refers to his next effort to rework material from “The Big Typescript”.
(2) The Großes Format (no. 140). This is a second reworking of a selection of material
from “The Big Typescript”, possibly done concurrently with the first reworking,
or soon thereafter. It is short – 39 pages – and is centred on the themes of the
understanding and use of words, propositions and language. It includes references to
(1), in which directions are given for inserting material from this reworking into that
one, and vice versa. New material, including some remarks from TS 212 not included
in “The Big Typescript”, is interspersed among the remarks from “The Big
Typescript”, but the latter appear in the same order as in the original.
“The Big Typescript” is a fruitful source for many less inclusive selections of remarks in
Wittgenstein’s later manuscripts. It is a source for many of the passages in the various versions
of Philosophical Investigations, and its themes forecast much of what is to follow in his
writings.
Editing the Text
The version of “The Big Typescript” that we are presenting here is rife with Wittgenstein’s
alterations, and it may interest the reader to know how we have dealt with them.
We have been guided in general by the desire to present a scholarly yet readable text –
i.e. one whose alterations are accessible to the scholar, but that does not require the reader
to choose among multiple alternatives in order to create intelligible sentences. As a result
we have selected among (sometimes numerous) unrejected alternatives (i.e. ones that
Wittgenstein did not strike through) to present a coherent set of remarks in the main body
of the text, and we have included the remainder (including rejected alternatives) in footnotes.
Most often we have selected for the main body what we took to be the last unrejected
alternative Wittgenstein entered into the manuscript. When this was not our choice, we often
EDITORS’ AND TRANSLATORS’ INTRODUCTION viiie
7 The pasted material occurs between pages 95 and
100 of Volume XI and consists of pages that were
themselves included and renumbered in “The Big
Typescript”.
WBTA01 30/05/2005 16:12 Page 11
frühere gewählt, die grammatisch mit den anderen Teilen der betreffenden Bemerkung übereinstimmt.
Manchmal fiel unsere Wahl auch auf eine frühere Variante, weil eine spätere in
eine Richtung zu führen schien, die von Wittgenstein an dieser Stelle nicht weiterverfolgt
wurde. Wir haben uns jedenfalls bemüht, unsere Entscheidungen strengeren Kriterien zu
unterwerfen als denen, die Wittgenstein aufgestellt haben soll. Auf die Frage Frau Professor
Anscombes hin, wie sie bei der Herausgabe seines Werkes hinsichtlich seiner Varianten verfahren
solle, soll er gesagt haben: „Toss a coin“.8
Neben den Varianten finden sich im „Big Typescript“ viele handschriftliche Zusätze verschiedener
Länge: Worte, Sätze, Absätze und ganze Bemerkungen. In den meisten Fällen
ist es klar, wo diese hingehören, bei den zusätzlichen handschriftlichen Bemerkungen aber
manchmal nicht. Im Fall der Bemerkungen, die auf den (leeren) Rückseiten der getippten
Blätter stehen, haben wir angenommen, daß sie zu den ihnen jeweils gegenüberliegenden
(getippten) Bemerkungen auf der Vorderseite des nächsten Blattes gehören und dementsprechend
plaziert. Weitere handschriftliche Bemerkungen, oft allgemeiner Art und eher einen
Kommentar zum Thema eines Kapitels als eine Auseinandersetzung mit spezifischen
Anliegen darstellend, finden sich bei den Kapitelüberschriften. In der Regel setzen diese
Bemerkungen unmittelbar unter der Kapitelüberschrift, aber oberhalb der ersten (getippten)
Bemerkung ein, um dann, wenn da der Platz knapp wurde, oberhalb der Kapitelüberschrift
fortgesetzt zu werden. In diesen Fällen weichen wir von einer einfachen Wiedergabe der
Seite „von oben nach unten“ ab und geben zuerst die Kapitelüberschrift, dann die sie
umgebende(n) handschriftliche(n) Bemerkung(en) in der oben erörterten Abfolge und dann
erst die erste getippte Bemerkung.
Die allen unseren Fußnoten vorangehenden Buchstaben in runden Klammern dienen zur
genaueren Kennzeichnung des jeweils folgenden Materials. (V) bedeutet, daß die Fußnote
eine Variante oder mehrere Varianten (eines Wortes, Satzes oder einer Bemerkung) anführt.
Von Wittgenstein durchgestrichene Varianten sind in den Fußnoten durch eine einfache waagrechte
Durchstreichung gekennzeichnet. Folgt auf das (V) ein einziges Wort, ist es eine
Alternative zu dem Wort im Text, das der Fußnote unmittelbar vorausgeht. Im Fall von
mehreren auf das (V) folgenden Worten haben wir diese um die ihnen im Text unmittelbar
vorausgehenden ein oder zwei Worte erweitert, damit deutlich wird, an welcher Stelle die
Variante einzusetzen ist. (M) bedeutet, daß das darauffolgende Material eine (meist handschriftlich
eingesetzte) Randbemerkung ist. (O) deutet auf Tippfehler, aber auch auf einige
orthographische Fehler im Original hin, die wir in unserem Text korrigiert haben. Wo die
Setzung von einfachen, bzw. doppelten Anführungszeichen im Original nicht den einschlägigen
Regeln entsprach, haben wir sie stillschweigend umgestellt. Soweit es dem Verständnis nicht
abträglich war, haben wir aber sonst Wittgensteins oft eigenwillige, unorthodoxe Schreibweise
– sowohl was Orthographie (etwa Groß- und Kleinschreibung) als auch Interpunktion
(insbesondere Beistrichsetzung) betrifft – in unseren Text übernommen. Wohl wegen der
englischen Tastatur der Schreibmaschine, auf der das „Big Typescript“ getippt wurde, sind
die Buchstaben ß, Ä, Ö und Ü jeweils durch ss, Ae, Oe, und Ue ersetzt worden. In unserem
Text haben wir das nicht nachvollzogen. Im „Big Typescript“ finden sich an zahlreichen
Stellen Verweiszeichen. Sie zeigen Umstellungen an, die Wittgenstein im Zuge seiner Überarbeitung
dieses Textes ins Auge faßte. Wir haben sie mit (R) gekennzeichnet. Unseren
ix EINLEITUNG DER HERAUSGEBER
8 Siehe Michael Nedo (Hg.), Ludwig Wittgenstein,
Wiener Ausgabe, Einführung – Introduction (Wien:
Springer-Verlag, 1993) S. 75.
WBTA01 30/05/2005 16:12 Page 12
made it on the basis of grammatical coherence. That is to say, when a later alternative required
grammatical changes in the rest of the sentence that Wittgenstein had neglected to make,
we chose an earlier alternative which was grammatical. Sometimes we chose an earlier alternative
because a later one seemed to be incomplete, in the sense that it took the discussion
in a new direction that Wittgenstein never continued in this text. In all cases, our decision
as to which alternative should appear, and where, has been more principled than the advice
that Wittgenstein himself is said to have given to Professor Anscombe. When she pressed
him for advice as to how in editing his work she should choose among alternatives, he reportedly
told her, “Toss a coin”.8
In addition to editing the text by providing alternative formulations of words and phrases,
over the years Wittgenstein also added numerous words, sentences, paragraphs and
remarks. It is usually clear where these are to be placed, but sometimes the remarks require
a decision as to where they belong. We have used two principles to decide where in the body
of the typescript to place Wittgenstein’s additional handwritten remarks. When he wrote them
on the backs of typed pages, he usually placed them opposite the point at which they were
to be inserted on the next page. We have placed them there. In addition, he often inserted
remarks at the beginning of chapters. These remarks, incidentally, are often of a general nature,
as if he were commenting on the topic of the chapter rather than entering into a discussion
of the issues. Often he seems to have begun writing these additions just below the chapter
title, but before the first typed remark. When he ran out of space he then continued writing
remarks above the chapter title. Thus we take the apparent order, beginning at the top
of the page, to be misleading, and so we begin the chapter with the first handwritten remark
below the title, continuing on to handwritten remarks above the title, and only then to the
first typed remark.
All of our footnotes are preceded by a letter in parentheses indicating the sort of material
that is contained therein. (V) indicates that the footnote contains the variants of a word, sentence,
or remark, with a horizontal line through the words he struck through. A single word
following a (V) is obviously a variant for the single word preceding the footnote number as
it occurs in the text, but for multiple words we have preceded them in the footnote with a
word or two of preceding text so the reader can see where they fit. (M) indicates the marginal
comments that Wittgenstein often wrote on the typescript. (O) indicates typographical errors
and misspellings that we have corrected in the main body of this text. For the most part we
have not corrected non-standard punctuation in the German text, such as a lack of initial
word capitalization in a sentence, and we have not changed many of Wittgenstein’s idiosyncratic
spellings. We have substituted ß for “ss” and have inserted umlauts above capitalized
vowels that require them, assuming that these symbols were lacking on the typewriter he
used. We have also standardized Wittgenstein’s sometimes idiosyncratic use of question marks.
EDITORS’ AND TRANSLATORS’ INTRODUCTION ixe
8 Recounted in Michael Nedo’s Ludwig Wittgenstein,
Wiener Ausgabe, Einführung – Introduction
(Vienna: Springer-Verlag, 1993), p. 75.
WBTA01 30/05/2005 16:12 Page 13
editorischen Anmerkungen haben wir ein (E) vorangestellt. Viele der ins „Big Typescript“
eingegangenen Bemerkungen waren in ihrer handschriftlichen Version mit Zeichnungen versehen.
Diese letzteren wurden von Wittgenstein nur selten in die Maschinenschrift mit
übernommen, obwohl für sie manchmal Platz ausgespart ist. Wir haben sie an den entsprechenden
Stellen in unseren Text eingesetzt. Die auf (F) folgende Signatur zeigt, auf welcher
Seite welchen Manuskripts (in der von Wright’schen Zählung) die betreffende Zeichnung
zu finden ist.
Nicht nur um der Zeichnungen willen haben wir auf die dem „Big Typescript“ vorausgehenden
Manuskripte zurückgegriffen. Zuweilen haben wir daraus auch Material eingesetzt,
das irrtümlich vom Typisten ausgelassen worden war oder das den für das Verständnis einer
Stelle nötigen Zusammenhang herstellt. Andererseits haben wir uns aber bei der Entscheidung,
welche Variante einer Stelle in den Text eingehen sollte und welche in eine Fußnote, nicht
von den auf das „Big Typescript“ folgenden Manuskripten leiten lassen. Aus den späteren
Manuskripten geht nur selten eine bevorzugte Alternative hervor, was damit zusammenhängt,
daß die Wahl einer bestimmten Variante oft durch einen neuen Kontext bestimmt (und daher
auch für das „Big Typescript“ nur wenig aufschlußreich) ist.
Wir haben versucht, handschriftliches Material diskret, durch eine „sans serif“ Version
unserer Schriftart, als solches kenntlich zu machen.
Die von Wittgenstein verwendeten Randzeichen, die sich alle auf die daneben stehenden
Bemerkungen in ihrer Gesamtheit beziehen, geben wir im Apparat so genau wie möglich
wieder: das„∫“, das anzeigt, daß Wittgenstein die damit versehene Bemerkung für schlecht
hielt, den Haken „3“, den einfachen Schrägstrich „/“, sowie das Fragezeichen „?“. Auch
die Zeichen „∀“ und „1“, denen meistens eine Seiten- bzw. Absatzzahl folgt, führen wir
da an. Sie weisen darauf hin, daß die bezeichnete(n) Stelle(n) hier eingesetzt werden sollte(n).
Die vielen Ausstreichungen, die Wittgenstein an seinen Bemerkungen vornahm, haben
wir weniger „naturgetreu“ wiedergegeben. Einfach, bzw. kreuzweise ausgestrichene Absätze
oder Teile von Absätzen haben wir dort, wo die Ausstreichung beginnt, mit „///“, bzw.
mit „XXX“ bezeichnet, wobei das jeweilige Zeichen bis zum Ende des Absatzes gilt. Wurden
ganze Bemerkungen einfach oder kreuzweise ausgestrichen, so haben wir dies am Anfang
der Bemerkung durch „////“, bzw. „XXXX“ angezeigt.
Die Paginierung des Originals findet sich in unserem deutschen Text am linken Rand.
Ein auf die Seitenzahl folgendes „v“ bezeichnet die Rückseite (das Verso) der betreffenden
Seite.
Weitere Zeichen, die sich im Wittgenstein’schen Original finden, haben wir unserem
Verständnis gemäß interpretiert und wie folgt wiedergegeben: die Sperrung von Worten
verstehen wir als eine Form der Hervorhebung und geben sie durch Kursivschrift wieder.
(Aus „V e r s t e h e n“ wird also „Verstehen“.) Handschriftlich unterstrichenes Material
verstehen wir ähnlich und geben auch dieses kursiv wieder. Sowohl die gesperrt getippten
einfachen als auch die handgeschriebenen wellenförmigen Unterstreichungen haben wir als
Ausdruck des Zweifels an dem darüber stehenden Material gelesen. So gekennzeichnete Worte
haben wir in ,Blaßdruck’ wiedergegeben. Manchmal findet sich ein (getipptes oder handgeschriebenes)
Fragezeichen über einem Wort, manchmal wird ein ganzer Satzteil unter
Fragezeichen gestellt, indem ein „?-“ über dessen Anfang gesetzt wird und ein „-?“ über
dessen Ende. Auch damit scheint uns Wittgenstein an Worten oder Wortgruppen Zweifel
ausdrücken zu wollen. Es handelt sich aber hierbei wohl um eine andere Form des Zweifels
als im Fall von gesperrter oder wellenförmiger Unterstreichung und wir haben daher das
betreffende Material auch anders, nämlich durch punktierte Unterstreichung, gekennzeichnet.
Welcher Art dieser Unterschied ist, wissen wir nicht. Ausgestrichene Worte und Satzteile,
die Wittgenstein durch Unterpunktieren wiederherstellte, haben wir nicht eigens gekennzeichnet,
sondern so wiedergegeben, als ob sie nie ausgestrichen gewesen wären.
x EINLEITUNG DER HERAUSGEBER
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(R) indicates the references Wittgenstein entered into the text indicating where in this or
later manuscripts he wished to have passages moved from or to. (E) refers to our editorial
comments and references. The first versions of many remarks appear in their respective
manuscripts accompanied by illustrations. When he had those remarks typed in this text
Wittgenstein sometimes left a space for them, but seldom bothered to transfer the illustrations.
We have done so. (F) gives our references to page numbers of manuscripts preceding
“The Big Typescript” in which he drew illustrations for various passages.
Not only have we used predecessor manuscripts to obtain illustrations, but sometimes also
to include material that was mistakenly omitted by the typist of TS 213, or material that
otherwise provides a useful context for understanding this text. On the other hand, we have
not used successor manuscripts to decide which alternatives to include in the main body and
which to put into footnotes, for two reasons. First, it sometimes happens that in one successor
manuscript Wittgenstein preferred one of the alternatives, but in an even later manuscript
he used the other (or another) alternative. This is connected with our second reason, which
is that the context in which a remark is used in a successor manuscript is often quite different
from that in which it appeared in “The Big Typescript”, and so what is done in a
successor manuscript often seems simply to be irrelevant to this text.
We have tried to make the distinction between typed and handwritten material distinguishable
but unobtrusive, by presenting handwritten material in a sans serif font.
Wittgenstein himself used several symbols in the marginalia, which we indicate as follows:
“∫” is Wittgenstein’s sign for Schlecht, presumably indicating what he thought a “bad” remark;
“3” is obviously a tick mark, “/” is a small slash mark that usually precedes its remark in
Wittgenstein’s left margin, but “///” and “////” are different things. They represent large
slashes that he used to mark through whole paragraphs and remarks, respectively. “XXX”
and “XXXX” indicate paragraphs and remarks, respectively, that he crossed through in both
directions. “///” and “XXX” are to be understood as extending only to the end of a paragraph,
whether they occur at its beginning or within it. “////” and “XXXX”, on the other
hand, extend through the whole remark, as does “∫”, “3”, and a “?” in a marginal note, unless
we indicate otherwise. “∀” (and sometimes “1”) is a symbol more or less resembling what
Wittgenstein used to indicate the need to move remarks or paragraphs.
In the German text we have included the page numbers of the original text, placing them
in the left margin of our text, and using a “v” to indicate the back (verso) side.
Wittgenstein used several other symbols that we have interpreted and chosen to represent
as follows. We have used italics to represent his broken spacing of words, which we
understand to be a mark of emphasis (for example, we write his “V e r s t e h e n” as
“Verstehen”). We understand single handwritten underlines the same way, and represent them
similarly. He had the typist insert broken underlinings below several words, and he also wrote
in wavy underlinings under both handwritten and typed words. We understand these to be
equivalent markings, which indicate a degree of doubt about a word or phrase. We represent
both of them with a fainter font than the rest of the text. Sometimes Wittgenstein typed
or wrote in a question mark above a word, and sometimes over a word followed by a dash
(“?-”), and then a dash followed by a question mark over a later word (“-?”). We understand
these, too, to be indications of doubt about a word or phrase, and we represent them with
small dotted underlines. Presumably they represent a different level of doubt than broken
underlines, but we do not know what that might be. Wittgenstein used handwritten dots
under the letters of crossed-through words to indicate that those words should be retained
(stet marks). We have not indicated these, choosing instead just to retain the words.
EDITORS’ AND TRANSLATORS’ INTRODUCTION xe
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Es folgt nun eine schematische Zusammenfassung der oben besprochenen Zeichen.
Editorische Abkürzungen in den Fußnoten
(V) Variante eines Wortes, Satzes oder einer Bemerkung
(M) Randbemerkung
(O) Tippfehler oder orthographischer Fehler im Original
(R) Hinweis auf eine mögliche Umstellung
(F) Seitenangaben früherer Manuskripte, denen Zeichnungen entnommen wurden
(E) Editorische Bemerkungen
Randzeichen
∫ Zeichen für „schlecht“
3 Haken
/ einfacher Schrägstrich
∀ oder 1 Einsetzungszeichen
/// Absatz ganz oder teilweise schräg durchgestrichen
//// Ganze Bemerkung schräg durchgestrichen
XXX Absatz ganz oder teilweise kreuzweise durchgestrichen
XXXX ganze Bemerkung kreuzweise durchgestrichen
Im Text erscheinende Zeichen
Im Original im vorliegenden Text
keine keine
keine keine
k e i n e keine
keine keine
?- -?
keine Absicht keine Absicht
xi EINLEITUNG DER HERAUSGEBER
~~~~
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The following charts summarize these symbols:
Editors’ footnote abbreviations
(V) variant of a word, sentence or remark
(M) marginal comments
(O) original of a typographical error or misspelling
(R) directions to move passage
(F) page numbers of passages in which illustrations occur
(E) comments and references by the editors
Marginal marks
∫ Schlecht mark
3 tick mark
/ slash mark
∀ or 1 move mark
/// paragraph slashed through
//// remark slashed through
XXX paragraph crossed through
XXXX remark crossed through
Wittgenstein’s emphasis marks
In original This edition
keine keine
keine keine
k e i n e keine
keine keine
?- -?
keine Absicht keine Absicht
EDITORS’ AND TRANSLATORS’ INTRODUCTION xie
~~~~
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Inhaltsverzeichnis
Verstehen. 1
1 Das Verstehen, die Meinung, fällt aus unsrer Betrachtung heraus. 2
2 „Meinen“ amorph gebraucht. „Meinen“ mehrdeutig. 5
3 Das Verstehen als Korrelat einer Erklärung. 8
4 Das Verstehen des Befehls, die Bedingung dafür, daß wir ihn befolgen
können. Das Verstehen des Satzes, die Bedingung dafür, daß wir uns nach
ihm richten. 12
5 Deuten. Deuten wir jedes Zeichen? 16
6 Man sagt: ein Wort verstehen heißt, wissen, wie es gebraucht wird.
Was heißt es, das zu wissen? Dieses Wissen haben wir sozusagen im Vorrat. 18
6a Einen Satz im Ernst oder Spaß meinen, etc. 21
Bedeutung. 22
7 Der Begriff der Bedeutung stammt aus einer primitiven philosophischen
Auffassung der Sprache her. 23
8 Bedeutung, der Ort des Wortes im grammatischen Raum. 26
9 Die Bedeutung eines Wortes ist das, was die Erklärung der Bedeutung
erklärt. 29
10 „Die Bedeutung eines Zeichens ist durch seine Wirkung (die Assoziationen,
die es auslöst, etc.) gegeben.“ 33
11 Bedeutung als Gefühl, hinter dem Wort stehend; durch eine Geste
ausgedrückt. 37
12 Man tritt mit der hinweisenden Erklärung der Zeichen nicht aus der
Sprachlehre heraus. 38
13 „Primäre und sekundäre Zeichen“. Wort und Muster. Hinweisende
Definition. 40
14 Das, was die Philosophie am Zeichen interessiert, die Bedeutung, die für sie
maßgebend ist, ist das, was in der Grammatik des Zeichens niedergelegt ist. 48
Satz. Sinn des Satzes. 49
15 „Satz“ und „Sprache“ verschwimmende Begriffe. 50
16 Die Logik redet von Sätzen und Wörtern im gewöhnlichen Sinn, nicht von
Sätzen und Wörtern in irgend einem abstrakten Sinn. 57
17 Satz und Satzklang. 59
xii
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Table of Contents
Understanding. 1
1 Understanding, Meaning, Drop Out of Our Considerations. 2
2 “Meaning” Used Amorphously. “Meaning” Used Equivocally. 5
3 Understanding as a Correlate of an Explanation. 9
4 Understanding a Command the Condition for Our Being Able to Obey It.
Understanding a Proposition the Condition for Our Acting in Accordance
with It. 12
5 Interpreting. Do We Interpret Every Sign? 16
6 One Says: Understanding a Word Means Knowing How it is Used.
What Does it Mean to Know That? We Have this Knowledge in Reserve,
as it Were. 18
6a Meaning a Proposition Seriously or in Jest, etc. 21
Meaning. 22
7 The Concept of Meaning Originates in a Primitive Philosophical Conception
of Language. 23
8 Meaning, the Position of the Word in Grammatical Space. 26
9 The Meaning of a Word is What the Explanation of its Meaning Explains. 29
10 “The Meaning of a Sign is Given by its Effect (the Associations that it
Triggers, etc.).” 33
11 Meaning as Feeling, Standing Behind the Word; Expressed with a Gesture. 37
12 In Giving an Ostensive Explanation of Signs one Doesn’t Leave Grammar. 38
13 “Primary and Secondary Signs”. Word and Sample. Ostensive Definition. 40
14 What Interests Philosophy About the Sign, the Meaning That is Decisive
for it, is What is Laid Down in the Grammar of the Sign. 48
Proposition. Sense of a Proposition. 49
15 “Sentence” and “Language” Blurred Concepts. 50
16 Logic Talks about Sentences and Words in the Ordinary Sense, not in
Some Abstract Sense. 57
17 Sentence and Sentence-Sound. 59
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18 Was als Satz gelten soll, ist in der Grammatik bestimmt. 61
19 Die grammatischen Regeln bestimmen den Sinn des Satzes; und ob eine
Wortzusammenstellung Sinn hat oder nicht. 63
20 Der Sinn des Satzes, keine Seele. 65
21 Ähnlichkeit von Satz und Bild. 67
22 Sätze mit Genrebildern verglichen. (Verwandt damit: Verstehen eines
Bildes.) 69
23 Mit dem Satz scheint die Realität wesentlich übereinstimmen oder nicht
übereinstimmen zu können. Er scheint sie zu fordern, sich mit ihm zu
vergleichen. 70
24 Das Symbol (der Gedanke), scheint als solches unbefriedigt zu sein. 73
25 Ein Satz ist ein Zeichen in einem System von Zeichen. Er ist eine
Zeichenverbindung von mehreren möglichen und im Gegensatz zu den
andern möglichen. Gleichsam eine Zeigerstellung im Gegensatz zu andern
möglichen. 76
26 Sich vorstellen können, „wie es wäre“, als Kriterium dafür, daß ein Satz
Sinn hat. 78
27 „Logische Möglichkeit und Unmöglichkeit“. – Das Bild des „Könnens“
ultraphysisch angewandt. (Ähnlich: „Das ausgeschlossene Dritte“.) 80
28 Elementarsatz. 82
29 „Wie ist die Möglichkeit von p in der Tatsache, daß ~p der Fall ist,
enthalten?“ „Wie enthält z.B. der schmerzlose Zustand die Möglichkeit der
Schmerzen?“ 83
30 „Wie kann das Wort ,nicht‘ verneinen?“ Das Wort „nicht“ erscheint uns
wie ein Anstoß zu einer komplizierten Tätigkeit des Verneinens. 87
31 Ist die Zeit den Sätzen wesentlich? Vergleich von: Zeit und
Wahrheitsfunktionen. 91
32 Wesen der Hypothese. 94
33 Wahrscheinlichkeit. 98
34 Der Begriff „ungefähr“. Problem des „Sandhaufens“. 105
Das augenblickliche Verstehn und die Anwendung des Worts in der Zeit. 109
35 Ein Wort verstehen = es anwenden können. Eine Sprache verstehen:
Einen Kalkül beherrschen. 110
36 Wie begleitet das Verstehen des Satzes das Aussprechen oder Hören des
Satzes? 113
37 Zeigt sich die Bedeutung eines Wortes in der Zeit? Wie der tatsächliche
Freiheitsgrad eines Mechanismus? Enthüllt sich die Bedeutung des Worts
erst nach und nach wie seine Anwendung fortschreitet? 115
38 Begleitet eine Kenntnis der grammatischen Regeln den Ausdruck des Satzes,
wenn wir ihn – seine Worte – verstehn? 117
39 Die grammatischen Regeln – und die Bedeutung eines Wortes. Ist die
Bedeutung, wenn wir sie verstehen, „auf einmal“ erfaßt; und in den
grammatischen Regeln gleichsam ausgebreitet? 121
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18 What is to Count as a Proposition is Determined in Grammar. 61
19 Grammatical Rules Determine the Sense of a Proposition; and Whether a
Combination of Words Makes Sense. 63
20 The Sense of a Proposition not a Soul. 65
21 Similarity of Proposition and Picture. 67
22 Propositions Compared to Genre-Paintings. (Related to This:
Understanding a Picture.) 69
23 Reality Seems Inherently Able Either to Agree with a Proposition or not to
Agree with it. A Proposition Seems to Challenge Reality to Compare Itself
to it. 70
24 A Symbol (a Thought) as Such Seems to be Unfulfilled. 73
25 A Sentence is a Sign within a System of Signs. It is a Combination of Signs
from among Several Possible Ones and in Contrast to Other Possible Ones.
One Position of the Pointer, as it Were, in Contrast to Other Possible Ones. 76
26 Being Able to Imagine “What it Would be Like” as a Criterion for
a Proposition Having a Sense. 78
27 “Logical Possibility and Impossibility”. – The Picture of “Being Able To”
Applied Ultraphysically. (Similar to: “The Excluded Middle”.) 80
28 Elementary Proposition. 82
29 “How is the Possibility of p Contained in the Fact that ~p is the Case?”
“How Does, for Example, a Pain-free State Contain the Possibility of Pain?” 83
30 “How Can the Word ‘Not’ Negate?” The Word “Not” Seems to Us Like an
Impetus to a Complicated Activity of Negating. 87
31 Is Time Essential to Propositions? Comparison between Time and
Truth-Functions. 91
32 The Nature of Hypothesis. 94
33 Probability. 98
34 The Concept “Roughly”. Problem of the “Heap of Sand”. 105
Immediate Understanding and the Application of a Word in Time. 109
35 To Understand a Word = To Be Able to Use It. To Understand a Language:
To Have Command of a Calculus. 110
36 How Does Understanding a Sentence Accompany Uttering or Hearing it? 113
37 Is the Meaning of a Word Shown in Time? Like the Actual Degree of
Freedom in a Mechanism? Is the Meaning of a Word Only Revealed
in the Course of Time as its Use Develops? 115
38 Does a Knowledge of Grammatical Rules Accompany the Expression of a
Sentence when We Understand it – Its Words? 117
39 The Rules of Grammar – and the Meaning of a Word. Is Meaning,
When We Understand it, Grasped “all at once”? And Unfolded, as it Were,
in the Rules of Grammar? 121
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Wesen der Sprache. 128
40 Lernen, Erklärung, der Sprache. Kann man die Sprache durch die
Erklärung gleichsam aufbauen, zum Funktionieren bringen? 129
41 Wie wirkt die einmalige Erklärung der Sprache, das Verständnis? 132
42 Kann man etwas Rotes nach dem Wort „rot“ suchen? Braucht man ein Bild,
ein Erinnerungsbild, dazu? Verschiedene Suchspiele. 136
43 „Die Verbindung zwischen Sprache und Wirklichkeit“ ist durch die
Worterklärungen gemacht, welche wieder zur Sprachlehre gehören. So daß
die Sprache in sich geschlossen, autonom, bleibt. 141
44 Die Sprache in unserem Sinn nicht als Einrichtung definiert, die einen
bestimmten Zweck erfüllt. Die Grammatik kein Mechanismus, der durch
seinen Zweck gerechtfertigt ist. 144
45 Die Sprache funktioniert als Sprache nur durch die Regeln, nach denen wir
uns in ihrem Gebrauch richten, wie das Spiel nur durch seine Regeln ein
Spiel ist. 150
46 Funktionieren des Satzes an einem Sprachspiel erläutert. 156
47 Behauptung, Frage, Annahme, etc. 160
Gedanke. Denken. 164
48 Wie denkt man den Satz „p“, wie erwartet (glaubt, wünscht) man, daß p
der Fall sein wird? Mechanismus des Denkens. 165
49 „Was ist ein Gedanke, welcher Art muß er sein, um seine Funktion erfüllen
zu können?“ Hier will man sein Wesen aus seinem Zweck, seiner Funktion
erklären. 168
50 Ist die Vorstellung das Porträt par excellence, also grundverschieden, etwa,
von einem gemalten Bild und durch ein solches oder etwas Ähnliches nicht
ersetzbar? Ist sie das, was eigentlich eine bestimmte Wirklichkeit darstellt, –
zugleich Bild und Meinung? 170
51 Ist das Denken ein spezifisch organischer Vorgang? Ein spezifisch
menschlich-psychischer Vorgang? Kann man ihn in diesem Falle durch
einen anorganischen Vorgang ersetzen, der denselben Zweck erfüllt,
also sozusagen durch eine Prothese? 172
52 Ort des Denkens. 173
53 Gedanke und Ausdruck des Gedankens. 175
54 Was ist der Gedanke? Was ist sein Wesen? „Der Gedanke, dieses seltsame
Wesen“. 178
55 Zweck des Denkens. Grund des Denkens. 179
Grammatik. 183
56 Die Grammatik ist keiner Wirklichkeit Rechenschaft schuldig.
Die grammatischen Regeln bestimmen erst die Bedeutung
(konstituieren sie) und sind darum keiner Bedeutung verantwortlich
und insofern willkürlich. 184
57 Regel und Erfahrungssatz. Sagt eine Regel, daß Wörter tatsächlich so und so
gebraucht werden? 189
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TABLE OF CONTENTS xive
The Nature of Language. 128
40 Learning, Explanation, of Language. Can We Use Explanation to Construct
Language, so to Speak, to Get it to Work? 129
41 What Effect Does a Single Explanation of Language Have, What Effect
Understanding? 132
42 Can One Use the Word “Red” to Search for Something Red? Does One
Need an Image, a Memory-Image, for This? Various Searching-Games. 136
43 “The Connection between Language and Reality” is Made Through
Explanations of Words, which Explanations Belong in Turn to Grammar.
So that Language Remains Self-contained, Autonomous. 141
44 Language in our Sense not Defined as an Instrument for a Particular
Purpose. Grammar is not a Mechanism Justified by its Purpose. 144
45 Language Functions as Language only by Virtue of the Rules We Follow in
Using it, just as a Game is a Game only by Virtue of its Rules. 150
46 The Functioning of a Proposition Explained with a Language-Game. 156
47 Assertion, Question, Assumption, etc. 160
Thought. Thinking. 164
48 How does one Think the Proposition “p”, how does one Expect
(Believe, Wish) that p will be the Case? Mechanism of Thinking. 165
49 “What is a Thought, What Must it be Like for it to Fulfill its Function?”
Here one Wants to Explain its Essence by its Purpose, its Function. 168
50 Is a Mental Image a Portrait Par Excellence, and thus Fundamentally
Different from, say, a Painted Picture, and not Replaceable by one or by
any such Thing? Is it a Mental Image that Really Represents a Particular
Reality – Simultaneously Picture and Meaning? 170
51 Is Thinking a Specifically Organic Process? A Process Specific to Human
Psychology? If so, can one Replace it with an Inorganic Process that Fulfills
the Same Purpose, that is, by a Prosthesis, as it Were? 172
52 Location of Thinking. 173
53 Thought and Expression of Thought. 175
54 What is Thought? What is its Essence? “Thought, this Peculiar Being.” 178
55 The Purpose of Thinking. The Reason for Thinking. 179
Grammar. 183
56 Grammar is not Accountable to any Reality. The Rules of Grammar
Determine Meaning (Constitute it), and Therefore they are not Answerable
to any Meaning and in this Respect are Arbitrary. 184
57 Rule and Empirical Proposition. Does a Rule Say that Words are Actually
Used in Such and Such a Way? 189
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58 Die strikten grammatischen Spielregeln und der schwankende
Sprachgebrauch. Die Logik normativ. Inwiefern reden wir von idealen
Fällen, einer idealen Sprache? („Logik des luftleeren Raums“.) 195
59 Wortarten werden nur durch ihre Grammatik unterschieden. 206
60 Sage mir, was Du mit einem Satz anfängst, wie Du ihn verifizierst, etc.,
und ich werde ihn verstehen. 207
Intention und Abbildung. 213
61 Wenn ich mich abbildend nach einer Vorlage richte, also weiß, daß ich
jetzt den Stift so bewege, weil die Vorlage so verläuft, ist hier eine mir
unmittelbar bewußte Kausalität im Spiel? 214
62 Wenn wir „nach einer bestimmten Regel abbilden“, ist diese Regel in dem
Vorgang des Kopierens (Abbildens) enthalten, also aus ihm eindeutig
abzulesen? Verkörpert der Vorgang des Abbildens sozusagen diese Regel? 216
63 Wie rechtfertigt man das Resultat der Abbildung mit der allgemeinen Regel
der Abbildung? 219
64 Der Vorgang der absichtlichen Abbildung, der Abbildung mit der Intention
abzubilden ist nicht wesentlich ein psychischer, innerer. Ein Vorgang der
Manipulation mit Zeichen auf dem Papier kann dasselbe leisten. 221
65 Wie hängen unsre Gedanken mit den Gegenständen zusammen, über
die wir denken? Wie treten diese Gegenstände in unsre Gedanken ein.
(Sind sie in ihnen durch etwas Andres – etwa Ähnliches – vertreten?)
Wesen des Porträts; die Intention. 225
Logischer Schluß. 229
66 Wissen wir, daß p aus q folgt, weil wir die Sätze verstehen? Geht das
Folgen aus einem Sinn hervor? 230
67 „Wenn p aus q folgt, so muß p in q schon mitgedacht sein“. 233
68 Der Fall: unendlich viele Sätze folgen aus einem. 235
69 Kann eine Erfahrung lehren, daß dieser Satz aus jenem folgt? 238
Allgemeinheit. 240
70 Der Satz „der Kreis befindet sich im Quadrat“ in gewissem Sinne
unabhängig von der Angabe einer bestimmten Lage (er hat, in gewissem
Sinne, nichts mit ihr zu tun). 241
71 Der Satz „der Kreis liegt im Quadrat“ keine Disjunktion von Fällen. 244
72 Unzulänglichkeit der Frege- und Russell’schen Allgemeinheitsbezeichnung. 247
73 Kritik meiner früheren Auffassung der Allgemeinheit. 249
74 Erklärung der Allgemeinheit durch Beispiele. 251
75 Bildungsgesetz einer Reihe. „U.s.w.“ 257
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58 The Strict Grammatical Rules of a Game and the Fluctuating Use of
Language. Logic as Normative. To what Extent do we Talk about Ideal
Cases, an Ideal Language? (“The Logic of a Vacuum.”) 195
59 Kinds of Words are Distinguished only by their Grammar. 206
60 Tell me What you do with a Proposition, How you Verify it, etc., and
I Shall Understand It. 207
Intention and Depiction. 213
61 If in Copying I am Guided by a Model and thus Know that I am Now
Moving my Pencil in such a Way because the Model Goes that Way, is a
Causality Involved Here of which I am Immediately Aware? 214
62 If We “Depict in Accordance with a Particular Rule”, is this Rule
Contained in the Process of Copying (Depicting), and can it Therefore be
Read out of it Unambiguously? Does the Process of Depicting Embody
this Rule, as it Were? 216
63 How Does one Use a General Rule of Representation to Justify the Result of
Representation? 219
64 The Process of Copying on Purpose, of Copying with the Intention to Copy,
is not Essentially a Psychological, Inner Process. A Process of Manipulating
Signs on a Piece of Paper can Accomplish the Same Thing. 221
65 How are our Thoughts Connected with the Objects we Think about?
How do these Objects Enter our Thoughts? (Are they Represented in our
Thoughts by Something Else – Perhaps Something Similar?) The Nature of
a Portrait; Intention. 225
Logical Inference. 229
66 Do we Know that p Follows from q because we Understand the
Propositions? Is Entailment Implied by a Sense? 230
67 “If p Follows from q, then p Must Have Been Mentally Included in q.” 233
68 The Case of Infinitely Many Propositions Following from a Single One. 235
69 Can an Experience teach us that one Proposition Follows from Another? 238
Generality. 240
70 In a Certain Sense the Proposition “The Circle is in the Square” is
Independent of the Indication of a Particular Position (in a Certain Sense
it has Nothing to do with It). 241
71 The Proposition “The Circle is in the Square” not a Disjunction of Cases. 244
72 The Inadequacy of Frege’s and Russell’s Notation for Generality. 247
73 Criticism of my Earlier Understanding of Generality. 249
74 Explanation of Generality by Examples. 251
75 The Law of the Formation of a Series. “Etc.”. 257
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Erwartung. Wunsch. etc. 263
76 Erwartung: der Ausdruck der Erwartung. Artikulierte und unartikulierte
Erwartung. 264
77 In der Erwartung wurde das erwartet, was die Erfüllung brachte. 266
78 „Wie kann man etwas wünschen, erwarten, suchen, was nicht da ist?“
Mißverständnis des „Etwas“. 270
79 Im Ausdruck der Sprache berühren sich Erwartung und Erfüllung. 275
80 „Der Satz bestimmt, welche Realität ihn wahr macht“. Er scheint einen
Schatten dieser Realität zu geben. Der Befehl scheint seine Ausführung in
schattenhafter Weise vorauszunehmen. 277
81 Intention. Was für ein Vorgang ist sie? Man soll aus der Betrachtung dieses
Vorgangs ersehen können, was intendiert wird. 280
82 Kein Gefühl der Befriedigung (kein Drittes) kann das Kriterium dafür sein,
daß die Erwartung erfüllt ist. 284
83 Der Gedanke – Erwartung, Wunsch, etc. – und die gegenwärtige Situation. 286
84 Glauben. Gründe des Glaubens. 289
85 Grund, Motiv, Ursache. 295
Philosophie. 299
86 Schwierigkeit der Philosophie, nicht die intellektuelle Schwierigkeit der
Wissenschaften, sondern die Schwierigkeit einer Umstellung. Widerstände
des Willens sind zu überwinden. 300
87 Die Philosophie zeigt die irreführenden Analogien im Gebrauch unsrer
Sprache auf. 302
88 Woher das Gefühl des Fundamentalen unserer grammatischen
Untersuchungen? 304
89 Methode der Philosophie: die übersichtliche Darstellung der grammatischen
Tatsachen. Das Ziel: Durchsichtigkeit der Argumente. Gerechtigkeit. 306
90 Philosophie. Die Klärung des Sprachgebrauches. Fallen der Sprache. 311
91 Die philosophischen Probleme treten uns im praktischen Leben gar nicht
entgegen (wie etwa die der Naturlehre), sondern erst, wenn wir uns bei der
Bildung unserer Sätze nicht vom praktischen Zweck, sondern von gewissen
Analogien in der Sprache leiten lassen. 314
92 Methode in der Philosophie. Möglichkeit des ruhigen Fortschreitens. 316
93 Die Mythologie in den Formen unserer Sprache. ((Paul Ernst.)) 317
Phänomenologie. 319
94 Phänomenologie ist Grammatik. 320
95 Kann man in die Eigenschaften des Gesichtsraumes tiefer eindringen?
etwa durch Experimente? 323
96 Gesichtsraum im Gegensatz zum euklidischen Raum. 325
97 Das sehende Subjekt und der Gesichtsraum. 334
98 Der Gesichtsraum mit einem Bild (ebenen Bild) verglichen. 336
99 Minima visibilia. 338
100 Farben und Farbenmischung. 340
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Expectation. Wish. etc. 263
76 Expectation: the Expression of Expectation. Articulate and Inarticulate
Expectation. 264
77 What Fulfillment Brought: that was what was Expected in Expectation. 266
78 “How can one Wish for, Expect, Look for, Something that isn’t There?”
Misunderstanding of the “Something”. 270
79 Expectation and Fulfillment Make Contact in Linguistic Expression. 275
80 “The Proposition Determines which Reality Makes it True.” It Seems
to Provide a Shadow of this Reality. A Command Seems to Anticipate
its Execution in a Shadowy Way. 277
81 Intention. What Kind of a Process is it? From an Examination of this
Process one is Supposed to be Able to See What is Being Intended. 280
82 No Feeling of Satisfaction (no Third Thing) Can Be the Criterion that
Expectation has been Fulfilled. 284
83 Thought – Expectation, Wish, etc. – and the Present Situation. 286
84 Belief. Grounds for Belief. 289
85 Reason, Motive, Cause. 295
Philosophy. 299
86 Difficulty of Philosophy not the Intellectual Difficulty of the Sciences,
but the Difficulty of a Change of Attitude. Resistance of the Will Must be
Overcome. 300
87 Philosophy Points out the Misleading Analogies in the Use of our Language. 302
88 Whence the Feeling that our Grammatical Investigations are Fundamental? 304
89 The Method of Philosophy: the Clearly Surveyable Representation of
Grammatical Facts. The Goal: the Transparency of Arguments. Justice. 306
90 Philosophy. The Clarification of the Use of Language. Traps of Language. 311
91 We Don’t Encounter Philosophical Problems at all in Practical Life (as we do,
for Example, Those of Natural Science). We Encounter them only When
we are Guided not by Practical Purpose in Forming our Sentences, but by
Certain Analogies within Language. 314
92 Method in Philosophy. The Possibility of Quiet Progress. 316
93 The Mythology in the Forms of our Language. ((Paul Ernst.)) 317
Phenomenology. 319
94 Phenomenology is Grammar. 320
95 Can one Penetrate More Deeply into the Properties of Visual Space?
Say through Experiments? 323
96 Visual Space in Contrast to Euclidean Space. 325
97 The Seeing Subject and Visual Space. 334
98 Visual Space Compared to a Picture (Two-Dimensional Picture). 336
99 Minima Visibilia. 338
100 Colors and the Mixing of Colors. 340
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Idealismus, etc. 346
101 Die Darstellung des unmittelbar Wahrgenommenen. 347
102 „Die Erfahrung im gegenwärtigen Moment, die eigentliche Realität.“ 351
103 Idealismus. 354
104 „Schmerzen haben.“ 356
105 Gedächtniszeit. 363
106 „Hier“ und „Jetzt“. 366
107 Farbe, Erfahrung, etc. als formale Begriffe. 369
Grundlagen der Mathematik. 370
108 Die Mathematik mit einem Spiel verglichen. 371
109 Es gibt keine Metamathematik. 376
110 Beweis der Relevanz. 378
111 Beweis der Widerspruchsfreiheit. 380
112 Die Begründung der Arithmetik, in der diese auf ihre Anwendungen
vorbereitet wird. (Russell, Ramsey.) 382
113 Ramsey’s Theorie der Identität. 388
114 Der Begriff der Anwendung der Arithmetik (Mathematik). 391
Über Kardinalzahlen. 392
115 Kardinalzahlenarten. 393
116 2 + 2 = 4. 399
117 Zahlangaben innerhalb der Mathematik. 410
118 Zahlengleichheit. Längengleichheit. 412
Mathematischer Beweis. 417
119 Wenn ich sonst etwas suche, so kann ich das Finden beschreiben,
auch wenn es nicht eingetreten ist; anders, wenn ich die Lösung eines
mathematischen Problems suche. Mathematische Expedition und
Polarexpedition. 418
120 Beweis, und Wahrheit und Falschheit eines mathematischen Satzes. 423
121 Wenn Du wissen willst, was bewiesen wurde, schau den Beweis an. 425
122 Das mathematische Problem. Arten der Probleme. Suchen. „Aufgaben“
in der Mathematik. 430
123 Eulerscher Beweis. 434
124 Dreiteilung des Winkels, etc. 437
125 Suchen und Versuchen. 441
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Idealism, etc. 346
101 The Representation of what is Immediately Perceived. 347
102 “The Experience at the Present Moment, Actual Reality.” 351
103 Idealism. 354
104 “Having Pain.” 356
105 Memory-Time. 363
106 “Here” and “Now”. 366
107 Color, Experience, etc., as Formal Concepts. 369
Foundations of Mathematics. 370
108 Mathematics Compared to a Game. 371
109 There is no Metamathematics. 376
110 Proof of Relevance. 378
111 Consistency Proof. 380
112 Laying the Foundations for Arithmetic as Preparation for its Applications.
(Russell, Ramsey.) 382
113 Ramsey’s Theory of Identity. 388
114 The Concept of the Application of Arithmetic (Mathematics). 391
On Cardinal Numbers. 392
115 Kinds of Cardinal Numbers. 393
116 2 + 2 = 4. 399
117 Statements of Number within Mathematics. 410
118 Sameness of Number. Sameness of Length. 412
Mathematical Proof. 417
119 If I am Looking for Something in Other Cases I Can Describe Finding it,
Even if it Hasn’t Happened; it is Different if I am Looking for the Solution
to a Mathematical Problem. Mathematical Expeditions and Polar Expeditions. 418
120 Proof, and the Truth and Falsity of Mathematical Propositions. 423
121 If you Want to Know What was Proved, Look at the Proof. 425
122 Mathematical Problems. Kinds of Problems. Searching. “Tasks” in
Mathematics. 430
123 Euler’s Proof. 434
124 The Trisection of an Angle, etc. 437
125 Trying to Find and Trying. 441
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Induktionsbeweis. Periodizität. 443
126 Inwiefern beweist der Induktionsbeweis einen Satz? 444
127 Der rekursive Beweis und der Begriff des Satzes. Hat der Beweis einen Satz
als wahr erwiesen und sein Gegenteil als falsch? 445
128 Induktion, (x)φx und (∃x)φx. Inwiefern erweist die Induktion den
allgemeinen Satz als wahr und einen Existentialsatz als falsch? 448
129 Wird aus der Anschreibung des Rekursionsbeweises noch ein weiterer Schluß
auf die Allgemeinheit gezogen, sagt das Rekursionsschema nicht schon alles
was zu sagen war? 452
130 Inwiefern verdient der Rekursionsbeweis den Namen eines „Beweises“?
Inwiefern ist der Übergang nach dem Paradigma A durch den Beweis von B
gerechtfertigt? 454
131 Der rekursive Beweis reduziert die Anzahl der Grundgesetze nicht. 464
132 Periodizität. 1 : 3 = 0,3. 466
133 Der rekursive Beweis als Reihe von Beweisen. 468
134 Ein Zeichen auf bestimmte Weise sehen, auffassen. Entdecken eines
Aspekts eines mathematischen Ausdrucks. „Den Ausdruck in bestimmter
Weise sehen“. Hervorhebungen. 473
135 Der Induktionsbeweis, Arithmetik und Algebra. 480
Das Unendliche in der Mathematik. Extensive Auffassung. 482
136 Allgemeinheit in der Arithmetik. 483
137 Zur Mengenlehre. 489
138 Extensive Auffassung der reellen Zahlen. 496
139 Arten irrationaler Zahlen. (π′, P, F) 499
140 Regellose unendliche Dezimalzahl. 504
Anhang I 506
Register 507
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TABLE OF CONTENTS xviiie
Inductive Proofs. Periodicity. 443
126 To what Extent does a Proof by Induction Prove a Proposition? 444
127 Recursive Proof and the Concept of a Proposition. Did the Proof Prove a
Proposition True and its Contradictory False? 445
128 Induction, (x).φx and (∃x).φx. To what Extent does Induction Prove a
Universal Proposition True and an Existential Proposition False? 448
129 Is a Further Inference to Generality Drawn from Writing Down
the Recursive Proof? Doesn’t the Recursion Schema Already Say all that
Needed to be Said? 452
130 To what Extent does a Recursive Proof Deserve the Name “Proof”?
To what Extent is a Step in Accordance with the Paradigm A Justified by
the Proof of B? 454
131 The Recursive Proof Doesn’t Reduce the Number of Fundamental Laws. 464
132 Periodicity. 1 ÷ 3 = 0·3. 466
133 The Recursive Proof as a Series of Proofs. 468
134 Seeing and Understanding a Sign in a Particular Way. Discovering an
Aspect of a Mathematical Expression. “Seeing an Expression in a Particular
Way.” Marks of Emphasis. 473
135 Proof by Induction, Arithmetic and Algebra. 480
The Infinite in Mathematics. The Extensional Viewpoint. 482
136 Generality in Arithmetic. 483
137 On Set Theory. 489
138 The Extensional Conception of the Real Numbers. 496
139 Kinds of Irrational Numbers. (π′, p, f) 499
140 Irregular Infinite Decimals. 504
Appendix I 506
Index 507
WBTA01 30/05/2005 16:12 Page 31
Verstehen.0
WBTC001-06 30/05/2005 16:18 Page 2
Understanding.
WBTC001-06 30/05/2005 16:18 Page 3
1
Das Verstehen, die Meinung, fällt aus
unsrer Betrachtung heraus.
1
Kann man denn etwas Anderes als2
einen Satz verstehen?
Oder aber: Ist es nicht erst ein Satz, wenn man es versteht. Also: Kann man Etwas anders,
als als Satz verstehen?
3
Man möchte4
davon reden, ”einen Satz zu erleben“.
Läßt sich dieses Erlebnis niederschreiben?5
6
Da ist es wichtig, daß es in einem gewissen Sinne keinen halben Satz gibt.
Das heißt, vom halben Satz gilt, was vom Wort gilt, daß er7
nur im Zusammenhang des
Satzes Sinn8
hat.
9
”Das Verstehen fängt aber erst mit dem Satz an. (Und darum interessiert es uns nicht.)“10
11
Wie es keine Metaphysik gibt, so gibt es keine Metalogik. Das Wort ”Verstehen“, der
Ausdruck ”einen Satz verstehen“, ist auch nicht metalogisch, sondern ein Ausdruck wie jeder
andre der Sprache.
12
Man könnte sagen: Was soll uns das Verstehen bekümmern? Wir müssen ja den Satz verstehen,
daß er für uns ein Satz ist!13
Es wäre ja auch seltsam, daß die Wissenschaft und die Mathematik die Sätze gebraucht,
aber von ihrem Verstehen nicht spricht.
14
Man sieht in dem Verstehen das Eigentliche, im Zeichen das Nebensächliche. –
Übrigens, wozu dann das Zeichen überhaupt? – Nur um sich Andern verständlich zu
machen? Aber wie geschieht dies?15
– Man sieht hier das Zeichen als eine Medizin an,16
die im
Andern die gleichen Zustände17
hervorrufen soll, wie ich sie habe.
1 (M): 3
2 (V): als nur
3 (M): ( (R): S 7
4 (V): könnte
5 (M): )
6 (R): [Zu: &das Wort hat nur im Satz Sinn“]
7 (V): es
8 (V): Bedeutung
9 (M): Prüfen: Überlegen: 3
10 (V): Das Verstehen . . . an (und . . . uns nicht).
// Das Verstehen fängt aber erst mit dem Satz
an. //
11 (M): 3
12 (M): 555
13 (V): 555 Wir haben es also in unsern
Betrachtungen mit dem Verstehen des Satzes
nicht zu tun; denn wir selbst müssen ihn
verstehen, damit er für uns ein Satz ist.
14 (M): 3
15 (V): Aber wie ist das // dies // möglich?
16 (V1): – Hier wird das Zeichen als eine Medizin
betrachtet // angesehen (V2): – Wir sehen
hier . . . an, (V3): Man sieht da . . . an,
17 (V): Magenschmerzen // Schmerzen
1
2
2
WBTC001-06 30/05/2005 16:18 Page 4
1
Understanding, Meaning, Drop Out
of Our Considerations.
1
Can one understand something other than2
a proposition?
Or, conversely: Doesn’t it only become a proposition when one understands it? So: Can
one understand something other than as a proposition?
3
One would like to4
talk about “experiencing a proposition”.
Can this experience be written down?5
6
Here it is important that in a certain sense there is no half a proposition.
That is, what is true of a word holds true for half a proposition: it7
only has a sense8
in
the context of the proposition.
9
“Understanding doesn’t begin until there’s a proposition. (And therefore it doesn’t interest
us.)”10
11
Just as there is no metaphysics, there is no metalogic; and the word “understanding”,
the expression “understanding a proposition”, aren’t metalogical. They are expressions of
language, just like all others.
12
Someone might say: Why should understanding be of any concern to us? After all, we have to
understand the proposition for it to be a proposition for us!13
Indeed it’s strange that science and mathematics use propositions, but don’t talk about
understanding them.
14
Understanding is seen as essential, the sign as incidental. – By the way, in that case what
is the point of the sign, anyway? – Only to make oneself understood to others? But how does
this happen?15
– Here the sign is seen as16
a drug that is supposed to produce in someone else
the same feelings17
that I’m experiencing.
1 (M): 3
2 (V): than only
3 (M): ( (R): P 7
4 (V): One could
5 (M): )
6 (M): [To: “a word has sense only in a proposition”]
7 (V): proposition: a word
8 (V): meaning
9 (M): Try out: Consider: 3
10 (V): Understanding . . . (and . . . doesn’t interest
us). // Understanding doesnFt begin until
thereFs a proposition. //
11 (M): 3
12 (M): 555
13 (V): 555 So in our considerations we don’t deal
with understanding a proposition; for we
ourselves have to understand it for it to be a
proposition for us.
14 (M): 3
15 (V): But how is that // this // possible?
16 (V1): – Here the sign is viewed // seen // as
(V2): – Here we see the sign as (V3): Here one
sees the sign as
17 (V): stomach-ache // pains //
2e
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18
Auf die Frage ”was meinst du“, kommt zur Antwort: ,,ich meine p“, & nicht19
”ich meine
das, was ich mit ,p‘ meine“.
20
Die gesamte Sprache kann nicht mißverstanden werden; sonst gäbe es21
zu diesem
Mißverständnis wesentlich keine Aufklärung.22
Die Sprache23
muß für sich selbst sprechen.24
25
Man kann es auch so sagen: wenn man sich immer in einem Sprachsystem ausdrückt
und also, was ein Satz meint, nur durch Sätze dieses Systems erklärt, so fällt am Schluß
die Meinung26
ganz aus der Sprache, also aus der Betrachtung, heraus und es bleibt die Sprache,
das Einzige, was wir betrachten können.
Was ein Satz meint, sagt eine Erklärung.
27
Gesprochenes erklärt man durch die Sprache; darum28
kann man die Sprache (in diesem Sinne)
nicht erklären.
29
Ich will doch sagen: Die ganze Sprache kann man nicht interpretieren.
Eine Interpretation ist immer nur eine im Gegensatz zu einer andern. Sie hängt sich an
das Zeichen und reiht es in ein weiteres System ein.
30
Alles was ich in der Sprache tun kann, ist etwas sagen: das eine sagen. (Das eine sagen
im Raume der Möglichkeiten dessen, was ich hätte sagen können.) (Keine Metalogik.)
31
Wenn Frege gegen die formale Auffassung der Arithmetik spricht, so sagt er gleichsam:
diese kleinlichen Erklärungen, die Symbole betreffend, sind müßig, wenn wir diese verstehen.
Und das Verstehen ist quasi das Sehen32
eines Bildes, aus dem dann alle Regeln folgen
(wodurch sie verständlich werden). Frege sieht aber nicht, daß dieses Bild nur wieder ein
Zeichen ist, oder ein Kalkül, der uns den geschriebenen Kalkül erklärt.
Und, was wir Verstehen einer Sprache nennen, gleicht überhaupt dem Verständnis, welches33
wir34
für einen Kalkül kriegen, wenn wir die Gründe seiner Entstehung,35
oder seine praktische
Anwendung kennen lernen. Und auch da lernen wir nur einen übersichtlichern Symbolismus
statt36
des fremden kennen. (Verstehen heißt hier etwa übersehen.)
37
Wenn komplizierte psychische38
Vorgänge hinter der Front der Symbole beim Verstehen des
Wortes ”und“ eine Rolle spielen und das Verstehen etwas für uns Wesentliches ist, wie39
kommt es, daß40
von ihnen in der Logik nie die Rede ist, noch sein braucht?
18 (M): 3
19 (V): du“, muß zur Antwort kommen: p; und nicht
20 (M): u (R): S. 11 oder S. 172
21 (V): mißverstanden werden. Denn sonst gäbe es
22 (V): Erklärung.
23 (V): Das heißt eben, die ganze Sprache
24 (R): [dazu 3/1]; dazu 3/1
25 (M): u 3
26 (V): am Schluß Meinung
27 (M): u 3
28 (V): Gesprochenes kann man nur durch die
Sprache erklären, darum
29 (M): u 3
30 (M): 3 (R): Zu S. 2/3 etwa zu S. 94
31 (M): 33
32 (V): das Verstehen besteht quasi im Sehen
33 (V): Aber das Verständnis gleicht überhaupt
immer dem, welches
34 (V): wir z.B.
35 (V): wenn wir z.B. seine Entstehung // Genesis,
36 (V): Und natürlich lernen wir auch da wieder
nur einen uns übersichtlichern statt
37 (M): 3 ( (R): Zu S. 108 oder zum Kapitel:
&Begleitet eine Kenntnis der gr. Regeln den Ausdr. d.
Satzes wenn etc.“
38 (V): seelische
39 (V): spielen, wie
40 (V): kommt es, daß diese Vorgänge in der symbolischen
Logik nie erwähnt werden? Wie
kommt es, daß (M): )
3
3 Das Verstehen, die Meinung . . .
WBTC001-06 30/05/2005 16:18 Page 6
18
The question “What do you mean?” is answered by “I mean p”,19
and not “I mean what I
mean by ‘p’.”
20
Language cannot be misunderstood in its entirety;if it could,21
it would be inherently impossible
to clear up22
this sort of misunderstanding.
Language23
must speak for itself.24
25
It can also be put this way: If one always expresses oneself in a system of language and
so uses only propositions of this system to explain what a proposition means, then in the
end meaning drops out of language completely, and thus out of consideration; what remains
is language, the only thing we can consider.
An explanation says what a proposition means.
26
What is spoken is explained by means of language;27
therefore one cannot explain language
(in this sense).
28
I want to say: one can’t interpret language in its entirety.
An interpretation is always just one interpretation, in contrast to another. It attaches itself
to a sign and integrates it into a wider system.
29
All I can do in language is to say something: one thing. (To say one thing within the realm
of the possibilities of what I could have said.) (No metalogic.)
30
When Frege argues against a formal conception of arithmetic he is saying, as it were:
These pedantic explanations of symbols are idle if we understand the symbols. And understanding
is like31
seeing a picture from which all the rules follow (and by means of which
they become understandable). But Frege doesn’t see that this picture is in turn nothing but
a sign, or a calculus, that explains the written calculus to us.
And in general, what we call “understanding a language” is like the understanding32
we get33
of
a calculus when we come to know the reasons for its existence34
or its practical application. In
that case too, we merely come to know a more surveyable symbolism35
in place of the strange
one. (Here “understanding” means something like “having an overview”.)
36
If in understanding the word “and”, complicated psychological37
processes play a role behind
the facade of the symbols, and understanding is something that is essential for us, then38
why
are they never talked about in logic, nor do they need to be?39
18 (M): 3
19 (V): mean?” must be answered by: p;
20 (M): r (R): P. 11 or P. 172
21 (V): entirety. For otherwise
22 (V): impossible to explain
23 (V): And this means, language in its entirety
24 (M): [to 3/1]; to 3/1
25 (M): r 3
26 (M): r 3
27 (V): What is spoken can only be explained with
language,
28 (M): r 3
29 (M): 3 (R): To p. 2/3 perhaps to p. 94
30 (M): 3 3
31 (V): understanding consists in
32 (V): But understanding in general is always like
the understanding
33 (V): get for example
34 (V): know, for example, its origin // genesis
35 (V): application. And of course we come to
know in turn a more surveyable symbolism
for us
36 (M): 3 ( (R): To p. 108 or to the chapter: “Does
a knowledge of gr. rules accompany the expr. of a
sentence when etc.”
37 (V): mental
38 (V): symbols, then
39 (V): why are these processes never mentioned in
symbolic logic? Why are they never talked about
in logic, nor do they need to be? (M): )
Understanding, Meaning . . . 3e
WBTC001-06 30/05/2005 16:18 Page 7
41
Wenn42
ich jemandem einen Befehl gebe, so ist es mir ganz43
genug, ihm Zeichen zu geben.
Und ich würde nie sagen: das sind ja nur Worte, und ich muß hinter die Worte dringen.
Ebenso, wenn ich jemand etwas gefragt hätte und er gibt mir eine Antwort (also ein Zeichen),
bin ich zufrieden – das war gerade, was ich erwartete – und wende nicht ein: das ist ja eine
bloße Antwort. Es ist klar, daß nichts anderes erwartet werden konnte, und daß die Antwort
den Gebrauch des bestimmten Sprachspiels voraussetzte; wie alles, was wir sagen können.44
45
Wenn man aber sagt ”wie soll ich wissen, was er meint, ich sehe ja nur seine Zeichen“,
so sage ich: ”wie soll er wissen, was er meint, er hat ja auch nur seine Zeichen“.
46
”Etwas habe ich aber doch gemeint, als ich das sagte!“ – Gut, aber wie können wir, was
es ist, herausbringen? Doch wohl nur dadurch, daß er es uns sagt. Wenn wir nicht sein übriges
Verhalten als47
Kriterium nehmen sollen, dann also das, was er uns erklärt.
Du meinst, was Du sagst.
41 (M): 33
42 (V): (Im gewöhnlichen Leben$) wenn
43 (V): gan
44 (V): den Gebrauch der Sprache // einer Sprache
// voraussetzte. Wie alles, was zu sagen ist.
45 (M): 3
46 (M): 3
47 (V): zum
4
4 Das Verstehen, die Meinung . . .
WBTC001-06 30/05/2005 16:18 Page 8
40
If 41
I give someone an order then it is quite enough for me to give him signs. And I would
never say: These are mere words, and I have to get behind them. Likewise, if I’ve asked
someone something and he gives me an answer (i.e. a sign), then I’m content – that was
exactly what I expected – and I don’t object: But that is merely an answer. It is clear that
nothing else could be expected, and that the answer presupposed the use of a particular
language-game; as does everything we can say.42
43
But if someone says “How am I supposed to know what he means, all I see are his signs?”,
then I say: “How is he supposed to know what he means? – He too has only his signs.”
44
“But I meant something when I said that!” – Fine, but how can we recover what it was?
Surely only by his telling us. If we’re not to take his other behaviour as45
a criterion, then
we’ll have to take the explanation he gives us.
You mean what you say.
40 (M): 3 3
41 (V): (In ordinary life,) if
42 (V): the use of language. // a language. // As does
everything we say.
43 (M): 3
44 (M): 3
45 (V): for
Understanding, Meaning . . . 4e
WBTC001-06 30/05/2005 16:18 Page 9
2
”Meinen“1
amorph gebraucht.
”Meinen“2
mehrdeutig.
3
”Du hast mit der Hand eine Bewegung gemacht; hast Du etwas damit gemeint? – Ich
dachte, Du meintest, ich solle zu Dir kommen.“
Wie meinte er etwas? Hat er also etwas4
Anderes gemeint, als, was er zeigte.5
Oder ist die Frage
nur: hat er gemeint was er zeigte?
Also er konnte etwas meinen, oder auch nichts meinen. Und wenn er etwas meinte, war es eben
was er zeigte oder etwas Anderes?
Darf man hier fragen: ”was hast Du gemeint“?6
– Auf diese Frage7
kommt ein Satz zur
Antwort. Darf man so nicht fragen, so ist das Meinen – sozusagen – amorph.8
Und ”ich meine
etwas mit dem Satz“ ist dann von ähnlicher Form wie:9
”dieser Satz ist nützlich“, oder ”dieser
Satz greift in mein Leben ein“.
Könnte man auch10
antworten: ”ich habe etwas mit dieser Bewegung gemeint, was ich nur
durch diese Bewegung ausdrücken kann“?11
12
Wir unterscheiden13
Sprache, von dem, was nicht Sprache ist, Schrift von dem, was keine
Schrift ist. Wir sehen Striche etwa auf einer Mauer und sagen,14
wir verstehen sie; und wir sehen
andere, und15
sagen, sie bedeuten nichts (oder, uns nichts). Damit ist doch eine sehr allgemeine
Erfahrung charakterisiert, die wir nennen könnten: ”etwas als Sprache verstehen“ –
ganz abgesehen von dem,16
was wir aus den Strichen (etc.)17
herauslesen. – (Vergleiche: die
Handlungen18
zweier Personen als Züge (Handlungen) eines Spiels verstehen.)
19
Ich sehe eine deutsche Aufschrift und eine chinesische: Ist die chinesische etwa
ungeeignet etwas mitzuteilen?
– Ich sage, ich habe Chinesisch nicht gelernt. Aber dies fällt als20
Ursache, Geschichte, aus
der gegenwärtigen Situation21
heraus. Nur auf seine Wirkungen kommt es an, und die sind
1 (V): ”Verstehen“
2 (V): ”Verstehen“
3 (M): 33
4 (V): Hat er etwas
5 (V): ausdrückte.
6 (V): Die Frage ist, ob man fragen darf, ”was hast
Du gemeint“.
7 (V): Frage (aber)
8 (V): Während, wenn man so nicht fragen darf,
das Meinen – sozusagen – amorph ist.
9 (V): von derselben Form, wie:
10 (V): aber
11 (V): (Könnte . . . ausdrücken kann“?)
12 (M): 3
13 (V): unterscheiden doch
14 (V1): Striche und sagen, (V2): Striche etwa
Folgen von Strichen auf einer Mauer stehen und
sagen,
15 (V): sie; und andere, und wir
16 (V): abgesehen davon,
17 (V): aus dem gegebenen Gebilde
18 (V): herauslesen. (Die Handlungen
19 (M): 3
20 (V): Aber das Lernen der Sprache fällt als
bloße
21 (V): aus der Gegenwart
5
5
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2
“Meaning”1
Used Amorphously.
“Meaning”2
Used Equivocally.
3
“You made a motion with your hand; did you mean something by it? – I thought you
meant I should come to you.”
How did he mean something? Did he mean something other than what he indicated?4
Or is the
question just: Did he mean what he indicated?
So he was able to mean something, or nothing. And if he meant something, was it exactly what
he indicated, or something else?
Is it legitimate to ask here:5
“What did you mean?”? This6
question is answered by a proposition.
If this kind of question is not legitimate7
then meaning is – so to speak – amorphous.
And then “I mean something by the proposition” is similar in form to:8
“This proposition
is useful”, or “This proposition affects my life.”
Could one also9
answer: “I meant something by this motion that I can express only with
this motion”?10
11
We distinguish12
language from what is not language, writing from what is not writing.
We see lines, say on a wall, and we say13
we understand them; and we see other lines and
say14
that they mean nothing (or nothing to us). Surely this sums up a very general common
experience that we might call: “Understanding something as language” – leaving aside what
it is we read out of the lines (etc.).15
(Compare: understanding16
the actions of two people as
moves (actions) in a game.)
17
I see an English inscription and a Chinese one: Is the Chinese one perhaps unsuited for
communicating something?
– I say that I haven’t learned Chinese. But as a cause, or background, this is imma-
terial18
to the present situation.19
All that matters are the effects, and they are phenomena
1 (V): “Understanding”
2 (V): “Understanding”
3 (M): 3 3
4 (V): expressed?
5 (V): The question is whether one can ask:
6 (V): (But) this
7 (V): proposition. Whereas, if one isnFt allowed
to ask this
8 (V): is of the same form as:
9 (V): But could one
10 (V): (Could one also answer: “I meant something
by this motion that I can express only with this
motion”?)
11 (M): 3
12 (V): do distinguish
13 (V1): lines and we say (V2): see sequences of
lines on a wall and we say
14 (V): and we say
15 (V): out of the given structure.
16 (V): (Understanding
17 (M): 3
18 (V): But as a mere cause learning a language is
immaterial
19 (V): to the present.
5e
WBTC001-06 30/05/2005 16:18 Page 11
Phänomene, die eben nicht eintreten, wenn ich das Chinesische anschaue.23
(Warum sie nicht
eintreten, ist24
gleichgültig.)
Geben wir denn den Worten, die uns gesagt werden, willkürliche Interpretationen? Kommt
nicht das Erlebnis des Verstehens mit dem Erlebnis des Hörens der Zeichen, wenn wir ”die
Sprache der Andern verstehen“?25
Wenn mir jemand etwas sagt und ich verstehe es, so geschieht mir dies ebenso, wie, daß
ich, was er sagt, höre.26
27
Und hier ist Verstehen das Phänomen, welches28
sich einstellt, wenn ich einen
deutschen Satz höre, und welches dieses Hören vom Hören eines Satzes einer mir fremden29
Sprache unterscheidet.
30
Denken wir an eine Chiffre: Ein Satz sei mir31
in der Chiffre gegeben und auch der
Schlüssel, dann ist mir32
natürlich, in einer Beziehung,33
alles zum Verständnis der Chiffre
gegeben. Und doch würde ich, gefragt: ”verstehst Du diesen Satz in der Chiffre“,34
antworten: Nein, ich muß ihn erst entziffern; und erst, wenn ich ihn z.B. ins Deutsche übertragen
hätte, würde ich sagen ”jetzt verstehe ich ihn“.
Wenn man hier die Frage stellte: ”In welchem Augenblick der Übertragung (aus der Chiffre
ins Deutsche) verstehe ich den Satz“, so würde man einen Einblick in das Wesen des
Verstehens erhalten.35
36
Ich sage einen Satz ”ich sehe einen schwarzen Kreis“; aber auf die Wörter37
kommt es
doch nicht an; setzen38
wir also statt dieses Satzes ”a b c d e“. Aber nun kann ich nicht ohne
weiteres mit diesem Zeichen den oberen Sinn verbinden (es sei denn, daß ich es als ein Wort
auffasse und dies als Abkürzung des oberen Satzes). Diese Schwierigkeit ist doch aber
sonderbar. Ich könnte sie so ausdrücken: Ich bin nicht gewöhnt statt ”ich“ ”a“ zu sagen und
statt ”sehe“ ”b“, und statt ”einen“ ”c“, etc. Aber damit meine ich nicht, daß ich, wenn ich
daran gewöhnt wäre, mit dem Worte ”a“ sofort das Wort ”ich“ assoziieren würde; sondern,
daß ich nicht gewöhnt bin, ”a“ an der Stelle von ”ich“ zu gebrauchen – in der Bedeutung
von ”ich“.
39
”Ich sage das nicht nur, ich meine auch etwas damit“. – Wenn man sich überlegt,
was dabei in uns vorgeht, wenn wir Worte meinen (und nicht nur sagen), so ist es uns, als
wäre dann etwas mit diesen Worten gekuppelt, während sie sonst leer liefen. – Als ob sie
gleichsam in uns eingriffen.
40
Ich verstehe einen Befehl als Befehl, d.h., ich sehe in ihm nicht nur diese Struktur von
Lauten oder Strichen, sondern sie hat – sozusagen – einen Einfluß auf mich. Ich reagiere
auf einen Befehl (auch ehe ich ihn befolge) anders, als etwa auf eine Mitteilung oder Frage.
(Ich lese ihn in anderem Tonfall mit anderer Geste.)
22 (E): 6a ist die spätere von zwei Versionen von
S. 6.
23 (V): sehe.
24 (V): ist ganz
25 (V): @Geben wir denn . . . wenn wir ,die
Sprache der Andern verstehen‘?!
26 (V): ebenso, wie, daß ich höre, was er sagt.
27 (M): 3
28 (V): Verstehen die Phänomene welche
29 (V): einer mir nicht geläufigen // bekannten
30 (M): 3
31 (V): uns
32 (V): uns
33 (V): in gewisser Beziehung, // in gewissem Sinne,
34 (V): Chiffre“, etwa // vielleicht
35 (V): in das Wesen dessen erhalten, was wir
”verstehen“ nennen.
36 (M): 3
37 (V): Worte
38 (V): sagen
39 (M): / 33
40 (M): / 33 (R): ∀ S. 1/2
6a22
7
6 ”Meinen“ amorph gebraucht. . . .
WBTC001-06 30/05/2005 16:18 Page 12
that simply don’t occur when I look at20
Chinese script. (Why they don’t occur is
irrelevant.21
)
Do we assign arbitrary interpretations to the words that are addressed to us? Doesn’t
the experience of understanding accompany the experience of hearing the signs when we
“understand other people’s language”?22
If someone tells me something and I understand it, then this is as much something that
happens to me as is hearing what he says.23
24
And here understanding is the phenomenon25
that occurs when I hear an English
sentence, and that distinguishes this type of hearing from hearing a sentence in a foreign26
language.
27
Let’s think about a code: Say I’ve28
been given a sentence in the code as well as the key
to it; then of course, in one respect,29
I’ve been given everything necessary to understand the
sentence. And yet if I were asked: “Do you understand this sentence in the code?” I would
answer:30
“No, first I have to decode it”; and only after having transcribed it into English,
for example, would I say “Now I understand it”.
If one were now to ask: “At what moment in the transcribing (from the code into
English) do I understand the sentence?”, that would give us an insight into the nature of
understanding.31
32
I utter the sentence “I see a black circle”; but it’s not the words that matter, so instead
of that sentence let’s write33
“a b c d e”. But now I can’t connect the above sense with this
sign straightaway (unless I understand it as one word, and this as an abbreviation of the
sentence above). But this awkwardness is certainly strange. I could express it this way: I’m
not used to saying “a” instead of “I” and “b” instead of “see” and “c” instead of “a”, etc.
But I don’t mean by this that if I were used to it I would immediately associate the word
“I” with the word “a”; rather, that I am not used to using “a” in place of “I” – in the
sense of “I”.
34
“I’m not merely saying that, I mean something by it.” – When we consider what goes
on in us when we mean words (and don’t just say them), then it seems as if they are geared
with something, whereas otherwise they run in neutral. – As if, so to speak, they meshed
gears with us.
35
I understand a command as a command, i.e. I see in it not merely this structure of sounds
or lines, but this structure influences me, so to speak. I react differently to a command (even
before I obey it) than, for example, to a piece of information or a question. (I read it as
having a different intonation, a different gesture.)
20 (V): I see
21 (V): is entirely irrelevant.
22 (V): !Do we really assign . . . when we ‘understand
other people’s language’?E
23 (V): me as is hearing what he says.
24 (M): 3
25 (V): the phenomena
26 (V): in an unfamiliar // unknown
27 (M): 3
28 (V): we’ve
29 (V): in a certain respect, // in a certain sense,
30 (V) would answer, say: // would maybe
answer:
31 (V): of what we call “understanding”.
32 (M): 3
33 (V): let’s say
34 (M): / 3 3
35 (M): / 3 3 (R): ∀ p. 1/2
“Meaning” Used Amorphously. . . . 6e
WBTC001-06 30/05/2005 16:18 Page 13
41
Der Satz, wenn ich ihn verstehe, bekommt für mich Tiefe.
42
Ich sage: Das Verstehen bestehe darin, daß ich eine bestimmte Erfahrung habe. –
Daß diese Erfahrung aber ein Verstehen ist43
besteht darin, daß diese Erfahrung ein Teil
meiner Sprache ist.
44
In einer Erzählung steht: ”Nachdem er das gesagt hatte, verließ er sie, wie am vorigen Tage“.
Fragt45
man mich, ob ich diesen Satz verstehe, so ist nicht leicht,46
darauf zu antworten. Es
ist ein deutscher Satz und insofern verstehe ich ihn. Ich wüßte, wie man diesen Satz etwa
gebrauchen könnte, ich könnte selbst einen Zusammenhang für ihn erfinden. Und doch
verstehe ich ihn nicht so, wie ich ihn verstünde, wenn ich die Erzählung47
bis zu dieser
Stelle gelesen hätte. (Vergleiche Sprachspiele.)
48
Was heißt es, ein gemaltes Bild zu verstehen? Auch da gibt es Verstehen und
Nichtverstehen; und auch da kann ”verstehen“49
und ”nicht verstehen“ verschiedenerlei heißen.
– Das Bild stellt eine Anordnung von Gegenständen im Raum dar,50
aber einen Teil des Bildes
bin ich unfähig, körperlich zu sehen; sondern sehe dort nur Farbflecke auf der Bildfläche.51
Wir können dann sagen, ich verstehe52
diese Teile des Bildes nicht. Es53
können aber auch
Gegenstände54
auf dem Bild dargestellt sein, die wir noch nie gesehen haben. Und da gibt
es wieder den Fall, wo etwas (z.B.) wie ein Vogel ausschaut,55
nur nicht wie einer, dessen
Art ich kenne; oder aber ein räumliches Gebilde ist dargestellt,56
desgleichen57
ich58
nie
gesehen habe. Vielleicht aber kenne ich alle Gegenstände, verstehe aber – in anderem Sinne –
ihre Anordnung nicht.59
60
Angenommen,61
das Bild stellte62
Menschen dar63
und die Menschen darauf wären
etwa ein Zoll64
lang. Angenommen nun, es gäbe Menschen, die diese Länge hätten, so
könnten wir diese65
in dem Bild erkennen und es würde uns nun einen ganz andern
Eindruck machen, als den gewöhnlichen.66
D.h.67
es spielt in diesen Eindruck nicht die Erinnerung
41 (M): 33 (R): 1 S. 19/2
42 (M): ////
43 (V): aber das Verstehen dessen ist – was ich
verstehe –
44 (M): / 33
45 (V): Bedenke auch: Man kann manchen Satz nur
im Zusammenhang mit anderen verstehen.
Wenn ich z.B. irgendwo lese: ”nachdem er das
gesagt hatte, verließ er sie, wie am vorigen
Tag“; fragt
46 (V): so wäre (es) nicht ganz leicht,
47 (V): ich das Buch
48 (M): 3
49 (V): gibt es Verständnis und Nichtverstehen.
Und auch hier kann ”Verstehen“
50 (V): – Wir können uns ein Bild denken, das eine
Anordnung von Gegenständen im dreidimensionalen
Raum darstellen soll, // Das Bild soll eine
Anordnung von Gegenständen im dreidimensionalen
Raum darstellen,
51 (V): aber wir sind // ich bin // für einen Teil
des Bildes unfähig, Körper im Raum darin
zu sehen; sondern sehen nur die gemalte
Bildfläche.
52 (V): sagen, wir verstehen
53 (V): nicht. Es kann sein, daß die räumlichen
Gegenstände, die dargestellt sind, uns bekannt,
d.h. Formen sind, die wir aus der Anschauung
von Körpern her kennen; es
54 (V): Formen
55 (V): wo etwas – z.B. – wie ein Vogel aussieht,
56 (V): oder aber$ wo ein räumliches Gebilde
dargestellt ist,
57 (O): dergleichen
58 (V): ich noch
59 (V): habe. Auch in diesen Fällen kann man von
einem Nichtverstehen des Bildes reden, aber in
einem anderen Sinne als im ersten Fall.
60 (M): 3
61 (V): Aber noch etwas: Angenommen,
62 (O): stellte den
63 (V): dar, wäre aber klein
64 (V): etwa einen Meter
65 (V): so würden wir sie
66 (V): machen, obwohl doch die Illusion der
dreidimensionalen Gegenstände ganz dieselbe
wäre.
67 (O): d.h.
8
7 ”Meinen“ amorph gebraucht. . . .
WBTC001-06 30/05/2005 16:18 Page 14
36
When I understand it, a proposition acquires depth for me.
37
I say: Understanding consists in my having a particular experience. – –
But that this experience is understanding38
consists in its being a part of my language.
39
In a story it says:40
“After he said that he left her, as he had done the day before”. If I am
asked whether I understand this sentence, there’s no easy answer.41
It’s an English sentence
and in that respect I understand it. I would know, for instance, how one could use this sentence,
I could come up with a context of my own for it. And yet I don’t understand it in the
way I would understand it if I had read the story42
up to that point. (Cf. language-games.)
43
What does it mean to understand a painted picture? Here too there is understanding and
a failure to understand! And here too “understanding” and “failure to understand” can mean
different things. – The picture represents an arrangement of objects in space,44
but I am incapable
of seeing a part of the picture three-dimensionally; rather, in that part I see only patches
of colour on the surface of the picture.45
So we can say that I46
don’t understand those parts
of the picture. But it’s47
also possible that objects48
we’ve never seen are portrayed in the picture.
And under this rubric there’s the case where something looks like a bird, for instance,
but just not like one whose species I know; or on the other hand a49
three-dimensional object
is represented, the likes of which I’ve never seen.50
Or maybe I know all of the objects, but – in
another sense – don’t understand how they’re arranged.51
52
Let’s assume53
that the picture portrayed people,54
and the people were about an inch55
tall. Now let’s further assume that there were people of this height: then we could56
recognize
them in the picture, but the impression it would make on us would be quite different
36 (M): 3 3 (R): 1 p. 19/2
37 (M): ////
38 (V): is an understanding of – what I understand
–
39 (M): / 3 3
40 (V): Also consider: Some propositions can only be
understood in connection with others. If for
instance I read somewhere:
41 (V): sentence, (it) wouldnFt be very easy to
answer.
42 (V): book
43 (M): 3
44 (V): – We can imagine a picture that is supposed
to represent an arrangement of objects in threedimensional
space, // The picture is supposed to
represent an arrangement of objects in threedimensional
space,
45 (V): but we are // I am // incapable of seeing bodies
in space in one part of the picture; rather,
we see only the painted surface.
46 (V): we
47 (V): picture. ItFs possible for the three-dimensional
objects that are represented to be known
to us, i.e. to be forms that we know from
observing bodies; but it’s
48 (V): forms
49 (V): hand where a
50 (V): yet seen.
51 (V): arranged. In these cases too one can speak
of not understanding the picture, but in a different
sense than in the first case.
52 (M): 3
53 (V): But something else: let’s assume
54 (V): people, but that it were small,
55 (V): about a metre
56 (V): would recognize them
“Meaning” Used Amorphously. . . . 7e
WBTC001-06 30/05/2005 16:18 Page 15
hinein,68
daß ich einmal Menschen in der gewöhnlichen Größe, und nie Zwerge, gesehen
habe, wenn auch dies die Ursache des Eindrucks ist.
69
Dieses Sehen der gemalten Menschen als Menschen (im Gegensatz etwa zu Zwergen)
ist ganz analog dem Sehen der Zeichnung70
als dreidimensionalem71
Gebilde. Wir können
hier nicht sagen, wir sehen immer dasselbe und fassen es nachträglich, einmal als das Eine,72
einmal als das Andre auf, sondern wir sehen jedes Mal etwas Anderes.
73
Und so auch, wenn wir einen Satz mit Verständnis und ohne Verständnis lesen.
(Erinnere Dich daran, wie es ist, wenn man einen Satz mit falscher Betonung liest, ihn daher
nicht versteht, und nun auf einmal daraufkommt, wie er zu lesen ist.)
74
(Lesen einer schleuderhaften Schrift.)75
76
Wenn man eine Uhr abliest, so sieht man einen Komplex von Strichen, Flecken etc.,
aber auf ganz bestimmte Weise, wenn man ihn als Uhr und Zeiger auffaßt.77
78
Wir könnten uns den Marsbewohner denken, der auf der Erde erst nach und nach den
Gesichtsausdruck der Menschen als solchen verstehen lernte und den drohenden erst nach
gewissen Erfahrungen als solchen empfinden lernt. Er hätte bis dahin diese Gesichtsform
angesehen,79
wie wir die Form eines Steins betrachten.
80
Kann ich81
nicht sagen: er lernt erst die befehlende Geste in einer gewissen Satzform
verstehen?
82
Chinesische Gesten verstehen wir so wenig, wie chinesische Sätze. [D.h. es gibt nicht nur83
Unverständnis für Sätze. Wie aber lernen wir die Sprache fremder Gesten? Sie können uns durch Worte
erklärt werden. Man kann uns sagen ,,das ist bei diesem Volk eine höhnische Gebärde“, etc. Oder
aber wir lernen die Gebärden verstehen wie wir als Kind die Gebärden & Mienen der Erwachsenen
– ohne Erklärung – verstehen lernen. Und verstehen lernen heißt eben in diesem Sinne nicht erklären
lernen & wir verstehen dann die Miene, können sie aber nicht durch einen andern Ausdruck erklären.]
68 (V1): gewöhnlichen. Und doch ist // besteht // der
tatsächliche // dieser tatsächliche // Eindruck, wie
er da ist, unabhängig davon, (V2): Und doch
spielt in den Eindruck, den ich habe // den ich beim
Anblick des Bildes habe // nicht die Erinnerung hinein,
69 (M): 3
70 (V): dem Sehen des Bildes
71 (O): dreidimensionales
72 (V): Eine und
73 (M): 3
74 (M): 3
75 (V): (Beim Lesen einer schleuderhaften Schrift
kann man erkennen, was es heißt, etwas in das
gegebene Bild hineinsehen.)
76 (M): 3
77 (V): Zeiger auffassen will.
78 (R): Zu &lernen der Sprache“
79 (V): angeschaut,
80 (R): Zu &lernen der Sprache“
81 (V): ich so
82 (R): Zu: &lernen der Sprache“
83 (V): nur für Sätze
9
10
8 ”Meinen“ amorph gebraucht. . . .
WBTC001-06 30/05/2005 16:18 Page 16
from the usual one.57
That is, my memory that I have seen humans of normal size and never
dwarfs is not a factor in this impression58
, even if it is its cause.
59
Seeing people in paintings as people (as opposed, for example, to dwarfs) is completely
analogous to seeing a drawing60
as something three-dimensional. Here we can’t say that each
time we see the same thing and subsequently understand it now as one thing, now61
as another;
rather, we see something different each time.
62
And that’s the way it is when we read a sentence with and without understanding.
(Remember how it is when you read a sentence with the wrong intonation and as a result
you don’t understand it, and then suddenly you discover how it ought to be read.)
63
(Reading a sloppy handwriting.)64
65
When one reads a clock one sees an aggregate of lines, spots, etc., but in a very particular
way if one takes the aggregate as66
a clock and hands.
67
We can imagine a Martian who on earth only gradually learns to understand human facial
expressions for what they are and doesn’t learn to sense a threatening one until after he has
suffered certain experiences. Until then he would have looked at this facial shape the way
we regard the shape of a stone.
68
Can’t I say:69
He doesn’t learn to understand the gesture of commanding until it appears
in a certain sentential form?
70
We don’t understand Chinese gestures any better than Chinese sentences. [That is, the
failure to understand isn’t limited71
to sentences. For how do we learn the language of foreign
gestures? They can be explained to us in words. We can be told “Among these people this is a
derisive gesture”, etc. Or, on the other hand, we learn to understand these gestures the way we
learned as children to understand the gestures and facial expressions of grown-ups – without
explanation. And in this sense learning to understand does not mean learning to explain, and so
we understand the facial expression, but can’t explain it by any other means.]
57 (V): usual one, even though the illusion of the
three-dimensional objects would be precisely the
same.
58 (V1): And yet the // this // actual impression, as
it occurs, is independent // exists independently
// of the fact that I have seen humans of normal
size, and never dwarfs. (V2): And yet my
memory is not a factor in the impression that I have
// that I have when I see the picture // that I have
seen humans of normal size, and never dwarfs.
59 (M): 3
60 (V): a picture
61 (V): and now
62 (M): 3
63 (M): 3
64 (V): (In reading a sloppy handwriting one can
see what it means to project something into a
given image.)
65 (M): 3
66 (V): one wants to take it as
67 (R): To “learning a language”
68 (R): To “learning a language”
69 (V): therefore say:
70 (R): To: “learning a language”
71 (V): limited to sentences
“Meaning” Used Amorphously. . . . 8e
WBTC001-06 30/05/2005 16:18 Page 17
3
Das Verstehen als Korrelat einer
Erklärung.
1
Ich meine mit dem Wort &Verstehen“ 2
ein Korrelat der Erklärung des Sinnes, nicht einer –
etwa medizinischen – Beeinflussung.
Mit dem Worte ”Mißverständnis“ meine ich also wesentlich etwas, was sich durch Erklärung
beseitigen läßt. Eine andere Nichtübereinstimmung nenne ich nicht ”Mißverständnis“.
3
Verständnis entspricht der Erklärung; soweit es aber der Erklärung nicht entspricht, ist
es unartikuliert und interessiert uns darum nicht;4
oder es ist artikuliert und entspricht dem
Satz selbst, dessen Sinn wir wiedergeben wollen.5
6
Wissen, was der Satz besagt, kann nur heißen: die Frage beantworten können ”was sagt er?“.
7
Den Sinn eines Satzes kennen,8
kann nur heißen:9
die Frage ”was ist sein Sinn“ beantworten
können.
10
Denn ist hier ”Sinn haben“ quasi intransitiv gebraucht, so daß man also nicht den Sinn
eines Satzes von dem eines anderen Satzes unterscheiden kann, dann ist das Sinnhaben ein11
den Gebrauch des Satzes begleitender Vorgang, der12
uns nicht interessiert.
13
Das Triviale, was ich zu sagen habe, ist, daß auf den Satz ”ich sage das nicht nur, ich
meine etwas damit“ und die Frage ”was?“, ein weiterer Satz, in irgend welchen Zeichen,
zur Antwort kommt.
14
Aber man kann fragen: Ist denn das Verständnis nicht etwas anderes als der Ausdruck
des Verständnisses? Ist es nicht so, daß der Ausdruck des Verständnisses eben ein unvollkommener
Ausdruck ist?
Das heißt doch wohl, ein Ausdruck, der etwas ausläßt, was wesentlich unausdrückbar ist.
Denn sonst könnte ich ja15
einen bessern finden. Also wäre der Ausdruck ein vollkommener
Ausdruck. –
1 (M): u
2 (V): ”Verstehen“, damit meine ich
3 (M): u (R): S. 2/3 ∀ ?
4 (V): und geht uns deswegen nichts an;
5 (V): selbst, dessen Verständnis wir beschreiben
wollten.
6 (M): 3
7 (M): 3
8 (V): verstehen,
9 (V): kennen, soll heißen:
10 (M): u
11 (V): ein Vorgang
12 (V): Sinnhaben eine, den Gebrauch des Satzes
begleitende, Angelegenheit, die
13 (M): ∫
14 (M): ? /
15 (V): ja eben
11
12
9
WBTC001-06 30/05/2005 16:18 Page 18
3
Understanding as a Correlate of an
Explanation.
1
I mean by the word “understanding”2
a correlate of an explanation of sense, not of an – say,
a medicinal – influence.
So by the word “misunderstanding” I essentially mean something that can be removed
with an explanation. I don’t call just any sort of lack of agreement a “misunderstanding”.
3
Understanding correlates with explanation; and in so far as it doesn’t, it is unarticulated
and therefore doesn’t interest us;4
or it is articulated and correlates with the proposition itself,
whose sense we want to render.5
6
To know what a proposition says can only mean: to be able to answer the question “What
does it say?”.
7
To know8
the sense of a proposition can only mean:9
being able to answer the question
“What is its sense?”.
10
For if “to have sense” is used intransitively, as it were, so that one can’t distinguish
the sense of one proposition from that of another, then having sense is a process that
accompanies the use of the proposition,11
and this process doesn’t interest us.
12
The trivial thing I have to say is that the sentence “I’m not only saying this, I mean something
by it” and the question “What?” are answered by a further sentence that is expressed
in some sort of signs.
13
But one can ask: Isn’t understanding something different from the expression of understanding?
Isn’t it the case that the expression of understanding is inherently an incomplete
expression?
And that surely means – an expression that omits something that is essentially inexpressible.
For otherwise I could find a better expression. And such an expression would be a complete
expression. –
1 (M): r
2 (V): “Understanding” – by this I mean
3 (M): r (R): P 2/3 ∀?
4 (V): therefore is of no concern to us;
5 (V): itself, whose understanding we wanted to
describe.
6 (M): 3
7 (M): 3
8 (V): understand
9 (V): proposition is supposed to mean:
10 (M): r
11 (V): sense is a matter accompanying the use of
the sentence,
12 (M): ∫
13 (M): ? /
9e
WBTC001-06 30/05/2005 16:18 Page 19
16
Es ist eine sehr häufige Auffassung: daß Einer gleichsam nur unvollkommen sein
Verständnis zeigen kann.17
Daß er gleichsam nur immer aus der Ferne darauf deuten, auch sich ihm nähern, es aber
nie mit der Hand ergreifen18
kann. Und das Letzte immer ungesagt bleiben muß.
19
Man will etwa sagen: Er versteht was Du ihm befohlen hast20
zwar ganz, kann dies aber
nicht ganz zeigen, da er sonst schon tun müßte, was ja erst in21
Befolgung des Befehls geschehen
darf.22
So kann er also nicht zeigen, daß er es ganz versteht. D.h. also, er weiß immer mehr,
als er zeigen kann.
23
Man möchte sagen: er ist mit seinem Verständnis bei24
der Ausführung,25
aber die Erklärung
kann nie die Ausführung enthalten.
Aber das Verständnis enthält nicht die Ausführung, sondern ist nur das Symbol, das bei
der Ausführung übersetzt wird.
26
Der Weg dazu, die Grammatik des Wortes ”meinen“ klar zu sehen, führt über die Fragen
”welches27
ist das Kriterium dafür, daß wir etwas so meinen“ und welcher Art ist der Ausdruck,
den dieses ”so“ vertritt. 28
Die Antwort auf die Frage ”wie ist das gemeint“ stellt29
die
Verbindung zwischen zwei Sprachen30
her. Also fragt auch die Frage nach dieser
Verbindung. Der Gebrauch der Hauptwörter ”Sinn“, ”Bedeutung“, ”Auffassung“ und anderer
Wörter verleitet uns zu glauben, daß dieser Sinn etc. dem Zeichen so gegenübersteht, wie
das Wort – der Name – dem Ding,31
das sein Träger ist. So daß man sagen könnte: ”Das
Zeichen hat eine ganz bestimmte Bedeutung,32
ist in einer ganz bestimmten Weise gemeint,
die ich nur faute de mieux wieder durch ein Zeichen ausdrücken muß“. Die Meinung, die
Intention wäre quasi seine Seele, die ich am liebsten direkt zeigen möchte, aber auf die ich
leider nur indirekt durch ihren Körper hinweisen kann. –
33
Wenn ich um den Sinn eines Pfeils zu erklären sage: ”ich meine diesen Pfeil so, daß man
ihm durch eine Bewegung in der Richtung vom Schwanz zur Spitze folgt“, so gebe ich eine
Definition (ich setze ein Zeichen für ein andres), während es scheint, als hätte ich sozusagen
die Angabe die der Pfeil meint34
ergänzt. Ich habe den Pfeil durch ein neues Zeichen ersetzt,
das wir statt des Pfeiles gebrauchen können. – Gebrauchen können – . Während es scheint, als
wäre der Pfeil selbst wesentlich unvollständig,35
ergänzungsbedürftig, und als hätte ich ihm
nur36
die nötige Ergänzung gegeben. Wie man eine Beschreibung eines Gegenstandes als unvollkommen
erkennt und vervollständigen kann.37
Als hätte der Pfeil die Beschreibung angefangen
und wir sie durch den Satz vollendet. – Auch so: Wenn ich, wie oben, sage ”ich meine
16 (M): /
17 (V): Es ist eine häufige // geläufige //
Auffassung, daß Einer gleichsam nur unvollkommen
zeigen kann$ ob er einen Satz [ein
Zeichen (einen Befehl)] verstanden hat.
18 (V): berühren
19 (M): ? /
20 (V): versteht es // den Befehl
21 (V): die
22 (V): soll.
23 (M): ////
24 (V): bei
25 (V): bei der Tatsache,
26 (M): ∫
27 (V): Die Schwierigkeit ist, die Grammatik des
Wortes ”meinen“ klar zu sehen. Aber der Weg
dazu ist nur der$ über die Antwort auf die
Frage ”welches
28 (M): /
29 (V): hält
30 (V): zwischen zwei sprachlichen Ausdrücken
31 (V): das Wort$ – der Name$ – dem Ding,
32 (V): könnte: @Der Pfeil hat eine ganz bestimmte
Bedeutung!,
33 (M): ? /
34 (V1): sozusagen die Aussage // Angabe // des
Pfeils (V2): sozusagen die Aussage die der
Pfeil meint
35 (V): unvollkommen,
36 (V): nun
37 (V): und vervollständigt.
13
10 Das Verstehen als Korrelat . . .
WBTC001-06 30/05/2005 16:18 Page 20
14
It’s an often held view that one can show one’s understanding only incompletely, as it were.15
That one can only point to it from afar, as it were, can get closer to it, but can never grab16
it with one’s hand. And that what finally matters must always remain unsaid.
17
One wants to say, for instance: There’s no doubt he understands completely what you
told him to do,18
but he can’t show this completely. If he could, he would have to do what
isn’t allowed19
to happen until he obeys the order. So he cannot show that he understands
it completely. That is to say, he always knows more than he can show.
20
One would like to say: His understanding is right next to21
the execution of an order22
,
but his explanation can never contain this execution.
And understanding an order doesn’t contain its execution; understanding is just a
symbol that is translated in the process of the execution.
23
The road to a clear view of the grammar of the word “mean” goes via the questions24
“What
is the criterion for our meaning something thus?”, and what sort of thing is the expression
for which this “thus” stands?25
The answer to the question “How is that meant?” establishes
the connection between two languages.26
Therefore the question too asks about this connection.
The use of the nouns “sense”, “meaning”, “understanding”, and of other words seduces us
into believing that this sense, etc., stands opposite the sign in the same way as a word – a
name – stands opposite the thing that is its bearer. So one could say: “The sign27
has a very
specific meaning, is meant in a very specific way, which I have to express through yet another
sign, only because I lack something better.” Meaning, intention, is as it were its soul, which
I would prefer to point at directly, but to which, unfortunately, I can only point indirectly,
via its body. –
28
If in order to explain the sense of an arrow, I say: “This is how I mean this arrow: you
follow it by moving in the direction from the tail to the point”, then I’m giving a definition
(I’m substituting one sign for another), whereas it seems as if I had supplemented the
indication that the arrow intended29
, so to speak. I have replaced the arrow with a new sign
that we can use instead of the arrow. – Can use –. Whereas it seems as if the arrow itself
were essentially incomplete30
, requiring completion, and as if all I had done31
was to give it
the requisite completion. As one recognizes a description of an object as incomplete and can
complete it.32
As if the arrow had begun the description and we had completed it with our
sentence. – Here is another way of putting it: If I say, as I did above, “This is how I mean
this arrow: . . .”, then this gives the impression that only now have I described what is essential,
the meaning; as if the arrow were merely a musical instrument, but the meaning was
the music, or better still: as if the arrow were the sign – i.e. in this case, the cause of my
14 (M): /
15 (V): It’s an often // ordinary // view that one can,
as it were, only show incompletely whether he
has understood a proposition [a sign (a command)].
16 (V): touch
17 (M): ? /
18 (V): understands it completely // understands the
command completely
19 (V): isn’t supposed
20 (M): ////
21 (V): right next to
22 (V): right next to the fact
23 (M): ∫
24 (V): The difficulty lies in seeing clearly the
grammar of the word “mean”. But the only way
there is via the answer to the question
25 (M): /
26 (V): two linguistic expressions.
27 (V): “The arrow
28 (M): ? /
29 (V1): supplemented the statement // indication
// of the arrow (V2): supplemented the
statement that the arrow makes
30 (V): essentially imperfect
31 (V): had done then
32 (V): and completes it.
Understanding as a Correlate of an Explanation. 10e
WBTC001-06 30/05/2005 16:18 Page 21
diesen Pfeil so, daß . . .“, so macht es den Eindruck, als hätte ich jetzt erst das Eigentliche
beschrieben, die Meinung; als wäre der Pfeil gleichsam nur das Musikinstrument, die Meinung
aber die Musik, oder besser: der Pfeil, das Zeichen – das heißt in diesem Falle – die Ursache
des inneren, seelischen, Vorgangs, und die Worte der Erklärung erst die Beschreibung dieses
Vorgangs. Hier spukt die Auffassung des Satzes als eines Zeichens des Gedankens; und des
Gedankens als eines Vorgangs in der Seele, oder im Kopf.
38
Was die Erklärung des Pfeiles betrifft, so ist es klar, daß man sagen kann: ”Dieser Pfeil
sagt39
nicht, daß Du dorthin (mit der Hand zeigend) gehen sollst, sondern dahin.“ – und daß
diese Erklärung verstanden werden könnte.40
38 (M): ? / ///
39 (V): bedeutet
40 (V): dahin.“ – Und ich würde diese Erklärung
natürlich verstehen. – @Das müßte man aber
dazuschreiben!.
14
11 Das Verstehen als Korrelat . . .
WBTC001-06 30/05/2005 16:18 Page 22
inner mental process, and its description had to wait for the words of the explanation. Here
the idea of a proposition as a sign of a thought is haunting us; as is that of a thought as a
process in one’s soul, or in one’s head.
33
As far as the explanation of the arrow is concerned, it’s clear that one can say: “This
arrow doesn’t say34
that you should go that way (motioning with one’s hand), but this way”
– and that this explanation could be understood.35
Understanding as a Correlate of an Explanation. 11e
33 (M): ? / ///
34 (V): mean
35 (V): but this way.” – And of course I would
understand this explanation. – !But one would
have to add that in writing.E
WBTC001-06 30/05/2005 16:18 Page 23
4
Das Verstehen des Befehls, die
Bedingung dafür, daß wir ihn
befolgen können.
Das Verstehen des Satzes, die
Bedingung dafür, daß wir
uns nach ihm richten.
1
”Das Verständnis eines Satzes kann nur die Bedingung dafür sein, daß wir ihn anwenden
können. D.h., es kann nichts sein, als die2
Bedingung und es muß die Bedingung der
Anwendung sein.“
3
Wenn ”einen Satz verstehen“ heißt, in bestimmter Weise4
nach ihm handeln, dann kann
das Verstehen nicht die logische Bedingung dafür sein, daß wir nach ihm handeln.
5
Das Kriterium des Verstehens6
ist manchmal ein Vorgang des Übersetzens7
des Zeichens in eine8
Handlung; wir übertragen den Satz in andere Zeichen,9
wir zeichnen nach der Beschreibung ein Bild
oder stellen uns eins vor; etc.10
11
Das Verstehen einer Beschreibung kann man mit12
dem Zeichnen eines Bildes nach dieser
Beschreibung vergleichen. (Und hier ist wieder das Gleichnis ein besonderer Fall dessen,
wofür es ein Gleichnis ist.) Und es wird13
auch in vielen Fällen als das Kriterium14
des
Verständnisses aufgefaßt.
15
Wir reden von dem Verständnis eines Satzes vielfach als der Bedingung dafür, daß wir ihn anwenden
können. Wir sagen &Wir können einen Befehl nicht befolgen, wenn wir ihn nicht verstehen“ oder
&ehe wir ihn verstehen“. (das Wort &können“, &muß“ verdächtig)16
17
Ich verstehe dieses Bild genau, ich könnte es in Ton kneten.18
– Ich verstehe diese
Beschreibung genau, ich könnte eine Zeichnung nach ihr machen.
1 (M): ? / 3
2 (V): diese
3 (M): / 3
4 (V): in gewissem Sinn
5 (M): ? /
6 (V): Was wir (verstehen lernenA nennen
7 (V): des Übertragens% Übersetzens%
8 (V): eine andere
9 (V): Satz in eine andere Sprache,
10 (R): 1 15/3
11 (M): / 3
12 (V): man$ mit
13 (V): würde
14 (V): als der Beweis
15 (M): ? / (R): Zu § 4
16 (O): verdachtig] (M): 1 15/4 1 17/1,2
17 (M): / 3
18 (V): es plastisch wiedergeben.
15
14v
15
14v
15
12
WBTC001-06 30/05/2005 16:18 Page 24
4
Understanding a Command
the Condition for Our Being
Able to Obey It.
Understanding a Proposition
the Condition for Our Acting
in Accordance with It.
1
“Understanding a proposition can only be the condition for our being able to apply it.
That is, it can’t be anything other than the2
condition, and it must be the condition for the
application.”
3
If “understanding a proposition” means acting in accordance with it in a particular way4
,
then understanding cannot be the logical condition for our acting in accordance with it.
5
The criterion of understanding6
is sometimes a process of translating7
a sign into an8
action;
we transcribe the sentence into other signs9
, we draw a picture based on a description, or we
imagine a picture; etc.10
11
One can compare understanding a description with drawing a picture based on that description.
(And here again the simile is a particular case of what it is a simile of.) And indeed in
many cases it is12
taken as the criterion13
of understanding.
14
We often speak of understanding a proposition as the prerequisite for being able to apply it.
We say “We can’t obey a command if we don’t understand it” or “until we understand it”. (The
words “can”, “must”, are fishy.)15
16
I understand this picture exactly. I could knead it in clay.17
– I understand this description
exactly, I could make a drawing from it.
1 (M): ? / 3
2 (V): this
3 (M): / 3
4 (V): in a certain sense
5 (M): ? /
6 (V): What we call !understandingE
7 (V): transcribing$ translating$
8 (V): a different
9 (V): into another language
10 (R): 1 15/3
11 (M): / 3
12 (V): it would be
13 (V): proof
14 (M): ? / (R): To § 4
15 (M): 1 15/4 1 17/2
16 (M): / 3
17 (V): I could reproduce it as a sculpture.
12e
WBTC001-06 30/05/2005 16:18 Page 25
19
Wenn hier unter dem &Verstehen“ ein psychischer Vorgang gemeint ist20
& gesagt werden soll,
daß dieser Vorgang erfahrungsgemäß eintreten muß21
ehe ein Mensch einen Befehl befolgen kann,
so interessiert uns diese Aussage nicht. – Sollte definiert werden, den Befehl befolgen22
heiße man
es nur, wenn jener psychische Vorgang eingetreten sei, so wäre diese Definition müßig.
Soll aber &verstehen“ hier heißen: erklären können, – warum sollte das notwendig sein um den
Befehl zu befolgen. Natürlich handelt es sich hier nicht um logische Notwendigkeit.23
24
Man könnte es in gewissen Fällen25
als Kriterium des Verstehens26
festsetzen, daß man
den Sinn des Satzes muß zeichnerisch darstellen können.
27
Es ist sonderbar: eine Geste möchten wir durch Worte erklären und Worte durch diesen
entsprechende Gesten.28
29
Und wirklich werden wir Worte durch eine Geste und eine Geste durch Worte erklären.
30
Wenn man mir sagt ”bringe eine gelbe Blume“ und ich stelle mir vor, wie ich eine gelbe
Blume hole, so kann das zeigen, daß ich den Befehl verstanden habe. Aber ebenso, wenn
ich ein Bild des Vorgangs male. – Warum? Wohl, weil das, was ich tue, mit Worten des
Befehls beschrieben werden muß. Oder soll ich sagen, ich habe tatsächlich einen (dem ersten)
verwandten Befehl ausgeführt.
31
Nun ist die Frage: Muß ich wirklich in so einem Sinne das Zeichen verstehen, um etwa
darnach handeln zu können? – Wenn jemand sagt: ”gewiß! sonst wüßte ich ja nicht, was ich
zu tun habe“, so würde ich antworten: Aber vom Wissen zum Tun ist ja doch wieder ein Sprung.32
33
Was heißt dann also der Satz: ”Ich muß den Befehl verstehen, ehe ich nach ihm handeln
kann“? Denn dies zu sagen, hat34
natürlich einen Sinn. Aber jedenfalls35
wieder keinen
metalogischen.
36
Die Idee, die man von dem Verstehen hat, ist etwa, daß man dabei von dem Zeichen
näher an die verifizierende Tatsache kommt, von den Worten des Befehls näher zur Ausführung,
etwa durch die Vorstellung. Und wenn man auch nicht wesentlich, d.h. logisch, näher kommt,
so ist doch etwas an der Idee richtig, daß das Verstehen in dem Vorstellen der Tatsache
besteht. Die Sprache der Vorstellung ist in dem gleichen Sinne wie die Gebärdensprache
primitiv.
19 (M): ////
20 (V): Wenn hier das Verstehen ein psychischer
Vorgang ist der
21 (V): erfahrungsgemäß eintritt
22 (V): werden, befolgen
23 (O): Notwendigkeit
24 (M): / 3
25 (V): Fällen geradezu
26 (V): Verständnisses
27 (M): / 3 (R): Zu S. 42
28 (V1): Es ist sehr sonderbar: Das Verstehen
einer Geste möchten // werden // wir durch ihre
// mit Hilfe ihrer // Übersetzung in Worte
erklären und das Verstehen von Worten durch
eine Übersetzung in Gesten. (V2): Es ist sehr
sonderbar: Wir sind versucht, das Verstehen
einer Geste durch ihr entsprechende Worte
zu erklären, und das Verstehen von Worten
durch diesen entsprechende Gesten. // . . .
das Verstehen einer Geste als Fähigkeit zu
erklären, sie in Worte zu übersetzen, . . .
29 (M): / 3 (R): Zu S. 42
30 (M): 555
31 (M): umgearb. / 3
32 (V): antworten: ”Aber es gibt ja keinen Übergang
vom Wissen zum Tun. /// Und keine
prinzipielle Rechtfertigung dessen, daß es das
war, was dem Befehl entsprach“.
33 (M): ∫ 3
34 (V): Denn dieser Satz hat
35 (V): gewiß
36 (M): überarb. / 3
14v
15
16
17
13 Das Verstehen des Befehls . . .
WBTC001-06 30/05/2005 16:18 Page 26
18
If what is meant by “understanding” is a psychological process19
, and if it’s insisted that
empirically this event must occur20
before someone can obey a command, then such a statement
wouldn’t interest us. – If it were so defined that one could only call it obeying the command if21
that psychological process had taken place, then such a definition would be idle.
But if in this context “understanding” is to mean: to be able to explain – why should this be
necessary in order to obey the command? Of course, here it isn’t a matter of logical necessity.
22
In certain cases one could stipulate that the ability to represent the sense of a proposition
with a drawing is a criterion of understanding it.
23
It’s strange: we’d like to explain a gesture with words, and words with the gestures that
correspond to them.24
25
And indeed, we do explain words with a gesture, and a gesture with words.
26
If I’m told “Bring me a yellow flower” and I imagine myself getting a yellow flower,
that can show that I understood the command. But so can my painting a picture of this
process. – Why? Most likely because what I’m doing has to be described in the words of
the command. Or should I say that what I’ve really done is to carry out a command that’s
related (to the original one)?
27
So the question is: Do I really have to understand the sign in a particular way if I’m to
be able to follow it? – If someone says: “Certainly! For otherwise I wouldn’t know what I’m
supposed to do”, then I’d answer: “But there’s still another leap to be made from knowing
to doing.”28
29
So what does this proposition mean: “I have to understand the command before I can
follow it”? For saying this30
does of course have a sense. But once again, not31
a metalogical
one.
32
The idea one has about understanding is roughly that it is a process that takes one –
let us say through one’s imagination – from the sign closer to the fact that verifies it, from
the words of the command closer to its execution. And even if one doesn’t get essentially, i.e.
logically, closer there still is something right about the idea that understanding consists
in imagining the fact. The language of imagining is primitive, in the same sense as the
language of gestures is.
18 (M): ////
19 (V): If understanding is a psychological process
20 (V): event occurs
21 (V): it obeying if
22 (M): / 3
23 (M): / 3 (R): To p. 42
24 (V1): It’s very strange: we’d like to // will // explain
the understanding of a gesture with its // with
the help of its // translation into words, and the
understanding of words through a translation
into gestures. (V2): It’s very strange: we’re
tempted to explain the understanding of a
gesture with words that correspond to it, and
the understanding of words with gestures
that correspond to them. // the understanding
of a gesture as the ability to translate it into
words . . .
25 (M): / 3 (R): To p. 42
26 (M): 555
27 (M): Reworked / 3
28 (V): answer: “But there is no transition from
knowing to doing. /// And there is no justification
in principle for that being what corresponded
to the command.”
29 (M): ∫ 3
30 (V): this proposition
31 (V): again, certainly not
32 (M): Rework / 3
Understanding a Command the Condition . . . 13e
WBTC001-06 30/05/2005 16:18 Page 27
37
”Aber ich muß doch einen Befehl verstehen, um nach ihm handeln zu können.“ Hier
ist das ”muß“ verdächtig. Wenn das wirklich ein Muß ist – ich meine – wenn es ein logisches
Muß ist, so handelt es sich hier um eine grammatische Anmerkung.
38
Auch wäre da die Frage möglich: Wie lange vor dem Befolgen mußt Du denn den Befehl
verstehen?
39
(Es kann keine notwendige Zwischenstufe zwischen dem Auffassen eines Befehls und
dem Befolgen geben.)
Das Verstehen, wenn es eine Vorbereitung des Befolgens war, kann man so auffassen, daß es dem
Zeichen (des Befehls) etwas hinzufügt; aber etwas was40
jedenfalls nicht die Ausführung war.
41
Wenn gesagt würde, daß der, der den Befehl erhält, wenn er ihn versteht eben außer den
Worten Vorstellungen erhält, die der Ausführung des Befehls ähnlich sind (während es die
Worte nicht sind), so will ich noch weiter gehen & annehmen,42
daß der Befehl dadurch gegeben
wird, daß wir den Andern veranlassen die Bewegungen, die er in 5 Minuten ausführen soll,
jetzt durch mechanische Beeinflussung auszuführen;43
und näher kann ich doch wohl der
Ausführung des Befehls in seinem Ausdruck44
nicht kommen. Dann haben wir die Ähnlichkeit
der Vorstellung durch eine viel größere Ähnlichkeit ersetzt. Und der Weg vom Zeichen zur
wirklichen Ausführung45
scheint nun46
sehr verkürzt zu sein.47
Es ist damit auch gezeigt, wie48
Phantasiebilder, Vorstellungen, für den Gedanken
unwesentlich sind.49
50
Ich könnte auch sagen: Es scheint uns, als ob51
wir dem Befehl durch
das Verstehen etwas hinzufügen, was52 53
die Lücke zwischen Befehl &
Ausführung füllt. Das heißt doch: was den Befehl in schattenhafter Weise ausführt.
So daß wir dem, der sagt &aber Du verstehst ihn ja, er ist also nicht unvollständig“54
antworten
können: ”Ja, aber ich verstehe ihn nur, weil55
ich noch etwas hinzufüge: die Deutung nämlich.“56
37 (M): ü / 3
38 (M): / 3
39 (M): ///
40 (V1): ∫Wenn das Verstehen eine notwendige
Vorbereitung des Folgens war, so muß es dem
Zeichen etwas hinzugefügt haben; aber etwas,
was // , so hat es wohl dem Zeichen // dem
Zeichen des Befehls // etwas hinzugefügt. – Aber
etwas, was (V2): Wenn das Verstehen eine
Vorbereitung des Befolgens war, so kann man es //
das Verstehen // so auffassen, daß es dem Zeichen (des
Befehls) etwas hinzufügt; aber etwas was
41 (M): ? / (R): [Zu: Die Kluft zwischen Befehl
& Ausführung nicht durch Ähnlichkeit überbrücken]
42 (V): so gehe ich noch weiter und nehme an,
43 (V): wird, daß wir den Andern die
Bewegungen, die er etwa in 5 Minuten ausführen
soll, jetzt durch mechanische Beeinflussung
(etwa indem wir seine Hand führen)
auszuführen veranlassen;
44 (V): Befehls im Ausdruck des Befehls
45 (V): vom Symbol zur Wirklichkeit
46 (V): hier
47 (V): sein. (Ebenso könnte ich$ um zu
beschreiben$ in welcher Stellung ich mich bei der
und der Gelegenheit befunden habe$ diese
Stellung einnehmen.) (R): (Siehe: Erwarten,
Wünschen, etc.)
48 (V): daß
49 (V): Es ist damit auch gezeigt, daß das
Vorkommen von Phantasiebildern, &
Vorstellungen, für den Gedanken ganz
unwesentlich ist. // Es ist damit auch das
Unwesentliche der Phantasiebilder für den
Gedanken gezeigt.
50 (M): / 3 (R): [siehe S. 89/4] Zu: &Deuten“?
51 (V): ob das Verstehen
52 (V): sagen: Es scheint uns, als ob, wenn wir den
Befehl –
A
C
z.B.
D
F
– verstehen,
wir etwas hinzufügen, was
53 (F): MS 114, S. 128.
54 (V): sagt ”aber Du verstehst ihn ja also ist er ja
vollkommen // vollständig //“
55 (V): können: ”Ja, aber nur, weil
56 (R): siehe: Erwarten, etc.
16v
17
18
17v
18
17v, 18,
17v, 18
14 Das Verstehen des Befehls . . .
x
x2
1 2 3 4
x
x2
1 2 3 4
WBTC001-06 30/05/2005 16:18 Page 28
33
“But I have to understand a command in order to be able to follow it.” Here the “have
to” is suspicious. If this really is a Have To – I mean, if it is a logical Have To, then we’re
dealing with a grammatical remark.
34
This question is also possible here: How long before you obey the command do you have
to understand it?
35
(There can be no intermediate step required between grasping a command and following
it.)
Understanding – when it’s a preparation for obeying – can be understood as adding something
to the sign (of the command); but36
that something is certainly not its execution.
37
If it were said that when a person who receives a command understands it, he receives,
in addition to the words, mental images that are similar to the execution of the command
(whereas the words are not), then I am prepared to go even further and assume38
that the command
is given by our mechanically causing the other person to carry out now the movements
that he is supposed to carry out in five minutes;39
and surely in expressing the command
I can’t get any closer to its execution. Then we have replaced the similarity of the mental
image with a much greater similarity. And the path from the sign to its actual execution40
now
seems41
to have been shortened considerably.42
This also shows how43
phantasms, mental images, are inessential to a thought.44
45
I could also say: It seems to us that by understanding the command
we add something to it 46
that47
fills the gap between the command
and its execution. And surely that means, something that executes the
command in a shadowy way. So that to someone who says “But you do understand it, so it isn’t
incomplete,”48
we can answer: “Yes, but I understand it only49
because I add something to it:
namely, the interpretation.”50
33 (M): r / 3
34 (M): / 3
35 (M): ///
36 (V1): ∫ When it is a necessary preparation for
obeying, understanding must have added
something to the sign; but // understanding
most likely has added something to the sign // of
the command // . – But (V2): If understanding
is a preparation for obeying, then one can conceive
of it // understanding // in such a way that it adds
something to the sign (of the command); but
37 (M): ? / (R): [To: don’t bridge the gap
between command and execution with similarity]
38 (V): then I am going even further and assuming
39 (V): is given by our mechanically causing the
other person (say by our moving his hand) to
carry out now the movements that he is to
carry out say in five minutes;
40 (V): from the symbol to reality
41 (V): seems here
42 (V): considerably. (Likewise$ in order to
describe the position I was in on this or that occasion$
I could assume that position.)
(R): (See: expect, wish, etc.)
43 (V): that
44 (V): This also shows that the occurrence of phantasms
& mental images is utterly inessential to
a thought. // This also shows how inessential
phantasms are for a thought.
45 (M): /3 (R): [see p. 89/4] To:
“Interpreting” ?
46 (F): MS 114, p. 128.
47 (V): say: It seems to us that when we understand
the command –
A
C
for example,
D
F
we add something
to it that
48 (V): “But you do understand it, therefore it is
complete // perfect // ,”
49 (V): “Yes, but only
50 (R): see: expecting, etc.
Understanding a Command the Condition . . . 14e
x
x2
1 2 3 4
x
x2
1 2 3 4
WBTC001-06 30/05/2005 16:18 Page 29
57
Aber58
was veranlaßt Dich gerade zu dieser59
Deutung? Ist es der Befehl, dann war er ja
schon eindeutig, da er diese Deutung befahl. Oder hast60
Du die Deutung willkürlich hinzugefügt
–, dann hast Du ja auch den Befehl nicht verstanden, sondern erst das, was Du aus ihm61
gemacht hast.
62
Eine ”Interpretation“ ist doch wohl etwas, was in Zeichen63
gegeben wird. Es ist diese
Interpretation im Gegensatz zu einer anderen (die anders lautet). – Wenn man also sagen
wollte64
”jeder Satz bedarf noch einer Interpretation“, so hieße das: kein Satz kann ohne einen
Zusatz verstanden werden.
65
”Ich kann den Befehl nicht ausführen, weil ich nicht verstehe, was Du meinst. – Ja, jetzt
verstehe ich Dich“.
Was ging da vor, als ich plötzlich den Andern verstand?66 67
Da gab es viele Möglichkeiten:
Der Befehl konnte z.B. mit68
falscher Betonung gegeben worden sein, & es fiel mir plötzlich die richtige
Betonung ein. Einem Dritten würde ich dann sagen: &jetzt verstehe ich ihn, er meint: . . .“ & nun würde
ich den Befehl in richtiger Betonung wiederholen. Und in der richtigen Betonung verstünde ich
nun den Befehl, das heißt:69
ich müßte nun nicht noch einen Abstrakten Sinn erfassen sondern es
genügt mir vollkommen der wohlbekannte deutsche Wortlaut.70
– Oder71
der Befehl ist72
mir in
verständlichem Deutsch gegeben worden schien mir aber ungereimt, da ich ihn in irgend einer
Weise mißverstand;73
dann fiel mir eine Erklärung ein &ach, er meint . . .“ & nun kann ich den
Befehl ausführen. Oder es konnten mir auch vor diesem Verstehen ”mehrere Deutungen
vorschweben“,74
für deren eine ich mich endlich entscheide.75
76
Wer zwischen zwei77
Arten einen B. zu verst. schwankt, der schwankt78
zwischen zwei
Deutungen, zwischen zwei Erklärungen.
79
Was heißt es: verstehen, daß etwas ein Befehl ist, wenn man auch den Befehl selbst noch
nicht versteht? (”Er meint: ich soll etwas tun, aber was er wünscht, weiß ich nicht.“)
57 (M): / 3 (R): Zu: &Deuten“
58 (V): Nun müßte man allerdings darauf sagen:
Aber
59 (V): veranlaßt Dich denn zu gerade dieser
60 (V): da er nur diese Deutung befahl. Oder$ hast
61 (V): ihm (auf eigene Faust)
62 (M): / (R): Zu: &Deuten“
63 (V): Worten
64 (V): also sagt
65 (M): / (R): (Dieser Satz bleibt im §)
66 (V): verstand? Ich konnte mich natürlich irren$
und daß ich den andern verstand$ war eine
Hypothese. Aber es fiel mir etwa plötzlich eine
Deutung ein$ die mir einleuchtete. Aber war
diese Deutung etwas anderes als ein Satz der
Sprache? // sprachlicher Ausdruck? // als eine
Erklärung?
67 (M): /
68 (V): konnte mit
69 (V): ich ihn nun; d.h.,
70 (V): vollkommen den wohlbekannten deutschen
Wortlaut zu haben^
71 (V): Oder aber
72 (V): wäre
73 (V): worden schiene mir aber ungereimt, da ich ihn
in auf irgend eine Weise mißverstehe;
74 (V): Es konnten mir . . . mehrere Deutungen
vorschweben, // Oder es schwebten mir . . .
75 (V): entscheide. Aber das Vorschweben der
Deutungen // von Bedeutungen // war das
Vorschweben von Ausdrücken einer Sprache.
(M): ∀ S. 20/4
76 (M): /
77 (O): zei
78 (V): Wer zwischen zwei Arten schwankt einen
Befehl zu verstehen, schwankt
79 (M): / (R): →S. 7/2
17v
19
18v
19
15 Das Verstehen des Befehls . . .
WBTC001-06 30/05/2005 16:18 Page 30
51
But52
what causes you to arrive at this particular interpretation? Is it the command? – If
so, then it was already unambiguous, since it commanded this interpretation.53
Or did you
add the interpretation arbitrarily? – In that case you didn’t understand the command, but
only what you then made of it.54
55
To be sure, an “interpretation” is something that is given in signs.56
It is this interpretation,
as opposed to another (which reads differently). – So if one wanted to say57
“Every
proposition needs an interpretation,” that would mean: No proposition can be understood
without a rider.
58
“I can’t execute the order because I don’t understand what you mean. – Oh, now I understand
you.”
What was going on when I suddenly understood the other person?59 60
Many things could
have been happening: for example, the61
order could have been given with an improper intonation,
and suddenly the right one occurred to me. In that case I would say to a third person: “Now I understand
him, he means: . . .” and then I’d repeat the command with the right intonation. And now,
given the right intonation I’d understand the command, that is:62
now I wouldn’t have to grasp an
additional abstract sense, but the familiar English wording would be quite adequate for me.63
– Or64
I was65
given the command in understandable English, but it seemed to make no sense to me because
I misunderstood66
it in some way; then I hit upon an explanation: “Oh, he means . . .” and then I could
carry out the command. Or possibly before understanding it in this way “several interpretations
were in my mind”67
, one of which I finally decided upon.68
69
Whoever wavers between two ways of understanding a command wavers between two interpretations,
two explanations.
70
What does this mean: Understanding that something is a command before understanding
the command itself? (“He means: I should do something, but I don’t know what it is he
wants.”)
51 (M): / 3 (R): To: “Interpreting”
52 (V): Now$ to be sure$ one ought to respond to
this by saying: But
53 (V): commanded only this interpretation.
54 (V): it (going it alone).
55 (M): / (R): To: “Interpreting”
56 (V): words.
57 (V): So if one says
58 (M): / (R): (This sentence stays in the §)
59 (V): person? Of course possibly I was mistaken$
and that I understood the other person was a
hypothesis. But$ say% suddenly an interpretation
came to my mind that made sense to me. But
was this interpretation something other than a
linguistic proposition? // a linguistic expression? //
other than an explanation?
60 (M): /
61 (V): happening: the
62 (V): I’d now understand it; i.e.,
63 (V): but it would be quite sufficient for me to have
the familiar English wording.
64 (V): Or on the other hand
65 (V): Or say I were
66 (V): but it would seem to make no sense to me
because I misunderstand
67 (V): Possibly before . . . in my mind // Or,
before understanding . . . in my mind
68 (V): upon. But the “being in mind” of the
interpretations // of meanings // was the
“being in mind” of expressions in a language.
(M): ∀ p. 20/4
69 (M): /
70 (M): / (R): → p. 7/2
Understanding a Command the Condition . . . 15e
WBTC001-06 30/05/2005 16:18 Page 31
5
Deuten.
Deuten wir jedes Zeichen?
1
Deuten wir denn etwas, wenn uns jemand einen Befehl gibt? Wir fassen auf, was wir
hören oder sehen; oder: wir sehen, was wir sehen.
2
Ein Zeichen deuten, ihm eine Deutung hinzufügen, ist ein Vorgang der wohl in manchen3
Fällen
geschieht aber durchaus nicht immer wenn ich ein Zeichen verstehe.
4
Es gibt Fälle, in denen wir einen erhaltenen Befehl deuten und Fälle, in denen wir es
nicht tun.
Eine Deutung ist eine Ergänzung des gedeuteten Zeichens durch ein Zeichen.
5
Wenn mich jemand fragt: ”wieviel Uhr ist es“, so geht in mir dann keine Arbeit des Deutens
vor. Ich6
reagiere unmittelbar auf das, was ich sehe und höre.
7
Der Zerstreute der8
auf den Befehl ”rechtsum“ sich nach links gedreht hätte und nun, an
die Stirne greifend, sagte ”ach so – ,rechtsum‘!“ und rechtsum machte.9
Ist ihm eine Deutung
eingefallen?
10
Ich deute die Worte; wohl; aber deute ich auch die Mienen? Deute ich, etwa, einen
Gesichtsausdruck als drohend,11
oder freundlich? – Auch das kann übrigens geschehen.12
13
Wenn ich nun sagte: Es ist nicht genug, daß ich das drohende Gesicht wahrnehme,
sondern ich muß es erst deuten. – Es zückt jemand das Messer und ich sage: ”ich verstehe
das als eine Drohung“.
14
Welchen Sinn hat es, jemandem zu befehlen,15
einen Satz zu verstehen?
Hier muß man verschiedene Fälle unterscheiden.
16
(Denken wir an verschiedene Befehle, die wir nicht ausführen können:
ein Gewicht zu heben das uns zu schwer ist,
einen Arm zu heben der gelähmt17
ist,
1 (M): ///
2 (M): /
3 (V): gewissen
4 (M): ? /
5 (M): /
6 (V): vor. Sondern ich
7 (M): / (R): Zu S. 18
8 (V): Denken wir uns einen Zerstreuten, der
9 (R): [gehört eigentlich zu einer Bemerkung: &das
Wort, wenn wir es verstehen gewinnt Tiefe“]
10 (M): /
11 (V): drohend?,
12 (V): freundlich? – Es kann geschehen.
13 (M): /
14 (M): / (R): [Zu: &Behauptung, Frage, etc.“] § 47
15 (V): Kann man jemandem befehlen,
16 (M): ü /
17 (O): gelämt
20
19v
20
21
16
WBTC001-06 30/05/2005 16:18 Page 32
5
Interpreting.
Do We Interpret Every Sign?
1
Do we really interpret something when someone gives us an order? We grasp what we
hear or see; or: We see what we see.
2
Interpreting a sign, adding an interpretation to it, is a process that does take place in some3
cases,
but certainly not every time I understand a sign.
4
There are cases where we interpret an order we have been given and cases where we don’t.
An interpretation is a supplementation of the interpreted sign with another sign.
5
If someone asks me: “What time is it?” then no work of interpretation goes on inside me.
I6
react immediately to what I see and hear.
7
The absent-minded person who8
, responding to the order “Right turn”, turns left, and then,
hitting himself on the forehead, says “Oh – ‘right turn’!” and turns right.9
Did he think of an
interpretation?
10
I interpret the words; fine; but do I also interpret the facial expressions? Do I, for example,
interpret a facial expression as threatening11
, or friendly? – That too can happen, by
the way.12
13
What if I were to say: It isn’t enough for me to perceive a threatening face – first I have
to interpret it. – Someone pulls a knife and I say: “I understand this as a threat”.
14
What sense is there to ordering someone15
to understand a proposition?
Here one has to distinguish various cases.
16
(Let’s think about various commands that we cannot carry out:
To lift a weight that is too heavy for us,
to raise an arm that is paralysed,
1 (M): ///
2 (M): /
3 (V): certain
4 (M): ? /
5 (M): /
6 (V): me. Rather, I
7 (M): / (R): To p. 18
8 (V): Let’s imagine an absent-minded person
who
9 (R): [really belongs to the remark: “a word acquires
depth when we understand it”]
10 (M): /
11 (V): threatening?
12 (V): friendly? – That can happen.
13 (M): /
14 (M): / (R): [To: “statement, question, etc.”] §
47
15 (V): Can one order someone
16 (M): r /
16e
WBTC001-06 30/05/2005 16:18 Page 33
ein Haar aufzustellen,
sich eines Namens zu erinnern der uns entfallen ist,
einen Satz zu verstehen).
Kann man sagen, daß man den Befehl, den gelähmten18
Arm zu heben in gewissem Sinne nicht
versteht? (Bewegen der Finger bei verschränkten Händen.) Den Befehl verstehen, heißt etwa
darstellen können wie es wäre wenn er ausgeführt würde. Und nun kann ich mir wohl vorstellen
oder zeichnen etc. wie es wäre wenn sich die Bewegung des Arms vollzöge; aber, wenn er sich
auf den Befehl hin höbe, so würden wir doch nicht sagen, wir haben ihn gehoben. Wir hätten also
den Befehl nicht ausgeführt. Denken wir an die Befehle: &habe Schmerzen!“ & &rufe Dir Schmerzen
hervor!“ Ferner: &Stelle19
Dir einen roten Kreis vor!“
18 (O): gelämten 19 (V): stelle
17 Deuten. . . .
WBTC001-06 30/05/2005 16:18 Page 34
to stand a hair on its end,
to remember a name we’ve forgotten,
to understand a proposition.)
Can one say that, in a certain sense, one doesn’t understand the command to raise one’s paralysed
arm? (Moving one’s fingers when one’s hands are intertwined.) Understanding a command means,
for example, being able to show what it would be like if it were carried out. And I can very well imagine
or draw, etc., what it would be like if the movement of the arm took place; but if it were to rise
at the command, then surely we wouldn’t say that we had raised it. So we wouldn’t have carried out
the command. Let’s think of the commands: “Have pain!” and “Elicit pain in yourself!” Further:
“Imagine a red circle!”
Interpreting. . . . 17e
WBTC001-06 30/05/2005 16:18 Page 35
6
1
Man sagt: ein Wort verstehen heißt,
wissen, wie es gebraucht wird.
Was heißt es, das zu wissen?
Dieses Wissen haben wir
sozusagen im Vorrat.
2
Wissen, wie ein Wort gebraucht wird = Es anwenden können.
3
Vergleiche:
&Ich sehne mich nach ihm“
&Ich erwarte ihn“
&Ich weiß, daß er kommen wird“
oder auch:
1 &ich habe mich vom Morgen an4
nach ihm gesehnt“
2 &ich habe ihn vom Morgen an5
erwartet“
3 &ich wußte vom Morgen an daß er kommen werde“
4 &ich hatte vom Morgen an Zahnschmerzen“
Kann man sagen &ich wußte vom Morgen an ununterbrochen daß er kommen werde“?
Vergleiche N° 4 mit jedem der anderen Sätze.
5 &Ich konnte von meinem 10ten Jahr an Schachspielen“
6 &Ich konnte seit damals nicht mehr hoch springen“
6
Es ist merkwürdig, daß wir uns bei dem Gedanken, daß es jetzt 3 Uhr sein dürfte, die
Zeigerstellung meist gar nicht genau oder überhaupt nicht vorstellen, sondern das Bild wie
in7
einem Werkzeugkasten der Sprache haben, aus dem wir wissen, das Werkzeug jederzeit
herausnehmen8
zu können, wenn wir es brauchen. – Dieser Werkzeugkasten, ist er aber nicht
die Grammatik mit ihren Regeln?9
(Denken wir aber, welcher Art dieses Wissen ist.)
1 (R): gehört zu § 35 (p. 134)
2 (M): /
3 (M): ? /
4 (V): mich den ganzen Tag
5 (V): ihn den ganzen Tag
6 (M): ? / (R): [Zu: &das augenblickliche
Verstehen etc.“]
7 (V): Bild, gleichsam, in
8 (V): hervorziehen
9 (V): Dieser Werkzeugkasten scheint mir die
Grammatik mit ihren Regeln zu sein.
22
21v
22
18
WBTC001-06 30/05/2005 16:18 Page 36
6
1
One Says: Understanding a Word
Means Knowing How it is Used.
What Does it Mean to Know That?
We Have this Knowledge in Reserve,
as it Were.
2
Knowing how a word is used = being able to use it.
3
Compare:
“I long for him”
“I’m expecting him”
“I know that he’ll come”
or also:
1 “I’ve longed for him ever since this morning4
”
2 “I’ve expected him ever since this morning5
”
3 “I’ve known ever since this morning that he’d come”
4 “I’ve had a toothache ever since this morning”
Can one say “Ever since this morning I knew continuously that he would come?”
Compare no. 4 with each of the other sentences.
5 “Ever since I was ten I’ve been able to play chess”
6 “Ever since that time I’ve no longer been able to jump very high”
6
It’s remarkable that in thinking that now it’s probably three o’clock we usually don’t picture
the position of the hands exactly, or don’t do so at all; rather, we have the image as7
in
a toolbox of language, from which we know that we can take out8
the tools any time we need
them. – But this toolbox – isn’t it grammar, with its rules? 9
(But let’s think about what kind
of knowledge this is.)
1 (R): belongs to § 35 (p. 134)
2 (M): /
3 (M): ? /
4 (V): him the entire day
5 (V): him the entire day
6 (M): ? / (R): [To: “immediate understanding
etc.”]
7 (V): as it were
8 (V): can pull out
9 (V): toolbox seems to me to be grammar, with
its rules.
18e
WBTC001-06 30/05/2005 16:18 Page 37
10
Es ist so, wie wenn ich mir im Werkzeugkasten der Sprache Werkzeuge zum künftigen
Gebrauch herrichtete. Oder im Malkasten Farben. (Ein Werkzeug ist ja auch das Abbild seines
Zwecks.)11
12
Was heißt es, zu sagen ”ich sehe zwar kein Rot, aber wenn Du mir einen Farbkasten
gibst, so kann ich es Dir darin zeigen“? Wie kann man wissen, daß man es zeigen kann, wenn
. . . ; daß man es also erkennen kann, wenn man es sieht?
13
Betrachte nun den Satz: Weißt Du,14
welche Farbe &rot“ bedeutet? Ja, wenn hier etwas rotes
wäre so15
könnte ich es erkennen.
16
&Ich könnte Dir die genaue Farbe der Tapete zeigen, wenn hier etwas wäre was diese Farbe hat“.
– &Wie weißt Du, daß Du sie erkennen würdest?“ – &Weil ich sie jetzt vor mir sehe.“17
Anderseits:18
&ich kann mir jederzeit wenn ich will einen roten Kreis vorstellen.“19
– &Wie weißt
Du, daß Du es20
kannst?“
21
Es ist etwa dies mein Wörterbuch und ich übersetze mit ihm22
den Satz bdca
in fhge. Nun habe ich im gewöhnlichen Sinne gezeigt, daß ich den Gebrauch
des Wörterbuchs verstehe und kann sagen, daß ich auf gleiche Weise den Satz
cdab übersetzen kann, wenn ich will. – Wenn also der Satz cdab ein Befehl ist, den
entsprechenden Satz in der zweiten Sprache hinzuschreiben, so verstehe ich diesen
Befehl, wie ich etwa den Befehl verstehe, |||||| Schritte zu gehen, wenn mir gezeigt wurde,
wie die entsprechenden Befehle mit den Zahlen |, ||, |||, ausgeführt werden.
23
Aber natürlich kann das nicht anders sein, als wenn ich z.B. sage ”ich will diesen
Fleck rot anstreichen“, eine Vorstellung von der Farbe habe und nun ”weiß“, wie diese
Vorstellung in die Wirklichkeit zu übersetzen ist.
24
Ja, das ganze Problem ist schon darin enthalten: Was heißt es, zu wissen, wie der Fleck
aussähe, wenn er meiner Vorstellung entspräche?
&Du weißt, wie er aussähe? – Nun wie sieht er aus?“
25
Wenn ich die Vorstellung, die bei der Erwartung etc. im Spiel ist, durch ein wirklich
gesehenes Bild ersetzen will, so scheint etwa folgendes zu geschehen: Ich sollte einen dicken
schwarzen Strich ziehen und habe als Bild einen dünnen gezogen. Aber die Vorstellung geht
noch weiter und sagt, sie weiß auch schon, daß der Strich dick sein soll. So ziehe ich einen
dicken, aber etwas blasseren Strich; aber die Vorstellung sagt, sie weiß auch schon, daß er
nicht grau sondern schwarz sein sollte. (Ziehe ich aber den dicken schwarzen Strich, so ist
das kein Bild mehr.)
10 (M): ? / /// (R): [Zu: &das augenbl.
Verstehen etc.“] Zu MS p. 21/1?
11 (R): [Dazu: Hypothese &ich sehe eine Kugel“]
Verwendung der Vorstellg. des Bildes einer Kugel.
12 (M): /
13 (M): /
14 (V): Weißt Du,
15 (V): Ich sage: Hier ist zwar nichts Rotes um
mich, aber wenn hier etwas wäre, so
16 (M): ? /
17 (V): &Weil ich sie mir jetzt vorstellen kann //
vorstelle //.“
18 (V): Anderseits aber:
19 (V): Anderseits: &Ich kann mir jederzeit einen roten
Kreis vorstellen% wenn ich will“.
20 (V): das
21 (M): ? / ///
22 (V): übersetze darnach
23 (M): ∫ ///
24 (M): ? / //// (R): [Zu: Erwartung] S. 364
25 (M): ? / /// (R): [Zu: Die Erwartung erwartet
das was sie erfüllen wird] § 77
21v
22
23
19 Man sagt: ein Wort verstehen heißt . . .
a
b
c
d
e
f
g
h
WBTC001-06 30/05/2005 16:18 Page 38
10
It’s as if, in the toolbox of language, I lay out tools for future use. Or colours in a paint
box. (A tool, after all, is also an illustration of its purpose.)11
12
What does it mean to say “I don’t see any red, but if you give me a box of paints, I can
point it out to you in there”? How can one know that one can point it out if . . . ; i.e. that
one can recognize it when one sees it?
13
Now look at the sentence: Do you14
know what colour is meant by “red”? Yes, if there were
something red here, I15
could recognize it.
16
“I could show you the exact colour of the wallpaper if there were something here of that colour.”
– “How do you know that you’d recognize it?” – “Because it’s now before my mind’s eye.”17
On18
the other hand: “If I want to I can imagine a red circle at any time”.19
– “How do you know
you can do that?”
20
Say this is my dictionary, and with it21
I translate the sentence bdca into fhge.
I have shown in the ordinary sense that I know how to use a dictionary and I can
say that I’m able to translate the sentence cdab in the same way, if I want to. – So
if the sentence cdab is a command to write down the corresponding sentence in
the second language, then I understand this command, as for example I understand
the command to walk |||||| steps if I’m shown how the corresponding commands are
carried out with the numbers |, ||, |||.
22
But of course that can’t be different from my saying, for example, “I want to paint this
spot red”, having a mental image of the colour, and “knowing” how this mental image is to
be translated into reality.
23
Indeed the whole problem is already included in: What does it mean to know what the
patch would look like if it corresponded to my mental image?
“You know what it would look like? – Well, what does it look like?”
24
If I want to replace the mental image that is involved in expectation, etc., with a picture
I really saw, then something like the following seems to happen: I was supposed to draw a
thick black line and drew a thin one as a picture of it. But my mental image jumps ahead
and says it knows that the line should have been thick. So then I draw a thick but somewhat
paler line; but my mental image says that it knows that it shouldn’t have been grey,
but black. (But if I do draw the thick black line then that isn’t a picture any more.)
10 (M): ? / /// (R): [To: “immediate understanding
etc.”] To MS p. 21/1?
11 (R): [To: Hypothesis “I see a sphere”] Use of the
mental image of the picture of a sphere.
12 (M): /
13 (M): /
14 (V): Do you
15 (V): I say: To be sure, there’s nothing red
around me here, but if something were here, I
16 (M): ? /
17 (V): “Because I can now imagine it.” // “Because I am
now imagining it.”
18 (V): But on
19 (V): On the other hand: “I can imagine a red circle
any time I want to”.
20 (M): ? / ///
21 (V): and following it
22 (M): ∫ ///
23 (M): ? / //// (R): [To: Expectation] P. 364
24 (M): ? / /// (R): [To: Expectation expects
what will fulfil it] § 77
Understanding a Word Means Knowing How it is Used. . . . 19e
a
b
c
d
e
f
g
h
WBTC001-06 30/05/2005 16:18 Page 39
26
Etwas wissen, ist damit27
zu vergleichen:28
einen Zettel in meiner Tasche29
zu haben, auf dem
es aufgeschrieben ist.30
31
Wie ist es, wenn ich jemandem den Befehl gebe &stelle Dir einen roten Fleck vor“ & nun sage:
den Befehl verstehen heiße, wissen wie es ist, wenn er ausgeführt ist; oder gar sich vorstellen
können, wie es ist, wenn. . . .
26 (M): ? /
27 (V): darin
28 (V): wissen, ist von der Art dessen
29 (V): in der Lade meines Schreibtisches
30 (V): steht.
31 (R): [Zu S. 182]
20 Man sagt: ein Wort verstehen heißt . . .
WBTC001-06 30/05/2005 16:18 Page 40
25
To know something can be compared to:26
having a slip of paper in my pocket27
on which
it is written down.
28
What’s it like if I give someone the command “Imagine a red patch” and then say:
Understanding that command means knowing what it’s like when it has been carried out; or even
being able to imagine what it’s like when. . . .
25 (M): ? /
26 (V): something is of the nature of:
27 (V): in my desk drawer
28 (R): [To p. 182]
Understanding a Word Means Knowing How it is Used. . . . 20e
WBTC001-06 30/05/2005 16:18 Page 41
6a1
Einen Satz im Ernst oder
Spaß meinen, etc.
Man wird sagen: der Maler der ”Malheurs de Chasse“ hat nicht gemeint, daß es wirklich
so zugeht; hätte er aber seine Bilder lehrhaft (um zu zeigen, wie es zugeht) gemeint, so wäre
er im Unrecht gewesen.
”Hast Du das im Ernst oder im Spaß gemeint?“ – Das ”im Ernst Meinen“ besteht nicht
darin, daß zu dem ausgesprochenen Satz im Stillen noch etwas hinzugesetzt wird, etwa die
Worte ”ich meine das im Ernst“. Von dem ganzen Satz, dem ausgesprochenen mit den
dazugedachten Worten, könnte man wieder fragen: wie war er gemeint? Von Ernst oder Spaß
kann man das aber nicht fragen. Also ist die Meinung (Auffassung) in diesem Sinne ein
bestimmtes Erlebnis, das mit dem Aussprechen2
des Satzes Hand in Hand geht, aber an
dem Sinn des Satzes nichts ändert, ob es nun so oder anders ist.
Wie geht das vor sich, wenn man einen Satz ausspricht und dabei den anderen nur
aufsitzen lassen will? Man spricht, lächelt, sieht zu, was der Andere macht,3
fühlt eine
Spannung.
Aber nirgends ist der amorphe Sinn.4
Diesen5
stellt man sich gleichsam vor, wie den Inhalt
eines Tiegels dessen Aufschrift der Satz ist.
”Ich habe gesagt ,sie ist nicht zu Hause‘, habe aber dabei gewußt, daß sie zu Hause war.“
Wie geht dieses Wissen zeitlich mit dem Sagen des Satzes zusammen? Wie eine kontinuierliche
Begleitung, ein Orgelpunkt, zu einem Thema?
Hast Du es in jedem Augenblick gewußt, und braucht das Wissen keine Zeit?
Ein falsches Bild verführt uns.
1 (E): Der nun folgende nicht numerierte
Abschnitt weist im Typoskript (TS) die
Seitenzahlen 1 bzw. 2 auf und unterbricht damit
die fortlaufende Seitenzählung. Wir haben ihn
eingereiht, indem wir ihm die Nummer 6a
gegeben haben und den Seiten, auf denen er sich
befindet, die Zahlen 23a bzw. 23b.
2 (V): mit den Zeichen
3 (V): lächelt, beobachtet den andern,
4 (V): ist die amorphe Meinung.
5 (V): Diese
23a
23b
23b
21
WBTC001-06 30/05/2005 16:18 Page 42
6a1
Meaning a Proposition Seriously
or in Jest, etc.
It will be said: the painter of “Malheurs de Chasse” didn’t mean that things really go on
like that; but if he had intended his pictures to be didactic (in order to show what does go
on), he would have been wrong.
“Did you mean that seriously or in jest?” – “Meaning something seriously” does not consist
in something’s being silently added to a spoken sentence, say the words “I’m serious
about that”. As for the whole sentence, the one uttered along with the mentally added words,
one could ask of it as well: How was it meant? But one can’t ask this about seriousness
or jest. So intention (understanding) in this sense is a specific experience that goes hand in
hand with the utterance2
of a sentence, but leaves the sense of the sentence unchanged, no
matter the kind of experience.
What happens when you utter a sentence just to make someone else look foolish? You
speak, smile, watch the other person react3
, feel a tension in the air.
But nowhere is there the amorphous sense.4
One imagines this sense as being, as it were,
like the contents of a jar, with the sentence as its label.
“I said ‘She’s not at home’, knowing all the while that she was at home.” How does this
knowing fit together chronologically with the utterance of the sentence? Like a continuous
accompaniment, a pedal point to a theme?
Did you know it at every moment, and doesn’t knowing take time?
A false image is leading us astray.
1 (E): The pages of the following (unnumbered)
section are numbered 1 and 2 in the TS, thus
interrupting the continuous numbering of
pages. We have inserted this section here, and
given it the number 6a.
2 (V): signs
3 (V): smile, observe the other person
4 (V): the amorphous meaning.
21e
WBTC001-06 30/05/2005 16:18 Page 43
Bedeutung.24
WBTC007-10 30/05/2005 16:18 Page 44
Meaning.
WBTC007-10 30/05/2005 16:18 Page 45
7
Der Begriff der Bedeutung stammt
aus einer primitiven philosophischen
Auffassung der Sprache her.1
2
Augustinus, wenn er vom Lernen der Sprache redet, redet ausschließlich davon, wie wir
den Dingen Namen beilegen, oder die Namen der Dinge verstehen. Hier scheint also das
Benennen Fundament und Um und Auf der Sprache zu sein.
Diese Betrachtungsweise der Sprache ist wohl die, welche die3
Erklärungsform ”das ist . . .“
als fundamental auffaßt. – Von einem Unterschied der Worte redet Augustinus nicht,
meint also mit ”Namen“ offenbar Wörter, wie ”Baum“, ”Tisch“, ”Brot“, und gewiß die
Eigennamen der Personen; dann aber wohl auch ”essen“, ”gehen“, ”hier“, ”dort“; kurz, alle
Wörter. Gewiß aber denkt er zunächst an Hauptwörter und an die übrigen als etwas, was
sich finden wird. (Und Plato sagt, daß der Satz aus Haupt- und Zeitwörtern besteht.)4
Sie beschreiben eben das Spiel einfacher, als es ist.
Dieses Spiel kommt aber wohl in der Wirklichkeit vor. – Nehmen wir etwa an, ich wollte
aus Bausteinen,5
die mir ein Andrer zureichen soll, ein Haus aufführen, so könnten wir erst
ein Übereinkommen dadurch treffen, daß ich auf einen Stein zeigend sagte ”das ist eine Säule“,
auf einen andern zeigend, ”das heißt Würfel“, – ”das heißt Platte“ u.s.w. Und nun bestünde
die Anwendung im Ausrufen jener Wörter ”Säule“, ”Platte“, etc. in der Ordnung, wie ich
die Bausteine brauche. Und ganz ähnlich ist ja das Übereinkommen
und etwa eines, das mit Farben arbeiten würde.
6
Augustinus beschreibt wirklich einen Kalkül; nur ist nicht alles, was wir Sprache
nennen, dieser Kalkül.
(Und das muß man in einer großen Anzahl von Fällen sagen, wo es sich fragt: ist diese
Darstellung brauchbar oder unbrauchbar. Die Antwort ist dann: ”ja, brauchbar; aber nur
dafür, nicht für das ganze Gebiet, das Du darzustellen vorgabst“.)7
1 (V): / primitiven Philosophie der Sprache her.
2 (M): ////
3 (V): Diese Auffassung des Fundaments der
Sprache ist offenbar aequivalent mit der, die die
4 (E): Vgl. Sophist, 261e.
5 (V): Bausteinen ein Haus aufführen,
6 (M): (
7 (M): )
25
26
24v
a
b
c
d
↓
↑
→
←
23
WBTC007-10 30/05/2005 16:18 Page 46
7
The Concept of Meaning Originates
in a Primitive Philosophical
Conception of Language.1
2
When Augustine talks about learning language he talks exclusively about how we attach
names to things, or understand the names of things. So naming here appears as the foundation,
the be-all and end-all of language.
This way of looking at language is the one that3
takes the form of explanation “That is . . .”
as fundamental. – Augustine does not speak of there being any difference among words, and
so by “names” he evidently means words such as “tree”, “table”, “bread”, and certainly the
proper names of people; but probably also “eat”, “go”, “here”, “there”; in short, all words.
Certainly he’s thinking primarily of nouns, and of the remaining words as something that
will take care of itself. (And Plato says that a sentence consists of nouns and verbs.)4
They just describe the game as simpler than it is.
But this game does occur in reality. – Let’s assume for instance that I wanted to build a
house out of building blocks that someone else is to pass to me; we might first create a convention
by my pointing to one block and saying “That’s a pillar” and to another saying “That’s
called a cube”, “That’s called a slab”, and so on. And now the application would consist in
calling out the words “pillar”, “slab”, etc. in the order in which I need the building blocks.
And indeed, the convention
is quite similar to this, as is one that works with colours.
5
Augustine really does describe a calculus; it’s just that not everything that we call language
is this calculus.
(And that is what one has to say in a great many cases where the question is: Is this
portrayal fitting, or not? The answer to that is: “Yes, it is fitting; but only here, not for the
whole region you claimed you were portraying.”)6
1 (V): in a Primitive Philosophy of Language.
2 (M): ////
3 (V): This conception of the foundation of language
is obviously equivalent to the one that
4 (E): Cf. Sophist, 261e.
5 (M): (
6 (M): )
a
b
c
d
↓
↑
→
←
23e
WBTC007-10 30/05/2005 16:18 Page 47
8
Es ist so, als erklärte jemand:9
”spielen besteht darin, daß man Dinge, gewissen Regeln
gemäß, auf einer Fläche verschiebt . . .“ und wir ihm antworteten: Du denkst da gewiß an
die Brettspiele, und auf sie ist Deine Beschreibung auch anwendbar. Aber das sind nicht die
einzigen Spiele. Du kannst also Deine Erklärung richtigstellen, indem Du sie ausdrücklich
auf diese Spiele einschränkst.
Man könnte also sagen, Augustinus stelle die Sache zu einfach dar;10
aber auch: er stelle
eine einfachere Sache dar.
(Wer das Schachspiel einfacher beschreibt – mit einfacheren Regeln – als es ist,
beschreibt damit dennoch ein Spiel, aber ein anderes.)11
12
Wie13
Augustinus das Lernen der Sprache beschreibt, das kann uns zeigen, von
welchem primitiven Bild14
sich diese Auffassung der Bedeutung eigentlich schreibt.15
Man könnte den Fall mit dem einer Schrift vergleichen, in der Buchstaben zum
Bezeichnen von Lauten benützt würden, aber auch zur Bezeichnung der Betonung16
und
als Interpunktionszeichen. Fassen wir dann diese Schrift als eine Sprache zur Beschreibung
des Lautbildes auf, so könnte man sich denken, daß Einer diese Schrift so auffaßte, als
entspräche einfach jedem Buchstaben ein Laut und als hätten die Buchstaben nicht auch
ganz andere Funktionen. – Und so einer – zu einfachen – Beschreibung der Schrift gleicht
Augustinus’ Beschreibung der Sprache völlig.
17
Hierher gehört auch: Man kann – für Andere verständlich – 18
von Kombinationen von Farben
mit Formen sprechen (etwa der Farben rot und blau mit den Formen Quadrat und Kreis)19
ebenso wie von Kombinationen verschiedener Formen oder Körper. Und hier haben wir die
Wurzel meines20
irreleitenden Ausdrucks, die Tatsache sei ein Komplex von Gegenständen.
Es wird also, daß ein Mensch krank ist, verglichen mit der Zusammenstellung zweier Dinge,
wovon das eine der Mensch,21
das andere die Krankheit wäre. Vergessen wir nicht, daß das nur
ein Gleichnis ist.22
Oder man muß sagen, es verhält sich hier mit dem Wort ”Kombination“, oder
”Komplex“, wie mit dem Wort ”Zahl“, das auch in verschiedenen – mehr oder weniger logisch
ähnlichen – Weisen (Bedeutungen) gebraucht wird.
23
”Bedeutung“ kommt von ”deuten“.
Was wir Bedeutung nennen, muß mit der primitiven Gebärdensprache (Zeigesprache)
zusammenhängen.
24
Wenn ich etwa die wirkliche Sitzordnung an einer Tafel nach einer Aufschreibung
kollationiere, so hat es einen guten Sinn, beim Lesen jedes Namens auf einen bestimmten
Menschen zu zeigen. Sollte ich aber etwa die Beschreibung eines Bildes mit dem Bild
8 (M): (
9 (V): Es ist also so, wie wenn jemand erklärte:
10 (V): stelle das Lernen der Sprache zu einfach dar;
11 (M): )
12 (M): (
13 (V): Ich wollte ursprünglich sagen: Wie
14 (V): von welcher primitiven Anschauung
15 (V): zeigen, woher sich diese Auffassung überhaupt
// eigentlich // schreibt.
16 (V): Stärke
17 (R): [Vielleicht auch zu &Komplex & Tatsache“]
[MS Gretl S. 113]
18 (V): Man kann z.B. – für Andere verständlich
–
19 (V): Kreis) zusammengeschlossen, (E)
darüber: unleserliches Wort in eckiger
Klammer.
20 (V): des
21 (V): Mensch, ist
22 (V): Hüten wir uns davor, // Hüten wir uns davor,
// zu vergessen, daß das nur ein Gleichnis ist.
23 (M): 555
24 (M): ////
27
26v, 27
24 Der Begriff der Bedeutung . . .
WBTC007-10 30/05/2005 16:18 Page 48
7
It’s as if someone were to declare:8
“Playing a game consists in moving objects about on
a surface according to certain rules . . .” and we replied: You must be thinking of board games,
and your description is quite applicable to them. But they are not the only games. So you
can make your definition correct by expressly restricting it to those games.
So one could say that Augustine was portraying the matter too simply;9
but also: that he
was portraying a simpler matter.
(Someone who describes the game of chess as simpler than it is – with simpler rules – is
still describing a game, but a different one.)10
11
The12
way Augustine describes the learning of language can show us how primitive the
picture13
is from which this conception of meaning is really derived.14
One could compare this case to a writing system in which letters were used to signify sounds,
but also to signify intonation,15
and as punctuation marks. If we then conceive of this system
as a language for describing phonetic patterns, then one could imagine that someone might
understand this system as simply linking a sound to each letter – as if the letters didn’t have
completely different functions as well. And Augustine’s description of language is just like
such an – oversimple – description of this system of writing.
16
This is also relevant here: One can – in a way others can understand –17
speak of combinations
of colours with shapes (for example, of the colours red and blue with the shapes square and
circle)18
just as one can speak of combinations of various shapes or bodies. And here we’re
at the root of my19
misleading claim that a fact is a complex of objects. So the fact that a
person is ill is compared with a conjunction of two things, one of them the person, the other
the illness. Let’s not forget that that’s only a simile.20
Or one has to say that here the word “combination”, or “complex”, is like the word
“number”, which is also used in various – logically more or less similar – ways (meanings).
21
“Meaning” (“Bedeutung”) comes from “point” (“deuten”).
What we call meaning must be connected with the primitive language of gestures
(pointing-language).
22
If for example I correlate the actual seating order at a dinner table with a written seating
plan, then pointing to a particular person when reading each name makes good sense.
But if I were to compare, say, the description of a picture with the picture itself, and if in
7 (M): (
8 (V): Thus it’s as if someone were to declare:
9 (V): portraying learning a language too simply;
10 (M): )
11 (M): (
12 (V): Originally I wanted to say: The
13 (V): view
14 (V): show us whence this conception is derived
in the first place // is really derived.
15 (V): intensity,
16 (R): [Perhaps also cf. “complex and fact”] [MS Gretl
p. 113]
17 (V): One can for example – in a way others can
understand –
18 (V): circle) all brought together (E): Above
this: unreadable word in brackets.
19 (V): the
20 (V): Let’s beware of // Let’s beware of // forgetting
that that’s only a simile.
21 (M): 555
22 (M): ////
The Concept of Meaning Originates in a Primitive Conception of Language. 24e
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vergleichen und außer dem Personenverzeichnis sagte die Beschreibung auch, daß eine gewisse
Person eine andere küßt, so wüßte ich nicht, worauf ich als Korrelat des Wortes ”küssen“
zeigen sollte. Oder, wenn etwa stünde ”A ist größer als B“, worauf soll ich beim Wort ”größer“
zeigen? – Ganz offenbar kann ich ja gar nicht auf etwas diesem Wort entsprechendes in dem
Sinne zeigen, wie ich etwa auf die Person A im Bilde zeige.
Es gibt freilich einen Akt ”die Aufmerksamkeit auf die Größe der Personen richten“,
oder auf ihre Tätigkeit, und in diesem Sinne kann man auch das Küssen und die
Größenverhältnisse kollationieren. Das zeigt, wie der allgemeine Begriff der Bedeutung
entstehen konnte. Es geschieht da etwas Analoges, wie wenn25
das Pigment an Stelle der
Farbe tritt.
Und der Gebrauch des Wortes ”kollationieren“ ist hier so schwankend, wie der
Gebrauch des Wortes ”Bedeutung“.26
27
Die Wörter haben offenbar ganz verschiedene Funktionen im Satz und diese
Funktionen erscheinen uns ausgedrückt in den Regeln, die von den Wörtern gelten.28
Die Bedeutg. des Wortes – & auf die Bedeutg. zeigen.
29
Wie in einem Stellwerk mit Handgriffen die verschiedensten Dinge ausgeführt werden,
so mit den Wörtern der Sprache, die Handgriffen entsprechen. Ein Handgriff ist der einer
Kurbel und diese kann kontinuierlich verstellt werden; einer gehört zu einem Schalter und
kann nur entweder umgelegt oder aufgestellt werden; ein dritter gehört zu einem Schalter,
der drei oder mehr Stellungen zuläßt; ein vierter ist der Handgriff einer Pumpe und wirkt
nur, solange30
er auf- und abbewegt wird; etc.: aber alle sind Handgriffe, werden mit der
Hand angefaßt.31
32
Vergleich der Linien mit verschiedenen Funktionen33
auf der Landkarte mit den
Wortarten im Satz. Der Unbelehrte sieht eine Menge Linien und weiß nicht, daß sie sehr
verschiedene Bedeutungen haben. Grenzen, Meridiane, Straßen, Schichtenlinien, Buchstaben.
Ein solches Zeichen sei durch einen Strich durchstrichen um zu zeigen daß es falsch ist.34
Auf dem
Plan sind viele Striche gezogen, aber der, der ihn durchstreicht, hat eine gänzlich andere
Funktion als die anderen.
35
Der Unterschied der Wortarten ist wie der Unterschied der Spielfiguren, oder, wie der
noch größere, einer Spielfigur und des Schachbrettes.36
25 (V): wenn man
26 (M): )
27 (M): ( ///
28 (M): )
29 (M): (
30 (V): wenn
31 (M): )
32 (M): / (R): ∀ S. 42/1
33 (V): Vergleich der verschiedenen Arten von
Linien
34 (V): Denken wir uns den Plan eines Weges gezeichnet
und mit einem Strich durchstrichen,
der anzeigen soll, daß dieser Plan nicht auszuführen
ist. // daß dieser Weg nicht zu gehen ist.
35 (M): ? /
36 (R): 1 S. 42/1
29
25 Der Begriff der Bedeutung . . .
28
WBTC007-10 30/05/2005 16:18 Page 50
addition to giving a list of people the description also said that one particular person was
kissing another, then I wouldn’t know what I should point to in the picture as a correlate of
the word “kiss”. Or if the description said, e.g., “A is taller than B” what should I point to
when I come to the word “taller”? – Quite obviously, I can not point to something corresponding
to this word in the sense that I point to person A in the picture.
To be sure, there is an act of “directing attention to the size of people” or to their actions,
and in this sense one can also give correlatives for kissing and size-relationships. This shows
how it was possible for the general concept of meaning to come about. What happens here
is analogous to pigment taking the place of colour.
And here the use of the word “correlate” fluctuates as much as the use of the word
“meaning”.23
24
Obviously, words have completely different functions in a sentence, and these functions
appear to us as expressed in the rules that apply to the words.25
The meaning of a word – and pointing to the meaning.
26
As in a signal tower, where the most varied things are done with handles – that’s how
it is in language with words, which correspond to handles. One handle is that of a crank and
it can be adjusted continuously; another is part of a switch and can only be turned on or off;
a third is part of a switch that allows for three or more positions; a fourth is the handle of
a pump and only has an effect so long as27
it is moved up and down; etc.: but all of them
are handles, are gripped by hand.28
29
Compare lines on a map that have different functions30
with the types of words in a
sentence. Someone who hasn’t been taught sees a large number of lines and doesn’t know
that they have very different meanings. Borders, meridians, streets, strata-lines, letters.
Let such a sign be crossed out by a line in order to show that it is wrong.31
Many lines are drawn
on the map, but the one that crosses out the sign has an entirely different function from
the others.
32
The difference between kinds of words is like the difference between pieces in games,
or the even greater one between a chess piece and a chess-board.33
23 (M): )
24 (M): ( ///
25 (M): )
26 (M): (
27 (V): effect when
28 (M): )
29 (M): / (R): ∀ p. 42/1
30 (V): Compare different kinds of lines
31 (V): Let’s imagine a drawing of a path on a map,
and it is crossed out with a line to show that
this map cannot be followed. // that this path
cannot be taken.
32 (M): ? /
33 (R): 1 p. 42/1
The Concept of Meaning Originates in a Primitive Conception of Language. 25e
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8
Bedeutung, der Ort des Wortes im
grammatischen Raum.
Wir können in der alten Ausdrucksweise sagen: Das Wesentliche ist die Bedeutg. des Wortes,
nicht das Wort. Wir können also das Wort durch ein anderes ersetzen, das die gleiche Bedeutg. hat.1
Damit ist gleichsam ein Platz für das Wort fixiert und man kann ein Wort für das andere
setzen, wenn man es an den gleichen Platz setzt.
2
Kann man aber in diesem Sinne in einem Gedicht Worte durch andere ersetzen? Welche Art
Unterschied macht es, wenn ich in einer Betrachtung der Gesetze des freien Falls3
&Schnelligkeit“ statt
&Geschwindigkeit“ sage oder statt des Buchstabens v4
einen Hebräischen gebrauche; anderseits
aber, wenn ich ein Wort eines Gedichts durch das Zeichen A ersetze, wobei ich erkläre, A solle
die Bedeutung5
des Wortes haben. Das wäre als wollte ich ein finsteres Gesicht machen & dazu
sagen, daß es das gleiche bedeuten solle wie ein freundliches Lächeln.
6
Wenn ich mich entschlösse (in meinen Gedanken) statt ”rot“ ein neues Wort zu sagen,
wie würde es sich zeigen, daß dieses an dem Platze des Wortes ”rot“ steht? Wodurch ist der
Platz7
eines Wortes bestimmt? Angenommen etwa, ich wollte auf einmal alle Wörter meiner
Sprache durch andere ersetzen, wie könnte ich wissen, an welcher Stelle eines der neuen Worte
steht?8
Sind es etwa immer die Vorstellungen, die9
den Platz des Wortes halten? 10
So daß an
einer Vorstellung quasi ein Haken ist, – und hänge ich an den ein Wort, so ist ihm dadurch11
der Platz angewiesen?
Oder: Wenn ich mir den Platz merke, was merke ich mir da?
12
Man könnte z.B. ausmachen, im Deutschen statt ”nicht“ immer ”non“13
zu setzen und
dafür statt ”rot“ ”nicht“. So daß das Wort ”nicht“ in der Sprache bliebe und doch könnte
man nun sagen, daß ”non“14
so gebraucht wird, wie früher ”nicht“, und daß jetzt ”nicht“
anders gebraucht wird als früher.
15
Die Bedeutung könnte ich den Ort eines Wortes in der Grammatik nennen.16
1 (V): /// Wir können in der alten
Ausdrucksweise sagen: das Wesentliche am
Wort ist seine Bedeutung.
/ Wir sagen: das Wesentliche am Wort ist
seine Bedeutung: wir können das Wort durch ein
anderes ersetzen, das die gleiche Bedeutung
hat.
2 (M): ? / ///
3 (V): Falls das Wo
4 (V): v etwa
5 (V): solle die gleiche Bedeutung haben wie^
6 (M): / (R): ∀ S. 31/1
7 (V): ist die Stelle
8 (V): wissen, welches Wort an der Stelle eines
früheren steht. // steht?
9 (V): die bleiben und
10 (M): ////
11 (V): damit
12 (M): ? /
13 (V): ,,not“
14 (V): ,,not“
15 (M): ? /
16 (V): Der Ort eines Wortes in der Sprache //
Grammatik // ist seine Bedeutung.
30
29v, 30, 29v
30
29v
30
31
26
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8
Meaning, the Position of the Word in
Grammatical Space.
In the old way of putting it we can say: What really counts is the meaning of a word, not the
word. So we can replace the word with another that has the same meaning.1
A place has thereby
been set for the word, as it were, and we can substitute one word for another if we put it in
the same place.
2
But in a poem, can one in this sense replace words with other words? What sort of difference
does it make if in an examination of the laws of free fall I say “speed” instead of “velocity”, or use
a Hebrew letter instead of v?3
On the other hand, what if I replace a word in a poem by the sign A,
explaining that A is to have the meaning of4
the word? That would be like frowning and saying that
this is to mean the same thing as a friendly smile.
5
If I decided to use a new word instead of “red” (in my mind), how would it show that
it stood in the place of the word “red”? What determines the place of a word? Say, for
example, I wanted simultaneously to replace all the words in my language with others, how
could I know at which point one of the new words stood?6
Is it perhaps mental images that
always reserve7
the place of a word? 8
So that a mental image is equipped with a hook, as it
were – and if I attach a word to the hook then the word has thereby been assigned a place?
Or: When I remember the place, what am I remembering?
9
One could agree for instance always to put “non”10
in place of “not” in English, and in
turn to put “not” in place of “red”. So that the word “not” would remain in the language,
and yet one could now say that “non”11
is used in such a way as “not” was before, and that
“not” is now used differently from before.
12
I could call “meaning” the location of a word in grammar.13
1 (V): /// In the old way of putting it we can say:
what really counts about a word is its meaning.
/ We say: what really counts about a word
is its meaning: we can replace the word with
another that has the same meaning.
2 (M): ? / ///
3 (V): v perhaps?
4 (V): have the same meaning as
5 (M): / (R): ∀ p. 31/1
6 (V): know which word stands in the place of a
previous one. // one?
7 (V): always remain and reserve
8 (M): ////
9 (M): ? /
10 (V): “nicht”
11 (V): “nicht”
12 (M): ? /
13 (V): The location of a word in language //
grammar // is its meaning.
26e
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17
Wäre es nicht ähnlich, wenn ich mich entschlösse, die Formen der Schachfiguren zu
ändern, oder etwa die Figur eines Pferdchens als König zu nehmen?18
Wie würde es sich
nun zeigen, daß das hölzerne Pferdchen Schachkönig ist? Kann ich hier nicht sehr gut von
einem Wechsel der Bedeutung reden?
Unter der Bedeutung eines Namens wird nicht sein Träger verstanden.19
20
Man kann sagen, daß die Worte ”der Träger des Namens ,N‘“dieselbe Bedeutung haben
wie der Name ”N“ – also für einander eingesetzt werden können.
Aber heißt es nicht dasselbe, zu sagen ”zwei Namen haben einen Träger“ und ”zwei Namen
haben ein- und dieselbe Bedeutung“? (Morgenstern, Abendstern, Venus.)
21
Wenn mit dem Satz ”,A‘ und ,B‘ haben denselben Träger“ gemeint ist: ”der Träger
des Namens22
,A‘“ bedeutet dasselbe wie ”der Träger des Namens23
,B‘“, so ist alles in
Ordnung, weil das dasselbe heißt wie A = B. Ist aber mit dem Träger von ”A“ etwa der
Mensch gemeint, von dem es sich feststellen läßt, daß er auf den Namen ”A“ getauft ist;
oder der Mensch, der das Täfelchen mit dem Namen ”A“ um den Hals trägt; etc., so ist
es gar nicht gesagt, daß ich mit ”A“ diesen Menschen meine, und daß die Namen, die den
gleichen Träger haben, dasselbe bedeuten.24
25
Aber zeigen wir nicht zur Erklärung der Bedeutung auf den Gegenstand, den der Name
vertritt? Ja; aber dieser Gegenstand ist nicht ”die Bedeutung“, obwohl sie durch das Zeigen
auf diesen Gegenstand bestimmt wird.
Aber es bestimmt hier schon das richtige Verstehen des Wortes ”Träger“ in dem
besondern Fall (Farbe, Gestalt, Ton, etc.) die Bedeutung sozusagen bis auf eine letzte
Bestimmung. D.h. der26
erklärende Hinweis auf den Träger entscheidet nur noch eine Frage
nach der Bedeutung von der Art: &Welcher dieser Leute ist Herr N?“, &Welche Farbe heißt
,lila‘?“, &Welcher Ton ist das hohe C?“.27
28
Man kann sagen: Die Bedeutung eines29
Wortes lehren, heißt seinen Gebrauch lehren & das kann
man durch Hinweisen auf den Träger eines Namens tun, wenn dieser Gebrauch, sozusagen, schon
bis auf eine letzte Bestimmung bekannt ist.
Erinnere Dich daran, daß durch die selbe hinweisende Geste auf denselben Körper die Bedeutung
von Worten verschiedener Art erklärt werden kann.30
Z.B.: &das heißt ,Holz‘“, &das heißt ,braun‘“,
&das heißt31
,Stab‘“, &das heißt32
,Federstiel‘“.
Denken wir aber dagegen33
an das Zeigen & Benennen von Gegenständen, wie34
man Kindern die
Anfänge der Sprache lehrt. Hier kann man natürlich nicht sagen, diese Erklärung (wenn man das eine
Erklärung nennen will) gebe noch eine letzte Bestimmung über den Gebrauch des Wortes, & das Kind
17 (M): ? /
18 (V): oder etwa eine Figur, die wir jetzt
”Rössel“ nennen würden, als Königsfigur zu
nehmen?
19 (V): Wir verstehen unter ”Bedeutung des
Namens“ nicht den Träger des Namens. //
Unter &Bedeutung des // eines // Namens“ wird
nicht . . . verstanden.
20 (M): (
21 (M): – /
22 (V): Träger von
23 (V): Träger von
24 (M): )
25 (M): (
26 (V): der H
27 (M): )
28 (M): ( ////
29 (V): des
30 (V): Körper Worte verschiedener Art erklärt werden
können.
31 (V): heißt ein
32 (V): heißt ein
33 (V): wieder
34 (V): Gegenständen, durch das
30v
31
32
31v
31
27 Bedeutung . . .
WBTC007-10 30/05/2005 16:18 Page 54
14
Wouldn’t it be similar if I decided to change the shapes of chess pieces or, say, to treat
a knight as the king?15
Then how would it be shown that the wooden knight is a king? Can’t
I quite rightly speak of a change in meaning here?
By meaning of a name we don’t understand its bearer.16
17
One can say that the words “The bearer of the name ‘N’” have the same meaning as
the name “N” – and thus can be used for each other.
But doesn’t it mean the same thing to say “Two names have one bearer” and “Two names
have one and the same meaning”? (Morning star, evening star, Venus.)
18
If what is meant by the sentence “‘A’ and ‘B’ have the same bearer” is: “The bearer of
the name ‘A’”19
means the same as “The bearer of the name ‘B’”20
, then everything is all
right because that means the same as A = B. But if, for instance, “the bearer of ‘A’” means
the person of whom it can be ascertained that he was given the name “A” at his baptism,
or the person who is wearing a little plaque with the name “A” around his neck, etc., then
it is by no means certain that by “A” I mean this person, and that names that have the same
bearer mean the same thing.21
22
But when we want to explain the meaning of a name, don’t we point to the object it
stands for? Yes, but this object isn’t “the meaning”, even though that is specified by pointing
to the object.
But here understanding the word “bearer” correctly in a particular case (colour, shape,
sound, etc.) determines the meaning, except for one final determination, so to speak. That
is, what is left for the explanatory reference to the bearer to decide is this sort of question about
the meaning: “Which of these people is Mr N?”, “Which colour is called ‘lilac’?”, “Which note is
high C?”.23
24
One can say: To teach the meaning of a word means teaching its use, and one can do this by
pointing to the bearer of a name, if this use is already known except for one final determination,
as it were.
Remember that by using the same ostensive gesture toward the same physical object the meaning
of different kinds25
of words can26
be explained. For example: “That means ‘wood’”, “That means
‘brown’”, “That means ‘stick’”, “That means ‘penholder’”.
But on the other hand,27
let’s think about pointing to and naming objects, as28
one teaches
children the beginnings of language. Of course one can’t say that this explanation (if one wants
to call it an explanation) provides one last determination of the use of a word; and the child can’t
14 (M): ? /
15 (V): or, say, to treat the piece that we now call
“knight” as the king?
16 (V): We don’t understand by “meaning of the
name” its bearer. // By “meaning of the // a // name”
we don’t understand its bearer.
17 (M): (
18 (M): – /
19 (V): bearer of ‘A’”
20 (V): bearer of ‘B’”
21 (M): )
22 (M): (
23 (M): )
24 (M): ( ////
25 (V): object, different kinds
26 (V): could
27 (V): Then again,
28 (V): objects, by means of which
Meaning, the Position of the Word in Grammatical Space. 27e
WBTC007-10 30/05/2005 16:18 Page 55
kann auch noch nicht fragen &wie heißt das?“. (D.h., diese &Erklärung“35
ist nicht die Antwort auf
die Frage &wie heißt dieser Gegenstand“.)36
37
Wenn ich sage ”die Farbe dieses Gegenstands heißt ,violett‘“, so muß ich die Farbe mit
den ersten Worten ”die Farbe dieses Gegenstands“ schon benannt haben, sie schon zur Taufe
gehalten haben, damit die Namengebung geschehen kann.38
Denn ich könnte auch sagen ”der
Name dieser Farbe (der Farbe dieses Dings) ist von Dir zu bestimmen“, und der den Namen
gibt, müßte nun schon wissen, wem er ihn gibt (an welchen Platz der Sprache er ihn stellt).
39
Ich könnte so40
erklären, die Farbe dieses Flecks heißt ”rot“, die Form ”Kreis“. Und
hier stehen die Wörter ”Farbe“ und ”Form“ für Anwendungsarten (grammatische Regeln)
und bezeichnen41
in Wirklichkeit Wortarten, wie ”Eigenschaftswort“, ”Hauptwort“. Man
könnte sehr wohl in der deutschen Grammatik die Bezeichnungen42
”Farbwort“, ”Formwort“,
”Klangwort“ einführen. (Aber mit demselben Recht auch ”Baumwort“, ”Buchwort“?)
43
Der Name, den ich einem Körper gebe, einer Fläche, einem Ort, einer Farbe, hat in jedem
dieser Fälle eine44
andere Grammatik. ”A“45
in ”A ist gelb“ hat eine andere Grammatik, wenn
es46
der Name eines Körpers als47
wenn es der Name der Fläche eines Körpers ist, ob nun
der48
Satz ”dieser Körper ist gelb“ sagt, daß die Oberfläche des Körpers gelb ist, oder daß
er durch und durch gelb ist. Und man zeigt in anderem Sinne auf einen Körper; auf seine
Länge, & auf seine Farbe.49
D.h. es ist etwa50
eine Definition möglich: auf eine Farbe zeigen heißt
auf den Körper zeigen der sie hat. 51
Und so hat auch das hinweisende Fürwort ”dieser“
andere Bedeutung (d.h. Grammatik), wenn es sich auf Hauptwörter mit verschiedener
Grammatik52
bezieht. [Worin soll der Unterschied dieser Grammatiken liegen?]
53
Man kann sagen &dieser Körper ist durch & durch gelb“ aber nicht, &seine Oberfläche ist durch
& durch gelb“.
Auf eine Zahl deuten.
54
Und wer55
auf einen Körper zeigt, zeigt dadurch, aber eben darum in anderem Sinne, auf seine
Farbe, seine Gestalt, den Ort an dem er sich befindet. Wie der, welcher jemand Klavier spielen hört,
dadurch in anderem Sinne das Musikstück hört, welches gespielt wird & in noch anderem Sinne die
Schönheit des Stückes. – Aber was heißt es &er hört in anderem Sinne“, &er zeigt in anderem Sinne“.
Was ich meine wäre jedenfalls in einer Definition ausgedrückt die etwa sagte: auf eine Farbe zeigen,
heißt: auf einen Körper zeigen der die Farbe hat. Also etwa F(φ) = (∃x).φx · Fx.56
Daß F von φ in anderm
Sinne ausgesagt wird als von x heißt, daß ich statt Fx nicht wieder einen Ausdruck wie die rechte Seite
setzen kann.
35 (V): (Diese &Erklärung“
36 (M): )
37 (M): ? /
38 (V): damit der Akt der Namengebung das sein
kann, was er ist.
39 (M): ? /
40 (V): also
41 (V): sind
42 (V): wohl in der (gewöhnlichen) Grammatik
neben diesen Wörtern die Wörter
43 (M): ü ? /
44 (V): hat jedes Mal
45 (V): Der Name ”A“
46 (V): A
47 (V): und
48 (V): ein
49 (V): durch gelb ist. ”Ich zeige auf A“ hat
verschiedene Grammatik, je nachdem A ein
Körper, eine Fläche, eine Farbe ist etc. // Und
man zeigt in anderem Sinne auf den Körper A, auf die
Länge A eines Körpers, & auf die Farbe A.
50 (V): ist z.B.
51 (M): ? ///
52 (V): Hauptwörter verschiedener Grammatik
53 (M): ∫
54 (M): ∫
55 (V): wer (mit der Hand)
56 (O): F(ϕ) = (∃x) · ϕx · Fx
32
33
32v
33
33
32v
28 Bedeutung . . .
WBTC007-10 30/05/2005 16:18 Page 56
yet ask “What’s the name for that?”. (That is, such an “explanation”29
is not the answer to the
question “What’s the name of this object?”.)30
31
If I say “The colour of this object is ‘violet’”, then I must already have named the colour
when I said “the colour of this object”; I must already have presented the colour at the
baptismal font so that the naming can take place.32
For I could also say “The name of this
colour (of the colour of this thing) is for you to decide”, and then the person who is to give
the name would have to know to what he is going to give it (at what location in the language
he is going to position it).
33
I could explain things this way: the colour of this patch is called “red”, the shape
“circle”. And here the words “colour” and “shape” stand for kinds of application (grammatical
rules) and really signify34
kinds of words, such as “adjective”, “noun”. One could
perfectly well introduce the terms “colour-word”, “shape-word”, “sound-word” into English35
grammar.36
(But also, with the same justification, “tree-word”, “book-word”?)
37
The name I give to an object, a surface, a place, a colour, has a different grammar in
each of these cases.38
“A”39
in “A is yellow” has a different grammar when it40
is the name of
an object from41
when it is the name of the surface of an object, regardless of whether the42
proposition “This object is yellow” says that the surface of the object is yellow or that it is
yellow through and through. And one points in different senses to an object, to its length and to
its colour.43
That is to say, one could come up with a definition: pointing to a colour means pointing
to the object that has that colour. 44
Likewise the demonstrative pronoun “this” also has different
meanings (i.e. a different grammar) when it refers to nouns with different grammars.
[What is the difference between these grammars supposed to consist in?]
45
One can say “This object is yellow through and through” but not “Its surface is yellow through
and through”.
To point to a number.
46
And a person who points to an object47
thereby points to its colour, its shape, the place where
it is; but for that very reason he is pointing to it in a different sense in each case. Just as whoever
hears someone playing a piano hears the piece that is being played in a different sense, and he hears
the beauty of the piece in yet another sense. – But what does “He hears in a different sense”, “He
points to in a different sense”, mean? Whatever the case, what I mean can be expressed by a definition
something like this: Pointing to a colour means: pointing to an object that has that colour. So for
example: F(φ) = (∃x).φx · Fx.48
That F is asserted of φ in a different sense than it is of x means that
I can’t replace Fx with another expression such as that on the right side.
29 (V): (This “explanation”
30 (M): )
31 (M): ? /
32 (V): so that the act of name-giving can be what
it is.
33 (M): ? /
34 (V): really are
35 (V): (ordinary)
36 (V): grammar to stand alongside the latter
terms.
37 (M): r ? /
38 (V): grammar every time.
39 (V): The name “A”
40 (V): A
41 (V): and
42 (V): a
43 (V): through. “I’m pointing to A” has a different
grammar depending on whether A is a
solid, a surface, a colour, etc. // And one points
in a different sense to the object A, to the length A
of an object, and to the colour A.
44 (M): ? ///
45 (M): ∫
46 (M): ∫
47 (V): physical object (with his hand)
48 (O): F(ϕ) = (∃x) · ϕx · Fx
Meaning, the Position of the Word in Grammatical Space. 28e
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9
Die Bedeutung eines Wortes ist das,
was die1
Erklärung der
Bedeutung erklärt.2
3
&Bedeutung, das was die Erklärung der B. erklärt“ d.h.: Fragen wir nicht was4
Bedeutung sei,
sondern sehen wir nach5
was man die &Erklärung6
der B.“ nennt.
7
Man sagt dem Kind: ”nein, kein Stück Zucker mehr!“ und nimmt es ihm weg. So lernt
das Kind die Bedeutung des Wortes ”kein“.
Hätte man ihm mit denselben Worten ein Stück Zucker gereicht, so hätte es gelernt, das
Wort anders zu verstehen. Es hat damit gelernt, das Wort gebrauchen, aber auch ein bestimmtes
Gefühl mit ihm zu verbinden, es in bestimmter Weise zu erleben.
8
Veranlassen wir es dadurch nicht, Worten einen Sinn beizulegen, ohne daß wir sie durch
ein anderes Zeichen ersetzen, also ohne diesen Sinn auf andere Weise auszudrücken?
Veranlassen wir es nicht gleichsam, für sich etwas zu tun, dem kein äußerer Ausdruck gegeben
wird, oder wozu der äußere Ausdruck nur im Verhältnis einer Hindeutung steht? Die
Bedeutung ließe sich nicht aussprechen, sondern nur auf sie von ferne hinweisen. Sie ließe
sich gleichsam nur verursachen. Aber welchen Sinn hat es dann überhaupt, wenn wir von
dieser Bedeutung reden? (Schlag und Schmerz)
9
Was wollen wir unter &Bedeutung“ eines Worts verstehen? Ein charakteristisches Gefühl, das das
Aussprechen (Hören) des Wortes begleitet? (Das und-Gefühl, wenn-Gefühl James’s) Oder wollen wir
das Wort &Bedeutung“ ganz anders gebrauchen; &, z.B., sagen zwei Worte haben die gleiche
Bedeutung wenn dieselben gramm. Regeln von beiden gelten? Wir können es halten, wie wir
wollen, müssen aber wissen10
daß dies zwei gänzlich verschiedene Gebrauchsweisen (Bedeutungen)
des Wortes &Bedeutung“ sind. (Man kann vielleicht auch von einem spezifischen Gefühl reden welches
der Schachspieler bei Zügen mit dem König empfindet.)
11
Gibt mir die Erklärung des Wortes die Bedeutung, oder verhilft sie mir nur zur
Bedeutung? Ist die Bedeutung das Gefühl, dann ist die Bedeutung in der Erklärung nicht
niedergelegt, aber durch sie etwa bewirkt wie12
die Krankheit durch eine Speise.
1 (V): die (grammatische)
2 (R): [Dazu der letzte Satz dieses §] Siehe auch §41
S. 179
3 (M): 3
4 (V): was die
5 (V): wir uns an
6 (V): die (ErklärungA
7 (M): ? / 3
8 (M): ? / 555
9 (M): 3
10 (V): wollen, aber wir müssen wissen
11 (M): / 3
12 (V): Bedeutung? So daß also das Verständnis in
der Erklärung nicht niedergelegt wäre, sondern
durch sie nur äußerlich bewirkt, wie
34
33v
34
33v
34
33v
34
29
WBTC007-10 30/05/2005 16:18 Page 58
9
The Meaning of a Word is
What the1
Explanation of its
Meaning Explains.2
3
“Meaning: what the explanation of meaning explains”, that is: Let’s not ask what4
meaning is,
but instead let’s examine5
what is called the “explanation of meaning”.
6
One says to a child: “Stop, no more sugar!”, and takes the sugar cube away from him.
That’s the way a child learns the meaning of the word “no”.
Had one said the same words while handing him a sugar cube, he would have learned to
understand the word differently. In this way he has learned to use the word, but also to associate
a particular feeling with it, to experience it in a particular way.
7
Isn’t this the way we cause the child to attribute sense to words, without substituting
another sign for them, and thus without expressing the sense in a different way? Aren’t we
causing him, as it were, to do something for himself, which is given no outward expression,
or for which the outward expression serves only as a suggestion? As if the meaning
couldn’t be uttered – one could merely point to it from afar. One could merely trigger it, as
it were. But then what’s the point of talking about meaning at all? (Blow and Pain)
8
What do we want to understand by the “meaning” of a word? A characteristic feeling that
accompanies the uttering (hearing) of the word? (James’s and-feeling, if-feeling.) Or do we want
to use the word “meaning” completely differently; and say, for instance, that two words have the
same meaning if the same grammatical rules apply to both? We can do as we like, but we must9
be
aware that these are two completely different uses (meanings) of the word “meaning”. (Perhaps
one can also speak of a specific feeling felt by a chess player when he moves his king.)
10
Does an explanation of a word give me its meaning, or does it only help me find it? If
meaning is a feeling, then meaning isn’t established by the explanation, but is brought about by it,
say as11
an illness is by a certain kind of food.
1 (V): the (Grammatical)
2 (R): [To the last sentence of this §] See also §41
p. 179
3 (M): 3
4 (V): what the
5 (V): let’s look at
6 (M): ? / 3
7 (M): ? / 555
8 (M): 3
9 (V): but we must
10 (M): / 3
11 (V): it? So that understanding would thus not
be recorded in the explanation, but would
merely be externally caused by it, as
29e
WBTC007-10 30/05/2005 16:18 Page 59
34
35
36
30 Die Bedeutung eines Wortes . . .
13
In einem Sinn kann man die Erklärung der Bedeutung die Ausschließung von Mißverständnissen
nennen.14
Sie sagt, das Wort hat diese Bedeutung, nicht jene.
15
Und &Erklärung der Bedeutung“ nennen wir vielerlei.
16
Das Problem äußert sich auch in der Frage: Wie erweist sich ein Mißverständnis? Denn
das ist dasselbe wie das Problem: Wie zeigt es sich, daß ich richtig verstanden habe? Und
das ist: Wie kann ich die Bedeutung erklären?
Es fragt sich nun: Kann sich ein Mißverständnis darin äußern, daß, was der Eine bejaht,
der Andere verneint?
17
Nein, denn dies ist eine Meinungsverschiedenheit und kann als solche aufrecht erhalten
werden. Bis wir annehmen, der Andere habe Recht. . . .
18
Wenn ich also, um das Wort ”lila“ zu erklären, auf einen Fleck zeigend sage ”dieser Fleck
ist lila“, kann diese Erklärung dann auf zwei Arten funktionieren? einerseits als Definition,
die den Fleck als Zeichen gebraucht, anderseits als Erläuterung? Und wie das letztere? Ich
müßte annehmen, daß der Andere die Wahrheit sagt und dasselbe sieht, was ich sehe. Der
Fall, der wirklich vorkommt, ist etwa folgender: A erzählt dem B in meiner Gegenwart,
daß ein bestimmter Gegenstand lila ist. Ich höre das, habe den Gegenstand auch gesehen
und denke mir: ”jetzt weiß ich doch, was ,lila‘ heißt“. Das heißt, ich habe aus jener
Beschreibung19
eine Worterklärung gezogen.
Ich könnte sagen: Wenn das, was A dem B erzählt, die Wahrheit ist, so muß das Wort
”lila“ diese Bedeutung haben.
Ich kann diese Bedeutung also auch quasi hypothetisch annehmen und sagen: wenn ich
das Wort so20
verstehe, hat A Recht. Aber dem &so“ entspricht eine Hinweisende Definition.
21
Man sagt: ”Ja, wenn das Wort das bedeutet, so ist der Satz wahr“.
22
Nehmen wir an, die Erklärung der Bedeutung war nur eine Andeutung: konnte man da
nicht sagen: Ja, wenn diese Andeutung so verstanden wird, dann gibt das Wort in dieser
Verbindung einen wahren Satz etc. Aber dann muß23
nun dieses ”so“ ausgedrückt sein.
Die Erklärung immer nur eine Andeutung.
24
Die Erklärung eines Zeichens kann jede Meinungsverschiedenheit in Bezug auf seine
Bedeutung beseitigen.25
Und ist dann noch eine Frage nach der Bedeutung zu entscheiden?
26
Mißverständnis nenne ich das, was durch eine Erklärung zu beseitigen ist. Die
Erklärung der Bedeutung eines Wortes schließt Mißverständnisse aus.
27
Die Aufklärung wird in einer Spr. gegeben, die28
unabhängig von dem Mißverständnis
besteht.
13 (M): 3
14 (V): In einem Sinn ist die Erklärung der Bedeutung die
Aufklärung von Mißverständnissen.
15 (M): 3
16 (M): / 3 ////
17 (M): ? / /// 3
18 (M): / (R): [Zu S. 48]
19 (V): aus jenen Sätzen
20 (O): Wort ,so
21 (M): /
22 (M): ? / 5555
23 (V): muß man
24 (M): //// 3
25 (V): Zeichens muß jede Meinungsverschiedenheit
in Bezug auf seine Bedeutung beseitigen
können.
26 (M): ///
27 (M): ∫
28 (V): Die Aufklärung kann nur verstanden werden,
wenn sie in einer Sprache gegeben wird, die
33v
WBTC007-10 30/05/2005 16:18 Page 60
12
In one sense, you can call an explanation of a meaning the exclusion of misunderstandings.13
It
says that the word has this but not that meaning.
14
And there are all sorts of things we call “explanation of meaning”.
15
The problem also emerges in this question: How does a misunderstanding become
evident? For that’s the same as the problem: How does it become evident that I have
understood correctly? And that means: How can I explain the meaning?
Now the question is: Can a misunderstanding be revealed in one person’s affirming what
another denies?
16
No, because that’s a difference of opinion and it can be adhered to as such. Until we
assume that the other person is right . . . .
17
So if to explain the word “lilac” I point to a patch and say “This patch is lilac”, can this
explanation then work in two ways – on the one hand as a definition that uses the patch
as a sign, and on the other as an elucidation? And how is the latter possible? I would have
to assume that the other person is telling the truth and seeing the same thing I’m seeing.
A case that really occurs is something like this: In my presence A tells B that a certain
object is lilac. I hear this, and have also seen the object, and think to myself: “Now I know
for sure what ‘lilac’ means”. That is, I have extracted an explanation of the word from that
description.18
I could say: If what A told B is the truth, then the word “lilac” must have this meaning.
So I can also assume this meaning quasi-hypothetically, and say: if I understand the word
in that way, then A is right. But an ostensive definition corresponds to the “in that way”.
19
We say: “Yes, if the word means that, then the proposition is true”.
20
Let’s assume that an explanation of meaning were only an allusion. Then couldn’t one
say: Well, if this allusion is understood in such a way, then the word in this context produces
a true proposition, etc.? But then the “in such a way” has to be expressed.
An explanation is always just an allusion.
21
The explanation of a sign can22
remove every disagreement about its meaning.
And then – is there still a question about meaning that needs deciding?
23
What can be removed by an explanation I call a misunderstanding. The explanation of
the meaning of a word excludes misunderstandings.
24
Clarification is given in a language that25
is free of the misunderstanding.
12 (M): 3
13 (V): In one sense the explanation of a meaning is the
elucidation of misunderstandings.
14 (M): 3
15 (M): / 3 ////
16 (M): ? / /// 3
17 (M): / (R): [To p. 48]
18 (V): from those sentences.
19 (M): /
20 (M): ? / 5555
21 (M): //// 3
22 (V): sign must be able to
23 (M): ///
24 (M): ∫
25 (V): Clarification can only be understood if it is
given in a language that
The Meaning of a Word is What the Explanation of its Meaning Explains. 30e
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29
Was für Konsequenzen will ich daraus ziehen?!30
Hängt damit zusammen daß die Erklärung an Stelle des Zeichens gebraucht werden kann. Der
Satz sollte sagen daß die Erklärung nur innerhalb der schon ihrem Wesen nach verstandenen
Sprache geschieht. Die Erklärung entscheidet nur zwischen Möglichkeiten die der Fragende
selbst voraussehen konnte. Nicht die Sprache als solche wird31
für ihn aufgebaut, sondern nur diese
Ausdrucksweise. Da die Aufklärung ja verstanden wird so konnte sie auch als Möglichkeit schon früher
ins Auge gefaßt werden; es konnte auch nach ihr unmittelbar gefragt werden, so daß der Erklärende
nur mehr &ja“ oder &nein“ zu antworten hatte. Und mit &ja“ & &nein“ konnte er nicht das Wesen
der Sprache erklären.
32
Wie kann Einer nach der Erklärung einer Wortbedeutung fragen? – Z.B. so: &Welche Farbe heißt
,violett‘?“, oder:33
&welches heißt34
das 3 gestrichene C?“; aber auch so: &was heißt das Wort ,nefas‘?“.
35
Auf die erste & zweite Frage36
wird man durch ein Zeigen antworten & die Frage hatte das auch
vorausgesehen. Die dritte Frage könnte man durch eine Übersetzung ins Deutsche beantworten (oder
auch durch Beispiele der Anwendung). – Wie aber, wenn ein mathematisch nicht vorgebildeter fragte
&Was bedeutet das Wort ,Integral‘?“. Da müßte man wohl antworten: das ist ein mathematischer
Ausdruck, den ich Dir erst dann erklären kann,37
wenn Du mehr Mathematik verstehen wirst.
38
Ich habe einmal als Kind nach der Bedeutung des Wortes &etwas“ gefragt [oder war es &vielleicht“?].
Man antwortete mir: &das verstehst Du noch nicht“ Wie aber hätte man es erklären sollen!39
Durch eine Definition? oder sollte man sagen,40
das Wort sei undefinierbar? Wie ich es später
verstehen gelernt habe, weiß ich nicht; aber ich habe wohl Phrasen, worin das Wort vorkommt
anwenden gelernt. Und dieses Lernen hatte wohl am meisten Ähnlichkeit mit einem Abrichten
[abgerichtet Werden].
Ich wollte hier [auf dieser41
Seite] das Wesen des Mißverständnisses im Gegensatz zum
Unverständnis der Sprache darstellen.
42
Ein Mißverständnis ist:43
”Ist das eine Orange? ich dachte das sei eine“.
Wie ist es mit diesem: ”Ist44
das rot?45
ich dachte, das sei ein Sessel46
“?
Kann47
man sich nicht einbilden (wenn man48
nicht deutsch versteht) ”rot“ heiße laut
(werde49
so gebraucht, wie tatsächlich das Wort ”laut“ gebraucht wird)? Wie wäre50
die
Aufklärung dieses Mißverständnisses? Etwa so: ”rot ist diese51
Farbe, – keine Tonstärke“52
– Eine solche Erklärung könnte man natürlich geben, aber sie wäre nur dem verständlich,
der sich bereits53
in der Grammatik auskennt.
29 (M): ////
30 (E): Pfeilspitze eines von der vorhergehenden
Bemerkung herführenden Pfeiles.
31 (V): Nicht die Sprache wird
32 (M): ? /
33 (V): ,violett‘?“; aber au:
34 (V): ist
35 (M): ? /
36 (V): Auf die erste Frage
37 (V): antworten: das kann ich Dir erst dann erklären,
38 (M): ∫ ///
39 (V): sollen?
40 (V): oder hätte man mir sagen sollen,
41 (E): Pfeil verweist auf TS Seite 36.
42 (M): ∫
43 (V): Das sind Mißverständnisse:
44 (V): Kann man sagen: ”Ist
45 (V): rot?
46 (V): Sessel
47 (V): Aber kann
48 (V): man etwa
49 (V): laut) (d. h. werde
50 (V): wäre aber
51 (V): eine
52 (V): Tonstärke“?
53 (V): bereits ganz
35v
36
31 Die Bedeutung eines Wortes . . .
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26
What sorts of consequences do I want to draw from this?!27
It’s connected with the fact that an explanation can be used in place of a sign. That sentence was
supposed to say that an explanation takes place only within a language whose nature is already understood.
The explanation only decides between possibilities that the person asking the question could
himself foresee. It is not language as such that28
is being constructed for him, just this mode of expression.
Since the clarification is in fact understood, it could also have been considered as a possibility
earlier; someone could also have asked for the clarification directly, such that the person doing the
explaining had only to answer “Yes” or “No”. And he couldn’t explain the nature of language with
“Yes” and “No”.
29
How can someone ask for an explanation of a word’s meaning? – Like this, for instance: “Which
colour is called ‘violet’?”, or “Which note is called30
C with three lines?”; but also like this: “What
does ‘nefas’ mean?”.
31
The first and second questions32
will be answered by pointing, and that is what the questions
had anticipated. One could answer the third question by translating the word into English (or also
by giving examples of its usage). – But what if someone who had no previous mathematical training
were to ask “What does the word ‘integral’ mean?”. One would probably have to answer:
That is a mathematical expression that I won’t be able to explain to you33
until you understand
more mathematics.
34
As a child I once asked about the meaning of the word “something” (or was it “perhaps”?). I
was given the answer: “You don’t understand that yet”. But how should it have been explained!35
By a definition? Or should it be said36
that the word was indefinable? I don’t know how I learned to
understand it later; but probably I learned how to use phrases in which the word appeared. And this
learning most closely resembled training [being trained].
Here [on this page]37
I wanted to portray the nature of misunderstanding, as opposed to a lack of
understanding, of a language.
38
A misunderstanding is:39
“Is this an orange? I thought that was one.”
What about this:40
“Is that red?41
I thought that was a chair.42
”?
Can’t43
one believe (if one44
doesn’t understand English) that “red” means loud (is45
used as the word “loud” actually is)? How46
would one clear up this misunderstanding?
Something like this: “Red is this47
colour – not a volume.”48
– Of course one could give such
an explanation, but it would only be understandable to someone who already49
knows his
way around in grammar.
26 (M): ////
27 (E): An arrow leads here from the previous
remark.
28 (V): not language that
29 (M): ? /
30 (V): note is
31 (M): ? /
32 (V): The first question
33 (V): answer: I won’t be able to explain that to you
34 (M): ∫ ///
35 (V): explained?
36 (V): Or should it have been said to me
37 (E): An arrow points to p. 36 of the TS.
38 (M): ∫
39 (V): These are misunderstandings:
40 (V): Can one say:
41 (V): red?
42 (V): chair.
43 (V): But can’t
44 (V): (if perhaps one
45 (V): i.e., is
46 (V): But how
47 (V): a
48 (V): sound?”
49 (V): already completely
The Meaning of a Word is What the Explanation of its Meaning Explains. 31e
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54
Der Satz ”ist das rot? ich dachte, das sei ein Sessel“ hat nur Sinn, wenn das Wort ”das“
beide Male im gleichen Sinn gebraucht wird und dann muß ich entweder ”rot“ als
Substantiv, oder ”ein Sessel“ als Adjektiv auffassen.
&Heißt ,weak‘ schwach? ich dachte, es heiße Woche.“
55
Ist es denn nicht denkbar, daß ein grammatisches System in der Wirklichkeit zwei (oder
mehr) Anwendungen hat?
Ja, aber wenn wir das überhaupt sagen können, so müssen wir die beiden Anwendungen
auch durch eine Beschreibung unterscheiden können.
56
Zu sagen, daß das Wort ”rot“ mit allen Vorschriften, die von ihm gelten, das bedeuten
könnte, was tatsächlich das Wort ”blau“ bedeutet; daß also durch diese Regeln die
Bedeutung nicht fixiert ist, hat nur einen Sinn, wenn ich die beiden Möglichkeiten der
Bedeutung ausdrücken kann und dann sagen, welche die von mir bestimmte ist.
(Diese letztere Aussage ist aber eben die Regel, die vorher zur Eindeutigkeit gefehlt hat.)
57
Die Grammatik erklärt die Bedeutung der Wörter, soweit sie zu erklären ist. Und zu
erklären ist sie soweit, als nach ihr gefragt werden kann; und nach ihr fragen kann man soweit,
als sie zu erklären ist. Die Bedeutung ist das, was wir in der Erklärung der Bedeutung eines
Wortes erklären.
58
”Das, was 159
cm3
Wasser wiegt, hat man ,1Gramm‘ genannt“ – ”Ja, was wiegt er denn?“
(Bedeutung eines Wortes). 60
Mißverständnis. Unverständnis. Die Erklärg der Bedeutg immer nur61
eine Andeutg.
54 (M): ∫
55 (M): ∫ (R): Zu S. 43
56 (M): ? / //// [Vagueheit des Wortes &Wortart“.]
(R): Zu S. 43
57 (M): ? / ///
58 (M): / 3
59 (V): ein
60 (V): (@Bedeutung eines Wortes!).
61 (V): der Bedeutg nur
36
37
32 Die Bedeutung eines Wortes . . .
WBTC007-10 30/05/2005 16:18 Page 64
50
The expression “Is that red? I thought that was a chair” only makes sense if the word
“that” is used in the same sense both times, and then I have to understand either “red” as
a noun or “a chair” as an adjective.
“Does ‘Wagen’ mean ‘cart’? I thought it meant ‘dare’.”
51
Isn’t it conceivable that in reality a grammatical system has two (or more) applications?
Yes, but if we can say that at all, then we have to be able to differentiate between the two
applications with a description.
52
To say that the word “red”, along with all of the rules that apply to it, could mean what
in fact the word “blue” means – i.e. that its meaning isn’t fixed by these rules – only makes
sense if I can express the two possibilities of meaning and then say which is the one I’ve
specified.
(But this latter statement is precisely the rule that prevented them from meaning the
same thing.)
53
Grammar explains the meaning of words to the extent that it can be explained. And
it can be explained to the extent that questions about it can be asked: and one can ask
questions about it to the extent that it can be explained. Meaning is what we explain in
explaining the meaning of a word.
54
“What 155
c.c. of water weighs is called ‘1 gram’ ” – “Yes, but what does it weigh?”
(Meaning of a word).56
Misunderstanding. Lack of understanding. The explanation of meaning is always just57
an allusion.
50 (M): ∫
51 (M): ∫ (R): To p. 43
52 (M): ? / //// [Vagueness of the words “type of
word”.] (R): To p. 43
53 (M): ? / ///
54 (M): / 3
55 (V): a
56 (V): (!Meaning of a wordE).
57 (V): is just
The Meaning of a Word is What the Explanation of its Meaning Explains. 32e
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10
”Die Bedeutung eines Zeichens
ist durch seine Wirkung
(die Assoziationen,
die es auslöst, etc.) gegeben.“
1
Man möchte mit dem Gedächtnis & der Assoziation den Mechanismus des Bedeutens erklären.
2
Aber wir fühlen, daß es uns nicht auf eine3
Erklärung eines Mechanismus ankommen kann. Denn
diese Erklärung ist wieder eine Beschreibung von Phänomenen in der4
Sprache. Sie sagt, etwa:5
wenn
das Wort &rot“ gehört wird, springt die Vorstellung rot hervor. (Eine Tafel durch den Druck6
eines
Knopfes) Nun, wenn das eintritt, – was weiter? Wir wollen7
eben die8
Erklärung eines Kalküls9
hören.
Und die Erklärung des Mechanismus stellt sich außerhalb des Kalküls.10
Sie hat mit dem, was uns interessiert,
nichts zu tun. Sie ist selbst eine Beschreibung in der Sprache & eine die in den Kalkül, der uns
etwa erklärt werden soll, nicht eingreift. Während wir eine Erklärung brauchen, die ein Teil dieses
Kalküls ist.
11
Wenn ich sage, das Symbol ist das, was diesen Effekt hervorruft, so fragt es sich eben,
wie ich von diesem Effekt reden kann, wenn er gar nicht da ist. Und wie ich weiß, daß es
der ist, den ich gemeint habe, wenn er eintritt.12
13
Der Ausdruck ”das was diesen Effect hervorruft“ ist14
ja wieder ein Symbol. Und15
erklärt daher
das Wesen des Symbols nicht.
16
Es ist darum keine Erklärung, zu sagen: sehr einfach, wir vergleichen die Tatsache mit
unserem Erinnerungsbild, – weil vergleichen eine bestimmte Vergleichsmethode voraussetzt,
die wieder nur beschrieben ist.17
18
Wie soll er wissen, welche Farbe er zu wählen hat, wenn er das Wort ”rot“ hört? –
Sehr einfach: er soll die Farbe nehmen, deren Bild ihm beim Hören des Wortes einfällt. –
1 (M): || ? /
2 (M): /
3 (V): die
4 (V): Phänomenen durch die
5 (V): etwa%
6 (V): Druck auf
7 (V): wollen ja
8 (V): Wir wollen die
9 (V): Kalküls %nicht eines Mechanismus
10 (V): Kalküls auf.
11 (M): /
12 (V): kommt.
13 (M): /
14 (V): &Das was diesen Effect hervorruft“ ist // Die
Worte &das was diesen Effect hervorruft“ sind
15 (V): Und dieser Satz
16 (M): /
17 (V): voraussetzt, die nicht gegeben ist. // die nur
wieder beschrieben ist.
18 (M): /
38
37v
38
37v
38
33
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10
“The Meaning of a Sign
is Given by its Effect
(the Associations that
it Triggers, etc.).”
1
One would like to use memory and association to explain the mechanism of meaning.
2
But we feel that an explanation3
of a mechanism can’t be what is important to us. For once
again this explanation is a description of linguistic phenomena.4
It says, for example: when the word
“red” is heard the mental image of red pops up. (A tab on5
a register by pressing a button.) Well,
when that happens – what of it? What we want to hear is the explanation of a calculus.6
And the
explanation of a mechanism situates itself outside the calculus.7
It has nothing to do with what we’re
interested in. It is itself a description within language and one that doesn’t engage with the calculus
that’s to be explained to us. Whereas we need an explanation that is a part of this calculus.
8
If I say that a symbol is what produces this particular effect, then the question is how I
can talk about this effect if it isn’t even there? And when it does occur how do I know that
it is the one I meant?
9
The expression10
“what produces this particular effect” is11
a symbol itself, and12
therefore
doesn’t explain the nature of a symbol.
13
Therefore it’s no explanation to say: “It’s quite simple, we compare the fact with the
image in our memory” – because comparing presupposes a particular method of comparison,
which in turn is only described.14
15
How is he to know which colour to choose when he hears the word “red”? – Quite
simple: He’s to pick the colour whose image comes to his mind when he hears the word. –
1 (M): || ? /
2 (M): /
3 (V): that the explanation
4 (V): of phenomena via language.
5 (V): of
6 (V): calculus% not a mechanism.
7 (V): calculus of.
8 (M): /
9 (M): /
10 (V): phrase
11 (V): “What produces this effect” is
12 (V): and this proposition
13 (M): /
14 (V): comparison, which is not given. // which is
only in turn described.
15 (M): /
33e
WBTC007-10 30/05/2005 16:18 Page 67
Aber wie soll er wissen, was die ”Farbe“ ist, ”deren Bild ihm einfällt“?19
Braucht es dafür
ein weiteres Kriterium? u.s.f.
(Es gibt übrigens auch ein Spiel: die Farbe wählen, die einem beim Wort ”rot“ einfällt.)
20
Man kann aber auch sagen, daß dieser Satz (die Bedeutung des Zeichens &rot“ sei die Farbe,
die ich mit dem Wort assoziiere)21
die22
Erklärung einer bestimmten Bedeutung, d.h. eine Definition,
ist; aber nicht die Erklärung23
des Begriffs der Bedeutung.
&,rot‘ bedeutet die Farbe, die mir beim Hören des Wortes ,rot‘ einfällt“ ist eine Definition.
24
Bezieht sich auf das, was Frege, & gelegentlich Ramsey,25
vom Wiedererkennen als einer
Bedingung des Symbolisierens sagte.
Was ist denn das Kriterium dessen, daß ich die Farbe rot richtig wiedererkannt habe? Etwa so etwas26
wie das Erlebnis der Freude beim Wiedererkennen?
27
(Die psychologischen – trivialen – Erörterungen über Erwartung, Assoziation, etc.
lassen immer das eigentlich Merkwürdige aus und man merkt ihnen an, daß sie herumreden,
ohne den springenden Punkt zu berühren.) Und umso mehr, als es nie notwendig ist
die Wirkungsweise eines Wortes durch Assoziation & Gedächtnis zu erklären & weil man statt28
der Vorstellungsbilder immer wirkliche (gemalte) Bilder verwenden könnte.
29
Wenn ich Worte wählen kann, daß sie der Tatsache – in irgend einem Sinne passen,
dann muß ich also schon vorher einen Begriff dieses Passens gehabt haben. Und nun fängt
das Problem von Neuem an, denn, wie weiß ich, daß dieser Sachverhalt dem Begriff vom
”Passen“ entspricht.
30
Aber warum beschreibe ich dann die Tatsache gerade so? Was ließ Dich diese Worte sagen?
31
Und wenn ich nun sagen würde: ”alles was geschieht, ist eben, daß ich auf diese
Gegenstände sehe und dann diese Worte gebrauche“, so wäre die Antwort: ”also besteht das
Beschreiben in weiter nichts? und ist es immer eine Beschreibung, wenn Einer . . . ?“ Und
darauf müßte ich sagen: ”Nein. Nur kann ich den Vorgang nicht anders, oder doch nicht
mit einer andern Multiplizität beschreiben, als, indem ich sage: ,ich beschreibe, was ich
sehe‘; und darum ist keine Erklärung mehr möglich, weil mein Satz bereits die richtige
Multiplizität hat.“
32
Ich könnte33
fragen: Warum verlangst Du Erklärungen? Wenn diese gegeben sein
werden,34
wirst Du ja doch wieder vor einem Ende stehen. Sie können Dich nicht weiter
führen, als Du jetzt bist. (&Nähmaschine“)
35
In welchem Sinne sagt man, man kennt die Bedeutung des Wortes A, noch ehe man
den Befehl, in dem es vorkommt, befolgt hat? Und inwiefern kann man sagen, man hat
die Bedeutung durch die Befolgung des Befehls kennengelernt? Können die beiden
Bedeutungen miteinander in Widerspruch stehen?
19 (V): wissen, was das ist: &die Farbe, die ihm
einfällt“?
20 (M): /
21 (V): assoziiere) zur
22 (V): eine
23 (O): Erklarung
24 (M): /
25 (V): Ramsey, ab
26 (V): Etwa etwas
27 (M): /
28 (V): durch Assoziation zu erklären & man statt
29 (M): /
30 (M): /
31 (M): ∫ ///
32 (M): /
33 (V): könnte auch so
34 (V): würden,
35 (M): ∫ 555
37v
38
37v
38, 39
34 Die Bedeutung eines Zeichens . . .
WBTC007-10 30/05/2005 16:18 Page 68
But how is he to know what the “colour” is “whose image comes to his mind”?16
Is a
further criterion necessary for this?, and so forth.
(By the way, there also is a game: choosing the colour that comes to mind on hearing the
word “red”.)
17
But one can also say that this proposition (that the meaning of the sign “red” is the colour that
I associate with the word)18
is the19
explanation of a specific meaning, i.e. a definition; but not the
explanation of the concept of meaning.
“‘Red’ means the colour that comes to mind when I hear the word ‘red’” is a definition.
20
Refers to what Frege and occasionally Ramsey said about recognizing as a prerequisite for
symbolizing.
What is the criterion for my having correctly recognized the colour red? Maybe something like
experiencing joy upon recognizing it?
21
(The psychological – trivial – discussions about expectation, association, etc. always
leave out what is really remarkable, and you can tell that they’re beating around the bush
without touching upon the essential point.) And all the more so because it’s never necessary
to use association and memory to explain how a word has an effect, and because you could22
always use real (painted) pictures instead of mental images.
23
If I can choose words so that in some sense they fit a particular fact, that means that I
must already have had a concept of that fitting. And now the problem begins anew, because
how do I know that this state of affairs corresponds to the concept of “fitting”?
24
But why do I describe the fact just this way? What caused you to say these words?
25
And now if I were to say: “All that happens is that I look at these objects and then use
these words,” the response would be: “So there’s nothing more to describing? And is it always
a description if someone . . . ?” And to this I would have to say: “No. It’s just that I can’t
describe the process differently, or at any rate with any other multiplicity, than by saying:
‘I describe what I see’; and for that reason no further explanation is possible, because my
proposition already has the right multiplicity.”
26
I could ask:27
Why are you demanding explanations? When these are given you’ll28
just
be facing a dead end again. They can’t take you any further than you are now. (“Sewing
machine”)
29
In what sense does one say that one knows the meaning of word A, even before one
has followed the command in which it occurs? And to what extent can one say that one has
come to know the meaning by following the command? Can the two meanings contradict
each other?
16 (V): what “the colour that comes to mind” is?
17 (M): /
18 (V): word) to the
19 (V): an
20 (M): /
21 (M): /
22 (V): association to explain . . . effect, and one could
23 (M): /
24 (M): /
25 (M): ∫ ///
26 (M): /
27 (V): I could also ask this:
28 (V): If these were given you’d
29 (M): ∫ 555
The Meaning of a Sign is Given by its Effect . . . 34e
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36
Ich wünsche mir, einen Apfel zu bekommen. In welchem Sinne kann ich sagen, daß ich
noch vor der Erfüllung des Wunsches die Bedeutung des Wortes ”Apfel“ kenne? Wie äußert
sich denn die Kenntnis der Bedeutung? d.h., was versteht man denn unter ihr.
Offenbar wird das Verständnis des Wortes durch eine Worterklärung gegeben, welche nicht
die Erfüllung des Wunsches ist.
37
Die Bedeutung ist eine Festsetzung, nicht Erfahrung. Und damit nicht Kausalität. Was
das Zeichen suggeriert, findet man durch Erfahrung. Es ist die Erfahrung, die uns lehrt, welche
Zeichen am seltensten mißverstanden werden. Das Zeichen, soweit es suggeriert, also soweit
es wirkt, interessiert uns nicht. Es interessiert uns nur als Zug in einem Spiel: Glied in einem
System, das selbstbedeutend ist; das seine Bedeutung in sich selbst hat.38
39
Unsere Weise von den Wörtern zu reden, können wir durch das beleuchten, was Sokrates
im ”Kratylos“ sagt. Kratylos: ”Bei weitem und ohne Frage ist es vorzüglicher, Sokrates, durch
ein Ähnliches darzustellen, was jemand darstellen will, als durch das erste beste.“ –
Sokrates: ”Wohl gesprochen, . . .“.40
41
Es ist eine Funktion des Wortes &rot“ uns die Farbe in Erinnerung zu rufen & es könnte z.B.
gefunden werden, daß sich dazu das Wort &rot“ besser eignet als ein anderes (daß seine Bedeutung
etwa schwerer vergessen oder verwechselt wird).42
Aber wir hätten uns, wie gesagt, statt des
Mechanismus der Assoziation43
einer Tabelle (oder dergl.) bedienen können; & nun müßte unser Kalkül
eben mit dem assoziierten oder gemalten Bild [Muster] weiterschreiten. Die Zweckmäßigkeit eines
Zeichens in jenem Sinne interessiert uns nicht. (Im Gegensatz dazu: Kratylos: &Bei weitem . . . erste
beste“.)
44
Ich könnte mir denken, daß ein Philosoph glaubte, einen Satz45
in46
roter Farbe drucken lassen
zu müssen, da er erst so ganz das ausdrücke, was der Autor sagen will. (Hier hätten wir die
magische Auffassung der Zeichen statt der logischen.)47
48
Aber wäre das wirklich so unsinnig, verwenden wir denn nicht wirklich Sperrdruck? – Ich wollte
sagen: die Wirkung eines Satzes auf das Gemüt ist nicht sein Sinn.
49
Die Untersuchung, ob die Bedeutung eines Zeichens seine Wirkung ist, ist eine grammatische
Untersuchung.
50
Ich glaube, auf die kausale Theorie der Bedeutung kann man einfach antworten, daß
wir, wenn Einer einen Stoß erhält und umfällt, das Umfallen nicht die Bedeutung des Stoßes
nennen.51
36 (M): ∫
37 (M): ∫ ///
38 (V): Spiel: Glied in einem System, das selbständig
ist. // Glied in einem System; das seine
Bedeutung in sich selbst hat.
39 (M): / 3
40 (E): Siehe: Platon, Kratylos, 434a.
41 (M): ? /
42 (V): besser eignet (daß seine Bedeutung etwa
schwerer vergessen wird). // wird). als
43 (V): Assoziation das
44 (M): ? / (R): [Zu § 14 S. 58 oder § 89 S. 414]
45 (V): Es wäre charakteristisch für eine bestimmte
irrige Auffassung, wenn ein Philosoph
glaubte, einen Satz // Ein Philosoph könnte
glauben einen Satz
46 (V): mit
47 (V): logischen.) 555 (Das magische
Zeichen würde wirken wie eine Droge, und für
sie wäre die kausale Theorie richtig.)
48 (M): ? /
49 (M): /
50 (M): /
51 (V): nennen.
40
39v
40
39v
40
41
35 Die Bedeutung eines Zeichens . . .
WBTC007-10 30/05/2005 16:18 Page 70
30
I wish to have an apple. In what sense can I say that I know the meaning of the word
“apple” even before my wish is fulfilled? How does the knowledge of the meaning manifest
itself? That is to say, what are we to understand by this knowledge?
Obviously the understanding of the word is given by a verbal explanation. But that isn’t
the fulfilment of the wish.
31
Meaning is a stipulation, not an experience. And therefore it is not causality. What the
sign suggests is found through experience. It is experience that teaches us which signs are
least frequently misunderstood. In so far as the sign is suggestive, i.e. in so far as it has an
effect, it doesn’t interest us. It only interests us as a move in a game: as a component in a
system that is self-meaning; that means all by itself.32
33
We can shed light on the way we talk about words by considering what Socrates says
in the Cratylus. Cratylus: “By far and without question it is more excellent, Socrates, to
portray what someone wants to portray by using something similar to it than by using the
next best thing.” – Socrates: “Well spoken, . . .”.34
35
One function of the word “red” is to recall the colour to our memory, and it might be found,
for instance, that the word “red” is better suited for this than another one, (that, say, its meaning
is less readily forgotten or confused).36
But, as mentioned above, we could have used a table (or
something like it) instead of the mechanism of association; and then our calculus would simply
have to proceed using the associated or painted image [sample]. We’re not interested in the suitability
of a sign in that sense. (In contrast to this: Cratylus: “By far . . . next best thing”.)
37
I can imagine that a philosopher might believe38
he had to have a sentence printed in39
red,
since only in this way would it express completely what the author wanted to say. (Here we
would have a magical, rather than logical, understanding of signs.)40
41
But would that really be so nonsensical? Don’t we in fact use spaced type? – I wanted to say:
The effect of a sentence on one’s feelings is not its sense.
42
Investigating whether the meaning of a sign is its effect is a grammatical investigation.
43
I believe that we can respond to the causal theory of meaning simply by saying that –
if someone is pushed and falls down, we do not call44
the falling down the meaning of the
push.
30 (M): ∫
31 (M): ∫ ///
32 (V): as a component in a system that is selfsupporting.
// as a component in a system that
means all by itself.
33 (M): / 3
34 (E): Cf. Plato, Cratylus, 434a.
35 (M): ? /
36 (V): better suited (that, say, its meaning is less
readily forgotten).
37 (M): ? / (R): [To § 14 p. 58 or § 89 p. 414]
38 (V): It would be characteristic of a certain erroneous
view if a philosopher were to believe // A
philosopher might believe
39 (V): printed with
40 (V): signs.) 555 (The magical sign would
have the effect of a drug, and for it the causal
theory would be correct.)
41 (M): ? /
42 (M): /
43 (M): /
44 (V): call
The Meaning of a Sign is Given by its Effect . . . 35e
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52
Die Verwendung einer Landkarte ist, daß wir uns in irgend einer Weise nach ihr53
richten. Daß
wir ihr Bild in unsere Handlungen übertragen.54
Es ist klar daß hier kausale Zusammenhänge
stattfinden; aber würden wir sagen, sie sind es die den Plan zum Plan machen?55
56
Der Sinn der Sprache ist nicht durch ihre Wirkung57
bestimmt. Oder: Was man den Sinn,
die Bedeutung, in der Sprache nennt, ist nicht ihre Wirkung.58
Damit meinte ich,59
daß, was wir Sinn eines Satzes nennen & durch eine sprachliche Erklärung
erklärt wird, nichts mit dem zu tun hat, was die beabsichtigte Wirkung der Sprache60
hervorrufen
hilft.
61
Es ist wirklich ”the meaning of meaning“, was wir untersuchen: Oder62
die Grammatik
des Wortes ”Bedeutung“.
52 (M): ∫
53 (V): ihm
54 (V): Die Verwendung eines Plans ist eine
Übersetzung in unsere Handlungen. Eine
Übertragung in unsere Handlungen.
55 (V): Es ist klar, daß da kausale Zusammenhänge
gesehen werden, aber es wäre
komisch, die als das Wesen eines Planes
auszugeben. // aber würde man sagen, sie sind es die
den Plan zum Plan machen?
56 (M): ∫ ///
57 (V): durch ihren Zweck
58 (V): nicht ihr Zweck.
59 (V): ich, ursprünglich
60 (V): hat, was diese Wirkung
61 (M): ∫
62 (V): Nämlich
40v
41
36 Die Bedeutung eines Zeichens . . .
WBTC007-10 30/05/2005 16:18 Page 72
45
The use of a map is that in some way we take our bearings from it. That we transfer its image
into our actions.46
It’s clear that causal connections take place here; but would we say that they are
what make the map a map?47
48
The sense of language is not determined by its effect.49
Or: What one calls sense, meaning,
in language is not its effect.50
By this I mean51
that what we call the sense of a proposition, what is explained by an explanation
of language, has nothing to do with what helps produce the intended effect52
of language.
53
It really is “the meaning of meaning” we’re investigating: or54
the grammar of the word
“meaning”.
45 (M): ∫
46 (V): The use of a map is a translation into our
actions. A transferral into our actions.
47 (V): It’s clear that causal connections are seen
here, but it would be comic to represent them
as the essence of a map. // but would one say that
they make the map into a map?
48 (M): ∫ ///
49 (V): by its purpose.
50 (V): its purpose.
51 (V): mean primarily
52 (V): produce this effect
53 (M): ∫
54 (V): namely,
The Meaning of a Sign is Given by its Effect . . . 36e
WBTC007-10 30/05/2005 16:18 Page 73
6 (V): verneinende
7 (V): das heißt hier: es greift in mich ein:
8 (M): / ///
9 (V): Gebärde würde die Bedeutung
10 (R): ∀ S. 16/1,2 ∀
11 (M): 555
12 (M): [Wo anders besser]
41v
42
11
Bedeutung als Gefühl,
hinter dem Wort stehend;
durch eine Geste ausgedrückt.
1
Jeder, der einen Satz einer ihm geläufigen Sprache liest, nimmt die Worte der verschiedenen
Wortarten in anderer Weise auf2
obwohl sich ihr Bild & Klang3
der Art nach nicht unterscheidet.
Wir vergessen ganz, daß ”nicht“ und ”Tisch“ und ”grün“, als Laute oder Schriftbilder
betrachtet sich nicht wesentlich voneinander unterscheiden und sehen es nur klar in einer
uns fremden Sprache. (James.) (Bedeutungskörper.)4
5
Das ”Nicht“ macht eine abwehrende6
Geste.
Nein, es ist eine abwehrende Geste.
”Das Verstehen der Verneinung ist dasselbe, wie das Verstehen einer abwehrenden Geste.“
Den Kopf schütteln.
Verstehen des Wortes &nicht“ im Sinne von &wissen, wie es gebraucht wird“ & dagegen das
Verstehen einer Geste, der Eindruck den mir die Geste macht.
Anderseits sagt man: ich verstehe diese Geste, wie: ich verstehe dieses Thema, es sagt mir etwas
& das heißt hier: ich erlebe es, es greift in mich ein:7
Ich folge ihm mit bestimmtem Erlebnis.
Wie lernt man eine Geste verstehen, die uns nicht durch Worte erklärt (definiert) wird?
8
Gefragt, was ich mit ”und“ im Satze ”gib mir das Brot und die Butter“ meine, würde
ich mit einer Gebärde antworten, und diese Gebärde würde, was ich meine9
illustrieren.
Wie das grüne Täfelchen ”grün“ illustriert und wie die W-F-Notation ”und“, ”nicht“, etc.
illustriert.10
11
Die Geste des Wortes &vielleicht“; des Wortes &bitte“ & &danke“ als Erklärung der Bedeutung
dieser Wörter.12
42
41v
42
1 (R): Zu: S. 29 (M): als Zitat / (
2 (V): liest, sieht die Worte . . . in anderer Weise
3 (V): Jeder, der einen Satz liest und versteht, sieht
die Worte // die verschiedenen Wortarten // in
verschiedener Weise, obwohl sich ihr Bild und
Klang
4 (M): )
5 (M): ∫
37
WBTC011-14 30/05/2005 16:18 Page 74
1 (R): To: p. 29; (M): as quotation / (
2 (V): sees
3 (V): Anyone reading and understanding a
sentence sees the words // the various types of
words // in different ways even though their
appearance and sound
4 (M): )
5 (M): ∫
6 (V): denying
7 (V): means: it engages me:
8 (M): / ///
9 (V): illustrate its meaning.
10 (R): ∀ p. 16/1,2; ∀
11 (M): 555
12 (M): [Better somewhere else]
11
Meaning as Feeling,
Standing Behind the Word;
Expressed with a Gesture.
1
Anyone reading a sentence in a familiar language takes2
the various types of words in different
ways even though their appearance and sound3
don’t differ in kind. We completely forget that
“not” and “table” and “green”, seen as sounds or written images, don’t differ from each other
very much, and we see this clearly only in a foreign language. (James.) (Meaning-body.)4
5
“Not” makes a rebuffing6
gesture.
No, it is a rebuffing gesture.
“Understanding a negation is the same thing as understanding a rebuffing gesture.”
To shake one’s head.
Understanding the word “not” in the sense of “knowing how it is used” and, in contrast, understanding
a gesture, the impression a gesture makes on me.
On the other hand one says “I understand this gesture” in the same way as “I understand this theme;
it speaks to me”, and here that means: I am involved with it, it engages me:7
I am involved in a
particular way as I follow it.
How do we learn to understand a gesture that isn’t explained to us (defined for us) in words?
8
If I were asked what I mean by “and” in the sentence “Give me the bread and the
butter,” I’d answer with a gesture, and this gesture would illustrate what I mean.9
Just as
the green colour chip illustrates “green” and the truth-table illustrates “and”, “not”, etc.10
11
The gestures for the words “maybe”, “please”, and “thank you” as an explanation of the meaning
of these words.12
37e
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12
Man tritt mit der hinweisenden
Erklärung der Zeichen nicht aus der
Sprachlehre heraus.
1
Zur Grammatik gehört nur das nicht, was die Wahrheit und Falschheit eines Satzes
ausmacht. Nur darum kümmert sich die Grammatik nicht. Zu ihr gehören alle
Bedingungen des Vergleichs des Satzes mit den Tatsachen.2
Das heißt, alle Bedingungen
des Verständnisses. (Alle Bedingungen des Sinnes.)
3
Die Deutung der Schrift & Lautzeichen durch hinweisende Erklärungen ist nicht Anwendung der
Sprache sondern ist ein Teil der Sprachlehre.4
Die Deutung vollzieht sich noch im Allgemeinen, als
Vorbereitung auf jede Anwendg. Sie geht in der Sprachlehre vor sich und nicht im Gebrauch
der Sprache.
5
Soweit sich die Bedeutung der Wörter in der eingetroffenen Erwartung, in der Befolgung des
Befehls zeigt,6
kommt sie7
in der Beschreibung der Tatsache zum Vorschein. (Sie wird also
ganz in der Sprachlehre bestimmt.)
(In dem, was sich hat voraussehen lassen; worüber man schon vor dem Eintreffen der
Tatsache reden konnte.)
8
&Das nennt man einen Krautkopf“ ist eine hinw. Def., & gehört zur Sprachlehre. &Gib mir diesen
Krautkopf“ ist ein Satz der Sprache, der die Wortsprache verläßt da er eine Gebärde & ein Object
verlangt worauf gezeigt wird.9
10
Der Grund, warum wir glauben, mit der hinweisenden Erklärung das Gebiet der Sprache,
des Zeichensystems, zu verlassen, ist,11
daß12
wir dieses Heraustreten aus den Schriftzeichen
mit einer Anwendung der Sprache, etwa mit einer Beschreibung dessen, was wir sehen,13
verwechseln.
1 (M): ∫ /// (R): ∀ S. 36/6, S. 37/1
2 (V): Satzes mit der Wirklichkeit.
3 (M): ? /
4 (V): Die Anwendung der Sprache geht über
diese hinaus, aber nicht die Deutung der
Schrift- und oder Lautzeichen. // Die Deutung der
Schrift & Lautzeichen durch hinweisende erklärungen
gehört nicht in die Anwendung der Sprache sondern
zu ihrer Grammatik . . .
5 (M): ∫ / ////
6 (V): Soweit die Bedeutung der Wörter in der
Tatsache (Handlung) zum Vorschein kommt,
7 (V): sie (schon)
8 (M): ? /
9 (V): & ein Object worauf gezeigt wird verlangt.
10 (M): ∫
11 (E): ”ist“ sinngemäß eingesetzt.
12 (V): Ist nicht der Grund, warum wir glauben,
. . . zu verlassen, daß
13 (V): was ich sehe,
43
42v
43
42v, 43
42v
43, 44
38
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12
In Giving an Ostensive Explanation
of Signs one Doesn’t Leave
Grammar.
1
The only thing that doesn’t belong to grammar is what makes a proposition true or false.
That’s the only thing grammar is not concerned with. Everything that’s required for comparing
the proposition with the facts2
belongs to grammar. That is, all the requirements for
understanding. (All the requirements for sense.)
3
The interpretation of written and oral signs by ostensive explanations is not an application of
language, but a part of grammar.4
The interpretation still takes place in a general realm, as a
preparation for any application. It takes place in grammar and not in the use of language.
5
In so far as the meaning of words appears in the fulfilment of an expectation, in the carrying
out of a command,6
it7
makes its appearance in the description of a fact. (Thus it is completely
determined within grammar.)
(In what could be foreseen, in what one could talk about, even before the fact occurred.)
8
“This is called a cabbage” is an ostensive definition, and it belongs to grammar. “Give me this
cabbage” is a sentence in a language, and it goes beyond word-language, since it calls for a gesture
and an object to which one points.
9
The reason we believe that in using an ostensive explanation we are leaving the province
of language, of a system of signs, is that10
we confuse this stepping outside the written signs
with an application of language, for instance with a description of what we see.11
1 (M): ∫ /// (R): ∀ p. 36/6, p. 37/1
2 (V): with reality
3 (M): ? /
4 (V): The application of language goes beyond
language, but the interpretation of written and
or oral signs does not. // The interpretation of written
and oral signs by ostensive explanations does not
fall under “application of language”, but belongs to
the grammar of language.
5 (M): ∫ / ////
6 (V): appears in a fact (action)
7 (V): it (already)
8 (M): ? /
9 (M): ∫
10 (V): Isn’t the reason we believe that . . . signs,
that
11 (V): what I see.
38e
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14
Man könnte fragen wollen: Ist es denn aber ein Zufall, daß ich zur Erklärung von15
Zeichen,
also zur Vervollständigung des Zeichensystems aus den16
Schrift- oder Lautzeichen heraustreten
muß? Trete ich damit nicht eben in das Gebiet, in dem17
sich dann das Beschriebene18
abspielt? Aber ist es nicht seltsam, daß ich dann überhaupt mit dem Schriftzeichen etwas anfangen
kann?19
– Man sagt etwa, daß20
die Schriftzeichen bloß die Vertreter jener Dinge sind,
auf die man zeigt. – Aber wie seltsam, daß so eine Vertretung möglich ist. Und es wäre nun
das Wichtigste, zu verstehen, wie denn Schriftzeichen die andern Dinge vertreten können.
Welche Eigenschaft müssen sie haben, die sie zu dieser Vertretung befähigt. Denn ich
kann nicht sagen: statt Milch trinke ich Wasser und esse statt Brot Holz, indem ich das Wasser
die Milch und Holz das Brot vertreten lasse. (Erinnert an Frege.)
21
Ich kann nun freilich doch sagen, daß das Definiendum das Definiens vertritt; und
hier steht dieses hinter jenem, wie die Wählerschaft hinter ihrem Vertreter. Und in diesem
Sinne kann man auch sagen, daß das in der hinweisenden Definition erklärte Zeichen den
Hinweis vertreten kann, da man ja diesen wirklich in einer Gebärdensprache für jenes
setzen könnte. Aber doch handelt es sich hier um eine Vertretung im Sinne einer Definition,
denn die Gebärdensprache bleibt22
eine Sprache.
Ich möchte sagen: Von einem Befehl in der Gebärdensprache zu seiner Befolgung ist es
ebenso weit, wie von diesem Befehl in der Wortsprache.
Denn auch die hinweisenden Erklärungen müssen ein für allemal gegeben werden.
D.h., auch sie gehören zu dem Grundstock von Erklärungen, die den Kalkül vorbereiten,
und nicht zu seiner Anwendung ad hoc.
14 (M): ∫ (R): [Zu § 13]
15 (V): vom
16 (V): dem
17 (V): Gebiet, worin
18 (V): dann das zu Beschreibende
19 (V): abspielt? Aber dann ist // erscheint // es
seltsam, daß ich überhaupt . . . kann.
20 (V): Man faßt es etwa so auf, daß
21 (M): ∫
22 (V): ist
45
39 Man tritt mit der hinweisenden Erklärung . . .
WBTC011-14 30/05/2005 16:18 Page 78
12
One might want to ask: But is it really a coincidence that in order to explain signs, i.e.
in order to complete the system of signs, I have to step outside the written or oral signs? In
doing this, am I not entering the very area in which13
what was described14
takes place? But
in that case isn’t it odd15
that I can use the written sign at all? – One says, for instance, that16
the written signs are merely substitutes for the things that one points to. – But how strange
that such a substitution is possible. And now the most important thing would be to understand
how written signs can substitute for other things.
What property must they have that qualifies them for this substitution? For, letting water
stand for milk and wood for bread, I can’t say: I’ll drink water instead of milk and eat wood
instead of bread. (Reminiscent of Frege.)
17
But I can say that the definiendum represents the definiens; and here the latter stands
behind the former as does the electorate behind its representative. And in this sense one can
also say that the sign explained by an ostensive definition can represent the ostension, since
in a language of gestures one really could replace the ostension with the sign. And yet what
we have here is a matter of representation in the sense of a definition, for the language of
gestures is still18
a language.
I’d like to say: The distance from a command to its being followed is just as great in the
language of gestures as it is in word-language.
For ostensive explanations too have to be given once and for all.
That is, they too belong to the basic stock of explanations that prepare the calculus, and
not to its ad hoc application.
12 (M): ∫ (R): [To § 13]
13 (V): where
14 (V): described then
15 (V): But then it’s // it seems // odd
16 (V): One conceives of it in such a way that
17 (M): ∫
18 (V): is
An Ostensive Explanation of Signs Doesn’t Leave Grammar. 39e
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13
”Primäre und sekundäre Zeichen“.
Wort und Muster.
Hinweisende Definition.
1
Der falsche Ton in der Frage, ob es nicht primäre Zeichen (hinweisende Gesten) geben
müsse, während unsere Sprache auch ohne die andern, die Worte, auskommen könnte, liegt
darin, daß man eine Erklärung der bestehenden Sprache zu erhalten erwartet, statt der
einfachen2
Beschreibung.
3
Nicht die Farbe Rot tritt an Stelle des Wortes ”rot“, sondern die Gebärde, die auf einen
roten Gegenstand hinweist, oder das rote Täfelchen.
Man kann nun sagen: ein rotes Täfelchen ist ein primäres4
Zeichen für rot, ein Wort5
ein
sekundäres, weil6
es die Bedeutung des Wortes &rot“7
erklärt wenn ich auf ein rotes Täfelchen zeige8
etc., dagegen nicht, wenn ich sage &rot“ heiße soviel wie &rouge“. Aber ist dies unter allen
Umständen so? Muß immer ein roter Gegenstand oder ein rotes Vorstellungsbild gegenwärtig sein,
wenn ich das Wort rot verstehen soll? Denke an den Befehl &stelle Dir einen roten Fleck auf blauem
Grund vor“. Und wie ist es mit9
Bindewörtern, Präpositionen etc.?
Ist es nicht (für mich) ein Kriterium meines10
Verständnisses des Wortes &perhaps“ daß ich es ins
Wort &vielleicht“ übersetzen kann?
Und wenn ein Befehl lautet &stell’ Dir einen roten Kreis vor“, muß ich da wirklich das Wort rot
zuerst in ein Farbmuster übersetzen ehe ich den Befehl befolgen kann?11
12
Wenn Einer sagte: &Es gilt mit Recht als ein Zeichen des Verständnisses13
des Wortes ”rot“, daß
Einer einen roten Gegenstand auf Befehl aus anders gefärbten herausgreifen kann; dagegen
ist das richtige Übersetzen des Wortes ”rot“ ins Englische oder Französische kein Beweis
des Verstehens. Darum ist das rote Täfelchen ein primäres Zeichen für ”rot“, dagegen jedes
Wort ein abgeleitetes14
Zeichen“, – so könnte15
ich antworten: das zeigt nur16
was Du17
mit &verstehen“
meinst.18
Und was heißt ”es gilt mit Recht . . .“? Heißt es: Wenn ein Mensch einen
1 (M): /
2 (V): bloßen
3 (M): ∫ 3
4 (V): ist das primäre
5 (V): rot. das Wort (rotA
6 (V): weil ich ich
7 (V): (rougeA
8 (V): auf ein Täfelchen zeige
9 (V): mit anderen Wortarten
10 (V): des
11 (V): Befehl verstehe?
12 (M): Nur als Probe des Puzzlements ? /
13 (V): Nun sage ich aber: ”Es gilt mit Recht als ein
Kriterium des Verstehens // Verständnisses //
14 (V): sekundäres
15 (V): würde
16 (V): nur das
17 (V): Du hier
18 (V): Zeichen.“ ((Aber das zeigt nur, was ich
mit dem ”Verstehen des Wortes ,rot‘“ meine.
46
45v
46
45v
46
40
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13
“Primary and Secondary Signs”.
Word and Sample.
Ostensive Definition.
1
The false ring to the question: “Don’t primary signs (ostensive gestures) have to exist,
whereas our language could manage without the others – the words?”, lies in the fact that
we expect to be given an explanation of an existing language instead of a simple2
description.
3
It isn’t the colour red that takes the place of the word “red”, but the gesture that points
to a red object or to a red colour chip.
One can say: A red colour chip is a4
primary sign for red, the word5
a secondary one, because6
pointing to a red chip7
, etc., explains the meaning of the word “red”8
, but saying that “red” is the
equivalent of “rouge” doesn’t. But is this the way it is in all situations? Does a red object or a mental
image of red always have to be present for me to understand the word “red”? Think of the command
“Imagine a red patch on a blue background”. And how about9
conjunctions, prepositions, etc.?
Isn’t it (as far as I’m concerned) a criterion of my understanding10
the word “vielleicht” that I can
translate it into the word “perhaps”?
And if a command says “Imagine a red circle” do I really have to translate the word “red” into a
colour sample before I can obey11
it?
12
Suppose someone were to say:13
“If someone who is ordered to do so can pick out a red
object from among others coloured differently, that rightly counts as a sign that he understood14
the word “red”, whereas correctly translating the word “red” into German or French is no
proof of his understanding. Therefore the red colour chip is a primary sign for “red”, whereas
any word is a derivative15
sign.” – Then I could16
respond: That only17
shows what you mean18
by
“understanding”.19
And what does “rightly counts . . .” mean? Does it mean: If someone who
1 (M): /
2 (V): mere
3 (M): ∫ 3
4 (V): the
5 (V): red, the word AredC
6 (V): because I I
7 (V): to a chip
8 (V): the word ArougeC
9 (V): about other kinds of words
10 (V): criterion of understanding
11 (V): understand
12 (M): Just as a proof of the puzzlement ? /
13 (V): But now I say:
14 (V): that rightly counts as a criterion for
understanding
15 (V): secondary
16 (V): would
17 (V): only that
18 (V): you mean here
19 (V): sign.” ((But that only shows what I mean
by “understanding the word ‘red’”.
40e
WBTC011-14 30/05/2005 16:18 Page 81
41 ”Primäre und sekundäre Zeichen“. . . .
roten Gegenstand auf Befehl etc. etc., dann hat er erfahrungsgemäß auch das Wort ”rot“
verstanden. Wie man sagen kann, gewisse Schmerzen gelten mit Recht als Symptom dieser
und dieser Krankheit? So ist es natürlich nicht gemeint. Also soll es wohl heißen, daß die
Fähigkeit, rote Gegenstände herauszugreifen, der spezifische Test19
dessen ist, was wir
Verständnis des Wortes ”rot“ nennen. Dann bestimmt diese Angabe also, was wir mit20
diesem
Verständnis meinen. Aber dann fragt es sich noch: wenn wir das Übersetzen ins Englische
etc. als Kriterium ansähen, wäre es nicht auch das Kriterium von dem, was wir ein
Verständnis des Wortes nennen? Es gibt nun den Fall, in welchem wir sagen: ich weiß nicht,
was das Wort ”rouge“21
bedeutet, ich weiß nur, daß es das Gleiche bedeutet, wie das Englische
”red“. So ist es, wenn ich die beiden Wörter in einem Wörterbuch auf der gleichen Zeile
gesehen habe, und dies ist die Verifikation des Satzes und sein Sinn. Wenn ich dann sage
”ich weiß nicht, was das Wort ,rouge‘22
bedeutet“, so bezieht sich dieser Satz auf eine
Möglichkeit der Erklärung dieser Bedeutung und ich könnte, wenn gefragt ”wie stellst Du
Dir denn vor, daß Du erfahren könntest, was das Wort bedeutet“, Beispiele solcher
Erklärungen geben (die die Bedeutung des Wortes ”Bedeutung“ beleuchten würden). Diese
Beispiele wären dann entweder der Art, daß statt des unverstandenen Worts ein verstandenes
– etwa das deutsche – gesetzt würde, oder, daß die Erklärung von der Art wäre ”diese 23
Farbe
heißt ,violett‘“. Im ersten Falle wäre es für mich ein Kriterium dafür, daß er das Wort ”rouge“
versteht:24
daß er sagt, es entspreche dem deutschen ”rot“. ”Ja“, wird man sagen, ”aber nur,
weil Du schon weißt, was das deutsche ,rot‘ bedeutet“. – Aber das bezieht sich ja ebenso
auf die hinweisende Definition. Das Hinweisen auf das rote Täfelchen ist auch nur dann25
ein Zeichen des Verständnisses, wenn26
vorausgesetzt wird, daß er die Bedeutung dieses Zeichens
kennt,27
was etwa soviel heißt, als daß er das Zeichen auf bestimmte Weise verwendet. – Es
gibt also allerdings28
den Fall, wo Einer sagt ”ich weiß, daß dieses Wort dasselbe bedeutet,
wie jenes, weiß aber nicht, was es bedeutet (sie bedeuten)“. Willst Du den ersten Teil dieses
Satzes verstehen, so frage Dich: ”wie konnte er es wissen?“ – willst Du den zweiten Teil
verstehen, so frage: ”wie kann er erfahren, was das Wort bedeutet?“ –
29
Welches ist das Kriterium unseres Verständnisses: das Wort richtig gebrauchen, oder, seine
Definition geben?30
Das Auswählen eines roten Gegenstands aus anderen wenn es verlangt wird,31
oder, die hinweisende Erklärung des W. &rot“ geben?32
Die Lösung beider Aufgaben betrachten wir als Zeichen des Verständnisses. Hören wir
jemand das Wort ”rot“ gebrauchen und zweifeln daran, daß er es versteht, so können wir
ihn zur Prüfung fragen ”welche Farbe nennst Du33
,rot‘ “. Anderseits,34
wenn wir jemandem
die hinweisende Erklärung gegeben hätten35
und nun sehen wollten, ob er sie36
richtig
19 (V): Probe
20 (V): unter
21 (V): ”rot“
22 (V): ”rot“
23 (F): MS 112, S. 74r.
24 (V): versteht$
25 (V): darum
26 (V): weil
27 (V): versteht,
28 (V): wohl
29 (M): /
30 (V): Verständnisses: das richtige Gebrauchen des
Wortes oder das Definieren?
31 (V): geben? Einen roten Gegenstand . . .
auswählen . . . ,
32 (V1): oder, die hinweisende Erklärung geben? //
oder das hinweisende Erklären des Wortes &rot“.
(V2): ? / Welches ist denn das Kriterium
unseres Verständnisses: das Aufzeigen des
roten Täfelchens, wenn gefragt wurde
”welches von diesen Täfelchen ist rot“, –
oder, das Wiederholen der hinweisenden
Definition ”das ist ,rot‘“?
33 (V): Farbe nennen wir
34 (V): Anderseits:
35 (V): hätten @diese Farbe heißt $rot#!
36 (V): er diese Erklärung
48
47
47v
48
WBTC011-14 02/06/2005 14:36 Page 82
is ordered to do so can pick out a red object, etc., etc., then experience shows us that he
has also understood the word “red”? As one can say that certain pains are rightly counted
as symptoms of this or that illness? Of course that’s not the way it’s meant. It’s probably
supposed to mean that the ability to pick out red objects is the specific test for20
what we
call understanding the word “red”. Then this specification determines what we mean by this
understanding. But then a further question remains: if we were to view translating into German,
etc. as a criterion, wouldn’t that also be a criterion for what we call understanding a word?
Now there is the case where we say: I don’t know what the word “rouge”21
means, I only
know that it means the same thing as the German “rot”. That’s the way it is when I’ve seen
the two words on the same line in a dictionary, and this is the verification of the proposition
and its sense. And if I then say: “I don’t know what the word ‘rouge’22
means”, this
sentence refers to the possibility of explaining the meaning, and if asked “Well, how do you
suppose you could find out what the word means?”, I could give examples of such explanations
(which would illuminate the meaning of the word “meaning”). These examples would
either be of the kind that explains something by putting a word that is understood – say the
English one – in place of the one that isn’t, or of a kind that explains something by pointing,
such as in: “This 23
colour is called ‘violet’.” In the first case his saying that “rouge”
corresponds to the English “red” would serve as my criterion for his understanding “rouge”.
“Granted,” one will say, “but only because you already know what the English ‘red’ means.”
– But that goes for the ostensive definition as well. Pointing to the red colour chip is only a
sign of understanding if24
we presuppose that he knows25
the meaning of this sign, which
means more or less the same thing as that he uses the sign in a certain way. – So, to be sure,
there is the case where someone says “I know that this word means the same thing as that,
but I don’t know what the latter means (they mean)”. If you want to understand the first
part of the sentence ask yourself: “How could he know that?” – If you want to understand
the second part, ask: “How can he find out what the word means?” –
26
Which is the criterion for our understanding: using the word correctly, or defining it?27
Picking
out a red object from among others when asked to do so, or giving an ostensive explanation of the
word “red”?28
We view the performance of both tasks as a sign of understanding. If we hear someone
using the word “red” and doubt that he understands it, then, as a test, we can ask him “Which
colour are you calling29
‘red’?”. On the other hand,30
if we had given someone the ostensive
explanation31
and now wanted to see whether he had understood it32
correctly, we wouldn’t
20 (V): specific proof of
21 (V): “rot”
22 (V): “rot”
23 (F): MS 112, p. 74r.
24 (V): because
25 (V): understands
26 (M): /
27 (V): understanding: the correct use of the word, or
defining it?
28 (V1): or giving an ostensive explanation? // or the
ostensive explaining of the word “red”. (V2): ?
/ Which is the criterion for our understanding:
pointing to the red chip when the question is:
“Which one of these chips is red?” – or repeating
the ostensive definition “That is ‘red’”?
29 (V): colour do we call
30 (V): On the other hand:
31 (V): explanation !this colour means #redFE
32 (V): understood this explanation
“Primary and Secondary Signs”. . . . 41e
WBTC011-14 30/05/2005 16:18 Page 83
verstanden hat,37
würden wir nicht von ihm verlangen, daß er sie wiederholt, sondern wir
gäben ihm etwa die Aufgabe, aus einer Anzahl von Dingen die roten herauszusuchen. 38
In
jedem Fall ist das, was wir ”Verständnis“ nennen, eben durch das39
bestimmt, was wir als
Probe des Verständnisses ansehen (durch die Aufgaben bestimmt, die wir zur Prüfung des
Verständnisses stellen).40
41
Ist denn das &primäre Zeichen“ unmißdeutbar?42
43
Kann man sagen es müsse eigentlich nicht mehr verstanden werden?
Denken wir auch an den Fall, wenn wir sagen: &Ja, wenn das Wort das bedeutet (bedeuten soll),
ist der Satz wahr.“
44
Wie ist es, wenn ich eine Bezeichnungsweise festsetze; wenn ich z.B. für den eigenen
Gebrauch gewissen Farbtönen Namen geben will: Ich45
werde das etwa mittels einer Tabelle
tun (es kommt immer auf derlei hinaus). Und nun werde ich doch nicht den Namen zur
falschen Farbe schreiben (zu der Farbe der ich ihn nicht geben will). Aber warum nicht?
Warum soll nicht ”rot“ gegenüber dem grünen Täfelchen stehen und ”grün“ gegenüber dem
roten, etc.? – Ja, aber dann müssen wir doch wenigstens wissen, daß ”rot“ nicht das gegenüberliegende
Täfelchen meint. – Aber was heißt es ”das wissen“, außer, daß wir uns etwa neben
der geschriebenen Tabelle noch eine andere vorstellen, in der die Ordnung richtiggestellt
ist. – ”Ja aber dieses Täfelchen ist doch rot, und nicht dieses!“ – Gewiß; und das ändert sich
ja auch nicht, wie immer ich die Täfelchen und Wörter setze; und es wäre natürlich falsch,
auf das grüne Täfelchen zu zeigen und zu sagen ”dieses ist rot“. Aber das ist auch keine
Definition, sondern eine Aussage. – Gut, dann nimmt aber doch unter allen möglichen
Anordnungen die gewöhnliche (in der das rote Täfelchen dem Wort ”rot“ gegenübersteht)
einen ganz besonderen Platz ein. – ((Da gibt es jedenfalls zwei verschiedene Fälle: Es kann
die Tabelle mit grün gegenüber ”rot“ etc. so gebraucht werden, wie wir die Tabelle in der
gewöhnlichen Anordnung gewöhnlich gebrauchen. Wir würden also etwa den,46
der sie
gebraucht, von dem Wort ”rot“ nicht auf das gegenüberliegende Täfelchen blicken sehen,
sondern auf das rote, das schräg darunter steht (aber wir müßten auch diesen Blick nicht
sehen) und finden, daß er dann statt des Wortes ”rot“ in einem Ausdruck das rote
Täfelchen einsetzt. Wir würden dann sagen, die Tabelle sei nur anders angeordnet (nach
einem andern räumlichen Schema), aber sie verbinde die Zeichen, wie die gewohnte. – Es
könnte aber auch sein, daß der, welcher die Tabelle benützt, von der einen Seite horizontal
zur andern blickt und nun in irgend welchen Sätzen das Wort ”rot“ durch ein grünes
Täfelchen ersetzt; aber nicht etwa auf den Befehl ”gib mir das rote Buch“ ein grünes bringt,
sondern ganz richtig das rote (d.h. das, welches auch wir ”rot“ nennen). Dieser hat nun
die Tabelle anders benützt, als der Erste, aber doch so, daß das Wort ”rot“ die gleiche
Bedeutung für ihn hatte, wie für uns. (Zu einer Tabelle gehört übrigens wesentlich die Tätigkeit
des Aufsuchens47
in der Tabelle.) Es ist nun48
der zweite Fall, der49
uns interessiert und die
Frage ist: kann ein grünes Täfelchen als Muster der roten Farbe dienen? Und da ist es klar,
daß dies (in einem Sinn) nicht möglich ist. Ich kann mir eine Abmachung denken, wonach
37 (V): hat, so
38 (M): ///
39 (V): eben dadurch
40 (V): stellen).) )
41 (M): /
42 (V): unmißverständlich?
43 (M): /
44 (M): ? / (R): ∀ S. 35/2, 3
45 (V): geben will. Ich
46 (O): dem,
47 (V): Nachschauens
48 (V): nun offenbar
49 (V): welcher
47v
48
49
50
42 ”Primäre und sekundäre Zeichen“. . . .
WBTC011-14 30/05/2005 16:18 Page 84
require him to repeat it, but would, for example, give him the task of picking out the red
objects from among several others. 33
In each case what we call “understanding” is determined
by what we view as the test of understanding (is determined by the tasks that we set in order
to test understanding).34
35
Is it really impossible to misinterpret the “primary sign”?36
37
Can one say that really there’s no further need to understand it?
Let’s also think of the case where we say: “Ah, if the word means (is to mean) that, then the proposition
is true.”
38
What’s it like when I set up a system of notation? If for example I want to assign names
to certain shades of colour for my own use: I’ll39
do this, say, by using a table (it always boils
down to something like that). And now I’m certainly not going to write the name next to the
wrong colour (next to the colour I don’t want to assign it to). But why not? Why shouldn’t
“red” stand across from the green chip and “green” across from the red one, etc.? – Fine,
but then at least we have to know that “red” doesn’t mean the chip across from it. – But
what does “know that” mean other than that, for example, aside from the written table, we’re
imagining another one in which the pairing is corrected? – “All right, but it’s this patch that’s
red and not that one!” – Certainly; and that doesn’t change, no matter how I place the chips
and the words; and of course it would be wrong to point to the green chip and say “This
one’s red”. But then again, that isn’t a definition, but a statement. – Fine, but then the usual
arrangement (in which the red chip is across from the word “red”) occupies a very special
place among all possible ones. – ( (At any rate, there are two different cases here: the table
with green across from “red” etc. can be used in the same way that we usually use the table
in the normal arrangement. So, for example, we wouldn’t see the person using it looking
from the word “red” to the chip across from it, but rather to the red one situated diagonally
below it (although it’s not necessary that we see him looking that way) and we’d find that
in that case he inserts the red chip instead of the word “red” in an expression. Then we’d say
that the table was merely arranged differently (in accordance with a different spatial
schema), but that the signs were connected in the same way as in the usual one. – But it
could also be that the user of the table looks horizontally from one side to the other and
replaces the word “red” by a green chip in some sentences; but when told, e.g., “Give me
the red book” he doesn’t bring a green one, but rather, quite rightly, the red one (i.e. the
one we too call “red”). Now this person has used the table differently from the first person
but still in such a way that the word “red” has the same meaning for him as for us. (By the
way, an essential part of a table is the activity of looking something up40
in it.) Now it’s41
the second case that interests us, and the question is: Can a green chip serve as a sample of
the colour red? And here it’s clear that (in one sense) this isn’t possible. I can imagine an
understanding according to which someone to whom I show a green colour chip and say,
“Paint me this colour”, paints a shade of red; when I say the same thing and show him blue,
he is supposed to paint yellow, etc. – each time the complementary colour; and therefore I
33 (M): ///
34 (V): understanding).) )
35 (M): /
36 (V): Is the ‘primary sign’ really unambiguous?
37 (M): /
38 (M): ? / (R): ∀ p. 35/2, 3
39 (V): own use. I’ll
40 (V): of checking something
41 (V): it’s obviously
“Primary and Secondary Signs”. . . . 42e
WBTC011-14 30/05/2005 16:18 Page 85
Einer, dem ich eine grüne Tafel zeige und sage, male mir diese Farbe, mir ein Rot malt;
wenn ich dasselbe sage und zeige ihm blau, so hat er gelb zu malen u.s.w., immer die komplementäre
Farbe; und daher kann ich mir auch denken, daß Einer meinen Befehl auch ohne
eine vorhergehende Abmachung so deutet. Ich kann mir ferner denken, daß die Abmachung
gelautet hätte ”auf den Befehl ,male mir diese Farbe‘, male immer eine gelblichere, als ich
Dir zeige“; und wieder kann ich mir die Deutung auch ohne Verabredung denken. Aber
kann man sagen, daß Einer ein rotes Täfelchen genau kopiert, indem er einen bestimmten
Ton von grün (oder ein anderes Rot als das des Täfelchens) malt und zwar so, wie er eine
gezeichnete Figur, nach verschiedenen Projektionsmethoden, verschieden und genau
kopieren kann? – Ist also hier der Vergleich zwischen Farben und Gestalten richtig, und kann
ein grünes Täfelchen einerseits als der Name einer bestimmten Schattierung von rot stehen
und anderseits als ein Muster dieses Tones? wie ein Kreis als der Name einer bestimmten
Ellipse verwendet werden kann, aber auch als ihr Muster. – Kann man also dort wie hier
von verschiedenen Projektionsmethoden sprechen, oder gibt es für das Kopieren einer Farbe
nur eine solche: das Malen der gleichen Farbe? Wir meinen diese Frage so, daß sie nicht
dadurch verneint wird, daß uns die Möglichkeit gezeigt wird, mittels eines bestimmten
Farbenkreises und der Festsetzung eines Winkels von einem Farbton auf irgend einen andern
überzugehn. Das, glaube ich, zeigt nun, in wiefern das rote Täfelchen gegenüber dem Wort
”rot“ in einem andern Fall ist, als das grüne. Übrigens bezieht sich, was wir hier für die
Farben gesagt haben, auch auf die Formen von Figuren, wenn das Kopieren ein Kopieren
nach dem Augenmaß und nicht eines mittels Meßinstrumenten ist. – Denken wir uns nun
aber doch einen Menschen, der vorgäbe ”er könne die Schattierungen von Rot in Grün
kopieren“ und auch wirklich beim Anblick des roten Täfelchens mit allen (äußeren) Zeichen
des genauen Kopierens einen grünen Ton mischte und so fort bei allen ihm gezeigten roten
Tönen. Dem50
gegenüber wären wir in der gleichen Lage, wie einem, der51
auf die gleiche Weise
(52
durch genaues Hinhorchen) Farben nach Violintönen mischte. Wir würden in dem Fall
sagen: ”Ich weiß nicht, wie er es macht“; aber nicht in dem Sinne, als verstünden wir nicht
die verborgenen Vorgänge in seinem Gehirn oder seinen Muskeln, sondern, wir verstehen
nicht, was es heißt ”dieser Farbton sei eine Kopie dieses Violintones“. Es sei denn, daß damit
nur gemeint ist, daß ein bestimmter Mensch erfahrungsgemäß einen bestimmten Farbton
mit einem bestimmten Klang assoziiert (ihn zu sehen behauptet, malt, etc.). 53
Anderseits wäre
ich vielleicht befriedigt, wenn man mir sagte, der Mann kopiere insofern, als er einen tiefern Violin
Ton54
dunkler male & die sieben Töne der Oktave in den &sieben Farben des Regenbogens“. Der
Unterschied zwischen dieser Assoziation und dem Kopieren, auch wenn ich selbst beide
Verfahren kenne, zeigt sich darin,55
daß es für die assoziierte Gestalt keinen Sinn hat, von
Projektionsmethoden zu reden, und daß ich von dem assoziierten Farbton sagen kann ”jetzt fällt
mir bei dieser Farbe (oder diesem Klang) diese Farbe ein, vor 5 Minuten war es eine andere“.
Etc. Wir könnten auch niemandem sagen ”Du hast nicht richtig assoziiert“, wohl aber ”Du hast
nicht richtig kopiert“. Und die Kopie einer Farbe – wie ich das Wort gebrauche – ist nur
eine; und es hat keinen Sinn, (hier) von verschiedenen Projektionsmethoden zu reden.))56
57
Es ist die Frage: Wenn sich die Regel, das Muster stehe für die Komplementärfarbe,
ihrem Wesen nach nur auf die Farben (oder Wörter) blau, rot, grün, gelb bezieht, ist sie
50 (V): Diesem
51 (V): Der wäre für uns auf derselben Stufe, wie
Einer, der
52 (V): Weise (auch
53 (M): ∫ – Regenbogens“.
54 (V): einen dunkleren Ton
55 (V): kenne, besteht darin,
56 (M): Falsch, aber kein uninteressantes Denken.
57 (M): ∫ Besser auslassen! (R): [Zu: Begriff der
Mischfarbe] S. 473 § 100]
51
50v
51
43 ”Primäre und sekundäre Zeichen“. . . .
WBTC011-14 30/05/2005 16:18 Page 86
can also imagine that someone might interpret my order in this way, even without a prior
understanding. Furthermore, I can imagine that there might have been the understanding
“When you are ordered ‘Paint this colour for me’, always paint one that is yellower than
the one I show you”; and again I can imagine such an interpretation even without the understanding.
But can we say that someone is copying a red chip exactly by painting a certain
shade of green (or a different red from that of the colour chip), and that he is doing that
just as, following different methods of projection, he might copy a drawn figure differently
but exactly? – So is the comparison between colours and shapes correct here, and can a
green colour chip stand for the name of a certain shade of red on the one hand, and on the
other as a sample of this shade? Just as a circle can be used as the name of a particular ellipse,
but also as its sample? – Can one therefore speak of different methods of projection in both
cases, or is there only one such method for copying a colour: painting the same colour? This
question is phrased in such a way that the answer needn’t be negative if we are shown how
we might move from one shade of colour to another by using a particular colour circle and
stipulating what angle is to be used. Now this shows, I believe, the way in which the red
chip lying across from the word “red” is different from the green one lying there. By the
way, what we have said here about colours also applies to the shapes of figures, if copying
them is done by eye and not with measuring instruments. – But now, in spite of all this,
let’s imagine someone who claims that “he can copy shades of red in green”, and who when
he sees the red chip mixes in a shade of green, continuing to do this with every shade of red
he’s shown, all the while showing all the (outward) signs of copying exactly. We’d be in the
same position with him as with someone who42
in the same way (by43
listening carefully) were
to mix colours together in accordance with notes played on a violin. In this case we’d say:
“I don’t know how he does it”; but not in the sense that we don’t understand the hidden
processes in his brain or muscles. Rather, we don’t understand what it means to say that
“This shade of colour is a copy of this note on the violin”. Unless all that is meant is that
a certain person as a matter of experience associates a particular shade of colour with a particular
sound (claims to see it, paints it, etc.). 44
On the other hand I might be satisfied if I were
told that the man was copying in so far as he was painting a low45
note on the violin darker, and was
painting the seven notes of an octave in the “seven colours of the rainbow”. The difference between
this association and copying, even if I myself am acquainted with both processes, is shown46
in the fact that it makes no sense to talk about methods of projection for the associated shape,
and that I can say of an associated shade of colour “Now this colour occurs to me upon
seeing this colour (or hearing this sound), but five minutes ago it was a different one”. Etc.
Neither could we tell someone “You didn’t associate correctly”, but we could very well tell
him “You didn’t copy correctly”. And – as I am using the word – there’s only one copy of
a colour; and (here) it makes no sense to talk about different methods of projection.))47
48
The question is: If the rule that the sample stands for the complementary colour refers
essentially only to the colours (or words) blue, red, green, yellow, then isn’t it identical to
42 (V): For us this person would be on the same
level as someone who
43 (V): also by
44 (M): ∫ – rainbow”.
45 (V): darker
46 (V): processes, consists
47 (M): False, but not uninteresting thinking.
48 (M): ∫ Better leave it out! (R): [To: concept of
mixed colour] p. 473 §100]
“Primary and Secondary Signs”. . . . 43e
WBTC011-14 30/05/2005 16:18 Page 87
44 ”Primäre und sekundäre Zeichen“. . . .
dann nicht identisch mit der, welche das grüne Zeichen als Wort für ”rot“, und umgekehrt,
etc. festsetzt? Denn eine Allgemeinheit,58
die ihrem logischen Wesen nach einem logischen
Produkt äquivalent ist, ist nichts anderes, als dieses logische Produkt. (Denn man kann nicht
sagen: hier ist das grüne Zeichen; nun hole mir ein Ding von der komplementären Farbe,
welche immer das sein mag. D.h., ”die komplementäre Farbe von rot“ ist keine Beschreibung
von grün; wie ”das Produkt von 2 und 2“ keine Beschreibung von 4.) Die Bestimmung, die
Komplementärfarbe59
als Bedeutung des Täfelchens zu nehmen, ist dann wie ein Querstrich
in einer Tabelle60
; ein Querstrich in der Grammatik der Farben gezogen. Es ist
klar, daß ich mit Hilfe einer solchen Regel eine Tabelle konstruieren61
kann,
ohne noch aus der Grammatik herauszutreten, also vor62
jeder Anwendung der
Sprache. Anders wäre es, wenn die Regel (R) hieße: das Täfelchen bedeutet immer
einen etwas dunkleren Farbton, als der seine63
ist. Man muß nur wieder auf den
verschiedenen Sinn der Farb- und der Gestaltprojektion achten (und bei der letzteren wieder
auf den Unterschied der Abbildung nach visuellen Kriterien und64
der Übertragung mit
Meßinstrumenten). Das Kopieren nach der Regel R ist ”kopieren“ in einem andern Sinne
als dem, in welchem das Hervorbringen des gleichen Farbtons so genannt wird. Es handelt
sich also nicht um zwei Projektionsmethoden, vergleichbar etwa der Parallel- und der
Zentralprojektion, durch die ich eine geometrische Figur mit Zirkel und Lineal in eine andere
projizieren kann. (Die Metrik der Farbtöne.)
Wenn ich das berücksichtige, so kann ich also in dem veränderten Sinn des Wortes ”Muster“
(der dem veränderten Sinn des Worts ”kopieren“ entspricht) das65
hellere Täfelchen zum
Muster des dunkleren Gegenstandes nehmen.
66
”Könnten wir nicht zur hinweisenden Erklärung von ,rot‘ ebensowohl auf ein grünes,
wie auf ein rotes Täfelchen zeigen? denn, wenn diese Definition nur ein Zeichen statt des
andern setzt, so sollte dies doch keinen Unterschied machen.“67
– Wenn die Erklärung nur
ein Wort für ein andres setzt, so macht es auch keinen.68
Bringt aber die Erklärung das Wort
mit einem Muster in Zusammenhang, so ist es nun nicht unwesentlich, mit welchem Täfelchen
das Zeichen verbunden wird (denke auch wieder daran, daß eine Farbe der andern nicht
im gleichen Sinn zum Muster dienen kann, wie ihr selbst). ”Aber dann gibt es also
willkürliche Zeichen und solche, die nicht willkürlich sind!“ – Aber denken wir nur an die
Verständigung durch Landkarten, Zeichnungen, und Sätze anderseits: die Sätze sind so
wenig willkürlich, wie die Zeichnungen. Aber die Worte sind willkürlich. (Vergleiche die
Abbildung | = 0, − = X69
.) Wird denn aber ein Wort eigentlich als Wort gebraucht, wenn
ich es nur in Verbindung mit einer Tabelle gebrauche, die den Übergang zu Mustern
macht? Ist es also nicht falsch, zu sagen, ein Satz sei ein Bild, wenn ich doch nur ein Bild
nach ihm und der Tabelle zusammenstelle? Aber so ist also doch der Satz und die Tabelle
58 (V): Regel,
59 (O): Komplementärfarbe zu nehmen,
60 (F): MS 112, S. 83r.
61 (V): herstellen
62 (O): also von
63 (V): als sein eigener
64 (V): von
65 (O): entspricht)$ das
66 (M): Als Erwägung nicht uninteressant. ? /
67 (V): doch aufs gleiche hinauslaufen.“
68 (V): setzt, ist es auch gleichgültig.
69 (E): In einer früheren Version dieser
Bemerkung (in MS 112, S. 83v) erscheint die
folgende Chiffre: | = 0, − = X, | − || −.
52
53
a
b
B
A
WBTC011-14 30/05/2005 16:18 Page 88
49 (V): rule
50 (F): MS 112, p. 83r.
51 (V): produce
52 (V): than its own.
53 (V): from
54 (M): Not uninteresting as a consideration. ? /
55 (V): then this should amount to the same
thing.”
56 (V): then it’s really irrelevant.
57 (E): An earlier version of this remark (in MS 112,
p. 83v) has the following code: | = 0, − = X, |
− || −.
“Primary and Secondary Signs”. . . . 44e
the one that establishes the green sign as a word for “red” and vice versa, etc.? For a
generality49
that, by virtue of its logic, is equivalent to a logical product, is nothing other
than the logical product. (For you can’t say: Here is the green sign – now get me an object
of the complementary colour, whatever that may be. That is, “the complementary colour of
red” isn’t a description of green; just as “the product of 2 and 2” isn’t a description of 4.)
Then the stipulation that we take the complementary colour as the meaning of the colour
chip is like a transverse line in a table,50
a transverse line drawn in the grammar
of colours. It’s clear that with the help of such a rule I can construct51
a table
while still remaining within grammar, i.e. before any application of language. It
would be different if the rule (R) were: the colour chip always means a somewhat
darker shade of colour than the one it has.52
Again, one just has to pay attention
to the different senses of colour and spatial projection (and, in the case of the latter, to the
difference between a depiction that follows visual criteria and53
a duplication with measuring
instruments). Copying following rule R is “copying” in a different sense from that in
which producing the same shade of colour is called copying. So it isn’t a case of two methods
of projection, comparable, say, to parallel and centre projection, in which I can project
one geometric figure onto another using a compass and a ruler. (The metrics of the shades
of colour.)
If I take this into account, then, in the new sense of “sample” (which corresponds to
the new sense of the word “copying”), I can take the lighter colour chip as a sample of the
darker object.
54
“In giving an ostensive explanation of ‘red’ couldn’t we just as well point to a green colour
chip as to a red one? For if this definition merely puts one sign in place of another, then
it shouldn’t make any difference.”55
– If the explanation only puts one word in place of
another, then it really doesn’t.56
But if the explanation connects a word to a sample, then
it isn’t irrelevant to which chip the sign is connected (remember again that one colour
cannot serve as a sample of another in the same sense that it can for itself). “But some
signs are arbitrary, and some aren’t!” – But let’s think about communicating with maps
and drawings, on the one hand, and sentences, on the other: sentences are no less arbitrary
than drawings. But the words are arbitrary. (Cf. the diagram | = 0, − = X57
.) But then
is a word really used as a word if I use it only together with a table that makes the transition
to samples? So isn’t it wrong to say that a proposition is a picture, if all I do is
follow it and the table to create a picture? But then the proposition and the table together
a
b
B
A
WBTC011-14 30/05/2005 16:18 Page 89
45 ”Primäre und sekundäre Zeichen“. . . .
zusammen ein Bild. Also zwar nicht adbcb allein, aber dieses Zeichen zusammen mit
Aber es ist offenbar, daß auch adbcb ein Bild von → ← ↑ ↓ ↑70
genannt werden
kann. Ja aber, ist nicht doch das Zeichen adbcb ein willkürlicheres Bild von
→ ← ↑ ↓ ↑ als dieses Zeichen von der Ausführung der Bewegung? Etwas ist
auch an dieser Übertragung willkürlich (die Projektionsmethode) und wie sollte
ich bestimmen, was willkürlicher ist.
Ich vergleiche also die Festsetzung der Wortbedeutung durch die hinweisende Definition,
der Festsetzung einer Projektionsmethode zur Abbildung räumlicher Gebilde. Dies ist aber
freilich71
nicht mehr, als72
ein Vergleich. Ein ganz guter Vergleich, aber er enthebt uns nicht
der Untersuchung des Funktionierens der Worte, getrennt von dem Fall der räumlichen
Projektion. Wir können allerdings sagen – d.h. es entspricht ganz dem Sprachgebrauch –,
daß wir uns durch Zeichen verständigen, ob wir Wörter oder Muster gebrauchen; aber das
Muster ist kein Wort, und das Spiel, sich nach Worten zu richten, ein anderes als das, sich
nach Mustern (zu) richten. (Wörter sind der Sprache nicht wesentlich.) Kann man aber
vielleicht sagen, daß Muster ihr wesentlich wären? (Muster sind dem Gebrauch73
von Mustern
wesentlich, Worte, dem Gebrauch74
von Worten.)
75
Vergiß hier auch nicht, daß die Wortsprache nur eine unter vielen möglichen Sprachen ist
und es Übergänge von ihr in die andern gibt. Untersuche die Landkarte auf das76
hin, was
in ihr dem Ausdruck der Wortsprache entspricht.
77
”Primär“ müßte eigentlich heißen: unmißverständlich.
78
Es klingt wie eine lächerliche Selbstverständlichkeit, wenn ich sage, daß der, welcher
glaubt die Gebärden79
seien die primären Zeichen, die allen andern zu Grunde liegen, außer
Stande wäre, den gewöhnlichsten Satz durch Gebärden zu ersetzen.
80
Regeln der Grammatik, die eine ”Verbindung zwischen Sprache und Wirklichkeit“
herstellen, und solche, die es nicht tun. Von der ersten Art etwa: ”diese Farbe nenne ich
,rot‘ “, – von der zweiten: ”~~p = p“. Aber über diesen Unterschied besteht ein Irrtum:
der Unterschied scheint prinzipieller Art zu sein; und die Sprache81
etwas, dem eine
Struktur gegeben, und das82
dann der Wirklichkeit aufgepaßt wird.
83
”Ich will nicht verlangen, daß in der erklärenden Tabelle das rote Täfelchen horizontal
gegenüber dem Wort ,rot‘ stehen soll, aber irgend ein Gesetz des Lesens der Tabelle
muß es doch geben. Denn sonst verliert ja die Tabelle ihren Sinn.“ Ist es aber gesetzlos,
wenn die Tabelle so aufgefaßt wird, wie die Pfeile andeuten?84
”Aber muß dann nicht
eben das Schema85
der Pfeile vorher gegeben werden?“ Nur, sofern auch das
Schema 86
früher gegeben wird.87
70 (V):
71 (V): allerdings
72 (V): wie
73 (V): sind der Benützung
74 (V): Worte, der Benützung
75 (M): ∫
76 (V): Landkarte darauf
77 (M): /
78 (M): / 3
79 (V): Gesten
80 (M): ∫ 3
81 (V): Sprache wesentlich
82 (V): was
83 (M): ∫ (
84 (F): MS 112, S. 100r.
85 (F): MS 112, S. 100r.
86 (F): MS 112, S. 100r.
87 (M): )
54
a
b
c
d
→
↑
↓
←.
a
b
A
B
c C
WBTC011-14 30/05/2005 16:18 Page 90
are a picture, after all. So, not adbcb alone, to be sure, but this sign together with
But it’s obvious that adbcb too can be called a picture of → ← ↑ ↓ ↑58
. Yes,
but isn’t the sign adbcb a more arbitrary depiction of → ← ↑ ↓ ↑ than this
sign is for carrying out the moves? There is something arbitrary about this translation
too (the method of projection), and how should I determine which is more
arbitrary?
So I am comparing establishing a meaning for a word via ostensive definition to establishing
a method of projection for the depiction of three-dimensional objects. But59
to be sure,
this is nothing more than a comparison. A pretty good comparison, but it doesn’t spare us
from having to investigate how words function, leaving aside the case of spatial projection.
To be sure we can say – i.e. it accords completely with how we use language – that we communicate
through signs, whether we use words or samples; but samples aren’t words and
the game of acting in accordance with words is different from that of acting in accordance
with samples. (Words aren’t essential to language.) But can one perhaps say that samples
are essential to it? (Samples are essential to the use of samples, words to the use of words.)60
61
Likewise, don’t forget here that word-language is only one among many possible languages
and that there are bridges from it to the others. Look at a map to see what in it corresponds
to an expression in word-language.
62
“Primary” really should be: “unambiguous”.
63
It sounds like a ridiculous truism if I say that anyone who believed that gestures are the
primary signs that underlie all others would be incapable of replacing the most ordinary
sentence with gestures.
64
Rules of grammar that establish a “connection between language and reality”, and those
that don’t. “I call this colour ‘red’ ” is an example of the first kind, for instance. – “~~p =
p” is of the second. But there’s a misconception about this difference: the difference seems
to be one of principle; and language65
seems to be something that is given a structure and
then superimposed on reality.
66
“I don’t want to demand that in the explanatory table the red chip has to be right next to
the word ‘red’, but there does have to be some sort of rule for reading the table. Otherwise the
table loses its sense.” But is there no rule if the table is understood as the arrows indicate?
67
“But in that case, mustn’t the schema for the arrows68
already have been given?”
Only in so far as the schema 69
has already been given.70
58 (V):
59 (V): But of course
60 (V): essential to the utilization of samples,
words to the utilization of words.)
61 (M): ∫
62 (M): /
63 (M): / 3
64 (M): ∫ 3
65 (V): language essentially
66 (M): ∫ (
67 (F): MS 112, p. 100r.
68 (F): MS 112, p. 100r.
69 (F): MS 112, p. 100r.
70 (M): )
“Primary and Secondary Signs”. . . . 45e
a
b
A
B
c C
a
b
c
d
→
↑
↓
← .
WBTC011-14 30/05/2005 16:18 Page 91
88
»Wird89
aber dann nicht wenigstens eine gewisse Regelmäßigkeit im Gebrauch
gefordert?! Würde es angehen, wenn wir einmal eine Tabelle nach diesem, einmal nach jenem
Schema zu gebrauchen hätten? Wie soll man denn wissen, wie man diese Tabelle zu
gebrauchen hat?« – Ja, wie weiß man es denn heute? Die Zeichenerklärungen haben doch
irgendwo90
ein Ende.
91
Nun gebe ich aber natürlich zu, daß ich, ohne vorhergehende Abmachung einer Chiffre,
ein Mißverständnis hervorrufen würde, wenn ich, auf den Punkt A zeigend, sagte, dieser
Punkt heißt ”B“. Wie ich ja auch, wenn ich jemandem den Weg weisen will, mit dem Finger
in der Richtung weise, in der er gehen soll, und nicht in der entgegengesetzten. Aber auch
diese Art des Zeigens könnte richtig verstanden werden, und zwar ohne daß dieses
Verständnis das gegebene Zeichen durch ein weiteres ergänzte. Es liegt in der menschlichen
Natur, das Zeigen mit dem Finger so zu verstehen. Und so ist die menschliche
Gebärdensprache primär in einem psychologischen Sinne.
92
Ist das Zeigen mit dem Finger unserer Sprache wesentlich? Es ist gewiß ein merkwürdiger
Zug unserer Sprache, daß wir Wörter hinweisend erklären: das ist ein Baum, das ist ein Pferd,
das ist grün, etc. (Überall bei den Menschen93
finden sich Brettspiele, die mit kleinen Klötzchen
auf Feldern gespielt werden. Überall auf der Erde findet sich eine Zeichensprache,94
die aus
geschriebenen Zeichen auf einer Fläche besteht.)
95
Ich bestimme die Bedeutung eines Worts, indem ich es als Name eines Gegenstandes
erkläre, und auch, indem ich es als gleichbedeutend mit einem andern Wort erkläre. Aber
habe ich denn nicht gesagt, man könne ein Zeichen nur durch ein anderes Zeichen erklären?
Und das ist gewiß so, sofern ja die hinweisende Erklärung 96
ein Zeichen ist.
Aber ferner bildet hier auch der Träger von ”N“, auf den gezeigt wird, einen Teil des Zeichens.
Denn: = 97
(N hat es getan). Dann heißt aber ”N“ der Name von diesem
Menschen, nicht vom Zeichen98
, von dem ein Teil auch dieser Mensch ist. Und
zwar spielt der Träger in dem Zeichen eine ganz besondere Rolle, verschieden von der eines
andern Teiles eines Zeichens. (Eine Rolle, nicht ganz ungleich der des Musters.)
99
Die hinweisende Erklärung eines Namens ist nicht nur äußerlich verschieden von einer
Definition wie ”1 + 1 = 2“, indem etwa das eine Zeichen in100
einer Geste meiner Hand,
statt in einem Laut- oder Schriftzeichen besteht, sondern sie unterscheidet sich von dieser
logisch; wie die Definition, die das Wort dem Muster beigesellt, von der eines Wortes durch
ein Wort. Es wird von ihr in andrer Weise Gebrauch gemacht.
Wenn ich also einen Namen hinweisend definiere und einen zweiten durch den ersten,101
so ist dieser zu jenem in anderer Beziehung,102
als zum Zeichen, das in der hinweisenden
Definition gegeben würde. D.h., dieses letztere ist seinem Gebrauch nach wesentlich von
dem Namen verschieden und daher sind die103
Verbaldefinition und die hinweisende
Definition, ”Definitionen“ im verschiedenen Sinne des Worts.
”dieser “
(dieser hat es getan)
”daß ist N“
88 (M): ? /
89 (V): »Wird da
90 (V): doch irgend einmal
91 (M): ü ? /
92 (M): ? /
93 (V): (Überall auf der Erde
94 (V): Schrift,
95 (M): ///
96 (F): MS 112, S. 108v.
97 (F): MS 112, S. 108v.
98 (F): MS 112, S. 108v.
99 (M): ∫
100 (V): aus
101 (V): durch ihn,
102 (V): so steht dieser zu jenem in anderem
Verhältnis,
103 (O): daher die
55
56
46 ”Primäre und sekundäre Zeichen“. . . .
WBTC011-14 30/05/2005 16:18 Page 92
71
»But isn’t72
at least a certain regularity of use called for?! Would it do if we had to use
a table now in accordance with this, now in accordance with that, schema? How is one to
know how to use this table?« – Well, how does one know this today? Explanations of signs,
after all, come to an end somewhere.73
74
Now of course I admit that, without prior agreement on a code I would cause a misunderstanding
if, pointing to point A, I said “This is called ‘B’ ”. Just as, if I want to show
a person the way I point my finger in the direction he is to follow, and not in the opposite
one. But this kind of pointing could also be understood correctly without such an understanding,
which supplements the given sign with an additional one. It’s in human nature to
understand pointing a finger in this way. And thus the human language of gestures is in a
psychological sense primary.
75
Is pointing a finger essential to our language? It certainly is a noteworthy trait of our
language that we explain words ostensively: that’s a tree, that’s a horse, that’s green, etc.
(Everywhere there are humans76
you can find board games played with little blocks on squares.
Everywhere on earth you can find a language of signs77
, consisting of written signs on a
surface.)
78
I specify the meaning of a word by declaring it to be the name of an object, and also
by declaring it to have the same meaning as another word. But didn’t I say that you can only
explain a sign with another sign? And that’s certainly the case, in so far as the ostensive
explanation 79
is indeed a sign. But what’s more, in this case the bearer of
“N” that is being pointed to constitutes a part of the sign as well. For: =
80
(N did it). But then “N” is the name of this person, and not of the sign ,81
a
part of which is also this person. In fact the bearer plays a very special role in the sign, a
role different from that of any other part of a sign. (A role not entirely dissimilar from that
of a sample.)
82
The ostensive explanation of a name is not only outwardly different from a definition
such as “1 + 1 = 2” (since the one sign consists of a gesture with my hand, the other of an
oral or written sign), but it differs logically from the latter, just as a definition that assigns
a word to a sample differs from one that defines a word with a word. The former is used in
a different way.
So if I define one name ostensively, and a second by using the first83
, then the latter is in
a different relationship to the former than to the sign that’s given in the ostensive definition.
That is, the use of this latter sign is essentially different from the name, and therefore
verbal and ostensive definitions are “definitions” in different senses of the word.
“This ”
(This did it)
“That is N”
71 (M): ? /
72 (V): isn’t there
73 (V): end some time.
74 (M): r ? /
75 (M): ? /
76 (V): (Everywhere in the world
77 (V): find a script
78 (M): ///
79 (F): MS 112, p. 108v.
80 (F): MS 112, p. 108v.
81 (F): MS 112, p. 108v.
82 (M): ∫
83 (V): a second via it
“Primary and Secondary Signs”. . . . 46e
WBTC011-14 30/05/2005 16:18 Page 93
104
Ich kann von primären und sekundären Zeichen sprechen – in einem bestimmten Spiel,
einer bestimmten Sprache. – Im Musterkatalog kann ich die Muster die primären Zeichen
und die Nummern die sekundären nennen. Was soll man aber in einem Fall, wie dem der
gesprochenen und geschriebenen Buchstaben sagen? Welches sind hier die primären,
welches die sekundären Zeichen?
105
Der Begriff vom sekundären Zeichen ist doch dieser: Sekundär ist ein Zeichen dann,
wenn, um mich nach ihm zu richten, ich eine Tabelle brauche, die es mit einem andern
(primären) Zeichen verbindet, über welches ich mich erst nach dem sekundären richten kann.
106
&Primär, das Zeichen, welches allein genügt hätte, wenn es nicht zu unbequem wäre es immer
mitzuführen.“
Die Tabelle garantiert mir die Gleichheit aller Übergänge nicht, denn sie zwingt mich ja
nicht, sie immer gleich zu gebrauchen. Sie ist da wie ein Feld, durch das Wege führen, aber
ich kann ja auch querfeldein gehen.
Ich mache den Übergang in der Tabelle bei jeder Anwendung von Neuem. Er ist nicht,
quasi, ein für allemal in der Tabelle gemacht. (Die Tabelle verleitet mich höchstens, ihn zu
machen.)
Wie ist es aber, wo keine Tabelle gebraucht wird wie z.B. im Fall107
der gesprochenen &
geschriebenen Buchstaben?
Lautes108
Lesen & anderseits Abschreiben eines geschriebenen Satzes.
109
Welcher Art ist denn meine Aussage über die Tabelle: ”daß sie mich nicht zwingt, sie
so und so zu gebrauchen“? Und: &daß die Anwendung durch die Regel (oder die Tabelle)
nicht anticipiert wird“? Wohl von derselben Art wie die Bemerkung, daß die Zeichenerklärungen
doch einmal110
ein Ende haben. Und das ist ähnlich, wie wenn man sagt: &Was nützt Dir die
Annahme eines Schöpfers, sie schiebt doch das Problem nur hinaus“. Diese Bemerkung hebt einen
Aspekt meiner Erklärung hervor, den ich vielleicht früher nicht gesehen hatte. Man könnte auch sagen:
&Sieh Deine Erklärung111
doch so an! – bist Du jetzt noch immer von ihr befriedigt?“
104 (M): ? /
105 (M): ∫
106 (M): ? / – mitzuführen.“
107 (V): wie im Fall
108 (V): Vor Lautes
109 (M): /
110 (V): die Zeichenerklärungen einmal
111 (V): Theorie
57
56v
57
56v
57
47 ”Primäre und sekundäre Zeichen“. . . .
WBTC011-14 30/05/2005 16:18 Page 94
84
I can speak of primary and secondary signs – in one particular game, in a particular
language. – In a catalogue of samples I can call the samples the primary signs and the
numbers the secondary ones. But what should one say in a case such as that of spoken and
written letters? Which are the primary signs there, which the secondary?
85
Surely this is the concept of a secondary sign: A sign is secondary when, in order to
follow it, I need a table that connects it to another (primary) sign, and only after using this
am I able to follow the secondary sign.
86
“A sign is primary that would have been sufficient by itself, if it weren’t too inconvenient always
to carry it along.”
The table gives me no guarantee that all of its links are the same, for it doesn’t force me
always to use it the same way. It’s there like a field criss-crossed by paths, but I can also
walk cross-country.
With each application I make a new link in the table. The link is not, as it were, made
once and for all in the table. (The most we can say is that the table seduces me into making
the link.)
But how about where no table is used, as, for example, in the case87
of spoken and written
letters?
Reading aloud and, on the other hand, copying a written sentence.
88
What sort of statement is mine about the table: “it doesn’t force me to use it in such
and such a way”? And: “the application isn’t anticipated by the rule (or the table)”? Very
likely of the same sort as the remark that explanations of signs do at89
some point come to an end.
And that is similar to someone’s saying: “What good does the assumption of a Creator do you, since
it just postpones the problem?” This remark emphasizes an aspect of my explanation that perhaps I
hadn’t seen before. One could also say: “Look at your explanation90
this way! – are you still satisfied
with it?”
84 (M): ? /
85 (M): ∫
86 (M): ? / – along.”
87 (V): example, as in the case
88 (M): /
89 (V): signs at
90 (V): theory
“Primary and Secondary Signs”. . . . 47e
WBTC011-14 30/05/2005 16:18 Page 95
14
Das, was die Philosophie1
am
Zeichen interessiert, die2
Bedeutung,
die für sie3
maßgebend ist, ist das,
was in der Grammatik des Zeichens
niedergelegt ist.
4
Wir fragen: Wie gebrauchst Du das Wort, was machst Du damit, – das wird mich lehren, wie Du
es verstehst.5
6
Die Grammatik, – könnte man sagen7
– das sind die Geschäftsbücher8
der Sprache; aus
denen alles über unsere9
Transaktionen zu ersehen sein muß10
, was nicht Gefühle betrifft,
sondern Tatsachen.11
Man könnte in gewissem Sinne sagen, daß es uns auf Nuancen nicht ankommt.
12
Ich will also eigentlich sagen: es gibt nicht Grammatik und Interpretation der Zeichen.
Sondern, soweit von einer Interpretation, also von einer Deutung13
der Zeichen, die Rede
sein kann, soweit muß sie die Grammatik selbst besorgen.
Denn ich brauchte nur zu fragen: Soll die Interpretation durch Sätze erfolgen? Und in
welchem Verhältnis sollen diese Sätze zu der Sprache stehen, die sie schaffen?
Gilt besonders für sogenannte &Deutungen“14
mathematischer Theoreme.
15
Wenn ich sage, daß ein Satz, der Mengenlehre etwa, in Ordnung ist, aber eine neue
Interpretation erhalten muß, so heißt das nur,16
dieser Teil der Mengenlehre bleibt in sich
unangetastet, muß aber in eine andere grammatische Umgebung gerückt werden.
1 (V): was uns
2 (V): interessiert; die
3 (V): uns
4 (M): ? /
5 (V): Wie gebrauchst . . . verstehst.
6 (M): ? / (R): ∀ S. 40/3
7 (V): Grammatik, – möchte ich sagen
8 (V): Die Grammatik ist das Geschäftsbuch
9 (V): die
10 (V): Sprache: aus denen alles zu ersehen sein
muß // woraus alles zu ersehen sein muß
11 (V): was nicht vage Gefühle betrifft, sondern
wesentliche Fakten.
12 (M): ? /
13 (V): Erklärung
14 (V): Ist besonders wichtig für die Deutungen
15 (M): ∫ (R): [Zu den Bemerkungen über die
Mengenlehre]
16 (V): nur, daß
58
57v
58, 57v
58
57v
58
48
WBTC011-14 30/05/2005 16:18 Page 96
14
What Interests Philosophy About
the Sign, the Meaning That
is Decisive for it1
, is What is
Laid Down in the Grammar
of the Sign.
2
We ask: How do you use the word, what do you do with it? – that will teach me how you
understand it.3
4
Grammar – one could say5
– that’s the ledger6
of language, from which everything about our7
transactions must be ascertainable8
– everything that concerns facts rather than feelings.9
In a certain sense one could say that we’re not concerned with nuances.
10
So what I really want to say is: There is no such thing as grammar as well as an interpretation
of signs. Rather, in so far as one can talk about an interpretation, i.e. an explica-
tion11
of signs, it is grammar itself that has to take care of that.
For I’d need only ask: Is the interpretation to take place using propositions? And in what
relationship are these propositions supposed to stand to the language they create?
Is particularly valid for so-called “interpretations”12
of mathematical theorems.
13
If I say that a proposition, say in set theory, is in order, but that it has to be given a new
interpretation, then all this means is that this part of set theory stays undisturbed internally,
but has to be moved to a different grammatical neighbourhood.
1 (V): What Interests Us About the Sign, the
Meaning That is Decisive for Us
2 (M): ? /
3 (V): How do you . . . understand it.
4 (M): ? / (R): ∀ p. 40/3
5 (V): Grammar – I would like to say
6 (V): Grammar is the ledger
7 (V): the
8 (V): from which everything must be ascertainable
9 (V): everything that concerns essential facts
rather than vague feelings.
10 (M): ? /
11 (V): explanation
12 (V): Is particularly valid for interpretations
13 (M): ∫ (R): [To the remarks on set theory]
48e
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59 Satz.
Sinn des Satzes.
WBTC015-21 30/05/2005 16:18 Page 98
Proposition.
Sense of a
Proposition.
WBTC015-21 30/05/2005 16:18 Page 99
15
”Satz“ und ”Sprache“
verschwimmende Begriffe.
1
Was ist ein Satz? Wovon unterscheide ich denn einen Satz? Oder, wovon will ich ihn denn
unterscheiden? Von Satzteilen in seinem grammatischen System (wie die Gleichung vom
Gleichheitszeichen), oder von allem, was wir nicht Satz nennen, also diesem Sessel, meiner
Uhr, etc. etc.? Denn, daß es Schrift- oder Lautbilder gibt, die Sätzen besonders ähnlich sind,
braucht uns eigentlich nicht zu kümmern.
2
Oder wir müssen sagen: Vom Satz kann nur in der Erklärung eines grammatischen Systems
die Rede sein.3
4
Es geht mit dem Wort ”Satz“ wie mit dem Wort ”Gegenstand“ und andern: Nur auf
eine beschränkte Sphäre angewandt sind sie zulässig und dort sind sie natürlich. Soll die
Sphäre ausgedehnt werden, damit der Begriff ein philosophischer wird, so verflüchtigt sich
die Bedeutung der Worte und es sind leere Schatten. Wir müssen sie dort aufgeben und
wieder in den Grenzen benützen.
5
Nun möchte man aber sagen: ”Satz ist alles, womit ich etwas meine“. Und gefragt ”was heißt
das, ,etwas‘ meinen“, würde6
ich Beispiele anführen. Nun haben diese Beispiele zwar ihren
Bereich, auf den sie ausgedehnt werden können, aber weiter führen sie mich doch nicht. Wie
ich ja in der Logik nicht ins Blaue verallgemeinern kann. Hier handelt es sich aber nicht um
Typen, sondern darum, daß die Verallgemeinerung selbst etwas bestimmtes ist; nämlich ein
Zeichen mit vorausbestimmten grammatischen Regeln. D.h., daß die Unbestimmtheit der
Allgemeinheit keine logische Unbestimmtheit ist. So als hätten wir nun nicht nur Freiheit
im logischen Raum, sondern auch Freiheit, diesen Raum zu erweitern, oder zu verändern.
Also nicht nur Bewegungsfreiheit, sondern eine Unbestimmtheit der Geometrie.
7
Über sich selbst führt uns kein Zeichen hinaus; und auch kein Argument.
8
(Wenn wir sagen, Satz ist jedes Zeichen, womit wir etwas meinen, so könnte man fragen:
was meinen wir und wann meinen wir es? Während wir das Zeichen geben? u.s.w., u.s.w.)
9
Wenn ich frage ”was ist die allgemeine Form des Satzes“, so kann die Gegenfrage lauten:
”haben wir denn einen allgemeinen Begriff vom Satz, den wir nur10
exakt fassen wollen?“ –
So wie: Haben wir einen allgemeinen Begriff von der Wirklichkeit?
1 (M): ? /
2 (M): ∫
3 (V): Vom Satzbegriff kann nur in einem //
innerhalb eines // grammatischen System // s
// gesprochen werden.
4 (M): ? /
5 (M): ∫ 3
6 (V): müßte
7 (M): / ∫ 3
8 (M): ? /
9 (M): ? / 3
10 (V): nun
60
61
50
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15
“Sentence” and “Language”
Blurred Concepts.
1
What is a sentence? From what do I distinguish a sentence? Or from what do I want to
distinguish it? From parts of sentences in its system of grammar (like the equation from the
equals sign), – or from everything we don’t call a sentence, i.e. from this chair, my watch,
etc., etc.? For that there are written images or sound patterns that are especially similar to
sentences really need not concern us.
2
Or do we have to say: Only in explaining a system of grammar can we speak of sentences.3
4
The same thing goes for the word “sentence” as for the word “object”, and others: They’re
only permitted when applied to a limited sphere, and there they’re natural. If the sphere is
to be expanded so that the concept can become philosophical, then the meaning of the words
evaporates and they are empty shadows. We have to abandon them there and use them back
within their boundaries.
5
But now we’d like to say: “A proposition is everything with which I mean something”.
And if asked “What does to mean ‘something’ mean?”, I’d give6
examples. Now these examples,
to be sure, do have a realm of their own into which they can be expanded; but they
don’t take me any further. Just as in logic I can’t generalize into the blue. And here it isn’t
a matter of types, but of the fact that generalization itself is something specific, i.e. a sign
with predetermined grammatical rules. That means that the indeterminacy of generality is
not a logical indeterminacy. As if now we had freedom not only within logical space, but
also the freedom to expand or to change this space.
So not only freedom of movement, but an indeterminacy of geometry.
7
No sign leads us beyond itself; nor does any argument.
8
(If we say that a proposition is any sign with which we mean something, then one could
ask: What do we mean and when do we mean it? While we’re using the sign? Etc., etc.)
9
If I ask “What is the general form of a proposition?”, then the counter-question can be:
“Do we really have a general concept of a proposition, which we just10
want to formulate
exactly?” – Just like: Do we have a general concept of reality?
1 (M): ? /
2 (M): ∫
3 (V): Only in // within // a grammatical system
can we speak of the concept of a sentence.
4 (M): ? /
5 (M): ∫ 3
6 (V): I’d have to give
7 (M): / ∫ 3
8 (M): ? /
9 (M): ? / 3
10 (V): we now
50e
WBTC015-21 30/05/2005 16:18 Page 101
51 ”Satz“ und ”Sprache“ verschwimmende Begriffe.
11
Die Frage kann auch lauten: Was geschieht, wenn ein neuer Satz in die Sprache aufgenommen
wird: Was ist das Kriterium dafür, daß er ein Satz ist? oder, wenn das Aufnehmen in
die Sprache ihn zum Satz stempelt, worin besteht diese Aufnahme? Oder: was ist Sprache?
12
Da scheint es nun offenbar, daß man das Zeichengeben von anderen Tätigkeiten
unterscheidet. Ein Mensch schläft, ißt, trinkt, gibt Zeichen (bedient sich einer Sprache).
13
Was ist ein Satz? wodurch ist dieser Begriff bestimmt? – Wie wird dieses Wort (”Satz“)
in der nicht-philosophischen Sprache gebraucht? Satz, im Gegensatz wozu?
14
Ich kenne einen Satz, wenn ich ihn sehe.
15
Diese Frage ist fundamental: Wie, wenn wir eine neue Erfahrung machen, etwa einen
neuen Geschmack oder einen neuen Hautreiz kennen lernen: woher weiß ich, daß, was diese
Erfahrung beschreiben wird,16
ein Satz sein wird?17
Oder, warum soll ich das einen Satz
nennen? Nun, mit demselben Recht, mit dem ich vom Beschreiben oder von einer neuen
”Erfahrung“18
gesprochen habe.19
Aber warum habe ich das Wort Erfahrung gebraucht, im
Gegensatz wozu?
Wie kann ich überhaupt von einem neuen &Geschmack“20
reden? Ich kann ihn mir ja nicht vorstellen!
– Antwort: Wie wird21
so ein Ausdruck gebraucht?
22
Habe ich denn, was geschehen ist, schon bis zu einem Grade damit charakterisiert, daß
ich sagte, es sei eine Erfahrung? Doch offenbar gar nicht. Aber es scheint doch, als hätte ich
es schon getan, als hätte ich davon schon etwas ausgesagt: ”daß es eine Erfahrung ist“. In
diesem falschen Schein liegt unser ganzes Problem. Denn, was vom Prädikat ”Erfahrung“
gilt, gilt vom Prädikat ”Satz“.
23
Das Wort ”Satz“ und das Wort ”Erfahrung“ haben schon eine bestimmte Grammatik.
24
Das heißt, ihre Grammatik muß im Vorhinein bestimmt sein und hängt nicht von irgend
einem künftigen Ereignis ab.
25
Hier ist auch der Unsinn in der ”experimentellen Theorie der Bedeutung“ ausgesprochen.
Denn die Bedeutung ist in der Grammatik festgelegt.
26
Wie verhält sich die Grammatik des Wortes ”Satz“ zur Grammatik der Sätze?
27
”Satz“ ist offenbar die Überschrift der Grammatik der Sätze. In einem Sinne aber auch
die Überschrift der Grammatik überhaupt, also äquivalent den Worten ”Grammatik“ und
”Sprache“.28
29
Wenn ich nun sage: aber die Sprache kann sich doch ausdehnen, so ist die Antwort: Gewiß,
aber wenn dieses Wort ”ausdehnen“ hier einen Sinn hat, so muß ich jetzt schon wissen, was ich
11 (M): ? / 3 ∫
12 (M): ? / ∫
13 (M): ? / 555
14 (M): 555 3
15 (M): / 3
16 (V): Erfahrung beschreibt,
17 (V): Satz ist?
18 (V): Wohl mit demselben Recht, womit // mit
welchem // ich von einer neuen Erfahrung
19 (V): habe. Denn Erfahrung und Satz sind
äquivalent.
20 (V): von einer möglichen neuen Sinneserfahrg.
21 (V): vorstellen! – Wie wird
22 (M): ? / 3
23 (M): ///
24 (M): ///
25 (M): ///
26 (M): 555 (
27 (M): 555
28 (M): )
29 (M): ? / F.u.i.
62
63
WBTC015-21 30/05/2005 16:18 Page 102
11
The question can also be phrased: What happens when a new sentence is admitted into
a language? What is the criterion for its being a sentence? Or if being admitted into language
labels it as a sentence, what does this admission consist in? Or: What is language?
12
Now here it seems obvious that one distinguishes using signs from other activities. A person
sleeps, eats, drinks, uses signs to communicate (makes use of a language).
13
What is a proposition? What defines this concept? – How is this word (“proposition”)
used in non-philosophical language? Proposition, as opposed to what?
14
I know a proposition when I see it.
15
This question is fundamental: What if we have a new experience, say getting to know a
new taste or a new sensation on our skin: how do I know that what will describe this experience
will be a proposition?16
Or why should I call it a proposition? Well, with the same
justification that allowed me to talk about describing it, or to talk about a new “experience”.17
But why did I use the word “experience” – in contrast to what?
How can I talk about a new “taste”18
in the first place? After all, I can’t imagine it! – Answer: how
is19
such an expression used?
20
Have I to a certain extent already characterized what has happened in saying that it’s
an experience? Obviously, not at all. But still it looks as if I had already done that, as if I
had already stated something about it: “that it is an experience”. Our entire problem lies in
this false appearance. For what holds true of the predicate “experience” also holds for the
predicate “proposition”.
21
The word “proposition” and the word “experience” already have a particular grammar.
22
That is, their grammar must be established in advance. It doesn’t depend on some future
event.
23
This also clearly articulates the nonsense of the “experimental theory of meaning”. For
meaning is laid down in grammar.
24
How does the grammar of the word “proposition” relate to the grammar of propositions?
25
“Proposition” is obviously the heading for the grammar of propositions. But in a sense
it’s also the heading for grammar in general, and is therefore equivalent to the words
“grammar” and “language”.26
27
If I say: “But language can expand”, then the answer is: Certainly, but if the word “expand”
makes any sense here, then I have to know now what I mean by it, have to be able to state
11 (M): ? / 3 ∫
12 (M): ? / ∫
13 (M): ? / 555
14 (M): 555 3
15 (M): / 3
16 (V): that what describes this experience is a
proposition?
17 (V): proposition? Most probably the same
justification that allowed me to talk about a new
experience. For experience and proposition
are equivalent.
18 (V): about a possible new sense-experience
19 (V): imagine it! – How is
20 (M): ? / 3
21 (M): ///
22 (M): ///
23 (M): ///
24 (M): 555 (
25 (M): 555
26 (M): )
27 (M): ? / F.u.i.
“Sentence” and “Language” Blurred Concepts. 51e
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64
63v
64
52 ”Satz“ und ”Sprache“ verschwimmende Begriffe.
damit meine, muß angeben können, wie ich mir so eine Ausdehnung vorstelle. Und was ich
jetzt nicht denken kann, das kann ich jetzt auch nicht ausdrücken, und auch nicht andeuten.30
31
Es scheint unsere Frage noch zu erschweren, daß auch die Worte ”Welt“ und
”Wirklichkeit“ Äquivalente des Wortes ”Satz“ sind.
32
Aber es ist doch lächerlich, die Welt, oder die Wirklichkeit, abgrenzen zu wollen. Wem
soll man sie denn entgegenstellen. Und so ist es mit der Bedeutung des Wortes ”Tatsache“.
Aber man gebraucht ja diese Wörter auch nicht als Begriffswörter.
33
Etwas ist ein Satz nur in einer Sprache.34
35
Das ist es auch, was damit gemeint ist, daß es in der Welt zwar Überraschungen gibt,
aber nicht in der Grammatik.
36
Und das Wort ”jetzt“ bedeutet hier: ”in dieser Grammatik37
“, oder: ”wenn die Worte
mit diesen grammatischen Regeln gebraucht werden“.
38
Hier haben wir dieses bohrende Problem: wie es denn möglich ist, auch nur auf den
Gedanken zu kommen!39
&Wie konnte ich nur auf den Ged. kommen“ heißt hier: &was bedeutet
denn der Gedanke, inwiefern ist er denn ein Ged. da ihm doch nichts entspricht?
40
Als wäre der Gedanke ein Zauber.
Was meinen wir denn mit der Existenz von Dingen, d.h. welche Anwendung hat denn dieser Begriff.
Ein Gedanke ist ja bloß ein Ausdruck & hinter dem kann kein Zauber stecken.41
Was dieser Ausdruck
leistet muß sich an seiner Anwendung zeigen.
42
Hierher gehört die alte Frage: ”wie bin ich dann aber überhaupt zu diesem Begriff gekommen“
(etwa zu dem der außer mir liegenden Gegenstände). (Es ist ein Glück, eine solche
Frage aus der Entfernung als alte Gedankenbewegung betrachten zu können; ohne in ihr
verstrickt zu sein.) Zu dieser Frage ist ganz richtig der Nachsatz zu denken: ”ich konnte
doch nicht mein eigenes Denken transcendieren“, ”ich konnte doch nicht sinnvoll das
transcendieren, was für mich Sinn hat“. Es ist das Gefühl, daß ich nicht auf Schleichwegen
(hinterrücks) dahin kommen kann, etwas zu denken, was zu denken mir eigentlich verwehrt
ist. Daß es hier keine Schleichwege gibt, auf denen ich weiter kommen könnte, als auf dem
direkten Weg.
43
Wir haben es natürlich wieder mit einer falschen Analogie zu tun: Es hat guten Sinn
zu sagen ”ich weiß, daß er in diesem Zimmer ist, weil ich ihn höre, wenn ich auch nicht
hineingehen und ihn sehen kann“. Es gibt in der Grammatik nicht direktes & indirektes Wissen.
44
”Satz“ ist so allgemein wie z.B. auch ”Ereignis“. Wie kann man ”ein Ereignis“ von dem
abgrenzen, was kein Ereignis ist?
Ebenso allgemein ist aber auch ”Experiment“, das vielleicht auf den ersten Blick
spezieller zu sein scheint.
30 (M): Bezieht sich auf die Kontroverse über die
Möglichkeit einer neuen Sinneswahrnehmung &
über ungelöste Probleme in der Mathematik.
31 (M): ? /
32 (M): ////
33 (M): / 3
34 (R): [Zu S. 93]
35 (M): / 3
36 (M): ? /
37 (V): ”in diesem Kalkül
38 (R): [Zu S. 79]
39 (V): wie es möglich ist, an die Existenz von
Dingen auch nur zu denken, wenn wir immer
nur Vorstellungen – ihre Abbilder – sehen.
40 (M): /
41 (V): sein.
42 (M): / 3 (R): [Zu S. 79]
43 (M): 3 ///
44 (M): ////
63
WBTC015-21 30/05/2005 16:18 Page 104
how I envisage such an expansion. And what I can’t think now I also can’t express now, nor
can I allude to it.28
29
That the words “world” and “reality” are also equivalents of the word “proposition” seems
to make our question even more difficult.
30
But it’s ridiculous to want to delimit the world or reality. With what should we contrast
them? And that’s how it is with the meaning of the word “fact”.
But then again we don’t use these words as concept-words.
31
Something is a proposition only in a language.32
33
That is also what is meant by saying that surprises do occur in the world, but not in
grammar.
34
And here the word “now” means: “in this grammar35
” or: “when the words are used
with these grammatical rules”.
36
Here we have this nagging problem: How it is possible even to come up with a
thought!37
Here “How could I have come up with a thought” means: “What does the thought mean,
to what extent is it a thought anyway, since nothing corresponds to it, after all?”
38
As if a thought were magic.
What do we mean by the existence of things, i.e. what use does this concept have? After all, a
thought is just an expression, and there can be no magic hidden behind39
an expression. What this
expression accomplishes can only be seen by looking at its use.
40
This is the place for the old question: “But how did I arrive at this concept anyway?”
(say that of external objects). (What a blessing it is to be able to look at such a question from
afar, as an old movement of thought, without being enmeshed in it.) What we really need
to do is to follow up this question with the thought: “I can’t transcend my own thinking,
after all”, “After all, I can’t meaningfully transcend what makes sense for me”. It’s the feeling
that I can’t get to the point of thinking something that in fact I’m barred from thinking
by using secret paths (through a back door). That there are no secret paths for getting further
than one gets on the direct path.
41
Of course once again we’re dealing with a false analogy: It makes good sense to say “I
know that he’s in this room because I hear him, even though I can’t go inside and see him”.
In grammar there is no such thing as direct and indirect knowledge.
42
“Proposition” is as general as, for example, “event”. How can we differentiate “an event”
from what isn’t one?
But “experiment” too, which at first sight might seem to be more specific, is just as
general.
28 (M): Refers to the controversy about the possibility
of a new sense-perception and to the one about
unsolved problems in mathematics.
29 (M): ? /
30 (M): ////
31 (M): / 3
32 (R): [To p. 93]
33 (M): / 3
34 (M): ? /
35 (V): calculus
36 (R): [To p. 79]
37 (V): problem: How is it possible even to think
about the existence of things if all we ever see
are mental images – their likenesses.
38 (M): /
39 (V): magic behind
40 (M): / 3 (R): [To p. 79]
41 (M): 3 ///
42 (M): ////
“Sentence” and “Language” Blurred Concepts. 52e
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65
53 ”Satz“ und ”Sprache“ verschwimmende Begriffe.
45
”Da geschah ein Ereignis . . .“: das heißt nicht ”ein Ereignis“ im Gegensatz zu etwas
Anderem.
46
Rechtmäßiger Gebrauch des Wortes ”Sprache“: Es bedeutet entweder die Erfahrungstatsache,
daß Menschen reden (auf gleicher Stufe mit der, daß Hunde bellen), oder es bedeutet:
festgesetztes System von Wörtern und grammatischen Regeln47
in den Ausdrücken ”die
englische Sprache“, ”deutsche Sprache“, ”Sprache der Neger“ etc. ”Sprache“ als logischer
Begriff könnte nur mit ”Satz“ äquivalent, und dann die48
Überschrift eines Teiles der
Grammatik sein.
49
Könnten wir etwas ”Sprache“ nennen, was nicht wirklich angewandt würde? Könnte
man von Sprache reden, wenn nie eine gesprochen worden wäre? (Ist denn Sprache ein Begriff,
vergleichbar mit dem Begriff ”Centaur“,50
der besteht, auch wenn es nie ein solches Wesen
gegeben hat?)
(Vergleiche damit ein Spiel, das nie gespielt wurde, eine Regel, nach der nie gehandelt
wurde.)
51
Was tut der, der eine neue Sprache konstruiert (erfindet)? nach welchem Prinzip geht
er vor? Denn dieses Prinzip ist der Begriff ”Sprache“.
52
Eine Sprache erfinden, heißt, eine Sprache konstruieren. Ihre Regeln aufstellen. Ihre
Grammatik verfassen.
53
Verändert54
jede erfundene Sprache den Begriff der Sprache?
55
Überlege, welches Verhältnis sie zum früheren Begriff hat. Denke56
an das Verhältnis der
komplexen Zahlen zum älteren Zahlbegriff;57
anderseits an den Fall, wenn zum ersten Mal gewisse
(etwa sehr große) Kardinalzahlen angeschrieben & miteinander multipliziert werden & an das Verhältnis
dieser neuen Multiplikation zu dem allgemeinen Begr. der Multiplikation von Kardinalzahlen.58
59
Was für das Wort ”Sprache“ gilt, muß auch für den Ausdruck ”System von Regeln“
gelten. Also auch für das Wort ”Kalkül“.
60
Wie bin ich denn zum Begriff ”Sprache“ gekommen? Doch nur durch die Sprachen,
die ich gelernt habe.
Aber die haben mich in gewissem Sinne über sich hinausgeführt, denn ich wäre jetzt im
Stande, eine neue Sprache zu konstruieren, z.B. Wörter zu erfinden. Also gehört diese Methode
der Konstruktion noch zum Begriff der Sprache. Aber nur, wenn ich ihn so festlege. Immer
wieder hat mein ”u.s.w.“ eine Grenze.
61
Der Begriff: sich einander etwas mitteilen. Wenn ich z.B. sage: ”Sprache“ werde ich
jedes System von Zeichen nennen, das Menschen untereinander vereinbaren, um sich
45 (M): /
46 (M): ///
47 (V): festgesetztes System der Verständigung
48 (V): eine
49 (M): ? /
50 (V): Ist denn Sprache ein Begriff, wie ”Centaur“,
51 (M): ? / 3 |
52 (M): 555
53 (M): ? / 3
54 (V): Erweitert
55 (M): ? /
56 (V): Denke einerseits
57 (V): zum Zahlbegriff von;
58 (V): anderseits an das Verhältnis einer Multiplikation
von Kardinalzahlen die zum ersten mal
hingeschrieben wird // einer neu aufgeschriebenen
Multiplikation von Kardinalzahlen // zum Begriff //
zum allgemeinen Begriff // dieser // der //
Multiplikation von Kardinalzahlen.
59 (M): 3 ///
60 (M): ∫ 3
61 (M): 555
64v
65
66
WBTC015-21 30/05/2005 16:18 Page 106
43
“At that moment an event took place . . .” That does not mean “an event”, as opposed
to something else.
44
Legitimate use of the word “language”: Either it means the empirical fact that humans
talk (on the same level as the fact that dogs bark), or it means an established system of words
and grammatical rules45
in such expressions as “the English language”, “the German language”,
“the language of Negroes”, etc. Taken as a logical concept, “language” would have
to be equivalent to “sentence”, and then it could be the heading46
for a part of grammar.
47
Could we call something that wasn’t really used, “language”? Could one speak of
language if none had ever been spoken? (Is language a concept comparable to the concept48
“centaur”, which exists even if there never was such a creature?)
(Compare to this a game that was never played, a rule that was never followed.)
49
What does a person do who constructs (invents) a new language? According to what
principle does he proceed? For this principle is the concept “language”.
50
To invent a language means to construct a language. To set up its rules. To compose
its grammar.
51
Does every invented language change52
the concept of language?
53
Think about the relationship it has to the earlier concept. Think about54
the relationship of
complex numbers to the older concept55
of numbers; on the other hand, think about the case in
which certain (say very large) cardinal numbers are written down and multiplied for the first time,
and think about how this new multiplication relates to the general concept of the multiplication of
cardinal numbers.56
57
What holds for the word “language” must also hold for the expression “system of rules”.
And therefore also for the word “calculus”.
58
How did I arrive at the concept “language”? Surely only through the languages that I
learned.
But in a certain sense they led me beyond themselves, for now I’m able to construct a
new language, can invent words, for instance. So this method of construction still belongs
to the concept of language. But only if I define the concept this way. Again and again my
“etc.” has a limit.
59
The concept: to communicate something to each other. If I say, for example: “I shall
call any system of signs that people agree upon in order to communicate with each other a
43 (M): /
44 (M): ///
45 (V): system of communication
46 (V): be a heading
47 (M): ? /
48 (V): language a concept like
49 (M): ? / 3 |
50 (M): 555
51 (M): ? / 3
52 (V): expand
53 (M): ? /
54 (V): about on the one hand
55 (V): the concept
56 (V): on the other hand, think about the relationship
of a multiplication of cardinal numbers that is
written down for the first time // of a newly written
down multiplication of cardinal numbers // to the
//general // concept of the multiplication of cardinal
numbers.
57 (M): 3 ///
58 (M): ∫ 3
59 (M): 555
“Sentence” and “Language” Blurred Concepts. 53e
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miteinander zu verständigen, so könnte man hier schon fragen: Und was schließt Du unter
dem Begriff ”Zeichen“ ein?
62
Was nenne ich ”Handlung“, was ”Sinneswahrnehmung“?
63
Die Worte ”Welt“, ”Erfahrung“, ”Sprache“, ”Satz“, ”Kalkül“, ”Mathematik“ können
alle nur für triviale Abgrenzungen stehen, wie ”essen“, ”ruhen“, etc.
64
Denn, wenn auch ein solches Wort der Titel unserer Grammatik wäre – etwa das
Wort ”Grammatik“ – so hätte doch dieser Titel nur dieses Buch von andern Büchern zu
unterscheiden.
65
Allgemeine Ausführungen über die Welt und die Sprache gibt es nicht.
66
Aber warum zerbreche ich mir über den Begriff ”Sprache“ den Kopf, statt Sprache zu
gebrauchen?!
Dieses Kopfzerbrechen ist nur dann berechtigt, wenn wir einen allgemeinen Begriff haben.
67
Ich finde bei Plato auf eine Frage wie ”was ist Erkenntnis“ nicht die vorläufige Antwort:
Sehen wir einmal nach, wie dieses Wort gebraucht wird. Sokrates weist es immer zurück,
von Erkenntnissen statt von der Erkenntnis zu reden.68
69
Aber wenn so der allgemeine Begriff der Sprache sozusagen zerfließt, zerfließt da nicht
auch die Philosophie? Nein, denn ihre Aufgabe ist es nicht etwas Neues an Stelle unserer70
Sprache
zu setzen sondern bestimmte71
Mißverständnisse in unserer Sprache zu beseitigen.72
73
Der, welcher darauf aufmerksam macht, daß ein Wort in zwei verschiedenen
Bedeutungen gebraucht wurde, oder daß bei dem Gebrauch eines74
Ausdrucks uns dieses
Bild vorschwebt, und der überhaupt die Regeln feststellt (tabuliert), nach welchen Worte
gebraucht werden, hat garnicht die Pflicht übernommen,75
eine Erklärung (Definition) des
Wortes ”Regel“ (oder ”Wort“, ”Sprache“, ”Satz“, etc.) zu geben.
76
So ist es mir erlaubt, das Wort ”Regel“ zu verwenden, ohne notwendig erst die Regeln
über dieses Wort zu tabulieren. Und diese Regeln sind nicht Über-Regeln.
77
Die Philosophie hat es auch in demselben Sinn mit Kalkülen zu tun, wie sie es mit
Gedanken zu tun hat (oder mit Sätzen und Sprachen). Hätte sie’s aber wesentlich mit dem
Begriff des Kalküls zu tun, also mit dem Begriff des Kalküls vor allen Kalkülen, so gäbe es
eine Metaphilosophie. Und die gibt es nicht. (Man könnte alles, was wir zu sagen haben, so
darstellen, daß das als ein leitender Gedanke erschiene.)
78
Das Wort ”Regel“ muß in der Erklärung eines Spiels nicht gebraucht werden (natürlich
auch kein äquivalentes).
62 (M): ///
63 (M): /
64 (M): ? /
65 (M): ///
66 (M): ////
67 (M): ( 3
68 (M): )
69 (M): ? / 3
70 (V): der
71 (V): einzelne
72 (V1): Sprache aufzuklären. (V2): Nein, denn
ihre Aufgabe ist es nicht, eine neue ideale
Sprache zu schaffen, sondern die zu reinigen,
die vorhanden ist.
73 (M): ? / 3
74 (V): dieses
75 (V): hat gar keine Pflicht,
76 (M): ? / 3
77 (M): ? / 3
78 (M): 555
67
66v
67
54 ”Satz“ und ”Sprache“ verschwimmende Begriffe.
WBTC015-21 30/05/2005 16:18 Page 108
‘language’”, then at that point one might ask: And what are you including in the concept
“sign”?
60
What do I call an “act”, what a “sense-perception”?
61
Each of the words “world”, “experience”, “language”, “proposition”, “calculus”,
“mathematics” can stand only for trivial demarcations, similar to “eat”, “rest”, etc.
62
For even if such a word were the title of our grammar – say the word “grammar” – then
still this title would only serve to distinguish this book from other books.
63
There are no such things as general discourses about the world and language.
64
But why am I wracking my brains about the concept “language” instead of using
language?!
This wracking of brains is only justified if we have a general concept.
65
In Plato when a question like “What is knowledge?” gets asked, I don’t find as a provisional
answer: “Let’s look and see how this word is used”. Socrates always rejects talking
about particular instances of knowing, in favour of talking about knowledge.66
67
But if the general concept of language thus dissolves, so to speak – doesn’t philosophy
dissolve as well? No, for its task isn’t to put something new in place of our language, but to remove
certain misunderstandings from it.68
69
Someone who points out that a word was used with two different meanings, or that this
picture is before our inner eye when an70
expression is used, and who in general ascertains
(tabulates) the rules in accordance with which words are used, has in no way assumed
the obligation71
of providing an explanation (definition) of the word “rule” (or “word”,
“language”, “proposition”, etc.).
72
Therefore I’m allowed to use the word “rule” without first having to tabulate the rules
for this word. And these rules are not super-rules.
73
Philosophy has to do with calculi in the same sense as it has to do with thoughts (or
with propositions and languages). But if it had to do essentially with the concept of a
calculus, i.e. with the concept of a calculus before there were any individual calculations,
then there would be a metaphilosophy. And that doesn’t exist. (Everything we have to say
could be presented so that this appeared as a guiding thought.)
74
The word “rule” doesn’t have to be used in explaining a game (and of course neither
does any equivalent word).
60 (M): ///
61 (M): /
62 (M): ? /
63 (M): ///
64 (M): ////
65 (M): ( 3
66 (M): )
67 (M): ? / 3
68 (V1): but to clarify individual misunderstandings
in it. (V2): No, for its task isn’t to create a
new ideal language but to cleanse the one that
we have.
69 (M): ? / 3
70 (V): this
71 (V): has no obligation at all
72 (M): ? / 3
73 (M): ? / 3
74 (M): 555
“Sentence” and “Language” Blurred Concepts. 54e
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79
Wie gebrauchen wir denn80
das Wort ”Regel“, wenn wir etwa von Spielen reden?81
Im
Gegensatz wozu? Wir sagen z.B. ”das folgt aus dieser Regel“, aber dann könnten wir ja die
Regel des Spiels zitieren, und so das Wort ”Regel“ ersetzen. Oder wir sprechen von ”allen
Regeln des Spiels“ und müssen sie dann entweder aufgezählt haben (und dann liegt wieder82
der erste Fall vor), oder wir sprechen von den Regeln als einer Gruppe,83
die auf bestimmte
Art aus bestimmten Grundregeln84
erzeugt werden und dann steht das Wort ”Regel“ für den
Ausdruck dieser Grundregeln85
und Operationen. Oder wir sagen ”Das ist eine Regel, das nicht“,
wenn etwa das Zweite nur ein einzelnes Wort ist, oder eine Konfiguration der Spielsteine.
(Oder: ”nein, das ist nach der neuen Abmachung auch eine Regel“.) Wenn wir etwa das
Regelverzeichnis des Spiels aufzuschreiben hätten, so könnte so etwas gesagt werden und
dann hieße es: Das gehört hinein, das nicht. Aber nicht vermöge einer bestimmten
Eigenschaft (nämlich der, eine Regel zu sein), wie wenn man etwa lauter Äpfel in eine Kisten
packen möchte und sagt ”nein, das gehört nicht hinein, das ist eine Birne“.
Ja aber wir nennen doch manches ”Spiel“, manches nicht, und manches ”Regel“, und
manches nicht! Aber auf die Abgrenzung alles dessen, was wir Spiel nennen gegen86
alles
andere, kommt es ja nie an. Die Spiele sind für uns die Spiele, von denen wir gehört haben,
die wir aufzählen können, und etwa noch einige nach Analogie anderer neu gebildete; und
wenn jemand etwa ein Buch über die Spiele schriebe, so brauchte er eigentlich das Wort
”Spiel“ auch im Titel nicht, sondern als Titel könnte eine Aufzählung der Namen der einzelnen
Spiele stehen. Und gefragt: Was ist denn aber das Gemeinsame aller dieser Dinge, dessent-
wegen87
Du sie zusammenfaßt? könnte er sagen: ich weiß es nicht in einem Satz anzugeben,
– aber Du siehst ja viele Analogien. Im übrigen scheint mir diese88
Frage müßig, da ich auch
wieder, nach Analogien fortfahrend, durch unmerkbare Stufen, zu Gebilden kommen kann,
die niemand mehr im gewöhnlichen Leben ”Spiel“ nennen wollte. Ich nenne daher ”Spiel“
das, was auf dieser Liste steht, wie auch, was diesen Spielen bis zu einem gewissen (von mir
nicht näher bestimmten) Grade ähnlich ist.89
Im übrigen behalte ich mir vor, in jedem neuen
Fall zu entscheiden, ob ich etwas zu den Spielen rechnen will oder nicht.
90
Es ist, wie wenn man für gewisse Spiele einen Strich mitten durchs Spielfeld zieht um die
Parteien zu scheiden, das Feld aber im Übrigen91
nicht begrenzt, da es nicht nötig ist.
Wenn Frege sagt, mit unscharfen Begriffen wisse die Logik nichts anzufangen so ist das insofern
wahr,92
als gerade die Schärfe der Begriffe zur Methode der Logik gehört. Das ist es was der
Ausdruck, die93
Logik sei normativ, bezeichnen kann.
94
Und so verhält es sich mit dem Begriff &Regel“.95
Nur in ganz besonderen96
Fällen d.h.: nicht
immer, wenn wir das Wort &Regel“ gebrauchen handelt es sich uns darum, die Regeln von etwas
abzugrenzen, was nicht Regel ist, und in allen diesen Fällen ist es leicht, ein unterscheidendes
Kriterium zu geben. Das heißt, wir brauchen das Wort ”Regel“ im Gegensatz zu ”Wort“,
”Konfiguration der Steine“ und einigem Andern, und diese Grenzen können leicht klar
79 (M): ? / 3
80 (V): denn auch
81 (V): ”Regel“ (wenn wir . . . reden)?
82 (V): (wieder)
83 (V): Regeln, als einer Gruppe,
84 (V): aus gegebenen Grundpositionen
85 (V): Grundpositionen
86 (V): nennen, gegen
87 (V): weshalb
88 (V): übrigen ist diese
89 (M): Wohl auszulassen! schon anders & vielleicht
besser gesagt.
90 (M): /
91 (V): aber weiter
92 (V): insofern eine die Wahrheit,
93 (V): Ausdruck ( die
94 (M): ? /
95 (V): Ebenso verhält es sich nun auch mit dem
Begriff der Regel.
96 (V): speziellen
68
68v
69
69
55 ”Satz“ und ”Sprache“ verschwimmende Begriffe.
WBTC015-21 30/05/2005 16:18 Page 110
75
How do we76
use the word “rule” when we talk about games, for instance?77
In contrast
to what? We say, for example, “This follows from that rule”, but in that case we could cite
the rule itself and thus get rid of the word “rule”. Or we speak of “all the rules of the game”,
and then we must either have enumerated them (and then it’s another instance of the first
case), or we speak of the rules as a group that are generated in a certain way from certain
basic rules,78
and then the word “rule” stands for the expression of these basic rules79
and
operations. Or we say “This is a rule, that isn’t,” if, perhaps, the latter is just a single word
or a configuration of game pieces. (Or: “No, according to our new understanding this is a
rule too.”) If for example we were supposed to write down the list of the rules of a game,
then something like this might be said, and then our list would read: This belongs here, that
doesn’t. But not by virtue of a particular property (i.e. that of being a rule), as is the case
for example when one wants to pack a crate with apples and says “No, this doesn’t belong
here, it’s a pear”.
Yes, but we do call some things “games” and some things not, and some things “rules”
and others not! But differentiating everything we call a game from everything else is never
important. For us games are the games we have heard of, that we can enumerate, and perhaps
some other ones, newly formed by analogy to ours. And if perhaps someone were to
write a book about games, he wouldn’t actually need the word “game”, not even in the title;
rather, an enumeration of the names of the individual games could serve as the title. And if
asked: But what is it that all of these things have in common, the reason you’re bringing them
together, he might say: I don’t know how to state it in a sentence – but surely you see many
analogies. Furthermore, this question strikes me as80
pointless, since, by using analogies and
proceeding by imperceptible steps, I can come up with entities that no one in everyday life
would want to call “games”. Therefore I call what is on this list a “game”, as well as what
is similar to these games up to a certain point (which I do not define more precisely). As for
the rest, I retain the right to decide in each new case whether I want to count something as
a game or not.81
82
It is as if for certain games one draws a line right through the middle of the playing field in order
to separate the teams, but doesn’t otherwise83
mark off the field, since this isn’t necessary.
When Frege says that logic doesn’t know what to do with vague concepts, this is true84
in so
far as it is precisely the sharpness of concepts that belongs to the method of logic. That is what the
expression “Logic is normative” can refer to.
85
And this is the way it is with the concept “rule”.86
Only in very particular87
cases – i.e.
not every time we use the word “rule” – are we concerned with differentiating rules from
something that isn’t a rule, and in all of these cases it is easy to come up with a distinguishing
criterion. For example, we use the word “rule” as opposed to “word”, “configuration of game
pieces” and several other things, and it’s easy to draw these boundaries clearly.88
On the other
75 (M): ? / 3
76 (V): we also
77 (V): “rule” (when we talk about games, for
instance)?
78 (V): from basic positions that are given,
79 (V): basic positions
80 (V): question is
81 (M): Better to leave out! Already expressed differently
and possibly better.
82 (M): /
83 (V): but beyond that doesn’t
84 (V): is a the truth
85 (M): ? /
86 (V): Now this is exactly the way it is with the
concept of rule.
87 (V): special
88 (V): and these boundaries are clearly drawn.
“Sentence” and “Language” Blurred Concepts. 55e
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gezogen werden.97
Dagegen ziehen wir dort nicht Grenzen, wo98
wir sie nicht brauchen. Wir
gebrauchen das Wort &Pflanze“99
in bestimmtem Sinne, aber, im Falle einzelliger Lebewesen
war die Frage eine Zeit lang schwebend, ob man sie Tiere oder Pflanzen nennen solle, und
es ließen sich auch beliebig viel andere Grenzfälle konstruieren, für die die Entscheidung,
ob etwas noch unter den Begriff Pflanze falle, erst zu treffen wäre. Ist aber darum die Bedeutung
des Wortes ”Pflanze“ in allen anderen Fällen verschwommen, sodaß man sagen könnte, wir
gebrauchen das Wort, ohne es zu verstehen? Ja, würde uns eine Definition, die den Begriff
nach verschiedenen Seiten begrenzte, die Bedeutung des Wortes in allen Sätzen klarer machen,
sodaß wir auch alle Sätze, in denen es vorkommt, besser verstehen würden? Offenbar nein.
Wenn wir sagen &der Boden war ganz mit Pflanzen bedeckt“ so meinen wir gewöhnlich nicht Bakterien
(d.h. wir würden diese Deutung wenn sie vorgeschlagen würde, ablehnen). Wir würden, müßten wir
bestimmte Grenzen ziehen, in den verschiedenen Fällen wenn wir das Wort im gewöhnlichen Leben
gebrauchen verschiedene Grenzen ziehen. Und manchmal müßten wir auch Grenzen andeuten.
&ein großes Stück Kuchen“, &ein großer Kirchturm“, &ein großer Hund“
Die Logik zieht ihrem Wesen nach Grenzen aber in der Sprache die wir sprechen sind solche Grenzen
nicht gezogen. Das heißt aber nicht daß nun die Logik die Sprache falsch darstellt oder eine ideale
Sprache. Sie portraitiert die Farbige verschwommene Wirklichkeit als Federzeichnung das ist ihre
Aufgabe.
100
(Sokrates stellt die Frage, was Erkenntnis sei und ist nicht mit der Aufzählung von
Erkenntnissen zufrieden. Wir aber kümmern uns nicht viel um diesen allgemeinen Begriff
und sind froh, wenn wir Schuhmacherei, Geometrie etc. verstehen.)101
102
Wir glauben nicht, daß nur der ein Spiel wirklich versteht, der eine Definition des Begriffs
”Spiel“ geben kann.
103
(Ich mache es mir in der Philosophie immer leichter und leichter. Aber die
Schwierigkeit ist, es sich leichter zu machen und doch exakt zu bleiben.)
104
Der Gebrauch des Worts &Spiel“ &Satz“,105
&Sprache“ etc. hat die Verschwommenheit des
normalen Gebrauchs aller Begriffswörter unserer Sprache. Zu glauben sie wären darum unbrauchbar
oder doch nicht ideal ihrem Zweck entsprechend wäre, als wollte man sagen &der Lichtschein meiner
Lampe ist unbrauchbar,106
weil man nicht weiß, wo er107
anfängt & wo er108
aufhört“.
109
Will ich zur Aufklärung & Vermeidung110
von Mißverständnissen im Gebiet eines (solchen) verschwommenen
Sprachgebrauchs scharfe111
Grenzen ziehen, so werden sich die scharf umgrenzten
Bezirke zu dem wirklichen Sprachgebrauch verhalten wie112
Konturen in einer Federzeichnung zu
allmählichen Übergängen von Farbflecken in der Wirklichkeit die dargestellt ist.113
69
56 ”Satz“ und ”Sprache“ verschwimmende Begriffe.
97 (V): Grenzen sind klar gezogen.
98 (V): Dagegen ist es müßig, Grenzen dort zu
ziehen, wo
99 (V): brauchen. Verhält es sich hier nicht
ebenso, wie mit dem Begriff @Pflanze!? Wir
gebrauchen dieses das Wort
100 (M): ( 3
101 (M): )
102 (M): /
103 (M): ? / 3
104 (M): /
105 (V): des Worts &Satz“,
106 (V): sagen &. . . ist unbrauchbar,
107 (V): es
108 (V): es
109 (M): ?
110 (V): zur Aufklärung & zur Vermeidung
111 (V): verschwommenen Begriffs klare
112 (V): wie die scharfen
113 (V): zu den allmählichen Farbübergängen im
Gesicht in der Wirklichkeit die sie darstellt. // zu
allmählichen Übergängen von Farbflecken in der
Wirklichkeit die die Zeichnung darstellt.
68v
70
WBTC015-21 30/05/2005 16:18 Page 112
hand, we don’t draw boundaries where89
we don’t need them. We use the word “plant”90
in a
specific sense, but in the case of unicellular organisms the question whether one should call
them animals or plants was undecided for a while, and one could also construct any number
of additional borderline cases, in which whether something belonged to the concept “plant”
would have to be decided. But does that make the meaning of the word “plant” blurred in
all other cases, so that it could be said that we were using the word without understanding
it? Indeed, would a definition that delimited the concept in various ways clarify the meaning
of the word in all propositions, so that we would better understand all the propositions
in which it appeared? Obviously not.
If we say “The ground was completely covered with plants”, then usually we don’t mean bacteria.
(That is, we would reject this interpretation if it were suggested to us). If we had to draw definite
boundary lines we would draw different ones for the different instances of the everyday use of the
word. And sometimes we’d have to imply boundary lines.
“A big piece of cake”, “a big steeple”, “a big dog”.
It is essential to logic to draw boundaries, but no such boundaries are drawn in the language we
speak. But this doesn’t mean that logic represents language incorrectly, or that it represents an ideal
language. Its task is to portray a colourful, blurred reality as a pen-and-ink drawing.
91
(Socrates asks the question what knowledge is and he isn’t content with an enumeration
of instances of knowledge. But we don’t pay much attention to that general concept, and are
happy if we understand shoe-making, geometry, etc.)92
93
We don’t believe that only someone who can provide a definition of the concept “game”
really understands a game.
94
(I’m making it easier and easier for myself in philosophy. But the difficulty is to make
it easier for oneself and yet to remain precise.)
95
The use of the words “game”, “proposition”,96
“language” etc. has the blurriness of the
normal use of all concept-words in our language. To believe that therefore they are useless or in any
case don’t ideally match up to their purpose is like saying “The light of my lamp is useless97
because
one doesn’t know where it begins and where it ends”.
98
If I want to draw sharp99
boundaries in the area of (such) blurred language use100
in order to
clear things up and to avoid misunderstandings, then the sharply demarcated areas will relate to real
language use like the101
contours of a pen-and-ink drawing to the gradual transitions of colour patches
in the reality that has been sketched.102
89 (V): hand, it is pointless to draw boundaries
where
90 (V): them. Isn’t it the same here as with the
concept #plantF? We use this that word
91 (M): ( 3
92 (M): )
93 (M): /
94 (M): ? / 3
95 (M): /
96 (V): of the word “proposition”,
97 (V): saying “. . . is useless
98 (M): ?
99 (V): clear
100 (V): of (such) a blurred concept
101 (V): the sharp
102 (V): to the gradual transitions of colour in the face
in the reality that it sketches. // to gradual transitions
of colour patches in the reality that the drawing
sketches.
“Sentence” and “Language” Blurred Concepts. 56e
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72
71v
16
Die Logik redet von Sätzen und
Wörtern im gewöhnlichen Sinn,
nicht von Sätzen und Wörtern in
irgend einem abstrakten Sinn.
1
Ich glaube nicht, daß die Logik in einem andern Sinne von Sätzen reden kann, als wir
für gewöhnlich tun, wenn wir sagen ”hier steht ein Satz aufgeschrieben“ oder ”nein, das
sieht nur aus wie ein Satz, ist aber keiner“, etc. etc.
2
Die Frage ”was ist ein Wort“ ist ganz analog der ”was ist eine Schachfigur“.
3
Wir reden natürlich von dem räumlichen und zeitlichen Phänomen der Sprache. Nicht
von einem unräumlichen und unzeitlichen Unding. Aber wir reden von ihr so, wie von
den Figuren des Schachspiels, von ihrem Gebrauch im Spiel, nicht von ihren physikalischen
Eigenschaften.4
5
Wir können in der Philosophie auch keine größere Allgemeinheit erreichen, als in dem,
was wir in Leben und Wissenschaft aussprechen.6
(D.h., auch hier lassen wir alles, wie es
ist.)
7
So ist eine aufsehenerregende Definition der Zahl nicht die8
Sache der Philosophie.
9
Die Philosophie hat es mit den bestehenden Sprachen zu tun und nicht vorzugeben, daß
sie von einer abstrakten Sprache handeln müsse.
Wir können leicht in der Untersuchung der Spr. & der Bed. dahin kommen zu denken,10
wir dürften
eigentlich nicht von Wörtern & Sätzen im ganz hausbackenen Sinn reden sondern von Wörtern etc.
in einem sublimierteren Sinn, abstrakteren Sinn. So als wäre ein bestimmter Satz nicht eigentlich was
irgend ein Mensch ausspricht sondern ein Idealwesen (die Klasse aller gleichbedeutenden Sätze
oder dergleichen.) Aber ist auch der Schachkönig von dem die Schachregeln handeln ein solches
Idealding ein abstraktes Wesen?
1 (M): / 3
2 (M): / 3
3 (M): ? / 3 (R): Gehört eigentl. zu:
&Verstehen“ kein Akt während des Redens etc.
4 (V): Schachspiels, indem wir Regeln für sie
tabulieren, nicht ihre physikalischen
Eigenschaften beschreiben.
5 (M): / 3
6 (V): sagen.
7 (M): /
8 (V): Zahl keine
9 (M): 555
10 (V): Wir fühlen beim Nachdenken über das Problem
// beim Studium des Problems // der Sprache & der
Bedeutung leicht die Versuchung anzunehmen //
Wir können leicht dahin kommen zu denken,
71
57
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16
Logic Talks about Sentences
and Words in the
Ordinary Sense, not in
Some Abstract Sense.
1
I don’t think that logic can talk about sentences in any other sense than we ordinarily do
when we say “Here’s a sentence that’s been written down” or “No, that only looks like a
sentence but isn’t one”, etc. etc.
2
The question “What is a word?” is completely analogous to “What is a chess piece?”.
3
Of course we are talking about the spatial and temporal phenomenon of language. And not
about a non-spatial, non-temporal absurdity. But we are talking about it as we do about the
pieces in a chess game, about its use in the game, and not about its physical properties.4
5
In philosophy too we cannot achieve any greater generality than in what we express6
in
life and in science. (That is, here too we leave everything as it is.)
7
So a spectacular definition of “number” is not the business of philosophy.
8
Philosophy is concerned with existing languages and shouldn’t pretend that it has to deal
with an abstract language.
In investigating language and meaning we can easily get to the point of thinking9
that really we
shouldn’t talk about words and sentences in their quite ordinary sense, but about words etc. in a more
sublimated, more abstract, sense. As if a particular sentence were really not something uttered by a
human being, but were an ideal entity (the class of all synonymous sentences or some such thing).
But is the chess king, with which the rules of chess deal, also such an ideal thing, an abstract entity?
1 (M): / 3
2 (M): / 3
3 (M): ? / 3 (R): Really belongs to: “Understanding”
isn’t an act that occurs while talking etc.
4 (V): chess game, by tabulating rules for it, not
by describing its physical properties.
5 (M): / 3
6 (V): say
7 (M): /
8 (M): 555
9 (V): In thinking about // studying // the problem
of language and meaning we are easily seduced
into assuming // We can easily get to the point of
thinking
57e
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11
Wenn ich nämlich über die Sprache – Wort, Satz, etc. – rede, muß ich die Sprache des
Alltags reden. – Aber gibt es denn eine andere?
12
Ist diese Sprache etwa zu grob, materiell, für das, was wir sagen wollen? Und kann es
eine andere geben? Und wie merkwürdig, daß wir dann mit der unseren überhaupt13
etwas
anfangen können.
14
Daß ich beim Erklären der Sprache (in unserem Sinne) schon die volle Sprache (nicht
etwa eine vorbereitende, vorläufige) anwenden muß, zeigt schon, daß ich nur Äußerliches
über die Sprache vorbringen15
kann.
16
Ja, aber wie können uns diese Ausführungen dann befriedigen? – Nun, Deine Fragen
waren ja auch schon in dieser Sprache abgefaßt; mußten in dieser Sprache ausgedrückt
werden, wenn etwas zu fragen war!
17
Und Deine Skrupel sind Mißverständnisse.
18
Deine Fragen beziehen sich auf Wörter, so muß ich von Wörtern reden.19
20
Man sagt: Es kommt doch nicht auf’s21
Wort an, sondern auf seine Bedeutung, und denkt
dabei immer an die Bedeutung, als ob sie nun eine Sache von der Art des Worts wäre,
allerdings vom Wort verschieden. Hier ist das Wort, hier die Bedeutung. (Das Geld, und
die Kuh, die man dafür kaufen kann. Anderseits aber: das Geld, und sein Nutzen.)
22
Über unsere Sprache sind nicht mehr Bedenken gerechtfertigt,23
als ein Schachspieler über das
Schachspiel hat, nämlich keine. 24
((Hier ist nicht gemeint ”über den Begriff der Sprache“.
Sondern es heißt eher: ”sprich ruhig darauf los, wie ein Schachspieler spielt, es kann Dir
nichts passieren, Deine Skrupel sind ja nur Mißverständnisse, ,philosophische‘ Sätze.“))
11 (M): / 3
12 (M): / 3
13 (V): dennoch
14 (M): / 3 (
15 (V): sagen
16 (M): / 3
17 (M): / 3
18 (M): / 3
19 (M): )
20 (M): / 3
21 (V): auf das
22 (M): / 3
23 (V): Über die Sprache des Alltags sind nicht mehr
Skrupeln Bedenken berechtigt,
24 (M): 555
72
73
58 Die Logik redet von Sätzen . . .
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10
For when I speak about language – word, sentence, etc. – I have to speak in everyday
language. – But is there any other?
11
Is this language perhaps too coarse, too material, for what we want to say? And can there
be another one? And in that case how remarkable it is that we can do anything at all12
with
ours.
13
That in explaining language (in our sense) I already have to use full-blown language (and
not, say, a preparatory, provisional one) shows that all I can do is to present14
external facts
about language.
15
Yes, but how can these elaborations then satisfy us? – Well, after all, your questions too
were formulated in this language. They had to be expressed in this language for something
to be asked!
16
And your scruples are misunderstandings.
17
Your questions refer to words, so I have to talk about words.18
19
It is said: It’s not the word that counts, but its meaning, and in saying this one always
thinks of meaning as if it were a thing of the same kind as the word, yet different from it.
Here is the word, here the meaning. (Money, and the cow that one can buy with it. But on
the other hand: money and its profit.)
20
No more misgivings are justified about our language21
than a chess player has about chess
games, namely none. 22
( (This doesn’t mean “about the concept of language”. It just means:
“Go ahead and talk away, just as a chess player plays. Nothing can happen to you. Your
scruples, after all, are only misunderstandings, ‘philosophical’ propositions.”) )
10 (M): / 3
11 (M): / 3
12 (V): that nevertheless we can do something
13 (M): / 3 (
14 (V): state
15 (M): / 3
16 (M): / 3
17 (M): / 3
18 (M): )
19 (M): / 3
20 (M): / 3
21 (V): No more misgivings scruples are justified
about our everyday language
22 (M): 555
Logic Talks About Sentences and Words in the Ordinary Sense . . . 58e
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17
Satz und Satzklang.
Was ist ein Satz? – Vor allem gibt es in unseren Sprachen einen Satzklang. (Daher Unsinngedichte
wie die Lewis Carroll’s).1
Und was wir oft Unsinn nennen ist nicht ein beliebiger.2
3
Bei der Frage nach der allgemeinen Satzform bedenken wir, daß die gewöhnliche
Sprache zwar einen bestimmten Satzrhythmus hat, aber nicht alles, was diesen Rhythmus
hat, ein Satz ist.
D.h. wie ein Satz klingt und keiner ist. – Daher die Idee vom sinnvollen und unsinnigen ”Satz“.
4
Anderseits ist dieser Rhythmus aber natürlich nicht wesentlich. Der Ausdruck ”Zucker
Tisch“ klingt nicht wie ein Satz, kann aber doch sehr wohl den Satz ”auf dem Tisch liegt
Zucker“ ersetzen. Und zwar nicht etwa so, daß wir uns etwas Fehlendes hinzudenken müßten,
sondern, es kommt wieder nur auf das System an, dem der Ausdruck ”Zucker Tisch“
angehört.
5
Es fragt sich also, ob wir außer diesem irreführenden Satzklang noch einen allgemeinen
Begriff vom Satz haben. (Ich rede jetzt von dem, was durch ”•“, ”1“, ”⊃“, zusammengehalten
wird.)
6
Denken wir uns, wir läsen die Sätze eines Buches verkehrt, die Worte in umgekehrter
Reihenfolge; könnten wir nicht dennoch den Satz verstehen? Und klänge er jetzt nicht ganz
unsatzmäßig?
7
Hat es Sinn,8
zu sagen: ”Ich habe so viele Schuhe, als eine Lösung der Gleichung x3
+ 2x
− 3 = 0 ergibt“.9
Hier könnte es scheinen, als hätten wir eine Notation, deren Grammatik allein
nicht bestimmt was ein sinnvoller Satz ist & was nicht.10
Daß es also von vornherein nicht bestimmt
wäre.
Wenn der Ausdruck ”die Wurzel der Gleichung F(x) = 0“ eine Beschreibung im
Russell’schen Sinne wäre, so hätte der Satz ”ich habe n Äpfel und n + 2 = 6“ einen andern
Sinn, als der: ”ich habe 4 Äpfel“.
Wir haben in dem ersten Satz ein außerordentlich lehrreiches Beispiel dafür, wie eine
Notation auf den ersten Blick einwandfrei erscheinen kann, nämlich so, als verstünden wir
sie; und daß wir in Wirklichkeit einen unsinnigen Satz nach Analogie eines sinnvollen gebildet
haben und nur glauben, die Regeln des ersteren zu übersehen. So ist ”ich habe n Schuhe
und n2
= 4“ ein sinnvoller Satz; aber nicht ”ich habe n Schuhe und n2
= 2“.
1 (O): Carolls’) (V): Carolls’) Daher reden wir
von Unsinng
2 (O): nicht eine Beliebige.
3 (M): ü / 3
4 (M): a ? / 3
5 (M): / 3
6 (M): ∫ 3
7 (M): / 3 (R): Zu § 18 S. 76 § 19 S. 79
8 (V): Hat es einen Sinn,
9 (V): als eine Wurzel der Gleichung x3
+ 2x − 3
= 0 Einheiten hat“?
10 (V): bestimmt ob ob ein Satz Sinn hat oder nicht. //
Notation, der wir es eventuell nicht ansehen können,
ob sie Sinn hat oder nicht.
74
75, 74v, 75
74v, 75, 74v
75
75
75
59
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17
Sentence and Sentence-Sound.
What is a sentence? – First of all, there is a sound to the sentences in our languages. (Hence nonsense
poems like those of Lewis Carroll.)1
And often what we call nonsense is not something that’s
arbitrary.
2
In dealing with the question of the general form of a proposition, we should bear in mind
that ordinary language does have a particular sentence-rhythm, but that not everything that
has this rhythm is a sentence.
That is, it sounds like a sentence but isn’t one. – Hence the idea of a meaningful and a
nonsensical “sentence”.
3
But on the other hand, this rhythm is of course inessential. The expression “sugar table”
doesn’t sound like a sentence, yet it can easily replace the sentence “Sugar is lying on the
table”. And it’s not necessary that mentally we supply something that is missing; rather, once
again it all depends on the system that the expression “sugar table” belongs to.
4
So the question arises whether, aside from this misleading sound of a sentence, we have
another general concept of a proposition. (Now I’m talking about what is connected by “&”,
“1”, “⊃”.)
5
Let’s imagine reading the sentences in a book backwards, the words in reverse order;
couldn’t we still understand the sentence? And wouldn’t it now sound completely unlike a
sentence?
6
Does it make sense to say: “I have as many shoes as the solution to the equation7
x3
+ 2x
− 3 = 0”? Here it might seem that we have a notation whose grammar alone doesn’t determine
what is a meaningful proposition and what isn’t.8
So that this wouldn’t be determined in advance.
If the expression “the root of the equation F(x) = 0” were a description in Russell’s sense,
then the proposition “I have n apples and n + 2 = 6” would have a different sense from: “I
have 4 apples”.
In the first sentence we have an extraordinarily instructive example of how a notation can
seem flawless at first sight, i.e. understandable; whereas in reality we have formed a nonsensical
sentence analogously to one that makes sense, and we only believe we have an overview
of the rules for the former. Thus “I have n shoes and n2
= 4” is a meaningful sentence, but
not: “I have n shoes and n2
= 2”.
1 (V): Carroll.) Hence we talk of nonsen
2 (M): r / 3
3 (M): a ? / 3
4 (M): / 3
5 (M): ∫ 3
6 (M): / 3 (R): To § 18 p. 76 §19 p. 79
7 (V): shoes as there are roots of the equation
8 (V): determine whether whether a proposition has
sense or not. // notation which, when we look at
it, we may not be able to tell whether it has sense
or not.
59e
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11
Dies gibt ein herrliches Beispiel dafür, was es heißt, einen Satz verstehen (meinen).12
13
Inwiefern ist das Verstehen – das augenblickliche Verstehen – des Satzes ein Kriterium dafür, daß
der Satz Sinn hat?
60 Satz und Satzklang.
11 (M): /
12 (V): einen Satz verstehen.
13 (M): /
74v
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9
This provides a splendid example of what it means to understand (to mean) a proposition.10
11
To what extent is understanding – immediate understanding – of a proposition a criterion for its
making sense?
9 (M): /
10 (V): means to understand a proposition.
11 (M): /
Sentence and Sentence-Sound. 60e
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18
Was als Satz gelten soll,
ist in der Grammatik bestimmt.1
2
Die Erklärung: &Satz ist alles, was wahr oder falsch sein kann“3
bestimmt den Begriff des
Satzes in einem bestimmten Sprachsystem als das was in diesem System Argument einer
Wahrheitsfunktion ist.4
Und wenn wir von dem sprechen, was der Satzform als solcher wesentlich ist so sind es
manchmal5
die Wahrheitsfunktionen. Wenn ich sagte die allgemeine Form des Satzes sei &es verhält
sich so & so“ so war eben das gemeint.
6
”p“ ist wahr = p. Man gebraucht das Wort ”wahr“ in Zusammenhängen wie ”was er sagt
ist wahr“, das aber sagt dasselbe wie ”er sagt ’p‘, und p ist der Fall“.
7
”Wahr“ und ”falsch“ sind tatsächlich nur Wörter einer bestimmten Notation der
Wahrheitsfunktion.
8
Wenn man sagt, Satz sei alles, was wahr oder falsch sein könne, so heißt das dasselbe wie:
Satz ist alles, was sich verneinen läßt.
9
Wenn wir von dem sprechen, was der Satzform als solcher wesentlich ist, so meinen wir
die Wahrheitsfunktion.10
11
Man kann natürlich12
nicht sagen, ”Satz“ sei dasjenige, wovon man ”wahr“ und ”falsch“
aussagen könne, in dem Sinn, als könnte man versuchen, zu welchen Symbolen die Wörter
”wahr“ und ”falsch“ paßten und danach entscheiden, ob etwas ein Satz ist. Denn das würde
nur dann etwas bestimmen, wenn diese Worte in einer bestimmten Weise gemeint sind, d.h.
bereits eine bestimmte Grammatik haben.13
Und eben im Zusammenhang mit einem Satz.
Alles, was man machen kann, ist hier, wie in allen diesen Fällen, das grammatische Spiel
bestimmen, seine Regeln angeben und es dabei bewenden lassen.
1 (R): ∀ S. 75/1 ∀ Anfang des § 40 S. 171 & 170v.
3 ∀ S. 114/3 3
2 (M): ? /
3 (V): Die Erklärung: (Satz sei alles, was wahr oder
falsch sein könne
4 (V1): System als Argument einer Wahrheitsfunktion
auftritt. (V2): ∫ Die Erklärung:, die man
erhält, wenn man nach dem Wesen des Satzes
fragt: Satz sei alles, was wahr oder falsch
sein könne – ist nicht so ganz unrichtig. Es
ist die Form der Wahrheitsfunktion (in
welcher Form der Zeichengebung immer
ausgedrückt), die das logische Wesen des
Satzes ausmacht.
5 (V): so meinen wir oft
6 (M): ∫ 3
7 (M): ? / 3
8 (M): a ? /
9 (M): ∫ 3
10 (V): Wahrheitsfunktionen.
11 (M): ∫ 3
12 (V): natürlich auch
13 (V): gemeint sind, das aber können sie nur im
Zusammenhang sein.
76
75v
76
77
76
61
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18
What is to Count as a Proposition
is Determined in Grammar.1
2
The statement: “A proposition is everything that can be true or false”3
defines the concept of a
proposition, within a specific system of language, as what is4
an argument of a truth-function in that
system.5
And when we speak of what is essential to the form of a proposition as such, then sometimes it
is6
truth-functions. When I said that the general form of a proposition is “Such and such is the case”
that was precisely what I meant.
7
“p” is true = p. The word “true” is used in contexts such as “What he says is true”, but
that says the same thing as “He says ‘p’, and p is the case”.
8
“True” and “false” are in point of fact only words in a particular notation for the truth-
function.
9
When one says that a proposition is everything that can be true or false, that means the
same as: a proposition is everything that can be denied.
10
When we speak of what is essential to the form of a proposition as such, what we mean
is the truth-function.11
12
Of course one cannot13
say that a proposition is what one can declare to be “true” or
“false” in the sense in which one could try out which symbols the words “true” and “false”
fit and decide accordingly whether something is a proposition. For that would only decide
something if these words are meant in a certain way, i.e. already have a certain grammar.14
That is, in connection with a proposition. All one can do here, as in all of these cases, is to
define the grammatical game, state its rules and leave it at that.
1 (R): ∀ p. 75/1 ∀ beginning of § 40 p. 171 & 170v.
3 ∀ p. 114/3 3
2 (M): ? /
3 (V): The statement: AA proposition is everything
that could be true or false
4 (V): what makes its appearance as
5 (V): ∫ The statement:, that one gets when one
asks about the nature of a proposition: A
proposition is everything that can be true or false
– is not as completely incorrect as it seems. It
is the form of the truth-function (no matter what
its form of signification) that constitutes the
logical essence of a proposition.
6 (V) then we frequently mean
7 (M): ∫ 3
8 (M): ? / 3
9 (M): a ? /
10 (M): ∫ 3
11 (V): is truth-functions.
12 (M): ∫ 3
13 (V): also
14 (V): way, but they can only be meant in a certain
way in a context.
61e
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78
62 Was als Satz gelten soll . . .
14
Was ein Satz ist, wird durch die Grammatik bestimmt. D.h., innerhalb der Grammatik.
(Dahin zielte auch meine ”allgemeine Satzform“.)
15
Man kann nicht sagen ”dieser Struktur fehlt noch etwas, um ein Satz zu sein“. Sondern
es fehlt ihr etwas, um in dieser Sprache ein Satz zu sein. Man kann sagen:16
dem
Zeichenausdruck ”2 + 2 4“ fehlt etwas, um eine Gleichung zu sein.
17
Den Russen, welche statt ”er ist gut“ sagen ”er gut“ geht nichts verloren, und sie denken
sich auch kein Verbum dazu.
18
Den kompletten Satz zu charakterisieren ist so unmöglich, wie die komplette Tatsache.
19
Kann man den Begriff des ”Satzes“ festlegen? oder die allgemeine Form des Gesetzes?
– Warum nicht! Wie man ja auch den Begriff ”Zahl“ festlegen könnte, etwa durch das Zeichen
”|0, ξ, ξ+1|“. Es steht mir ja frei, nur das Zahl zu nennen; und so steht es mir auch frei,
eine analoge Vorschrift zur Bildung von Sätzen oder Gesetzen20
zu geben und das Wort ”Satz“
oder ”Gesetz“ [Ramsey] als ein Äquivalent dieser Vorschrift zu gebrauchen. Wehrt man sich
dagegen und sagt, es sei doch klar, daß damit nur gewisse Gesetze von andern abgegrenzt
worden seien, so antworte ich: Ja, Du kannst freilich nicht eine Grenze ziehen, wenn
Du von vornherein entschlossen bist, keine anzuerkennen! – 21
Sollen die ”Sätze“ den
unendlichen logischen Raum erfüllen, so kann von keiner allgemeinen Satzform die Rede
sein. Es fragt sich dann natürlich: Wie gebrauchst Du nun das Wort ”Satz“? im Gegensatz
wozu? – Etwa im Gegensatz zu ”Wort“, ”Satzteil“, ”Buchtitel“, Erzählung“, etc.
22
(Ein Satz, der von allen Sätzen oder allen Funktionen handelt. Was meint man damit?23
Denkt man an einen Satz der Logik?24
Denken wir nun daran, wie der Satz ~ ~p = p25
bewiesen
wird.)
26
Wenn ich ”es verhält sich so und so“ als allgemeine Satzform gelten lasse, dann muß
ich 2 + 2 = 4 unter die Sätze rechnen, denn es ist grammatisch richtig, zu sagen: ”es verhält
sich so, daß 2 + 2 gleich 4 ist“. Es braucht weitere Regeln, um die Sätze der Arithmetik
auszuschließen.
27
Falsche Ideen über das Funktionieren der Sprache: Broad, der sagte, etwas werde eintreffen,
sei kein Satz. Was spricht man dieser Aussage damit ab? Etwas anderes, als, daß sie
Gegenwärtiges oder Vergangenes beschreibt? – Die Magie mit Wörtern. Ein solcher Satz,
wie der Broads, kommt mir so vor, wie ein Versuch, eine chemische Änderung magisch zu
bewirken; indem man den Substanzen, quasi, zu verstehen gibt, was sie tun sollen (wenn
man etwa Eisen in Gold überführen wollte, indem man ein Stück Eisen mit der rechten und
zugleich ein Stück Gold mit der linken Hand faßte).
14 (M): ? / 3
15 (M): ∫ (R): [Zu: &Was ist ein Erfahrungssatz“]
16 (V): sein. Wie man sagen kann:
17 (M): ∫ ///
18 (M): ? ∫ (R): [Zu: &Was ist ein Erfahrungssatz“]
19 (M): / 3
20 (E): Möglicherweise wollte Wittgenstein die
untenstehende Tabelle, die sich auf der
Rückseite dieser Seite findet, hier einsetzen.
O’ a11 a12 a13 . . .
O’ a21 a22 a23 . . .
O’ a31 a32 a33 . . .
. . .
. . .
21 (M): 555
22 (M): / 3
23 (V): handelt. Was stellt man sich darunter vor?
24 (V): Es wäre wohl ein Satz der Logik.
25 (V): Satz ~2n
p = p
26 (M): ? / 3
27 (M): ? / (R): [Zu: &Was ist ein Erfahrungssatz“]
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15
What a proposition is is determined by grammar. That is, within grammar.
(This is also what my “general form of the proposition” was aiming at.)
16
One cannot say “This structure still needs something in order to be a proposition”. Rather,
it needs something in order to be a proposition in this language. One can say:17
the expression
“2 + 2 4” needs something in order to be an equation.
18
Russians who say “He good” instead of “He is good” lose nothing, nor do they mentally
add a verb.
19
It is as impossible to characterize a complete proposition as it is a complete fact.
20
Can one define the concept of a “proposition”, or the general form of laws? – Why not!
In the same way as one could also define the concept “number”, say by the sign “|0, ξ,
ξ + 1|”. I am free, after all, to call only that a number; and likewise I am free to issue an
analogous prescription for the formation of propositions or laws,21
and to use the word
“proposition” or “law” (Ramsey) as equivalent to this prescription. If someone resists this
and says that it’s perfectly clear that in so doing only certain laws have been marked off
from others, then I answer: Well, of course you can’t draw a boundary line if you are determined
from the outset not to acknowledge any! – 22
If “propositions” are to fill up infinite
logical space, then there can’t be any talk of a general propositional form. Then, of course,
the question arises: How do you now use the word “proposition”? In contrast to what? –
For instance, in contrast to “word”, “part of a proposition”, “book title”, “story”, etc.
23
(A proposition that is about all propositions or all functions. What does one mean24
by this? Is one thinking of a logical proposition?25
Let’s now think about how the proposition
~~p = p26
is proved.)
27
If I allow “This and that is the case” to count as the general form of the proposition,
then I have to include 2 + 2 = 4 among the propositions, for it is grammatically correct to
say “It is the case that 2 + 2 is equal to 4”. Additional rules are needed to exclude arithmetical
propositions.
28
False ideas about how language functions: Broad, who said that “Something will happen”
is not a proposition. What does one deny of this statement by saying that? Something
other than that it describes something present or past? – Magic with words. A proposition
such as Broad’s strikes me as being like an attempt to effect a chemical change by magic; by
letting the substances know, as it were, what they are supposed to do (as if one wanted to
transform iron into gold by grabbing a piece of iron with one’s right hand and at the same
time a piece of gold with the left).
15 (M): ? / 3
16 (M): ∫ (R): [To: “What is an empirical
proposition”]
17 (V): As one can say:
18 (M): ∫ ///
19 (M): ? ∫ (R): [To: “What is an empirical
proposition”]
20 (M): / 3
21 (E): Perhaps this is where Wittgenstein wanted
to insert the following table, which appears on
the back of this page:
O′ a11 a12 a13 . . .
O′ a21 a22 a23 . . .
O′ a31 a32 a33 . . .
. . .
. . .
22 (M): 555
23 (M): / 3
24 (V): imagine
25 (V): this? It probably would be a logical
proposition.
26 (V): proposition ~2n
p = p
27 (M): ? / 3
28 (M): ? / (R): [To: “What is an empirical
proposition”]
What is to Count as a Proposition . . . 62e
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19
Die grammatischen Regeln
bestimmen den Sinn des Satzes;
und ob eine Wortzusammenstellung
Sinn hat oder nicht.
1
Man könnte sagen: ”Wie mach ich’s denn, um ein Wort immer sinnvoll2
anzuwenden,
schau ich immer in der Grammatik nach? Nein, daß ich etwas meine – was ich meine, hindert
mich Unsinn zu sagen.“ Aber was meine ich denn? Ich sage: ich rede vom Teilen eines
Apfels, aber nicht vom Teilen der Farbe Rot, weil ich beim ”Teilen eines Apfels“ mir etwas
denken kann, etwas vorstellen, etwas wollen kann; beim Ausdruck ”Teilen einer Farbe“ nicht.
Oder ist es so,3
daß man bei diesem Wort nur noch keine Wirkung auf andere Menschen
beobachtet hat?! Richtiger wäre es zu sagen daß ich mir bei den Worten &Teilen eines A.“ etwas
denke, vorstelle, will; beim Ausdruck Teilen der Farbe rot nicht.
Wie mach ich’s denn, etwas mit ihm meinen? Ich stelle mir wohl etwas bei meinen Worten vor,4
will etwas mit ihnen,5
treibe etwas mit ihnen,6
kurz verwende sie in einem Sprachspiel.
Ich brauche das Wort zu einem Zweck & darum nicht unsinnig.
7
Was machen wir nun wenn wir der Wortgruppe &ich teile rot“ einen Sinn geben? Ja wir könnten
doch ganz verschiedenes aus ihr machen: Einen Satz der Arithmetik, einen Ausruf, einen
Erfahrungssatz;8
einen unbewiesenen Satz der Mathematik. Ich habe also eine beliebige Auswahl.
Und wie ist die begrenzt? Das ist schwer zu sagen: durch allerlei Arten von Nützlichkeit & auch durch
die Formelle Ähnlichkeit der Gebilde mit gewissen primitiven Satzformen & alle diese Grenzen sind
verschwimmend.
9
Der Satz &ich teile rot“ kann doch einen Sinn haben10
(z.B. kann er dasselbe sagen wie ich teile
etwas Rotes). Was, wenn ich fragte: welches11
Wort welcher Fehler macht den Satz zum Unsinn?
Warum soll es gerade das Wort &Rot“ sein? Da sieht man daß wir bei diesem Satz auch in seiner unsinnigen
Gestalt an ein ganz bestimmtes gramm. System denken.12
Daher sagen wir auch &rot kann man
nicht teilen“ geben also eine Antwort; während man auf eine Wortzusammenstellung wie &ist hat
gut“ nichts antworten würde. Denkt man nun aber an ein bestimmtes vorhandenes Sprachspiel13
&
1 (M): ü / 3 (R): ∀ S. 75/1 ∀ S. 64/1, 2, 3 3
2 (V): richtig
3 (V): nicht. Und ist es etwa so,
4 (V): wohl etwas vor,
5 (V): etwas damit,
6 (V): etwas damit,
7 (M): /
8 (V): Erfahrungssatz; etc.
9 (M): ü ? /
10 (V): kann ich doch einen Sinn geben
11 (V): fragte% welches
12 (V): an ein ganz bestimmtes System sinnvoller Sätze
denken.
13 (O): vorhandenes System Spr
79
78v
63
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19
Grammatical Rules Determine
the Sense of a Proposition; and
Whether a Combination of
Words Makes Sense.
1
One could ask: “How do I go about always using words meaningfully?2
Do I consult a
grammar each time? No, that I mean something – what I mean – prevents me from talking
nonsense.” But what do I mean? I say: I talk about dividing an apple but not about dividing
the colour red, because in the case of “dividing an apple” I can think something, can
imagine something, can want something; and I can’t with the expression “dividing a
colour”. Or is it3
just that one hasn’t yet observed any effect of the latter phrase on other
people?! It would be more correct to say that in the case of “dividing an apple” I think, imagine,
want something; in the case of “dividing the colour red”, I don’t.
How do I go about meaning something by a word? Most likely I imagine something when
using my words, want something with them, do something with them, in short use them in a
language-game.
I use the word for a purpose and therefore not without a sense.
4
What do we do to give a sense to the group of words “I divide red”? Well, we could turn it into
completely different things: an arithmetical proposition, an exclamation, an empirical proposition,5
an unproven mathematical proposition. Thus I have any number of choices. And how is this number
limited? That’s difficult to say: by various kinds of usefulness and also by the formal similarity of these
creations to certain primitive propositional forms; and all of these boundaries are fluid.
6
But the sentence “I divide red” can have a sense7
(for example, it can say the same thing as “I
divide something that is red”). What if I were to ask: Which8
word, which mistake, turns this
sentence into nonsense? Why should it be the word “red” and no other? Here one sees that even
when we encounter this sentence in its nonsensical form we think about a very specific grammatical
system.9
And that’s why we say “You can’t divide red”, i.e. why we give an answer; whereas we
would say nothing in response to a combination of words such as “is has good”. But if one thinks
1 (M): r / 3 (R): ∀ p. 75/1 ∀ p. 64/1, 2, 3 3
2 (V): correctly?
3 (V): And is it
4 (M): /
5 (V): proposition; etc.
6 (M): r ? /
7 (V): But I can give a sense to the sentence “I divide
red”
8 (V): ask) which
9 (V): a very specific system of meaningful sentences.
63e
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seine Anwendung dann sagt der Satz daß &ich teile rot“ unsinnig ist vor allem, daß er nicht zu dem
bestimmten Spiel gehört zu dem er seiner Erscheinung nach zu gehören scheint.
&Rot kann man nicht teilen“ heißt also: Erinnere Dich daran daß Du in dem Spiel zu welchem dieser
Satz seiner Form nach zu gehören scheint nichts anzufangen weißt.14
15
”Woher weiß ich, daß man16
Rot nicht teilen kann?“ – Die Frage selbst heißt nichts. Ich
möchte sagen: Man17
muß mit der Unterscheidung von Sinn und Unsinn anfangen. Vor ihr
ist nichts möglich. Ich kann sie nicht begründen.
18
Welcher Art19
sind die Regeln, welche sagen, daß die und die Zusammenstellungen von
Wörtern keinen Sinn haben? Sind sie von der Art derjenigen Vorschriften, welche20
sagen,
daß es keine Spielstellung im Schach ist, wenn zwei Figuren auf dem gleichen Feld stehen,
oder wenn eine Figur auf der Grenze zwischen zwei Feldern steht, etc.? Diese Sätze sind
wieder ähnlich gewissen21
Handlungen, wie Wenn man z.B. ein Schachbrett aus einem größeren
Stück eines karierten Papiers herausschnitte.22
Sie ziehen eine Grenze. – Was heißt es denn,
zu sagen: ”diese Wortzusammenstellung heißt nichts“?23
Von einem Namen kann man sagen
”diesen Namen habe ich niemandem gegeben“ und das Namengeben ist eine bestimmte
Handlung (wie das Umhängen eines Täfelchens).
Denken wir an eine24
Darstellung einer Reise auf der Erde durch eine Linie die in den Proj.
der zwei H. gezogen ist. 25
Wir können nun sagen:26
ein Linienstück, das auf der Zeichenebene
die Grenzkreise dieser27
Projektionen
verläßt, ist in dieser Darstellung sinnlos. D.h.:28
nichts ist darüber ausgemacht worden.
29
Gesichtsraum und Retina. Es ist, wie wenn man eine Kugel orthogonal auf eine Ebene
projiziert, etwa in der Art, wie die beiden Halbkugeln der Erde in einem Atlas dargestellt
werden, und nun könnte einer glauben, daß, was auf der Ebene außerhalb der beiden
Kugelprojektionen vor sich geht, immerhin noch einer möglichen Ausdehnung dessen
entspricht, was sich auf der Kugel befindet. Hier wird eben ein kompletter Raum auf einen
Teil eines andern Raumes projiziert; und analog ist es mit den Grenzen der Sprache im
Wörterbuch.30
14 (V): nichts anfangen kannst.
15 (M): ? /
16 (V): ich
17 (V): Ich
18 (M): / 3
19 (V): Art nun
20 (V): welche etwa
21 (V): wieder wie gewisse
22 (V): wie wenn man etwa ein Schachbrett
aus einem größeren Stück karierten Papiers
herausschneidet.
23 (O): nichts“.
24 (V): die
25 (F): MS 113, S. 30r.
26 (V): Linie in der Projektion der zwei
Halbkugeln und daß wir sagen:
27 (V): der
28 (V): sinnlos. Man könnte auch sagen:
29 (M): ? / (R): [Zu: &und auf gleiche Weise . . .“]
30 (V): Sprache in der Grammatik. (R): Für S.
124 MS?
79
80
79v, 80
64 Die grammatischen Regeln bestimmen den Sinn des Satzes . . .
79
78v
WBTC015-21 30/05/2005 16:18 Page 128
of a particular existing language-game and its use, then the proposition that says “‘I divide red’
is nonsensical” says more than anything else that it doesn’t belong to the particular game that,
judging from its appearance, it seems to belong to.
Thus “one can’t divide red” means: Remember that you don’t know what to do10
in the game to
which, based on its form, this proposition seems to belong.
11
“How do I know that one12
can’t divide red?” – The question itself means nothing. I’d
like to say: One has13
to begin with the distinction between sense and nonsense. Nothing is
possible before that distinction; nor can I justify it.
14
Of what kind15
are the rules that state that such and such combinations of words make
no sense? Are they of a piece with those regulations that16
say that two pieces on one square,
or one piece between two squares, etc., is not a position in chess? These propositions are
similar to17
certain actions – as if, for example, one were to cut a chess board18
out of a larger piece
of graph paper. These propositions draw a boundary. – What does it mean, anyway, to say:
“This combination of words means nothing”? Of a name one can say “I haven’t given this
name to anyone”, and name-giving is a specific action (like attaching a label).
Let’s think of a19
representation of a journey on earth by a line that is drawn in the
projections of the two hemispheres. 20
Now we can say:21
In this representation that part of the line that
extends beyond the boundaries of these22
circular projections on the surface of
the drawing is senseless. That is:23
nothing
about it has been agreed upon.
24
Visual space and retina. It is like projecting a sphere orthogonally onto a plane, say
in the way in which the two hemispheres of the earth are represented in an atlas, and now
someone could think that what is going on on the plane outside the two spherical projections
corresponds to a possible extension of what is on the sphere. Here it’s simply that a
complete space is projected onto a part of another space; and the boundaries of language in a
dictionary25
are analogous to this.26
10 (V): that you cannot do anything
11 (M): ? /
12 (V): I
13 (V): say: I have
14 (M): / 3
15 (V): kind then
16 (V): that perhaps
17 (V): are again like
18 (V): as if, for example, one cut a chess board
19 (V): the
20 (F): MS 113, p. 30r.
21 (V): a line in the projections of the two hemispheres,
and that we say:
22 (V): the
23 (V): One could also say:
24 (M): ? / (R): [To: “and in like manner . . .”]
25 (V): in grammar
26 (R): For p. 124 MS?
Grammatical Rules Determine the Sense of a Proposition . . . 64e
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20
Der Sinn des Satzes, keine Seele.
1
Die Methode des Messens, z.B. des räumlichen Messens, verhält sich zu einer bestimmten
Messung genau so, wie der Sinn eines Satzes zu seiner Wahr- oder Falschheit.
Der Sinn einer Längenangabe wird durch die Beschreibung der Meßmethode erklärt; die
Wahrheit der Längenangabe.
2
Der Sinn des3
Satzes ist nicht pneumatisch, sondern ist das, was auf die Frage nach der
Erklärung des Sinnes zur Antwort kommt. Und – oder – der eine Sinn unterscheidet sich
vom andern, wie die Erklärung des einen von der Erklärung des andern. Also auch: der Sinn
des einen Satzes unterscheidet sich vom Sinn des andern wie der eine Satz vom andern.
4
Welche Rolle der Satz im Kalkül spielt, das ist sein Sinn.
5
Der Sinn (also) nicht hinter ihm (wie der psychische Vorgang der Vorstellung etc.).
6
Was heißt es denn: ”entdecken, daß ein Satz keinen Sinn hat“?
Und was heißt das: ”wenn ich etwas damit meine, muß es doch Sinn haben“? Worin besteht
dieses Meinen?
”Wenn ich etwas damit meine. . . .“ – wenn ich was damit meine?!7
8
Was heißt es: ”Wenn ich mir etwas dabei vorstellen kann, muß es doch Sinn haben“?
Wenn ich mir was dabei vorstellen kann? Das, was ich sagte?9
– Dann heißt dieser Satz
nichts.10
– Und ”Etwas“? Das würde heißen: Wenn ich die Worte auf diese Weise benützen
kann, dann haben sie Sinn. Oder eigentlich: wenn ich sie zum Kalkulieren benütze, dann
haben sie Sinn.
Die Antwort wäre: wenn der Sinn ist daß ich mir etwas vorstelle. Aber es heißt wohl auch: Wenn
ich mir ein Bild danach machen kann so garantiert das mir andere Anwendungen.
11
Man könnte auch so fragen: Ist der ganze Satz nur ein unartikuliertes Zeichen, in dem
ich erst nachträglich Ähnlichkeiten mit anderen Sätzen erkenne?
1 (M): ∫ 3 2 (R): ∀ S. 75/1 siehe § 25 S. 93
2 (M): ? / 3 3
3 (V): eines
4 (M): ? / 3 1
5 (M): /// 3 5
6 (M): / 3
7 (R): [dazu S. 75/1] 3
8 (M): ü / 3
9 (V): sage?
10 (V): sagte? – Das heißt nichts.
11 (M): ∫ 3
81
80v
81
80v
81
82
65
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20
The Sense of a Proposition
not a Soul.
1
The method of measuring, e.g. of spatial measuring, is related to a particular measurement
in exactly the same way as the sense of a proposition is to its truth or falsity.
The sense of a statement of length is given by describing the method of measuring; the truth of
a statement of length.
2
The sense of the3
proposition isn’t soul-like; it’s what is given as an answer to a request
for an explanation of the sense. And – or – one sense differs from the other as does the explanation
of the one from the explanation of the other. Therefore: the sense of one proposition
differs from the sense of another as does the one proposition from the other.
4
Whatever role the proposition plays in the calculus, that is its sense.
5
(Therefore) its sense is not located behind it (like the psychological process of imagination,
etc.).
6
What does “Discovering that a proposition has no sense” mean?
And what does this mean: “If I mean something by it, then it must have sense”? What
does this meaning something consist in?
“If I mean something by it, . . .”. – if I mean what by it?!7
8
What does this mean: “If I can imagine something by it, then it must have a sense”?
If I can imagine what by it? What I said?9
– Then this proposition means10
nothing. – And
“something”? That would mean: If I can use the words in this way, then they have a sense.
Or actually: If I use them to calculate, then they have a sense.
The answer would be: If the sense is that I am picturing something, then the proposition does have
sense. But I suppose it also means: If I can use it as a model for a picture, then this guarantees that I
can apply it in other ways.
11
One could also put the question like this: Is the entire sentence nothing but an inarticulate
sign in which only later do I discover similarities with other sentences?
1 (M): ∫ 3 2 (R): ∀ p. 75/1 see §25 p. 93
2 (M): ? / 3 3
3 (V): a
4 (M): ? / 3 1
5 (M): /// 3 5
6 (M): / 3
7 (R): [To p. 75/1] 3
8 (M): r / 3
9 (V): say?
10 (V): – That means
11 (M): ∫ 3
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Als12
wäre er eine Flüssigkeit deren chemische Analyse uns erst13
gemeinsame Bestandteile mit
anderen Flüssigkeiten erkennen läßt.14
15
Ja, man könnte unsere Frage in einer sehr elementaren Form stellen: Warum eine Sprache
nicht mit bloß einem Wort auskommen könnte,16
da es ja doch vorkommt, daß ein Wort (in
einer Sprache) mehrere Bedeutungen hat. (Warum also nicht alle?)
66 Der Sinn des Satzes, keine Seele.
12 (V): Wie Als
13 (V): Analyse erst
14 (V1): mit anderen Substanzen aufzeigt. (V2):
Das wäre etwa so, wenn jeder Satz eine Droge
// Medizin // mit bestimmter Wirkung wäre
und man käme erst nachträglich durch Analyse
darauf, daß zwei Medizinen gewisse Ingredientien
mit einander gemein hätten.
15 (M): ? / ///
16 (V): Wort möglich ist,
81v
82
WBTC015-21 30/05/2005 16:18 Page 132
As if it were a liquid that required chemical analysis for us to12
recognize13
the ingredients it has in
common with other liquids.14
15
Indeed, one could ask our question in a very elementary form: Why couldn’t a language
make do with only one word,16
since a word (in a language) often has several meanings? (So
why not all of them?)
12 (V): analysis to
13 (V): show
14 (V): That would be similar to every proposition
being a drug // medicine // with a particular
effect, and our not discovering that two
medicines had certain ingredients in common
until we carried out an analysis.
15 (M): ? / ///
16 (V): form: Why isn’t a language with only one
word possible,
The Sense of a Proposition not a Soul. 66e
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21
Ähnlichkeit von Satz und Bild.
1
In welchem Sinne kann ich sagen, der Satz sei ein Bild? Wenn ich darüber denke, möchte
ich sagen: er muß ein Bild sein, damit er mir zeigen kann, was ich tun soll, damit ich mich
nach ihm richten kann. Aber dann willst Du also bloß sagen,2
daß Du Dich nach dem Satz
richtest in demselben Sinne, in dem Du Dich nach einem Bild richtest. Das Bild ist eine
Beschreibung.
3
Ist jedes Bild ein Satz? Und was heißt es, etwa zu sagen, daß jedes als ein Satz gebraucht
werden kann?
4
Ich kann die Beschreibung des Gartens in ein gemaltes Bild, das Bild in eine
Beschreibung übersetzen.
5
Zu sagen, daß der Satz ein Bild ist, hebt gewisse Züge in der Grammatik des Wortes
”Satz“ hervor.
6
Das Denken ist ganz dem Zeichnen7
von Bildern zu vergleichen.
Man kann aber auch sagen: Das Denken ist (wesentlich) mit keinem Vorgang zu vergleichen,
und was wie ein Vergleichsobjekt scheint, ist in Wirklichkeit ein Beispiel.
8
Wenn ich den Satz mit einem Maßstab verglichen habe, so habe ich, strenggenommen,
nur einen Satz, der mit Hilfe eines Maßstabes eine Länge aussagt, als Beispiel eines Satzes
herangezogen.9
10
Wenn man die Sätze als Vorschriften auffaßt, um Modelle zu bilden, wird ihre
Bildhaftigkeit noch deutlicher.
11
Die Sprache muß von der Mannigfaltigkeit eines Stellwerks sein, das die Handlungen
veranlaßt, die ihren Sätzen entsprechen.
12
Die Übereinstimmung von Satz und Wirklichkeit ist der Übereinstimmung zwischen
Bild und Abgebildetem nur so weit ähnlich, wie der Übereinstimmung zwischen einem
Erinnerungsbild und dem gegenwärtigen Gegenstand.
13
Der Satz ist der Tatsache so ähnlich wie das Zeichen ”5“ dem Zeichen ”3 + 2“. Und
das gemalte Bild der Tatsache, wie ”| | | | |“ dem Zeichen ”| | + | | |“.
1 (R): ∀ § 43 S. 189/1 3 & S 188v 3 ∀ S^ 289/1%2 ;
S. 217/1 3 (M): ///
2 (V): Aber) dann willst Du bloß sagen,
3 (M): ///
4 (M): ///
5 (M): vielleich unnütz / 3
6 (M): ∫ 3
7 (O): Zeichen
8 (M): 3
9 (V): Maßstabes die Länge eines Gegenstands
aussagt beschreibt, als Beispiel für alle Sätze
herangezogen.
10 (M): 555
11 (M): 555
12 (M): 555
13 (M): 555
83
82v
83
84
67
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21
Similarity of Proposition and Picture.
1
In what sense can I say that a proposition is a picture? If I think about it, I’m inclined
to say: It has to be a picture so that it can show me what I am supposed to do, so that I can
be guided by it. But2
then all you really want to say is that you are guided by the proposition
in the same sense in which you are guided by a picture. A picture is a description.
3
Is every picture a proposition? And what does it mean to say, for example, that every
picture can be used as a proposition?
4
I can translate the description of a garden into a painted picture, the picture into a
description.
5
Saying that a proposition is a picture emphasizes certain features of the grammar of the
word “proposition”.
6
Thinking is comparable in every way to drawing pictures.
But one can also say: (At bottom) thinking can’t be compared to any process, and what
looks like an object of comparison is really an example.
7
When I compared a proposition to a yardstick, then strictly speaking all I did was to
take a proposition that relies on a yardstick to state a length8
and use it as an example of a
proposition.9
10
If one conceives of propositions as instructions for making models, their picture-like
quality becomes even clearer.
11
Language has to have the multiple controls of a railway signal tower, arranging for those
actions that correspond to its propositions.
12
The agreement between proposition and reality resembles the agreement between a
picture and what it depicts only in so far as it resembles the agreement between a memoryimage
and the present object.
13
A proposition is as similar to a fact as the sign “5” is to the sign “3 + 2”. And a painted
picture is as similar to a fact as “|||||” is to the sign “|| + |||”.
1 (R): ∀ § 43 p. 189/1 3 & p. 188v 3 ∀ p^
289/1)2 ; p. 217/1 3 (M): ///
2 (V): But$
3 (M): ///
4 (M): ///
5 (M): maybe useless / 3
6 (M): ∫ 3
7 (M): 3
8 (V): to describe state the length of an object
9 (V): example for all propositions.
10 (M): 555
11 (M): 555
12 (M): 555
13 (M): 555
67e
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14
Z.B. a, b, c, d bedeuten Bewegungen und zwar a = ↓, b = ↑, c = →, d = ←. Also heißt
z.B. bccbda der Linienzug. Nun, ist der Satz ”bccbda“15
nicht ähnlich jenem
Linienzug? Offenbar ja, in gewisser Weise. (Ist es nicht genau die Ähnlichkeit einer
Photographie und des photographierten Gegenstandes?)
14 (M): 555 15 (V): ,,bccbad“
68 Ähnlichkeit von Satz und Bild.
WBTC015-21 30/05/2005 16:18 Page 136
14
For example, let a, b, c, d mean movements, specifically a = ↓, b = ↑, c = →, d = ←.
So “bccbda” for instance means this line sequence. Well, isn’t the sentence
“bccbda”15
similar to that line sequence? Obviously, in a certain way, it is.
(Isn’t it exactly the similarity of a photograph to the photographed object?)
14 (M): 555 15 (V): “bccbad”
Similarity of Proposition and Picture. 68e
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1 (R): ∀ S. 289 (M): ? / ∫ 3
2 (M): /
3 (M): ∫ 3
4 (O): denn!?”
5 (M): ///
6 (M): ∫ / ///
7 (M): / 3
8 (M): ? / 3
9 (R): ∀ S. 390
84v
85
86
85
22
Sätze mit Genrebildern verglichen.
(Verwandt damit:
Verstehen eines Bildes.)
1
Wie ist es mit den Sätzen, die in Dichtungen vorkommen. Hier kann doch gewiß von
einer Verifikation nicht geredet werden und doch haben diese Sätze Sinn. Sie verhalten sich
zu den Sätzen, für die es (eine) Verifikation gibt, wie ein Genrebild zu einem Portrait. Und
dieses Gleichnis dürfte wirklich die Sache vollständig darstellen.
2
Die Beschreibung eines wirklichen Gegenstandes verhält sich zu der Beschreibung in einer
Dichtung wie ein Portrait zu einem Genrebild.
3
Wenn ich ein Bild anschaue, so sagt es mir etwas, auch wenn ich keinen Augenblick glaube
(mir einbilde), die Menschen seien wirklich oder es habe wirkliche Menschen gegeben,
von denen dies ein verkleinertes Bild sei. ”Es sagt mir etwas“ kann aber hier nur heißen,
”es bringt eine bestimmte Einstellung in mir hervor“.
Denn wie, wenn ich fragte: ”was sagt es mir denn?“4
5
Meine Stellung gegen das Bild ist auch keine hypothetische, so daß ich mir etwa sagte
”wenn es solche Menschen gäbe, dann . . .“
6
Wenn ich ein Genrebild ansehe, so halte ich die gemalten Menschen darin nicht für
wirkliche Menschen, andererseits ist ihre Ähnlichkeit mit Menschen für das Verständnis des
Bildes wesentlich.
7
Wenn man es für selbstverständlich hält, daß sich der Mensch an seiner Phantasie vergnügt,
so bedenke man, daß diese Phantasie nicht wie ein gemaltes Bild oder ein plastisches Modell
ist, sondern ein kompliziertes Gebilde aus heterogenen Bestandteilen: Wörtern und Bildern.
Man wird dann das Operieren mit Schrift- und Lautzeichen nicht mehr in Gegensatz stellen
zu dem Operieren mit ”Vorstellungsbildern“ der Ereignisse.
8
Die Illustration in einem Buch ist dem Buch nichts fremdes, sondern gesellt sich hinzu
wie ein verwandter Behelf einem anderen, – wie etwa eine Reibahle dem Bohrer.
(Wenn einen die Häßlichkeit eines Menschen abstößt, so kann sie im Bild, im gemalten,
gleichfalls abstoßen, aber auch in der Beschreibung, in den Worten.)9
69
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1 (R): ∀ p. 289 (M): ? / ∫ 3
2 (M): /
3 (M): ∫ 3
4 (M): ///
5 (M): ∫ / ///
6 (M): / 3
7 (M): ? / 3
8 (R): ∀ p. 390
22
Propositions Compared to
Genre-Paintings.
(Related to This:
Understanding a Picture.)
1
How about sentences that occur in works of literature? Here one certainly can’t speak of
verification, and yet these sentences have sense. They relate to propositions for which there
is (a) verification as does a genre-painting to a portrait. And this simile probably does
portray the matter completely.
2
A description of a real object is related to a description in a literary work as a portrait is to a
genre-painting.
3
If I look at a picture it speaks to me, even if I don’t believe (imagine) for a moment that
the people in it are real or that there used to be real people of whom this is a scaled-down
picture. Here, however, “it speaks to me” can only mean “it begets a certain attitude in me”.
For what if I were to ask: “What does it say to me?”
4
Nor is my attitude toward the picture hypothetical, so that I’d say to myself, for instance,
“If there were such people, then . . .”.
5
When I look at a genre-painting I don’t take the people painted in it for real; on the other
hand their similarity to real people is essential to understanding the picture.
6
If one takes it to be self-evident that human beings take pleasure in their imaginings,
then one ought to consider that what is imagined is not like a painted picture or a threedimensional
model, but is a complicated amalgam made up of heterogeneous parts: words
and pictures. Then one will no longer set up a contrast between operating with written and
phonetic signs and operating with “mental images” of events.
7
An illustration in a book is nothing foreign to it; rather it goes with it as one tool goes
with a related one – as, for example, a hand reamer goes with a drill.
(If someone’s ugliness revolts us it can likewise revolt us in a painted picture, but also in
a description, in words.)8
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23
Mit dem Satz scheint die Realität
wesentlich übereinstimmen oder nicht
übereinstimmen zu können.
Er scheint sie zu fordern, sich mit
ihm zu vergleichen.
1
”Meine Erwartung ist so gemacht, daß, was immer kommt, mit ihr übereinstimmen muß,
oder nicht.“
2
”Der Satz ist als Richter hingestellt und wir fühlen uns vor ihm verantwortlich.“
3
Ich sage, die Hand über dem4
Tisch haltend, ”ich wollte, dieser Tisch wäre so hoch“.
Nun ist das Merkwürdige: die Hand über dem Tisch an und für sich drückt gar nichts aus.
D.h., sie ist eine Hand über einem Tisch, aber kein Symbol (wie der Pfeil, der etwa die
Gehrichtung anzeigen soll, an sich nichts ausdrückt).
5
”Die Hand zeigt dahin“. Aber in wiefern zeigt sie dahin? Einfach, weil sie sich in einer
Richtung verjüngt? (Zeigt ein Nagel in die Wand?) D.h., ist es dasselbe zu sagen ”sie zeigt
etc.“ und6
”sie verjüngt sich in dieser Richtung“?
7
Man kann eine Lehne auf das Maß eines Körpers einstellen, vorbereiten. Dann liegt in
dieser Einstellung zwar das eingestellte Maß, aber in keiner Weise, daß ein bestimmter Körper
es hat. Ja vor allem liegt darin keine Annahme darüber, ob der Körper dieses Maß hat, oder
nicht hat.
8
Ich sagte, der Satz wäre wie ein Maßstab an die Wirklichkeit angelegt: Und9
der Maßstab
ist, wie alle10
Gleichnisse des Satzes, ein besonderer Fall eines Satzes. 11
Und auch er
bestimmt nichts, solange man nicht mit ihm mißt. Aber Messen ist Vergleichen (und muß
heißen, Übersetzen).
12
Man möchte sagen: Lege den Maßstab an einen Körper an; er sagt nicht, daß der Körper
so lang ist. Vielmehr ist er an sich ich möchte sagen13
tot und leistet nichts von dem, was
1 (M): / 3
2 (M): / 3
3 (M): ? / ///
4 (V): den
5 (M): ? / ///
6 (V): oder
7 (M): ///
8 (M): /// 3
9 (V): Aber
10 (V): alle richtigen
11 (M): ||
12 (M): / 3
13 (V): sich gleichsam
87
88
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23
Reality Seems Inherently Able
Either to Agree with a Proposition or
not to Agree with it. A Proposition
Seems to Challenge Reality to
Compare Itself to it.
1
“My expectation is so constructed that whatever happens has to accord with it, or not.”
2
“The proposition has been set up as a judge, and we feel answerable to him.”
3
Holding my hand above the table I say, “I wish this table were this high”. Now the remarkable
thing is: the hand above the table doesn’t express anything by and of itself. That is, it
is a hand above a table, not a symbol (just as an arrow that is supposed to indicate the direction
one is to follow expresses nothing by itself).
4
“The hand is pointing in that direction.” But in what respect is it pointing in that
direction? Simply because it gets narrower in one direction? (Does a nail point into the
wall?) That is, is it the same thing to say “It is pointing etc.” and5
“It gets narrower in this
direction”?
6
One can adjust a back-rest to prepare it for the dimensions of a body. Then, to be sure,
these dimensions are contained in this setting, but in no way does the setting indicate that
a particular body has them. Indeed, this setting doesn’t contain any supposition as to whether
a body has these dimensions or not.
7
I said that a proposition is laid alongside reality like a yardstick: And8
the yardstick, like
all9
similes for a proposition, is a particular case of a proposition.10
And like a yardstick,
neither does it determine anything, so long as one doesn’t measure with it. But measuring
is comparing (and needs to be called translating).
11
One wants to say: Lay the yardstick alongside a body; it doesn’t say that the body is
such-and-such a length. Rather, I want to say, in and of itself it is dead, and12
it achieves
1 (M): / 3
2 (M): / 3
3 (M): ? / ///
4 (M): ? / ///
5 (V): or
6 (M): ///
7 (M): /// 3
8 (V): But
9 (V): all correct
10 (M): ||
11 (M): / 3
12 (V): it is so to speak dead and
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der Gedanke leistet. Es ist, als hätten wir uns eingebildet, das Wesentliche am lebenden
Menschen sei die äußere Gestalt, und hätten nun einen Holzblock von dieser Gestalt hergestellt
und sähen mit Enttäuschung den toten Klotz, der auch keine Ähnlichkeit mit dem Leben hat.
14
Man könnte sagen, ”die Erwartung ist kein Bild, sie bedient sich nur eines Bildes“. Ich
erwarte etwa, daß meine Uhr jetzt auf 7 zeigen wird und drücke dies durch ein Bild der
Zeigerstellung aus. Dieses Bild kann ich nun mit der wirklichen Stellung vergleichen; die
Erwartung aber nicht.
15
Mein Gedanke ist immer: wenn einer die Erwartung sehen könnte, daß er erkennen16
müßte, was erwartet wurde.
Aber so ist es ja auch: wer den Ausdruck der Erwartung sieht, sieht was erwartet wird. Und wie
könnte man es auf andere Weise, in anderem Sinne sehen?!
17
Gut, ich sage: wenn ich meine Uhr herausziehe, wird sie mir jetzt entweder dieses Bild
der Zeigerstellung bieten, oder nicht. Aber wie kann ich es ausdrücken, daß ich mich für
eine dieser Annahmen entscheide?
Jeder Gedanke ist der Ausdruck eines Gedankens.
18
Ich könnte mein Problem so darstellen: Wenn ich untersuchen wollte, ob die Krönung
Napoleons so und so stattgefunden hat, so könnte ich mich dabei, als einer Urkunde, des
Bildes bedienen, statt einer Beschreibung. Und es frägt sich nun, ist die ganze Vergleichung
der Urkunde mit der Wirklichkeit von der Art, wie der Vergleich der Wirklichkeit mit dem
Bild, oder gibt es dabei noch etwas Andres, von andrer Art?
19
Aber womit soll man die Wirklichkeit vergleichen, als20
mit dem Satz? Und was soll man
andres tun, als21
sie mit ihm zu vergleichen?
22
Wenn man das Beispiel von dem, durch Gebärden mitgeteilten Befehl betrachtet,
möchte man einerseits immer sagen: Ja, dieses Beispiel ist eben unvollkommen, die
Gebärdensprache zu roh, darum kann sie den beabsichtigten Sinn nicht vollständig ausdrücken“
– aber tatsächlich ist sie so gut wie jede denkbare andere, und erfüllt ihren Zweck so vollständig,
wie es überhaupt denkbar ist.
(Es ist eine der wichtigsten Einsichten, daß es keine Verbesserung der Logik gibt.)
23
Der Befehl die Zahlen 1 bis 4 zu quadrieren24
kommt uns unvollständig vor. Es scheint uns,
als wäre etwa nur angedeutet, was nicht ausgesprochen ist.
25
Angedeutet aber ist etwas nur insofern, als ein System nicht ausdrücklich, oder
unvollkommen festgelegt ist. Wir möchten sagen, es sei uns unvollkommen angedeutet
oder, das Zeichen suggeriere nur undeutlich, was wir zu tun hätten. 26
Es sei etwa undeutlich
in dem Sinn, in welchem wir der Deutlichkeit halber Zeichen ausführlicher geben.27
14 (M): ///
15 (M): / 3
16 (V): sehen
17 (M): ? / 5555
18 (M): ///
19 (M): ///
20 (V): vergleichen: als
21 (V): tun: als
22 (M): ////
23 (M): / 3 (R): S. 92
24 (V): Der Befehl x 1 2 3 4
x2
25 (M): 3
26 (M): ///
27 (V): /// Es sei etwa in dem Sinn undeutlich,
wie eine Tafel mit der Aufschrift ”Links
Gehen“ deutlicher wird, wenn zugleich ein Pfeil
die Richtung zeigt.
89
88v, 89
90
89
71 Mit dem Satz scheint die Realität . . .
89v
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nothing of what thought achieves. It is as if we had imagined that the essential thing about
a living human being was his outer shape, that we had fashioned a block of wood into this
shape, and then were disappointed to view the dead block that bears no similarity to life.
13
One could say, “Expectation isn’t a picture, it just uses a picture”. I expect, for example,
that my watch will now point to 7, and I express this with a picture of the position
of the hands. Now I can compare this picture with the real position; but I can’t compare my
expectation with it.
14
My thought is always: if someone could see the expectation he would have to perceive15
what was expected.
But that is the way it is: whoever sees the expression of expectation sees what is being expected.
And how could one see it in a different way, in a different sense?!
16
It’s all very well to say: if I pull out my watch, I will either be shown this picture of the
position of the hands, or I won’t. But how can I express the fact that I am deciding in favour
of one of these suppositions?
Every thought is the expression of a thought.
17
I could present my problem this way: if I wanted to investigate whether Napoleon’s
coronation took place in such and such a way I could use a picture rather than a description
as a document. And now the question is: Is the whole process of comparing the document
with reality like comparing reality with the picture, or is something additional, of a
different kind, involved?
18
But what should one compare reality to other than a proposition? And what else should
one do other than compare it to a proposition?
19
If one looks at the example of a command that is communicated through gestures, one
is always inclined on the one hand to say: “All right, this example is just incomplete; the
language of gestures is too coarse, so it can’t express the intended sense completely.” – But
in fact it is as good as any other conceivable language, and it fulfils its purpose as completely
as anyone can conceive.
(It is one of the most important insights that there is no such thing as an improvement
upon logic.)
20
The command to square the numbers 1 to 421
strikes us as incomplete. It seems to us that
something not stated is perhaps only hinted at.
22
But something is hinted at only in so far as a system is not expressly established, or is
so incompletely. We’d like to say that what we’re to do is incompletely suggested to us or
that the sign only vaguely suggests it. 23
That the sign is vague in the sense in which we might
give more detailed signs for the sake of clarity.24
13 (M): ///
14 (M): / 3
15 (V): see
16 (M): ? / 5555
17 (M): ///
18 (M): ///
19 (M): ////
20 (M): / 3 (R): p. 92
21 (V): The command x 1 2 3 4
x2
22 (M): 3
23 (M): ///
24 (V): /// sense in which a sign saying “keep left”
becomes clearer if at the same time an arrow
points in that direction.
Reality Seems . . . 71e
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28
Aber für uns ist der Befehl deutlich, der unzweideutig ist; und einen deutlicheren gibt
es nicht.
29
Eindeutig aber kann er nur werden, dadurch, daß in dem System von Befehlen eine
Unterscheidung gemacht wird, die, wenn sie fehlt, eben die Zweideutigkeit hervorruft. (Wenn
also das System die richtige Mannigfaltigkeit erhält.)
30
Was, in der Logik, nicht nötig ist, ist auch nicht von Nutzen.31
Was nicht nötig ist, ist überflüssig.
32
Die Unbeholfenheit, mit der das Zeichen wie ein Stummer durch allerlei suggestive
Gebärden sich verständlich zu machen sucht, – sie verschwindet, wenn wir erkennen, daß
das Wesentliche am Zeichen das System ist, dem es zugehört und sein übriger Inhalt wegfällt.
Man möchte sagen nur der Gedanke kann es ganz sagen, das Zeichen33
nicht.
72 Mit dem Satz scheint die Realität . . .
28 (M): ? / 3
29 (M): ///
30 (M): ∫ 3
31 (V): nötig ist, hilft auch nicht.
32 (M): ? / 3
33 (V): sagen, der G das Zeichen
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25
But for us a command that is unambiguous is clear; and there is no clearer one.
26
But a command can become unambiguous only if the system of commands makes a
distinction which, when it is lacking, produces ambiguity. (That is, when the system is endowed
with the right degree of multiplicity.)
27
What isn’t necessary in logic isn’t useful.28
What isn’t necessary is superfluous.
29
The clumsiness with which a sign seeks to make itself understood, like someone who is
mute, through all sorts of suggestive gestures – this vanishes when we recognize that the
essential thing about a sign is the system to which it belongs and that the rest of its content
is unimportant.
One would like to say, only a thought can say it completely, not a sign.
25 (M): ? / 3
26 (M): ///
27 (M): ∫ 3
28 (V): isn’t helpful.
29 (M): ? / 3
Reality Seems . . . 72e
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24
Das Symbol (der Gedanke), scheint
als solches unbefriedigt zu sein.
Der Wunsch scheint schon zu wissen was ihn erfüllen wird oder würde, der Satz1
was ihn wahr
macht auch wenn es gar nicht da ist! Woher dieses Bestimmen, dessen, was noch nicht da ist? – dieses
despotische Verfügen?2
Und woher diese seltsame Sinnestäuschung? Wir sagen der Satz sagt etwas, der Wunsch
wünscht der Befehl befiehlt3
etwas.4
Aber wie benützen5
wir denn diese Aussagen, wann benützen
wir sie in welchem weitern Zusammenhang? Was ist es, was ein Satz sagt, was setzen wir statt dem
&etwas“ ein? Dieser Satz sagt: daß . . . & nun folgt ein weiterer Satz.6
&Der Satz sagt etwas“ darauf ist die Ergänzung entweder die Frage &Was?“ & ein andrer Satz –
oder &sagt etwas“ ist gar keine7
Variable, heißt nicht: sagt8
dies, oder jenes.9
Wir sagen:10
Der Befehl befiehlt11
dies, & tun es; aber auch &der Befehl befiehlt dies: ich soll12
das
& das tun;
Wir übersetzen ihn einmal in13
einen andern Satz, einmal in14
eine Demonstration & einmal15
in
die Tat.
Ja er befiehlt16
ja schon – möchte ich sagen – daß ich das tun soll! Aber was ist denn das das? Ich
werde von der Form: &Er befiehlt17
das“ hypnotisiert.
&Der Befehl befiehlt18
seine Befolgung“. Ja also kennt er seine Befolgung schon ehe sie da ist! –
Aber der Satz ist ja nur ein grammatischer über die Worte &Befehl“ & &Befolgung“. Er sagt: Wenn
ein Befehl lautet &Tue das & das“ dann nennt man &das & das tun“ das Befolgen dieses19
Befehls.
Jener Satz ist von der Art grammatikalischer Sätze wie: Der Hund hat &Beine“ der Hase &Läufe“.20
21
Jedes Symbol scheint als solches etwas offen zu lassen.
22
Der Plan ist als Plan etwas Unbefriedigtes. (Wie der Wunsch, die Erwartung, die
Vermutung u.s.f.)
1 (V): Gedanke
2 (V): Verlangen?
3 (O): befielt
4 (V): der Wunsch wünscht etwas.
5 (V): verwenden
6 (V): Ausdruck.
7 (V): ist keine
8 (V): nicht: (sagt
9 (V): Satz – oder es hieß man könnte dafür setzen (der
Satz sagtA.
10 (V): sagen auch:
11 (O): befielt
12 (V): dies: Du sollst
13 (V): ihn in
14 (V): Satz, in
15 (V): Demonstration, oder
16 (O): befielt
17 (O): befielt
18 (O): befielt
19 (V): tun” die Befolgung des
20 (V): hat einen (SchwanzA der Fuchs eine (RuteA.
21 (M): / 3 (R): 1 S. 163/5
22 (M): / 3 – u. s. f.)
91
90v
91
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24
A Symbol (a Thought) as Such
Seems to be Unfulfilled.
A wish seems already to know what will or would fulfil it, a proposition1
what makes it true, even
when it isn’t here! Whence this determining of what isn’t yet here? – this despotic dictate?2
And where does this strange illusion come from? We say a proposition says something, a wish
wishes, a command commands something.3
But how do we use these statements, when do we
use them, and in what larger context? What is it that a proposition says, what do we substitute for
the “something”? This proposition says that . . . , and now an additional proposition4
follows.
“The proposition says something”; the follow-up to this is either the question “What?” and another
proposition – or else “says something” is not a variable at all,5
and does not mean: says6
this or that.7
We8
say: the command commands this, and we do it; but also “The command commands this:
I’m9
supposed to do this and that”.
At times we translate it into another proposition, at other times into10
a demonstration, and at
still others into11
an action.
Indeed, I’d like to say, the command does command that I am supposed to do that! But what is
the that? I’m hypnotized by the form: “It commands that”.
“The command commands that it be carried out.” So it already knows how it is to be carried out
even before it happens! – But this sentence is only a grammatical one about the words “command”
and “carry out”. It says: If a command says “Do such-and-such” then we call “doing such-and-such”
the carrying out of this command. Our initial sentence resembles grammatical sentences such as: “A
dog has ‘paws’, a rabbit ‘feet’”.12
13
By itself every symbol seems to leave something open.
14
A plan as such is something unsatisfied. (As is a wish, an expectation, a conjecture, etc.)
1 (V): a thought
2 (V): decree?
3 (V): something, a wish wishes something.
4 (V): expression
5 (V): is no variable,
6 (V): A says
7 (V): proposition – or it meant that one could replace
it with Athe proposition saysC.
8 (V): We also
9 (V): You’re
10 (V): proposition in
11 (V): demonstration, or into
12 (V): “A dog has a BtailD% a fox a BbrushD.”
13 (M): / 3 (R): ∀ p. 163/5
14 (M): / 3 – etc.)
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Und hier meine ich, die Erwartung ist unbefriedigt weil sie die Erwartung von etwas ist;23
der Glaube,
die Meinung unbefriedigt weil sie die Meinung ist daß etwas der Fall ist, etwas Wirkliches, etwas
außerhalb des Vorgangs24
der Meinung.
25
Ich möchte manchmal mein Gefühl dem Plan gegenüber als eine Innervation bezeichnen.
Aber auch die Innervation an sich ist nicht unbefriedigt, ergänzungsbedürftig.
26
In wiefern kann man den Wunsch als solchen, die Erwartung ”unbefriedigt“ nennen?
Was ist das Vorbild27
der Unbefriedigung? Ist es der leere Hohlraum (in den etwas
hineinpaßt)? Und würde man von einem leeren Raum sagen, er sei unbefriedigt? Wäre das
nicht auch eine Metapher? Ist es nicht ein gewisses Gefühl, das wir Unbefriedigung nennen?
Etwa der Hunger. Aber der Hunger enthält nicht das Bild seiner Befriedigung.
28
Die Hohlform ist nur unbefriedigt in dem System, in dem auch die Vollform
vorkommt.29
30
Ich meine, man kann das Wort ”unbefriedigt“ nicht schlechtweg von einer Tatsache
gebrauchen. Es kann aber in einem System eine Tatsache beschreiben helfen. Ich könnte
z.B. festsetzen, daß ich den Hohlzylinder ”den unbefriedigten Zylinder“ nennen werde, den
entsprechenden Vollzylinder, seine Befriedigung.
31
Aber man kann nicht sagen, daß der Wunsch ”p möge der Fall sein“, durch die Tatsache
p befriedigt wird, es sei denn als Zeichenregel: |der Wunsch p möge der Fall sein| =
|der Wunsch, der durch die Tatsache p befriedigt wird|.32
Man könnte auch so sagen: Dieser Befehl befiehlt33
dies (& tut es). – Aber hat er dies nicht schon
früher befohlen? (Er hat doch früher nichts34
anderes befohlen!) Also hat er diese Tat befohlen, ehe
es sie noch gab. Inwiefern hat er aber früher dies befohlen? – Ist35
denn Befehlen eine Tätigkeit, die
er auch früher ausübte? Und wie hat er sie ausgeübt? Der Befehl36
befiehlt37
das & das enthält ja
die Zeit gar nicht sowenig wie 2 + 2 ist 4.
Ich habe auch früher dies gemeint enthält wohl die Zeit. Aber was ist denn hier das Kriterium dafür
daß ich dies meinte.38
Heißt es ich habe schon früher den Dieb gehangen ehe ich ihn noch hatte.
Wie kann man meinen was noch nicht geschehen ist. Worin bestand aber dies Meinen damals.
Was nennen wir also jetzt dies was wir jetzt tun gemeint39
zu haben? Worin besteht die Identität:
dasselbe jetzt tun, was ich früher meinte. Worin besteht es: dieselbe Speise jetzt zubereiten, die40
ich
später esse.
Ja ich meine ja jetzt schon das was ich später ausführe.41
Ja manchmal42
meine ich jetzt dasselbe;
– manchmal43
etwas anderes!44
In welchem Falle sagen wir das eine, in welchem Falle das andre? In
23 (V): meine ich: &die Erwartung ist . . . Erwartung von
etwas istA;
24 (O): außerhalb des Vorgang (V): Wirkliches,
etwas außer dem Vorgang
25 (M): 3
26 (M): ? / 3
27 (V): Urbild
28 (M): ? / 3
29 (V): in dem auch die entsprechende Vollform
vorkommt.
30 (M): / 3
31 (M): ? / 3
32 (R): ∀ S. 17/5, 18/1,2 3, dazu erstes MS S. 130 139/3
= S. 89/4 3 & S. 90/4 3
33 (O): befielt
34 (V): hat doch nichts
35 (V): befohlen? – Wa Ist
36 (V): ausgeübt? Er Der befehl
37 (O): befielt
38 (V): meine.
39 (V): dies gemeint
40 (V): es: dasselbe jetzt kochen, was
41 (V): tue.
42 (V): manchmal
43 (O): dasselbe; M manchmal
44 (V): anderes:
74 Das Symbol (der Gedanke) . . .
90v
91
92
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And here I mean: an expectation is unsatisfied because it is an expectation of something;15
a belief,
an opinion is unsatisfied because it is the opinion that something is the case, something real,
something external to the occurrence of the opinion.
16
Sometimes I am inclined to call my feeling towards a plan an innervation. But an innervation
as such is not unfulfilled, in need of supplementation.
17
In what respect can we call a wish as such, an expectation, “unsatisfied”? What is the
model18
for non-satisfaction? Is it an empty cavity (into which something fits)? And would
one say of an empty space that it was unsatisfied? Wouldn’t that be a metaphor too? Isn’t it
a certain feeling that we call non-satisfaction? Say hunger. But hunger doesn’t contain the
image of its satisfaction.
19
A hollow form is only unsatisfied in a system that also contains the solid20
form.
21
I’m of the opinion that one can’t simply apply the word “unsatisfied” to a fact. But it can
help describe a fact within a system. I could stipulate for example that I will call the hollow
cylinder “the unsatisfied cylinder”, and the corresponding solid cylinder its satisfaction.
22
But we can’t say that the wish that “p be the case” is satisfied by the fact p, except as a
rule for symbols: |the wish that p be the case| = |the wish that is satisfied by the fact p|.23
One could also put it this way: This command commands this (and one does it). – But didn’t it command
this earlier? (Certainly it didn’t command anything different earlier!24
). So it commanded that
this deed be done before it existed. But in what way did it command this earlier? – Is commanding
perhaps an activity that it also carried out earlier? And how did it carry it out? “A command commands
this and that” doesn’t contain time, any more than 2 + 2 = 4.
“I also meant this earlier” does contain time, to be sure. But what is the criterion here for my
having meant25
this? Does it mean that I hanged the thief before I caught him?
How can one mean what hasn’t happened yet? But what did meaning this consist in back then?
So what do we now call: to have meant precisely what26
we’re now doing? In what does the
identity consist between doing the same thing now that I meant earlier? In what does this consist:
to prepare the same food now that27
I will eat later?
I mean now what I carry out28
later. Well, sometimes29
I mean the same thing now, but sometimes
something else!30
In which case do we say the one thing, in which case the other? In which
15 (V): mean: “an expectation is . . . somethingC;
16 (M): 3
17 (M): ? / 3
18 (V): archetype
19 (M): ? / 3
20 (V): contains the corresponding solid
21 (M): / 3
22 (M): ? / 3
23 (R): ∀ p. 17/5, 18/1,2 3, to first MS p. 130 139/3 =
p. 89/4 3 & p. 90/4 3
24 (V): different!
25 (V): my meaning
26 (V): to have meant this?
27 (V): to cook the same thing now that
28 (V): I do
29 (V): sometimes
30 (V): else:
A Symbol as Such Seems to be Unfulfilled. 74e
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welchem45
Falle sage ich, daß ich etwas anderes getan habe als ich meinte – & in welchem dasselbe.
Und wenn der Befehl nicht befolgt wird, – wo ist dann der Schatten seiner46
Befolgg. den Du zu sehen
meintest, weil Dir die Form vorschwebte: Er befiehlt47
das & das. Wie macht man es denn: das & das48
zu befehlen? Man sagt: man befiehlt:49
den Befehl und auch man befiehlt:50
die Handlung (die
Befolgung).
Man möchte sagen: ich befehle mehr als die Worte & weniger als die Handlung.
Wir identifizieren den Satz &daß . . .“ mit der Handlung.
Er hat das getan was ich ihm befohlen habe – Warum soll man hier nicht51
sagen es sei eine Identität
der Handlung52
& der WORTE?! Wozu soll ich53
einen Schatten zwischen die beiden stellen? Wir haben
ja eine Projektionsmethode. Nur ist es eine andere Identität: Ich habe das getan54
was er getan hat
& ich habe getan das was55
er befohlen hat.
45 (O): welche
46 (V): der
47 (O): befielt
48 (V): denn: etwas
49 (O): befielt:
50 (O): befielt:
51 (V): nicht von einer
52 (V): es b sei eine Identität der Handlung
53 (V): ich mich
54 (V): Ich habe getan
55 (V): habe getan was
75 Das Symbol (der Gedanke) . . .
92
WBTC022-29 30/05/2005 16:18 Page 150
case do I say that I did something other than what I meant – and in which case the same thing? And
if the command is not obeyed, then where is the shadow of its31
observance that you thought you
saw because the form “he commands this and that” was in your mind? How does one go about:
commanding this and that?32
We say: We command the command, and also: We command the action
(the carrying out of the command).
One wants to say: I command more than the words and less than the action.
We identify the proposition “that . . .” with the action.
He did what I ordered him to do – Why shouldn’t one say here33
that there is an identity of action34
and WORDS?! Why should I35
place a shadow between these two? After all, we do have a method
of projection. Except that there’s a difference in identity between: I did what he did and I did what
he commanded.
31 (V): the
32 (V): commanding something?
33 (V): here of a
34 (V): identity of action
35 (V): I me
A Symbol as Such Seems to be Unfulfilled. 75e
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25
Ein Satz ist ein Zeichen in einem
System von Zeichen. Er ist eine
Zeichenverbindung von mehreren
möglichen und im Gegensatz zu den
andern möglichen. Gleichsam eine
Zeigerstellung im Gegensatz zu
andern möglichen.1
2
Einen Satz verstehen heißt, eine Sprache verstehen.
3
Jeder Satz einer Sprache hat nur Sinn im Gegensatz zu anderen Wortzusammenstellungen
derselben Sprache.
4
Wenn ein Satz nicht eine mögliche Verbindung unter anderen wäre, so hätte er keine
Funktion.
D.h.: Wenn eine Beschreibung5
nicht das Ergebnis einer Entscheidung wäre, hätte sie6
nichts
zu sagen.
Sprache die nur aus einem Signal besteht das, immer gegeben wird, wenn eine bestimmte
Handlung vollführt werden soll.
Abrichten.
7
Denken ist Pläne machen.
Wenn Du Pläne machst, so machst Du einen Plan im Gegensatz zu8
andern Plänen.
9
↑ im Gegensatz zu 2 ist ein anderes Zeichen als ↑ im Gegensatz zu .
10
”Geh so → nicht so 2“ hat nur Sinn, wenn es die Richtung ist, die dem Pfeil hier
wesentlich ist, und nicht, etwa nur die Länge.
1 (M): 3 ||
2 (M): ? / 3 (R): siehe S. 196
Wort nur im Satzzusammenhang Bedeutg.
Satz komplex. siehe S. 82/2 auch §23 S. 87 ff
∀ S. 63/6 3
3 (M): ∫ 3
4 (M): ? /
5 (V): Wenn ein Satz
6 (V): er
7 (M): 3 ////
8 (V): Plan zum Unterschied von
9 (M): /// 3
10 (M): /// (F): MS 110, S. 76
93
92v
93
94
76
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25
A Sentence is a Sign within a System
of Signs. It is a Combination of Signs
from among Several Possible Ones
and in Contrast to Other Possible
Ones. One Position of the Pointer,
as it Were, in Contrast to
Other Possible Ones.1
2
To understand a sentence means to understand a language.
3
Each sentence of a language has sense only in contrast to other combinations of words
in the same language.
4
If a sentence were not one possible combination among others, then it wouldn’t have any
function.
That is to say: If a description5
were not the result of a decision, it would have nothing
to say.
A language consisting of just one signal that is always given when a particular action is to be
carried out.
Animal-training.
6
To think is to make plans.
When you make plans, you make one plan in contrast to7
others.
8
↑ in contrast to 2 is a different sign from ↑ in contrast to .
9
“Go this way → – not this 2” makes sense only if it is the direction that is essential to
the arrow and not, say, just its length.
1 (M): 3 ||
2 (M): ? / 3 (R): see p. 196
A word has meaning only in the context of a sentence.
Sentence complex.
See p. 82/2 also § 23 pp. 87ff.
∀ p. 63/6 3
3 (M): ∫ 3
4 (M): ? /
5 (V): a proposition
6 (M): 3 ////
7 (V): plan as opposed to
8 (M): /// 3
9 (M): /// (F): MS 110, p. 76.
76e
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77 Ein Satz ist ein Zeichen . . .
11
Man muß wissen, worauf im Zeichen man zu sehen hat. Etwa: auf welcher Ziffer der
Zeiger steht, nicht darauf, wie lang er ist.
12
”Geh’ in der Richtung, in der der Zeiger zeigt“.
”Geh’ so viele Meter in der Sekunde, als der Pfeil cm lang ist“.
”Mach’ so viele Schritte, als ich Pfeile zeichne“.
”Zeichne diesen Pfeil nach“.
Für jeden dieser Befehle kann der gleiche Pfeil stehen. –
13
”Ich muß auf die Länge achten“, ”ich muß auf die Richtung achten“, das heißt schon:
auf die Länge im Gegensatz zu anderen, etc.
14
Wie soll ich mich nach der Uhr richten? Wie kann ich mich nach diesem Bild
15
richten? (Wie nach jedem andern.)
16
Es zeigt mir jemand zum ersten Mal eine Uhr und will, daß ich mich nach ihr richte.
Ich frage nun: worauf soll ich bei diesem Ding achten. Und er sagt: auf die Stellung der
Zeiger.
17
Natürlich, das Zeichen eines Systems bezeichnet es nur im Gegensatz zu anderen Systemen
und setzt selbst ein System voraus. (Interne Relation, die nur besteht, wenn ihre Glieder da
sind.)18
11 (M): ///
12 (M): ? / 3 ////
13 (M): ///
14 (M): ///
15 (F): MS 110, S. 138.
16 (M): ///
17 (M): ///
18 (R): ∀ S. 3/1
WBTC022-29 30/05/2005 16:18 Page 154
10
One has to know what one is to look for in a sign. For example: what number the hand
is on, and not how long it is.
11
“Go in the direction in which the arrow is pointing.”
“Walk as many metres per second as the length of the arrow in centimetres.”
“Walk as many steps as I draw arrows.”
“Copy this arrow.”
The same arrow can be used in each of these commands. —
12
“I have to pay attention to the length”, “I have to pay attention to the direction” already
means: to that length in contrast to other lengths, etc.
13
How am I to be guided by a clock? How can I be guided by this picture?
14
(As with any other.)
15
Someone shows me a clock for the first time and wants me to be guided by
it. So I ask: What about this thing should I pay attention to? And he says: To the position
of the hands.
16
Of course a sign in a system only signifies that system in contrast to others, and itself
presupposes a system. (An internal relation that exists only if its components are present.)17
10 (M): ///
11 (M): ? / 3 ////
12 (M): ///
13 (M): ///
14 (F): MS 110, p. 138.
15 (M): ///
16 (M): ///
17 (R): ∀ p. 3/1
A Sentence is a Sign Within a System of Signs. . . . 77e
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26
Sich vorstellen können,
”wie es wäre“, als Kriterium dafür,
daß ein Satz Sinn hat.
1
Was heißt es, wenn man sagt: ”ich kann mir das Gegenteil davon nicht vorstellen“, oder
”wie wäre es denn, wenn’s anders wäre“; z.B. wenn jemand gesagt hat, daß meine
Vorstellungen privat seien, oder daß nur ich selbst wissen kann, ob ich Schmerzen
empfinde, und dergleichen.
2
Wenn ich mir nicht vorstellen kann, wie es anders wäre, so kann ich mir auch nicht
vorstellen, wie es so sein kann.
”Ich kann mir nicht vorstellen“ heißt nämlich hier nicht, was es im Satz ”ich kann mir
keinen Totenkopf vorstellen“ heißt. Ich will damit nicht auf eine mangelnde
Vorstellungskraft deuten.
3
Überlege: ”Ich habe4
tatsächlich nie gesehen, daß ein schwarzer Fleck nach und nach immer
heller wird, bis er weiß ist, und dann immer rötlicher, bis er rot ist; aber ich weiß, daß es
möglich ist, weil ich es mir vorstellen kann. D.h., ich operiere mit meinen Vorstellungen
im Raume der Farben und tue mit ihnen, was mit den Farben möglich wäre.“ ((Siehe
”Logische Möglichkeit“.))
5
Es scheint, als könnte man sagen:6
Die Wortsprache läßt unsinnige Wortzusammenstel-
lungen7
zu, die Sprache der Vorstellung aber nicht unsinnige Vorstellungen. (Natürlich kann
das, so wie es da steht, nichts heißen.)
Also8
die Sprache der Zeichnung auch nicht unsinnige Zeichnungen; – aber so ist es nicht, denn
eine Zeichnung kann in demselben Sinne unsinnig sein wie ein &Satz“. Denken wir uns eine
Zeichnung nach der Körper modelliert werden sollen9
dann hätte z.B. die Zeichnung eines Würfels
Sinn aber nicht die eines Sechsecks mit seinen Diagonalen. Und denken wir an das10
Beispiel vom
Einzeichnen einer Reiseroute in die beiden Erdprojektionen.
11
Was heißt es denn ”entdecken, daß ein Satz keinen Sinn hat“? Oder fragen wir so: Wie
kann man denn die Unsinnigkeit eines Satzes (etwa:12
”dieser Körper ist ausgedehnt“) dadurch
bekräftigen, daß man sagt: ”Ich kann mir nicht vorstellen, wie es anders wäre“?
1 (M): 3
2 (M): ü / 3
3 (M): ? / 3
4 (V): ”Ich habe
5 (M): / 3 ||
6 (V): man so etwas sagen wie:
7 (V): Ausdrücke
8 (V): Aber so ist es nicht Also
9 (V): sollen // Denken wir uns nach
10 (V): das sinnlose Stück in der Zeichnung einer
Reiseroute
11 (M): / 3
12 (V): Satzes (etwa):
95
96
95v
96
78
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26
Being Able to Imagine “What it
Would be Like” as a Criterion for
a Proposition Having a Sense.
1
What does it mean when one says: “I can’t imagine the opposite of that” or “What would
it be like if it were otherwise?”?. For example, when someone has said that my mental images
are private or that only I can know whether I am feeling pain, and things like that.
2
If I can’t imagine how it would be otherwise then I also can’t imagine how it can be like
this.
For here “I can’t imagine” doesn’t mean what it means in the sentence “I can’t imagine
a skull”. In saying the former, I’m not trying to indicate a lack of imagination.
3
Consider: “In fact4
I’ve never seen a black spot gradually turning lighter and lighter
until it was white, and then turning more and more reddish until it was red; but I know
that that is possible, because I can imagine it. That is, using my mental images I operate
within the realm of colours and do with them what can be done with colours.” ((Cf.
“logical possibility”.))
5
It seems that one could say:6
Word-language allows nonsensical combinations7
of words,
but the language of imagining doesn’t allow nonsensical mental images. (Of course this can’t
mean anything as it stands.)
So8
too the language of drawing doesn’t allow nonsensical drawings. – But that’s not so, for a
drawing can be nonsensical in the same way as a “proposition”. Let’s imagine a drawing that is to
serve as a model for three-dimensional objects;9
then a drawing of a cube for instance would make
sense, but not one of a hexagon with its diagonals. And let’s consider the example of drawing a route10
on the two hemispheric projections of the earth.
11
What does “to discover that a sentence has no sense” really mean? Or let’s put the question
this way: How can one reinforce the senselessness of a sentence (say: “This body is
extended”) by saying “I can’t imagine how it could be otherwise”?
1 (M): 3
2 (M): r / 3
3 (M): ? / 3
4 (V): “In fact
5 (M): / 3 ||
6 (V): say something like:
7 (V): expressions
8 (V): But that’s not the way it is So
9 (V): objects; // Let’s imagine in accordance with
10 (V): consider the senseless portion in the drawing of
a route
11 (M): / 3
78e
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Denn, kann ich etwa versuchen, es mir vorzustellen? Heißt es nicht: Zu sagen, daß ich es
mir vorstelle, ist sinnlos? Wie hilft mir dann also diese Umformung von einem Unsinn in
einen andern? – Und warum sagt man gerade: ”ich kann mir nicht vorstellen, wie es anders
wäre“? und nicht – was doch auf dasselbe hinauskommt – ”ich kann mir nicht vorstellen,
wie das wäre“?
Man erkennt scheinbar in dem unsinnigen Satz etwas, wie eine Tautologie, zum
Unterschied von einer Kontradiktion. Aber das ist ja auch falsch. – Man sagt gleichsam: ”Ja,
er13
ist ausgedehnt, aber wie könnte es denn anders sein? also, wozu es sagen!“
Es ist dieselbe Tendenz, die uns auf den Satz ”dieser Stab hat eine bestimmte Länge“
nicht antworten läßt ”Unsinn!“, sondern ”Freilich!“.
Was ist aber der Grund (zu) dieser14
Tendenz? Sie könnte auch so beschrieben werden:
wenn wir die beiden Sätze ”dieser Stab hat eine Länge“ und seine Verneinung ”dieser Stab
hat keine Länge“ hören, so sind wir parteiisch und neigen dem ersten Satz zu (statt beide
für Unsinn zu erklären).
Der Grund hievon ist aber eine Verwechslung: Wir sehen den ersten Satz verifiziert (und
den zweiten falsifiziert) dadurch, ”daß der Stab 4m hat“. Und man wird sagen: ”und 4m ist
doch eine Länge“ und vergißt, daß man hier einen Satz der Grammatik hat.
15
Warum sieht man es als Beweis dafür an, daß ein Satz Sinn hat: daß16
ich mir, was er
sagt, vorstellen kann? Ich könnte sagen: Weil ich diese Vorstellung mit einem dem ersten
verwandten Satz beschreiben müßte.
17
Könnte ich durch eine Zeichnung darstellen, wie es ist, wenn es sich so verhält, wenn
es keinen Sinn hätte, zu sagen ”es verhält sich so“?
Zu sagen ”ich kann aufzeichnen wie es ist, wenn es sich so verhält“ ist18
hier eine grammatische
Bestimmung über den betrachteten Satz (denn ich will ja nicht sagen, ich könne
es zeichnen, etwa weil ich zeichnen gelernt habe u.s.w.). Wie, wenn ich sagte: ”ist das kein
Spiel, da ich doch darin gewinnen und verlieren kann?“ – Nun, wenn das Dein Kriterium
eines Spieles ist, dann ist es ein Spiel.
19
”Ich weiß, daß es möglich ist, weil . . .“. Diese Ausdrucksform ist von Fällen hergenommen,
wie: ”Ich weiß, daß es möglich ist, die Tür mit diesem Schlüssel aufzusperren, weil
ich es schon einmal getan habe“. Vermute ich also in dem Sinn, daß dieser Farbenübergang
möglich sein wird, weil ich mir ihn vorstellen kann?! Muß es nicht vielmehr heißen: der
Satz ”der Farbenübergang ist möglich“ heißt dasselbe wie der: ”ich kann ihn mir
vorstellen“, oder: der erste Satz folgt aus dem zweiten? – Wie ist es damit: ”Das ABC läßt
sich laut hersagen, weil ich es mir im Geiste vorsagen kann“?
”Ich kann mir vorstellen, wie es wäre“, oder – was wieder ebenso gut ist – : ”ich kann es
aufzeichnen, wie es wäre, wenn p der Fall ist“ gibt eine Anwendung des Satzes. Es sagt etwas
über den Kalkül, in welchem wir p verwenden.
13 (V): es
14 (V): dieser E
15 (M): / 3
16 (V): hat, daß
17 (M): 5555
18 (O): verhält ”ist
19 (M): ü / 3
97
79 Sich vorstellen können, ”wie es wäre“ . . .
WBTC022-29 30/05/2005 16:18 Page 158
For can I possibly try to imagine it? Doesn’t it mean: To say that I am imagining it is
senseless? So how then does this transformation from one piece of nonsense into another
help me? – And why does one say: “I can’t imagine how it could be otherwise” and not “I
can’t imagine what that would be like” – which, after all, amounts to the same thing?
Seemingly one discovers something like a tautology, as opposed to a contradiction, in the
nonsensical sentence. But that too is false. – One says, as it were, “Yes, it is extended, but
how could it be otherwise? So why say it!”
It is the same tendency that causes us to respond to the sentence “This rod is of a certain
length” by saying “Certainly!” rather than “Nonsense!”.
But what’s the reason for this tendency? It could also be described this way: if we hear
the two sentences “This rod has a length” and its denial “This rod has no length”, then we
take sides and favour the former (rather than declaring both to be nonsense).
But the reason for this is a confusion: We consider the first sentence verified (and the
second falsified) by “The rod has a length of 4 metres”. And we’ll say: “After all, 4 metres
is a length”, forgetting that what we have here is a grammatical proposition.
12
Why does one view being able to have a mental image of what a proposition says as proof
that it makes sense? I could say: Because I would have to describe this mental image with a
proposition that’s related to the original.
13
Could I make a drawing to show what it’s like if that’s the way it is, if it made no sense
to say “That’s the way it is”?
To say “I can draw how it is if that’s the way it is” is a grammatical stipulation about the
proposition under consideration (for I don’t want to say that I could draw this, say, because
I had learned to draw, etc.). What if I said: “Isn’t this a game, since I can win and lose at
it?” – Well, if that is your criterion for a game then it is a game.
14
“I know that it’s possible, because . . .”. This form of expression derives from cases such
as: “I know that it’s possible to unlock the door with this key because I’ve done it once before”.
So is this the sense in which I surmise that this colour transition will be possible, because I
can imagine it?! Mustn’t I say instead: The sentence “The colour transition is possible” means
the same thing as: “I can imagine it”, or: “The first sentence follows from the second”? –
How about this: “The ABC’s can be said aloud because I can recite them in my mind”?
“I can imagine what it would be like” or – what is just as good – “I can draw what it
would be like, if p is the case” gives me an application of the sentence. It says something
about the calculus in which we use p.
12 (M): / 3
13 (M): 5555
14 (M): r / 3
Being Able to Imagine “What it Would be Like” . . . 79e
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27
”Logische Möglichkeit und
Unmöglichkeit“. – Das Bild
des ”Könnens“ ultraphysisch
angewandt. (Ähnlich: ”Das
ausgeschlossene Dritte“.)
1
Wenn man sagt, die Substanz ist unzerstörbar, so meint man, es ist sinnlos, in irgend einem
Zusammenhang – bejahend oder verneinend – von dem ”Zerstören einer Substanz“ zu reden.
2
Man kann auch3
zeigen, daß ein Satz metaphysisch gemeint ist indem man fragt: Ist das4
eine
Erfahrungstatsache? Kannst Du Dir denken (vorstellen) daß es anders wäre. Willst Du sagen
Substanz sei noch nie zerstört worden oder es sei undenkbar daß sie zerstört werde?
Undenkbar
5
Seltsam daß man sollte sagen können das & das sei undenkbar! Auch wenn wir im Denken
wesentlich eine Begleitung des Ausdrucks sehen so müssen6
also doch die Worte &das & das“ in diesem
Satz unbegleitet sein. Was soll er also für einen Sinn haben? Es sei denn daß er aussagen soll diese
Worte seien sinnlos. Aber dann ist nicht quasi ihr Sinn sinnlos sondern sie werden aus unserer Sprache
ausgeschaltet wie irgend ein beliebiges Geräusch & der Grund ihrer ausdrücklichen Ausschließung
kann nur darin liegen daß wir aus irgend einem Grunde versucht sind das Gebilde mit einem Satz
unserer Sprache zu verwechseln.
7
Ich versuche etwas, kann es aber nicht. – Was heißt es aber: ”etwas nicht versuchen
können“?
”Wir können auch nicht einmal versuchen, uns ein rundes Viereck vorzustellen.“
8
Logische Möglichkeit und Sinn. Kann man fragen: ”wie müssen die grammatischen Regeln
für die Wörter beschaffen sein, damit sie einem Satz Sinn geben“?
9
Der Gebrauch des Satzes, das ist sein Sinn.
10
Ich sage z.B. ”auf diesem Tisch steht jetzt keine Vase, aber es könnte eine da stehn; dagegen
ist es unsinnig11
zu sagen, der Raum könnte vier Dimensionen haben.“ Aber wenn der
1 (M): /// 3
2 (M): /
3 (V): auch einem
4 (V): das nun
5 (M): ? /
6 (V): so ist // sind
7 (M): / 3
8 (M): ∫ 3
9 (M): ∫
10 (M): ∫ 3
11 (V): sinnlos
98
97v
98
80
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27
“Logical Possibility and
Impossibility”. – The Picture
of “Being Able To” Applied
Ultraphysically. (Similar to:
“The Excluded Middle”.)
1
If we say that substance is indestructible, we mean that it is senseless to talk of “destroying
a substance” in any context – either by affirming or denying it.
2
One can also show3
that a proposition is meant metaphysically by asking: Is this4
an empirical fact?
Can you think (imagine) that it were otherwise? Do you want to say that substance has never been
destroyed, or that it is unthinkable that it be destroyed?
Unthinkable
5
Strange, that one should be able to say “This and that is unthinkable”! Even if we regard thought
essentially as an accompaniment to an expression, in this sentence the words “this and that” still have
to be6
unaccompanied. So what sense is the sentence to have? Unless it is supposed to state that
these words are senseless. But then it’s not their sense that is senseless, as it were. Rather they are
eliminated from our language like some random noise, and the reason for their explicit exclusion
can only be that for some reason we are tempted to confuse this structure with a sentence in our
language.
7
I try something but can’t do it. – But what does this mean: “not to be able to try
something”?
“We can’t even try to imagine a round rectangle”.
8
Logical possibility and sense. Can one ask: “What must the grammatical rules for words
be like for them to give sense to a sentence?”
9
The use of a proposition – that is its sense.
10
I say for instance, “There is no vase standing on this table now, but there could be; on
the other hand it is nonsensical11
to say that space could have four dimensions”. But if a
1 (M): /// 3
2 (M): /
3 (V): show someone
4 (V): this now
5 (M): ? /
6 (V): still is // are
7 (M): / 3
8 (M): ∫ 3
9 (M): ∫
10 (M): ∫ 3
11 (V): senseless
80e
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Satz dadurch sinnvoll wird, daß er mit den grammatischen Regeln im Einklang ist, nun, so
machen wir eben die Regel, die den Satz, unser Raum habe vier Dimensionen, erlaubt. Wohl,
aber damit ist nun die Grammatik dieses Ausdrucks noch nicht festgelegt. Nun müssen erst
noch weitere Bestimmungen darüber getroffen12
werden, wie ein solcher Satz zu gebrauchen
ist, wie er etwa verifiziert wird.
13
Wenn man auch den Satz als Bild des beschriebenen Sachverhalts auffaßt und sagt, der
Satz zeige eben wie es ist, wenn er wahr wäre, er zeige also die Möglichkeit des behaupteten
Sachverhalts, so kann der Satz doch bestenfalls tun, was ein gemaltes oder modelliertes Bild
tut, und er kann also jedenfalls nicht das erzeugen,14
was15
nicht der Fall ist. Also hängt es
ganz von unserer Grammatik ab, was möglich genannt wird und was nicht, nämlich eben,
was sie zuläßt. Aber das ist doch willkürlich! – Gewiß, aber nicht mit jedem Gebilde kann
ich etwas anfangen; d.h.: nicht jedes Spiel ist nützlich, und wenn ich versucht bin, etwas
ganz Unnützes einen Satz zu nennen,16
so geschieht es, weil ich mich durch eine Analogie dazu
verleiten lasse und nicht sehe, daß mir für meinen Satz noch die wesentlichen Regeln der
Anwendung fehlen.17
So ist es z.B., wenn man von einer unendlichen Baumreihe redet
und sich nicht fragt, wie es denn zu verifizieren sei, daß eine Baumreihe unendlich ist, und
was etwa die Beziehung dieser Verifikation zu der des Satzes ”die Baumreihe hat 100
Bäume“ ist.18
12 (V): gemacht
13 (M): ? / 3
14 (V): hinstellen,
15 (V): was nun eben (M): [Frege]
16 (V): ganz Nutzloses als Satz zuzulassen,
17 (V1): gewiß, aber nicht jeder Kalkül der dem, mit
gewissen unserer Erfahrungssätzen, analog ist,
ist irgendwie n ist irgendwie von Nutzen. Nicht
jedes Gebilde das in sei so einem Kalkül jenen
Erfahrungssätzen entspricht werden wir Satz nennen
wollen. (V2): Gewiß aber unsere Erfahrungssätze
z.B. die, welche sich durch ein gemaltes Bild
ersetzen ließen weil sie eine sichtbare Verteilung von
Körpern beschreiben haben eher eine bestimmte
Anwendung einen bestimmten Nutzen. Aber nicht
18 (R): ∀ 145/3
99
98v
81 ”Logische Möglichkeit und Unmöglichkeit“. . . .
WBTC022-29 30/05/2005 16:18 Page 162
proposition becomes meaningful by conforming to grammatical rules, well, let’s simply fabricate
the rule that allows for the proposition that our space has four dimensions. Very well,
but this still doesn’t determine the grammar of this expression. Further stipulations still have
to be made about how such a proposition can be used, how for instance it is to be verified.
12
Even if one conceives of a proposition as a picture of the state of affairs it describes and
says that what a proposition shows is what it’s like if it were true, and that thus it shows the
possibility of the state of affairs that’s being asserted, still the most a proposition can do is
what a painted or sculpted image does. So in any case it can’t produce13
what14
is not the
case. Therefore what we call possible and what not depends entirely on our grammar, i.e.
on what it permits. But that’s arbitrary! – Certainly, but I can’t do something with just any
structure; that is: not every game is useful, and if I am tempted to call something completely
useless a proposition,15
then this happens because I allow myself to be seduced by an analogy,
and I don’t see that I am still lacking the essential rules of application for my proposition.16
That’s how it is, for example, when we talk about an endless row of trees and don’t ask ourselves
how it can be verified that a row of trees is endless, and what, say, the relationship of
this verification is to that of the proposition “The row consists of 100 trees”.17
12 (M): ? / 3
13 (V): can’t set down
14 (V): what simply (M): [Frege]
15 (V): tempted to allow something completely
useless as a proposition,
16 (V1): Certainly, but not every calculus that corresponds
to certain empirical propositions is analogous,
is of some sort n is of some sort of use. But we
won’t want to call every structure that corresponds
to those empirical propositions in such a calculus a
proposition. (V2): Certainly, but our empirical
propositions, e.g. those that describe a visible distribution
of objects and thus could be replaced by a
painted picture, tend to have a particular application,
a particular use. But we won’t . . .
17 (R): ∀ 145/3
“Logical Possibility and Impossibility”. . . . 81e
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28
Elementarsatz.
1
Kann ein logisches Produkt in einem Satz verborgen sein? Und wenn, wie erfährt man
das, und was für Methode haben wir, das im Satz Verborgene ans Licht2
zu ziehen? Haben
wir noch keine Methode,3
(es zu finden,) dann können wir auch nicht davon reden, daß etwas
verborgen ist, oder verborgen sein könnte. Und haben wir eine Methode des Suchens, so
kann das logische Produkt, z.B.,4
im Satz nur so verborgen sein, wie es etwa der Quotient
753 : 3 ist solange die Division noch nicht ausgeführt ist.5
Die Frage ob ein log. Produkt in einem
Satz versteckt sei ist ein mathematisches Problem. Denn, das verborgene logische Produkt
finden, ist eine mathematische Aufgabe.
6
Also ist Elementarsatz ein solcher, der sich in dem Kalkül, wie ich ihn jetzt7
benütze, nicht
als Wahrheitsfunktion anderer Sätze darstellt.
Die Idee, Elementarsätze zu konstruieren (wie dies z.B. Carnap versucht hat), beruht auf
einer falschen Auffassung der logischen Analyse. Das Problem dieser Analyse ist nicht: es sei8
eine Theorie der Elementarsätze zu finden.9
Als seien Prinzipien der Mechanik zu finden.10
11
Meine Auffassung in der log. – phil. Abhandlg.12
war falsch: 1.)13
weil ich mir über den
Sinn der Worte ”in einem Satz ist ein logisches Produkt versteckt“ (und ähnlicher) nicht
klar war, 2.)14
weil auch ich dachte, die logische Analyse müsse verborgene Dinge an den
Tag bringen (wie es die chemische und physikalische tut).
15
Man kann den Satz ”dieser Ort ist jetzt rot“ (oder ”dieser Kreis ist jetzt rot“, etc.) einen
Elementarsatz nennen, wenn man damit sagen will, daß er weder eine Wahrheitsfunktion
anderer Sätze ist, noch als solche definiert (ist). (Ich sehe hier von Verbindungen der Art
p & (q .1. ~q) und analogen ab.)
Aus ”a ist jetzt rot“ folgt aber ”a ist jetzt nicht grün“ und die Elementarsätze in diesem
Sinn sind also nicht von einander unabhängig, wie die Elementarsätze in meinem seinerzeit
beschriebenen Kalkül, von dem ich annahm, der ganze Gebrauch der Sätze müsse sich auf
ihn zurückführen lassen; – verleitet durch einen falschen Begriff von dieser Zurückführung.16
1 (M): /
2 (V): Tageslicht
3 (V): keine sicheren Methoden,
4 (V): kann – das logische Produkt, etwa,
5 (V): es etwa die Teilbarkeit durch 3 in der Zahl
753 ist, solange ich das Kriterium noch nicht
angewandt habe, oder aber auch die solange
ich sie noch nicht ausgerechnet habe.
6 (M): /
7 (V): ich es heute
8 (V): Analyse besteht nicht darin%
9 (V1): logischen Analyse. Sie betrachtet das
Problem dieser Analyse als das, eine Theorie
7
. . . zu finden. (V2): Elementarsätze zu
finden^
10 (V): Sie lehnt sich an das an, was, in der
Mechanik z.B., geschieht, wenn eine Anzahl von
Grundgesetzen gefunden wird, aus denen das
ganze System von Sätzen hervorgeht.
11 (M): /
12 (V): Meine eigene Auffassung
13 (V): falsch: Teils,
14 (V): war, zweitens,
15 (M): ü /
16 (V): von diesem ”zurückführen“.
100
101
82
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28
Elementary Proposition.
1
Can a logical product be hidden in a proposition? And if so, how does one find this out,
and what methods do we have of pulling what is hidden in the proposition into the open?2
So long as we don’t have any3
methods (for finding it), then we can’t speak of something
being hidden or possibly hidden. And if we have a method for looking for it, then a logical
product, for instance, can be hidden in a proposition only in the way that, say, the quotient
of 753 ÷ 3 is hidden so long as the division hasn’t been carried out.4
The question whether a logical
product is hidden in a proposition is a mathematical problem. For finding the hidden logical product
is a mathematical task.
5
So an elementary proposition is one that, in the calculus as I am now6
using it, doesn’t
represent itself as a truth-function of other propositions.
The idea of constructing elementary propositions (as Carnap, for example, tried to do) is
based on a false conception of logical analysis. The problem of that analysis is not:7
a theory
of elementary propositions must be discovered.8
As if principles of mechanics had to be discovered.9
10
My view in the Tractatus Logico-Philosophicus11
was wrong: 1.12
because I didn’t clearly understand
the sense of the words “a logical product is hidden in a proposition” (and similar words),
2.13
because I too thought that logical analysis would have to bring hidden things to light (as
do chemical and physical analysis).
14
One can call the proposition “This place is now red” (or “This circle is now red”, etc.)
an elementary proposition, if one means by this that it is neither a truth-function of other
propositions nor (is it) defined as such. (Here I am disregarding combinations of the sort
p & (q .1. ~q), and ones like it.)
But from “a is now red”, “a is now not green” follows, and so elementary propositions in
this sense aren’t independent of each other like the elementary propositions in the calculus
I described earlier – a calculus to which I assumed the entire use of propositions must be
reducible – seduced by a false concept of such a reduction.15
1 (M): /
2 (V): into daylight?
3 (V): any certain
4 (V): say, the divisibility by 3 is hidden in the
number 753 so long as I havenFt yet applied the
criterion, or also the so long as I havenFt yet
calculated it.
5 (M): /
6 (V): I am today
7 (V): analysis doesnDt consist of%
8 (V): conception of logical analysis. It views the
problem of this analysis as discovering a theory
of elementary propositions.
7
9 (V): It relies on what happens in mechanics, for
instance when a number of basic laws are found
from which the whole system of propositions
emerges.
10 (M): /
11 (V): My own view
12 (V): wrong: in part,
13 (V): words), second,
14 (M): r /
15 (V): of “reducing”.
82e
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1
29
”Wie ist die Möglichkeit von
p in der Tatsache, daß ~p der
Fall ist, enthalten?“
”Wie enthält z.B. der schmerzlose
Zustand die Möglichkeit der
Schmerzen?“
2
Man scheint etwas über den Zustand der Schmerzlosigkeit zu sagen wenn man sagt daß er die
Möglichkeit des Schmerzes enthalten3
muß. Man redet aber nur vom System der Bilder das wir
verwenden.4
5
”
Über6
den schmerzlosen Zustand sinnvoll reden7
setzt die Fähigkeit voraus, Schmerzen zu
fühlen und das kann keine ,physiologische Fähigkeit‘8
sein, – denn wie wüßte man sonst,
wozu es die Fähigkeit ist – sondern eine logische Möglichkeit. – Ich beschreibe meinen gegenwärtigen
Zustand durch die Anspielung auf Etwas, was nicht der Fall ist. Das zu sagen9
ist
irreführend, denn es klingt,10
als sei es eine Anspielung auf einen nicht Existierenden, während es eine
Anspielung auf einen Abwesenden ist. Aber auch das ist irreführend. Wenn diese Hinweisung zu
der Beschreibung nötig ist (und nicht bloß eine Verzierung), so muß in meinem gegenwärtigen
Zustand etwas liegen, was diese Erwähnung (Hinweisung) nötig macht. Ich vergleiche
diesen Zustand mit einem anderen, also muß er mit ihm vergleichbar sein. Er muß auch im
Schmerzraum liegen, wenn auch an einer andern Stelle. – Sonst würde mein Satz etwa heißen,
mein gegenwärtiger Zustand hat mit einem schmerzhaften nichts zu tun; etwa, wie ich sagen
würde, die Farbe dieser Rose hat mit der Eroberung Galliens durch Cäsar nichts zu tun. D.h.
es ist kein Zusammenhang vorhanden. Aber ich meine gerade, daß zwischen meinem jetzigen
Zustand und einem schmerzhaften ein Zusammenhang besteht.“ Ich meine nur, was ich sage.
In wiefern ist aber Schmerzlosigkeit ein Zustand. Was nenne ich einen ”Zustand“?
11
Wenn ich sage, ich habe heute Nacht nicht geträumt, so muß ich doch wissen, wo nach
dem Traum zu suchen wäre (d.h., der Satz ”ich habe geträumt“ darf, auf die Situation
angewendet, nur falsch, aber nicht unsinnig sein).
1 (R): Siehe Sinn & Grammatik
2 (M): ü / 3
3 (O): enhalten
4 (R): ∀ S. 388, 389
5 (M): ∫ 3 //// Vielleicht lehrreich. Sonst U.
6 (O): Uber
7 (V): Der schmerzlose Zustand
8 (V): keine @physiologische Disposition!
9 (V): Diese Ause
10 (V): ist,
11 (M): ///
102
101v
102
103
83
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1
29
“How is the Possibility of
p Contained in the Fact that
~p is the Case?”
“How Does, for Example,
a Pain-free State Contain
the Possibility of Pain?”
2
We seem to be saying something about not being in pain when we say that this state has to contain
the possibility of pain. But all we’re talking about is the system of pictures that we use.3
4
“To talk meaningfully about a painless state5
presupposes the capacity to feel pain, and that
cannot be a ‘physiological capacity’.6
Rather, it has to be a logical possibility – for how else
would one know what it is the capacity for? – I describe my present state by alluding to something
that is not the case. To say that is misleading, because it sounds like7
an allusion to a
non-existent state, whereas it is an allusion to one that is absent. But that too is misleading. If this
reference is necessary for the description (and isn’t merely an embellishment), then there
must be something in my present state that makes this mention (reference) necessary. I am
comparing this state with another, so it must be comparable to it. It too must be situated
within pain-space, although at a different point. – Otherwise my sentence would read, for
example, ‘My present state has nothing to do with a painful one’ – as I might say the colour
of this rose has nothing to do with Caesar’s conquest of Gaul. That is, there is no connection.
But what I mean is precisely that there is a connection between my present state and
a painful one.” I only mean what I say.
But how is the absence of pain a state? What is it that I’m calling a “state”?
8
If I say I did not dream last night, still I must know where one could look for the dream
(that is, when it is used in this situation the proposition “I had a dream” can only be false,
not nonsensical).
1 (R): See Sense and Grammar
2 (M): r / 3
3 (R): ∀ pp. 388, 389
4 (M): ∫ 3 //// Perhaps instructive. Otherwise
useless
5 (V): A painless state
6 (V): a !physiological dispositionE.
7 (V): it is
8 (M): ///
83e
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12
Ich drücke die gegenwärtige Situation durch eine Stellung – die negative – der
Signalscheibe ”Träume – keine Träume“ aus. Ich muß sie aber trotz ihrer negativen
Stellung von andern Signalscheiben unterscheiden können. Ich muß wissen, daß ich diese
Signalscheibe in der Hand habe.
13
Man könnte nun fragen: Heißt das, daß Du doch etwas gespürt hast, sozusagen die
Andeutung eines Traums, die Dir die Stelle zum Bewußtsein bringt, an der ein Traum
gestanden wäre? 14
Oder, wenn ich sage ”ich habe keine Schmerzen im Arm“, heißt das, daß
ich eine Art schattenhaftes Gefühl habe, welches die Stelle andeutet, in die der Schmerz
eintreten würde? Doch offenbar, nein. Aber muß ich nicht wissen, wie es wäre wenn ich Schm.
hätte?15
16
In wiefern enthält der gegenwärtige, schmerzlose Zustand17
die Möglichkeit der
Schmerzen?
Wenn einer sagt: ”Damit das Wort Schmerzen Bedeutung habe, ist es notwendig, daß
man Schmerzen als solche erkennt, wenn sie auftreten“, so kann man antworten: ”Es ist nicht
notwendiger, als daß man das Fehlen von Schmerzen erkennt“.
18
”Schmerzen“ heißt sozusagen der ganze Maßstab und nicht einer seiner Teilstriche. Daß
er auf einem bestimmten Teilstrich steht, ist durch einen Satz auszudrücken.
19
”Was wäre das für eine Frage: ,Könnte denn Alles nicht der Fall sein, und nichts der –
Fall – sein‘? Könnte man sich einen Zustand einer Welt denken, in dem mit Wahrheit nur
negative Sätze zu sagen wären? Ist das nicht offenbar alles Unsinn? Gibt es denn wesentlich
negative und positive Zustände?“ Nun, es kommt darauf an, was man ”Zustände“ nennt.
Man kommt nicht davon weg, daß die Benützung des Satzes darin besteht daß man sich bei jedem
Wort etwas vorstellt.20
Die Anwendg. des Satzes ist nicht die, die eine solche Vorstellung fordert. Immer wieder möchte
man sich den Sinn eines Satzes, also seine Verwendung21
(seinen Nutzen) in einem22
Geisteszustand
des Redenden23
konzentriert denken. Man denkt nicht, daß man mit ihm rechnet, operiert, ihn mit
der Zeit durch dies oder jenes Bild ersetzt. Sondern sein Sinn, d.i. aber sein24
Zweck, soll in einem Zustand
liegen.25
Wie weiß Einer daß er nicht taub ist wenn er kein Geräusch hört, & daß er nicht innerlich taub ist,
wenn er sich keins vorstellen kann.
26
Ist absolute Stille zu verwechseln mit innerer Taubheit, ich meine der Unbekanntheit
mit dem Begriff des Tones? Wenn das der Fall wäre, so könnte man den Mangel des
Gehörssinnes nicht von dem Mangel eines andern Sinnes unterscheiden.
27
Ist das aber nicht genau dieselbe Frage wie: Ist der Mann, der jetzt nichts Rotes um sich
sieht, in derselben Lage, wie der, der unfähig ist, rot zu sehen?
12 (M): ///
13 (M): / /// – ”gestanden wäre?“
14 (M): 3
15 (O): hatte?
16 (M): /
17 (V): schmerzlose, Zustand
18 (M): ∫ ///
19 (M): ∫ 555
20 (V): etwas vorstellen muß.
21 (V): Anwendung
22 (V): dem
23 (O): redenden
24 (V): d.i. sein
25 (V): soll in einer Art Bild liegen die . . .
26 (M): ∫ 3 ///
27 (M): ///
104
84 die Möglichkeit von p . . .
WBTC022-29 30/05/2005 16:18 Page 168
9
I express the present situation through a position – the negative position – of the signal
disk that signifies “dreams – no dreams”. But in spite of its negative position, I must be able
to distinguish it from other signal disks. I must know that I am holding this signal disk in
my hand.
10
Now one could ask: Does that mean that you did feel something after all, the hint of a
dream, as it were, which makes you conscious of the place the dream would have occupied?
11
Or if I say “I have no pain in my arm” does that mean that I have a kind of shadowy feeling
that indicates the place where pain might occur? Obviously not. But don’t I have to know
what it would be like if I had pain?
12
In what way does my present painless state contain the possibility of pain?
If someone says: “For the word ‘pain’ to have a meaning it’s necessary that pain be
recognized as such when it occurs”, then one can respond: “It’s no more necessary than
that one recognize the absence of pain”.
13
“Pain” means the entire measuring stick, as it were, and not one of its graduation marks.
That it is located on a certain mark is to be expressed by a proposition.
14
“What kind of a question would this be: ‘Mightn’t everything not be the case and
nothing the case’? Can one imagine a state of the world in which only negative propositions
could be uttered truthfully? Isn’t all of this obviously nonsense? Are there such things as
essentially negative or positive states?” Well, it depends on what one calls “states”.
It’s hard to leave the path that the use of a sentence consists in imagining15
something in response
to every word.
It is not the use of a proposition that demands such an imagining. Over and again we’re inclined
to think of the sense of a proposition, i.e. its application16
(its use), as concentrated in a speaker’s
mental state. We don’t think about calculating with it, operating with it, replacing it with this or that
picture as time goes by. Rather its sense – and that17
is its purpose – is supposed to be contained in
a state.18
How does someone know that he isn’t deaf if he doesn’t hear noises, and that he isn’t deaf inside
if he can’t imagine any?
19
Can absolute silence be confused with inner deafness – I mean with unfamiliarity with
the concept of a sound? If that happened one couldn’t distinguish the lack of the sense of
hearing from the lack of any other sense.
20
But isn’t this exactly the same question as: Is the man who presently sees nothing red
around him in the same situation as one who is incapable of seeing red?
9 (M): ///
10 (M): / /// – “occupied?”
11 (M): 3
12 (M): /
13 (M): ∫ ///
14 (M): ∫ 555
15 (V): in having to imagine
16 (V): its employment
17 (V): sense – that
18 (V): contained in a kind of image that . . .
19 (M): ∫ 3 ///
20 (M): ///
How is the Possibility of p Contained . . . 84e
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28
Man kann natürlich sagen: Der Eine kann sich rot doch vorstellen, aber das vorgestellte
Rot ist ja nicht dasselbe, wie das gesehene.
29
Nun, worin äußert sich denn die Fähigkeit, rot zu sehen und worin die Bekanntschaft
mit dem Begriff des Tons? Man wird sagen: Er muß wissen was &Ton“ heißt. Aber was heißt es
das zu wissen? – Ich sage &ich weiß was ,rot‘ heißt“ – Jemand fragt: &Bist Du sicher?“ – Was würde
ich da tun um mich davon zu überzeugen?30
31
Wenn ich nur etwas Schwarzes sehe und sage, es ist nicht rot, wie weiß ich, daß ich
nicht Unsinn rede, d.h. daß es rot sein kann,32
daß es Rot gibt? Wenn nicht rot eben ein
anderer Teilstrich auf dem Maßstab ist, auf dem auch schwarz einer ist. Was ist der
Unterschied zwischen ”das ist nicht rot“ und ”das ist nicht abracadabra“? Ich muß offenbar
wissen, daß ”schwarz“, welches den tatsächlichen Zustand beschreibt (oder beschreiben
hilft) das ist, an dessen Stelle in der Beschreibung ”rot“ steht.33
34
Das Gefühl ist, als müßte ~p, um p zu verneinen, es erst in gewissem Sinne wahr machen.
Man fragt ”was ist nicht der Fall“. Dieses muß dargestellt werden, kann aber doch nicht so
dargestellt werden, daß p wirklich wahr gemacht wird.
35
”Das Grau muß bereits im Raum von dunkler und heller vorgestellt sein, wenn ich davon
reden will, daß es dunkler oder heller werden kann.“ D.h.: es kann zum Verständnis des Satzes
gehören, daß man etwas helleres &36
dunkleres vor sich sieht & man sagt dann etwa:37
&dieses Grau
kann so oder auch so werden“.
38
Man könnte also vielleicht auch sagen: Der Maßstab muß schon angelegt sein, ich kann
ihn nicht – willkürlich – anlegen, ich kann nur einen Teilstrich darauf hervorheben.
Das kommt auf Folgendes hinaus: Wenn es um mich her vollkommen still ist, so kann
ich an diese Stille den Gehörsraum nicht willkürlich anbringen (aufbauen),39
oder nicht
anbringen. D.h., es ist für mich entweder still im Gegensatz zu einem Laut, oder das
Wort ”still“ hat keine Bedeutung für mich. D.h. ich kann nicht wählen zwischen innerem
Gehör und innerer Taubheit.
Und ebenso kann ich, wenn ich Grau sehe, nicht zwischen normalem innerem Sehen,
partieller oder vollkommener Farbenblindheit wählen.“
40
”Kann ich mir Schmerzen in der Spitze meines Nagels denken, oder in meinen
Haaren? Sind diese Schmerzen nicht ebenso wohl, und ebenso wenig vorstellbar, wie die an
irgend einer Stelle des Körpers, wo ich gerade keine Schmerzen habe und mich an keine
erinnere?“ 41
42
Wir43
sind versucht zu sagen; ”ich habe jetzt in der Hand keine Schmerzen“ heißt nur
etwas, wenn ich weiß wie es ist, wenn man Schmerzen in der Hand hat. Was heißt es, das
zu wissen? Was ist unser Kriterium dafür, daß man es weiß? Nun, ich würde sagen: ”ich
habe schon öfters Schmerzen gehabt“, ”ich habe öfters Schmerzen an dieser Stelle gehabt“,
oder ”ich habe zwar nicht an dieser Stelle Schmerzen gehabt, aber an andern Stellen meines
28 (M): ///
29 (M): |
30 (M): gut darüber nachzudenken.
31 (M): ∫ ///
32 (V): sein kann,
33 (M): vielleicht lehrreich
34 (M): 3
35 (M): vielleicht lehrreich ∫ 3
36 (V): helleres oder
37 (V): sieht & etwa sagt:
38 (M): ////
39 (V): anbringen, (aufbauen),
40 (M): ? / 3 ///
41 (R): (Siehe: Sinn & Grammatik)
42 (M): v.l. ∫ ///
43 (V): Sehen wir die Sache vom Standpunkt des
gesunden Menschenverstandes an. Wir
105
85 die Möglichkeit von p . . .
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21
Of course you can say, “But the former can imagine red”; but the imagined red isn’t the
same as the one that’s seen.
22
Well, how does the ability to see red manifest itself, and how does the familiarity with
the concept of sound? It will be said: He has to know what “sound” means. But what does it
mean to know that? – I say “I know what ‘red’ means”. – Someone asks: “Are you sure?” –
What would I do to convince myself of this?23
24
If all I see is something black and I say it isn’t red, how do I know that I’m not talking
nonsense, i.e. that it can25
be red, that there is red? Unless red is simply another graduation
mark on a yardstick on which black is situated. What’s the difference between “That
isn’t red” and “That isn’t abracadabra”? Obviously I must know that “black” – which
describes (or helps to describe) the actual state of affairs – is what is replaced by “red” in
a description.26
27
One has the feeling that in some sense ~p first has to make p true in order to negate it.
We ask “What is not the case?”. This has to be represented, but it can’t be represented in
such a way that p is actually made true.
28
“Grey must already have been imagined within the space of darker and lighter if I want
to speak about the fact that it can become darker or lighter.” That is: it can be part of understanding
a proposition that we see something lighter and29
darker in front of us, and then say, for
example:30
“This grey can become like this, or that.”
31
So perhaps one could also say: “The yardstick must already have been applied. I can’t
– arbitrarily – apply it; I can only emphasize a graduation mark on it”.
This amounts to the following: If everything around me is absolutely quiet, I can’t arbitrarily
attach (construct) or not attach the space of hearing to this silence. That is, as far as
I’m concerned either it’s quiet as opposed to there being a sound, or the word “quiet” has
no meaning. That is, I can’t choose between inner hearing and inner deafness.
And likewise, when I see grey I cannot choose between normal inner sight, and partial or
complete colour-blindness.
32
“Can I imagine pain in the tip of my nail or in my hair? Isn’t this pain just as much and
as little imaginable as a pain in any other part of my body where for the moment I don’t
have and don’t remember any pain?”33
34
We35
are tempted to say: “I don’t have any pain in my hand now” only means something
if I know what it’s like to have a pain in my hand. What does knowing this mean?
What is our criterion for knowing it? Well, I would say: “I’ve had pain several times”, “I’ve
had pain in that place several times”, or “I haven’t had pain in this place, but I have had it
21 (M): ///
22 (M): |
23 (M): Good to think about.
24 (M): ∫ ///
25 (V): it can
26 (M): Perhaps instructive
27 (M): 3
28 (M): Perhaps instructive ∫ 3
29 (V): or
30 (V): us, and say% for example:
31 (M): ////
32 (M): ? / 3 ///
33 (R): (See: Sense and Grammar)
34 (M): p. i. [E: perhaps instructive] ∫ ///
35 (V): LetFs look at the matter from the point of
view of common sense. We
How is the Possibility of p Contained . . . 85e
WBTC022-29 30/05/2005 16:18 Page 171
Körpers“. Es könnte gefragt werden: worin besteht die Erinnerung an Deine vergangenen
Schmerzen? fühlst Du sie in einer Art schattenhafter Weise wieder? Aber sei diese
Erfahrung (des Sich-Erinnerns) wie immer, sie ist eine bestimmte Erfahrung und ich nenne
sie die Erinnerung ”an Schmerzen, die ich gehabt habe“ und dies zeigt eben, wie ich das
Wort ”Schmerzen“ und den Ausdruck der Vergangenheit gebrauche.
44
Die Verneinung enthält eine Art Allgemeinheit durch das Gebiet von Möglichkeiten,
die sie offen läßt.
Aber freilich muß auch die Bejahung sie enthalten und nur einen andern Gebrauch von
ihr machen.
45
”~p“ schließt einfach p aus. Was dann statt p der Fall sein kann, folgt aus dem Wesen
des Ausgeschlossenen.46
47
Ist die Verneinung identisch einer Disjunktion der ausgeschlossenen Fälle?
Sie ist es in manchen Fällen & in manchen ist sie es nicht.
&Die Permutation48
von ABC die ich sah war nicht ACB.“
86 die Möglichkeit von p . . .
44 (M): ? / ////
45 (M): ∫
46 (V): ~p schließt p aus; was es dann zuläßt, hängt
von der Natur, d.h. der Grammatik, des p ab.
(R): ∀ S. 21 Anmerkung
47 (M): /
48 (V): &Die Komb
106
WBTC022-29 30/05/2005 16:18 Page 172
in other parts of my body.” The question could be raised: What does the memory of your
past pains consist in? Do you feel them again in a shadowy sort of way? But be this experience
(of remembering) as it may, it is a particular one, which I call the memory “of pain
that I have had”, and this serves to show how I use the word “pain” in conjunction with the
expression of the past.
36
Negation contains a kind of generality because of the range of possibilities that it leaves
open.
But of course affirmation must also contain this range. It just makes a different use of it.
37
“~p” simply excludes p. So what can be the case instead of p follows inherently from
what was excluded.38
39
Is negation identical to a disjunction of the excluded cases?
In some cases it is, and in some not.
“The permutation40
of ABC that I saw was not ACB.”
How is the Possibility of p Contained . . . 86e
36 (M): ? / ////
37 (M): ∫
38 (V): ~p excludes p; what it then allows depends
on the nature, i.e. the grammar, of p. (R):
∀ p. 21 note
39 (M): /
40 (V): comb
WBTC022-29 30/05/2005 16:18 Page 173
1 (O): Tatigkeit
2 (V): was hilft // nützt // das?
3 (O): nutzlicher
4 (M): /
5 (V): denn abgesehen von der Verneinung
6 (M): /
7 (V): einrahmen,
8 (V): Satz. Wie gesagt, das scheint ein
9 (V): anders. Ich glaube aber,
10 (V): und dem Sprechen – in dem wir ja doch
denken
11 (M): /
12 (V): Noch einmal, der
13 (V): Sprache
14 (V): Sprache
108
107
30
”Wie kann das Wort ,nicht‘
verneinen?“ Das Wort ”nicht“
erscheint uns wie ein Anstoß zu
einer komplizierten Tätigkeit
des Verneinens.
Verneinen, eine &geistige Tätigkeit1
“.
Verneine etwas & beobachte was Du tust. Du schüttelst etwa innerlich den Kopf. Nun und was
weiter?2
Ist das nützlicher3
als ein &~“ vor einen Satz schreiben?
4
”Wie kann das Wort ,nicht‘ verneinen?“ Ja, haben wir denn außer der Verneinung5
durch
ein Zeichen, noch einen Begriff von der Verneinung?
Doch es fällt uns dabei etwas ein, wie: Hindernis, abwehrende Geste, Ausschluß. Aber
das alles (ist) doch immer in einem Zeichen verkörpert.
6
Was ist der Unterschied zwischen: Wünschen, daß etwas geschieht und Wünschen, daß
dasselbe nicht geschieht?
Wollte man es bildlich darstellen, man würde mit dem Bild der Handlung etwas
vornehmen: es durchstreichen, in bestimmter Weise einzäunen,7
und dergleichen. Aber das
erscheint uns als eine rohe Methode des Ausdrucks; aber ich glaube, daß jede wesentlich ebenso
sein muß; in der Wortsprache setze ich das Zeichen ”nicht“ in den Satz. Das scheint uns wie
ein8
ungeschickter Behelf und man meint etwa, im Denken geschieht es schon anders. Aber,9
im Denken, Erwarten, Wünschen, geschieht es ganz ebenso. Sonst würde ja auch die
Diskrepanz zwischen dem Denken und der Sprache – in der wir doch denken10
– unerträglich
sein.
11
Der12
Ausdruck der Verneinung, den wir gebrauchen, wenn wir uns irgendeiner
Schrift13
bedienen, erscheint uns primitiv; als gäbe es einen richtigeren, der mir nur in den
rohen Verhältnissen dieser Ausdrucksform14
nicht zur Verfügung steht.
87
WBTC030-34 30/05/2005 16:17 Page 174
1 (V): instance. And how does that help? // And of what
use is that?
2 (M): /
3 (V): negation aside from
4 (M): /
5 (V): frame
6 (V): sentence. As we’ve said, this seems to be
7 (V): But I believe
8 (V): thinking and speaking – during which
9 (M): /
10 (V): To repeat, the
11 (V): language
12 (V): this language.
30
“How Can the Word ‘Not’
Negate?” The Word “Not”
Seems to Us Like an Impetus
to a Complicated Activity
of Negating.
Negating, a “mental activity”.
Negate something and notice what you’re doing. You shake your head inwardly, for instance. Well,
so what?1
Is that more effective than writing “~” in front of a sentence?
2
“How can the word ‘not’ negate?” Do we even have a concept of negation other than3
negation with a sign?
Yes, we can think of something like: impediment, rejecting gesture, exclusion. But all of
them as well (are) always embodied in a sign.
4
What’s the difference between: wishing that something will happen and wishing that the
same thing won’t happen?
If you wanted to portray it graphically you’d do something to the picture of the action:
cross it out, enclose5
it in a certain way, and other such things. This seems to us to be a crude
method of expression; but I think that every such method must essentially be just like it; in
word-language I place the sign “not” in the sentence. This seems to us6
a clumsy makeshift
and we think, e.g., that it happens differently in thinking. But7
in thinking, expecting, wishing,
it happens in exactly the same way. Otherwise the discrepancy between thinking and
language – in which8
we think, after all – would be intolerable.
9
The10
way we express negation in any sort of writing11
strikes us as primitive; as if there
were a more appropriate way that just isn’t at our disposal given the crude circumstances of
this form of expression.12
87e
WBTC030-34 30/05/2005 16:17 Page 175
109
88 ”Wie kann das Wort ,nicht‘ verneinen?“ . . .
15
Diesem Primitiven der Ausdrucksform, das uns bei der Verneinung aufgefallen ist, sind
wir auch anderwärts16
begegnet;17
wenn man nämlich etwa einem Menschen begreiflich machen
will, daß er einen gewissen Weg gehen soll, so kann man ihm den Weg aufzeichnen, und
hierin mit beliebig weitgehender Genauigkeit verfahren. Die Andeutung jedoch, die ihm
verständlich machen soll, daß er den Weg gehen soll, ist wieder von der primitiven Art,
die man gerne verbessern möchte.
18
”Was hilft es, daß als Negationszeichen nur ein Haken vor dem Satz p steht, ich muß
ja doch die ganze Negation denken.“
19
”Das Zeichen ,nicht‘20
deutet an, Du sollst,21
was darauf folgt, negativ auffassen.“22
23
”Es deutet an“24
heißt aber, daß dieses Zeichen der Verneinung nicht25
der letzte sprachliche
Ausdruck ist; daß26
das nicht das Bild des Gedankens ist. Daß mehr in der Negation
ist, als das.
27
Man möchte sagen: das Zeichen der Verneinung28
ist nur eine Veranlassung um etwas
sehr Kompliziertes29
zu tun; – aber was? Läßt sich die Frage nicht beantworten,30
so ist sie
unsinnig, und dann ist es auch jener erste Satz.
Es ist, als veranlaßte uns das Zeichen der Negation zu etwas; aber wozu? das31
wird32
nicht
gesagt. Es ist, als brauchte es nur angedeutet werden; als wüßten wir es schon. Als wäre eine
Erklärung33
unnötig, da wir die Sache ohnehin schon kennen.34
35
Gäbe es eine explizitere Ausdrucksweise der Negation, so müßte sie sich doch in die
andere abbilden lassen und könnte darum nicht von anderer Multiplizität sein.
36
Nun wäre aber die Frage: wie zeigt sich das uns bekannte Spezifische der Negation in
den Regeln, die vom Negationszeichen handeln.37
Daß z.B. ein gezeichneter Plan eines Weges
ein Bild des Weges ist, verstehen wir ohne weiteres; wo sich der gezeichnete Strich nach
links biegt, biegt38
auch der Weg nach links, etc. etc. Daß aber das Zeichen ”nicht“ den Plan
ausschließt, sehen wir nicht. Eher noch, wenn wir etwas ausgeschlossenes mit einem Strich
umfahren, gleichsam abzäunen.39
Aber so könnte man ja das ”~“ als eine Tafel auffassen
”Verbotener Weg“.
Denken wir aber daran, wie jemandem wirklich die Bedeutung so einer Tafel gelehrt würde.
Man würde ihn etwa zurückhalten, den Weg zu gehen.
40
”Ich sage doch diese Worte nicht bloß, sondern ich meine auch etwas mit ihnen“. Wenn
ich z.B. sage ”Du darfst nicht hereinkommen“, so ist es der natürliche Akt, zur Begleitung
15 (M): ∫ /
16 (V): wir schon früher
17 (O): Dieses Primitive der Ausdrucksform,
das . . . ist, haben wir . . . begegnet;
18 (M): /
19 (M): /
20 (V): Zeichen ” “
21 (V): sollst das,
22 (O): auffassenG“.
23 (M): 555
24 (V): Es deutet an
25 (V): daß das nicht
26 (V): ist. Daß
27 (M): /
28 (V): Ich möchte sagen, die Verneinung
29 (V): etwas viel Komplexeres // Komplizierteres
30 (V): beantworten (und das eine Symbol der
Negation durch ein anderes zu ersetzen, ist
keine Antwort),
31 (V): aber was, das
32 (V): wird scheinbar
33 (V): Erklärung jetzt
34 (R): ∀ S. 3/2
35 (M): ? / ///
36 (M): ? / ////
37 (V): gelten.
38 (V): links biegt, biegt sich
39 (O): abzäumen.
40 (M): ∫ ////
WBTC030-34 30/05/2005 16:17 Page 176
“How Can the Word ‘Not’ Negate?” . . . 88e
13
We’ve run across this primitiveness of form of expression, which we’ve noticed in
negation, in other places as well14
; for instance, when you want to get across to someone that
he is to take a certain path, you can draw the path for him as precisely as you wish. But the
indication that is to make him understand that he should take the path is once again of a
primitive nature, on which one would like to improve.
15
“What good is it that only a squiggle is in front of the sentence p as a negation sign? I
still have to think the whole negation.”
16
“The sign ‘not’17
indicates that you’re supposed to take what follows it negatively.”
18
However, “indicates”19
means that this sign for negation isn’t20
the final linguistic expression
of negation; that21
this isn’t the picture of the thought. That there is more to negation
than this.
22
We want to say: The sign for negation23
is no more than a prompt for doing something
very complicated24
– but what? If this question can’t be answered25
it is nonsense, and then
so is our initial sentence.
It’s as if the negation sign prompted us to do something; but what? That isn’t said.26
It’s
as if it only had to be alluded to; as if we already knew what it was. As if an explanation
were unnecessary27
because we already knew about the matter.28
29
If there were a more explicit way of expressing negation, we’d still have to be able
to portray it in terms of the other one, and therefore it couldn’t be of a different order of
multiplicity.
30
But now the question is: How is the specificity of negation that we’re familiar with shown
in the rules for the negation sign? We understand without ado that a hand-drawn map of a
path is a picture of the path; where the hand-drawn line turns left the path turns left as well,
etc. etc. But we don’t see that the “not” sign pertains to the map. We have a better chance
at seeing this if we draw a line around something that is ruled out, fence it off, as it were.
Thus one could take the “~” as a “No Trespassing” sign.
But let’s consider how someone is actually taught the meaning of such a sign. We’d restrain
him from taking the path, for instance.
31
“After all, I don’t just utter these words, but mean something by them.” If I say, for
example, “You’re not allowed to come in”, then it’s natural to accompany these words by
13 (M): ∫ /
14 (V): negation, earlier
15 (M): /
16 (M): /
17 (V): sign “ ”
18 (M): 555
19 (V): However, indicates
20 (V): that that isn’t
21 (V): expression of negation. That
22 (M): /
23 (V): I want to say the negation
24 (V): something much more complex // more
complicated
25 (V): answered (and replacing one symbol of
negation with another isn’t an answer)
26 (V): but what, apparently isn’t said.
27 (V): unnecessary now
28 (R): ∀ p. 3/2
29 (M): ? / ///
30 (M): ? / ////
31 (M): ∫ ////
WBTC030-34 30/05/2005 16:17 Page 177
dieser Worte, mich vor die Tür zu stellen und sie zuzuhalten. Aber es wäre nicht so
offenbar naturgemäß, wenn ich sie ihm bei diesen Worten öffnen würde. Diese Worte
haben, wie sie hier verstanden werden, offenbar etwas mit jenem Akt zu tun.41
Der Akt ist sozusagen eine Illustration zu ihnen – müßte als Sprache aufgefaßt werden
können. Andrerseits ist er aber auch der Akt, den ich abgesehen von jedem Symbolismus
aus meiner Natur tue.42
Wie ist es aber mit diesem Gedanken: Wenn ”~p“ ein Bild sein soll, wäre es da nicht am
natürlichsten, wenn es das Gegenteil von p durch das Gegenteil des Zeichens43
&p“ darstellte. Man
würde dann, daß zwei Menschen nicht miteinander kämpfen dadurch abbilden44
daß man sie nicht
miteinander kämpfend abbildete. Ich sagte einmal, ein solcher negativer Symbolismus wäre schon
möglich,45
er sei nur darum nicht zu gebrauchen, weil man aus ihm nicht erfahren könne,
was verneint sei. Dann ist er eben kein Symbolismus der Negation, wenn er uns nicht das
Nötige mitteilt. Und dann fehlt es ihm am Wesentlichen.
Es hat ja seinen Grund, warum in gewissen Fällen der negative Symbolismus funktioniert
und z.B. keine Antwort auch eine Antwort ist. In diesen Fällen ist eben der Sinn des
Schweigens eindeutig bestimmt.
46
Es wird eine andere Art Porträt entworfen, durch ein Bild, was zeigen soll, wie es
sich nicht verhält, als durch eines, was zeigt wie es sich verhält. Würde man es ein Porträt
nennen?
47
Die Farbangabe, daß etwas nicht rot ist, ist von anderer Art als die, daß etwas rot (oder
blau) ist. D.h. sie ist nicht in dem gleichen Sinn eine Farbangabe.
48
Aber es kann49
die Negation eines Satzes eine Angabe gleicher Art sein, wie der negierte
Satz.
50
”Ich brauche im negativen Satz das intakte Bild des positiven Satzes.“
51
”Ich kann ein Bild davon zeichnen, wie Zwei miteinander fechten; aber doch nicht davon,
wie Zwei miteinander nicht fechten (d.h. nicht ein Bild, das bloß dies darstellt).
,Sie fechten nicht miteinander‘ heißt nicht, daß davon nicht die Rede ist, sondern es ist
eben davon die Rede und wird (nur) ausgeschlossen.“
52
Die Idee der Negation ist nur in einer Zeichenerklärung verkörpert und soweit wir
eine solche Idee besitzen, besitzen wir sie nur in der Form so einer Erklärung. Denn wenn
man fragen kann ”was meinst Du mit diesem Zeichen53
“, so ist die Antwort nur eine
Zeichenerklärung (irgendeiner Art).
41 (M): überlegen
42 (V): Natur tun will.
43 (V): Zeichens von
44 (V): darstellen
45 (V): sein soll, wäre, was es bedeutet, nicht am
besten dadurch darzustellen, daß das im
Zeichen nicht der Fall ist, was, wenn es der
Fall wäre, darstellen würde, daß p der Fall ist.
Es ist aber klar, daß so ein Symbolismus nicht
funktioniert.
Es ist dafür keine Erklärung, zu sagen
(was ich einmal sagte), ein solcher negativer
Symbolismus ginge schon,
46 (M): /
47 (M): ? ∫ ///
48 (M): /
49 (V): Dagegen kann
50 (M): ///
51 (M): ∫ ////
52 (M): ∫
53 (V): Du damit
108v
110
111
110
89 ”Wie kann das Wort ,nicht‘ verneinen?“ . . .
WBTC030-34 30/05/2005 16:17 Page 178
stepping in front of the door and holding it closed. But it would not be so obviously natural
if I uttered those words as I opened the door for him. As they are understood in this
case, these words obviously have something to do with that action.32
The action is an illustration of the words, so to speak – it should be possible to understand
it as a language. On the other hand, it is also the action that I undertake33
naturally,
apart from all symbolism.
But what about this thought: If “~p” is to be a picture, wouldn’t it most naturally represent
the opposite of p by using the opposite of the sign “p”? In that case, we’d depict34
the fact that
two people are not fighting each other by depicting them as not fighting each other. Once I said
that this kind of negative symbolism would be possible, but just not35
usable, because one
couldn’t find out from it what was negated. If it doesn’t tell us what we need, then it just
isn’t a symbolism of negation. And then it is essentially deficient.
There is a reason, after all, why negative symbolism works in certain cases, and why, for
example, no answer is also an answer. It’s because in these cases the meaning of silence has
been established unambiguously.
36
A different type of portrait is created by a picture that is to show how things aren’t than
by one that shows how things are. Would we call it a portrait?
37
The statement that something isn’t red is of a different kind from the one that something
is red (or blue). That is, the former isn’t a statement about colour in the same sense
as the latter.
38
But39
the negation of a proposition can be a statement of the same kind as the negated
proposition.
40
“In the negative proposition I need the complete, accurate picture of the positive
proposition.”
41
“I can draw a picture of two people fencing with each other; but not of two people not
fencing with each other (i.e. not a picture that shows only that).
‘They’re not fencing with each other’ doesn’t mean that that isn’t being talked about. That’s
precisely what is being talked about; it’s (just) that it’s being ruled out.”
42
The idea of negation is embodied only in the explanation of a sign, and in so far as we
have such an idea, we have it only in the form of such an explanation. For when one asks
“What do you mean by this sign43
”, the answer is nothing other than an explanation of a
sign (of some kind).
32 (M): consider
33 (V): I want to undertake
34 (V): represent
35 (V): to be a picture, wouldn’t its meaning be best
represented by that not being the case which,
if it were the case, would represent that p is
the case? But it’s clear that such a symbolism
doesn’t work.
It is no explanation of this to say (as I once
said) that such a negative symbolism would
work, but just not be
36 (M): /
37 (M): ? ∫ ///
38 (M): /
39 (V): On the other hand,
40 (M): ///
41 (M): ∫ ////
42 (M): ∫
43 (V): by this
“How Can the Word ‘Not’ Negate?” . . . 89e
WBTC030-34 30/05/2005 16:17 Page 179
112
90 ”Wie kann das Wort ,nicht‘ verneinen?“ . . .
54
Den Begriff der Negation55
besitzen wir nur in einem Symbolismus. Und darum kann
man nicht sagen: ”auf die und die Art kann man die Negation nicht darstellen, weil diese
Art nicht eindeutig wäre“ – als handelte es sich um die Beschreibung eines Gegenstandes,
die nicht eindeutig gegeben worden wäre. Wenn der Symbolismus nicht erkennen läßt, was
verneint wurde, so verneint er nicht; wie ein Schachbrett ohne Felder kein schlechtes, d.h.
unpraktisches Schachbrett ist, sondern keins. Und wenn ich glaubte, auf56
einem Brett ohne
Felder Schach spielen zu können, so habe ich das Spiel einfach mißverstanden und werde
etwa jetzt auf das Mißverständnis57
aufmerksam gemacht.
Ein Symbolismus, der die Negation ”nicht darstellen kann“, ist kein Symbolismus der
Negation.
58
Ich glaube, ein Teil der Schwierigkeit rührt vom Gebrauch der Wörter ”ja“ und ”nein“
her (auch ”wahr“ und ”falsch“). Diese beiden lassen es so erscheinen, als wäre ein Satz
und sein Gegenteil im Verhältnis zweier Pole zueinander oder zweier entgegengesetzter
Richtungen. Während schon, daß ~~p = p ist, eine doppelte Bejahung aber keine
Verneinung ist, zeigen kann, daß dieses Bild falsch ist.
59
Wenn gefragt würde: ist die Verneinung60
in der Mathematik, etwa in ~(2 + 2 = 5),
die gleiche, wie die nicht-mathematischer Sätze? so müßte erst bestimmt werden, was
als Charakteristikum dieser61
Verneinung als solcher aufzufassen ist. Die Bedeutung eines
Zeichens liegt ja in den Regeln, die seinen Gebrauch vorschreiben.62
Welche dieser Regeln
machen das Zeichen ”~“ zur Verneinung? Denn es ist klar, daß gewisse Regeln, die sich auf
”~“ beziehen, für beide Fälle die gleichen sind; z.B. ~~p = p. Man könnte ja auch fragen:
ist die Verneinung eines Satzes ”ich sehe einen roten Fleck“ die gleiche, wie die von ”die
Erde bewegt sich in einer Ellipse63
um die Sonne“; und die Antwort müßte auch sein: Wie
hast Du ”Verneinung“ definiert, durch welche Klasse von Regeln? – daraus wird sich ergeben,
ob wir in beiden Fällen ”die gleiche Verneinung“ haben. Wenn die Logik allgemein von
der Verneinung redet, oder einen Kalkül mit ihr treibt, so ist die Bedeutung des
Verneinungszeichens nicht weiter festgelegt, als die Regeln seines Kalküls. Wir dürfen hier
nicht vergessen, daß ein Wort seine Bedeutung nicht als etwas, ihm ein für allemal verliehenes,
mit sich herumträgt, sodaß wir sicher sind, wenn wir nach dieser Flasche greifen, auch die
bestimmte Flüssigkeit, z.B. Spiritus, in der Hand zu halten.64
54 (M): ? /
55 (V): Verneinung
56 (V): mit
57 (V): jetzt darauf
58 (M): /
59 (M): ü /
60 (V): Negation
61 (V): der
62 (V): in den Regeln, nach denen es verwendet
wird.
63 (O): Elipse
64 (V): Flüssigkeit, etwa Spiritus, zu erwischen.
(R): ∀ Siehe S. 106 letzter Satz
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44
Only within a system of symbols do we possess the concept of negation. And therefore
we can’t say: “You can’t represent negation in such-and-such a way because that wouldn’t
be unambiguous” – as if it were a matter of not giving an unambiguous description of an
object. If the system of symbols doesn’t allow one to recognize what was negated, then it
doesn’t negate; just as a chess board without squares is not a bad, i.e. an impractical, chess
board, but no chess board at all. And if I thought I could play chess on45
a board without
squares, then I have simply misunderstood the game and am now, perhaps, having my mis-
understanding46
pointed out to me.
A symbolism that “cannot represent” negation is not a symbolism of negation.
47
I think part of the difficulty derives from the use of the words “yes” and “no” (“true”
and “false” as well). These two words make it appear as if a proposition and its contradictory
were related to each other like two poles or two opposite directions. Whereas the
simple fact that ~~p = p, but that a double affirmation is not a negation, shows that this
picture is false.
48
If someone asked: Is negation in mathematics, say in ~(2 + 2 = 5), the same as the negation
of non-mathematical propositions? – then first of all we’d have to determine what is to
be taken as characteristic of this49
negation as such. After all, the meaning of a sign lies in
the rules that prescribe its use.50
Which of these rules turn the sign “~” into negation? For
it is clear that certain rules that pertain to “~” are the same for both cases; e.g. ~~p = p.
One could also ask: Is the negation of the proposition “I see a red patch” the same as that
of “The earth travels around the sun in an ellipse”? And again the answer would have to
be: How, by what set of rules, did you define “negation”? – This will show whether we have
“the same negation” in both cases. If logic talks about negation in general, or carries out a
calculus with it, then the meaning of the sign for negation is not specified further than are
the rules of its calculus. Here we mustn’t forget that a word doesn’t carry its meaning around
with it as something that it was bequeathed once and for all, allowing us to be certain that
when we reach for a particular bottle we’ll have a specific liquid in hand, say alcohol.51
44 (M): ? /
45 (V): with
46 (V): having this
47 (M): /
48 (M): r /
49 (V): the
50 (V): rules according to which it is used.
51 (V): bottle that we’re taking hold of a specific
liquid, say alcohol. (R): ∀ see p. 106 last
sentence
“How Can the Word ‘Not’ Negate?” . . . 90e
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31
Ist die Zeit den Sätzen wesentlich?
Vergleich von: Zeit und
Wahrheitsfunktionen.
Tritt1
die Zeit in ein Landschaftsbild ein? oder in ein Stilleben?
Literatur die aus Landschaftsschilderungen besteht.
2
Die Grammatik, wenn sie in der Form eines Buches uns vorläge, bestünde nicht aus einer
Reihe bloß nebengeordneter Artikel, sondern würde eine andere Struktur zeigen. Und in
dieser müßte man – wenn ich Recht habe – auch den Unterschied zwischen Phänomenologischem
und Nicht-Phänomenologischem sehen. Es wäre da etwa ein Kapitel von den Farben,
worin der Gebrauch der Farbwörter geregelt wäre; aber dem vergleichbar wäre nicht, was über
die Wörter ”nicht“, ”oder“, etc. (die ”logischen Konstanten“) in der Grammatik gesagt würde.
3
Es würde z.B. aus den Regeln hervorgehen, daß diese letzteren Wörter in jedem Satz
anzuwenden seien (nicht aber die Farbwörter). Und dieses ”jedem“ hätte nicht den
Charakter einer erfahrungsmäßigen Allgemeinheit; sondern der inappellablen Allgemeinheit
einer obersten Spielregel. Es scheint mir ähnlich, wie das Schachspiel wohl ohne gewisse
Figuren zu spielen (oder doch fortzusetzen) ist, aber nie ohne das Schachbrett. [Das ist nicht
wahr, man könnte ganz gut mit einem Teil des Brettes auskommen.]
4
Wie offenbart sich die Zeitlichkeit der Tatsachen, wie drückt sie sich aus, als dadurch,
daß gewisse Ausdrücke5
in unsern Sätzen vorkommen müssen. D.h.: Wie drückt sich die
Zeitlichkeit der Tatsachen aus, als grammatisch? &Zeitlichkeit“ damit ist nicht gemeint daß ich
um 5h komme sondern daß ich irgendwann komme d.h. daß mein Satz die Struktur hat, die er hat.
6
Woher – möchte ich fragen – die Allgemeinheit der Zeitlichkeit der Erfahrungssätze?7
Könnte8
man auch so fragen: &Wie9
kommt es daß man alle Erfahrungstatsachen mit dem was
eine Uhr zeigt in Verbindung bringen kann?“?
10
Negation und Disjunktion, möchten wir sagen, hat mit dem Wesen des Satzes zu tun,
die Zeit aber nicht, sondern mit seinem Inhalt.
Wie aber kann es sich in der Grammatik zeigen, daß Etwas mit dem Wesen des Satzes
zusammenhängt und etwas anderes nicht, wenn sie beide gleich allgemein sind?
1 (V): Landsch Tritt
2 (M): ///
3 (M): /// – Schachbrett.
4 (M): ?
5 (V): Wendungen
6 (M): /
7 (V): (WoherA – möchte ich fragen – (die ganz
allgemeine Zeitlichkeit der Sätze?A
8 (O): Konnte
9 (V): &Woher
10 (M): /
113
114
113v
114
91
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31
Is Time Essential to Propositions?
Comparison between Time and
Truth-Functions.
Does time enter into a painting of a landscape? Or into a still life?
Literature consisting of descriptions of landscapes.
1
If grammar were available to us in the form of a book it wouldn’t consist of a series of
equivalent items one after another, but would have a different structure. And it is in this
structure – if I am right – that one could see the difference between the phenomenological
and the non-phenomenological. For instance, there would be a chapter about colours, in which
the rules for the use of colour-words would be laid down; but what would be said in this
grammar about the words “not”, “or”, etc. (the “logical constants”) wouldn’t be comparable
to this.
2
The rules would show, e.g., that these latter words (but not the colour-words) could
be used in every sentence. And this “every” wouldn’t have the character of an empirical
generality, but of the unappealable generality of a cardinal rule of the game. This strikes me
as similar to how chess can be played (in its later stages) without certain pieces, but never
without a chess board. [That isn’t true; one could manage quite well with only part of the board.]
3
How does the temporality of facts manifest itself, express itself, other than by certain
expressions4
having to occur in our sentences? That is: How does the temporality of facts
express itself other than grammatically? “Temporality” doesn’t mean that I’m coming at
5 o’clock, but that I’m coming at some time, i.e. that my sentence has the structure it has.
5
Whence – I’m inclined to ask – does it arise that, in general, empirical propositions are temporal?6
Could one also put the question this way: “How does it come about7
that we can connect all
empirical facts with what a clock shows?”?
8
Negation and disjunction, we’re inclined to say, have to do with the essence of a proposition,
but not time, which has to do with its content.
But if two things are equally universal, how can it be shown in grammar that one thing
is connected with the essence of the proposition and the other isn’t?
1 (M): ///
2 (M): /// – chess board.
3 (M): ?
4 (V): certain turns of phrases
5 (M): /
6 (V): AWhenceC – I’m inclined to ask – Adoes it
arise that, completely generally, propositions are
temporal?C
7 (V): way: “Where does it come from
8 (M): /
91e
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Oder sollte ich sagen, die geringere Allgemeinheit wäre auf seiten der Zeit, da die mathematischen
Sätze negiert und disjungiert werden können, aber nicht zeitlich sind? Ein
Zusammenhang ist wohl da, wenn auch diese Form, die Sache darzustellen, irreführend ist.11
Das zeigt eben was ich unter &Satz“ oder dem &Wesen des Satzes“ verstehe.
12
Wie unterscheidet die Grammatik zwischen Satzform und Inhalt? Denn dies soll ja ein
grammatikalischer Unterschied sein. Wie sollte man ihn beschreiben können, wenn ihn die
Grammatik nicht zeigt?
13
Was hat es mit dem Schema ”Es verhält sich so und so“ für eine Bewandtnis? Man
könnte sagen, das ”Es verhält sich“ ist der Angriff für die Wahrheitsf.14
”Es verhält sich“ ist also nur ein Ausdruck aus einer Notation der Wahrheitsfunktionen.
Ein Ausdruck, der uns zeigt, welcher Teil der Grammatik hier in Funktion tritt.
Jene Zweifache Art der Allgemeinheit wäre so seltsam, wie wenn von zwei Regeln eines
Spiels, die beide gleich ausnahmslos gelten, die eine als die fundamentalere angesprochen
würde. Als könnte man also darüber reden,15
ob der König oder das Schachbrett für
das Schachspiel essentieller wäre. Welches von beiden das Wesentlichere, welches das
Zufälligere wäre.
16
Zum mindesten scheint eine Frage berechtigt: Wenn ich die Grammatik aufgeschrieben
hätte und die verschiedenen Kapitel, über die Farbwörter, etc. etc. der Reihe nach da
stünden, wie Regeln über alle die Figuren des Schachspiels, wie wüßte ich dann, daß dies
nun alle Kapitel sind? Und wenn sich nun in allen vorhandenen Kapiteln eine gemeinsame
Eigentümlichkeit findet, so haben wir es hier scheinbar mit einer logischen Allgemeinheit,
aber keiner wesentlichen, d.h. voraussehbaren Allgemeinheit, zu tun. Man kann aber doch
nicht sagen, daß die Tatsache, daß das Schachspiel mit 16 Figuren gespielt wird, ihm weniger
wesentlich ist, als, daß es auf dem Schachbrett gespielt wird.
17
Da Zeit und Wahrheitsfunktionen so verschieden schmecken und da sie ihr Wesen
allein und ganz in der Grammatik offenbaren, so muß die Grammatik den verschiedenen
Geschmack erklären.
Das eine schmeckt nach Inhalt, das andere nach Darstellungsform.
Sie schmecken so verschieden, wie der Plan und der Strich durch den Plan.
18
Es kommt mir so vor, als wäre die Gegenwart, wie sie in dem Satz ”der Himmel ist blau“
steht (wenn dieser Satz nicht-hypothetisch gemeint ist), keine Form der Zeit. Als ob also
die Gegenwart in diesem Sinne unzeitlich wäre.
19
Es ist merkwürdig, daß die Zeit, von der ich hier rede, nicht die im physikalischen Sinne
ist. Es handelt sich hier nicht um eine Zeitmessung. Und es ist verdächtig, daß etwas, was
mit einer solchen Messung nichts zu tun hat, in den Sätzen eine ähnliche Rolle spielen soll,
wie die physikalische Zeit in den Hypothesen der Physik.
11 (M): Lehrreich
12 (M): 555
13 (M): ? / 3 (R): [Zu § 18]
14 (V): ist die Handhabe für den Angriff der
Wahrheitsfunktionen.
15 (V): also fragen,
16 (M): ///
17 (M): /
18 (M): ? /
19 (M): /
115
92 Ist die Zeit den Sätzen wesentlich? . . .
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Or should I say that the lesser universality is on the side of time since mathematical propositions
can be negated and can be stated as disjunctions, but are not temporal? There is a
connection here, even if this form of representing the matter is misleading.9
That just shows what I understand by “proposition” or “essence of a proposition”.
10
How does grammar distinguish between the form of a proposition and its content?
For this is supposed to be a grammatical difference. How can one describe it, if grammar
doesn’t show it?
11
What’s special about the schema “Things are so and so”? One could say, “Things are”
is the starting-point for truth-functions.12
So “Things are” is only one expression from a notation of truth-functions. An expression
that shows us which part of grammar is here coming into play.
Having two kinds of generality would be as strange as if two rules of a game were equally
valid without exception, but one were pronounced the more fundamental. As if there could
be a debate about13
whether the king or the chess board is more essential to chess; about
which of the two is more essential, which is more accidental.
14
At the least, one question seems justified: If I had written the book of grammar, and the
various chapters on colour words, etc., etc., were there, one after the other, like the rules
for all the chess pieces, how would I know that those were all the chapters? And if a common
characteristic shows up in all of the existing chapters, then we seem to be dealing with
a logical generality, but not one that’s essential, i.e. foreseeable. But we can’t say that the
fact that chess is played with 16 pieces is any less essential to it than its being played on a
chessboard.
15
Since time and truth-functions have such different flavours and since they manifest their
nature solely and completely in grammar, grammar has to explain their difference in flavour.
One tastes like content, the other like form of representation.
They taste as different as a map and a line crossing out the map.
16
It seems to me that the present, as it occurs in the proposition “The sky is blue” (if this
proposition is meant non-hypothetically) is not a temporal form. It’s as if in this sense the
present were atemporal.
17
It’s remarkable that time, as I am talking about it here, is not time in a physical sense.
Here it’s not a matter of measuring time. And it’s suspicious that something that has nothing
to do with such measurement should play a role in propositions similar to that which
physical time plays in the hypotheses of physics.
9 (M): Instructive
10 (M): 555
11 (M): ? / 3 (R): [To § 18]
12 (V): is the handle for activating truth-functions.
13 (V): As if one could ask
14 (M): ///
15 (M): /
16 (M): ? /
17 (M): /
Is Time Essential to Propositions? . . . 92e
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20
Diskutiere:
Der Unterschied zwischen der Logik des Inhalts und der Logik der Satzform überhaupt.
Das eine erscheint gleichsam bunt, das andere matt. Das eine scheint von dem zu handeln,
was das Bild darstellt, das andere wie21
der Rahmen des Bildes ein Charakteristikum der
Bildform zu sein.22
23
Daß alle Sätze die Zeit in irgend einer Weise enthalten, scheint uns zufällig, im
Vergleich damit, daß auf alle Sätze die Wahrheitsfunktionen anwendbar sind.
Das scheint mit ihrem Wesen als Sätzen zusammenzuhängen, das andere mit dem Wesen
der vorgefundenen Realität.24
Ein Satz kann in sehr verschiedenem Sinne die Zeit enthalten.
Du25
tust mir weh!
Es ist herrliches Wetter draußen.
Der Inn fließt in die Donau.
Wasser gefriert bei 0°.
Ich verschreibe mich oft.
Vor einiger Zeit. . . .
Ich hoffe er wird kommen.
Um 5 Uhr. . . .
Diese Stahlsorte ist sehr gut.
Unsere Erde war einmal ein Gasball.
20 (M): ? /
21 (O): das andere, wie
22 (M): überlege
23 (M): ? /
24 (M): überlegen
25 (V): enthalten.
Ich habe Zahnschmerzen!
Du
116
93 Ist die Zeit den Sätzen wesentlich? . . .
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18
Discuss:
The difference between the logic of content and the logic of propositional form in general.
The one seems to be bright, as it were, the other dull. The one seems to be about what
the picture represents, the other seems to be a characteristic of the form of the picture, like
its frame.19
20
That all propositions in some way contain time seems to us accidental in comparison to
the fact that truth-functions can be applied to them all.
The latter seems to be connected with their nature as propositions, the former with the
nature of the reality we encounter.21
A proposition can contain time in very different senses.
You’re22
hurting me!
The weather outside is gorgeous.
The Inn flows into the Danube.
Water freezes at O°.
I often make mistakes when I’m writing.
A while ago. . . .
I hope he’ll come.
At 5 o’clock. . . .
This kind of steel is very good.
Our earth once was a ball of gas.
18 (M): ? /
19 (M): consider
20 (M): ? /
21 (M): consider
22 (V): senses.
I have a toothache!
You’re
Is Time Essential to Propositions? . . . 93e
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32
Wesen der Hypothese.
1
Eine Hypothese könnte man offenbar durch Bilder erklären. Ich meine, man könnte z.B.
die Hypothese ”hier liegt ein Buch“ durch Bilder erklären, die das Buch im Grundriß, Aufriß
und verschiedenen Schnitten zeigen.
2
Eine solche Darstellung gibt ein Gesetz. Wie die Gleichung einer Kurve ein Gesetz gibt,
nach der die Ordinatenabschnitte aufzufinden sind, wenn man in verschiedenen Abszissen
schneidet.3
Die fallweisen Verifikationen entsprechen dann solchen
wirklich ausgeführten Schnitten.
Wenn unsere Erfahrungen die Punkte auf einer Geraden
ergeben, so ist der Satz, daß diese Erfahrungen die verschiedenen
Ansichten einer Geraden sind, eine Hypothese.
Die Hypothese ist eine Art der Darstellung dieser Realität,
denn eine neue Erfahrung kann mit ihr übereinstimmen oder nicht-übereinstimmen, bezw.
eine Änderung der Hypothese nötig machen.
4
Drücken wir z.B. den Satz, daß eine Kugel sich in einer bestimmten Entfernung von
unseren Augen befindet, mit Hilfe eines Koordinatensystems und der Kugelgleichung aus,
so hat diese Beschreibung eine größere Mannigfaltigkeit, als die einer Verifikation durch
das Auge. Jene Mannigfaltigkeit entspricht nicht einer Verifikation, sondern einem Gesetz,
welchem Verifikationen gehorchen.
5
Eine Hypothese ist ein Gesetz zur Bildung von Sätzen.
Man könnte auch sagen: Eine Hypothese ist ein Gesetz zur Bildung von Erwartungen.
Ein Satz ist sozusagen ein Schnitt durch eine Hypothese in einem bestimmten Ort.
6
Nach meinem Prinzip müssen die beiden Annahmen7
ihrem Sinne nach identisch sein,
wenn alle mögliche Erfahrung, die die eine bestätigt, auch die andere bestätigt. Wenn also
keine Entscheidung zwischen beiden durch8
die Erfahrung denkbar ist.
Darstellung einer Linie als Gerade mit Abweichungen. Die Gleichung der Linie
enthält einen Parameter, dessen Verlauf die Abweichungen von der Geraden ausdrückt.
1 (M): ∫
2 (M): ? /
3 (F): MS 107, S. 253.
4 (M): ? /
5 (M): ∫
6 (M): ∫
7 (E): In MS 107 (S. 287) steht Folgendes vor
dieser Bemerkung: ”Ist aber nicht doch ein
Unterschied zwischen den Annahmen daß die
Anderen Schmerzen haben und daß sie keine
haben & sich nur so benehmen wie ich, wenn
ich welche habe?“ Weiter unten (Kapitel 104,
S. 357) werden die beiden Bemerkungen
folgendermaßen zusammengefaßt: ”Die beiden
Hypothesen, daß die Anderen Schmerzen
haben, und die, daß sie keine haben, und sich
nur so benehmen wie ich, wenn ich welche
habe, müssen ihrem Sinne nach identisch sein,
wenn . . . denkbar ist.“
8 (O): zwischen durch
117
118
94
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32
The Nature of Hypothesis.
1
Obviously you could explain a hypothesis with pictures. I mean you could for example
explain the hypothesis “There’s a book lying here” with pictures showing the book in plan,
elevation and various cross-sections.
2
Such a representation produces a law. Just as the equation of a curve produces a law, an
equation you can use to discover the intercepts on the ordinate when you make cuts at various
points on the abscissa.3
The verifications of particular cases then correspond to such
cuts as are actually made.
If our experiences produce points on a straight line, the proposition
that these experiences are various sections of a straight line
is a hypothesis.
The hypothesis is a way of representing this reality, for a new
experience may tally with it or not – or it may force us to modify the hypothesis.
4
If for instance we use a system of coordinates and the equation for the sphere to express
the proposition that a sphere is located at a certain distance from our eyes, this description
has a greater multiplicity than one verified by eye. The first multiplicity corresponds not to
a single verification but to a law that governs verifications.
5
A hypothesis is a law for forming propositions.
One could also say: A hypothesis is a law for forming expectations.
A proposition is, so to speak, a section of a hypothesis at a certain point.
6
According to my principle the two suppositions7
must be identical in sense if any possible
experience that confirms the one also confirms the other. In other words, if no distinction
between the two is conceivable based on experience.
Representation of any line as a straight line with deviations. The equation for the line contains
a parameter, the course of which expresses the deviations from a straight line. It’s not
1 (M): ∫
2 (M): ? /
3 (F): MS 107, p. 253.
4 (M): ? /
5 (M): ∫
6 (M): ∫
7 (E): In MS 107 (p. 287) this remark is preceded
by the remark: “But still, isn’t there a difference
between the suppositions that others are in
pain, and that they aren’t but only behave the
way I do when I am in pain?” Later in this TS
(Ch. 104, p. 357e) the two remarks get combined:
“The two hypotheses, the one that others are in
pain, the other that they are not in pain but are
only behaving the way I do when I am in pain,
must be identical . . . between the two.”
94e
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119
120
95 Wesen der Hypothese.
Es ist nicht wesentlich, daß diese Abweichungen ”gering“ seien. Sie können so groß sein, daß
die Linie einer Geraden nicht ähnlich sieht.9
Die ”Gerade mit Abweichungen“ ist nur
eine Form der Beschreibung. Sie erleichtert es mir, einen bestimmten Teil der Beschreibung
auszuschalten, zu vernachlässigen, wenn ich will. (Die Form ”Regel mit Ausnahmen“.)
10
Was heißt es, sicher zu sein, daß man Zahnschmerzen haben wird. (Kann man nicht
sicher sein, dann erlaubt es die Grammatik nicht, das Wort ”sicher“ in dieser Verbindung
zu gebrauchen.)
Grammatik des Wortes ”sicher sein“.
11
Man sagt: ”Wenn ich sage, daß ich einen Sessel dort sehe, so sage ich mehr, als ich sicher
weiß“. Und nun heißt es meistens: ”Aber eines weiß ich doch sicher“. Wenn man aber nun
sagen will, was das ist, so kommt man in eine gewisse Verlegenheit.
”Ich sehe etwas Braunes – das ist sicher“; damit will man eigentlich sagen, daß die braune
Farbe gesehen, und nicht vielleicht auch bloß aus anderen Anzeichen vermutet ist.12
Und
man sagt ja auch einfach: ”Etwas Braunes sehe ich“.
13
Wenn mir gesagt wird: ”Sieh in dieses Fernrohr und zeichne mir auf, was Du siehst“,
so ist, was ich zeichne, der Ausdruck eines Satzes, nicht einer Hypothese.
14
Wenn ich sage ”hier steht ein Sessel“, so ist damit – wie man sagt – ”mehr“ gemeint,
als die Beschreibung dessen, was ich wahrnehme. Und das kann nur heißen, daß dieser Satz
nicht wahr sein muß, auch wenn die Beschreibung des Gesehenen stimmt. Unter welchen
Umständen werde ich nun sagen, daß jener Satz nicht wahr war? Offenbar: wenn gewisse
andere Sätze nicht wahr sind, die in dem ersten mit beinhaltet waren. Aber es ist nicht so,
als ob nun der erste ein logisches Produkt gewesen wäre.
15
Das beste Gleichnis für jede Hypothese, und selbst ein Beispiel, ist ein Körper mit seinen
nach einer bestimmten Regel konstruierten Ansichten aus den verschiedenen Punkten des
Raumes.
16
Der Vorgang einer Erkenntnis in einer wissenschaftlichen Untersuchung (in der
Experimentalphysik etwa) ist freilich nicht der einer Erkenntnis im Leben außerhalb des
Laboratoriums;17
aber er ist ein ähnlicher und kann, neben den andern gehalten,18
diesen
beleuchten.
19
Es ist ein wesentlicher Unterschied zwischen Sätzen wie ”das ist ein Löwe“, ”die
Sonne ist größer als die Erde“, die alle ein ”dieses“, ”jetzt“, ”hier“ enthalten und also an
die Realität unmittelbar anknüpfen, und Sätzen wie ”Menschen haben zwei Hände“ etc.
Denn, wenn zufällig keine Menschen in meiner Umgebung wären, wie wollte ich diesen Satz
kontrollieren?
9 (F): MS 107, S. 224.
10 (M): ? /
11 (M): ü /
12 (V): und nicht vielleicht auch nur // bloß //
vermutet ist (wie etwa in dem Fall, wo ich es
// sie // aus gewissen anderen Anzeichen
vermute).
13 (M): ∫
14 (M): ∫ /
15 (M): ? /
16 (M): /
17 (V): außerhalb dem Laboratorium;
18 (V): gestellt,
19 (M): ∫
A B
WBTC030-34 30/05/2005 16:17 Page 190
essential that these deviations be “minor”. They can be so large that the line looks nothing
like a straight line.8
“Straight line with deviations”
is only a form of description. It
makes it easier for me to eliminate a certain part of the description, to neglect it, if I want
to. (The form “rule with exceptions”.)
9
What does it mean to be certain that one will have a toothache? (If one cannot be certain,
then grammar doesn’t allow the use of the word “certain” in this connection.)
The grammar of the expression “to be certain”.
10
We say: “If I say that I see a chair over there, I’m saying more than I know for certain”.
And then we usually go on: “But one thing I do know for certain”. But if we now want to
say what this is, we get into a predicament of sorts.
“I see something brown – that’s for certain”: this really means that the brown colour has
been seen and not, say, merely been surmised from other indications.11
And indeed we do
simply say: “I see something brown”.
12
If I am told: “Look into this telescope and draw what you see”, then what I draw is the
expression of a proposition, not of a hypothesis.
13
If I say “Here there’s a chair” then “more” is meant by this – as we say – than the description
of what I perceive. And that can only mean that this proposition doesn’t have to be
true, even if the description of what I saw is correct. Well, under what circumstances am
I going to say that that proposition wasn’t true? Obviously: if certain other propositions
that were contained in the original are not true. But it isn’t as if the original were a logical
product.
14
The best simile for any hypothesis – which is itself an example of a hypothesis – is
a solid, along with various views of it that are constructed using a particular rule, from
various points in space.
15
To be sure, the process that leads to a piece of knowledge in a scientific investigation
(say in experimental physics) is not the same as one that leads to a piece of knowledge in
life outside the laboratory; but it is similar and, when held16
next to the latter, can shed light
on it.
17
There is an essential difference between propositions such as “That’s a lion”, and “The
sun is larger than the earth”, all of which contain a “this”, “now”, “here”, and therefore
connect to reality directly, and propositions such as “Humans have two hands”, etc. For if
by chance no human beings were around, how would I go about checking this proposition?
8 (F): MS 107, p. 224.
9 (M): ? /
10 (M): r /
11 (V): not, say, // merely // been surmised (as in
the case where I surmise it from certain other
indications).
12 (M): ∫
13 (M): ∫ /
14 (M): ? /
15 (M): /
16 (V): placed
17 (M): ∫
The Nature of Hypothesis. 95e
A B
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20
Es werden immer Facetten21
der Hypothese verifiziert.
22
Ist es nun nicht etwa so, daß das, was die Hypothese erklärt, selbst nur wieder durch
eine Hypothese ausdrückbar ist. Das heißt natürlich: gibt es überhaupt primäre Sätze; die
also endgültig verifizierbar sind, und nicht die Facetten23
einer Hypothese sind? (Das ist etwa,
als würde man fragen ”gibt es Flächen, die nicht Oberflächen von Körpern sind?“)
24
Es kann jedenfalls kein Unterschied sein zwischen einer Hypothese, als Ausdruck einer
unmittelbaren Erfahrung gebraucht, und einem Satz im engeren Sinne.
25
Es ist ein Unterschied zwischen einem Satz wie ”hier liegt eine Kugel vor mir“ und
”es schaut so aus, als läge eine Kugel vor mir“. – Das zeigt sich auch so: man kann sagen
”es scheint eine Kugel vor mir zu liegen“, aber es ist sinnlos zu sagen: ”es schaut so aus,
als schiene eine Kugel hier zu liegen“. Wie man auch sagen kann ”hier liegt wahrscheinlich
eine Kugel“, aber nicht ”wahrscheinlich scheint hier eine Kugel zu liegen“. Man würde
in so einem Falle sagen: ”ob es scheint, mußt Du doch wissen“.
26
In dem, was den Satz mit der gegebenen Tatsache verbindet, ist nichts Hypothetisches.
27
Es ist doch klar, daß eine Hypothese von der Wirklichkeit – ich meine von der unmittelbaren
Erfahrung – einmal mit ja, einmal mit nein beantwortet wird; (wobei freilich das
”ja“ und ”nein“ hier nur Bestätigung und Fehlen der Bestätigung ausdrückt) und daß man
dieser Bejahung und Verneinung Ausdruck verleihen kann.
28
Die Hypothese wird, mit der Facette29
an die Realität angelegt, zum Satz.
30
Ob der Körper, den ich sehe, eine Kugel ist, kann zweifelhaft sein, aber, daß er von hier
etwa eine Kugel zu sein scheint, kann nicht zweifelhaft sein. – Der Mechanismus der Hypothese
würde nicht funktionieren, wenn der Schein noch zweifelhaft wäre; wenn also auch nicht
eine Facette31
der Hypothese unzweifelhaft verifiziert würde. Wenn es hier Zweifel gäbe,
was könnte den Zweifel heben? Wenn auch diese Verbindung locker wäre, so gäbe es auch
nicht Bestätigung einer Hypothese, die Hypothese hinge dann gänzlich in der Luft und wäre
zwecklos (und damit sinnlos).
32
Wenn ich sagte ”ich sah einen Sessel“; so widerspricht dem (in einem Sinne) nicht der
Satz ”es war keiner da“. Denn den ersten Satz würde ich auch in der Beschreibung eines
Traums verwenden und niemand würde mir dann mit den Worten des zweiten widersprechen.
Aber die Beschreibung des Traums mit jenen Worten wirft ein Licht auf den Sinn der Worte
”ich sah“.
In dem Satz ”es war ja keiner da“ kann das ”da“ übrigens verschiedene Bedeutung haben.
33
Ich stimme mit den Anschauungen neuerer Physiker überein,34
wenn sie sagen, daß die
Zeichen in ihren Gleichungen keine ”Bedeutungen“ mehr haben, und daß die Physik zu keinen
solchen Bedeutungen gelangen könne, sondern bei den Zeichen stehen bleiben müsse: sie sehen
121
96 Wesen der Hypothese.
20 (M): ! /
21 (O): Fassetten
22 (M): ∫
23 (O): Fassetten
24 (M): /
25 (M): ? /
26 (M): ∫
27 (M): ∫
28 (M): ? ∫
29 (O): Fassette
30 (M): ∫
31 (O): Fassette
32 (M): ? /
33 (M): ? /
34 (E): In einem früheren Manuskript (107, S.
223) heißt es spezifischer: ”Die Anschauungen
neuerer Physiker (Eddington) stimmen ganz
mit der meinen überein, . . .“
122
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18
It’s always facets of a hypothesis that are verified.
19
Isn’t it more or less the case that what a hypothesis explains can itself be expressed only
by a hypothesis? Of course this means: Are there such things as primary propositions at all;
that is to say, propositions that are definitively verifiable, and are not facets of a hypothesis?
(That is somewhat like asking “Are there surfaces that aren’t surfaces of solids?”.)
20
In any case, there can be no difference between a hypothesis used as an expression of
an immediate experience and a proposition in the narrower sense.
21
There is a difference between a proposition such as “There’s a ball lying in front of me”
and “It looks as if there were a ball lying in front of me”. – This also comes out this way:
One can say: “There seems to be a ball lying in front of me”, but it makes no sense to say:
“It looks as if there seemed to be a ball lying here”. Just as one can also say “There’s probably
a ball lying here”, but not “Probably a ball seems to be lying here”. In such a case one
would say: “Surely you must know whether it seems”.
22
There is nothing hypothetical in what connects a proposition to a given fact.
23
It’s clear that a hypothesis is answered by reality – I mean by immediate experience –
sometimes with a “yes” and sometimes with a “no” (here “yes” and “no” only express
confirmation and lack of confirmation, to be sure), and that one can give expression to this
affirmation and denial.
24
When this facet of a hypothesis is laid alongside reality, the hypothesis turns into a
proposition.
25
Whether the body I see is a sphere may be doubtful, but that from here, say, it seems
to be a sphere cannot be doubtful. – The mechanism of hypothesis would not work if appearance
were still in doubt, i.e. if not even one facet of the hypothesis were verified beyond a
doubt. If there were doubt there, what could remove it? If this connection too were loose,
then there also wouldn’t be any such thing as the confirmation of a hypothesis; it would be
completely suspended in air and would be pointless (and thus senseless).
26
If I were to say “I saw a chair”, then (in one sense) the proposition “There wasn’t any
chair there” doesn’t contradict that. For I could also use the first proposition in describing
a dream, and then nobody who used the words of the second would contradict me. But the
description of the dream that uses those words sheds light on the sense of the words “I saw”.
In the proposition “But there wasn’t any chair there”, “there” can have various meanings,
by the way.
27
I agree with the views of contemporary physicists28
when they say that the signs in
their equations no longer have any “meanings” and that physics cannot arrive at any such
meanings, but has to stop at the signs: for they don’t see that these signs have meaning in
18 (M): ! /
19 (M): ∫
20 (M): /
21 (M): ? /
22 (M): ∫
23 (M): ∫
24 (M): ? ∫
25 (M): ∫
26 (M): ? /
27 (M): ? /
28 (E): In an earlier version of this remark (MS 107,
p. 223) Wittgenstein has: “My view agrees
completely with those of modern physicists
(Eddington) . . .”.
The Nature of Hypothesis. 96e
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nämlich nicht, daß diese Zeichen insofern Bedeutung haben – und nur insofern – als ihnen,
auf welchen Umwegen immer, das beobachtete Phänomen entspricht, oder nicht entspricht.
35
Denken wir uns, daß das Schachspiel nicht als Brettspiel erfunden worden wäre, sondern
als Spiel, das mit Ziffern und Buchstaben auf Papier zu spielen ist und so, daß sich
niemand dabei ein Quadrat mit 64 Feldern etc. vorgestellt hätte. Nun aber hätte jemand die
Entdeckung gemacht, daß dieses Spiel ganz einem entspricht, das man auf einem Brett in
der und der Weise spielen könnte. Diese Erfindung wäre eine große Erleichterung des Spiels
gewesen (Leute, denen es früher zu schwer gewesen wäre, könnten es nun spielen). Aber es
ist klar, daß diese neue Illustration der Spielregeln nur ein neuer, leichter übersehbarer,
Symbolismus wäre, der übrigens mit dem Geschriebenen auf gleicher Stufe stünde.
Vergleiche nun damit das Gerede darüber, daß die Physik heute nicht mehr mit mechanischen
Modellen, sondern ”nur mit Symbolen“ arbeitet.
35 (M): /
97 Wesen der Hypothese.
WBTC030-34 30/05/2005 16:17 Page 194
so far – and only in so far – as a phenomenon that has been observed corresponds to them
or doesn’t, no matter how roundabout the way.
29
Let’s imagine that chess hadn’t been invented as a board game, but as a game to be played
with numbers and letters on paper, and in such as way that nobody had imagined a square
subdivided into 64 smaller squares, etc. But now someone has discovered that this game corresponds
exactly to one that could be played on a board in such and such a way. This invention
has made the game much easier (people for whom it had been too difficult previously
can now play it). But it is clear that this new illustration of the rules of the game would only
be a new, more easily surveyable symbolism, which in other respects would be on the same
level as the game that’s written. Now compare with this the prattle that physics nowadays
no longer works with mechanical models but “only with symbols”.
29 (M): /
The Nature of Hypothesis. 97e
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33
Wahrscheinlichkeit.
1
Die Wahrscheinlichkeit einer Hypothese hat ihr Maß darin, wieviel Evidenz nötig ist,
um es vorteilhaft zu machen, sie umzustoßen.
Nur in diesem Sinne kann man sagen, daß wiederholte gleichförmige Erfahrung in der
Vergangenheit das Andauern dieser Gleichförmigkeit in der Zukunft wahrscheinlich macht.
Wenn ich nun in diesem Sinne sage: Ich nehme an, daß morgen die Sonne wieder aufgehen
wird, weil das Gegenteil zu unwahrscheinlich ist, so meine ich hier mit ”wahrscheinlich“
oder ”unwahrscheinlich“ etwas ganz Anderes, als mit diesen Worten im Satz ”es
ist gleich wahrscheinlich, daß ich Kopf oder Adler werfe“ gemeint ist. Die beiden
Bedeutungen des Wortes ”wahrscheinlich“ stehen zwar in einem gewissen Zusammenhang,
aber sie sind nicht identisch.
2
Man gibt die Hypothese nur um einen immer höheren Preis auf.
3
Die Induktion ist ein Vorgang nach einem ökonomischen Prinzip.
4
Die Frage der Einfachheit der Darstellung durch eine bestimmte angenommene
Hypothese hängt, glaube ich, unmittelbar mit der Frage der Wahrscheinlichkeit zusammen.
5
Man kann einen Teil einer Hypothese vergleichen mit
der Bewegung eines Teils eines Getriebes, einer Bewegung,
die man festlegen kann, ohne dadurch die bezweckte
Bewegung zu präjudizieren. Wohl aber hat man dann das
übrige Getriebe auf eine bestimmte Art einzurichten, daß es
die gewünschte Bewegung hervorbringt. Ich denke an ein
Differentialgetriebe. – Habe ich die Entscheidung getroffen,
daß von einem gewissen Teil meiner Hypothese nicht
abgewichen werden soll, was immer die zu beschreibende
Erfahrung sei, so habe ich eine Darstellungsweise festgelegt
und jener Teil der Hypothese ist nun ein Postulat. Ein
Postulat muß von solcher Art sein, daß keine denkbare Erfahrung es widerlegen kann, wenn
es auch äußerst unbequem sein mag, an dem Postulat festzuhalten. In dem Maße, wie man
hier von einer größeren oder geringeren Bequemlichkeit reden kann, gibt es eine größere
oder geringere Wahrscheinlichkeit des Postulats.
6
Von einem Maß dieser Wahrscheinlichkeit zu reden, ist nun vor7
der Hand sinnlos.
Es verhält sich hier ähnlich wie im Falle, etwa, zweier Zahlenarten, wo wir mit einem
1 (M): ? /
2 (M): ? /
3 (M): ? /
4 (M): ∫
5 (M): / (F): MS 108, S. 109.
6 (M): ? /
7 (V): von
123
124
T
rieb
Postulat
b
ezweckte Bewegung
98
WBTC030-34 30/05/2005 16:17 Page 196
33
Probability.
1
The probability of a hypothesis is measured in terms of how much evidence is needed
to make it profitable to overturn it.
Only in this sense can we say that repeated uniform experience in the past makes the continuance
of this uniformity probable in the future.
Now if in this sense I say: “I assume that the sun will rise again tomorrow because the
opposite is too improbable”, then here I mean something entirely different by “probable”
or “improbable” from what I mean by these words in the proposition “It’s equally probable
that I’ll throw heads or tails”. The two meanings of the word “probable” have a certain connection,
to be sure, but they are not identical.
2
We give up a hypothesis only for an ever higher gain.
3
Induction is a process based on an economic principle.
4
The question, how simple is the account that results from assuming a particular hypothesis,
is directly connected, I believe, with the question of probability.
5
One can compare a part of a hypothesis with the movement
of a part of a gear train, a movement that can be established
without thereby prejudging the intended motion. But
then of course one does have to make appropriate adjustments
to the rest of the train for it to produce the desired motion.
I’m thinking of differential gearing. – If I’ve decided that there
is to be no deviation from a certain part of my hypothesis, no
matter what the experience to be described may be, then I’ve
stipulated a mode of representation, and that part of the
hypothesis is now a postulate. A postulate must be irrefutable
by any conceivable experience, even though clinging to it
may be extremely inconvenient. To the extent that one can speak here of greater or lesser
convenience there is a greater or lesser probability of the postulate.
6
At this point it makes no sense to talk about a measure for this probability. The situation
here is like that, say, of two kinds of numbers, where we can say with a certain amount
1 (M): ? /
2 (M): ? /
3 (M): ? /
4 (M): ∫
5 (M): / (F): MS 108, p. 109.
6 (M): ? /
D
irection of Movemen
t
Postulate
D
rive
98e
WBTC030-34 30/05/2005 16:17 Page 197
gewissen Recht sagen können, die eine sei der andern ähnlicher (stehe ihr näher) als einer
dritten, ein zahlenmäßiges Maß der Ähnlichkeit aber nicht existiert. Man könnte sich
natürlich auch in solchen Fällen ein Maß konstruiert denken, indem man etwa die Postulate
oder Axiome zählt, die beide Systeme gemein haben, etc. etc.
8
Ich gebe jemandem die Information und nur diese:
Du wirst um die und die Zeit auf der Strecke AB einen
Lichtpunkt erscheinen sehen. Hat nun die Frage einen
Sinn, ”ist es wahrscheinlicher, daß dieser Punkt im Interval AC erscheint, als in CB“? Ich
glaube, offenbar nein. – Ich kann freilich bestimmen, daß die Wahrscheinlichkeit, daß das
Ereignis in CB eintritt, sich zu der, daß es in AC eintritt, verhalten soll, wie , aber das
ist eine Bestimmung, zu der ich empirische Gründe haben kann, aber a priori ist darüber
nichts zu sagen. Die beobachtete Verteilung von Ereignissen kann mich9
zu dieser
Annahme führen. Die Wahrscheinlichkeit, wo unendlich viele Möglichkeiten in Betracht
kommen, muß natürlich als Limes betrachtet werden. Teile ich nämlich die Strecke AB
in beliebig viele, beliebig ungleiche Teile und betrachte die Wahrscheinlichkeiten, daß
das Ereignis in irgend einem dieser Teile stattfindet, als untereinander gleich, so haben wir
sofort den einfachen Fall des Würfels vor uns. Und nun kann ich ein Gesetz – willkürlich
– aufstellen, wonach Teile gleicher Wahrscheinlichkeit gebildet werden sollen. Z.B., das
Gesetz, daß gleiche Länge der Teile gleiche Wahrscheinlichkeit bedingt.10
Aber auch jedes
andere Gesetz ist gleichermaßen erlaubt.
Könnte ich nicht auch im Fall des Würfels etwa 5 Flächen zusammennehmen als eine
Möglichkeit und sie der sechsten als der zweiten Möglichkeit gegenüberstellen? Und was,
außer der Erfahrung, kann mich hindern, diese beiden Möglichkeiten als gleich wahrscheinlich
zu betrachten?
Denken wir uns etwa einen roten Ball geworfen, der nur eine ganz kleine grüne Calotte
hat. Ist es in diesem Fall nicht viel wahrscheinlicher, daß er auf dem roten Teil auffällt,
als auf dem grünen? – Wie würde man aber diesen Satz begründen? Wohl dadurch, daß
der Ball, wenn man ihn wirft, viel öfter auf die rote, als auf die grüne Fläche auffällt. Aber
das hat nichts mit der Logik zu tun. – Man könnte die rote und grüne Fläche und die
Ereignisse, die auf ihnen stattfinden, immer auf solche Art auf eine Fläche projizieren, daß
die Projektion der grünen Fläche gleich oder größer wäre als die der roten; so, daß die
Ereignisse, in dieser Projektion betrachtet, ein ganz anderes Wahrscheinlichkeitsverhältnis
zu haben scheinen, als auf der ursprünglichen Fläche. Wenn ich z.B. die Ereignisse in einem
geeigneten gekrümmten Spiegel sich abbilden lasse und mir nun denke, was ich für das
wahrscheinlichere Ereignis gehalten hätte, wenn ich nur das Bild im Spiegel sehe.
Dasjenige, was der Spiegel nicht verändern kann, ist die Anzahl bestimmt umrissener
Möglichkeiten. Wenn ich also auf meinem Ball n Farbenflecke habe, so zeigt der Spiegel
auch n, und habe ich bestimmt, daß diese als gleich wahrscheinlich gelten sollen, so kann ich
diese Bestimmung auch für das Spiegelbild aufrecht erhalten.
Um mich noch deutlicher zu machen: Wenn ich das Experiment im Hohlspiegel ausführe,
d.h. die Beobachtungen im Hohlspiegel mache, so wird es vielleicht scheinen, als fiele der
Ball öfter auf die kleine Fläche, als auf die viel größere und es ist klar, daß keinem der
Experimente – im Hohlspiegel und außerhalb – ein Vorzug gebührt.
CB
AC
8 (M): / (F): MS 108, S. 110.
9 (O): nicht (E): Auf Grund von MS 108
(S. 111) geändert.
10 (V): bedingt,.
125
126
99 Wahrscheinlichkeit.
A C B
WBTC030-34 30/05/2005 16:17 Page 198
of justification that one of them is more like (is closer to) the other than a third, but where
there isn’t any numerical measure of this likeness. Of course one could imagine a measure
being constructed in such cases too, say by counting the postulates or axioms common to
the two systems, etc. etc.
7
I give someone this – and only this – information: At
such and such a time you will see a point of light appear
on the segment AB. Now does this question make sense:
“Is it more probable that this point will appear in the segment AC than in CB”? Obviously
not, I believe. – To be sure, I can stipulate that the probability of the event happening in
CB should relate to its probability of occurring in AC as , but that is a stipulation for
which I might have empirical grounds; nothing can be said about it a priori. The observed
distribution of events can lead me to8
this assumption. Where an infinite number of possibilities
come into consideration, probability must of course be viewed as a limit. For if I
divide the segment AB into an arbitrary number of arbitrarily unequal parts and view the
probabilities that the event will take place in any one of these parts as equal to every other,
then we are immediately faced with the simple case of a die. And now I can – arbitrarily –
posit a law for constructing parts that have an equal probability. For instance, the law that
parts of equal length are equally probable. But any other law is just as permissible.
In the case of the die, couldn’t I also subsume, say, 5 faces as one possibility and oppose
them to the sixth as a second possibility? And what, apart from experience, can prevent me
from regarding these two possibilities as equally probable?
Let’s imagine, for example, a red ball with just a very small green patch on it being thrown
in the air. In this case isn’t it much more probable that it will land on the red part than on
the green? – But how would one justify this proposition? Presumably by pointing out that
when the ball is thrown it lands much more often on its red than on its green surface. But
that’s got nothing to do with logic. – One could always project the red and green surfaces
and the incidences of their landings onto another surface in such a way that the projection
of the green surface would be equal to or greater than that of the red; so that the incidences
as viewed in this projection seem to have a totally different probability ratio than on the original
surface. As would happen, for example, if I have the incidences reflected in a suitably
curved mirror, and now imagine what I would have taken to be the more probable event if
I had seen only the image in the mirror.
What the mirror cannot change is the number of clearly defined possibilities. So if I have
n colour patches on the ball then the mirror too shows n, and if I have stipulated that they
are to count as equally probable, then I can also maintain this stipulation for the image in
the mirror.
To make myself even clearer: If I conduct the experiment with a concave mirror, i.e. make
the observations in a concave mirror, then it might seem as if the ball were falling more often
on its smaller surface than on its much larger one, and it’s clear that neither the experiment
in the concave mirror nor the one outside it is privileged.
CB
AC
7 (M): / (F): MS 108, p. 110. 8 (O): cannot lead to (E): We have made this
correction on the basis of the corresponding
passage in MS 108 (p. 111).
Probability. 99e
A C B
WBTC030-34 30/05/2005 16:17 Page 199
11
Wir können unser altes Prinzip auf die Sätze, die eine Wahrscheinlichkeit ausdrücken,
anwenden und sagen, daß wir ihren Sinn erkennen werden, wenn wir bedenken, was sie
verifiziert.
Wenn ich sage ”das wird wahrscheinlich eintreffen“, wird dieser Satz durch das
Eintreffen verifiziert, oder durch das Nichteintreffen falsifiziert? Ich glaube, offenbar nein.
Dann sagt er auch nichts darüber aus. Denn, wenn ein Streit darüber entstünde, ob es
wahrscheinlich ist oder nicht, so würden immer nur Argumente aus der Vergangenheit herangezogen
werden. Und auch dann nur, wenn es bereits bekannt wäre, was eingetroffen ist.
12
Die Kausalität beruht auf einer beobachteten Gleichförmigkeit. Nun ist zwar nicht gesagt,
daß eine bisher beobachtete Gleichförmigkeit immer so weiter gehen wird, aber, daß die
Ereignisse bisher gleichförmig waren, muß feststehen; das kann nicht wieder das unsichere
Resultat einer empirischen Reihe sein, die selbst auch wieder nicht gegeben ist, sondern von
einer ebenso unsicheren abhängt, u.s.f. ad inf.
13
Wenn Leute sagen, der Satz ”es ist wahrscheinlich, daß p eintreffen wird“ sage etwas
über das Ereignis p, so vergessen sie, daß es auch wahrscheinlich bleibt, wenn das Ereignis
p nicht eintrifft.
14
Wir sagen mit dem Satz ”p wird wahrscheinlich eintreffen“ zwar etwas über die
Zukunft, aber nicht15
etwas ”über das Ereignis p“, wie die grammatische Form der Aussage
uns glauben macht.
16
Wenn ich nach dem Grund einer Behauptung frage, so ist die Antwort auf diese Frage
nicht für den Gefragten und eben diese Handlung (die Behauptung), sondern allgemein gültig.
17
Wenn ich sage: ”das Wetter deutet auf Regen“, sage ich etwas über das zukünftige Wetter?
Nein, sondern über das gegenwärtige, mit Hilfe eines Gesetzes, welches das Wetter zu einer
Zeit mit dem Wetter in einer früheren18
Zeit in Verbindung bringt. Dieses Gesetz muß bereits
vorhanden sein, und mit seiner Hilfe fassen wir gewisse Aussagen über unsere Erfahrung
zusammen. –
Aber dasselbe könnte man dann auch für historische Aussagen behaupten. Aber es war
ja auch vorschnell, zu sagen, der Satz ”das Wetter deutet auf Regen“ sage nichts über das
zukünftige Wetter. Das kommt darauf an, was man darunter versteht, ”etwas über etwas
auszusagen“. Der Satz sagt eben seinen Wortlaut!
Er sagt19
nur etwas über die Zukunft in einem Sinn, in welchem seine Wahr- und Falschheit
gänzlich unabhängig ist von dem, was in der Zukunft geschehen wird.
20
Wenn wir sagen, ”das Gewehr zielt jetzt auf den Punkt P“, so
sagen wir nichts darüber, wohin der Schuß treffen wird. Der Punkt
auf den es zielt, ist ein geometrisches Hilfsmittel zur Angabe seiner
Richtung. Daß wir gerade dieses Mittel verwenden, hängt allerdings
mit gewissen Beobachtungen21
zusammen (Wurfparabel, etc.),
aber diese treten jetzt nicht in die Beschreibung der Richtung ein.
127
100 Wahrscheinlichkeit.
11 (M): /
12 (M): ∫
13 (M): ∫
14 (M): ∫
15 (V): Zukunft, nicht
16 (M): ∫
17 (M): ∫ /
18 (V): mit dem Wetter zu einer späteren
19 (V): Der Satz ”p wird wahrscheinlich eintreten“
sagt
20 (M): ? / (F): MS 113, S. 2r.
21 (V): Erfahrungen
128
Parabel
P
WBTC030-34 30/05/2005 16:17 Page 200
9
We can apply our old principle to propositions that express a probability, and say that
we shall discover their sense when we consider what verifies them.
If I say “That will probably occur”, is this proposition verified by its occurrence or falsified
by its non-occurrence? Obviously not, I believe. Then neither does it say anything about it.
For if a disagreement developed as to whether it is probable or not, then only arguments
from the past would be marshalled. And even this would happen only if what has occurred
were already known.
10
Causality depends on an observed regularity. Now to be sure, it isn’t certain that a
regularity so far observed will continue this way forever, but what has to be certain is that
the events have been regular up until now; that cannot be the uncertain result of an empirical
series that in turn isn’t a given, but depends on one that is equally uncertain, and so on
ad infinitum.
11
When people say that the proposition “It is probable that p will occur” says something
about the event p, they forget that it remains probable even if the event p does not occur.
12
When we utter the sentence “p will probably occur” we do say something about the future,
to be sure, but not13
something “about the event p”, as the grammatical form of the statement
would have us believe.
14
If I ask for the grounds of an assertion, the answer to this question is not valid for the
person I asked and for this specific action (the assertion), but in general.
15
If I say “The weather looks like rain” am I saying anything about future weather? No,
rather something about the present weather, with the help of a law that connects the weather
at a certain time with the weather at an earlier16
time. This law must already exist, and with
its help we summarize certain statements about our experience. –
But one could also say the same for historical statements. So it really was rash to say that
the proposition “The weather looks like rain” doesn’t say anything about future weather.
It depends on what you understand by “saying something about something”. A sentence
simply says its wording!
It says17
something about the future only in the sense in which its truth and falsehood are
completely independent of what will happen in the future.
18
When we say, “The gun is now aiming at point P”, we’re not
saying anything about where the shot will hit. The point the gun is
aiming at is a geometric aid for indicating its direction. To be sure,
our use of just this aid is connected with certain observations19
(trajectory parabola, etc.), but these observations don’t enter into
our description of the direction at this time.
9 (M): /
10 (M): ∫
11 (M): ∫
12 (M): ∫
13 (V): to be sure, not
14 (M): ∫
15 (M): ∫ /
16 (V): at a later
17 (V): The proposition “p will probably take
place” says
18 (M): ? / (F): MS 113, p. 2r.
19 (V): experiences
Probability. 100e
Parabola
P
WBTC030-34 30/05/2005 16:17 Page 201
22
Die Galton’sche23
Photographie, das Bild einer Wahrscheinlichkeit.24
Das Gesetz der
Wahrscheinlichkeit, das Naturgesetz, was man sieht, wenn man blinzelt.
25
Was heißt es: ”die Punkte, die das Experiment liefert, liegen durchschnittlich auf einer
Geraden“? oder: ”wenn ich mit einem guten Würfel würfle, so werfe ich durchschnittlich
alle 6 Würfe eine 1“? Ist dieser Satz mit jeder Erfahrung, die ich etwa mache, vereinbar?
Wenn er das ist, so sagt er nichts. Habe ich (vorher) angegeben, mit welcher Erfahrung er
nicht mehr vereinbar ist, welches die Grenze ist, bis zu der die Ausnahmen von der Regel
gehen dürfen, ohne die Regel umzustoßen? Nein. Hätte ich aber nicht eine solche Grenze
aufstellen können? Gewiß. – Denken wir uns, die Grenze wäre so gezogen: wenn unter 6
aufeinander folgenden Würfen 4 gleiche auftreten, ist der Würfel schlecht. Nun fragt man
aber: ”Wenn das aber nur selten genug geschieht, ist er dann nicht doch gut?“26
– Darauf
lautet die Antwort: Wenn ich das Auftreten von 4 gleichen Würfen unter 6 aufeinander folgenden
für eine bestimmte Zahl von Würfen erlaube, so ziehe ich damit eine andere Grenze,
als die erste war. Wenn ich aber sage ”jede Anzahl gleicher aufeinander folgender Würfe ist
erlaubt, wenn sie nur selten genug auftritt, dann habe ich damit die Güte des Würfels im
strengen Sinne als unabhängig von den Wurfresultaten erklärt. Es sei denn, daß ich unter
der Güte des Würfels nicht eine Eigenschaft des Würfels, sondern eine Eigenschaft einer
bestimmten Partie im Würfelspiel verstehe. Denn dann kann ich allerdings sagen: Ich nenne
den Würfel in einer Partie gut, wenn unter den N Würfen der Partie nicht mehr als logN
gleiche aufeinander folgende vorkommen. Hiermit wäre aber eben kein Test zur Überprüfung
von Würfeln gegeben, sondern ein Kriterium zur Beurteilung einer Partie des Spiels.
27
Man sagt, wenn der Würfel ganz gleichmäßig und sich selbst überlassen ist, dann muß
die Verteilung der Ziffern 1, 2, 3, 4, 5, 6 unter den Wurfresultaten gleichförmig sein, weil
kein Grund vorhanden ist, weshalb die eine Ziffer öfter vorkommen sollte als die andere.
28
Stellen wir nun aber die Wurfresultate statt durch die Ziffern 1 bis 6 durch die Werte29
der Funktion (x − 3)2
für die Argumente 1 bis 6 dar, also durch die Ziffern 0, 1, 4, 9. Ist
ein Grund vorhanden, warum eine dieser Ziffern öfter in den neuen Wurfresultaten
fungieren soll, als eine andere? Dies lehrt uns, daß das Gesetz a priori der Wahrscheinlichkeit
eine Form von Gesetzen ist, wie die der Minimumgesetze der Mechanik etc. Hätte man durch
Versuche herausgefunden, daß die Verteilung der Würfe 1 bis 6 mit einem regelmäßigen
Würfel so ausfällt, daß die Verteilung der Werte (x − 3)2
eine gleichmäßige wird, so hätte
man nun diese Gleichmäßigkeit als die Gleichmäßigkeit a priori erklärt.
So machen wir es auch in der kinetischen Gastheorie: wir stellen die Verteilung der
Molekülbewegungen in der Form irgend einer gleichförmigen Verteilung dar; was aber
gleichförmig verteilt ist – so wie an andrer Stelle was zu einem Minimum wird – wählen
wir so, daß unsere Theorie mit der Erfahrung übereinstimmt.
30
”Die Moleküle bewegen sich bloß nach den Gesetzen der Wahrscheinlichkeit“, das soll
heißen: die Physik tritt ab, und die Moleküle bewegen sich jetzt quasi bloß nach Gesetzen
der Logik. Diese Meinung ist verwandt der, daß das Trägheitsgesetz ein Satz a priori ist;
und auch hier redet man davon, was ein Körper tut, wenn er sich selbst überlassen ist. Was
22 (M): ? /
23 (O): Gallstone’sche
24 (E): Siehe: Francis Galton, Inquiries into
Human Faculty and Its Development, London,
1883, 1. Kap. und Anhang A, ”Composite
Portraiture“.
25 (M): /
26 (V): gut!?“
27 (M): /
28 (M): /
29 (O): Worte
30 (M): /
129
130
101 Wahrscheinlichkeit.
WBTC030-34 30/05/2005 16:17 Page 202
20
Galtonian photography, the picture of a probability.21
The law of probability, the law of
nature, what you see when you screw up your eyes.
22
What does this mean: “The points that result from the experiment lie on the average on
a straight line”? or: “If I play dice with a good die, then on average I get a 1 every sixth
try”? Is this proposition compatible with every experience I might have? If it is, it says nothing.
Have I indicated (in advance) which experience is incompatible with it, and the limit
that exceptions to the rule may approach without overthrowing the rule? No. But couldn’t
I have set up such a limit? Certainly. – Let’s imagine it were set up this way: if among 6
consecutive throws 4 turn out the same, the die is bad. But now someone asks: “But if that
happens only rarely, isn’t the die still good?”23
– The answer to this is: If within a certain
number of throws I allow 4 throws to be the same out of 6 consecutive ones, then the limit
I am setting is different from the first one. But if I say “Any number of consecutive equal
throws is allowed if this only occurs sufficiently rarely”, then in a strict sense I have declared
the quality of the die to be independent of the results of the throws. Unless by “quality of
the die” I understand not a property of the die, but of a particular round in a game of dice.
For then I really can say: I call the die in a round “good” if among N throws in that round
not more than logN equal results occur consecutively. But this wouldn’t give us a test for
checking dice, but a criterion for judging a particular round of a game.
24
We say, if the die is completely regular and is left to itself the distribution of the numbers
1, 2, 3, 4, 5, 6 that results from throwing it must be uniform, because there is no reason
why one number should occur more frequently than another.
25
But now, instead of representing the results of the throws by the numbers 1 to 6, let’s
represent them by the values of the function (x − 3)2
for the arguments 1 to 6, i.e. by the
numbers 0, 1, 4, 9. Is there a reason why one rather than another of these numbers ought to
figure more frequently among the new results? This teaches us that the a priori law of probability
is a form of law, like that of the minimum-principles of mechanics, etc. If we had
discovered experimentally that the distribution of the throws 1 to 6 with a regular die turned
out such that the distribution of the values (x − 3)2
became uniform, then this regularity would
have been declared to be the a priori regularity.
That is the way we proceed in kinetic gas theory as well: We represent the distribution
of the molecular movements in the form of some uniform distribution; but we choose what
is distributed uniformly – just as elsewhere we choose what turns into a minimum – so that
our theory agrees with experience.
26
“Molecules move only according to the laws of probability” is supposed to mean: physics
exits and now the molecules move, as it were, purely according to the laws of logic. This
idea is related to the idea that the law of inertia is an a priori proposition; and there too one
speaks of what a body does when left to itself. What’s the criterion for its being left to itself?
20 (M): ? /
21 (E): Cf. Francis Galton, Inquiries into Human
Faculty and Its Development, London, 1883,
Ch. I and Appendix A, “Composite
Portraiture”.
22 (M): /
23 (V): good!?”
24 (M): /
25 (M): /
26 (M): /
Probability. 101e
WBTC030-34 30/05/2005 16:17 Page 203
ist das Kriterium dafür, daß er sich selbst überlassen ist? Ist es am Ende das, daß er sich
gleichförmig in einer Geraden bewegt? Oder ist es ein anderes. Wenn das letztere, dann ist
es eine Sache der Erfahrung, ob das Trägheitsgesetz stimmt; im ersten Fall aber war es gar
kein Gesetz, sondern eine Definition. Und Analoges gilt von einem Satz: ”wenn die
Teilchen sich selbst überlassen sind, dann ist die Verteilung ihrer Bewegungen die und die“.
Welches ist das Kriterium dafür, daß sie sich selbst überlassen sind? etc.
31
|Wenn die Messung ergibt, daß der Würfel genau und homogen ist, – ich nehme an,
daß die Ziffern auf seinen Flächen die Wurfresultate nicht beeinflussen – und die werfende
Hand bewegt sich regellos – folgt daraus die durchschnittlich gleichmäßige Verteilung der
Würfe 1 bis 6? Woraus sollte man die schließen? Über die Bewegung beim Werfen hat man
keine Annahme gemacht und die Prämisse32
der Genauigkeit des Würfels ist doch von ganz
anderer Multiplizität,33
als eine durchschnittlich gleichförmige Verteilung von Resultaten.
Die Prämisse ist gleichsam einfärbig, die Konklusion gesprenkelt. Warum hat man gesagt,
der Esel werde zwischen den beiden gleichen Heubündeln verhungern, und nicht, er werde
von beiden durchschnittlich gleich oft fressen?34
|
35
Zu sagen, die Punkte, die dieses Experiment liefert, liegen durchschnittlich auf dieser
Linie, z.B. einer Geraden, sagt etwas Ähnliches wie: ”aus dieser Entfernung gesehen, scheinen
sie in einer Geraden zu liegen“.
Ich kann von einer Strecke36
sagen, der allgemeine Eindruck ist der einer Geraden; aber nicht:
”die Strecke37
schaut gerade aus, denn sie kann das Stück einer Linie sein, die mir als Ganze38
den Eindruck der Geraden macht“. (Berge auf der Erde und auf dem Mond. Erde eine Kugel.)
39
Das Experiment des Würfelns dauert eine gewisse Zeit, und unsere Erwartungen über
die zukünftigen Ergebnisse des Würfelns können sich nur auf Tendenzen gründen, die wir
in den Ergebnissen des Experiments wahrnehmen. D.h., das Experiment kann nur die
Erwartung begründen, daß es so weitergehen wird, wie (es) das Experiment gezeigt hat. Aber
wir können nicht erwarten, daß das Experiment, wenn fortgesetzt, nun Ergebnisse liefern
wird, die mehr als die des wirklich ausgeführten Experiments mit einer vorgefaßten
Meinung über seinen Verlauf übereinstimmen. Wenn ich also z.B. Kopf und Adler werfe
und in den Ergebnissen des Experiments keine Tendenz der Kopf- und Adler-Zahlen finde,
sich weiter einander zu nähern, so gibt das Experiment mir keinen Grund zur Annahme,
daß seine Fortsetzung eine solche Annäherung zeigen wird. Ja die Erwartung dieser
Annäherung muß sich selbst auf einen bestimmten Zeitpunkt beziehen, denn man kann nicht
sagen, man erwarte, daß ein Ereignis einmal – in der unendlichen Zukunft – eintreten werde.
40
Alle ”begründete Erwartung“ ist Erwartung, daß eine bis jetzt beobachtete Regel
weiter41
gelten wird.
(Die Regel aber muß beobachtet worden sein und kann nicht selbst wieder bloß erwartet
werden.)
42
Die Logik der Wahrscheinlichkeit hat es mit dem Zustand der Erwartung nur soweit zu
tun, wie die Logik überhaupt, mit dem Denken.
31 (M): /
32 (V): Annahme
33 (V): Art
34 (V): und nicht, er werde durchschnittlich so oft
von dem einen, wie von dem andern fressen?
35 (M): /
36 (V): Linie
37 (V): Linie
38 (V): Ganzes
39 (M): /
40 (M): /
41 (V): weiterhin
42 (M): /
131
132
102 Wahrscheinlichkeit.
WBTC030-34 30/05/2005 16:17 Page 204
Possibly that it moves uniformly in a straight line? Or is it a different criterion? If the
latter, then it’s an empirical matter whether the law of inertia is correct; but if the former,
then it wasn’t a law at all, but a definition. And something analogous is true of this proposition:
“When particles are left to themselves then the distribution of their movements is
such and such”. What is the criterion for their being left to themselves? Etc.
27
|If a measurement proves that the die is exact and homogenous – I’m assuming that the
numbers on its surfaces don’t influence the results of the throws – and the hand performing
the throws moves arbitrarily, does an average uniform distribution of the throws 1 to 6
result? How is one to deduce that? No assumption was made about the throwing motion,
and the premiss28
that the die is accurate is of an entirely different multiplicity29
than an
average uniform distribution of results. The premiss is monochrome, as it were, the conclusion
mottled. Why was it said that the ass would starve between two identical bundles of
hay and not that he would eat from both with an equal average frequency?30
|
31
To say that the points produced by this experiment lie on average on a line, e.g. a straight
line, is to say something like: “Viewed from this distance, they seem to lie in a straight line”.
I can say of a line segment32
that it gives the general impression of a straight line; but not
“The line segment33
looks straight because it can be part of a line that, as a whole, seems to
me to be straight.” (Mountains on earth and on the moon. The earth a sphere.)
34
The experiment of throwing the die lasts a certain time, and our expectations about future
results of casting the die can only be based on tendencies that we observe in the results
of the experiment. That is, the experiment can only justify the expectation that things will
continue in the way shown by the experiment. But we can’t expect that the experiment, if
continued, will now produce results that tally better with a preconceived idea of its course
than did those of the experiment that was actually carried out. So if I am flipping a coin,
for instance, and discover no tendency in the results of the experiment for the number of
heads and tails to continue to approximate to each other more closely, then the experiment
gives me no reason to assume that continuing it will show such a convergence. Indeed, the
expectation of this convergence must itself refer to a definite point in time, for we can’t say
that we expect that an event will occur eventually – in the infinite future.
35
Any “reasonable expectation” is an expectation that a rule that has been followed up until
now will continue to hold.
(But the rule must have been followed, and can’t itself be merely expected.)
36
The logic of probability is concerned with a state of expectation only to the extent that
logic in general is concerned with thinking.
27 (M): /
28 (V): assumption
29 (V): of a different kind
30 (V): and not that on average he will eat from the
one as frequently as from the other?
31 (M): /
32 (V): line
33 (V): line
34 (M): /
35 (M): /
36 (M): /
Probability. 102e
WBTC030-34 30/05/2005 16:17 Page 205
43
Von der Lichtquelle Q wird ein Lichtstrahl
ausgesandt, der die Scheibe AB trifft, dort einen
Lichtpunkt erzeugt und dann die Scheibe AC
trifft. Wir haben nun keinen Grund, anzunehmen,
daß der Lichtpunkt auf AB eher auf der einen
Seite der Mitte M, als auf der andern liegen wird;
aber auch keinen Grund, anzunehmen, der
Lichtpunkt auf AC werde auf der einen und nicht
auf der andern Seite der Mitte m liegen.44
Das
gibt also widersprechende Wahrscheinlichkeiten.
Wenn ich nun eine Annahme über den Grad der Wahrscheinlichkeit mache, daß der eine
Lichtpunkt im Stück AM liegt, – wie wird diese Annahme verifiziert. Wir meinen45
doch,
durch einen Häufigkeitsversuch. Angenommen nun, dieser bestätigt die Auffassung, daß
die Wahrscheinlichkeiten für das Stück AM und BM gleich sind (also für Am und Cm
verschieden), so ist sie damit als die richtige erkannt und erweist sich also als eine
physikalische Hypothese. Die geometrische Konstruktion zeigt nur, daß die Gleichheit
der Strecken AM und BM kein Grund zur Annahme gleicher Wahrscheinlichkeit war.
46
Wenn ich annehme, die Messung ergebe, daß der Würfel genau und homogen ist, und
die Ziffern auf seinen Flächen die Wurfresultate nicht beeinflussen, und die Hand, die ihn
wirft, bewegt sich ohne bestimmte Regel; folgt daraus eine47
durchschnittlich gleichförmige
Verteilung der Würfe 1 bis 6 unter den Wurfergebnissen? – Woraus sollte sie hervorgehen?
Daß der Würfel genau und homogen ist, kann doch keine durchschnittlich gleichförmige
Verteilung von Resultaten begründen. (Die Voraussetzung ist sozusagen homogen, die
Folgerung wäre gesprenkelt.) Und über die Bewegung beim Werfen haben wir ja keine
Annahme gemacht. (Mit der Gleichheit der beiden Heubündel hat man zwar begründet,
daß der Esel in ihrer Mitte verhungern (werde); aber nicht, daß er ungefähr gleich oft von
jedem fressen werde.) – Mit unseren Annahmen ist es auch vollkommen vereinbar, daß mit
dem Würfel 100 Einser nacheinander geworfen werden, wenn Reibung, Handbewegung,
Luftwiderstand so zusammentreffen. Die Erfahrung, daß das nie geschieht, ist eine
diese Faktoren betreffende.48
Und die Vermutung der gleichmäßigen Verteilung der
Wurfergebnisse ist eine Vermutung über das Arbeiten dieser Faktoren.49
Wenn man sagt, ein gleicharmiger Hebel, auf den symmetrische Kräfte wirken, müsse
in Ruhe bleiben, weil keine Ursache vorhanden ist, weshalb er sich eher auf die eine als
auf die andre Seite neigen sollte, so heißt das nur, daß, wenn wir gleiche Hebelarme und
symmetrische Kräfte konstatiert haben und nun der Hebel sich nach der einen Seite neigt,
wir dies aus den uns bekannten – oder von uns angenommenen – Voraussetzungen nicht
erklären können. (Die Form, die wir ”Erklärung“ nennen, muß auch asymmetrisch sein; wie
die Operation, die aus ”a + b“ ”2a + 3b“50
macht.) Wohl aber können wir die andauernde
Ruhe des Hebels aus unsern Voraussetzungen erklären. – Aber auch eine schwingende
Bewegung, die durchschnittlich gleich oft von der Mitte51
nach rechts und nach links
gerichtet ist? Die schwingende Bewegung nicht, denn in der ist ja wieder Asymmetrie. Nur
43 (F): MS 114, S. 18v.
44 (V): Wir haben nun keinen Grund zur
Annahme, der Lichtpunkt auf AB werde rechts
von der Mitte M liegen, noch zur entgegengesetzten;
aber auch keinen Grund anzunehmen,
der Lichtpunkt auf AC werde auf der und nicht
auf jener Seite von der Mitte m liegen.
45 (V): denken
46 (M): ? /
47 (V): die
48 (V): ist eine, die diese Faktoren betrifft.
49 (V): Einflüsse.
50 (V): @a + b! @2a! @2a + 3b!
51 (V): Mittellage
133
134
103 Wahrscheinlichkeit.
B
Q
A
m
C
M
WBTC030-34 30/05/2005 16:17 Page 206
37
A ray of light, emitted from a light source Q,
falls upon a disc AB, creating a point of light there,
and then falls on disc AC. We have no reason to
assume that the point of light on AB is more likely
to lie on one side of the centre M than on the other;
and also none to assume that the point of light on
AC will be located on one but not on the other side
of the centre m.38
Thus we have contradictory probabilities.
Now if I make an assumption about the
degree of probability that the one point of light will
be situated in segment AM – how is this assumption verified? By a frequency experiment,
we would think. Now, assuming that this experiment confirms the view that the probabilities
for segments AM and BM are equal (and are therefore different for Am and Cm), then
this view is thereby acknowledged as correct, and thus proves to be a hypothesis of physics.
The geometric construction merely shows that the equality of the segments AM and BM
gave us no reason to assume equal probability.
39
If I assume that a measurement proves a die to be accurate and homogeneous and that
the numbers on its surfaces don’t influence the results of the throws and that the hand that
throws it moves without following any definite pattern, does it follow that among the results
of the throws there will be on average a40
uniform distribution of throws 1 to 6? – From
what should this distribution follow? That the die is accurate and homogeneous can’t be the
basis for an average uniform distribution of results. (The premiss is homogeneous, so to speak,
the conclusion mottled.) And we haven’t made any assumption about the movements in throwing.
(To be sure, the equality of the two bundles of hay has been given as the reason that
an ass (will) starve between them; but not as a reason for his eating from both bundles with
more or less equal frequency.) – It is also completely compatible with our assumptions that
100 ones will be cast in succession if friction, movements of the hand, and air-resistance combine
in a certain way. Our experience that this never happens is related to these factors. And
the conjecture about the regular distribution of the results of the throws is a conjecture about
the workings of these factors.41
If someone says that an equal-armed lever, when affected by symmetrical forces, must
remain at rest because there is no reason for it to tip toward one side rather than the other,
then all that means is that, if we have established that the lever has equally long arms and
that the forces are symmetrical, but now the lever leans to one side, we cannot explain this
from the presuppositions we know – or have assumed. (The form that we call “explanation”
must also be asymmetrical; like the operation that makes “2a + 3b”42
out of “a + b”.) What
we can explain on the basis of our presuppositions is the lever’s constantly remaining at rest.
– But can we also explain a swinging motion towards the left and then right of the middle43
with equal average frequency? No, because once again there is an asymmetry in it. All we
can explain is the symmetry in this asymmetry. If the lever had uniformly swung right, then
37 (F): MS 114, p. 18v.
38 (V): We have no reason to assume that the
point of light on AB will lie on the right side
of the centre M, nor do we have any reason
for assuming the opposite; but we also have no
reason to assume that the point of light on AC
will be located on this and not on that side of
the centre m.
39 (M): ? /
40 (V): the
41 (V): influences.
42 (V): !2aE
43 (V): middle position
Probability. 103e
B
Q
A
m
C
M
WBTC030-34 30/05/2005 16:17 Page 207
die Symmetrie in dieser Asymmetrie. Hätte sich der Hebel gleichförmig nach rechts
gedreht, so könnte man analog sagen: Mit der Symmetrie der Bedingungen kann ich
die Gleichförmigkeit der Bewegung, aber nicht ihre Richtung erklären.
Eine Ungleichförmigkeit der Verteilung der Wurfresultate ist mit der Symmetrie des Würfels
nicht zu erklären. Und nur insofern erklärt diese Symmetrie die Gleichförmigkeit der
Verteilung. – Denn man kann natürlich sagen: Wenn die Ziffern auf den Würfelflächen
keine Wirkung haben, dann kann ihre Verschiedenheit nicht eine Ungleichförmigkeit
der Verteilung erklären; und gleiche Umstände können selbstverständlich nicht
Verschiedenheiten erklären; soweit also könnte man auf eine Gleichförmigkeit schließen.
Aber woher dann überhaupt verschiedene Wurfresultate? Was diese52
erklärt, muß nun
auch ihre durchschnittliche Gleichförmigkeit erklären. Die Regelmäßigkeit des Würfels
stört nur eben diese Gleichförmigkeit nicht.
53
Angenommen, Einer der täglich im Spiel würfelt, würde etwa eine Woche lang nichts
als Einser werfen, und zwar mit Würfeln, die nach allen anderen Methoden54
der Prüfung55
sich als gut erweisen, und wenn ein Andrer sie wirft, auch die gewöhnlichen Resultate liefern.56
Hat er nun Grund, hier ein Naturgesetz anzunehmen, dem gemäß er immer Einser werfen
muß;57
hat er Grund zu glauben, daß das nun so weiter gehen wird, – oder (vielmehr) Grund
anzunehmen, daß diese Regelmäßigkeit nicht lange mehr andauern wird?58
Hat er also
Grund das Spiel aufzugeben, da es sich gezeigt hat, daß er nur Einser werfen kann; oder
weiterzuspielen, da es jetzt nur um so wahrscheinlicher ist, daß er beim nächsten Wurf eine
höhere Zahl werfen wird? – In Wirklichkeit wird er sich weigern, die Regelmäßigkeit als
ein Naturgesetz anzuerkennen; zum mindesten wird sie lang andauern müssen, ehe er diese
Auffassung in Betracht zieht. Aber warum? – Ich glaube, weil so viel frühere Erfahrung seines
Lebens gegen ein solches Gesetz spricht, die alle – sozusagen – erst überwunden werden
muß, ehe wir eine ganz neue Betrachtungsweise annehmen.
59
Wenn wir aus der relativen Häufigkeit eines Ereignisses auf seine relative Häufigkeit in
der Zukunft Schlüsse ziehen, so können wir das natürlich nur nach der bisher tatsächlich
beobachteten Häufigkeit tun. Und nicht nach einer, die wir aus der beobachteten durch
irgend einen Prozess der Wahrscheinlichkeitsrechnung erhalten haben. Denn die berechnete
Wahrscheinlichkeit stimmt mit jeder beliebigen tatsächlich beobachteten Häufigkeit überein,
da sie die Zeit offen läßt.
60
Wenn sich der Spieler, oder die Versicherungsgesellschaft, nach der Wahrscheinlichkeit
richten, so richten sie sich nicht nach der Wahrscheinlichkeitsrechnung, denn nach dieser
allein kann man sich nicht richten, da, was immer geschieht, mit ihr in Übereinstimmung
zu bringen ist; sondern die Versicherungsgesellschaft richtet sich nach einer tatsächlich
beobachteten Häufigkeit. Und zwar ist das natürlich eine absolute Häufigkeit.
135
104 Wahrscheinlichkeit.
52 (V): Wurfresultate? Gewiß, was diese
53 (M): /
54 (V): Arten
55 (V): Untersuchung
56 (V): geben.
57 (V): Einser wirft;
58 (V): kann?
59 (M): /
60 (M): /
WBTC030-34 30/05/2005 16:17 Page 208
analogously we could say: Given the symmetry of the conditions we can explain the uniformity
of the motion, but not its direction.
A lack of uniformity in the distribution of the results of the throws cannot be explained
by the symmetry of the die. And this is the extent to which this symmetry explains the uniformity
of the distribution. – For of course one can say: If the numbers on the surfaces of
the die have no effect, then their being different can’t explain a lack of uniformity in the
distribution; and of course, identical circumstances can’t explain differences; so to this extent
one could infer uniformity. But then why do we have different results of throws at all? Whatever
explains44
them must also explain their uniformity on average. It’s just that the regularity of
the die doesn’t upset this uniformity.
45
Let’s assume that someone playing dice every day were to throw, say, nothing but ones
for a whole week, and that he does this with dice that turn out to be good when subjected
to all other methods of testing,46
and that also produce47
the normal results when someone
else throws them. Does he now have reason to assume a natural law here, according to which
he always has to throw48
ones? Does he have reason to believe that things will continue in
this way – or (rather) to assume that this regularity won’t49
last much longer? So does he
have reason to quit the game since it has turned out that he can throw only ones; or to continue
playing, because now it is just all the more likely that on the next try he’ll throw a
higher number? – In actual fact he’ll refuse to acknowledge the regularity as a law of nature;
at least it will have to last for a long time before he’ll consider this view of regularity. But
why? – I think it’s because so much of his previous experience in life refutes such a law,
experience that has to be, so to speak – vanquished – before we accept a totally new way of
looking at things.
50
If we infer the relative frequency of an event in the future from its relative frequency,
then of course we can do this only based on the frequency that has in fact been observed
so far. We can’t do this based on a frequency we have got by applying some process of
the probability calculus to the one we’ve observed. For since it leaves the time open, the
probability we calculate agrees with any frequency we actually observe.
51
When gamblers or insurance companies base their actions on probability, they don’t
base them on the probability calculus. For you can’t base them on that alone, since whatever
happens can be made to agree with it. Rather, insurance companies base their actions
on an actually observed frequency. And that, of course, is an absolute frequency.
44 (V): Certainly whatever explains
45 (M): /
46 (V): all other kinds of investigation,
47 (V): give
48 (V): he always throws
49 (V): can’t
50 (M): /
51 (M): /
Probability. 104e
WBTC030-34 30/05/2005 16:17 Page 209
136
137
34
Der Begriff ”ungefähr“.
Problem des ”Sandhaufens“.
1
”Er kam ungefähr von dort →.“
”Ungefähr da ist der hellste Punkt des Horizontes.“
”Mach’ das Brett ungefähr 2m lang.“
Muß ich, um das sagen zu können, Grenzen wissen, die den Spielraum dieser Länge
bestimmen? Offenbar nicht. Genügt es nicht z.B. zu sagen: ”der Spielraum ± 1 cm ist
ohneweiteres erlaubt; ± 2 cm wäre schon zu viel“? – Es ist doch dem Sinn meines Satzes
auch wesentlich, daß ich nicht imstande bin, dem2
Spielraum ”genaue“ Grenzen zu geben.
Kommt das nicht offenbar3
daher, daß der Raum, in dem ich hier arbeite, eine andere Metrik4
hat, als der Euklidische?
Wenn man nämlich den Spielraum genau durch den Versuch feststellen wollte,
indem man die Länge ändert und sich den Grenzen des Spielraums nähert und5
immer
fragt, ob diese Länge noch angehe oder schon nicht mehr, so käme man nach einigen
Einschränkungen zu Widersprüchen, indem einmal ein Punkt noch als innerhalb der
Grenzen liegend bezeichnet würde, ein andermal ein weiter innerhalb gelegener als schon
unzulässig erklärt würde; beides etwa mit der Bemerkung, die Angaben6
seien nicht mehr
(ganz) sicher.
7
Die Unsicherheit ist von der Art, wie die der Angabe8
des höchsten Punktes einer Kurve.
Wir sind eben nicht im euklidischen Raum und es gibt hier nicht im euklidischen Sinne
einen höchsten Punkt. Die Antwort wird heißen: ”der höchste Punkt ist ungefähr da“, und
die Grammatik des Wortes ”ungefähr“ – in diesem Zusammenhang – gehört dann zur
Geometrie unseres Raumes.
9
Ist es denn nicht so, wie man etwa beim Fleischhauer nur auf Deka genau abwiegt,
obwohl das anderseits willkürlich ist, und nur bestimmt durch die herkömmlichen
Messinggewichte?10
Es genügt hier zu wissen: mehr als P1 wiegt es nicht und weniger als
P2 auch nicht. Man könnte sagen: die Gewichtsangabe besteht hier prinzipiell nicht aus
einer Zahlangabe, sondern aus der Angabe eines Intervalls, und die Intervalle bilden eine
diskontinuierliche Reihe.
11
Man könnte doch sagen: ”halte Dich jedenfalls innerhalb ± 1 cm“, damit12
eine willkürliche
Grenze setzend. – Würde nun gesagt: ”gut, aber dies ist doch nicht die wirkliche Grenze
1 (M): ? /
2 (O): den
3 (V): das offenbar
4 (V): Metrik is
5 (O): ändert // und . . . nähert // und
6 (V): Antworten
7 (M): ? /
8 (O): die, der Angabe
9 (M): ∫ /
10 (O): Messinggewichte.
11 (M): ? /
12 (O): 1 cm“ damit
105
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34
The Concept “Roughly”.
Problem of the “Heap of Sand”.
1
“He came from roughly there → .”
“Roughly there is the brightest point on the horizon.”
“Make the board roughly 2 m long.”
In order to say this do I have to know the limits that specify the margins of tolerance for
this length? Obviously not. Isn’t it enough to say, for example: “A margin of ± 1 cm is
perfectly all right; ± 2 cm would be too much”? After all, it’s essential to the sense of my
sentence that I can’t assign “exact” limits to the margin. Isn’t that obviously2
because the
space in which I am working here doesn’t have3
the same metric as the Euclidean one?
Suppose we wanted to establish the margin exactly by experiment – we alter the length
and each time we approach the limits of the margin we ask whether such a length is still
acceptable or not. Then, after some narrowing down of the length we would arrive at
contradictions, where a point was at one time designated as falling within the limits, but
one that lay closer to the true length was declared at another time to be impermissible; and
both responses might be accompanied by the remark that the statements4
were no longer
(completely) certain.
5
This uncertainty is like that of specifying the highest point of a curve. We’re simply not
in Euclidean space and here there is no highest point in the Euclidean sense. The answer
will be: “The highest point is roughly there”, and then the grammar of the word “roughly”
– in this context – belongs to the geometry of our space.
6
Isn’t it just like at the butcher’s where the weighing is only accurate to within 10 grams,
though from another point of view this is arbitrary, merely the result of the customary brass
weights. Here all we need to know is: it doesn’t weigh more than P1, and not less than P2.
One could say: In principle specifying the weight is not a specification of a number, but of
an interval, and intervals form a discontinuous series.
7
Yet one might say: “In any case stay within ± 1cm”, thus setting an arbitrary limit. –
If someone now said: “Fine, but that isn’t the real limit of the permissible margin; so what
1 (M): ? /
2 (V): Is that obviously
3 (V): working here is
4 (V): answers
5 (M): ? /
6 (M): ∫ /
7 (M): ? /
105e
WBTC030-34 30/05/2005 16:17 Page 211
des zulässigen Spielraums: welche ist es also?“, so13
wäre etwa die Antwort ”ich weiß keine,
ich weiß nur, daß ± 2 cm schon zu viel wäre“.
14
Denken wir uns folgendes psychologisches Experiment:
Wir zeigen dem Subjekt zwei Linien G1, G2, durch
welche quer die Gerade A gezogen ist. Das Stück dieser
Geraden, welches zwischen G1 und G2 liegt, werde ich
die Strecke a nennen. Wir ziehen nun in beliebiger
Entfernung von a und parallel dazu b und fragen, ob er
die Strecke b größer sieht als a, oder die beiden Längen
nicht mehr unterscheidet. Er antwortet, b erscheine
größer als a. Darauf nähern wir uns a, indem wir die Distanz von a zu b mit unsern
Meßinstrumenten halbieren und ziehen c. ”Siehst Du c größer als a?“ – ”Ja“. Wir halbieren
die Distanz c–a und ziehen d. ”Siehst Du d größer als a?“ – ”Ja“. Wir halbieren a–d. ”Siehst
Du e größer als a?“ – ”Nein“. Wir halbieren daher e–d. ”Siehst Du f größer als e?“ – ”Ja“.
Wir halbieren also e–f und ziehen h. Wir könnten uns so auch von der linken Seite der Strecke
a nähern, und dann sagen, daß einer gesehenen Länge a im euklidischen Raum nicht eine
Länge, sondern ein Intervall von Längen entspricht, und in ähnlicher Weise einer gesehenen
Lage eines Strichs (etwa des Zeigers eines Instruments) ein Intervall von Lagen im euklidischen
Raum; aber dieses Intervall hat nicht scharfe Grenzen. Das heißt: es ist nicht von Punkten
begrenzt, sondern von konvergierenden Intervallen, die nicht gegen einen Punkt konvergieren.
(Wie die Reihe der Dualbrüche, die wir durch Werfen von Kopf und Adler erzeugen.) Das
Charakteristische zweier Intervalle, die so nicht durch Punkte sondern unscharf begrenzt
sind, ist, daß auf die Frage, ob sie einander übergreifen oder getrennt voneinander liegen,
in gewissen Fällen die Antwort lautet: ”unentschieden“. Und daß die Frage, ob sie einander
berühren, einen Endpunkt miteinander gemein haben, immer sinnlos ist, da sie ja keine
Endpunkte haben. Man könnte aber sagen: sie haben vorläufige Endpunkte. In dem Sinne,
in welchem die Entwicklung von π ein vorläufiges Ende hat. An dieser Eigenschaft des
”unscharfen“ Intervalls ist natürlich nichts geheimnisvolles, sondern das etwas Paradoxe klärt
sich durch die doppelte Verwendung des Wortes ”Intervall“ auf.
Es ist dies der gleiche Fall, wie der der doppelten Verwendung des Wortes ”Schach“,
wenn es einmal die Gesamtheit der jetzt geltenden Schachregeln bedeutet, ein andermal: das
Spiel, welches N.N. in Persien erfunden hat und welches sich so und so entwickelt hat. In
einem Fall ist es unsinnig, von einer Entwicklung15
der Schachregeln zu reden, im andern
Fall nicht. Wir können ”Länge einer gemessenen Strecke“ entweder das nennen, was bei
einer bestimmten Messung, die ich heute um 5 Uhr durchführe, herauskommt, – dann gibt
es für diese Längenangabe kein ”± etc.“ – oder16
etwas, dem sich Messungen nähern etc.;
in den zwei Fällen wird das Wort ”Länge“ mit ganz verschiedener Grammatik gebraucht.
Und ebenso das Wort ”Intervall“, wenn ich einmal etwas Fertiges, einmal etwas sich
Entwickelndes ein Intervall nenne.
13 (O): es also?“ so
14 (M): / (F): MS 114, S. 7r.
15 (V): Änderung
16 (V): ”± etc.“, oder
138
139
106 Der Begriff ”ungefähr“. . . .
h f
ae d c b
G1
G2
WBTC030-34 30/05/2005 16:17 Page 212
is?” then the answer might be: “I don’t know of any, I just know that ± 2 cm would be
too much”.
8
Let’s imagine the following psychological experiment:
We show our subject two lines G1, G2 across
which the straight line a has been drawn. I’ll call the part
of this line that lies between G1 and G2 segment a. Now,
at an arbitrary distance from a and parallel to it, we draw
b and ask whether he sees the line segment b as being
longer than a or whether he no longer distinguishes
between the two lengths. He answers that b seems
longer than a. Then we move closer to a by halving the distance between a and b with our
measuring instruments, and draw c. “Do you see c as longer than a?” – “Yes”. We halve
the distance c–a and draw d. “Do you see d as longer than a?” – “Yes”. We halve a–d. “Do
you see e as longer than a?” – “No”. So we halve e–d. “Do you see f as longer than e?” –
“Yes”. So we halve e–f and draw h. In the same way we could also approach segment a from
the left side and then say that what corresponds to a perceived length a in Euclidean space
is not one length, but an interval of lengths, and similarly, that what corresponds to one
perceived position of a line (say of the pointer of an instrument) is an interval of positions
in Euclidean space; but this interval doesn’t have sharply defined limits. That means: it isn’t
bounded by points, but by converging intervals that don’t converge toward a single point.
(Like the series of binary fractions that we generate by throwing heads and tails.) What is
characteristic of two intervals that are bounded in this way – not by points but by blurred
boundary lines – is that in certain cases the answer to the question whether they overlap or
are separate from each other is: “undecided”. And that the question whether they touch,
have an end point in common, is always senseless, because of course they have no end points.
But one could say: They have temporary end points, in the sense in which the expansion
of π has a temporary end. Of course there is nothing mysterious about this property of
“blurred” intervals; rather, its somewhat paradoxical nature is explained by the ambiguity
of the word “interval”.
Here we have the same type of ambiguity as that of the word “chess”; at times that means
the totality of currently valid chess rules, and at times the game that N N invented in Persia
and that has developed in such and such a way. In the one case it is nonsensical to speak
about a development of9
the rules of chess, and in the other it is not. We can either call
what results from a particular measurement that I carry out today at 5 o’clock “the length
of a measured line segment” – in which case there is no “± etc.” for this specification of
length – or something to which measurements approximate, etc. In the two cases the word
“length” is used with completely different grammars. So too is the word “interval”, where
at some times I call something that has been completed an interval and at others something
that’s in flux.
8 (M): / (F): MS 114, p. 7r. 9 (V): about a change in
The Concept “Roughly”. . . . 106e
h f
ae d c b
G1
G2
WBTC030-34 30/05/2005 16:17 Page 213
17
I) die Intervalle liegen getrennt
II) sie liegen getrennt und berühren sich vorläufig
III)18
unentschieden
IV) unentschieden
V) unentschieden
VI) sie übergreifen
VII) sie übergreifen
Wir können uns aber nicht wundern, daß nun ein Intervall so seltsame Eigenschaften haben
soll; da wir eben das Wort ”Intervall“ jetzt in einem nicht gewöhnlichen Sinn gebrauchen.
Und wir können nicht sagen, wir haben neue Eigenschaften gewisser Intervalle entdeckt.
Sowenig wie wir neue Eigenschaften des Schachkönigs entdecken würden, wenn wir die
Regeln des Spiels änderten, aber die Bezeichnung ”Schach“ und ”König“ beibehielten. (Vergl.
dagegen Brouwer, über das Gesetz des ausgeschlossenen Dritten.)
Jener Versuch ergibt also wesentlich, was wir ein ”unscharfes“ Intervall genannt haben;
dagegen wären natürlich andere Experimente denkbar,19
die statt dessen ein scharfes
Intervall ergeben. Denken wir etwa, wir bewegten ein Lineal von der Anfangsstellung b, und
parallel zu dieser, gegen a hin, bis in unserm Subjekt irgend eine bestimmte Reaktion einträte:
dann könnten wir den Punkt, an dem die Reaktion beginnt, die Grenze unseres Streifens
nennen. – So könnten wir natürlich auch ein Wägungsresultat ”das Gewicht eines Körpers“
nennen und es gäbe dann in diesem Sinn eine absolut genaue Wägung, d.i. eine, deren Resultat
nicht die Form ”G ± g“ hat. Wir haben damit unsere Ausdrucksweise geändert, und müssen
nun sagen, daß das Gewicht des Körpers schwankt und zwar nach einem uns unbekannten
Gesetz. (Der Unterschied20
zwischen ”absolut genauer“ Wägung und ”wesentlich ungenauer“
Wägung ist ein grammatischer21
und bezieht sich auf zwei verschiedene Bedeutungen des
Ausdrucks ”Ergebnis der Wägung“.)
107 Der Begriff ”ungefähr“. . . .
17 (F): MS 114, S. 8r.
18 (V): III) sie lie
19 (V): möglich,
20 (V): Gesetz. (Die Unterscheidung
21 (V): ist eine grammatische
140
I II
III IV V
VI VII
WBTC030-34 30/05/2005 16:17 Page 214
10
(I) the intervals are separate
(II) they are separate and touch each other temporarily
(III)11
undecided
(IV) undecided
(V) undecided
(VI) they overlap
(VII) they overlap
But we shouldn’t be surprised that an interval can have such strange properties; for now
we’re using the word “interval” in an unusual sense. And we can’t say that we’ve discovered
new properties of certain intervals. Any more than we’d discover new properties
of the chess king if we changed the rules of the game but kept the designations “chess” and
“king”. (On the other hand cf. Brouwer on the law of excluded middle.)
So in essence our experiment results in what we have called a “blurred” interval; on
the other hand of course other experiments are conceivable12
that result instead in a sharp
interval. Let’s imagine for instance that we move a ruler from the starting position b and
parallel to it towards a, until some particular reaction occurs in our subject; then we could
call the point at which the reaction first occurs the limit of our strip. – Of course, in the
same way we could also call the result of a single weighing “the weight of a body”, and in
that sense there would be an absolutely accurate weight, i.e. one whose result would not have
the form “G ± g”. We would thus have altered our form of expression, and we would now
have to say that the weight of a body varies, and does so in accordance with a law of which
we are unaware. (The difference13
between an “absolutely exact” weighing and an “inherently
inexact” one is grammatical, and refers to two different meanings of the expression
“result of a weighing”.)
10 (F): MS 114, p. 8r.
11 (V): (III) they ar
12 (V): possible
13 (V): differentiation
The Concept “Roughly”. . . . 107e
I II
III IV V
VI VII
WBTC030-34 30/05/2005 16:17 Page 215
22
Die Unbestimmtheit des Wortes ”Haufen“. Ich könnte definieren: ein Körper von gewisser
Form und Konsistenz etc. sei ein Haufe, wenn sein Volumen K m3
beträgt, oder mehr; was
darunter liegt, will ich ein Häufchen nennen. Dann gibt es kein größtes Häufchen; das heißt:
dann ist es sinnlos, von dem ”größten Häufchen“ zu reden. Umgekehrt könnte ich bestimmen:
Haufe solle alles das sein, was größer als K m3
ist, und dann hätte der Ausdruck ”der
kleinste Haufe“ keine Bedeutung. Ist aber diese Unterscheidung nicht müßig? Gewiß, – wenn
wir unter dem Resultat der Messung des Volumens einen Ausdruck von der Form ”V ± v“
verstehen.23
Sonst aber könnte die Unterscheidung nicht müßiger sein als24
die, zwischen
einem Schock Äpfel und 61 Äpfeln.
25
Zu dem Problem vom ”Sandhaufen“: Man könnte sich hier, wie in ähnlichen Fällen,
denken, daß es einen offiziellen Begriff, wie den einer Schrittlänge gäbe,26
etwa: Haufe ist
alles, was über einen halben m3
groß ist. Dieser wäre aber dennoch nicht unser gewöhnlich
gebrauchter Begriff. Für diesen liegt keine Abgrenzung vor (und bestimmen wir eine, so ändern
wir den Begriff); sondern es liegen nur Fälle vor, welche wir zu27
dem Umfang des Begriffs28
rechnen und solche, die wir nicht mehr zu dem Umfang des Begriffs rechnen.
29
”Mach’ mir hier einen Haufen Sand“. – ”Gut, das nennt er gewiß noch einen Haufen“.
Ich konnte dem Befehl Folge leisten, also war er in Ordnung. Wie aber ist es mit diesem
Befehl: ”Mach’ mir den kleinsten Haufen, den Du noch so nennst“? Ich würde sagen: das
ist Unsinn; ich kann nur eine vorläufige obere und untere Grenze bestimmen.
141
108 Der Begriff ”ungefähr“. . . .
22 (M): /
23 (V): Gewiß, – wenn wir unter dem Volumen ein
Messungsresultat im gewöhnlichen Sinne verstehen;
denn dieses Resultat hat die Form ”V± v“.
24 (V1): Unterscheidung so unbrauchbar sein,
wie (V2): aber wäre diese Unterscheidung
so unbrauchbar, wie // Unterscheidung nicht
müßiger als
25 (M): /
26 (V): Fällen, einen offiziellen // offiziell festgesetzten
// Begriff denken
27 (O): zum
28 (V): zu den Haufen
29 (M): /
WBTC030-34 30/05/2005 16:17 Page 216
14
The indeterminacy of the word “heap”. I could give as a definition: A body of a certain
form and consistency etc. is a heap if its volume is K m3
or more; anything less I shall call
a heaplet. In that case there is no such thing as a largest heaplet; that is: then it makes no
sense to speak of the “largest heaplet”. Conversely, I could stipulate: everything that is greater
than K m3
is to be a heap, and then the expression “the smallest heap” would have no meaning.
But isn’t this distinction idle? Certainly – if by “the result of measuring the volume”
we understand an expression of the form “V ± v”.15
But otherwise this distinction could16
be no more idle than17
the one between threescore apples and 61 apples.
18
Concerning the problem of the “heap of sand”: Here, as in similar cases, one could imagine
that there were an official19
concept like that of the length of a pace: say, anything that
is larger than half a m3
is a heap. Still, this wouldn’t be the concept that we normally use.
For this there is no delimitation (and if we specify one we change the concept); rather there
are only cases that we include within the extension of that concept20
and cases that we no
longer include within the extension of that concept.
21
“Put a heap of sand here for me”. – “Good, surely he’ll still call that a heap”. I was able
to carry out the command, so it was in order. But what about this command: “Make me the
smallest heap that you’d still call by that name”? I would say: “That’s nonsense; I can only
specify a temporary upper and lower limit”.
14 (M): /
15 (V): Certainly – if by “volume” we understand
the result of a measurement in the usual sense;
for this result has the form “V ± v”.
16 (V): would
17 (V): be as useless as
18 (M): /
19 (V): an officially determined
20 (V): cases that we count as heaps
21 (M): /
The Concept “Roughly”. . . . 108e
WBTC030-34 30/05/2005 16:17 Page 217
142 Das augenblickliche
Verstehn und die
Anwendung des Worts
in der Zeit.
WBTC035-39 30/05/2005 16:17 Page 218
Immediate
Understanding and
the Application of
a Word in Time.
WBTC035-39 30/05/2005 16:17 Page 219
35
Ein Wort verstehen = es anwenden
können. Eine Sprache verstehen:
einen Kalkül beherrschen.
1
Man könnte sagen: Uns2
interessiert nur der Inhalt des Satzes [d.h. nur das was der Satz
selber sagt3
]; und4
der Inhalt des Satzes ist in ihm. Seinen Inhalt hat der Satz als Glied des
Kalküls.
5
Ist also ”einen Satz verstehen“ von der gleichen Art, wie ”einen Kalkül beherrschen“?
Also wie: multiplizieren können?6
7
Die Bedeutung eines Worts verstehen, heißt, seinen Gebrauch kennen, verstehen.
8
”Ich kann das Wort ,gelb‘ anwenden“ – ist das auf einer anderen Stufe als ”ich kann Schach
spielen“, oder ”ich kann den König im Schachspiel verwenden“?
9
Die Frage, die unmittelbar mit unserer in Beziehung steht, ist die nach dem Sinn der
Aussage ”ich kann Schach spielen“?
”Ich weiß, wie ein Bauer ziehen darf.“
”Ich weiß, wie das Wort ,Kugel‘ gebraucht werden darf.“
10
Wenn ich sage ”ich kann dieses Gewicht heben“, so kann man antworten: ”das wird sich
zeigen, wenn Du es versuchst“; und geht es dann nicht, so kann man sagen ”siehst Du, Du
konntest es nicht“; und ich kann darauf nicht antworten ”doch, ich konnte es, als ich es sagte,
nur als es zum Aufheben kam, konnte ich es nicht“. Ob man es kann, wird die Erfahrung
zeigen. Anders ist es, wenn ich sage ”ich verstehe diesen Befehl“; dies ist, oder scheint ein
Erlebnis zu sein. ”Ich muß wissen, ob ich ihn (jetzt) verstehe“ – aber nicht: Ich muß
wissen, ob ich das Gewicht jetzt heben kann. – Wie ist es nun in dieser Hinsicht mit dem
Satz ”ich kann Schach spielen“? Ist das etwas, was sich zeigen wird, oder kann man sagen
”als ich es behauptete, konnte ich Schach spielen, nur jetzt kann ich es nicht“.
Ist nicht das, was mich rechtfertigt, nur, daß ich mich erinnere, früher Schach gespielt
zu haben? Und etwa, daß ich, aufgefordert zur Probe die Regeln im Geiste durchfliegen
kann?
1 (M): ? / (R): dazu S. 2/3
2 (V): Kann ich sagen$ mich // uns
3 (V): Satz sagt
4 (O): Und
5 (M): ? /
6 (R): S. 190 siehe S. 148/4 (V): können? Das
glaube ich.
7 (M): ∫ 3
8 (M): / 3
9 (M): /
10 (M): ?
143
144
110
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35
To Understand a Word =
To Be Able to Use It.
To Understand a Language:
To Have Command of a Calculus.
1
One could say:2
we’re only interested in the content of a proposition [that is, only in what
the proposition itself says3
]. And the content of a proposition lies within it. A proposition has
its content as part of a calculus.
4
So is “understanding a proposition” like “having command of a calculus”? And so like:
being able to multiply?5
6
Understanding the meaning of a word means being familiar with, understanding, its use.
7
“I can use the word ‘yellow’” – is that on a different level from “I can play chess” or “I
can use the king in a chess game”?
8
The question immediately related to ours is: What is the sense of the statement “I can
play chess”?
“I know how a pawn can move.”
“I know how the word ‘sphere’ can be used.”
9
If I say “I can lift this weight” then a possible response is: “That will become apparent
when you try”; and then if I fail it can be said: “See, you couldn’t do it”; and I can’t respond
to this, “Oh, yes I could when I said I could; it was just when it came to the lifting that I
couldn’t do it”. Experience will show whether one can do it. Things are different when I say
“I understand this command”; this is, or seems to be, an experience. “I must know whether
I understand it (now)” – but not: I must know whether I can lift the weight now. – And
what about the proposition “I can play chess”? Is that something that will become apparent,
or can one say: “When I made the claim I could play chess, it’s only now that I can’t”?
Isn’t the only thing that vindicates me that I remember having played chess before? And
that, say, I can run over the rules in my mind if I’m asked to prove it?
1 (M): ? / (R): to p. 2/3
2 (V): Can I say$ I’m
3 (V): the proposition says
4 (M): ? /
5 (R): P. 190 see p. 148/4 (V): multiply? I
believe so.
6 (M): ∫ 3
7 (M): / 3
8 (M): /
9 (M): ?
110e
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11
Ist es nicht auch so beim Gebrauch des Wortes ”Kugel“? Ich gebrauche das Wort
instinktiv. Aufgefordert aber, Rechenschaft darüber zu geben, ob ich es verstehe, rufe ich
mir, gleichsam zur Probe, gewisse Vorstellungen hervor.
(Es kann nicht darauf ankommen, ob die Sprache instinktiv oder halbinstinktiv gebraucht
wird. Wir sind hier im Sumpf der graduellen Unterschiede, nicht auf dem festen Grund
der Logik.) (Ist es nicht das Schachspiel, wenn es automatisch gespielt wird?)
12
Wenn ich auf die Frage: &kannst Du dieses Gewicht heben“, antworte &ja“, ist damit ein
Zustand13
nicht-hypothetisch beschrieben, ähnlich14
also, wie wenn man mich gefragt hätte &hast
Du Kopfschmerzen15
“?
16
Man kann das Wort können so verstehen daß die Ausführung als Kriterium der Fähigkeit gilt aber
auch so daß dies nicht der Fall ist.
Fragt man mich kannst Du diese Kugel heben & ich sage ja dann versuche ich, sie zu heben, aber
es gelingt mir nicht, so gibt es einen Fall in welchem ich sagen werde &ich hatte mich geirrt, ich
konnte sie nicht heben“ aber auch den &jetzt kann ich sie nicht heben, weil ich schon müde bin, als
ich aber sagte ich könne sie heben, da konnte17
ich es“, ebenso18
&als ich sagte ich könne Schach
spielen konnte ich’s, aber durch diesen Schreck habe ich es vergessen“, etc. Gefragt &wie weißt
Du daß Du es damals konntest“ würde19
man etwa sagen: &ich habe es immer gekonnt“, &ich hatte
gerade zuvor eine Partie gespielt“, &mein Gedächtnis ist doch nicht so schlecht“, &ich hatte20
gerade
die Regeln hergesagt“ und Ähnliches.
In keinem Fall ist die Fähigkeit ein bewußter Zustand wie Zahnschmerzen.
Was gilt alles als Kriterium dafür daß man ein Recht hat zu sagen man könne es.
21
&Wenn ich sage ,sieh’, dort ist eine Kugel‘, oder ,dort ist eine Halbkugel22
‘, so kann die
Ansicht23
zu beidem24
passen, und wenn ich sage ,ja, ich sehe sie25
‘, so unterscheide ich doch
zwischen den beiden Hypothesen. Wie ich im Schachspiel zwischen einem Bauer und dem
König unterscheide, auch wenn der gegenwärtige Zug einer ist, den beide machen könnten,
und wenn selbst eine Königsfigur als Bauer fungierte.
Das Wort ,Kugel‘ ist mir bekannt und steht in mir für etwas; d.h., es bringt mich in eine
gewisse Stellung zu sich (wie ein Magnet eine Nadel in seine Richtung bringt).“
26
Man ist in der Philosophie immer in der Gefahr, eine Mythologie des Symbolismus zu
geben, oder der Psychologie. Statt einfach zu sagen, was jeder weiß und zugeben muß.
27
Wenn ich gefragt würde ”kannst Du das Alphabet hersagen“, so würde ich antworten:
ja. – ”Bist Du sicher“ – ”Ja“. Wenn ich nun aber im Hersagen steckenbliebe und nicht
weiter wüßte, so gibt es doch einen Fall, in welchem ich sagen würde ”ja, als ich sagte, ich
könne es hersagen, da konnte ich es“, und zwar dann, wenn ich es mir damals ”im Geiste“
hergesagt hätte. Ich würde dies auch als Beweis angeben. Das heißt aber, daß das
Hersagen im Geiste die Fähigkeit zum wirklichen Hersagen – so wie wir hier das Wort
Fähigkeit verstehen – enthält.
11 (M): ∫ 5555
12 (M): /// (R): An den Anfang des § 6 (p. 22)
13 (V): Zustand beschrieben
14 (O): ahnlich
15 (V): Kopfschmerzen;
16 (M): /
17 (O): könnte
18 (O): Ebenso
19 (O): konntest würde
20 (V): ”ich habe mir
21 (M): //// (R): Zu § 38 S. 165
22 (V): ist ein Kegel
23 (V): Ansicht (ein Kreis)
24 (V): Ansicht (ein Kreis) auf beides
25 (V): es
26 (M): ? / 3
27 (M): ? / 3
143v
144
145
111 Ein Wort verstehen . . .
WBTC035-39 30/05/2005 16:17 Page 222
10
Isn’t this the way it is when I use the word “ball”? I use the word instinctively. But if
I’m asked to demonstrate whether I understand it, then as a kind of check, I call up certain
mental images.
(It can’t matter whether language is used instinctively or semi-instinctively. Here we’re
in the swamp of gradual differences and not on the firm ground of logic.) (Isn’t it still chess,
even if it’s played automatically?)
11
If I answer “Yes” to the question: “Can you lift this weight”, have I described my state unhypothetically,
i.e. as if I had been asked “Do you have a headache”?
12
One can understand the words “be able to” in such a way that performance counts as the
criterion for the ability, but also in such a way that it doesn’t.
If I’m asked whether I can lift this ball and I say yes, and then I try to lift it but don’t succeed, then
in one case I’ll say “I was wrong, I wasn’t able to lift it”; but in another case I might say, “I can’t lift
it now because I’m tired, but when I said I could, I could”, and similarly, “When I said I could play
chess I was able to, but because of this scare I’ve forgotten how to”, etc. When asked “How do
you know that you were able to do this earlier?”, one might say: “I’ve always been able to do it”,
“I’d just finished playing a game”, “My memory isn’t all that bad”, “I’d just recited the rules”,13
and
such things.
In no case is the ability a conscious state, like a toothache.
What counts as a criterion for being justified in saying that one is able to do something?
14
“If I say ‘Look, there’s a sphere over there’, or ‘There’s a hemisphere15
over there’, then
what is seen16
can fit both cases, and if I say ‘Yes, I see it’, I am distinguishing between the
two hypotheses. Just as in chess I distinguish between a pawn and a king, even if the current
move is one that both could make, and even if the king were to function like a pawn.
The word ‘sphere’ is known to me and stands for something within me; i.e. it brings me
into a certain position with respect to itself (as a magnet draws a needle in its direction).”
17
In philosophy one is always in danger of creating a mythology of symbolism, or of psychology.
Instead of simply saying what everybody knows and has to admit.
18
If I were asked “Can you recite the alphabet?”, I’d answer “Yes”. – “Are you certain?”
– “Yes”. But if in reciting it I were to get stuck and couldn’t continue, then there is a case
in which I might say “When I said I could recite it I was able to”, i.e. if at that time I had
recited it “in my head”. And I would cite this as proof. But that means that reciting in one’s
head includes the ability actually to recite – as we understand the word “ability” here.
10 (M): ∫ 5555
11 (M): /// (R): In the beginning of § 6 (p. 22)
12 (M): /
13 (V): “I’ve just recited the rules to myself”,
14 (M): //// (R): To § 38 p. 165
15 (V): cone
16 (V): seen (a circle)
17 (M): ? / 3
18 (M): ? / 3
To Understand a Word . . . 111e
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28
Etwas tun können hat ja eben jenen schattenhaften Charakter, das heißt, es erscheint als29
ein Schatten des tatsächlichen30
Tuns, gerade wie der Sinn des Satzes als Schatten seiner
Verifikation31
erscheint; oder das Verständnis des Befehles als Schatten seiner Ausführung.
Der Befehl ”wirft, gleichsam, seinen Schatten schon voraus“, oder, ”im32
Befehl wirft die
Tat ihren Schatten voraus“. – Dieser Schatten aber, was immer er sein mag, ist, was er ist,
und nicht das Ereignis. Er ist in sich selbst abgeschlossen und weist nicht weiter als er selbst
reicht.
33
Das ist doch der gleiche Fall wie: ”Kannst Du Deinen Arm heben?“ In welchem Falle
würde ich dies verneinen müssen, oder bezweifeln? Solche Fälle sind leicht zu denken.
Als Bestätigung dessen, daß wir den Arm heben können, sehen wir etwa34
ein Zucken mit
den Muskeln an, oder eine kleine Bewegung des Arms.35
Oder die geforderte Bewegung selbst,
jetzt ausgeführt, als Kriterium dafür, daß ich sie gleich darauf ausführen kann.
112 Ein Wort verstehen . . .
28 (M): /// 3 (R): Zu S. 99
29 (V): wie
30 (V): wirklichen
31 (V): Schatten einer Tatsache
32 (O): oder, im
33 (M): ? / 3
34 (V): etwas
35 (V): Die Bestätigung dessen, daß . . . können,
sehen wir etwa in einem Zucken mit den Muskeln
oder einer kleinen Bewegung des Arms.
146
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19
Being able to do something has a shadowy quality, i.e. it seems like a shadow of actu-
ally20
doing it, just as the sense of a proposition seems like the shadow of its verification21
;
or the understanding of a command the shadow of its being carried out. The command “casts
its shadow ahead itself, as it were”, or “The act casts its shadow ahead of itself” in the command.
– But whatever this shadow may be, it is what it is, and is not the event. It is contained
within itself, and doesn’t point any further than it reaches.
22
But that’s the same as: “Can you lift your arm?”. When would I have to say “No” to
this, or doubt it? Such cases are easy to imagine.
As confirmation that we can raise our arm we see, say, a quick jerking of our muscles, or
a slight movement of our arm.23
Or the requisite motion itself is now carried out, and we
see that as a criterion that it can be carried out in the immediate future.
19 (M): /// 3 (R): To p. 99
20 (V): really
21 (V): of a fact
22 (M): ? / 3
23 (V): We see confirmation that we can raise our
arm, say, in a quick jerking of our muscles, or
in a slight movement of our arm.
To Understand a Word . . . 112e
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36
Wie begleitet das Verstehen
des Satzes das Aussprechen
oder Hören des Satzes?
Wann kann das Gefäß . . . enthalten?
1
Das schwierigste Problem scheint der Gegensatz, das Verhältnis zu sein zwischen dem
Operieren mit der Sprache im Lauf der Zeit2
und dem momentanen Erfassen des Satzes.
3
Aber wann erfassen oder verstehen wir den Satz?! Nachdem wir ihn ausgesprochen haben?
– Und wenn: während wir ihn aussprechen, –4
ist das Verstehen ein artikulierter Vorgang,
wie das Bilden des Satzes, oder ein unartikulierter? Und wenn ein artikulierter: muß er nicht
projektiv mit dem andern verbunden sein? Denn sonst wäre seine Artikulation von der ersten
unabhängig.
5
”Er sagt das, und meint es“: Vergleiche das einerseits mit: ”er sagt das, und schreibt es
nieder“; anderseits mit: ”er schreibt das und unterschreibt es“.
6
Man könnte fragen: Wie lange braucht es,7
um einen Satz zu verstehen. Und wenn man
ihn eine Stunde lang versteht, beginnt man da immer wieder von frischem?8
9
Ist das Verstehen nicht das Erfassen des Satzes, so kann es auch nach diesem (und warum
nicht auch vorher) vor sich gehen.
10
Ist das Verstehen eines Satzes dem Verstehen eines Schachzuges, als Schachzuges,11
nicht
analog? Wer das Schachspiel gar nicht kennt und sieht jemand einen Zug machen, der wird
ihn nicht verstehen, d.h. nicht als Zug eines Spieles verstehen. Und es ist etwas anderes,
dem Zug12
mit Verständnis zu folgen, als ihn13
bloß zu sehen.
14
Was ist es aber dann, was15
uns immer das Gefühl gibt, daß das Verstehen eines Satzes
das Verstehen von etwas außerhalb ihm Liegendem ist und zwar nicht von der Welt
außerhalb des Zeichens, wie sie eben ist, sondern von der Welt, wie das Zeichen sie –
gleichsam – wünscht.
1 (M): / 3
2 (V): Sprache in der Zeit
3 (M): / 3
4 (V): Und wenn, während . . . aussprechen:
5 (M): / 3
6 (M): / 3
7 (V): man,
8 (O): wieder vom frischen?
9 (M): ∫ 3
10 (M): / 3 ///
11 (V): Schachzuges, als solchen,
12 (V): Spiel
13 (V): es
14 (M): / 3 ///
15 (V): das
147
148
113
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36
How Does Understanding a Sentence
Accompany Uttering or Hearing it?
When can the vessel contain . . . ?
1
The most difficult problem seems to be the contrast, the relationship, between carrying
out linguistic operations over time2
and the instantaneous grasping of a sentence.
3
But when do we grasp or understand a sentence?! After we have uttered it? – And if it’s
while we’re uttering it: is this understanding an articulated event, like the formation of a
sentence, or an unarticulated one? And if it’s articulated: doesn’t it have to be connected
with the process of sentence-formation like a projection? For otherwise its articulation
would be independent of that of the process of sentence-formation.
4
Compare “He says that and means it” with: “He says that and writes it down”, on the
one hand, and with: “He writes that and affirms it”, on the other.
5
One could ask: How long does it6
take to understand a sentence? And if one understands
it for one hour, does one then always start out afresh?
7
If understanding is not the grasping of the sentence then it can also take place after it
(and why not also before it?).
8
Isn’t understanding a sentence analogous to understanding a chess move as a chess
move?9
Someone who doesn’t know anything about chess and sees someone making a move
won’t understand it, i.e. won’t understand it as a move of a game. And following a move10
with understanding is different from simply seeing it.
11
But what is it that always gives us the feeling that understanding a sentence means understanding
something that lies outside it, and that that something is not the world outside the
sign, as it happens to be, but the world as the sign – as it were – wishes it to be?
1 (M): / 3
2 (V): operations in time
3 (M): / 3
4 (M): / 3
5 (M): / 3
6 (V): one
7 (M): ∫ 3
8 (M): / 3 ///
9 (V): understanding a chess move as such?
10 (V): following a game
11 (M): / 3 ///
113e
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16
Man würde etwa (so) sagen: Ich sage ja nicht nur ”zeichne einen Kreis“, sondern ich
wünsche doch, daß der Andre etwas tut. (Gewiß!) Und, was der Andre tut, ist doch außerhalb
dessen, was ich sage.17
18
Das Verstehen eines Satzes der Wortsprache ist dem Verstehen eines musikalischen
Themas (oder Musikstückes) viel verwandter, als man glaubt. Und zwar so, daß das
Verstehen des sprachlichen Satzes näher als man denkt dem liegt, was man gewöhnlich das
Verständnis des musikalischen Ausdrucks nennt. – Warum pfeife ich das gerade so? warum
will ich den Wechsel der Stärke und des Zeitmaßes gerade auf dieses ganz bestimmte
Ideal bringen?19
Ich möchte sagen: ”weil ich weiß, was das alles heißt“ – aber was heißt es
denn? – Ich wüßte es nicht zu sagen, außer durch eine Übersetzung in einen Vorgang vom
gleichen Rhythmus.
20
Das Können und Verstehen wird von der Sprache scheinbar als Zustand dargestellt, wie
der Zahnschmerz, und das ist die falsche Analogie, unter der ich laboriere.
21
Wie, wenn man fragte: Wann kannst Du Schach spielen? Immer? Während Du spielst? &
während jedes Zuges? – Und wie seltsam22
, daß Schachspielen-Können so kurze Zeit dauert23
und eine Schachpartie so viel länger!
24
Wenn nun ”das Wort ,gelb‘ verstehen“ heißt, es anwenden können, so ist25
die gleiche
Frage: Wann kannst Du es anwenden. Redest Du von einer Disposition? Ist es eine
Vermutung?
26
Augustinus: ”Wann messe ich einen Zeitraum?“ Ähnlich meiner Frage: Wann kann ich
Schach spielen.
16 (M): / 3 ///
17 (V): (Gewiß!) Und dieses Tun ist doch etwas
anderes als das Sagen, und ist eben das
Außerhalb, worauf ich weise // worauf der Satz
weist //.
18 (M): / / 3
19 (V): warum bringe ich den Wechsel . . . Ideal?
20 (M): / 3
21 (M): / 3
22 (V): Immer? oder während Du es sagst? // einen
Zug machst? // aber während des ganzen Satzes?
– Und wie seltsam
23 (V): braucht
24 (M): ? / 3
25 (V): besteht
26 (M): / 3
149
148v
149
114 Wie begleitet das Verstehen des Satzes das Aussprechen . . .
WBTC035-39 30/05/2005 16:17 Page 228
12
One might put it somewhat (like this): I don’t just say “Draw a circle”; no, I want the
other person to do something. (Of course!) And what he does is, after all, outside what I say.13
14
Understanding a sentence in word-language is much more closely related than one may
think to understanding a musical theme or a piece of music. Specifically, understanding a
sentence in language lies closer than one may think to what one ordinarily calls understanding
a musical expression. – Why am I whistling this just this way? Why do I want to make my
change in volume and tempo fit this very specific ideal? I’m inclined to say: “Because I know
what it all means” – but what does it mean? – I couldn’t say other than by translating it into
a sequence that had the same rhythm.
15
Language seemingly represents being able to, and understanding, as if they were states,
like a toothache, and that is the false analogy under which I am labouring.
16
What if one were to ask: When are you able to play chess? Always? While you are playing
it? And during each move? – And how strange17
that being able to play chess lasts18
such a short
time and a game of chess so much longer!
19
Now if “understanding the word ‘yellow’” means being able to use it, then there’s the
same question:20
When are you able to use it? Are you talking about a disposition? Is it a
hunch?
21
Augustine: “When do I measure a length of time?” Similar to my question: When am I
able to play chess?
12 (M): / 3 ///
13 (V): And this doing is, after all, something
different from the saying, and is precisely
the “outside” that I’m pointing to. // that the
proposition points to.
14 (M): / / 3
15 (M): / 3
16 (M): / 3
17 (V): Always? Or while you are saying it? // you
are making a move? // and during the entire
sentence? – And how strange
18 (V): requires
19 (M): ? / 3
20 (V): then the same question exists:
21 (M): / 3
How Does Understanding a Sentence . . . 114e
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37
Zeigt sich die Bedeutung eines
Wortes in der Zeit? Wie der
tatsächliche Freiheitsgrad eines
Mechanismus?1
Enthüllt sich die
Bedeutung des Worts erst nach und
nach wie seine Anwendung
fortschreitet? 2
3
Es ist eine sehr merkwürdige Tatsache, daß ich mich bei dem Gebrauch der Sprache nicht
erinnere, wie ich sie gelernt habe.4
Ich sage ”hier sehe ich eine schwarze Kugel“. Ich weiß
nicht, wie ich ”schwarz“ und ”Kugel“ gelernt habe. Meine Anwendung der Wörter ist unabhängig
von diesem Erlernen. Es ist so, als hätte ich die Wörter selbst geprägt. Und hier werden
wir wieder zu der Frage geführt: Wenn die Grammatik, die von den Wörtern handelt,
für ihre Bedeutung wesentlich ist, muß ich die grammatischen Regeln, die von einem Wort
handeln, alle im Kopf haben, wenn es für mich etwas bedeuten soll? Oder ist es hier, wie
im Mechanismus: Das Rad, das stillsteht, oder auch sich dreht, das Rad in einer Lage, weiß,
gleichsam, nicht, welche Bewegung ihm noch erlaubt ist, der Kolben weiß nicht, welches
Gesetz seiner Bewegung vorgeschrieben ist; und doch wirkt das Rad und der Kolben nur
durch jenes Gebundensein.5
Soll ich also sagen: Die grammatischen Regeln wirken in der Zeit? (Wie jene Führung.)6
7
Unterscheidung zwischen Regel & Erfahrungssatz. Wenn ein Satz der Grammatik ein Naturgesetz
der Anwendung des Wortes wäre, so gäbe es grammatische Hypothesen; & wie ein Wort gebraucht
werden kann zeigt sich dadurch, wie es gebraucht wird.
8
Wie seltsam: es9
scheint als ob zwar eine physische (mechanische) Führung versagen,
unvorhergesehenes zulassen könnte,10
aber eine Regel nicht! Sie wäre sozusagen die einzig verläßliche
1 (O): Mechanismus.
2 (R): gehört zu &Bedeutung“ § 9
3 (M): ? / (R): [Zu: &Lernen der Sprache“ &Wie
wirkt das einmalige . . .“]
4 (E): Siehe: MS 110, S. 89: ”Drury sagte mir
heute, er habe überlegt, daß man sich nicht des
Zustandes erinnern könne wo man noch nicht
sprechen konnte. // daß es unmöglich sei sich des
Zust. zu erinnern vor der Erlernung der
Sprache.“
5 (V): durch jene Gebundenheit.
6 (M): Vielleicht lehrreich
7 (M): ///
8 (M): /
9 (V): Wie selstsam) es es
10 (O): zulassen) könnte,
150
149v
115
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37
Is the Meaning of a Word Shown in
Time? Like the Actual Degree of
Freedom in a Mechanism?
Is the Meaning of a Word Only
Revealed in the Course of Time as
its Use Develops?1
2
It’s a very remarkable fact that when I use language I don’t remember how I learned it.3
I say “Here I see a black ball”. I don’t know how I learned “black” and “ball”. My use of
these words is independent of such learning. It’s as if I had coined the words myself. And
here again we’re led to the question: If grammar, which deals with words, is essential to their
meaning, do I have to have all the grammatical rules for a word in my head if it is to have
any meaning for me? Or is it here as it is in a mechanism: A wheel that’s either standing
still or turning, a wheel in one position, doesn’t know, as it were, what additional movements
it’s allowed, a piston doesn’t know what law governs its movement; and yet the wheel and
the piston are effective only by being constrained in this way.4
So should I say: Grammatical rules work temporally? (Like those controls.)5
6
Distinction between a rule and an empirical proposition. If a grammatical proposition were a
natural law for the use of a word there would be grammatical hypotheses; but how a word can be
used is shown by how it is used.
7
How strange: it8
seems that a physical (mechanical) control could fail and allow unforeseen things
to happen, but not a rule! That would make a rule the only reliable control, so to speak. But what’s
1 (R): belongs to “meaning” § 9
2 (M): ? / (R): [To: “learning a language”
“What effect does a one-time . . . have”] ?
3 (E): Cf. MS 110, p. 89: “Drury said to me
today that he had concluded that one couldn’t
remember the state in which one couldn’t yet
speak. // that it would be impossible to remember
the state prior to learning language.”
4 (V): only by that constraint.
5 (M): Perhaps instructive
6 (M): ///
7 (M): /
8 (V): How strange) it it
115e
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Führung. Aber worin besteht es, daß eine Führung eine Bewegung nicht zuläßt, & worin, daß eine
Regel sie nicht zuläßt? – Wie weiß man das eine, & wie das andere?
11
Also: Das Wort ”Kugel“ wirkt nur durch die Art12
seiner Anwendung. Wenn aber &die
Bedeutung eines Wortes verstehen“ heißt seine grammatische Anwendung kennen (die
Möglichkeit seiner gr. Anw.), dann kann man fragen: ”Wie13
kann ich denn gleich wissen was
ich mit ,Kugel‘ meine14
, ich kann doch nicht die ganze Art der Anwendung des Wortes auf
einmal im Kopfe haben?“
15
Und ist es nicht ähnlich mit dem Schachspiel: in irgend einem Sinne kann man sagen,
ich wisse die Regeln des Schachspiels (”habe sie im Kopf“), wenn16
ich spiele. Aber ist dieses
”sie im Kopf haben“ nicht wirklich nur eine Hypothese. Habe ich sie nicht nur insofern
im Kopf, als ich sie in jedem besondern Falle anwende? – Gewiß, dies Wissen17
ist nur das
hypothetische Reservoir, woraus das wirklich gesehene Wasser fließt.
18
Das Verständnis der Sprache – quasi des Spiels – scheint wie ein Hintergrund, auf dem
der einzelne Satz erst Bedeutung gewinnt.19
20
Die allgemeine Regel erst enthüllt den Freiheitsgrad, die Beweglichkeit des
Mechanismus. Das Bild des Mechanismus in einer seiner Stellungen enthält hievon nichts.
21
Soll ich nun sagen, der Freiheitsgrad des Mechanismus kann sich nur mit der Zeit
enthüllen? Aber wie kann ich dann je wissen, daß er gewisse Bewegungen nicht machen kann
(und daß er gewisse Bewegungen machen kann, die er gerade noch nicht gemacht hat).
22
Das Verständnis als eine Disposition der Seele, oder des Gehirns, geht uns nichts an.
Das Bild des Mechanismus kann wohl ein Zeichen des Freiheitsgrades sein. D.h. als Ausdruck dafür
gebraucht werden, welche Bewegungen etwas23
ausführen soll (meiner Meinung nach ausführen
wird, ausgeführt hat, etc.). Wenn ich aber sage das Bild kann ein Zeichen des Freiheitsgrades sein, –
was heißt das? Was macht ein Bild zum Zeichen des Freiheitsgrades? Doch nicht daß man ihm etwas
anderes, quasi einen existierenden Freiheitsgrad zuordnet. Außer indem man zur Erklärung des Zeichens
auf einen Mechanismus zeigt & diesen gewisse Bewegungen ausführen läßt. Aber dann liegt darin
keine Prophezeiung über das Verhalten dieses Mechanismus! Seine24
vorgeführten Bewegungen waren
vielmehr nur ein Zeichen, womit wir ein anderes erklärten.
150v
150, 151
116 Zeigt sich die Bedeutung eines Wortes in der Zeit? . . .
11 (M): || (
12 (V): nur in der Art
13 (O): fragen: Wie
14 (V): seiner Anwendung. Und es wäre die seltsame
Frage denkbar: ”wie kann ich denn dann
gleich wissen, was ich mit ’Kugel‘ meine
15 (M): / 3 ///
16 (V): während
17 (O): wissen
18 (M): / 3
19 (M): )
20 (M): ∫ ///
21 (M): ∫ / 555
22 (M): ∫ 555
23 (V): Bewegungen es // er
24 (V): Mechanismus! Es // s // seine
150, 150v
151
WBTC035-39 30/05/2005 16:17 Page 232
involved in a control not allowing a movement, and what in a rule not doing so? – How do you know
when it’s the one, and when the other?
9
So: The word “ball” works only because of10
the way it is used. But if “understanding the
meaning of a word” means knowing its grammatical use (the possibility of its grammatical use)
then it can be asked: “How can I know straightaway what I mean by ‘ball?’.11
After all, I can’t have
the complete range of the use of this word in my head all at once.”
12
And isn’t this similar to chess: in some sense one can say that I know the rules of chess
(“have them in my head”) when13
I’m playing. But isn’t this “having them in my head” really
just a hypothesis? Don’t I have them in my head only in so far as I apply them in each specific
case? – Certainly; this knowledge is only the hypothetical reservoir from which flows the
water that is actually seen.
14
An understanding of language – of a game, as it were – seems like a background against
which the individual sentence then gains meaning.15
16
Only the general rule reveals the degree of freedom, the mobility, of a mechanism. A
picture of the mechanism in one of its positions contains nothing of this.
17
So am I to say that a mechanism’s degree of freedom can only become apparent as time
goes by? But in that case how can I ever know that it cannot perform certain movements
(and that it can perform certain movements it just hasn’t performed yet)?
18
Understanding as a disposition of the mind or the brain is none of our concern.
The picture of a mechanism can indeed be a sign of a degree of freedom. That is, it can be used
to show what movements something19
is supposed to perform (in my opinion will perform, has performed,
etc.). But when I say the picture can be a sign of a degree of freedom, what does that mean?
What turns a picture into a sign for a degree of freedom? Certainly not that one attributes something
else to it, an already existing degree of freedom, as it were. Except where, as an explanation
of a sign, one points to a mechanism and lets it perform certain movements. But that contains no
prediction about the behaviour of this mechanism! Rather, the movements we had it perform were
only a sign we used to explain a different sign.
9 (M): || (
10 (V): only in
11 (V): it is used. And one could imagine the
strange question: “Then how can I know
straightaway what I mean by ‘ball’?;
12 (M): / 3 ///
13 (V): while
14 (M): / 3
15 (M): )
16 (M): ∫ ///
17 (M): ∫ / 555
18 (M): ∫ 555
19 (V): movements it
Is the Meaning of a Word Shown in Time? . . . 116e
WBTC035-39 30/05/2005 16:17 Page 233
152
153
38
Begleitet eine Kenntnis der
grammatischen Regeln den
Ausdruck des Satzes, wenn wir ihn
– seine Worte – verstehn?
&Gedanke“ nennen wir einen Vorgang1
der den Satz begleitet aber auch den Satz2
selbst3
im System
der Sprache.
4
Kann ich nicht sagen: ich meine die Verneinung, welche verdoppelt eine Bejahung gibt?
5
Wäre das nicht, als würde man sagen: Ich meine die Gerade, deren zwei sich in einem
Punkt schneiden.
6
Wenn7
Du von Rot gesprochen hast, hast Du dann das gemeint, wovon man sagen kann,
es sei hell, aber nicht, es sei grün, auch wenn Du an diese Regel nicht gedacht, noch8
von
ihr Gebrauch gemacht hast? – Hast Du das ”~“ verwendet, wofür ~~~p = ~p ist? auch
wenn Du diese Regel nicht verwendet hast? Ist es etwa eine Hypothese, daß es das ~ war?
Kann es zweifelhaft sein, ob es dasselbe war, und durch die Erfahrung bestätigt werden.
9
Was heißt die Frage: Ist das dasselbe ”~“, für welches die Regel ~~~p = ~p gilt?
10
”Meinst Du das ,~‘ so, daß ich aus ~p ~~~p schließen kann?“
11
Das Schachspiel ist gewiß durch seine Regeln (sein Regelverzeichnis) charakterisiert. Und
wir sagen, daß Einer, der eine Partie Schach spielt und jetzt einen Zug macht, etwas anderes
tut, als der, der nicht Schach spielen kann (d.h. das Spiel nicht kennt) und nun eine Figur
in die Hand nimmt und sie zufällig der Regel gemäß bewegt. Anderseits ist es klar, daß
der Unterschied nicht darin bestehen muß,12
daß der Erste in irgend einer Form die Regeln
des Schachspiels vor sich hersagt oder überdenkt. – Wenn ich nun sage: ”daß er Schach
spielen kann, (wirklich Schach spielt, die Absicht hat, Schach zu spielen) besteht darin,
daß er die Regeln kennt“, ist diese Kenntnis der Regeln in jedem Zuge in irgendeiner Form
enthalten?
1 (V): nennen wir etwas
2 (O): auch der Satz
3 (V): &Gedanke“ ist ein Vorgang der den Satz
begleitet aber auch der Satz selbst
4 (M): ? / /// (R): ∀ S. 144/4
5 (M): ? / ///
6 (M): / 3 |
7 (V): Das heißt: Wenn
8 (V): oder
9 (M): /
10 (M): /
11 (M): ∫ / 5555
12 (V): darin besteht,
117
WBTC035-39 30/05/2005 16:17 Page 234
38
Does a Knowledge of Grammatical
Rules Accompany the Expression of
a Sentence when We Understand it –
Its Words?
We call “thought” a process1
that accompanies a sentence, but also the sentence itself in the
system of language.
2
Can’t I say: I mean the negation which, when doubled, gives me an affirmation?
3
Wouldn’t that be like saying: I mean the straight line, two of which intersect at a point?
4
When5
you spoke of red did you mean that of which one can say that it’s bright, but not
that it’s green, even if you were neither thinking of this rule nor6
had you made use of it? –
Did you use the “~” for which ~~~p = ~p, even if you didn’t use this rule? Is it perhaps
a hypothesis that it was that ~? Can it be doubted whether it was the same ~, and can this
then be confirmed by experience?
7
What’s the meaning of the question: Is this the same “~” for which the rule ~~~p = ~p
is valid?
8
“Do you mean the ‘~’ in such a way that I can conclude ~~~p from ~p?”
9
Chess is certainly characterized by its rules (its list of rules). And we say that someone
who makes a move in playing a game of chess is doing something different from someone
else who can’t play (i.e. doesn’t know the game) and now picks up a piece and by chance
moves it in accordance with the rules. On the other hand it’s clear that the difference
need not10
consist in the first person reciting or thinking over the rules of chess in some
way. – Now if I say “That he can play chess (really is playing chess, intends to play chess)
consists in his knowing the rules”, is this knowledge of the rules contained in some way in
his every move?
1 (V): We call something “thought” // “Thought” is a
process
2 (M): ? / /// (R): ∀ p. 144/4
3 (M): ? / ///
4 (M): / 3 |
5 (V): That means: when
6 (V): were not thinking of this rule nor
7 (M): /
8 (M): /
9 (M): ∫ / 5555
10 (V): difference does not
117e
WBTC035-39 30/05/2005 16:17 Page 235
Was heißt das: ”er tut etwas anderes“? Hierin liegt schon die Verwendung eines
irreführenden13
Bildes. Worin besteht der Unterschied? Man denkt da wieder an
Gehirnvorgänge.
Das Schachspiel ist gewiß durch seine Regeln (sein Regelverzeichnis) charakterisiert. Wenn ich Schach
nur durch14
seine Regeln definiere, so gehören diese Regeln zur Gr. des Wortes &Schach“.15
Worin besteht es die Absicht zu haben eine Partie Sch. zu spielen.16
17
Kann man eine Intention haben, ohne sie auszudrücken? Kann man die Absicht haben,
Schach zu spielen (in dem Sinne, in welchem man apodiktisch sagt ”ich hatte die Absicht
Schach zu spielen; ich muß es doch wissen“), ohne einen Ausdruck dieser Absicht? – Könnte
man da nicht fragen: Woher weißt Du, daß das, was Du hattest, diese Absicht war? D.h. wie
unterscheidet sich diese Absicht von der, Dame18
zu spielen.
Ist die Absicht Schach zu spielen etwa wie die Vorliebe für das Spiel, oder für eine19
Person.
Wo man auch fragen könnte: Hast Du diese Vorliebe die ganze Zeit oder etc., und die Antwort
ist, daß ”eine Vorliebe haben“ gewisse Handlungen, Gedanken und Gefühle einschließt und
andere ausschließt.20
Das Sch. ist doch durch seine Regeln definiert.
21
Muß22
ich nicht sagen: ”Ich weiß, daß ich die Absicht hatte, denn ich habe mir gedacht
,jetzt komme ich endlich zum Schachspielen‘ “ oder etc. etc.
23
Es würde sich mit der Absicht in diesem Sinne auch vollkommen vertragen, wenn ich
beim ersten Zug darauf käme, daß ich alle Schachregeln vergessen habe, und zwar so, daß
ich nicht etwa sagen könnte ”ja, als ich den Vorsatz faßte,24
da habe25
ich sie noch gewußt“.
26
Es wäre wichtig, den Fehler allgemein auszudrücken, den ich in allen diesen
Betrachtungen zu machen geneigt bin.27
Die falsche Analogie, aus der er entspringt.
28
Ich glaube, jener Fehler liegt in der Idee, daß die Bedeutung eines Wortes eine
Vorstellung ist, die das Wort begleitet.
Und diese Conception steht wieder mit der des Bewußt-Seins in Verbindung.29
Dessen,
was ich immer ”das Primäre“ nannte.
30
Es stört uns quasi, daß der Gedanke eines Satzes in keinem Moment ganz vorhanden
ist. Hier sehen wir, daß wir den Gedanken mit einem Ding vergleichen, welches wir erzeugen,
und das wir nie als Ganzes besitzen; sondern, kaum entsteht ein Teil, so verschwindet
ein andrer. Das hat gewissermaßen etwas unbefriedigendes, weil wir – wieder durch ein
Gleichnis31
verführt – uns etwas Anderes erwarten.
13 (V): falschen
14 (V): ich das Schach durch
15 (V): ∫ 3 Wenn das Schachspiel durch seine
Regeln definiert ist, so gehören diese Regeln zur
Grammatik des Wortes ”Schach“.
16 (R): [Zu: S. 354]
17 (M): / (
18 (O): von der Dame
19 (V): seine
20 (M): )
21 (M): /
22 (O): Muß
23 (M): ? / /// (R): [Zu: S. 354]
24 (V): hatte,
25 (V): hatte
26 (M): ? ∫
27 (V): zu machen neige.
28 (M): 5555
29 (V): Und diese Conception hat wieder mit . . .
zu tun.
30 (M): / (R): Zu S. 223
31 (V): durch eine Erklärung
154
153
118 eine Kenntnis der grammatischen Regeln . . .
152v
153
WBTC035-39 30/05/2005 16:17 Page 236
What does it mean: “He’s doing something different”? This already uses a misleading11
image. What does the difference consist in? Here again one thinks of processes in the brain.
Chess is certainly characterized by its rules (its list of rules). If I define chess only by its rules, then
these rules belong to the grammar of the word “chess”.12
What does having the intention to play a game of chess consist in?13
14
Can one have an intention without expressing it? Can one have the intention to play chess
(in the sense in which one says apodeictically “I had the intention to play chess; I ought to
know, after all”) without expressing this intention? – Couldn’t one then ask: How do you
know that what you had was this intention? That is, how does this intention differ from the one
to play draughts?
Is the intention to play chess like a fondness for that game, or for a15
person? If so, we
could ask again: Do you have this fondness the whole time, or, etc.? And the answer is that
“having a fondness” includes certain actions, thoughts and feelings, and excludes others.16
Chess is defined by its rules.
17
Mustn’t I say: “I know I had the intention because I thought to myself ‘Now I’m finally
going to get to play chess’” or etc., etc.
18
It would also be entirely consistent with intention in this sense if, as I made my first
move, I were to discover that I had forgotten all the rules of chess, in fact had forgotten
them so completely that I couldn’t say, for instance, “Well, when I formed19
the intention
I still knew them”.
20
It’s important to express the mistake that I am prone to21
make in all of these observations
in general terms. To express the false analogy that is its source.
22
I believe that mistake lies in the idea that the meaning of a word is a mental image that
accompanies the word.
And this conception is connected23
in turn with that of consciousness. With the conception
of what I always called “the primary”.
24
It bothers us, as it were, that the thought expressed by a sentence isn’t present in its
entirety at any given moment. Here we see that we are comparing the thought with some
thing we fabricate and that we never possess in its entirety; rather, as soon as one part comes
into being, another one disappears. There is something disappointing about that, in a way,
because – seduced once again by a simile25
– we expect something else.
11 (V): false
12 (V): ∫ 3 If chess is defined by its rules, then these
rules belong to the grammar of the word
“chess”.
13 (R): [To: p. 354]
14 (M): / (
15 (V): for its
16 (M): )
17 (M): /
18 (M): ? / /// (R): [To: p. 354]
19 (V): had
20 (M): ? ∫
21 (V): I tend to
22 (M): 5555
23 (V): conception has to do
24 (M): / (R): To p. 223
25 (V): by an explanation
Knowledge of Grammatical Rules . . . 118e
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32
Der Spieler, der die Intention hatte, Schach zu spielen, hatte sie schon dadurch, daß
er zu sich etwa die Worte sagte ”jetzt wollen wir Schach spielen“. Und etwa durch gewisse
Gefühle die die Worte begleiteten.
Ich will sagen, daß das Wort ”Schach“ eben auch (nur) ein Stein in einem Spiel33
ist, das
wir fortlaufend spielen.34
Wird der Kalkül beschrieben, so müssen wir die Regeln tabulieren,35
wird er aber angewandt, so wird jetzt gemäß der einen, dann gemäß der andern Regel vorgegangen,
dabei kann uns ihr Ausdruck vorschweben, oder auch nicht.
36
Muß denn dem, der das Wort ”Schach“ gebraucht, eine Definition des Wortes
vorschweben? Gewiß nicht. – Gefragt, was er unter ”Schach“ versteht, wird er erst eine geben.
Diese Definition ist (selber) eine Handlung im Kalkül, den wir betreiben.37
38
Wenn ich39
aber nun fragte: Wie Du das Wort ausgesprochen hast, was hast Du damit
gemeint? – Wenn er mir darauf antwortet: ”ich habe das Spiel gemeint, das wir so oft gespielt
haben etc. etc.“, so weiß ich, daß ihm diese Erklärung in keiner Weise beim Gebrauch des
Wortes vorgeschwebt hatte, und daß seine Antwort meine Frage nicht in dem Sinn beantwortet,
daß sie mir sagt, was40
”in ihm vorgegangen ist41
“, als er dieses Wort sagte.
42
Statt &ich habe das Spiel gemeint, welches . . .“ hätte ich auch sagen können: ich setze jetzt statt
des Wortes Schach das ich vorhin43
gebraucht habe den Ausdruck: &das Spiel, was wir so oft . . .“
44
Denn die Frage ist eben, ob unter der ”Bedeutung, in der man ein Wort gebraucht“ ein
Vorgang verstanden werden soll, den wir beim Sprechen oder Hören des Wortes erleben.
45
Die Quelle der Verwirrung ist vielleicht der Begriff vom Gedanken, der46
den Satz begleitet. (Oder47
seinem Ausdruck vorangeht.) Dem Wortausdruck kann natürlich ein andrer Ausdruck
vorangehen, aber für uns kommt der Artunterschied48
dieser beiden Ausdrücke – oder
Gedanken – nicht in Betracht. Und es kann der Gedanke unmittelbar in seiner Wortform
gedacht werden.
49
(”Er hat diese Worte gesagt, sich aber dabei gar nichts gedacht.“ – ”Doch, ich habe mir
etwas gedacht50
“. – ”Und zwar was denn?“ – ”Nun,51
was ich gesagt habe“.)
52
Man könnte sagen: auf die Aussage ”dieser Satz hat Sinn“ kann man nicht wesentlich
fragen ”welchen?“ So wie man ja auch53
auf den Satz ”diese Worte sind ein Satz“ nicht
fragen kann ”welcher?“
54
”Dieses Wort hat doch eine ganz bestimmte Bedeutung“. Wie ist sie denn (ganz)
bestimmt?
32 (M): ∫ ∫ //// (R): [Zu: S. 354]
33 (V): Kalkül
34 (V): das wir spielen.
35 (V): die Regeln tabuliert vor uns haben,
36 (M): / 3 ///
37 (V): ist selber ein bestimmter Schritt in seinem
Kalkül. // im Kalkulieren.
38 (M): / 3
39 (V): ich ihn
40 (V): was, quasi,
41 (V): ”in ihm vorging
42 (M): /
43 (V): früher
44 (M): ∫ 3
45 (M): / 3 ///
46 (V): Die Quelle des Fehlers scheint die Idee vom
Gedanken zu sein, der // ist der Begriff vom
Gedanken, der
47 (V): (Oder der
48 (V): Unterschied
49 (M): / 3
50 (V): habe etwas dabei gedacht
51 (V): ”Nun, das$
52 (M): / 3
53 (V): wie man auch
54 (M): 555
155
154v
155
154v, 155
156
119 eine Kenntnis der grammatischen Regeln . . .
WBTC035-39 30/05/2005 16:17 Page 238
26
The player who had the intention of playing chess had it merely by saying to himself,
for example, “Now let’s play chess”. And perhaps by having certain feelings that accompanied
those words.
I’m trying to say that the word “chess” too is (nothing more than) a piece in a game27
that
we continually play.28
To describe the calculus we have to tabulate the rules,29
but to apply it
we proceed now in accordance with one, now in accordance with another rule, the expression
of that rule being in our minds in the process, or not.
30
Does someone who uses the word “chess” have to have a definition of the word in mind?
Certainly not. – Only when he’s asked what he understands by “chess” will he give one.
This definition is (itself) an action in the calculus that we’re carrying out.31
32
But suppose I were now to ask:33
When you uttered the word what did you mean by it?
– If he answers: “I meant the game that we played so often, etc., etc.” then I know that in
no way did he have this explanation in mind when he used the word, and that his response
doesn’t answer my question in this sense – that it tells me what34
“was35
going on inside him”
when he said this word.
36
Instead of “I meant the game that . . .” I could also have said: Now I am replacing the word
“chess”, which I just used,37
with the expression “the game that we so often . . .”.
38
For the question is precisely whether the “sense in which one uses a word” is supposed
to mean an event we experience when we speak or hear the word.
39
Perhaps the source of the confusion is the concept of the thought that40
accompanies the sentence.
(Or precedes its expression.) Of course another expression can precede the verbal
expression, but the difference in kind41
between these two expressions – or thoughts –
is immaterial to us. And the thought can be thought immediately in its verbal form.
42
(“He said these words, but gave them no thought whatsoever.” – “Oh yes, I was43
thinking
something”. – “And exactly what was that?” – “Well, what I said”.)
44
We could say: In response to the statement “This sentence has a sense” you have missed
something essential if you ask “Which one?”. Just as in response to the sentence “These words
are a sentence” you can’t ask “Which one?”.
45
“But this word, after all, has a quite specific meaning.” So how is it (quite) specific?
26 (M): ∫ ∫ //// (R): [To: p. 354]
27 (V): calculus
28 (V): we play.
29 (V): we have to have the rules tabulated in
front of us,
30 (M): / 3 ///
31 (V): definition is itself a particular step in its
calculus. // in calculating.
32 (M): / 3
33 (V): ask him:
34 (V): what, for example,
35 (V): is
36 (M): /
37 (V): which I used earlier,
38 (M): ∫ 3
39 (M): / 3 ///
40 (V): The source of the mistake seems to be the
idea of the thought that // is the concept of the
thought that
41 (V): the difference
42 (M): / 3
43 (V): yes, in saying them I was
44 (M): / 3
45 (M): 555
Knowledge of Grammatical Rules . . . 119e
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55
”Ich habe etwas bestimmtes damit gemeint, als ich sagte . . .“. – ”Hast56
Du bei jedem
Wort etwas anderes gemeint, oder während des ganzen Satzes dasselbe?“
57
(Übrigens seltsam: wenn58
man bei jedem – sagen wir, deutschen – Wort etwas meint,
daß eine Zusammenstellung dieser59
Wörter Unsinn sein kann!)
60
”Ich meine aber doch mit diesen Worten etwas“. Gewiß: im Gegensatz61
zu dem Falle,
wo ich nichts meine, wo ich etwa Silben ihres komischen Klangs wegen aneinander reihe.
Ich will eigentlich sagen, daß ”ich meine etwas mit den Worten“ nur heißt: ich unterscheide
doch diesen Fall von dem des sinnlosen Plapperns etc. Und das ist zugegeben. Aber
es ist damit noch keine besondere Theorie des Meinens gegeben.
62
Und so geht es in allen solchen Fällen. Wenn etwa jemand sagt: ”aber ich meine doch
wirklich, daß der Andere Zahnschmerzen hat; nicht, daß er sich bloß so benimmt“. Immer muß
man antworten: ”Gewiß“ und zugeben, daß auch wir diese Unterscheidung machen müssen.63
64
”Jetzt sehe ich’s erst, er zeigt immer auf die Leute, die dort vorübergehen.“ Er hat ein
System verstanden: wie Einer, dem ich die Ziffern 1, 4, 9, 16 zeige und der sagt ”ich versteh’
jetzt das System, ich kann jetzt selbst weiterschreiben“. Aber was ist diesem Menschen
geschehen, als er das System plötzlich verstand?
65
Es handelt sich beim Verstehen, Meinen, nicht um einen Akt des momentanen,
sozusagen nicht diskursiven,66
Erfassens der Grammatik. Als könnte man sie gleichsam auf
einmal herunterschlucken.
67
Das also, was der macht,68
der auf einmal die Bewegung des Andern deutet69
(ich sage
nicht ”richtig deutet“), ist ein Schritt in einem Kalkül. Er tut ungefähr, was er sagt, wenn
er seiner Deutung70
Ausdruck gibt.71
Und wenn ich sage ”was er macht, ist der Schritt eines
Kalküls“, so heißt das, daß ich diesen Kalkül schon kenne; in dem Sinne, in dem ich die
deutsche Sprache kenne, oder das Einmaleins. Welche ich ja auch nicht so in mir habe, als
wären72
die ganze deutsche Grammatik und die Einmaleins-Sätze zusammengeschoben auf
etwas, was ich nun als Ganzes besitze.73
74
Gewiß, der Vorgang des ”jetzt versteh’ ich . . . !“ ist ein ganz spezifischer, aber es ist
eben auch ein ganz spezifischer Vorgang, wenn wir auf einen bekannten Kalkül stoßen, wenn
wir ”weiter wissen“.
Aber dieses Weiter-Wissen ist eben auch diskursiv (nicht intuitiv).
75
”Intuitives Denken“, das wäre so, wie ”eine Schachpartie auf die Form eines dauernden,
gleichbleibenden Zustandes gebracht“.76
120 eine Kenntnis der grammatischen Regeln . . .
55 (M): / (R): Zu S. 224
56 (V): sagte . . .“. – @Wann hast Du es gemeint und
wie lange hat es gebraucht. Und hast
57 (M): / (R): Zu S. 224
58 (V): Übrigens komisch, daß, wenn
59 (V): solcher
60 (M): / 3
61 (V): Gegensatz z.B. etwa
62 (M): ∫ ///
63 (V): daß diese Unterscheidung besteht.
64 (M): 555
65 (M): / 3
66 (V): momentanen, sozusagen nicht diskursiven,
67 (M): ? / 3
68 (V): tut,
69 (V): auf einmal das Zeichen, das im [sic] der
Andere gegeben // gegeben hat, // versteht
70 (V): er seinem Verständnis
71 (V): gibt. – Und das ist ja immer unser Prinzip.
–.
72 (V): wäre
73 (V): auf Etwas, was man auf einmal, als
Ganzes, erfassen kann. // was ich nun auf
einmal, als Ganzes$ besitze.
74 (M): 5555
75 (M): / (R): Zu S. 223
76 (V): gebracht“. (ebenso undenkbar).
157
158
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46
“I meant something specific when I said . . . .” – Did47
you mean something different
as you got to each word, or the same thing throughout the entire sentence?
48
(Strange, by the way:49
if one means something by each – let’s say English – word, that
a combination of these words50
can be nonsense!)
51
“But I do mean something by these words.” Certainly; as opposed52
to the case where I
don’t mean anything, where, say, I string syllables together because of their amusing sound.
Really all I am trying to say is that “I mean something by the words” means: “I distinguish
this case from one of senseless chatter, etc.” Granted. But this doesn’t constitute a
distinctive theory of meaning.
53
And that’s the way it is in all of these cases. As when someone says, for instance: “But
I really do mean that that person has a toothache, not that he is merely acting as if he did”.
One always has to answer: “Of course”, and admit that we too must make this distinction.54
55
“Now I see it, he keeps pointing to the people passing by over there.” This person has
understood a system: like someone to whom I show the numbers 1, 4, 9, 16 and who says
“Now I understand the system, now I can continue on my own”. But what happened to this
person when he suddenly understood the system?
56
In understanding, or meaning, something, it isn’t a matter of an act of grasping the
grammar instantaneously, and so to speak, non-discursively.57
As if one could swallow it
down whole.
58
So what someone takes59
who suddenly interprets someone else’s movement60
(I’m not
saying that he “interprets it correctly”) is a step in a calculus. He does more or less what he
says when he expresses his interpretation.61
And when I say that “what he is doing is a step
in a calculus” that means that I already know this calculus; in the sense in which I know
English or the multiplication tables. Which of course I also don’t have in me, as if all of
English grammar and the propositions of the multiplication tables were telescoped into
something that I now possess as a whole.62
63
To be sure, the process of “Now I understand . . . !” is quite specific, but hitting upon
a familiar calculus, “knowing how to go on”, is also a quite specific process.
But the point is that this knowing-how-to-go-on is discursive (not intuitive).
64
“Intuitive thinking.” That would be like “a game of chess transformed into a perpetual,
unchanging state”.65
46 (M): / (R): To p. 224
47 (V): said . . . ”–“When did you mean it and how
long did it take? And did
48 (M): / (R): To p. 224
49 (V): Funny, by the way, that
50 (V): of such words
51 (M): / 3
52 (V): opposed for example
53 (M): ∫ ///
54 (V): that this distinction exists.
55 (M): 555
56 (M): / 3
57 (V): instantaneously, and so to speak$ non-
discursively.
58 (M): ? / 3
59 (V): does
60 (V): suddenly understands the sign that someone
else gives // has given // him
61 (V): expresses his understanding. – And to be
sure that is always our princip -. –.
62 (V): something that one can grasp all at once,
as a whole. // something that now all of a
sudden I possess, as a whole.
63 (M): 5555
64 (M): / (R): To p. 223
65 (V): state”. (Just as inconceivable.)
Knowledge of Grammatical Rules . . . 120e
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39
Die grammatischen Regeln – und die
Bedeutung eines Wortes. Ist die
Bedeutung, wenn wir sie verstehen,
”auf einmal“ erfaßt; und in den
grammatischen Regeln gleichsam
ausgebreitet?
1
Und doch ist noch etwas nicht klar,2
was sich z.B. in der dreifachen Verwendung des
Wortes ”ist“ zeigt. Denn, was heißt es, wenn ich sage, daß im Satz ”die Rose ist rot“ das
”ist“ eine andere Bedeutung hat, als in ”zweimal zwei ist vier“? Wenn man sagt, es heiße,
daß verschiedene Regeln von diesen beiden Wörtern gelten, so muß man zunächst sagen,
daß wir hier nur ein Wort haben. Zu sagen aber: von diesem gelten in einem Fall die
Regeln, im anderen jene, ist Unsinn.
Und das hängt wieder mit der Frage zusammen, wie wir uns denn aller Regeln bewußt
sind, wenn wir ein Wort in einer bestimmten Bedeutung gebrauchen, und doch die Regeln
die Bedeutung ausmachen?
3
Wenn ich nun aber das Wort ”ist“ betrachte: Wie kann ich hier zwei verschiedene
Anwendungsarten unterscheiden, wenn ich nur auf die grammatischen Regeln achte?4
Denn
diese erlauben ja eben die Verwendung des Wortes im Zusammenhang ”die Rose ist rot“
und ”zweimal zwei ist vier“. An diesen Regeln sehe ich nicht, daß wir hier zwei verschiedene
Wörter haben.5
– Ich ersehe es aber z.B. wenn ich versuche, in beiden Sätzen statt ”ist“ ”ist
gleich“ einzusetzen6
(oder auch den Ausdruck ”hat die Eigenschaft“). Aber nur wieder,
weil ich für den Ausdruck ”ist gleich“ die Regel kenne, daß er in ”die Rose . . . rot“ nicht
stehen darf.7
8
Wenn ich mich weigere ein Wort, z.B. das Wort ”ist gleich“, in zwei Zusammenhängen
zu gebrauchen, so ist der Grund das, was wir mit den Worten beschreiben das Wort werde
in diesen Fällen in verschiedenem Sinn gebraucht.9
1 (M): / 3
2 (V): etwas unklar,
3 (M): / 3
4 (V): sehe?
5 (V): daß es sich um zwei verschiedene Wörter
handelt.
6 (V): ”ist gleich“ zu setzen
7 (V): nicht eingesetzt werden darf.
8 (M): ///
9 (V): beschreiben ”das Wort habe in den beiden
Fällen verschiedene Bedeutung“.
159
160
160
121
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39
The Rules of Grammar –
and the Meaning of a Word.
Is Meaning, When We Understand
it, Grasped “all at once”? And
Unfolded, as it Were, in the Rules
of Grammar?
1
And yet something else, which shows itself for example in the three uses of the word
“is”, is not clear.2
For what does it mean when I say that “is” in the sentence “The rose is
red” has a different meaning than in “Twice two is four”? If we say that this means that
different rules are valid for these two words, then the first thing to be said is that we have
only one word here. But to say that in one case these rules are valid for it, and in another,
those, is nonsense.
And this in turn is connected with the question of how we can be aware of all the rules
when we use a word with a certain meaning, considering that the rules, after all, constitute
the meaning?
3
Now if I look at the word “is”: How can I distinguish between two different kinds of its
use if I pay attention only to4
the grammatical rules? For it’s precisely these rules that allow
the use of the word in “The rose is red” and “Twice two is four”. Looking at these rules,
I don’t see that here we have5
two different words. – But I do see it, for example, if I try to
substitute6
“equals” for “is” in both sentences (or “has the property”). But again only because
I know the rule for the expression “equals”, know that it can’t appear in7
“The rose . . . red”.
8
If I refuse to use a word, e.g. the word “equals”, in two contexts, the reason for this is
what is described in saying that the word is used in a different sense in these two cases.9
1 (M): / 3
2 (V): is unclear.
3 (M): / 3
4 (V): if I only look at
5 (V): that here it is a matter of
6 (V): posit
7 (V): it can’t be inserted into
8 (M): ///
9 (V): saying “the word has different meanings
in both cases”.
121e
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10
Kann ich nun aber das, was die grammatischen Regeln von einem Worte sagen, auch
anders beschreiben, nämlich durch die Beschreibung des Vorgangs, der beim Verstehen des
Wortes stattfindet?
11
Wenn also die Grammatik – z.B. – die Geometrie der Verneinung ist, kann ich sie durch
eine Beschreibung dessen ersetzen, was bei der Verwendung sozusagen hinter dem Wort
”nicht“ steht?
12
Aber so eine Beschreibung wäre doch – wie gesagt – ein Ersatz für das Wort13
”nicht“,
etwa wie und könnte die Grammatik nicht ersetzen.14
15
Es können die gr. Regeln als die Auseinanderlegung dessen erscheinen, was ich im Gebrauch
des Wortes auf einmal erlebe.16
Sozusagen (nur) Folgen, Äußerungen,
der Eigenschaften, die ich beim Verstehen auf einmal erlebe?17
Das muß natürlich ein
Unsinn sein.
18
Man möchte19
sagen: die20
Verneinung hat die Eigenschaft, verdoppelt21
eine Bejahung zu
ergeben.22
Während die Regel die Verneinung nicht näher beschreibt, sondern konstituiert.
23
Daß wir dieses Wort dieser Regel gemäß gebrauchen, das dafür einsetzen etc., damit
dokumentieren wir, wie wir es meinen.
24
”Wie ich einen Körper durch seine verschiedenen Ansichten geben kann und er mit diesen
äquivalent ist, so offenbart sich die Natur der Negation in den verschiedenen, grammatisch
erlaubten Anwendungen des Negationszeichens.“
25
”Die doppelte Negation gibt eine Bejahung“, das klingt so wie: Kohle und Sauerstoff
gibt Kohlensäure. Aber in Wirklichkeit gibt die doppelte Negation nichts, sondern ist etwas.
26
”Wer die Negation versteht, der weiß, daß die doppelte Negation . . .“
27
Es täuscht uns da etwas eine physikalische Tatsache vor.
So, als sähen wir ein Ergebnis des logischen Prozesses. Während das Ergebnis nur das des
physikalischen Prozesses ist.
28
Das Wort ”nicht“ in der grammatischen Regel hat keine Bedeutung, sonst könnte das
nicht von ihm ausgesagt werden.
29
Die Negation hat keine andere Eigenschaft, als etwa die, in gewissen Sätzen, die
Wahrheit zu ergeben.
159v, 160
161
162
122 Die grammatischen Regeln . . .
10 (M): / Ib
11 (M): / Ic
12 (M): 555 (F): MS 110, S. 101.
13 (V): Ersatz des Wortes
14 (M): (?)
15 (M): ? / Ia
16 (V): In meiner Darstellung schienen doch
die grammatischen Regeln die
Auseinanderlegung dessen, was ich . . . erlebe.
// Sind die grammatischen Regeln die
Auseinanderlegung dessen, was . . . erlebe?
17 (V): erlebe.
18 (M): ? / Id
19 (V): Man würde ja geradezu
20 (V): eine
21 (O): Eigenschaft, sie verdoppelt
22 (V1): die Eigenschaft, daß sie verdoppelt eine
Bejahung ergibt. (V2): zu ergeben. (Etwa
wie: Eisen hat die Eigenschaft$ mit
Schwefelsäure Eisensulfat zu geben.)
23 (M): ///
24 (M): ///
25 (M): / Ib
26 (M): / Ia
27 (M): / Ic
28 (M): ∫ /// If
29 (M): ? / Ie
P
W
F
F
W
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10
Now can I give a different description of what the rules of grammar say about a word
by describing the process that takes place when we understand it?
11
So if the grammar is, for example, the geometry of negation, can I replace it with a description
of what stands behind the word “not”, so to speak, when it is applied?
12
But – as we have said – such a description would be a replacement for13
the word “not”,
as would be, for instance; and it couldn’t replace the grammar.14
15
Grammatical rules can seem like an explication of16
what I experience all at
once when I use a word. (Merely) consequences, expressions, so to speak, of the
properties I experience17
all at once when I understand? Of course that has to be nonsense.
18
One is inclined to say:19
negation has the property of resulting in an affirmation when it
is doubled.20
Whereas the rule for it doesn’t describe negation any better. Rather, it
constitutes it.
21
In using this word in accordance with this rule, in substituting this for it, etc., we
document how we mean it.
22
“Just as I can portray an object through its various projections, and it is equivalent to
these, the nature of negation manifests itself in the various grammatically permissible uses
of the negation sign.”
23
“A double negation yields an affirmation” sounds like: Carbon and oxygen yield carbon
dioxide. But in reality a double negation doesn’t yield anything; it is something.
24
“Anyone who understands negation knows that a double negation . . . .”
25
Something here is giving us the illusion of a physical fact.
As if we were seeing the result of a logical process. Whereas results only come from physical
processes.
26
The word “not” in the grammatical rule has no meaning; otherwise that statement could
not be made about it.
27
Negation has no other property than, for example, that of producing the truth in certain
propositions.
10 (M): / Ib
11 (M): / Ic
12 (M): 555 (F): MS 110, p. 101.
13 (V): of
14 (M): (?)
15 (M): ? / Ia
16 (V): In my presentation grammatical rules
seemed to be the explication of // Are grammatical
rules the explication of
17 (V): experience
18 (M): ? / Id
19 (V): Indeed$ one would flatly say:
20 (V): doubled. (More or less like: iron has the
property of resulting in iron sulphate when
combined with sulphuric acid.)
21 (M): ///
22 (M): ///
23 (M): / Ib
24 (M): / Ia
25 (M): / Ic
26 (M): ∫ /// If
27 (M): ? / Ie
The Rules of Grammar . . . 122e
P
T
F
F
T
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Und ebenso hat ein Kreis die Eigenschaft, da oder dort zu stehen, diese Farbe zu haben,
von einer Geraden tatsächlich geschnitten zu werden; aber nicht, was ihm die Geometrie
zuzuschreiben scheint. (Nämlich diese Eigenschaften haben zu können.)
30
Was heißt es: ”Dieses Papier ist nicht schwarz und ,nicht‘ ist hier so31
gebraucht, daß
eine dreifache Verneinung eine Verneinung ergibt“? Wie hat sich denn das im Gebrauch
geäußert?
Oder: ”Dieses Papier ist nicht schwarz und zwei von diesen Verneinungen geben eine
Bejahung“. Kann ich das sagen?
Oder: ”Dieses Buch ist rot und die Rose ist rot und die beiden Wörter ,rot‘ haben die
gleiche Bedeutung“. (Dieser Satz ist von gleicher Art wie die beiden oberen.) Was ist denn
das für ein Satz? ein grammatischer? Sagt er etwas über das Buch und die Rose?
Ist der Zusatz zum Verständnis des ersten Satzes nicht nötig, so ist er Unsinn, und wenn
nötig, dann war das erste noch kein Satz; und dasselbe gilt in den oberen Fällen.
32
”Daß 3 Verneinungen wieder eine Verneinung ergeben, muß doch schon in der einen
Verneinung, die ich jetzt gebrauche, liegen.“ Aber deute ich hier nicht schon wieder? (D.h.
bin ich nicht im Begriffe, einen Mythus33
zu erfinden?)
34
Heißt es etwas, zu sagen, daß drei solche Verneinungen eine Verneinung ergeben. (Das
erinnert immer an ”drei solche Pferde können diesen Wagen fortbewegen“.) Aber, wie gesagt,
in jenem logischen Satz ist gar nicht von der Verneinung die Rede (von der Verneinung
handeln nur Sätze wie: es regnet nicht) sondern nur vom Wort ”nicht“, und es ist eine Regel
über die Ersetzung eines Zeichens durch ein anderes.
35
Aber können wir die Berechtigung dieser Regel nicht einsehen, wenn wir die
Verneinung verstehen? Ist sie nicht eine Folge aus dem Wesen der Verneinung? Sie ist nicht
eine Folge, aber ein Ausdruck dieses Wesens.
36
Was wir sehen, wenn wir einsehen, daß eine doppelte Verneinung etc. . . . muß von der
Art dessen sein, was wir im Zeichen 37
wahrnehmen.38
39
Die Geometrie spricht40
so wenig von Würfeln, wie die Logik von der Verneinung.
41
(Man möchte hier vielleicht einwenden, daß die Geometrie vom Begriff des Würfels und
die Logik vom Begriff der Negation handelt. Aber diese Begriffe gibt es nicht.)
42
Man kann einen Würfel, aber nicht die Würfelform beschreiben.43
Aber kann ich denn nicht
beschreiben, wie man z.B. eine Kiste macht? Und ist darin44
nicht eine Beschreibung der
Würfelform enthalten?45
Das Wesentliche am Würfel ist damit nicht beschrieben, das steckt
vielmehr in der Möglichkeit dieser Beschreibung, d.h. darin, daß sie eine Beschreibung ist;
nicht darin, daß sie zutrifft.
30 (M): / 1 (R): ∀
31 (V): hier in dem Sinne
32 (M): / 1
33 (V): Begriffe, eine Mythologie
34 (M): 1
35 (M): /
36 (M): / ///
37 (F): MS 110, S. 104.
38 (M): (?)
39 (M): /
40 (V): spricht aber
41 (M): ///
42 (M): / 3
43 (V): einen Würfel – ich meine das
Wesentliche des Würfels – nicht beschreiben.
44 (V): damit
45 (V): Beschreibung des // eines // Würfels
gegeben?
163
123 Die grammatischen Regeln . . .
P
W
F
F
W
W
F
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And likewise a circle has the property of being situated here or there, of having this colour,
of actually being bisected by a straight line; but it doesn’t have what geometry seems to ascribe
to it. (Namely, the ability to have these properties.)
28
What does this mean: “This paper is not black, and here ‘not’ is used in such a way29
that a triple negation yields a single one”? How did that get expressed in its use?
Or: “This paper is not black, and two of these negations result in an affirmation”. Can I
say this?
Or: “This book is red and a rose is red and the two words ‘red’ have the same meaning”.
(This proposition is the same kind as the two above.) What kind of a proposition is that,
anyway? A grammatical one? Does it say anything about the book and the rose?
If the addendum isn’t necessary for understanding the first proposition then it’s nonsense,
and if it is necessary then the first proposition wasn’t yet a proposition; and the same holds
for the other cases above.
30
“That three negations result in a single negation must already be contained in the
single negation that I’m now using.” But isn’t this interpretation, once again? (That is, am
I not about to invent a myth31
?)
32
Does it mean anything to say that three such negations result in a single negation? (This
is always reminiscent of “Three such horses can move this wagon”.) But, as mentioned above,
negation isn’t even being talked about in that proposition of logic (only sentences like “It’s
not raining” are about negation); only the word “not” is being talked about, and it is a rule
for the replacement of one sign by another.
33
But can’t we see the justification for this rule when we understand negation? Isn’t the
rule a consequence of the nature of negation? It isn’t a consequence, but an expression, of
that nature.
34
What we see when we realize that a double negation, etc. . . . must be of the same kind
as what we perceive in the table .35
36
Geometry37
no more speaks of cubes than logic does of negation.
38
(One might be inclined to object here that geometry deals with the concept of a cube
and logic with the concept of negation. But there are no such concepts.)
39
One can describe a cube, but not the form of a cube.40
But can’t I describe how to make
a box, for instance? And doesn’t this contain a description of the form of a cube?41
What
is essential to a cube isn’t described by it. On the contrary, the essence is contained in the
possibility of this description, i.e. in the fact that it is a description; not in the fact that the
description is accurate.
28 (M): / 1 (R): ∀
29 (V): used in the sense
30 (M): / 1
31 (V): inventing a mythology
32 (M): 1
33 (M): /
34 (M): / ///
35 (M): (?) (F): MS 110, p. 104.
36 (M): /
37 (V): But geometry
38 (M): ///
39 (M): / 3
40 (V): One cannot describe a cube (I mean what
is essential to a cube).
41 (V): And isn’t a description of // of a // cube
given by this?
The Rules of Grammar . . . 123e
P
T
F
F
T
T
F
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46
Nun kann ich doch aber sagen: ”Ich sehe die Figur 3-dimensional“. Aber dieser
Satz entspricht der Beschreibung einer Kiste. Er beschreibt einen bestimmten Würfel, nicht
die Würfelform. Freilich kann ich das Wort ”Würfelform“ definieren. D.h. Zeichen geben,
durch die es ersetzt werden kann.47
48
Man kann eine geometrische Figur nicht beschreiben. Auch die Gleichung beschreibt
sie nicht, sondern vertritt sie durch die Regeln, die von ihr gelten.
49
Und haben wir hier nicht das Wort ”Figur“ so angewandt,50
wie in unseren
Betrachtungen so oft das Wort ”Gedanke“ oder ”Symbol“? Die Art der Anwendung dieses
Wortes, von welcher ich sagte, es bedeute dann kein Phänomen, sondern sei quasi ein unvollständiges
Symbol51
und entspreche eher einer Funktion.
52
Man kann auch nicht sagen, die Würfelform habe die Eigenschaft, lauter gleiche
Seiten zu besitzen. Wohl aber hat ein Holzklotz diese Eigenschaft. (Noch hat ”die Eins die
Eigenschaft, zu sich selbst addiert, zwei zu ergeben“.)
53
Ich sagte doch: Es schien, als wären die grammatischen Regeln die ”Folgen in der Zeit“
dessen, was wir in einem Augenblick wahrnehmen, wenn wir eine Verneinung verstehen.
Und als gebe es also zwei Darstellungen des Wesens der Verneinung: Den Akt (etwa
den seelischen Akt) der Verneinung selbst, und seine Spiegelung in dem System der
Grammatik.
54
Man könnte sagen:55
die Gestalt eines Würfels wird doch sowohl durch die Grammatik
des Wortes ”Würfel“, als auch durch einen Würfel, dargestellt.
56
In ”~p & (~~p = p)“ kann der zweite Teil nur eine Spielregel sein.
Wenn Du weißt was ich mit einer halben Drehung meine so wirst Du verstehen daß zwei
halbe Drehungen einander aufheben.
57
Es hat den Anschein, als könnte man aus der Bedeutung der Negation schließen, daß
~~p, p heißt.
58
Als würden aus der Natur der Negation die Regeln über das Negationszeichen folgen.
59
So daß, in gewissem Sinne, die Negation zuerst vorhanden ist60
und dann die Regeln
der Grammatik.
61
Es ist also, als hätte das Wesen der Negation einen zweifachen Ausdruck in der Sprache:
Dasjenige, was ich sehe, wenn ich die Negation verstehe, und die Folgen dieses Wesens in
der Grammatik.
62
Zu sagen, daß eine Vierteldrehung ein Quadrat mit sich selbst zur Deckung bringt, heißt
doch offenbar nichts andres als: Das Quadrat ist um zwei zueinander senkrechte Achsen
46 (M): ///
47 (V): darf.
48 (M): /// (R): Zu S. 91
49 (M): ∫ /// (R): Zu S. 91
50 (V): angewendet,
51 (V): Zeichen
52 (M): / / 3 ///
53 (M): ∫ / 3 ////
54 (M): / ∫ / ///
55 (V): Man ist versucht zu sagen:
56 (M): ∫ ///
57 (M): / / 3
58 (M): / 3 ///
59 (M): ∫
60 (V): wäre
61 (M): / 3
62 (M): ∫ ///
164
165
124 Die grammatischen Regeln . . .
WBTC035-39 30/05/2005 16:17 Page 248
42
Still, I can say: “I see the figure 3-dimensionally”. But this sentence corresponds
to the description of a box. It describes a particular cube, not the form of a cube. To be
sure, I can define the words “form of a cube”. That is, produce signs with which the words
can43
be replaced.
44
One can’t describe a geometric figure. And its equation doesn’t describe it. Rather, it
stands in for the figure via the rules that are valid for it.
45
And haven’t we used the word “figure” here the same way we’ve so often used the words
“thought” and “symbol” in our examinations? The way of using this word where I said
that it didn’t mean a phenomenon, but that it was like an incomplete symbol,46
and that it
corresponded more closely to a function.
47
Likewise, you can’t say that the form of a cube has the property of having all sides equal.
But a block of wood does. (Neither does “the number one have the property of resulting in
two when added to itself”.)
48
But I did say: It seemed as if the grammatical rules were the “consequences in time” of
what we perceive in an instant when we understand a negation.
As if there were therefore two ways of representing the nature of negation: The act (say,
the psychological act) of negation itself, and its reflection in the system of grammar.
49
One could say:50
The form of a cube is represented both by the grammar of the word
“cube” and by a cube.
51
In “~p & (~~p = p)” all the second part can be is a rule of the game.
If you know what I mean by half a turn then you’ll understand that two half turns
cancel each other.
52
It seems as if one could infer from the meaning of negation that ~~p means p.
53
As if the rules for the negation sign followed from the nature of negation.
54
So that in a certain sense there is55
first of all negation, and only then the rules of
grammar.
56
So it is as if the nature of negation had a dual expression in language: What I see when
I understand a negation, and the consequences of this nature in grammar.
57
To say that a quarter turn will bring a square into coincidence with itself obviously means
nothing more than: A square is symmetrical around two axes that are perpendicular with
42 (M): ///
43 (V): may
44 (M): /// (R): To p. 91
45 (M): ∫ /// (R): To p. 91
46 (V): sign,
47 (M): / / 3 ///
48 (M): ∫ / 3 ////
49 (M): / ∫ / ///
50 (V): One is tempted to say:
51 (M): ∫ ///
52 (M): / / 3
53 (M): / 3 ///
54 (M): ∫
55 (V): there would be
56 (M): / 3
57 (M): ∫ ///
The Rules of Grammar . . . 124e
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symmetrisch, und das wieder, daß es Sinn hat, von zwei senkrechten Achsen zu reden, ob
sie vorhanden sind oder nicht. Dies ist ein Satz der Grammatik.
63
Die Schwierigkeit ist wieder, daß es so scheinen kann,64
als wäre in einem Satz, der, z.B.,
das Wort ”Quadrat“65
enthält, schon der Schatten anderer Sätze, die mit diesem Wort gebildet
sind, enthalten.66
– Nämlich eben die Möglichkeit andere Sätze67
zu bilden, die ja, wie ich sagte,
im Sinn des Wortes ”Quadrat“ liegt.
Und doch kann man eben nur sagen, der andere Satz ist nicht mit diesem ausgesprochen,
auch nicht in einer schattenhaften Weise.68
69
Statt der Betrachtung der Negation, könnte ich auch die eines Pfeiles setzen und z.B.
sagen: wenn ich ihn zweimal um 180° drehe, zeigt er wieder, wohin er jetzt zeigt; welcher
Satz dem ~ ~p = p entspricht. Wie ist es nun hier mit der Darstellung des Wesens dieses
Pfeils durch die Sprache? Jener Satz muß doch unmittelbar von diesem Wesen abgelesen70
sein und es also darstellen.
Oder nehmen wir den Fall eines Quadrats und eines Rechtecks und die Sätze, daß das
Quadrat durch eine Vierteldrehung mit sich selbst zur Deckung gebracht werden kann; das
Rechteck aber erst durch eine halbe Drehung.
71
Es frägt sich: Was ist das für ein Satz ”das Wort ,ist‘ in ,die Rose ist rot‘ ist dasselbe
wie in ,das Buch ist rot‘, aber nicht dasselbe, wie in ,zweimal zwei ist vier‘ “? Man kann
nicht antworten, es heiße, verschiedene Regeln gelten von den beiden Wörtern, denn damit
geht man im Zirkel. Wohl aber heißt es, das Wort ist in seinen verschiedenen Verbindungen
durch zwei Zeichen ersetzbar, die nicht für einander einzusetzen sind. Ersetze ich dagegen
das Wort in den beiden ersten Sätzen durch zwei verschiedene Wörter, so darf ich sie für
einander einsetzen.72
73
Nun könnte ich wieder fragen: ist diese Regel74
nur eine Folge des Ersten: daß im einen
Fall die beiden Wörter ”ist“ die gleiche Bedeutung haben, im andern Fall nicht? Oder ist
es so, daß diese Regel eben der sprachliche Ausdruck dafür ist, daß die Wörter das Gleiche
bedeuten?
75
Ich möchte die Metapher gebrauchen,76
daß das Wort ”ist“ einen andern Bedeutungskörper
hinter sich hat77
wenn es einmal für &=“ einmal für &e“ steht.78
Daß es beide Male die gleiche
Vorderfläche79
ist, die einem andern Körper angehört, wie wenn ich ein Dreieck im
Vordergrund sehe, das das eine Mal die Endfläche eines Prismas, das andre Mal eines
Tetraeders ist.
80
Oder denken wir uns diesen Fall: Wir hätten vollkommen81
durchsichtige Glaswürfel, deren
eine Seitenfläche82
rot gefärbt wäre. Wenn wir sie aneinander reihen, so werden im Raum
63 (M): ∫ / ////
64 (V): daß es scheint,
65 (V): der etwa das Wort &Kugel“
66 (V): schon der Schatten eines andern Satzes mit
diesem Worte enthalten.
67 (V): Möglichkeit jenen anderen Satz
68 (V): auch nicht schattenhaft. (Und wird
vielleicht nie ausgesprochen werden.)
(R): Siehe: S. 144/2
69 (M): ////
70 (V): abgeleitet
71 (M): ? / /// (
72 (M): )
73 (M): ∫ / ///
74 (V): fragen: sind diese Regeln
75 (M): /
76 (V): Ich will es damit vergleichen,
77 (V): andern Wortkörper hinter sich hat.
78 (O): steht“. (V): einmal = einmal e bedeutet
79 (V): Fläche
80 (R): ∀ S. 145/1
81 (V): ganz
82 (V): Seite
166
125 Die grammatischen Regeln . . .
WBTC035-39 30/05/2005 16:17 Page 250
respect to each other, and that in turn means that it makes sense to talk about two perpendicular
axes, whether they are present or not. This is a grammatical proposition.
58
Once again, the difficulty is that it can look59
as if a sentence containing the word “square”,
for example,60
already contained the shadows of other sentences that are formed with this word.61
– That is to say, the possibility of forming sentences62
, which, as I said, is contained in the
sense of the word “square”.
And yet all one can say is that the other sentence is not uttered by uttering this one, not
even in a shadowy way.63
64
Instead of considering negation I might also consider an arrow and say, for example: “If
I turn it 180° twice then it points once again where it’s pointing now; which proposition
corresponds to ~~p = p.” Now how does representing the nature of this arrow through language
work here? For presumably the latter proposition must have been read off65
this nature
directly, and therefore must represent it.
Or let’s take the case of a square and a rectangle, and the propositions that a square can
be brought into coincidence with itself by a quarter turn, but that it takes half a turn to do
this with a rectangle.
66
The question arises: What kind of a sentence is “The word ‘is’ in ‘The rose is red’ is
the same as in ‘The book is red’ but not the same as in ‘Twice two is four’”? The answer
can’t be that this means that different rules are valid for the two words, for that would be
arguing in a circle. What it means is that the word is replaceable in its different contexts by
two signs that cannot be substituted for each other. On the other hand, if I substitute two
different words for a word that is the same in the first two sentences, then I am allowed to
substitute them for each other.67
68
But again I could ask: Is this rule69
just a consequence of the first proposition: that in the
one case the two words “is” have the same meaning, and not in the other? Or is this rule
simply the linguistic expression of the fact that the words mean the same thing?
70
I’d like to use the metaphor71
that the word “is” has a different meaning-body72
behind it
when at one time it stands for “=”, at another for “e”.73
That in both cases it is the same outer
surface74
but one that belongs to different bodies, as when I see a triangle in the foreground
that is at one time the end surface of a prism, at another that of a tetrahedron.
75
Or let’s imagine this: we have absolutely76
transparent cubes of glass, one surface77
of which
is coloured red. If we line them up against each other, then because of their cubical shapes
58 (M): ∫ / ////
59 (V): it looks
60 (V): word “sphere”, for example$
61 (V): contained the shadow of another sentence
that contains this word.
62 (V): forming that other sentence
63 (V): not even in a shadowy way. (And perhaps
will never be uttered.) (R): See: p. 144/2
64 (M): ////
65 (V): must be derived from
66 (M): ? / /// (
67 (M): )
68 (M): ∫ / ///
69 (V): ask: Are those rules
70 (M): /
71 (V): I’d like to compare this to the fact
72 (V): word-body
73 (V): it when it first means “=”, then “e”.
74 (V): the same surface
75 (R): ∀ p. 145/1
76 (V): completely
77 (V): side
The Rules of Grammar . . . 125e
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nur ganz bestimmte Anordnungen83
roter Quadrate entstehen können, bedingt durch die
Würfelform der Körper. Ich könnte nun die Regel, nach der hier rote Quadrate angeordnet
sein können, auch ohne Erwähnung der Würfel geben,84
aber in ihr wäre doch bereits das
Wesen der Würfelform präjudiziert. Freilich nicht, daß wir gläserne Würfel haben,85
wohl
aber die Geometrie des Würfels.
86
Wenn wir nun aber einen Würfel87
sehen, sind damit wirklich schon alle Gesetze der
möglichen Zusammenstellung gegeben?! Also die88
Geometrie?
Kann ich die Geometrie des Würfels von einem Würfel ablesen?
89
Der Würfel ist dann eine Notation der Regel.
Und hätten wir eine solche Regel gefunden, so könnten wir sie wirklich nicht besser
notieren, als durch die Zeichnung eines Würfels (und daß es hier eine Zeichnung tut, ist
wieder ungemein bedeutsam90
).
91
Und nun ist die Frage: Wie kann aber der Würfel,92
oder die Zeichnung (denn die beiden
kommen hier auf dasselbe93
hinaus) als Notation der geometrischen Regeln dienen?
94
Doch95
nur, sofern er einem System angehört:96
nämlich der Würfel mit der einen roten
Endfläche wird etwas anderes notieren, als eine Pyramide mit quadratischer roter Basis, etc.
D.h., er97
wird dasjenige Merkmal der Regeln notieren, worin sich z.B. der Würfel von der
Pyramide unterscheidet.
98
Jedes Zeichen der Negation ist gleichwertig jedem andern, denn 99
ist ebenso ein
Komplex von Strichen, wie das Wort ”nicht“, und zur Negation wird es nur durch die Art,
wie es ”wirkt“ d.h. wie es im Spiel gebraucht wird.100
101
Ich möchte sagen: Nur dynamisch wirkt das Zeichen, nicht statisch. Der Gedanke ist
dynamisch.
102
Daß die Tautologie und Kontradiktion nichts sagen, geht nicht etwa aus dem W-FSchema
hervor, sondern muß festgesetzt werden. Und die Schemata machen nur die
Festsetzung der Form einfach.103
104
Du sagst, das Hinweisen auf einen roten Gegenstand ist das primäre Zeichen für
”rot“. Aber das Hinweisen auf einen roten Gegenstand ist nicht mehr, als die bestimmte
Handbewegung gegen einen roten Gegenstand, und ist außer in einem System105
gar kein
Zeichen. Wenn Du sagst, Du meinst: das Hinweisen auf den roten Gegenstand als Zeichen
126 Die grammatischen Regeln . . .
83 (V): so wird im Raum nur (eine) ganz bestimmte
Anordnung
84 (V): angeben,
85 (V): daß es gläserne Würfel sind,
86 (M): //
87 (V): einen solchen Würfel
88 (V): die ganze
89 (M): //
90 (V): wichtig
91 (M): /
92 (V): Frage: in wiefern // wie // kann der
Würfel,
93 (V): auf eines (O): auf eine
94 (M): ? /
95 (V): Doch auch
96 (V): nur, sofern er als Satz einem System von
Sätzen angehört.
97 (O): es
98 (M): /
99 (F): MS 110, S. 116.
100 (V): es ”wirkt“. Hier aber ist nicht die
Wirkung im Sinne der Psychologie (das Wort
@Wirkung! also nicht kausal gemeint$ sondern
die Form seiner Wirkung.
101 (M): /
102 (M): / 3
103 (V1): der Form leicht. (V2): machen nur die
Form der allgemeinen Festsetzung einfach.
104 (M): ? / 555 – ehe sie gezogen wurden.
(R): [Zu S. 93]
105 (V): und ist vorläufig
167
168
P
W
F
F
W
WBTC035-39 30/05/2005 16:17 Page 252
only certain arrangements78
of red squares can come about. Now I could give79
the rule for
the possible arrangement of the red squares without even mentioning the cubes, but the nature
of the form of a cube would already be prejudged in it. But this rule would prejudge not
that we have80
glass cubes, but the geometry of a cube.
81
When we see a82
cube, does this really give us all the laws for its possible combinations?!
That is, its geometry83
?
Can I read the geometry of a cube off a cube?
84
In that case a cube is a notation for its rule.
And if we had discovered such a rule, we really couldn’t record it better than by drawing
a cube (and it is extraordinarily significant85
that it is a drawing that does this here).
86
And now the question arises: How87
can a cube, or a drawing of one (for here the two
amount to the same thing88
) serve as a notation for its geometrical rules?
89
Surely90
only in so far as it belongs to a system91
: for the cube with the one red surface
will record something different from a pyramid with a square red base, etc. That is, the
cube will record that particular feature of the rules by which it differs, for example, from a
pyramid.
92
Every negation sign is equivalent to every other one, for 93
is as much a complex
of lines as is the word “not”, and it becomes a negation only by the kind of “effect” it has,
i.e. by how it is used in a game.94
95
I would like to say: A sign only works dynamically, not statically. Thought is dynamic.
96
That tautology and contradiction say nothing in no way emerges from the T–F schema,
but has to be stipulated. And all the schemata do is to make the stipulation of the form
simple.97
98
You say that pointing to a red object is the primary sign for “red”. But pointing to
a red object is nothing more than a particular motion of the hand towards a red object,
and is no sign at all except within a system.99
If you say you mean: pointing to a red object
understood as a sign – then I say: The understanding that is our concern is not a process that
78 (V): only a certain arrangement
79 (V): state
80 (V): that they are
81 (M): / /
82 (V): see such a
83 (V): its entire geometry
84 (M): / /
85 (V): important
86 (M): /
87 (V): arises: To what extent
88 (V): to one thing
89 (M): ? /
90 (V): Surely also
91 (V): only in so far as a proposition it belongs to a
system of propositions
92 (M): /
93 (F): MS 110, p. 116.
94 (V): it has. But what is meant here isnFt
!effectE in the psychological sense$ i.e. the
word !effectE isnFt meant in its causal sense)
but rather the form of its effect.
95 (M): /
96 (M): / 3
97 (V1): easy (V2): make the form of a general
stipulation simple.
98 (M): ? / 555 – were drawn. (R): [To
p. 93]
99 (V): and is for the time being no sign at all.
The Rules of Grammar . . . 126e
P
T
F
F
T
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verstanden – so sage ich: das Verständnis, auf das es uns ankommt, ist kein Vorgang, der das
Hindeuten begleitet (etwa ein Vorgang im Gehirn) und wenn Du doch so einen Vorgang
meinst, so ist dieser an sich wieder kein Zeichen. Die Idee ist hier immer wieder, daß die
Meinung, die Interpretation, ein Vorgang sei, der das Hinweisen begleitet und ihm
sozusagen die Seele gibt (ohne welche es tot wäre).
Es scheint hier als ob das Zeichen die ganze Gr. zusammenfaßte, daß sie in ihm enthalten wäre,
wie eine Perlenschnur in einer Schachtel und wir sie nur herausziehen müßten.106
(Aber dieses
Bild ist es eben, welches107
uns irreführt.) Als wäre108
das Verständnis ein momentanes Erfassen
von etwas, wovon später nur die Konsequenzen gezogen werden; und zwar so, daß diese
Konsequenzen bereits in einem ideellen Sinn existieren, ehe sie gezogen wurden. 109
Als ob110
der Würfel111
schon die ganze Geometrie des Würfels enthielte und ich sie nun nur noch
auszubreiten hätte.112
Aber welcher Würfel? Der Gesichtswürfel, oder ein Eisenwürfel?
Oder gibt es einen idealen geometrischen Würfel?113
– Offenbar schwebt uns der Vorgang
vor, wenn wir aus einer Zeichnung, Vorstellung (oder einem Modell) Sätze der Geometrie
ableiten.114
Aber welche Rolle spielt dabei das Modell? Doch wohl die des Zeichens.115
Des
Zeichens, welches eine bestimmte Verwendungsart hat und nur durch diese116
bezeichnet.
Es ist allerdings interessant und merkwürdig, wie dieses Zeichen verwendet wird, wie wir,
etwa, die Zeichnung des Würfels wieder und wieder bringen mit immer anderen Zutaten.
Einmal sind die Diagonalen gezogen, einmal Würfel aneinander gereiht, etc. etc. Und es ist
dieses Zeichen (mit der Identität eines117
Zeichens), welches wir für jenen Würfel nehmen,
in dem die geometrischen Gesetze bereits liegen. (Sie liegen in ihm so wenig, wie im
Schachkönig die Dispositionen, in gewisser Weise benützt zu werden.)118
106 (V1): Das scheint besonders dort so, wo ein
Zeichen die ganze Grammatik zusammenzufassen
scheint, daß wir sie aus ihm ableiten
könnten$ und es scheint$ daß, daß sie in
ihm enthalten wäre, wie . . . und wir sie nur
herausziehen müßten. (V2): Der Würfel
scheint seine ganze Gr. zusammenzufassen, // Es
scheint, als ob der Würfel seine ganze Grammatik
zusammenfaßte, // daß wir sie aus ihm ableiten
könnten, daß sie in ihm enthalten wäre, wie
die Perlenschnur . . . müßten. (V3): Es
scheint hier als ob das Zeichen die ganze Gr.
zusammenfaßte, daß wir sie aus ihm ableiten
könnten, & sie in ihm wie eine Perlenschnur in
einer Schachtel wäre, die wir nur herausziehen
müßten.
107 (V): was
108 (V): wäre also
109 (M): /
110 (V): ob also
111 (V): Würfel -z.B.-
112 (V): habe.
113 (V): einen ideellen Würfel?
114 (V): der Vorgang vor, aus einer Zeichnung,
. . . Geometrie abzuleiten.
115 (V): Zeichens!
116 (V): dieses
117 (V): des
118 (M): überprüfen (R): [dazu S. 145/1] ∀
S. 22/1, 2 (V): werden.) /// Die
geometrischen Gesetze konstituieren den
Begriff des Würfels (sie geben eine
Konstitution, eine Verfassung). Was ich seinerzeit
über den ”Wortkörper“ geschrieben
habe, ist der klare Ausdruck des besprochenen
Irrtums.) )
167v
169
168
167v
169
127 Die grammatischen Regeln . . .
WBTC035-39 30/05/2005 16:17 Page 254
accompanies the pointing (say, a process in the brain), and if you do mean such a process
after all, then it too is not inherently a sign. Again and again the idea here is that meaning,
interpretation, is a process that accompanies the pointing and provides it with a soul, as it
were (without which it would be dead).
Here it seems as if the sign were a summary of all of grammar – that the latter is contained in it
like a string of pearls in a box and that all we have to do is pull it out.100
(But it is precisely
this picture that leads us astray.) As if understanding were an101
instantaneous grasping of
something, and all one had to do was then to draw out its consequences; so that these consequences
already existed in an ideal sense before they were drawn. 102
As if the cube already103
contained the entire geometry of a cube, and now I had104
only to unfold it. But which cube?
The visual cube or a metal cube? Or is there such a thing as an ideal geometrical cube105
? –
Evidently we have this process in mind when we derive geometrical propositions from a drawing,
a mental image (or a model). But what role does the model play in this derivation? Surely
that of a sign.106
Of a sign that has a particular kind of use and that signifies only because of
it. It is interesting, though, and remarkable, how this sign is used, how, for example, we
break out the drawing of a cube again and again, each time with different accessories. One
time the diagonal lines are drawn, another time cubes are set in a row next to each other,
etc., etc. And it is this sign (with the identity of a sign107
) that we take to be that cube in which
the geometric laws are already contained. (They are no more contained in it than the dispositions
to be used in a certain way are contained in the chess king.)108
100 (V1): That seems to be the case, particularly
where a sign seems to summarize all of grammar,
so that we can deduce grammar from the
sign, and it seems that, that grammar were contained
in the sign like a string of pearls . . . it
out. (V2): The cube seems to summarize its
entire grammar, // It seems as if the cube summarized
its entire grammar, // so that we could deduce the
grammar from it, that grammar were contained in
it like a string of pearls . . . it out. (V3): It
seems here as if the sign summarized the entire
grammar, so that we could deduce the grammar
from it, and it were contained in it like a string of
pearls in a box that we have only to pull out.
101 (V): were thus an
102 (M): /
103 (V): the cube$ for example$ thus already
104 (V): have
105 (V): an ideal cube
106 (V): sign!
107 (V): of signs
108 (M): Check (R): [to p. 145/1] ∀ p. 22/1, 2
(V): king.) /// The geometric laws constitute
the concept of a cube (they provide a constitution,
a fundamental law). What I once wrote
about a “word-body” is a clear expression of
the error I’ve been discussing.))
The Rules of Grammar . . . 127e
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Wesen der Sprache.170
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The Nature of
Language.
WBTC040-45 30/05/2005 16:17 Page 257
40
Lernen, Erklärung, der Sprache.
Kann man die Sprache durch die
Erklärung gleichsam aufbauen,
zum Funktionieren bringen?1
2
Wenn ich erkläre ”’~p‘ ist wahr, wenn ’p‘ nicht wahr ist“, so setzt das voraus, daß ich
verstehe, was es heißt, ”p“ sei nicht wahr. Dann habe ich aber nichts getan als zu definieren:
~p Def.
”p“ ist falsch.
3
Es kommt nämlich wesentlich darauf an, daß es nicht möglich ist, das Zeichen ”p“ auf
der rechten Seite der Definition auszulassen, bezw. durch ein anderes zu ersetzen (es sei denn
wieder mit Hilfe einer4
Definition). Solange das nicht möglich ist, kann und muß man auch
die rechte Seite als Funktion auffassen von p, nämlich: ”( )“ ist falsch.
5
Das hängt auch damit zusammen, daß ja der Tintenstrich nicht falsch ist. Wie er schwarz
oder krumm ist.
6
Ist es denn richtig zu schreiben &*p’ ist falsch“? Muß es nicht heißen &p ist falsch“?
7
Sagt denn &*p’ ist wahr“ etwas über das Zeichen &p“ aus? Man sagt: &ja, es sagt daß *p’ mit der
Wirklichkeit übereinstimmt“. Denken wir uns – statt eines8
Satzes der Wortsprache ein nach einer
exakten Projektionsmethode gezeichnetes Bild der betreffenden Wirklichkeit. Hier muß es sich
gewiß am deutlichsten zeigen, was & *p’ ist wahr“ von dem Bild &p“ aussagt. Man kann also den Satz
&*p’ ist wahr“ mit dem vergleichen: &Dieser Gegenstand hat9
die Länge dieses Meterstabes10
“ & &p“
dem Satz: &dieser Gegenstand ist 1m lang“. Aber der Vergleich ist falsch denn &dieser Meterstab“
ist eine Beschreibung weil11
&Meterstab“ eine Begriffsbestimmung ist. Dagegen tritt in &*p’ ist
wahr“ der Maßstab unmittelbar in den Satz ein. &p“ repräsentiert12
hier einfach die Länge & nicht
den Stab.13
Denn14
die projizierte Figur ist ja auch gar nicht wahr außer nach einer bestimmten
Projektionsmethode die den Meterstab15
zu einem rein – geometrischen Anhängsel der gemessenen
Strecke macht.
16
Wenn ich also auch dem Schriftzug ”p“ den Namen A gebe und daher schreibe: ”~p
Def.
A ist falsch“, so hat das nur einen Sinn, d.h. die rechte Seite kann nur verstanden
1 (R): [Zu § 18 S. 76]
2 (M): ? 3
3 (M): |
4 (V): wieder durch eine
5 (M): / |
6 (M): / 3
7 (M): Überprüfe
8 (V): eines gewohnl
9 (V): hat zweimal
10 (V): Maßstabes
11 (V): während
12 (O): representiert
13 (V): Meterstab.
14 (V): Denn p
15 (V): Maßstab
16 (M): ∫ 3
171
170v
171
129
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40
Learning, Explanation, of Language.
Can We Use Explanation to
Construct Language, so to Speak,
to Get it to Work?1
2
If I declare that “‘~p’ is true if ‘p’ is not true” then this presupposes that I understand
what “‘p’ is not true” means. But then all I’ve done is to give this definition: ~p Def.
”p“ is
false.
3
For it’s important that it isn’t possible to omit the sign “p” on the right side of the definition
or to replace it with another sign (except once again with the help of4
a definition). So long
as that isn’t possible, one can and must understand the right side as a function of p as well,
that is to say: “( )” is false.
5
This is also connected with the fact that a line drawn in ink isn’t false. In the way that
it’s black or crooked.
6
Is it correct to write “‘p’ is false”? Mustn’t this read “p is false”?
7
Does “‘p’ is true” say something about the sign “p”? We say: “Yes, it says that ‘p’ agrees with
reality.” Let’s imagine a picture of a particular reality that’s drawn using a precise method of projection,
instead of a sentence8
belonging to word-language. Surely this will show most clearly what “‘p’
is true” says about the picture “p”. So one can compare the proposition “‘p’ is true” with “This object
is as long9
as this yardstick10
” and “p” with the proposition: “This object is one yard long.” But the
comparison is wrong, for “this yardstick” is a description, because11
“yardstick” is a definition of a
concept. In “‘p’ is true”, on the other hand, the measuring stick enters the proposition immediately.
Here “p” represents simply the length and not the stick.12
For the projected figure isn’t true at all,
except in accordance with a particular method of projection that turns the yardstick13
into a purely
geometric appendage to the distance that was measured.
14
So even if I give the name A to the written character “p”, and therefore write “~p Def.
A is false”, then that only makes sense, i.e. the right side can only be understood, if for
1 (R): [To § 18 p. 76]
2 (M): ? 3
3 (M): |
4 (V): again via
5 (M): / |
6 (M): / 3
7 (M): Check
8 (V): of an ordina sentence
9 (V): is twice as long
10 (V): this measuring stick
11 (V): whereas
12 (V): yardstick.
13 (V): measuring stick
14 (M): ∫ 3
129e
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werden, wenn A für uns als Satzzeichen steht. Dann aber ist nichts gewonnen; zum mindesten
keine Erklärung des Mechanismus der Negation.
17
Und dasselbe muß der Fall sein, wenn man erklärt, ”(x).fx“ sei wahr, wenn f( ) für alle
Substitutionen wahr ist. Jeder dieser beiden Sätze folgt aus dem anderen, drum sind sie identisch.
Man muß auch dazu schon den logischen Mechanismus der Verallgemeinerung verstehen.
Es ist (auch) nicht so, daß man erst ahnungslos ist, und die Verallgemeinerung nun durch
die Erklärung erst zum Funktionieren gebracht wird. Wie wenn man in eine Maschine ein
Rad einsetzt und sie dann18
erst funktioniert (oder, die Maschine erst in zwei getrennten Teilen
da ist und sie nun erst durch das Zusammensetzen als diese Maschine funktionieren).
19
Die Erklärung einer Sprache (der Zeichen einer Sprache) führt uns nur von einer Sprache
in eine andere.20
21
Wie schaut die Erklärung eines Zeichens aus? Das müßte doch eine für die Sprache
außerordentlich wichtige Form sein, sei dieser Behelf nun ein Satz oder nicht.
22
Denken wir uns eine Sprache, in der ich ”A ist größer als B“ nicht nur so ausdrücke:
”↑ ist größer als 2“ 23
sondern in der ich auch statt des Wortes ”größer“ eine Geste mache,
die die Bedeutung des Wortes zeigt. – Wie könnte ich nun so eine Sprache erklären? (Wie
könnte ich die Zeichen so einer Sprache erklären?) Und würde ich nun noch das frühere
Bedürfnis nach einer Erklärung fühlen?
Eine Erklärung für die Bedeutung eines Zeichens tritt an Stelle des erklärten Zeichens.
24
Auch das Kind lernt durch Erklärungen25
nur eine Sprache vermittels einer anderen. Die
Wortsprache durch die Gebärdensprache.
26
Die Gebärdensprache ist eine Sprache und wir haben sie nicht – im gewöhnlichen Sinne
– gelernt. Das heißt: sie wurde uns nicht geflissentlich gelehrt. – Und jedenfalls nicht durch
Zeichenerklärungen.
27
Man kann sich das Lernen einer Sprache in anderm Sinne aber analog dem
Fingerhutsuchen vorstellen, wo die gewünschte Bewegung durch ”heiß, heiß“, ”kalt, kalt“
herbeigeführt wird. Man könnte sich denken, daß der Lehrende statt dieser Worte auf
irgendeine Weise (etwa durch Mienen) angenehme und unangenehme Empfindungen
hervorruft, und der Lernende nun dazu gebracht wird, die Bewegung auf den Befehl hin
auszuführen, die regelmäßig von der angenehmen Empfindung begleitet wird (oder zu
ihr führt).28
29
Verbindung von Wort und Sache durch das Lehren der Sprache30
hergestellt. Was ist
das für eine Verbindung, welcher Art? Was für Arten von Verbindungen gibt es?
Eine elektrische, mechanische, psychische Verbindung kann funktionieren oder nicht
funktionieren: Anwendung auf die Verbindung, die die Worterklärung herstellt.
17 (M): ∫ 3
18 (V): nun
19 (M): ∫ 3 ///
20 (R): ∀ S. 2/3 & S. 3/1
21 (M): /// 3
22 (M): ? / 3
23 (F): MS 109, S. 145.
24 (M): ? /
25 (V): lernt in diesem Sinne
26 (M): ? / 3
27 (M): ∫ (R): [Zu S. 201]
28 (M): Abrichten
29 (M): ∫ 3 (
30 (V): durch die Erklärung
130 Lernen, Erklärung, der Sprache. . . .
172
171v
172
173
WBTC040-45 30/05/2005 16:17 Page 260
us A is a sign for a proposition. But then nothing is gained; at least no explanation of the
mechanism of negation.
15
And the same thing must be the case if one declares that “(x).fx” is true if f( ) is true
for all substitutions. Each of these two propositions follows from the other, so they’re identical.
But even this requires an understanding of the logical mechanism of generalization.
(Neither) is it true that at first one has no clue, and that the generalization doesn’t work until
the explanation gets it going. As when one inserts a wheel into a machine and only then16
does it work (or as when the machine comes initially in two separate parts and they don’t
function as this machine until they are assembled).
17
The explanation of a language (of the signs of a language) merely leads us from one
language to another.18
19
What does the explanation of a sign look like? Surely that ought to be an extraordinarily
important form for a language, whether this makeshift is a sentence or not.
20
Let’s imagine a language in which I express “A is larger than B” not only with “↑ is
larger than 2”,21
but in which, instead of the word “larger”, I make a gesture that shows
the meaning of the word. – How could I explain such a language? (How could I explain
the signs of such a language?) And would I then still feel my previous need for an
explanation?
An explanation of the meaning of a sign takes the place of the sign that is explained.
22
It is only through explanations that a child23
learns one language by means of another.
The language of words through the language of gesture.
24
The language of gesture is a language, but we haven’t learned it in the usual sense. That
is: we weren’t taught it deliberately. – And in any case not by having its signs explained.
25
But in another sense one can think of learning a language as analogous to the game of
searching for a thimble, where the desired movement is brought about by “hot, hot”, “cold,
cold”. One could imagine a teacher, instead of using these words, evoking pleasant and
unpleasant feelings in some way (say with facial expressions), and that when the student
is ordered to do so he is induced to make that movement that is regularly accompanied by
a pleasant feeling (or leads to one).26
27
The connection of word and object that is established by teaching a language.28
What
sort of connection is this, of what kind is it? What kinds of connections are there?
An electrical, mechanical, or psychological connection can work or not work: application
to the connection that explaining a word establishes.
15 (M): ∫ 3
16 (V): now
17 (M): ∫ 3 ///
18 (R): ∀ p. 2/3 & p. 3/1
19 (M): /// 3
20 (M): ? / 3
21 (F): MS 109, p. 145.
22 (M): ? /
23 (V): It is only in this sense that a child
24 (M): ? / 3
25 (M): ∫ (R): [To p. 201]
26 (M): Training an animal
27 (M): ∫ 3 (
28 (V): by an explanation.
Learning, Explanation, of Language. . . . 130e
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174
131 Lernen, Erklärung, der Sprache. . . .
31
Die Zuordnung von Gegenstand und Name ist keine andere, als die durch die Worte
”das ist . . .“ oder eine Tabelle erzeugte etc. Sie ist ein Teil des Symbolismus. Es ist
daher unrichtig zu sagen, die Beziehung zwischen32
Name und Gegenstand sei eine
psychologische.
33
Das Wort ”Teekanne“ hat doch Bedeutung; gewiß, im Gegensatz zum Worte
”Abracadabra“, nämlich in der deutschen Sprache. Aber wir könnten ihm natürlich auch
eine Bedeutung geben; das wäre ein Akt ganz analog dem, wenn ich ein Täfelchen mit der
Aufschrift ”Teekanne“ an eine Teekanne hänge. Aber was habe ich hier anderes34
als eine
Teekanne mit einer Tafel, auf der Striche zu sehen sind? Also wieder nichts logisch
Interessantes. Die Festsetzung der Bedeutung eines Wortes kann nie (wesentlich) anderer
Art sein.35
31 (M): ∫ 3
32 (V): von
33 (M): ∫ 3
34 (O): anders
35 (M): )
WBTC040-45 30/05/2005 16:17 Page 262
29
The assignment of name to object is nothing other than that produced by the words “That
is . . .” or by a table, etc. It is a part of the symbolism. Therefore it’s incorrect to say that
the relationship between30
a name and an object is psychological.
31
The words “tea kettle” do have meaning – certainly in contrast to the word
“Abracadabra” – i.e. in the English language. But of course we could also give it a meaning;
doing so would be completely analogous to my hanging a sign reading “tea kettle” on a tea
kettle. But then what do I have other than a tea kettle with a sign that has lines on it? So
once again, nothing that is logically interesting. Establishing the meaning of a word can never
be (essentially) different from this.32
29 (M): ∫ 3
30 (V): of
31 (M): ∫ 3
32 (M): )
Learning, Explanation, of Language. . . . 131e
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41
Wie wirkt die einmalige
Erklärung der Sprache,
das Verständnis?
1
Vielleicht ist die eigentliche Schwierigkeit die: daß ich das Wort ”rot“ erkläre, indem ich
auf etwas Rotes zeige und sage ”das ist rot“, während doch dieses Rote später meinem Blick
entschwindet. Und nun scheinbar etwas Anderes an seine Stelle tritt (die Erinnerung oder
wie man es heißen mag).
2
”Also so wird dieses Wort gebraucht!“ Aber wie bewahre ich denn dieses So in der
Erinnerung?
3
Die4
Weise, wie wir die Sprache lernen,5
ist in ihrem Gebrauch6
nicht enthalten.7
(Wie die
Ursache eben nicht in ihrer Wirkung.)
8
Ich kann die Regel selbst festsetzen und mich eine9
Sprache lehren. Ich gehe spazieren
und sage mir: Wo immer ich einen Baum treffe, soll mir das das Zeichen sein, bei der
nächsten Kreuzung links zu gehen, und nun richte ich mich nach den Bäumen in dieser
Weise (fasse ihre Stellung als einen Befehl auf).
10
Wie kann ich mir vornehmen, einer Regel zu folgen?
Nicht nur soweit, als ich die Regel ausdrücken kann?
11
Welche Wirkung hatte nun die hinweisende Erklärung? Hatte sie sozusagen nur eine
automatische Wirkung? Das heißt aber, wird sie nun immer wieder benötigt, oder hatte sie
eine12
Wirkung, wie etwa eine Impfung, die uns ein für allemal, oder doch bis auf weiteres,
geändert hat.
13
Ich sage ”wähle alle blauen Kugeln aus“; er aber weiß nicht, was ”blau“ heißt. Nun zeige
ich und sage ”das ist blau“. Nun versteht er mich und kann meinem Befehl folgen.
14
Ich setze ihn in Stand, dem Befehl zu folgen. Was geschieht nun aber, wenn er in Zukunft
diesen Befehl hört? Ist es nötig, daß er sich jener Erklärung, d.h. des einmaligen Ereignisses
jener Erklärung erinnert? Ist es nötig, daß das Vorstellungsbild des blauen Gegenstands oder
1 (M): ? / 3
2 (M): / 3
3 (M): ? / 3
4 (V): Die (Art &)
5 (V1): lernten, (V2): Das Lernen der Sprache
// Die Weise des Lernens der Sprache
6 (V): in ihrer Benützung
7 (V): nicht en enthalten.
8 (M): ∫ ///
9 (V): die
10 (M): ∫ ////
11 (M): ? / 3
12 (V): eine ursächliche
13 (M): ? / 555
14 (M): ? / 555
175
174v, 175
176
175
132
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41
What Effect Does a Single
Explanation of Language Have,
What Effect Understanding?
1
Perhaps the real difficulty is as follows: that I explain the word “red” by pointing to something
red and saying “That is red”, whereas the red thing later disappears from my sight.
And now seemingly something else takes its place (memory or whatever you might call it).
2
“So that is how that word is used!” But how do I keep the that in my memory?
3
The way4
we learn language5
is not contained in its use. (Just as a cause isn’t contained in
its effect.)
6
I can establish a rule myself and teach myself a7
language. I take a walk and tell myself:
“Wherever I come upon a tree I’ll take that as a sign to turn left at the next crossroads”,
and then I let the trees so guide me (I take their location as a command).
8
How can I resolve to follow a rule?
Can’t I follow it only so far as I can express it?
9
So what effect does an ostensive explanation have? Is it automatic, so to speak? That is
to say, is it needed over and over, or is its effect10
like, say, an inoculation that changes us
once and for all, or at least for the time being?
11
I say “Pick out all the blue balls”; but he doesn’t know what “blue” means. Then I point
and say “That is blue”. Now he understands me and can follow my order.
12
I enable him to follow the order. What’s going to happen, though, when he hears this
command in the future? Does he need to remember that explanation, i.e. the one-time occurrence
of that explanation? Does the mental image of the blue object or of a blue object have
1 (M): ? / 3
2 (M): / 3
3 (M): ? / 3
4 (V): The (manner and) way
5 (V): The way we learned language // The learning
of language // The way language is learned
6 (M): ∫ ///
7 (V): the
8 (M): ∫ ///
9 (M): ? / 3
10 (V): is it a causal effect
11 (M): ? / 555
12 (M): ? / 555
132e
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eines blauen Gegenstands vor seine Seele tritt? Alles das scheint nicht nötig zu sein, obwohl
es möglicherweise geschieht. Und doch scheint das Wort ”blau“ jetzt einen anderen
Aspekt15
für ihn zu haben, als da es ihm noch nicht erklärt war. Es gewinnt gleichsam Tiefe.
16
In wiefern hilft die hinweisende Erklärung ”das ist ’rot‘ “ zum Verständnis des Wortes.
(Sie ”hilft“ gar nicht, sondern ist eben eine der symbolischen Regeln für den Gebrauch
des Wortes ”rot“.)
17
Eine Erklärung kann nicht in die Ferne wirken. Ich meine: sie wirkt nur, wo sie angewandt
wird. Wenn sie außerdem noch eine ”Wirkung“ hat, dann nicht18
als Erklärung.
19
Ist es so, daß eine Erklärung, eine Tabelle z.B., zuerst so gebraucht werden kann,20
daß
man sie ”nachschlägt“; daß man sie dann gleichsam im Kopf nachschlägt, d.h., sie sich vor
das innere Auge ruft (oder dergleichen); und daß man endlich ohne diese Tabelle arbeitet,
also so, als wäre sie nie da gewesen. In diesem letzten Fall spielt man also offenbar ein anderes
Spiel. Die Tabelle21
ist aus unserm Spiel ausgeschieden und wenn ich auf sie zurückgreife,22
so tue ich, was der Erblindete tut, der etwa auf den Tastsinn zurückgreift. Eine Erklärung
fertigt eine Tabelle an und sie wird zur Geschichte, wenn23
ich die Tabelle nicht mehr benütze.
24
Ich muß unterscheiden zwischen den Fällen: wenn ich mich einmal nach einer Tabelle
richte, und ein andermal in Übereinstimmung mit der Tabelle (der Regel, welche die Tabelle
ausdrückt) handle, ohne die Tabelle zu benützen. – Die Regel, deren Erlernung uns veranlaßte,
jetzt so und so zu handeln, ist als Ursache unserer Handlungsweise Geschichte25
ohne
Interesse für uns.26
Sofern sie aber eine allgemeine Beschreibung unserer Handlungsweise ist,
ist sie eine Hypothese. Die27
Hypothese, daß diese zwei Leute, die am28
Schachbrett sitzen,
so und so handeln werden 29
(wobei auch ein Verstoß gegen die Spielregeln unter die Hypothese
fällt, denn diese sagt dann etwas darüber aus, wie sich die Beiden benehmen werden,
wenn sie auf den30
Verstoß aufmerksam werden). Die Spieler können aber die Regel auch
benützen, indem sie in jedem besonderen Fall nachschlagen, was zu tun ist; hier tritt die
Regel in die Spielhandlungen selbst ein und verhält sich zu ihnen31
nicht, wie eine Hypothese
zu ihrer Bestätigung. ”Hier gibt es aber eine Schwierigkeit. Denn der Spieler, welcher ohne
Benützung des Regelverzeichnisses spielt, ja, der nie eins gesehen hätte, könnte dennoch,
wenn es verlangt würde, ein Regelverzeichnis anlegen und zwar meine ich nicht, –
behaviouristisch – indem er durch wiederholte Beobachtung feststellte, wie er in diesem
und in jenem Fall handelt,32
sondern, indem er, vor einem Zug stehend, sagt: ’in diesem
Fall zieht man so‘“. – Aber, wenn dies33
so ist, so zeigt es doch nur, daß er unter gewissen
15 (O): Aspect
16 (M): ? /
17 (M): ? 3
18 (V): nicht die
19 (M): ? / 3
20 (V): gebraucht wird,
21 (V): Spiel. Denn es ist nun nicht so, daß jene
Tabelle ja doch im Hintergrund steht und man
immer auf sie zurückgreifen kann; sie
22 (V): auf sie @zurückgreife!,
23 (V): Eine Erklärung ist das Anlegen // die
Konstruktion // Anfertigung // das Anfertigen
einer Tabelle und sie die Erklärung wird
Geschichte, wenn
24 (M): / 3
25 (V): Handlungsweise als ihre Gesch Vorgeschichte
Geschichte
26 (V1): Geschichte. (V2): Geschichte und
(für uns) ohne Interesse.
27 (V): Hypothese. Es ist die
28 (V): die // uber dem //
29 (M): /// – aufmerksam werden).
30 (V): diesen
31 (V): ihr
32 (V): Fall gehandelt hat,
33 (V): das
177
176v
177
178
133 die einmalige Erklärung der Sprache . . .
WBTC040-45 30/05/2005 16:17 Page 266
to appear in his mind? None of that seems necessary, although it could happen. Yet the word
“blue” seems now to have a different aspect for him from what it had before it was explained
to him. It gains depth, as it were.
13
How does the ostensive explanation “That is ‘red’” help us to understand the word?
(It doesn’t “help” at all. It simply is one of the symbolic rules for the use of the word “red”.)
14
An explanation can’t work at a distance. I mean: it only works where it is applied. If
aside from that it has an “effect”, then that’s not because it was an explanation.
15
Is this the way it is: an explanation, e.g. a table, can16
be used at first by “looking something
up” in it; then one looks something up in it in one’s head, as it were, i.e. summons it
before the mind’s eye (or some such thing); and then finally one proceeds without this table,
i.e. as if it had never been there. So in this last case one is obviously playing a different game.
The table17
has been eliminated from our game, and if I fall back18
on it then I’m doing what
a person does who has gone blind and who falls back on, say, his sense of touch. An explanation
constructs a table, and when I no longer consult the table the explanation19
becomes
a thing of the past.
20
I need to distinguish between these cases: sometimes I am guided by a table and other
times I act in conformity with the table (with the rule expressed by the table) without consulting
it. – The rule – the learning of which caused us just now to act in such and such a
way – is, as the root cause of our behaviour, a matter of history,21
and it is of no interest to us.
In so far as the rule is a general description of the way we act, however, it is a hypothesis –
the22
hypothesis that these two people sitting at the chess board will act in such and such
a way 23
(in which case even a violation of the rules of the game falls within the range of
the hypothesis, for it then states something about how they’ll behave when they notice the24
violation). But the players can also use the rule by looking up what is to be done in each
particular case; here the rule enters into the moves of the game itself, and it doesn’t relate
to them25
as a hypothesis does to its confirmation. “But there is a problem here. For a player
who plays without using the list of rules, who indeed might never have seen such a list, could
nevertheless, when called upon, construct one. And he would do this, in my opinion, not –
behaviouristically – by ascertaining through repeated observations how he acts26
in this or
that case, but rather by saying, when he is about to make a move: ‘Here one moves this way’.”
– But if this27
is so it merely shows that he will utter a rule under certain circumstances, and
not that he has used it (explicitly) when making the move. It is a hypothesis that he will28
13 (M): ? /
14 (M): ? 3
15 (M): ? / 3
16 (V): will
17 (V): game. For it is not the case that the table
is in the background, and that one can always
fall back on it; it
18 (V): !fall backE
19 (V): An explanation is the creation // the construction
// the manufacturing // the manufacture
of a table, and when I no longer consult the table,
it the explanation
20 (M): / 3
21 (V): as the root cause of our behaviour, as its hist
prehistory history (V2): history, and without
interest (for us).
22 (V): hypothesis. It is the
23 (M): /// – violation).
24 (V): this
25 (V): it
26 (V): acted
27 (V): if that
28 (V): would
What Effect Does a Single Explanation of Language Have . . . 133e
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Umständen eine Regel aussprechen wird, nicht, daß er von ihr beim Zug (expliciten) Gebrauch
gemacht hat. Daß er ein Regelverzeichnis anlegen wird,34
wenn man es verlangt,35
ist eine
Hypothese und wenn man eine Disposition, ein Vermögen, ein Regelverzeichnis anzulegen
annimmt, so ist es eine psychische Disposition auf gleicher Stufe mit einer physiologischen.
Wenn gesagt wird, diese Disposition charakterisiert den Vorgang des Spiels, so charakterisiert
sie ihn als einen psychischen oder physiologischen, was er tatsächlich ist. (Im
Studium36
des Symbolismus gibt es keinen Vordergrund und Hintergrund, nicht ein greif-
bares37
Zeichen und ein es begleitendes ungreifbares38
Vermögen, oder Verständnis.)
39
Wie wirkt nun die hinweisende Erklärung? Sie lehrt den Gebrauch eines Zeichens;
und das Merkwürdige ist nur, daß sie ihn auch für die Fälle zu lehren scheint, in denen
ein Zurückgehen auf das hinweisende Zeichen nicht möglich ist. Aber geschieht das nicht,
indem wir, quasi, die in der hinweisenden Definition gelernte Regel40
in bestimmter Weise
transformieren? Ich mache von einer Zeichenerklärung Gebrauch41
für Transformationen, deren
Paradigma sie ist.42
Wenn43
z.B. der Mann, der mir vorgestellt wurde, abwesend ist und ich
nun trotzdem seinen Namen gebrauche,44
in wiefern mache ich da von der hinweisenden Erklärung
Gebrauch?45
Es gibt ein Spiel, worin ich immer statt des Namens das hinweisende Zeichen
geben kann, und eins, in welchem das nicht46
möglich ist. (Und wir müssen nur daran
festhalten, daß die Erklärung, als fortwirkende Ursache unseres Gebrauchs von Zeichen,
uns nicht interessiert, sondern nur, sofern wir von ihr in unserm Kalkül Gebrauch machen
können.) Es macht eine Schwierigkeit in der Erklärung des Gebrauchs der hinweisenden
Definition, daß wir47
verschiedene Kriterien der Identität anwenden (also das Wort &iden-
tisch“48
in verschiedener Weise gebrauchen), je nachdem, ob ein Ding sich vor unsern Augen
bewegt, oder unserm Blick entschwindet und vielleicht wieder erscheint. Das ist wichtig,
denn für den zweiten Fall gibt uns die hinweisende Definition49
ein Muster50
und tut,51
was
auch der Hinweis auf ein Bild tut. Denn die gegebene hinw. Erkl. nützt nichts, wenn52
wir vergessen
haben, wie der Mensch, auf den gezeigt wurde, aussah.
53
Es ist möglich, daß Einer die Bedeutung des Wortes ”blau“ vergißt. Was hat er da
vergessen? –54
Wie äußert sich das?
55
Da gibt es verschiedene Fälle.56
Er zeigt etwa auf verschieden gefärbte Täfelchen und sagt:
”ich weiß nicht mehr, welche von diesen man ’blau‘ nennt“. Oder aber, er weiß überhaupt
nicht mehr, was das Wort57
bedeutet, und nur, daß es ein Wort der deutschen Sprache ist.58
34 (V): würde,
35 (V): verlangte,
36 (V): Stadium
37 (V): sichtbares
38 (V): unsichtbares
39 (M): ? / /// (R): Zu § 13 S. 46
40 (V): Definition gelernten Regeln
41 (V): Ich gebrauche eine Zeichenerklärung
42 (V): sie mir gibt.
43 (O): (Wenn
44 (V): gebrauche, dessen Gebrauch mir durch
die Vorstellung – hinweisende Erklärung – erklärt
wurde.) Wenn ich ihn nun brauche,
45 (V): Gebrauch? Offenbar nicht in der Weise, in
welcher ich in der Anwesenheit des Menschen
von ihr Gebrauch machen konnte.
46 (V): nicht mehr
47 (V): Eine Schwierigkeit in der Erklärung des
Gebrauchs der hinweisenden Definition macht
es daß wir
48 (V): ”Identität“
49 (V): Definition eigentlich nur
50 (V): Muster
51 (V): tut nur,
52 (V): Bild tut. Das drückt sich darin aus, daß die
gegebene hinweisende Erklärung nichts nützt,
wenn
53 (M): ? / 3
54 (O): :
55 (M): ///
56 (V): Fälle:
57 (V): was es
58 (V): daß es ein deutsches Wort ist.
177v
178
179
179
134 die einmalige Erklärung der Sprache . . .
WBTC040-45 30/05/2005 16:17 Page 268
make up a list of rules if he’s asked to, and if one assumes that there’s a disposition here, a
capacity to set up a list of rules, then it is a psychological disposition, on the same level as
a physiological one. If it is said that this disposition characterizes the process of the game,
then it characterizes it as a psychological or physiological process, which in fact it is. (In the
study of symbolism there is no foreground and background, no tangible sign plus an intan-
gible29
capacity or understanding that accompanies it.)
30
So how does an ostensive explanation work? It teaches the use of a sign; and the odd
thing is that it seems to teach us to use it even in those cases where we can’t return to the
ostensive sign. But doesn’t this happen by our transforming, as it were, the rule31
we learned
in the ostensive definition in a particular way? I use an explanation of a sign for transformations
for which it is the paradigm.32
If for example a man to whom I was introduced is now absent,
but nevertheless I use his name,33
in what way am I using the ostensive explanation?34
There
is a game in which I can always substitute an ostensive sign for a name, and another where
that isn’t possible. (And we have to remain firm in maintaining that an explanation doesn’t
interest us as a continuous cause of our use of signs, but only in so far as we can use it in
our calculus.) A problem in explaining the use of an ostensive definition arises from the fact
that we apply different criteria of identity (i.e. use the word “identical”35
in different ways)
depending on whether an object moves in front of our eyes, or vanishes and maybe reap-
pears.36
This is important, for the ostensive definition furnishes37
us a sample38
for the latter
case, and it does39
what reference to a picture would do. For an ostensive explanation that we’re
given is useless if40
we have forgotten what the person who was pointed to looked like.
41
Someone can forget the meaning of the word “blue”. What has he forgotten? – How
does this become apparent?
42
Here we have different cases.43
He might point to colour chips of different colours and
say: “I no longer know which of these are called ‘blue’.” Or he no longer knows what the
word44
means at all, only that it’s a word in the English language.45
29 (V): no visible sign plus an invisible
30 (M): ? / /// (R): To § 13 p. 46
31 (V): rules
32 (V): I use an explanation of a sign for transformations,
whose paradigm the explanation gives me.
33 (V): I use his name, the use of which was
explained to me by the presentation – the
ostensive explanation. Now if I use it,
34 (V): explanation? Obviously not in the way I was
able to use it when the man was present.
35 (V): “identity”
36 (V): A problem in explaining the use of the
ostensive definition causes us to apply different
criteria of identity . . . reappears.
37 (V): really only furnishes
38 (V): a sample
39 (V): does only
40 (V): do. This is made clear by the fact that a
given ostensive definition is of no use if
41 (M): ? / 3
42 (M): ///
43 (V): cases:
44 (V): what it
45 (V): it’s an English word
What Effect Does a Single Explanation of Language Have . . . 134e
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59
Wenn wir ihn nun fragen: ”weißt Du, was das Wort ’blau‘ bedeutet“, und er sagt ”ja“;
da konnte er verschiedene Kriterien anwenden, um sich ”zu überzeugen“, daß er die Bedeutung
wisse. (Denken wir wieder an die entsprechenden Kriterien dafür, daß er das Alphabet
hersagen kann.) Vielleicht rief er sich ein blaues Vorstellungsbild vor die Seele, vielleicht
sah er nach einem blauen Gegenstand im Zimmer, vielleicht fiel ihm das englische Wort
”blue“ ein, oder er dachte an einen ”blauen Fleck“, den er sich geholt hatte, etc., etc.
Wenn nun gefragt würde: wie kann er sich denn zur Probe seines Verständnisses ein blaues
Vorstellungsbild vor die Seele rufen, denn, wie kann ihm das Wort ”blau“ zeigen, welche
Farbe aus dem Farbenkasten seiner Vorstellung er zu wählen hat, – so ist zu sagen, daß
es sich da60
eben zeigt,61
daß das Bild vom Wählen, etwa, eines blauen Gegenstands mittels
eines blauen Mustertäfelchens hier ungeeignet62
ist. Und der Vorgang eher mit dem zu
vergleichen ist, wenn beim Drücken eines Knopfes, auf dem das Wort ”blau“ geschrieben
steht, automatisch ein blaues Täfelchen vorspringt, oder, wenn der Mechanismus versagt,
nicht vorspringt.
Man könnte nun sagen: Der, welcher die Bedeutung des Wortes ”blau“ vergessen hat
und aufgefordert wurde, einen blauen Gegenstand aus anderen auszuwählen, fühlt beim
Ansehen dieser Gegenstände, daß die Verbindung zwischen dem Wort ”blau“ und jenen
Farben nicht mehr besteht (unterbrochen ist). Und die Verbindung wird wieder hergestellt,
wenn wir ihm die Erklärung des Wortes wiederholen. Aber wir konnten die Verbindung
auf mannigfache Weise wieder herstellen: Wir konnten63
einen blauen Gegenstand zeigen
und die hinweisende Definition geben, oder ihm sagen ”erinnere Dich an Deinen ’blauen
Fleck‘“, oder wir konnten ihm das Wort ”blue“ zuflüstern, etc. etc. Und wenn ich sagte,
wir konnten die Verbindung auf diese verschiedenen Arten herstellen, so liegt nun der
Gedanke nahe, daß ich ein bestimmtes Phänomen, welches ich die Verbindung zwischen
Wort und Farbe, oder das Verständnis des Wortes nenne, auf alle diese verschiedenen Arten
hervorgerufen habe; wie ich etwa sage, daß ich die Enden zweier Drähte durch Drahtstücke
verschiedener Länge und Materialien leitend miteinander verbinden kann. 64
Aber von so
einem Phänomen, etwa dem Entstehen eines blauen Vorstellungsbildes, muß keine Rede
sein und das Verständnis wird sich dann dadurch zeigen, daß er etwa die blaue Kugel aus
den andern tatsächlich auswählt, oder sagt, er könne es nun tun, wolle es aber nicht; etc.,
etc. etc. Wir können dann immer ein Spiel festsetzen, welches eine Möglichkeit so eines
Vorgangs darstellt, 65
und müssen nicht vergessen, daß in Wirklichkeit hundert verschiedene
und ihre Kreuzungen mit den Worten ”die Bedeutung vergessen“, ”sich an die Bedeutung
erinnern“, ”die Bedeutung kennen“ beschrieben werden.66
181
135 die einmalige Erklärung der Sprache . . .
59 (M): ///
60 (V): so
61 (V): daß es sich ebenso so zeigt,
62 (V): unpassend
63 (V): konnten ihm
64 (M): ///
65 (M): /
66 (R): Siehe auch Notizbuch
180
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46
Now if we ask him: “Do you know what the word ‘blue’ means?” and he says “Yes”,
then he could apply various criteria in order to “satisfy” himself that he knows the meaning.
(Let’s think again of the respective criteria for his being able to recite the alphabet.)
Maybe he called to mind a mental image of blue, maybe he looked at a blue object in his
room, maybe the German word “blau” came to mind, or he thought of a “blue bruise” that
he had suffered, etc., etc.
Now suppose it were asked, how can he call to mind a mental image of blue as a test of
his understanding? – For how can the word “blue” show him which colour he’s to choose
from the paint set in his imagination? – The proper answer to this is: it is precisely this
problem that shows how unsuitable47
the picture is of choosing, say, a blue object with the
help of a blue sample colour chip. And that the process is better compared to what occurs
when, upon pressing a button with the word “blue” on it, a blue colour chip automatically
pops up, or doesn’t pop up if the mechanism fails.
Now one could say: Someone who has forgotten the meaning of the word “blue” and is
asked to choose a blue object from among others, feels, when looking at these objects,
that the connection between the word “blue” and those colours no longer exists (has been
severed). And the connection is reestablished when we repeat the explanation of the word
to him. But we’re able to reestablish the connection in a multitude of ways: We could show
him a blue object and give him the ostensive definition, or say to him “Remember your ‘blue
bruise’”, or we could whisper the word “blau” in his ear, etc., etc. And when I say that we
could establish the connection in these different ways then it’s tempting to think that I have
elicited one particular phenomenon – which I am calling the connection between word and
colour or the comprehension of the word – in all of these different ways; just as I might say
that I can make an electrical connection between the ends of two wires using pieces of wire
of different lengths and materials. 48
But there need be no talk about such a phenomenon as
the formation of a mental image of blue, and his understanding will be shown by his actually
choosing the blue ball from among the others, or by his saying that although he could
do this he didn’t want to; etc., etc., etc. We can always establish a game that represents one
possibility for such a process, 49
and we shouldn’t forget that in reality hundreds of different
possibilities of this kind can be described, along with their intersections with the words “to
forget the meaning”, “to recall the meaning”, “to know the meaning”.50
46 (M): ///
47 (V): inappropriate
48 (M): ///
49 (M): /
50 (R): Also see notebook
What Effect Does a Single Explanation of Language Have . . . 135e
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182
183
182v
182v
183
42
Kann man etwas Rotes nach dem
Wort ”rot“ suchen? Braucht man ein
Bild, ein Erinnerungsbild, dazu?
Verschiedene Suchspiele.
1
Man könnte eine wesentliche Frage auch so stellen: Wenn ich jemandem sage ”male diesen
Kreis rot“, wie entnimmt er aus dem Wort ”rot“, welche Farbe er zu nehmen hat?
2
Heißt es etwas, zu sagen, daß das Wort ”rot“, um ein brauchbares Zeichen zu sein, ein
Supplement – etwa im Gedächtnis – braucht?
D.h., inwiefern ist es allein nicht Zeichen?
3
Wenn ich eine Erfahrung mit den Worten beschreibe ”vor mir steht ein blauer Kessel“,
ist die Rechtfertigung dieser Worte, außer der Erfahrung die in den Worten beschrieben
wird, noch eine andere, etwa die Erinnerung, daß ich das Wort ”blau“ immer für diese Farbe
verwendet habe, etc.?
4
Wenn ich jemandem sage ”wenn ich läute, komm’ zu mir“, so wird er zuerst, wenn er
läuten hört, sich diesen Befehl (das Läuten) in Worte übersetzen und erst den übersetzten
befolgen. Nach einiger Zeit aber wird er das Läuten ohne Intervention anderer Zeichen in
die Handlung übersetzen.
Und so, wenn ich sage ”zeige auf einen roten Fleck“, befolgt er diesen Befehl, ohne daß
ihm dabei zuerst das Phantasiebild eines roten Flecks als Zeichen für ”rot“ erscheint.
Der Witz muß sein, daß die Erinnerung (wie das Wissen) dem verglichen wird, was irgendwo
aufgeschrieben steht.
5
Wenn er läutet, so komme ich zu ihm, ohne mir erst ein Bild meiner Bewegungen
vorzustellen, wonach ich (dann) handle.
Dies bezieht sich auf den Fall vom Läuten, damit jemand kommt & die Weise wie er die
Bedeutung dieses Zeichens lernt.
Ich glaube die Frage war: Muß er, wenn er sich das Läuten nicht in eine Erklärung6
übersetzt, sich
nicht nach der Erinnerung an die letzte Befolgung des Befehls richten?
1 (M): / 3 (R): ∀ S. 23/5
2 (M): / 3
3 (M): / 3
4 (M): ? / ////
5 (M): /
6 (O): nicht eine Erklärung (V): nicht in Worte
136
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42
Can One Use the Word “Red” to
Search for Something Red? Does One
Need an Image, a Memory-Image,
for This? Various Searching-Games.
1
An important question could be put this way: If I tell someone “Paint this circle red”
how does he infer from the word “red” which colour he’s supposed to pick?
2
Does it mean anything to say that the word “red” needs a supplement – perhaps in one’s
memory – in order to be a usable sign?
That is, in what respect isn’t it a sign by itself?
3
If I use the words “There’s a blue vat in front of me” to describe an experience, is the
justification for these words something in addition to the experience they describe, for instance
the memory that I’ve always used the word “blue” for this colour, etc.?
4
If I tell someone “When I ring, come”, then when he hears the ring, at first he’ll translate
this command (the ring) into words, and he won’t obey it until it’s been translated. After
a while, though, he’ll translate the ringing into action without the intervention of other signs.
And thus when I say “Point to a red patch”, he’ll obey this order without first having an
imaginary picture of a red patch appear to him as a sign for “red”.
The joke is always played when memory (like knowledge) gets compared to what is written down
somewhere.
5
If he rings, I come to him without first imagining a picture of my movements, which I
(then) translate into action.
This refers to the case of ringing to get someone to come, and the way he learns the meaning of
this sign.
I believe the question was: If he doesn’t translate the ringing into an explanation6
, doesn’t he have
to be guided by his memory of the last time he obeyed the order?
1 (M): / 3 (R): ∀ p. 23/5
2 (M): / 3
3 (M): / 3
4 (M): ? / ////
5 (M): /
6 (V): ringing into words
136e
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7
Muß er sich daran erinnern, wie er den Befehl gestern befolgt hat? Ich kann auch den ausdrückl.
Befehl geben:8
”Tu jetzt, was Du, Deiner Erinnerung nach, heute vor einem Jahr9
getan hast“.
Und wenn er sich daran erinnert, kann er seiner Erinnerung folgen. Erinnert er sich aber
nicht, so hat der Befehl keinen Sinn für ihn. [D.h. die Erinnerung wirkt automatisch.]
Und wie weiß er dann, was die Worte dieses Befehls (Tu was Du Deiner Erinnerung nach . . . ) von
ihm wollen – wenn wir annehmen es sei immer ein Erinnerungsbild das den Worten ihre Bedeutung
gibt.
[Lösung ↓]
10
Dieser Bef. ist also ähnlich wie der:11
”Tu, was auf dem Zettel in dieser Lade
aufgeschrieben steht“. Wenn in der Lade kein Zettel ist, so ist das kein Befehl. (Oder denken
wir uns, daß auf dem Zettel eine sinnlose12
Wortverbindung steht etwa:13
&Kaufe n Pflaumen
& n2
+ 2n + 2 = 0“.)
14
Wenn ich jemandem sage ”male das Grün Deiner Zimmertür nach dem Gedächtnis“,
so bestimmt das, was er zu tun hat, nicht eindeutiger, als der Befehl ”male das Grün, was
Du auf dieser Tafel siehst“. Denn er wird auch im zweiten Fall für gewöhnlich nicht nach der
Projectionsmethode fragen.
15
Wenn es bei der Bedeutung des Wortes ”rot“ auf das Bild ankommt, das mein
Gedächtnis beim Klang dieses Wortes automatisch reproduziert, so muß ich mich auf
diese Reproduktion gerade so verlassen, als wäre ich entschlossen, die Bedeutung durch
Nachschlagen in einem Buche zu bestimmen, wobei ich mich diesem Buche, dem
Täfelchen, das ich darin fände, quasi auf Gnade und Ungnade ergeben würde.
16
Ich bin dem Gedächtnis ausgeliefert.
17
Freilich kann man sagen: das rote Täfelchen ist in Wirklichkeit auch nicht maßgebend,
weil das Gedächtnis immer als Kontrolle des Täfelchens verwendet wird.
18
Die Frage aber ist: Ist im Falle einer relativen Veränderung der Farbe des Täfelchens
zu meinem Gedächtnis (ein gewagter Ausdruck) in irgend einem Sinne unbedingt der Deutung
der Vorzug zu geben, das Täfelchen habe sich geändert und ich müsse mich also nach
dem Gedächtnis richten? Offenbar nein. Übrigens besagt die ”Deutung“, das Täfelchen und
nicht das Gedächtnisbild habe sich verändert, nichts als eine Worterklärung der Wörter
”verändern“ und ”gleichbleiben“.
19
Könnte ich behaupten, daß mein Gedächtnis immer etwas nachdunkle?
Jedenfalls könnte ich sagen: ”wähle die Farbe, die Du im Gedächtnis hast“ und auch ”wähle
eine etwas dunklere Farbe, als die Du im Gedächtnis hast“. Von einem Nachdunkeln kann
man natürlich nur im Vergleich zu etwas andrem20
sprechen und es genügt nicht, zu sagen
”nun, mit der Farbe, wie sie wirklich war“, weil hier die besondere Art der Verifikation,
d.h., die (besondere) Grammatik der Worte ”wie sie war“ noch nicht festgelegt ist, diese
Worte (also) noch mehrdeutig sind.
182v
183
184
137 Kann man etwas Rotes nach dem Wort ”rot“ suchen? . . .
7 (M): / 3
8 (V): Ich kann gewiß sagen:
9 (V): nach, gestern um diese Zeit
10 (M): / 3 ///
11 (V): Wäre dieser Befehl also wie der:
12 (V): unsinnige
13 (O): steht. oder etwa:
14 (M): ? / 3 ///
15 (M): ? / 3
16 (M): ? /
17 (M): /// 3
18 (M): / 3
19 (M): / 3
20 (V): zu Etwas
183
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7
Does he have to remember how he obeyed the order yesterday? I can also give the explicit order8
:
“Do now what you remember doing a year ago today9
”. And if he remembers, he can follow
his memory. But if he doesn’t, the command makes no sense to him. (That is, memory works
automatically.)
And how does he know what the words of this order (“Do what you remember . . .”) ask of him
– if we assume that it is always a memory-image that gives words their meaning?
[Solution ↓]
10
So that order is similar to this:11
“Do what is written on the piece of paper in this drawer”.
If there is no piece of paper in the drawer then that’s not a command. (Or let’s imagine that
there is a senseless12
combination of words on the piece of paper, say: “Buy n plums and
n2
+ 2n + 2 = 0”.)
13
If I tell someone “Paint the green of the door to your room from memory” then this
doesn’t specify what he is supposed to do any more unequivocally than the command “Paint
the green that you see on this colour chart”. For in the latter case too he will usually not ask about
the method of projection.
14
If determining the meaning of the word “red” requires the image that my memory automatically
reproduces at the sound of that word, then I have to rely on this reproduction in
just the same way as if I had decided to determine the meaning by looking it up in a book
– in which case I’d surrender myself to the book, to the colour chip I’d find in it, for
better or worse.
15
I am at the mercy of my memory.
16
To be sure, one can say: Really the red colour chip isn’t decisive either, because memory
is always used as a check on it.
17
But the question is: Supposing the colour of the chip changes relative to my memory (a
risky expression), do I in some sense have to prefer the interpretation that it is the colour
chip that has changed, and that therefore I have to be guided by my memory? Obviously
not. Incidentally, the “interpretation” that it is the chip and not my memory-image that has
changed is nothing more than a definition of the words “change” and “remain the same”.
18
Could I claim that my memory always darkens things a bit?
In any case I could say: “Choose the colour that’s in your memory”, but also “Choose a
somewhat darker colour than the one that’s in your memory”. Of course one can only talk
about a darkening in comparison to something else19
, and it is not enough to say “Well,
in comparison to the colour as it really was”, because that particular type of verification,
i.e. the (particular) grammar of the words “as it was”, hasn’t yet been established here, and
(therefore) these words are still ambiguous.
7 (M): / 3
8 (V): I can certainly say
9 (V): doing this time yesterday
10 (M): / 3 ///
11 (V): So would that order be like this:
12 (V): nonsensical
13 (M): ? / 3 ///
14 (M): ? / 3
15 (M): ? /
16 (M): /// 3
17 (M): / 3
18 (M): / 3
19 (V): to something
Can One Use the Word “Red” to Search for Something Red? . . . 137e
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21
Mit einem Draht nach einem Kurzschluß suchen: er ist gefunden, wenn es läutet. Aber
suche ich dabei auch nach etwas, was der Idee des Klingelns gleich ist?
22
Der Befehl sei: ”Stelle Dir einen roten Kreis vor“. Und ich tue es. Wie konnte ich den
Worten auf diese Weise folgen?
Das ist doch ein Beweis23
dafür, daß wir den Worten auch ohne Vorstellungen gehorchen
können.
24
Wie kann ich es rechtfertigen, daß ich mir auf diese Worte hin diese Vorstellung mache?
Was heißt denn hier &diese Vorstellung“? Kann ich denn auf sie zeigen? Dies hängt unmittelbar
mit25
dem Problem zusammen &ob & woher ich26
denn wissen kann ob & was der Andre fühlt,
sieht etc.“.
27
Hat mir jemand die Vorstellung der blauen Farbe gezeigt und gesagt, daß sie das ist?
28
Es ist also richtig: ”Ich erinnere mich daran“, an das, was ich hier vor mir sehe. Das
Bild ist dann in einem gewissen Sinne sowohl gegenwärtig und vergangen.
29
Der Vorgang des Vergleiches eines Bildes mit der Wirklichkeit ist also der Erinnerung
nicht wesentlich.
30
Es ist instruktiv zu denken, daß, wenn wir mit einem gelben Täfelchen die Blume suchen,
uns jedenfalls nicht die Relation der Farbengleichheit in einem weiteren Bild gegenwärtig
ist. Sondern wir sind mit dem einen ganz zufrieden.
31
(So wie wir nicht für einen Augenblick daran dächten, ein Kind die Gebärdensprache
zu lehren.)
32
Ich kann die Bedeutung der Zeichen , , 33
durch die Tabelle
erklären; aber diese Tabelle wieder erklären, indem ich sie so schreibe
und sie einer anderen entgegenstelle:
Aber konnte denn auch die erste Erklärung wegbleiben? Gewiß, wenn die Zeichen ,
, , uns (etwa) ursprünglich ebenso beigebracht worden wären, wie die Wörter
”Kirche“, ”Haus“, ”Stadt“. Aber diese mußten uns doch erklärt werden! – Soweit sie uns
überhaupt ”erklärt“ wurden, geschah es durch eine Gebärdensprache, die uns nicht erklärt
Kirche
Haus
Stadt
Kirche
Haus
Stadt
Kirche
Haus
Stadt
21 (M): / 3
22 (M): / 3 ////
23 (V): Zeichen
24 (M): / 3
25 (V): hängt mit
26 (V): &ob ich
27 (M): // (R): [Zu § 77 S. 357]
28 (M): ∫ 3 ///
29 (M): / 3
30 (M): /// 3
31 (M): /// 3
32 (M): ///
33 (F): MS 110, S. 278–279.
185
186
138 Kann man etwas Rotes nach dem Wort ”rot“ suchen? . . .
WBTC040-45 30/05/2005 16:17 Page 276
20
Searching for a short circuit with a wire: it’s found when it rings. But in doing this, am
I also searching for something that’s just like the idea of ringing?
21
Let the order be: “Imagine a red circle”. And I do it. How was I able to follow those
words in this way?
Surely that is proof22
that we can follow words even without mental images.
23
How can I justify the fact that in response to these words I conjure up this mental image
for myself?
What does “this mental image” mean here anyway? Can I point to it? This is directly connected24
with the problem “whether and how I25
can know whether and what someone else feels, sees, etc.”.
26
Did someone show me the mental image of the colour blue and tell me it was that?
27
So it is correct to say: “I remember this” – what I see here in front of me. Therefore in
a certain sense the image is both present and past.
28
Therefore the process of comparing an image with reality is not essential to memory.
29
It is instructive to realize that when we look for a flower using a yellow colour chip, the
relationship of sameness of colour is not present in an additional image. Rather, we are quite
content with the one.
30
(Just as we wouldn’t for a moment think of teaching a child a language of gestures.)
31
I can explain the meaning of the signs , , ,32
using the table
but I can explain this table in turn by writing it like this
and contrasting it with another one:
But could the first explanation be omitted as well? Certainly, if (say) we had originally
been taught the signs , , , in the same way as the words “church”, “house”, “city”.
But these had to be explained to us! – In so far as they were “explained” to us at all, this
Church
House
City
Church
House
City
Church
House
City
20 (M): / 3
21 (M): / 3 ////
22 (V): is a sign
23 (M): / 3
24 (V): is connected
25 (V): “whether I
26 (M): / / (R): [To § 77 p. 357]
27 (M): ∫ 3 ///
28 (M): / 3
29 (M): /// 3
30 (M): /// 3
31 (M): ////
32 (F): MS 110, pp. 278–279.
Can One Use the Word “Red” to Search for Something Red? . . . 138e
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wurde. – 34
Aber wäre denn diese Gebärdensprache einer Erklärung fähig gewesen? – Gewiß;
z.B. durch eine Wortsprache.
35
Denken wir an das laute Lesen nach der Schrift (oder das Schreiben nach dem Gehör).
Wir könnten uns natürlich eine Art Tabelle denken, nach der wir uns dabei richten könnten.
Aber wir richten uns nach keiner. Kein Akt des Gedächtnisses, nichts, vermittelt zwischen
dem geschriebenen Zeichen und dem Laut.
36
(Das Wort ”rot“ ist ein Stein in einem Kalkül und das rote Täfelchen ist auch einer.)
37
Es ist ein anderes Spiel, mit einem Täfelchen herumgehen, es an die Gegenstände
anzulegen und so die Farbengleichheit zu prüfen; und anderseits: ohne ein solches Muster
nach Wörtern in einer Wortsprache handeln.
Man denkt nun: ”Ja, das erste Spiel verstehe ich; das ist ja ganz einfach: Der erste Schritt
ist der, von einem geschriebenen Wort auf das gleiche geschriebene Wort des Musters; der
zweite ist der Übergang von dem Wort auf dem Mustertäfelchen zu der Farbe auf dem
gleichen Täfelchen; und der dritte, das Vergleichen von Farben. Jeden Schritt dieses Kalküls
gehen wir also auf einer Brücke. (Wir sind geführt, der Schritt ist vorgezeichnet.)“
38
Aber wir sind doch hier nur insofern39
geführt, als wir uns so führen lassen. Auf diese
Weise kann40
ich alles, und muß ich nichts eine Führung nennen. – Und am Schluß tu ich,
was ich tue und das ist Alles.
Man möchte Gründe & Gründe & Gründe angeben. In dem Gefühl: solange ein Grund da ist, ist
alles richtig.41
Wir möchten42
nicht aufhören zu erklären; & nicht einfach beschreiben. Wie kann denn
das interessant sein, was eben geschieht,43
uns interessiert doch nur44
immer die Rechtfertigung das
Warum! Das ist doch nicht Mathematik zu sagen was die Menschen tun.
45
Aber ein Unterschied bleibt doch: Wenn ich gefragt werde ”warum nennst Du gerade
diese Farbe ’rot‘“, so würde ich tatsächlich antworten: weil sie auf dem gleichen Täfelchen
mit dem Wort ”rot“ steht. Würde ich aber in dem zweiten Spiel gefragt ”warum nennst Du
diese Farbe ’rot‘“, so gäbe es darauf keine Antwort und die Frage hätte keinen Sinn. – Aber
im ersten Spiel hat die46
Frage keinen Sinn: ”warum nennst Du die Farbe ’rot‘, die auf dem
gleichen Täfelchen mit dem Wort ’rot‘ steht“. So handle ich eben (und man kann dafür
wohl eine Ursache angeben, aber keinen Grund). Das Gedächtnis ist jedenfalls nicht immer die
letzte Instanz.
47
Bedenke vor allem: Wie weiß man, daß das Täfelchen rot bleibt? Braucht man dazu wieder
ein Bild? Und wie ist es mit dem? etc. Woran erkennt er das Vorbild als Vorbild?
&Eine Häuserreihe48
ist eigentlich unendlich, denn man könnte immer noch weitere Häuser
bauen.“
49
(Ein Grund läßt sich nur innerhalb eines Spiels angeben.)
50
Die Kette der Gründe kommt zu einem Ende und zwar dem Ende in diesem Spiel.51
187
139 Kann man etwas Rotes nach dem Wort ”rot“ suchen? . . .
34 (M): /
35 (M): / 3
36 (M): 555
37 (M): / 3
38 (M): ∫ / 555
39 (V): insofern so
40 (V): Weise ka kann
41 (V1): alles in Ordnung. (V2): Gefühl: wo ein
Grund ist, ist alles in Ordnung.
42 (V): Man möchte
43 (V): was geschieht,
44 (V): geschieht, wir wollen doch nur
45 (M): ///
46 (V): die
47 (M): /
48 (O): Hauserreihe
49 (M): ? / 3
50 (M): ∫ 3 ///
51 (V): und zwar (an) der Grenze des Spiels.
WBTC040-45 30/05/2005 16:17 Page 278
took place via a language of gestures, which wasn’t explained to us. – 33
But could this
gesture-language have been explained? – Certainly; with a word-language, for instance.
34
Let’s think about reading aloud something that’s written (or about writing something as
we hear it). Of course we could imagine a kind of table to guide ourselves with. But we’re
not guided by any. No act of memory, nothing, mediates between the written sign and the
sound.
35
(The word “red” is a building stone in a calculus, as is the red colour chip.)
36
It’s one kind of game: to walk around with a colour chip and check on the sameness of
the colour of objects by putting it next to them; and it’s another kind: to act in accordance
with the words of a word-language without such a sample.
Now we think: “Fine, I understand the first game; that’s quite simple: The first step
is the one from a written word to the same written word on the sample; the second is the
transition from the word on the sample colour chip to the colour on that same chip; and the
third is comparing the colours. So it is that we take each step of this calculus across a bridge.
(We are guided, our step has been marked out ahead of time.)”
37
But here we’re really only being guided in so far as we allow ourselves to be so guided.
In this respect I can call everything and don’t have to call anything “guidance”. – And in
the end I do what I do and that is all.
We would like to give reason after reason after reason. Because we feel: so long as there is a
reason, everything is all right.38
We don’t39
want to stop explaining – and simply describe. How can
what is happening right now40
be interesting? All that we’re ever interested in41
is the justification,
the why! It isn’t mathematics, after all, to say what people do.
42
But there’s still a difference: If I am asked “Why do you call this particular colour ‘red’?”
then in fact I’d answer: Because it’s on the same colour chip that has the word “red” on it.
But if in the second game I were asked “Why do you call this colour ‘red’?”, there would
be no answer, and the question would make no sense. – But in the first game this question43
makes no sense: “Why do you call that colour ‘red’ that’s on the same colour chip with the
word ‘red’ on it?” That’s simply the way I act (and one can cite a cause for this, but not a
reason). In any case, memory is not always the final authority.
44
Above all, consider this: How does one know that the colour chip remains red? Does
this once again require a picture? And what is it like? Etc. How does he recognize the model
as a model?
“Strictly speaking, a row of houses is infinite because one could always build additional houses.”
45
(A reason can be given only within a game.)
46
The chain of reasons comes to an end – to its end within this game.47
33 (M): /
34 (M): / 3
35 (M): 555
36 (M): / 3
37 (M): ∫ / 555
38 (V1): feel: so long as there is a reason, everything is
in order. (V2): feel: where there is a reason,
everything is in order.
39 (V): One doesn’t
40 (V): what is happening
41 (V): interesting? All that we ever want
42 (M): ///
43 (V): this question
44 (M): /
45 (M): ? / 3
46 (M): ∫ 3 ///
47 (V): comes to an end – and it does so (at) the
game’s limits.
Can One Use the Word “Red” to Search for Something Red? . . . 139e
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Und, wenn man sich daran erinnert,52
&daß die Tabelle uns nicht zwingt“, sie so & so zu
benützen, noch, sie immer auf die gleiche Weise zu benützen, so wird es Jedem klar,53
daß unser
Gebrauch des Wortes54
&Regel“ & des Wortes &Spiel“ ein schwankender ist. (Nach den Rändern zu
verschwimmt.)
55
Man kann sagen: Die Regeln des Spiels sind die, die gelehrt werden, wenn das Spiel
gelehrt wird. – Nun wird z.B. dem Menschen, der lesen lernt, tatsächlich gelehrt: das ist
ein a, das ein e, etc.; also, könnte man sagen, gehören diese Regeln, gehört diese Tabelle mit
zum Spiel. – Aber erstens: lehrt man denn auch den Gebrauch dieser Tabelle? und könnte
man ihn, anderseits, nicht lehren? Und zweitens kann doch das Spiel wirklich auf zwei
verschiedene Arten gespielt werden.
Man kann nun fragen: ist es denn aber auch noch ein Spiel, wenn Einer die Buchstaben
abbc sieht und irgend etwas macht? Und wo hört das Spiel auf, und wo fängt es an?
Die Antwort ist natürlich: Spiel ist es, wenn es nach einer Regel vor sich geht. Aber was
ist noch eine Regel und was keine mehr?
Eine Regel kann ich nicht anders geben, als durch ihren Ausdruck; denn auch Beispiele,
wenn sie Beispiele sein sollen, sind ein Ausdruck für die Regel, wie jeder andre.
Wenn ich also sage: Spiel nenne ich es nur, wenn es einer Regel gemäß geschieht und die
Regel ist eine Tabelle, so kann ich nicht die Art des Gebrauches56
dieser Tabelle garantieren,
denn ich kann sie nur durch eine weitere Tabelle festlegen, oder durch Beispiele. Diese
Beispiele tragen nicht weiter, als sie selbst reichen57
und die zweite Tabelle ist im gleichen
Fall wie die erste.
Ich könnte auch sagen: was ist das Schachspiel andres (oder was ist vom Schachspiel andres
vorhanden), als Regelverzeichnisse (gesprochen, geschrieben, etc.) und die Beschreibung einer
Anzahl von Schachpartien?
Es steht mir (danach) natürlich frei, ”Spielregel“ nur ein Ding von bestimmt festgelegter
Form zu nennen.
52 (V): sich in die Erinnerung ruft,
53 (V): wird es ganz klar,
54 (V): Wortes (SielA
55 (M): ? / 3 ////
56 (V): die Verwendungsart
57 (V): gehen
186v
187
188
140 Kann man etwas Rotes nach dem Wort ”rot“ suchen? . . .
WBTC040-45 30/05/2005 16:17 Page 280
And if we remember48
“that the table doesn’t force us” to use it this or that way, nor to use it in
the same way every time, then it becomes clear to everybody49
that our use of the word “rule” and
the word “game” vacillates. (Blurs as it approaches the edges.)
50
One can say: The rules of the game are those that are taught when the game is taught.
– Now a person learning to read, for instance, is actually taught: This is an a, that an e, etc.;
so, one could say, these rules, this table, also belong to the game. – But first of all: Does
anyone ever really teach the use of this table? But isn’t it possible to teach it? And second:
the game really can be played in two different ways.
Now one can ask: But is it really still a game if someone sees the letters abbc and does
just any old thing? And where does the game end and where does it begin?
Of course the answer is: It’s a game if it follows a rule. But what is still a rule, and what
no longer one?
I cannot produce a rule in any other way than through its expression; for even examples,
if they are to be examples, are an expression of the rule just like any other expression.
So if I say: “I call it a game only if it is played according to a rule”, and if the rule is a
table, then I can’t guarantee the way the table is used. I can only establish this by way of a
further table or by way of examples. These examples don’t carry me any further than they
themselves extend,51
and the second table is in the same situation as the first.
I could also say: What is chess (or what do we have in chess) other than lists of rules
(spoken, written, etc.) and the description of a number of chess games?
(Having said that) I am free, of course, to call only something that has a specifically established
form a “rule of the game”.
48 (V): we call to mind
49 (V): it becomes completely clear
50 (M): ? / 3 ////
51 (V): go,
Can One Use the Word “Red” to Search for Something Red? . . . 140e
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43
”Die Verbindung1
zwischen Sprache
und Wirklichkeit“ ist durch die
Worterklärungen gemacht,2
welche
wieder zur Sprachlehre gehören.
So daß die Sprache in sich
geschlossen, autonom, bleibt.
3
Übereinstimmung von Gedanke und Wirklichkeit. Wie alles Metaphysische ist die
(prästabilierte) Harmonie zwischen Gedanken und Wirklichkeit in der Grammatik der
Sprache aufzufinden.
4
Was macht uns glauben daß so etwas wie eine Übereinstimmung des Gedankens5
mit der
Wirklichkeit besteht? – Statt6
Übereinstimmung könnte man hier ruhig7
setzen:8
&Bildhaftigkeit“.
Ist aber die Bildhaftigkeit eine Übereinstimmung? In der &Log. phil. Abh.“ habe9
ich so etwas gesagt,
wie: sie sei10
eine Übereinstimmung der Form. Das ist aber eine Irreführung.11
12
Alles kann ein Bild von allem sein: wenn wir den Begriff des Bildes entsprechend ausdehnen. Und
sonst müssen wir13
sagen, was wir noch ein Bild von etwas nennen wollen & damit auch was14
wir
noch die Übereinstimmung der Bildhaftigkeit, die Übereinstimmung der Formen nennen wollen.
Denn was ich sagte kommt ja eigentlich darauf hinaus: jede Projektion, nach welcher Methode
immer, müsse etwas mit dem Projizierten gemeinsam haben.15
Aber das sagt nur, daß ich16
den Begriff
1 (V): Beziehung
2 (V): hergestellt,
3 (M): ? / 3 (R): [Zu § 21 S. 76 83]
4 (M): / (R): [Zu § 21 S. 83]
5 (V): übereinstimmung von den Gedanken
6 (V): besteht? – Die Ub Statt
7 (V): man ruhig
8 (V): sagen:
9 (V): In der &Abhandlung“ hätte
10 (V): ist
11 (V): aber irreführend.
12 (M): überlege
13 (V): wir eben
14 (V): sagen, was wir ein Bild von etwas nennen & auch
was
15 (V1): nach welcher Methode immer, habe etwas
mit dem Projizierten gemeinsam. (V2): darauf
hinaus: zu sagen daß jede Projektion, nach welcher
Methode immer, etwas mit dem Projizierten gemeinsam
haben muß.
16 (V): ich hier
189
188v
141
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43
“The Connection1
between Language
and Reality” is Made2
Through
Explanations of Words, which
Explanations Belong in Turn to
Grammar. So that Language Remains
Self-contained, Autonomous.
3
Agreement of thought and reality. Like everything metaphysical the (pre-established)
harmony between thoughts and reality is to be discovered in the grammar of language.
4
What makes us think that there is something like an agreement of thought5
and reality? – Here
one could6
safely put7
“pictoriality” instead of agreement.
But is pictoriality an agreement? In Tractatus Logico-Philosophicus I said8
something like: pictoriality
is an agreement of form. But that is misleading.
9
Anything can be a picture of anything – if we expand the concept of picture enough. Otherwise
we have to say what it is that we still want to call10
a picture of something, and hence too what we
want to call “agreement of pictoriality”, “agreement of forms”.
For what I said really boils down to this: any projection, regardless of its method, has to have
something in common with what is projected.11
But that only means that12
I am expanding
1 (V): Relation
2 (V): Established
3 (M): ? / 3 (R): [To § 21 p. 76 83]
4 (M): / (R): [To § 21 p. 83]
5 (V): of thoughts
6 (V): One could
7 (V): safely say
8 (V): In the Tractatus I had said
9 (M): Think about
10 (V): that we call
11 (V1): regardless of its method, has something in
common with what is projected. (V2): boils
down to this: to say that any projection, regardless of
its method, has to have something in common with
what is projected.
12 (V): that here
141e
WBTC040-45 30/05/2005 16:17 Page 283
des &gemeinsam-habens“ ausdehne & ihn dem allgemeinen Begriff des Projizierens äquivalent
mache.17
Es drängt sich mir also eine bestimmte Form18
der Verallg. auf, eine Form der Darstellung, ein
bestimmter Aspekt.19
20
Es ist21
auch unrichtig22
zu sagen, die Übereinstimmung (und Nichtübereinstimmung)
zwischen Satz und Realität23
sei willkürlich durch eine Zuordnung geschaffen. Denn, wie
ist die Zuordnung auszudrücken? Sie besteht darin, daß der Satz ”p“ sagt, es sei gerade das
der Fall. Aber wie ist dieses ”gerade das“ ausgedrückt?24
Wenn durch einen andern Satz, so
gewinnen wir nichts dabei; wenn aber durch die Realität, dann muß diese schon in bestimmter
Weise – artikuliert – aufgefaßt sein. Das heißt: man kann nicht auf einen Satz und auf eine
Realität deuten und sagen: ”das entspricht dem“. Sondern, dem Satz entspricht nur wieder
das schon Artikulierte. D.h., es gibt keine hinweisende Erklärung für Sätze.
25
Um im Chinesischen26
einen Satz bilden zu können, dazu genügt es nicht, die Lautreihe
zu lernen und zu wissen, daß sie, etwa in der Fibel neben einem bestimmten Bild steht. Denn
das befähigt mich nicht, die Tatsache auf Chinesisch27
zu porträtieren.
Ja, wenn es mir im Deutschen so geschähe, daß ich die ganze Sprache vergäße,28
mir
aber bei einer bestimmten Gelegenheit doch die Lautverbindung des Satzes einfiele, die
man in diesem Falle zu gebrauchen pflegt,29
so würde ich diese Lautverbindung damit30
nicht
verstehen.
Denk aber etwa an den Satz: &Komm!“
31
Wenn man jemanden fragt ”wie weißt Du, daß die Worte dieser Beschreibg.32
wiedergeben,33
was Du siehst“, so könnte er etwa antworten ”ich meine das mit diesen
Worten“. Aber was ist dieses ”das“, wenn es nicht (selbst) wieder artikuliert, also schon
Sprache ist? Also ist ”ich meine das“ gar keine Antwort. Die Antwort ist eine Erklärung
der Bedeutung der Worte.
34
Wenn ich die Beschreibung nach Regeln bilde (sie mit der Wirklichkeit kollationiere),35
dann übersetze ich sie als eine Sprache aus einer anderen. Und das kann ich natürlich mit
Grammatik und Wörterbuch tun und so rechtfertigen. – Aber dann ist die Übertragung von
17 (V): daraus hinaus: jedes Bild müsse etwas mit der
Welt des Dargestellten gemeinsam haben um ein
Bild von etwas in dieser Welt sein darstellen zu
können. Was aber nur heißt:
Das Bild habe sozusagen die Projektionsmethode
mit dem Dargestellten gemeinsam.
Wie könnte etwas ein Befehl sein wenn ich mich
nicht danach richten könnte. Und wie könnte ich
mich nach ihm richten, wenn ihm nicht die Form
meiner einer Handlung eigen wäre.
Es kann mich nun reizen den Begriff &gem. h.“ so
weit auszudehnen, daß man dies sagen kann.
18 (V): Art
19 (V1): ein Aspekt. (V2): äquivalent mache.
Ich mache also nur auf eine Möglichkeit der
Verallgemeinerung aufmerksam (was freilich sehr
wichtig sein kann). (V3): äquivalent mache. Es
schwebt // drängt sich // mir also eine bestimmte
mogliche Verallgemeinerung vor eine Form der
Darstellung // eine Form der Darstellung //, ein
Aspekt.
20 (M): ∫ 3 ///
21 (V): ist wohl
22 (V): Unsinn
23 (V): Welt
24 (V): gegeben?
25 (M): ? / 3 ////
26 (V): Um in einer Sprache
27 (V): Tatsache in jener Sprache
28 (V): verlernte,
29 (V): Falle gebraucht,
30 (V): Lautverbindung in diesem Falle
31 (M): ? / || 3
32 (V): daß diese Beschreibung
33 (O): wiedergibt,
34 (M): ∫ 3
35 (V): kollationiere), was auch möglich ist,
142 Die Verbindung zwischen Sprache und Wirklichkeit . . .
189
190
WBTC040-45 30/05/2005 16:17 Page 284
the concept of “having in common” and am making it equivalent to the general concept of
projecting.13
So a particular form14
of generalizing forces itself on me, a form of representation, a particular
aspect.15
16
It is also17
incorrect18
to say that the agreement (and lack of agreement) between proposition
and reality19
is created arbitrarily by coordinating them. For how is this coordination
to be expressed? It consists of the proposition “p” saying that this particular thing is the case.
But how does this “this particular thing” get expressed?20
If through another proposition,
then we gain nothing in the process; but if through reality, then the latter must already have
been articulated – understood – in a particular way. This means: one cannot point to a proposition
and reality and say: “This agrees with that”. Rather, only what has been articulated
agrees with a proposition. That is to say, there is no ostensive explanation of propositions.
21
To be able to form a sentence in Chinese22
, it is not enough to learn a sequence of sounds
and to know that, say in my primer, it is next to a certain picture. For this doesn’t enable
me to portray a fact in Chinese.23
Indeed if it happened that I forgot24
all my English, but nevertheless on a certain occasion
remembered the combination of sounds of the sentence that one usually uses25
in a
particular case, then in so remembering26
I wouldn’t understand this combination of sounds.
But think of the sentence “Come!”, for instance.
27
If you ask someone “How do you know that the words of your description capture28
what
you see?”, he could answer, for instance, “I mean that by my words”. But what is this “that”
if it isn’t (itself) articulated in turn, and is therefore already language? So “I mean that” is
not an answer at all. The answer is an explanation of the meaning of the words.
29
If I form a description in accordance with rules (collate it with reality),30
then I’m translating
from one language into another. And of course I can do that with a grammar and a
dictionary, and use these to justify it. – But then the transposition is from one articulated
13 (V): equivalent to this: every picture has to have
something in common with the world of what
it represents in order to represent a picture of
something in this world. But this means only:
The picture and what it represents have their
method of projection in common, so to speak.
How could something be an order if I couldn’t
comply with it? And how could I comply with it if it
didn’t contain the form of my an action?
It can be tempting to stretch the concept of
“having in common” so far that one can say this.
14 (V): kind
15 (V1): representation, an aspect. (V2): of projecting.
Therefore I am simply pointing out a possibility
of generalization (which, to be sure, can be very
important). (V3): of projecting. Therefore a
particular possible generalization is in my mind’s eye,
// forces itself upon me, // a form of representation,
// a form of representation % // an aspect.
16 (M): ∫ 3 ///
17 (V): also quite
18 (V): nonsense
19 (V): world
20 (V): how is this “this particular thing” given
21 (M): ? / 3 ////
22 (V): in a language
23 (V): in that language.
24 (V): unlearned
25 (V): one uses
26 (V): then in this case
27 (M): ? / || 3
28 (V): that this description captures
29 (M): ∫ 3
30 (V): reality), which is indeed possible,
The Connection Between Language and Reality . . . 142e
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Artikuliertem in Artikuliertes. Und wenn ich sie durch Berufung auf die Grammatik und
das Wörterbuch rechtfertige, so tue ich nichts, als eine Beziehung zwischen Wirklichkeit
und Beschreibung (eine projektive Beziehung) festzustellen, von der Intention aber, meiner
Beschreibung ist hiebei keine Rede. (D.h., ich kann eben nur die Ähnlichkeit des Portraits36
prüfen, nichts weiter.)37
36 (V): Bildes 37 (R): ∀ S. 143/1, 2, 3
143 Die Verbindung zwischen Sprache und Wirklichkeit . . .
WBTC040-45 30/05/2005 16:17 Page 286
thing to another. And if I justify my description by appealing to grammar and a dictionary
then all I’m doing is establishing a relationship between reality and a description (a projective
relationship); nothing is said here, though, about what I intend with my description.
(That is, all I can do is to check the portrait’s31
similarity, nothing more.)32
31 (V): picture’s 32 (R): ∀ p. 143/1, 2, 3
The Connection Between Language and Reality . . . 143e
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44
1
Die Sprache in unserem Sinn
nicht als Einrichtung definiert, die
einen bestimmten Zweck erfüllt.
Die Grammatik kein
Mechanismus, der durch seinen
Zweck gerechtfertigt ist.
2
Kann man sagen: &Die Grammatik ist die richtige, die die gewünschte Wirkung hat.“?
Wir möchten3
dann sagen: die Wirkung interessiert uns nicht (wir erlauben uns, irgend eine zu
erdichten), sondern4
nur die Form der Wirkung. D.h. was wir als Wirkung von Etwas auffassen können.
5
Oder auch: Wir erlauben uns irgend welche Erfahrungstatsachen zu erdichten & die Grenze ist
für uns nur dort gezogen wo das aufhört was wir6
Erdichtung nennen; wo der7
Sinn aufhört.
8
Ich muß nun so etwas sagen, wie: Was ein Zeichen sein kann, kann auch eine Ursache sein;
aber nicht immer umgekehrt. Eine Ursache muß die Multiplizität eines Zeichens haben. (Ein Zeichen,
die Multiplizität einer möglichen Ursache.) (Vergleiche Gesetz der Symmetrie,9
Gleichgewicht des
symmetrischen10
Hebels.)
11
Kann ich eine grammatische Regel durch ihren Zweck rechtfertigen?
12
Ich kann sagen, ich gebrauche zwei verschiedene Wörter hier um eine Verwechslung zu vermeiden.
13
Aber sind die Grammatischen Regeln so durch ihren Zweck gerechtfertigt wie die Regeln über
den Bau einer Dampfmaschine durch die beabsichtigte Wirkungsweise der Dampfmaschine?
Sind sie die Regeln nach denen ein Mechanismus konstruiert sein muß um die & die Bewegungen
& . . . hervorzubringen?
14
Man kann eine bestimmte Zeichengebung damit rechtfertigen, daß ein Anderer danach
gewisse Handlungen ausführen soll.15
1 (M): 3
2 (M): ? / ////
3 (V): würden
4 (V): sondern uns interessiert
5 (M): ? /
6 (V): was wir noch
7 (V): der Uns
8 (M): ∫ / ///
9 (O): Symetrie,
10 (O): symetrischen
11 (M): ? / ///
12 (M): ///
13 (M): ///
14 (M): /
15 (V): Wie wäre es, wenn man eine bestimmte
Zeichengebung damit rechtfertigte, daß ein
Anderer danach die & die Handlungen ausführen
soll?
191
190v
144
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44
1
Language in our Sense not
Defined as an Instrument
for a Particular Purpose.
Grammar is not a
Mechanism Justified
by its Purpose.
2
Can one say: “The grammar that has the desired effect is the right one”?
In that case we are inclined to3
say: The effect doesn’t interest us (we take the liberty of inventing
some random one), only its form.4
That is, what we can conceive of as the effect of something.
5
Or alternatively: We take the liberty of inventing some random empirical facts, and the boundary
is drawn for us only where what we call “invention” ends, where sense ends.
6
Now I have to say something like: What can be a sign can also be a cause; but not always the
other way around. A cause must have the multiplicity of a sign. (A sign the multiplicity of a possible
cause.) (Cf. the law of symmetry, the equilibrium of a symmetrical lever.)
7
Can I justify a grammatical rule by its purpose?
8
I can say that here I am using two different words in order to avoid a mix-up.
9
But are grammatical rules justified by their purpose in the same way that the rules for building a
steam engine are justified by the way it’s intended to operate? Are they the rules to be followed in
constructing a mechanism for producing such and such movements and . . . ?
10
One can justify a particular use of signs by saying that they are supposed to induce someone to
perform certain actions.11
1 (M): 3
2 (M): ? / ////
3 (V): we would
4 (V): form interests us.
5 (M): ? /
6 (M): ∫ / ///
7 (M): ? / ///
8 (M): ///
9 (M): ///
10 (M): /
11 (V): What would it be like if one justified a . . . ?
144e
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Man würde die Wirkung der verschiedenen Zeichen auf ihn beschreiben. Man würde vielleicht sagen,
daß dieses Zeichen die gewünschte Wirkung hat16
ein anderes nicht. Man würde also etwa sagen
das Zeichen17
→ bewirkt daß er nach rechts geht dieses ← daß er nach links geht. Gäbe man
Erklärungen der Bedeutung, so würde man sagen:18
das Zeichen ←19
bedeutet &Geh nach links“,
etc. Es ist klar, daß es so eine kausale Rechtfertigung der20
Zeichen gibt.21
22
Könnte ich nicht die Sprache als soziale Einrichtung betrachten, die gewissen Regeln
unterliegt, weil sie sonst nicht wirksam wäre.23
Aber hier liegt es: dieses Letzte24
kann ich
nicht sagen; eine Rechtfertigung der Regeln kann ich, auch so, nicht geben. Ich könnte sie
nur als ein Spiel, das die Menschen spielen, beschreiben.
25
Aber wie ist es: Ich gehe diesen Weg, um dorthin zu kommen; ich drehe den Hahn
auf, um Wasser zu erhalten, ich winke, damit jemand zu mir kommt und endlich teile ich
ihm meinen Wunsch mit, damit er ihn erfüllt! ((D.h.: War also die Mitteilung meines
Wunsches nicht nur das Ziehen eines Hebels und der Sinn meiner Mitteilung ihr Zweck
aber nicht ihre Wirkung?))26
27
Aber was geht vor sich, wenn ich den Hahn aufdrehe, damit Wasser herausfließt? Was
geschieht, ist, daß ich den Hahn aufdrehe, und daß dann Wasser herauskommt, oder nicht.
Was geschieht, ist also, daß ich den Hahn aufdrehe.
Was auf das Wort ”damit“ folgt, die Absicht, ist darin nicht enthalten. Ist sie vorhanden,
so muß sie ausgedrückt sein und sie kann nur dann bereits durch das Aufdrehen des Hahnes
ausgedrückt sein, wenn das Teil einer Sprache ist.
28
Die Rechtfertigung würde etwa lauten: wenn ich das sagen will, muß ich nach solchen Regeln
vorgehen.29
Aber was ich sagen will (ich meine der Ausdruck für das &das“) ist ja erst durch die Regeln
bestimmt.
30
Die grammatischen Regeln sind nicht diejenigen (natürlich erfahrungsmäßigen) Regeln nach denen
die Sprache gebaut sein muß um ihren Zweck zu erfüllen. Um diese Wirkung zu haben.
Vielmehr sind sie die Beschreibung davon, wie die Sprache es macht, – was immer sie macht.
D.h. die Grammatik beschreibt nicht die Wirkungsweise der Sprache sondern nur das Spiel der
Sprache, die Sprachhandlungen.
31
Eine Sprache erfinden um mit ihr etwas Bestimmtes auszudrücken: aber dieses etwas muß schon32
ausgedrückt sein, wenn33
ich sagen kann, daß ich es ausdrücken will.
34
Man könnte z.B. Einem ein Bild zeigen, damit er tut, was auf dem Bild dargestellt ist. Hätte man
durch Erfahrung gefunden, daß dieser Behelf ihn zu gewissen Handlungen bringen kann, so könnte
man nun eine Sprache, wie einen Mechanismus konstruieren um ihn damit zu lenken.
16 (V): hat jenes
17 (V): sagen dieser Pfeil
18 (V): sagen: dieses
19 (O): Zeichen →
20 (V): unserer
21 (V): gibt & auch% daß sie uns nicht interessiert.
22 (M): ///
23 (V): nicht wirken würde.
24 (V): dieses Letztere
25 (M): ? / 555
26 (O): Mitteilung, Ihr Zweck?) ) aber nicht ihre
Wirkung.
27 (M): 5555
28 (M): ? /
29 (V): will, muß ich solche // diese // Regeln geben.
30 (M): /
31 (M): / ///
32 (V): schon vorher
33 (V): ehe
34 (M): ? / ///
191
192
191v
145 Die Sprache in unserem Sinn . . .
WBTC040-45 30/05/2005 16:17 Page 290
One would describe the effect of the various signs on him. One might say that this sign has the
desired effect, but another one doesn’t. Thus one might say that the sign12
→ has the effect that he
goes to the right, this one ← that he goes to the left. If we were giving explanations of meaning,
we’d say: “The13
sign ← means ‘Go to the left’, etc.” It’s clear that there is such a causal justification
for14
signs.15
16
Couldn’t I look at language as a social institution that is subject to certain rules because
otherwise it wouldn’t be effective?17
But here’s the problem: I cannot make this last18
claim;
I cannot give any justification of the rules, not even like this. I can only describe them as a
game that people play.
19
But what about this: I take this path in order to get there; I turn on the tap in order to
get water, I beckon so that someone will come to me, and finally I tell him my wish so that
he’ll fulfil it! ((That is to say: Wasn’t communicating my wish merely pulling a lever and
the sense of my communication its purpose, but not its effect?))
20
But what is going on when I turn on the tap so that water will flow out? What happens
is that I turn on the tap and then water comes out, or doesn’t. So what happens is that I
turn on the tap.
What follows the words “so that”, the intention, is not contained in the turning on of
the tap. If the intention is present it must be expressed; and it can have been expressed by
turning on the tap only if that is part of a language.
21
The justification might run something like this: If I want to say this I have to proceed according
to such rules.22
But what I want to say (I mean the spelling out of “this”) isn’t determined until the
rules determine it.
23
Grammatical rules are not those (it goes without saying: empirical) rules in accordance with which
language has to be constructed to fulfil its purpose. In order to have a particular effect.
Rather they are the description of how language does it – whatever it does.
That is, grammar doesn’t describe the way language takes effect but only the game of language,
the linguistic actions.
24
Inventing a language in order to express something specific with it: but this something must already
have been expressed25
for me to say that I want to express it.
26
One could show someone a picture, for instance, so that he’ll do what’s on it. If one had
discovered through past experience that this makeshift can get him to carry out certain actions,
then one could go on to construct a language as we do a mechanism, and control him with it.
12 (V): say that this arrow
13 (V): This
14 (V): for our
15 (V): signs and also% that it doesn’t interest us.
16 (M): ///
17 (V): wouldn’t have any effect?
18 (V): latter
19 (M): ? / 555
20 (M): 5555
21 (M): ? /
22 (V): this I have to establish such // these // rules.
23 (M): /
24 (M): / ///
25 (V): must have been expressed beforehand
26 (M): ? / ///
Language in our Sense not Defined as an Instrument . . . 145e
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35
Wenn man sagte: Sprache ist alles, womit man sich verständigen kann, so müßte36
man
fragen: Aber worin besteht es, ”sich verständigen“?
Ich könnte als Antwort darauf einen realen oder fiktiven Fall einer Verständigung von
Menschen oder andern Lebewesen beschreiben. In dieser Beschreibung werden dann
fingierte kausale Verbindungen eine Rolle spielen. Aber wenn der Begriff Sprache durch
solche bestimmt ist, so interessiert er uns nicht. Und abgesehen von jenen empirischen
Regelmäßigkeiten der Ereignisse, haben wir dann nur noch einen beliebigen37
Kalkül. – Aber
worin besteht denn das Wesentliche eines Kalküls?
38
”Sprache“ und ”Lebewesen“. Der Begriff des Lebewesens ist so unbestimmt wie39
der
der Sprache.
40
”Ein Zeichen ist doch immer für ein lebendes Wesen da, also muß das etwas dem Zeichen
Wesentliches sein“. Gewiß: auch ein Sessel ist immer nur für einen Menschen da, aber er
läßt sich beschreiben, ohne daß wir von seinem Zweck reden. Das Zeichen hat nur einen
Zweck in der menschlichen Gesellschaft, aber dieser Zweck kümmert uns gar nicht.
Ja am Schluß sagen wir überhaupt keine Eigenschaften von den Zeichen aus – denn diese
interessieren uns nicht – sondern nur die (allgemeinen) Regeln ihres Gebrauchs. Wer das
Schachspiel beschreibt, gibt weder Eigenschaften der Schachfiguren an, noch redet er vom
Nutzen und Gebrauch des Schachspiels.
41
Denken wir uns den Standpunkt eines Forschers: er findet, daß in der Sprache der
Erde ein Zeichen benützt wird, das nach diesen und diesen Regeln (etwa nach denen der
Negation) gebraucht wird, und fragt sich: Wozu können sie das brauchen? Die Antwort wäre
aber: Wenn immer ein Zeichen mit diesen Regeln zu gebrauchen ist.
42
Es wäre ja auch möglich daß man fände, daß nur die deutsche Sprache dazu geeignet wäre von
Menschen verstanden zu werden. Und wenn es sich um Menschen handelt die nur Deutsch gelernt
haben, so ist das ja wirklich so. Man würde dann sagen: nur mit diesem Zeichensystem kann man
Menschen beeinflussen.
Wäre dann das aber die einzig richtige Grammatik?
43
Die Grammatik ist die Beschreibung der Sprache.
Aber sie teilt nicht mit, ob jemand die Sprache versteht, wer sie versteht, oder ob44
ein Befehl dieser
Sprache befolgt wird.
45
Die Sprache ist Teil eines Mechanismus (oder46
zu mindest kann sie so aufgefaßt werden47
). Mit
ihrer Hilfe beeinflussen wir die Handlungen anderer Menschen & werden wir beeinflußt.
Als Teil des Mechanismus, kann man sagen, hat die Sprache einen Zweck. Aber die Grammatik
kümmert sich nicht um den Zweck der Sprache & ob sie ihn erfüllt. Sowenig wie die Arithmetik um
die Anwendung der Addition.
Sind die Regeln des Schachspiels willkürlich? Denken wir uns den Fall, es stellte sich heraus, daß
nur das Schachspiel mit seinen gegenwärtigen Regeln die Menschen zerstreute & befriedigte. Dann
192v
146 Die Sprache in unserem Sinn . . .
35 (M): ∫ ////
36 (V): muß
37 (V): willkürlichen
38 (M): ? / 3
39 (V): Lebewesens hat die gleiche
Unbestimmtheit wie
40 (M): ∫ 3 ////
41 (M): ? / 555 überlege
42 (M): /
43 (M): ////
44 (V): ob etwa
45 (M): ////
46 (V): oder man
47 (V): kann man sie so auffassen
192
193
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27
If it were said: “Language is everything one can use to communicate with”, then it needs
to be asked: What does “communicating” consist in?
In answer to this, I could describe a real or fictional case of humans or other living beings
communicating. Fictitious causal connections will then play a role in this description. But
if the concept of language is determined by such connections, then we’re not interested in
it. And aside from those empirical regularities of the events, all we have left is an arbitrary
calculus. – But what does the essence of a calculus consist in?
28
“Language” and “living being”. The concept of a living being is as indeterminate as29
the concept of language.
30
“A sign is always intended for a living being, so that must be something essential to a
sign.” Certainly: a chair too is always intended just for a human being, but it can be described
without our talking about its purpose. A sign has a purpose only in human society, but this
purpose is of absolutely no concern to us.
Indeed, in the final analysis we don’t make any statements about the properties of signs
– for these don’t interest us – but only about the (general) rules for their use. Someone describing
the game of chess neither lists the properties of the chess men nor does he talk about
the usefulness and use of the game of chess.
31
Let’s imagine the point of view of an explorer. He finds that in a language spoken on
earth a sign is used in accordance with such and such rules (say, with those of negation),
and he asks himself: What can they use that for? But the answer would be: Whenever a sign
with such rules is of use.
32
One might also find that only the English language were suitable to be understood by human
beings. And as concerns people who have learned only English, that’s the way it really is. In this case
one would say: You can only influence people with this system of signs.
But would that then be the only correct grammar?
33
Grammar is the description of language.
But it doesn’t tell us whether someone understands the language, who understands it, or34
whether a command in this language is obeyed.
35
Language is part of a mechanism (or at least it can be understood36
that way). With its help we
influence the actions of other people, and are influenced by them.
As a part of a mechanism, one can say, language has a purpose. But grammar isn’t concerned
with the purpose of language and whether it fulfils it. Any more than arithmetic is with the uses of
addition.
Are the rules of chess arbitrary? Let’s imagine that it turned out that only chess with its
present rules entertained and satisfied people. Then in so far as the purpose of the game is to be
27 (M): ∫ ////
28 (M): ? / 3
29 (V): living being has the same indeterminacy as
30 (M): ∫ 3 ////
31 (M): ? / 555 Think about
32 (M): /
33 (M): ////
34 (V): or perhaps
35 (M): ////
36 (V): at least one can understand it
Language in our Sense not Defined as an Instrument . . . 146e
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wären doch diese Regeln, wenn der Zweck des Spiels erfüllt werden soll, nicht willkürlich. Wenn man
aber von diesem Zweck absähe,48
könnte man sie willkürlich nennen.
49
Eine Sprache erfinden, heißt nicht auf Grund von Naturgesetzen (oder in Überein-
stimmung50
mit ihnen) eine Vorrichtung zu einem bestimmten Zweck erfinden. Wie es etwa
die Erfindung des Benzinmotors oder der Nähmaschine ist. Auch die Erfindung eines Spiels
ist nicht in diesem Sinne eine Erfindung, aber vergleichbar der Erfindung einer Sprache.
51
Ich brauche nicht zu sagen, daß ich nur die Grammatik des Wortes ”Sprache“ weiter
beschreibe, indem ich sie mit der Grammatik des Wortes ”Erfindung“ in Verbindung bringe.
52
Ist alles, was ich sagen kann53
damit gesagt: Man kann nicht von den grammatischen
Regeln sagen, sie seien eine Einrichtung dazu, daß die Sprache ihren Zweck erfüllen
könne. Wie man etwa sagt: wenn die Dampfmaschine keine Steuerung hätte, so könnte der
Kolben nicht hin und her gehen, wie er soll. Als könne man sich eine Sprache auch ohne
Grammatik denken.
54
Die grammatischen Regeln sind, wie sie nun einmal da sind, Regeln des Gebrauchs
der Wörter. Übertreten wir sie, so können wir deswegen die Wörter dennoch mit Sinn
gebrauchen. Wozu wären dann die grammatischen Regeln da? Um den Gebrauch der
Sprache im Ganzen gleichförmig zu machen? (etwa aus ästhetischen Gründen?) Um den
Gebrauch der Sprache als gesellschaftliche Einrichtung zu ermöglichen? also wie eine
Verkehrsordnung, damit keine Kollision entsteht?55
(Aber was geht es uns an,56
wenn eine
entsteht?) Die Kollision, die nicht entstehen57
darf, darf nicht entstehen können! D.h.,
ohne Grammatik ist es nicht eine schlechte Sprache, sondern keine Sprache.
58
Anderseits muß man doch sagen, die Grammatik einer Sprache als allgemein anerkannte
Institution ist eine Verkehrsordnung. Denn, daß man das Wort ”Tisch“ immer in dieser Weise
gebraucht, ist nicht der Sprache als solcher wesentlich, sondern quasi nur eine praktische
Einrichtung.
59
Wie unterscheiden sich die Sprachregeln von denen des Anstandes?
Wenn man kein Ziel angeben kann, das nicht erreicht würde, wenn diese Regeln anders
wären.
60
Denken wir61
uns einen Baukasten62
zur Errichtung63
von Mechanismen. Es gäbe da64
Zahnräder, Hebel, Wellen65
Lager, etc. Man könnte nun Regeln geben, wie diese Bestandteile
aneinander gefügt werden dürften,66
abgesehen davon, welchen Zweck der zusammengestellte
Mechanismus haben soll.
48 (V): absähe, wären
49 (M): ? / 3
50 (V): (oder im Einklang
51 (M): / 3
52 (M): ///
53 (V): darf
54 (M): ///
55 (V): geschieht?
56 (V): was macht es uns,
57 (V): geschehen
58 (M): / ///
59 (M): ////
60 (M): ? / ///
61 (V): wir uns an
62 (V): uns eine Art Baukasten
63 (V): Herstellung
64 (V): Mechanismen. Er enthält
65 (V): Stäbe,
66 (V): dürften ganz
193
194
193v
147 Die Sprache in unserem Sinn . . .
WBTC040-45 30/05/2005 16:17 Page 294
achieved, these rules wouldn’t be arbitrary. But if we disregarded this purpose we could37
call them
arbitrary.
38
Inventing a language does not mean inventing a device for a particular purpose on the
basis of laws of nature, or in agreement39
with them. As is the case, for instance, with the
invention of the gasoline engine or the sewing machine. The invention of a game is also
not an invention in this sense, but is comparable to the invention of a language.
40
I don’t need to say that I am merely further describing the grammar of the word
“language” when I connect it with the grammar of the word “invention”.
41
Is everything I can42
say said by: One cannot say of grammatical rules that they are an
instrument that enables language to fulfil its purpose? As one might say: If a steam engine
didn’t have a governor, the piston couldn’t go back and forth as it is supposed to. As if one
could imagine a language without grammar.
43
Grammatical rules, as they currently exist, are rules for the use of words. Even if we
transgress them we can still use words meaningfully. Then what do they exist for? To make
language-use as a whole uniform? (Say for aesthetic reasons?) To make possible the use of
language as a social institution? And thus – like a set of traffic rules – to prevent a collision?
(But what concern is it of ours44
if that happens?) The collision that mustn’t come about45
must be the collision that cannot come about! That is to say, without grammar it isn’t a bad
language, but no language.
46
On the other hand, one does have to admit that the grammar of a language as a generally
recognized institution is a set of traffic rules. For it isn’t essential to language as such
that we always use the word “table” this way; rather, this is just a practical arrangement,
as it were.
47
How do the rules of language differ from those of decorum?
When one can’t specify a goal we’d fail to achieve if these rules were different.
48
Let’s imagine a construction set for building mechanical devices.49
Let it contain cogwheels, levers,
shafts,50
bearings, etc. Now one could issue rules for how these parts can be joined together, without
regard to what purpose the assembled device is to have.
37 (V): would
38 (M): ? / 3
39 (V): harmony
40 (M): / 3
41 (M): ///
42 (V): I am allowed to
43 (M): ///
44 (V): what does it matter to us
45 (V): mustn’t happen
46 (M): / ///
47 (M): ////
48 (M): ? / ///
49 (V): a kind of construction set for the manufacture
of mechanical devices.
50 (V): bars,
Language in our Sense not Defined as an Instrument . . . 147e
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67
Es ist klar, daß es einer Verwechslung entspringt, z.B., zu sagen: die Grammatik müsse von vier
primären Farbwörtern68
reden, weil es vier primäre Farben gäbe. Als wäre der Fall vergleichbar dem:
die Astronomie muß von vier Jupitermonden sprechen, weil es vier Jupitermonde gibt.
69
Man kann also sagen, die Grammatik läßt sich nicht mit der Wirklichkeit rechtfertigen. Aber es
ist ein andrer Satz, daß sie sich nicht als Teil eines psychologischen Mechanismus rechtfertigen läßt.
Ja es wäre eben der Fall70
denkbar – daß sie sich durch die psychologische Erfahrung rechtfertigen
ließe, wenn sich z.B. die deutsche Sprache als die einzige erwiese die ein Mensch lernen kann. Aber
diese Rechtfertigung interessiert uns nicht.
71
So könnte es sein daß ein Mensch das Zeigen einer Richtung72
(etwa der, in welcher er gehen
soll) nur verstünde, wenn es mit der Hand oder einem Pfeil in der73
gewöhnlichen Weise geschähe,
aber nicht wenn man mit dem Ellbogen74
in diese75
Richtung wiese. Und verstehen heißt hier auf das
Zeichen in bestimmter Weise reagieren.76
77
Der Zweck der Grammatik ist nur der Zweck der Sprache.
Der Zweck der Grammatik ist der Zweck der Sprache.
78
Woher die Bedeutung der Sprache? Kann man denn sagen: Ohne79
Sprache könnten wir
uns nicht miteinander verständigen. Nein, das ist ja nicht so, wie: ohne Telephon könnten
wir nicht von Amerika nach Europa reden. (Es sei denn, daß wir unter ”Telephon“ jede
Vorrichtung verstehen, welche etc. etc.)
Ohne Sprache könnten wir nicht Gedanken austauschen. Ja was heißt das Gedanken
austauschen? Und übrigens was heißt denn Gedanken lesen?
80
Wir können aber sagen: Ohne Sprache könnten wir die Menschen nicht beeinflussen.
Oder, nicht trösten. Oder: nicht ohne eine Sprache Häuser und Maschinen bauen.
Ohne Sprache könnten wir die Menschen nicht bewegen unseren Willen zu tun.
81
Denken wir daran wie ein Mensch durch die Sprache die Tätigkeiten einer Schar von Arbeitern
lenkt – beim Bau einer Pyramide etwa; und wie Worte & Menschen durch Maschinen zu ersetzen
wären. Aber es ist auch eine Maschine denkbar82
in die man Befehle hineinspricht & die auf dieses
System von Einwirkungen durch Befolgung der Befehle reagiert. – Und83
nun kann man fragen: Welches
Interesse hat nun dieser Mechanismus für die Philosophie?
84
Es ist auch sinnvoll85
zu sagen, ohne den Gebrauch des Mundes oder der Hände
können sich Menschen nicht verständigen.
86
Die Worte, die Einer bei gewisser Gelegenheit sagt, sind insofern nicht willkürlich,
als gerade diese in der Sprache, die er sprechen will (oder muß),87
das meinen, was er sagen
will; d.h., als gerade für sie diese grammatischen Regeln gelten. Was er aber meint, d.h. das
67 (M): /
68 (V): Farben
69 (M): ∫
70 (V): wäre – wie ich oben gesagt habe – ein der Fall
71 (M): ? / ///
72 (V): Richtung nur
73 (V): mit der Hand in der
74 (V): Fuß
75 (O): dieser
76 (V1): hier reagieren wie ein Verstehender. (V2):
hier auf das Zeichen reagieren wie ein Verstehender.
77 (M): ? / ////
78 (M): ? /
79 (V): Sprache? Nicht: Ohne
80 (M): /
81 (M): / ///
82 (V): denkbar die auf
83 (V): reagiert. – Welches Und
84 (M): / 3
85 (V): richtig
86 (M): ///
87 (O): muß)
194
193v
194v
195
148 Die Sprache in unserem Sinn . . .
WBTC040-45 30/05/2005 16:17 Page 296
51
It’s clear that the following statement, e.g., stems from a confusion: Grammar has to speak of
four primary colour words52
because there are four primary colours. As if that were comparable to:
Astronomy has to talk about four moons of Jupiter because there are four moons of Jupiter.
53
So one can say, grammar can’t be justified by reality. But that it can’t be justified as part of a
psychological mechanism is a different proposition.
In fact this very case – that grammar could be justified by psychological discoveries – is conceivable
if54
the English language, for example, turned out to be the only one that a human could learn.
But this justification doesn’t interest us.
55
Thus someone might understand the indication of a direction56
(say, of the one in which he is
supposed to go) only if it were carried out in the usual way by hand or by an arrow, but57
not if one
were to point in that direction with an elbow.58
And here “understanding” means reacting to the
sign in a specific way.59
60
The purpose of grammar is nothing other than the purpose of language.
The purpose of grammar is the purpose of language.
61
Whence the significance of language? Can one say: “Without62
language we couldn’t
communicate with each other”? No, that’s not like: Without a telephone we couldn’t speak
from America to Europe. (Unless by “telephone” we understand any device that etc., etc.)
Without language we couldn’t exchange thoughts. Well, what does “exchanging thoughts” mean,
anyway? And by the way, what does “reading thoughts” mean?
63
But we can say: Without language we couldn’t influence people. Or console them. Or
build houses and machines.
Without language we couldn’t get people to do our will.
64
Let’s think about how someone uses language to direct the activities of a number of workers –
say, when building a pyramid; and how words and people could be replaced by machines. But we
can also imagine a machine into which65
one speaks commands and which reacts to this system of
inputs by obeying the commands. – And then one can ask: Of what66
interest is this mechanism to
philosophy?
67
It also makes sense68
to say that people can’t communicate without using their mouths
or their hands.
69
The words that someone utters on a specific occasion are not arbitrary, in so far as it is
just these words that mean what he wants to say in the language that he wants (or has) to
speak; i.e. in so far as these grammatical rules apply precisely to these words. And what he
51 (M): /
52 (V): primary colours
53 (M): ∫
54 (V): conceivable – as I said above – if
55 (M): ? / ///
56 (V): direction only
57 (V): by hand, but
58 (V): with a foot.
59 (V1): reacting as one does who understands.
(V2): reacting to the sign as one does who understands
it.
60 (M): ? / ////
61 (M): ? /
62 (V): language? Not: “Without
63 (M): /
64 (M): / ///
65 (V): machine from which
66 (V): commands. – Of what
67 (M): / 3
68 (V): It is also correct
69 (M): ///
Language in our Sense not Defined as an Instrument . . . 148e
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grammatische Spiel, das er spielt, ist insofern nicht willkürlich, als er etwa seinen Zweck
nur so glaubt erreichen zu können.
88
Die Sprache mit den Bärten von Schlüsseln zu vergleichen. Ebenso kann ich sie aber auch mit der
Perforation der Pianolarolle vergleichen.
89
Man kann sagen, daß die gramm. R. den Bau der Spr. beschreiben; ihre Möglichkeiten
beschreiben.
90
Wie wäre es wenn ein Mensch die Sprache erfände wie man eine Maschine erfindet? Könnte
er denn nicht das Abrichten von Tieren oder Menschen erfinden, entdecken, daß91
sie auf gewisse
Signale reagieren & dies dazu benützen sie gewisse Arbeiten verrichten zu lassen?
Wenn ich aber eine Notation erfinde so ist das eine &Erfindung“ in einem andern Sinn des Wortes.
92
Uns interessiert die Sprache als Phänomen, nicht als die Maschine, die einen bestimmten
Zweck erfüllt.
93
Sprache ist für uns: die deutsche Sprache, die englische Sprache, etc., etc. & ähnliche Systeme.
94
Warum interessiert uns aber das Phänomen der Sprache? Gewisser Mißverständnisse halber. –
Aber was sind Mißverständnisse?
Worin besteht das Sich-nicht-auskennen? Es findet scheinbar ja auch95
seinen Ausdruck in der
Sprache.
96
Könnte sich die Philosophie auch für andere Mechanismen als den der Sprache interessieren?
Denken wir es würde uns beunruhigen, daß Handgriffe, deren Verrichtungen ganz verschieden,97
gleich geformt sind. Wäre es nicht auch eine Philosophische Tat die gleichen Handgriffe durch
verschiedene zu ersetzen. Denken wir an die Handgriffe beim Automobil: das Volant,98
eine Pumpe,
einen Hahn, die Bremse, etc. Könnte es nicht einen Menschen stutzig machen99
daß man aus
einem Rohr Flüssigkeit stetig erhalten100
kann indem man eine einzige Bewegung macht (einen
Hahn aufdreht) & aus einem andern nur, indem man einen Handgriff solange bewegt als man
Flüssigkeit erhalten will (Pumpe)?
194v
149 Die Sprache in unserem Sinn . . .
88 (M): a ? / ///
89 (M): ∫ ///
90 (M): ∫ ////
91 (V): erfinden, daß
92 (M): ∫ ///
93 (M): ∫ ///
94 (M): ∫ ////
95 (V): Es findet ja auch
96 (M): ∫
97 (V): verschieden, sind
98 (O): Vollan,
99 (V): Menschen beunruhigen
100 (V): Flüssigkeit erhalten
194v
195
WBTC040-45 30/05/2005 16:17 Page 298
means, i.e. the grammatical game he is playing, is not arbitrary, in so far as he may believe
that this is the only way to achieve his purpose.
70
Language can be compared to the wards of keys. But I can equally well compare it to the
perforations on a pianola’s cylinder.
71
One can say that grammatical rules describe the structure of language; describe its possibilities.
72
What would it be like if someone invented language, as one invents a machine? Couldn’t he invent
the training of animals or of humans, discover that73
they react to certain signals, and use this to have
them perform certain chores?
But if I invent a notation, this is an “invention” in a different sense of the word.
74
We’re interested in language as a phenomenon, not as a machine that fulfils a particular
purpose.
75
For us language is: the German language, the English language, etc., etc., and similar systems.
76
But why are we interested in the phenomenon of language? Because of certain misunderstandings.
– But what are misunderstandings?
What does not-knowing-one’s-way-about consist in? Seemingly, it77
also finds expression in
language.
78
Could philosophy also take an interest in mechanisms other than that of language? Let’s imagine
that it bothered us that handles that do entirely different things were shaped the same way.
Wouldn’t it also be a philosophical task to replace the identical handles with different ones? Let’s
think about the handles of a car: the steering wheel, a pump, a tap, the brake, etc. Mightn’t it
puzzle79
someone that from one pipe you can get a steady stream80
of liquid with a single motion
(by turning on a tap) and from another you get the liquid only by moving a handle for as long as
you want (a pump)?
70 (M): a ? / ///
71 (M): ∫ ///
72 (M): ∫ ////
73 (V): humans, that
74 (M): ∫ ///
75 (M): ∫ ///
76 (M): ∫ ////
77 (V): in? It
78 (M): ∫
79 (V): bother
80 (V): a stream
Language in our Sense not Defined as an Instrument . . . 149e
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45
Die Sprache funktioniert als
Sprache nur durch die Regeln,
nach denen wir uns in ihrem
Gebrauch richten, wie das Spiel nur
durch seine Regeln ein Spiel ist.
1
Das ist insofern nicht richtig, als für die Sprache keine Regeln niedergelegt sein müssen; sowenig
wie für’s Spiel. Aber man kann die Sprache (& das Spiel) vom Standpunkt eines Vorgangs nach Regeln
betrachten.
2
Grammatik besteht aus Vereinbarungen; anderseits kann die Sprache Teil3
eines Mechanismus
sein. Wie tritt nun das, was einer Vereinbarung entspricht, in einen Mechanismus, z.B. das Pianola,
ein? Nun eine Vereinbarung ist doch z.B. eine Tabelle & eine Tabelle könnte ganz gut auch Teil eines
Mechanismus von der Art des Pianolas4
sein.
5
Warum interessiere ich mich denn so sehr für die Sprache? Kann man einen Grund dafür
angeben, oder ist es eben eine Tatsache deren Ursachen mich einfach nicht interessieren?
Und könnten Bilder nicht als Befehle6
verwendet werden? und warum sollte ich mich nicht für Bilder
interessieren, wenn sie im menschlichen Leben eine überragende Rolle spielen würden?
7
Und könnten philosophische Probleme, Beunruhigungen, auch in gemalten8
Bildern entstehen?
So etwas ließe sich schon ausdenken.
9
Wie, wenn eine Sprache aus lauter einfachen und unabhängigen Signalen bestünde?! Denken
wir uns diesen Fall: Es handle sich etwa um die Beschreibung einer Fläche, auf der in schwarz
und weiß sich allerlei Figuren zeigen können. Wäre es nun möglich, alle möglichen Figuren
durch unabhängige Symbole zu bezeichnen?10
(Ich nehme dabei an, daß ich nur über, sagen
wir 10000 Figuren reden will.) Wenn ich Recht habe, so muß die ganze Geometrie in den
Regeln über die Verwendung dieser 10000 Signale wiederkehren. (Und zwar ebenso, wie
die Arithmetik, wenn wir statt 10 unabhängiger Zahlzeichen eine Billion verwendeten.)
11
Um eine Abhängigkeit auszudrücken, bedarf es einer Abhängigkeit.
1 (M): 3
2 (M): ? / ///
3 (V): teil
4 (O): Pianola
5 (M): ? /
6 (V): Befehle gebra
7 (M): ∫
8 (V): gezeichneten
9 (M): ///
10 (V): kennzeichnen?
11 (M): 555
196
195v
196
150
WBTC040-45 30/05/2005 16:17 Page 300
45
Language Functions as
Language only by Virtue of
the Rules We Follow in Using it,
just as a Game is a Game
only by Virtue of its Rules.
1
That is not correct, in so far as no rules have to have been laid down for language; no more than
for a game. But one can look at language (and a game) from the standpoint of a process that uses
rules.
2
Grammar consists of arrangements; on the other hand, language can be part of a mechanism.
Now how does something corresponding to an arrangement enter into a mechanism, for example,
a pianola? Well, a chart, for example, is an arrangement, and a chart could easily be part of a pianolatype
mechanism.
3
Why am I so interested in language? Can a reason be given, or is it simply a fact whose causes
just don’t interest me?
And couldn’t pictures be used as commands? And why shouldn’t I be interested in pictures if they
were to play a paramount role in human life?
4
And could philosophical problems, concerns, originate in painted pictures5
as well? Something
like that could be imagined.
6
What if a language consisted of nothing but simple and independent signs?! Imagine this
case: It’s a matter of describing a surface on which a variety of figures can appear in black
and white. Now would it be possible to refer to7
all possible figures using independent
symbols? (Here I’m assuming that I only want to talk about, say, 10,000 figures.) If I’m
right, then all of geometry has to recur in the rules for the use of these 10,000 signs. (And
it would have to recur in the same way as would arithmetic if, instead of 10 independent
numerical symbols, we used a trillion.)
8
To express a dependence you need a dependence.
1 (M): 3
2 (M): ? / ///
3 (M): ? /
4 (M): ∫
5 (V): in drawings
6 (M): ///
7 (V): to designate
8 (M): 555
150e
WBTC040-45 30/05/2005 16:17 Page 301
12
Denken wir uns ein Tagebuch mit Signalen geführt. Etwa die Seite in Abschnitte
für jede Stunde eingeteilt und nun heißt ”ד ich schlafe, ”|“ ich stehe auf, ”⎡“ ich schreibe,
etc.13
14
Muß denn nicht die Regel der Sprache – daß also dieses Zeichen das bedeutet – irgendwo
niedergelegt sein?
Freilich auch: Mehr als die Regel niederlegen, kann ich nicht.
Ist die Regel niedergelegt, so ist es eben eine andere Sprache, als wenn sie nicht
niedergelegt ist.
15
&Contrat16
sociale“ auch hier ist in Wirklichkeit kein Vertrag geschlossen worden; aber die Situation
ist17
mehr oder weniger ähnlich, analog, der in welcher wir wären, wenn. . . . Und sie ist mit großem
Nutzen18
unter dem Gesichtspunkt eines solchen Vertrages zu betrachten.
19
Und warum soll ich, daß ”ד in dieser Zeile steht, nicht ein Bild dessen nennen, daß
ich dann schlafen gehe? Freilich, daß es die Multiplizität dessen wiedergeben soll die in jenen
Worten liegt, kann ich nicht verlangen.
Der Akt des Schlafengehens war ja auch nicht dadurch bestimmt.
Denken wir, ich zeichne einen Sitzplan, 20
ist ein Kreuzchen das Bild eines
Menschen oder nicht? –
21
Wie kann ich denn kontrollieren, daß es immer dasselbe ist, was ich ”ד nenne. Es sei
denn, daß ich etwa ein Erinnerungsbild zuziehe. Das aber dann zum Zeichen gehört.
22
Wenn z.B. Einer fragte: wie weißt Du, daß Du jetzt dasselbe tust wie vor einer Stunde,
und ich antwortete: ich habe mir’s ja aufgeschrieben, hier steht ja ein ”ד!
23
Wenn ich mich in dieser Sprache ausdrücke, so werde ich also mit ”⎡“ immer dasselbe
meinen. Es kann keinen24
Sinn haben, zu sagen, daß ich beide Male dasselbe tue, wenn ich
den Befehl ”⎡“ befolge (oder dasselbe getan habe, als ich tat, was ich durch ”⎡“ bezeichnete).
⎡ = ⎡
25
D.h. die Sprache funktioniert als Sprache nur durch die Regeln, nach denen wir uns in
ihrem Gebrauch richten. (Wie das Spiel nur durch Regeln als Spiel funktioniert.)
26
Und zwar, ob ich zu mir oder Andern rede. Denn auch mir teile ich nichts mit, wenn
ich Lautgruppen ad hoc mit irgend welchen Fakten associiere.
27
Ich könnte auch sagen, daß, wenn die Zeichen ad hoc erfunden sind, eben ein System eine Regel
erfunden werden muß.
28
Ich muß, auch wenn ich zu mir rede, schon auf einem bestehenden29
Sprachklavier spielen.
197
196v
197
198
197v
198
151 Die Sprache funktioniert als Sprache . . .
12 (M): 3
13 (E): Statt der Symbole ×, |, und ⎡ setzt
Wittgenstein in TS 213 die Buchstaben A, B
und C. Wir haben hier auf die Symbole
zurückgegriffen, die MS 110 (S. 208)
entstammen.
14 (M): ////
15 (M): 3 /
16 (O): &Contract
17 (V): ist ähnlich
18 (V): Nutzen vom Ge
19 (M): ////
20 (F): TS 212, S. 600.
21 (M): / ///
22 (M): ∫ 3
23 (M): ∫ 3
24 (V): einen
25 (M): ? / 3 ///
26 (M): ∫ / /// 3
27 (M): 3 ///
28 (M): ∫ / 3 ///
29 (V): gegebenen
+ +
+ + +
WBTC040-45 30/05/2005 16:17 Page 302
9
Let’s imagine a diary kept in signs. For instance, each page is subdivided into portions
for each hour and “×” means I’m sleeping, “|” I’m getting up, “⎡” I’m writing, etc. 10
11
Doesn’t a rule of language – i.e. that this sign means this – have to be laid down
somewhere?
To be sure, though: I can’t do more than lay down the rule.
When the rule is laid down, it’s just a different language from when it isn’t.
12
“Contrat sociale” – here too, no actual contract was ever concluded; but the situation is more
or less similar,13
analogous, to the one we’d be in, if. . . . And there’s much to be gained in viewing
it in terms of such a contract.
14
And why shouldn’t I call the fact that “×” is written on this line a picture of my going
to sleep then? To be sure, I can’t expect that it will reproduce the multiplicity of what is
contained in those words.
After all, the act of going to sleep was not determined by my diary.
Let’s imagine that I draw a seating plan. 15
Is a little cross a picture of a person,
or not? –
16
How can I verify that what I call “×” is always the same thing? Unless I
bring in, say, a memory-image. But then that is part of the sign.
17
If someone were to ask, for instance: How do you know that you are now doing the same
thing as an hour ago, and I were to answer: Because I wrote it down – there’s an “×” here!
18
If I express myself in this language, I’ll always mean the same thing by “⎡”. It can’t make
sense19
to say that I do the same thing both times when I obey the command “⎡” (or did the
same thing when I did what I designated by “⎡”).
⎡ = ⎡
20
That means that language functions as language only by virtue of the rules we follow in
using it. (Just as a game functions as a game only by virtue of its rules.)
21
And this holds, as a matter of fact, regardless of whether I talk to myself or to others.
For neither do I communicate anything to myself if I just associate groups of sounds with
random facts on an ad hoc basis.
22
I could also say that if the signs have been invented ad hoc, then a system, a rule, has to be invented.
23
Even when I’m talking to myself, I have to be playing on an existing24
language-piano.
+ +
+ + +
9 (M): 3
10 (E): In TS 213 Wittgenstein uses A, B, and C,
instead of the symbols ×, |, and ⎡ . We have
taken these symbols from the corresponding
passage in MS 110 (p. 208).
11 (M): ////
12 (M): 3 /
13 (V): is similar,
14 (M): ////
15 (F): TS 212, p. 600.
16 (M): / ///
17 (M): ∫ 3
18 (M): ∫ 3
19 (V): It can make sense
20 (M): ? / 3 ///
21 (M): ∫ / /// 3
22 (M): 3 ///
23 (M): ∫ / 3 ///
24 (V): on a given
Language Functions as Language . . . 151e
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30
Man kann sagen: die Grammatik erklärt die Bedeutungen der Zeichen & dadurch macht sie
die Sprache bildhaft.
Man würde nicht sagen daß ich aus den Signalen im Tagebuch die Ereignisse eines Tages ableiten
kann d.h. z.B. Bilder nach den31
Aufzeichnungen entwerfen kann, wenn zu den Signalen nicht
noch eine Erklärung tritt.
Es handelt sich um den Begriff des Ableitens. Man spricht vom Ableiten wo eine
allgemeine32
Regel, also ein Ausdruck einer solchen Regel, gegeben ist.
Wir würden33
nämlich nicht sagen aus a b b c ließe sich die Figur ableiten
wohl aber aus abbc & der Tabelle
Die Erklärung der Bedeutung bestimmt wie ein Wort beim portraitieren eines Sachver-halts
zu verwenden ist.
34
Man kann sagen: die Grammatik bestimmt die Bedeutg. der Wörter & bestimmt ihnen damit den
Platz den sie beim Portraitieren eines Sachverhaltes einnehmen dürfen. Denn wonach richte ich mich
wenn ich hier &rot“ & nicht &gelb“ verwende, hier &aber“35
& nicht &oder“? Doch wohl nach der
Bedeutg. der Wörter nach dem was in Übereinkommen über sie in der36
Grammatik festgehalten ist;
denn warum sollte ich sonst das eine Wort dem andern vorziehen.
37
”Ich verstehe diese Worte“ (die ich etwa zu mir selbst sage), ”ich meine etwas damit“,
”sie haben einen Sinn“ muß immer dasselbe heißen wie: ”sie sind nicht ad hoc erfundene
Laute, sondern Zeichen aus einem vorbereiteten System38
“. Aber da könnte man fragen: Tut es
jedes System? Ist es nicht eben das System unserer Sprache was ich meine? Ich spiele ein Spiel mit
ihnen. [d.h. ein schon bestehendes Spiel.]
39
Etwa, wie die Teilstriche auf einem Maßstab nur solche sind, wenn sie ein System bilden.
40
Denn, wenn wir einen Befehl befolgen, so deuten wir die Worte nicht willkürlich.
D.h. wieder, wir müssen die Unterscheidung anerkennen zwischen dem ”Befolgen eines
Befehls“ und einem ”willkürlichen Zuordnen einer Handlung“.
41
Das Aussprechen eines Satzes wäre kein Porträtieren, wenn ich meine Worte nicht aus
einem System wählte, so daß man sagen kann, ich wähle sie im Gegensatz zu anderen.
Aber die Worte, wenn sie nicht in einem grammatischen System stehen, sind ja alle
gleichwertig und also wäre es dann ganz gleichgültig,42
welche ich wählte, ja, – man könnte
sagen – als Worte würden sie sich (dann) voneinander gar nicht unterscheiden.
Man muß die Worte wählen, in demselben Sinne wie man43
die Striche und Farben wählt,
mit denen man einen Körper abbildet.
44
Warum wir ein Wort – und nicht ein anderes – an dieser Stelle gebrauchen, erfahren
wir, wenn wir jemand fragen: warum gebrauchst Du hier das Wort A. Die Antwort wird
sein: das und das heißt A. Und das ist eine Regel der Grammatik, die die Position des Wortes
in der Sprache bestimmt.45
Und (zum Zeichen, daß es sich hier wirklich um Grammatik
30 (M): 3 ////
31 (V): dem
32 (V): Allgemeine
33 (V): werden
34 (M): ///
35 (V): verwende – &aber“
36 (V): was in der
37 (M): ∫ ∫ ? / ///
38 (M): [Zeichen, über die eine Konvention besteht?]
39 (M): ∫ ///
40 (M): ∫ ? ∫∫ 3
41 (M): ∫ 3
42 (V): dann gleichgültig,
43 (V): wählen, wie man
44 (M): ///
45 (V): bestimmt. (Und
198
199
198v
152 Die Sprache funktioniert als Sprache . . .
197v
a
b
c
d
d
g
h
b
WBTC040-45 30/05/2005 16:17 Page 304
25
One can say: Grammar explains the meanings of the signs and thereby makes language pictorial.
One wouldn’t say that I can deduce the events of a day from the signs in my diary, i.e. that I can
draw pictures based on the entries, unless an explanation were added to the signs.
It’s a matter of the concept of deduction. We talk about deduction where a general rule, i.e. an
expression of such a rule, is given.
For we wouldn’t26
say that the figure can be deduced from abbc, but we would say
that it certainly can be deduced from abbc plus the table
The explanation of its meaning determines how a word is to be used in portraying a
state of affairs.
27
One can say: Grammar determines the meaning of words, and in so doing determines the place
they are allowed to occupy when they portray a state of affairs. For what am I taking as a guideline
when I use “red” and not “yellow” here, and “but”28
rather than “or” there? Surely the meaning
of the words, what has been agreed about them and has been established in grammar. For why else
should I prefer one word to another?
29
“I understand these words” (which I say to myself, for instance), “I mean something by
them”, “They make sense”, must always mean the same as: “They are not sounds invented
ad hoc but signs from a prepared system30
”. But here one could ask: Will any system do? Isn’t
what I mean the system of our language? I play a game with the signs. [That is, an already
existing game.]
31
As, say, the graduation marks on a measuring stick are such marks only if they form a
system.
32
For in obeying a command we don’t interpret the words arbitrarily.
That is, once again we have to acknowledge the distinction between “obeying a command”
and an “arbitrary pairing of an action with words”.
33
Uttering a sentence wouldn’t be a portrayal if I didn’t choose my words from a system,
so that it can be said that I choose them as opposed to others.
For words are all of equal value if they’re not situated in a grammatical system; if they
weren’t it would be completely irrelevant34
which ones I chose. Indeed – one could say – as
words they wouldn’t (then) differ from each other at all.
We have to choose words in the same sense as35
we choose the lines and colours with which
we portray a body.
36
We find out why we use one word at a particular point – and not another – when we
ask someone: Why are you using the word “A” here? The answer will be: “‘A’ means such
and such”. And that is a rule of grammar, which determines the position of the word in
25 (M): 3 ////
26 (V): won’t
27 (M): ///
28 (V): here – “but”
29 (M): ∫ ∫ ? / ///
30 (M): system [signs, about which there is a
convention?]
31 (M): ∫ ///
32 (M): ∫ ? ∫ ∫ 3
33 (M): ∫ 3
34 (V): be irrelevant
35 (V): words as
36 (M): ///
Language Functions as Language . . . 152e
a
b
c
d
d
g
h
b
WBTC040-45 30/05/2005 16:17 Page 305
handelt) wenn A das Wort ”und“ gewesen wäre, so könnte man weiter nichts tun, als die
Regeln für ”und“ angeben.
46
Sage ich jemandem ”bringe eine rote Blume“ und er bringt eine, und nun frage ich ”warum
hast Du mir eine von dieser Farbe gebracht?“ – und er: ”diese Farbe heißt doch47
’rot‘ “: so
ist dies Letzte ein Satz der Grammatik. Er rechtfertigt eine Anwendung des Worts.
48
Zusammenhang der Grammatik mit der Bildhaftigkeit der49
Sprache.
50
Fehlt diese Regel,51
so ist die Grammatik des Worts (seine Bedeutung) eine andere.
52
Wenn man einen Satz braucht, so muß er schon irgendwie funktionieren. Das heißt, man
gebraucht ihn nicht, um einer Tatsache einen Lärm beizuordnen.
53
Wir vergleichen den tatsächlichen Vorgang mit dem in welchem die Rechtfertigung ausgeführt
ist.
54
Wir ergänzen das tatsächlich Ausgeführte zu einem bestimmten Kalkül, um es dadurch zu
beleuchten. Ähnlich wie die Grammatik einen elliptischen55
Satz zu einem vollständigen56
ergänzt,
d.h. ein gewisses Gebilde als elliptischen57
Satz auffaßt.
58
Es wäre doch nicht, einen Tatbestand porträtieren, wenn ich etwa beliebige Striche
auf das Papier kritzelte und sagte ”es gibt gewiß eine Projektionsmethode, die diesen
Tatbestand in diese Zeichnung projiziert“.59
60
Ja auch hier (beim Porträtieren61
) fühle ich mich schon beim ersten Strich verpflichtet
– d.h. er ist nicht willkürlich. Jedenfalls aber fängt das Bild erst dort an, wo die Verpflichtung
anfängt.
62
Wenn ich die Achsel zucke, könnte man da sagen: ich meine etwas damit? Gewiß,63
es könnte
mich doch jemand fragen: &hast Du mit der Achsel nur zufällig gezuckt64
oder hast Du es als
Achselzucken gemeint? Und was ist der wesentliche Unterschied zwischen diesen Fällen;65
worin
besteht es, diese Bewegung als Achselzucken zu meinen? Ist es ein besonderes Gefühl was66
die
Bewegung begleitet? Ist es nicht vielmehr die ganze Umgebung in der sie67
liegt? Was sozusagen
aus ihr folgt, was ich zu ihrer Erklärung sagen würde, oder was ich zu ihrer Ergänzung sage oder denke.
Würden wir etwa von der Bedeutung der Meinung des Achselzuckens reden wenn es isoliert von aller
andern Ausdrucksweise aufträte?68
Sagen wir daß der Hund etwas mit dem Wedeln des Schweifs
meint? Wir werden69
vielleicht auch da von einer Meinung reden, wenn wir den Fall des Wedelns
aus Freude von dem aus einer anderen Ursache unterscheiden wollen, & doch ist das natürlich ein
anderer Fall als der, in welchem70
das Kriterium der Meinung der Ausdruck einer Sprache ist.
46 (M): ///
47 (V): ”diese Farbe nenne ich
48 (M): 3 3
49 (V): einer
50 (M): ///
51 (V): Fehlt dieser Satz,
52 (M): ∫ 3 ///
53 (M): 3
54 (M): 3
55 (O): eliptischen
56 (O): vollstandigen
57 (O): eliptischen
58 (M): /// 3
59 (O): projiziert.
60 (M): ? / ∫
61 (V): Abbilden
62 (M): v 3 /
63 (V): Gewiß, man
64 (V): nur gezuckt weil
65 (V): Und worin unterscheiden sich diese Fälle;
66 (V): das
67 (V): die
68 (V): geschähe?
69 (V): werden auch da
70 (V): welchem wir
198v
199
198v
199v
199
200
199v
153 Die Sprache funktioniert als Sprache . . .
WBTC040-45 30/05/2005 16:17 Page 306
language. And37
(as a sign that it really is a matter of grammar here) if “A” had been the
word “and”, one could do no more than state the rules for “and”.
38
If I tell someone “Bring me a red flower” and he brings me one, and then I ask “Why
did you bring me one with this colour?” – and he says: “Well, because this colour is called39
‘red’”, then this last statement is a grammatical proposition. It justifies a use of the word.
40
The connection between grammar and the pictorial nature of language.41
42
If this rule43
is missing, then the grammar of the word (its meaning) is different.
44
When one uses a proposition it must already function in some way. That is to say, one
doesn’t use it to assign a noise to a fact.
45
We compare the actual process with the one in which the justification has been carried out.
46
We expand what has actually been carried out into a particular calculus, so as to cast light on it.
This is similar to grammar expanding an elliptical sentence into a complete one, i.e. conceiving of
a particular structure as an elliptical sentence.
47
After all, I wouldn’t be portraying a factual situation if, for example, I scribbled random
lines on a piece of paper and said “Surely there’s a method of projection that projects that
situation into this drawing”.
48
Indeed, here too (in portraying49
something) I already feel committed when I draw the
first line – i.e. it isn’t arbitrary. In any case, though, the picture begins at the point of commitment
– and not before.
50
If I shrug my shoulders, could it be said that I mean something by this? Certainly. Someone could
ask me, after all: “Did you just happen to shrug your shoulders51
, or did you mean it as a shrugging
of your shoulders?” And what’s the essential difference between these cases?52
What does meaning
this movement as a shrugging of the shoulders consist in? Is what accompanies the movement,
when we mean it this way, a certain feeling? Isn’t it rather the entire context in which the movement
is situated? What follows from it, so to speak – what I would say to explain it, or what I say or think
to complement it? Would we talk about the meaning, the intention of shrugging one’s shoulders,
for instance, if it occurred53
in isolation from all other modes of expression? Do we say that a dog
means something by wagging its tail? Maybe in this case too we’ll speak of a meaning, if we want
to distinguish wagging for joy from wagging for other reasons. But of course that’s a different case
from the one in which the criterion of meaning is a linguistic expression.
37 (V): (And
38 (M): ///
39 (V): “I call this colour
40 (M): 3 3
41 (V): of a language.
42 (M): ///
43 (V): proposition
44 (M): ∫ 3 ///
45 (M): 3
46 (M): 3
47 (M): /// 3
48 (M): ? / ∫
49 (V): depicting
50 (M): v 3 /
51 (V): shoulders because
52 (V): And where’s the difference between these
cases?
53 (V): happened
Language Functions as Language . . . 153e
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71
Wir würden kaum fragen, ob das Krokodil etwas damit meint wenn es mit offenem Rachen
auf einen Menschen72
zukommt. Und wir würden erklären, das Krokodil könne nicht denken &
darum sei eigentlich hier von einem Meinen keine Rede.
73
&Meinen“ ist so vieldeutig wie &Zeichen“74
oder das Wort &ausdrücken75
“.
76
Wie unterscheidet sich eine Geste von irgend einer anderen Bewegung? Dadurch daß sie etw.
ausdrückt? –
77
Denken wir es würde uns Einer vorschlagen: &Meine einmal mit der Lautreihe *ber‘ die
Negation, statt mit dem Wort *nicht‘“.78
Wie mache ich das? Besteht es darin daß ich in mir ein
bestimmtes Gefühl hervorrufe wenn ich das Wort &ber“ ausspreche?
79
Man könnte sagen,80
daß das Wort je nach der Wortart in einem anderen Sinne &bedeutet“.
81
Wenn das Achselzucken ein Zeichen ist, – kann man es durch ein beliebiges anderes ersetzen?
& wie kann man das andere an die Stelle des ersten setzen?
82
Ist das Gähnen unbedingt ein Zeichen der Langenweile & nicht meist nur83
ein Anzeichen von
ihr? Und wie wird etwas zum Zeichen, sagen wir, des Zweifels? Wenn ich etwa von heute an,
immer wenn ich im Zweifel wäre statt meinen Kopf zu schütteln mit der Hand wackeln würde
(erfahrungsgemäß), würde dann die Handbewegung dadurch zum Zeichen des Zweifels? Und
kann man das Gähnen auch meinen?
84
Kann man sagen der Unterschied zwischen dem Gähnen & dem Achselzucken ist Unwillkürlichkeit
& Willkürlichkeit?
85
Oder könnte man eine Gebärde, für die es Sinn hat zu sagen, ich trachtete sie nicht zu machen
aber mein Körper hat gegen meine Anstrengung sie gemacht, könnte man diese Gebärde auch meinen
wenn sie in diesem Sinne gegen den Willen geschieht?
86
Wenn ich mich kratze, nenne ich das ein Zeichen, daß es mich juckt? Gewiß, ich kann es als Zeichen
dafür gebrauchen, aber auch nicht.
87
Ich kann in einem Gespräch ein88
trauriges Gesicht machen als Zeichen der Trauer aber es kann
auch nur ein Anzeichen sein. Worin besteht es nun in diesem Falle das Gesicht als Zeichen der Trauer
zu verziehen? Ich würde sagen: &ich habe es absichtlich getan & ihm auch89
90
Kann ich alle die91
Gesten die bei uns92
im Zusammenhang mit der Sprache stehn, mir auch
ohne diesen Zusammenhang richtig vorstellen?93
Würde ich sie außer diesem Zusammenhang
auch Zeichen nennen?
71 (M): v 3 /
72 (V): auf uns
73 (M): ? / 555
74 (V): wie &ein Zeichen geben“
75 (O): &ausdrucken
76 (M): ∫ ? / 555
77 (M): ? v 3 // ///
78 (V): &Meine einmal mit dem Wort D der Lautreihe
*ber‘ die Negation %nichtB“.
79 (M): v ∫ ///
80 (V): Es ist zu bedenken,
81 (M): ? / 555
82 (M): v / 555
83 (V): nicht nur
84 (M): ///
85 (M): ∫
86 (M): ∫ ///
87 (M): ∫ ///
88 (V): kann ein
89 (E): Satz nicht vollendet.
90 (M): / 3 ///
91 (V): ich die
92 (V): mir
93 (V1): diesen Zusammenhang denken? (V2):
Ich kann mir kaum alle die Gesten . . . stehn, auch
ohne diesen Zusammenhang denken // richtig
vorstellen.
200
200v
154 Die Sprache funktioniert als Sprache . . .
WBTC040-45 30/05/2005 16:17 Page 308
54
We’d hardly ask whether a crocodile means something when it approaches a person55
with its
jaws wide open. And we’d explain that the crocodile can’t think, and so here it’s really not a matter
of meaning at all.
56
“To mean” is as ambiguous as “sign”57
or “to express”.
58
How does a gesture differ from some other movement? In that it expresses something? –
59
Suppose someone were to suggest to us: “Try meaning negation using the sound sequence
‘ber’ instead of the word ‘not’”.60
How do I do that? By evoking a particular feeling in myself when
I utter the word “ber”?
61
One could say62
that a word “has meaning” in different senses, depending on the kind of word
it is.
63
If shrugging one’s shoulders is a sign – can you arbitrarily replace it with a different one? And
how can you put that in place of the first?
64
Is yawning necessarily a sign of boredom, and not just a symptom of it? And how does something
come to be a sign, let’s say of doubt? If, e.g., starting today I were to wiggle my hand (as I
sometimes do) instead of shaking my head whenever I was in doubt, would that hand movement
thereby turn into a sign of doubt? And can one also mean a yawn?
65
Can one say that the difference between yawning and shrugging one’s shoulders is that
between involuntariness and voluntariness?
66
Or take a gesture of which it makes sense to say that I didn’t intend to make it, but my body
made it in spite of my efforts: if in this sense it occurs against one’s will, could one also mean this
gesture?
67
When I scratch, do I call this a sign that I itch? To be sure, I can use it as a sign for this, but I don’t
have to.
68
In a conversation I can make a sad face as a sign of sadness, but it can also be just a symptom.69
Now in this case what does contorting one’s face as a sign of sadness consist in? I would say: “I did
it on purpose and also . . . to him70
71
All the gestures that are connected with our language: Can I imagine them accurately without
that context?72
Outside this context would I still call them signs?
54 (M): v 3 /
55 (V): approaches us
56 (M): ? / 555
57 (V): as “giving a sign”
58 (M): ∫ ? / 555
59 (M): ? v 3 // ///
60 (V): “Try using the word the sound sequence ‘ber’
to mean the negation BnotD.”
61 (M): v ∫ ///
62 (V): We ought to consider
63 (M): ? / 555
64 (M): v / 555
65 (M): ///
66 (M): ∫
67 (M): ∫ ///
68 (M): ∫ ///
69 (V): I can make a sad face as a sign of sadness, but
it can also be just a symptom.
70 (E): Sentence incomplete in TS.
71 (M): / 3 ///
72 (V1): conceive of them without that context?
(V2): I can barely imagine // get an accurate mental
picture of // all those gestures that are connected with
my language without that context.
Language Functions as Language . . . 154e
WBTC040-45 30/05/2005 16:17 Page 309
94
Man sagt: die Henne lockt ihre Jungen durch Glucken,95
– aber liegt dem nicht schon die Vorstellung
unserer Sprache zugrunde? Wird nämlich der Aspekt nicht ganz verändert indem man sich vorstellt
durch irgend eine96
Physikalische Einwirkung ziehe das Glucken die Kücklein zur Henne?
97
Wenn aber im Fall der menschlichen Sprache gezeigt würde, daß das Wort &komm zu mir“ auf
die Menschen eine Anziehung im physikalischen Sinne bewirkt, würde damit die Sprache den
Charakter der Sprache verlieren?
98
Ich will doch immer wieder sagen der Apparat unserer Sprache, unserer Wortsprache, ist99
vor allem das was wir Sprache nennen, & dann anderes nach seiner Analogie oder Vergleichbarkeit
mit ihr.
100
Wir benützen das Wort &Sprache“, &Meinen“, etc. nach sehr verschiedenen Kriterien.
101
Und das Achselzucken ist natürlich gar nicht wesentlich verschieden von einem Wort, ja einem
Satz, etwa: &Ich weiß nicht!“ oder &Weiß Gott!“. Diese Worte können gewiß so unwillkürlich
ausgesprochen werden wie eine Geste gemacht werden kann.
102
Die Zeichen, will103
ich sagen, haben ihre Bedeutung nicht durch das was sie begleitet, noch
durch das, was sie hervorruft, sondern104
durch ein System dem sie zugehören wovon aber beim
Aussprechen des Worts nichts andres als dieses Wort vorhanden sein braucht.105
94 (M): v 3 /
95 (V): sagt: der Hahn ruft die Hühner durch sein
Krähen herbei,
96 (V): eine Me
97 (M): v 3 / (V): Lernen der Sprache als ein
Abrichten
98 (M): v 3 /
99 (V): sagen unsere Sprache, unsere Wortsprache, ist
100 (M): ///
101 (M): ///
102 (M): ///
103 (V): wollte
104 (V): Bedeutung nicht durch etwas was sie
begleitet; sondern
105 (R): [siehe Notizbuch]
201
155 Die Sprache funktioniert als Sprache . . .
WBTC040-45 30/05/2005 16:17 Page 310
73
We say: A hen clucks to call its young74
– but isn’t this already based on a conception we have
of our language? For isn’t the aspect changed completely if we imagine that the clucking draws the
chicks to her because of some physical influence?
75
But if it were shown in the case of human speech that the words “come here” effect an attraction
(in a physical sense) on humans, would language thereby lose the character of language?
76
I want to say over and again that primarily it is the apparatus of our77
language, our wordlanguage,
that we call language; and only then do we call other things language, depending on their
analogy or comparability to word-language.
78
We use the words “language”, “to mean”, etc., according to very different criteria.
79
And of course shrugging one’s shoulders is not so very different from a word, indeed a sentence,
such as “I don’t know!” or “Who knows!”. To be sure, these words can be uttered just as involuntarily
as a gesture can be made.
80
I want81
to say that signs have their meanings neither by virtue of what accompanies them,82
nor
because of what evokes them – but by virtue of a system to which they belong – one, however, in
which when a word is uttered nothing need be present other than that word.83
73 (M): v 3 /
74 (V): say: A cock crows to call the hens
75 (M): v 3 / (V): Learning language as a training
76 (M): v 3 /
77 (V): it is our
78 (M): ///
79 (M): ///
80 (M): ///
81 (V): wanted
82 (V): by something that accompanies them,
83 (R): [See notebook]
Language Functions as Language . . . 155e
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1 (R): ∀ S^ 173/3 Lernen der Sprache durch Abrichten^
2 (M): 3
3 (V): ich den Satz
4 (V): als wäre der ganze Satz nur
5 (O): (Eliptischer
6 (M): 3
7 (V): daß es mitteilt, ob es licht oder finster ist.
8 (M): 3
9 (V): Spiele
10 (V): die wirklichen
11 (M): 3
12 (V): eben
13 (V): Aber wenn ich
14 (V): ”Licht“, – nun was
15 (V): des Spiels, auf das wir gefaßt sind.
202
201
46
Funktionieren des Satzes an einem
Sprachspiel erläutert.1
2
Ich halte meine Wange, und jemand fragt, warum ich es tue und ich antworte:
”Zahnschmerzen“. Das heißt offenbar dasselbe, wie ”ich habe Zahnschmerzen“, aber weder
stelle ich mir die fehlenden Worte im Geiste vor, noch gehen sie mir im Sinn irgendwie ab.
”Daher ist es auch möglich, daß ich die Worte3
”ich habe Zahnschmerzen“ in dem Sinne
ausspreche, als sagte ich nur das letzte Wort oder, als wären die drei nur4
ein Wort“.
(Elliptischer5
Satz. Was tut die Grammatik, wenn sie sagt: ”,Hut und Stock!‘ heißt eigentlich
,gib mir meinen Hut und meinen Stock!‘“)
6
Ein einfaches Sprachspiel ist z.B. dieses: Man spricht zu einem Kind (es kann aber auch
ein Erwachsener sein), indem man das elektrische Licht in einem Raum andreht: ”Licht“,
dann, indem man es abdreht: ”Finster“; und tut das etwa mehrere Male mit Betonung und
variierenden Zeitlängen. Dann geht man etwa in das Nebenzimmer, dreht von dort aus das
Licht im ersten an und bringt das Kind dazu, daß es mitteilt: ”Licht“, oder: ”Finster“.7
Soll ich da nun ”Licht“ und ”Finster“ ”Sätze“ nennen? Nun, wie ich will. – Und wie ist
es mit der ”Übereinstimmung mit der Wirklichkeit“?
8
Wenn ich bestimmte einfache Sprachspiele9
beschreibe, so geschieht es nicht, um mit ihnen
nach und nach die10
Vorgänge der ausgebildeten Sprache – oder des Denkens – aufzubauen
(Nicod, Russell), was nur zu Ungerechtigkeiten führt, – sondern ich stelle die Spiele als solche
hin, und lasse sie ihre aufklärende Wirkung auf die besonderen Probleme ausstrahlen.
11
Man könnte aber12
sagen: ”die Worte ,Licht‘, ,Finster‘ sind hier als Sätze gemeint und
sind nicht einfach Wörter“. Das heißt, sie sind hier nicht so gebraucht, wie wir sie in der
gewöhnlichen Sprache gebrauchen (obwohl wir tatsächlich auch oft so sprechen).
Wenn ich13
plötzlich ohne sichtbaren Anlaß das Wort ”Licht“ isoliert ausspreche, so
wird man allerdings sagen: ”was heißt das? das ist doch kein Satz“ oder: ”Du sagst ,Licht‘,
– was14
soll’s damit?“ Das Aussprechen des Wortes ”Licht“ ist in diesem Fall sozusagen noch
kein (kompletter) Zug des Spiels, das, wie wir annehmen, der Andre spielt.15
Ebenso aber
auch der Satz &er darf nicht kommen“.
156
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1 (R): ∀ p^ 172/3 Learning language through training^
2 (M): 3
3 (V): sentence
4 (V): as if the whole sentence were only one word.”
5 (M): 3
6 (V): tell you whether it is light or dark.
7 (M): 3
8 (V): games,
9 (V): the real
10 (M): 3
11 (V): One could just
12 (V): But if
13 (V): ‘light’ – now what
14 (V): move in the game that we are expecting.
46
The Functioning of a Proposition
Explained with a Language-Game.1
2
I hold my cheek, and someone asks why I do this and I answer: “Toothache”. Obviously
that means the same thing as “I have a toothache”, but I neither imagine the missing words,
nor do I miss them in terms of their meaning. “Therefore it’s also possible that I could utter
the words3
‘I have a toothache’ in that sense – as if I were only saying the last word or as if
the four words were only one.”4
(Elliptical sentence. What is grammar doing when it says: “‘Hat and cane!’ actually means
‘Give me my hat and my cane!’”?)
5
Here, for example, is a simple language-game: Turning on the electric light in a room,
you say “light” to a child (but it can also be an adult), then turning it off you say “dark”;
and you might do that several times, emphasizing your words and doing it for varying lengths
of time. Then you might go into the adjoining room, from there turn on the light in the
first room and get the child to tell you “light” or “dark”.6
So should I call “light” and “dark” “sentences”? Well, as I like. – And what about their
“agreement with reality”?
7
When I describe certain simple language-games,8
I don’t do this so I can use them to construct
gradually the9
processes of a fully developed language – or of thinking – (Nicod, Russell),
for this only results in injustices. – Rather, I present the games as games and allow them to
shine their illuminating effects on particular problems.
10
But one could11
say: “Here the words ‘light’, ‘dark’ are meant as sentences and are not
simply words.” That is, here they haven’t been used as we use them in ordinary language
(although often we really do speak this way as well).
If12
I suddenly utter the word “light”, in isolation, without any discernible reason, then
of course someone will say: “What does that mean? That’s no sentence” or “You say ‘light’
– what13
about it?”. In this case, uttering the word “light” is, as it were, not yet a (complete)
move in the game we assume the other person is playing.14
But neither is the sentence “He’s
not allowed to come”.
156e
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157 Funktionieren des Satzes an einem . . .
16
Wie unterscheidet sich aber17
”Licht“, wenn es den Wunsch nach Licht ausdrückt, von
”Licht“, wenn es konstatiert, daß es im Zimmer licht ist? 18
Daß wir es in jedem Fall anders
meinen? Und worin besteht das? In bestimmten Vorgängen, die das Aussprechen begleiten,
oder in einem bestimmten Benehmen, das ihm vorangeht, eventuell es begleitet, und ihm
folgt? Wir können es einmal in anderm Ton aussprechen als das andere mal, oder mit anderer
Empfindung (Meinung im andern Sinn). Oder es kommt bloß in einem anderen Spielzusammenhang
vor. (Vielleicht antwortet er das einemal auf die Frage was meinst Du: ”ich meinte, Du sollst Licht
machen“.)19
Denken wir uns die Frage: Wie unterscheidet sich ein Zug im Damespiel von der gleichen
Bewegung in Fuchs & Jäger?
20
Wenn ein Mann im Ertrinken ”Hilfe!“ schreit, – konstatiert er die Tatsache, daß er Hilfe
bedarf? daß er ohne Hilfe ertrinken wird? – Dagegen gibt es den Fall, in dem man, quasi
sich beobachtend, sagt ”ich hätte (oder: habe) jetzt den Wunsch nach . . .“.
21
Ich sage das Wort ”Licht!“, – der Andere fragt mich: ”was meinst Du?“ – und ich
antworte:22
”Ich meinte, Du sollst Licht machen“. – Wie war das, als ich es meinte? Sprach
ich den ”kompletten Satz“ in der Vorstellung unhörbar aus, oder den entsprechenden in
einer andern Sprache? (Ja, das kann vorkommen oder auch nicht.) Die Fälle, die man alle
mit dem Ausdruck ”ich meinte“ zusammenfaßt, sind sehr mannigfach.
23
Nun kann man ruhig annehmen: ”ich meinte, Du solltest Licht machen“ heißt, daß mir
dabei ein Phantasiebild von Dir in dieser Tätigkeit vorgeschwebt hat, und ebensogut: der
Satz heißt, daß mir dabei die Worte des vollständigen Satzes in der Phantasie gegenwärtig
waren, oder, daß eins von diesen beiden der Fall war; – nur muß ich wissen, daß ich damit
eine Festsetzung über die Worte ”ich meinte“ getroffen habe und eine engere als die ist,
welche dem tatsächlichen allgemeinen Gebrauch des Ausdrucks entspricht.
24
Wenn das Meinen für uns irgend eine Bedeutung, Wichtigkeit, haben soll, so muß dem
System der Sätze ein System der Meinungen zugeordnet sein, was immer für Vorgänge die
Meinungen sein mögen.25
Modepuppen
26
Aber reden wir doch nicht vom Meinen als einem unbestimmten & uns nicht genau bekannten27
Vorgang sondern vom (tatsächlichen), &praktischen“, Gebrauch des Wortes von den Handlungen,
die wir mit ihm28
ausführen.
Reden wir vom Meinen nur wenn es ein Teil des Sprachkalküls ist (etwa der Teil der aus
Vorstellungsbildern29
besteht). Und30
dann brauchen wir eigentlich das Wort &Meinen“ nicht denn
das suggests immer, daß es sich um Vorgänge handelt die der Sprache nicht angehören sondern ihr
gegenüberstehen & daß es Vorgänge von wesentlich anderer Natur als der sprachlichen sind.
16 (M): 3
17 (V): nun
18 (M): 3 /// – folgt?
19 (E): Dieser Satz wurde von uns (nach dem
Doppelpunkt), einem Verweis auf die
übernächste Bemerkung folgend, ergänzt.
20 (M): 3 ///
21 (M): 3 ///
22 (V): sage:
23 (M): ///
24 (M): 3 3 ///
25 (V): sollen.
26 (M): 3
27 (V): uns unbekannten
28 (O): ihnen
29 (V): aus Bildern unserer Vorstellung
30 (V): &
203
202
203
201v
WBTC046-55 30/05/2005 16:17 Page 314
15
But how16
does “light”, when it expresses the desire for light, differ from “light” when
it states that it is light in the room? 17
By our meaning it differently in each case? And what
does that consist in? In certain processes that accompany the utterance, or in a particular
behaviour that precedes it, perhaps accompanies it, and follows it? We can utter it with different
inflections depending on which is the case, or with a different feeling (meaning in the other
sense). Or it just occurs in a different context in the game. (Perhaps in the one case he’ll answer the
question “What do you mean?” with “I meant that you should turn on the light”.)18
Let’s think of the question: How does a move in draughts differ from the same move in fox and
hunter?
19
If a drowning man calls out “Help!” – is he stating the fact that he needs help? That
without help he’ll drown? – On the other hand there is the case in which, observing oneself,
as it were, one says, “Now I’d like (or: want) . . .”.
20
I utter the word “light!” – someone else asks me: “What do you mean?” – and I answer:21
“I meant that you should turn on the light”. – What was going on when I meant that? Did
I silently utter the “complete sentence” in my mind, or one corresponding to it, but in a
different language? (That can happen, or not.) All the cases that one lumps together in the
expression “I meant” are extremely diverse.
22
One can easily assume: “I meant that you should turn on the light” means that an image
of you performing this action was present in my mind when I said those words, and with
equal validity: this sentence means that when I said those words the words of the complete
sentence were present in my imagination, or that one of these two things happened. – I
just have to be aware that in assuming this I have set up a definition of the words “I meant”
– and that it is narrower than the one that corresponds to the actual common use of the
expression.
23
If meaning something is to have any significance or importance for us, then a system of
meanings must be assigned to the system of propositions, no matter what sorts of processes
meanings might be.24
Mannequins
25
But let’s not talk about “meaning something” as an indefinite process that we don’t know
very well,26
but about the (actual), “practical” use of the word, about the actions that we carry out
with it.
Let’s talk about “meaning something” only when it is part of the language-calculus (say the part
that consists of mental images27
). And then we really don’t need the words “meaning something”,
for that always suggests that we are dealing with processes that don’t belong to language but stand
apart from it, processes whose nature is essentially different from that of language.
The Functioning of a Proposition . . . 157e
15 (M): 3
16 (V): Now how
17 (M): 3 /// – follows it?
18 (E): Wittgenstein has drawn an arrow after
“mean?’” that points to “I meant . . .” in the second
remark below, and we have inserted that
sentence here.
19 (M): 3 ///
20 (M): 3 ///
21 (V): say:
22 (M): ///
23 (M): 3 3 ///
24 (V): meanings are supposed to be.
25 (M): 3
26 (V): don’t know,
27 (V): of images in our imagination
WBTC046-55 30/05/2005 16:17 Page 315
31
Inwiefern stimmt nun das Wort ”Licht“ im obigen Symbolismus oder Zeichenspiel mit
einer Wirklichkeit überein, – oder nicht überein?
Wie gebrauchen wir32
das Wort ”übereinstimmen“? – Wir sagen ”die beiden Uhren stimmen
überein“, wenn sie die gleiche Zeit zeigen, ”die beiden Maßstäbe stimmen überein“,
wenn gewisse Teilstriche zusammenfallen, ”die beiden Farben stimmen überein“, wenn etwa
ihre Zusammenstellung uns angenehm ist oder manchmal wenn wir sagen wollen die beiden
Dinge haben dieselbe Farbe. Wir sagen ”die beiden Längen stimmen überein“, wenn sie gleich
sind, aber auch, wenn sie in einem von uns gewünschten Verhältnis stehen. Und, daß sie
”übereinstimmen“ heißt dann nichts andres, als daß sie in diesem Verhältnis – etwa 1 : 2 –
stehen. So muß also in jedem Fall erst festgesetzt werden, was unter ”Übereinstimmung“
zu verstehen ist. – So ist es nun auch mit der Übereinstimmung einer Längenangabe mit
der Länge eines Gegenstandes.33
Wenn ich sage: ”dieser Stab ist 2m lang“, so kann ich z.B.
erklären,34
wie man nach diesem Satz mit einem Maßstab die Länge des Stabes kontrolliert,
wie man etwa nach diesem Satz einen Meßstreifen für den Stab erzeugt.35
Und ich sage nun,
der Satz stimmt mit der Wirklichkeit überein, wenn der auf diese Weise konstruierte
Meßstreifen mit dem Stab übereinstimmt. Diese Konstruktion eines Meßstreifens illustriert
übrigens, was ich in der ”Abhandlung“ damit meinte, daß der Satz bis an die Wirklichkeit
herankommt. – Man könnte das auch so klar machen: Wenn ich die Wirklichkeit daraufhin
prüfen will, ob sie mit einem Satz übereinstimmt, so kann ich das auch so machen, daß
ich sie nun beschreibe und sehe, ob der gleiche Satz herauskommt. Oder: ich kann die
Wirklichkeit nach grammatischen Regeln in die Sprache des Satzes übersetzen und nun im
Land der Sprache den Vergleich durchführen.
Als ich nun dem Andern erklärte: ”Licht“ (indem ich Licht machte), ”Finster“ (indem
ich es auslöschte), hätte ich auch sagen können und mit genau derselben Bedeutung:
”das heißt36
,Licht‘“ (wobei ich Licht mache) und ”das heißt37
,Finster‘“ etc., und auch
ebensogut:38
”das stimmt mit ,Licht‘ überein“, ”das stimmt mit ,Finster‘ überein“.39
40
Es kommt eben wieder auf die Grammatik des Wortes ”Übereinstimmung“ an, auf seinen
Gebrauch. Und hier liegt die Verwechslung mit ”Ähnlichkeit“ nahe, in dem Sinn, in dem
zwei Personen einander ähnlich sind, wenn ich sie leicht miteinander verwechseln kann.
Ich kann auch wirklich nach der Aussage über die Gestalt eines Körpers eine Hohlform
konstruieren, in die nun der Körper paßt, oder nicht paßt, je nachdem die Beschreibung
richtig oder falsch war, und die konstruierte Hohlform gehört dann in dieser Auffassung
noch zur Sprache (die bis an die Wirklichkeit herankommt).
Aber auch die Hohlform macht kein finsteres Gesicht, wenn der Körper nicht in sie paßt.
41
Wenn das Wort ”Übereinstimmung mit der Wirklichkeit“ gebraucht werden darf,42
dann nicht als metalogischer Ausdruck, sondern als (ein)43
Teil der gewöhnlichen, praktischen,
31 (M): 3 3
32 (V): wir überhaupt
33 (V): mit einer Länge.
34 (V): z.B. eine Erklärung geben,
35 (V): einen 2m langen Stab erzeugt.
36 (V): ist
37 (V): ist
38 (V): und warum hätte ich nicht sagen sollen:
39 (M): passen. Der Plan stimmt mit der Wirklichkeit
überein.
Was er spielt stimmt mit den Noten überein.
40 (M): 3
41 (M): 3
42 (V): gebraucht wird,
43 (O): (einen)
204
205
204v
158 Funktionieren des Satzes an einem . . .
203
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28
Now to what extent does the word “light” in the symbolism or sign-game above agree
– or not agree – with reality?
How do we use29
the word “agree”? – We say “The two clocks agree” when they show
the same time, “The two yardsticks agree” if certain graduation marks coincide, “The two
colours agree” if, say, we like the way they go together or sometimes if we want to say that two
things have the same colour. We say “The two lengths agree” if they are the same, but also if
they are in a desired ratio to each other. And that they “agree” then means nothing more
than that they are in that ratio to each other – say 1 : 2. So in each case what is to be understood
by “agreement” has to be established first. – Now that’s also the way it is with the
agreement between a specification of length and the length of an object.30
If I say: “This rod
is 2 metres long,” I can explain for example how,31
in accordance with this proposition, to
check the length of the rod with a measuring stick, how to use this proposition, say, to produce
a measuring tape for the rod.32
And now I say that the proposition agrees with reality
if the measuring tape that was constructed in this way agrees with the rod. Incidentally, this
construction of a measuring tape illustrates what I meant in the Tractatus when I said that
the proposition comes right up to reality. – One could also make it clear this way: If I want
to test reality to see whether it agrees with a proposition, then I can also do this by describing
it and seeing whether the same proposition results. Or: I can – in accordance with grammatical
rules – translate reality into the language of the proposition and then carry out the
comparison within the domain of language.
So when I was explaining to someone: “Light” (by turning on the light), “Dark” (by turning
it off), I could also have said – with exactly the same meaning: “This means33
‘light’”
(turning on the light) and “This means34
‘dark’” etc., and I could just as well have said:35
“This agrees with ‘light’”, “This agrees with ‘dark’”.36
37
Again it simply depends on the grammar of the word “agreement”, on its use. And
in using it it’s easy to confuse it with “resemblance”, in the sense in which two people
resemble each other if I can easily confuse them.
Using a statement about the shape of a body, I can actually construct a mould, into which
the body fits or doesn’t fit, depending on whether the description was true or false; and then
– following this way of looking at it – the mould I constructed is still a part of language
(which extends right up to reality).
But the mould doesn’t frown if the body doesn’t fit it.
38
If the phrase “agreement with reality” can be39
used, then it is not as a metalogical
expression, but as (a) part of normal, practical language.40
One can say, for instance:
28 (M): 3 3
29 (V): we actually use
30 (V): specification of length and a length.
31 (V): can give an explanation of how,
32 (V): produce a rod 2 metres long.
33 (V): is
34 (V): is
35 (V): and why shouldn’t I have said:
36 (M): To fit. The plan agrees with reality.
What he is playing agrees with the notes.
37 (M): 3
38 (M): 3
39 (V): reality” is
40 (V1): but as part of a calculus, part of normal
language. // but only as a coin in the practical traffic
with money. (V2): but // only // as a part
of speech in the practical use of our ordinary
language.
The Functioning of a Proposition . . . 158e
WBTC046-55 30/05/2005 16:17 Page 317
Sprache.44
Man kann etwa sagen: Im Sprachspiel ”Licht! – Finster!“ kommt der Ausdruck
”Übereinstimmung mit der Wirklichkeit“ nicht vor.
45
In dem Sprachspiel ”Licht – Finster“ kommt keine Frage vor. – Aber wir könnten es
auch mit Fragen spielen.
Das Sprachspiel &eine Geschichte erfinden“ anderseits ein wirkliches Erlebnis erzählen.
So lernt der Maler portraitieren, aber auch ein Bild aus der Phantasie entwerfen, oder ein
Ornament erfinden, etc.
159 Funktionieren des Satzes an einem . . .
205
44 (V1): sondern als Teil eines Kalküls, als Teil der
gewöhnlichen Sprache. // sondern nur als eine
Münze im praktischen Geldverkehr. (V2): sondern
// sondern nur // als Redeteil im praktischen
Gebrauch unserer gewöhnlichen Sprache.
45 (M): ///
WBTC046-55 30/05/2005 16:17 Page 318
In the language-game “Light! – Dark!” the expression “agreement with reality” doesn’t
appear.
41
No question occurs in the language-game “Light! – Dark!”. – But we could just as well
play it with questions.
The language-game “inventing a story”, and on the other hand, narrating a real experience.
Likewise, a painter learns how to paint portraits, but also how to sketch a picture from his
imagination, or how to make an embellishment, etc.
41 (M): ///
The Functioning of a Proposition . . . 159e
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47
Behauptung, Frage, Annahme, etc.
1
Das Frege’sche Behauptungszeichen ist am Platze, wenn es nichts weiter bezeichnen soll,
als den Anfang des Satzes. Man könnte sagen ”den Anfang der Behauptung“, im Gegensatz
zu den Sätzen, die in der Behauptung vorkommen können. Das Behauptungszeichen dient
dann demselben Zweck, wie der Schlußpunkt des vorhergehenden Satzes.
”Ich denke p“ hat dann mit ”4p“ eben nur das Zeichen ”p“ gemeinsam.2
3
Was zum Wesen des Satzes gehört, kann die Sprache schon darum nicht ausdrücken,
weil es für jeden Satz das Gleiche wäre; und ein Zeichen, das in jedem Satz vorkommen
muß, logisch eine bloße Spielerei wäre. Die Zeichen des Satzes sind ja nicht Talismane oder
magische Zeichen, die auf den Betrachter einen bestimmten Eindruck hervorrufen sollen.
Gäbe es philosophische Zeichen im Satz, so müßte ihre Wirkung4
eine solche unmittelbare
sein.
5
Man hat natürlich das Recht, ein Behauptungszeichen zu verwenden, wenn man es im
Gegensatz etwa zu einem Fragezeichen gebraucht. Irreleitend ist es nur, wenn man meint,
daß die Behauptung nun aus zwei Akten bestehe, dem Erwägen und dem Behaupten (Beilegen
des Wahrheitswertes, oder dergl.) und daß wir diese Akte nach dem geschriebenen Satz
ausführen, ungefähr wie wir nach Noten Klavier spielen.6
Mit dem Klavierspielen nach Noten ist nun allerdings das laute oder auch leise, Lesen
nach dem geschriebenen oder gedruckten Satz zu vergleichen und ganz analog. 7
Aber die Zeichen
des Satzes sind nicht Signale zu psychischen Tätigkeiten des Meinens.8
Daher ist auch das B.Zeichen9
in der Fregeschen Schreibweise ganz überflüssig. Es wird (ja) vor alle Sätze gesetzt, kann also in allen
weggelassen werden. Dort wo er einmal einen Satz ohne dieses Zeichen hinschreibt tut er es mit einer
1 (M): 3 ////
2 (V): gemein.
3 (M): 3 ////
4 (V): Funktion
5 (M): 3
6 (V): Noten singen.
7 (M): /// – kann
8 (V1): analog; aber nichts, was wir ”denken“
nennen. // aber nicht das Denken des Satzes. //
Ist also z.B. ein Behauptungszeichen im
geschriebenen Satz, so wird wieder ein
Behauptungszeichen im gelesenen sein (etwa
die Betonung, oder der Stimmfall). Aber nicht,
als ob im geschriebenen Satz die // das //
Zeichen, im gedachten aber die Bedeutung
anwesend wäre. – (V2): /// aber nicht
etwas, was wir ein Denken oder Meinen des Satzes
nennen würden. (V3): /// Aber nicht so als
ob das Denken des Satzes darin bestünde // Aber
das Denken des Satzes besteht [nicht darin – E]
// daß wir nach den Signalen des Satzes
Gedankenoperationen, – u.a. auch das Behaupten –,
ausführten. Als // Und als // seien im Satz die Zeichen
& die Bedeutungen im Denken. (V4): /// aber
nicht Akte des Denkens oder Meinens. (V5):
/// Aber nicht das Denken; insofern da wir nicht
(V6): /// aber nicht eine psychologische Tätigkeit //
Reihe seelischer Akte // des Denkens oder Meinens
des Satzes. (V7): /// Aber die Zeichen des
Satzes sind nicht Signale nach welchen // denen //
wir psychische Operationen ausführen // vornehmen
// – u.a. auch das Behaupten. –
9 (V): das Zeichen
206
207
206v
207
207
206v
160
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47
Assertion, Question, Assumption, etc.
1
Frege’s assertion-sign is properly placed if it’s supposed to indicate nothing more than
the beginning of a sentence. One could say “the beginning of the assertion”, as opposed to
the sentences that can occur within the assertion. Then the assertion-sign serves the same
purpose as the period in the previous sentence.
Then “I think p” and “4p” have only the sign “p” in common.
2
The first and most obvious reason language cannot express what belongs to the essence
of a sentence is that this would be the same for every sentence; and that from a logical point
of view a sign that has to appear in every sentence would be nothing but a frivolous game.
After all, the signs in a sentence aren’t talismans or magical symbols, which are supposed to
cast a certain spell on the beholder.
If there were philosophical signs in a sentence, then their effect3
would have to be an immediate
one like that.
4
Of course one has the right to use an assertion-sign if one uses it in contrast, say, to a question
mark. But it is misleading to believe that therefore an assertion consists of two acts, contemplating
and asserting (attaching a truth-value, or some such thing), and that we perform these
acts following the written sentence, more or less as we play the piano5
following the score.
Now to be sure reading – aloud, or to oneself – in accordance with a written or printed
sentence can be compared to using a score to play the piano, and it is completely analogous
to it. 6
But the signs of a sentence are not signals that trigger psychological acts of meaning.7
Therefore
the assertion-sign8
in Frege’s notation is completely superfluous. (After all), it’s placed in front of
all sentences and can therefore be omitted in all of them. Where he occasionally writes a sentence
1 (M): 3 ////
2 (M): 3 ////
3 (V): function
4 (M): 3
5 (V): sing
6 (M): /// – all sentences
7 (V1): analogous to it; but nothing that we call
“thinking” is. // but the thinking of a sentence
isn’t. // So if, say, there is an assertion-sign in a
written sentence, then in turn there will be an
assertion-sign in the one that is read (say, an
intonation or a cadence of voice). But it’s not as
if the sign // signs // is present in the written sentence,
and the meaning in the one that is
thought. – (V2): /// but something that we
would call a thinking or meaning of a sentence isn’t.
(V3): /// But it’s not as if thinking a sentence consisted
in // But thinking a sentence [doesn’t – Eds.]
consist // in our carrying out thought-operations –
including among others the act of assertion – following
the signals in the sentence. As if // And as if // the
signs were in the sentence & the meanings in thinking.
(V4): /// but acts of thinking or meaning
aren’t. (V5): /// But thinking isn’t; since in so
far as we do not (V6): /// but psychological
activity //a series of psychic acts //of thinking or meaning
a proposition isn’t // aren’t. (V7): /// But
the signs of a proposition are not signals according to
which we carry out // undertake // psychic operations
including, among others, the act of assertion. –
8 (V): sign
160e
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gewissen Unsicherheit die zeigt, daß er eigentlich nicht darauf gefaßt war es je wegzulassen also10
Gebrauch im Kalkül davon zu machen.
11
Freges Ansicht daß in der Behauptung eine Annahme steckt läuft12
darauf hinaus zu sagen daß
jede Behauptung in der Form geschrieben werden könne
Ich behaupte, daß. . . .
13
Eine Sprache (ich meine eine Sprechart) ist denkbar, in der es keine Behauptungssätze
gibt, sondern nur Fragen und die Bejahung und Verneinung.
14
Behauptung, Annahme, Frage. Man kann auf dem Schachbrett einen Zug in einer
Schachpartie machen, – aber auch während eines Gesprächs über ein Schachproblem zur
Illustration, oder wenn man jemand das Spiel lehrt, – etc. Man sagt dann auch etwa: ”angenommen,
ich zöge so, . . .“. So ein Zug hat Ähnlichkeit mit dem, was man in der Sprache
”Annahme“ nennt. Ich sage nun etwa ”im Nebenzimmer ist ein Dieb“, – der Andre fragt
mich ”woher weißt Du das?“ und ich antworte: ”oh ich15
wollte nicht sagen, daß wirklich
ein Dieb im Nebenzimmer ist, ich habe es nur in Erwägung gezogen“. – Möchte man da
nicht fragen: Was hast Du erwogen? wie Du Dich benehmen würdest, wenn ein Dieb da
wäre, oder, was für ein Geräusch es machen würde, oder, was er Dir wohl stehlen würde?
Freges Anschauung könnte man so wiedergeben: daß die Annahme (so wie er das
Wort gebraucht) das ist, was die Behauptung, daß p der Fall ist, mit der Frage, ob p der
Fall ist, gemeinsam hat. Oder auch, daß die Annahme dasselbe ist wie die Frage. Man
könnte auch eine Behauptung immer als eine Frage mit einer Bejahung darstellen. Statt ”Es
regnet“: ”Regnet es? Ja!“16
17
Wenn es so etwas gäbe, wie eine Annahme im Sinne Freges, müßte dann nicht die
Annahme daß p der Fall ist gleich der sein, daß ~p der Fall ist?
18
In dem Sinn, in welchem die Frage ”ist p der Fall?“ die gleiche ist wie ”ist p nicht der Fall?“.
19
Es gibt wirkliche Annahmen, die wir eben durch Sätze von der Form ”angenommen p
wäre (oder: ist) der Fall“ ausdrücken. Aber solche Sätze nennen wir nicht vollständig und
sie erinnern uns an Sätze der Form20
”wenn p der Fall ist, . . .“.
21
Ist nun aber eine solche Annahme ein Teil einer Behauptung? Ist das nicht, als sagte man,
die Frage, ob p der Fall ist, sei ein Teil der Behauptung, daß p der Fall ist?
22
Ist es aber nicht auffällig, daß wir es in unsern gewöhnlichen23
philosophischgrammatischen
Problemen nie damit zu tun haben, ob sie sich auf Behauptungen oder Fragen
beziehen? (Etwa in dem Problem vom Idealismus und Realismus.)
24
Und welcher Art ist ein Satz, wenn sich Einer eine mögliche Situation, etwa ihrer
Seltsamkeit wegen, notiert? Oder: die Erzählung eines Witzes?
Annahme
10 (V): also wirklichen
11 (M): ////
12 (V): kommt
13 (M): 3
14 (M): ////
15 (V): antworte: @ic ”oh ich
16 (V): Ja!” ”Ich behaupte daß es regnet“
17 (M): ///
5
4
6
4
7
18 (M): ///
19 (M): ///
20 (V): sie scheinen sehr ähnlich den Sätzen der
Form
21 (M): ///
22 (M): ///
23 (O): gewöhnlich
24 (M): 3
161 Behauptung, Frage, Annahme, etc.
206v
207
208
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without this sign he does so with some uncertainty, which shows that actually he was never prepared
to leave it out, i.e. to9
use it in a calculus.
10
Frege’s view that an assumption is contained within an assertion amounts to saying that every
assertion can be written in the form
“I assert that . . .”
11
A language (I mean a way of speaking) can be imagined in which there are no assertoric
sentences, but only questions and their affirmation and denial.
12
Assertion, assumption, question. One can make a move on a chess board in a game of
chess – but also in a conversation to illustrate a chess problem, or when one is teaching someone
the game – etc. There one says, for instance: “Say I were to move this way, . . .”. Such
a move is similar to what in language is called an “assumption”. Suppose I say, for instance,
“There’s a thief next door” – the other person asks me “How do you know that?” and I
answer: “Oh, I13
didn’t want to say that there really was a thief next door, I was just turning
it over in my mind”. – Isn’t one then inclined to ask: What were you turning over in
your mind? How you would behave if a thief were there, or what sort of noise he would
make, or what he might steal from you?
One could restate Frege’s view this way: An assumption (as he uses the word) is what the
assertion that p is the case has in common with the question whether p is the case. Or: An
assumption is the same thing as a question. One could also represent every assertion as
a question together with an affirmation. Instead of “It’s raining”: “Is it raining? Yes!”14
15
If there were such a thing as an assumption in Frege’s sense, wouldn’t the assumption
that p is the case have to be the same as that ~p is the case?
16
In the sense in which the question “Is p the case?” is the same as “Is p not the case?”.
17
There are real assumptions, which we express in sentences of the form “Let’s assume
that p were (or: is) the case”. But we don’t call such sentences complete, and they remind
us of sentences18
of the form “If p is the case, . . .”.
19
But now: is such an assumption part of an assertion? Isn’t that as if one were to say that
the question whether p is the case is part of the assertion that p is the case?
20
But isn’t it striking that in our usual philosophical-grammatical problems we never
deal with whether they refer to assertions or questions? (For example, in the problem of
idealism and realism.)
21
And what sort of sentence is it if someone jots down a possible situation, say because of
its oddity? Or: the narration of a joke?
Assumption
9 (V): i.e. really to
10 (M): ////
11 (M): 3
12 (M): ////
13 (V): answer: !I “oh, I
14 (V): Yes!” “I assert that it’s raining”
15 (M): ///
16 (M): ///
17 (M): ///
18 (V): they seem to be quite similar to sentences
19 (M): ///
20 (M): ///
21 (M): 3
Assertion, Question, Assumption, etc. 161e
5
4
6
4
7
WBTC046-55 30/05/2005 16:17 Page 323
25
Sprachspiel: eine Geschichte erfinden. Oder: eine Geschichte erfinden und zeichnen. –
Etc.
26
Es ist uns, als könnten wir sagen,27
der fragende Tonfall sei dem Sinn der Frage angemessen.
28
Wir können29
uns auch eine Sprache denken, die nur aus Befehlen besteht. So eine Sprache
verhält sich zu der unseren, wie eine primitive Arithmetik zu unserer. Und wie jene Arithmetik
nicht wesentlich unvollständig ist, so ist es auch die primitivere Form der Sprache nicht.30
31
Denken wir an die große Mannigfaltigkeit32
der Sprachspiele:
Eine Mitteilung machen wie &Licht“, &Finster“.33
Einen Befehl geben &mach Licht!“ &Lösch aus“.
Auf Fragen Licht? Finster? mit ja & nein antworten.
Einen Befehl ausführen.
Fragen & die Antwort auf ihre Richtigkeit prüfen.34
Negative & positive Befehle ausführen. Disjunktive.
Eine Vermutung aussprechen (Aufschlagen von Karten) & sie verifizieren.
Die Form eines Satzes vereinfachen (~~~p = ~p), Schließen.
Ein35
angewandtes Rechenexempel lösen.
Eine Zeichnung36
herstellen & sie beschreiben.
Einen Hergang erzählen.
Eine Erzählung erfinden.
Eine Hypothese aufstellen & prüfen.
Eine Tabelle anlegen.
Grüßen.
37
Es hilft hier immer sich den Fall des Kindes vorzustellen welches38
sprechen lernt oder auch den39
eines Volkes40
das nur eine41
primitive Sprache besitzt.
Aber auch der Erwachsene lernt neue Sprachspiele42
wenn er die Form des Beweises kennen lernt
oder lernt Tabellen43
anzulegen oder abzulesen oder graphische Darstellungen zu machen & zu
verwenden.
44
Geschicklichkeits- & Hasardspiele sie sind viel fundamentaler verschieden als ihre Bezeichnung
erkennen läßt.
45
Richtig & falsch spielen.
Richtig & falsch vermuten wobei man richtig spielt.
46
Der Tonfall der Frage angelernt & instinktiv. Und was macht es aus, ob er angelernt ist oder nicht;
da wir, wenn wir ihn einmal gebrauchen, doch nicht auf das Lernen zurückgreifen. Später ist er
jedenfalls instinktiv, was immer sein Ursprung ist.
25 (M): 3
26 (M): 3
27 (V): sagen;
28 (M): 3
29 (V): könnten
30 (R): S^ 21/1 & Bemerkung gehört nicht hierher. ∀
31 (M): | 3
32 (V): die Mannigfaltigkeit
33 (V): &Finster“, einen
34 (V): Antwort kontrollieren.
35 (V): Eine Rechnung machen^ Ein
36 (O): Zeichung
37 (M): 3
38 (V): welches di
39 (V): den Fall
40 (O): Volkesstammes (V): Volksstammes
41 (V): das eine
42 (V): neue Formen der Sprache
43 (V): Tabellen oder
44 (M): 3
45 (M): 3
46 (M): ///
209
208v
209
162 Behauptung, Frage, Annahme, etc.
WBTC046-55 30/05/2005 16:17 Page 324
22
Language-game: inventing a story. Or inventing and drawing a story. – Etc.
23
We feel that we could say that the inflection used in a question is appropriate for the sense of
the question.
24
We can also imagine a language that consists only of commands. Such a language relates
to ours as a primitive arithmetic does to ours. And just as that arithmetic is not essentially
incomplete, neither is the more primitive form of language.25
26
Let’s think about the great variety27
of language-games:
Making a report, such as “light”, “dark”.28
Issuing a command “Turn on the light!”, “Lights out”.
Answering the questions “Light?”, “Dark?” with yes and no.
Carrying out an order.
Asking a question and checking the correctness of the answer to it.29
Carrying out negative and positive orders. Disjunctive ones.
Uttering a hunch (turning up cards) and verifying it.
Simplifying the form of a proposition (~~~p = ~p), drawing conclusions.
Solving30
a problem of applied mathematics.
Making a drawing and describing it.
Narrating a course of events.
Inventing a story.
Setting up and testing a hypothesis.
Compiling a table.
Greeting someone.
31
Here it’s always helpful to imagine the case of a child who is learning to talk, or that32
of a
people who have only a33
primitive language.
But a grown-up too learns new language-games34
when he becomes acquainted with the form
of a proof or learns how to construct or read tables or how to draw and use graphs.
35
Games of skill and games of chance: they differ more fundamentally than their names reveal.
36
Playing correctly and incorrectly.
Guessing correctly and incorrectly, while playing correctly.
37
The inflection of a question, learned and instinctive. And what’s the difference whether it’s been
learned or not? Because once we use it we don’t fall back on this learning. In any case it’s instinctive
later on, whatever its origin.
22 (M): 3
23 (M): 3
24 (M): 3
25 (R): p^ 21/1 & remark don’t belong here. ∀
26 (M): | 3
27 (V): the variety
28 (V): “dark”, a
29 (V): Questions and checking their answers.
30 (V): Making a calculation^ Solving
31 (M): 3
32 (V): or the case
33 (V): of a tribe that has a
34 (V): new forms of language
35 (M): 3
36 (M): 3
37 (M): ///
Assertion, Question, Assumption, etc. 162e
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47
Man sagt: Die Affen sprechen nicht, weil ihnen die geistigen Fähigkeiten dazu fehlen. Das heißt:
sie denken nicht, darum sprechen sie nicht. Aber sie sprechen eben nicht & das ist alles. Befehlen,
fragen, erzählen,48
plauschen, sind so natürliche Handlungen wie gehen, trinken, spielen.
49
Das hängt damit zusammen daß man meint das Lernen der Sprache bestehe darin daß man
Gegenstände benennt & zwar: Menschen, Gattungen, Farben, Schmerzen, Stimmungen, Zahlen, etc.
50
Wie gesagt, das Benennen ist etwas Ähnliches wie einem Ding eine Namenstafel51
umhängen.
Man kann es eine Vorbereitung zum weiteren Gebrauch eines52
Worts nennen. Aber worauf ist es
eine Vorbereitung?!
53
&Wir benennen die Dinge, & können nun über sie reden, uns54
in der Rede auf sie beziehen“:
als ob mit dem Akt des Benennens schon das was wir weiter tun gegeben sei. Als ob es nur Eines
gäbe was heißt: von Dingen reden. Während wir das Verschiedenartigste mit unseren Sätzen tun.
55
Denken wir nur an die Ausrufe56
mit ihren ganz verschiedenen Funktionen: Wasser!57
Weg!58
Au! – Hilfe! – Schön! – Nicht! –
59
Bist Du nun noch geneigt diese Wörter &Namen“ zu nennen?
60
Es sagte mir einmal jemand: &Wie wäre es, wenn die Menschen ihre Schmerzen nicht äußerten
(nicht stöhnten, etc.), – dann könnte man einem Kind nicht das Wort *Zahnschmerz‘ beibringen!“
– Denken wir uns nun, es würde Einer sagen: &Ich nehme an, das Kind sei ein Genie & erfinde selbst
einen Namen für den Schmerz, obwohl ihm keiner gelehrt wurde. – Aber nun könnte er sich freilich
mit diesem Wort nicht verständlich machen!“ Also versteht es ihn, kann aber seine61
Bedeutung niemandem
erklären? Aber was heißt es denn, daß er &seinen Schmerz62
benannt hat“? Was ist die
Verbindung des Wortes das63
er ausspricht mit dem Schmerz? Und was für eine Funktion hat dieses
Wort? Wie hat er das gemacht den Schmerz zu benennen?? Und was immer er getan hat, was hat
es für einen Zweck? – Wenn man sagt &er hat dem Schmerz einen Namen gegeben“ so vergißt man
daß sozusagen schon64
alles mögliche in der Sprache vorbereitet sein muß damit das bloße
Benennen65
einen Sinn hat. Und wenn wir davon reden daß er dem Schmerz einen Namen gibt ist
die66
Grammatik des Wortes &Schmerz“ hier das Vorbereitete:67
es zeigt den Posten an an den das
neue Wort gestellt wird.
209v
163 Behauptung, Frage, Annahme, etc.
47 (M): 3
48 (V): beschreiben,
49 (M): 3
50 (M): 3
51 (V): wie einem Ding Mineral eine Etiquette
52 (V): des
53 (M): 3
54 (V): redenA, uns
55 (M): 3
56 (V): die Verschiedenen Ausrufe
57 (V): Trinken! // (Ich will trinken)
58 (V): (Geh weg)
59 (M): 3
60 (M): 3
61 (V): kann seine
62 (V): er &seine Schmerzen
63 (O): daß
64 (V): daß schon
65 (O): benennen
66 (O): gibt die
67 (O): Vorbereitete
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38
One says: Monkeys don’t talk because they lack the mental abilities for it. That is: They don’t think,
therefore they don’t talk. But they simply don’t talk, and that’s it. Commanding, asking,39
narrating,
chatting are activities that are as natural as walking, drinking, playing.
40
This is connected with the view that learning a language consists in naming objects, that is to
say: humans, genres, colours, pains, moods, numbers, etc.
41
As mentioned above, naming is something like hanging a name-plate on a thing.42
One can call
it a preparation for the further use of a word.43
But what is it a preparation for?!
44
“We name things, and then we can talk about them, can refer to them in speech.” As if what
we do afterwards were already given in the act of naming. As if there were only one thing that means:
talking about things. Whereas we do the most varied things with our sentences.
45
Just think of exclamations46
with their completely different functions: Water!47
Leave48
! Ow! –
Help! – Beautiful! – Don’t! –
49
Now are you still inclined to call these words “names”?
50
Once someone said to me: “What would it be like if humans didn’t express their pains (didn’t
groan, etc.)? – then we couldn’t teach a child the word ‘toothache’!” – Now let’s imagine that someone
were to say: “I’m assuming that the child is a genius and invents a name for the pain himself,
even though he wasn’t taught one. – But then, of course, he couldn’t make himself understood when
he used this word!” So he understands the name, but can’t51
explain its meaning to anybody? But
what does it mean to say that he “has named his pain52
”? What is the connection between the word
he utters and the pain? And what kind of function does this word have? How did he accomplish naming
the pain?? And whatever he did, what purpose does it have? – When one says “He has given a
name to his pain” then one forgets that all sorts of things in the language have to have been prepared
in advance, as it were, for the mere act of naming to make sense. And when we speak about
his giving a name to a pain, then here it is the grammar of the word “pain” that has been prepared;
it shows the post where the new word is stationed.
38 (M): 3
39 (V): describing%
40 (M): 3
41 (M): 3
42 (V): like labeling a thing mineral.
43 (V): the word.
44 (M): 3
45 (M): 3
46 (V): of different exclamations
47 (V): Drinking! // (I want to drink)
48 (V): (Go away)
49 (M): 3
50 (M): 3
51 (V): name, can’t
52 (V): pains
Assertion, Question, Assumption, etc. 163e
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210 Gedanke.
Denken.
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Thought.
Thinking.
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48
Wie denkt man den Satz ”p“,
wie erwartet (glaubt, wünscht) man,
daß p der Fall sein wird?
Mechanismus des Denkens.
Das Unverständnis der Grammatik des Wortes &Denken“ & psychologisches Unverständnis angesehen
als nicht Verstehen eines komplizierten mechanischen Vorgangs.
Die Idee ist: Denken, Glauben, etc. als Tätigkeiten in denen der Satz vorkommt, etwa wie1
die Karten
in den Operationen des Musterwebstuhls.
2
Man ist3
versucht, zu fragen: ”wie denkt man den Satz p,4
wie erwartet man, daß das
und das eintreffen wird“ (wie macht man das).5
6
”Wie arbeitet der Gedanke, wie bedient er sich seines Ausdrucks?“ – dies scheint analog
der Frage:7
”wie arbeitet der Musterwebstuhl, wie bedient er sich der Karten?“
Das Gefühl ist, daß in8
dem Satz &ich glaube, daß p der Fall ist“ etwas Wesentliches, der
eigentliche Vorgang9
des Glaubens, nur angedeutet sei; daß sich diese Andeutung durch eine10
Beschreibung dieses11
Mechanismus müsse ersetzen lassen. Eine Beschreibung worin &p ist der
Fall“ wie12
die Karten in der Beschreibung des Musterwebstuhls vorkäme. Und daß nun diese
Beschreibung erst der komplette Ausdruck des Glaubens wäre.
Vergleichen wir nun das Glauben mit dem Aussprechen eines Satzes. Es gehen da auch sehr
komplizierte Vorgänge in unseren Sprechmuskeln, Nerven etc., etc., vor sich. Diese begleiten den
ausgesprochenen Satz & er bleibt das Einzige was uns interessiert. Und zwar als Bestandteil13
eines
Kalküls,14
nicht eines Mechanismus.15
1 (V): vorkommt, wie
2 (M): 3
3 (V): ist (durch die irreführende Grammatik) //
(durch unsere Grammatik irregeführt)
4 (V): Satz . . . ,
5 (V): das). Und in dieser falschen Frage liegt wohl
die ganze Schwierigkeit in nuce enthalten.
6 (M): 3
7 (V): Ausdrucks?” – das ist // klingt // dies
möchte man fragen analog der Frage:
8 (V): mit
9 (V): der Vorgang
10 (V): eine eigentliche
11 (V): des
12 (V): worin p wie
13 (V): Und ist Bestandteil
14 (O): Kalkül,
15 (V): 3 Das Gefühl ist, daß mit dem Satz ”ich
glaube, daß p der Fall ist“ der Vorgang des
Glaubens nicht beschrieben sei (daß vom
Webstuhl nur die Karten gegeben seien und
alles übrige bloß angedeutet ist). Daß man
die Beschreibung ”ich glaube p“ durch die
Beschreibung eines Mechanismus ersetzen
könnte, worin dann p, d.h. jetzt die Wortfolge
”p“, wie die Karten im Webstuhl nur als ein
Bestandteil vorkommen würde. Aber hier ist
der Irrtum: Was immer diese Beschreibung
enthielte, wäre für uns wertlos, außer eben der
Satz p mit seiner Grammatik. Sie ist quasi der
eigentliche Mechanismus, in welchem // dem //
er eingebettet liegt.
211
210v
211
210v
211
212
165
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48
How does one Think the Proposition
“p”, how does one Expect (Believe,
Wish) that p will be the Case?
Mechanism of Thinking.
A lack of understanding of the grammar of the word “thinking” and a lack of psychological understanding
seen as not understanding a complicated mechanical process.
The idea is: thinking, believing, etc. as activities in which the proposition occurs more or less like1
the punch cards in the workings of a pattern-loom.
2
One3
is tempted to ask: “How does one think the proposition p, how does one expect
that such and such will happen?” (How does one do that?)4
5
“How does thought work, how does it make use of its expression?” – this seems analogous
to the question:6
“How does the pattern-loom work, how does it make use of the punch cards?”
The feeling is that something essential, the actual process7
of believing, is only alluded to in8
the sentence “I believe that p is the case”; that this allusion must be replaceable by a9
description of
this10
mechanism. A description in which “p is the case” would11
occur like the punch cards in the
description of a pattern-loom. And that then only this description would be the complete expression
of believing.
Let’s now compare believing with uttering a sentence. Here too very complicated processes take
place in the muscles and nerves in our throat, etc., etc. These processes accompany the spoken
sentence, yet it remains the only thing that we are interested in – the sentence as part12
of a
calculus, not of a mechanism.13
1 (V): occurs like
2 (M): 3
3 (V): (Because of misleading grammar) one //
(Misled by our grammar) one
4 (V): that?) And in this false question the entire
difficulty is contained in a nut-shell.
5 (M): 3
6 (V): expression?” – that is // sounds // one is
inclined to ask this analogously to the question:
7 (V): the process
8 (V): with
9 (V): by an actual
10 (V): of the
11 (V): in which p would
12 (V): are interested in. And it is part
13 (V): 3 The feeling is that the process of believing
is not described by the sentence “I believe
that p is the case” (that only the punch cards in
a pattern-loom are given, and that everything else
is merely alluded to). That one could replace the
description “I believe p” with the description
of a mechanism, in which p, i.e. the wordsequence
“p”, occurred only as one component
part, like the punch cards in a pattern-loom. But
here is the mistake: whatever this description
contained would be worthless to us, except for
the sentence “p”, with its grammar. Grammar
is as it were the true mechanism in which the
proposition is embedded.
165e
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16
Wenn man fragt ”wie macht der Satz17
das, daß er darstellt?“ So könnte die Antwort
sein: ”Weißt Du es denn (wirklich) nicht? Du siehst es doch, wenn Du ihn benützt18
“. Es
ist ja nichts verborgen.
Wie macht der Satz das? – Weißt Du es denn nicht? Es ist ja nichts versteckt.
19
Daß ”alles fließt“, scheint uns am Ausdruck der Wahrheit zu hindern, denn es ist, als
ob wir sie nicht auffassen könnten, da sie uns entgleitet.
Aber es hindert uns eben nicht am Ausdruck. – Was es heißt, etwas Entfliehendes in der
Beschreibung festhalten zu wollen, wissen wir. Das geschieht etwa, wenn wir das Eine
vergessen, während wir das Andere beschreiben wollen. Aber darum handelt es sich doch
hier nicht. Und so ist das Wort20
”entfliehen“ anzuwenden.21
22
Aber auf die Antwort ”Du weißt ja, wie es der Satz macht, es ist ja nichts verborgen“,
möchte man sagen: ”ja, aber es fließt alles so rasch vorüber und ich möchte es gleichsam
breiter auseinander gelegt sehen“.
23
Aber auch hier irren wir uns. Denn es geschieht dabei auch nichts, was uns durch die
Geschwindigkeit entgeht.
24
Warum können wir uns keine Maschine mit einem Gedächtnis denken? Es wurde
oft gesagt, daß das Gedächtnis darin besteht, daß Ereignisse Spuren hinterlassen, in denen
nun gewisse Vorgänge vor sich gehen müßten. Wie wenn Wasser sich ein Bett macht und
das folgende Wasser in diesem Bett fließen muß; der eine Vorgang fährt das Geleise25
aus,
das den andern führt.26
Geschieht dies nun aber in einer Maschine, wie es wirklich
geschieht, so sagt niemand, die Maschine habe Gedächtnis, oder habe sich den einen
Vorgang gemerkt.
27
Nun ist das aber ganz so, wie wenn man sagt, eine Maschine kann nicht denken, oder
kann keine Schmerzen haben. Und hier kommt es darauf an, was man darunter versteht
”Schmerzen zu haben“. Es ist klar, daß ich mir eine Maschine denken kann, die sich genau
so benimmt (in allen Details), wie ein Mensch der Schmerzen hat. Oder vielmehr: ich kann
den Andern eine Maschine nennen, die Schmerzen hat, d.h.: den andern Körper. Und ebenso,
natürlich, meinen Körper. Dagegen hat das Phänomen der Schmerzen, wie es auftritt, wenn
”ich Schmerzen habe“, mit meinem Körper, d.h. mit den Erfahrungen, die ich als Existenz
meines Körpers zusammenfasse, gar nichts zu tun. (Ich kann Zahnschmerzen haben ohne
Zähne.) Und hier hat nun die Maschine gar keinen Platz. – Es ist klar, die Maschine kann
nur einen physikalischen Körper ersetzen. Und in dem Sinne, wie man von einem solchen
sagen kann, er ”habe“ Schmerzen, kann man es auch von einer Maschine sagen. Oder wieder,
die Körper, von denen wir sagen, sie hätten Schmerzen, können wir mit Maschinen
vergleichen, und auch Maschinen nennen.
Könnte eine Maschine denken? – Könnte sie Schmerzen haben? In dem Sinne in welchem der
tierische Körper Schmerzen hat – ja.28
Wenn ich diesen eine Maschine nennen will.29
Hier kommt es
16 (M): ← 3
17 (V): Gedanke
18 (V): wenn Du denkst
19 (M): ( (R): [gehört nicht hierher, sondern zur
Betrachtg. der Zeit oder zu Solipsismus.]
20 (V): ist der Ausdruck
21 (M): )
22 (M): ← 3
23 (M): ///
24 (M): 3 ///
25 (O): das Gleise
26 (V): der eine Vorgang fährt für den nächsten das
Gleise aus.
27 (M): 3
28 (V): ja. warum
29 (O): will:
212
213
212v
166 Wie denkt man den Satz ”p“ . . .
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14
If one asks “How does a sentence15
go about representing?”, the answer could be:
“Do you (really) not know this? After all, you see it when you use one16
“. For nothing is
concealed.
How does a sentence do that? – Do you really not know this? After all, nothing is hidden.
17
That “everything is in flux” seems to hinder us in expressing the truth, for it’s as if we
can’t grasp it, since it slips away from us.
But (and this is the point) this doesn’t prevent us from expressing something. – We know
what it means to want to pin down something in a description that’s fleeting. This happens,
for instance, when we forget the one thing as we’re trying to describe the other. But that
isn’t what this is all about. And that’s the way the word18
“fleeting” is to be used.19
20
But one is inclined to respond to the answer “You know how the proposition does it;
after all, nothing is concealed” by saying: “Yes, but everything goes by so quickly, and I’d
like to see it with all of its parts spread out, as it were”.
21
But here too we’re mistaken. For in this process nothing that happens escapes us because
of speed.
22
Why can’t we imagine a machine with a memory? It was often said that memory consists
in events leaving behind traces, in which certain events then have to occur. As when water
erodes a bed for itself and the water that follows has to flow in this bed; the first process
lays down the track that guides the latter.23
But if this happens in a machine, as indeed it
does, nobody says that the machine has a memory or that it remembered the first process.
24
But this is exactly like saying that a machine cannot think or cannot have pain. And here
it depends on what one understands by “having pain”. It’s clear that I can imagine a machine
that behaves exactly (in all details) like a human in pain. Or rather: I can call someone else
– i.e. his body – a machine that has pain. And of course my own body as well. On the other
hand, the phenomenon of pain as it occurs when “I’m in pain” has nothing at all to do with
my body, i.e. with the experiences that I sum up as the existence of my body. (I can have a
toothache without teeth.) And here the machine has absolutely no place. – It is clear that
a machine can only replace a physical body. And in the sense in which one can say of such
a body that it “has” pain, one can also say it of a machine. Or, once again, we can compare
the bodies of which we say that they are in pain to machines, and can also call them machines.
Could a machine think? – Could it have pain? In the sense in which an animal’s body feels pain –
yes.25
If I want to call the animal’s body a machine. Here it depends on how the expression “have
14 (M): ← 3
15 (V): thought
16 (V): when you think one
17 (M): ( (R): [doesn’t belong here, but rather to
the considerations of time or to solipsism.]
18 (V): expression
19 (M): )
20 (M): ← 3
21 (M): ///
22 (M): 3 ///
23 (V): the first process lays down the track for the
next.
24 (M): 3
25 (V): yes. Why
How Does one Think the Proposition “p” . . . 166e
WBTC046-55 30/05/2005 16:17 Page 333
drauf an, wie der Ausdruck &Schmerzen haben“ angewandt wird. Aber im Satz &Ich habe
Schmerzen“ bezeichnet &ich“ keinen Körper also30
auch keine Maschine.
31
Und ganz ebenso verhält es sich mit dem Denken und dem Gedächtnis.
32
Es ist uns – wie gesagt – als ginge es uns mit dem Gedanken so, wie mit einer Landschaft,
die wir gesehen haben und beschreiben sollen, aber wir erinnern uns ihrer nicht genau genug,
um sie in allen ihren Zusammenhängen beschreiben zu können. So, glauben wir, können
wir das Denken nachträglich nicht beschreiben, weil uns alle die33
feineren Vorgänge dann
verloren gegangen sind.
Diese feinen Verhäkelungen möchten wir sozusagen unter der Lupe sehen.
167 Wie denkt man den Satz ”p“ . . .
30 (V): also nicht
31 (M): ///
32 (M): ←
33 (V): die vielen
213
214
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pain” is used. But in the sentence “I’m in pain”, “I” doesn’t signify a body, and therefore neither
does it signify26
a machine.
27
And that’s exactly the way it is with thinking and memory.
28
It seems to us – as we’ve said – as if we fared with thought the way we do with a landscape
we’ve seen and are supposed to describe, but which we don’t remember well enough
to be able to describe in all of its detail. Likewise, we believe, we can’t describe thinking
after it’s happened, because by then all of the29
more subtle processes have escaped us.
We would like to see these fine interconnections under a magnifying glass, so to speak.
26 (V): therefore not
27 (M): ///
28 (M): ←
29 (V): the many
How Does one Think the Proposition “p” . . . 167e
WBTC046-55 30/05/2005 16:17 Page 335
49
”Was ist ein Gedanke, welcher Art
muß er sein, um seine Funktion
erfüllen zu können?“
Hier will man sein Wesen aus seinem
Zweck, seiner Funktion erklären.
1
Wir fragen: Was ist ein Gedanke, welcher Art muß etwas sein, um die Funktion des
Gedankens verrichten zu können? Und diese Frage ist2
analog der: Was ist, oder, wie funktioniert,
eine Nähmaschine. ”Wie macht sie das?“ Aber die Antwort könnte sein: Schau den
Stich an; alles, was der Nähmaschine wesentlich3
ist, ist in ihm zu sehen; alles andre kann
so, oder anders sein.
4
Wir fragen, wie muß der Gedanke beschaffen sein, um seine Funktion5
zu erfüllen; aber
was ist denn seine Funktion?6
Wenn sie nicht in ihm selbst liegt (d.h. wenn sie nicht ist,
(das) zu sein, was er ist), liegt sie in seiner Wirkung; aber die interessiert uns nicht.
7
Wir sind nicht im Bereiche der Erklärungen und jede Erklärung klingt uns trivial.
8
Aber dieser Verzicht auf die Erklärung macht es so schwer zu sagen, was der Gedanke
uns eigentlich bedeutet.
9
Man kann etwa sagen: er rechnet auf Grund von Gegebenem und endet in einer
Handlung.
10
Willst Du sehen wie der Gedanke verwendet wird: die Berechnung der Wandstärke eines
Kessels und die dieser entsprechende11
Verfertigung12
muß ein Beispiel des Denkens &
seiner Verwendung sein.13
14
Der Schritt, der von der Berechnung auf dem Papier zur Handlung führt, ist noch ein
Schritt der Rechnung.
1 (M): 3
2 (V): ist ganz
3 (V): wesentlich
4 (M): 3
5 (V): Bestimmung
6 (V): Bestimmung?
7 (M): 3
8 (M): ///
9 (M): 3
10 (M): 3
11 (O): entsprechenden
12 (V): Die Berechnung der Wandstärke eines
Kessels und, der entsprechenden, Verfertigung
13 (V): Verfertigung ist ein sicheres Beispiel des
Denkens.
14 (M): ///
215
216
168
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49
“What is a Thought, What
Must it be Like for it to
Fulfil its Function?”
Here one Wants to Explain its
Essence by its Purpose, its Function.
1
We ask: What is a thought, what does something have to be like in order to carry out the
function of a thought? And this question is2
analogous to: What is a sewing machine, or
how does one work? “How does it do that?” But the answer could be: Look at the stitch;
everything that is essential3
to the sewing machine can be seen in it; everything else can
be this way or that.
4
We ask: How must a thought be constituted in order to fulfil its function?5
But what is
its function,6
anyway? If it isn’t contained in the thought itself (i.e. if the function isn’t to
be what the thought is), then it’s contained in its effect; but that doesn’t interest us.
7
We are not in the realm of explanations, and every explanation sounds trivial to us.
8
But this renunciation of an explanation makes it so difficult to say what “thought” actually
means to us.
9
One can say, for example: Thought calculates on the basis of what it’s been given, and
ends up in an action.
10
Do you want to see how thought gets used? The calculation of the thickness of the wall of
a boiler, and the11
construction that conforms to it is an example of thinking and its use.12
13
The step that leads from a calculation on paper to an action is still a step within the
calculation.
1 (M): 3
2 (V): is completely
3 (V): essential
4 (M): 3
5 (V): purpose?
6 (V): purpose,
7 (M): 3
8 (M): ///
9 (M): 3
10 (M): 3
11 (V): and of the
12 (V): to it is a good example of thinking.
13 (M): ///
168e
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15
Wir sagen, wir werden das Denken untersuchen von dem Standpunkt aus, daß es auch
von einer Maschine ausgeführt werden könnte.
Aber hier befinden wir uns in einer falschen Betrachtungsweise. Wir sehen das Denken
als16
einen Vorgang wie das Schreiben an, oder das Weben, das17
Erzeugen eines Stoffes,
etc. Und dann läßt sich natürlich sagen, daß dieser Vorgang der Erzeugung sich im
Wesentlichen auch maschinell muß denken lassen.
169 ”Was ist ein Gedanke . . .
15 (M): ////
16 (V): für
17 (O): Weben das
WBTC046-55 30/05/2005 16:17 Page 338
14
We say that we’ll investigate thinking from the standpoint of its also being carried out
by a machine.
But here we’ve taken a false approach. We’re viewing thinking as a process like writing,
or weaving, making cloth, etc. And then of course it will be said that by its very nature this
process of making things must also be conceivable as something a machine can do.
14 (M): ////
“What is a Thought . . . 169e
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50
1
Ist die Vorstellung das Porträt par
excellence, also grundverschieden,
etwa, von einem gemalten Bild und
durch ein solches oder etwas
Ähnliches nicht ersetzbar? Ist sie das,
was eigentlich eine bestimmte
Wirklichkeit darstellt, – zugleich
Bild und Meinung?2
Denn so ein Wunderding, scheint es, brauchen wir.
Und die Vorstellung scheint dies3
zu sein: Denn4
wir können uns nicht fragen, ob, z.B., unsre
Vorstellung von diesem Menschen wirklich die Vorstellung von diesem Menschen sei, & nicht
vielleicht von5
einem Andern der ihm nur ähnlich sieht.
6
Aber ist nicht der Satz dieses Wunderding – der7
sagt, was er meint?
8
Sokrates zu Theaitetos: ”Und wer vorstellt, sollte nicht etwas vorstellen?“
Th.: ”Notwendig“.
Sok.: ”Und wer etwas vorstellt, nichts Wirkliches?“
Th.: ”So scheint es“.9
Und wer malt sollte nicht etwas malen – & wer etwas malt, nichts wirkliches? – Ja, was10
meinst
Du: das Bild was er malt oder den Gegenstand etwa den Menschen den es darstellt?
11
”Ist die Vorstellung nur die Vorstellung, oder ist sie Vorstellung von Etwas in der
Wirklichkeit?“
12
Und von dieser Frage aus13
könnte man auch die Beziehung der Vorstellung zum gemalten
Bild erfassen. Denn ich kann nicht zweifeln, wenn ich mir Napoleon vorstelle, ob es wirklich
Napoleon ist den ich mir vorstelle oder nicht nur jemand der ihm ähnlich sieht!
1 (M): v
2 (V): Intention?
3 (V): es
4 (V): sein% denn
5 (V): sei, oder von
6 (M): v
7 (V): Wunderding? der
8 (M): v 5555 (R): [Zu § 21]
9 (E): Vgl. Platon, Theaitetos, 189a.
10 (V): was
11 (M): 5555
12 (M): v
13 (V): aus
217
170
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50
1
Is a Mental Image a Portrait Par
Excellence, and thus Fundamentally
Different from, say, a Painted Picture,
and not Replaceable by one or by any
such Thing? Is it a Mental Image
that Really Represents a Particular
Reality – Simultaneously Picture
and Meaning?2
For it seems that we need such a marvel.
And a mental image seems to be just that3
: For we can’t ask ourselves whether, for example,
our mental image of this person is really the mental image of him and not perhaps4
of someone
else who just looks like him.
5
But isn’t a proposition this marvel – that6
says what it means?
7
Socrates to Theaetetus: “And he who thinks, doesn’t he think something?”
Theaetetus: “Necessarily so.”
Socrates: “And he who thinks something, doesn’t he think something real?”
Theaetetus: “So it seems.”8
And he who paints, doesn’t he paint something – and he who paints something, doesn’t he
paint something real? – Well, which9
do you mean: the picture he’s painting, or the object – say the
person it represents?
10
“Is a mental image only a mental image or is it a mental image of something real?”
11
And using this question as a starting-point, one can also grasp the relationship of a
mental image to a painted picture. For if I imagine Napoleon I can’t be in doubt whether it really
is Napoleon I am imagining or only someone who looks like him!
1 (M): v
2 (V): Intention?
3 (V): be it
4 (V): him, or
5 (M): v
6 (V): marvel? One that
7 (M): v 5555 (R): [To § 21]
8 (E): Cf. Plato, Theaetetus, 189a.
9 (V): which
10 (M): 5555
11 (M): v
170e
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14
Die Frage könnte aber nicht heißen: ”Ist die Vorstellung immer Vorstellung von etwas,
was in der Wirklichkeit existiert“ – denn das ist sie offenbar nicht immer –; sondern, es müßte
heißen: bezieht sich die Vorstellung immer, wahr oder falsch, auf Wirklichkeit. – Denn das
kann man von einem gemalten Bild nicht sagen. – Aber worin besteht dieses ”sich auf die
Wirklichkeit beziehen“? Es ist doch wohl die Beziehung des Porträts zu seinem Gegenstand.
15
Aber warum sollte man dann nicht sagen, daß eine Vorstellung Vorstellung eines
Traumes sei?
Wenn mir heute geträumt hat, daß N mich besuche und N besucht mich nun wirklich,
so war darum jene Traumphantasie keine Erwartung, und die Tatsache, daß N mich besuchte,
keine Erfüllung einer16
Erwartung.
Wie kommt es daß es diese Situation nicht gibt:17
Ich habe irgend ein Vorstellungsbild vor
mir und sage: ”jetzt weiß ich nicht, ist das eine Erwartung oder eine Erinnerung, oder nur
ein Bild ohne jede Beziehung zur Wirklichkeit“.
18
Denn ich erwarte ebenso wirklich, wie ich warte.
14 (M): 555
15 (M): ///
16 (V): der
17 (V): Diese Situation ist nicht denkbar:
18 (M): ?
218
171 Ist die Vorstellung das Porträt par excellence . . .
WBTC046-55 30/05/2005 16:17 Page 342
12
But the question couldn’t be: Is a mental image always a mental image of something
that exists in reality? – For obviously it isn’t always that. Rather, the question should be:
Does a mental image, true or false, always refer to reality? – For that it does cannot be said
of a painted picture. – But what does this “referring to reality” consist in? Surely it is the
relationship of a portrait to its subject.
13
But then why shouldn’t one say that a mental image is a mental image of a dream?
If I dreamed last night that N was visiting me and now N really does visit me, then that
doesn’t make that dream fantasy an expectation, and it doesn’t make the fact that N visited
me the fulfilment of an14
expectation.
How come this situation doesn’t exist:15
I have some mental image in front of me and I say
“I’m at a loss – is this an expectation, or a memory, or only an image with no relationship
to reality?”?
16
For my expecting is just as real as my waiting.
12 (M): 555
13 (M): ///
14 (V): of the
15 (V): This situation is not conceivable:
16 (M): ?
Is a Mental Image a Portrait Par Excellence . . . 171e
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51
Ist das Denken ein spezifisch
organischer Vorgang? Ein spezifisch
menschlich-psychischer Vorgang?
Kann man ihn in diesem Falle durch
einen anorganischen Vorgang
ersetzen, der denselben Zweck erfüllt,
also1
sozusagen durch eine Prothese?
Wenn man an den Gedanken als etwas spezifisch menschliches, organisches denkt, möchte man
fragen: könnte es eine Gedankenprothese geben?2
Der &Gedanke“, das ist nichts organisches, sollte nicht mit etwas organischem verglichen werden;3
das sich dann etwa durch4
Anorganisches5
wie durch eine Prothese ersetzen ließe.6
7
Eine Gedankenprothese ist darum nicht möglich, weil der Gedanke für uns nichts spezifisch
Menschliches ist.
Wir könnten die Rechenmaschine als eine Prothese statt der 10 Finger ansehen, aber die
Rechnung ist nichts spezifisch Menschliches und für sie gibt es keine Prothese.8
1 (V): aber
2 (V): fragen: kann man sich eine Gedankenprothese
denken?
3 (V): organisches, läßt sich mit nichts organischem
vergleichen;
4 (V): durch ein // etwas
5 (V): Totes
6 (V): Anorganisches // Totes // ersetzen läßt.
7 (M): ////
8 (V): es keinen Ersatz.
219
218v
219
172
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51
Is Thinking a Specifically Organic
Process? A Process Specific to Human
Psychology? If so, can one Replace it
with an Inorganic Process that Fulfils
the Same Purpose, that is,1
by a
Prosthesis, as it Were?
If one thinks of thought as something specifically human, organic, one is inclined to ask: Could
there be2
a thought-prosthesis?
“Thought” is nothing organic, and it shouldn’t3
be compared to something organic, which
could then be replaced, say, by4
something inorganic, as by a prosthesis.5
6
A thought-prosthesis is not possible because for us thought is nothing specifically
human.
We could view an adding machine as a prosthesis in place of our 10 fingers, but a calculation
is nothing specifically human, and there is no prosthesis7
for it.
1 (V): Purpose, but
2 (V) Can one conceive of
3 (V): can’t
4 (V): by a // something
5 (V): something inorganic. // something dead.
6 (M): ////
7 (V): is no substitute
172e
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52
Ort des Denkens.
Das Denken: ein Vorgang im Gehirn & Nervensystem; im Geist; im1
Mund & Kehlkopf; auf dem
Papier.
2
Eine der gefährlichsten Ideen ist, merkwürdigerweise, daß wir mit dem Kopf, oder im
Kopf denken.
3
Die Idee vom Denken als einem4
Vorgang im Kopf, in dem gänzlich abgeschlossenen Raum,
gibt ihm5
etwas Okkultes.6
7
”Das Denken geht im Kopf vor sich“ heißt eigentlich nichts anderes, als, unser Kopf hat
etwas mit dem Denken zu tun. Man sagt freilich auch: ”ich denke mit der Feder auf dem
Papier“ und diese Ortsangabe ist mindestens so gut, wie die erste.
8
Wenn wir fragen ”wo geht das Denken vor sich“, so ist dahinter immer die Vorstellung
eines maschinellen Prozesses, der in einem abgeschlossenen Raum vor sich geht, sehr ähnlich,
wie die Vorgänge9
in der Rechenmaschine.
10
Schon die Bezeichnung ”Tätigkeit“ für’s Denken ist in einer Weise irreführend. Wir
sagen: das Reden ist eine Tätigkeit unseres Mundes. Denn wir sehen dabei unseren Mund
sich bewegen und fühlen es, etc. In diesem11
Sinne kann man nicht sagen, das Denken sei
eine Tätigkeit unseres Gehirns.
Und kann man sagen, das Denken sei eine Tätigkeit des Mundes oder des Kehlkopfs oder
der Hände (etwa, wenn wir schreibend denken)?
12
Zu sagen, Denken sei eben eine Tätigkeit des Geistes, wie Sprechen des Mundes, ist
eine Travestie (der Wahrheit).
Wir gebrauchen eben ein Bild, wenn wir von der Tätigkeit des Geistes reden.
13
Das Denken ist nicht mit der Tätigkeit eines Mechanismus zu vergleichen, den wir von
außen sehen, in dessen Inneres wir aber erst dringen müssen.14
1 (V): Geist; auf dem
2 (M): ← 3
3 (M): ← 3
4 (V): Die Idee von einem // Die Idee es sei ein
5 (V): gibt dem Denken
6 (O): Okultes.
7 (M) 3
8 (M): ///
9 (V): wie der Vorgang
10 (M): /// – Mundes
11 (V): demselben
12 (M): 3
13 (M): ///
14 (V1): Das Denken ist nicht mit der Tätigkeit
eines Mechanismus zu vergleichen, die wir von
außen sehen // der wir von außen zuschauen //,
deren Inneres aber wir sehen müßten // müssen
// um sie zu verstehen. (V2): Das Denken ist
nicht die Tätigkeit eines Mechanismus, der
wir von außen zusehen, deren Inneres aber
erforscht werden muß.
220
221
173
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52
Location of Thinking.
Thinking: a process in the brain and nervous system; in the mind; in the1
mouth and larynx; on
paper.
2
Remarkably, one of the most dangerous ideas is that we think with or in our heads.
3
The idea of thinking as a4
process in the head, in that completely closed-off space, endows
it with5
an occult quality.
6
“Thinking takes place in the head” really means nothing other than: Our head has something
to do with thinking. To be sure, one also says: “I think with pen on paper”, and this
specification of location is at least as good as the first one.
7
The question “Where does thinking take place?” always implies the mental image of
a mechanical process taking place in an enclosed space, very similar to the processes8
in an
adding machine.
9
In a way, the designation “activity” for thinking is already misleading. We say: Speaking
is an activity of our mouth. For we see and feel our mouth move in the process, etc. We
can’t say in this sense10
that thinking is an activity of our brain.
And can one say that thinking is an activity of our mouth or larynx or our hands (say,
when we think while writing)?
11
To say that thinking is simply an activity of the mind, as speaking is of the mouth, is a
travesty (of the truth).
When we talk about the activity of the mind all we’re doing is using an image.
12
Thinking is not to be compared to the activity of a mechanism that we see from the outside,
but into whose inner workings we have yet to penetrate.13
1 (V): mind; from the
2 (M): ← 3
3 (M): ← 3
4 (V): The idea of a // The idea that there is a
5 (V): space, gives thinking
6 (M): 3
7 (M): ///
8 (V): the process
9 (M): /// – of our mouth.
10 (V): in the same sense
11 (M): 3
12 (M): ///
13 (V1): Thinking is not to be compared to the activity
of a mechanism, which activity we see from
the outside, // we observe from the outside,
// but whose interior we would have to see //
we have to see // in order to understand it.
(V2): Thinking is not the activity of a mechanism,
which activity we observe from the outside, but
whose interior must be investigated.
173e
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15
Die Wendung ”daß etwas in unserem Geist vor sich geht“, soll, glaube ich, andeuten,
daß es im physikalischen Raum nicht lokalisierbar ist. Von Magenschmerzen sagt man nicht,
daß sie in unserem Geist vor sich gehen, obwohl der physikalische Magen ja nicht der unmittelbare
Ort der Schmerzen ist, in dem Sinn, in welchem er der Ort der Verdauung ist.
15 (M): ///
174 Ort des Denkens.
WBTC046-55 30/05/2005 16:17 Page 348
14
The expression “that something is going on in our mind” is supposed to suggest, I believe,
that it can’t be situated in physical space. One doesn’t say of a stomach-ache that it is going
on in our mind, even though, as we know, the physical stomach isn’t the immediate location
of the pain, in the sense in which it is the location of digestion.
14 (M): ///
Location of Thinking. 174e
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53
Gedanke und Ausdruck
des Gedankens.
Das Denken ein Vorgang in einem ätherischen Mechanismus.
Der Schrei als Ausdruck des Schmerzes, der Satz als Ausdr. des Gedankens.
Denken nennen wir einen bestimmten Gebrauch von Symbolen.
Der Gedanke ist nicht eine Art von Stimmung die durch seinen1
Ausdruck wie durch eine Droge
hervorgerufen wird.
Und die Verständigung, die Vermittlung des Gedankens durch die Sprache2
ist nicht der Vorgang
daß ich durch ein Gift im Andern die gleichen Schmerzen hervorrufe wie ich sie habe.
Was für einen Vorgang3
könnte man &Gedankenlesen“ nennen.
4
Der Gedanke ist wesentlich das, was durch den Satz ausgedrückt ist, wobei ”ausgedrückt“
nicht heißt ”hervorgerufen“. Ein Schnupfen wird durch ein kaltes Bad hervorgerufen, aber
nicht5
ausgedrückt.
6
Man hat nicht den Gedanken, und daneben die Sprache. – Es ist also nicht so, daß man
für den Andern die Zeichen, für sich selbst aber einen stummen Gedanken hat. Gleichsam
einen gasförmigen oder ätherischen Gedanken, im Gegensatz zu sichtbaren, hörbaren
Symbolen.
7
Man könnte so sagen, am Gedanken ist nichts wesentlich privat. – Es kann jeder in ihn
Einsicht nehmen.
8
Man hat nicht den Zeichenausdruck und daneben, für sich selbst, den (gleichsam
dunkeln) Gedanken. Dann wäre es doch auch zu merkwürdig, daß man den Gedanken durch
die Worte sollte wiedergeben können.
9
D.h.: wenn der Gedanke nicht schon artikuliert wäre, wie könnte der Ausdruck durch
die Sprache ihn artikulieren. Der artikulierte Gedanke aber ist in allem Wesentlichen ein
Satz.
1 (V): den
2 (V): die Verständigung mittels der Sprache
3 (V): Welchen Vorgang
4 (M): ///
5 (V): nicht durch ein kaltes Bad
6 (M): 3
7 (M): ///
8 (M): ///
9 (M): ///
222
221v
222
223
175
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53
Thought and Expression
of Thought.
Thinking: a process in an ethereal mechanism.
The scream as an expression of pain, the proposition as an expression of thought.
We call a specific use of symbols thinking.
A thought is not a kind of mood that is triggered by expressing it, as by a drug.
And communication, transmitting1
a thought through language, is not the same thing as my using
a poison to elicit in someone else the same pain that I feel.
What kind of a process2
could one call “thought-reading”?
3
A thought is essentially what is expressed by a proposition, in which context “expressed”
does not mean “brought about”. A cold is brought about, but not expressed, by a cold bath.4
5
One doesn’t have a thought and apart from it language. – So it’s not that one has the signs
for someone else, but only a mute thought for oneself. A gaseous or ethereal thought, as it
were, as opposed to visible, audible symbols.
6
One could put it this way: There is nothing essentially private about a thought. – Everybody
can look into it.
7
One doesn’t have an expressive symbol and apart from it, for oneself, the (as it were,
“dark”) thought. For if that were the case, it would certainly be strange beyond belief that
one should be able to render a thought with words.
8
That is to say: if a thought were not already articulated, how could expressing it by means
of language articulate it? Rather, an articulated thought is essentially a proposition.
1 (V): communicating
2 (V): Which process
3 (M): ///
4 (V): brought about by a cold bath, but not
expressed by a cold bath.
5 (M): 3
6 (M): ///
7 (M): ///
8 (M): ///
175e
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Um einzusehen, wie Gedanke & Rede sich zu einander verhalten bedenke,10
ob11
das
”Verständnis“ (der Gedanke) einer Rechnung (12
einer Multiplikation z.B.) als gesonderter
Prozeß neben dem Rechnungsvorgang einherläuft.
13
Wenn man das Verstehen, Wissen, etc., als Zustand auffaßt, dann nur hypothetisch im
Sinne einer psychischen Disposition, welche auf derselben Stufe steht, wie eine physiologische
Disposition.
14
”Dachtest Du denn, als Du den Satz sagtest, daran, daß Napoleon . . .“ – ”ich dachte
nur, was ich sagte“.
15
Plato nennt die Hoffnung eine Rede. (Philebos)16
17
Der Gedanke ist kein geheimer – und verschwommener – Prozeß, von dem wir nur
Andeutungen in der Sprache sehen, als wäre die Negation ein Stoß und der Gedanke
darauf wie ein unbestimmter Schmerz, von diesem Stoß hervorgerufen, aber gänzlich von
ihm verschieden.
18
Gedankenlesen kann nur darin bestehen, daß wir Zeichen interpretieren, also einfach
lesen (nur vielleicht andere Zeichen). Oder aber es besteht darin, daß Einem, wenn man des
Anderen Hand hält (oder in andrer Art mit ihm in Kontakt steht) Gedanken kommen, die
durch nachträgliche19
Fragen als die Gedanken auch des Anderen erkannt werden. Aber da
handelt es sich überhaupt um kein Lesen, sondern es wäre nur die Hypothese erlaubt, daß
zwei Leute unter gewissen Umständen das Gleiche dächten.
20
Ist das Denken ein augenblicklicher Vorgang oder etwa ein andauernder Zustand, wovon
die Worte, der Satz, nur eine ungeschickte Wiedergabe sind (sodaß man etwa sagen könnte,
wie von dem Eindruck einer Landschaft: Worte können das gar nicht wiedergeben)? Der
Gedanke braucht solange wie sein Ausdruck. Weil der Ausdruck der Gedanke ist.
21
Ich habe (einmal) diesen Ausspruch eines französischen22
Politikers gelesen,23
die französische
Sprache sei dadurch ausgezeichnet, daß in ihr die Wörter in der Ordnung folgen, wie
man wirklich denkt.
24
Niemand würde fragen, ob die Multiplikation zweier Zahlen (etwa nach der gewöhnlichen
Art durchgeführt) gleichläuft mit dem Denkvorgang.25
Man betrachtet eben die Mult. als
ein Instrument.26
Während man den Satz27
nicht als ein Instrument ansieht.
28
Die Idee, daß eine Sprache eine Wortfolge haben kann, die der Reihenfolge des
Denkens entspricht, im Gegensatz zu anderen Sprachen,29
rührt von der Auffassung her, daß
das Denken vom Ausdruck der Gedanken getrennt vorgeht. Also ein wesentlich anderer
10 (V): bedenke man,
11 (V): Wie sich der Gedanke zur Rede verhält,
kann man am besten verstehen, wenn man
bedenkt, ob etwa
12 (V): Rechnung (etwa // z.B.
13 (M): ///
14 (M): /// 3 (R): ∀ S. 154/5 3 S. 158/1 3
15 (M): ///
16 (E): Platon, Philebos, 40a.
17 (M): ///
18 (M): 3
19 (V): nachträgliches
20 (M): ///
21 (M): 3
22 (O): französischer
23 (V): Ich habe einmal gelesen, daß ein französischer
Politiker gesagt hat,
24 (M): 3
25 (V): Gedanken. // Gedankenprozeß.
26 (V): Weil jeder die Multiplikation als ein
Instrument ansieht. // betrachtet
27 (V): Gedanken
28 (M): 3
29 (V): zu einer anderen Sprache,
224
176 Gedanke und Ausdruck des Gedankens.
WBTC046-55 30/05/2005 16:17 Page 352
To understand how thought and speech relate to each other, consider whether9
the “understanding”
(the thought) of a calculation (10
one of multiplication, for example) runs alongside
the process of calculation as a separate process.
11
If one conceives of understanding, knowing, etc. as a state, then one does so only hypothetically,
in the sense of a psychological disposition that’s on the same level as a physiological
one.
12
“When you uttered that sentence, were you thinking of the fact that Napoleon . . . ?” –
“I only thought what I said.”
13
Plato calls hope a speech. (Philebus.)14
15
A thought is no secret – and blurred – process of which we only see hints in language,
as if negation were a jolt and the thought following it like an indefinite pain, elicited by this
jolt, but completely different from it.
16
Reading another’s thoughts can only consist in our interpreting signs, i.e. in our simply
reading (perhaps only different signs). Or it consists in having thoughts – when one holds
someone else’s hand or is in contact with him in some other way – that, when we ask him
later, we can recognize as his thoughts. But here it is not a matter of reading at all; here the
only legitimate thing would be the hypothesis that under certain circumstances two people
might think the same thing.
17
Is thinking a momentary process or, say, a continuing state, of which the words, the
sentence, are only a clumsy rendering (so that one might say, as of the impression of a
landscape: Words can’t even come close to expressing that)? A thought takes as long as its
expression. Because the expression is the thought.
18
(Once) I read this statement by a French politician:19
The French language has the special
merit that words in it follow upon each other in the order in which one really thinks.
20
Nobody would ask whether the multiplication of two numbers (carried out, say, in the
usual way) runs parallel with the thought process.21
Multiplication is simply regarded22
as an instrument.
Whereas one doesn’t view a proposition23
as an instrument.
24
The idea that one language, as opposed to other languages,25
can have a sequence of words
that corresponds to the sequence of thinking originates in the view that thinking takes place
separately from the expression of thoughts. And is therefore an essentially different process.
9 (V): One can best understand how a thought
relates to speech if one considers whether
perhaps
10 (V): (for example // e.g.
11 (M): ///
12 (M): /// 3 (R): ∀ p. 154/5 3 P. 158/1 3
13 (M): ///
14 (E): Plato, Philebus, 40a.
15 (M): ///
16 (M): 3
17 (M): ///
18 (M): 3
19 (V): Once I read that a French politician had
said:
20 (M): 3
21 (V): with thought.
22 (V): Because everyone looks at // regards // multi-
plication
23 (V): thought
24 (M): 3
25 (V): to another language,
Thought and Expression of Thought. 176e
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Vorgang ist. Nach dieser Auffassung könnte man nun freilich sagen: Die wesentlichen
Eigenschaften des Negationszeichens offenbaren sich freilich erst nach und nach im
Gebrauch, aber ich denke die Negation auf einmal. Das Zeichen ”nicht“ ist ja nur ein Hinweis
auf den Gedanken ”nicht“. Es stößt mich nur, daß ich das Rechte denke. (Es ist nur Signal.)
30
Willkürlichkeit des sprachlichen Ausdrucks: Könnte man sagen: das Kind muß das Sprechen
einer bestimmten Sprache zwar lernen, aber nicht das Denken, d.h. es würde von selber denken,
auch ohne irgend eine Sprache zu lernen? ( (D.h. Willkürlichkeit, wie sie gewöhnlich
aufgefaßt wird. Sozusagen: ”auf den Gedanken kommt es an, nicht auf die Worte“.) )
Ich meine aber, wenn es denkt, so macht es sich eben Bilder und diese sind in einem
gewissen Sinne willkürlich, insofern nämlich, als andere Bilder denselben Dienst geleistet
hätten. Und andererseits ist ja die Sprache auch natürlich entstanden, d.h., es muß wohl
einen ersten Menschen gegeben haben, der einen bestimmten Gedanken zum ersten Mal in
gesprochenen Worten ausgedrückt hat. Und übrigens ist das Ganze gleichgültig, weil jedes
Kind, das die Sprache lernt, sie nur in dieser Weise lernt, daß es anfängt in ihr zu denken.
Plötzlich anfängt; ich meine: Es gibt kein Vorstadium, in welchem das Kind die Sprache
zwar schon gebraucht, sozusagen zum Zweck der Verständigung31
gebraucht, aber noch nicht
in ihr denkt.
Lernt das Kind nur sprechen, & nicht auch denken? Lernt es den Sinn des Multiplizierens vor, oder
nach dem32
Multiplizieren?
Oft könnte33
man auf die Frage &was meinst Du, wenn Du sagst . . . ?“ nur antworten: ich meine
nur was ich sage.
&Ich bin nicht ganz sicher, aber ziemlich sicher daß er kommen wird.“34
Ich meine, was ich sage.
Ist es quasi eine Verunreinigung des Sinnes, daß wir ihn in einer bestimmten Sprache,
mit ihren Zufälligkeiten, ausdrücken und nicht gleichsam körperlos und rein? 35
Nein,
denn es ist wesentlich, daß ich die Idee der Übersetzung von einer Sprache in die andere
verstehe.
Spiele ich eigentlich doch nicht das Schachspiel selbst, da die Figuren ja auch andere
Formen haben könnten?36
37
Da der Sinn eines Satzes ganz in der Sprache fixiert ist, und es auf den Sinn ankommt,
so ist jede Sprache gleich gut. Der Sinn aber ist, was Sätze, die in einander übersetzbar sind,
gemein haben.
Beweise die das Dez. Syst. verwenden.
225
224v
177 Gedanke und Ausdruck des Gedankens.
30 (R): ∀ S. 156/4,5 3
31 (V): sozusagen zur Verständigung
32 (V): oder nach% dem
33 (V): müßte
34 (V): &Ich weiß ziemlich sicher daß er kommen
wird%“ nicht ganz sicher
35 (M): ∫
36 (V): ja auch anders sein könnten?!
37 (M): ∫
225
224v
WBTC046-55 30/05/2005 16:17 Page 354
Now according to this view one could undoubtedly say: True, the essential properties of the
negation sign become apparent only gradually as it’s being used, but I think the negation
all at once. The sign “not”, after all, is just a reference to the thought “not”. It just jolts
me into thinking the right thing. (It’s only a signal.)
26
Arbitrariness of linguistic expression. Could one say: A child does have to learn to speak
a particular language, but not to think, i.e. it would think of its own accord, even without
learning any language? ((That is, arbitrariness as it is usually understood. As if to say: “The
thought is what counts, not the words”.))
But I believe that when a child thinks, it simply creates images for itself, and in a certain
sense these images are arbitrary, i.e. other images would have served as well. And on the
other hand, of course, language also came about naturally, i.e. there must have been a first
human who expressed a particular thought in spoken words for the first time. And furthermore,
all of this is irrelevant because every child who learns a language only learns it by beginning
to think in it. By beginning straight off; I mean: there is no preliminary stage in which
the child already uses the language, uses it for the purpose of communication,27
as it were,
but doesn’t yet think in it.
Does a child learn only how to speak and not also how to think? Does it learn the meaning of
multiplying before or after multiplying?
When asked “What do you mean when you say . . . ?” often the only answer that could be28
given
is: I mean only what I say.
“I’m not completely sure, but fairly sure that he will come.”29
I mean what I say.
Is it so to speak a pollution of sense that we express it in a particular language with
its contingencies, and not, as it were, disembodied and pure? 30
No, for it’s essential that I
understand the idea of translating from one language into another.
Am I really not playing the game of chess itself, since to be sure the figures could be
shaped differently31
?!
32
Since the sense of a proposition is completely fixed in a language, and since sense is what
counts, each language is equally good. But sense is what sentences that can be translated into
each other have in common.
Proofs that use the decimal system.
26 (R): ∀ p. 156/4,5 3
27 (V): for communication,
28 (V): that has to be
29 (V): “I know pretty surely that he will come, but not
completely surely.”
30 (M): ∫
31 (V): could be different
32 (M): ∫
Thought and Expression of Thought. 177e
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54
Was ist der Gedanke?
Was ist sein Wesen?
”Der Gedanke, dieses
seltsame Wesen“.
Sage Dir (beim Philosophieren) immer wieder:1
daß es eine Verführung ist, die2
Dich das Denken
als einen geheimn. Vorg. sehen läßt.3
Der Gedanke, soweit man überhaupt von ihm reden kann, muß etwas ganz hausbackenes
sein. (Man pflegt, sich ihn als etwas Ätherisches, noch Unerforschtes, zu denken; als
handle es sich um Etwas, dessen Außenseite bloß wir kennen, dessen Wesen aber noch
unerforscht ist, etwa wie unser Gehirn.)4
5
Der Gedanke hat aber nur eine Außenseite und kein Innen. Und ihn analysieren heißt
nicht in ihn dringen.
Man kann wieder nur die Grammatik des Wortes ”denken“6
explicit machen. (Und so
des Wortes ”erwarten“,7
etc.)
1 (V): Sage Dir: // Sage Dir (beim Philosophieren):
2 (V): was
3 (V1): daß Denken // denken // etwas hausbackenes
// etwas ganz hausbackenes // sein muß. Daß es sich
nicht darum handelt // Es handelt sich nicht darum //,
ein geheimnisvolles Wesen zu studieren. (V2): ||
daß // Daß // Du verführt bist, – wenn Du meinst
denkst, daß es hier // da // ein seltsamer Vorgang
vorliegt // ist //. (V3): – daß Du verführt bist,
sowie wenn Du das Denken als seltsamen Vorgang
(an)siehst. (V4): daß Du verführt bist, wenn
Dir das Denken als ein seltsamer // als seltsamer
// Vorgang erscheint. (V5): ; daß es eine
Verführung ist, zu meinen, // wenn wir denken, // daß
hier etwas Geheimnisvolles vorliegt // daß uns hier
ein geheimnisvoller Vorgang vorliegt.
4 (V): wie das unseres Gehirns.)
5 (M): ///
6 (V): ”erwarten“
7 (V): ”denken“,
226
178
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54
What is Thought?
What is its Essence?
“Thought, this
Peculiar Being.”
Tell yourself over and over (when doing philosophy):1
that it is a seduction that causes you to see
thinking as a mysterious process.2
A thought, in so far as one can talk about it at all, has to be something utterly prosaic.
(Usually one thinks of it as something ethereal, something still unexplored; as if it were a
matter of something of which we know only the outside, whose essence, however, is still
unexplored, say like our brain.)3
4
But a thought has only an exterior and no inside. And to analyse it does not mean to
penetrate into it.
Once again, all one can do is to make the grammar of the word “think”5
explicit. (And so
too of the word “expect”,6
etc.).
1 (V): Tell yourself: // Tell yourself (when doing
philosophy):
2 (V1): : that thinking has to be something // utterly //
prosaic. That it is not a matter // It is not a matter //
of studying a mysterious being (V2): || : that
you have been led astray, – if you think believe, that
here // there // is a strange process // a strange process
is going on. (V3): philosophy) – that you have
been led astray as soon as if you see (look at) thinking
as a strange process. (V4): : you have been
led astray if thinking looks like a strange process to you.
(V5): philosophy); that it is a seduction to believe //
if we think // that something mysterious is the case //
that here we have a mysterious process.
3 (V): like that of our brain.)
4 (M): ///
5 (V): “expect”
6 (V): “think”,
178e
WBTC046-55 30/05/2005 16:17 Page 357
55
Zweck des Denkens.
Grund des Denkens.
Wozu denkt der Mensch? Weil Denken sich bewährt hat?
Denkt man, weil man denkt, es sei vorteilhaft zu denken?
Erzieht er seine Kinder weil sich das1
bewährt hat?
2
Wie wäre herauszubringen,3
warum er denkt?
Wozu denkt der Mensch? wozu ist es nütze? Wozu berechnet er Dampfkessel und überläßt
ihre Wandstärke4
nicht dem Zufall?5
Es ist doch nur Erfahrungstatsache, daß Kessel, die
so berechnet wurden, nicht so oft explodieren.6
Aber so, wie er alles eher täte, als die Hand
ins Feuer stecken, das ihn früher gebrannt hat, so wird er alles eher tun, als den Kessel nicht
berechnen. Da7
uns aber Ursachen nicht interessieren,8
können9
wir nur sagen: die
Menschen denken tatsächlich – 10
sie gehen (z.B.) auf diese Weise vor, wenn sie einen
Dampfkessel bauen. Und dieses Vorgehen hat sich bewährt. – Kann nun ein so erzeugter Kessel
nicht explodieren? Oh freilich.11
– Warum denn nicht?12
Und doch kann man sagen, das Denken habe sich bewährt.
Es seien jetzt weniger Kesselexplosionen als früher seit man die Dimensionen etwa nicht mehr
nach dem Gefühl bestimmt sondern auf die & die Weise berechnet. Oder, seit man jede Rechnung
unabhängig von zwei Leuten ausführen läßt.13
Manchmal, also, denkt man weil es sich bewährt hat.
14
Sich etwas überlegen. Ich überlege, ob ich jetzt ins Kino gehen soll. Ich mache mir
ein Bild der Zeiteinteilung des Abends. Aber wozu tue ich das?? Ich mache ja kein
”Gedankenexperiment“!
1 (V): weil es sich
2 (M): §
3 (V): herauszubringen:
4 (V): überläßt die Dimensionen
5 (V): und überläßt es nicht dem Zufall wie stark
er ihre Wand // Wände // macht // wie stark die
Wand des Kessels wird //?
6 (V): explodierten.
7 (V): Wenn
8 (V): interessieren, so
9 (V): werden
10 (V): tatsächlich:
11 (V): explodieren? Oh ja. // Doch!
12 (V): Warum sollte er nicht?
13 (V):
Denkt der Mensch also, weil Denken sich
bewährt hat?
Weil er denkt, es sei vorteilhaft zu denken?
In gewissen speziellen Fällen wird man das sagen
können.
In gewissen, speziellen, Fällen aber wird man //aber
// sagen können: Heute berechnet man dies, weil &
überläßt es nicht // nicht mehr // dem Gefühl (oder //
oder dem // Zufall) weil es sich // sich das // bewährt
hat.
Man kann auch sagen es hat sich bewährt diese
Berechnungen immer genau kontrollieren zu lassen.
14 (M): ∫
227
227v
227
227
179
WBTC046-55 30/05/2005 16:17 Page 358
55
The Purpose of Thinking.
The Reason for Thinking.
Why do humans think? Because it has proved its worth?
Does one think because one thinks that it is advantageous to think?
Do humans raise their children because that1
has proved its worth?
2
How could one find out3
why humans think?
Why do humans think? What is it good for? Why do they make calculations for boilers
and not leave the thickness of their walls4
to chance?5
After all, it is only an empirical fact that
boilers calculated this way don’t explode6
so often. But just as they will do anything rather
than put their hands into the fire that once burned them, they will do anything rather than
not calculate for a boiler. But as causes don’t interest us, we can only say7
: Humans do in
fact think – this, for instance, is the way they proceed when they build a boiler, and this procedure
has proved its worth. – Now can’t a boiler produced this way explode? Oh, certainly. –
And why not?8
And yet one can say that thinking has proved its worth.
That there are fewer boiler explosions now than before – ever since we stopped determining their
dimensions instinctively, and started calculating them in such and such a way. Or ever since we have
had every calculation worked out independently by two people.9
So sometimes one does think because it has proved its worth.
10
To think about something. I’m thinking about whether to go to the movies now. I
create a mental image of the evening’s schedule. But what do I do that for?? I’m not, after
all, performing any “thought-experiment”!
1 (V): it
2 (M): §
3 (V): could it be brought out:
4 (V): leave the dimensions
5 (V): and not leave it to chance how thick they
make their wall // walls // how thick the wall of
the boiler is to be made // ?
6 (V): way didn’t explode
7 (V): But if causes don’t interest us, we will
only say
8 (V): explode? Oh, yes. // Sure! //Why should it not?
9 (V): So do humans think because thinking has
proved its worth?
Because they believe it is advantageous to
think?
In certain special cases one can say that.
In certain special cases // however // one can
say: Nowadays one calculates this because and does
not leave // and no longer leaves // it to instinct (or //
to // chance) because it // that // has proven its
worth.
One can also say that it has proven its
worth always to have these calculations accurately
checked.
10 (M): ∫
179e
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Wir verstehen alle, was es heißt, in einem Kalender nachschlagen, an welchem Tag der
Woche wir frei sind. Das Bild, das wir sehen, ist etwa
und wir sagen nun, wir seien nur Freitag frei, und handeln
demgemäß. Mit welcher Berechtigung handeln wir nach dem Bild?
Fahrplan
Wir erwarten etwas und handeln der Erwartung gemäß. Muß die Erwartung eintreffen?
– Nein. Warum aber handeln wir nach der Erwartung? Weil wir dazu getrieben werden,
wie dazu, einem Automobil auszuweichen, uns niederzusetzen, wenn wir müde sind, und15
aufzuspringen, wenn wir uns auf16
einen Dorn gesetzt haben.
17
Die Natur des Glaubens an die Gleichförmigkeit des Geschehens wird vielleicht am
klarsten im Falle, in dem wir Furcht vor dem Erwarteten18
empfinden. Nichts könnte mich
dazu bewegen, meine Hand in die Flamme zu stecken, obwohl ich mich doch nur in der
Vergangenheit verbrannt habe.
19
Daß20
mich das Feuer brennen wird, wenn ich die Hand hineinstecke: das ist Sicherheit.
D.h., da siehst Du,21
was Sicherheit bedeutet. (Nicht nur was das Wort ”Sicherheit“ bedeutet,
sondern auch, was es mit ihr auf sich hat.)
22
Der Glaube, daß mich das Feuer brennen wird, ist von der Natur der Furcht, daß es
mich brennen wird.
23
Wenn man mich ins Feuer zöge, so würde ich mich wehren und nicht gutwillig gehn;
und ebenso würde ich schreien: ”es24
wird mich brennen!“ und ich würde nicht schreien:
”vielleicht wird es ganz angenehm sein!“
Ich kalkuliere so, weil ich nicht anders kalkulieren kann. (Ich glaube das, weil ich nicht
anders glauben kann.)
Was sollte ich als Grund angeben dafür, weswegen25
man denken soll.
Es sei denn einen26
Grund von der Art dessen, weswegen man essen soll.
Es ist eines: einen Gedanken aus anderen begründen, – ein anderes: das Denken begründen.27
Das,
glaube ich, ist es, was unsere Untersuchung rein beschreibend macht.
28
Ich weiß nicht, warum ich denken sollte. Aber ich denke.
Es läßt sich kein rationaler Grund angeben, weshalb wir denken sollten.29
30
Ich nehme an, daß dieses Haus nicht in einer halben Stunde zusammenstürzen wird.
Wann nehme ich das an? Die ganze Zeit? und was ist dieses Annehmen für eine Tätigkeit?
Heißt, das annehmen, nicht (wieder) zweierlei? Einmal bezeichnet es eine hypothetische
psychologische Disposition; einmal den Akt des Denkens, Ausdrückens, des Satzes ”das Haus
15 (O): sind und
16 (V): auf de
17 (M): ←
18 (V): dem erwarteten Ereignis
19 (M): ←
20 (V): Daß mei
21 (V): da sehe ich, // da sehen wir,
22 (M): ←
23 (M): ←
24 (V): schreien: ”das Feuer
25 (V): Es läßt sich kein // Man kann keinen //
Grund angeben, weswegen // Was sollte ich für
einen Grund angeben, weswegen
26 (V): ein
27 (V): Man kann einen Gedanken aus anderen
begründen, aber nicht das Denken.
28 (M): v
29 (V): müßten.
30 (M): Fahrplan
228
229
180 Zweck des Denkens. . . .
M D M D F S S
WBTC046-55 30/05/2005 16:17 Page 360
All of us understand what it means to check in a calendar to see on which day of the
week we are free. The image that we see is, say , and now we
say that we’re free only on Friday and act accordingly.
What justification do we have for acting in accordance with that image?
A travel timetable.
We expect something and act in accordance with our expectation. Does the expectation
have to come about? – No. So why do we act in accordance with an expectation? Because
we are driven to do this as we are driven to dodge a car, to sit down when we’re tired and
to jump up when we’ve sat on a thorn.
11
Possibly the essence of our belief in the uniformity of nature comes out most clearly when
we fear what we expect.12
Nothing could get me to put my hand in a flame even though it
was only in the past, after all, that I got burned.
13
The fire will burn me if I stick my hand in it: that is certainty.
That is to say: here you see14
what certainty means. (Not only what the word “certainty”
means, but also what certainty is all about.)
15
The belief that fire will burn me is of the same nature as the fear that it will burn me.
16
If I were dragged into a fire I would resist and not go willingly; likewise I would shout:
“It’s17
going to burn me!”, not: “Maybe it will be quite pleasant!”
I’m calculating thus because I cannot calculate otherwise. (I believe this because I cannot
believe otherwise.)
What should I list as a reason for18
why one should think?
Unless it is a reason of the sort why one should eat.
It is one thing to justify a thought on the basis of other thoughts – something else to justify
thinking.19
It is this, I believe, that makes our investigation purely descriptive.
20
I don’t know why I should think. But I do think.
It is not possible to give a rational basis for why we should21
think.
22
I assume that this house won’t collapse in half an hour. When do I assume that? All the
time? And what kind of an activity is this assuming? Doesn’t assuming that (again) mean
two different things? On the one hand it designates a hypothetical psychological disposition;
on the other an act of thinking, of expressing the sentence “The house won’t collapse”.23
In
the first sense the criterion for my assuming that24
is what I normally say, feel and do; in
M T W T F S S
11 (M): ←
12 (V): fear the event we expect.
13 (M): ←
14 (V): here I see // here we see
15 (M): ←
16 (M): ←
17 (V): shout: “The fire’s
18 (V): No reason can be given // One can give no
reason // What kind of a reason should I give
19 (V): One can justify a thought on the basis of
other thoughts, but not thinking.
20 (M): v
21 (V): we have to
22 (M): Travel timetable
23 (V): of expressing that sentence.
24 (V): for my making that assumption
The Purpose of Thinking. . . . 180e
WBTC046-55 30/05/2005 16:17 Page 361
wird nicht einstürzen“.31
Im ersten Sinne ist das Kriterium dafür, daß ich das annehme,32
das,33
was ich sonst sage, fühle und tue; im andern Sinn, daß ich einen Satz sage, der wieder
ein Glied einer Kalkulation34
ist. Nun sagt man: Du mußt aber doch einen Grund haben,
das anzunehmen, sonst ist die Annahme ungestützt und wertlos (erinnere Dich daran, daß
wir zwar auf der Erde stehen, die Erde aber nicht wieder auf irgend etwas; und Kinder glauben,
sie müsse fallen, wenn sie nicht gestützt ist). Nun, ich habe auch Gründe zu meiner Annahme.
Sie lauten etwa: daß das Haus schon jahrelang gestanden hat, aber nicht so lang, daß es schon
baufällig sein könnte, etc. etc. Was ein Grund wofür ist (Was als Grund wofür gilt), kann
von vornherein angegeben werden und bestimmt35
einen Kalkül, in welchem36
eben das
eine ein Grund des andern ist. Soll aber nun ein Grund für diesen ganzen Kalkül gegeben
werden, so sehen wir, daß er fehlt. Fragt man aber, ob der Kalkül also eine willkürliche
Annahme ist, so ist die Antwort, daß er es so wenig ist, wie die Furcht vor dem Feuer oder
einem wütenden Menschen, der sich uns nähert. Ist es Willkür,37
daß wir das als Grund von dem
betrachten? Ist es Willkür,38
daß wir auf die Erzählung39
dieser Hund habe40
gebissen, diesem Hund
nicht in die Nähe gehen wollen?
Wenn man nun sagt: gewiß sind doch die Regeln der Grammatik, nach denen wir vorgehen
und operieren, nicht willkürlich; so müßte man zur Antwort fragen: Gut also, warum
denkt denn ein Mensch, wie er denkt? warum geht er denn durch diese Denkhandlungen?
(gefragt ist hier natürlich nach den Gründen,41
nicht Ursachen). Nun, da lassen sich Gründe
in dem Kalkül angeben; und ganz zum Schluß ist man dann versucht zu sagen: ”es ist eben
sehr wahrscheinlich, daß sich das Ding jetzt so verhalten wird, wie es sich immer verhalten
hat“,42
– oder dergleichen. Eine Redensart, die den Anfang des Raisonnements verhüllt und
an diesem Anfang43
eine ähnliche Rolle spielt, wie der Schöpfer am Anfang44
der Welt, welcher45
zwar in Wirklichkeit nichts erklärt, aber ein den Menschen acceptabler Anfang ist.46
Das, was so schwer einzusehen ist: Solange wir im Bereich der W.F. Spiele bleiben, kann uns
eine Änderung der Gramm. nur von einem solchen Spiel zu einem andern führen, aber nicht47
von
etwas Wahrem zu etwas Falschem. Und wenn wir anderseits aus dem Bereich dieser Spiele
heraustreten, so nennen wir es eben nicht mehr Grammatik, und zu einem Widerspruch mit
der Wirklichkeit kommen wir wieder nicht.
Denken wir uns die Tätigkeit in einem Haus, in einer Werkstätte. Da wird gehobelt, gesägt,
gestrichen, etc. etc.; und außerdem gibt es da eine Tätigkeit, die man ”Rechnen“48
nennt,
und die sich scheinbar von allen den andern unterscheidet,49
besonders, was ihren50
Grund
anbelangt. Wir machen da etwa ein Bild, die Tätigkeit des Rechnens (Zeichnens, etc.) verbindet
Teile der andern Tätigkeit. Er setzt aus, rechnet etwas, dann mißt er und arbeitet mit dem
31 (V): Ausdrückens, jenes Satzes.
32 (V): ich jene Annahme mache,
33 (O): annehme das, // ich jene Annahme mache
das,
34 (V): Rechnung
35 (V): beschreibt
36 (V): dem
37 (V): willkürlich,
38 (V): willkürlich,
39 (V): Erzählung des A
40 (V): habe ihn
41 (V): Gründen,
42 (V): daß das Ding jetzt das gleiche Verhalten
zeigen wird, das es immer gezeigt hat“,
43 (V): und hier
44 (V): Beginn
45 (V): der
46 (V): aber einen den Menschen acceptablen
Anfang macht.
47 (V): Das, was so schwer einzusehen ist, ist,
eigentlich, daß, // Das was so schwer einzusehen ist,
lautet eigentlich // etwa //: daß, solange wir ein
Wahr-Falsch-Spiel spielen // daß, solange wir
im Bereich der Wahr-Falsch-Spiele bleiben //,
eine Änderung der Grammatik uns nur von
einem solchen Spiel zu einem andern führen
kann, aber nicht
48 (V): ”rechnen“
49 (V): von allen diesen unterscheidet,
50 (V): den
230
229v
230
231
181 Zweck des Denkens. . . .
WBTC046-55 30/05/2005 16:17 Page 362
the second the criterion for my assumption is that I utter a sentence that is in turn a step
in a calculation.25
Now it is said: But you must have a reason for assuming that, otherwise
the assumption is unsupported and worthless (remember that, to be sure, we stand on the
earth, but that the earth doesn’t rest on anything in turn; and children believe that it has to
fall if it isn’t supported). Well, I do have reasons for my assumption. They run something
like this: that the house has been standing for years, but not long enough for it to be ramshackle,
etc., etc. What is a reason for what (what counts as a reason for what) can be stated from the
outset, and this determines26
a calculus in which27
one thing is a reason for another. But if
one is then to give a reason for this whole calculus, we see that it’s missing. And if it’s asked
whether the calculus is therefore an arbitrary assumption, the answer is that it is so no more
than fear of fire or of an angry person approaching us. Is it arbitrariness for us to look at this as
a reason for that? Is it arbitrariness28
that we don’t want to get close to a dog after we’ve heard that
it bit someone?
Now if one says: “But certainly the rules of grammar according to which we proceed and
operate are not arbitrary”, then we’d have to respond by asking: All right, then, why does
someone think as he thinks? Why does he go through these acts of thought? (Of course the
reasons29
are in question here, not the causes). Well, reasons for this can be given within the
calculus; and at the very end, one is tempted to say “It is simply quite probable that this
thing will now behave as it has always behaved”30
– or something like that. A phrase that
veils the beginning of the reasoning process and plays a similar role at this beginning31
to
that played by the Creator at the beginning of the world. He doesn’t really explain anything,
but is a beginning32
that is acceptable to humans.
What is so hard to understand: So long as we stay within the realm of True–False games, all that
a change in grammar can do is to lead us from one such game to another, not33
from something
true to something false. And if, alternatively, we step outside the realm of these games, we
then no longer call it grammar, and once again we don’t get to the point of contradicting
reality.
Let’s think of the activity in a house, in a workshop. Planing, sawing, painting, etc., etc.
is going on there; and in addition there’s an activity called “calculating”, which seemingly
differs from all of the others, especially34
as concerns its35
reason. Here we might create a
picture: the activity of calculating (drawing, etc.) connects parts of the other activities. He
stops, calculates something, then he measures something and continues working with a plane.
25 (V): computation.
26 (V): describes
27 (V): calculus for which
28 (V): approaching us. Is it arbitrary for us to look at
this as a reason for that? Is it arbitrary
29 (V): reasons
30 (V): will now show the same behaviour that it
has always shown”
31 (V): role here
32 (V): but does create a beginning
33 (V): What is so hard to accept is actually that, //
What is so hard to accept is actually that // is, say:
that // so long as we are playing a True–False
game // we remain in the realm of True–False
games //, a change in grammar can only lead
us from one such game to another, but not
34 (V): from all of them, especially
35 (V): the
The Purpose of Thinking. . . . 181e
WBTC046-55 30/05/2005 16:17 Page 363
Hobel weiter. Er setzt auch manchmal aus, um das Hobelmesser zu schleifen; aber ist diese
Tätigkeit analog der andern des Kalkulierens? – ”Aber Du glaubst doch auch, daß mehr
Kesselexplosionen wären,51
wenn die Kessel nicht berechnet würden.“ Ja, ich bin überzeugt
davon;52
– aber was will das sagen?53
Folgt daraus, daß weniger sein werden? Und was ist
denn die Grundlage dieses Glaubens?
Wenn man nun nach dem Grund einer einzelnen Denkhandlung (Kalkülhandlung)
fragt, so erhält man als Antwort die Auseinandersetzung eines Systems dem die Handlung
angehört.
&Ist also daß es sich in der Vergangenheit bewährt hat kein guter Grund54
anzunehmen daß es in
Zukunft so sein wird?“ – Das ist, was wir einen guten Grund nennen.55
51 (V): mehr Kessel explodieren würden,
52 (V): ich glaube es;
53 (V): @Ja, . . . sagen?!
54 (V ): Grund zu
55 (V): nennen. ?
182 Zweck des Denkens. . . .
WBTC046-55 30/05/2005 16:17 Page 364
Sometimes he also stops to sharpen the plane’s blade; but is this activity analogous to the
other one, to calculating? – “But you too believe, don’t you, that there would be more explosions
of boilers36
if the boilers weren’t calculated?” Yes, I’m convinced of it;37
– but what is
that supposed to say?38
Does it follow from this that there will be fewer explosions? And
what is the basis for this belief?
If one asks for the reason behind an individual act of thought (act of calculation), the answer
one gets is an analysis of a system to which the act belongs.
“So is the fact that it has proved its worth in the past no good reason to assume that that’s the
way it will be in the future?” – That’s what we call a good reason.
36 (V): that more boilers would explode
37 (V): Yes, I believe it;
38 (V): !Yes, I’m convinced of it; – but what is that
supposed to say?E
The Purpose of Thinking. . . . 182e
WBTC046-55 30/05/2005 16:17 Page 365
232 Grammatik.
WBTC056-59 30/05/2005 16:16 Page 366
Grammar.
WBTC056-59 30/05/2005 16:16 Page 367
56
Die Grammatik ist keiner
Wirklichkeit Rechenschaft schuldig.
Die grammatischen Regeln
bestimmen erst die Bedeutung
(konstituieren sie) und sind darum
keiner Bedeutung verantwortlich
und insofern willkürlich.
Angenommen, wir lassen die Übersetzung in die Gebärdensprache fort; zeigt es sich dann
in der Anwendung (ich meine, in den grammatischen Regeln der Anwendung), daß diese
Übersetzung möglich ist?
Und kann es sich nur zeigen, daß sie1
möglich ist, oder auch, daß sie notwendig ist?
Wenn sie notwendig ist, so heißt das, daß die Sprache vermittels des roten Täfelchens in
irgend einem Sinn notwendig ist; und nicht gleichberechtigt der Wortsprache.
Aber wie könnte das sein? denn dann wären ja die hinweisenden Erklärungen überflüssig:
das heißt aber schon, implicite in den andern enthalten. Wie kann denn eine Regel eines
Spiels überflüssig sein, wenn es eben das Spiel sein soll, was auch durch diese Regel charakterisiert
wird.
Mein2
Fehler besteht hier immer wieder darin, daß ich vergesse daß erst alle Regeln
das Spiel, die Sprache, charakterisieren, und daß diese Regeln nicht einer Wirklichkeit
verantwortlich sind, so daß sie von ihr kontrolliert würden, und so daß man von einer
Regel bezweifeln könnte, daß sie notwendig, oder richtig wäre. (Vergleiche das Problem
der Widerspruchsfreiheit der Nicht-euklidischen Geometrie.)
3
Die Grammatik ist keiner Wirklichkeit verantwortlich.
4
(Die Grammatik ist5
der Wirklichkeit nicht Rechenschaft schuldig.)
6
Kann7
diese hinweisende Erklärung mit den übrigen Regeln der Verwendung des Worts
kollidieren? Denn eigentlich können ja Regeln nicht kollidieren, außer sie widersprechen
1 (V): die
2 (V): Der
3 (M): 3
4 (M): 3
5 (V): ist kei
6 (M): ← 3
7 (V): Kann man
233
234
184
WBTC056-59 30/05/2005 16:16 Page 368
56
Grammar is not
Accountable to any Reality.
The Rules of Grammar Determine
Meaning (Constitute it), and
Therefore they are not Answerable
to any Meaning and in this Respect
are Arbitrary.
Let’s set the translation into a language of gestures to one side; then does it show in the
use (I mean in the grammatical rules for the use) that such a translation is possible?
And can it show only that such a translation is possible, or also that it’s necessary?
If such a translation is necessary, this means that the language that uses the red colour chip
is in some sense necessary; and that it isn’t on an equal footing with a word-language.
But how could that be? For then ostensive explanations would be superfluous; but that
means they’d be implicitly contained in other explanations. How can a rule of a game be
superfluous if it is supposed to be precisely the game that is characterized by that rule?
Again and again my1
mistake consists in forgetting that it is all its rules that characterize
a game, a language, and that these rules are not answerable to a reality in the sense that they
are controlled by it, and that we could have doubts whether a particular rule is necessary or
correct. (Compare the problem of consistency in non-Euclidean geometry.)
2
Grammar is not answerable to any reality.
3
(Grammar is not accountable to reality.)
4
Can this ostensive explanation collide with the rest of the rules for the use of the word?
Actually, rules can’t collide, unless they contradict each other. What is more, they determine
1 (V): the
2 (M): 3
3 (M): 3
4 (M): ← 3
184e
WBTC056-59 30/05/2005 16:16 Page 369
einander. Denn im Übrigen bestimmen sie ja eine Bedeutung und sind nicht einer verantwortlich,
so daß sie ihr widersprechen könnten. ( (Dazu eine Bemerkung, daß die hinweisende
Erklärung eine der Regeln ist, die von einem Wort gelten.) )
Eine Sprache ist, was sie ist, und eine andere Sprache ist nicht diese Sprache. Ich gebrauche
also die Nummern des Musterkataloges anders, als die Wörter ”rot“, ”blau“, etc.
8
Wie kann es eine Diskussion darüber geben, ob diese Regeln oder andere die richtigen
für das Wort ”nicht“ sind?9
Denn das Wort hat ohne diese 10
Regeln noch keine Bedeutung,
und wenn wir die Regeln ändern, so hat es nun eine andere Bedeutung (oder keine) und wir
können dann ebensogut auch das Wort ändern. Daher sind diese Regeln willkürlich, weil
die Regeln erst dem Zeichen die Bedeutung geben.11
12
Überlege: ”Das einzige Korrelat in der Sprache zu13
einer Naturnotwendigkeit ist eine
willkürliche Regel. Sie ist das einzige, was man von dieser Notwendigkeit in einen Satz14
abziehen kann.“ Bezieht sich auf Sätze wie ~~p = p.
15
Wenn man fragt ”warum gibst Du Eier in diesen Teig“, so ist die Antwort etwa ”weil
der Kuchen dann besser schmeckt“. Also, man hört16
eine Wirkung und sie wird als Grund
gegeben.
17
Wenn ich dem Holzblock eine bestimmte Form geben will, so ist der Hieb der richtige,
der diese Form erzeugt. – Ich nenne aber nicht das Argument das richtige, das die erwünschten
Folgen hat. Vielmehr nenne ich die Rechnung falsch, auch wenn18
die Handlungen,
die dem Resultat entspringen, zum gewünschten Ende geführt haben. Vergl. den Witz: A 19
erzählt
dem B20
er habe in der Lotterie den Haupttreffer gewonnen;21
er habe auf der Straße eine Kiste liegen
sehen22
& drauf23
die Zahlen 5 & 7. Er24
habe25
gerechnet, 5 × 7 ist 64 – & habe 64 gesetzt.26
B: Aber
5 × 7 ist27
doch nicht 64! A:28
Ich mach den Haupttreffer & er will mich belehren.29
Das zeigt, daß
die ”Rechtfertigung“ in den beiden Fällen von verschiedener Art ist.30
In einem Fall kann
man sagen: ”Wart’ nur, Du wirst schon sehen, daß das Richtige (d.h. hier: Gewünschte)
herauskommt“; im andern ist dies keine Rechtfertigung.31
32
Wenn man nun von der33
Willkürlichkeit der grammatischen Regeln spricht, so kann
das nur bedeuten, daß es die Rechtfertigung, die in der Grammatik als solcher liegt, nicht
für die Grammatik gibt. Und wenn man das Rechnen aber34
nicht das Kochen dem Spiel
vergleicht, so ist es aus eben35
diesem Grund.36
Das ist aber auch der Grund, warum man
8 (M): / ← 3
9 (V): Es kann keine Diskussion darüber geben,
. . . sind.
10 (V): die
11 (V): erst das Zeichen machen.
12 (M): ← 3
13 (V): Korrelat$ in der Sprache$ zu
14 (V): in Sätze
15 (M): 3 555
16 (V): erfährt
17 (M): / (
18 (V): falsch, obwohl
19 (O): Witz A
20 (V): Ein Jude erzählt einem andern
21 (V): gemacht:
22 (V): gesehen
23 (V): darauf
24 (V): & auf der seien die Zahlen 5 & 7 gestanden, er
25 (V): habe nun
26 (V): gerechnet; 5 × 7 ist 64% & habe auf die Nummer
64 gesetzt.
27 (V): Der Andre: Aber 5 × 7 ist
28 (V): 64! Der Erste:
29 (V): haben. (Vergleich den Witz @Ich machF den
Haupttreffer$ und er will mich belehren!!)
30 (V): daß die Rechtfertigungen in den beiden
Fällen verschiedene sind. // sind$ und also
@Rechtfertigung! verschiedenes in beiden
bedeutet.
31 (M): )
32 (M): 555
33 (V): der Wirklich
34 (V): und
35 (V): es eben aus
36 (V): Grunde.
235
234v
235
234v
235
235
185 Die Grammatik ist keiner Wirklichkeit Rechenschaft schuldig. . . .
WBTC056-59 30/05/2005 16:16 Page 370
a meaning; they are not answerable to one and thus can’t contradict it. ( (Cf. the remark that
an ostensive explanation is one of the rules that are valid for a word.) )
A language is what it is, and no other one is this language. So I use the numbers of a
sample-catalogue differently from the words “red”, “blue”, etc.
5
How can there be a discussion whether these rules, or others, are the correct ones for the
word “not”?6
For without these7
rules the word doesn’t as yet have any meaning, and if we
change the rules it has another meaning (or none), and then we might just as well change
the word. Thus these rules are arbitrary, because it is the rules that first give meaning to
the sign.8
9
Consider: “In language the only correlate to natural necessity is an arbitrary rule. It is
the only thing one can remove from this necessity and put into a proposition.”10
Refers to
propositions such as ~~p = p.
11
If someone asks “Why do you put eggs into this dough?” the answer might be “Because
then the cake tastes better”. So one hears12
an effect and it is given as a reason.
13
When I want to shape a block of wood a certain way, the stroke that produces that shape
is the right one. – But I don’t call an argument that has the desired consequences the right
one. Rather, I call a calculation wrong even if14
the actions stemming from its result have
led to the desired end.15
Cf. the joke: A tells B16
that he won17
the jackpot in the lottery: he saw a
crate lying in the street and on it the numbers 5 and 7. He18
calculated that 5 × 7 is 64 and filled
in 64. B: But 519
× 7 isn’t 64! A20
: I win the jackpot and he wants to give me lessons. This shows
that “justification” is of a different kind in the two cases.21
In the one case it can be said:
“Just wait, you’ll see that the right thing (i.e. here: the desired thing) will result”; in the
other case that is no justification.22
23
Now when one talks about the arbitrariness of grammatical rules this can only mean that
the justification that is inherent in grammar as such doesn’t exist for grammar. And when
one compares calculating, but24
not cooking, to a game, one does so for this very reason. And
that’s also the reason one wouldn’t call cooking a calculus. But how about tidying up a room
5 (M): ← 3
6 (V): There can be no discussion . . . “not”.
7 (V): the
8 (V): first create the sign.
9 (M): ← 3
10 (V): into propositions.”
11 (M): 3 555
12 (V): one is told
13 (M): / (
14 (V): wrong, although
15 (V): end. (Cf. the joke !I win the jackpot and he
wants to give me lessons!E)
16 (V): One Jew tells another
17 (V): got
18 (V): street & on it the numbers 5 & 7 were written,
he
19 (V): The other person: But 5
20 (V): 64! The first person
21 (V): that the justifications in the two cases are
different. // are different and that therefore
!justificationE means something different in
each.
22 (M): )
23 (M): 555
24 (V): and
Grammar is not Accountable to any Reality. . . . 185e
WBTC056-59 30/05/2005 16:16 Page 371
das Kochen keinen Kalkül nennen würde. Wie ist es aber mit dem Aufräumen eines Zimmers,
oder dem Ordnen eines Bücherschrankes, – oder dem Stricken eines bestimmten Musters?
Diese Dinge kommen dem Spiel in irgendeiner Weise näher. Ich glaube, der Grund, warum
man das Kochen kein Spiel zu nennen versucht ist, ist der: es gibt natürlich auch für das
Kochen Regeln, aber ”Kochen“ bezeichnet nicht wesentlich eine Tätigkeit nach diesen Regeln,
sondern eine Tätigkeit, die ein bestimmtes Resultat hat. Es ist etwa37
eine Regel, daß man
Eier 3 Minuten lang kocht, um weiche Eier zu erhalten; wird aber durch irgendwelche38
Umstände das gleiche Ergebnis durch 5 Minuten langes Kochen erreicht, so sagt man nun
nicht ”das heißt dann nicht ,weiche Eier kochen‘“. Dagegen heißt ”Schachspielen“ nicht
die Tätigkeit, die ein bestimmtes Ergebnis hat, sondern dieses Wort bedeutet eine Tätigkeit,
die den & den Regeln entspricht.39
Die Regeln der Kochkunst hängen mit der Grammatik des
Wortes ”kochen“ anders zusammen, als die Regeln des Schachspiels mit der Grammatik des
Wortes ”Schach spielen“ und als die Regeln des Multiplizierens mit der Grammatik des Wortes
”multiplizieren“.
Die Regeln der Grammatik sind in demselben Sinne40
willkürlich, & in demselben Sinne
nicht willkürlich wie die Wahl einer Maßeinheit. Man drückt dies auch so aus: diese Regeln
seien &praktisch“ oder &unpraktisch“, &brauchbar“ oder &unbrauchbar“, aber nicht &wahr“ oder
&falsch“.41
In diesem Sinn würde man es eine willkürliche Regel nennen, die Ingredientien beim Kochen nach
Pfund zu wägen, aber nicht, Eier 3 Minuten lang kochen zu lassen.
42
&Die Maßeinheit ist willkürlich“ (wenn dies nicht heißen soll: &wähle in diesem Falle die Einheit
ganz wie Du willst“) sagt nichts anderes, als daß die Angabe der Maßeinheit keine Längenangabe
ist (obwohl sie so klingt). Und zu sagen, die Regeln der Grammatik sind willkürlich, sagt bloß: Verwechsle
eine Regel43
über den Gebrauch des Wortes A nicht mit einem Satz, in dem vom Wort A Gebrauch
gemacht wird. Denke nicht die Regel sei in ähnlicher Weise einer Realität verantwortlich, mit einer
Realität zu vergleichen, wie der Erfahrungssatz der von A handelt.44
45
Die grammatischen Regeln sind zu vergleichen Regeln über das46
Vorgehn beim Messen von
Zeiträumen,47
von Entfernungen, Temperaturen, Kräften48
etc. etc. Oder auch: diese methodologischen
Regeln sind selbst Beispiele49
grammatischer Regeln. Grammatische Regeln wird man mit Vorteil
Übereinkommen vergleichen.
Diese Regeln des Vorgehens sind willkürlich kann nur heißen:50
Wenn Dein Zweck nur der ist, so
kannst Du ihn auf alle diese Weisen erreichen.
236v
236
235v
186 Die Grammatik ist keiner Wirklichkeit Rechenschaft schuldig. . . .
37 (V): z.B.
38 (O): irgend welche
39 (V): die nach gewissen Regeln ausgeführt wird.
// die gewissen Regeln entspricht.
40 (V): sind so (d.h. in demselben Sinne )
41 (V): Maßeinheit. Aber das kann doch nur
heißen$ daß sie von der Länge des
Zumessenden unabhängig ist. Und daß nicht die
Wahl der einen Einheit @wahr!$ der andern
@falsch! ist$ wie die Angabe der Länge wahr
oder falsch ist. Was natürlich nur eine
Bemerkung über die Grammatik des Wortes
@Längeneinheit! ist.
42 (M): v
43 (V): Regel nicht mit einem
44 (V): G Denke nicht daß die Regel in über (AA in
ähnlichem Sinne einer Realität verantwortlich ist,
wie der Erfahrungssatz der (AA enthält.
45 (M): v 3
46 (V): das Messen
47 (V): beim Messen der Zeit, d
48 (V): von Entfernungen, Kräften
49 (V): sind Beispiele
50 (V): willkürlich heißt:
WBTC056-59 30/05/2005 16:16 Page 372
or arranging a bookshelf – or knitting a particular pattern? In some way these things come
closer to a game. I believe the reason one isn’t tempted to call cooking a game is this: of
course there are also rules governing cooking, but in essence “cooking” doesn’t refer to an
activity that follows those rules – rather, to an activity that has a particular result. It’s a rule,
for example, that you cook eggs for 3 minutes in order to get soft-boiled eggs; but if, because
of some set of circumstances, the same result is achieved by boiling them for 5 minutes, you
don’t say “That doesn’t mean ‘cooking soft-boiled eggs’”. On the other hand “playing chess”
doesn’t mean an activity that has a particular result, but rather an activity that corresponds
to such and such25
rules. The rules of culinary art have a connection to the grammar of
the word “cook” that is different from the connection between the rules of chess and
the grammar of the words “play chess”, different from the connection between the rules of
multiplication and the grammar of “multiply”.
The rules of grammar are26
arbitrary and not arbitrary, in the same sense as is the choice
of a unit of measurement. This is also expressed by saying that these rules are “practical” or
“impractical”, “useful” or “useless”, but not “true” or “false”.27
In this sense one would call it an arbitrary rule of cooking to weigh the ingredients out in pounds,
but not to let eggs cook for three minutes.
28
“The unit of measurement is arbitrary” (if this is not to mean “Choose the unit any way you want
in this case”) means nothing other than that the specification of the unit of measurement is not
a specification of length (even though it sounds like one). And to say that the rules of grammar
are arbitrary just means: Don’t confuse a rule for the use of the word A with29
a sentence in which
the word A is used. Don’t think that a rule is answerable to a reality, is comparable to a reality, in
more or less the way an empirical proposition about A is.30
31
The rules of grammar can be compared to rules for procedures to measure32
periods of
time33
, distances, temperatures, forces34
, etc, etc. Or: these methodological rules are themselves
examples35
of grammatical rules. We’ll profit by comparing grammatical rules to agreements.
All that “These rules of procedure are arbitrary” can mean is:36
If your only purpose is this, you
can attain it in all of these ways.
25 (V): that is carried out according to certain // that
corresponds to certain
26 (V): are thus (i.e. in the same sense)
27 (V): measurement. But all that can mean is that
this unit of measurement is independent of the
length of what is to be measured. And that it
isnFt the case that the choice of one unit is !trueE
and that of the other !falseE, as a measurement
of length is true or false. All of which is, of
course, only a remark about the grammar of the
phrase !unit of lengthE.
28 (M): v
29 (V): a rule with
30 (V): Don’t think that the rule in about AAC is
answerable to a reality in a similar sense as is the
empirical proposition that contains AAC.
31 (M): v 3
32 (V): rules for the measure of
33 (V): measure time
34 (V): distances, forces
35 (V): are examples
36 (V): arbitrary” means is:
Grammar is not Accountable to any Reality. . . . 186e
WBTC056-59 30/05/2005 16:16 Page 373
187 Die Grammatik ist keiner Wirklichkeit Rechenschaft schuldig. . . .
51
&Wenn Du mit diesem Zeichen die Negation ausdrücken willst, so mußt Du von ihm die52
Regeln
gelten lassen.“ – Was für eine Art Satz53
ist das?
Man ist versucht, Regeln54
der Grammatik durch Sätze zu rechtfertigen von der Art: ”Aber
es gibt doch wirklich 4 primäre Farben“; und gegen die Möglichkeit dieser Rechtfertigung,
die nach dem Modell der Rechtfertigung eines Satzes durch (den) Hinweis auf seine
Verifikation gebaut ist, richtet sich das Wort, daß die Regeln der Grammatik willkürlich sind.
Kann man aber nicht doch in irgend einem Sinne sagen, daß die Grammatik der
Farbwörter die Welt, wie sie tatsächlich ist, charakterisiert? Man möchte sagen: kann ich
nicht wirklich vergebens nach einer fünften primären Farbe suchen?55
Nimmt man nicht die
primären Farben zusammen, weil sie eine Ähnlichkeit haben, oder zum mindesten die Farben,
im Gegensatz z.B. zu den56
Formen oder Tönen, weil sie eine Ähnlichkeit haben? Oder habe
ich, wenn ich diese Einteilung der Welt als die richtige hinstelle, schon eine vorgefaßte Idee
als Paradigma57
im Kopf? Von der ich dann etwa nur sagen kann: ”ja, das ist die Art,58
wie
wir die Dinge betrachten“, oder ”wir wollen eben ein solches Bild (von der Wirklichkeit)
machen“. Wenn ich nämlich sage: ”die primären Farben haben doch eine bestimmte
Ähnlichkeit miteinander“59
– woher nehme ich den Begriff dieser Ähnlichkeit?60
Ist nicht so,
wie der Begriff ”primäre Farbe“ nichts andres ist, als ”blau oder rot oder grün oder gelb“,
– auch der Begriff jener Ähnlichkeit nur durch die vier Farben gegeben? Ja, sind sie nicht
die gleichen! – ”Ja, könnte man denn auch rot, grün und kreisförmig zusammenfassen?“ –
Warum nicht?!
Die Wichtigkeit eines Spiels61
liegt darin, daß wir dieses Spiel spielen. Daß wir diese
Handlungen ausführen. Es verliert seine Wichtigkeit nicht dadurch, daß es selbst nicht wieder
eine Handlung in einem andern (übergeordneten) Spiel ist.
62
Warum nenne ich die Regeln des Kochens nicht willkürlich; und warum bin ich versucht,
die Regeln der Grammatik willkürlich zu nennen? Weil ”Kochen“63
durch seinen Zweck
definiert ist, dagegen das Sprechen der Sprache64
nicht. Darum ist der Gebrauch der Sprache
in einem gewissen Sinne autonom, in dem das Kochen und Waschen es nicht ist. Denn, wer
sich beim Kochen nach andern als den richtigen Regeln richtet, kocht schlecht; aber wer
sich nach andern Regeln als denen des Schachs richtet, spielt ein65
anderes Spiel und wer sich
nach andern grammatischen Regeln richtet, als denen und denen,66
spricht darum nichts
Falsches, sondern von etwas Anderem.
67
Könnte ich den Zweck der grammatischen Konventionen dadurch beschreiben, daß ich
sagte, ich müßte sie machen, weil etwa die Farben gewisse Eigenschaften haben, so wären
damit diese Konventionen überflüssig, denn dann könnte ich eben das sagen, was die
Konventionen gerade ausschließen. Umgekehrt, wenn die Konventionen nötig waren, also
gewisse Kombinationen der Wörter als unsinnig ausgeschlossen werden mußten, dann kann
51 (M): a
52 (V): mußt Du davon die & die
53 (V): Was für ein Satz
54 (V): versucht, die Regeln
55 (V): suchen? (Und wenn man suchen kann$
dann ist ein Finden denkbar.)
56 (V): z.B. von
57 (V): Paradigma der
58 (V): Weise,
59 (V): Wenn ich nämlich sage: @die primären
Farben haben doch eine bestimmte Ähnlichkeit
mit-einander“
60 (V): Ähnlichkeit? D.h.: habe ich hier eine
Funktion @x ähnlich mit y!$ in die ich die
Farben als Argumente einsetzen kann?
61 (V): Wichtigkeit in einem Spiel
62 (M): 3
63 (V) Weil das Kochen
64 (V): dagegen der Gebrauch der Sprache //
dagegen die Regeln des Gebrauchs der Sprache
65 (V): ein
66 (O): als den und den,
67 (M): ///
236
237
238
WBTC056-59 30/05/2005 16:16 Page 374
37
“If you want to express negation with this sign, then you have to have it conform to these38
rules.”
What kind of a proposition39
is that?
One is tempted to justify grammatical rules with sentences such as: “But there really are
4 primary colours”; and the statement that the rules of grammar are arbitrary is directed
against the possibility of this justification, which is constructed on the model of justifying a
proposition by referring to its verification.
But still, can’t one in some sense say that the grammar of the colour-words characterizes
the world as it actually is? One is inclined to say: Don’t I really search in vain for a fifth
primary colour?40
Don’t we group the primary colours together because they are similar,
or at least group colours together, as opposed to, say, shapes or tones, because they are
similar? Or do I already have a preconceived idea in my head as a paradigm when I posit
this classification of the world as the correct one? – An idea about which I can only say, for
example: “Yes, that’s the way we look at things” or “What we want is to create this kind
of an image (of reality)”. For if I say: “But the primary colours do have a certain similarity
to each other”41
– where do I get the concept of this similarity from?42
Just as the concept
of primary colour is nothing more than “blue or red or green or yellow” – isn’t the concept
of that similarity also only given via the four colours? Indeed, aren’t these concepts the
same! – “Well, could one also combine red, green and circular?” – Why not?!
The importance of a game43
lies in the fact that we play this game. That we carry out these
actions. It doesn’t lose its importance by not being an action in another (superior) game.
44
Why don’t I call the rules of cooking arbitrary; and why am I tempted to call the rules
of grammar arbitrary? Because “cooking” is defined by its end, whereas speaking a
language45
isn’t. Therefore the use of language is autonomous in a certain sense in which
cooking and washing aren’t. For anyone guided by other than the correct rules when he
cooks, cooks badly; but anyone guided by rules other than those for chess plays a46
different
game, and anyone guided by grammatical rules other than such and such doesn’t as a
result say anything that is false, but is talking about something else.
47
If I could describe the purpose of grammatical conventions by saying that I had to
create them because, for instance, colours have certain properties, that would make these
conventions unnecessary, because then I could say precisely what it was the conventions were
excluding. Conversely, if the conventions were necessary, i.e. if certain combinations of words
had to be excluded as being nonsensical, then for that very reason I couldn’t name a single
37 (M): a
38 (V): to such and such
39 (V): What sort of a proposition
40 (V): colour? (And if one can search, then
finding is conceivable.)
41 (V): For if I say: !But the primary colours do
have a certain similarity to each other”
42 (V): from? That is, do I have a function !x
similar to yE here into which I can insert the
colours as arguments?
43 (V): importance in a game
44 (M): 3
45 (V): whereas the use of language // whereas the
rules for the use of language
46 (V): plays a
47 (M): ///
Grammar is not Accountable to any Reality. . . . 187e
WBTC056-59 30/05/2005 16:16 Page 375
ich eben darum nicht eine Eigenschaft der Farben angeben, die die Konventionen nötig machte,
denn dann wäre es denkbar, daß die Farben diese Eigenschaft nicht hätten und das könnte
nur entgegen den Konventionen ausgedrückt werden.
Angenommen man wollte eine grammatische68
Konvention damit rechtfertigen, daß – z.B. –
die Farben die & die Eigenschaften haben & daher gewisse Regeln für den Gebrauch der Farbwörter
gelten müßten.69
Dann wäre es nach dieser Grammatik auch denkbar, nämlich sagbar,70
daß die Farben
jene71
Eigenschaften nicht hätten & es müßte sich nach ihr alles sagen lassen, was dann der Fall72
wäre.73
74
&Die grammatischen Regeln sind willkürlich“ heißt: 75
Ich 76
nenne die Vorschriften77
der
Darstellung keine Konvention, die sich durch Sätze rechtfertigen läßt, Sätze, welche das
Dargestellte beschreiben und zeigen, daß die Darstellung adäquat ist. Die Konventionen
der Grammatik lassen sich nicht durch eine Beschreibung des Dargestellten rechtfertigen.
Jede solche Beschreibung setzt schon die Regeln der Grammatik voraus. D.h., was in der
zu rechtfertigenden Grammatik als Unsinn gilt, kann78
in der Grammatik der rechtfertigenden
Sätze auch nicht als Sinn gelten, u.u.
79
Wer etwas dagegen hat, daß man sagt, die Regeln der Grammatik seien Spielregeln, hat
in dem Sinne Recht, daß das, was das Spiel zum Spiel macht die Konkurrenz von Spielern,
der Zweck der Unterhaltung und Erholung, in der Grammatik abwesend ist, etc. Aber
niemand wird leugnen, daß das Studium des Wesens der Spielregeln für das Studium
der grammatischen Regeln nützlich sein muß, da (irgend) eine Ähnlichkeit80
offenbar81
besteht. Es ist überhaupt besser, ohne ein gefaßtes Urteil oder Vorurteil über die Analogie
zwischen Grammatik und Spiel, und nur getrieben von dem sicheren Instinkt, daß hier eine
Verwandtschaft vorliegt, die Spielregeln zu betrachten. Und hier wieder soll man einfach
berichten, was man sieht und nicht fürchten, daß man82
damit eine wichtige Anschauung
untergräbt, oder auch, seine Zeit mit etwas Überflüssigem verliert.
Man sieht dann vor allem, wie der Begriff des Spiels und damit der Spielregel ein an den
Rändern verschwimmender ist.
Ferner sieht man etwa Folgendes, wenn man die Regeln z.B. des Schachspiels betrachtet:
Es gibt hier Sätze, die die Züge der einzelnen Figuren beschreiben; allgemeiner ausgedrückt,
Regeln über Spielhandlungen. Dann aber gibt es doch die Sätze, die die Grundstellung
beschreiben und solche, die das Schachbrett beschreiben.
83
Dieser Kalkül, die Zahlentheorie etwa, zeigt84
nicht, welche wunderbaren85
Eigenschaften86
Gott
den Zahlen gegeben hat; sondern, welche Eigenschaften er uns & den Dingen gegeben hat, daß dieser
Kalkül nützlich, interessant &, mit unsern Schreibmaterialien, leicht ausführbar87
ist.
68 (O): Grammatische
69 (V): Farbwörter gelten.
70 (V): denkbar, sagbar,
71 (V): diese
72 (V): was der Fall
73 (V): wäre% wenn
74 (M): v
75 (M) 3
76 (V): Ich mus
77 (V): Regel
78 (V): kann ich
79 (M): 3
80 (V): da eine Analogie
81 (V): zweifellos
82 (V): man ei
83 (M): v
84 (V): Dieser Kalkül zeigt
85 (O): wunderbare
86 (V): welche Eigenschaften
87 (V): interessant% &, mit unsern Schreibmaterialien,
ausführbar
237v
238
239
188 Die Grammatik ist keiner Wirklichkeit Rechenschaft schuldig. . . .
WBTC056-59 30/05/2005 16:16 Page 376
feature of the colours that would make the conventions necessary, for then it would be conceivable
that the colours might not have that feature, and that could only be expressed by
contravening the conventions.
Let’s assume that someone wanted to justify a grammatical convention by saying, for example,
that colours have such and such qualities and that therefore certain rules had to be valid48
for the
use of colour words. Then, in accordance with this grammar it would also be conceivable, i.e. sayable49
,
that colours don’t have those50
qualities, and, in accordance with this grammar, everything that would
then be the case51
would have to be sayable.
52
“The rules of grammar are arbitrary” means: 53
I don’t call the regulations54
for representing
something a convention that can be justified by propositions, propositions that describe what
is represented and show that a representation is adequate. The conventions of grammar can’t
be justified by a description of what is represented. Any description of that kind already presupposes
the rules of grammar. That is, what counts as nonsense within the grammar that
is to be justified can’t55
count as making sense in the grammar of the justifying propositions,
and so on.
56
Anyone who objects to the statement that the rules of grammar are rules of a game is
right, in the sense that what makes a game a game – the competition of players, the purpose
of entertainment and recreation – is absent in grammar, etc. But nobody will deny that studying
the nature of the rules of games must be useful for the study of grammatical rules, since
there obviously57
is (some sort of ) similarity.58
In general it is better to look at the rules of
games without a preconceived judgement or prejudice as to the analogy between grammar
and games, and without being driven solely by the infallible instinct that there is a relationship
here. And here again one should simply report what one sees and not be afraid that in
doing so one is undermining an important view, or wasting one’s time with something
superfluous.
More than anything else, one then sees how the concept of a game and therefore of a rule
of a game is blurred at its edges.
Furthermore, one sees something like this when one looks at the rules, say, of chess: Here
there are propositions that describe the moves of the individual pieces; more generally
expressed, here there are rules for moves in the game. But then there are those propositions
that describe the starting position and ones that describe the chess board.
59
This calculus, say the theory of numbers, does60
not show what wonderful properties61
God
has given to numbers, but what properties He has given to us and to things, with the result that this
calculus is useful, interesting and easily carried62
out with our writing materials.
48 (V): rules are valid
49 (V): conceivable, sayable
50 (V): these
51 (V): would be the case% if
52 (M): v
53 (M): 3
54 (V): rule
55 (V): I can’t
56 (M): 3
57 (V): unquestionably
58 (V): is an analogy.
59 (M): v
60 (V): This calculus does
61 (V): what properties
62 (V): and carried
Grammar is not Accountable to any Reality. . . . 188e
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57
Regel und Erfahrungssatz.
Sagt eine Regel, daß Wörter
tatsächlich so und so
gebraucht werden?
1
Eine Regel verglichen mit einem Weg. Sagt ein Weg man solle auf ihm (& nicht auf dem Rasen)
gehen? Sagt er aus die2
Menschen gingen meistens so?
3
Die Regel ist eine Anwendung der Satzform;4
sie geht nach verschiedenen Seiten in Befehl, Rat,
Vorschlag, Erklärung eines Spiels, Erfahrungssatz u.a. über.
5
Was ist eine Regel? Ist sie ein6
Erfahrungssatz – (z.B.)7
über den Gebrauch der Sprache? Ist
eine Regel des Schachspiels ein Satz darüber, wie die Menschen seit dem Ereignis der
Erfindung des Schachspiels es gespielt haben; d.h. etwa mit so geformten Figuren gezogen
haben? Denn, wenn davon die Rede ist, daß die Menschen das Schachspiel so gespielt haben,
so muß das Schachspiel so definiert sein, daß es Sinn hat, davon auszusagen, es sei anders
gespielt worden. Sonst nämlich gehören die Regeln zur Definition des Schachspiels. Daß
jemand der Regel . . . gemäß spielt, das ist eine Erfahrungstatsache; oder: ”A spielt der
Regel . . . gemäß“, ”die meisten Menschen spielen der Regel . . . gemäß“, ”niemand spielt
der Regel . . . gemäß“ sind Erfahrungssätze. Die Regel ist kein Erfahrungssatz, sondern nur
der Teil eines solchen Satzes.
Ist eine Regel ein Befehl? oder eine Bitte?8
& was ist eine Bitte? Schau wie der Satz wirklich im
Verkehr gebraucht wird.
Die Regel setzt die Maßeinheit fest,9
und der Erfahrungssatz sagt, wie lang ein
Gegenstand ist. (Und hier sieht man, wie logische Gleichnisse funktionieren, denn die
Festsetzung der Maßeinheit ist wirklich eine grammatische Regel und die Angabe einer
Länge in dieser Maßeinheit ein Satz, der von der Regel Gebrauch macht.)
10
Wenn man die Regel dem Satz beifügt, so ändert sich der Sinn des Satzes nicht. Wenn
die Definition des Meters ist, es sei die Länge des Pariser Urmeters,11
so sagt der Satz
”dieses Zimmer ist 4m lang“ dasselbe wie, ”dieses Zimmer ist 4m lang,12
und 1m = die
Länge des Pariser Urmeters“.
1 (M): ∫ ///
2 (V): Sagt er die
3 (M): ∫ ///
4 (V): Satzform; sie grenzt
5 (M): 3
6 (V): Regel und Erfahrungssatz. Ist eine Regel ein
7 (V): Erfahrungssatz – etwa
8 (V): Bitte? Ist eine Bitte
9 (V): Die Regel ist die Festsetzung der
Maßeinheit,
10 (M): ← 3
11 (V): Wenn die Definition des Meters die Länge
des Pariser Urmeters ist$
12 (V): lang;
240
241
189
WBTC056-59 30/05/2005 16:16 Page 378
57
Rule and Empirical Proposition.
Does a Rule Say that Words
are Actually Used in Such and
Such a Way?
1
A rule compared to a path. Does a path say that one is to walk on it (and not on the grass)? Does
it state2
that people usually go that way?
3
A rule is one application of the form of a proposition;4
it takes different directions in a command,
a piece of advice, a suggestion, an explanation of a game, an empirical proposition, among other
propositional forms.
5
What is a rule? Is it an6
empirical proposition – (e.g.),7
about the use of language? Is a rule
of chess a proposition about how people have played since the game was invented; that is, say,
how they’ve moved with pieces shaped in such and such a way? For if it is said that people
played chess in a particular way, then the game of chess has to be defined so that it makes
sense to say that it had been played differently. If this doesn’t make sense, then the rules are
part of the definition of the game of chess. That someone plays in accordance with the rule
. . . is an empirical fact; or: “A plays in accordance with the rule . . .”, “Most people play in
accordance with the rule . . .”, “Nobody plays in accordance with the rule . . .” are empirical
propositions. A rule is not an empirical proposition, but only part of such a proposition.
Is a rule a command? Or a request?8
And what is a request? Look at how the sentence is actually
used in communicating.
A rule establishes a unit9
of measurement, and an empirical proposition says how long
an object is. (And here we see how logical similes work, for the establishment of a unit of
measurement really is a grammatical rule, and an indication of length in terms of this unit
of measurement is a proposition that makes use of that rule.)
10
When one adds the rule to the proposition, the sense of the proposition doesn’t change.
If the definition of a metre is that it is the length11
of the standard metre in Paris, then the
proposition “This room is 4 m long” says the same thing as “This room is 4 m long,12
and
1 m = the length of the standard metre in Paris”.
1 (M): ∫ ///
2 (V): say
3 (M): ∫ ///
4 (V): proposition; it borders
5 (M): 3
6 (V): Rule and empirical proposition. Is a rule an
7 (V): empirical proposition – perhaps
8 (V): request? Is a request
9 (V): A rule is the establishment of a unit
10 (M): ← 3
11 (V): metre is the length
12 (V): long;
189e
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Die Legende zu einer Landkarte ist so eine Anweisung zum Gebrauch – oder zum
Verständnis13
– einer Beschreibung.14
15
Diese Legende sagt jedenfalls nichts über die Geographie des Landes aus. So wenig, wie
der Satz ”1 m ist die Länge des Pariser Urmeters16
“ die Länge eines Gegenstandes angibt.17
18
Ferner bezieht sich die Regel auf ihre Anwendung in der Beschreibung (der
Wirklichkeit).19
Denn, was hat es für einen Sinn von einem Stab zu sagen ”das ist das
Urmeter“, wenn sich diese Aussage nicht auf Messungen mit dem Metermaß bezieht. Insofern
könnten wir uns die Regel jedem Satz beigefügt denken.
Die Regel ist eine Art vorgezeichneter Route; ein vorgezeichneter Weg.
20
Die Regel möchte ich ein Instrument nennen.
21
Wenn eine Regel ein Satz ist, dann wohl einer, der von den Wörtern der Sprache
handelt. Aber was sagt so ein Satz von den Wörtern aus? Daß sie in dem und dem
Zusammenhang gebraucht werden? Aber von wem und wann? Oder, daß jemand wünscht,
daß sie so gebraucht werden? Und wer? – Vielmehr ist die Regel von allen solchen22
Aussagen
ein Teil.
23
Die Regel ”links gehen!“ oder einfach ein Pfeil. Wie, wenn ich mir in meinem Zimmer
einen Pfeil an die Wand malte – wäre der auch der Ausdruck eines Gesetzes, wie es der Pfeil
auf einem Bahnhof wohl sein könnte? Um ihn zu einem Gesetz zu machen, gehört wohl24
noch der übrige Apparat, dessen einer Teil der Pfeil nur ist.
25
(Sraffa) Ein Ingenieur baut eine Brücke; er schlägt dazu in mehreren Handbüchern nach;
in technischen Handbüchern und in juridischen. Aus dem einen erfährt er, daß die Brücke
zusammenbrechen würde, wenn er diesen Teil schwächer machen würde als etc. etc.; aus
den andern, daß er eingesperrt würde, wenn er sie so und so bauen würde.26
– Stehn nun
die beiden Bücher nicht auf gleicher Stufe? – Das kommt drauf an, was für eine Rolle sie
in seinem Leben spielen. Das juridische Handbuch kann ja für ihn einfach ein Buch über
die Naturgeschichte der ihn umgebenden Menschen sein. Vielleicht muß er auch ein Buch
über das Leben der Biber nachschlagen, um zu erfahren, wie er die Brücke streichen muß,
daß die Biber sie nicht annagen. – Gibt es aber nicht noch eine andere Weise, die Gesetze
zu betrachten? Fühlen wir nicht sogar deutlich, daß wir sie nicht so betrachten? – Ist dies
nicht die gleiche Frage, wie: – Ist ein Vertrag nur die Feststellung, daß es für die Parteien
nützlich ist, so und so zu handeln? Fühlen wir uns nicht in manchen Fällen (wenn auch
nicht in allen) auf andre Weise ”durch den Vertrag gebunden“? – Kann man nun sagen:
”Wer sich durch einen Vertrag oder ein Gesetz gebunden fühlt, stellt sich irrtümlicherweise
das Gesetz als einen Menschen (oder Gott) vor, der ihn mit physischer Gewalt zwingt“?
– Nein; denn, wenn er handelt, als ob ihn jemand zwänge, so ist doch seine Handlung
jedenfalls Wirklichkeit und auch die Vorstellungsbilder, die er etwa dabei hat, sind nicht
Irrtümer; und er braucht sich in nichts irren und kann doch handeln wie er handelt und
242
190 Regel und Erfahrungssatz. . . .
13 (V): oder Verständnis
14 (M): |[Ende]|
15 (M): ( 555
16 (V): die Länge des Urmeters in Paris
17 (M): ) (V): beschreibt.
18 (M): ////
19 (V): Ferner muß sich die Regel auf die
Anwendung in der Beschreibung (der
Wirklichkeit) beziehen.
20 (M): v ← ///
21 (M): ///
22 (V): diesen
23 (M): ///
24 (V): doch
25 (M): 555
26 (V): wollte.
WBTC056-59 30/05/2005 16:16 Page 380
In the same way the key to a map is an instruction in the use of, or for the understand-
ing13
of – a description.14
15
In any case, this key states nothing about the geography of a country. Any more than
the proposition “The length of the standard metre in Paris16
is 1 m” indicates17
the length of
an object.18
19
Furthermore, the rule refers to its application in the description (of reality).20
For what kind
of sense does it make to say of a rod “This is the standard metre”, if this statement doesn’t
refer to measurements with the metre stick? As far as that goes, we could imagine the rule
conjoined to every proposition.
A rule is a kind of marked-out route; a marked-out path.
21
I’m inclined to call a rule an instrument.
22
If a rule is a proposition, then most likely it is one about the words of a language. But
what does such a proposition state about the words? That they are used in this or that context?
But by whom and when? Or that someone wishes that they be used in such a way?
Who? – Rather, a rule is a part of all such23
statements.
24
The rule “Keep left!”, or simply an arrow. What if I painted an arrow on the wall of
my room – would it too be the expression of a law, as an arrow in a railway station might
well be? In order to turn it into a law, I undoubtedly need the rest of the apparatus, of which
the arrow is only one part.
25
(Sraffa) An engineer is building a bridge. To do this, he checks in several handbooks;
in technical handbooks and in legal ones. From the one he finds out that the bridge would
collapse if he were to make this part weaker than etc., etc.; from the other that he would be
locked up if he built26
it in such and such a way. – Now are the two books not on the same
level? – That depends on what kind of a role they play in his life. After all, for him the legal
handbook can simply be a book about the natural history of the people around him. Maybe
he also has to check in a book on the life of beavers in order to find out how he has to paint
the bridge so that beavers won’t gnaw at it. – But isn’t there still another way of looking at
laws? In fact don’t we even have the distinct feeling that this is not the way we look at them?
– Isn’t this the same question as: – Is a contract merely a statement that it is useful for the
parties to act in such and such a way? Don’t we in some cases (even if not in all) feel “bound
by a contract” in a different way? – Now can one say: “Whoever feels bound by a contract
or a law erroneously pictures the law as a person (or God) who constrains him with physical
force”? – No; because even if he acts as if someone were constraining him, then still his
action is real, and likewise the mental images that might occur to him in the process are not
erroneous; he doesn’t have to err in any way, and he can still act the way he acts and also
13 (V): or understanding
14 (M): |[End]|
15 (M): ( 555
16 (V): standard metre in Paris
17 (V): describes
18 (M): )
19 (M): ////
20 (V): rule must refer to the application in the
description (of reality).
21 (M): v ← ///
22 (M): ///
23 (V): these
24 (M): ///
25 (M): 555
26 (V): he wanted to build
Rule and Empirical Proposition. . . . 190e
WBTC056-59 30/05/2005 16:16 Page 381
243
242v
243
244
191 Regel und Erfahrungssatz. . . .
sich auch vorstellen, was er sich etwa vorstellt. Die Worte ”der Vertrag bindet mich“ sind
zwar eine bildliche Darstellung und daher mit der gewöhnlichen Bedeutung des Wortes
”binden“ ein falscher Satz: aber, richtig aufgefaßt, sind sie wahr (oder können es sein) und
unterscheiden einen Fall von dem, in welchem der Vertrag mir bloß sagt, was zu tun mir
nützlich ist. Und wenn man etwas gegen die Worte einwendet ”der Vertrag (oder das Gesetz)
bindet mich“, so kann man nichts sagen gegen die Worte: ”ich fühle mich durch den Vertrag
gebunden“.
27
Die Regel – wie ich sie verstehe – ist wie ein Weg in einem Garten. Oder wie die vorgezeichneten
Felder auf einem28
Schachbrett, oder die Linien einer Tabelle. Von diesen Linien
etc. wird man nicht sagen, daß sie uns etwas mitteilen (obwohl sie ein Teil einer Mitteilung
sein können, ja auch selbst Mitteilungen). Ich lege in einer Abmachung mit jemandem eine
Regel fest. In dieser Abmachung teile ich ihm etwa die Regel (einer künftigen Darstellung)
mit. Ich sage ihm etwa: ”der Plan, den ich Dir von meinem Haus zeichne, ist im Maßstab
1 : 10“. Das ist eigentlich ein Teil der Beschreibung des Hauses. Und wenn ich schreibe ~p
& (~~p = p) so ist das wirklich ähnlich, wie wenn ich dem Plan den Maßstab beifüge.
29
Kann man &~“ nicht in der selben30
Bedeutung gebrauchen ob man nun definiert ~~p = p oder
~~p = ~p? Denn warum sollen wir nicht wie in vielen Wortsprachen eine doppelte Negation als
Negation verwenden? Man könnte dann etwa unterscheiden zwischen ~(~p) = p & ~~p = ~p
aber eine Schreibweise ~(~p) braucht es gar nicht in unserer Sprache zu geben31
& die Schreibweise
~~p = ~p multipliziert zwar unnötig Zeichen der Negation, aber mehr kann man ihr nicht zum
Vorwurf machen. Wie ist mir aber dann die Bedeutung des Verneinungszeichens32
gegeben?
Durch das Kopfschütteln, die abwehrende Bewegung? (Aber diese bestimmen keinen Kalkül.)
Oder durch eine Reihe besonderer Erklärungen wie etwa die &der Fleck befindet sich nicht innerhalb
dieser Figur heißt . . .“?
33
Ich könnte auch so sagen: Ich will nur das mitteilen, was der Satz der Sprache mitteilt;
und die Regel ist nichts als ein Hilfsmittel dieser Mitteilung (so wie ich sie, die Regel, verstehe).
Schon deshalb kann34
die Regel nicht selbst eine Mitteilung sein; denn sonst würde
der Sinn des Satzes irgendwie zugleich den Sinn der Mitteilung über den Sprachgebrauch
beinhalten.
35
Wir müssen uns vergegenwärtigen, wie wir in der Philosophie, d.h. beim Klären grammatischer
Fragen, wirklich von Regeln reden; – damit wir auf der Erde bleiben und nicht
nebelhafte Konstruktionen bauen.36
Ich gebe z.B. Regeln wie: (∃x).φx :1: φa :1: φb = (∃x).φx
oder ~~p = p, oder ich sage, daß es sinnlos ist von einem ”rötlichen Grün“ zu reden, oder
von ”schwärzlichem37
Schwarz“, oder ich sage, daß ”a = a“ sinnlos ist, oder beschreibe eine
Notation die dieses Gebilde und ”(∃x).x = x“ vermeidet, oder sage, es habe keinen Sinn zu
sagen, etwas ”scheine rot zu scheinen“, oder es habe Sinn zu sagen, daß im Gesichtsraum
eine krumme Linie aus geraden Stücken zusammengesetzt sei, oder es habe den gleichen
Sinn, zu sagen ”der Stein falle, weil er von der Erde angezogen werde“ und ”der Stein müsse
fallen, weil er von der Erde etc.“.
Ich biete dem Verwirrten eine Regel an und er nimmt sie an. Ich könnte auch sagen: ich
biete ihm eine Notation an.
27 (M): 555
28 (V): dem
29 (M): 3
30 (V): gegenwärtigen
31 (V): nicht zu geben
32 (V): Bedeutung des Zeichens „nicht“ der Verneinung
33 (M): 3
34 (V): kann // darf
35 (M): ////
36 (V): machen.
37 (O): ”schwärzlichen
WBTC056-59 30/05/2005 16:16 Page 382
imagine what he imagines. The words “the contract binds me” are, to be sure, a pictorial
representation, and are therefore a false proposition, given the usual meaning of the word
“to bind”; but understood correctly, they are true (or can be true), and they distinguish the
one case from another in which a contract merely tells me what it is useful for me to do.
And even if one objects to the words “the contract (or the law) binds me”, one can’t object
to the words: “I feel bound by the contract”.
27
A rule – as I understand it – is like a path in a garden. Or like the pre-established
squares on a28
chess board or the lines in a table. One doesn’t say of these lines, etc., that
they communicate anything to us (even though they can be part of a communication, even
communications themselves). In an agreement with someone I set down a rule. In this
agreement I apprise him of the rule (for a future representation). I might say to him, for
instance: “The blueprint of my house that I’m drawing for you is on a scale of 1 : 10”.
Actually that is part of the description of the house. And if I write ~p & (~~p = p), then
that really is similar to my appending the scale to the blueprint.
29
Can’t one use “~” with the same30
meaning, whether one defines ~~p = p or ~~p = ~p?
For why shouldn’t we use a double negation as a negation, as is done in many word-languages?
Then one could differentiate, say, between ~(~p) = p and ~~p = ~p, but the symbolic form ~(~p)
doesn’t even have to exist in our language,31
and although the symbolic form ~~p = ~p multiplies
negation signs unnecessarily, one cannot blame it for more. But how then is the meaning of
the negation sign32
given to me? By a shaking of the head, a rebuffing movement? (But these
don’t determine a calculus.) Or by a series of particular explanations, such as, say, “‘The spot is
not located within this figure’ means . . .”?
33
I could also put it this way: All I want to report is what a proposition of language reports;
and a rule (as I understand it) is nothing but an aid for this report. For this reason alone
a rule cannot34
itself be a report; because otherwise the sense of the proposition would
somehow also contain the sense of the report about the use of language.
35
We need to call to mind how we actually talk about rules in philosophy, i.e. when
we are clarifying grammatical questions; so that we keep our feet on the ground and don’t
construct36
buildings in the mist. For example, I set up rules such as: (∃x).φx :1: φa :1: φb
= (∃x).φx or ~~p = p, or I say that it makes no sense to speak about a “reddish green”
or about a “blackish black”, or I say that “a = a” makes no sense, or I describe a notation
that avoids this formation, as well as “(∃x).x = x”, or I say that it makes no sense to say
that something “seems to seem red”, or that it makes sense to say that in the visual field a
curved line is made up of straight pieces, or that it makes equally good sense to say “The
stone falls because it is attracted by the earth” and “The stone has to fall because it . . . by
the earth, etc.”.
I present a rule to someone who is confused and he accepts it. I could also say: I present
him with a notation.
27 (M): 555
28 (V): the
29 (M): 3
30 (V): current
31 (V): have to exist,
32 (V): sign A~C of negation
33 (M): 3
34 (V): cannot // rule may not
35 (M): ////
36 (V): create
Rule and Empirical Proposition. . . . 191e
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Wie schaut nun so eine Notation aus? Nun, in den meisten Fällen werde ich Sätze der
alten Notation (etwa der Wortsprache) in die entsprechenden Sätze der neuen Schreibweise
übersetzen; etwa indem ich schreibe:
alt: neu:
(∃x, y).φ(x, y) . . . . . . . . (∃x, y).φ(x, y) :1: (∃x).φ(x, x)
(∃x, y).φ(x, y) .&. x ≠ y38
. . . . . . . . (∃x, y).φ(x, y)
etc.
39
Die Regel entspricht aber in gewissem Sinne dem, was man eine ”Annahme“ genannt
hat. Sie ist quasi ein Satzradikal (chemisch gesprochen). Und es ist charakteristisch für die
Art unserer Untersuchung, daß wir uns nicht für die Sätze interessieren, die mit diesem Radikal
gebildet werden (können). Im Mittelpunkt der Betrachtung steht die Regel; nicht, daß ich
sie jemandem anbiete, nicht, daß jemand sie benützt, etc. Sie könnte, glaube ich, verglichen
werden dem Plan eines Hauses, ich meine einer Zeichnung, die als Plan eines Hauses gebraucht
werden kann, der aber kein existierendes Haus entspricht und von der auch nicht gesagt
wird, daß ihr einmal eines entsprechen soll, etc.
40
Die Beschreibung einer neuen, etwa übersichtlicheren, Notation (denn auf die Übersichtlichkeit
kommt es uns an) ist dann von der gleichen Art, wie die Beschreibung einer
jener Sprachen, die die Kinder erfinden oder von einander lernen, worin z.B. jeder Vokal
der gewöhnlichen Wörter41
verdoppelt und zwischen die Teile der Verdoppelung ein b gestellt
wird. Hier sind wir ganz nah an’s Spiel herangekommen. So eine Beschreibung oder ein
Regelverzeichnis kann man als Definiens des Namens der Sprache oder des Spiels auffassen.
Denken wir auch an die Beschreibung des Zeichnens, Konstruierens, irgend einer Figur,
etwa eines Sternes (welches auch in Spielen eine Rolle spielt). Sie lautet etwa so: ”Man
zieht eine Gerade von einem Punkt A nach einem Punkt B, etc. etc.“. Diese Beschreibung
könnte ich offenbar einfach42
durch eine Vorlage, d.h. Zeichnung, ersetzen.
Das, was hier irrezuführen scheint, ist ein Doppelsinn des Wortes ”Beschreibung“,
wenn man einmal von der Beschreibung eines wirklichen Hauses oder Baumes etc. spricht,
einmal43
von der Beschreibung einer Gestalt, Konstruktion, etc., einer Notation, eines
Spiels. Worunter aber eben nicht ein Satz gemeint ist der sagt, daß ein solches Spiel
irgendwo wirklich gespielt, oder eine solche Notation wirklich verwendet wird; vielmehr
steht die Beschreibung statt der hier gebrauchten Wörter ”ein solches Spiel“ und ”eine
solche Notation“.
Die Beschreibung einer Notation fängt (man) charakteristisch (erweise) oft mit den
Worten an: ”Wir können auch so schreiben: . . .“. Man könnte fragen: ”was ist das für eine
Mitteilung ,wir können . . .‘“? etc.44
Man schreibt auch etwa: ӟbersichtlicher wird unsere
Darstellung, wenn wir statt . . . schreiben: . . . ; und die Regeln geben . . .“; und hier stehen
die Regeln in einem Satz.
45
Denken wir uns etwa ein Bild, einen Boxer in bestimmter Kampfstellung darstellend.
Dieses Bild kann nun dazu gebraucht werden um jemandem mitzuteilen, wie er stehen, sich
halten soll; oder, wie er sich nicht halten soll; oder, wie ein bestimmter Mann dort und dort
gestanden ist;46
etc. etc. Man könnte dieses Bild ein Satzradikal nennen.
245
246
192 Regel und Erfahrungssatz. . . .
38 (O): .&. x ≠
39 (M): 3 ///
40 (M): 3
41 (V): Sprache
42 (V): auch
43 (V): spricht, ein andermal
44 (V): ‘wir können . . .! ? etc.
45 (M): 3
46 (V): hat;
WBTC056-59 30/05/2005 16:16 Page 384
Now what does such a notation look like? Well, in most cases I’ll translate propositions
expressed in the old notation (say in a word-language) into corresponding propositions
expressed in the new way of writing; for example by writing:
Old: New:
(∃x, y).φ(x, y) . . . . . . . . (∃x, y).φ(x, y) :1: (∃x).φ(x, x)
(∃x, y).φ(x, y) .&. x ≠ y . . . . . . . . (∃x, y).φ(x, y)
etc.
37
But in a certain sense a rule corresponds to what has been called an “assumption”. It is
as it were a propositional radical (in the chemical sense of that term). And it is characteristic
of the nature of our investigation that we are not interested in the propositions that can
be formed with this radical. It is the rule that stands at the centre of our examination; not
the fact that I offer it to someone, not the fact that someone uses it, etc. I believe that it
could be compared to the blueprint for a house, i.e. to a drawing that can be used as a blueprint
for a house, but to which no existing house corresponds, and about which no one is saying
that some day a house will correspond to it, etc.
38
The description of a new notation, say one that is more easily surveyed (for it is surveyability
that is our concern), is like the description of one of those languages that children
invent or learn from each other, in which for example each vowel of our ordinary words39
is doubled and a “b” is inserted between the two parts of the doubling. Here we’ve come
very close to a game. One can take such a description or list of rules as the definiens of the
name of a language or a game. Let’s also think of describing the drawing or construction of
some figure (which also plays a part in games), for example a star. The description might
run something like this: “You draw a straight line from point A to point B, etc., etc.”. Obviously
I could also just40
replace this description with a model, i.e. a drawing.
What seems to lead us astray here is the ambiguity of the word “description”. Sometimes
we speak of the description of a real house or tree, etc., and sometimes41
of the description
of a shape, a structure, etc., of a notation, a game. By which, however, we do not mean a
proposition that says that such a game is actually played somewhere, or that such a notation
is actually used; rather, the description takes the place of the words we used above: “such a
game” and “such a notation”.
Characteristically, one often begins the description of a notation with the words: “We can
also write it like this: . . .”. It can be asked: “What kind of a report is ‘We can . . .’?” etc.
One might also write: “Our presentation becomes clearer if instead of . . . we write: . . . ;
and set up these rules . . .”; and here the rules occur in propositional form.
42
Let’s think of a picture, for instance, that represents a boxer in a certain fighting stance.
Now this picture can be used to inform someone how he is to stand, to carry himself; or
how he is not to carry himself; or how a particular man stood in a certain place; etc., etc.
One could call this picture a propositional radical.
37 (M): 3 ///
38 (M): 3
39 (V): vowel in our ordinary language
40 (V): Obviously I could also
41 (V): and at another time
42 (M): 3
Rule and Empirical Proposition. . . . 192e
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47
”Regel“ ist in demselben Sinne ein Begriff mit verschwommenen Rändern, wie ”Blatt“
oder ”Stiel“ oder ”Tisch“, etc.
48
Wenn man eine Notation beschreibt, sagt man etwa: ”ich werde49
in diesem Buch statt
,p oder q‘ ,p 1 q‘ schreiben“, und das ist natürlich ein kompletter Satz. Das aber, was ich
”Regel“ nennen will, und etwa ”p oder q . = . p 1 q“ geschrieben wird, ist keiner. – Was
ich ”Regel“ nenne, soll nichts von einer bestimmten (oder auch unbestimmten) Zeit oder
einem Ort der Anwendung enthalten, sich auf keine bestimmten (oder unbestimmten) Personen
beziehen; sondern nur Instrument der Darstellung sein.
Wir sagen nun: ”wir gebrauchen die Wörter ,rot‘ und ,grün‘ in solcher Weise, daß es als
sinnlos gilt (kontradiktorisch ist) zu sagen, am selben Ort sei zu gleicher Zeit rot und grün“.
Und dies ist natürlich ein Satz, Erfahrungssatz über unsere tatsächliche Sprache.
50
Eine Regel, könnte man sagen, ist kein Befehl, sondern quasi ein51
Vorschlag.
Man könnte sich die Regeln eines Spiels auch in der Form gegeben denken: &Willst Du nicht
folgendes Spiel spielen: . . .“
52
Die Stellung der Spielregeln zu den Sätzen. Eine Regel verhält sich zu einem
Erfahrungssatz ähnlich, wie die Zeichnung, die die charakteristischen Merkmale eines
Wohnhausplanes hat, zu der Beschreibung, welche sich einer solchen Zeichnung bedient,
und welche sagt, daß so ein Haus dort und dort stehe.53
54
Der Respekt, den man vor den Regeln (z.B.55
des Schachspiels) hat, warum wir ihre
Autorität – sozusagen – nicht in Frage ziehen,56
kommt57
daher, daß die Spiele, die von
ihnen58
beschrieben59
werden,60
uns in vielerlei Hinsicht61
gemäß sind. Denken wir uns aber,
ich beschriebe62
ein Spiel, ich will es etwa ”Abracadabra“ nennen,63
indem ich64
die Regel
gebe:65
”Man lege einen Feldstein in eine viereckige Kiste, nagle die Kiste zu und werfe
mit einem andern Stein nach ihr“ – gewiß hat dieses Gebilde auch das Recht, eine Regel
genannt zu werden. Man wird nur fragen: ”was soll das alles? wozu sollen wir das machen?“
Aber auf solche Fragen geben ja auch die Schachregeln keine Antwort. Aber in dem Fall
der eben gegebenen Regel66
fällt das Wort auf ”man lege . . . und werfe“,67
nämlich die
imperative Form;68
man möchte fragen: warum soll ich . . . legen etc., oder in welchem Fall?
Was muß mein Zweck sein, damit ich das tun soll? Das heißt, der Imperativ scheint uns
hier unsinnig. Aber er ist es ebensowenig, wie in einer gewöhnlichen Spielregel. Nur sieht
47 (M): ///
48 (M): ////
49 (V): will
50 (M): ////
51 (V): sondern ein
52 (M): 555
53 (V): existiere.
54 (M): ///
55 (V): (z. B. denen
56 (V1): Der Respekt, den man vor den Regeln des
Schachspiels – etwa // (z^B^) // – hat, ich meine,
daß man sie annimmt ohne sich über sie zu wundern,
// sich nicht über sie . . . (V2): Der Respekt, den
man vor den Regeln, // – des Schach z.B. – daß hat,
– daß man wir sie annehmen, uns nicht über sie
wundern – ,
57 (V): entspringt
58 (V): von den Regeln
59 (O): beschreiben
60 (V): die Spiele, die diese Regeln charakterisieren,
// beschreiben,
61 (V): Beziehung // Weise
62 (V): erfände
63 (V): ein Spiel, es soll etwa ”Abracadabra“
heißen,
64 (V): ich dafür
65 (V): ein Spiel, das ich etwa ”Abracadabra“
nenne und gebe dafür die Regel:
66 (V): Fall jener Regel
67 (V): fällt das Wort ”man lege . . . und werfe“ auf,
68 (V): nämlich der Imperativ;
245v
246
247
193 Regel und Erfahrungssatz. . . .
WBTC056-59 30/05/2005 16:16 Page 386
43
“Rule” is a concept with blurred edges in the same sense as “leaf” or “stem” or “table”,
etc.
44
In describing a notation one might say: “In this book I shall45
write ‘p 1 q’ instead of
‘p or q’”, and of course that is a complete sentence. But what I want to call a “rule” that
might be written “p or q . = . p 1 q” is not a complete sentence. – What I am calling a
“rule” must not contain anything about a particular (or even general) time or place for its
application, and must not refer to particular people (or people in general); it is to serve only
as an instrument of representation.
We say: “We use the words ‘red’ and ‘green’ in such a way that it is considered senseless
(is contradictory) to say that there is red and green in the same place at the same time”. And
of course this is a proposition, an empirical proposition, about our actual language.
46
One could say that a rule is not a command, but as it were a47
suggestion.
One could also imagine the rules of a game given this way: “Don’t you want to play the following
game: . . . ?”
48
The status of the rules for a game relative to propositions. A rule relates to an empirical
proposition very much like a drawing that has the characteristic features of a blueprint of a
house to the description that uses such a drawing, and that says that there is such a house49
in such and such a place.
50
The respect that one has for the rules (e.g.51
of chess), why – in a manner of speaking – we
don’t question their authority52
, comes53
from the fact that the games they54
describe55
are suited
to us in many different respects.56
But let’s imagine that I were to describe57
a game I want
to call, say, “Abracadabra”58
, by setting up this rule:59
“Put a fieldstone into a rectangular crate,
nail the crate shut and then throw another stone at it” – certainly this concoction also has
the right to be called a rule. The only thing is, someone will ask: “What’s all this good for?
Why should we do that?” But neither are such questions answered by the rules of chess.
But in the case of the rule just mentioned60
the words “Put . . . and throw” catch one’s
attention, specifically their imperative form;61
one is inclined to ask: “Why should I
put . . . etc., or when? What’s the point of doing that?” That is, the imperative strikes us
here as nonsensical. But it isn’t, any more so than in a normal rule of a game. It’s just that
43 (M): ///
44 (M): ////
45 (V): I want to
46 (M): ////
47 (V): but a
48 (M): 555
49 (V): that such a house exists
50 (M): ///
51 (V): (e.g. those
52 (V1): The respect that one has for the rules of
chess – perhaps // (for example) // – I mean that one
accepts them without being surprised about them //
accepts them and is not surprised about them
(V2): The respect that one has for rules, // – of
chess, for example – that has, – that one we accept
them, and are not surprised about them –
53 (V): derives
54 (V): the rules
55 (V): games that these rules characterize //
describe
56 (V): ways.
57 (V): invent
58 (V): a game – let it be called “Abracadabra”
59 (V): game that I call, say, “Abracadabra” and I
give this rule for the game:
60 (V): of this rule
61 (V): specifically the imperative;
62 (V): It’s just that here
Rule and Empirical Proposition. . . . 193e
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man in diesem Fall69
klar, daß man es nicht mit einem Befehl70
zu tun hat. Höchstens
mit der Definition von ”Abracadabra“,71
nämlich: ”Abracadabra spielen“72
heißt, einen
Feldstein in eine Kiste legen, etc.
73
Kaufe Dir in einer Spielwarenhandlung ein Spiel. Du erhältst eine Schachtel darin die
Implemente des Spiels & ein Regelverzeichnis. Was sind die Regeln dieses Verzeichnisses für74
Sätze?
Wird Dir vom Erzeuger des Spiels befohlen so & so zu handeln, oder wird es Dir angeraten?75
Oder
wird Dir mitgeteilt daß die & die Menschen, oder alle Menschen, so handeln?76
Nun, sieh doch nur
nach wie das Regelverzeichnis gebraucht wird!77
Die meisten Leute die das Spiel kaufen lesen die
Regeln & spielen nach ihnen. –
&Wenn der Satz von den Spielfiguren handelt so ist er also ein Erfahrungssatz!“ – So ist also auch
dies ein Erfahrungssatz: &Alle Wohlgerüche Arabiens . . .“?78
Dieser Satz kann die Rolle eines Erfahrungssatzes spielen & die Sätze des Regelverzeichnisses
könnten die Rolle von Befehlen, von Ratschlägen oder von Erfahrungssätzen spielen, (sie) tun es
aber nicht.
194 Regel und Erfahrungssatz. . . .
69 (V): man hier
70 (V): einem kompletten Satz
71 (O): ”Abracadabra;
72 (V): ”Abracadabra! spielen“
73 (M): v
74 (V): die Regeln für
75 (V): befohlen so zu handeln? Oder angeraten?
76 (V): so gehandelt haben?
77 (V): sieh doch nach wie . . . gebraucht wird^
78 (E): Shakespeare, Macbeth, 5. Akt, 1. Aufzug.
246v
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in this case62
one sees clearly that one is not dealing with a command63
, but at most with the
definition of “Abracadabra” – namely, “‘Playing Abracadabra’ means putting a field stone
into a crate, etc.”.
64
Buy a game in a toy store. You get a box and in it the implements of the game and a list of
rules. What kinds of propositions are the rules on this list?65
Are you told by the manufacturer of
the game to act in such and such a way, or66
is this suggested to you? Or are you informed that
such and such people, or all people, act67
in this way? Well, just check and see how the list of
rules is used! Most people who buy the game read the rules and play in accordance with them. –
“If the proposition is about the pieces of the game, then it’s an empirical proposition!” – So is this
too an empirical proposition: “All the perfumes of Arabia . . .”68
?
This proposition can play the role of an empirical proposition, and the propositions listing the
rules could play the roles of commands, pieces of advice or empirical propositions, but (they)
don’t do so.
63 (V): a complete sentence
64 (M): v
65 (V): the rules?
66 (V): to act thus? Or
67 (V): people have acted
68 (E): Shakespeare, Macbeth, Act V, Scene 1.
Rule and Empirical Proposition. . . . 194e
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58
Die strikten grammatischen
Spielregeln und der schwankende
Sprachgebrauch.
Die Logik normativ.
Inwiefern reden wir von idealen
Fällen, einer idealen Sprache?1
(”Logik des luftleeren Raums“.)
2
Was heißt es, zu wissen, was eine Pflanze ist?
Was heißt es, es zu wissen und es nicht sagen zu können?
”Du weißt es und kannst hellenisch reden, also mußt Du es doch sagen können.“3
Müßigkeit
einer Definition, etwa der, des Begriffs ”Pflanze“. Aber ist die Definition kein Erfordernis
der Exaktheit? ”Der Boden war ganz mit Pflanzen bedeckt“:4
damit meinen wir nicht Bacillen.
Ja, wir denken dabei vielleicht an grüne Pflanzen einer bestimmten Größenordnung. 5
Wer
uns sagen würde, wir wissen nicht, was wir reden, ehe wir eine6
Definition der Pflanze gegeben
haben, den würden7
wir mit Recht für verrückt halten. Ja, wir könnten auch mit einer solchen
Definition uns in den gewöhnlichen Fällen nicht besser verständigen. Ja, es scheint sogar,
in gewissem Sinne schlechter, weil gerade das Undefinierte in diesem Fall zu unserer Sprache
zu gehören scheint.
8
Eine9
richtige Erklärung könnte in so einem Falle durch ein gemaltes Bild gegeben werden und
mit den Worten10
&so ähnlich hat der Boden ausgesehen“. Denken wir uns aber nun wir wollten11
die Erklärung12
exakt machen indem wir sagen: &der Boden hat genau so ausgesehen“. Also13
genau
diese Gräser & Blätter in diesen Lagen waren dort? Das ist es offenbar nicht was ich meinte. Welche
exakte Erklärung immer mir Einer gäbe, ich könnte keine anerkennen.
1 (O): Sprache.
2 (M): 3
3 (E): Vgl. Platon, Charmides, 159a.
4 (V): bedeckt“;
5 (M): ///
6 (V): keine
7 (O): haben, würden
8 (M): 3
9 (V): Denken wir Eine
10 (O): und den Worten (V): werden zusammen
mit den Worten
11 (V): nun wir es wollte jemand
12 (O): Erklarung
13 (V): ausgesehen“. Nun d Also
248
195
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58
The Strict Grammatical Rules
of a Game and the Fluctuating
Use of Language.
Logic as Normative.
To what Extent do we Talk about
Ideal Cases, an Ideal Language?
(“The Logic of a Vacuum.”)
1
What does it mean to know what a plant is?
What does it mean to know it and not be able to say it?
“You know it and you can speak Greek, so surely you must be able to say it.”2
The
uselessness of a definition, say of the concept “plant”. But doesn’t precision require a
definition? “The ground was all covered with plants”: here we don’t mean bacilli. Indeed,
in this context we might think of green plants of a particular size. 3
We’d rightly consider
someone insane who told us that we don’t know what we’re talking about until we’ve given
a definition of a plant. Under ordinary circumstances we couldn’t communicate any better
with such a definition. Indeed, it even seems worse, in a certain sense, because precisely what
is undefined seems in this case to belong to our language.
4
In5
such a case a correct explanation could be given using a painting and6
the words “That’s more
or less what the ground looked like”. But now suppose we7
wanted to make the explanation exact,
by saying: “The ground looked exactly like this.” So precisely these grasses and leaves were there in
those positions? That is obviously not what I meant. No matter which exact explanation I’m given,
I couldn’t accept it.
1 (M): 3
2 (E): Cf. Plato, Charmides, 159a.
3 (M): ///
4 (M): 3
5 (V): We think In
6 (V): painting together with
7 (V): someone
195e
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14
Ist eine scharfe Photographie immer & für alle Zwecke besser15
als eine unscharfe,
verschwommene? Was, wenn Einer16
sagen würde: &ein unscharfes Bild ist eigentlich gar kein Bild“?!
17
Denken wir uns in dem Satz einer Erzählung ”der Boden war ganz mit Gräsern und
Kräutern bedeckt“ die Wörter ”Gräser“ und ”Kräuter“ durch Definitionen ersetzt. Es ist
klar, daß diese Definitionen lange und komplizierte Ausdrücke sein werden;18
und nun ist
die Frage, ob wir denn wirklich mit dem Satz das gemeint haben, was jetzt in dem ungleich
viel komplizierteren steht. Wir würden – glaube ich – sagen, daß wir an alles das gar
nicht gedacht hätten.
19
Kann man nun aber auf eine solche Sprache die Idee des Kalküls anwenden? Und ist
das nicht so, als wollte man in einem Bild, worin alle Farbflecken ineinander verlaufen, von
Farbgrenzen reden? Oder liegt die Sache so: Denken wir uns ein Spiel, etwa das Tennis, in
dessen Regeln nichts über die Höhe gesagt ist, die ein Ball im Flug nicht übersteigen darf.
Und nun sagte Einer: Das Spiel ist ja gar nicht geregelt, denn, wenn Einer den Ball so hoch
wirft, daß er nicht wieder auf die Erde zurückfällt, oder so weit, daß er um die Erde herumfliegt,
so wissen wir nicht, ob dieser Ball als ”out“ oder ”in“ gelten soll. Man würde ihm – glaube
ich – antworten, wenn ein solcher Fall einträte, so werde man Regeln für ihn geben, jetzt
sei es nicht nötig.
20
So können doch grammatische Regeln über den Gebrauch des Wortes ”Pflanze“
gegeben werden und wir können also auf Fragen von der Art ”folgt aus diesem Sachverhalt,
daß dort eine Pflanze steht“ Bescheid geben. Auf andere solche Fragen aber sind wir nicht
gerüstet und können antworten: Ein solcher Fall ist noch nie vorgekommen und es wäre für
uns müßig, für ihn vorzusorgen. (Wenn es etwa gelänge, ein Lebewesen halb maschinell und
halb auf organischem Weg zu erzeugen, und nun gefragt würde: ist das nun noch ein Tier
(oder eine Pflanze).)
Wenn etwa beim Preisschießen für gewisse Grenzfälle keine Bestimmung getroffen wäre,
ob dieser Schuß noch als Treffer ins Schwarze gelten soll (oder nicht). Nehmen wir nun
aber an, ein solcher Schuß komme bei unserem Preisschießen gar nicht vor; könnte man
dann dennoch sagen, die ganze Preisverteilung gelte nichts, weil für diesen Fall nicht
vorgesorgt21
war?
22
Ich mache mich doch anheischig, das Regelverzeichnis unserer Sprache aufzustellen: Was
soll ich nun in einem Fall wie dem des Begriffes ”Pflanze“, tun?
23
Soll ich sagen, daß für diesen und diesen Fall keine Regel aufgestellt ist? Gewiß, wenn
es sich so verhält. Soll ich aber solches sagen wie, es gibt24
kein Regelverzeichnis unserer
Sprache und das ganze Unternehmen, eins aufzustellen, ist Unsinn? – Aber es ist ja klar,
daß es nicht unsinnig ist, denn wir stellen ja mit Erfolg Regeln auf, und wir müssen uns nur
enthalten, Dogmen aufzustellen. (Was ist das Wesen eines Dogmas? Ist es nicht die
Behauptung eines naturnotwendigen Satzes über alle möglichen Regeln?)25
196 Die strikten grammatischen Spielregeln . . .
14 (M): 3
15 (V): immer besser
16 (V): man
17 (M): 3
18 (V): müssen;
19 (M): ///
20 (M): /// – vorzusorgen.
21 (V): vorgesehen
22 (M): ///
23 (M): /// – aufzustellen.
24 (O): aber solche sagen, es gibt
25 (V): Dogmas? Besteht es nicht darin,
naturnotwendige Sätze über alle möglichen
Regeln zu behaupten?)
250
248v
249
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8
Is a sharp photograph always and for all purposes better9
than an out-of-focus, blurry one? What
if someone10
were to say: “An out-of-focus picture really isn’t a picture at all”?!
11
Let’s imagine that in this sentence of a story, “The ground was all covered with grass
and herbs”, the words “grass” and “herbs” are replaced by definitions. It is clear that these
definitions will be12
long and complicated expressions; and now the question is whether we
really did mean by this sentence what is now contained in the immeasurably more complicated
one. We would say – I believe – that we hadn’t even thought of all that.
13
So can one now apply the idea of a calculus to such a language? And wouldn’t that be
like trying to talk about colour boundaries in a picture in which all the patches of colour
bleed into each other? Or is it like this: Let’s think of a game, say tennis, whose rules say
nothing about the maximum height for a ball’s trajectory. And now imagine that someone
were to say: This game isn’t regulated at all, for if someone throws the ball so high that it
doesn’t return to earth, or so far that it flies around the earth, then we don’t know whether
this ball is to count as “out” or “in”. We’d answer him – I believe – that if such a case were
to come about we’d set up rules for it, but that for now it isn’t necessary.
14
Thus we can give grammatical rules for the use of the word “plant”, and so we can answer
questions of this kind: “Does it follow from this state of affairs that there is a plant over
there?”. But we aren’t prepared for other such questions, and can answer: Such a case has
never happened, and it would be pointless for us to prepare for it. (Say if someone succeeded
in creating a living being that was half-machine and half-organic, and then the question were:
Now is that still an animal (or a plant)?)
If, say at a shooting match, no regulation had been made for certain borderline cases, whether
this shot is to count as hitting the bull’s eye (or not). But now let’s assume that at our match
such a shot never occurs; then could one say that the entire distribution of prizes was invalid,
because no provisions had been made15
for that case?
16
I’m undertaking the establishment of the list of rules for our language: Now what am I
to do in a case like that of the concept “plant”?
17
Should I say that for this or that case no rule has been established? Certainly, if that’s
the way it is. But should I say such things as: There is no list of rules for our language and
the whole venture of setting one up is nonsense? – But it’s clear that this isn’t nonsensical,
for after all, we do successfully set up rules, and we just have to refrain from setting
up dogmas. (What’s the nature of a dogma? Isn’t it the assertion that there’s a naturally
necessary proposition for every possible rule?18
)
8 (M): 3
9 (V): purposes always better
10 (V): one
11 (M): 3
12 (V): will have to be
13 (M): ///
14 (M): /// – for it.
15 (V): because nothing had been planned
16 (M): ///
17 (M): /// – dogmas.
18 (V): dogma? Doesn’t it consist in asserting
that there are naturally necessary propositions
for all possible rules?
The Strict Grammatical Rules of a Game . . . 196e
WBTC056-59 30/05/2005 16:16 Page 393
26
Ich mache mich nicht anheischig ein Regelverzeichnis aufzustellen das alle unsere Sprachhandlungen
regelt; sowenig ein Jurist es versucht für sämtliche Handlungen der Menschen Gesetze zu geben.
27
Was ist eine exakte Definition im Gegensatz zu einer unexakten? Nun etwa; eine Def. in der nicht
das Wort &ungefähr“, &beiläufig“ &28
ähnliche vorkommen.
29
”Ich weiß, was eine Pflanze ist, kann es aber nicht sagen.“ Hat dieses Wissen die
Multiplizität eines Satzes, der nur nicht ausgesprochen wurde? So daß, wenn der Satz ausgesprochen
würde, ich ihn als den Ausdruck meines Wissens anerkennen würde? – Ist es
nicht vielmehr wahr, daß jede exakte Definition als Ausdruck unseres Verstehens abgelehnt
werden müßte? D.h., würden wir nicht von so einer sagen müssen, sie bestimme zwar einen,
dem unseren verwandten, Begriff, aber nicht diesen selbst? Und die Verwandtschaft sei etwa
die, zweier Bilder, deren eines aus unscharf begrenzten Farbflecken, das andere aus ähnlich
geformten und verteilten, aber scharf begrenzten, bestünde. Die Verwandtschaft wäre dann
ebenso unleugbar, wie die Verschiedenheit.
30
Die Frage ist nun: kannst Du bei dem ersten Bild auch von Flecken reden? Gewiß, nur
in einem anderen, aber verwandten, Sinn.
31
Das heißt: die unscharfen Grenzen gehören zu meinem Begriff der Pflanze, so wie er
jetzt ist, d.h. so, wie ich dieses Wort jetzt gebrauche, und es charakterisiert diesen Begriff,
daß ich z.B. sage: ich habe darüber keine Bestimmung getroffen, ob dieses Ding eine Pflanze
heißen soll oder nicht.
32
Es verhält sich doch mit dem Begriff ”Pflanze“ ähnlich, wie mit dem der Eiförmigkeit,
wie wir sie im gewöhnlichen Leben meinen. Die Grenzen dieses Begriffs sind
nicht scharf bestimmt und wir würden z.B. ein Osterei von dieser Form nicht als solches
gelten lassen und doch nicht sagen können, bei welchem Verhältnis der Länge und
Breite etwas anfängt, ein Osterei zu sein. Ja, wenn Einer nun ein solches Verhältnis
angäbe, was es auch sei, so könnten wir es nicht als die richtige Begrenzung unseres Begriffs
anerkennen. Sondern wir müßten entweder sagen: nein, das nenne ich kein Osterei, es
ist zu schlank, oder zu dick etc., oder: ja, das ist auch ein Osterei, aber der Grenzfall ist es
nicht gerade. Diesen gibt es eben nicht in unserm Kalkül und wer einen Grenzfall einführt,
führt einen andern Kalkül ein.
33
Wenn man sagt ”N. existiert nicht“, so kann das verschiedenerlei bedeuten. Es kann heißen,
daß ein Mann, der, als er lebte, diesen Namen trug, nicht, oder nicht zu einer gewissen
Zeit, in einem gewissen Land existiert hat; aber auch, daß spätere Geschichtsschreiber den
Charakter, den wir so (etwa ”Moses“) nennen, erfunden haben, daß die und die Ereignisse
nie stattgefunden haben und ihr Held also nie gelebt hat. 34
&Moses hat nicht existiert“ – das35
kann heißen: Es hat nicht einen Menschen gegeben der alle die Taten die von Moses berichtet
werden getan hat. Es hat keinen Mann mit Namen &Moses“ gegeben der die Israeliten durch die
Wüste36
geführt hat. Es hat so einen Mann gegeben aber er hat nicht &Moses“ geheißen.
37
Russell würde sagen daß wir den Namen Moses durch verschiedene Beschreibungen definieren
können (”der Mann, welcher ,Moses‘ hieß und zu dieser Zeit an diesem Ort lebte“, oder
26 (M): 3 ///
27 (M): 3
28 (V): &beläufig“ od
29 (M): 3 ///
30 (M): ///
31 (M): ///
32 (M): 3
33 (M): 3
34 (M): /// – geheißen.
35 (V): existiert“; das
36 (V): Israeliten aus Ägypten
37 (M): /// – erhält
249v
250
251
252
251v
197 Die strikten grammatischen Spielregeln . . .
WBTC056-59 30/05/2005 16:16 Page 394
19
I am not undertaking the establishment of a list of rules that regulates all of our speech acts; any
more than a jurist tries to make laws for all human actions.
20
What is an exact definition, as opposed to an inexact one? Well, for instance: a definition in which
the words “about”, “approximately” and so forth do not occur.
21
“I know what a plant is, but I can’t say it.” Does this knowledge have the multiplicity
of a sentence that just didn’t get expressed? So that if the sentence were expressed, I
would acknowledge it as the expression of my knowledge? – Isn’t it true, rather, that every
exact definition would have to be rejected as an expression of our understanding? That is,
wouldn’t we have to say of such a definition that although it does determine a concept
related to ours, it doesn’t determine our concept? And that the connection was, say, like
two pictures, one of which consisted of colour patches with blurred borderlines, and the
other of ones that were similarly shaped and distributed, but clearly demarcated. Then the
connection would be just as undeniable as the difference.
22
Now the question is: Can you talk about patches in the first picture as well? Certainly,
but only in another though related sense.
23
That means: The blurred borderlines are part of my concept of a plant as it now stands,
i.e. as I now use this word, and it is characteristic of this concept that I say, for example: I
haven’t determined whether this thing is to be called a plant or not.
24
After all, the concept “plant” is similar to that of “ovoid”, as we understand it in everyday
life. The boundaries of this concept aren’t sharply determined, and we wouldn’t
acknowledge an Easter egg that had this shape , for example, as being an Easter egg,
and yet we couldn’t say at what ratio of length and width something begins to be one.
Indeed, if someone were to state such a ratio, whatever it might be, we could not acknowledge
it as the correct delimitation of our concept. Rather, we would either have to say:
No, I don’t call that an Easter egg, it is too narrow or too fat, etc., or: Yes, that’s an Easter
egg too, but it isn’t exactly a borderline case. The borderline case simply doesn’t exist in
our calculus, and whoever introduces a borderline case introduces a different calculus.
25
If someone says “N doesn’t exist”, this can mean a variety of things. It can mean that a
man who went by this name when he was alive didn’t live in a certain country or didn’t live
there at a certain time; but also that later historians invented the character that we call by
that name (say “Moses”), that such and such events never took place and that therefore
their hero never lived. 26
“Moses didn’t exist” – that can mean: There wasn’t any one person
who did all of the deeds that are reported of Moses. There was no man by the name of “Moses”
who led the Israelites through the desert.27
There was such a man, but his name wasn’t “Moses”.
28
Russell would say that we can define the name Moses by different descriptions (“the man whose
name was ‘Moses’ and who lived at that time in that place”, or “the man – whatever he was
19 (M): 3 ///
20 (M): 3
21 (M): 3 ///
22 (M): ///
23 (M): ///
24 (M): 3
25 (M): 3
26 (M): /// – wasn’t “Moses”.
27 (V): Israelites out of Egypt.
28 (M): /// – another definition
The Strict Grammatical Rules of a Game . . . 197e
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”der Mann – wie immer er damals genannt wurde – welcher die Israeliten durch die Wüste
führte“, oder ”der Mann, der als kleines Kind von der Königstochter aus dem Nil gefischt
wurde“, etc. etc.). Jenachdem wir die eine oder andere Def. annehmen erhält38
der Satz ”Moses
hat existiert“ einen andern Sinn und ebenso jeder andere Satz, der von Moses handelt. 39
Man
könnte40
auch immer, wenn uns jemand sagte ”N existiert nicht“, fragen: ”was meinst Du?
willst Du sagen, daß . . . , oder daß . . . etc.?“ – Wenn ich nun sage: ”N ist gestorben“ so
kann es mit ”N“ etwa folgende Bewandtnis haben:41
Ich glaube, daß ein Mensch N gelebt
hat: den ich 1.) dort und dort gesehen habe, der 2.) so und so ausschaut, 3.) das und das
getan hat und 4.) in der bürgerlichen Welt den Namen ”N“ führt. 42
Gefragt, was ich unter
”N“ verstehe, würde ich alle diese Dinge, oder einige von ihnen, und bei verschiedenen
Gelegenheiten verschiedene, aufzählen. Meine Definition von ”N“ wäre also: der Mann,
von dem alles das stimmt. Wenn aber nun einiges davon sich als falsch erwiese, – wäre der
Satz ”N43
ist gestorben“ nun als falsch anzusehen? auch, wenn nur etwas vielleicht ganz
Nebensächliches, was ich von dem Menschen glaubte, nicht stimmen würde; – wo aber44
fängt das Nebensächliche45
an? Das kommt nun darauf hinaus, daß wir den Namen ”N“ in
gewissem Sinne ohne feste Bedeutung gebrauchen, oder: daß wir bereit sind, die Spielregeln
nach Bedarf zu verändern (make the rules as we go along). Das erinnert an das, was ich früher
einmal über die Benützung der Begriffswörter, z.B. des Wortes ”Blatt“ oder ”Pflanze“,
geschrieben habe. – Und hier erinnere ich mich daran, daß Ramsey einmal betont hat, die
Logik sei eine ”normative Wissenschaft“. 46
Wenn man damit meint, sie stelle ein Ideal auf,
dem sich die Wirklichkeit nur nähere, so muß gesagt werden, daß dann dieses ”Ideal“ uns
nur als ein Instrument der annähernden Beschreibung der Wirklichkeit interessiert. &Die
Logik ist eine normative Wissenschaft“ sollte doch wohl heißen47
sie stelle Ideale auf48
denen wir
nachstreben sollen. Aber so ist es ja nicht. Die Logik stellt exacte Kalküle auf. 49
Es ist allerdings
möglich, einen Kalkül genau zu beschreiben und zwar zu dem Zweck, um dadurch eine Gruppe
anderer Kalküle beiläufig zu charakterisieren. Wollte z.B. jemand wissen, was ein Brettspiel
ist, so könnte ich ihm zur Erklärung das Damespiel genau beschreiben und dann sagen:
siehst Du, so ungefähr funktioniert jedes Brettspiel. – War es nun nicht ein Fehler von mir
(denn so scheint es mir jetzt) anzunehmen, daß der, der die Sprache gebraucht, immer ein
bestimmtes Spiel spiele? Denn, war das nicht der Sinn meiner Bemerkung, daß alles an einem
Satz – wie beiläufig immer er ausgedrückt sein mag – ”in Ordnung ist“? Aber wollte ich
38 (V): nie gelebt hat. D.h. also: kein Mensch hat
Moses geheißen und diese Taten vollbracht;
oder: das Ding, das Dir als Herr N vorgestellt
wurde, war eine Puppe; etc. Denken wir uns,
es sagte uns Einer, er habe Moses auf der
Straße gesehen. Wir würden ihn dann fragen:
”wie meinst Du das: Du hast ihn gesehen?
Wie wußtest Du denn, daß er es war?“ und
nun könnte der Andre sagen: ”er hat es mir
gesagt“, oder ”er sah so aus, wie ich mir
Moses vorstelle“, oder ”er hatte diese und
diese Merkmale“, etc. Ich will doch wohl
das sagen, was Russell dadurch ausdrückt,
daß der Name Moses durch verschiedene
Beschreibungen definiert sein kann (”der
Mann, welcher ,Moses‘ hieß und zu dieser Zeit
an diesem Ort lebte“, oder ”der Mann – wie
immer er damals genannt wurde – welcher die
Israeliten durch die Wüste führte“, oder ”der
Mann, der als kleines Kind von der
Königstochter aus dem Nil gefischt wurde“,
etc. etc.). Und je nachdem wir die eine
oder andere Definition annehmen, bekommt
39 (M): | – daß . . . etc.?“
40 (V): würde
41 (V): so hat es mit ”N“ gewöhnlich etwa
folgende Bewandtnis:
42 (M): | – as we go along).
43 (V): @N!
44 (V): würde; – und wo
45 (V): Hauptsächliche
46 (M): /// – interessiert.
47 (V): Wissenschaft“ heißt eigentlich
48 (V): auf nach
49 (M): ///
252
252v
253
198 Die strikten grammatischen Spielregeln . . .
WBTC056-59 30/05/2005 16:16 Page 396
then called – who led the Israelites through the desert” or “the man who as a baby was fished
out of the Nile by the pharaoh’s daughter”, etc., etc.). Depending on whether we accept one
or another definition,29
the sentence “Moses existed” acquires a different sense, and so too does
every other sentence about Moses. 30
In addition, every time someone tells us “N doesn’t exist”
we could31
ask: “What do you mean? Do you want to say that . . . , or that . . . etc.?” – Now
if I say: “N has died”, then, for example, the situation with “N” can be as follows32
: I believe
that a person N used to live: 1. whom I saw in such and such a place, 2. who has such and
such an appearance, 3. who has done this and that, and 4. who is called “N” in bourgeois
society. 33
If I am asked what I understand by “N” I’d enumerate all of these things, or some
of them, and different ones on different occasions. So my definition of “N” would be: the
man of whom all of this is correct. But if some of these things now turned out to be false –
would the proposition “N34
has died” now have to be viewed as false? Even if only something
possibly quite trivial that I believed about this person were incorrect? – But where35
does the trivial36
begin? What this amounts to is that, in a certain sense, we use the name
“N” without a fixed meaning, or that we are ready to alter the rules of the game as we need
(make up the rules as we go along). This brings to mind what I once wrote about the use of
concept-words, for example, of the word “leaf” or “plant”. – And here I recall that Ramsey
once emphasized that logic is a “normative discipline”. 37
If what’s meant by this is that
logic establishes an ideal to which reality can only approximate, then it must be said that
this “ideal” interests us only as an instrument for the approximate description of reality.
“Logic is a normative discipline” presumably is supposed to mean38
that it sets up ideals towards39
which we are supposed to strive. But that’s not the way it is. Logic sets up exact calculi. 40
Of course
it is possible to describe a calculus exactly: for the specific purpose of giving a rough
characterization of a different group of calculi. If for instance someone wanted to know
what a board game was, then I could explain by describing draughts to him in detail
and then say: “See, that’s more or less how every board game works”. – Now wasn’t
it a mistake of mine (for that’s what it strikes me as now) to assume that whoever uses
language always plays a particular game? For wasn’t that the sense of my remark that
everything about a proposition – no matter how off-the-cuff it is – “is in order”? But
wasn’t what I wanted to say: Everything has to be in order when someone utters a
29 (V): never lived. That is: There was never a
human Moses who accomplished these deeds;
or: the object that was introduced to you as Mr
N was a doll; etc. Let’s imagine that someone
were to tell us that he had seen Moses on the
street. Then we would ask him: “What do you
mean: did you see him? How did you know that
it was he?” And then the other person could
say: “He told me so”, or “He looked the way
I imagine Moses”, or “He had such and such
characteristics”, etc. What I want to say is what
Russell expresses by saying that the name
“Moses” can be defined with various descriptions
(“the man whose name was ‘Moses’ and
who lived at that time in that place”, or “the
man – whatever he was called then – who led
the Israelites through the desert”, or “the
man who as a baby was fished out of the
Nile by the pharaoh’s daughter”, etc., etc.).
And depending on whether we accept one
or another definition,
30 (M): | – etc.?”
31 (V): would
32 (V): with “N” is usually as follows
33 (M): | – go along).
34 (V): !NE
35 (V): incorrect; – and where
36 (V): essential
37 (M): /// – reality.
38 (V): discipline” actually means
39 (V): ideals according to
40 (M): ///
The Strict Grammatical Rules of a Game . . . 198e
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nicht sagen: alles müsse in Ordnung sein, wenn Einer einen Satz sage und ihn anwende?
Aber daran ist doch weder etwas in Ordnung noch in Unordnung. – In 50
Ordnung wäre es,
wenn man sagen könnte: auch dieser Mann spielt ein Spiel nach einem bestimmten, festen
Regelverzeichnis.51
Denn ich habe zur Feststellung der Regel, nach der er handelt, zwei Wege angegeben.52
Der eine, der hypothetische, bestand in der Beobachtung seiner Handlungen und die Regel
war dann von der Art eines naturwissenschaftlichen Satzes. Der andere war, ihn zu fragen,
nach welcher Regel er vorgehe. Wie aber, wenn der erste Weg kein klares Resultat ergibt
und die Frage keine Regel zu Tage fördert, wie es im Fall ”N53
ist gestorben“ geschieht.
Denn, wenn wir den, der das sagte, fragen ”was ist N?“, so wird er zwar ”N“ durch eine
Beschreibung erklären, wird aber bereit sein, diese Beschreibung zu widerrufen und
abzuändern, wenn wir ihm den einen oder andern Satz entziehen.54
Wie soll ich also die
Regel bestimmen,55
nach der er spielt? er weiß sie selbst nicht. Ich könnte eine Regel nur
nach dem bestimmen, was er auf die Frage ”wer ist N“ in diesem Fall gerade antwortet.
56
Unsre Untersuchung trachte nicht die exacte57
Bedeutung der Wörter zu finden;58
wohl aber geben
wir den Wörtern im Verlauf unsrer Untersuchung oft exacte Bedeutungen.
59
Steckt uns da nicht die Analogie der Sprache mit dem Spiel ein Licht auf? Wir können
uns doch sehr wohl denken, daß sich Menschen auf einer Wiese damit unterhielten, mit einem
Ball zu spielen; und zwar so, daß sie verschiedene bestehende Spiele der Reihe nach anfingen,
nicht zu Ende spielten und etwa dazwischen sogar planlos den Ball würfen, auffingen, fallen
ließen, etc. Nun sagte Einer: die ganze Zeit hindurch spielen die Leute ein Ballspiel und
richten sich daher bei jedem Wurf nach bestimmten60
Regeln. – Aber – wird man einwenden
– der den Satz ”N ist gestorben“ gesagt hat, hat doch nicht planlos Worte aneinander
gereiht (und darin besteht es ja, daß er ”etwas mit seinen Worten gemeint hat“). – Aber
man kann wohl sagen: er sagt den Satz planlos, was sich eben in der beschriebenen
Unsicherheit zeigt. Freilich ist der Satz von irgendwo hergenommen und wenn man will,
so spielt er nun auch ein Spiel mit sehr primitiven Regeln; denn es bleibt ja wahr, daß ich
auf die Frage ”wer ist N“ eine Antwort bekam, oder eine Reihe von Antworten, die nicht
gänzlich regellos waren. – Wir können sagen: Untersuchen wir die Sprache auf ihre Regeln
hin. Hat sie dort und da keine Regeln, so ist das das Resultat unsrer Untersuchung.61
Wenn aber der Träger dem Namen abhanden kommen, oder nie existiert haben kann,
so mußte man beim Gebrauch des Namens von vornherein damit rechnen. Das mußte
in seiner Bedeutung liegen. ((Es sei denn, daß wir diese Bedeutung geändert haben, oder,
daß das Wort keine bestimmte Bedeutung hatte; denn welches ist die Bedeutung, wenn er sie
nicht angeben kann? Nun, wir werden sein tatsächliches Verhalten durch ein ”Schwanken
zwischen mehreren Bedeutungen“ beschreiben können. Es ist wohl wesentlich, daß ich ihn
fragen kann: was hast Du eigentlich gemeint. Und als Antwort wird er mir vieles sagen, und
sich vielleicht62
an mich wenden, daß ich ihm das Regelverzeichnis einrichte, das seinem Zweck
50 (V): Unordnung$ – in (O): Unordnung. –
in
51 (V): Regelverzeichnis. Und setzt das nicht
wieder voraus$ daß dieses
52 (O): angeben.
53 (V): ”N!
54 (V): widerlegen.
55 (V): auffassen,
56 (M): 3
57 (V): eigentliche
58 (V): finden; so
59 (M): 3
60 (V): gewissen
61 (R): ∀ S. 250v
62 (V): etwa
254
253v
254
255
199 Die strikten grammatischen Spielregeln . . .
WBTC056-59 30/05/2005 16:16 Page 398
sentence and uses it? But there is nothing either in order or in disorder about this. – It41
is
in order if one can say: This man too is playing a game in accordance with a particular, fixed
set of rules.42
For I have indicated two ways of ascertaining the rule according to which he acts. One of
them, the hypothetical, consisted in observing his actions, and then the rule was of the same
kind as a proposition of science. The other was to ask him which rule he was following. But
what if the first approach leads to no clear result, and the question as well turns up no rule,
as happens in the case of “N43
has died”. For if we ask the person who said that, “Who is
N?” then, to be sure, he’ll explain “N” by describing him, but he’ll be ready to recant this
description and to change it if we deprive him of this44
or that proposition. So how should
I determine45
the rule in accordance with which he is playing? He doesn’t know it himself.
I could only ascertain the rule based on what he happens to answer when he is asked
“Who is N?”.
46
Our investigation shouldn’t endeavour to discover the exact47
meaning of words; but frequently,
in the course of our investigation, we give exact meanings to words.
48
Doesn’t the analogy of language to a game enlighten us here? We can very well imagine
that people in a meadow might entertain themselves by playing with a ball; and that they
might do this in such a way that they’d begin various existing games one after the other without
finishing them, and in between would even throw, catch, drop the ball, etc., aimlessly.
Now someone might say: These people are playing a ball game the entire time, and that is
why at each throw they comply with specific49
rules. – But – it will be objected – whoever
spoke the sentence “N has died” didn’t string words together aimlessly (and that’s what
“having meant something by his words” consists in). – But one can say: He utters the
sentence aimlessly, which is shown precisely in the uncertainty described above. To be sure,
the sentence was taken from somewhere, and if you like, now he’s also playing a game
with very primitive rules; for it remains true that I got an answer to the question “Who is
N?”, or a series of answers that were not entirely erratic. – We can say: Let’s investigate
language with regard to its rules. If here and there it doesn’t have any rules, then that is the
result of our investigation.50
But if it’s possible that a name can lose its bearer or that that bearer never existed, then
this had to be taken into consideration when it was first used. It had to be contained within
its meaning. ( (Unless we changed this meaning or the word had no particular meaning; for
what is the meaning if the speaker can’t state it? Well, we’ll be able to describe his actual
behaviour as “fluctuating between several meanings”. It’s without a doubt essential that I
can ask him: What did you actually mean? And in answering me he’ll tell me a lot of things,
and maybe ask me to set up a list of rules for him that suits his purpose. Then the expression
“So what you really wanted to say, was . . .” will often come up in our conversation
(and once again, this expression can be completely misunderstood – it isn’t a description of
41 (V): about this$ – this
42 (V): rules. And doesnFt this in turn presuppose
that this
43 (V): “NE
44 (V): we refute this
45 (V): understand
46 (M): 3
47 (V): essential
48 (M): 3
49 (V): certain
50 (R): ∀ p. 250v
The Strict Grammatical Rules of a Game . . . 199e
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entspricht. Es wird sich dann in unserm Gespräch oft die Redeweise finden ”Du wolltest
also eigentlich sagen . . .“ (und diese kann wieder ganz mißverstanden werden – sie ist keine
Beschreibung des damaligen Geisteszustands des Sprechenden; als ob das, ”was er sagen wollte“
irgendwo in seinem Geist ausgedrückt gewesen wäre).63
Hier64
ist eine Gefahr: Es scheint nämlich dann oft, als erreichten wir endlich etwas,65
was
wir mit unserer gewöhnlichen Sprache gar nicht mehr ausdrücken können. Das ist aber das
sicherste Zeichen (dafür), daß wir fehl gegangen sind; aus unserm Spiel herausgetreten sind.
– Was versteht man unter ”allen Regeln des Tennisspiels“? Alle Regeln, die in einem bestimmten
Buche stehen, oder alle die der Spieler im Kopf hat, oder alle die je ausgesprochen
wurden, oder gar: alle die sich angeben lassen?! – Daher wollen wir lieber nicht so vag von
”allen Regeln“ reden, sondern nur von bestimmten Regeln, oder allen Regeln eines
Verzeichnisses, etc. Und das gleiche gilt von den Regeln über die Verwendung eines
Wortes. Wenn Einer mich, z.B., etwas fragt, so will ich, wenn ich ihm antworte, wissen,
ob diese Antwort in seinem Spiel als Antwort auf seine Frage gilt; ob in seinem Spiel dieser
Satz aus dem, was er gesagt hat, folgt.66
Für uns ist es genügend, daß es eine Frage gibt: ”wie meinst Du das?“ und daß als Antwort
auf diese Frage das zuerst gegebene Zeichen durch ein neues ersetzt wird. – Der Einwand
dagegen ist, daß mir eine Erklärung ja nichts zum Verständnis hilft, wenn sie nicht die letzte
ist, und daß sie nie67
die letzte ist. Ich kann zwar erklären: unter ”Moses“ verstehe ich den
Mann, wenn es einen solchen gegeben hat, der die Israeliten aus Ägypten geführt hat, wie
immer er damals genannt worden sein mag und was immer er sonst getan oder nicht getan
haben mag –, aber ähnliche Fragen ergeben sich nun in Bezug auf die Wörter dieser Erklärung68
(was nennst Du ”Ägypten“? wen, ”die Israeliten“? etc.). Ja, diese Fragen kommen auch nicht
zu einem Ende, wenn wir etwa bei Wörtern69
wie ”rot“, ”dunkel“, ”süß“, angelangt wären.
Unrichtig war es nur, zu sagen, daß mir deshalb eine dieser Erklärungen nichts hilft.
Im Gegenteil, sie ist es gerade, was ich brauche, ja alles, was ich brauchen, und auch geben
kann. Als ich nach einer Erklärung fragte, war es gerade das was ich brauchte. Und wenn ich
auf eine solche Erklärung hin sage ”jetzt versteh’70
ich, was Du meinst“, so kann man nicht
einwenden, das könne ich ja doch nie verstehen; sondern seine Erklärung hat mir eben das
gegeben, was ich Verständnis nenne; sie hat die Schwierigkeit beseitigt, die ich hatte. Was
uns quälte, ist, glaube ich, ganz in dem Pseudoproblem ausgedrückt: Das Schachspiel ist
doch durch die Gesamtheit der Schachregeln konstituiert, – was macht dann das Rücken
einer Figur im Spiel zu einem Schachzug, da doch dabei in keiner Weise alle Regeln des
Schachspiels beteiligt sind.))
71
Es ist nicht unsere Aufgabe unsere Sprache zu verbessern,72
exakter zu machen, oder etwa (gar)
zu versuchen an ihre Stelle eine &ideal exakte“ zu setzen.73
Wir haben74
von einer solchen gar keinen
Begriff. Damit meine ich nicht, daß wir für unsere Zwecke nicht auf preciseren75
Ausdruck dringen.76
200 Die strikten grammatischen Spielregeln . . .
63 (M): 3 (R): [Siehe Notizbuch: was geschieht,
wenn man sagt ich kann nicht gut ausdrücken was
ich denke.]
64 (V): Aber hier
65 (V): Es scheint nämlich dann (leicht), als
landeten wir am Schlusse bei etwas,
66 (V): dieser Satz aus jenem folgt.
67 (V): nicht
68 (V): Wörter dieses Satzes
69 (V): Worten
70 (V): weiß
71 (M): 3
72 (V): Sprache wesentlich zu verbessern,
73 (V): Unsere Aufgabe ist es nicht eine Sprache zu konstruieren
deren sämtliche
74 (V): Ich habe
75 (V): exa
76 (V): dringen. /// Wer die eine
Verkehrsregelung an Stellen starken Verkehrs
verbessern oder strenger gestalten will
256
255v
WBTC056-59 30/05/2005 16:16 Page 400
the speaker’s mental state at that time; as if “what he wanted to say” had been expressed
somewhere in his mind).51
Here52
there is a danger: It frequently seems as if we were finally reaching something53
that
we can no longer express with our ordinary language. But that is the surest sign that we
have gone astray, have stepped outside our game. – What is to be understood by “all the
rules of tennis”? All rules that are in a particular book, or all that are in a player’s head,
or all that have ever been uttered, or even: all that can be listed?! – Therefore we’d better
not speak so loosely about “all the rules”, but only about certain rules, or all the rules in
a list, etc. And the same goes for the rules for the use of a word. If, for example, someone
asks me something, I want to know, when I answer him, whether in his game my answer
counts as an answer to his question; whether in his game my proposition follows from what
he has said.54
For us it is sufficient that there is a question: “How do you mean that?” and that, as an
answer the first sign that was given is replaced by a new one. – The objection to this is that,
after all, a single explanation is no help to my understanding unless it is the final one, and that
it never is55
the final one. To be sure, I can explain: By “Moses” I understand the man, if
there was such a man, who led the Israelites out of Egypt, whatever he might have been
called at that time and whatever else he might or might not have done. – But now similar
questions arise with respect to the words in this explanation56
(What do you call “Egypt”?,
Whom “the Israelites”?, etc.). Indeed these questions don’t come to an end even when we
get down to words like, say, “red”, “dark”, “sweet”. The only thing that was wrong was to
say that that is why a single explanation doesn’t help me. On the contrary, this single explanation
is precisely what I need. Indeed, it is all that I can need – and all that I can give.
When I asked for an explanation, that was exactly what I needed. And if I say, in answer to such
an explanation, “Now I understand57
what you mean”, it can’t be objected that I can never
understand that; rather, his explanation gave me precisely what I call “understanding”;
it removed the difficulty that I had. I believe that what was torturing us is completely
expressed in this pseudo-problem: Chess is, after all, constituted by the totality of its rules
– so what makes moving one piece in the game into a chess move, since this in no way involves
all of the rules of chess?))
58
Our task is not to improve our language,59
to make it more exact, or possibly even to try to replace
it with an “ideally exact” one.60
We have61
absolutely no concept of such a language. But by this
I don’t mean that, for our purposes, we don’t push for more precise expressions.62
51 (M): 3 (R): [Cf. Notebook: what is going on
when someone says I can’t easily express what I’m
thinking.]
52 (V): But here
53 (V): danger: it (easily) seems as if we were
ending up with something
54 (V): from that one.
55 (V): that it is not
56 (V): in this proposition
57 (V): know
58 (M): 3
59 (V): our language essentially,
60 (V): Our task isnDt to construct a language all of
whose
61 (V): I have
62 (V): expressions.
/// Whoever wants to improve or strengthen
traffic regulations // a traffic regulation // in places
where there is heavy traffic
The Strict Grammatical Rules of a Game . . . 200e
WBTC056-59 30/05/2005 16:16 Page 401
77
Die Verkehrsregelung in den Straßen erlaubt & verbietet gewisse Verkehrshandlungen.78
Aber
sie versucht nicht sämtliche Bewegungen der Fußgänger & Fahrzeuge durch Vorschriften zu
regeln.79
Und es wäre unsinnig, von einer idealen Verkehrsordnung zu reden die das täte. Wir
wüßten nicht wie wir uns dieses Ideal zu denken hätten.80
Wünscht einer die Verkehrsordnung
in irgendwelchen Punkten strenger zu gestalten so bedeutet das nicht daß er sich so einem Ideal zu
nähern wünscht.
81
Was wir Regeln nennen bilden wir nach Analogie von bestehenden Regeln.
82
Wir wissen alle was es heißt83
daß eine Taschenuhr auf die genaue Stunde gestellt wird,84
oder
gerichtet wird damit85
sie genau geht. Wie aber wenn man fragte: ist diese Genauigkeit eine ideale
Genauigkeit oder86
wie weit nähert sie sich ihr?
Wir können freilich von Zeitmessungen87
reden bei denen es eine andere & in gewissem Sinne
größere Genauigkeit gibt. Bei denen die Worte die Uhr auf die genaue Stunde stellen eine andere
Bedeutung haben. Wo die Uhr ablesen ein anderer Prozess ist etc. Wenn ich nun jemandem sage Du
solltest pünktlicher zum Mittagessen kommen, Du weißt daß es genau um 2h anfängt – ist88
die
Genauigkeit von der89
hier die Rede ist eine unvollkommene im Vergleich zu jener andern. Und gibt
es90
ein Ideal der Genauigkeit.
91
Was bedeutet ”undefinierbar“? Dieses Wort ist offenbar irreführend, denn es erweckt
den Anschein, als könnten wir hier etwas versuchen, was sich dann als unausführbar
erwiese. Als wäre also das Undefinierbare etwas, was sich nicht weiter definieren ließe, wie
sich ein zu großes Gewicht nicht heben läßt. Wir könnten sagen: ”Wie denn ,undefinierbar‘?!
Könnten wir denn versuchen, es zu definieren?“
92
Ist &rot“ undefinierbar. Undefinierbar darunter stellt man sich etwas vor wie unanalysierbar &
zwar so als wäre der betreffende Gegenstand unanalysierbar (wie ein chem. Element). Dann wäre
die Logik also doch eine Art sehr allgemeiner Naturwissenschaft. Aber die Unmöglichkeit der
Analyse entspricht einer von uns festgesetzten Form der Darstellung.93
94
Nun könnte man freilich sagen: die Definition ist ja etwas Willkürliches, d.h., wie ich
ein Wort definiere, so ist es definiert. Aber darauf kann geantwortet werden: Es kommt darauf
an, es so zu definieren, wie wir das Wort meinen. Also so, daß wir zur Definition des Wortes
”Tisch“, z.B., sagen: ja, das ist es, was ich mit dem Wort meine. – Ja hat Dich nun aber die
Definition dahin gebracht, das mit dem Wort zu meinen oder willst Du sagen, daß Du das
schon immer gemeint hast? Und wenn das Letztere, so hast Du also immer das gemeint,
was die Definition sagt (im Gegensatz zu etwas Anderem, was sie auch sagen könnte).
D.h.: die Definition ist auch eine Beschreibung dessen, was Du schon früher gemeint hast.
Du warst also auch früher schon im Besitz einer Übersetzung dieser Definition; sie hat
sozusagen nur laut gesagt, was Du schon im Stillen wußtest. Sie hat also auch wesentlich
77 (M): 3
78 (V): Handlungen.
79 (V): sämtliche Handlungen der gehenden &
fahrenden Fußgänger & Fahrzeuge durch Regeln
zu leiten.
80 (V): nicht was wir unter diesem Ideal vorstellen
sollten.
81 (M): ///
82 (M): 3
83 (V): heißt daß der
84 (V): heißt eine Uhr auf die genaue Stunde zu stellen,
85 (V): daß
86 (V): &
87 (O): Zeitmessen
88 (O): anfängt ist
89 (V): der ich
90 (V): es zetwa
91 (M): 3 3
92 (M): /// 3
93 (V): Analyse ist eine logische entspringt aus der
von uns festgesetzten Darstellungsform.
94 (M): /// 3
256v
256
257
201 Die strikten grammatischen Spielregeln . . .
WBTC056-59 30/05/2005 16:16 Page 402
63
Traffic regulations allow and forbid certain movements of traffic64
in the streets. But they don’t
attempt to regulate all movements of pedestrians and vehicles with ordinances.65
And it would be
nonsensical to speak of an ideal set of traffic rules that would do that. We wouldn’t know how
to imagine66
this ideal. If someone wants to strengthen traffic regulations in some respects this
doesn’t mean that he wants to approximate to such an ideal.
67
What we call rules we form in analogy to existing rules.
68
All of us know what it means that a pocket watch is set69
to the exact hour or that it is adjusted
so that it runs accurately. But what if one were to ask: Is this accuracy an ideal accuracy, or70
how
closely does it approximate to one?
To be sure, we can talk about time-measurements of different and in some sense greater
accuracy. Where the words “setting a watch to the exact hour” have a different meaning. Where
reading the clock is a different process, etc. Now if I tell someone “You should be more punctual
in arriving for lunch, you know that it starts exactly at 2 o’clock”, is that accuracy imperfect as
compared to the other kind? And is there one ideal of accuracy?
71
What does “indefinable” mean? This word is obviously misleading, for it gives the impression
that we could attempt something here that would later prove impossible to carry out.
As if therefore the indefinable were something that couldn’t be further defined, as a weight
that’s too heavy can’t be lifted. We could say: “‘Indefinable’ in what way?! Could we try to
define it?”
72
Is “red” indefinable? By “indefinable” one imagines something like “unanalysable” – as if the
object in question were unanalysable (like a chemical element). But then logic would be a kind of
very general natural science. But the impossibility of analysis corresponds to a form of representation
that we have set down.73
74
Now of course one could say: A definition is something arbitrary, i.e. a word is defined
as I define it. But this can be answered: What matters is to define a word the way we mean
it. That is, in such a way that when we come upon the definition of the word “table”, for
instance, we say: Yes, that’s what I mean by that word. – Fine, but did the definition get
you to mean that by the word or do you want to say that you’ve meant that from the very
start? And if the latter, then you’ve always meant what the definition says (as opposed
to something else that it might say as well). That is: the definition is also a description of
what you already meant. So at an earlier point as well you were already in possession of a
translation of this definition; it merely said aloud, as it were, what you already knew in your
63 (M): 3
64 (V): certain actions
65 (V): to guide all movements of the walking and
driving pedestrians & vehicles with rules.
66 (V): know what we should imagine by
67 (M): ///
68 (M): 3
69 (V): means to set a pocket watch
70 (V): and
71 (M): 3 3
72 (M): /// 3
73 (V): of analysis is a logical one originates from the
form of representation that we have set down.
74 (M): /// 3
The Strict Grammatical Rules of a Game . . . 201e
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nichts zergliedert. (Vergleiche: Begriff der 3 Teilung des Winkels vor & nach der Betrachtung die
die Unmöglichkeit der 3 Teilung zeigt.)
95
Gibt es ein komplettes Regelverzeichnis für die Verwendung eines Wortes?
Gibt es ein komplettes Regelverzeichnis für die Verwendung einer Figur im Schachspiel?
96
Denken wir uns Jemand, der alle97
Formen in diesem Zimmer beschreibt, indem er sie
mit ebenflächigen geometrischen Formen vergleicht. Gibt es in diesem Zimmer nur solche
Formen? Nein. – Muß der, der die Formen unter dem Gesichtspunkt der ebenflächigen
Körper beschreibt, behaupten, es gäbe nur solche Formen im Zimmer? Auch nicht. Kann
man sagen, daß das einseitig ist, weil er alle Formen durchgängig nach diesem Schema
auffaßt? Und sollte es ihn an98
dieser Auffassung irre99
machen, wenn er bemerkt, daß auch
runde Körper vorhanden sind? Nein. Es wäre auch irreführend, den ebenflächigen Körper
ein ”Ideal“ zu nennen, dem sich die Wirklichkeit nur mehr oder weniger nähert. Aber die
Geometrie der ebenflächigen Körper könnte man mit Bezug auf diese Darstellung100
eine
normative Wissenschaft nennen. (Eine, die das Darstellungsmittel darstellt; gleichsam eine,
die die Meßgläser eicht.)
101
Man kann fragen: Wenn wir nicht eine ideale102
Exactheit anstreben, im Gegensatz zu der
alltäglichen, wozu103
hantieren104
wir an105
der Grammatik unserer Sprache überhaupt herum. Und
die Antwort ist:106
Wir suchen uns von philos. Beunruhigungen zu befreien107
& das tun wir indem
wir Unterscheidungen welche die Grammatik der gewöhnlichen Sprache verschleiert, hervorheben.
Sozusagen Regeln die mit verblaßter Tinte geschrieben sind, stark nachziehen und anderes mehr.
Dadurch kann es allerdings den Anschein haben als reformierten wir die Sprache.
108
Ich habe ein Bild mit verschwommenen Farben und komplizierten Übergängen. Ich stelle
ein einfaches mit klargeschiedenen Farben, aber mit dem ersten verwandtes, daneben. Ich sage
nicht, daß das erste eigentlich das zweite109
sei; aber ich lade den Andern ein, das einfache anzusehen,
und verspreche mir davon, daß gewisse Beunruhigungen für ihn verschwinden werden.
110
Wer etwa . . . einführte könnte im Interesse der Chemie die Sprache verbessern . . .
111
So eine Reform für gewisse praktische112
Zwecke ist wohl denkbar die Verbesserung unserer
Terminologie zur Vermeidung von Mißverständnissen. (Wenn zwei Mitglieder einer Familie &Paul“
heißen, so ist es manchmal zweckmäßig den einen von ihnen bei einem andern Namen zu nennen.)
Aber das sind nicht die Fälle mit denen wir es zu tun haben. Die Konfusionen mit denen wir’s zu tun
haben113
entstehen, gleichsam, wenn die Sprache feiert, nicht wenn sie arbeitet. (Man könnte
sagen: wenn114
sie leer läuft.)
115
Behandle die deutlichen Fälle in der Philosophie, nicht die undeutlichen. Diese werden
sich lösen, wenn jene gelöst sind.
95 (M): 3
96 (M): ///
97 (V): die
98 (V): in
99 (V): irrefü
100 (V): Darstellungsweise
101 (M): 3
102 (O): idealen
103 (V): Wenn wir nicht nach einer idealen Exaktheit
streben, wozu
104 (V): arbeiten
105 (V): mit
106 (V): Antwort ist: Unsere Aufgabe ist es gewisse
Beunruhigungen zu beseitigen und wir suchen
nach dem erlösenden Wort
107 (V): Wirsuchenphilos.Beunruhigungenzubeseitigen
108 (M): ///
109 (V): andere
110 (M): 555
111 (M): 3
112 (V): für praktische
113 (V): Konfusionen die uns beschäftigen
114 (V): sagen, wenn
115 (M): ////
256v
257
256v
257v
258
258
257v
202 Die strikten grammatischen Spielregeln . . .
WBTC056-59 30/05/2005 16:16 Page 404
heart. So, essentially it didn’t analyse anything either. (Compare: The concept of the trisection
of an angle before and after the investigation that shows the impossibility of trisection.)
75
Is there a complete list of rules for the use of a word?
Is there a complete list of rules for the use of a chess piece in a game of chess?
76
Let’s imagine someone who describes all the77
shapes in this room by comparing them
to plane geometric shapes. Are there only such shapes in this room? No. – Does the person
who describes the shapes from the viewpoint of plane figures have to claim that there are
only such shapes in the room? No again. Can one say that this procedure is one-sided, because
he is viewing all shapes uniformly, according to this schema? And should his also noticing
the presence of round objects make him question this view? No. It would also be misleading
to call the plane figures an “ideal” that reality more or less closely approximates.
But with respect to this representation,78
one could call the geometry of plane shapes a
normative science. (One that represents the means of representation; one that calibrates
the measuring beakers, as it were.)
79
One can ask: If we’re not striving for an ideal accuracy as opposed to the everyday kind, why80
are we fiddling around with81
the grammar of our language at all? And the answer is: We’re trying
to free ourselves from philosophical82
anxieties, and we do this by emphasizing distinctions that the
grammar of everyday language obscures. By retracing in bold the rules that are written in faded ink,
as it were, and other such things. This can indeed make it seem as if we were reforming language.
83
I have a picture with blurred colours and complicated transitions. Next to it I place a
simple one, with clearly separated colours, but related to it. I don’t say that the first is actually
the second84
; but I invite someone to look at the simple one, and expect that in doing
this certain worries he has will disappear.
85
Whoever introduced, say . . . , could improve language in the interest of chemistry . . . .
86
Such a reform for certain practical87
purposes is conceivable, to be sure – the improvement
of our terminology to avoid misunderstandings. (If two members of a family are called “Paul” then
sometimes it’s practical to call one of them by another name.) But those are not the cases we’re
dealing with. The confusions that concern us88
come about, as it were, when language is on
holiday, not when it’s working. (One could say: when89
it is idling.)
90
Deal with the clear cases in philosophy, not the ones that are unclear. They will be solved
when the former are.
75 (M): 3
76 (M): ///
77 (V): who describes the
78 (V): this mode of representation,
79 (M): 3
80 (V): If we’re not striving for an ideal precision, why
81 (V): we belabouring
82 (V1): answer is: It is our task to get rid of certain
anxieties and we search for the liberating word
(V2): We’re trying to get rid of philosophical
83 (M): ///
84 (V): the other
85 (M): 555
86 (M): 3
87 (V): for practical
88 (V): that are occupying us
89 (V): say, when
90 (M): ////
The Strict Grammatical Rules of a Game . . . 202e
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Die Tendenz mit der Untersuchung eines Satzes da anzufangen, wo seine Anwendung
ganz nebelhaft und unsicher ist (der Satz der Identität ist ein gutes Beispiel), anstatt diese
Fälle vorläufig beiseite zu lassen und den Satz dort anzugehen, wo wir mit gesundem
Menschenverstand über ihn reden können, diese Tendenz ist für die aussichtslose Methode
der meisten Menschen, die philosophieren, bezeichnend.
116
Ich betrachte die Sprache und Grammatik als Kalkül d.h. als Vorgang nach festgesetzten
Regeln.117
118
Wir wollen nicht das Regelsystem in unerhörter Weise verfeinern oder vervollständigen.119
Wir wollen Verwirrungen & Beunruhigungen beseitigen die aus der Schwierigkeit herrühren, das
System120
zu übersehen.121
122
Es ist als wäre dieses Regelsystem in einem Buch niedergelegt; wir zögen aber dieses Buch in
praktischen Fällen beinahe nie zu Rate. Hie & da aber wären wir versucht123
darin zu lesen. Dann aber
verwirrt es uns gänzlich. Vieles124
darin ist so vergilbt daß wir es kaum lesen können anderes steht
klar125
da, ist aber ohne die nötigen Qualificationen falsch & irreführend.
126
Untersuchen wir unsere127
Sprache auf ihre Regeln hin.
128
Gibt es so etwas, wie eine komplette Grammatik, z.B., des Wortes ”nicht“?
129
Es ist von der größten Bedeutung, daß wir uns zu einem Kalkül der Logik immer ein
Beispiel denken, auf welches der Kalkül wirklich angewandt wird, und nicht Beispiele, von
denen wir sagen, sie seien eigentlich nicht die idealen, diese aber hätten wir noch nicht.
Das ist das Zeichen einer ganz falschen Auffassung.130
Kann ich den Kalkül überhaupt
verwenden, dann ist dies131
auch die ideale Verwendung und die Verwendung, um die
es sich handelt. Man geniert sich nämlich einerseits, das Beispiel als das eigentliche
anzuerkennen, weil man in ihm noch eine Komplikation erkennt, auf die der Kalkül sich
nicht bezieht.132
Aber es ist das Urbild133
des Kalküls und er davon hergenommen, & dies
ist kein Fehler, keine134
Unvollkommenheit des Kalküls. Der Fehler liegt darin seine Anwendung in
nebelhafter Ferne zu versprechen.135
258
203 Die strikten grammatischen Spielregeln . . .
116 (M): ///
117 (V): Grammatik unter dem Gesichtspunkt des
Kalküls, // unter der Form des Kalküls, // d.h.
des Operierens nach festgelegten Regeln.
118 (M): 3
119 (V): komplettieren.
120 (V): Regelsystem
121 (V): die aus der Unübersichtlichkeit des
Regelsystems herrühren.
122 (M): ///
123 (V): verleitet
124 (V): gänzlich; denn Vieles
125 (V): klar
126 (M): ///
127 (V): die
128 (M): ///
129 (M): 3
130 (E): In MS 115 (S. 55–56) folgt auf diesen Satz:
”(Russell und ich haben, in verschiedener
Weise an ihr laboriert. Vergleiche was ich
in der ,Logisch-philosophischen Abhandlung‘
über Elementarsätze und Gegenstände sage.)“
131 (V): das
132 (V): weil man in ihm eine Komplication sieht für die
der Kalkül nicht aufkommt.
133 (V): bezieht; anderseits ist es doch das Urbild
// bezieht; aber es ist das Urbild
134 (V): oder
135 (V1): hergenommen, und auf eine geträumte
Anwendung kann man nicht warten. Man
muß sich also eingestehen, welches das
eigentliche Urbild des Kalküls ist. (V2): 3
Das ist aber kein Eingeständnis – als habe man
damit einen Fehler gemacht // begangen //, den
Kalkül von daher genommen zu haben, sondern
der Fehler liegt darin, ihn jetzt in nebelhafter
Weise anzuwenden, oder eine Anwendung
zu versprechen. // oder eine Anwendung in
nebuloser Ferne zu versprechen.
257v
259
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The tendency to start investigating a proposition at the point where its application is completely
nebulous and uncertain (the law of identity is a good example), instead of leaving
these cases aside for the moment and approaching the proposition at the point where we can
use our common sense to talk about it – this tendency is typical of the futile methods most
people use who do philosophy.
91
I view language and grammar as a calculus, i.e. as a process that follows92
fixed rules.
93
We don’t want to refine the system of rules in fantastic ways, nor do we want to complete it.
We want to remove the confusions and anxieties that stem from the difficulty of seeing the
system94
all at a glance.
95
This is the way it seems to be: this system of rules has been set down in a book; but we almost
never consult this book in practical cases. Every now and then, though, we’re tempted to read96
in
it. And then it completely confuses us. Much97
in it is so yellowed that we can barely read it, other
things are clearly98
visible, but without the necessary qualifications they are false and misleading.
99
Let’s examine our language with respect to its rules.
100
Is there such a thing as a complete grammar, e.g., of the word “not”?
101
It is of the utmost importance that for a logical calculus we always think of an example
to which the calculus is actually applied, and not of examples of which we say: “These really
aren’t the ideal ones – we don’t have those yet”. That is the sign of a totally false view.102
If I can use the calculus at all then this103
is also the ideal use and the use that is at issue.
For on the one hand we’re embarrassed to acknowledge our example as the proper one
because we recognize a complication in it to which the calculus doesn’t apply.104
But it
is the archetype105
for the calculus and the latter is derived from it, and this is no mistake,
no imperfection106
in the calculus. The mistake lies in promising its use in the nebulous future.107
91 (M): ///
92 (V): grammar from the point of view of a
calculus, // in the form of a calculus, // i.e. as
operating according to
93 (M): 3
94 (V): stem from the inability to see the system of rules
95 (M): ///
96 (V): we’re misled into reading
97 (V): us; for much
98 (V): clearly
99 (M): ///
100 (M): ///
101 (M): 3
102 (E): In MS 115 (pp. 55–56), this sentence is
followed by: “(Russell and I have in different
ways laboured under it. Compare what I
say in the Tractatus about elementary propositions
and objects.)”
103 (V): that
104 (V): because one sees a complication in it for
which the calculus doesn’t account.
105 (V): apply; on the other hand it is, after all, the
archetype // apply; but it is the archetype
106 (V): no mistake or imperfection
107 (V1): from it, and one cannot wait for an
application one dreams up. Therefore one has
to admit to oneself which application is the
actual archetype of the calculus. (V2): 3 But
it isn’t an admission – as if one had made //
committed // an error in having derived the
calculus from that; rather the mistake lies in
applying it now in a nebulous way, or in
promising a use. // or in promising a use in the
nebulous future.
The Strict Grammatical Rules of a Game . . . 203e
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136
(So könnte Spengler besser verstanden werden, wenn er sagte: ich vergleiche verschiedene
Kulturperioden dem Leben von Familien; innerhalb der Familie gibt es eine Familienähnlichkeit,
während es auch zwischen den Mitgliedern verschiedener Familien eine Ähnlichkeit
gibt; die Familienähnlichkeit unterscheidet sich von der andern Ähnlichkeit so und so etc.
Ich meine: das Vergleichsobjekt, der Gegenstand, von welchem diese Betrachtungsweise
abgezogen ist, muß uns angegeben werden, damit nicht in die Diskussion immer
Ungerechtigkeiten einfließen. Denn nun wird alles137
das vom Objekt der Betrachtung behauptet
was für das Urbild stimmt138
: und behauptet ”es müsse immer . . .“.139
140
Das kommt nun daher, daß man den Merkmalen des Urbilds einen Halt in der
Betrachtung geben will. Da man aber Urbild und Objekt vermischt, dem Objekt dogmatisch
beilegen muß, was nur das Urbild charakterisieren soll.141
Anderseits glaubt man, die
Betrachtung habe nicht die142
Allgemeinheit, die man ihr geben will, wenn sie nur für den
einen besondern Fall wirklich stimmt. Aber das Urbild soll ja eben als solches hingestellt
werden; so, daß es die ganze Betrachtung charakterisiert, ihre Form bestimmt. Es steht
also an der Spitze und ist dadurch ausgezeichnet aber nicht dadurch, daß alles, was nur von
ihm gilt, von allen Objekten der Betrachtung ausgesagt wird.143
Der Syllogismus144
ist ein Kalkül der auf Sätze angewandt werden kann.145
(Wie das Einmaleins
auf Pflaumen.)146
Der Syllogismus wartet nicht auf eine zukünftige exacte147
Anwendung.
Fragen wir uns: Was ist148
die praktische Anwendung des Syllogismus.
149
Wie seltsam, wenn sich die Logik mit einer ”idealen“ Sprache befaßte, und nicht mit
unserer, denn woher sollten wir diese ideale Sprache nehmen? Und was sollte diese ideale
Sprache ausdrücken? Doch wohl das, was wir jetzt in unserer gewöhnlichen Sprache
ausdrücken; dann muß die Logik also diese untersuchen. Oder etwas anderes: aber wie soll
ich dann überhaupt wissen, was das ist. – Die logische Analyse ist die Analyse von etwas,
was wir haben, nicht von etwas, was wir nicht haben. Sie ist also die Analyse der Sätze wie
sie sind. (Es wäre seltsam, wenn die menschliche Gesellschaft bis jetzt gesprochen hätte,
ohne einen richtigen Satz zusammenzubringen.)
136 (M): 3
137 (V): alles was
138 (V): Denn da wird dann alles, was für das
Urbild der Betrachtung stimmt // gilt //, auch
von dem Objekt, worauf wir die Betrachtung
anwenden, behauptet // ausgesagt
139 (M): Schenkersche Betrachtungsweise der Musik
140 (M): ///
141 (V): muß.
142 (V): Betrachtung ermangle ja der
143 (M): [dieser Satz Absatz vom Typisten falsch
kopiert.] (E): Der Typist hat an dieser
Stelle einen Teil des letzten Satzes ausgelassen.
Um dem Satz Sinn zu verleihen, hat
Wittgenstein hier eine handschriftliche
Ergänzung eingesetzt. Im Manuskript (MS
111, S. 120), das dieser Stelle als Grundlage
dient, heißt es: ”Es steht also an der Spitze und
ist dadurch allgemein gültig, daß es die Form
der Betrachtung bestimmt, nicht dadurch,
daß alles was nur von ihm gilt von allen
Objekten der Betrachtung ausgesagt wird.“
144 (O): Sylogismus
145 (V): Die Aristotelische Logik // Der Syllogismus
// ist ein Spiel, das // Kalkül, der // sich auf Sätze
anwenden läßt.
146 (M): [Gehört an eine andere Stelle]
147 (V): eine exacte
148 (V): Was ist
149 (M): ///
258v
259
260
259v
260
259
260
260
204 Die strikten grammatischen Spielregeln . . .
WBTC056-59 02/06/2005 14:38 Page 408
108
(Spengler could be better understood if he were to say: I am comparing different cultural
periods to the lives of families; within families there are family resemblances, though
there also are resemblances between the members of different families; family resemblances
differ from these other resemblances in such and such a way, etc. I mean: The object of
comparison, the object from which this way of looking at things is derived, has to be given
to us, so that injustices won’t constantly flow into the discussion. For everything that holds
true for the archetype is now being claimed for the object under examination109
: and it is claimed
that “it always has to . . .”.110
111
This comes from wanting to give the characteristics of the archetype a foothold in the
investigation. We conflate the archetype and the object, and then we have to dogmatically
attribute to the object what should112
be ascribed only to the archetype. On the other hand,
we think the investigation doesn’t have113
the generality we want to give it, if it really holds
true only for the one particular case. But the archetype should be presented as precisely that;
in such a way that it characterizes the whole investigation, determining its form. So it stands
at the apex of the investigation and for that reason is superior, but not because everything that
holds true only of it is predicated of all of the objects being investigated.114
The syllogism is a calculus that can be applied to propositions.115
(As the multiplication tables to
plums.)116
A syllogism doesn’t wait for an exact application in the future.117
Let’s ask ourselves: What is118
the practical application of a syllogism?
119
How strange if logic were to occupy itself with an “ideal” language and not with ours.
For where should we get this ideal language from? And what should this ideal language express?
Presumably, what we now express in our everyday language; so logic has to investigate that.
Or investigate something else: But in that case, how am I to have any idea what that is? –
Logical analysis is the analysis of something that we have, not of something we don’t have.
Therefore it is the analysis of propositions as they are. (It would be strange if humans had
been speaking all this time without managing to utter a single correct sentence.)
108 (M): 3
109 (V): For in that case everything that is correct
// is valid // for the archetype that governs the
examination is also claimed // predicated // of
the object we’re examining
110 (M): Schenker’s way of looking at music.
111 (M): ///
112 (V): what has to
113 (V): investigation indeed lacks
114 (M): [This sentence paragraph was falsely copied by
the typist] (E): The person who typed TS
212, from which this remark is taken, left a
phrase out of the last sentence. Wittgenstein
inserted the handwritten phrase (“superior,
but not because”) in TS 213 to give sense to
the sentence. The original sentence (in MS 111,
p. 120) read: “So it stands at the apex of the
investigation and for that reason is generally
valid in that it determines the form of the investigation,
but not because everything that holds
true only of it is predicated of all of the
objects being investigated.”
115 (V): Aristotelian logic // The syllogism // is a game
that // is a calculus that // can be applied to
propositions.
116 (M): [Belongs somewhere else.]
117 (V): an exact application.
118 (V): What is
119 (M): ///
The Strict Grammatical Rules of a Game . . . 204e
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150
Nicht das ist wahr, daß, was wir sagen,151
nur für eine ”ideale Sprache“ gilt (oder Geltung
hätte); wohl aber kann man sagen, daß wir eine ideale Sprache konstruieren, in die aber dann
alles übersetzbar ist, was in unidealen152
Sprachen gesagt werden kann.
153
Wenn Einer von einer idealen Sprache redet, so müßte man fragen: in 154
welcher Beziehung
”ideal“?
155
(Es gibt keine Logik für den luftleeren Raum. Insofern es keine Hypothese in der
Logik gibt.)
150 (M): ///
151 (V): was ich sage,
152 (V): in den anderen
153 (M): /// 3
154 (E): Hier steht ein Hinweis eines Lesers auf
irrtümliches Auslassen von S. 261 bei der
Seitenzählung.
155 (M): /// 3
262
205 Die strikten grammatischen Spielregeln . . .
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120
It isn’t true that what we are121
saying is valid (or would have validity) only for an “ideal
language”; but it can be said that we are constructing an ideal language; but one into which
whatever can be said in non-ideal122
languages can then be translated.
123
If someone talks about an ideal language, one ought to ask: “ideal” in what respect?
124
(There is no logic for a vacuum. In so far as there is no hypothesis in logic.)
120 (M): ///
121 (V): what I am
122 (V): in other
123 (M): /// 3
124 (M): /// 3
The Strict Grammatical Rules of a Game . . . 205e
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59
Wortarten werden nur durch ihre
Grammatik unterschieden.
1
Es gibt nicht zwei Wortarten, die ich grammatisch (ganz) gleich behandeln kann, die
aber doch zwei Wortarten sind. Sondern die Regeln, die von ihnen handeln, machen die
Wortarten aus: dieselben Regeln, dieselbe Wortart. Das hängt damit zusammen, daß, wenn
sich ein Zeichen ganz so benimmt wie ein anderes, die beiden dasselbe Zeichen sind.
2
Verschiedenen Arten von Schachfiguren wie Läufer, Rössel etc. entsprechen verschiedene
Wortarten.
3
Ich komme hier auf jene Methode der Zeichenerklärung, über die sich Frege so lustig
gemacht hat. Man könnte nämlich die Wörter ”Rössel“, ”Läufer“, etc. dadurch erklären,
daß man die Regeln angibt, die von diesen Figuren handeln.
4
Genau dasselbe gilt in jeder Geometrie von den Ausdrücken ”Punkt“ und ”Gerade“ etc.
Was ein Punkt ist und was eine Gerade, sieht man nur daran, welche Plätze das eine und
das andere in dem System von Regeln einnimmt. Denken wir uns etwa ein System von
Buchstaben von solcher Art, daß alle erlaubten Zeichen Gruppen von 3 Buchstaben
sind, und zwar derart, daß ein Buchstabe, der an einer Außenstelle stehen darf, nicht in
der Mittelstelle stehen darf und umgekehrt. Diese Regel würde zwischen zwei ”Wortarten“
unterscheiden und wir könnten das dadurch zum Ausdruck bringen, daß wir für die
Außenglieder große, für die Innenglieder kleine Buchstaben verwenden. – Andrerseits aber
hat die Unterscheidung zweier Wortarten keinerlei Sinn, wenn sie nicht auf die obige Art
syntaktisch unterschieden sind, d.h. wenn sie nicht auch ohne die verschiedene Art der
Bezeichnung, bloß durch die von ihnen geltenden Regeln, als verschieden zu erkennen wären.
(Zwei Rössel könnten einander in keiner Hinsicht ähnlich sehen und wären, wenn man
die für sie geltenden Spielregeln kennt, doch als solche gekennzeichnet.) Damit hängt es
unmittelbar zusammen, daß das Einführen neuer Gattungsnamen in die Philosophie der Logik
uns um kein Haar weiterbringt, solange nicht die syntaktischen Regeln gegeben sind, die
den Unterschied machen.
Das Wort ”ein gewisser“ und seine Grammatik. Ein Beispiel, wie man Worte häuft, um
eine Bedeutung zu sichern, statt auf die Spielregeln zu achten. (Als wollte man dem
Schachkönig ein wirkliches Gesicht anmalen, um ihm die richtige Wirkung zu sichern.)
1 (M): ///
2 (M): ///
3 (M): ///
4 (M): ///
263
264
206
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59
Kinds of Words are Distinguished
only by their Grammar.
1
There is no such thing as two kinds of words that I can treat (completely) equally in terms
of grammar, and still have two kinds. Rather, the rules about kinds of words constitute them:
the same rules, the same type of word. This is connected to the fact that if one sign behaves
exactly like another, the two are the same sign.
2
Different kinds of words correspond to different kinds of chess pieces, such as bishop,
knight, etc.
3
Here I am touching on the way of explaining signs that Frege ridiculed so much. For
one could explain the words “knight”, “bishop”, etc. by citing the rules that apply to these
pieces.
4
Exactly the same thing holds for the expressions “point” and “straight line”, etc., in
a geometry. One sees what a point is, and what a straight line, only by the positions each
occupies in a system of rules. Let’s imagine, for instance, a system of letters in which all
allowable signs are groups of three letters and, more precisely, are arranged so that a letter
that is allowed on the outside is not allowed in the middle and vice versa. This rule would
differentiate between two “kinds of words”, and we could express this by using capital
letters for the outside parts and lower-case letters for the inside parts. – But on the other
hand distinguishing between two kinds of words makes no sense whatsoever if they are not
distinguished syntactically as described above, i.e. if they aren’t recognizable as different even
without the different names given to them, but simply by the rules that apply to them. (Two
knights might bear no resemblance to each other and still be designated as knights if one
knows the rules of the game that apply to them.) Directly connected with this is the fact
that introducing new names for categories into the philosophy of logic doesn’t get us a hair’s
breadth further, so long as the syntactical rules that make the difference aren’t given.
The expression “a certain” and its grammar. An example of how one amasses words in
order to safeguard a meaning, instead of paying attention to the rules of the game. (As if
one were to paint a real face on a chess-king to ensure that he has the right effect.)
1 (M): ///
2 (M): ///
3 (M): ///
4 (M): ///
206e
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60
Sage mir, was Du mit einem Satz
anfängst, wie Du ihn verifizierst, etc.,
und ich werde ihn verstehen.
Die einzige Funktion des Satzes scheint es, auf dem Gedankenklavier zu spielen; die Musik die er
darauf hervorbringt, – das Gedankengebilde, – das ist1
der Gegenstand unsrer Untersuchung. Die
Anwendung, die Tauschkraft des Satzes2
im Verkehr, mag uns wohl3
manchmal einen Wink geben:
aber4
das ist nicht der Sinn des Satzes.
Untersuche seine Nützlichkeit!
5
Die Frage &wie kann man das wissen“ fragt (in einer Bedeutung) nach einem logischen
Zusammenhang, wenn sie nach einer logischen Möglichkeit fragt.
6
Wie weiß man wenn es regnet. Wir sehen & fühlen etwa7
den Regen. Die Bedeutung des Wortes
&Regen“ wurde uns durch solche8
Erfahrungen erklärt. &Was ist Regen“ & &wie sieht Regen aus“
sind logisch verwandte Fragen. Die Erfahrung habe9
nun gelehrt daß ein plötzliches10
Fallen des
Barometers eintritt, wenn es regnet. Dann kann ich nun aus dem Fallen des Barometers entnehmen
daß es regnet. Ich nenne es ein Symptom dafür daß es regnet. Ob ein Phänomen ein Symptom
des Regens ist lehrt die Erfahrung. Was als Kriterium dafür gilt11
daß12
es regnet ist Sache der
Abmachung (Definition).
13
Die Beschreibung14
der Verifikation eines Satzes ist ein Beitrag zu seiner Grammatik.
15
Man kann nicht die Möglichkeit der Evidenz mit der Sprache überschreiten. D.h. eigentlich:
die Möglichkeit der Evidenz für einen Satz ist eine Angelegenheit der Grammatik.
16
Die Frage nach der Verifikation ist nur eine besondere17
Form der Frage &Was tut man
mit diesem Satz?“.18
1 (V): spielen; was er darauf hervorruft, das ist
2 (V): Untersuchung. Seine Anwendung, seine
Tauschkraft
3 (V): zwar
4 (V): geben% aber
5 (M): 3
6 (M): 3
7 (V): sehen etwa
8 (V): diese
9 (V): habe mich
10 (V): daß ein gewisses Phänomen etwa das plötzliche
11 (V): Was ein Kriterium dafür ist
12 (O): das
13 (M): 3
14 (V): Angabe
15 (M): 3
16 (M): 3
17 (V): andere
18 (V): der Frage ”wie meinst Du das?“.
265
264v
265
264v
265
207
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60
Tell me What you do with a
Proposition, How you Verify it, etc.,
and I Shall Understand It.
It seems that the only function of a proposition is to play on the thought-piano; the music that it
produces on it – the mental construct – is1
the object of our investigation. The use, the exchange
value a proposition has in commerce2
, might at times give us a hint, but that is not the sense of a
proposition.
Examine its usefulness!
3
(In one sense,) when it asks about a logical possibility the question “How can one know that?”
is asking about a logical connection.
4
How does one know when it’s raining? We might see and feel the5
rain. The meaning of the word
“rain” was explained to us via such6
experiences. “What is rain?” and “What does rain look like?”
are logically related questions. Now suppose experience has taught me that7
the barometer suddenly
falls whenever it rains. Then I can deduce from the falling of the barometer that it’s raining. I call this
a symptom that it’s raining. Experience teaches whether a phenomenon is a symptom of rain. What
counts as a8
criterion for rain is a matter of agreement (definition).
9
The description10
of how a proposition is verified is a contribution to its grammar.
11
In language, one cannot transcend the possibility of evidence. What this really means is:
The possibility of evidence for a proposition is a matter of grammar.
12
The question about verification is only a particular13
form of the question “What does one
do with this proposition?”14
1 (V): thought-piano; what it produces on it is
2 (V): Its use, its exchange value in commerce
3 (M): 3
4 (M): 3
5 (V): might see the
6 (V): these
7 (V): taught me that a certain phenomenon% for
instance that
8 (V): What is a
9 (M): 3
10 (V): statement
11 (M): 3
12 (M): 3
13 (V): only another
14 (V): question “How do you mean that?”
207e
WBTC060-68 30/05/2005 16:16 Page 415
Wie sich die Sprache von der Beschreibung der Verifikation entfernt. Wie sie abstrakt wird!
Man muß wieder entdecken, daß man die Zeit mit der Uhr mißt. – Und erkennt dabei nicht
einmal, daß man eine grammatische Entdeckung gemacht hat.
19
Wie ein Satz verifiziert wird, das sagt er. Vergleiche die Allgemeinheit in der Arithmetik
mit der Allgemeinheit von nicht arithmetischen Sätzen. Sie wird anders verifiziert und ist
darum eine Andere. Die Verifikation ist nicht bloß ein Anzeichen20
der Wahrheit, sondern
sie bestimmt den Sinn des Satzes. (Einstein: wie eine Größe gemessen wird, das ist sie.)
21
Was ist ein Sessel?
Wie sieht der Sessel aus?
Sind das etwa von einander unabhängige Fragen?
22
Man ist vielleicht geneigt zu denken: der Sessel steht da oder nicht;23
wie ich das weiß ist eine
andere Sache; wie mich die Kunde davon erreicht hat. Aber fragen wir also so? Was nenne ich denn
also eine Kunde daß ein Sessel dort steht? (Oder habe ich auch von24
dieser Kunde nur Kunde?) Und
was kennzeichnet denn diese Kunde als Kunde von etwas?25
Leitet uns da nicht unser sprachlicher
Ausdruck irre? Ist das eben nicht ein irreleitendes Bild: &mein Auge gibt mir Kunde davon daß dort
ein Sessel steht“?
26
&Der Sessel existiert unabhängig davon ob ihn jemand wahrnimmt27
.” Ist das ein Erfahrungssatz
oder eine verschleierte Festsetzung28
der Grammatik? 29
Soll es sagen die Erfahrung habe gelehrt daß
ein Sessel nicht verschwindet wenn man sich von ihm wegwendet?
30
Folgt nun, daraus, daß ich einen Mann dorten sehe, daß einer sich dort befindet?
Wir müssen31
hier nur stark schematisierte Fälle betrachten da die wirklichen zu mannifach sind.
32
Ob unsere Sinne uns belügen, davon rede ich nicht, sondern nur davon, daß wir ihre Sprache
verstehen.
33
Die Frage nach der Verification ist eine Frage nach der Methode.
34
Welches ist die ”wirkliche Lage“ des Körpers, den ich unter Wasser sehe, was die
”wirkliche Farbe“ des Tisches. Welches nennst Du &die wirkliche Lage“. Du selbst kannst es
entscheiden.35
Hier macht eben die Frage nach der Verifikation den Sinn dieser Ausdrücke36
klar.
37
Eigentlich hat ja schon Russell durch seine ”theory of descriptions“ gezeigt, daß
man sich nicht eine Kenntnis der Dinge von hinten herum erschleichen kann, und daß
es nur scheinen kann, als wüßten wir von den Dingen mehr, als sie uns auf geradem
Weg geoffenbart haben. Aber er hat durch die Idee der ”indirect knowledge“ wieder alles
verschleiert.
19 (M): ///
20 (V): ist nicht ein bloßes Anzeichen // Anzeigen
21 (M): 3
22 (M): 3
23 (V): denken // glauben // : es regnet oder nicht;
24 (V): ich von
25 (V): Und was macht denn diese Kunde zur Kunde?
26 (M): 3
27 (V): sieht
28 (V): oder ein Satz
29 (V): Grammatik? Sag
30 (M): 555
31 (V): können
32 (M): 3
33 (M): 3
34 (M): ∫ 3
35 (V): kannst entscheiden.
36 (V): den Sinn der Worte
37 (M): ∫
265
266
264v
265v
266
208 Sage mir, was Du mit einem Satz anfängst . . .
WBTC060-68 30/05/2005 16:16 Page 416
How language distances itself from a description of verification. How abstract it gets! We
have to rediscover that we measure time with a clock. – And in the process we don’t even
notice that we’ve made a grammatical discovery.
15
How a proposition is verified, that’s what it says. Compare generality in arithmetic with
the generality of non-arithmetical propositions. The latter is verified differently and is
therefore a different generality. Verification isn’t merely an16
indication of the truth; it
determines the sense of the proposition. (Einstein: How a quantity is measured, that’s
what it is.)
17
What is a chair?
What does a chair look like?
Are those really independent questions?
18
Perhaps one is inclined to think: the chair is either there or it isn’t;19
how I know this is another
matter: how this bit of news reached me. But is this the way we ask questions? What do I call
news that a chair is there? (Or do I only have news about that news?) And what characterizes this
news as20
news about something? Isn’t our linguistic expression misleading us here? Isn’t this really
a misleading image: “My eye gives me the news that there’s a chair there”?
21
“A chair exists independently of whether someone perceives22
it.” Is that an empirical proposition,
or a veiled regulation of23
grammar? Is it supposed to mean “Experience has taught us that a
chair doesn’t vanish when we turn away from it”?
24
Now does it follow from my seeing a man over there that one is there?
Here we have to look25
at heavily schematized cases since the real ones are too manifold.
26
I’m not talking about whether our senses lie to us, only about our understanding their language.
27
The question about verification is a question about method.
28
What is the “real position” of the body that I see under water, what is the “real colour”
of the table? What do you call “the real position”? You can decide this yourself.29
Here it is
precisely the question about verification that clarifies the sense of these expressions.30
31
Actually Russell has already shown in his “theory of descriptions” that one cannot gain
knowledge of things surreptitiously, as through a back door, and that it can only seem that
we knew more about things than they have revealed to us straightforwardly. But with the
idea of “indirect knowledge” he obscured everything again.
15 (M): ///
16 (V): isn’t a mere
17 (M): 3
18 (M): 3
19 (V): think // believe // : either it’s raining or it isn’t;
20 (V): And what makes this news into
21 (M): 3
22 (V): sees
23 (V): or a proposition of
24 (M): 555
25 (V): we can look
26 (M): 3
27 (M): 3
28 (M): ∫ 3
29 (V): decide yourself.
30 (V): of the words.
31 (M): ∫
Tell me What you do with a Proposition . . . 208e
WBTC060-68 30/05/2005 16:16 Page 417
38
Es ist gut sich zu sagen: Aus derselben Quelle fließt nur Eines.39
40
Welche Sätze aus ihm folgen und aus welchen Sätzen er folgt, das macht seinen Sinn
aus.41
Daher auch die Frage nach seiner Verifikation eine Frage nach seinem Sinn ist.
42
Betrachten wir den Satz:43
”es wird niemals Menschen mit 2 Köpfen geben“. Dieser Satz
scheint irgendwie ins Unendliche, Unverifizierbare zu reichen und sein Sinn von jeder
Verifikation unabhängig zu sein. Aber wenn wir seinen Sinn erforschen wollen, so meldet
sich ganz richtig die Frage: Können wir die Wahrheit eines solchen Satzes je wissen, und
wie können wir sie wissen; und welche Gründe können wir haben, was der Satz sagt
anzunehmen oder abzulehnen? Nun wird man vielleicht sagen: es ist ja nach dem Sinn
gefragt worden; und nicht danach, ob und wie man ihn wissen kann. Aber die Antwort auf
die Frage ”wie kann man diesen Satz wissen?“ ist nicht eine psychologische, sondern sie lehrt
Zusammenhänge im Kalkül.44
Und die Gründe, die möglich sind den Satz anzunehmen, sind
nicht persönliche Angelegenheiten, sondern Teile des Kalküls, zu dem der Satz gehört.
Wenn ich frage: wie kann ich den Satz ”jemand ist im Nebenzimmer“ verifizieren, oder
wie kann ich herausfinden, daß jemand im Nebenzimmer ist, so ist etwa eine Antwort: ”indem
ich ins Nebenzimmer gehe und nachsehe“.45
Wenn nun gefragt wird ”wie kann ich ins
Nebenzimmer kommen, wenn die Türe versperrt ist“, so ist dieses ”kann“ ein anderes
als das erste: Die erste Frage nach der Möglichkeit (der logischen) hatte eine Erklärung
über den Satzkalkül zur Antwort,46
die zweite Frage war eine nach der physikalischen
Möglichkeit und hatte einen Erfahrungssatz zur Antwort: daß man, etwa, die Mauer nicht
durchbrechen könne, weil sie zu stark sei, dagegen die Tür mit einem Sperrhaken öffnen
könne. Beide Fragen nun sind in gewissem Sinn, aber nicht im gleichen, Fragen nach der
Möglichkeit der Verifikation. Und, indem man die erste Art mit der zweiten verwechselt,
glaubt man, die Frage nach der Verifikation sei für den Sinn ohne Belang. Die Gründe
für die Annahme eines Satzes sind nicht zu verwechseln mit den Ursachen der Annahme.
Jene gehören zum Kalkül des Satzes.
So kann ja auch der Satz der Komet . . . bewege sich in einer Parabel nicht verifiziert werden.
Aber können wir ihn nicht verwenden?
Denke darüber nach, was wir mit so einem Satz machen.47
Wie er unsre Beobachtungen leitet.
48
Die Ursachen, warum wir einen Satz glauben, wären für die49
Frage, was es denn
ist, was wir glauben, allerdings irrelevant, aber nicht so die Gründe, die ja mit dem Satz
grammatisch verwandt sind und uns sagen, wer er ist.
38 (M): /
39 (R): [Gehört in einen größeren Zusammenhang
wohl zur Mathematik]
40 (M): 555
41 (E): Aus dem Zusammenhang dieser
Bemerkung in MS 113, S. 81 geht hervor, daß
”seinen Sinn“ ”den Sinn des Satzes“ bedeutet.
42 (M): 3
43 (V): Wende das auf einen Satz an, wie etwa
44 (V): sondern sagt, aus welchem andern Satz er
folgt, gehört also zur Grammatik des erstern. //
sondern sie sagt, mit welchen andern Sätzen er in
bestimmtem Zusammenhang des Kalküls steht //
sondern sie stellt einen Zusammenhang des Kalküls mit
andern Sätzen her. // sondern sie gibt // macht // erklärt
// den Zusammenhang eines Kalküls mit andern
Sätzen // sondern sie erklärt seinen Zusammenhang
mit andern Sätzen des Kalküls // Zusammenhang im
Kalkül mit andern Sätzen. // sondern sie stellt seinen
Zusammenhang mit andern Sätzen des Kalküls her.
45 (V): und ihn sehe“.
46 (V): Antwort, daß nämlich dieser Satz aus
jenem folgt;
47 (V): machen?
48 (M): 3
49 (V): bei der
267
266v
267
266v
267
267
266v
209 Sage mir, was Du mit einem Satz anfängst . . .
WBTC060-68 30/05/2005 16:16 Page 418
32
It is good to say to oneself: only one thing flows from the same spring.33
34
Which propositions follow from it and from which propositions it follows, constitute its
sense.35
Which is why the question about its verification is also a question about its sense.
36
Let’s look at the proposition:37
“There will never be people with two heads”. This proposition
seems somehow to reach into the infinite, the unverifiable, and its sense seems to be
independent of any verification. But if we want to explore its sense, then quite properly the
question arises: Can we ever know the truth of such a proposition, and how can we know it;
and what reasons can we have to accept or reject what the proposition says? Now someone
might say: After all, the question has been raised about its sense, and not about whether and
how one can know it. But the answer to the question “How can one know this proposition?”
is not a psychological one; rather, it teaches us connections in a calculus.38
And the possible
reasons for accepting the proposition are not personal matters, but parts of the calculus to
which the proposition belongs.
If I ask: How can I verify the proposition “Someone is in the room next door”?, or How
can I find out whether someone is in the room next door?, then an answer might be: “By
going into the room next door and checking”.39
But if the question is raised: “How can I get
into the room next door if the door is locked?”, this “can” is different from the first. The
first question, about (logical) possibility, was answered by an explanation of the sentence-
calculus,40
the second question was about physical possibility, and was answered by an empirical
proposition: say, that the wall could not be broken through because it was too thick, but
that the door could be opened with a skeleton key. Now in a certain – but not the same –
sense, both questions are questions about the possibility of verification. And, confusing the
first kind of question with the second, we believe that the question about verification is
irrelevant to sense. The reasons for accepting a proposition are not to be confused with the
causes for accepting it. The former belong to the calculus of the proposition.
So too the proposition that comet . . . travels in a parabola can’t be verified. But can’t we use it?
Think about what we do with such a proposition.41
How it guides our observations.
42
The causes for believing a proposition are indeed irrelevant to the question what it is
that we believe, but not so the reasons, which are grammatically related to the proposition,
and tell us who it is.
32 (M): /
33 (M): [Belongs in a larger context, probably to
mathematics]
34 (M): 555
35 (E): This remark has been separated from the
sentences preceding it in its source (MS 113,
p. 81), where it is clear that “its sense” refers
to “the sense of a proposition”.
36 (M): 3
37 (V): Apply that to a proposition such as:
38 (V): rather it says from which other proposition
it follows, and thus belongs to the grammar of
the first proposition. // rather it says with which
other propositions it stands in the particular connection
of a calculus. // rather it establishes // gives //
makes // explains // a // the // connection of a // of
the // calculus with other propositions. // rather it
explains its connection with other propositions of a
calculus. // rather it explains its connection in a calculus
with other propositions. // rather it establishes
a connection with other propositions of a calculus.
39 (V): and I see him”.
40 (V): sentence-calculus, i.e. that this proposition
follows from that;
41 (V): proposition?
42 (M): 3
Tell me What you do with a Proposition . . . 209e
WBTC060-68 30/05/2005 16:16 Page 419
Wenn Du erfahren50
willst, wie51
ein Mensch seinen Tag verbringt; frage nach seinem Beruf. Hat
jeder Mensch einen Beruf?52
Ist es klar was alles &Beruf“ zu nennen ist?
Die Lagrangeschen Gleichungen, die Keplerschen Gesetze, ein Satz aus der Naturgeschichte,53
der Satz &dort geht Herr N.N.“, sie haben54
alle verschiedene Art der Verwendung, wenn auch
Verwandschaft zwischen ihnen besteht. Es sind eben alles Instrumente zu verschiedenartigen
(wenn auch bis zu einem gewissen Grade verwandten) Zwecken.55
Und hier kann man ermessen welche unheilvolle Wirkung die Präokkupation56
mit dem &Sinn“
des Satzes, dem &Gedanken“, den er ausdrückt, gehabt hat. Denn so werden charakteristische
Vorstellungen die sich mit den Worten des Satzes57
verbinden für maßgebende angesehen,58
auch,59
wo sie es gar nicht sind & alles auf die Technik seiner Verwendung ankommt.60
– Und man
kann sagen der Satz habe einen anderen Sinn wenn er ein anderes Bild macht. Und wenn ich mir
erlauben darf Freges Grundgedanken61
in seiner Theorie vom Sinn & der Bedeutung62
zu erraten so
würde ich nun fortfahren: die Bedeutung des Satzes, im Sinne Freges, sei seine Verwendung.63
64
Und der Sinn des Satzes ist ja nicht etwas, was wir wie die Struktur der Materie erforschen
und was vielleicht zum Teil unerforschlich ist. So daß wir später erst noch einmal
daraufkommen könnten, daß dieser Satz von andern Wesen, als wir sind, auf eine andere
Art gewußt werden kann. So daß er dieser Satz mit diesem Sinn bliebe, dieser Sinn aber
Eigenschaften hätte, die wir jetzt nicht ahnen. Der Satz, oder sein Sinn, ist nicht das pneumatische
Wesen, was sein Eigenleben hat und nun Abenteuer besteht, von denen wir nichts
zu wissen brauchen. Wir hätten ihm quasi Geist von unserm Geist eingehaucht – seinen
Sinn – aber nun hat er sein Eigenleben – wie unser Kind – und wir können ihn (nur) erforschen
und mehr oder weniger verstehen.65
66
Der Instinkt leitet67
Einen richtig, der zur Frage führt: Wie kann man so etwas wissen;
was für einen Grund können wir haben, das anzunehmen; aus welchen Erfahrungen
würden wir so einen Satz ableiten; etc.
68
Der Sinn ist keine Seele des Satzes. Er muß, soweit wir an ihm interessiert sind, sich
gänzlich ausmessen lassen, sich ganz in Zeichen erschließen.69
70
Wenn man nun fragt: hat es Sinn zu sagen ”es wird nie das und das geben“?71
– Nun,
welche Evidenz gibt es dafür; und was folgt daraus? – Denn, wenn es keine Evidenz
dafür gibt – nicht, daß wir noch nicht im Stande waren sie zu kriegen72
sondern, daß keine73
im Kalkül vorgesehen wurde, – dann ist damit der Charakter dieses Satzes bestimmt. Wie
50 (V): wissen
51 (V): was
52 (V): Beruf?
53 (V): Naturgeschichte, oder
54 (V): NN“ haben
55 (V): zu verschiedenartigen Zwecken.
56 (O): Preokupation
57 (V): mit dem Satz
58 (V): werden den Satz begleitenden Empfindungen &
Bilder für wichtig angesehen, // werden charakteristische
Vorstellungen die mit dem Satz // mit den
Worten des Satzes // verbunden sind für das
Wichtige angesehen,
59 (V): auch dort,
60 (V): wo sie es nicht sind.
61 (V): Gedanken
62 (V): von Sinn & Bedeutung der Sätze
63 (V): Anwendung.
64 (M): 3
65 (M): Mathematik
66 (M): 3
67 (V): führt
68 (M): ///
69 (V): offenbaren.
70 (M): 3
71 (V): geben?“
72 (O): krigen
73 (V): zu kriegen – sondern, wenn keine
210 Sage mir, was Du mit einem Satz anfängst . . .
266v
267v
268
WBTC060-68 30/05/2005 16:16 Page 420
If you want to find out43
how44
someone passes the day, ask about his job. And does every
person have a job? Is it clear what all is to be called “a job”?
Lagrange’s equations, Kepler’s laws, a proposition from natural history,45
the proposition
“Mr N N is walking over there”, all have different kinds of use, even though there is a relationship
between them. They are all simply instruments for varied purposes (although these purposes are
related, up to a point).46
And here one can appreciate what a disastrous effect the preoccupation with the “sense” of a
proposition, with the “thought” that it expresses, has had. For as a result of this, characteristic
mental images that attach themselves to the words of a sentence47
are seen as decisive48
even when
they aren’t, and when everything depends on the technique for using the sentence.49
– And one
can say that the proposition has a different sense if it creates a different image. And if I might take
the liberty of guessing at Frege’s basic idea50
in his theory of sense and meaning,51
I would now
continue: that the meaning of a proposition, in Frege’s sense, is its use.52
53
And the sense of a proposition isn’t something that we explore, like the structure of
matter, and that is perhaps partially unexplorable. So that only later could we discover that
this proposition can be known in a different way by beings different from us. So that it
would remain this proposition with this sense, but that its sense would have qualities that
at this point we don’t dream of. The proposition, or its sense, is not a kind of breathing
organism that has a life of its own, and that carries out various exploits, about which we
need to know nothing. As if in a manner of speaking we had breathed a soul into it from
our soul – its sense – but now it has its own life – like our child – and all we can do is explore
it and more or less understand it.54
55
That instinct is guiding56
us rightly that leads to the questions: How can one know
something like that? What reason can we have to assume that? From what experiences would
we deduce such a proposition?, etc.
57
Sense is not the soul of a proposition. So far as we are interested in it, it must be completely
measurable, must disclose58
itself completely in signs.
59
What if one asks: Does it make sense to say “There will never be this or that”? – Well,
what evidence is there for this; and what follows from it? – For if there is no evidence for
it – and not because we haven’t been able to gather it yet, but because none was provided for
in the calculus – then this determines the character of that proposition. Just as the nature
43 (V): to know
44 (V): find out what
45 (V): history, or
46 (V): for varied purposes.
47 (V): to a sentence
48 (V): of this, sensations and images that accompany
the proposition are seen as important // of this,
characteristic mental images that are connected to
the proposition // to the words of the proposition //
are seen as what’s important
49 (V): even when they aren’t.
50 (V): Frege’s idea
51 (V): and meaning of propositions,
52 (V): application.
53 (M): 3
54 (M): Mathematics
55 (M): 3
56 (V): leading
57 (M): ///
58 (V): reveal
59 (M): 3
Tell me What you do with a Proposition . . . 210e
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das Wesen einer Zahlenart dadurch, daß kein Vergleich zwischen ihr und gewissen
Rationalzahlen möglich ist.
74
Übrigens: Eine Zahl, die heute auf bewußte Weise mittels des Fermat’schen Satzes
definiert ist, wird dadurch nicht geändert, daß der Beweis dieses Satzes, oder des
Gegenteils, gefunden wird. Denn der Kalkül dieser Zahl weiß von dieser Lösung des
Problems nichts (und wird auch dann nichts von ihr wissen).
75
”Ich werde nie einen Menschen mit 2 Köpfen sehen“; man glaubt, durch diesen Satz
irgendwie in die Unendlichkeit zu reichen. Quasi, zum mindesten eine Eisenbahn dorthin
gelegt zu haben, wenn wir auch noch nicht die ganze Strecke bereist haben.
Es liegt da die Idee zu Grunde, daß76
das Wort ”nie“ die Unendlichkeit schon77
mitbringe,
da das eben seine Bedeutung ist.78
Es kommt darauf an: Was fange ich mit diesem79
Satze an;80
denn, auf die Frage ”was bedeutet81
er?“ kommt ja wieder ein Satz zur Antwort, und der führt mich solange nicht weiter, als ich
aus der Erklärung nichts über die Züge erfahre, die ich mit den Figuren machen darf. (Als
ich, sozusagen, nur immer wieder die gleiche Spielstellung82
vor mir sehe und keine anderen,
die ich aus ihr bilden kann.) So höre ich z.B., daß keine Erfahrung diesen Satz beweisen
kann und das beruhigt mich über seine unendliche Bedeutung.
83
Aus keiner Evidenz folgt, daß dieser Satz wahr ist. Ja, aber ich kann doch glauben, daß es
so ist wie84
er sagt!85
Aber welcher Art ist86
das: ”glauben, daß das der Fall ist87
“? Reicht etwa
dieser Glaube in die Unendlichkeit; fliegt er der Verifikation voran? – Was heißt es, diesen
Satz88
glauben:89
Ihn90
mit bestimmten Gefühlen sagen? sich so & so verhalten, so & so zu
handeln?91
– Und diese Handlungen92
interessieren uns nur, sofern sie zeigen was wir mit dem Satz
anfangen wie wir ihn im Kalkül gebrauchen.93
94
Um den Sinn einer Frage zu verstehen, bedenken wir: Wie sieht denn die Antwort auf
diese Frage aus.
95
Auf die Frage ”ist A mein Ahne“ kann ich mir nur die Antwort denken ”A findet sich
in meiner Ahnengalerie“ oder ”A findet sich nicht in meiner Ahnengalerie“ (wo ich unter
Ahnengalerie die Gesamtheit aller Arten von Nachrichten über meine Vorfahren verstehe).
Dann konnte aber auch die Frage nur dasselbe heißen wie: ”Findet sich A in meiner
74 (M): ///
75 (M): 3
76 (V): daß z.B.
77 (V): bereits
78 (M):
79 (V): mit so einem
80 (V): Was kann ich mit so einem Satz tun //
anfangen;
81 (V): sagt
82 (V): Konfiguration
83 (M): 3
84 (V): daß es sich so verhält wie
85 (V): aber ich kann doch glauben, daß er // was
er sagt // wahr ist! // daß das der Fall ist, was er
sagt!
86 (V): Aber was heißt
87 (V): daß was er sagt sich so verhalt
88 (V): es das
89 (V): glauben?
90 (V): glauben: Diesen Satz
91 (V): sagen? ist es ein bestimmtes Benehmen?
denn etwas andres kann es doch nicht sein
92 (V): Diese Vorgänge
93 (V): handeln? – Und dann interessiert es uns nur
insofern, // Das alles interessiert uns alles nur
insofern es // Alles das interessiert uns nur insofern
als es // ein Kalkulieren mit dem Satz ist.
94 (M): ///
95 (M): 555
269
268v
269
268v
269
270
269
269
268v
269
211 Sage mir, was Du mit einem Satz anfängst . . .
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of a kind of number is determined by the fact that no comparison can be made between it
and certain rational numbers.
60
By the way: A number that today has been defined in the well-known way via Fermat’s
theorem is not changed by the discovery of a proof of this theorem or of its opposite.
That is because the calculus for this number knows nothing about such a solution to this
problem (and won’t know anything about it when it happens, either).
61
“I shall never see a person with two heads”; we think that somehow this proposition
allows us to reach into infinity. At least to have built a railway there, in a manner of speaking,
even if we haven’t yet travelled to the end of the line.
At the bottom of this is the idea that62
the word “never” is already accompanied by infinity,
since that is its very meaning.63
It depends on: what I do with this proposition.64
For in answer to the question “What does
it mean65
?”, yet another proposition is forthcoming, and it doesn’t get me any further so long
as the explanation doesn’t tell me anything about the moves I’m allowed to make with the
pieces. (So long, so to speak, as all that I see in front of me is the same configuration of the
game66
over and over, and no others that I can develop out of it.) Thus I hear, for instance,
that no experience can prove this proposition, and that reassures me about its infinite
meaning.
67
No evidence shows that this proposition is true. Yes, but surely I can believe that things
are as it says!68
But what is the nature of69
: “believing that that is the case”70
? Does this belief
perhaps extend into infinity; does it fly ahead of its verification? – What does it mean to
believe that proposition71
, to utter it72
with certain feelings? To behave in such and such a way,
to act in such and such a way?73
Such actions74
interest us only in so far as they show what we are
doing with the proposition, how we are using it in the calculus.75
76
To understand the sense of a question, we think: What does the answer to this question
look like?
77
As an answer to the question “Is A my ancestor?” all I can imagine is: “A is in my gallery
of ancestral portraits” or “A is not in my gallery of ancestral portraits” (by “gallery of ancestral
portraits” I understand the totality of all kinds of information about my ancestors). But
then all the question means is the same thing as: “Is A in my gallery of ancestral portraits?”.
60 (M): ///
61 (M): 3
62 (V): that for example
63 (M):
64 (V): with such a proposition. // on: what I can do
with such a proposition.
65 (V): say
66 (V): configuration
67 (M): 3
68 (V): Yes, but after all I can believe that it // what
it says // is true! // believe that what it says is the
case!
69 (V): But what does this mean
70 (V): “believing that what it says behaves this way”
71 (V): to believe that
72 (V): To utter this proposition
73 (V): feelings? Is this a particular form of
behaviour? For surely it canFt be something
else.
74 (V): These processes
75 (V): such a way? – And then it only interests us
in so far as it // All of that interests us only in so far
as it // is a calculating with the proposition
76 (M): ///
77 (M): 555
Tell me What you do with a Proposition . . . 211e
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Ahnengalerie“. (Eine Ahnengalerie hat ein Ende: das ist ein Satz der Syntax.) Wenn mir
ein Gott offenbarte, A sei mein Ahne, aber nicht, der wievielte, so könnte auch diese
Offenbarung für mich nur den Sinn haben, ich werde A unter meinen Ahnen finden, wenn
ich nur lang genug suche; da ich aber die Zahl N von Ahnen durchsuchen werde, so muß
die Offenbarung bedeuten, A sei unter jenen N Ahnen.
212 Sage mir, was Du mit einem Satz anfängst . . .
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(A gallery of ancestral portraits has an end: that is a syntactical proposition.) If a god revealed
to me that A is my ancestor, but not how many generations ago, then all that even this
revelation could mean is that I shall find A among my ancestors if only I look long enough;
but since I am going to search through N number of ancestors, the revelation must mean
that A is among those N ancestors.
Tell me What you do with a Proposition . . . 212e
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Intention und
Abbildung.
271
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Intention and
Depiction.
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61
Wenn ich mich abbildend nach
einer Vorlage richte, also weiß,
daß ich jetzt den Stift so bewege,
weil die Vorlage so verläuft, ist hier
eine mir unmittelbar bewußte
Kausalität im Spiel?
1
Wenn ich, den Regeln folgend, statt ”↑“ ”a“ schreibe, so ist es, als wäre hier eine Kausalität
im Spiel, die nicht hypothetisch, sondern unmittelbar erlebt, wäre. (Natürlich ist nichts
dergleichen der Fall.)
Die Verwechslung von Grund & Ursache.
Der Gegenstand meines Hasses ist nicht die Ursache meines Hasses.
2
Wenn ich mich aber nun ärgere, weil jemand zur Türe hereinkommt, kann ich mich hier
im Nexus irren, oder erlebe ich ihn wie den Ärger?
In einem gewissen Sinne kann ich mich irren, denn ich kann mir sagen ”Ich weiß nicht,
warum mich sein Kommen heute so ärgert“. Das heißt, über die Ursache meines Ärgers
läßt sich streiten. – Anderseits nicht darüber, daß der Gedanke an sein Kommen – wie man
sagt – unlustbetont ist.
Wie aber in dem Fall: Ich sehe den Menschen und der Haß gegen ihn steigt bei seinem
Anblick in mir gegen ihn auf. – Könnte man fragen: wie weiß ich, daß ich ihn hasse, daß er
die Ursache meines Hasses ist. Und wie weiß ich, daß sein Anblick diesen Haß neu erweckt?
Auf die erste Frage: – ”ich hasse ihn“ heißt nicht ”ich hasse und er ist die Ursache meines
Hasses“. Sondern er, beziehungsweise sein Gesichtsbild – etc. – kommt in meinem Haß
vor, ist ein Bestandteil meines Hasses. (Auch hier tut’s die Vertretung nicht, denn was
garantiert mir dafür, daß das Vertretene existiert.) Im zweiten Fall kommt eben unmittelbar
die Erscheinung des Menschen in meinem Haß vor, oder, wenn nicht, dann ist seine
Erscheinung wirklich nur die hypothetische Ursache meines Gefühls und ich kann mich darin
irren, daß sie es ist, die das Gefühl hervorruft.
1 (M): 3 (R): ∀ S. 281/4 3 2 (R): [Zu: Grund, Ursache, Motiv.]
272
271v
272
273
214
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61
If in Copying I am Guided by
a Model and thus Know that I am
Now Moving my Pencil in such
a Way because the Model Goes that
Way, is a Causality Involved Here of
which I am Immediately Aware?
1
If, following the rules, I write “a” instead of “↑” then it is as if a causality were involved
here that is not hypothetical, but experienced immediately. (Of course, nothing of the sort
is the case.)
The confusion of reason and cause.
The object of my hatred is not the cause of my hatred.
2
But if I am annoyed because someone is coming in, can I be wrong about this causal
connection, or do I experience this connection in the same way I do the annoyance?
In a certain sense I can be wrong, for I can say to myself “I don’t know why his coming
is annoying me so much today”. That is, the cause of my annoyance is debatable. – On the
other hand it is not debatable that the thought of his coming is tinged with aversion – as
one says.
But how about in this case: I see the person and my hatred for him starts to well up. –
Could one ask: How do I know that I hate him, that he is the cause of my hatred? And how
do I know that seeing him awakens this hatred anew? With regard to the first question: – “I
hate him” does not mean “I hate, and he is the cause of my hatred”. Rather he, or alternatively,
the image of him – etc. – occur within my hatred, are components of my hatred. (The
proxy doesn’t work here either, for what guarantee do I have that the thing it stands for
exists?) As for the second question, the appearance of the person simply occurs immediately
within my hatred or, if it doesn’t, then his appearance is really only the hypothetical cause
of my feeling, and I can be wrong in thinking that it is what elicits the feeling.
1 (M): 3 (R): ∀ p. 281/4 3 2 (R): [To: Reason, Cause, Motive.]
214e
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3
”Ganz ebenso muß es sich auch mit dem Handeln nach einem Zeichenausdruck
verhalten. Der Zeichenausdruck muß in diesem Vorgang involviert sein, während er nicht
involviert ist, wenn er bloß die Ursache meines Handelns ist.“
Wenn der Satz ”ich hasse ihn“ so aufgefaßt wird: ich hasse und er ist die Ursache; dann
ist die Frage möglich ”bist Du sicher, daß Du ihn haßt, ist es nicht vielleicht ein Anderer
oder etwas Anderes“ und das ist offenbarer Unsinn.
3 (M): ///
215 Wenn ich mich abbildend nach einer Vorlage richte . . .
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3
“It must be exactly the same so far as acting in accordance with any symbolization is
concerned. The symbolization must be involved in that kind of process, whereas if it is
merely the cause of my action, it is not involved.”
If the proposition “I hate him” is understood as: “I hate and he is the cause”, then the
question can arise, “Are you sure that you hate him, mightn’t it be somebody or something
else?”, and that is obvious nonsense.
3 (M): ///
If in Copying I am Guided by a Model . . . 215e
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62
Wenn wir ”nach einer bestimmten
Regel abbilden“, ist diese Regel in
dem Vorgang des Kopierens
(Abbildens) enthalten, also aus
ihm eindeutig abzulesen?
Verkörpert der Vorgang des
Abbildens sozusagen diese Regel?
Denken wir,1
daß jemand eine Strecke absichtlich im Maßstab 1 : 1 kopiert. Ist dann in
dem Vorgang des Kopierens schon das Verständnis der allgemeinen Regel2
enthalten? 3
D.h.
ist die Weise, in der mein Bleistift von der Strecke geführt wird, eben dieses allgemeine Gesetz?
Mein Stift wurde von mir quasi ganz voraussetzungslos gehalten und nur von der Länge der
Vorlage beeinflußt.4
Ich würde5
sagen: Wäre die Vorlage länger gewesen, so wäre ich mit meinem Bleistift noch
weitergefahren und wenn kürzer, weniger weit. Aber war, gleichsam, der Geist, der sich hierin
ausspricht, schon im Nachziehen des einen Strichs enthalten?
Ich kann mir vornehmen: Ich gehe solange, bis ich ihn finde (ich will etwa jemand auf
einer Straße treffen). Und nun gehe ich die Straße entlang und treffe ihn an einem
bestimmten Punkt und bleibe stehen. War in dem Vorgang des Gehens, oder irgend einem
andern gleichzeitigen, die Befolgung der allgemeinen Regel, die ich mir vorgesetzt hatte,
enthalten? Oder war der Vorgang nur in Übereinstimmung mit dieser Regel, aber auch mit
anderen entgegengesetzten Regeln?
Ich gebe jemandem den Befehl von A eine Linie parallel zu a zu ziehen.
Er versucht (beabsichtigt) es zu tun, aber mit dem Erfolg, daß die Linie
parallel zu b wird.6
War nun der Vorgang des Kopierens derselbe, als hätte er
beabsichtigt, parallel zu b zu ziehen und seine Absicht ausgeführt? Ich glaube offenbar,
nein. Er hat sich von der Linie a führen lassen.
1 (V): wir uns den einfachen Fall,
2 (V): Verständnis des Nachzeichnens irgendeiner
Strecke im Maßstab 1 : 1
3 (M): /// – Gesetz?
4 (V): geführt.
5 (V): würde dann
6 (F): MS 109, S. 238.
274
275
a
b
A
216
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62
If We “Depict in Accordance with
a Particular Rule”, is this Rule
Contained in the Process of Copying
(Depicting), and can it Therefore
be Read out of it Unambiguously?
Does the Process of Depicting
Embody this Rule, as it Were?
Let’s imagine1
that someone intentionally copies a line segment on a scale of 1 : 1. Is an
understanding of the general rule2
already contained in the process of copying? 3
That is, is
the way my pencil is guided by the line segment precisely this general law? I held my pencil
completely neutrally, as it were, and it was only influenced4
by the length of the original line.
I would5
say: If the drawing had been longer I would have moved my pencil further, and
if it had been shorter I would have moved it less far. But was the spirit, so to speak, that is
expressed in this contained in my copying of the line?
I can resolve: I’ll walk until I find him (say I want to meet someone on a street). And
now I go down the street and meet him at a certain point and stop. Was the observance
of the general rule to which I had resolved myself contained in the process of walking, or
in some other simultaneous process? Or was the process just in accordance with this rule,
and with differing rules as well?
I order someone to draw a line parallel to a, starting at A. 6
He tries (intends)
to do this, but with the result that the line turns out parallel to b. Now was
the process of copying the same as if he had intended to draw a line parallel
to b and had carried out his intention? I think: obviously not. He let himself be guided by
the line a.
1 (V): imagine the simple case
2 (V): understanding of copying some line segment
on a scale of 1 : 1
3 (M): /// – law?
4 (V): led
5 (V): would then
6 (F): MS 109, p. 238.
a
b
A
216e
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Wer liest, macht das, was er abliest abhängig von dem, was da steht.7
8
Was hätte übrigens eine9
allgemeine Regel überhaupt auszudrücken, wenn nicht das?10
Die Frage ist nun: wenn ich (nun) auf diese Weise eine Vorlage nachgezeichnet habe,
ist es dann möglich, den Vorgang des Nachzeichnens, wie er war, auch nach einer anderen
allgemeinen Regel richtig zu beschreiben? Oder kann ich so eine Beschreibung zurückweisen11
mit den Worten: ”nein, ich habe mich wirklich nur von dieser (allgemeinen) Regel leiten lassen
(und nicht von jener anderen, die hier12
allerdings auch dasselbe Resultat ergeben hätte)“.
Wenn ich absichtlich eine gewisse Form nachzeichne,13
so hat der Vorgang des Kopierens
mit der Wirklichkeit an einer bestimmten Stelle diese Form gemein. Sie ist eine Fassette
des Vorgangs des Kopierens. Eine Fassette, die an dem kopierten Gegenstand anliegt und
sich dort mit ihm deckt.
Man könnte dann sagen: wenn auch mein Bleistift die Vorlage nicht trifft, die Absicht
trifft sie immer.
Es ist nur die Absicht, die an das Modell heranreicht. Und das ist dadurch ausgedrückt,
daß der Ausdruck der Absicht die Beschreibung des Modells und den Ausdruck der
Projektionsregel enthält. Was ich tatsächlich spiele, ist gleichgültig; die Erfahrung wird es
lehren und die Beschreibung des Gespielten muß nichts mit der Beschreibung des
Notenbildes gemein haben. Wenn ich dagegen meine Absicht beschreiben will, so muß es
heißen, daß ich dieses Notenbild auf die Weise in Tönen abzubilden beabsichtige. Und nur
das kann der Ausdruck dafür sein, daß die Absicht an die Vorlage heranreicht und eine
allgemeine Regel enthält.
14
Wenn ich einen Apparat machte, der nach Noten spielen könnte, der also auf das Notenbild
in der Weise reagierte, daß er die entsprechenden Tasten einer Klaviatur drückte, und wenn
dieser Apparat bis jetzt immer klaglos funktioniert hätte, so wäre doch weder er, noch sein
Funktionieren der Ausdruck einer allgemeinen Regel. Ferner, dieses Funktionieren ist,
wie immer er funktioniert, an sich weder richtig noch falsch; d.h. weder der Notenvorlage
entsprechend, noch ihr nichtentsprechend. Kein Mechanismus, welcher Art immer, kann
eine solche Regel etablieren. Man kann nur sagen: der Mechanismus arbeitet bis jetzt
dieser Regel gemäß (was natürlich heißt, daß er auch anderen Regeln gemäß arbeitet). Das
Funktionieren des Apparates bis zum gegenwärtigen Zeitpunkt würde gewisse Regeln von15
seiner Beschreibung ausschließen, aber nie eine Regel eindeutig bestimmen.
16
Wir können wohl eine Maschine zur Illustration der Koordination zweier Vorgänge, der
Abbildung des einen in dem andern, verwenden, aber nur die Maschine wie sie funktionieren
soll, also die Maschine in ganz bestimmter Weise als Ausdruck aufgefaßt, also als Teil der
Sprache.
17
Nur in diesem Sinne bildet z.B. das Pianola die Loch-Schrift auf dem Streifen in die
Tonfolge ab. Oder der Musterwebstuhl die Sprache der gelochten Karten in das Muster des
gewebten Stoffes.
7 (V): steht. Aber die Abhängigkeit kann nur
durch eine Regel ausgedrückt werden.
8 (M): ///
9 (V): die
10 (V): wenn das nicht?
11 (V): ablehnen
12 (V): die in diesem Falle
13 (V): nachzieh,
14 (M): ///
15 (V): zu
16 (M): ///
17 (M): ///
276
277
217 Wenn wir ”nach einer bestimmten Regel abbilden“ . . .
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Whoever reads makes what he gathers from it dependent on what is there.7
8
By the way, what should a9
general rule express anyway, if not that?
Now the question is: If I have (now) copied a drawing in this way, is it possible to correctly
describe the process of copying, as it took place, in accordance with another general
rule as well? Or can I reject10
such a description, saying: “No, I really only let myself be
guided by this (general) rule (and not by that other one which, to be sure, would also have
had the same result here11
)”.
If I intentionally copy a particular shape, at a certain point the process of copying has this
shape in common with reality. The shape is a facet of the copying process. A facet that is
contiguous with the copied object and that coincides with it at these points of contact.
Then one could say: Even if my pencil doesn’t capture the original, my intention always
does.
Only intention can measure up to the original. And that is shown by the fact that the expression
of intention contains the description of the original as well as the expression of the rule
of projection. What I am actually playing is irrelevant; experience will teach me that, and
the description of what is being played doesn’t have to have anything in common with the
description of the layout of the notes. If on the other hand I want to describe my intention,
then I have to say that what I intend to depict in sounds is this layout of notes in this
way. And that’s the only way of expressing the fact that the intention comes right up to the
original, and that it contains a general rule.
12
If I built an apparatus that could play written music, i.e. that reacted to the depiction
of the notes in such a way that it depressed the respective keys of a keyboard, and if this
apparatus had always worked trouble-free until now, still, neither it nor its functioning
would be the expression of a general rule. Furthermore this functioning, however it works,
is inherently neither true nor false; that is, it neither corresponds nor fails to correspond to
the notes on the page. No mechanism, whatever its kind, can establish such a rule. One can
only say: Up until now the mechanism has been working in accordance with this rule (which
means of course that it also works in accordance with other rules). The functioning of the
apparatus up to the present point in time would exclude certain rules from13
its description,
but it would never determine a rule unambiguously.
14
To be sure, we can use a machine to illustrate the coordination of two processes – the
depiction of the one by the other. But only a machine as it is supposed to work, i.e. a machine
understood in a very particular way as an expression, i.e. as a part of language.
15
Only in this sense does the pianola, for example, copy the perforations on the strip into
a sequence of sounds. Or does the pattern loom copy the language on the punch cards into
the pattern of the woven material.
7 (V): is there. But the dependency can only be
expressed by a rule.
8 (M): ///
9 (V): the
10 (V): decline
11 (V): result in this case
12 (M): ///
13 (V): of
14 (M): ///
15 (M): ///
If we “Depict in Accordance With a Particular Rule” . . . 217e
WBTC060-68 30/05/2005 16:16 Page 435
In dem Ausdruck der Absicht muß ich die Vorlage beschreiben; in der Beschreibung des Abbildes
nicht. (Und das ist der Kern des ganzen Problems, & seine Lösung.)
Das Wort ”psychischer Vorgang“, ”mental process“, ist an vieler Verwirrung schuld. Wenn
wir sagen, der Gedanke, die Intention sind psychische Vorgänge, so stellen wir uns darunter
etwas ähnliches oder analoges vor, wie unter dem Wort chemischer Vorgang, oder physiologischer
Vorgang. – Und soweit das richtig ist, haben wir mit dem Gedanken und der Intention
nichts zu tun.
18
”Wenn man kopiert, d.h. überhaupt abbildet, sich von einer Vorlage leiten läßt, so ist
das Charakteristische daran, daß nur die Vorlage mir bewußt wird, dagegen nicht die
Projektionsart (Nach Noten spielen). Ich bin mir bewußt, daß mich die Vorlage einmal so,
einmal so lenkt, aber das Wie dieser Übertragung nehme ich sozusagen hin; ich bemerke es
weiter nicht. Und zwar, weil ich es nicht mit einem Anderen vergleiche. Ich befolge die
Projektionsregel, aber ich drücke sie nicht aus und sie fällt sozusagen aus der Betrachtung
heraus, weil sie mit nichts verglichen wird. Wenn ich sie beschreibe, so setzt das voraus, daß
ich sie mit anderen Regeln vergleiche.“
Was ist das Kriterium der Absicht? Kommt diese Frage in die Betrachtung dieser Seite hinein?
19
”Ja, in gewissem Sinne ist alles, was beim Nachbilden der Vorlage geschieht, daß diese
Vorlage an uns vorüberzieht und wir sie besser oder weniger gut treffen. D.h. es ist das Ende
der Kopiermaschine, das unserer Vorlage entlangläuft, was wir beobachten; die ganze übrige
Maschine nehmen wir als gegeben hin. Wir merken sozusagen nur, was sich ändert, nicht,
was gleichbleibt. Der Abbildungsweise haben wir durch eine Einstellung (die gleichbleibt)
(ein für allemal) Rechnung getragen. – Und was wir spüren, ist nur das Modell.“
20
”Darum, wenn wir falsch nach Noten singen oder spielen – so verschieden diese
Abbildung der Art nach von ihrem Vorbild ist – nennen wir es21
einen Verstoß gegen das
Vorbild.“22
18 (R): Zu § 63 Diese Bemerkung gehört nicht zu der,
daß die Rechtfertigungen der Abbildung irgendwo
aufhören. etc.
19 (R): Zu § 63
20 (R): Zu § 63
21 (V): ist – führen // fühlen // wir es als
22 (V): gegen das Modell.“ // gegen die Vorlage.“
276v
276v
277
277, 278
218 Wenn wir ”nach einer bestimmten Regel abbilden“ . . .
WBTC060-68 30/05/2005 16:16 Page 436
In an expression of intent I have to describe the original; I don’t have to do this in the description
of the depiction. (And that is the core of the whole problem, and its solution.)
The words “psychological process”, “mental process”, are responsible for a lot of confusion.
If we say that thought, intention, are psychological processes then we imagine something
similar or analogous to the words “chemical process” or “physiological process”. – And
in so far as this is correct we are in no way dealing with thought and intention.
16
“When one copies, i.e. in any way makes a likeness, allows oneself to be guided by an
original, then what’s characteristic of this is that I am only aware of the original, but not of
the mode of projection (reading music). I am conscious that the original guides me now this
way, now that, but how this transference takes place I take for granted, as it were; I don’t
take any further notice of it. And this is because I don’t compare it to anything else. I’m
following the projection rule, but I’m not expressing it, and it drops out of consideration,
so to speak, because it isn’t compared to anything. If I describe it, that presupposes that
I’m comparing it to other rules.”
What is the criterion of intent? Does this question enter into the considerations on this page?
17
“Indeed, in a certain sense, all that happens in copying an original is that it marches past
us and that we capture it more or less well. That is, what we observe is the tail end of the
copying machine that runs alongside our original; all the rest of the machine we accept as
given. We only notice what changes, as it were, not what stays the same. We have accounted
(once and for all) for the mode of depiction by an approach (that remains the same). – And
the only thing we sense is the original.”
18
“Therefore if in following the written music we sing or play incorrectly – no matter how
much this rendition differs in kind from the score – we call this an19
offence against the score.”20
16 (R): To § 63 This remark belongs does not belong to
the remark that justifications for making a likeness
come to an end at some point etc.
17 (R): To § 63
18 (R): To § 63
19 (V): we feel it as an
20 (V): against the model.”
If we “Depict in Accordance With a Particular Rule” . . . 218e
WBTC060-68 30/05/2005 16:16 Page 437
63
Wie rechtfertigt man das Resultat
der Abbildung mit der allgemeinen
Regel der Abbildung?
1
Ich kann 52
mittels x2
rechtfertigen, wenn ich dabei x2
einem x3
oder einem andern Zeichen
des Systems entgegenstelle.
2
Die Schwierigkeit ist offenbar, das nicht zu rechtfertigen versuchen, was keine
Rechtfertigung zuläßt.3
4
Wenn man fragt: ”warum schreibst Du 52
?“ und ich antworte ”es steht doch da, ich
soll quadrieren“, so ist das eine Rechtfertigung – und eine volle – . Eine Rechtfertigung
verlangen, in dem Sinne, in dem dies keine ist, ist sinnlos.
Ich hätte jemandem alle möglichen Erklärungen5
dafür gegeben, was der Befehl
”quadriere diese Zahlen“ heißt. (Und diese Erklärungen sind doch sämtlich Zeichen.)
Er quadriere darauf, und nun frage ich ihn ”warum tust Du das auf diese Erklärung hin?“
Dann hätte es keinen Sinn mir zu antworten: ”Du hast mir doch gesagt: (es folgt die
Wiederholung der Erklärungen)“. Eine andre Art der Antwort ist aber auf diese Frage auch
nicht möglich und die Frage heißt eben nichts. Sie müßte sinnvoll lauten: ”Warum tust Du
das und nicht jenes auf diese Erklärungen hin (ich habe Dir doch gesagt . . . )“.
6
Wenn man nun fragen würde: Wie lange vor der Anwendung der Regel muß die Disposition
”x2
“ gedauert haben? Eine Sekunde, oder zwei? Diese Frage klingt natürlich, und mit Recht,
wie eine Persiflage. Wir fühlen, daß es darauf gar nicht ankommen kann. Aber diese Art der
Frage taucht immer wieder auf.
7
Wenn man nach einer Regel einen Tatbestand abbildet, so ist dieser dabei die Vorlage.
Ich brauche keine weitere Vorlage, die mir zeigt, wie die Abbildung vor sich zu gehen hat,
wie also die erste Vorlage zu benützen ist, denn sonst brauchte ich auch eine Vorlage, um
mir die Anwendung der zweiten zu zeigen, u.s.f. ad infinitum. D.h. eine weitere Vorlage
nützt mir8
nichts, ich muß ja doch einmal ohne Vorlage handeln.
1 (M): ///
2 (M): ///
3 (V): verträgt.
4 (M): 3
5 (V): alle mögliche Erklärung
6 (M): ///
7 (M): 3 ///
8 (O): mich
279
280
219
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63
How Does one Use a General Rule of
Representation to Justify the Result of
Representation?
1
I can justify 52
by means of x2
if in so doing I contrast x2
with x3
, or another sign in the
system.
2
The difficulty is obviously not to try to justify what admits of no3
justification.
4
If one asks: “Why do you write 52
?” and I answer “It says I’m to square”, then that is
a justification – and a complete one – . To demand a justification in the sense in which this
isn’t one is senseless.
Say I have given someone all possible explanations5
for what the command “Square
these numbers” means. (And after all, these explanations are all signs.) Then he squares
them, and now I ask him, “Why, in response to this explanation, did you do that?”. Then
it wouldn’t make any sense for him to answer: “But you told me: (a repetition of the
explanations follows)”. But no different kind of answer to this question is possible, and the
question just doesn’t mean anything. To be meaningful, it would have to ask: “Why did
you do this and not that in response to these explanations? (I told you, after all . . . )”.
6
Now suppose one were to ask: How long must the disposition “x2
” have lasted before
the rule was applied? One second, or two? This question naturally, and justifiably, sounds
like a parody. We feel that it can’t possibly be relevant. Yet this kind of question pops up
again and again.
7
If one portrays a state of affairs in accordance with a rule then this state of affairs is the
model. I don’t need any further model to show me how the depiction is to proceed, i.e. how
the first model is to be used, because then I would also need a model in order to show me
the application of the second one, and so forth ad infinitum. In other words, an additional
model doesn’t do me any good, for at some point I’m going to have to act without a model.
1 (M): ///
2 (M): ///
3 (V): what tolerates no
4 (M): 3
5 (V): someone every possible explanation
6 (M): ///
7 (M): 3 ///
219e
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9
Wenn ich mich mit10
der Bewegung des Punktes P von A nach B
nach dem Pfeil richte, so ist das11
nur dadurch beschrieben, daß ich das
System von Pfeilen beschreibe, dem dieser angehört. – Ich könnte nun
wohl sagen: Ist das genug? muß ich nicht auch die Regel angeben,
nach der die Übersetzung geschieht, z.B. hier, daß ich mich parallel zum
Pfeil bewegen soll? Aber diese Übersetzungsregel könnte12
ich mir in
Gestalt etwa des Zeichens ”||“ (im Gegensatz etwa zu ”|–“) dem
Pfeile zugesetzt denken; aber dann würde das Zeichen ”2||“ auf
keiner andern Stufe stehen wie ”2“ und ich könnte doch jetzt nur das
System beschreiben, dem dieses Zeichen angehört, wenn ich nicht
ad infinitum, also erfolglos, weitere Zeichen zu den obigen setzen will.
Jedes Abbilden,13
(Handeln nach – nicht bloß in Übereinstimmung mit –14
einer Regel15
) Ableiten
einer Handlung aus einem Befehl, Rechtfertigen einer Handlung mit einem Befehl ist von der Art des
schriftlichen Ableitens eines Resultats aus einer Angabe dem Hinweis auf die Stellung von Zeichen in
einer Tabelle.16
17
Wir stoßen hier immer auf die peinliche Frage, ob denn nicht das Anschreiben des ”52
“
(z.B.) mehr oder weniger (oder ganz) automatisch erfolgt sein könne, und fühlen, daß das
der Fall sein mag und daß es uns gar nichts angeht. Daß wir hier auf ganz irrelevantem Boden
sind, wo wir nicht hingehören.
18
”Ich schreibe ,52
‘, weil hier ,x2
‘ steht.“ Was aber, wenn ich sagte: ”Ich schreibe ,+‘, weil
hier ,A‘ steht“? Man würde fragen: Schreibst Du denn überall ”+“ wo ”A“ steht? man19
würde
nach einer allgemeinen Regel fragen. Und das ”weil“ im letzten Satz hätte sonst keinen Sinn.
20
Warum schreibst Du 25? – Weil dort ”y2
“ steht. – Ja, ist das das Signal
für 25? – Nein, aber ich habe ”25“ geschrieben, weil dort ”y2
“ steht. –
Woher weißt Du denn, daß Du es deswegen21
geschrieben hast? Hier hätte
man das &weil“ als Einleitung einer Angabe der Ursache aufgefaßt statt des Grundes.
22
Was heißt es aber: Ich geh’ zur Tür, weil der Befehl gelautet hat ”geh’ zur Tür“?
Und wie vergleicht sich dieser Satz mit: ich geh’ zur Tür, obwohl der Befehl gelautet
hat ”geh’ zur Tür“. Oder: ich geh’ zur Tür, aber nicht weil der Befehl lautete ”geh’ . . .“,
sondern . . . . Oder: Ich geh’ nicht zur Tür, weil der Befehl gelautet hat ”geh’ z.T.“.
23
Das Phänomen der Rechtfertigung. Ich rechtfertige das Resultat 32
durch x2
.
So schaut jede Rechtfertigung aus.
In gewissem Sinn bringt uns das nicht weiter. Aber es kann uns ja auch nicht
weiter, d.h., zu dem Metalogischen,24
bringen.
9 (M): 3 (F): MS 110, S. 228.
10 (V): Wenn ich mit
11 (V): so ist, was hier geschieht,
12 (V): kann
13 (V): Abbilden, (Ableiten einer H
14 (V): nach (nicht bloß . . . mit)
15 (O): Regeln (V): mit – gewissen Zeichen //
Regeln
16 (V): Hinweis auf eine Tabelle.
17 (M): 3 ///
18 (M): 3
19 (V): steht? D.h., man
20 (M): 3
21 (V): deswegen
22 (M): 3 (R): Zu S. 272
23 (M): 3
24 (V): zu einem Fundament,
281
281
280v
282
220 Wie rechtfertigt man das Resultat der Abbildung . . .
P
A
B
P
y2
5
25
x2
3
32
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8
If in the movement of point P from A to B I take my cue from the
arrow, then that9
can only be described by describing the system of arrows
to which this one belongs. – Now I might well say: Is that enough?
Mustn’t I also cite the rule that governs the translation, e.g. that in this
case I am to move parallel to the arrow? I might10
imagine this rule of
translation appended to the arrow in the shape of, say, the sign “||”
(as opposed, say, to “|–”); but then the sign “2||” wouldn’t be on a
different level from “2”. And then I can only describe the system to
which this sign belongs, if I don’t want to add further signs to the ones
above – ad infinitum, and thus uselessly.
Any portrayal (action following – not merely in agreement with – a rule11
), or derivation of an action
from a command, or justification of an action by a command, is the same kind of thing as deriving
a written result from data, from pointing out the position of signs in a table.12
13
Here we always encounter the awkward question whether writing down “52
” (for
instance) couldn’t have taken place more or less (or completely) automatically, and we feel
that that might well be the case but that it is of absolutely no concern to us. That here we
are on completely irrelevant ground, where we don’t belong.
14
“I write ‘52
’ because ‘x2
’ is written here”. But what if I said: “I write ‘+’ because ‘A’ is
written here”? One would ask: Do you write “+” wherever “A” is written? One15
would ask
about a general rule. Otherwise the “because” in the second sentence would make no sense.
16
Why do you write 25? – Because “y2
” is written there. – Well, is that
the sign for 25? – No, I just wrote “25” because “y2
” was written there.
How do you know that that is why17
you wrote it? In this case one would
have taken the “because” as a prelude to an indication of a cause rather than of a reason.
18
But what does this mean: “I’m going to the door because the command was ‘Go to the
door’ ”?
And how does this proposition compare with: “I’m going to the door even though the
command was ‘Go to the door’ ”? Or: “I’m going to the door, but not because the command
was ‘Go . . .’, but rather . . .” Or: “I’m not going to the door because the command was
‘Go to the door’ ”.
19
The phenomenon of justification. I justify the result 32
by x2
. That’s what all
justifications look like.
In a certain sense this doesn’t get us any further. But of course it can’t get us
further, i.e. to a metalogical realm.20
8 (M): 3 (F): MS 110, p. 228.
9 (V): then what happens here
10 (V): can
11 (V): with – particular signs // rules
12 (V): from pointing to a table.
13 (M): 3 ///
14 (M): 3
15 (V): That is, one
16 (M): 3
17 (V): that is why
18 (M): 3 (R): To p. 272
19 (M): 3
20 (V): i.e. to a foundation.
How Does one Use a General Rule of Representation . . . 220e
x2
3
32
y2
5
25
P
A
B
P
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64
Der Vorgang der absichtlichen
Abbildung, der Abbildung mit
der Intention abzubilden ist nicht
wesentlich ein psychischer, innerer.
Ein Vorgang der Manipulation
mit Zeichen auf dem Papier
kann dasselbe leisten.
1
Kein psychischer Vorgang kann besser symbolisieren, als Zeichen, die auf dem Papier
stehen.
Der psychische Vorgang kann auch nicht mehr leisten, als die Schriftzeichen auf dem Papier.
Denn immer wieder ist man in der Versuchung, einen symbolischen Vorgang durch einen
besonderen psychischen Vorgang erklären zu wollen, als ob die Psyche in dieser Sache viel
mehr tun könnte, als das Zeichen.
2
Es mißleitet uns da die falsche Analogie mit einem Mechanismus, der mit anderen Mitteln
arbeitet, und daher eine besondere Bewegung3
erklären kann. Wie wenn wir sagen: diese
Bewegung kann nicht durch den Eingriff von Zahnrädern allein erklärt werden.
Hierher gehört, daß es eine wichtige Einsicht in das Wesen der Zeichenerklärung ist, die sie in
Gegensatz bringt zur Kausalerklärung,4
daß sich das Zeichen durch seine Erklärung ersetzen läßt.5
6
Die Beschreibung des Psychischen müßte sich7
wieder als Symbol verwenden lassen.
8
Das Behaviouristische an unserer Behandlung9
besteht nur darin, daß wir10
keinen
Unterschied zwischen ”außen“ und ”innen“ machen.11
Weil mich die Psychologie nichts
angeht.
1 (M): 3
2 (M): 3
3 (V): daher besondere Bewegungen
4 (V): in das Wesen der Zeichenerklärung, im
Gegensatz zur Kausalerklärung ist,
5 (V): Hierher gehört irgendwie: daß es nicht
selbstverständlich ist, daß sich das Zeichen
durch seine Erklärung ersetzen läßt. Sondern eine
merkwürdige, wichtige Einsicht in das Wesen
dieser (Art von) Erklärung. (Im Gegensatz einer //
zu einer // kausalen Erklärung.)
6 (M): 3
7 (V): sich ja doch
8 (M): 3
9 (V): an meiner Auffassung
10 (V): ich
11 (V): mache.
283
283v
284
284
221
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64
The Process of Copying on Purpose,
of Copying with the Intention
to Copy, is not Essentially a
Psychological, Inner Process.
A Process of Manipulating Signs
on a Piece of Paper can Accomplish
the Same Thing.
1
No psychological process can symbolize better than signs on paper.
A psychological process can’t accomplish any more than written signs on paper.
For again and again one is tempted to want to explain a symbolic process by a particular
psychological process, as if the psyche could do much more in this matter than signs.
2
Here we are being misled by a false analogy with a mechanism that uses a different means,
and can therefore explain a particular movement.3
As when we say: This movement can’t
be explained by the meshing of cog wheels alone.
This is relevant here: it is an important insight into the nature of explaining signs – one that places
it in opposition to a causal explanation – that4
a sign can be replaced by its explanation.5
6
The description of what is psychological ought to be7
usable in turn as a symbol.
8
The behaviourist aspect of our discussion consists only in our9
not distinguishing
between “outer” and “inner”. Because psychology is no concern of mine.
1 (M): 3
2 (M): 3
3 (V): explain particular movements.
4 (V): signs, in contrast to a causal explanation, that
5 (V): Here it is somehow relevant that it’s not
obvious that a sign can be replaced by its
explanation. Rather, it’s a strange, important
insight into the nature of this (kind of) explanation.
(In contrast to a causal explanation.)
6 (M): 3
7 (V): ought indeed still to be
8 (M): 3
9 (V): of my discussion consists only in my
221e
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12
Kann man etwas in einem wesentlich anderen13
Sinne ”offen lassen“, als man eine Klammer
leer läßt?
14
Es kann nie essentiell für unsere Betrachtung15
sein, daß ein symbolisierendes Phänomen
in der Seele sich abspielt und nicht auf dem Papier, für den Andern sichtbar.
16
Man kann sagen, daß, ob ich lese, oder nur Laute hervorbringe, während ein Text vor
meinen Augen ist, sich nicht durch die Beobachtung von außen entscheiden läßt. Aber das
Lesen kann nicht wesentlich eine innere Angelegenheit sein. Das Ableiten der Übersetzung
von17
Zeichen, wenn es überhaupt ein Vorgang ist, muß auch ein sichtbarer Vorgang sein
können. Man muß also z.B. auch den Vorgang dafür ansehen18
können, der sich auf dem
Papier abspielt, wenn die Glieder der Reihe 1, 4, 9, 16 (als Übersetzung von 1, 2, 3, 4) durch
die Gleichungen 1 × 1 = 1, 2 × 2 = 4, 3 × 3 = 9, etc. ausgerechnet erscheinen.
Man könnte dann vom Standpunkt des Behaviourism sagen: Wenn
ein Mensch das hinschreibt, dann hat er die untere Reihe durch
Rechnung gewonnen, schreibt er aber bloß die untere Reihe19
an, dann
nicht.
Schriebe er aber nun:
so würden wir sagen, er hat falsch gerechnet, weil 2 × 2 nicht 5 ist, etc.
Man könnte natürlich ebensogut schreiben x 1 2 3 4
x2
1 4 9 16
und diese Darstellung ist ganz gleichwertig mit der ersten, oder überhaupt jeder andern, wenn
eine Regel festgesetzt ist, die sie von einer anderen Darstellung unterscheidet.
20
Das Gefühl, welches man bei jeder solchen Darstellung hat, daß sie roh (unbeholfen)
ist, leitet irre, denn wir sind versucht, nach einer ”besseren“ Darstellung zu suchen. Die
gibt es aber gar nicht. Eine ist so gut wie die andere, solange die Multiplizität die richtige
ist; d.h., solange jedem Unterschied im Dargestellten ein Unterschied in der Darstellung
entspricht.
21
Und nun kann aber auch der Gedanke als psychischer Prozeß nicht mehr tun, als dieses
”rohe“ Zeichen.
22
Man kann nicht fragen: Welcher Art sind die geistigen Vorgänge, daß sie wahr und falsch
sein können, was die außergeistigen nicht können. Denn, wenn es die ”geistigen“ können,
so müssen’s auch die anderen können; und umgekehrt.
12 (M): 3
13 (O): anderem
14 (M): 3
15 (V): für uns
16 (M): 3
17 (O): vom
18 (V): nehmen
19 (O): Rechnung
20 (M): 3
21 (M): 3
22 (M): 3
285
286
222 Der Vorgang der absichtlichen Abbildung . . .
1 2 3 4
× × × ×
1 2 3 4
|| || || ||
1 4 9 16
1 2 3 4
× × × ×
1 2 3 4
|| || || ||
1 5 9 20
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10
Can something be left open in a sense that is essentially different from leaving parentheses
empty?
11
It can never be essential to our investigation12
that a phenomenon of symbolizing takes place
in the soul and not on paper, visible to others.
13
One can say that it cannot be decided by external observation whether I am reading or
only uttering sounds when a text is before my eyes. But reading cannot in essence be an
inner matter. Deriving a translation for signs, if it is a process at all, must also be possible
as a visible process. So one must also be able to view14
the process, for example, that occurs
on paper when the members of the progression 1, 4, 9, 16 (as a translation of 1, 2, 3, 4)
appear to be calculated from the equations 1 × 1 = 1, 2 × 2 = 4, 3 × 3 = 9, etc., as such a
derivation.
So, from the point of view of behaviourism, one could say: If someone
writes this down then he arrived at the lowest row through calculation,
but if he writes down only the lowest row15
then he didn’t.
But if he were to write:
then we would say that he had calculated incorrectly, because 2 × 2 isn’t 5, etc.
Of course, one could just as well write x 1 2 3 4
x2
1 4 9 16
and this representation is exactly equivalent to the first, or in general to any other, if a rule
has been established that distinguishes it from that other representation.
16
The feeling that one has with any such representation – that it is rough (awkward) – is
misleading, for we are tempted to seek a “better” representation. But there is no such thing.
One is as good as the next, so long as the multiplicity is correct; that is, so long as there’s a
difference in representation corresponding to every difference in what is represented.
17
And neither can thought as a psychological process do more than this “rough” sign.
18
One cannot ask: What is it about mental processes that allows them to be true and false
– something extra-mental ones cannot be? For if the “mental” ones can be true and false,
then the others must have that capacity as well; and vice versa.
10 (M): 3
11 (M): 3
12 (V): to us
13 (M): 3
14 (V): take
15 (O): calculation
16 (M): 3
17 (M): 3
18 (M): 3
The Process of Copying on Purpose . . . 222e
1 2 3 4
× × × ×
1 2 3 4
|| || || ||
1 4 9 16
1 2 3 4
× × × ×
1 2 3 4
|| || || ||
1 5 9 20
WBTC060-68 30/05/2005 16:16 Page 445
Denn, können es die geistigen23
Vorgänge, so muß es auch ihre Beschreibung können. Denn
in ihrer Beschreibung muß es sich zeigen, wie es möglich ist.
24
Wenn man sagt, der Gedanke sei eine seelische Tätigkeit, oder eine Tätigkeit des Geistes,
so denkt man an den Geist als an ein trübes, gasförmiges Wesen, in dem manches geschehen
kann, das außerhalb dieser Sphäre nicht geschehen kann. Und von dem man manches erwarten
muß,25
das sonst nicht möglich ist.
Als handle26
gleichsam die Lehre vom Gedanken vom organischen Teil, im Gegensatz zum
anorganischen des Zeichens.
Es wäre27
gleichsam der Gedanke der organische Teil des Symbols, das Zeichen der
anorganische. Und jener organische Teil kann Dinge leisten, die der anorganische nicht
könnte.
Als geschähe hinter dem Ausdruck noch etwas Wesentliches, was sich nicht durch den
Ausdruck ersetzen läßt28
– worauf29
sich etwa nur hinweisen läßt – was in dieser Wolke (dem
Geist) geschieht und den Gedanken erst zum Gedanken macht. Wir denken hier an einen
Vorgang analog dem Vorgang der Verdauung und die Idee ist, daß im Inneren des Körpers
andere chemische Veränderungen vor sich gehen, als wir sie außen produzieren können, daß
der organische Teil der Verdauung einen anderen Chemismus hat, als, was wir außen mit
den Nahrungsmitteln vornehmen könnten.
Abbilden ist kein metalogischer Begriff.
30
Das heißt, das Abbilden kann sich von einem andern Vorgang auch nur so unterscheiden,
wie eben ein Vorgang vom andern und das heißt, daß dieser Unterschied kein metalogischer
Vorgang ist.31
So wie ich früher einmal gesagt habe: Die Intention kann auch nur ein Phänomen wie
jedes andere sein, wenn ich überhaupt von ihr reden darf.
32
Das Wählen der Striche beim Abbilden einer Vorlage ist also allerdings ein anderer
Vorgang, als etwa das bloße Zeichnen dieser Striche, wenn ich mich ”nicht nach der Vorlage
richte“, aber der Unterschied ist ein äußerer, beschreibbarer, wie der Unterschied zwischen
den Zeichengruppen
2, 4, 6, 8 x, 2, 4, 6, 8
4, 16, 36, 64 und x2
4, 16, 36, 64
und steht mit diesem Unterschied auf einer Stufe.33
34
Und so steht es also auch mit dem Wählen der Worte, wenn ich etwas mit Worten
beschreibe: dieser Vorgang unterscheidet sich von dem, des willkürlichen Zuordnens
von Worten, aber eben nur (äußerlich), wie sich die beiden Zeichen im vorigen Satze
unterscheiden.
23 (V): seelischen
24 (M): 3
25 (V): kann
26 (V): Es handelt
27 (V): ist
28 (V): was sich nicht ausdrücken läßt
29 (V): auf das
30 (M): 3
31 (V): Unterschied nicht logische Bedeutung
haben kann.
32 (M): 3
33 (V): auf gleicher Stufe.
34 (M): 3
285v
287
223 Der Vorgang der absichtlichen Abbildung . . .
WBTC060-68 30/05/2005 16:16 Page 446
For if mental19
processes can be true and false then of necessity a description of them can
be as well. For it has to show in their description how this is possible.
20
If one says that thought is a psychological activity, or an activity of the mind, then one
thinks of the mind as a cloudy, gaseous entity in which a lot of things can happen that
can’t happen outside this sphere. And of which one must21
expect a lot of things that are
not possible otherwise.
As if the study of thought were22
so to speak about the organic – as opposed to the
inorganic – part of a sign.
The thought would be23
the organic part of a symbol, so to speak, the sign the inorganic.
And that organic part can accomplish things that the inorganic one can’t.
As if behind the expression there were still something essential going on, which can’t
be replaced by the expression24
– something that can, for example, only be pointed to –
something which happens in this cloud (the mind) and is what makes a thought a thought.
Here we’re thinking of a process analogous to that of digestion, and the idea is that within
the body chemical changes take place that are different from those we can produce on the
outside, that the organic part of digestion has a different chemistry from what we can do
with food on the outside.
Depiction is not a metalogical concept.
25
This means that depiction too can differ from another process only as one process does
from another, and that means that this difference is no metalogical process.26
Just as I once said: Intention too can only be a phenomenon like any other, when I’m
allowed to talk about it at all.
27
So choosing the lines when portraying a model is a different process, to be sure, from
simply drawing these lines when I am “not being guided by the model”, but this difference
is an external, describable one, like the difference between the groups of signs
2, 4, 6, 8 x, 2, 4, 6, 8
4, 16, 36, 64 and x2
4, 16, 36, 64
and it is on a level with this28
difference.
29
And that’s also the way it is with selecting words with which to describe something: this
process differs from that of arbitrarily assigning words, but only (externally), as the two signs
in the previous sentence differ from each other.
19 (V): psychological
20 (M): 3
21 (V): can
22 (V): The study of thought is
23 (V): thought is
24 (V): which can’t be expressed
25 (M): 3
26 (V): difference cannot have a logical meaning.
27 (M): 3
28 (V): on the same level as this
29 (M): 3
The Process of Copying on Purpose . . . 223e
WBTC060-68 30/05/2005 16:16 Page 447
35
”Wenn man einen Hund gelehrt hätte, den Zeichenverbindungen von a, b, c, d zu folgen
(wobei a = → , b = ↓ , c = ← , d = ↑), so mag er das mechanisch tun, aber, wenn ich
nun wissen will, welches Zeichen ich ihm geben muß, um ihn einen bestimmten Linienzug
36
laufen zu lassen, so muß ich das Zeichen von dem Linienzug nach der Regel
ableiten.“
35 (M): 3 /// 36 (F): MS 109, S.
224 Der Vorgang der absichtlichen Abbildung . . .
WBTC060-68 30/05/2005 16:16 Page 448
30
“If one had taught a dog to follow this assignment of signs: a, b, c, d (where a = → ,
b = ↓ , c = ← , d = ↑), then it may well do that mechanically, but if I now want to
know which sign I have to give it in order to get it to run a particular course,31
then I have to use the rule to derive the sign from the course.”
30 (M): 3 /// 31 (F): MS 109, p. 290.
The Process of Copying on Purpose . . . 224e
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65
Wie1
hängen unsre Gedanken mit
den Gegenständen zusammen, über
die wir denken? Wie treten diese
Gegenstände in unsre Gedanken ein?2
(Sind sie in ihnen durch etwas
andres – etwa ähnliches – vertreten?)
Wesen des Porträts; die Intention.
Die Vorstellung von ihm ist ein ungemaltes Portrait.
3
”Das soll4
er sein“ (dieses Bild stellt ihn vor) darin liegt das ganze Problem der
Darstellung.
Kann man sagen: &Mein Erinnerungsbild stellt ihn vor“?5
Kann man nicht statt6
&ich stelle mir ihn
vor“ sagen: ich habe ein Vorstellungsbild, welches ihn darstellt7
“?
Worin besteht es, daß ich mir ihn vorstelle? Daß mein Vorstellungsbild ihm ähnlich ist? Und
wie wenn es einem andern Menschen zufälligerweise noch ähnlicher wäre, – stellte ich mir dann den
andern vor?
Nun, die Vorstellung als Bild ist ihm nur ähnlich oder unähnlich.8
9
Wenn ich sage ”der Sinn eines Satzes ist dadurch bestimmt, wie er zu verifizieren ist“,
was muß ich dann von dem Sinn des Satzes sagen: daß dieses Bild das Porträt10
jenes
Gegenstandes sein soll? Wie ist das denn zu verifizieren?
Was heißt es: Ich kann mir vorstellen, daß der Fleck A sich an den Ort B
bewegt? 11
Die seltsame Täuschung, der man unterliegt, daß im Satze die
Gegenstände das tun, was der Satz sagt, muß sich aufhellen.
1 (O): Wir
2 (O): Gedanken ein.
3 (M): 3
4 (V): sol
5 (V): vor?“ oder: (ich habe ein Vorstellun
6 (V): man statt
7 (V): vorstellt
8 (V): als Bild kann ihm nur ähnlich oder unähnlich
sein.
9 (M): 3
10 (V): daß dieser Satz die Übersetzung
11 (F): MS 108, S. 192.
288
287v
288
287v
288
A B
225
WBTC060-68 30/05/2005 16:16 Page 450
65
How are our Thoughts Connected
with the Objects we Think about?
How do these Objects Enter our
Thoughts? (Are they Represented in
our Thoughts by Something Else –
Perhaps Something Similar?)
The Nature of a Portrait; Intention.
The mental image of him is an unpainted portrait.
1
“That is supposed to be he” (this picture portrays him); therein lies the entire problem
of portrayal.
Can one say: “My memory-image represents him”?2
Can’t one3
say: “I have a mental image that
represents4
him” instead of “I imagine him”?
In what does it consist that I imagine him? That my mental image is similar to him? And what if
by chance it were even more similar to someone else – would I then be imagining this other person?
Well, as an image the mental image is5
only similar or dissimilar to him.
6
If I say “The sense of a proposition is determined by how it is to be verified”, then what
must I say about the sense of the proposition: This image is supposed to be a portrait of that7
object? How is that to be verified?
What does this mean: I can imagine that spot A moves to location B?8
The
strange deception one is subject to – that in a proposition the objects do what
the proposition says – must be cleared up.
1 (M): 3
2 (V): him?” or: AI have a mental
3 (V): Can one
4 (V): signifies
5 (V): image can be
6 (M): 3
7 (V): This proposition is supposed to be a translation
of that
8 (F): MS 108, p. 192.
A B
225e
WBTC060-68 30/05/2005 16:16 Page 451
Es ist, als ob im Befehl bereits ein Schatten der Ausführung läge. Aber ein Schatten
eben dieser Ausführung. Du gehst im Befehl dort und dort hin. – Sonst wäre es aber eben
ein anderer Befehl. Gewiß die12
Identität ist die die der Differenz13
zweier verschiedener Befehle
entspricht.
14
”Der Satz ist ein Bild.“ Ein Bild wovon? Kann man sagen: ”von der Tatsache, die ihn
wahr macht, wenn er wahr ist und von der Tatsache, die ihn falsch macht, wenn er falsch
ist. Im ersten Fall ist er ein korrektes Bild, im zweiten ein unkorrektes“? ((Wenn ich bei
einem gemalten Bild frage: ”wovon ist das ein Bild“; was ist die Art der Antwort?)) Die
Antwort kann offenbar verschiedener Art sein: Sie ist anders für ein Porträt als Porträt & anders15
für ein Genrebild.
16
Wenn man mit Bild meint: die richtige, oder falsche Darstellung der Realität, dann muß
man wissen, welcher Realität, oder welches Teils der Realität. Ich kann dieses Zimmer richtig
oder falsch darstellen, aber um herauszufinden17
, ob richtig oder nicht, muß ich wissen, daß
dieses Zimmer gemeint ist.
18
Was heißt es: sich eine Vorstellung machen, die der Wirklichkeit nicht entspricht?
19
Man vergleiche das Vorstellen mit dem Malen eines Bildes. Er malt also ein Bild des
Menschen, wie dieser in Wirklichkeit nicht ist.
Sehr einfach. Aber warum nennen wir es das Bild dieses Menschen? Denn, wenn es das
nicht ist, ist es (ja) nicht falsch. – Wir nennen es so, weil er selbst es drübergeschrieben hat.
Also hat er nichts weiter getan, als jenes Bild zu malen, und jenen Namen drüberzuschreiben.
Und das tat er wohl auch in der Vorstellung.
20
Es muß uns klar sein, daß der Zusammenhang unseres Gedankens mit Napoleon nur
durch diesen selbst und durch kein Bild (Vorstellung, etc.) und sei es noch so ähnlich, gemacht
werden kann. Anderseits aber ist Napoleon für uns in seiner Abwesenheit nicht weniger
enthalten, als in seiner Anwesenheit.
21
”Der Plan besteht darin, daß ich mich das und das tun sehe.“ Aber wie weiß ich, daß
ich es bin. – Nun, ich bin es ja nicht, was ich sehe, sondern etwa ein Bild. Warum aber nenne
ich es mein Bild? Nicht etwa, weil es mir ähnlich sieht.
”Woher weiß ich, daß ich es bin“: Das ist ein gutes Beispiel einer falsch angebrachten
Frage. Die Frage hat nämlich Sinn, wenn es etwa heißt: Woher weiß ich, daß ich es bin,
den ich da im Spiegel sehe. Und die Antwort gibt dann Merkmale, nach denen ich zu
erkennen bin. –
22
Die Frage ”woher weiß ich, daß ich das bin“ oder richtiger ” . . . daß das mich vertritt“
ist Unsinn, denn, daß es mich vertritt, ist meine (eigene) Bestimmung. Ja, ich könnte
ebensogut fragen: ”woher weiß ich, daß das Wort ,ich‘ mich vertritt“, denn meine Figur im
Bild war nur ein anderes Wort ”ich“.
12 (V): diese
13 (V): Differenz entspricht
14 (M): 555 (R): [Zu § 21 S. 83]
15 (V): für ein Portrait & anders
16 (M): 555 (R): [Zu § 21 S. 83]
17 (O): heraus zu finden
18 (M): 3
19 (M): 3
20 (M): 3
21 (M): 3
22 (M): 3
289
288v
289
289
288v
290
226 Wie hängen unsre Gedanken mit den Gegenständen zusammen . . .
WBTC060-68 30/05/2005 16:16 Page 452
It is as if in the command there were already a shadow of its execution. But a shadow of
this execution. It is you who goes to this or that place in the command. – Otherwise it would
just be a different command. Certainly the9
identity is one that corresponds to the difference between
two different commands.
10
“A proposition is a picture”. A picture of what? Can one say: “Of the fact that makes
it true if it is true, and of the fact that makes it false if it is false. In the first case it is a
correct picture, in the second case an incorrect one”? ( (If in the case of a painted picture I
ask: “What is this a picture of?”, what kind of an answer do I get?) ) Obviously, the answer
can be varied: It is one thing for a portrait qua portrait and11
another for a genre painting.
12
If one means by a picture: the correct or false representation of reality, then one has
to know of what reality or of what part of reality. I can represent this room correctly or
incorrectly, but in order to find out whether my portrayal is correct or not, I have to know
that it is this room that is meant.
13
What does it mean: to imagine something that doesn’t correspond to reality?
14
Compare imagining with painting a picture. So he is painting a picture of a person that
is different from the way this person is in reality.
Very simple. But why do we call it the picture of this person? For if that isn’t what it is,
it (certainly) isn’t inaccurate. – We call it that because he himself wrote that over it.
So he didn’t do anything more than paint that picture and write that name over it. And
he probably also did that in his imagination.
15
It has to be clear to us that the connection of our thought with Napoleon can only be
made through Napoleon himself, and not through an image (mental image, etc.), be it ever
so similar. But on the other hand, Napoleon is embodied for us in his absence no less than
in his presence.
16
“Having a plan consists in my seeing myself doing this or that.” But how do I know that
it is I? – Well it isn’t I whom I’m seeing, but for instance a picture. But why do I call it my
picture? Not, say, because it looks like me.
“How do I know that it is I?”: That’s a good example of a misplaced question. For
the question makes sense if it goes something like: “How do I know that it is I that I’m
seeing here in the mirror?” And the answer then provides some features by which I can be
recognized. –
17
The question “How do I know that this is I?”, or more correctly, “. . . that this
represents me” is nonsense, because that it represents me is my (own) decision. Indeed, I
might just as well ask: “How do I know that the word ‘I’ represents me?”, for my shape in
the picture was merely a different word for “I”.
9 (V): this
10 (M): 555 (R): [To § 21 p. 83]
11 (V): for a portrait and
12 (M): 555 (R): [To § 21 p. 83]
13 (M): 3
14 (M): 3
15 (M): 3
16 (M): 3
17 (M): 3
How are our Thoughts Connected . . . 226e
WBTC060-68 30/05/2005 16:16 Page 453
23
Wohl aber könnte man fragen ”was hat denn24
der Name ,a‘ mit diesem Menschen zu
tun“. Und die Antwort wäre: Nun, er heißt a.25
26
”Diese Figur des Bildes bin ich“ ist ein Übereinkommen.
27
Ja, aber worin kommen wir überein? Welche Beziehung zwischen Zeichen und mir stellen
wir her? Nun, nur die, die etwa durch das Zeigen mit der Hand oder das Umhängen eines
Täfelchens besteht. Denn diese Relation ist nur durch das System bedeutungsvoll, dem sie
angehört.
28
Wenn man sagt: Ich stelle mir die Sonne vor, wie sie über den Himmel zieht; so ist doch
nicht die Vorstellung damit beschrieben, daß ”die Sonne über den Himmel zieht“! Nun
könnte ich einerseits fragen: ist nicht, was Du vor Dir siehst, eine gelbe Scheibe in Bewegung?
aber doch nicht gerade die Sonne. – Andrerseits, wenn ich sage ”ich stelle mir die Sonne in
dieser Bewegung vor“, so ist das nicht dasselbe, wie wenn ich (etwa kinematographisch) ein
solches Bild zu sehen bekäme.
Ja, es hätte Sinn, von diesem Bild zu fragen: ”stellt das die Sonne vor?“
29
Das Porträt ist nur ein dem N ähnliches Bild (oder auch das nicht), es hat aber nichts
in sich (wenn30
noch so ähnlich), was es zum Bildnis dieses Menschen, d.h. zum beabsichtigten
Bildnis machen würde. (Ja, das Bild, was dem Einen täuschend ähnlich ist, kann in
Wirklichkeit das schlechte Porträt eines Anderen sein.)
31
Nun kann man doch fragen: ”Wie zeigt sich denn das, daß er das Bild als Porträt des
N meint?“ – ”Nun, indem er’s sagt“ – ”Aber wie zeigt es sich denn, daß er das mit dem
meint, was er sagt?“ – ”Gar nicht!“ ( (Worauf bezieht sich denn dieses ”das“. Man kann
fragen: Wie zeigt sich, daß er meint, was er sagt. Antwort z.B. an seinem Gesicht.) )
32
”Ich war der Meinung,33
Napoleon sei 1805 gekrönt worden.“ – ”Warst Du die ganze
Zeit ununterbrochen dieser Meinung?“
34
Was hat aber Deine Meinung35
mit Napoleon zu tun? Welcher Zusammenhang36
besteht
zwischen Deiner Meinung37
und Napoleon?
Es kann, z.B., der sein, daß das Wort ”Napoleon“ in dem Ausdruck meiner Meinung
vorkommt, plus dem Zusammenhang, den dieses Wort mit seinem Träger hat. Also etwa,
daß er sich so unterschrieben hat, so angeredet wurde, etc. etc.
”Aber mit dem Wort ,Napoleon‘ bezeichnest Du doch, während Du es aussprichst, eben
diesen Menschen“. – ”Wie geht denn, Deiner Meinung nach, dieser Akt des Bezeichnens
vor sich? Momentan? oder braucht er Zeit?“ – ”Ja aber, wenn man Dich fragt ,hast Du jetzt
(eben) den Mann gemeint, der die Schlacht bei Austerlitz gewonnen hat?‘ wirst Du doch
sagen ,ja‘. Also hast Du diesen Mann gemeint, als Du den Satz, in dem sein Name vorkommt,
aussprachst!“ – Wohl, aber nur etwa in dem Sinne, in welchem ich damals auch wußte, daß
2 + 2 = 4 sei.38
Nämlich nicht so, als ob zu dieser Zeit ein besonderer Vorgang stattgefunden
23 (M): 3
24 (V): ”was denn
25 (V): Nun, das ist a.
26 (M): 3
27 (M): 3
28 (M): 3
29 (M): 3
30 (V): wenn auch
31 (M): 3
32 (M): 3
33 (V): Ich dachte,
34 (M): 3
35 (V): aber Dein Gedanke
36 (V): Welche Verbindung
37 (V): zwischen Deinem Gedanken
38 (V): ist.
291
292
227 Wie hängen unsre Gedanken mit den Gegenständen zusammen . . .
WBTC060-68 30/05/2005 16:16 Page 454
18
But one could ask “What does the name ‘a’ have to do with this person?”. And the answer
would be: Well, a is his name.19
20
“This figure in the picture is I” is an agreement.
21
Fine, but about what are we agreeing? What relation are we establishing between signs
and myself? Well, nothing other than the one that exists, say, by pointing with one’s hand or
attaching a label. For this relation is only meaningful because of the system to which it belongs.
22
If one says: “I imagine the sun as it moves across the sky”, then that surely doesn’t describe
the mental image that “the sun is moving across the sky”! Now on the one hand I could ask:
“Isn’t what you’re seeing in front of you a yellow disc in motion? But surely not the sun, of
all things”. – On the other hand, if I say “I’m imagining the sun moving like this”, then this
isn’t the same thing as if I got to see such an image (say, in a film).
Indeed, it would make sense to ask about this image: “Does that represent the sun?”
23
The portrait is only a picture similar to N (or not even that). But it contains nothing
(no matter how similar) that would make it a portrait of this person, i.e. the intended
portrait. (Indeed, the picture that looks virtually identical to one person can actually be
a bad portrait of someone else.)
24
But now one can ask: “How does it show that he means the picture to be a portrait of
N?” – “Well, by his saying this.” – “But how does it show that he means that by what he
says?” – “It doesn’t show at all!” ( (What does this “that” refer to? One can ask: How does
it show that he means what he is saying? Answer, for instance: By his face.) )
25
“I was of the opinion that26
Napoleon was crowned in 1805”. – “Were you of that opinion
the entire time, without interruption?”
27
But what does your opinion28
have to do with Napoleon? What connection is there between
your opinion29
and Napoleon?
It may, for example, consist in the fact that the word “Napoleon” occurs in the expression
of my opinion, plus the connection that word has with its bearer. So for instance that
he signed his name that way, was addressed that way, etc., etc.
“But with the word ‘Napoleon’, as you utter it, you are designating this very person”. –
“How does this act of designation take place, in your view? Instantaneously? Or does it take
time?” – “Yes. But if you’re asked ‘Did you just now mean the (very) man who won the
battle of Austerlitz?’ you’ll surely say ‘Yes’. So you meant this man when you uttered the sentence
in which his name appeared!” – True, but only in more or less the same sense in which
I also knew at that time that 2 + 2 = 4. That is, it’s not as if at that time a special process
had taken place that we could call “meaning”; even if certain images perhaps accompanied
the utterance of the sentence, images that are characteristic of this intention and might have
18 (M): 3
19 (V): Well, that is a.
20 (M): 3
21 (M): 3
22 (M): 3
23 (M): 3
24 (M): 3
25 (M): 3
26 (V): I thought that
27 (M): 3
28 (V): your thought
29 (V): thought
How are our Thoughts Connected . . . 227e
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hätte, den wir dieses ”Meinen“ nennen könnten; auch wenn vielleicht gewisse Bilder das
Aussprechen begleitet haben, die für diese Meinung charakteristisch sind und bei andrer
Bedeutung des Wortes ”Napoleon“ vielleicht andre gewesen wären. Vielmehr ist die
Antwort ”ja, ich habe den Sieger von Austerlitz gemeint“ ein weiterer Schritt im Kalkül.
Täuschend ist an ihr39
die vergangene Form, die eine Beschreibung dessen zu geben scheint,
was ”in mir“ während des Aussprechens des Satzes vorgegangen war. In Wirklichkeit knüpft
das Präteritum nur an den früher ausgesprochenen Satz an.
40
”Aber ich habe ihn gemeint“. Sonderbarer Vorgang, dieses Meinen! Kann man hier in
Europa jemanden meinen, auch wenn er in Amerika41
ist? Oder42
gar, wenn er schon tot ist?
43
Meine ganzen Überlegungen gehen immer dahin, zu zeigen, daß es nichts nützt, sich
das Denken als ein Halluzinieren44
vorzustellen. D.h., daß es überflüssig ist, die Schwierigkeit
aber bestehen bleibt. Denn auch die Halluzination45
, kein Bild, kann die Kluft zwischen dem
Bild und der Wirklichkeit überbrücken, und das eine nicht eher als das andere.
39 (V): ihm
40 (M): 3
41 (V): Amerika und man in Europa
42 (V): Und
43 (M): 3
44 (O): Haluzinieren
45 (O): Haluzination
228 Wie hängen unsre Gedanken mit den Gegenständen zusammen . . .
WBTC060-68 30/05/2005 16:16 Page 456
been different, given a different meaning of the word “Napoleon”. Rather, the answer
“Yes, I meant the victor at Austerlitz” is a further step in the calculus. What is deceptive
about this answer is the past tense, which seems to be giving a description of what went on
“inside me” while I was uttering the sentence. Actually, the preterite tense only picks up
the previously uttered sentence.
30
“But I meant him.” What a strange process, this meaning! Can we here in Europe mean
someone even if he is in America?31
Or32
what is more, if he is dead?
33
All my reflections are always directed towards showing that it doesn’t do any good to
conceive of thinking as hallucinating. In other words, that it is superfluous, and leaves the
problem unchanged. For no image, not even a hallucination, can bridge the gap between
image and reality, and no one image is better at this than another.
30 (M): 3
31 (V): Can you mean someone even if he is in
America and you are in Europe?
32 (V): And
33 (M): 3
How are our Thoughts Connected . . . 228e
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Logischer Schluß.293
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Logical Inference.
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66
Wissen wir, daß p aus q folgt,
weil wir die Sätze verstehen?
Geht das Folgen aus einem
Sinn hervor?
p · q = p heißt ”q folgt aus p“.
(∃x).fx 1 fa = (∃x).fx, (∃x).fx & fa = fa. Wie weiß ich das? (denn das Obere habe ich
sozusagen bewiesen). Man möchte etwa sagen: ”ich verstehe ,(∃x).fx‘ eben“. (Ein herrliches
Beispiel dessen, was ”verstehen“ heißt.)
Ich könnte aber ebensogut fragen ”wie weiß ich, daß (∃x).fx aus fa folgt“ und antworten:
”weil ich ,(∃x).fx‘ verstehe“. Wie weiß ich aber wirklich, daß es folgt? – Weil ich so kalkuliere.
Wie weiß ich, daß (∃x).fx aus fa folgt? Sehe ich quasi hinter das Zeichen ”(∃x).fx“, und
sehe den Sinn, der hinter ihm steht und aus ihm,1
daß er aus fa folgt? ist das das Verstehen?
Nein, jene Gleichung drückt einen Teil des Verständnisses2
aus (das so ausgebreitet vor
mir liegt).
Vergleiche die3
Auffassung des Verstehens, das ursprünglich ein Erfassen mit einem Schlag,4
erst so ausgebreitet werden kann.
Wenn ich sage ”ich weiß, daß (∃x).fx folgt, weil ich es verstehe“, so hieße das, daß ich,
es verstehend, etwas Anderes sehe, als das gegebene Zeichen, gleichsam eine Definition des
Zeichens, aus der das Folgen hervorgeht.
Wird nicht vielmehr die Abhängigkeit durch die Gleichung hergestellt und festgesetzt?
Denn eine verborgene Abhängigkeit gibt es eben nicht.
5
Aber, meinte ich, muß also nicht (∃x).fx eine Wahrheitsfunktion von fa sein,
damit das möglich ist? Damit diese Abhängigkeit möglich ist?
Ja sagt denn eben (∃x).fx 1 fa = (∃x).fx nicht, daß fa schon in (∃x).fx
enthalten ist? Zeigt es nicht die Abhängigkeit des fa vom (∃x).fx? Nein, außer,
wenn (∃x).fx als logische Summe definiert ist (mit einem Summanden fa). –
Ist das der Fall, so ist (∃x).fx (nichts als) eine Abkürzung.
1 (V): und daraus,
2 (V): Verstehens
3 (V): Denke an die
4 (V): ursprünglich mit einem Schlag erfaßbar,
5 (F): MS 111, S. 63.
294
295
(∃x).fx
W
W
F
F
fa
W
F
W
F
230
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66
Do we Know that p Follows
from q because we Understand
the Propositions?
Is Entailment Implied by a Sense?
p · q = p means “q follows from p”.
(∃x).(fx 1 fa) = (∃x).fx, (∃x).(fx & fa) = fa. How do I know that? (Because for the
equation above I have given a kind of proof). One is inclined to say something like “I just
understand ‘(∃x).fx’ ”. (A splendid example of what “understand” means.)
But I could just as well ask “How do I know that (∃x).fx follows from fa?” and answer:
“Because I understand ‘(∃x).fx’”. But how do I really know that it follows? – Because that
is the way I calculate.
How do I know that (∃x).fx follows from fa? Do I look behind the sign “(∃x).fx”, as it
were, and see the sense lying behind it, and see from that that (∃x).fx follows from fa? Is
that what understanding is?
No, that equation expresses a part of the understanding (that thereby lies spread out in
front of me).
Compare1
this conception of understanding with the one according to which it is originally
a grasping all at once,2
and only later can be spread out in this way.
If I say “I know that (∃x).fx follows because I understand it”, that would mean that when
I understand it, I see something other than the sign I’m given, a kind of definition of the
sign, from which the entailment results.
Isn’t it rather that the relation of dependence is created and defined by the equation? For
there just is no such thing as a hidden relation of dependence.
3
But, I used to think, doesn’t (∃x).fx have to be a truth-function of fa for
this to be possible? For this relation of dependence to be possible?
Indeed, isn’t it precisely (∃x).(fx 1 fa) = (∃x).fx that says that fa is already
contained in (∃x).fx? Doesn’t that show the dependence of fa on (∃x).fx?
No, unless (∃x).fx is defined as a logical sum (with fa as one of its terms). – If
that’s the case, then (∃x).fx is (nothing but) an abbreviation.
1 (V): Think of
2 (V): was graspable all at once,
3 (F): MS 111, p. 63.
(∃x).fx
T
T
F
F
fa
T
F
T
F
230e
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Einen verborgenen Zusammenhang gibt es in der Logik nicht.
Hinter die Regeln kann man nicht dringen, weil es kein Dahinter gibt.
fE & fa = fa Kann man sagen: das ist nur möglich, wenn fE aus fa folgt; oder muß man
sagen: das bestimmt, daß fE aus fa folgen soll.6
Wenn das erste, so muß es vermöge der Struktur folgen, etwa indem fE durch eine Definition
so bestimmt ist, daß es die entsprechende Struktur hat. Aber kann denn wirklich das folgen,
gleichsam aus der sichtbaren Struktur der Zeichen hervorgehen, wie ein physikalisches
Verhalten aus einer physikalischen Eigenschaft, und braucht etwa nicht vielmehr immer solche
Bestimmungen, wie die Gleichung fE & fa = fa? Ist es etwa dem7
p 1 q anzusehen, daß es aus
p folgt, oder auch nur den Regeln, welche Russell für die Wahrheitsfunktionen gibt?
Und warum sollte auch die Regel fE & fa = fa aus einer andern Regel hervorgehen und
nicht die primäre Regel sein?
Denn was soll es heißen ”fE muß doch fa in irgendeiner Weise enthalten“? Es enthält es
eben nicht, insofern wir mit fE arbeiten können, ohne fa zu erwähnen. Wohl, aber, insofern
eben die Regel fE & fa = fa gilt.
Die Idee8
ist nämlich, daß fE & fa = fa nur vermöge einer Definition von fE gelten kann.
Und zwar – glaube ich – darum, weil es sonst den falschen Anschein hat, als würde
nachträglich noch eine Bestimmung über fE getroffen, nachdem es schon in die Sprache
eingeführt sei. Es wird aber tatsächlich keine Bestimmung einer künftigen Erfahrung
überlassen.
Und die Definition des fE aus ”allen Einzelfällen“ ist ja ebenso unmöglich, wie die
Aufzählung aller Regeln von der Form fE & fx = fx.
Ja, die Einzelgleichungen fE & fx = fx sind eben gerade ein Ausdruck dieser
Unmöglichkeit.
Wenn man gefragt wird: ist es aber nun auch sicher, daß ein anderer Kalkül als dieser
nicht gebraucht wird, so muß man sagen: Wenn das heißt ”gebrauchen wir nicht in unserer
wirklichen9
Sprache noch andere Kalküle“, so kann ich nur antworten ”ich weiß (jetzt)
keine anderen (so, wie wenn jemand fragte ”sind das alle Kalküle der (gegenwärtigen)
Mathematik“, ich sagen könnte ”ich erinnere mich keiner anderen, aber ich kann etwa noch
genauer nachlesen). Die Frage kann aber nicht heißen ”kann kein anderer Kalkül gebraucht
werden?“ Denn wie sollte die Antwort auf diese Frage gefunden werden?10
Ein Kalkül ist ja da, indem man ihn beschreibt.
Kann man sagen: ”Kalkül“ ist kein mathematischer Begriff?
Wenn ich sagte: ”ob p aus q folgt, muß aus p und q allein hervorgehen11
“; so müßte es
heißen: daß p aus q folgt, ist eine Bestimmung, die den Sinn von p und q bestimmt; nicht
etwas, das, von dem Sinn dieser beiden ausgesagt, wahr ist. Daher kann man (sehr) wohl
die Schlußregeln angeben,12
gibt damit aber Regeln für die Benützung der Schriftzeichen
6 (V): aus fa folgt?
7 (O): den
8 (V): Meinung
9 (V): tatsächlichen
10 (V): Denn wie sollte ich diese Frage
beantworten?
11 (V): allein zu ersehen sein
12 (V): geben,
296
297
231 Wissen wir, daß p aus q folgt . . .
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In logic there is no such thing as a hidden connection.
You can’t get behind the rules, because there isn’t any “behind”.
fE & fa = fa. Can one say: That’s only possible if fE follows from fa? Or does one have to
say: That determines that fE is supposed to follow4
from fa?
If the former, then it has to follow by virtue of the structure, say because fE has been
defined in such a way that it has the appropriate structure. But can that really follow from,
be entailed, as it were, by the visible structure of the signs, like physical behaviour from a
physical property? Doesn’t it rather always need stipulations, such as the equation fE & fa =
fa? Can one perhaps tell by looking at p 1 q that it follows from p, or only by looking at
the rules that Russell gives for truth-functions?
And why should the rule fE & fa = fa be derived from another rule, and not be the primary
rule?
For what is “But fE must somehow contain fa” supposed to mean? It doesn’t contain it,
in so far as we can work with fE without mentioning fa. But it does in so far as the rule
fE & fa = fa is valid.
For the idea5
is that fE & fa = fa can only be valid by virtue of a definition of fE.
And this – I think – is because otherwise the false impression arises that an additional
stipulation about fE was subsequently made, after it had been introduced into language. But
in fact there is no stipulation left for future experience to make.
And the definition of fE based on “all particular cases” is just as impossible as the
enumeration of all rules of the form fE & fx = fx.
Indeed, the individual equations fE & fx = fx are the very expression of this impossibility.
If one is asked: “But is it really certain that no calculus other than this one is being used?”,
then one has to say: If that means “Don’t we use other calculi too in our real6
language?”,
I can only answer “I don’t know of any others (at present)”. (As when someone asks
“Are these all the calculi of (present-day) mathematics?”, I can say, “I don’t remember any
others, but perhaps I can check the literature on this more closely”.) But the question
cannot be “Can no other calculus be used?”. For how should we find the answer to this
question?7
After all, a calculus exists by virtue of its being described.
Can one say: “Calculus” is not a mathematical concept?
If I were to say: “Whether p follows from q must be implied solely by8
p and q”, then
this would have to mean: That p follows from q is a stipulation that determines the sense
of p and q; it is not something that, when stated about the sense of these two, is true. Therefore
one can certainly state the rules of inference, but in so doing one is stating rules for the use
of the written signs that determine their previously undetermined sense; and this means
4 (V): that fE follows
5 (V): the meaning
6 (V): actual
7 (V): how should I answer this question?
8 (V): be visible solely from
Do we Know that p Follows from q . . . 231e
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an, die deren Sinn erst bestimmen; was nichts andres heißt, als daß diese Regeln willkürlich
festzusetzen sind; d.h. nicht von der Wirklichkeit abzulesen, wie eine Beschreibung. Denn,
wenn ich sage, die Regeln sind willkürlich, so meine ich, sie sind nicht von der Wirklichkeit
determiniert, wie die Beschreibung dieser Wirklichkeit. Und das heißt: Es ist Unsinn, von
ihnen zu sagen, sie stimmen mit der Wirklichkeit überein; die Regeln über die Wörter ”blau“,
”rot“, etwa, stimmten mit den Tatsachen, die diese Farben betreffen, überein, etc.
Die Gleichung p & q = p zeigt eigentlich den Zusammenhang des Folgens und der
Wahrheitsfunktionen.
232 Wissen wir, daß p aus q folgt . . .
298
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nothing more than that these rules must be established arbitrarily; that is to say, are not to
be read off reality like a description. For when I say that the rules are arbitrary I mean that
they are not determined by reality, as is the description of this reality. And that means: It
is nonsense to say of them that they correspond to reality; that, say, the rules for the words
“blue” and “red” agree with the facts about those colours, etc.
What the equation p & q = p really shows is the connection between entailment and
truth-functions.
Do we Know that p Follows from q . . . 232e
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67
”Wenn p aus q folgt, so muß p
in q schon mitgedacht sein“.
Bedenke, daß aus dem allgemeinen Satz eine logische Summe von, sagen wir, hundert
Summanden folgen könnte, an die wir doch bestimmt nicht gedacht haben, als wir den
allgemeinen Satz aussprachen. Können wir nicht dennoch sagen, daß sie aus ihm folgt?
”Was aus einem Gedanken folgt, muß in ihm mitgedacht werden. Denn an einem Gedanken
ist nichts, was wir nicht wissen, während wir ihn denken. Er ist keine Maschine, deren
Untersuchung Ungeahntes zu Tage fördern kann, oder eine Maschine, die etwas leisten kann,
was man ihr zuerst nicht ansieht. D.h. er wirkt eben logisch überhaupt nicht als Maschine.
Als Gedanke liegt in ihm nicht mehr, als hineingelegt wurde.1
Als Maschine, d.h. kausal,
wäre ihm alles zuzutrauen; logisch ergibt er nur, was wir mit ihm gemeint haben.“
Wenn ich sage, das Viereck ist ganz weiß, so denke ich nicht an zehn kleinere, in ihm
enthaltene Rechtecke, die weiß sind; und an ”alle“ in ihm enthaltenen2
Rechtecke oder
Flecken, kann ich nicht denken. Ebenso denke ich im Satz ”er ist im Zimmer“ nicht an
hundert mögliche Stellungen, die er einnehmen kann, und gewiß nicht an alle.
”Wo immer Du die Scheibe triffst, hast Du gewonnen. – Du hast sie rechts oben getroffen,
also. . . .“
Auf den ersten Blick scheint es zwei Arten der Deduktion zu geben: in der einen ist in
der Prämisse von allem3
die Rede, wovon die Konklusion handelt, in der andern nicht. Von
der ersten Art ist der Schluß von p & q auf q. Von der anderen der Schluß: der ganze Stab
ist weiß, also ist auch das mittlere Drittel weiß. In dieser Konklusion wird von Grenzen
gesprochen, von denen im ersten Satz nicht die Rede war. (Das ist verdächtig.) Oder wenn
ich sage: ”wo immer in diesem Kreise Du die Scheibe triffst, wirst Du den Preis gewinnen“
und dann ”Du hast sie hier getroffen, also . . .“, so war dieser Ort im ersten Satz nicht
vorausgesehen. Die Scheibe mit dem Einschuß hat zu der Scheibe, wie ich sie früher gesehen
habe, eine bestimmte interne Beziehung und darin besteht es, daß das Loch hier unter die
vorausgesehene allgemeine Möglichkeit fällt. Aber es selbst war nicht vorausgesehen und
es kam4
in dem ersten Bild nicht vor. Oder mußte doch nicht darin vorkommen. Denn
selbst angenommen, ich hätte dabei an tausend bestimmte Möglichkeiten gedacht, so hätte
es zum mindesten geschehen können, daß die ausgelassen wurde, die später eintraf. Und wäre
das Voraussehen dieser Möglichkeit wesentlich gewesen, so hätte die Prämisse durch das
Übersehen dieser einen Möglichkeit den unrechten Sinn bekommen und die Konklusion würde
nun nicht aus ihr folgen.
1 (O): würde.
2 (O): enthaltene
3 (V): von dem
4 (V): und kam
299
300
233
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67
“If p Follows from q, then p Must
Have Been Mentally Included in q.”
Bear in mind that a logical sum of a hundred or so terms could follow from a general proposition,
terms that we certainly weren’t thinking of when we uttered the general proposition.
But can we not say that this sum follows from the proposition anyway?
“What follows from a thought must be mentally included in it. For there is nothing about
a thought that we don’t know while we are thinking it. It isn’t a machine we can investigate
to unearth things we had never dreamed of, or a machine that can do something that at first
glance one doesn’t think it capable of. That is, logically a thought doesn’t work like a machine
at all. Qua thought there is no more in it than was put into it. As a machine, i.e. causally, it
could be thought capable of anything; logically, all it produces is what we meant by it.”
If I say that a rectangle is completely white, I’m not thinking of ten smaller rectangles
contained in it which are white; and I can’t think of “all” the rectangles or patches that
are contained in it. Likewise, in the proposition “He is in the room” I’m not thinking of a
hundred possible locations he can be in, and certainly not of all of them.
“Wherever you hit the target, you’ve won. – You’ve hit it in the upper right hand
section, therefore. . . .”
At first glance there seem to be two kinds of deduction: in one everything1
in the conclusion
is mentioned in the premiss, in the other it isn’t. The inference from p & q to q is
of the first kind. The inference: the whole rod is white, therefore its middle third is also
white, is of the latter kind. In this conclusion boundaries are mentioned that were not mentioned
in the first proposition. (That’s suspicious.) Or if I say: “Wherever in this circle you
hit the target you will win the prize” and then “You’ve hit it here, therefore . . .”, this place
was not foreseen in the first proposition. The target with the hole shot in it has a certain
internal relation to the target as I saw it before the shot, and this relation consists in the
hole’s being included within the general possibility that was foreseen. But the hole itself
wasn’t foreseen, and didn’t occur in the first picture. Or didn’t have to occur in it. For even
assuming that I had been thinking about a thousand specific possibilities when I uttered the
first sentence, it was at least possible to omit the one that later occurred. And if foreseeing
this possibility had been essential, then overlooking this one possibility would have given
the premiss the wrong sense, and then the conclusion wouldn’t follow from it.
1 (V): in one, what is
233e
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Anderseits wird dem Satz ”wohin immer Du in diesem Kreis triffst . . .“ nichts
hinzugefügt, wenn man sagt: ”wohin immer Du in diesem Kreis triffst, und wenn
Du insbesondere den schwarzen Punkt triffst, . . .“. 5
Aber, war der schwarze Punkt
schon da, als man den ersten Satz aussprach, so war er natürlich mitgemeint; war
er aber nicht da, so hat sich durch ihn eben der Sinn des Satzes geändert.
Was soll es aber dann heißen, zu sagen: wenn ein Satz aus dem andern folgt, so muß der
erste im zweiten mitgedacht sein, da es doch nicht nötig ist, im Satz ”ich bin 170 cm hoch“
auch nur einen einzigen6
der aus ihm folgenden negativen Längenangaben mitzudenken.
”Das Kreuz liegt so auf der Geraden: “ – ”Es liegt also zwischen den
Strichen . . .“
”Es hat hier 161
/2°.“ – ”Es hat also jedenfalls mehr als 15°.“
Wenn man sich übrigens wundert, daß ein Satz aus dem andern folgt, obwohl man doch
bei diesem gar nicht an jenen dachte,7
so denke man nur daran, daß p 1 q aus p folgt, und
ich denke doch gewiß nicht alle Sätze p 1 ξ wenn ich p denke.
Die ganze Idee, daß man bei dem Satz, aus dem ein anderer folgt, diesen denken muß,
beruht auf einer falschen, und psychologisierenden, Auffassung. Wir haben uns ja nur um
das zu kümmern, was in den Zeichen und (ihren) Regeln liegt.
Wenn das Kriterium dafür, daß p aus q folgt, darin besteht, daß man ”beim Denken von
q p mitdenkt“, so denkt man wohl beim Denken des Satzes ”in dieser Kiste sind 105
Sandkörner“ die 105
Sätze: ”in dieser Kiste ist ein Sandkorn“, ” . . . 2 Sandkörner“,
etc., etc.? Was ist denn hier das Kriterium des Mitdenkens!
Und wie ist es mit einem Satz: ”ein Fleck (F) liegt
zwischen den Grenzen AA? 8
Folgt aus ihm nicht, daß F
auch zwischen BB und CC liegt, u.s.w.? Folgen hier aus
einem Satz unendlich viele? und ist er also unendlich vielsagend? – Aus dem Satz ”ein Fleck
liegt zwischen den Grenzen AA“ folgt jeder Satz von der Art ”ein Fleck liegt zwischen den
Grenzen BB“, den ich hinschreibe – und so viele, als ich hinschreibe. Wie aus p soviele Sätze
der Form p 1 ξ folgen, als ich hinschreibe (oder ausspreche, etc.). (Der Induktionsbeweis
beweist soviele Sätze von der Form . . . als ich hinschreibe.)
5 (F): MS 109, S. 10.
6 (V): einzigen aus
7 (V): daß dieser Satz aus jenem folgt, obwohl man
doch bei jenem gar nicht an ihn dachte,
8 (F): MS 113, S. 132r.
301
302
234 Wenn p aus q folgt . . .
·
C B A A B CF
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On the other hand, nothing is added to the sentence “Wherever you hit this
circle. . . .” by saying: “Wherever you hit the circle and if, in particular, you hit
the black dot, . . .”. 2
But if the black dot was already there when the first sentence
was uttered, then of course it was meant as well; but if it wasn’t there, then for
that reason the sense of the sentence changed.
But then what is it supposed to mean to say: “If one sentence follows from another then
the first must be mentally included in the second”?, since in the sentence “I am 170 cm tall”
it isn’t necessary to have in mind even one of the negative specifications of height that
follow from it?
“The cross is situated on the straight line like this: ” – “Thus it lies between
the strokes . . .”
“It’s 161
/2° here.” – “So at any rate it’s more than 15°.”
Incidentally, if you are surprised that one proposition follows from another3
even though
you don’t think of the latter when you think of the former, then just consider that p 1 q
follows from p, and I certainly don’t think all propositions of the form p 1 ξ when I’m
thinking p.
The whole idea – that when you think one proposition that entails another, you have to
think the latter – rests on a false, psychologizing conception. All that we need be concerned
with is what is contained in signs and (their) rules.
If the criterion for p following from q consists in “having p in mind when thinking q”,
then when you’re thinking the proposition “There are 105
grains of sand in this box”, are
you really thinking the 105
propositions: “There is one grain of sand in this box”, “. . . 2
grains of sand . . .”, etc., etc.? Then what is the criterion for “having in mind”, anyway!
And how about this proposition: “A spot (F) is situated
between the limits AA”? 4
Doesn’t it follow from this
proposition that F is also between BB and CC, etc.? Do
infinitely many propositions follow from one? And is this one thus infinitely suggestive? –
Every proposition of the kind “A spot is situated between the boundaries BB” that I write
down – and just as many as I write down – follows from the proposition “A spot is situated
between the limits AA”. Just as from p there follow as many propositions of the form
p 1 ξ as I write down (or utter, etc.). (An inductive proof proves as many propositions
of the form . . . as I write down.)
2 (F): MS 109, p. 10.
3 (V): that this proposition follows from that
4 (F): MS 113, p. 132r.
If p Follows from q . . . 234e
·
C B A A B CF
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68
Der Fall: unendlich viele
Sätze folgen aus einem.
Ist es unmöglich, daß aus einem Satz unendlich viele Sätze folgen, – in dem Sinn nämlich,
daß nach einer Regel immer neue Sätze aus dem einen gebildet werden könnten,1
ad infinitum?
Angenommen, die ersten tausend Sätze dieser Reihe schrieben wir in Konjunktion an.
Mußte der Sinn dieses Produktes dem Sinne des ursprünglichen Satzes nicht näherkommen,
als das Produkt der ersten hundert Sätze? Müßte man nicht eine immer bessere
Annäherung an den ersten Satz bekommen, je mehr man das Produkt ausdehnte und würde
das nicht zeigen, daß aus dem Satz nicht unendlich viele andere folgen können, da ich schon
nicht mehr im Stande bin, das Produkt aus 1010
Gliedern zu verstehen und doch den Satz
verstanden habe, dem das Produkt aus 10100
Gliedern noch näher kommt als das von 1010
Gliedern?
Man denkt sich wohl, der allgemeine Satz ist eine abgekürzte Ausdrucksweise des
Produkts. Aber was ist am Produkt abzukürzen, es enthält ja nichts Überflüssiges.
Wenn man ein Beispiel braucht dafür, daß unendlich viele Sätze aus einem folgen, so
wäre vielleicht das Einfachste das, daß aus ”a ist rot“ die Negation aller Sätze folgt, die a
eine andere Farbe zuschreiben. Diese2
negativen Sätze werden gewiß in dem einen nicht
mitgedacht. Man könnte natürlich sagen: wir unterscheiden doch nicht unendlich viele
Farbtöne; aber die Frage ist: hat die Anzahl der Farbtöne, die wir unterscheiden, überhaupt
etwas mit der Komplikation jenes ersten Satzes zu tun; ist er mehr oder weniger komplex,
je nachdem wir mehr oder weniger Farbtöne unterscheiden?
Müßte man nun nicht so sagen: Ein Satz folgt erst aus ihm, wenn er da ist. Erst wenn
wir zehn Sätze gebildet haben, die aus dem ersten folgen, folgen zehn Sätze aus ihm.
Ich möchte sagen, ein Satz folgt erst dann aus dem anderen, wenn er mit ihm konfrontiert
wird. Jenes ”u.s.w. ad infinitum“ bezieht sich nur auf die Möglichkeit der Bildung von
Sätzen, die aus dem ersten folgen, ergibt aber keine Zahl solcher Sätze.
Könnte ich also einfach sagen: Unendlich viele Sätze folgen darum nicht aus einem
Satz, weil es unmöglich ist, unendlich viele Sätze hinzuschreiben (d.h. ein Unsinn ist, das
zu sagen).
1 (V): können, 2 (V): Diese Negatien
303
304
235
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68
The Case of Infinitely
Many Propositions Following
from a Single One.
Is it impossible that infinitely many propositions should follow from a single one – i.e. to
the extent that we could use a rule to produce ever new propositions ad infinitum from
the one?
Let’s assume that we wrote down the first thousand propositions of this progression as a
conjunction. Wouldn’t the sense of this conjunct necessarily approximate the sense of the
original proposition more closely than the conjunct of the first hundred propositions? Wouldn’t
one necessarily arrive at an ever better approximation to the first proposition the more one
expanded the conjunct? And wouldn’t that show that infinitely many other propositions can
not follow from the original one, since I’m incapable of understanding the conjunct made
up of 1010
members, whereas I understood the proposition to which the conjunct made up
of 10100
members is a closer approximation than that made up of 1010
members?
One tends to think that the universal proposition is an abbreviated expression of the
conjunct. But what is there to abbreviate in the conjunct? It doesn’t contain anything
superfluous, after all.
If one needs an example of infinitely many propositions following from one, the simplest
might be that from “a is red” there follows the negation of all propositions that ascribe a
different colour to a. These1
negative propositions certainly are not mentally included in the
one. Of course it might be said: But we don’t differentiate between infinitely many shades
of colour. But the question is: Does the number of shades of colour that we do distinguish
have anything at all to do with the complexity of that first proposition; is it more or less
complex, depending on whether we distinguish more or fewer shades of colour?
So wouldn’t one have to put it like this: A proposition doesn’t follow from it until it exists.
It’s only when we have formed ten propositions that follow from the first one, that ten propositions
follow from it.
I’m inclined to say that one proposition doesn’t follow from another until it is confronted
with it. That “etc. ad infinitum” only refers to the possibility of forming propositions that
follow from the original; but it doesn’t produce a specific number of such propositions.
So could I simply say: The reason infinitely many propositions don’t follow from a single
proposition is that it is impossible to write down infinitely many propositions (i.e. that it is
nonsense to say that)?
1 (V): These negation
235e
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3
Wie verhält es sich nun mit dem Satz: ”die Fläche ist von A bis B weiß“?
Aus ihm folgt doch, daß sie auch von A′ bis B′ weiß ist. Es braucht sich da
nicht um gesehenes Weiß zu handeln; und der Schluß von dem ersten Satz auf
den zweiten wird jedenfalls immer wieder ausgeführt. Es sagt mir Einer ”ich
habe die Fläche von A bis B damit bestrichen“ und ich sage darauf ”also ist sie jedenfalls
von A′ bis B′ gestrichen“.
Man müßte a priori sagen können, daß F(A′B′) aus F(AB) folgen würde.
Sind die Striche A′ und B′ vorhanden, dann folgt allerdings jener zweite Satz aus dem
ersten (dann ist die Zusammengesetztheit schon in dem ersten Satz offenbar vorhanden) dann
folgen aber aus dem ersten Satz nur so viele Sätze, als seiner Zusammengesetztheit
entspricht (also nie unendlich viele).
”Eine ungeteilt gesehene Fläche hat keine Teile.“
Denken wir uns aber einen Maßstab an die Fläche angelegt, sodaß
wir etwa zuerst das Bild , dann das Bild , und
dann 5
vor uns hätten, dann folgt daraus, daß das erste
Band durchaus weiß ist durchaus nicht, daß im zweiten und dritten
alles mit Ausnahme der Teilstriche weiß ist.
”Wo immer, innerhalb dieses Kreises Du die Scheibe triffst, hast
Du gewonnen.“
”Ich denke, Du wirst die Scheibe irgendwo innerhalb dieses Kreises treffen.“
Was den ersten Satz betrifft, könnte man fragen: woher weißt Du das? Hast Du alle
möglichen Orte ausprobiert? Und die Antwort müßte dann lauten: das ist ja kein Satz,
sondern eine allgemeine Festsetzung.
Der Schluß lautet auch nicht so: ”wo immer auf der Scheibe der Schuß hintrifft, hast Du
gewonnen. Du hast auf der Scheibe dahin getroffen, also hast Du den Preis gewonnen“. Denn
wo ist dieses da? wie ist es außer dem Schuß bezeichnet, etwa durch einen Kreis? Und war
der auch schon früher auf der Scheibe? Wenn nicht, so hat die Scheibe sich ja verändert,
wäre er aber schon dort gewesen, dann wäre er als eine Möglichkeit des Treffens vorgesehen
worden. Es muß vielmehr heißen: ”Du hast die Scheibe getroffen, also . . .“.
Der Ort auf der Scheibe muß nicht notwendig durch ein Zeichen, einen Kreis, auf der
Scheibe angegeben sein. Denn es gibt jedenfalls die Beschreibung ”näher dem Mittelpunkt“,
”näher dem Rand“, ”rechts oben“ etc. Wie immer die Scheibe getroffen wird, stets muß
so eine Beschreibung möglich sein. (Aber von diesen Beschreibungen gibt es auch nicht
”unendlich viele“.)
Hat es nun einen Sinn zu sagen: ”aber wenn man die Scheibe trifft, muß man sie irgendwo
treffen“? Oder auch: ”wo immer er die Fläche trifft, wird es keine Überraschung sein, – so
3 (F): MS 109, S. 6.
4 (F): MS 109, S. 11.
5 (F): MS 109, S. 106.
305
306
236 Der Fall . . .
”Das Ganze ist weiß, folglich ist auch ein Teil, der durch eine solche
Grenzlinie charakterisiert ist, weiß.“ 4
”Das Ganze war weiß,
also war auch jener Teil davon weiß, auch wenn
ich ihn damals nicht begrenzt darin wahrgenommen
habe.“
A B
A′ B′
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2
Now what about the proposition: “The surface is white from A to B”?
It does follow from this that it is also white from A′ to B′. Here the white
doesn’t have to be seen; and in any case, the inference from the first proposition
to the second is made over and again. Someone tells me “I painted the
surface from A to B” and in response I say “So it’s painted at least from A′ to B′”.
It ought to be possible to say a priori that F(A′B′) would follow from F(AB).
If the lines A′ and B′ exist, then that second proposition does indeed follow from the first;
(then obviously there’s already a complexity in the first proposition); but then only as many
propositions follow from the first proposition as correspond to its complexity (that is to say,
never infinitely many).
2 (F): MS 109, p. 6.
3 (F): MS 109, p. 11.
4 (F): MS 109, p. 106.
The Case of Infinitely Many Propositions . . . 236e
A B
A′ B′
“The whole is white, therefore a part that is bounded by such a line
is also white.”3
“The whole was white, therefore that part
of it was also white, even if at the time I didn’t perceive it
as bounded within the whole.”
“A surface seen as undivided has no parts.”
But let’s imagine a ruler laid alongside the surface so that, say,
at first we have before us the picture , then the picture
, and then .4
Then in no way does it follow from
the first strip’s being completely white that everything except the
dividing lines is white in the second and third strips.
“No matter where you hit the target within this circle, you’ve won.”
“I think you’ll hit the target somewhere within this circle.”
Concerning the first proposition, one could ask: How do you know this? Did you try out
all possible places? And then the answer would have to be: That isn’t a proposition at all,
but a general stipulation.
And neither does the inference go like this: “Wherever the shot lands on the target, you
have won. You hit the target here, so you have won the prize”. For where is this here? How
is it designated other than by the shot – perhaps by a circle? And was that already on the
target beforehand? If not, then the target has changed, but if so, it would have been foreseen
as a possible place to hit. Rather, we must say: “You’ve hit the target, so . . .”.
The place on the target doesn’t necessarily have to be marked by a sign, say a circle, on
the target. For at any rate there are descriptions like “closer to the centre”, “closer to the
edge”, “on the upper right”, etc. However the target is hit, some such description must always
be possible. (But neither are there “infinitely many” such descriptions.)
Now does it make sense to say: “But if one hits the target, one has to hit it somewhere”?
Or: “Wherever he hits the surface, it won’t be a surprise – causing me to say, for instance,
WBTC060-68 30/05/2005 16:16 Page 473
daß man etwa sagen würde ,das habe ich mir nicht erwartet, ich habe gar nicht gewußt, daß
es diesen Ort gibt‘“. Das heißt aber doch, es kann keine geometrische Überraschung sein.
Was für eine Art Satz ist: ”Auf diesem Streifen sind alle Schattierungen von Grau
zwischen Schwarz und Weiß zu sehen“? Hier scheint es auf den ersten Blick, daß von
unendlich vielen Schattierungen die Rede ist.
Ja, wir haben hier scheinbar das Paradox, daß wir zwar nur endlich viele Schattierungen
von einander unterscheiden können und der Unterschied zwischen ihnen natürlich nicht
ein unendlich kleiner ist, und wir dennoch einen kontinuierlichen Übergang sehen.
Man kann ein bestimmtes Grau ebensowenig als eines der unendlich6
vielen Grau zwischen Schwarz und Weiß auffassen, wie man eine
Tangente t 7
als eine der unendlich vielen Übergangsstationen8
von t1 nach
t2 auffassen kann. Wenn ich etwa ein Lineal von t1 nach t2 am Kreis abrollen
sehe, so sehe ich – wenn es sich kontinuierlich bewegt – keine einzige der
Zwischenlagen in dem Sinne, in welchem ich t sehe, wenn die Tangente
ruht; oder aber ich sehe nur eine endliche Anzahl von Zwischenlagen. Wenn
ich aber in so einem Fall scheinbar von einem allgemeinen Satz auf einen
Spezialfall schließe, so ist die Quelle dieses allgemeinen Satzes nie die Erfahrung und der
Satz wirklich kein Satz.
Wenn ich9
z.B. sage: ”Ich habe das Lineal sich von t1 nach t2 bewegen sehen, also muß
ich es auch in t gesehen haben“, so haben wir hier keinen richtigen logischen Schluß. Wenn
ich nämlich damit sagen will, das Lineal muß mir10
in der Lage t erschienen sein – wenn ich
also von der Lage im Gesichtsraum rede, so folgt das aus dem Vordersatz durchaus nicht.
Rede ich aber vom physischen Lineal, so ist es natürlich möglich, daß das Lineal die Lage
t übersprungen hat und das Phänomen im Gesichtsraum dennoch kontinuierlich war.
6 (O): unendlichen
7 (F): MS 108, S. 108.
8 (O): Übergangsstation
9 (V): ich als
10 (O): mit
307
237 Der Fall . . .
t1
t2
t
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‘I didn’t expect that, I didn’t even know that this location existed’ ”? But what that means
is that it can’t be a geometric surprise.
What sort of a proposition is: “On this strip you can see all shades of grey between black
and white”? Here it seems at first glance that we’re talking about infinitely many shades.
Indeed, here we’re apparently confronted with the paradox that although we can only distinguish
a finite number of shades from each other – and of course the difference between
them is not infinitely slight – still we see a continuous transition.
It is just as impossible to conceive of a particular grey as being one of
the infinite number of greys between black and white, as it is to conceive
of a tangent t 5
as being one of the infinite number of transitional stages
in going from t1 to t2. If for instance I see a ruler hugging the edge of a
circle as it moves from t1 to t2 then – if it moves continuously – I don’t
see a single one of the intermediate positions in the sense in which I see
t, when the tangent is at rest; or else I see only a finite number of intermediate
positions. But if in such a case I seem to infer a particular case
from a universal proposition, then experience is never the source of this universal proposition,
and the proposition isn’t really a proposition.
If I6
say, for instance: “I saw the ruler move from t1 to t2, therefore I must have also
seen it at t”, we have no valid logical inference. For if I mean by this that the ruler must
have appeared to me at t – if therefore I am talking about a position within my visual space
– then this in no way follows from the premiss. But if I am talking about the physical ruler,
then of course it’s possible that the ruler skipped over position t and that nevertheless the
phenomenon in my visual field was continuous.
The Case of Infinitely Many Propositions . . . 237e
t1
t2
t
5 (F): MS 108, p. 108. 6 (V): I as
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69
Kann eine Erfahrung lehren,
daß dieser Satz aus jenem folgt?
Es ist nur wesentlich, daß wir (hier) nicht sagen können, wir sind durch Erfahrung
daraufgekommen, daß es auch noch diesen Fall der Grammatik gibt. Denn den müßten wir
in dieser statement1
beschreiben und diese Beschreibung, obwohl ich ihre Wahrheit erst jetzt
einsehe, hätte ich doch schon vor dieser Erfahrung verstehen können.
Es ist die alte Frage: inwiefern kann man jetzt von einer Erfahrung sprechen, die man
jetzt nicht hat.
Was ich nicht voraussehen kann, kann ich nicht voraussehen. Und wovon ich jetzt sprechen
kann, kann ich jetzt sprechen, unabhängig von dem, wovon ich jetzt nicht sprechen kann.
Die Logik ist eben immer komplex.
”Wie kann ich wissen, was alles folgen wird?“ – Was ich dann wissen kann, kann ich auch
jetzt wissen.
Aber gibt es denn auch allgemeine Regeln der Grammatik, oder nicht nur Regeln über
allgemeine Zeichen?
Was wäre etwa eine allgemeine und eine besondere Regel im Schachspiel (oder2
einem
andern)? Jede Regel ist ja allgemein.
Doch ist eine andere Art der Allgemeinheit in der Regel, daß p 1 q aus p folgt, als in
der, daß jeder Satz der Form p, ~~p, . . . aus p & q folgt. Ist aber nicht die Allgemeinheit
der Regel für den Rösselsprung eine andere als die, einer Regel für den Anfang einer Partie?
Ist das Wort ”Regel“ überhaupt vieldeutig? Und sollen wir also nicht von Regeln im
Allgemeinen reden, wie auch nicht von Sprachen im Allgemeinen? Sondern nur von Regeln
in besonderen Fällen.
”Wenn aus F1(a) (a hat die Farbe F1) folgt ~F2(a), so mußte in der Grammatik des ersten
Satzes auch schon die Möglichkeit des zweiten vorausgesehen sein (wie könnten wir auch
sonst F1 und F2 Farben nennen).“
”Wenn der zweite Satz dem ersten, sozusagen, unerwartet gekommen wäre, so könnte er
nie aus ihm folgen.“
”Der erste Satz muß den anderen als seine Folge anerkennen. Oder vielmehr es muß dann
beide eine Grammatik vereinigen und diese muß dieselbe sein, wie vor dem Schluß.“
(Es ist sehr schwer, hier keine Märchen von den Vorgängen im Symbolismus zu erzählen,
wie an anderer Stelle keine Märchen über die psychologischen Vorgänge. Denn alles ist ja
einfach und allbekannt (und nichts neues zu erfinden). Das ist ja eigentlich das Unerhörte
1 (V): Aussage 2 (V): Schachspiel &
308
309
238
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69
Can an Experience teach us that one
Proposition Follows from Another?
The only essential point is that we cannot say (here) that experience led us to discover
that an additional grammatical fact existed. For in making this statement we would have to
describe this fact, and surely I could have understood this description before having this experience,
even though it is only now that I realize that the description is true.
It’s the old question: To what extent can one now talk about an experience that one isn’t
now having?
What I cannot foresee, I cannot foresee. And what I can speak of now, I can speak of now,
independently of what I cannot speak of now.
Logic is just complex, always.
“How can I know everything that will follow?” – What I can know then, I can also know
now.
But are there general rules of grammar too, or only rules for general signs?
For instance, what might be a general versus a particular rule in chess (or1
some other
game)? After all, every rule is general.
Still, the generality in the rule that p 1 q follows from p is of a different kind from the
generality in the rule that every proposition of the form p, ~~p, . . . , follows from p & q.
But isn’t the generality of the rule for moving a knight different from that of the rule for
beginning a game?
Is the word “rule” ambiguous from the outset? And should we therefore not talk about
rules in general – as we also shouldn’t talk about languages in general – but only about rules
in particular cases?
“If ~F2(a) follows from F1(a) (a has the colour F1), then the possibility of the former proposition
had to have been foreseen in the grammar of the latter (otherwise, how could we
call F1 and F2 colours?).”
“If that first proposition had come as a surprise to the second, so to speak, it could never
follow from it.”
“That second proposition must acknowledge the first as its consequence. Or rather, a
single grammar must unite them both, and it must be the same as it was before the inference.”
(Here it is very difficult not to tell fairy tales about symbolic processes, as elsewhere about
psychological processes. For in fact everything is simple and known to everyone (and there
1 (V): chess &
238e
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an der Logik, daß ihre außerordentliche Schwierigkeit darauf beruht, daß nichts zu
konstruieren, sondern alles schon da und bekannt ist.)
”Welchen Satz p nicht als seine Folge erkennt, der ist nicht seine Folge.“
Aus der Grammatik des Satzes – und aus ihr allein, muß es hervorgehen, ob ein Satz aus
ihm folgt. Keine Einsicht in einen neuen Sinn kann das ergeben; – sondern nur die Einsicht
in den alten Sinn. – Es ist nicht möglich, einen neuen Satz zu bilden, der aus jenem folgt,
den man nicht hätte bilden können (wenn auch ohne zu wissen, ob er wahr oder falsch ist)
als jener gebildet wurde. Entdeckte man einen neuen Sinn und folgte3
dieser aus dem4
ersten
Satz, so hätte dieser Satz damit5
seinen Sinn geändert.
3 (O): folge
4 (V): jenem
5 (O): Satz dann nicht (E): Auf Grund von MS
109, S. 16 geändert.
310
239 Kann eine Erfahrung lehren . . .
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is nothing new to invent). Indeed the really incredible thing about logic is that its extraordinary
difficulty resides in the fact that nothing can be constructed, but that everything
is already present and well-known.)
“Any proposition that p does not recognize as following from it does not follow from it.”
Whether one proposition follows from another must emerge from the grammar of the
latter – and from that grammar alone. This cannot be the result of an insight into a new
sense – only of an insight into the old sense. – It is not possible to form a new proposition
that follows from the first, a proposition that couldn’t have been formed when the first was
formed (without knowing at that time whether it was true or false). If one were to discover
a new sense, and if it followed from the2
first proposition, then that first proposition would
thereby have3
changed its sense.
2 (V): from that 3 (O): would not then have (E): We have
made this change on the basis of the corresponding
passage in MS 109 (p. 16).
Can an Experience teach us . . . 239e
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Allgemeinheit.311
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Generality.
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70
Der Satz ”Der Kreis befindet sich im
Quadrat“ in gewissem Sinne
unabhängig von der Angabe einer
bestimmten Lage (er hat, in gewissem
Sinne, nichts mit ihr zu tun).
Ich möchte sagen: das allgemeine Bild |o| hat eine andre Metrik als das besondere.
Im allgemeinen Zeichen ”|o|“ spielen die Distanzen so wenig eine Rolle wie im Zeichen
”aRb“.
Wie man die Zeichnung | o| als eine Darstellung des ”allgemeinen Falls“ ansehen kann.
Quasi nicht im Maßraum, sondern so, daß die Distanzen des Kreises von den Geraden
garnichts ausmachen. Man sieht dann das Bild als Fall eines anderen Systems, wie wenn
man es als Darstellung einer besonderen Lage des Kreises zwischen den Geraden sieht.
Oder richtiger: Es ist dann Bestandteil1
eines andren Kalküls. Von der Variablen gelten
eben andre Regeln, als von ihrem besonderen Wert.
”Wie2
weißt Du, daß er im Zimmer ist?“ – ”Weil ich ihn hineingesteckt habe und er
nirgends heraus kann.“ – So ist also Dein Wissen der allgemeinen Tatsache, daß er
irgendwo im Zimmer ist, auch von der Multiplizität dieses Grundes.
Nehmen wir die besonderen Fälle des allgemeinen Sachverhalts, daß das Kreuz sich
zwischen den Grenzstrichen befindet:
Jeder dieser Fälle z.B. hat seine3
besondere Individualität. Tritt diese Individualität
irgendwie in den Sinn des allgemeinen Satzes ein? Offenbar nicht.
Es scheint uns aber das ”zwischen den Strecken, oder Wänden, Liegen“ etwas Einfaches,
wovon die verschiedenen Lagen (ob die Gesichtserscheinungen, oder die durch Messen
festgestellten Lagen) ganz unabhängig sind.
D.h., wenn wir von den einzelnen (gesehenen) Lagen reden, so scheinen wir von etwas
ganz Anderem zu reden, als von dem, wovon im allgemeinen Satz die Rede ist.
1 (O): Bestandteils
2 (V): ”Woher
3 (V): eine
312
313
241
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70
In a Certain Sense the Proposition
“The Circle is in the Square” is
Independent of the Indication of a
Particular Position (in a Certain Sense
it has Nothing to do with It).
I’d like to say: the general picture |o| has a different metric from a particular one.
In the general sign “|o|” distances play no more a role than they do in the sign “aRb”.
Just as one can view the drawing | o|as representing the “general case”. Not within the
measured space, as it were, but in such a way that the distances of the circle from the
straight lines make no difference at all. Taken that way, one sees the picture as belonging
to a different system than when one sees it as representing a particular position of the
circle between the straight lines. Or, more correctly: Taken that way, it is a part of a different
calculus. The rules that apply to a variable are simply different from those that apply
to its particular value.
“How1
do you know that he is in the room?” – “Because I put him in there and there is
no way for him to get out.” – So your knowledge of the general fact that he is somewhere
in the room has the same multiplicity as this reason.
Let’s take some particular cases of the general state of affairs in which the cross is to be
found between the end-lines:
Each of these cases, for instance, has its own2
particular individuality. Does this individuality
somehow enter into the sense of the general proposition? Obviously not.
But it seems to us that “lying between the line segments, or walls” is something simple,
of which the various positions are completely independent (whether they are visual appearances
or are ascertained by measurement).
That is, when we’re talking about the individual positions (that we’ve seen), we seem
to be talking about something entirely different from what is talked about in the general
proposition.
1 (V): “Whence 2 (V): has a
241e
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Es ist ein anderer Kalkül, zu dem unsere Allgemeinheitsbezeichnung gehört und ein
anderer, in dem es jene Disjunktion gibt. Wenn wir sagen, das Kreuz liegt zwischen
diesen Strichen, so haben wir keine Disjunktion bereit, die den Platz dieses4
allgemeinen
Satzes nehmen könnte.
Wenn man die allgemeinen Sätze von der Art ”der Kreis befindet sich im Quadrat“
betrachtet, so kommt es einem immer wieder so vor, als sei die Angabe der Lage im
Quadrat nicht eine nähere Bestimmung zur Angabe, der Kreis liege im Quadrat (wenigstens
nicht, soweit der Gesichtsraum5
in Betracht kommt), als sei vielmehr das ”im Quadrat“
eine komplette Bestimmung, die an sich nicht mehr näher zu beschreiben sei. So wie eine
Angabe der Farbe die Angabe der Härte eines Materials nicht näher bestimmt. – So ist nun
das Verhältnis der Angaben über den Kreis natürlich nicht, und doch hat das Gefühl einen
Grund.
In den grammatischen Regeln für die Termini des allgemeinen Satzes muß es liegen, welche
Mannigfaltigkeit er für mögliche Spezialfälle voraussieht.6
Was in den Regeln nicht liegt, ist
nicht vorhergesehen.
7
Alle diese Verteilungen könnten verschiedene Zerrbilder desselben Sachverhalts
sein. (Man denke sich die beiden weißen Streifen und den schwarzen Streifen in
der Mitte dehnbar.)
Ist denn in (x).fx von a die Rede, da fa aus (x).fx folgt? In dem Sinne des allgemeinen
Satzes, dessen Verifikation in einer Aufzählung besteht, ja.
Wenn ich sage ”in dem Quadrat ist ein schwarzer Kreis“ so ist es mir immer, als habe
ich hier wieder etwas Einfaches vor mir. Als müsse ich nicht an verschiedene mögliche Lagen8
oder Größen des Kreises denken. Und doch kann man sagen: wenn ein Kreis in dem Quadrat
ist, so muß er irgendwo und von irgend einer Größe sein. Nun kann aber doch auf keinen
Fall davon die Rede sein, daß ich mir alle möglichen Lagen und Größen zum voraus denke.
– In dem ersten Satz scheine ich sie vielmehr, sozusagen, durch ein Sieb zu fassen, sodaß
”Kreis innerhalb des Quadrats“ einem Eindruck zu entsprechen scheint, für den das Wo etc.
überhaupt noch nicht in Betracht kommt, als sei es (gegen allen Anschein) etwas, was mit
jenem ersten Sachverhalt nur physikalisch, nicht logisch verbunden sei.
Der Ausdruck ”Sieb“ kommt daher: wenn ich etwa eine Landschaft ansehe, durch ein
Glas, das nur die Unterschiede von Dunkelheit und Helligkeit durchläßt, nicht aber die
Farbunterschiede, so kann man so ein Glas ein Sieb nennen. Denkt man sich nun das
Quadrat durch ein Glas betrachtet, das nur den Unterschied ”Kreis im Quadrat, oder nicht
im Quadrat“ durchließe, nicht aber einen Unterschied der Lage oder Größe des Kreises,
so könnten wir auch hier von einem Sieb sprechen.
Ich möchte sagen, in dem Satz ”ein Kreis liegt im Quadrat“ ist von der besonderen Lage
überhaupt nicht die Rede. Ich sehe dann in dem Bild nicht die Lage, ich sehe von ihr ab.
So als wären etwa die Abstände von den Quadratseiten dehnbar und als gälten ihre Längen
nicht.
Ja, kann denn nicht der Fleck sich wirklich im Viereck bewegen? Ist das nicht nur ein
spezieller Fall von dem, im Viereck zu sein? Dann wäre es also doch nicht so, daß der Fleck
an einer bestimmten Stelle im Viereck liegen muß, wenn er überhaupt darin ist.
4 (V): des
5 (V): Gesichtsform
6 (V): vorsieht.
7 (F): MS 109, S. 5.
8 (V): Stellungen
314
315
242 Der Satz ”Der Kreis befindet sich im Quadrat“ . . .
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There is one calculus to which our designation of generality belongs, and another one in
which disjunction exists. If we say that the cross is situated between the lines, we don’t have
any disjunction ready that could take the place of this3
general proposition.
If one looks at general propositions of the kind “the circle is in the square”, it seems
time and again as if giving its position within the square is not a more precise specification
of the statement that the circle is situated in the square (at least not so far as visual space4
is concerned). As if, rather, “in the square” were a complete determination that inherently
couldn’t be described any more accurately. Just as the specification of a material’s colour
doesn’t more accurately determine the specification of its hardness. – Of course that isn’t
the same relationship as between the statements about the circle, and yet this feeling is not
groundless.
The multiplicity a general proposition anticipates5
for its possible particular cases has
to be located in the grammatical rules for its terms. What isn’t located in these rules isn’t
anticipated.
6
All these spatial distributions could be different distortions of the same state
of affairs. (Imagine that in each row the two white areas and the black area in the
middle were elastic.)
Is a mentioned in (x).fx, since fa follows from (x).fx? Yes, in the sense of a general proposition
whose verification consists in an enumeration.
If I say “There is a black circle in the square” then it always seems to me that here again
I am looking at something simple. As if I didn’t have to think of different possible positions7
or sizes of the circle. And yet one can say: If a circle is situated within a square, then it
has to be somewhere and of some size. But surely it’s entirely out of the question for me to
think in advance of all possible positions and sizes. – Rather, in the first proposition I seem
to be apprehending them through a sieve, as it were, so that “circle in a square” seems to
correspond to one impression, for which the “where”, etc. doesn’t enter into the picture. As
if (contrary to all appearances) it were something connected to that first state of affairs only
physically, and not logically.
The basis for the expression “sieve” is this: If I look at, say, a landscape through a lens
that lets through only differences in darkness and light, but not differences of colour, then
one can call such a lens a sieve. Now let’s imagine the square viewed through a lens that
would only let through the difference between “circle in the square or not in the square”,
but not any difference in positions or sizes of the circle. Then here too we might speak of
a sieve.
I’d like to say that in the proposition “A circle is situated within a square” the circle’s
particular position isn’t addressed at all. I don’t see the position in the picture – I disregard
it. As if the distances from the sides of the square were elastic and their lengths didn’t count.
Indeed, can’t the spot actually be moving within the square? Isn’t that just a special case
of being within the square? In that case it wouldn’t even be necessary for the spot to be
situated at a particular place within the square, in so far as it’s in it at all.
3 (V): the
4 (V): visual form
5 (V): provides
6 (F): MS 109, p. 5.
7 (V): places
The Proposition “The Circle is in the Square” . . . 242e
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Ich will sagen, daß es eine Beziehung des Flecks zum Rand zu geben scheint, die
unabhängig von dem Abstand ist. – Gleichsam als bediente ich mich einer Geometrie, in
der es keinen Abstand gibt, wohl aber ein Innen und Außen. So gesehen, sind allerdings
auch die Bilder und 9
gleich.
Der Satz ”der Fleck ist im Quadrat“ hält gleichsam selbst den Fleck bloß im Quadrat, das
heißt, er beschränkt die Freiheit des Flecks nur auf diese Weise und gibt ihm in dem Quadrat
volle Freiheit. Der Satz bildet dann einen Rahmen, der die Freiheit des Flecks beschränkt
und ihn innerhalb frei läßt, das heißt, mit seiner Lage nichts zu schaffen hat. – Dazu muß
aber der Satz (gleichsam eine Kiste, in der der Fleck eingesperrt ist) die logische Natur dieses
Rahmens haben und das hat er, denn ich könnte jemandem den Satz erklären und dann jene
Möglichkeiten auseinandersetzen und zwar unabhängig davon, ob ein solcher Satz wahr ist
oder nicht, also unabhängig von einer Tatsache.
”Wo immer der Fleck im Viereck ist . . .“ heißt ”solange er10
im Viereck ist . . .“ und hier
ist nur die Freiheit (Ungebundenheit) im Viereck gemeint, aber keine Menge von Lagen.
Es besteht freilich eine logische Ähnlichkeit (formelle Analogie) zwischen dieser Freiheit
und der Gesamtheit von Möglichkeiten, daher gebraucht man oft in beiden Fällen dieselben
Wörter (”alle“, ”jeder“, etc.).
”Alle Helligkeitsgrade unter diesem tun meinen Augen weh.“ Prüfe die Art der
Allgemeinheit.
”Alle Punkte dieser Fläche sind weiß“. Wie verifizierst Du das? – dann werde ich wissen,
was es heißt.
9 (F): MS 109, S. 113. 10 (V): heißt ”wenn er
316
243 Der Satz ”Der Kreis befindet sich im Quadrat“ . . .
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I want to say that there seems to be a relationship of the spot to the edge that is independent
of distance. – As if, so to speak, I were using a geometry in which distance didn’t
exist, but “inside” and “outside” did. Seen this way the pictures and 8
are the same.
By itself the proposition “The spot is in the square” only keeps the spot in the square, as
it were, i.e. it limits the freedom of the spot only in this way, but grants it full freedom within
the square. In that case the proposition constitutes a frame that limits the freedom of the
spot, and leaves it free within, i.e. it has nothing to do with its position. – But for this to
be so the proposition (a pen, as it were, in which the spot is locked up) must have the
same logical nature as this frame. And so it does, for I could explain the proposition to
someone and then expound on those possibilities, and do this independently of whether
such a proposition is true or not, i.e. independently of a fact.
“Wherever the spot is in the square . . . .” means “So long as it is9
within the
square . . . .” and all that is meant here is the freedom (lack of restraint) within the square,
not a set of positions.
There is, to be sure, a logical similarity (formal analogy) between this freedom and the
totality of possibilities; thus one often uses the same words in both cases (“all”, “every”, etc.).
“All degrees of brightness less than this hurt my eyes.” Look closely at this kind of
generality.
“All points on this surface are white.” How do you verify that? – then I’ll know what it
means.
8 (F): MS 109, p. 113. 9 (V): means “If it is
The Proposition “The Circle is in the Square” . . . 243e
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71
Der Satz ”Der Kreis liegt
im Quadrat“ keine Disjunktion
von Fällen.
Wenn ich sage, der Fleck liegt im Quadrat, so weiß ich – und muß wissen – daß es
verschiedene mögliche Lagen für ihn gibt. Aber auch, daß ich nicht eine bestimmte Zahl
aller solcher Lagen nennen könnte. Ich weiß von vornherein nicht, wieviele Lagen ”ich unterscheiden
könnte“. – Und ein Versuch darüber lehrt mich auch nicht das, was ich hier
wissen will.
Das Dunkel, welches über den Möglichkeiten der Lage etc. herrscht, ist die gegenwärtige
logische Situation. So wie trübe Beleuchtung auch eine bestimmte Beleuchtung ist.
Es ist da immer so, als könnte man eine logische Form nicht ganz übersehen, da man nicht
weiß, wieviel, oder welche möglichen1
Lagen es für den Fleck im Viereck gibt. Anderseits
weiß man es doch, denn man ist von keiner überrascht, wenn sie auftritt.
Es ist natürlich nicht ”Stellung des Kreises in diesem Quadrat“ ein Begriff, und die
besondere Stellung ein Gegenstand, der unter ihn fällt. So daß Gegenstände gefunden
würden, von denen man sich überzeugt, daß sie (auch) Stellungen des Kreises im Quadrat
sind, von denen man aber früher nichts gewußt hat.
Die Mittelstellung des Kreises und andere ausgezeichnete Stellungen sind übrigens ganz
analog den primären Farben in der Farbenskala. (Dieses Gleichnis könnte man mit Vorteil
fortsetzen.)
Der Raum ist sozusagen eine Möglichkeit. Er besteht nicht aus mehreren Möglichkeiten.
Wenn ich also höre, das Buch liegt – irgendwo – auf dem Tisch, und finde es nun
in einer bestimmten Stellung, so kann ich nicht überrascht sein und sagen ”ah, ich habe
nicht gewußt, daß es diese Stellung gibt“ und doch hatte ich diese besondere Stellung
nicht vorhergesehen, d.h., als besondere Möglichkeit vorher ins Auge gefaßt. Was mich
überrascht, ist eine physische Möglichkeit, nicht eine logische!
Was ist aber der Unterschied zwischen dem Fall ”das Buch liegt irgendwo auf dem
Tisch“ und dem ”das Ereignis wird irgendeinmal in Zukunft eintreten“? Offenbar der,
daß wir im einen Fall eine sichere Methode kennen zu verifizieren, ob das Buch auf dem
Tisch liegt, im anderen Fall eine analoge Methode nicht existiert. Wenn etwa ein bestimmtes
Ereignis bei einer der unendlich vielen Bisektionen einer Strecke eintreten sollte, oder besser:
wenn es eintreten sollte, wenn wir die Strecke in einem Punkt (ohne nähere Bestimmung)
1 (O): mögliche
317
318
244
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71
The Proposition “The Circle
is in the Square” not a Disjunction
of Cases.
If I say the spot is in the square, I know – and must know – that there are various possible
positions for it. But I also know that I couldn’t give a definite number for all such positions.
At the outset I don’t know how many positions “I could distinguish”. – And neither does
trying to do so teach me what I want to know here.
The darkness reigning over the possible positions, etc. is the present logical situation. Just
as dim lighting is nevertheless a particular kind of lighting.
Here it always seems as if we can’t quite get an overview of a logical form, since we don’t
know how many or what possible positions there are for the spot in the square. But on the
other hand we do know this, for we aren’t surprised by any of them when they occur.
Of course, “position of the circle in this square” is not a concept, and the particular position
isn’t an object that falls within its scope. It’s not that objects could be found that would
convince you they were (also) positions of the circle within the square, but of which you
didn’t know anything beforehand.
Incidentally, the central position of the circle and other special positions are completely
analogous to the primary colours on the colour scale. (This simile could be profitably
pursued.)
Space is one possibility, as it were. It doesn’t consist of several possibilities.
So if I hear that the book is lying – somewhere – on the table and now I find it in a
particular position, then I can’t be surprised and say “Ah, I didn’t know that this position
existed”; and yet I hadn’t foreseen this particular position, i.e. hadn’t envisaged it in advance
as a particular possibility. What surprises me is a physical possibility, not a logical one!
But what is the difference between “The book lies somewhere on the table” and this: “The
event will occur sometime in the future”? Obviously that in the one case we have a sure
method of verifying whether the book is on the table, while in the other case there is no
analogous method. If, say, a particular event were supposed to occur at one of the infinitely
many bisections of a line, or better yet: if it were supposed to occur when we cut the line at
244e
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schneiden und an diesem Punkt eine Minute verweilen, so ist diese Angabe ebenso sinnlos,
wie die über die unendliche Zukunft.
Angenommen, ich gäbe eine Disjunktion von so vielen Stellungen an, daß es mir
unmöglich wäre, eine Stellung von allen angegebenen als verschieden zu sehen;2
wäre nun
die Disjunktion der allgemeine Satz (∃x).fx? Wäre es nicht sozusagen Pedantrie, die
Disjunktion noch immer nicht als den allgemeinen Satz anzuerkennen? Oder besteht ein
wesentlicher Unterschied, und ist die Disjunktion vielleicht dem allgemeinen Satz gar nicht
ähnlich?
Das, was uns auffällt, ist, daß der eine Satz so kompliziert, der andere so einfach ist. Oder
ist der einfache nur eine kurze Schreibweise des komplizierteren?
Was ist denn das Kriterium dafür (für den allgemeinen Satz), daß der Kreis im Quadrat
ist? Entweder überhaupt nichts, was mit einer Mehrheit von Lagen (bezw. Größen) zu tun
hat, oder aber etwas, was mit einer endlichen Anzahl solcher Lagen zu tun hat.
Wenn man sagt, der Fleck A ist irgendwo zwischen den Grenzen B und
C, 3
ist es denn nicht offenbar möglich, eine Anzahl von Stellungen des A
zwischen B und C zu beschreiben oder abzubilden, sodaß ich die Succession aller dieser
Stellungen als kontinuierlichen Übergang sehe? Und ist dann nicht die Disjunktion aller
dieser N Stellungen eben der Satz, daß sich A irgendwo zwischen B und C befindet?
Aber wie verhält es sich mit diesen N Bildern? Es ist klar, daß ein Bild und das
unmittelbar folgende visuell nicht unterscheidbar sein dürfen, sonst ist der Übergang
visuell diskontinuierlich.
Die Stellungen, deren Succession ich als kontinuierlichen Übergang sehe, sind
Stellungen nicht im Gesichtsraum.
Wie ist der Umfang des Begriffs ”Dazwischenliegen“ bestimmt? Denn es soll doch im
Vorhinein festgelegt werden, welche Möglichkeiten zu diesem Begriff gehören. Es kann,
wie ich sage, keine Überraschung sein, daß ich auch das ”dazwischenliegen“ nenne. Oder:
wie können die Regeln für das Wort ”dazwischenliegen“ angegeben werden, da ich doch
nicht die Fälle des Dazwischenliegens aufzählen kann? Natürlich muß gerade das für die
Bedeutung dieses Worts charakteristisch sein.
Wir würden das Wort ja auch nicht durch Hinweisen auf alle besonderen Fälle jemandem
zu erklären suchen, sondern4
indem wir auf einen solchen Fall (oder einige) zeigten und in
irgendeiner Weise andeuteten, daß es auf den besonderen Fall nicht ankomme.
Das Aufzählen von Lagen ist nicht nur nicht nötig, sondern es kann hier wesentlich von
so einem Aufzählen keine Rede sein.
Zu sagen ”der Kreis liegt entweder zwischen den beiden Geraden oder hier“ (wo das5
”hier“ ein Ort zwischen den Geraden ist) heißt offenbar nur: ”der Kreis liegt zwischen den
beiden Geraden“, und der Zusatz ”oder hier“ ist6
überflüssig. Man wird sagen: in dem
”irgendwo“ ist das ”hier“ schon mitinbegriffen. Das ist aber merkwürdig, weil es nicht (darin)
genannt ist.
2 (V): erkennen;
3 (F): MS 108, S. 134.
4 (V): suchen, aber wohl,
5 (V): dieses
6 (V): erscheint
319
320
245 Der Satz ”Der Kreis liegt im Quadrat“ . . .
B A C
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a certain point (without further specification) and then wait at this point for one minute,
then this statement is just as senseless as the one about the infinite future.
Let’s assume that I were to state a disjunction of so many positions that it would be
impossible for me to see1
any one position as different from all the rest; would that disjunction
now be the general proposition (∃x).fx? Wouldn’t it be pedantry, as it were, not to acknowledge
this disjunction as the general proposition? Or is there an essential difference, and is
the disjunction perhaps not even similar to the general proposition?
What strikes us is that the one proposition is so complicated and the other so simple. Or
is the simple one just a short way of writing the more complicated one?
What then is the criterion (for the general proposition) that the circle is in the square?
Either nothing that pertains to a majority of positions (or sizes), or something pertaining to
a finite number of such positions.
If it’s said that patch A lies somewhere between the borders B and C,
2
isn’t it obviously possible to describe or portray a number of positions
of A between B and C so that I see the succession of all these positions as a continuous
transition? And in that case isn’t the disjunction of all of these N positions precisely the
proposition that A lies somewhere between B and C?
But what about these N pictures? It’s clear that a picture and its immediate successor must
not be visually distinguishable; otherwise the transition will be visually discontinuous.
The positions whose succession I see as a continuous transition are positions that are not
in visual space.
How is the scope of the concept “lying between” determined? For after all, the possibilities
that belong to this concept are supposed to be determined from the very start. It
cannot come as a surprise, as I like to say, that I also call this “lying between”. Or: How
can the rules for the words “lie between” be given, since there is no way I can enumerate
the cases of lying between? Of course it is precisely this that must characterize the meaning
of these words.
We wouldn’t try to explain the expression to someone by pointing to all particular instances.
Rather, we’d3
try to explain it by pointing to one such case (or several) and indicating in
some way that that particular case wasn’t the point.
Enumerating positions is not only unnecessary, but such an enumeration is by its very
nature out of the question here.
Saying “The circle lies either between the two straight lines, or here” (where “here”4
is
a location between the straight lines) obviously means nothing more than: “The circle lies
between the two straight lines”, and the addendum “or here” is5
superfluous. It will be
said: The “here” is already included in the “somewhere”. But that’s remarkable, because it
isn’t mentioned (in it).
1 (V): recognize
2 (F): MS 108, p. 134.
3 (V): instances, but we would
4 (V): where this “here”
5 (V): appears
The Proposition “The Circle is in the Square” . . . 245e
B A C
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Eine bestimmte Schwierigkeit besteht wenn die Zeichen7
das nicht zu sagen scheinen, was
der Gedanke erfaßt, oder: wenn die Worte das nicht sagen, was der Gedanke zu erfassen
scheint.
So, wenn wir sagen ”dieser Satz gilt von allen Zahlen“ und glauben in dem Gedanken
alle Zahlen wie die Äpfel in einer Kiste aufgefaßt8
zu haben.
Nun könnte man aber fragen: Wie kann ich (nun) im Voraus wissen, aus welchen Sätzen
dieser allgemeine Satz folgt? Wenn ich diese Sätze nicht angeben kann.
Kann man aber sagen: ”man kann nicht sagen, aus welchen Sätzen dieser Satz folgt“? Das
klingt so wie: man weiß es nicht. Aber so ist es natürlich nicht. Und ich kann ja Sätze sagen,
und im Vorhinein sagen, aus denen er folgt. – ”Nur nicht alle“. – Aber das heißt ja eben
nichts.
Es ist eben nur der allgemeine Satz und besondere Sätze (nicht die besonderen Sätze).
Aber der allgemeine Satz zählt besondere Sätze nicht auf. Aber was charakterisiert ihn denn
dann als9
allgemein, und was zeigt, daß er nicht einfach die10
besonderen Sätze umschließt,
von denen wir in diesem bestimmten Falle sprechen?
Er kann nicht durch seine Spezialfälle charakterisiert werden; denn wieviele man auch
aufzählt, so könnte er immer mit dem Produkt der angeführten Spezialfälle11
verwechselt
werden. Seine Allgemeinheit liegt also in einer Eigenschaft (grammatischen Eigenschaft)
der Variablen.
7 (V): besteht darin, daß die Worte
8 (V): gefaßt
9 (V): denn als
10 (V): diejenigen
11 (V): Fälle
321
246 Der Satz ”Der Kreis liegt im Quadrat“ . . .
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There is a particular problem when the signs don’t seem to say6
what the thought grasps,
or: when the words don’t say what the thought seems to grasp.
For instance, when we say “This proposition is valid for all numbers”, and think that in
this thought we have gathered up7
all numbers, like apples in a crate.
But now one could ask: How can I (now) know in advance from which propositions this
general proposition follows, if I can’t specify these propositions?
But can you say: “It cannot be said from which propositions this proposition follows”?
That sounds like: “We don’t know this”. But of course that isn’t how it is. And indeed I
can utter propositions from which it follows, and utter them in advance. – “Just not all of
them”. – But that doesn’t mean a thing.
There’s just the general proposition, and particular propositions (not the particular
propositions). But the general proposition doesn’t enumerate particular propositions. What,
then, characterizes it as general, and what shows that it doesn’t simply encompass those
particular propositions we are speaking of in this particular case?
It can’t be characterized by its special cases; for however many we enumerate it could
always still be confused with the product of those special cases8
we’ve listed. Thus, its
generality lies in a property (grammatical property) of the variables.
6 (V): A particular difficulty consists in the words
not seeming to say
7 (V): grasped
8 (V): those cases
The Proposition “The Circle is in the Square” . . . 246e
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72
Unzulänglichkeit der Fregeund
Russell’schen
Allgemeinheitsbezeichnung.
Die eigentliche Schwierigkeit liegt nämlich im Begriff des ”(∃n)“ und allgemein des ”(∃x)“.
Ursprünglich stammt diese Notation vom Ausdruck unsrer Wortsprache her: ”es gibt ein
. . . von der und der Eigenschaft“. Und was hier an Stelle der Punkte steht, ist etwa ”Buch
meiner Bibliothek“, oder ”Ding (Körper) in diesem Zimmer“, ”Wort in diesem Brief“, u.s.w.
Man denkt dabei an Gegenstände, die man der Reihe nach durchgehen kann. Durch einen,
so oft angewandten,1
Prozeß der Sublimierung wurde diese Form dann zu der: ”es gibt
einen Gegenstand, für welchen . . .“, und hier dachte man sich ursprünglich auch die
Gegenstände der Welt ganz analog den ”Gegenständen“ im Zimmer (nämlich den Tischen,
Stühlen, Büchern, etc.). Obwohl es ganz klar ist, daß die Grammatik dieses ”(∃x) etc.“ in
vielen Fällen eine ganz andere ist, als im primitiven und als Urbild dienenden Fall.
Besonders kraß wird die Diskrepanz zwischen dem ursprünglichen Bild und dem, worauf
die Notation nun angewendet wird,2
wenn ein Satz ”in diesem Viereck sind nur zwei Kreise“
wiedergegeben wird in der3
Form ”es gibt keinen Gegenstand, der die Eigenschaft hat, ein
Kreis in diesem Viereck, aber weder der Kreis a noch der Kreis b zu sein“, oder ”es gibt
nicht drei Gegenstände, die die Eigenschaft haben, ein Kreis in diesem Viereck zu sein“.
Der Satz ”es gibt nur zwei Dinge, die Kreise in diesem Viereck sind“ (analog gebildet dem
Satz ”es gibt nur zwei Menschen, die diesen Berg erstiegen haben“) klingt verrückt; und
mit Recht. D.h., es ist nichts damit gewonnen, daß wir den Satz ”in diesem Viereck sind
zwei Kreise“ in jene Form pressen; vielmehr hilft uns das nur zu übersehen, daß wir die
Grammatik dieses Satzes nicht klargestellt haben. Zugleich aber gibt hier die Russell’sche
Notation einen Schein von Exaktheit, der Manchen glauben macht, die Probleme seien dadurch
gelöst, daß man den Satz auf die Russell’sche Form gebracht hat. (Es ist das ebenso gefährlich,
wie der Gebrauch des Wortes ”wahrscheinlich“, ohne weitere Untersuchung darüber, wie
das Wort in diesem speziellen Fall gebraucht wird. Auch das Wort ”wahrscheinlich“ ist, aus
leicht verständlichen Gründen, mit einer Idee der Exaktheit verbunden.)
In allen den Fällen: ”Einer der vier Füße dieses Tisches hält nicht“, ”es gibt Engländer
mit schwarzen Haaren“, ”auf dieser Wand ist ein Fleck“, ”die beiden Töpfe haben das
gleiche Gewicht“, ”auf beiden Seiten stehen gleichviel Wörter“ – wird in der Russell’schen
Notation das ”(∃ . . . ) . . .“ gebraucht; und jedesmal mit anderer Grammatik. Damit will
ich also sagen, daß mit einer Übersetzung so eines Satzes aus der Wortsprache in die
Russell’sche Notation nicht viel gewonnen ist.
1 (V): verwendeten,
2 (V): angewendet werden soll,
3 (V): wird durch die
322
323
247
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72
The Inadequacy of Frege’s
and Russell’s Notation
for Generality.
The real difficulty lies in the concept of “(∃n)” and, in general terms, of “(∃x)”. This
notation originates in this expression of our word-language: “There is a . . . with such and
such a property”. And what replaces the dots here is, say, “book in my library” or “thing
(body) in this room”, “word in this letter”, etc. Here one thinks of objects that one can go
through, one after the other. Via a process of sublimation that is so often applied,1
that form
is then turned into this: “There is an object such that . . .”. And here too we originally thought
of all the objects in the world as completely analogous to the “objects” in a room (i.e. to
tables, chairs, books, etc.), even though it is quite clear that in many cases the grammar of
this “(∃x) etc.” is utterly different from the grammar of the primitive case that serves as the
archetype. The discrepancy between the original image and what the notation is now applied
to2
becomes particularly blatant when a proposition like “There are only two circles in this
square” is rendered formally as3
“There is no object that has the property of being a circle
in this square, and of being neither circle a nor circle b”, or “There are not three objects
that have the property of being a circle in this square”. The proposition “There are only
two things that are circles in this square” (formed in analogy with the proposition “There
are only two people who have climbed this mountain”) sounds insane; and rightly so. That
is, nothing is gained by our squeezing the proposition “There are two circles in this square”
into that form; on the contrary, this only helps us lose sight of the fact that we haven’t clarified
the grammar of this proposition. At the same time, however, Russell’s notation gives an appearance
of precision, which makes some people believe that problems are solved by having converted
a proposition into Russell’s form. (This is just as dangerous as the use of the word
“probably”, without any further investigation of how the word is used in particular cases.
For reasons that are easy to understand, the word “probably” is also connected with an idea
of precision.)
“One of the four legs of this table isn’t stable”, “There are Englishmen with black hair”,
“There’s a spot on this wall”, “The two pots have the same weight”, “There are the same
number of words on both pages”. – In all of these cases “(∃ . . . ) . . .” is used in Russellian
notation; and each time with a different grammar. So here I want to say that not much is
gained by translating such a proposition from word-language into Russellian notation.
1 (V): used,
2 (V): now supposed to be applied to
3 (V): is rendered via the form
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Unzulänglichkeit der Frege’schen und Russell’schen Allgemeinheitsbezeichnung.
Es hat Sinn, zu sagen ”schreib’ eine beliebige Kardinalzahl hin“, ist aber Unsinn zu sagen:
”schreib’ alle Kardinalzahlen hin“. ”In dem Viereck befindet sich ein Kreis“ ( (∃x).φx) hat
Sinn, aber nicht ~(∃x).~φx: ”in dem Viereck befinden sich alle Kreise“. ”Auf einem andersfarbigen
Hintergrund befindet sich ein roter Kreis“ hat Sinn, aber nicht ”es gibt keine von
rot verschiedene Farbe eines Hintergrundes, auf der sich kein roter Kreis befindet“.
”In diesem Viereck ist ein schwarzer Kreis“: Wenn dieser Satz die Form ”(∃x).x ist ein
schwarzer Kreis im Viereck“ hat, was4
ist so ein Ding x, das5
die Eigenschaft hat, ein schwarzer
Kreis zu sein (und also auch die haben kann, kein schwarzer Kreis zu sein)? Ist es etwa ein
Ort im Quadrat? dann aber gibt es keinen Satz ”(x).x ist ein schwarzer . . .“. Anderseits
könnte jener Satz bedeuten ”es gibt einen Fleck im Quadrat, der ein schwarzer Kreis ist“.
Wie verifiziert man diesen Satz? Nun, man geht die verschiedenen Flecken im Quadrat
durch und untersucht sie daraufhin, ob sie ganz schwarz und kreisförmig sind. Welcher
Art ist aber der Satz: ”Es ist kein Fleck in dem Quadrat“? Denn, wenn das ”x“ in ”(∃x)“
im vorigen Fall ”Fleck im Quadrat“ hieß, dann kann es zwar einen Satz ”(∃x).φx“ geben,
aber keinen ”(∃x)“ oder ”~(∃x)“. Oder, ich könnte wieder fragen: Was ist das für ein Ding,
das die Eigenschaft hat (oder nicht hat) ein Fleck im Quadrat zu sein?
Und wenn man sagen kann ”ein Fleck ist in dem Quadrat“, hat es dann6
auch schon Sinn,
zu sagen ”alle Flecken sind in dem Quadrat“? Welche alle?
Die gewöhnliche Sprache sagt ”in diesem Viereck ist ein roter Kreis“, die Russell’sche
Notation sagt ”es gibt einen Gegenstand, der ein roter Kreis in diesem Viereck ist“. Diese
Ausdrucksform ist offenbar nach dem Modell gebildet: ”es gibt eine Substanz, die im
Dunkeln leuchtet“, ”es gibt einen Kreis in diesem Viereck, der rot ist“. – Vielleicht ist
schon der Ausdruck ”es gibt“ irreführend. ”Es gibt“ heißt eigentlich soviel wie ”es findet
sich“, oder ”es gibt unter diesen Kreisen einen . . .“.
Wenn man also in größtmöglicher Annäherung an die Russell’sche Ausdrucksweise sagt
”es gibt einen Ort in diesem Viereck, wo ein roter Kreis ist“, so heißt das eigentlich, unter
diesen Orten gibt es einen, an welchem etc.
(Der schwierigste Standpunkt in der Logik ist der des gesunden Menschenverstandes.
Denn er verlangt zur Rechtfertigung seiner Meinung die volle Wahrheit und hilft uns nicht,
durch die geringste Konzession, oder Konstruktion.)
Der richtige Ausdruck dieser Art Allgemeinheit ist also der der7
gewöhnlichen Sprache
”in dem Viereck ist ein Kreis“, welcher die Lage des Kreises einfach offen läßt (unentschieden
läßt). (”Unentschieden“ ist ein richtiger Ausdruck, weil die Entscheidung einfach fehlt.)
4 (V): hat, welcher Art
5 (V): welches
6 (V): damit
7 (O): der, der
324
325
248 Unzulänglichkeit der Frege- und Russell’schen Allgemeinheitsbezeichnung.
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The inadequacy of Frege’s and Russell’s notation for generality.
It makes sense to say “Write down any cardinal number”, but it’s nonsense to say: “Write
down all cardinal numbers”. For “There is a circle in the square” ( (∃x).fx) makes sense,
but not ~(∃x).~φx: “All circles are in the square”. “There is a red circle on a background
of a different colour” makes sense, but not “There is no background colour other than red
on which there is no red circle”.
“There is a black circle in this square”: If this proposition has the form “(∃x).x is a black
circle in the square”, what is such a thing x that has4
the property of being a black circle
(and therefore can also have the property of not being a black circle)? Perhaps a location
in the square? But then there is no proposition “(x).x is a black . . .”. On the other hand,
that proposition could mean “There is a spot in the square that is a black circle”. How
does one verify this proposition? Well, one checks through the various spots in the square
and investigates them to see whether they’re completely black and circular. But what kind
of proposition is: “There is no spot in the square”? For if in the previous case the “x” in
“(∃x)” was called “spot in the square”, then, to be sure, there can be a proposition “(∃x).fx”,
but not “(∃x)” or “~(∃x)”. And once more I could ask: What kind of a thing is it that has
(or doesn’t have) the property of being a spot in the square?
And if one can say “There is a spot in the square” does it then5
also make good sense to
say “All spots are in the square”? Which all?
Ordinary language says “There’s a red circle in this square”, Russell’s notation says
“There is an object that is a red circle in this square”. This form of expression is obviously
modelled on: “There is a substance that glows in the dark”, “There is a circle in this square
that is red”. – Perhaps the expression “There is” is misleading from the start. “There is”
really means the same as “It is found” or “Among these circles there is one . . .”.
So if one approximates Russell’s mode of expression as closely as possible and says “There
is a place in this square where there is a red circle”, then what this actually means is: Among
these places there is one where, etc.
(The most difficult standpoint in logic is that of common sense. To live up to its name,
it demands the whole truth, and it doesn’t provide us with even the most minute concession,
or provision.)
So the correct expression of this kind of generality is that of ordinary language: “There’s
a circle in the square”, which simply leaves the position of the circle open (undecided).
(“Undecided” is the correct expression, because the decision is simply absent.)
4 (V): square”, of what nature is such a thing x that
has
5 (V): thereby
Frege’s and Russell’s Notation for Generality. 248e
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73
Kritik meiner früheren
Auffassung der Allgemeinheit.
Meine Auffassung des allgemeinen Satzes war, daß (∃x).fx eine logische Summe ist und
daß nur ihre Summanden hier nicht aufgezählt seien, sich aber aufzählen ließen (und zwar
aus dem Wörterbuch und der Grammatik der Sprache).
Denn ließen sie sich nicht aufzählen, so haben wir ja doch keine logische Summe.1
(Vielleicht
ein Gesetz, logische Summen zu bilden.)
Die Erklärung von (∃x).φx als einer logischen Summe und (x).φx als logischem Produkt
kann natürlich nicht aufrecht erhalten werden. Sie ging mit einer falschen Auffassung der
logischen Analyse zusammen, indem ich etwa dachte, das logische Produkt für ein bestimmtes
(x).φx werde sich schon einmal finden. – Es ist natürlich richtig, daß (∃x).φx irgendwie als
logische Summe funktioniert und (x).φx als Produkt; ja in einer Verwendungsart der Worte
”alle“ und ”einige“ ist meine alte Erklärung richtig, nämlich – z.B. – in dem Falle ”alle primären
Farben finden sich in diesem Bild“ oder ”alle Töne der C-Dur Tonleiter kommen in diesem
Thema vor“. In Fällen aber wie ”alle Menschen sterben, ehe sie 200 Jahre alt werden“ stimmt
meine Erklärung nicht. Daß nun aber (∃x).φx als logische Summe funktioniert, ist darin
ausgedrückt, daß es aus φa und aus φa .1. φb folgt, also in den Regeln:
(∃x).φx .&. φa = φa und
(∃x).φx :&: φa .1. φb = φa .1. φb.
Aus diesen Regeln ergeben sich dann die Grundgesetze Russells
φx .⊃. (∃z).φz und
φx .1. φy :⊃: (∃z).φz als Tautologien.
Für (∃x).φx, etc. brauchen wir auch die Regeln:
(∃x).φx 1 ψx = (∃x).φx .1. (∃x).ψx,
(∃x,y).φx & ψy) .1. (∃x).φx & ψx = (∃x).φx .&. (∃x).ψx.
Jede solche Regel ist ein Ausdruck der Analogie zwischen (∃x).φx und einer logischen Summe.
Man könnte übrigens wirklich eine Notation für (∃x).φx einführen, in der man es durch
ein Zeichen ”φα 1 φβ 1 φγ 1 . . .“ ersetzt und dürfte dann damit rechnen, wie mit
einer logischen Summe; es müßten aber die Regeln vorgesehen sein, nach denen ich
diese Notation immer in die von ”(∃x).φx“ zurücknehmen kann und die also das Zeichen
1 (V): so handelt es sich ja doch nicht um // um eine
// um keine // logische Summe.
326
327
249
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73
Criticism of my Earlier
Understanding of Generality.
My understanding of the general proposition was that (∃x).fx is a logical sum, and that
although its terms weren’t enumerated there, they could be enumerated (from the dictionary
and the grammar of language).
For if they couldn’t be enumerated, then we wouldn’t have a logical sum.1
(Perhaps a law
for forming logical sums.)
Of course, the explanation of (∃x).φx as a logical sum and of (x).φx as a logical product
cannot be maintained. It was linked to a false view of logical analysis, with my thinking,
for instance, that the logical product for a particular (x).φx would most likely be found
some day. – Of course it’s correct that (∃x).φx functions in some way as a logical sum, and
that (x).φx functions in some way as a product; indeed for one use of the words “all” and
“some” my old explanation is correct, namely, in a case like “All the primary colours can
be found in this picture”, or “All the notes of the C major scale occur in this theme”.
But in cases like “All people die before they are 200” my explanation is not correct. That
(∃x).φx functions as a logical sum, however, is expressed by its following from φa and from
φa .1. φb, i.e., in the rules:
(∃x).φx .&. φa = φa and
(∃x).φx :&: φa .1. φb = φa .1. φb.
From these rules Russell’s basic laws then follow as tautologies:
φx .⊃. (∃z).φz and
φx .1. φy :⊃: (∃z).φz.
For (∃x).φx, etc. we also need the rules:
(∃x).φx 1 ψx = (∃x).φx .1. (∃x).ψx,
(∃x,y).φx & ψy .1. (∃x).φx & ψx = (∃x).φx .&. (∃x).ψx.
Every such rule is an expression of the analogy between (∃x).φx and a logical sum.
Incidentally, we really could introduce a notation for (∃x).φx in which it is replaced with
a sign “φα 1 φβ 1 φγ 1 . . .” and then we could legitimately perform calculations with it
as with a logical sum; however, rules would have to be provided, according to which I can
always convert this notation back into that of “(∃x).φx”, and which therefore distinguish the
1 (V): then it isn’t at all a matter of a logical sum.
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”φa 1 φb 1 φc 1 . . .“ von dem einer logischen Summe unterscheiden. Der Zweck
dieser Notation wäre nur der, in gewissen Fällen leichter mit (∃x).φx rechnen zu können.
Wenn ich Recht habe, so gibt es keinen Begriff ”reine Farbe“; der Satz ”A hat eine reine
Farbe“ heißt einfach ”A ist rot, oder gelb, oder blau, oder grün“. ”Dieser Hut gehört entweder
A oder B oder C“ ist nicht derselbe Satz wie ”dieser Hut gehört einem Menschen in diesem
Zimmer“, selbst wenn tatsächlich nur A, B, C im Zimmer sind, denn das muß erst dazugesagt
werden. – Auf dieser Fläche sind zwei reine Farben, heißt: Auf dieser Fläche sind rot
und gelb, oder rot und blau, oder rot und grün, oder etc.
Wenn ich nun nicht sagen kann ”es gibt 4 reine Farben“, so sind die reinen Farben und
die Zahl 4 doch irgendwie miteinander verbunden und das muß sich auch irgendwie ausdrücken.
– Z.B. wenn ich sage ”auf dieser Fläche sehe ich 4 Farben: gelb, blau, rot, grün“.
Die Allgemeinheitsbezeichnung unserer gewöhnlichen Sprache faßt die logische Form
noch viel oberflächlicher, als ich früher geglaubt habe. Sie ist eben in dieser Beziehung
mit der Subjekt–Prädikat Form vergleichbar.
Die Allgemeinheit ist so vieldeutig, wie die Subjekt–Prädikat Form.
Es gibt so viel verschiedene ”alle“, als es verschiedene ”Eins“ gibt.2
Darum nützt es nichts, zur Klärung das Wort ”alle“ zu gebrauchen, wenn man seine
Grammatik in diesem Fall noch nicht kennt.
2 (V): Es gibt so viel verschiedene Allgemeinheiten,
als es verschiedene Zahlarten gibt.
328
250 Kritik meiner früheren Auffassung der Allgemeinheit.
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sign “φa 1 φb 1 φc 1 . . .” from the sign for a logical sum. The only purpose of this
notation would be to enable us to calculate more easily with (∃x).φx in certain cases.
If I am right, there is no concept “pure colour”; the proposition “A’s colour is a pure
colour” simply means “A is red, or yellow, or blue, or green”. “This hat belongs either to
A or to B or to C” is not the same proposition as “This hat belongs to a person in this room”,
even if in fact only A, B and C are in the room, for that is something that has to be added.
– “There are two pure colours on this surface” means: There are red and yellow or red and
blue or red and green or etc. on this surface.
Even if I can’t say “There are 4 pure colours”, still the pure colours and the number 4
are somehow connected with each other, and that has to be expressed somehow as well. –
For example, by saying “I see 4 colours on this surface: yellow, blue, red, green”.
In our ordinary language the notation for generality contains the logical form much more
superficially than I earlier believed. In this respect it’s comparable to the subject–predicate
form.
Generality is as ambiguous as the subject-predicate form.
There are as many different “all’s” as there are different “one’s”.2
So it’s no good using the word “all” for clarification if we don’t know its grammar in
this case.
2 (V): There are as many different generalities as
there are different kinds of numbers.
My Earlier Understanding of Generality. 250e
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74
Erklärung der Allgemeinheit
durch Beispiele.
Denken wir uns die Erklärung des Begriffs der Pflanze. Wir zeigen jemand mehrere
Gegenstände und sagen, das sind Pflanzen. Dann zeigt auch er auf einen weiteren
Gegenstand und sagt ”ist auch das eine Pflanze“ und wir antworten ”ja, das auch“, u.s.w.
Ich hätte nun einmal gesagt, er habe nun in dem Gezeigten den Begriff ”Pflanze“ – das gewisse
Gemeinsame – gesehen und er sehe1
die Beispiele der Erklärung anders, wenn er in ihnen
eben diesen Begriff sieht als, wenn er sie etwa als Repräsentanten dieser bestimmten
Gestalt2
und Farbe allein auffasse. (So wie ich auch sagte, er sähe in der Variablen, wenn er
sie als solche versteht, etwas, was er im Zeichen für den besonderen Fall nicht sieht.) Aber
der Gedanke des ”darin Sehens“ ist von dem Fall hergenommen, wo ich z.B. die Figur ||||
verschieden ”phrasiert“ sehe. Aber dann sehe ich eben in einem andern Sinn wirklich
verschiedene Figuren und, was diese gemein haben, ist außer ihrer Ähnlichkeit die
Verursachung durch das gleiche physikalische Bild.
Aber diese Erklärung ist doch nicht ohneweiteres auf den Fall des Verstehens der
Variablen oder der Beispiele für den Begriff ”Pflanze“ anzuwenden. Denn angenommen, wir
hätten wirklich etwas anderes in ihnen gesehen, als in Pflanzen, die nur um ihrer selbst willen
gezeigt wurden, so ist die Frage, kann denn dieses, oder irgendein anderes, Bild uns zu der
Anwendung als Variable berechtigen? Ich hätte einem also die Pflanzen zur Erklärung zeigen
können und ihm dazu einen Trank gegeben, durch den es verursacht wird, daß er die Beispiele
in der bestimmten Weise sieht. (Wie es möglich wäre, daß ein Alkoholisierter eine Gruppe
|||| immer als ||| | sieht.) Und damit wäre die Erklärung des Begriffs in eindeutiger Weise
gegeben und wer sie verstanden hat, hätte von den vorgezeigten Specimina und den begleitenden
Gesten dieses Bild empfangen. So ist es aber doch nicht. – Es ist nämlich wohl möglich,
daß der, welcher z.B. das Zeichen |||||| als Zahlzeichen für die 6 sieht, es anders sieht
(etwas anderes darin sieht) als der, welcher es nur als Zeichen für ”einige“ auffaßt, weil er
seine Aufmerksamkeit nicht auf das Gleiche richten wird; aber es kommt dann auf das System
von Regeln an, die von diesen Zeichen gelten und das Verstehen wird wesentlich kein Sehen
des Zeichens in gewisser Weise sein.
Es wäre also möglich, zu sagen ”jetzt sehe ich das nicht mehr als Rose, sondern nur noch
als Pflanze“!
Oder: ”Jetzt sehe ich es nur als Rose, nicht mehr als diese Rose“.
”Ich sehe den Fleck nur noch im Quadrat, aber nicht mehr in einer bestimmten Lage.“
Der seelische Vorgang des Verstehens interessiert uns eben gar nicht. (So wenig, wie der
einer Intuition.)
1 (V): sähe 2 (V): Form
329
330
251
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74
Explanation of Generality
by Examples.
Let’s imagine explaining the concept of a plant. We show someone several objects and
say they are plants. Then he points to another object and asks “Is that a plant as well?”, and
we answer “Yes, that too”, and so on. At one time I would have said that now he has seen
the concept “plant” – the specific common element – in what he was shown, and that he
sees1
the examples used in the explanation differently when he sees this concept in them
from when he understands them, say, just as representatives of this particular shape2
and
colour. (Just as I also used to say that he sees something in a variable – when he understands
it as such – that he doesn’t see in the sign for the particular case.) But the notion of
“seeing something in something” is taken from the case where I see the figure |||| , for
example, “phrased” differently. But in that case – and in a different sense – I really am seeing
different figures, and what they have in common, aside from their similarity, is that they
were caused by the same physical image.
But this explanation cannot simply be applied to the case of understanding variables, or
of examples of the concept “plant”. For let’s assume that we really had seen something else
in them than in plants that were shown only for their own sake; then the question is: Can
this or some other image justify us in applying them as variables? So I might have shown
someone the plants as an explanation, and also have given him a potion that causes him to
see the examples in that particular way. (Just as it’s possible that someone who is drunk
always sees the group|||| as||| | .) And in this way the explanation of the concept would
have been given unambiguously, and whoever understood it would have got this image from
the specimens shown to him and from the accompanying gestures. But that’s not the way it
is. – For it is quite possible that whoever sees, e.g., the sign |||||| as a sign for the number
6 sees it differently (sees something different in it) from someone who only understands
it as a sign for “some”, because he will not direct his attention towards the same thing; but
it depends on the system of rules that are valid for these signs, and understanding will not
amount to seeing the sign in a particular way.
So it’s possible to say “Now I no longer see this as a rose, but only as a plant”!
Or: “Now I see it only as a rose, but no longer as this rose”.
“I see the spot simply as in the square, and no longer in a particular position.”
The psychological process of understanding simply doesn’t interest us. (Any more than
the psychological process of an intuition.)
1 (V): he would see 2 (V): form
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”Es ist doch gar kein Zweifel, daß der, welcher die Beispiele als beliebige Fälle zur
Veranschaulichung des Begriffs versteht, etwas andres versteht, als der, welcher sie als
bestimmt begrenzte Aufzählung auffaßt“. Sehr richtig, aber was versteht der erste also, was
der zweite nicht versteht? Nun, er sieht eben nur Beispiele in den vorgezeigten Dingen, die
nur gewisse Züge aufweisen3
sollen, aber er meint nicht, daß ich ihm im Übrigen diese Dinge
um ihrer selbst willen zeige. –
Ich möchte die eine Klasse4
”logisch begrenzt“, die andere ”logisch nicht begrenzt“
nennen.
Ja, aber ist es denn so, daß er nun tatsächlich nur diese Züge an den Dingen sieht? Etwa
am Blatt nur das, was allen Blättern gemeinsam ist? Das wäre so, als sähe er alles übrige
”in blanco“. Also gleichsam ein unausgefülltes Formular, in dem die wesentlichen Züge
vorgedruckt sind. (Aber die Funktion ”f( . . . )“ ist ja so ein Formular.)
Aber was ist denn das für ein Prozeß, wenn mir Einer mehrere verschiedene Dinge
als Beispiele für einen Begriff5
zeigt, um mich darauf zu führen, das Gemeinsame in ihnen
zu sehen; und wenn ich es suche und nun wirklich sehe?6
Er kann mich auch auf das
Gemeinsame aufmerksam machen. – Bringt er aber dadurch hervor, daß ich den Gegenstand
anders sehe? Vielleicht auch, denn ich kann jedenfalls besonders auf einen seiner Teile schauen,
während ich sonst etwa alle gleichmäßig deutlich gesehen hätte. Aber dieses Sehen ist nicht
das Verstehen des Begriffs. Denn wir sehen nicht etwas mit einer leeren Argumentstelle.
Man könnte auch fragen: Sieht der, welcher das Zeichen ”||| . . .“ als Zeichen des
Zahlbegriffs (im Gegensatz zu ”|||“, welches 3 bezeichnen soll) auffaßt, jene erste Gruppe
von Strichen anders, als die zweite? Aber auch wenn er sie anders – gleichsam, vielleicht,
verschwommener – sieht, sieht er da etwa das Wesentliche des Zahlbegriffs? Hieße das nicht,
daß er dann ”||| . . .“ und ”|||| . . .“ tatsächlich nicht voneinander müßte unterscheiden
können? (Wenn ich ihm (nämlich) etwa den Trank eingegeben hätte, der ihn den Begriff
sehen macht.)7
Denn wenn ich sage: Er bewirkt dadurch, daß er uns mehrere Beispiele zeigt, daß wir
das Gemeinsame in ihnen sehen und von dem Übrigen absehen, so heißt das eigentlich,
daß das Übrige8
in den Hintergrund tritt, also gleichsam blasser wird (und warum soll es
dann nicht ganz verschwinden) und ”das Gemeinsame“, etwa die Eiförmigkeit, allein im
Vordergrund bleibt.
Aber so ist es nicht. Übrigens wären die mehreren Beispiele nur ein technisches
Hilfsmittel, und wenn ich einmal das Gewünschte gesehen hätte, so könnte ich’s auch in
einem Beispiel sehen. (Wie ja auch ”(∃x).fx“ nur ein Beispiel enthält.)
Es sind also die Regeln, die von dem Beispiel gelten, die es zum Beispiel machen. –
Nun genügt aber doch heute jedenfalls das bloße Begriffswort ohne eine Illustration, um
sich mir verständlich zu machen9
(und die Geschichte des Verständnisses interessiert uns ja
nicht) z.B., wenn mir Einer sagt ”forme ein Ei“; und ich will doch nicht sagen, daß ich etwa
dabei den Begriff des Ei’s vor meinem inneren Auge sehe, wenn ich diesen Befehl (und das
Wort ”Ei“) verstehe.10
3 (V): aufzeigen
4 (V): Aufzählung
5 (V): Beispiele eines Begriffes
6 (V): ich es nun suche und wirklich sehe?
7 (V): läßt.)
8 (V): übrige
9 (V): um sich mit mir zu verständigen
10 (O): (und das Wort ”Ei“ verstehe.
331
332
252 Erklärung der Allgemeinheit durch Beispiele.
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“There is no doubt whatsoever that someone who understands the examples as arbitrary
cases chosen to illustrate a concept understands something different from someone who
understands them as an enumeration with definite limits.” Quite true, but what does the
former understand that the latter doesn’t? Well, all he sees in the objects he is shown are
examples, examples that are supposed to exhibit3
only certain traits; he doesn’t think that in
addition I’m showing him these things for their own sake. –
I would like to call the one class4
“logically bounded”, and the other “logically
unbounded”.
Yes, but is it true that he really sees only these features in the things? In a leaf, say, does
he see only what is common to all leaves? That would be as if he saw everything else “as
blank”, i.e. as if he saw a form that listed only the essential features, but with the spaces
next to them not filled in. (But the function “f( . . . )” is just such a form.)
But what kind of a process is it when someone shows me several different things as
examples of a concept to get me to see what they have in common, and when I look for it
and then actually5
see it? He can also call my attention to what is common. – But in doing
this does he bring it about that I see the object differently? Perhaps he does this as well,
for in any case I can look at one of its parts in particular, whereas otherwise, say, I would
have seen all of them with equal clarity. But this seeing is not the understanding of the
concept. For we don’t see something with an empty argument place.
One might also ask: Does someone who understands the sign “||| . . .” as a sign for the
concept of number (in contrast to “||| ”, which is supposed to stand for 3) see that first
group of lines differently from the second? But even if he does see it differently – perhaps
more blurred, as it were – does he really see the essence of the concept of number there?
Wouldn’t that mean that he would actually have to be unable to distinguish “||| . . .”
from “|||| . . .”? (That is, if I had perhaps given him a potion that makes6
him see the
concept.)
For if I say: By giving us several examples he causes us to see the common element in
them and to disregard the rest, that really means that the rest recedes into the background,
i.e. becomes paler, as it were (and why shouldn’t it then disappear altogether?), and “the
common element”, say the oval shape, is the only thing that remains in the foreground.
But that isn’t the way it is. Apart from anything else, the multiple examples would be no
more than a technical aid, and once I had seen what I was supposed to see, I could also see
it in a single example. (As indeed “(∃x).fx” contains only one example.)
So it is the rules that are valid for the example that make it into an example. –
But in any case, at present the mere word for a concept, without any illustration, is enough
to get it across to me7
(for the history of my understanding doesn’t interest us); as when
someone says to me, for example, “Sculpt an egg”; I certainly don’t want to say that when
I understand this command (and the word “egg”) I see the concept of an egg before my
mind’s eye.
3 (V): show
4 (V): enumeration
5 (V): and when I now look for it and actually
6 (V): lets
7 (V): to communicate with me
Explanation of Generality by Examples. 252e
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Wenn wir eine Anwendung des Begriffes ”Ei“ oder ”Pflanze“ machen, so schwebt uns
gewiß nicht vorerst ein allgemeines Bild vor, oder bei dem Hören des Wortes ”Pflanze“ das
Bild des bestimmten Gegenstandes, den ich dann als eine Pflanze bezeichne. Sondern ich
mache die Anwendung sozusagen spontan. Dennoch gibt es eine Anwendung, von der ich
sagen würde: nein, das habe ich unter ”Pflanze“ nicht gemeint; oder anderseits ”ja, das habe
ich auch gemeint“. Aber heißt das, daß mir diese Bilder vorschwebten11
und ich sie in meinem
Geist ausdrücklich abgewiesen und zugelassen habe? – Und doch hat es diesen Anschein,
wenn ich sage: ”ja, das und das und das habe ich alles gemeint, aber das nicht“. Man könnte
aber fragen: Ja, hast Du denn alle diese Fälle vorausgesehen? und die Antwort würde dann
lauten ”ja“, oder ”nein,12
aber ich dachte mir, es sollte etwas zwischen dieser und dieser
Form sein“, oder dergleichen. Meistens aber habe ich in diesem Moment gar keine Grenzen
gezogen und diese ergeben sich nur auf einem Umweg durch eine Überlegung. Ich sage z.B.
”bring’ mir noch eine ungefähr so große Blume“ und er bringt eine und ich sage: Ja, so eine
habe ich gemeint. So erinnere ich mich vielleicht an ein Bild, was mir vorschwebte, aber aus
diesem geht nicht hervor, daß auch die herbeigebrachte Blume noch zulässig ist. Sondern
hier wende ich eben jenes Bild an. Und diese Anwendung war nicht anticipiert worden.
Was uns interessiert ist nur die exakte Beziehung des Beispiels zu dem Danachhandeln.13
Es wird aus dem Beispiel heraus wieder kalkuliert.
Beispiele sind ordentliche Zeichen, nicht Abfall, nicht Beeinflussung.
Denn uns interessiert nur die Geometrie des Mechanismus. (Das heißt doch, die
Grammatik seiner Beschreibung.)
Wie äußert es sich aber in unsern Regeln, daß die behandelten Fälle fx keine wesentlich
abgeschlossene Klasse sind? – Doch wohl nur durch die Allgemeinheit der allgemeinen
Regeln.14
– Daß sie nicht die Bedeutung für den Kalkül haben, wie eine abgeschlossene
Gruppe von Grundzeichen (etwa den Namen der 6 Grundfarben). Wie anders, als durch
die Regeln, die von ihnen ausgesagt sind. – Wenn ich etwa in einem Spiel die Erlaubnis habe,
eine gewisse Art von Steinen in beliebiger Anzahl zu borgen, andere aber in festgesetzter
Anzahl vorhanden sind, oder das Spiel zwar zeitlich unbegrenzt, aber räumlich begrenzt
ist, haben wir ja wohl denselben Fall. Und der Unterschied zwischen den einen und den
anderen Figuren des Spiels muß eben durch die Spielregeln festgesetzt sein. Es heißt dann
etwa von den15
einen: Du kannst soviele Steine dieser Art nehmen, als Du willst. – Und
nach einem anderen bindenderen16
Ausdruck dieser17
Regel darf ich nicht suchen.
Das heißt, daß der Ausdruck für die Unbegrenztheit der behandelten Einzelfälle (eben)
ein allgemeiner Ausdruck sein wird und kein andrer sein kann, kein Ausdruck, in dem18
die
anderen nicht behandelten Einzelfälle in schattenhafter Weise vorkämen.
Es ist ja klar, daß ich keine logische Summe als Definition des Satzes ”das Kreuz liegt
zwischen den Strichen“ anerkenne. Und damit ist doch alles gesagt.
Eines möchte ich immer sagen, um den Unterschied der Fälle zu erklären, die als Beispiele
für einen Begriff beigebracht werden, von denen, die in der Grammatik eine bestimmte
abgeschlossene Gruppe bilden. Wird nämlich zuerst erklärt ”a, b, c, d sind Bücher. – Nun
11 (V): Bilder vorgeschwebt haben
12 (O): ”nein“,
13 (V): Beispiels zum Folgen.
14 (O): Regel.
15 (O): der
16 (V): exakteren
17 (V): der
18 (O): Ausdruck, indem
333
334
335
253 Erklärung der Allgemeinheit durch Beispiele.
WBTC069-77 30/05/2005 16:16 Page 506
When we use the concept “egg” or “plant” we don’t start out by having a general picture
in mind, nor when I hear the word “plant” is there present a picture of the particular object
that I then call a plant. Rather, I apply the word spontaneously, as it were. Nevertheless,
there are uses where I would say “No, I didn’t mean that by ‘plant’ ”, or on the other hand
“Yes, I meant that too”. But does that mean that these pictures were before my mind and
that in my mind I expressly rejected and admitted them? – And yet that is what it looks like
when I say: “Yes, I meant all of those things, but not that.” But one could ask: “Did you
really foresee all of those cases?”, and the answer would be “Yes” or “No, but I imagined
it ought to be something between this shape and that one”, or the like. But usually at that
moment I haven’t drawn any boundaries, and they arise only in a roundabout way via
reflection. For instance, I say “Bring me another flower that is about this big”, he brings
one and I say: “Yes, I meant one like that”. Perhaps then I remember a picture that was
before my mind, but it doesn’t follow from this that the flower that was brought to me is
still acceptable. Rather, I am simply applying that picture here. And this application had
not been anticipated.
The only thing that interests us is the exact relationship of the example to acting in accordance
with it.8
An example is the point of departure for further calculation.
Examples are fully-fledged signs, not left-overs. They don’t pressure us.
For the only thing that interests us is the geometry of the mechanism. (And that means,
the grammar of its description.)
But how does it come out in our rules that the cases of fx that we dealt with are not an
essentially closed class? – Surely only in the generality of the general rules. – In that they
don’t have the same meaning for the calculus as a closed group of basic signs (as for instance
that of the names of the 6 primary colours). How else does it come out other than in the
rules that have been given for them? – If, say, in some game I am allowed to help myself to
as many pieces of a certain kind as I like, while only a set number of another kind is available,
or if the game is unlimited in time but limited in space, then we have the same case.
And the distinction between one kind of piece and another kind must have been established
by the rules of the game; for example, they will say about the one kind: you can take as many
pieces of that kind as you want. – And I ought not look for another more binding expression
of this rule.9
That means that the expression of the infinity of the individual cases in question will be
a general expression; it cannot be of any other kind; it cannot be an expression in which the
other individual cases that were not considered appear in a shadowy way.
Indeed it’s clear that I do not recognize any logical sum as a definition of the proposition
“The cross is between the lines”. And really that says everything that is to be said.
There is one thing I always want to say to explain the distinction between instances
that are brought forward as examples of a concept and those that make up a definite closed
group in grammar. For if I begin by explaining “A, B, C, D are books”, and then I say to
8 (V): to following it. 9 (V): more exact expression of the rule.
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bringe mir ein Buch“ und er bringt eines, das von allen gezeigten verschieden ist, so kann
dennoch gesagt werden, er habe ganz richtig nach der aufgestellten Regel gehandelt. Hätte
es aber geheißen ”a, b, c, d sind meine Bücher. – Bringe mir eines von meinen Büchern“,
so wäre es falsch gewesen, überhaupt ein weiteres19
zu bringen und die Antwort hätte gelautet:
Ich habe Dir doch gesagt, daß a, b, c, d meine Bücher sind. Im ersten Fall handelt der der
Regel nicht zuwider, der einen anderen Gegenstand bringt, als die in der Regel genannten,
im zweiten Fall würde er dadurch der Regel zuwider handeln. Wenn Du aber auch nur a,
b, c, d im Befehl nanntest, aber die Handlung f(e) als Befolgung des Befehls ansahst, heißt
das nicht, daß Du mit F(a, b, c, d . . . ) doch F(a, b, c, d, e) meintest? Oder, wie unterscheiden
sich diese Befehle, wenn sie doch von dem Selben befolgt werden? – Ja, aber es
hätte ja auch f(g) mit dem Befehl übereingestimmt und nicht nur f(e). – Gut, dann meintest
Du eben mit dem ersten Befehl: F(a, b, c, d, e, g) u.s.f.20
Was immer Du mir bringst,
ich hätte es doch in einer Disjunktion einschließen können. Wenn wir also eine Disjunktion
aller von uns tatsächlich gebrauchten Fälle konstruieren, wie würde sich die syntaktisch von
dem allgemeinen Satz unterscheiden? Denn wir dürfen nun nicht sagen: dadurch,21
daß der
allgemeine Satz auch noch durch r (das nicht in der Disjunktion steht) wahr gemacht wird.
Denn dadurch unterscheidet sich der allgemeine Satz nicht von einer Disjunktion, die r
enthält. (Und also ist auch jede andere ähnliche Antwort unmöglich.) Wohl aber wird es
einen Sinn haben, zu sagen: F(a, b, c, d, e) ist die Disjunktion aller tatsächlich von uns
gebrauchten Fälle, aber auch andere Fälle (es wird natürlich keiner erwähnt) machen den
allgemeinen Satz ”F(a, b, c, d, . . . )“ wahr. Während man hierin natürlich nicht den
allgemeinen Satz für F(a, b, c, d, e) einsetzen kann.
Es ist übrigens hier gerade wichtig, daß die Parenthese22
im vorigen Satz ”und also ist
auch jede andere ähnliche Antwort unmöglich“ unsinnig23
ist, weil man zwar verschiedene
besondere Fälle als Beispiele einer Allgemeinheit angeben24
kann, aber nicht verschiedene
Variable, da die Variablen r, s, t sich ihrer Bedeutung nach nicht unterscheiden.
Man könnte dann freilich nicht sagen, wir befolgen F(∃) anders, wenn wir f(d) tun, als
eine Disjunktion, in welcher25
f(d) vorkommt, denn F(∃) = F(∃) 1 f(d). Wem der Befehl
gegeben wird ”hole mir irgend eine Pflanze, oder diese“ (von welcher ihm ein Bild mitgegeben
wird), der wird dieses Bild ruhig beiseite legen und sich sagen ”da es irgend eine tut, so
geht mich dieses Bild nichts an“. Dagegen werden wir das Bild nicht einfach beiseite legen
dürfen, wenn es uns mit fünf anderen gegeben wurde und der Befehl lautete, eine von diesen
sechs Pflanzen zu bringen. (Es kommt also darauf an, in welcher Disjunktion sich der
besondere Befehl befindet.) Und nach dem Befehl ”f(a) 1 f(b) 1 f(c)“ wird man sich anders
richten, als nach dem Befehl ”f(∃)“ (= f(∃) 1 f(c)), auch wenn man jedes Mal f(c) tut. –
Das Bild f(c) geht in f(∃) unter. (Und es hilft uns ja nichts, in einem Kahn zu sitzen, wenn
wir mitsamt ihm unter Wasser sind und sinken.) Man möchte (uns) sagen: Wenn Du auf
den Befehl ”f(∃)“ f(c) tust, so hätte Dir ja auch f(c) ausdrücklich erlaubt sein können, und
wie hätte sich dann der allgemeine Befehl von einer Disjunktion unterschieden? – Aber
auf diese Erlaubnis hättest Du Dich eben, in der Disjunktion mit dem allgemeinen Satz,
gar nicht stützen können.
Ist es also so, daß der Befehl ”bringe mir eine Blume“ nie durch den Befehl ersetzt
werden kann von der Form ”bringe mir a oder b oder c“, sondern immer lauten muß ”bringe
mir a oder b oder c, oder eine andere Blume“?
19 (V): fünftes
20 (O): F(a, b, c, d, e, g) u.s.f.
21 (V): sagen, dadurch,
22 (O): Paranthese
23 (V): unmöglich“ ein Unsinn
24 (V): geben
25 (V): Disjunktion, worin
336
337
254 Erklärung der Allgemeinheit durch Beispiele.
WBTC069-77 30/05/2005 16:16 Page 508
someone – “Now bring me a book”, and he brings one that’s different from all of the ones
I’ve shown him, he can still be said to have complied completely with the given rule. But if
what had been said had been “A, B, C, D are my books. – Bring me one of my books”, it
would have been incorrect to bring any other10
one, and the response would have been: “I
told you that A, B, C, D are my books”. In the first case someone who brings an object that
is different from the ones named in the rule doesn’t violate the rule; in the second case doing
this very same thing would violate it. But even if in the command you named only a, b, c,
and d, and yet you regarded the act f(e) as obeying the command, doesn’t that mean that by
F(a, b, c, d, . . . ) you meant F(a, b, c, d, e) after all? Or how are these commands distinct
from each other, if the same thing obeys them? – Sure, but f(g) too would have been in accordance
with the command, not just f(e). Very well, then by your first command you simply
meant F(a, b, c, d, e, g), etc. Whatever you bring me I could have included in a disjunction.
So if we construct a disjunction of all the cases we actually use, how would it differ syntactically
from the general proposition? For we’re not allowed to say: By the fact that the general
proposition is also made true by r (which doesn’t occur in the disjunction). Because that
doesn’t distinguish the general proposition from a disjunction that contains r. (And thus every
other similar answer is impossible as well.) But it will make sense to say: F(a, b, c, d, e) is
the disjunction of all of the cases we have actually used, but other cases too (of course none
will be mentioned) make the general proposition “F(a, b, c, d, . . . )” true. Whereas of course
you can’t insert the general proposition here in place of F(a, b, c, d, e).
By the way, it is important at this point that the parenthesis in the previous paragraph
“(And thus every other similar answer is impossible as well.)” is nonsensical11
, because one
can point out12
different particular cases as examples of a generalization, but not different
variables, because the variables r, s, t don’t differ in their meaning.
Of course you couldn’t say that when we do f(d) we obey f(∃) differently from the way
we obey a disjunction containing f(d), because f(∃) = f(∃) 1 f(d). Whoever is given the
command “Bring me any plant, or this one” (giving him a picture of it) will put aside the
picture without any qualms and say to himself “Since any one will do, this picture doesn’t
matter”. By contrast, we wouldn’t be allowed simply to put the picture aside if we were given
it along with five others and the command was to bring one of these six plants. (So what
matters is which disjunction the particular command is located in.) And one will be guided
differently by the command “f(a) 1 f(b) 1 f(c)” than by the command “f(∃)” (= f(∃)1 f(c)),
even if in each case one does f(c). – The picture f(c) drowns in f(∃). (And it doesn’t help us
to be sitting in a boat, if we and it are under water and sinking.) One is inclined to say: “If
you do f(c) at the command f(∃) then you might also have been expressly permitted to do
f(c), and in that case how would the general command have differed from a disjunction?” –
But in the disjunction that contains the general proposition it is precisely this permission
that you could not have relied on.
So is it the case that the command “Bring me a flower” can never be replaced by a
command of the form “Bring me a or b or c”, but that it must always be “Bring me a or b
or c or some other flower”?
10 (V): bring a fifth
11 (V): nonsense
12 (V): give
Explanation of Generality by Examples. 254e
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Aber warum tut der allgemeine Satz so unbestimmt, wenn ich ja doch jeden Fall, der wirklich
eintritt, auch im Voraus hätte beschreiben können?
Aber auch das scheint mir noch nicht den wichtigsten Punkt dieser Sache zu treffen. Weil
es, wie ich glaube, nicht eigentlich auf die Unendlichkeit der Möglichkeiten ankommt,
sondern auf eine Art von Unbestimmtheit. Ja, gefragt, wie viele Möglichkeiten es denn für
einen Kreis im Gesichtsfeld26
gäbe, innerhalb eines bestimmten Vierecks zu liegen, könnte
ich weder eine endliche Zahl nennen, noch sagen, es gäbe unendlich viele (wie in der
euklidischen Ebene). Sondern wir kommen hier zwar nie zu einem Ende, aber die Reihe
ist nicht endlos im Sinne von |1, ξ, ξ + 1|.
Sondern, kein Ende, zu dem wir kommen, ist wesentlich das Ende. Das heißt, ich
könnte immer sagen: ich seh’ nicht ein, warum das alle Möglichkeiten sein sollen. – Und
das heißt doch wohl, daß es sinnlos ist, von ”allen Möglichkeiten“ zu sprechen. Der Begriff
”Pflanze“ und ”Ei“ wird also von der Aufzählung gar nicht angetastet.
Wenn wir auch sagen, wir hätten die besondere Befolgung f(a) immer als möglich
voraussehen können, so haben wir dies doch in Wirklichkeit nie getan. – Aber selbst, wenn
ich die Möglichkeit f(a) vorhersehe und ausdrücklich in meinen Befehl aufnehme, so verliert
sie sich neben dem allgemeinen Satz und zwar, weil ich eben aus dem allgemeinen Satz
ersehe, daß dieser besondere Fall erlaubt ist, und nicht einfach daraus, daß er im Befehl als
erlaubt festgesetzt ist. Denn, steht der allgemeine Satz da, so nützt mir das Hinzusetzen des
besonderen Falles nichts mehr (d.h. es macht den Befehl nicht expliciter). Denn nur aus
dem allgemeinen Satz leite ich ja die Rechtfertigung her, diesen besonderen Fall neben ihn
zu setzen. Man könnte nämlich glauben, und darauf geht ja meine ganze Argumentation
aus, daß durch das Hinzusetzen des besonderen Falles die – gleichsam verschwommene –
Allgemeinheit des Satzes aufgehoben wird; daß man sagen könnte27
”jetzt brauchen wir sie
nicht mehr, wir haben ja hier den bestimmten Fall“. Ja, aber wenn ich doch zugebe, daß ich
den besonderen Fall darum hierhersetze, weil er mit dem allgemeinen Satz übereinstimmt!
Oder, daß ich doch anerkenne, daß f(a) ein besonderer Fall von f(∃) ist! Denn nun kann
ich nicht sagen: das heißt28
eben, daß f(∃) eine Disjunktion ist, deren ein Glied f(a) ist.
Denn wenn dies so ist, so muß sich diese Disjunktion angeben lassen. f(∃) muß dann als
eine Disjunktion definiert sein. Eine solche Definition wäre auch ohne weiteres zu geben,
sie entspräche aber nicht dem Gebrauch von f(∃), den wir meinen. Nicht so, daß
die Disjunktion immer noch etwas übrig läßt; sondern, daß sie das Wesentliche der
Allgemeinheit gar nicht berührt, ja, wenn man sie dieser beifügt, ihre Rechtfertigung erst
von dem allgemeinen Satz bezieht.29
Ich befehle zuerst f(∃); er befolgt den Befehl und tut f(a). Nun denke ich, ich hätte ihm
ja gleich den Befehl ”f(∃) 1 f(a)“ geben können. (Denn, daß f(a) den Befehl f(∃) befolgt,
wußte ich ja früher und es kam ja auf dasselbe hinaus, ihm f(∃) 1 f(a) zu befehlen.) Und
dann hätte er sich also bei der Befolgung nach der30
Disjunktion ”tue Eines oder f(a)“ gerichtet.
Und ist es, wenn er den Befehl durch f(a) befolgt, nicht gleichgültig, was in Disjunktion mit
f(a) steht? Wenn er auf jeden Fall f(a) tut, so ist ja doch der Befehl befolgt, was immer die
Alternative ist.
Ich möchte auch sagen: In der Grammatik ist nichts nachträglich, keine Bestimmung nach
einer andern, sondern alles ist zugleich da.
26 (V): Gesichtsraum
27 (V): wird. Man könnte sagen
28 (V): beweist
29 (V): nimmt.
30 (V): einer
338
339
255 Erklärung der Allgemeinheit durch Beispiele.
WBTC069-77 30/05/2005 16:16 Page 510
But why does the general proposition pretend to be so indeterminate, since I really could
just as well have described in advance every case that actually occurs?
But even that still doesn’t seem to me to hit upon the most important point of this
matter. I believe that what matters isn’t really the infinity of possibilities, but a kind of
indeterminacy. Indeed, if I were asked how many possibilities a circle in my visual field13
had of being within a particular square, I could neither name a finite number, nor say that
there were infinitely many (as in a Euclidean plane). Rather, although we never come to an
end here, the series isn’t endless in the sense in which |1, ξ, ξ + 1| is.
Rather, no end to which we come is essentially the end. That is, I could always say: I
don’t see why these should be all the possibilities. – And that means, after all, that it makes
no sense to speak of “all of the possibilities”. So the concepts “plant” and “egg” are not even
touched by enumeration.
Even if we say that we could always have foreseen the particular compliance f(a) as
possible, still we never did so in fact. – But even if I do foresee the possibility f(a) and expressly
include it in my command, it gets lost in comparison to the general proposition, because I
can see from the general proposition itself that this particular case is permitted, and not
simply from the fact that it is stipulated as being permitted in the command. For if the
general proposition is there, then the addition of the particular case is of no further use to
me (that is, it doesn’t make the command more explicit). For it is only from the general
proposition that I derive the justification for placing this particular case next to it. For one
could believe – and this is the whole point of my argument – that the “blurred” generality
of the proposition is eliminated by adding on the specific case; that one14
could say “Now
we don’t need it any more, for here we have the particular case”. Yes, but I am admitting
that I’m putting the particular case over here because it agrees with the general proposition!
Or: I am acknowledging that f(a) is a particular case of f(∃)! For now I can’t say: That
simply means15
that f(∃) is a disjunction with f(a) as one of its terms. For if that is so, then
this disjunction must be capable of being stated. f(∃) must then be defined as a disjunction.
To be sure, such a definition could easily be given, but it wouldn’t correspond to the use
of f(∃) that we have in mind. It’s not that the disjunction always leaves something remaining;
rather, it’s that it doesn’t even touch what is essential to generality; indeed, if it is
added to generality it is only by the general proposition that it gets justified.
First I command f(∃); he obeys the command and does f(a). Then I think that I could
have given him the command “f(∃) 1 f(a)” in the first place. (For I knew in advance that
f(a) obeyed the command f(∃), and it amounted to the same thing to command him to do
f(∃) 1 f(a).) And then when he obeyed the command, he would have been guided by the16
disjunction “Do something or f(a)”. And if he obeys the command by doing fa, isn’t it
irrelevant what is in the disjunction with f(a)? If he does f(a) in any case, then the command
has been obeyed, whatever the alternative is.
I would also like to say: In grammar nothing comes later, no determination comes after
the first one; rather, everything is there all together.
13 (V): visual space
14 (V): case. One
15 (V): proves
16 (V): a
Explanation of Generality by Examples. 255e
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Insofern kann ich also (auch) nicht sagen, ich habe zuerst den Befehl f(∃) gegeben und
bin dann erst draufgekommen, daß f(a) ein Fall von f(∃) ist; jedenfalls aber war und blieb
mein Befehl f(∃), und f(a) setzte ich dazu in der Erkenntnis,31
daß f(a) mit f(∃) übereinstimmt.
Und diese Bestimmung, daß f(a) mit f(∃) übereinstimmt, setzt doch eben den Sinn des Satzes
f(∃) voraus, wenn er überhaupt selbständig festgehalten wird, und nicht erklärt wird, er sei
durch eine Disjunktion zu ersetzen. Und mein Satz ”jedenfalls war und blieb aber mein
Befehl f(∃) u.s.w.“ hieß nur, daß ich den allgemeinen Befehl nicht durch eine Disjunktion
ersetzt hatte.
Man kann sich nun denken, daß ich einen Befehl p 1 f(a) gebe und der Andre den ersten
Teil des Befehls nicht deutlich versteht, wohl aber, daß der Befehl ” . . . 1 f(a)“ lautet. Er
könnte dann f(a) tun und sagen ”ich weiß gewiß, daß ich den Befehl befolgt habe, wenn ich
auch den ersten Teil nicht verstanden habe“. So nun denke ich es mir auch, wenn ich sage,
es käme ja auf die andere Alternative nicht an. Aber dann hat er doch nicht den gegebenen
Befehl befolgt, sondern ihn als Befehl f(a) aufgefaßt.32
Man könnte fragen: Hat der, welcher
auf den Befehl ”f(∃) 1 f(a)“ f(a) tut, den Befehl darum (d.h. insofern) befolgt, weil der Befehl
von der Form ξ 1 f(a) ist, oder darum, weil f(∃) 1 f(a) = f(∃) ist? Wer f(∃) versteht, also
weiß, daß f(∃) 1 f(a) = f(∃) ist, der befolgt durch f(a) f(∃), auch wenn ich es ”f(∃) 1 f(a)“
schreibe, weil er ja doch sieht, daß f(a) ein Fall von f(∃) ist. – Und nun kann man uns
entgegenhalten: Wenn er sieht, daß f(a) ein Fall von f(∃) ist, so heißt das ja doch, daß f(a)
disjunktiv33
in f(∃) enthalten ist, daß also f(∃) mit Hilfe von f(a) definiert ist! Und – muß
er jetzt weiter sagen – die übrigen Teile der Disjunktion gehen mich eben nichts an, wenn
die Glieder, die ich sehe, alle sind, die ich jetzt brauche. ”Du hast eben mit der Erklärung
,daß f(a) ein Fall von f(∃) ist‘ nichts weiter gesagt, als daß f(a) in f(∃) vorkommt, und noch
andere Glieder.“ – Aber gerade das meinen wir nicht. Und es ist nicht so, als hätten wir
durch unsere Bestimmung f(∃) unvollständig34
definiert. Denn dann wäre ja eine vollständige
Definition möglich. Und es wäre diejenige Disjunktion, nach welcher das angehängte
”1 f(∃)“ gleichsam lächerlich wäre, weil ja doch nur die aufgezählten35
Fälle für uns in
Betracht kämen. Wie wir aber f(∃) auffassen, ist die Bestimmung, daß f(a) ein Fall von f(∃)
ist, keine unvollkommene, sondern gar keine Definition von f(∃). Ich nähere mich also auch
nicht dem Sinn von f(∃), wenn ich die Disjunktion der Fälle vermehre; die Disjunktion der
Fälle 1 f(∃) ist zwar gleich f(∃), aber niemals gleich der Disjunktion der Fälle, sondern ein
ganz anderer Satz.
Auf keinem Umweg kann, was über eine Aufzählung von Einzelfällen gesagt wird,36
die
Erklärung der Allgemeinheit sein.37
Kann ich denn aber die Regeln des Folgens in diesem Fall angeben? Denn, wie weiß ich,
daß gerade aus fa (∃x).fx folgt? ich kann ja doch nicht alle Sätze angeben, aus denen es
folgt. – Das ist aber auch gar nicht nötig; folgt (∃x).fx aus fa, so war das jedenfalls vor jeder
besonderen Erfahrung zu wissen, und möglich, es in der Grammatik anzugeben.
Ich sagte ”es war möglich, vor jeder Erfahrung zu wissen, daß (∃x).fx aus fa folgt und es
in der Grammatik anzugeben“. Es sollte aber heißen: ”(∃x).fx folgt aus fa“ ist kein Satz
(Erfahrungssatz) der Sprache, der ”(∃x).fx“ und ”fa“ angehören, sondern eine in ihrer
Grammatik festgesetzte Regel.
31 (V): dazu wissend,
32 (V): ihn als ”fa!“ aufgefaßt.
33 (V): disjunktiv ist
34 (V): f(∃) unvollkommen defin.
35 (V): genannten
36 (V): gesagt ist,
37 (V): ergeben.
340
256 Erklärung der Allgemeinheit durch Beispiele.
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And so neither can I say that I first gave the command f(∃) and only then realized that
f(a) is a case of f(∃); but at any rate, my command was and remained f(∃), and I added f(a),
having recognized17
that f(a) is in accordance with f(∃). And the stipulation that f(a) is in
accordance with f(∃) presupposes the sense of the proposition f(∃), when one states it
independently and doesn’t declare that it can be replaced by a disjunction. And my sentence
“At any rate, my command was and remained f(∃), etc.” meant only that I had not replaced
the general command with a disjunction.
Now one can imagine that I give a command p 1 f(a) and that the other person doesn’t
understand the first part of the command clearly, but does understand that the command is
“. . . 1 f(a)”. Then he could do f(a) and say “I know for certain that I obeyed the command,
even though I didn’t understand its first part”. And that is also how I imagine it when I say
that the other alternative is irrelevant. But then he hasn’t obeyed the given command after
all, but has taken it as the command f(a).18
One could ask: Has someone who does f(a) in
response to the command “f(∃) 1 f(a)” obeyed the command because (i.e. in so far as) it was
of the form ξ 1 f(a), or because f(∃) 1 f(a) = f(∃)? Whoever understands f(∃), and therefore
knows that f(∃) 1 f(a) = f(∃) is obeying f(∃) by means of f(a), even if I write it as “f(∃) 1
f(a)”, because he none the less sees that f(a) is a case of f(∃). – And now one can say to us in
rebuttal: If he sees that f(a) is a case of f(∃), then that does mean that f(a) is disjunctively
contained in f(∃), and that therefore f(∃) has been defined with the help of f(a)! And – he
now has to continue – the remaining parts of the disjunction simply are of no concern to
me if the terms that I see are all the ones that I now need. “In explaining ‘that f(a) is a case
of f(∃)’ you said nothing more than that f(a) occurs in f(∃), along with other terms as well.”
– But that is precisely what we don’t mean. And it isn’t as if in our stipulation we had incom-
pletely19
defined f(∃). For then a complete definition would be possible. And that would be the
disjunction in which the additional “1 f(∃)” would be ridiculous, as it were, because it would
only be the enumerated20
cases that would concern us. But according to our understanding
of f(∃), the stipulation that f(a) is a case of f(∃) is not an incomplete definition of f(∃), but
no definition of it at all. So neither do I get closer to the sense of f(∃) by increasing the number
of cases in the disjunction; although the disjunction of the cases 1 f(∃) is equivalent to
f(∃), it is never equivalent to the disjunction of the cases; it is a totally different proposition.
There is no detour to make what is said21
about an enumeration of individual cases into22
an explanation of generality.
But in this case can I specify the rules of entailment? For how do I know that it is precisely
fa from which (∃x).fx follows? After all, I can’t specify all the propositions from which
it follows. – But neither is that necessary; if (∃x).fx follows from fa, then that at any rate
was something that could be known before any particular experience, and it was possible to
state it in grammar.
I said: “It was possible to know before any experience that (∃x).fx follows from fa and to
state it in grammar”. But that should read: “ ‘(∃x).fx follows from fa’ is not a proposition
(empirical proposition) of the language to which ‘(∃x).fx’ and ‘fa’ belong; rather, it is a rule
laid down in their grammar”.
17 (V): f(a), knowing
18 (V): as “fa!”.
19 (V): imperfectly
20 (V): named
21 (V): what has been said
22 (V): cases produce
Explanation of Generality by Examples. 256e
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75
Bildungsgesetz einer Reihe.
”U.s.w.“
Man kann für den Gebrauch der Variablen wohl eine Regel aufstellen und es ist kein
Pleonasmus,1
daß wir dabei eben diese Art der Variablen gebrauchen. Denn brauchten wir
sie nicht, so wäre ja durch die Regeln die Variable definiert. Und wir nehmen ja nicht an,
daß sie sich definieren lasse, oder: daß sie definiert werden müsse (denn einmal nehmen die
Definitionen doch ein2
Ende).
Das heißt (nur), daß – z.B. – die Variable ”x2
“ keine Abkürzung ist (etwa für eine
logische Summe) und daß in unserm Gedanken auch nur ein Zeichen dieser Multiplizität
vorhanden ist.
Denn nehmen wir an, ich hätte 7 Spezialfälle3
aufgezählt und sagte ”ihre logische Summe
ist aber nicht der allgemeine Satz“, so ist das nicht genug und ich will noch sagen, daß auch
keine andere Zahl von Spezialfällen4
den allgemeinen Satz ergibt. Aber in diesem Zusatz scheine
ich nun wiederum eine Aufzählung, wenn auch nicht wirklich, so doch quasi schattenhaft
auszuführen. Aber so ist es nicht, denn in dem Zusatz kommen ganz andere Wörter als die
Zahlwörter vor.
”Wie aber soll ich es verbieten, daß ein Zahlwort dort und dort eingesetzt wird? Ich kann
doch nicht vorhersehen, welches Zahlwort Einer5
wird einsetzen wollen, um es zu verbieten“.
– Du kannst es ja verbieten, wenn es kommt. – Aber da sprechen wir ja schon, allgemein,
vom Zahlbegriff!
Was6
aber macht ein Zeichen zum Ausdruck der Unendlichkeit? Was gibt ihm den
eigentümlichen Charakter dessen, was wir unendlich nennen? Ich glaube, daß es sich
ähnlich verhält wie das Zeichen einer enormen Zahl. Denn das Charakteristische des
Unendlichen, wie man es so auffaßt, ist seine enorme Größe.
Aber es gibt nicht etwas, was eine Aufzählung ist und doch keine Aufzählung. Eine
Allgemeinheit, die quasi nebelhaft aufzählt, aber nicht wirklich und bis zu einer bestimmten
Grenze.
Die Punkte in ”1 + 1 + 1 + 1 . . .“ sind eben auch nur die vier Pünktchen. Ein Zeichen,
für das sich gewisse Regeln angeben lassen müssen. (Nämlich dieselben, wie für das Zeichen
”u.s.w. ad inf.“) Dieses Zeichen ahmt zwar die Aufzählung in gewisser Weise nach, ist
aber keine Aufzählung. Und das heißt wohl, daß die Regeln, die von ihm gelten, bis zu
1 (O): Pläonasmus,
2 (V): ihr
3 (V): Fälle
4 (V): Fällen
5 (V): Einer mir e
6 (V): Was macht
341
342
257
WBTC069-77 30/05/2005 16:16 Page 514
75
The Law of the Formation of a Series.
“Etc.”.
To be sure, we can set up a rule for the use of the variables, and the fact that in doing so
we use this same kind of variable is not a pleonasm. For if we didn’t use it, then the variable
would be defined by the rules. And we don’t assume that it can be defined, or that it
must be defined (for at some point definitions do come to an1
end).
This means (only) that – for example – the variable “x2
” is not an abbreviation (say for a
logical sum), and that in our thought as well there is only one sign for this multiplicity.
For let’s assume I had enumerated 7 particular instances2
and said “But their logical sum
isn’t the general proposition”. But that isn’t enough, and so I want to add that neither does
any other number of particular instances3
yield the general proposition. But when I add this
rider I seem once again to be carrying out an enumeration, if not in actuality, still in a kind
of shadowy way. But that’s not so, because words occur in the rider that are completely
different from the numerals.
“But how can I forbid that one numeral is to be inserted in such and such a place? Surely
I can’t foresee what numeral someone will want to insert, so that I can forbid it.” – Well,
you can forbid it when it appears. – But then we are already speaking generally about the
concept of number!
But what turns4
a sign into an expression of infinity? What gives it the peculiar character
of what we call “infinite”? I believe that it behaves like a sign for an enormous number. For
the characteristic feature of the infinite, as it is commonly conceived, is its enormous size.
But there is nothing that is an enumeration and yet not an enumeration – a generality that
enumerates in a cloudy way, as it were, but doesn’t really do so, nor up to a specified limit.
The dots in “1 + 1 + 1 + 1. . . .” are nothing but four little dots: a sign, for which it must
be possible to state certain rules. (Namely, the same rules as for the sign “etc., ad inf.”.)
This sign does imitate enumeration in a way, but it isn’t an enumeration. And that most
1 (V): its
2 (V): 7 instances
3 (V): number of instances
4 (V): What turns
257e
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einem Punkt mit denen, die von einer Aufzählung gelten, übereinstimmen, aber nicht ganz
übereinstimmen.
Es gibt kein Mittelding zwischen der7
bestimmten Aufzählung und dem allgemeinen
Zeichen.8
Man hat natürlich nur die Zahlen bis zu einer gewissen höchsten – sagen wir 1010
– hingeschrieben. Worin besteht nun die Möglichkeit, Zahlen hinzuschreiben, die man
noch nicht hingeschrieben hat? Wie seltsam dieses Gefühl, als wären sie doch schon alle
irgendwie vorhanden! (Frege sagte, eine Konstruktionslinie sei in gewissem Sinne schon
vorhanden, auch ehe sie gezogen wurde.)
Hier ist die Schwierigkeit, sich zu wehren gegen den Gedanken, die Möglichkeit sei eine
Art schattenhafter Wirklichkeit.9
In den Regeln für die Variable a kann eine Variable b vorkommen und auch besondere
Zahlzeichen; aber auch keine Gesamtheit von Zahlen.
Nun scheint es aber, als wäre damit etwas (aus der Logik) weggeleugnet. Etwa gerade die
Allgemeinheit; oder das, was die Punkte andeuten. Das Unfertige (Lockere, Dehnbare) der
Zahlenreihe.10
Und natürlich dürfen und können wir nichts wegleugnen. Wo kommt also
diese Unbestimmtheit zum Ausdruck? Etwa so: Wenn wir Zahlen anführen, die wir statt
der Variablen a einsetzen dürfen, so sagen wir von keiner, es sei die letzte, oder höchste.
Würde uns aber nun nach der Erklärung einer Rechnungsart jemand fragen ”und ist nun
103 das letzte Zeichen, welches ich benützen kann“; was sollen wir antworten? ”Nein, es ist
nicht das letzte“, oder ”es gibt kein letztes“? – Aber muß ich ihn nicht zurückfragen: ”Und
wenn es nicht das letzte ist, was käme dann noch?“ Und sagt er nun ”104“, so müßte ich
sagen: Ganz richtig, Du kannst die Reihe selber fortsetzen.
Von einem Ende der Möglichkeit kann ich überhaupt nicht reden.
(Nur vor dem Geschwätz muß man sich in der Philosophie hüten. Eine Regel aber, die
praktisch anwendbar ist, ist immer in Ordnung.)
Es ist klar, daß man einer Regel von der Art |a, ξ, ξ + 1| folgen kann; ich meine, ohne
schon von vornherein die Reihe hinschreiben zu können, sondern, indem man wirklich der
Bildungsregel folgt.11
Es ist ja dann dasselbe, wie wenn ich eine Reihe etwa mit der Zahl 1
anfinge und sagte: ”nun gib 7 dazu, multipliziere mit 5 und zieh’ die Wurzel, und diese
zusammengesetzte Operation wende immer wieder auf das12
Resultat an“. (Das wäre ja die
Regel |1, ξ, |.)
Schließlich ist ja das Wort ”u.s.w.“ nichts anderes, als das Wort ”u.s.w.“ (d.h. wieder als
ein Zeichen des Kalküls, das nicht mehr tun kann, als durch die Regeln zu bedeuten, die
von ihm gelten. Das nicht mehr sagen kann, als es zeigt.)
D.h. es wohnt13
dem Wort ”u.s.w.“ keine geheime Kraft inne, durch die nun die Reihe
fortgesetzt wird, ohne fortgesetzt zu werden.
Das wohl nicht, wird man sagen, aber eben die Bedeutung der unendlichen Fortsetzung.
( )ξ + ⋅7 5
7 (V): einer
8 (V): und der Variablen.
9 (V): Existenz.
10 (V): Reihe.
11 (V): indem man sich wirklich nach der
Bildungsregel richtet.
12 (V): ihr
13 (V): wohnt in
343
344
258 Bildungsgesetz einer Reihe. . . .
WBTC069-77 30/05/2005 16:16 Page 516
likely means that the rules that apply to it agree up to a point, but not completely, with those
that apply to an enumeration.
There is no intermediate thing between the particular enumeration and the general sign.5
Of course the natural numbers have only been written down up to a certain highest point,
let’s say 1010
. Now what constitutes the possibility of writing down numbers that haven’t yet
been written down? How odd is this feeling, as if in some way all of them already existed!
(Frege said that a construction line was in a certain sense already there, even before it was
drawn.)
The difficulty here is to defend oneself against the thought that possibility is a kind of
shadowy reality.6
In the rules for the variable a, a variable b can occur, and so can particular numerals; but
not any totality of numbers.
But now it seems as if in saying this, something (in logic) had been flatly denied. Say, generality
itself; or what the dots signify. What is unfinished (loose, elastic) in the number series.7
And of course we may not and cannot deny anything. So how does this indeterminacy find
expression? Perhaps thus: if we state the numbers we’re allowed to substitute for the variable
a, we don’t say of any of them that it is the last, or the highest.
But what if someone asks us, after we’ve explained a form of calculation, “And is 103 the
last sign I can use?”? What are we to answer? “No, it isn’t the last”, or “There isn’t any last
one”? – Mustn’t I ask him in turn: “And if it isn’t the last, what might come next?” And if
he says “104”, I’d have to say: “Quite right, you can continue the series yourself”.
In no way can I speak of an end to possibility.
(In philosophy the only thing one must guard against is prattle. But a rule that can be
applied in practice is always in order.)
It is clear that we can follow a rule such as |a, ξ, ξ + 1|. I mean without being able to
write down the series right from the start, but by actually following8
the rule for constructing
the series. For in that case it’s the same as if I were to begin a series, say, with the
number 1 and were then to say “Now add 7, multiply by 5, find the square root of this,
and apply this complex operation over and again to the9
result”. (That would be the rule
|1, ξ, |.)
After all, the word “etc.” is nothing but the word “etc.” (i.e. once again nothing but a sign
in a calculus, which sign can do no more than have meaning via the rules that are valid for
it. Which can’t say more than it shows).
That is, there is no secret power inherent in the expression “etc.”, by which the series is
then continued without being continued.
Of course not, one will say, but what is inherent in it is precisely the meaning of infinite
continuation.
( )ξ + ⋅7 5
5 (V): between a particular enumeration and the
variable.
6 (V): existence.
7 (V): in the series.
8 (V): actually taking as a guideline
9 (V): its
The Law of the Formation of a Series. . . . 258e
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Man könnte nun aber fragen: Wie kommt es, daß der, welcher die allgemeine Regel nun
auf eine weitere Zahl anwendet, nur dieser Regel folgt. Daß keine weitere Regel nötig war,
die ihm erlaubt, die allgemeine auch auf diesen Fall anzuwenden; und daß doch dieser Fall
in der (allgemeinen) Regel nicht genannt war.
Es wundert uns also, daß wir diesen Abgrund zwischen den einzelnen Zahlen und dem
allgemeinen Satz nicht überbrücken können.14
”Kann man sich einen leeren Raum vorstellen?“ (Diese Frage gehört merkwürdigerweise
hierher.)
Es ist einer der tiefstwurzelnden Fehler der Philosophie, die Möglichkeit als einen
Schatten der Wirklichkeit zu sehen.15
Anderseits aber kann es kein Irrtum sein. Und das ist es auch nicht, wenn man den Satz
diesen Schatten nennt.
Die Gefahr ist natürlich hier wieder, in einen Positivismus zu verfallen, nämlich in einen,
der einen eigenen Namen verdient und daher natürlich ein Irrtum sein muß. Denn wir
dürfen überhaupt keine Tendenz haben, keine besondere Auffassung der Dinge, sondern
müssen alles anerkennen, was jeder Mensch darüber je gesagt hat, außer soweit er selbst eine
besondere Auffassung16
oder Theorie hatte.
Denn das Zeichen ”u.s.w.“, oder ein ihm entsprechendes, ist wohl für die Bezeichnung
der Endlosigkeit wesentlich. Natürlich durch die Regeln, die von einem solchen Zeichen
gelten. D.h. wir können wohl das Reihenstück ”1, 1 + 1, 1 + 1 + 1“ unterscheiden von
der Reihe ”1, 1 + 1, 1 + 1 + 1, u.s.w.“. Und das letzte Zeichen und sein Gebrauch ist so
wesentlich für den Kalkül, als irgend ein andres.17
Das, was mich nun bedrückt, ist, daß das ”u.s.w.“ scheinbar auch in den Regeln für das
Zeichen ”u.s.w.“ vorkommen muß. Z.B. ist 1, 1 + 1, u.s.w. = 1, 1 + 1, 1 + 1 + 1, u.s.w.
u.s.w.
Aber haben wir denn hier nicht die alte Erkenntnis, daß wir die Sprache nur von außen
beschreiben können? Daß wir also nicht erwarten dürfen, durch eine Beschreibung der Sprache
in andere Tiefen zu dringen, als die Sprache selbst offenbart: Denn die Sprache beschreiben
wir mittels der Sprache.
Wir könnten sagen: Es ist ja gar kein Anlaß, zu fürchten, daß wir das Wort ”u.s.w.“ in
einer das Endliche übersteigenden Weise gebrauchen.
Übrigens kann der, für das ”u.s.w.“ charakteristische Teil seiner Grammatik nicht in
Regeln über die Verbindung von ”u.s.w.“ mit einzelnen Zahlzeichen (nicht: ”den einzelnen
Zahlzeichen“) bestehen – denn diese Regeln geben ja wieder ein beliebiges Stück einer Reihe
– sondern in Regeln der Verbindung von ”u.s.w.“ mit ”u.s.w.“.
Die Möglichkeit noch weitere Zahlen anzuführen. Die Schwierigkeit scheint uns die
zu sein, daß die Zahlen, die ich tatsächlich angeführt habe, ja gar keine wesentliche
Gruppe sind18
und nichts dies andeutet, daß sie eine beliebige Kollektion sind: die zufällig
aufgeschriebenen unter allen Zahlen.
14 (O): kann.
15 (V): der Philosophie: die Möglichkeit als ein
Schatten der Wirklichkeit.
16 (V): Auffassung de
17 (V): als eines der vorhergehenden.
18 (V): angeführt habe, ja gar nicht wesentlich
sind
345
346
259 Bildungsgesetz einer Reihe. . . .
WBTC069-77 30/05/2005 16:16 Page 518
But one could now ask: How does it happen that someone who applies the general
rule to a further number is following only this rule? How does it happen that no further rule
allowing him to apply the general rule to this additional case was necessary, even though
this case wasn’t mentioned in the (general) rule?
And so we are surprised that we can’t bridge this abyss between individual numbers and
the general proposition.
“Can one imagine an empty space?” (Strangely enough, this question belongs here.)
It is one of the most deep-rooted mistakes of philosophy to see possibility as a shadow of
reality.10
But on the other hand it can’t be an error. And neither is it an error if one calls the proposition
such a shadow.
Here again, of course, there is the danger of lapsing into a kind of positivism, namely one
that ought to have a name of its own and that consequently must certainly be an error. For
we must not have any slant at all, any particular take on things, but must instead acknowledge
everything that anyone has ever said on the topic, except in so far as he himself had
a particular conception or theory.
For the sign “etc.”, or one corresponding to it, is essential for indicating endlessness. Of
course, through the rules that are valid for such a sign. That is, we can distinguish the part
of the series, “1, 1 + 1, 1 + 1 + 1”, from the series “1, 1 + 1, 1 + 1 + 1, etc.”. And this last
sign and its use are as essential for the calculus as any other.11
What disturbs me now is that the “etc.” seemingly has to occur in the rules for the sign
“etc.” as well. For instance, 1, 1 + 1, etc. = 1, 1 + 1, 1 + 1 + 1 etc., etc.
But isn’t what we have here the old realization that we can describe language only from
the outside? That therefore we mustn’t expect that by describing language we shall penetrate
to depths deeper than language itself reveals: for we describe language with language.
We could say: There’s no reason to be afraid that we’ll use the word “etc.” in a way that
transcends the finite.
By the way: the part of its grammar that is distinctive of “etc.” can’t consist in rules for
connecting “etc.” with individual numerals (not: “the individual numerals”) – for such
rules in turn represent an arbitrary part of a series – but rather, in rules for connecting
“etc.” with “etc.”.
The possibility of listing additional numbers. The difficulty seems to be that the numbers
I’ve listed are not in fact in any way an essential group,12
yet nothing indicates that they
are an arbitrary collection – out of all the numbers, those that just happened to have been
written down.
10 (V): philosophy: possibility as a shadow of
reality.
11 (V): as one of the preceding ones.
12 (V): are in no way essential,
The Law of the Formation of a Series. . . . 259e
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(So, als hätte ich in einer Schachtel alle Steine eines Spiels und auf dem Tisch daneben
eine zufällige Auswahl aus dieser Schachtel.
Oder, als wären die einen Ziffern in Tinte nachgezogen, während sie alle schon gleichsam
blaß vorgezeichnet sind.)
Daß wir aber außer diesen zufällig benützten nur die allgemeine Form haben.
Haben wir hier übrigens nicht – so komisch das klingt – den Unterschied zwischen
Zahlzeichen und Zahlen?
Wenn ich z.B. sage ”,Kardinalzahlen‘ nenne ich alles, was aus 1 durch fortgesetztes Addieren
von 1 entsteht“, so vertritt das Wort ”fortgesetzt“ nicht eine nebelhafte Fortsetzung von
1, 1 + 1, 1 + 1 + 1, vielmehr ist auch das Zeichen ”1, 1 + 1, 1 + 1 + 1, . . .“ ganz exakt zu
nehmen; als verschieden von ”1, 1 + 1, 1 + 1 + 1“ anderen bestimmten Regeln unterworfen
und nicht ein Ersatz19
einer Reihe ”die sich nicht hinschreiben läßt“.
Das heißt: Mit dem Zeichen ”1, 1 + 1, 1 + 1 + 1, . . .“ wird auch gerechnet, wie mit (den)
Zahlzeichen, nur nach andern Regeln.
Was bildet man sich denn aber ein? Welchen Fehler macht man denn? Wofür hält man
das Zeichen ”1, 1 + 1, . . .“? D.h.: wo kommt denn das wirklich vor, was man in diesem
Zeichen zu sehen meint? Etwa, wenn ich sage ”er zählte 1, 2, 3, 4 und so weiter bis 1000“?
wo es auch möglich wäre, wirklich alle Zahlen hinzuschreiben.
Als was sieht man denn ”1, 1 + 1, 1 + 1 + 1, . . .“ an?
Als eine ungenaue Ausdrucksweise. Die Pünktchen sind so, wie weitere Zahlzeichen, die
aber undeutlich sind. So, als hörte man auf, Zahlzeichen hinzuschreiben, weil man ja doch
nicht alle hinschreiben kann, aber als seien sie wohl, gleichsam in einer Kiste vorhanden.20
Etwa auch, wie wenn ich von einer Melodie nur die ersten Töne deutlich singe und den
Rest nur noch andeute und in Nichts auslaufen lasse. (Oder wenn man beim Schreiben von
einem Wort nur wenige Buchstaben deutlich schreibt und mit einem unartikulierten Strich
endet.) Wo dann dem Kundeutlich“ ein Kdeutlich“ entspräche.
Ich habe einmal gesagt, es könne nicht Zahlen geben und den Begriff der Zahl. Und das
ist richtig, wenn es heißt, daß die Variable zur Zahl nicht so steht, wie der Begriff Apfel zu
einem Apfel (oder der Begriff Schwert zu Nothung).21
Anderseits ist die Zahlvariable kein Zahlzeichen.
Ich wollte aber auch sagen, daß der Zahlbegriff nicht unabhängig von den Zahlen
(gegeben) sein könnte, und das ist nicht wahr. Sondern die Zahlvariable ist in dem Sinne
von einzelnen Zahlen unabhängig, als es einen Kalkül mit einer Klasse unsrer Zahlzeichen,
und ohne die allgemeine Zahlvariable, wohl gibt. Freilich gelten dann eben nicht alle Regeln
von diesen Zahlzeichen, die von unsern gelten, aber doch entsprechen sie unseren, wie die
Damesteine im Damespiel denen im Schlagdamespiel.
Wogegen ich mich wehre, ist die Anschauung, daß eine22
unendliche Zahlenreihe etwas
uns Gegebenes sei, worüber es nun spezielle Zahlensätze und auch allgemeine Sätze über
alle Zahlen der Reihe gibt. So daß der arithmetische Kalkül nicht vollständig wäre, wenn er
nicht auch die allgemeinen Sätze über die Kardinalzahlen enthielte, nämlich allgemeine
Gleichungen der Art a + (b + c) = (a + b) + c. Während schon 1 : 3 = 0,3 einem andern
347
348
260 Bildungsgesetz einer Reihe. . . .
19 (V): Vertreter
20 (V): aber als seien sie allerdings, quasi, in einer
Kiste, vorhanden.
21 (E): Nothung ist das Schwert Siegfrieds im
Nibelungenlied.
22 (V): die
WBTC069-77 30/05/2005 16:16 Page 520
(As if I had all the pieces of a game in a box and next to it on the table an arbitrary selection
from the box.
Or as if all the numerals had been faintly outlined, as it were, and then some of them had
been filled in in ink.)
And nothing indicates that, apart from these numbers that we happen to have used, we
have only the general form.
By the way, don’t we have here the distinction between numerals and numbers, as funny
as that may sound?
If I say, for example, “I call everything that results from continually adding 1 to 1 ‘cardinal
numbers’”, then the word “continually” doesn’t stand for a nebulous continuation of
1, 1 + 1, 1 + 1 + 1; rather, the sign “1, 1 + 1, 1 + 1 + 1, . . .” is also to be taken as perfectly
exact; as differing from “1, 1 + 1, 1 + 1 + 1”, as governed by different definite rules, and
not as a replacement13
for a series “that can’t be written down”.
That means: We calculate with the sign “1, 1 + 1, 1 + 1 + 1, . . .” just as with (the)
numerals, but in accordance with different rules.
But what is it that we delude ourselves about? What is the mistake that we make? What
do we take the sign “1, 1 + 1, . . .” for? That is: Where does what we think we see in this
sign really occur? Perhaps when I say “He counted 1, 2, 3, 4 and so on up to 1,000”? Where
it would also be possible actually to write down all the numbers?
What does one see “1, 1 + 1, 1 + 1 + 1, . . .” as?
As an inexact form of expression. The dots are like additional numerals, but ones that are
indistinct. As if one stopped writing down numerals because to be sure one can’t write them
all down, but as if they were nevertheless there14
in a kind of box. Something like my singing
only the first notes of a melody distinctly, and then merely hinting at the rest and letting it
taper off into nothing. (Or if in writing a word one writes only a few letters distinctly and
ends with an inarticulate line.) Where a “distinctly” would be correlated with “indistinctly”.
I once said that there couldn’t be both numbers and the concept of number. And that
is correct, if it means that a variable doesn’t have the same relation to a number as the
concept apple has to an apple (or the concept sword to Nothung15
).
On the other hand, a number-variable is not a numeral.
But I also wanted to say that the concept of number couldn’t be (given) independently
of the numbers, and that isn’t true. Rather a number-variable is independent of particular
numbers in the same sense as a calculus with a class of our numerals and without the
general number-variable really does exist. In that calculus, to be sure, not all the rules that
are valid for our numerals are valid for these, but still the latter numerals correspond to
ours as the pieces in draughts do to those in losing draughts.
What I am resisting is the view that an16
infinite number series is something given to us
for which there are both particular number theorems and also general theorems about all
numbers of the series. So that the arithmetical calculus wouldn’t be complete if it didn’t also
contain the general theorems about cardinal numbers, i.e. general equations of the form
a + (b + c) = (a + b) + c. Whereas 1 ÷ 3 = 0·3 already belongs to a different calculus from
13 (V): proxy
14 (V): they were indeed there
15 (E): Nothung is Siegfried’s sword in the
Nibelungenlied.
16 (V): the
The Law of the Formation of a Series. . . . 260e
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Kalkül angehört als 1 : 3 = 0,3. Und so ist eine allgemeine Zeichenregel (z.B. rekursive
Definition), die für 1, (1) + 1, ((1) + 1) + 1, (((1) + 1) + 1) + 123
, u.s.w. gilt, etwas andres,
als eine spezielle Definition. Und die allgemeine Regel fügt dem Zahlenkalkül etwas neues
bei, ohne welches er ebenso vollständig gewesen wäre, wie die Arithmetik der Zahlenreihe
1, 2, 3, 4, 5.
Es fragt sich auch, wo denn der Zahlbegriff (oder Begriff der Kardinalzahl) unbedingt
gebraucht wird. Zahl, im Gegensatz wozu?
|1, ξ, ξ + 1| wohl im Gegensatz zu |5, ξ, | u.s.w. – Denn wenn ich so ein Zeichen
(wie ”|1, ξ, ξ + 1|“) wirklich einführe – und nicht nur als Luxus mitschleppe, so muß ich
auch etwas mit ihm tun, d.h., es in einem Kalkül verwenden, und dann verliert es seine
Alleinherrlichkeit und kommt in ein System ihm koordinierter Zeichen.
Man wird vielleicht sagen: aber ”Kardinalzahl“ steht doch im Gegensatz zu ”Rationalzahl“,
”reelle Zahl“ etc. Aber dieser Unterschied ist ein Unterschied der Regeln (der von ihnen
geltenden Spielregeln) – nicht einer, der Stellung auf dem Schachbrett – nicht ein
Unterschied, für den man im selben Kalkül verschiedene koordinierte Worte braucht.
Man sagt ”dieser Satz ist für alle Kardinalzahlen bewiesen“. Aber sehen wir doch nur
hin, wie der Begriff der Kardinalzahl in den24
Beweis eintritt. Doch nur, indem im Beweis
von 1 und der Operation ξ + 1 die Rede ist – aber nicht im Gegensatz zu Etwas, was den
Rationalzahlen entspräche. Wenn man also den Beweis in Prosa mit Hilfe des Begriffsworts
”Kardinalzahl“ beschreibt, so sehen wir wohl, daß kein Begriff diesem Wort entspricht.
Die Ausdrücke ”die Kardinalzahlen“, ”die reellen Zahlen“ sind außerordentlich
irreführend, außer, wo sie als Teil einer Bestimmung verwendet werden, wie in: ”die
Kardinalzahlen von 1 bis 100“, etc. ”Die Kardinalzahlen“ gibt es nicht, sondern nur
”Kardinalzahlen“ und den Begriff, die Form, ”Kardinalzahl“. Nun sagt man: ”die Zahl der
Kardinalzahlen ist kleiner, als die der reellen25
Zahlen“ und denkt sich, man könnte die
beiden Reihen etwa nebeneinander schreiben (wenn wir nicht schwache Menschen wären)
und dann würde die eine im Endlosen enden, während die andere ins wirklich-Unendliche
über sie26
hinaus liefe. Aber das ist alles Unsinn. Wenn von einer Beziehung, die man nach
Analogie ”größer“ und ”kleiner“ nennen kann, die Rede sein kann, dann nur zwischen
den Formen ”Kardinalzahl“ und ”reelle Zahl“.27
Was eine Reihe ist, erfahre ich dadurch,
daß man es mir erklärt und nur soweit, als man es erklärt. Eine endliche Reihe wurde mir
durch Beispiele der Art 1, 2, 3, 4 erklärt, eine endlose durch Zeichen der Art ”1, 2, 3, 4,
u.s.w.“ oder ”1, 2, 3, 4 . . .“.
Es ist wichtig, daß ich eine28
Projektionsregel verstehen (sehen) kann, ohne sie in einer
allgemeinen Notation vor mir zu haben. Ich kann aus der Reihe eine allgemeine
Regel entnehmen – freilich auch beliebig viele andere, aber doch auch eine bestimmte und das
heißt, daß für mich diese Reihe irgendwie der Ausdruck dieser einen Regel war.29
Hat man ”intuitiv“ das Bildungsgesetz einer Reihe, z.B. der Reihe m verstanden, so daß
man also im Stande ist, ein beliebiges m(v) zu bilden, so hat man das Bildungsgesetz ganz
verstanden, also so gut, wie es etwa30
eine algebraische Darstellung vermitteln könnte.
1
1
2
4
3
9
4
16
ξ
23 (O): ((1) + 1) + 1) + 1
24 (V): dem
25 (O): rellen
26 (V): die
27 (O): und reelle Zahl‘.
28 (V): die
29 (O): war.“
30 (V): es irgend
349
350
261 Bildungsgesetz einer Reihe. . . .
WBTC069-77 30/05/2005 16:16 Page 522
the one for 1 ÷ 3 = 0·3. And thus a general rule for a sign (e.g. a recursive definition) that
holds for 1, (1) + 1, ((1) + 1) + 1, (((1) + 1) + 1) + 1, etc., is something different from a
particular definition. And the general rule adds something new to the number calculus,
without which it would have been just as complete as the arithmetic of the number series
1, 2, 3, 4, 5.
There is also the question: Where is the concept of number (or of cardinal number) indispensable?
Number, in contrast to what?
|1, ξ, ξ + 1|, no doubt in contrast to |5, ξ, |, etc. – For if I really do introduce such
a sign (like “|1, ξ, ξ + 1|”) and don’t just bring it along as a luxury, then I must do something
with it, i.e. use it in a calculus, and then it loses its autocratic splendour and is put
into a system of signs that are coordinated with it.
Perhaps someone will say: But “cardinal number” stands in contrast to “rational number”,
“real number”, etc. But this distinction is a distinction among rules (the rules of games that
apply to them) – not a distinction among positions on the chess board – not a distinction
requiring different coordinated words in the same calculus.
We say: “This theorem has been proved for all cardinal numbers”. But let us just look at
how the concept of cardinal numbers enters into the proof. Clearly only because 1 and the
operation ξ + 1 are mentioned in the proof – and not in contrast to something that might
correspond to rational numbers. So when we describe the proof in prose, with the help of
the concept-word “cardinal number”, we see perfectly well that no concept corresponds to
that word.
The expressions “the cardinal numbers”, “the real numbers”, are extraordinarily misleading
except where they are used as part of a specification, such as in “the cardinal numbers from
1 to 100”, etc. There is no such thing as “the cardinal numbers”, but only “cardinal numbers”
and the concept, the form, “cardinal number”. Now we say “the number of cardinal
numbers is smaller than that of real numbers”, and we imagine that we could perhaps write
the two series side by side (if we just weren’t weak humans) and then the one series would
end in interminability, whereas the other would go beyond it into the actual infinite. But
this is all nonsense. If we can talk of a relationship which by analogy can be called “larger”
and “smaller”, it can only be a relationship between the forms “cardinal number” and “real
number”. I find out what a series is by having it explained to me, and only to the extent that
it is explained to me. A finite series was explained to me by examples of the type 1, 2, 3, 4,
an infinite one by signs of the type “1, 2, 3, 4, etc.” or “1, 2, 3, 4 . . .”.
It is important that I can understand (see) a17
rule of projection without having it in front
of me in a general notation. I can infer a general rule from the series , and to be
sure, I can infer any number of others too, but nevertheless I can also infer a particular rule,
and that means that for me this series was somehow the expression of this one rule.
If you have “intuitively” understood the law of the formation of a series, e.g., of the series
m, so that you are therefore able to construct an arbitrary term m(v), then you have understood
that law completely, that is to say, as well as an algebraic formulation, for example,
could convey it.18
That is, you can’t understand it any better with such a formulation. And
1
1
2
4
3
9
4
16
ξ
17 (V): the 18 (V): as well as any algebraic formulation could
convey it.
The Law of the Formation of a Series. . . . 261e
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D.h. man kann es durch eine solche Darstellung nicht mehr besser verstehen. Und diese
Darstellung ist daher insofern auch nicht strenger. Obwohl sie natürlich einprägsamer sein kann.
Man ist geneigt, zu glauben, daß die Notation, die eine Reihe durch Anschreiben einiger
Glieder mit dem Zeichen ”u.s.w.“ darstellt, wesentlich unexakt ist, im31
Gegensatz zur Angabe
des allgemeinen Gliedes. Dabei vergißt man, daß die Angabe des allgemeinen Gliedes sich
auf eine Grundreihe bezieht, welche nicht wieder durch ein allgemeines Glied beschrieben
sein32
kann. So ist 2n + 1 das allgemeine Glied der ungeraden Zahlen, wenn n die
Kardinalzahlen durchläuft, aber es wäre Unsinn zu sagen, n sei das allgemeine Glied der
Reihe der Kardinalzahlen. Wenn man diese Reihe erklären will, so kann man es nicht durch
Angabe des ”allgemeinen Gliedes n“, sondern natürlich nur durch eine Erklärung der Art
1, 1 + 1, 1 + 1 + 1, u.s.w. Und es ist natürlich kein wesentlicher Unterschied zwischen dieser
Reihe und der: 1, 1 + 1 + 1, 1 + 1 + 1 + 1 + 1, u.s.w., die ich ganz ebensogut als Grundreihe
hätte annehmen33
können (sodaß dann das allgemeine Glied der Kardinalzahlenreihe
gelautet hätte).
(∃x).φx & ~(∃x,y).φx & φy
(∃x,y).φx & φy .&. ~(∃x,y,z).φx & φy & φz
(∃x,y,z).φx & φy & φz .&. ~(∃x,y,z,u).φx & φy & φz & φu
»Wie müßte man es nun anfangen, die allgemeine Form solcher Sätze zu schreiben? Die
Frage hat offenbar einen guten Sinn. Denn, wenn ich nur einige solcher Sätze als Beispiele
hinschreibe, so versteht man, was das Wesentliche dieser Sätze sein soll.«
Nun, dann ist also die Reihe der Beispiele schon eine Notation; denn das Verstehen dieser
Reihe besteht doch in der Verwendung dieses Symbols und darin, daß wir es von andern in
demselben System unterscheiden, z.B. von:
(∃x).φx
(∃x,y,z).φx & φy & φz
(∃x,y,z,u,v).φx & φy & φz & φu & φv.
Warum sollen wir aber nicht das allgemeine Glied der ersten Reihe so schreiben:
(∃x1 . . . xn).(Πx1
xn
φx) & (∃x1 . . . xn+1).(Πx1
xn+1
φx)?
Ist diese Notation unexakt? Sie selbst soll ja nichts bildhaft machen, sondern nur auf die
Regeln ihres Gebrauchs, auf das System, in dem sie gebraucht wird, kommt es an.34
Die
Skrupel, die ihr anhaften, schreiben sich von einem Gedankengang her, der sich mit der
Zahl der Urzeichen in dem Kalkül der ”Principia Mathematica“ beschäftigte.
n − 1
2
31 (V): ist. Im
32 (V): werden
33 (V): nehmen
34 (V): Gebrauchs, das System in die [sic] sie
gebraucht wird, kommt es an.
351
352
262 Bildungsgesetz einer Reihe. . . .
WBTC069-77 30/05/2005 16:16 Page 524
therefore to that extent neither is this formulation more rigorous. Although of course it can
be more easily remembered.
We are inclined to believe that the notation that represents a series by writing down a few
terms along with the sign “etc.” is essentially inexact, as19
opposed to the statement of the
general term. Believing this, we forget that the statement of the general term refers to a basic
series that cannot in turn be described by a general term. Thus 2n + 1 is the general term
for the odd numbers, when n ranges over the cardinal numbers, but it would be nonsense
to say that n was the general term of the series of cardinal numbers. If you want to define
this series, you can’t do it by specifying “the general term n”, but only by an explanation of
the type 1 + 1, 1 + 1 + 1, etc. And of course there is no essential difference between this
series and 1, 1 + 1 + 1, 1 + 1 + 1 + 1 + 1, etc., which I could just as well have adopted20
as
the basic series (so that then the general term of the cardinal number series would have read
).
(∃x).φx & ~(∃x,y).φx & φy
(∃x,y).φx & φy .&. ~(∃x,y,z).φx & φy & φz
(∃x,y,z).φx & φy & φz .&. ~(∃x,y,z,u).φx & φy & φz & φu
»How would one now go about writing the general form of such propositions? The question
obviously makes good sense. For if I write down only a few such propositions as examples,
one understands what the essential element in these propositions is supposed to be.«
Well, in that case the series of examples is already a notation. For understanding this series,
after all, consists in the use of this symbol, and in our distinguishing it from others in the
same system, e.g., from:
(∃x).φx
(∃x,y,z).φx & φy & φz
(∃x,y,z,u,v).φx & φy & φz & φu & φv.
But why shouldn’t we write the general term of the first series this way:
(∃x1 . . . xn).(Πx1
xn
φx) & (∃x1 . . . xn+1).(Πx1
xn+1
φx)?
Is this notation inexact? After all, by itself it isn’t supposed to make anything graphic;
rather, the only things that matter are the rules for its use, the system21
in which it is used.
The scruples that adhere to it originate in a train of thought that occupied itself with the
number of primitive signs in the calculus of Principia Mathematica.
n − 1
2
19 (V): inexact. As
20 (V): taken
21 (V): only thing that matters is the system
The Law of the Formation of a Series. . . . 262e
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Erwartung.
Wunsch.
etc.
353
WBTC069-77 30/05/2005 16:16 Page 526
Expectation.
Wish.
etc.
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76
Erwartung: der Ausdruck
der Erwartung. Artikulierte
und unartikulierte Erwartung.
1
Kann man sagen, die Erwartung ist eine vorbereitende, erwartende, Handlung. – Es wirft
mir jemand einen Ball, ich strecke die Hände aus und richte sie zum Erfassen des Balls. Aber
sagen wir, ich hätte mich verstellt, ich hatte erwartet, daß er nicht werfen würde, wollte
aber so tun, als erwartete ich den Wurf. Worin besteht dann mein Erwarten, daß er nicht
werfen wird, wenn meine Handlung die gegenteilige Erwartung ausdrückt? Sie2
mußte doch
auch in etwas bestehen, was ich tat. Ich war also doch irgendwie nicht drauf vorbereitet,
daß der Ball kam.
3
Es ist sehr trivial, wenn ich sage, daß ich in der Erwartung eines Flecks die Erwartung
eines kreisförmigen von der eines elliptischen4
muß unterscheiden können und es überhaupt
so viele Unterschiede in der Erwartung geben muß, wie in den Erfüllungen der Erwartungen.
(Der Hunger und der Apfel, der ihn befriedigt, haben nicht die gleiche Multiplizität.)
Wenn wir5
wissen wollen was die Worte &ich erwarte daß er kommt“ bedeuten, – fragen wir uns:
Was ist das Kriterium dafür, daß, was wir tun ist, ihn zu erwarten.
Wie6
wissen wir, daß wir ihn erwarten?
7
Nehmen wir an, ich erwarte jemand: ich sehe auf die Uhr, dann zum Fenster hinaus,
richte etwas in meinem Zimmer zurecht, schaue wieder hinaus, etc. Diese Tätigkeit könnte
ich das Erwarten nennen. Denke ich nun die ganze Zeit dabei? (D.h. ist diese Tätigkeit
wesentlich eine Denktätigkeit, oder von ihr begleitet?) Letzteres bestimmt nicht. Und wenn
ich jene Tätigkeiten Denken nenne, welches wären die Worte, durch die dieser Gedanke
ausgedrückt würde? – Wohl aber werden auch Gedanken während dieses Wartens sich
einfinden. Ich werde mir sagen: ”vielleicht ist er zu Hause aufgehalten worden“, und drgl.
mehr; vielleicht auch die artikulierte Erwartung ”wenn er nur käme“.
In allen jenen erwartenden Handlungen ist nichts, was uns interessiert (die Erfüllung der
Erwartung in diesem Sinn ist nichts anderes, als die Stillung eines Hungers). Uns interessiert
nur das zu einem Zweck gemachte Bild. – Der artikulierte Gedanke.
8
Es ist – glaube ich, – wichtig zu erkennen, daß, wenn ich etwa glaube, daß jemand zu
mir kommen wird, mein Dauerzustand nichts mit dem Betreffenden und den übrigen
Elementen des Gedankens zu tun hat, d.h. sie nicht enthält. Das Gleiche gilt aber für
1 (M): 3 (R): ∀ S. 154/23, 63
2 (V): Diese
3 (M): ///
4 (O): eliptischen
5 (V): wir die Worte (ich
6 (V): Wie weiß
7 (M): 3
8 (M): 3
354
354v
355
264
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76
Expectation: the Expression
of Expectation. Articulate
and Inarticulate Expectation.
1
Can one say that expectation is a preparatory, expectant action? Someone throws me a
ball and I stretch out my hands preparing to catch it. But let’s say that I had dissimulated:
I had expected that he wouldn’t throw it, but wanted to act as if I were expecting the throw.
What does my expectation that he won’t throw it consist in when my action expresses the
opposite expectation? It2
too had to consist in something I did. So in some way I wasn’t
prepared for the ball coming towards me after all.
3
It’s quite trivial to say that in expecting a spot I have to be able to distinguish the expectation
of a circular spot from that of an elliptical one; and that in general there have to be
as many differences in expectation as there are in what fulfils the expectations. (Hunger and
the apple that satisfies it do not have the same multiplicity.)
If we4
want to know what the words “I’m expecting him to come” mean – we ask ourselves: What
is the criterion for what we’re doing being a case of expecting him?
How do we know that we’re expecting him?
5
Let’s assume that I’m expecting someone: I look at my watch, then out of the window,
straighten up something in my room, look out again, etc. I could call this activity “expecting”.
Now am I thinking all this while? (That is to say, is this activity essentially a mental
activity, or is it accompanied by one?) Certainly not the latter. And if I do call these
activities thinking, what words would be used to express this thought? – But thoughts will
no doubt turn up during this wait. I’ll say to myself: “Maybe he’s been held up at home”
and other such things; maybe I’ll also articulate my expectation: “If only he’d come”.
In all of these actions of expecting there is nothing that interests us (the fulfilment of the
expectation is in that sense no different from the satisfaction of hunger). All that interests
us is the image that was created for a purpose. – The articulated thought.
6
It is important to realize – I think – that when for example I believe that someone will
be coming to see me, my ongoing state has nothing to do with the person in question and
the other elements of the thought, i.e. that my occurrent state doesn’t contain them. But the
1 (M): 3 (R): ∀ p. 154/2 3, 6 3
2 (V): This
3 (M): ///
4 (V): If I
5 (M): 3
6 (M): 3
264e
WBTC069-77 30/05/2005 16:16 Page 529
Erwartung, Wunsch, etc. etc. Wenn ich jemand erwarte, so denke ich nicht während dieser
ganzen Zeit, daß er kommen wird, oder dergleichen. Ja selbst, wenn ich es gerade denke, so
ist ja dieser Vorgang kein amorpher, wie etwa der des Schmerzes, sondern besteht nur darin,
daß ich etwa jetzt gerade den Satz sage, ”er wird kommen“. Man kann nicht amorph sehen,
daß etwas der Fall ist, glauben, daß etwas der Fall ist, wünschen, befürchten, denken, etc.
9
Der Ausdruck der Erwartung ist die Erwartung.
Die Vorbereitung ist quasi selbst die Sprache und kann nicht über sich selbst hinaus.
(In dem ”nicht über sich selbst hinauskönnen“ liegt die Ähnlichkeit meiner Betrachtungen
und jener der Relativitätstheorie.)
10
Wenn ich früher gesagt habe, es kommt darauf an, ob dieses Bild erwartet wird, d.h., ob
wir gerade dieses Bild ”verwenden“ (”benützen“) so könnte ich jetzt sagen, es kommt darauf
an, ob gerade dieses Bild zu unserer Sprache gehört.11
Die Sprache als Ausdruck der Erwartung ist das Vorbereitete.
9 (M): 3
10 (M): ///
11 (V): Bild unsere Sprache ist.
356
265 Erwartung: der Ausdruck der Erwartung. . . .
WBTC069-77 30/05/2005 16:16 Page 530
same goes for expectation, wish, etc., etc. When I’m expecting someone I’m not thinking all
the while that he’ll come, or some such thing. Indeed, even when I am thinking this, this
process is not amorphous, like that of pain, but consists only in my saying, perhaps just now,
the sentence “He’ll come”. You can’t amorphously see that something is the case, believe that
something is the case, wish, fear, think, etc.
7
The expression of expectation is expectation.
In a way preparation is itself language, and it can’t get beyond itself. (The similarity of
my observations to those of the theory of relativity lies in this “not being able to get beyond
themselves”.)
8
If I said earlier that what is important is whether this image is expected, i.e. whether we
“use” (“utilize”) this particular image, I could say now that what is important is whether
this particular image belongs to our language.9
Language as an expression of expectation is what is prepared.
7 (M): 3
8 (M): ///
9 (V): image is our language.
Expectation . . . 265e
WBTC069-77 30/05/2005 16:16 Page 531
77
In der Erwartung wurde das erwartet,
was die Erfüllung brachte.
1
Die Erwartung und die Tatsache, die die Erwartung befriedigt, passen doch irgendwie
zusammen. Man soll nun eine Erwartung beschreiben, und eine Tatsache, die zusammenpassen,
damit man sieht, worin diese Übereinstimmung besteht. Da denkt man sofort an das
Passen einer Vollform in eine entsprechende Hohlform. Aber wenn man nun hier die beiden
beschreiben will, so sieht man, daß, soweit sie passen, eine Beschreibung für beide gilt.
Vergleiche das Passen eines Hutes zu einem Kleid.
2
Kann man den Vorgang des Verständnisses eines Befehls mit dem Vorgang der
Befolgung3
vergleichen, um zu zeigen, daß diese Befolgung diesem Verständnis, dieser
Auffassung, wirklich entspricht? und inwiefern sie übereinstimmen? Gewiß, – nämlich z.B.
die Auffassung ”p“4
mit der Befolgung p. ”Ich habe mir das heller vorgestellt.“ Aber nicht
die Vorstellung ist als solche heller als die Wirklichkeit.
5
Kann man denn die Erwartung mit der eingetroffenen Tatsache vergleichen? Man
sagt ja, die Tatsache stimme mit der Erwartung überein oder nicht überein. Aber dieses
Übereinstimmen bezieht sich nicht auf Eigenschaften der Erwartung als solcher (des
Vorgangs der Erwartung) und Eigenschaften des Ereignisses als Realität.
Kann man eine Hohlform mit einer Vollform vergleichen.
6
(Es ist aber nicht so als ob ich sagte: ”ich habe Lust auf einen Apfel, was immer also
diese Lust stillen wird, werde ich einen Apfel nennen“. (Also etwa auch ein Schlafmittel.))
7
Das Seltsame ist ja darin ausgedrückt, daß, wenn dies8
der Fleck ist, den ich erwartet
habe, er sich nicht von dem unterscheidet, den ich erwartet habe. Wenn man also fragt:
”Wie unterscheidet sich denn der Fleck von dem, den Du erwartet hast, denn in Deiner
Erwartung war doch der wirkliche Fleck nicht vorhanden, sonst hättest Du ihn nicht erwarten
können“, so ist die Antwort dennoch: der Fleck ist der, den ich erwartet habe.
9
Ich sage ”genau so habe ich mir’s vorgestellt“. Und jemand antwortet etwa ”das ist
unmöglich, denn das eine war eine Vorstellung und das andere ist keine; und hast Du etwa
Deine Vorstellung für Wirklichkeit gehalten?“
10
”Ich erwarte mir einen Schuß.“ Der Schuß fällt. Wie, das hast Du Dir erwartet; war
also dieser Krach irgendwie schon in Deiner Erwartung? &Der Knall ist leiser als ich mir ihn erwartet
1 (M): 3 (R): ∀ S. 185/2
2 (M): ///
3 (V): Befolgung d
4 (O): Auffassung p‘
5 (M): ///
6 (M): 3
7 (M): 3
8 (V): das
9 (M): 3
10 (M): 3
357
358
266
WBTC069-77 30/05/2005 16:16 Page 532
77
What Fulfilment Brought: that was
what was Expected in Expectation.
1
Expectation and the fact that fulfils expectation do, after all, go together. Now to see what
this correspondence consists in we are supposed to describe an expectation and a fact that
goes with it. In this situation we immediately think of a convex shape fitting into a corresponding
concave shape. But if we then want to describe the two shapes, we see that in so
far as they fit into each other, the same description is valid for both. Compare a hat matching
a dress.
2
Can one compare the process of understanding a command with that of executing it to
show that this act of execution really corresponds to this understanding, this interpretation –
and to show in which respect they correspond? Certainly – that is, one can for example
compare the understanding of “p” with the execution of p. “I imagined that as brighter.”
But it isn’t the mental image itself that is brighter than reality.
3
Can one really compare expectation with the fact that has occurred? One does say that a
fact matches or doesn’t match an expectation. But this matching doesn’t refer to properties
of expectation itself (properties of the process of expectation), nor to properties of the actual
event.
Can one compare a concave shape with a convex one?
4
(But it isn’t as if I said: “I have a craving for an apple, so whatever will satisfy this
craving I’ll call an apple”. (Say, a sleeping pill as well.))
5
The oddity is expressed in the fact that if this6
is the spot I expected, it doesn’t differ
from the one I expected. So if one asks: “How does the spot differ from the one you were
expecting? (for after all, the real spot wasn’t present in your expectation; otherwise you
wouldn’t have been able to expect it)”, then all the same the answer is: The spot is the one
I expected.
7
I say: “That’s exactly the way I imagined it”. And someone might respond: “That’s impossible,
because the one thing was a mental image and the other isn’t; and did you perhaps
take your mental image for reality?”
8
“I’m expecting a shot.” The shot rings out. So that was what you expected? So was this
report somehow already present in your expectation? “The report is quieter than I expected” –
1 (M): 3 (R): ∀ p. 185/2
2 (M): ///
3 (M): ///
4 (M): 3
5 (M): 3
6 (V): that
7 (M): 3
8 (M): 3
266e
WBTC069-77 30/05/2005 16:16 Page 533
habe“ – &Hat es also in Deiner Erwartung lauter geknallt?“ Oder stimmt Deine Erwartung
nur in anderer Beziehung mit dem Eingetretenen überein, war dieser Lärm nicht in
Deiner Erwartung enthalten und kam nur als Accidens hinzu, als die Erwartung erfüllt wurde?
Aber nein, wenn der Lärm nicht eingetreten wäre, so wäre meine Erwartung nicht erfüllt
worden; der Lärm hat sie erfüllt, er kam nicht zu der Erfüllung hinzu wie ein zweiter Gast
zu dem einen, den ich erwartete.
11
War das am Ereignis, was nicht auch in der Erwartung war, ein Accidens, eine Beigabe
der Schickung?12
Aber was war denn dann nicht Beigabe, kam denn irgend etwas vom Schuß
schon in meiner Erwartung vor? Und was war denn Beigabe, denn hatte ich mir nicht den
ganzen Schuß erwartet.
13
Unterscheidet sich etwa ein vorgestellter Ton von dem gleichen, wirklich gehörten
durch die Klangfarbe?!
14
Es hat auch einen Sinn zu sagen, es sei nicht das geschehen, was ich erwartet habe,
sondern etwas ähnliches; im Gegensatze aber zu dem Fall, wenn das geschieht, was erwartet
wurde. Und das zeigt, welcher Art der Mißbrauch der Sprache ist, zu dem15
wir hier
verleitet werden.
16
Wenn man nun sagte: Das Rot, das Du Dir vorstellst,17
ist doch gewiß nicht dasselbe
(dieselbe Sache) wie das, was Du wirklich vor Dir siehst, – wie kannst Du dann sagen ”das
ist dasselbe, was ich mir vorgestellt habe“? – Zeigt denn das nicht nur, daß, was ich ”dieses
Rot“ nenne, eben das ist, was meiner Vorstellung und der Wirklichkeit gemein ist? Denn
das Vorstellen des Rot ist natürlich anders als das Sehen des Rot, aber darum heißt ja auch
das eine ”Vorstellen eines roten Flecks“ und das andre ”Sehen eines roten Flecks“. In
beiden (verschiedenen) Ausdrücken aber kommt dasselbe Wort ”rot“ vor und so muß
dieses Wort nur das bezeichnen, was beiden Vorgängen zukommt.
Ist es denn nicht dasselbe in den Sätzen ”hier ist ein roter Fleck“ und ”hier ist kein roter
Fleck“? In beiden kommt das Wort ”rot“ vor, also kann dieses Wort nicht das Vorhandensein
von etwas Rotem bedeuten. – (Der Satz ”das ist rot“ ist nur eine Anwendung des Wortes
”rot“, gleichberechtigt mit allen anderen, wie mit dem Satz ”das ist nicht rot“.)
(Das Wort ”rot“ hat eben – wie jedes Wort – nur im Satzzusammenhang eine Funktion.
Und ist das Mißverständnis das, in dem Wort allein schon den Sinn18
eines Satzes zu sehen
glauben?)
19
Wie komisch wäre es, zu sagen: ein Vorgang sieht anders aus, wenn er geschieht, als
wenn er nicht geschieht. Oder: ”Ein roter Fleck sieht anders aus, wenn er da ist, als wenn
er nicht da ist, aber die Sprache abstrahiert von diesem Unterschied, denn sie spricht von
einem roten Fleck, ob er da ist oder nicht“.
20
Wie unterscheidet sich das Rot eines Flecks, den wir vor uns sehen, von dem dieses Flecks,
wenn wir ihn uns bloß vorstellen? – Aber wie wissen wir denn, daß es das Rot dieses Flecks
ist, wenn es (von dem Ersten) verschieden ist? – Woher wissen wir denn, daß es dasselbe
11 (M): 3
12 (V): Beigabe des Schicksals
13 (M): ///
14 (M): ///
15 (V): welchem
16 (M): 3
17 (O): vorstellt,
18 (V): Sinn d
19 (M): 3
20 (M): ///
359
360
267 In der Erwartung wurde das erwartet . . .
WBTC069-77 30/05/2005 16:16 Page 534
“So was there a louder report in your expectation?” Or is it only in a different respect that your
expectation corresponds to what happened? Was this noise not contained in your expectation,
and did it merely accidentally supervene when the expectation was fulfilled? Certainly
not – if the noise hadn’t occurred my expectation wouldn’t have been fulfilled; the noise
fulfilled it; it didn’t just accompany the fulfilment, as a second guest accompanies the one I
was expecting.
9
Was the part of the event that wasn’t also in my expectation an accidental property,
something extra contributed by providence10
? But if so, then what wasn’t extra? Was there
anything about the shot that had already occurred within my expectation? And what was
extra, anyway, for hadn’t I expected the whole shot?
11
Does an imagined tone perhaps differ in timbre from the same tone that’s actually heard?!
12
It also makes sense to say that what happened was not what I had expected, but something
similar; but this would be in contrast to the case where precisely what was expected
happens. And this shows into what kind of misuse of language we are here being lured.
13
Now if one were to say: But surely the red you are imagining isn’t the same (the same
thing) as the one you’re actually seeing in front of you – so how can you say “That is the
same red I imagined”? – Doesn’t that merely show that what I am calling “this red” is
precisely what is common to my mental image and reality? For of course imagining red is
different from seeing it, but that’s the very reason the one thing is called “imagining a red
patch” and the other “seeing a red patch”. But the same word “red” occurs in both (different)
expressions, and so this word has to designate what belongs to both processes.
Isn’t it the same in the sentences “There’s a red patch here” and “There’s no red patch
here”? The word “red” occurs in both sentences, and therefore it can’t mean the presence
of something red. – (The sentence “That is red” is only one application of the word “red”,
no more privileged than any other, including the sentence “That isn’t red”.)
(As with any word, the word “red” has a function only in the context of a sentence. And
is this the misunderstanding: To think you are already seeing the sense of a sentence in the
word alone?)
14
How funny it would be to say: An occurrence looks different when it happens from
when it doesn’t happen. Or: “A red patch looks different when it is there from when it
isn’t there, but language abstracts from this difference, for it talks about a red patch whether
it’s there or not”.
15
How does the red of a patch that we see in front of us differ from the red of this patch
when we are merely imagining it? – But how do we know, anyway, that it is the red of this
9 (M): 3
10 (V): fate
11 (M): ///
12 (M): ///
13 (M): 3
14 (M): 3
15 (M): ///
What Fulfilment Brought . . . 267e
WBTC069-77 30/05/2005 16:16 Page 535
Rot ist, wenn es nicht dasselbe ist?21
– Dieser Galimathias22
zeigt, daß hier ein Mißbrauch
der Sprache vorliegt.
23
Wie ist es möglich, daß ich erwarte, und das, was ich erwarte, kommt? Wie konnt’ ich
es erwarten, da es nicht da war?
Die Realität ist keine Eigenschaft, die dem Erwarteten noch fehlt und die nun hinzutritt,
wenn es eintritt. – Sie ist auch nicht wie das Tageslicht, das den Dingen erst ihre Farbe
gibt, wenn sie im Dunkeln schon gleichsam farblos vorhanden sind.
Wie konnte ich es erwarten, und es kommt dann wirklich; – als ob die Erwartung ein
dunkles Transparent wäre und mit der Erfüllung das Licht dahinter angezündet würde. –
Aber jedes solche Gleichnis ist falsch, weil es die Realität als einen24
beschreibbaren Zusatz
zur Erwartung25
darstellt; was unsinnig ist.
(Es ist das im Grunde derselbe Unsinn, wie der, der die vorgestellte Farbe als matt im
Vergleich zur wirklichen darstellt.)
26
Du siehst also, möchte ich sagen, an diesen Beispielen, wie die Worte wirklich
gebraucht werden.
27
Ich habe etwas vorausgesagt, es tritt nun ein, und ich sage nun einfach ”es ist eingetroffen“
und das beschreibt schon den Tatbestand vollkommen. Er ist also auch jetzt nur so
weit beschrieben, als man ihn auch hat beschreiben können, ehe28
er eingetreten war.
29
Wenn ich einfach sagen kann ”es ist eingetroffen“ so kann ich andrerseits nicht30
beschreiben, wie ein Tatbestand sein muß, um eine bestimmte Erwartung zu befriedigen.
31
Das Befolgen des Befehls liegt darin, daß ich etwas tue – Kann ich aber auch sagen, ”daß
ich das tue, was er befiehlt“? Gibt es ein Kriterium dafür, daß das die Handlung ist, die ihn
befolgt?
Was soll hier unter einem Kriterium verstanden werden.
32
Die Erwartung verhält sich eben zu ihrer Befriedigung nicht wie der Hunger zu seiner
Befriedigung. Ich kann sehr wohl den Hunger beschreiben und das, was ihn stillt, und sagen,
daß es ihn stillt.
33
Wenn ich ein Ereignis erwarte und es kommt dasjenige, welches meine Erwartung erfüllt,
hat es dann einen Sinn zu fragen, ob das wirklich das Ereignis ist, welches ich erwartet habe.
D.h., wie würde ein Satz, der das behauptet, verifiziert werden?
34
”Wie weißt Du, daß Du einen roten Fleck erwartest?“ – d.h. ”wie weißt Du, daß ein
roter Fleck die Erfüllung dessen ist, was Du Dir erwartest“. Aber ebensogut könnte man
fragen, ”wie weißt Du, daß das ein roter Fleck ist?“
Wie weißt Du, daß, was Du getan hast, wirklich war, das Alphabet im Geist herzusagen?
– Aber wie weißt Du, daß, was Du hersagst, nun wirklich das Alphabet35
ist?
21 (V): es verschieden ist?
22 (O): Callimathias
23 (M): 3
24 (V): Realität einen
25 (V): Zusatz zum Gedanken
26 (M): 3
27 (M): ///
28 (V): bevor
29 (M): ///
30 (V): andrerseits nicht auch
31 (M): ////
32 (M): 3
33 (M): ///
34 (M): 3
35 (V): Alphabet is?
361
362
268 In der Erwartung wurde das erwartet . . .
WBTC069-77 30/05/2005 16:16 Page 536
patch, if it is different (from the first)? – How do we know that it’s the same red if it isn’t
the same?16
– This gibberish shows that here there’s a misuse of language.
17
How is it possible for me to expect something, and for what I am expecting to happen?
How was I able to expect it, since it wasn’t here?
Reality isn’t a property that is missing so long as something is expected, and that then
joins it when it happens. – Neither is it like daylight, which endows things with their colour,
whereas in the dark they are actually in a colourless state, as it were.
How could I expect it and then it really happens? – as if expectation were a dark transparency
and in fulfilling it a light were switched on behind it. – But every simile of that sort
is wrong, because it portrays reality as a describable addendum to expectation;18
which is
nonsense.
(This is essentially the same kind of nonsense as representing the imagined colour as dull,
compared to the real one.)
19
So in these examples, I’m inclined to say, you see how words are really used.
20
I predicted something and then it happens, and now I simply say “It has happened”,
and that in itself describes the state of affairs perfectly. So even now it has only been described
to the extent that was possible before it had happened.
21
Although I can simply say “It has happened”, I can’t on the other hand22
describe what
a state of affairs must be like in order to satisfy a particular expectation.
23
Obeying a command consists in my doing something. – But can I also say “that I
am doing what it commands”? Is there a criterion for this being the act that obeys the
command?
What is to be understood here by a criterion?
24
Expectation doesn’t relate to its fulfilment the way hunger does to its satisfaction. I can
quite easily describe hunger and what satisfies it, and I can say that it satisfies it.
25
If I am expecting an event and what fulfils my expectation happens, does it then make
sense to ask whether that really is the event I was expecting? That is, how would a proposition
that makes that claim be verified?
26
“How do you know that you’re expecting a red patch?” – i.e. “How do you know that a
red patch is the fulfilment of what you’re expecting?” But one could just as well ask “How
do you know that that is a red patch?”
How do you know that what you did really was to recite the alphabet in your head? – But
how do you know that what you are reciting really is the alphabet?
16 (V): if it’s different?
17 (M): 3
18 (V): to thought;
19 (M): 3
20 (M): ///
21 (M): ///
22 (V): I can’t on the other hand also
23 (M): ////
24 (M): 3
25 (M): ///
26 (M): 3
What Fulfilment Brought . . . 268e
WBTC069-77 30/05/2005 16:16 Page 537
Das ist natürlich die gleiche Frage wie: Woher weißt Du, daß, was Du rot nennst,
wirklich dasselbe ist, was der Andre so nennt. Und die eine Frage ist ebenso unsinnig
wie die andere.36
37
Was immer ich über die Erfüllung der Erwartung sagen mag, was sie zur Erfüllung dieser
Erwartung machen soll, zählt sich zur Erwartung, ändert den Ausdruck der Erwartung. D.h.,
der Ausdruck der Erwartung ist der vollständige Ausdruck der Erwartung.
38
Wenn ich sage ”das ist dasselbe Ereignis, welches ich erwartet habe“ und ”das ist
dasselbe Ereignis, was auch an jenem Ort stattgefunden hat“, so bedeutet hier das Wort
”dasselbe“ jedesmal etwas anderes. (Man würde auch normalerweise nicht sagen ”das ist
dasselbe, was ich erwartet habe“, sondern ”das ist das, was ich erwartet habe“.)39
36 (O): wie andere.
37 (M): 555
38 (M): 555
39 (R): ∀ S. 23/3 3
269 In der Erwartung wurde das erwartet . . .
WBTC069-77 30/05/2005 16:16 Page 538
Of course that is the same question as: How do you know that what you are calling red
is really the same thing as what someone else calls red? And the one question is just as
nonsensical as the other.
27
Whatever I might say about the fulfilment of expectation, what is supposed to turn it
into the fulfilment of this expectation counts as part of the expectation. It changes the expression
of the expectation. That is, the expression of expectation is the complete expression of
expectation.
28
If I say “That’s the same event I was expecting” and “That’s the same event that also
happened at that other place”, then the words “the same” mean something different in the
two cases. (Furthermore, normally one wouldn’t say “That’s the same thing I was expecting”,
but “That’s what I was expecting”.)29
27 (M): 555
28 (M): 555
29 (R): ∀ p. 23/3 3
What Fulfilment Brought . . . 269e
WBTC069-77 30/05/2005 16:16 Page 539
78
”Wie kann man etwas
wünschen, erwarten, suchen,
was nicht da ist?“
Mißverständnis des ”etwas“.
1
Es könnte gesagt werden: Wie kann ich denn das Ereignis erwarten, es ist ja noch
garnicht da?
2
Man kann sich vorstellen, es sei etwas der Fall, was nicht ist: sehr merkwürdig! Denn,
daß die Vorstellung nicht mit der Wirklichkeit übereinstimmt, ist nicht merkwürdig, daß
sie sie aber dann repräsentiert, ist merkwürdig.
3
Sokrates: Wer also vorstellt, was nicht ist, der stellt nichts vor? – Theaitetos: So scheint
es. – S: Wer aber nichts vorstellt, der wird gewiß überhaupt garnicht vorstellen? – Th.:
Offenbar, wie wir sehen.4
Setzen wir in diesem Argument und dem ihm vorhergehenden statt5
”vorstellen“ etwa
”töten“,6
so läuft es auf eine Regel der Verwendung dieses Wortes hinaus. Es hat keinen
Sinn zu sagen7
”ich töte8
etwas, was nicht existiert“.
Ich kann mir einen Hirsch auf dieser Wiese vorstellen, der nicht da ist, aber keinen töten,
der nicht da ist. – 9
Und sich einen Hirsch vorstellen, der nicht da ist, heißt, sich vorstellen,
daß ein Hirsch da ist, obwohl keiner da ist. Einen Hirsch töten aber, heißt nicht: töten,
daß ein Hirsch da ist.10
Wenn aber jemand sagt: ”um mir einen Hirsch vorzustellen, muß
es ihn doch in einem gewissen Sinne geben“, so ist die Antwort: nein, es muß ihn dazu
in keinem Sinne geben. Und wenn darauf gesagt würde: Aber z.B. die braune Farbe muß
es doch geben, damit ich mir sie vorstellen kann, so ist zu sagen: ”,Es gibt die braune
Farbe‘ heißt überhaupt nichts, außer etwa, daß sie da oder dort11
als Färbung eines
Gegenstandes (Flecks) erscheint12
und das ist nicht nötig, damit ich mir einen braunen
Hirsch vorstellen kann.“
13
”Ich stelle mir vor, wie das sein wird“ (wenn der Sessel weiß gestrichen sein wird) –
wie kann ich es mir denn vorstellen, wenn es nicht ist?! Ist denn die Vorstellung eine Zauberei?
1 (M): 3
2 (M): ///
3 (M): 3
4 (E): Platon, Theaitetos, 189a.
5 (V): Setzen wir in diesem Argument statt
6 (V): ”zerschneiden“,
7 (V): hinaus. Man dürfe nicht sagen:
8 (V): zerschneide
9 (M): 555 – Hirsch da ist.
10 (V): ist (.also: verschiedene grammatische
Regeln).
11 (V): dort d
12 (V): auftritt
13 (M): ///
363
364
270
WBTC078-85 30/05/2005 16:16 Page 540
78
“How can one Wish for,
Expect, Look for, Something
that isn’t There?”
Misunderstanding of the
“Something”.
1
It could be said: How can I expect an event, since it isn’t even here yet?
2
One can imagine that something that is the case doesn’t exist: quite remarkable! For it
isn’t remarkable that a mental image doesn’t correspond to reality; what is, is that it then
represents it.
3
Socrates: So whoever imagines what doesn’t exist, imagines nothing?
Theaetetus: So it seems.
Socrates: But he who imagines nothing certainly doesn’t imagine at all?
Theaetetus: Evidently, as we see.4
If in this and in the previous argument5
we put, say, “kill”6
in place of “imagine”, then
that amounts to a rule for the use of this word. It makes no sense to say: “I’m killing7
something that doesn’t exist”.
I can imagine a stag on this meadow that isn’t there but I can’t kill one that isn’t there.
– 8
And to imagine a stag that isn’t there means to imagine that a stag is there even though
there is none there. But to kill a stag does not mean: to kill that a stag is there.9
But if
someone says: “In order for me to imagine a stag it must exist in a certain sense”, then
the answer is: No, for this to happen it doesn’t have to exist in any sense. And if in answer
to that someone said: But the colour brown, for example, must exist for me to be able to
imagine it, then one ought to say: “‘The colour brown exists’ means nothing at all, except
maybe that it appears here and there as the colour of an object (of a patch), but that isn’t
necessary for me to be able to imagine a brown stag.”
10
“I’m imagining what it will be like” (for the chair to be painted white) – how can I
imagine it if it isn’t the case?! Is imagination perhaps a magic trick?
1 (M): 3
2 (M): ///
3 (M): 3
4 (E): Plato, Theaetetus, 189a.
5 (V): If in this argument
6 (V): “cut up”
7 (V): word. One is not allowed to say: “I’m
cutting up
8 (M): 555 – is there.
9 (V): there. (.thus: various grammatical rules).
10 (M): ///
270e
WBTC078-85 30/05/2005 16:16 Page 541
14
Man möchte fragen: Welcher außerordentliche Prozeß muß das Wollen sein, daß ich
das schon jetzt wollen kann, was ich erst in 5 Minuten tun werde?!
15
Die Antwort: Wenn Dir das sonderbar vorkommt, so vergleichst Du es mit etwas, womit
es nicht zu vergleichen ist. – 16
Etwa damit: Wie kann ich jetzt dem Mann die Hand geben,
der erst in 5 Minuten hereintreten wird? (Oder etwa gar: Wie kann ich dem die Hand geben,
den es17
vielleicht gar nicht gibt?)
18
Das ”foreshadowing“ der Tatsache besteht offenbar darin, daß wir jetzt denken können,
daß das eintreffen wird, was erst eintreffen wird. Oder, wie das irreführend ausgedrückt
wird: daß wir (an) das denken können, was erst eintreffen wird.
19
”Wenn immer ich über die Erfüllung eines Satzes rede, rede ich über sie im
Allgemeinen. Ich beschreibe sie in irgendeiner Form. Ja, es liegt diese Allgemeinheit schon
darin, daß ich die Beschreibung zum Voraus geben kann und jedenfalls unabhängig von dem
Eintreten der Tatsache.“
Wir sagen, daß der Ausdruck der Erwartung die erwartete Tatsache beschreibt & denken an sie
wie an einen Gegenstand oder Komplex der mit20
der Erfüllung der Erwartung in die Erscheinung tritt.21
22
Wenn wir sagen, daß wir die Tatsachen auf ”allgemeine Art“ beschreiben,23
so setzen
wir diese Art im Geiste einer andern entgegen. (Diese Entgegenstellung nehmen wir aber
natürlich von wo anders her.) Wir denken uns, daß bei der Erfüllung etwas Neues entsteht
und nun da ist, was früher nicht da war. Das heißt, wir denken an einen Gegenstand oder
Komplex, auf den wir nun zeigen können, beziehungsweise, der sich nun selbst repräsentieren
kann, während die Beschreibung nur sein Bild war. Wie wenn ich den Apfel, der auf
diesem Zweig wachsen wird, zum Voraus gemalt hätte, nun aber er selber kommt. Man
könnte dann sagen, die Beschreibung des Apfels war allgemein, d.h. mit Wörtern, Farben,
etc. bewerkstelligt, die schon vor dem Apfel und nicht speziell für ihn da waren. Gleichsam
altes Gerümpel im Vergleich mit dem wirklichen Apfel. Vorläufer,24
die alle abdanken
müssen, wenn der Erwartete (selber) kommt.
25
Aber der Erwartete ist nicht die Erfüllung, sondern: daß er gekommen ist.
26
Dieser Fehler ist tief in unserer Sprache verankert: Wir sagen ”ich erwarte ihn“ und
”ich erwarte sein Kommen“ und ”ich erwarte, daß er kommt“.
27
Die Tatsache wird allgemein beschrieben heißt, sie wird aus alten Bestandteilen
zusammengesetzt.
Sie wird beschrieben, das ist so, als wäre sie uns, außer durch die Beschreibung, noch
anders gegeben.
28
Hier wird die Tatsache mit einem Haus oder einem sonstigen29
Komplex gleichgestellt.
14 (M): 3
15 (R): ∀ S. 23/2, 33
16 (M): ///
17 (V): sie
18 (M): 3
19 (M): ///
20 (V): bei
21 (V): Komplex der in die Erscheinung tritt.
22 (M): ///
23 (V): Wenn man sagt, daß die Tatsache auf
”allgemeine Art“ beschrieben wird,
24 (V): Vorbilder,
25 (M): 3
26 (M): 3
27 (M): ////
28 (M): ///
29 (V): andern
365
364v
365
366
271 Wie kann man etwas wünschen . . .
WBTC078-85 30/05/2005 16:16 Page 542
11
One would like to ask: What kind of an extraordinary process must wanting be for me
to be able to want now to do what I won’t do for another five minutes?!
12
The answer: If that seems strange to you then you’re comparing it to something it can’t
be compared to. – 13
Say, with: How can I now shake the hand of the man who won’t arrive
for another five minutes? (Or even: How can I shake the hand of someone who perhaps
doesn’t exist at all?)
14
The “foreshadowing” of a fact obviously consists in our being able to think now that
that will happen which only will happen. Or as this is misleadingly expressed: in that we can
think (of) what only will happen.
15
“Whenever I talk about the fulfilment of a proposition I am talking about it in general.
I’m describing it in some form. Indeed, this generality is already contained in my being able
to provide the description in advance, and in any case independently, of the occurrence of
the fact.”
We say that the expression of the expectation describes the expected fact, and we think of it as
we think of an object or complex that enters the picture when the expectation is fulfilled.16
17
When we say that we describe facts18
in a “general way”, then mentally we contrast that
way with a different one. (But of course we’re taking this contrast from somewhere else.)
We imagine that at the moment of fulfilment something new comes into being and is here
which wasn’t here before. That is, we think of an object or complex to which we can now
point, or which can now represent itself, whereas the description was merely its image. As
if I had painted the apple that will grow on this branch in advance, but now it appears
itself. Then one could say that the description of the apple was general, i.e. was brought
about with words, colours, etc. that were already here before the apple, and weren’t created
especially for it. Stuff, as it were, in comparison to the real apple. Precursors19
, all of whom
must abdicate when the Expected One (himself) arrives.
20
But the Expected One is not the fulfilment; rather: it is the fact that he has come.
21
This mistake is deeply rooted within our language: We say “I’m expecting him” and
“I’m expecting his arrival” and “I’m expecting that he’ll come”.
22
“The fact is described in general terms” means: it is cobbled together out of old parts.
It is described: that’s as if it were given to us in a way other than through description.
23
Here a fact is equated to a house, or to some other24
complex.
11 (M): 3
12 (R): ∀ p. 23/2, 3 3
13 (M): ///
14 (M): 3
15 (M): ///
16 (V): that enters the picture.
17 (M): ///
18 (V): When one says that the fact is described
19 (V): Prefigurations
20 (M): 3
21 (M): 3
22 (M): ////
23 (M): ///
24 (V): or with another
“How can one Wish for, Something that isn’t There?” . . . 271e
WBTC078-85 30/05/2005 16:16 Page 543
30
Noch einmal der Vergleich: der Mensch tritt ein – das Ereignis31
tritt ein: Als wäre das
Ereignis32
schon vorgebildet vor der Tür der Wirklichkeit und würde nun in diese eintreten,
wenn es33
eintritt.
34
Das ganze Problem der Bedeutung der Worte ist darin aufgerollt, daß ich den A suche,
ehe ich ihn gefunden habe. – Es ist darüber zu sagen, daß ich ihn suchen kann, auch wenn
er in gewissem Sinne nicht existiert.
Wenn wir sagen, ein Bild ist dazu35
nötig, wir müssen in irgend einem Sinne ein Bild von
ihm herumtragen, so sage ich: vielleicht; aber was hat es für einen Sinn, zu sagen, es sei ein
Bild von ihm. Das hat also auch nur einen Sinn, wenn ich ein weiteres Bild von ihm habe,
das dem Wort ”ihm“ entspricht.
36
Man sagt etwa: Wenn ich von der Sonne spreche, muß ich ein Bild der Sonne in mir
haben. – Aber wie kann man sagen, daß es ein Bild der Sonne ist. Hier wird doch die Sonne
wieder erwähnt, im Gegensatz zu ihrem Bilde. Und damit ich sagen kann: ”das ist ein Bild
der Sonne“, müßte ich ein weiteres Bild der Sonne besitzen. u.s.w.
37
Man könnte nur sagen: Wenn er von der Sonne spricht, muß er ein visuelles Bild (oder
Gebilde von der und der Beschaffenheit – rund, gelb, etc.) vor sich sehen. Nicht, daß das
wahr ist, aber es hat Sinn, und dieses Bild ist dann ein Teil des Zeichens.
38
Wie seltsam, ich kann ihn suchen, wenn er nicht da ist, aber ich kann nicht auf ihn zeigen,
wenn er nicht da ist. Das ist eigentlich das Problem des Suchens und zeigt den irreführenden
Vergleich.
Man könnte sagen wollen: da muß er doch auch dabei sein, wenn ich ihn suche. – Dann
muß er auch dabei sein, wenn ich ihn nicht finde, und auch, wenn es ihn nicht gibt.
39
Ihn (etwa meinen Stock) suchen, ist eine Art des Suchens und unterscheidet sich davon,
daß man etwas andres sucht, durch das, was man beim Suchen tut (sagt, denkt), nicht durch
das, was man findet.
40
Und trage ich beim Suchen ein Bild mit mir oder eine Vorstellung, nun gut. Und sage
ich, das Bild sei das Bild des Gesuchten, so sagt das nur, welchen Platz das Bild im Vorgang
des Suchens einnimmt. Und finde ich ihn und sage ”da ist er! den habe ich gesucht“, so
sind die letzten Worte nicht etwa eine Worterklärung für die Bezeichnung des gesuchten
Gegenstandes (etwa für die Worte ”mein Stock“), die erst jetzt, wo er gefunden ist, gegeben
werden kann.41
– Wie man das, was man wünscht, nach der Erfüllung des Wunsches nicht
besser weiß, oder erklären kann, als vorher.
42
Man kann den Dieb nicht hängen ehe man ihn hat, wohl aber schon suchen.
43
”Du hast den Menschen gesucht? Wie war das möglich, er war doch gar nicht da!“
44
”Ich suche meinen Stock. – Da ist er!“ Dies letztere ist keine Erklärung des Ausdrucks
”mein Stock“, die für das Verständnis des ersten Satzes wesentlich wäre und die ich daher
30 (M): 3
31 (V): Mensch tritt ein – die Tatsache
32 (V): wäre die Tatsache
33 (V): sie
34 (M): 3
35 (V): dann
36 (M): ///
37 (M): ///
38 (M): 3
39 (M): 3
40 (M): 3
41 (V): könnte.
42 (M): 3
43 (M): 3
44 (M): ///
367
368
272 Wie kann man etwas wünschen . . .
WBTC078-85 30/05/2005 16:16 Page 544
25
Once again the comparison: A person comes in – an event26
comes about. As if the event27
,
already pre-formed, were standing outside the door of reality, and then entered into reality
when it occurred.
28
The entire problem of the meaning of words is laid out in the fact that I look for A before
I have found him. – In this connection we can say that I can look for him even if in a
certain sense he doesn’t exist.
If we say that we need a picture to do this,29
that in some sense we have to be carrying a
picture of him around with us, then I say: Maybe, but what’s the point of saying that it’s a
picture of him? That too makes sense only if I have an additional picture of him that corresponds
to the word “him”.
30
One might say: If I’m going to talk about the sun I have to have a picture of the sun
within me. – But how can one say that it is a picture of the sun? For here the sun, as opposed
to its picture, is mentioned once again. And in order for me to say “That’s a picture of the
sun” I’d have to possess an additional picture of the sun. Etc.
31
All one can say is: If he’s talking about the sun he must be seeing a visual image (or a
configuration of such and such qualities – round, yellow, etc.) in his mind’s eye. Not that
this is true, but it makes sense, and then this image is part of the sign.
32
How strange that if he isn’t here I can look for him, but not point to him. That is really
the problem of looking for something, and it shows the misleading comparison.
One might want to say: If I’m looking for him then surely he too has to be part of the
process. – Then he also has to be part of the process if I don’t find him, and even if he
doesn’t exist.
33
Looking for it (say my cane) is a kind of looking, and it differs from looking for
something else because of what one does (says, thinks) while looking, not because of what
one finds.
34
And if while I’m looking I carry a picture or a mental image around with me, fine. And
if I say that that picture is a picture of what I’m looking for, then that simply expresses
the place that the picture occupies in the process of looking. And if I find the object and
say “Here it is! This is what I’ve been looking for”, then these last words are in no way a
verbal explanation of the term for the object I was looking for (say of the words “my
cane”), such that the explanation can35
only be given now that it has been found. – Just as
one doesn’t know or can’t explain what one wishes for any better after the wish has been
fulfilled than before.
36
You can’t hang the thief before you’ve caught him, but you can be looking for him.
37
“You were looking for that man? How was that possible, he wasn’t even there!”
38
“I’m looking for my cane. – Here it is!” The latter is not an explanation of the
expression “my cane” that’s essential to understanding the first sentence, and that therefore
25 (M): 3
26 (V): in – the fact
27 (V): the fact
28 (M): 3
29 (V): that a picture is then necessary,
30 (M): ///
31 (M): ///
32 (M): 3
33 (M): 3
34 (M): 3
35 (V): could
36 (M): 3
37 (M): 3
38 (M): ///
“How can one Wish for, Something that isn’t There?” . . . 272e
WBTC078-85 30/05/2005 16:16 Page 545
nicht hätte geben können, ehe mein Stock gefunden war. Vielmehr muß der Satz ”da ist
er“, wenn er nicht eine Wiederholung der (auch) früher möglichen Worterklärung ist, ein
neuer synthetischer Satz sein.
45
Das Problem entspricht einer Verwechslung eines46
Wortes oder Ausdrucks mit dem Satz,
der die Existenz, das Dasein, des Gegenstands behauptet.
47
”Den hast Du gesucht? Du konntest ja nicht einmal wissen, ob er da ist!“ (Vergleiche
dagegen das Suchen nach der Dreiteilung des Winkels.)
48
Auch haben wir hier die Verwechslung zwischen der Bedeutung und dem Träger eines
Wortes. Denn der Gegenstand, auf den ich bei dem Worte ”den“ zeige, ist der Träger des
Namens, nicht seine Bedeutung.
49
Kurz: ich suche den Träger des Namens, nicht dessen50
Bedeutung.51
Aber anderseits: ich suche und hänge den Träger des Namens.52
Man kann von dem Träger des Namens sagen, daß er (existiert oder) nicht existiert, und
das ist natürlich keine Tätigkeit, obwohl man es mit einer verwechseln könnte und sagen,
er müsse doch dabei sein, wenn er nicht existiert. (Und das ist von einem Philosophen
bestimmt schon einmal geschrieben worden.)
53
(”Ich suche ihn“. – ”Wie schaut er aus“. – ”Ich weiß es nicht, aber (ich bin sicher) ich
werde ihn wiedererkennen, wenn ich ihn sehe“.)
54
Der Gedanke, daß uns (erst) das Finden sagt,55
was wir erwartet haben, heißt, den Vorgang
so beurteilen, wie etwa die Symptome der Erwartung bei einem Andern. Ich sehe ihn etwa
unruhig auf und ab gehen; da kommt jemand zur Tür herein und er wird ruhig und gibt
Zeichen der Befriedigung; und nun sage ich: ”er hat offenbar diesen Menschen erwartet“.
56
Die ”Symptome der Erwartung“ sind nicht der Ausdruck der Erwartung.
Und zu glauben, ich wüßte erst nach dem Finden, was ich gesucht (nach der Erfüllung,
was ich gewünscht) habe, läuft auf einen unsinnigen ”behaviourism“ hinaus.
57
”Ich wünsche mir eine gelbe Blume“. – ”Ja, ich gehe und suche Dir eine gelbe Blume.
Hier habe ich eine gefunden“. – Gehört die Bedeutung von ”gelbe Blume“ mehr zum
letzten Satz, als zu den zwei vorhergehenden?
58
Die Bedeutung des Wortes ”gelb“ ist nicht die Existenz eines gelben Flecks: Das ist es,
was ich über das Wort ”Bedeutung“ sagen59
möchte.
60
»Die Vorstellung, die mit dem Wort rot verbunden ist, ist gewiß die, welche der Tatsache
entspricht, daß etwas rot ist, – nicht die, die der Tatsache entspricht, daß etwas blau, also
nicht rot ist. Statt der Worterklärung ”das ↑ ist rot“ sollte ich sagen ”so sieht es aus, wenn
etwas rot ist“. Ja, die Vorstellung rot ist die Vorstellung, daß etwas rot ist. Und darauf beruht
jene Verwechslung von Wort und Satz, von der ich früher sprach.«
45 (M): 3
46 (V): des
47 (M): 3
48 (M): ///
49 (M): ////
50 (V): seine
51 (V): nicht die Bedeutung des Namens.
52 (M): (?)
53 (M): ///
54 (M): | 3
55 (V): zeigt,
56 (M): 3
57 (M): ///
58 (M): ///
59 (V): suchen
60 (M): ///
369
370
273 Wie kann man etwas wünschen . . .
WBTC078-85 30/05/2005 16:16 Page 546
I couldn’t have given before my cane had been found. Rather, the sentence “Here it is”, if
it isn’t a repetition of a verbal explanation that could (also) have been given earlier, must
be a new synthetic proposition.
39
This problem is equivalent to confusing a word or expression with a proposition that
asserts the existence, the being, of an object.
40
“You were looking for him? You couldn’t even know whether he was there!” (Compare,
on the other hand, looking for the trisection of an angle.)
41
Here we also have the confusion of the meaning and the bearer of a word. For the object
to which I point when I say “that” is the bearer of the name, not its meaning.
42
In short: I’m looking for the bearer of the name and not its meaning.43
But on the other hand: I look for and hang the bearer of the name.44
You can say of the bearer of a name that he (exists or) doesn’t exist, and of course that’s no
activity, even though you could confuse it with one and say that he has to be involved if he
doesn’t exist. (And most certainly that has been written by some philosopher at some point.)
45
(“I’m looking for him.” – “What does he look like?” – “I don’t know, but (I’m sure) I’ll
recognize him when I see him.”)
46
The idea that it’s only finding something that tells47
us what we have been expecting
means that one judges this process as one judges, say, the symptoms of expectation in someone
else. I might see him nervously pacing up and down; then someone comes in the door
and he quietens down and shows signs of satisfaction; and then I say: “Evidently he was
expecting this person”.
48
The “symptoms of expectation” aren’t the expression of expectation.
And to think that I couldn’t know what I had been looking for until after I had found it
(what I had wished for until after it had been fulfilled) amounts to a nonsensical form of
“behaviourism”.
49
“I wish I had a yellow flower”. – “Fine, I’ll go and look for one for you. Here, I’ve found
one.” – Does the meaning of “yellow flower” belong more closely to the last sentence than
to the first two?
50
The meaning of the word “yellow” is not the existence of a yellow patch: That is what
I would like to say51
about the word “meaning”.
52
»The mental image that’s connected with the word red is certainly the one that corresponds
to the fact that something is red – and not the one that corresponds to the fact that
something is blue, i.e. not red. Instead of the verbal explanation “That ↑ is red” I should
say “This is what it looks like when something is red”. Indeed, the mental image red is the
mental image that something is red. And that is the basis of the confusion of word and proposition
that I spoke of earlier.«
39 (M): 3
40 (M): 3
41 (M): ///
42 (M): ////
43 (V): and not the meaning of the name.
44 (M): (?)
45 (M): ///
46 (M): | 3
47 (V): shows
48 (M): 3
49 (M): ///
50 (M): ///
51 (V): seek
52 (M): ///
“How can one Wish for, Something that isn’t There?” . . . 273e
WBTC078-85 30/05/2005 16:16 Page 547
Könnte man zur Erklärung des Wortes &rot“ auf etwas hinweisen61
was nicht rot ist? Wie wenn
man einem, der der deutschen Sprache nicht mächtig ist das Wort &bescheiden“ erklären sollte
& man zeigte dazu auf einen sehr arroganten62
Menschen & sagte zur Erklärung:63
&der ist das
Gegenteil von bescheiden“. Es ist kein Argument gegen diese Erklärungsweise daß sie64
vieldeutig
ist. Mißverstanden werden kann jede Erklärung.
Das Mißverständnis äußert sich auch darin, daß es doppelsinnig ist vom &Vorkommen von
rot“ zu reden.65
In dem einen Fall heißt es, daß sowohl da wie dort etwas rot ist – d.h. die
Eigenschaft rot hat. In dem andern handelt es sich nicht um eine Gemeinsamkeit der Farbe
(die ja durch eine Farbangabe ausgedrückt würde).
Diese Gemeinsamkeit ist eben die Harmonie von66
Wirklichkeit67
und Gedanken, der68
in
Wahrheit eine Form69
unserer Sprache70
entspricht.71
61 (V): weisen,
62 (V): unbescheidenen
63 (V): & sagte:
64 (V): sie mißverstan
65 (V): Und hier ist, glaube ich, ein Hauptanstoß
zum Mißverständnis, daß das ”Vorkommen
von rot“ in zwei Tatbeständen als deren
gemeinsamer Bestandteil einen doppelten Sinn
hat. // daß es einen doppelten Sinn hat, wenn ich vom
&Vorkommen . . .“ rede.
66 (V): zwischen
67 (V): Welt
68 (V): die
69 (V): Regel
70 (V): Ausdrucksweise
71 (V): und Gedanken, die nicht zu beschreiben ist.
369v
370
370
274 Wie kann man etwas wünschen . . .
WBTC078-85 30/05/2005 16:16 Page 548
Could you point to something that isn’t red to explain the word “red”? What if you were supposed
to explain the meaning of the word “modest” to someone who didn’t know English, and to
do this you were to point to a very arrogant53
person and explain:54
“He’s the opposite of modest”.
That this way of explaining is ambiguous55
isn’t an argument against it. Any explanation can be
misunderstood.
A misunderstanding also comes out in the fact that talking about the “occurrence of red” is
ambiguous.56
In the one case it means that both here as well as there, something is red – i.e.
has the property red. In the other case it is not a matter of a commonality of colour (which
would, after all, be expressed by a specification of colour).
This commonality is precisely the harmony of57
reality and58
thought, to which a form of
our language59
in fact corresponds.60
53 (V): immodest
54 (V): and say:
55 (V): explaining can be misundersto
56 (V): And here, I believe, is the main trigger of
a misunderstanding: that “The occurrence of
red”, as the common component of two sets
of circumstances, is ambiguous. // : that there
is an ambiguity in speaking about the “occurrence
of red”.
57 (V): between
58 (V): harmony between world and
59 (V): a rule of our mode of expression
60 (V): thought, which cannot be described.
“How can one Wish for, Something that isn’t There?” . . . 274e
WBTC078-85 30/05/2005 16:16 Page 549
79
1
Im Ausdruck der Sprache berühren
sich Erwartung und Erfüllung.
2
In der Sprache berühren sich Erwartung und Ereignis.
3
”Ich sagte, ,geh’ aus dem Zimmer‘ und er ging aus dem Zimmer“.
”Ich sagte, ,geh aus dem Zimmer‘ und er ging langsam aus dem Zimmer“.
”Ich sagte, ,geh aus dem Zimmer‘ und er sprang zum Fenster hinaus“.
Hier ist eine Rechtfertigung möglich, auch wo die Beschreibung der Handlung nicht die
ist, die der Befehl gibt.
4
Es ist doch offenbar nicht undenkbar,5
daß Einer die gelbe Blume so mit einem
Phantasiebild sucht, wie ein Anderer mit dem färbigen Täfelchen, oder ein Dritter in
irgendeinem Sinne, mit dem Bild einer Reaktion, die durch das, was er sucht, hervorgerufen
werden soll (Klingel).
Womit immer aber er suchen geht (mit welchem Paradigma immer), nichts zwingt
ihn, das als das Gesuchte anzuerkennen, was er am Schluß wirklich anerkennt, und
die Rechtfertigung in Worten, oder andern Zeichen, die er dann von dem Ergebnis6
gibt,
rechtfertigt wieder nur in7
Bezug auf eine andere Beschreibung in derselben Sprache.
8
Die Schwierigkeit ist aufzuhören, ”warum“ zu fragen (ich meine, sich dieser Frage zu
enthalten).
9
Du befiehlst mir ”bringe mir eine gelbe Blume“; ich bringe eine und Du fragst: ”warum
hast Du mir so eine gebracht?“ Dann hat diese Frage nur einen Sinn, wenn sie zu ergänzen
ist ”und nicht eine von dieser (andern) Art“.
D.h., diese Frage bezieht sich schon auf10
ein System; und die Antwort muß11
sich auf
das gleiche System beziehen.
12
Auf die Frage ”warum tust Du das auf meinen Befehl?“ kann man fragen: ”was?“
Da wäre es nun absurd zu fragen ”warum bringst Du mir eine gelbe Blume, wenn ich
Dir befohlen habe, mir eine gelbe Blume zu bringen“. Eher könnte man fragen ”warum
bringst Du eine rote Blume, wenn ich sagte, Du sollest13
eine gelbe bringen“ oder ”warum
bringst Du eine dunkelgelbe auf den Befehl ,bring’ eine gelbe‘?“
1 (M): 3
2 (M): 3
3 (M): ////
4 (M): ////
5 (V): unmöglich,
6 (V): Resultat
7 (V): im
8 (M): ///
9 (M): ////
10 (V): Frage gehört schon in
11 (V): muß man
12 (M): ////
13 (V): solltest
371
372
275
WBTC078-85 30/05/2005 16:16 Page 550
79
1
Expectation and Fulfilment Make
Contact in Linguistic Expression.
2
Expectation and event make contact in language.
3
“I said, ‘Leave the room’ and he left the room.”
“I said, ‘Leave the room’ and he left the room slowly.”
“I said, ‘Leave the room’ and he jumped out of the window.”
A justification is possible here, even when the description of the action isn’t the same as
that given by the command.
4
In looking for a yellow flower, it’s obviously in no way inconceivable5
that one person
might use a mental image, just as another might use a colour chip, or a third person might
– in some way – use the image of a reaction that will be triggered by what he is looking for
(a bell).
But whatever he uses in his search (whatever paradigm), nothing forces him to acknowledge
what in the end he actually acknowledges as the thing he was looking for, and the
justification he then gives for the result, whether in words or other signs, once again only
justifies that thing in reference to a different description in the same language.
6
The difficulty is to stop asking “Why?” (I mean, to abstain from this question).
7
You give me the command “Bring me a yellow flower”; I bring one and you ask: “Why
did you bring me one like that?” This question only makes sense if it is to be supplemented
by “and not one of this (other) kind”.
That is, this question already refers to8
a system; and the answer has to refer to the same
system.
9
When asked “Why are you doing this in response to my command?” one can respond:
“What?”
In this case it would be absurd to ask “Why are you bringing me a yellow flower when I
ordered you to bring me a yellow flower?”. More likely one could ask “Why are you bringing
a red flower when I said you should bring a yellow one?” or “Why are you bringing a
dark yellow flower in response to the order ‘Bring a yellow one’?”
1 (M): 3
2 (M): 3
3 (M): ////
4 (M): ////
5 (V): impossible
6 (M): ///
7 (M): ////
8 (V): already belongs in
9 (M): ////
275e
WBTC078-85 30/05/2005 16:16 Page 551
14
Noch einmal: was ist das Kriterium dafür, daß der Befehl richtig ausgeführt wurde? Was
ist das Kriterium, nämlich auch für den Befehlenden? Wie kann er wissen, daß der Befehl
nicht richtig ausgeführt wurde. Angenommen, er ist von der Ausführung befriedigt und sagt
nun: ”von dieser Befriedigung lasse ich mich aber nicht täuschen, denn ich weiß, daß doch
nicht das geschehen ist, was ich wollte“. Er erinnert sich in irgend einem Sinne daran, wie
er den Befehl gemeint hatte. – In welchem Sinne? Woran erinnere ich mich, wenn ich mich
erinnere, das gewünscht zu haben.
15
Man hat vielleicht das Gefühl: es kann doch nicht im Satz ”ich glaube, daß p der Fall
ist“ das ”p“ dasselbe bedeuten, wie in der Behauptung ”p“, weil ja in der Tatsache des
Glaubens, daß p der Fall ist, die Tatsache daß p der Fall ist, nicht enthalten ist.
16
Man hat das Gefühl, daß ich mich im Satz ”ich erwarte, daß er kommt“ der Worte ”er
kommt“ in anderem Sinne bediene, als in der Behauptung ”er kommt“. – Aber wäre es so,
wie könnte ich davon reden, daß meine Erwartung durch die Tatsache befriedigt ist?
17
Nun könnte man aber fragen: Wie schaut das aus, wenn er kommt? – ”Es geht die Tür
auf und ein Mann tritt herein, der . . .“. Wie schaut das aus, wenn ich erwarte, daß er kommt?
– ”Ich gehe auf und ab, sehe auf die Uhr, . . .“. – Aber der eine Vorgang hat ja mit dem
anderen18
nicht die geringste Ähnlichkeit! Wie kann man dann dieselben Worte zu ihrer
Beschreibung gebrauchen? Diesen Vorgang würde ich nicht mit den Worten &ich erwarte daß er
kommt“ beschreiben. Worin läge es denn z.B. daß ich gerade ihn erwarte? Ich sagte doch der Vorgang
der Erwartung sollte ein solcher sein, daß ich, ihn sehend, erkennen müßte was erwartet wird. Aber,
auf-und-abgehen konnte ich ja auch, ohne zu erwarten,19
daß er kommen werde, auf die Uhr
sehen auch, etc.; das ist also nicht das Charakteristische des Erwartens, daß er kommt. Das
Charakteristische aber ist nur eben durch diese Worte gegeben. Und ”er“ heißt dasselbe,
wie in der Behauptung ”er kommt“ und ”kommt“ heißt dasselbe, wie in der Behauptung,
und ihre Zusammenstellung bedeutet nichts anderes. D.h. z.B.: eine hinweisende Erklärung
des Wortes ”er“ gilt für beide Sätze.
20
Wenn ich ~p glaube, so glaube ich dabei nicht zugleich p, weil ”p“ in ”~p“ vorkommt.
p kommt in ~p in demselben Sinne vor, wie ~p in p.
Die Worte ”vorkommen“ etc. sind eben unbestimmt, wie alle solche Prosa. Exakt und
unzweideutig und unbestreitbar sind nur die grammatischen Regeln, die am Schluß zeigen
müssen, was gemeint ist.
14 (M): ///
15 (R): [Zu: Behauptung, etc.]
16 (M) 3
17 (M): 3
18 (O): mit anderen
19 (V): warten,
20 (R): [Zu: Behauptung, etc.]
373
372v
373
374
276 Im Ausdruck der Sprache berühren sich Erwartung und Erfüllung.
WBTC078-85 30/05/2005 16:16 Page 552
10
Once again: What is the criterion for the command having been carried out correctly?
That is, what is the criterion even for the person who gives the command? How can he know
that the command was not carried out correctly? Let’s assume that he’s satisfied with the
execution of his command, and he says: But I’m not going to be fooled by this satisfaction,
for I know that what I wanted still hasn’t happened. In some sense he remembers how he
had meant the command. – In what sense? What do I remember when I remember that I
wanted that?
11
Perhaps one has the feeling: In the sentence “I believe that p is the case”, “p” cannot
possibly mean the same thing as in the assertion “p”, because the fact that p is the case is
not contained in the fact that I believe that p is the case.
12
One has the feeling that in the sentence “I expect that he’s coming” I use the words
“he’s coming” in a different sense than in the assertion “He’s coming”. – But if this were
so, how could I talk about my expectation having been satisfied by the fact?
13
But now one could ask: What does it look like for him to come? – “The door opens and
a man comes in who . . .”. What does it look like when I’m expecting him to come? – “I
pace back and forth, look at my watch, . . .”. – But the one sequence of events doesn’t have
even the remotest similarity to the other! So how can one use the same words to describe
them? I wouldn’t use the words “I’m expecting that he is coming” to describe the one sequence
of events. What would my expecting him and no one else, for example, consist in? I did say, after
all, that the process of expectation should be of such a kind that in seeing it I would have to
recognize what is being expected. Surely I could pace back and forth without expecting14
him
to come, as well as look at my watch, etc.; so that isn’t what characterizes expecting him
to come. What does characterize it is given just by these very words. And “he” means the
same thing as in the assertion “He is coming”, and “is coming” means the same thing as in
that assertion, and their combination means nothing different. That is to say: one ostensive
explanation of the word “he” is good for both propositions.
15
If I believe ~p, then in that process I don’t simultaneously believe p, just because “p”
occurs in “~p”.
p occurs in ~p in the same sense as ~p occurs in p.
The words “to occur”, etc., are simply indefinite, as is all such prose. The only things
that are exact and unambiguous and indisputable are the grammatical rules, which in the
end must show what is meant.
10 (M): ///
11 (R): [To: Assertion, etc.]
12 (M): 3
13 (M): 3
14 (V): without waiting for
15 (R): [To: Assertion, etc.]
Expectation and Fulfilment . . . 276e
WBTC078-85 30/05/2005 16:16 Page 553
80
”Der Satz bestimmt, welche
Realität ihn wahr macht.“
Er scheint einen Schatten dieser
Realität zu geben. Der Befehl scheint
seine Ausführung in schattenhafter
Weise vorauszunehmen.
Denn es ist also als ob dieses Etwas, die Handlung, ein Ding wäre das1
in der Befolgung des Befehls
in die Existenz treten solle & als ob der Befehl uns eben dieses Ding kennen lehrte,2
zeigte, so daß er
es also schon in irgend einem Sinne in die Existenz riefe.3
Wie kann der Befehl die Erwartung uns den Menschen zeigen ehe er in unser Zimmer eingetreten
ist?!
4
Die Beschreibung der Sprache muß dasselbe leisten wie die Sprache.
5
Denn dann kann ich wirklich aus dem Satz, der Beschreibung der6
Wirklichkeit, ersehen,
wie es sich in der Wirklichkeit verhält.
7
(Aber nur das nennt man ja ”Beschreibung“ und nur das nennt man ja ”ersehen, wie es
sich verhält“!)
8
(Und etwas anderes ist es ja nicht, was wir alle damit sagen: daß wir aus der
Beschreibung ersehen, wie es sich in Wirklichkeit verhält.)
9
”Du beziehst von dem Befehl die Kenntnis dessen, was Du zu tun hast.
Und doch gibt Dir der Befehl nur sich selbst, und seine Wirkung ist gleichgültig.“
10
Das wird erst dann seltsam, wenn der Befehl etwa ein Glockenzeichen ist. – Denn, in
welchem Sinne mir dieses Zeichen mitteilt, was ich zu tun habe, außer daß ich es eben11
tue
und das Zeichen da war – . Denn es ist auch nicht das, daß ich es erfahrungsgemäß immer
tue, wenn das Zeichen gegeben wird.
1 (V): das wir
2 (V): lehrte, also
3 (V): Existenz rufen müßte.
4 (M): 3 (R): ∀ S. 384/1,2
5 (M): 3
6 (V): Beschreibung$ der
7 (M): 3
8 (M): 3
9 (M): 3
10 (M): ///
11 (V): einfach
375
374v
375
376
277
WBTC078-85 30/05/2005 16:16 Page 554
80
“The Proposition Determines
which Reality Makes it True.”
It Seems to Provide a Shadow
of this Reality. A Command
Seems to Anticipate its Execution
in a Shadowy Way.
For it’s as if this something, the action, were a thing that’s supposed to enter into existence as the
command is being obeyed, and as if the command were acquainting us with this very thing, were
showing it to us, so that in some sense it would already be1
calling it into existence.
How can the command, the expectation, show us a person before he has entered our room?!
2
The description of language must accomplish the same thing as language.
3
For then I really can learn from the proposition, from the description of reality, how things
are in reality.
4
(But that is really the only thing that is called “description”, and that is really the only
thing that is called “learning how things are”!)
5
(And what all of us mean by this is nothing more than: We learn from the description
how things are in reality.)
6
“You acquire the knowledge of what you are supposed to do from a command.
And yet all the command does is to present itself to you; its effect is irrelevant.”
7
That only becomes strange when the command is something like the ringing of a bell. –
For in what sense this sign tells me what I am supposed to do, except that I just8
do it and
the sign was there – . Nor is it the case that on the basis of experience I always do it when
I’m given the sign.
1 (V): would have to be
2 (M): 3 (R): ∀ p. 384/1, 2
3 (M): 3
4 (M): 3
5 (M): 3
6 (M): 3
7 (M): ///
8 (V): simply
277e
WBTC078-85 30/05/2005 16:16 Page 555
12
Darum hat es ja auch ohne weiteres keinen Sinn, zu sagen: ”Ich muß gehen, weil die
Glocke geläutet hat“. Sondern, dazu muß noch etwas anderes gegeben sein.
13
Wie kann man die Handlung von dem Befehl ”hole eine gelbe Blume“ ableiten? – Wie
kann man das Zeichen ”5“ aus dem Zeichen ”2 + 3“ ableiten?
14
Kann man denn, und in welchem Sinne kann man, aus dem Zeichen plus dem
Verständnis (also der Interpretation) die Ausführung ableiten, ehe sie geschieht? Alles was
man ableitet, ist doch nur eine Beschreibung der Ausführung und auch diese Beschreibung
war erst da, nachdem man sie abgeleitet hatte.
15
Die Ausführung des Befehls leiten wir von diesem erst ab, wenn wir ihn ausführen.
16
The bridge can only be crossed when we get there. (Gemeint ist die Brücke zwischen
Zeichen und Realität.)
17
Von der Erwartung zur Erfüllung ist ein Schritt einer Rechnung. Ja, die Rechnung
steht zu ihrem Resultat 625 genau im Verhältnis der Erwartung zur Erfüllung.
18
Und so weit – und nur so weit – als diese Rechnung ein Bild des Resultats ist,
ist auch die Erwartung ein Bild der Erfüllung.
19
Und so weit das Resultat von der Rechnung bestimmt ist, so weit20
ist die Erfüllung durch
die Erwartung bestimmt.
21
”Der Befehl nimmt die Ausführung voraus.“ Inwiefern nimmt er sie denn voraus?
Dadurch, daß er jetzt befiehlt,22
was später ausgeführt (oder nicht ausgeführt) wird. Oder:
Das, was wir damit meinen, wenn wir sagen, der Befehl nimmt die Ausführung voraus,
ist dasselbe, was dadurch ausgedrückt ist, daß der Befehl befiehlt, was später geschieht.
Aber richtig: ”geschieht, oder nicht geschieht“. Und das sagt nichts. (Der Befehl kann sein
Wesen eben nur zeigen.)
23
Aber, wenn auch mein Wunsch nicht bestimmt, was der Fall sein wird, so bestimmt er
doch sozusagen das Thema einer Tatsache, ob die nun den Wunsch erfüllt, oder nicht.
24
Muß er nun dazu etwas voraus wissen? Nein. p 1 ~p sagt wirklich nichts.
25
Wir wundern uns – sozusagen – nicht darüber, daß Einer die Zukunft weiß, sondern –
darüber, daß er überhaupt (richtig oder falsch) prophezeien kann.
26
Es ist, als würde die bloße Prophezeiung (gleichgültig ob richtig oder falsch) schon
einen Schatten der Zukunft vorausnehmen. – Während sie über die Zukunft nichts weiß,
und weniger als nichts nicht wissen kann. –
27
Worin besteht das Vorgehen nach einer Regel? – Kann man das fragen? –
25 × 25
50
125
12 (M): ///
13 (M): ///
14 (M): 3
15 (M): 3
16 (M): 3
17 (M): 3
18 (M): 3
19 (M): 3
20 (V): Und so weit das Resultat von der // durch
die // Rechnung, so weit
21 (M): 3
22 (V): daß er das befiehlt,
23 (M): 3
24 (M): ///
25 (M): 3
26 (M): 3
27 (M): ////
377
378
278 Der Satz bestimmt, welche Realität ihn wahr macht. . . .
WBTC078-85 30/05/2005 16:16 Page 556
9
Therefore without further specification it really makes no sense to say: “I’ve got to go
because the bell has rung”. Something else has to be given for that to make sense.
10
How can you derive the action from the command “Get a yellow flower”? – How can
you derive the sign “5” from the sign “2 + 3”?
11
Before it takes place, can you really derive the execution from the sign plus understanding
(i.e. interpretation) – and in what sense can you do this? After all, everything you
derive is just a description of the execution, and this description too was here only after
you derived it.
12
We don’t derive the execution of a command from the command until we execute it.
13
The bridge can only be crossed when we get there. (What is meant is the bridge between
sign and reality.)
14
It is a step in a calculation that leads from expectation to fulfilment. Indeed, the calculation
is related to the result 625 exactly as expectation is to fulfilment.
15
And to the extent – and only to the extent – that this calculation is a picture of
the result, is expectation also a picture of fulfilment.
16
And the extent to which the result is determined by17
the calculation – that’s the extent
to which fulfilment is determined by expectation.
18
“A command anticipates its execution.” In what way does it anticipate it? By ordering
now what19
is carried out (or not carried out) later. Or: What we mean when we say that a
command anticipates its execution is the same thing that is expressed by the command
commanding what happens later. But put correctly: “happens or doesn’t happen”. And that
doesn’t say anything. (All a command can do is to show its nature.)
20
But even if my wish doesn’t determine what is going to be the case, still it does determine
the theme of the fact, so to speak, whether or not this fact then fulfils the wish.
21
Now does my wish have to know something in advance in order to do this? No. p 1 ~p
really says nothing.
22
We are surprised – as it were – not that someone knows the future, but at his being able
to prophesy (correctly or incorrectly) at all.
23
It’s as if the mere prophecy (whether true or false) foreshadowed the future. – Whereas
it knows nothing about the future, and can’t know less than nothing.
24
What does acting in accordance with a rule consist in? – Can one ask that? –
25 × 25
50
125
9 (M): ///
10 (M): ///
11 (M): 3
12 (M): 3
13 (M): 3
14 (M): 3
15 (M): 3
16 (M): 3
17 (V): determined through
18 (M): 3
19 (V): now that which
20 (M): 3
21 (M): ///
22 (M): 3
23 (M): 3
24 (M): ////
The Proposition Determines which Reality Makes it True. . . . 278e
WBTC078-85 30/05/2005 16:16 Page 557
Ich gehe nach einer Regel vor heißt: ich gehe so vor, daß das, was herauskommt, . . . Daß
das, was herauskommt, dieser Regel genügt.
Nach der Regel vorgehen, heißt so vorgehen, und das ”so“ muß die Regel enthalten.
28
Wenn die Regel heißt ”wo Du ein ↑ siehst, schreib’ ein ,c‘“, so ist damit gegeben, was
ich tun soll, so weit es überhaupt gegeben sein kann.
Denn mehr bestimmt, als durch eine genaue Beschreibung, kann etwas nicht sein. Denn,
bestimmen kann nur heißen, es beschreiben.
Dann ist eine Handlung nicht bestimmt, wenn die Beschreibung noch etwas offen
gelassen hat29
(so, daß man sagen kann ”ich weiß noch nicht ob . . .“), was30
also die31
Beschreibung bestimmen kann. Ist die Beschreibung vollständig, so ist die Handlung
bestimmt. Und das heißt, es kann der Beschreibung nur eine Handlung entsprechen. (Nur
so können wir diesen Ausdruck32
gebrauchen.)
(Erinnern wir uns an die Argumentation über ”Zahnschmerzen“.)
33
Hier ist auch der Zusammenhang mit der Frage: ”sieht der Andere wirklich dieselbe
Farbe, wenn er blau sieht, wie ich?“ Freilich, er sieht blau! Das ist ja eben dieselbe Farbe.
– D.h., die Frage, ob er als blau dieselbe Farbe sieht, ist unsinnig, wenn angenommen ist,
daß wir das Recht haben, was er sieht und ich sehe, als ”blau“ zu bezeichnen. Läßt sich im
gewöhnlichen Sinne – d.h. nach der gewöhnlichen Methode – konstatieren, daß er nicht
dieselbe Farbe sieht, so kann ich nicht sagen, daß wir beide blau sehen. Und läßt es sich
konstatieren, daß wir beide blau sehen, dann ”sehen wir beide die gleiche Farbe“, denn dieser
Satz hat ja nur auf diese Proben Bezug.34
35
Und analog36
verhält es sich mit der Frage: ”ist das, was ich jetzt ,gelb‘ nenne, gewiß
die gleiche Farbe, die ich früher ,gelb‘ genannt habe?“ – Gewiß, denn es ist ja gelb. – Aber
woher weißt Du das? – Weil ich mich daran erinnere. – Aber kann die Erinnerung nicht
täuschen? – Nein. Nicht, wenn ihr Datum gerade das ist, wonach ich mich richte.
37
Wenn man nun fragt: Ist also die Tatsache durch die Erwartung auf ja und nein
bestimmt, oder nicht, d.h. ist es bestimmt, in welchem Sinne die Erwartung durch ein
Ereignis – welches immer eintrifft – beantwortet werden wird, so muß man antworten: ja!
Unbestimmt wäre es etwa im Falle einer Disjunktion im Ausdruck der Erwartung.
38
Wenn ich sage ”der Satz bestimmt doch schon im Voraus, was ihn wahr machen wird“:
Gewiß, der Satz ”p“ bestimmt, daß p der Fall sein muß, um ihn wahr zu machen; das ist
aber auch alles, was man darüber sagen kann, und heißt nur: ”der Satz p = der Satz, den
die Tatsache p wahr macht“.
In der Sprache wird alles ausgetragen.
28 (M): ///
29 (V): offen läßt
30 (O): ”ich weiß noch nicht ob . . .“ was
31 (V): eine
32 (V): wir das Wort
33 (M): ///
34 (O): bezug.
35 (R): [Zu: Erinnerungszeit]
36 (V): Und so
37 (M): 3
38 (M): 3
379
279 Der Satz bestimmt, welche Realität ihn wahr macht. . . .
WBTC078-85 30/05/2005 16:16 Page 558
I act in accordance with a rule means: I act in such a manner that what results, . . . . That
what results satisfies the rule.
To act in accordance with a rule means to act in such a way, and this “in such a way”
must contain the rule.
25
If the rule is “Wherever you see a ↑, write a ‘c’”, then what I’m supposed to do is thereby
given, in so far as it can be given at all.
For a thing cannot be specified any further than it is by an exact description. For to
specify something can only mean to describe it.
Then an action isn’t specified if its description has left26
something open (such that one
can say “I don’t know yet whether . . .”) that can specify the description.27
If the description
is complete, the action is specified. And that means, only one action can correspond to
the description. (This is the only way we can use this expression.28
)
(Let’s remember the argument about “toothache”.)
29
Here there is also a connection with the question: “When someone sees blue, does he
really see the same colour as I do?” Of course, he’s seeing blue! It’s exactly the same colour.
– That is to say, the question whether he sees the same colour as blue is nonsensical if it is
assumed that we have the right to call what he and I see “blue”. If it can be ascertained in
the usual sense – i.e. according to the usual method – that he isn’t seeing the same colour,
then I can’t say that both of us are seeing blue. And if it can be ascertained that both of us
are seeing blue then “Both of us are seeing the same colour”, for this proposition refers only
to such tests.
30
And this question is analogous:31
“Is it certain that what I am now calling ‘yellow’ is the
same colour that I called ‘yellow’ before?” – Certainly, because it is yellow. – But how do
you know that? – Because I remember it. – But can’t memory deceive? – No. Not when its
data are precisely what I’m being guided by.
32
Now if one asks: “So is a fact, or isn’t it, determined by an expectation right down to a
yes or a no, i.e. is it determined in which sense the expectation will be answered by an event
– whatever event occurs?” – then the answer has to be: Yes! It would be indeterminate,
say, if the expression of the expectation contained a disjunction.
33
If I say: “But the proposition determines in advance what will make it true”, the answer
is: To be sure, the proposition “p” determines that to make it true, p must be the case; but
that’s all that can be said about this, and it just means: “The proposition p = the proposition
that the fact p makes true”.
Everything is carried out in language.
25 (M): ///
26 (V): description leaves
27 (V): specify a description.
28 (V): use the word.
29 (M): ///
30 (R): [To: Memory-time]
31 (V): And that’s how it is with the question:
32 (M): 3
33 (M): 3
The Proposition Determines which Reality Makes it True. . . . 279e
WBTC078-85 30/05/2005 16:16 Page 559
81
Intention.
1
Was für ein Vorgang ist sie? Man
soll aus der Betrachtung dieses
Vorgangs ersehen können, was
intendiert wird.
2
Wenn eine Vorrichtung als3
Bremse wirken soll, tatsächlich aber aus irgendwelchen
Ursachen den Gang der Maschine beschleunigt, so ist die Absicht, der die Vorrichtung
dienen sollte, aus ihr allein nicht zu ersehen.
Wenn man sagt ”das ist der Bremshebel, er funktioniert aber nicht“, so spricht man von
der Absicht. Ähnlich ist es, wenn man eine verdorbene Uhr doch eine Uhr nennt.
4
Angenommen, das Anziehen des Bremshebels bewirkt manchmal das Abbremsen
der Maschine und manchmal nicht. So ist daraus allein nicht zu schließen, daß er als
Bremshebel gedacht war. Wenn nun eine bestimmte Person immer dann, wenn der Hebel
nicht als Bremshebel wirkt, ärgerlich würde –. So wäre damit auch nicht das gezeigt, was
ich zeigen will. Ja, man könnte dann sagen, daß der Hebel einmal die Bremse, einmal den
Ärger betätigt. – Wie nämlich drückt es sich aus,5
daß die Person darüber ärgerlich wird,
daß der Hebel die Bremse nicht betätigt hat?
6
(Dieses über etwas ärgerlich sein ist nämlich scheinbar von ganz derselben Art, wie: etwas
fürchten, etwas wünschen, etwas erwarten, etc.) Das ”über etwas ärgerlich sein“ verhält sich
nämlich zu dem, worüber man ärgerlich ist, nicht wie die Wirkung zur Ursache, also nicht
wie Magenschmerzen zu der Speise mit der man sich den Magen verdorben hat. Man kann
darüber im Zweifel sein, woran man sich den Magen verdorben hat und die Speise, die etwa
die Ursache ist, tritt in die Magenschmerzen nicht als ein Bestandteil dieser Schmerzen ein;
dagegen kann man, in einem gewissen Sinne, nicht zweifelhaft sein, worüber man sich
ärgert, wovor man sich fürchtet, was man glaubt. (Es heißt nicht ”ich weiß nicht, – ich glaube
heute, aber ich weiß nicht woran“!) – Und hier haben wir natürlich das alte Problem, daß
nämlich der Gedanke, daß das und das der Fall ist, nicht voraussetzt, daß es der Fall ist.
Daß aber anderseits doch etwas von der Tatsache für den Gedanken selbst Voraussetzung
sein muß. ”Ich kann nicht denken, daß etwas rot ist, wenn rot garnicht existiert.“ 7
Die Antwort
1 (M): 3
2 (M): 3
3 (V): der
4 (M): 3
5 (V): Wie drückt es sich nämlich aus,
6 (M): /// ?
7 (M): ||
380
381
280
WBTC078-85 30/05/2005 16:16 Page 560
81
Intention.
1
What Kind of a Process is it? From
an Examination of this Process one is
Supposed to be Able to See What is
Being Intended.
2
If a device is supposed to function as a brake, but for whatever reasons it actually accelerates
the workings of the machine, then the purpose the device was supposed to serve can’t
be understood from it alone.
If one says “That’s the brake lever, but it isn’t working”, one is speaking of intention. It’s
similar to calling a broken watch a watch anyway.
3
Let’s assume that pulling the brake lever sometimes brings about the braking of the machine
and sometimes doesn’t. Then one can’t conclude from this alone that it was intended as a
brake lever. Now if a particular person were to become annoyed whenever the lever didn’t
function as a brake lever – that still wouldn’t show what I want to show. Indeed, in that case
one could say that the lever sometimes actuates the brake, sometimes the annoyance. – For
how is it expressed that that person gets annoyed by the fact that the lever hasn’t activated
the brake?
4
(For this being annoyed about something seems to be exactly the same kind of thing as:
fearing something, wishing something, expecting something, etc.) For “being annoyed about
something” isn’t related to what one is annoyed about as effect is to cause, that is, not as a
stomach-ache is to the food that has upset one’s stomach. You can be in doubt about what
has upset your stomach, and the food that may be the cause doesn’t enter your stomachache
as part of that pain; on the other hand, in a certain sense you can’t be in doubt about
what annoys you, what you fear, what you believe. (We don’t say “I don’t know – I have a
belief today, but I don’t know in what”!) – And here of course we have the old problem,
namely, that although the thought that such and such is the case doesn’t presuppose that it
is the case, nevertheless some part of the fact must be a prerequisite for the thought itself.
“I can’t think that something is red if red doesn’t even exist.” 5
The answer to this is that
1 (M): 3
2 (M): 3
3 (M): 3
4 (M): ///?
5 (M): ||
280e
WBTC078-85 30/05/2005 16:16 Page 561
darauf ist, daß die Gedanken im selben8
Raum sein müssen, wie das Zweifelhafte, wenn auch
an einer andern Stelle. Im Raum der Sprache nämlich. In der Sprache wird alles ausgetragen.
Der Satz &ich könnte nicht denken daß etwas rot ist wenn Rot nicht existierte“ bezieht sich wirklich
auf die Vorstellung von etwas Rotem oder die Existenz eines roten Musters als Teil unserer Sprache.
Aber natürlich kann man auch nicht sagen, unsere Sprache müsse ein solches Muster enthalten. Enthält
sie es nicht so ist sie eben eine Andere. Aber man kann sagen & betonen daß sie es enthält.
9
Darin, und nur darin besteht auch die (prästabilierte) Harmonie zwischen Welt und
Gedanken.
Die Intention ist nun aber von genau derselben Art wie – z.B. – der Ärger. Und da scheint
es irgendwie, als würde man die Intention von außen betrachtet nie als Intention erkennen;
als müßte man sie selbst intendieren,10
um sie als Meinung zu verstehen.11
Das hieße aber,
sie nicht als Phänomen, nicht als Tatsache, zu betrachten! D.h. es hieße eine weitere (unklar
angedeutete) Bedingung der Erfahrung allem hinzufügen.12
Und freilich, wenn Meinen13
eine Erfahrung ist so muß man eben diese haben um14
wirklich zu
meinen & nicht eine andere die man nennen könnte die Meinung von außen sehen.
Und hier erinnert die Intention an den Willen (auch im Schopenhauerschen Sinn).
Woher die Idee15
man könne etwas &nicht als Phänomen“ betrachten? Wie kommt denn hier das
Subjekt in die Betrachtung?
Und einerseits ist das so als wollte man sagen, man könne Zahnschmerzen nur von innen
betrachtet als solche erkennen. Von außen betrachtet wären sie16
z.B. gar nicht unangenehm. Die
Zahnschmerzen geben einem17
aber gar kein solches Problem.
Das Problem18
aber ist: wie kann man die Intention, wenn man sie nun hat in die Worte übersetzen,
sie sei die Intention das & das zu tun? Denn daß die Intention nur kennt wer sie erlebt hat (siehe
Zahnschmerzen) gebe ich zu; warum aber nennst Du sie die Intention das zu tun. Das hat mit ihrem
unbeschreibbaren Charakter offenbar nichts zu tun.
Das ist natürlich wieder das vorige Problem, denn der Witz ist, daß man es dem19
Gedanken
(als selbständige Tatsache betrachtet) ansehen muß, daß er der Gedanke ist, daß das und
das der Fall ist. Kann man es ihm nicht ansehen (so wenig wie den Magenschmerzen woher
sie rühren), dann hat er kein logisches Interesse.20
– Das kommt auch darauf hinaus, daß
man den Gedanken mit der Realität muß unmittelbar vergleichen können und es nicht
erst einer Erfahrung bedürfen kann, daß diesem Gedanken diese Realität entspricht. (Darum
unterscheiden sich auch Gedanken nach ihrem Inhalt, aber Magenschmerzen nicht nach dem,
was sie hervorgerufen hat.)
21
Wie kommt es daß ich hier etwas der Erfahrung entgegensetzen will?
Meine Auffassung scheint unsinnig, wenn man sie so ausdrückt: man soll sehen können,
worüber Einer denkt, wenn man ihm den Kopf aufmacht; wie ist denn das möglich? Die
8 (V): Gedanken in demselben
9 (M): 3
10 (V): meinen.
11 (V): verstehen (von innen).
Da ist zuerst zu sagen daß es hier kein außen &
innen gibt.
12 (V): Bedingung hinzufügen.
13 (V): wenn die Meinung
14 (V): um zu
15 (V): Woher der Gedanke
16 (O): betrachtet wäre er
17 (O): eiem // Eiem
18 (V): Die Frage
19 (O): den
20 (V): Interesse, oder vielmehr, dann gibt es
keine Logik.
21 (M): 3 /
380v
381
380v
381
382
381v
382
281 Intention. . . .
WBTC078-85 30/05/2005 16:16 Page 562
thoughts must be in the same space as what admits of doubt, although at a different location.
That is, in the space of language. Everything is carried out in language.
The proposition “I couldn’t think that something was red if red didn’t exist” really does refer to
the mental image of something red, or to the existence of a red pattern, as a part of our language.
But of course on the other hand one can’t say that our language has to contain such a pattern. If it
doesn’t contain it, then it’s just another language. But one can say, and emphasize, that it does
contain this red pattern.
6
In the space of language, and only therein, does the (pre-established) harmony between
world and thought consist.
Intention, however, is exactly the same kind of thing as – for example – annoyance. And
here it seems somehow that one can never recognize an intention, when viewed from the
outside, as an intention; as if one has to intend7
it oneself in order to understand it as an
intention.8
But that would mean not to view it as a phenomenon, as a fact! That is to say,
that would mean adding a further (unclearly indicated) condition of experience to everything.9
And of course, if meaning something is an experience, then one must have precisely this
experience in order to really mean something, and not a different experience, which might be
called “seeing the meaning from the outside”.
And here intention reminds us of the will (in Schopenhauer’s sense as well).
Where does the idea10
come from that you can view something “not as a phenomenon”? For then
how does the subject enter into consideration?
On the one hand it’s as if you wanted to say that you could recognize a toothache as a toothache
only from the inside. Viewed from the outside, for instance, it wouldn’t be at all unpleasant. But a
toothache doesn’t present us with any such problem.
The problem,11
though, is: How can one translate intention, once one has it, into the words that
it is the intention to do this or that? For I admit that intention can only be known by someone who
has experienced it (cf.: toothache); but why do you call it the intention to do this? – Obviously that
has nothing to do with its being indescribable.
Of course this is the previous problem again, because the funny thing about it is that you
have to be able to tell by looking at a thought (viewed as an independent fact) that it is the
thought that such and such is the case. If you can’t tell this by looking at it (just as you can’t
tell from a stomach-ache what caused it) then it’s of no logical interest.12
– This also amounts
to saying that you have to be able to compare a thought to reality directly, and that there
can’t first be need of an experience for this reality to correspond to this thought. (That’s
also why thoughts differ according to their content, but stomach-aches don’t vary according
to what caused them.)
13
Why is it that I want to oppose something to experience here?
My conception seems nonsensical if one expresses it this way: One is supposed to be
able to see what someone is thinking about by opening up his head; how is that possible?
6 (M): 3
7 (V): mean
8 (V): intention (from the inside).
At this point the first thing to be said is that there is
no outside and inside here.
9 (V): condition to everything.
10 (V): the thought
11 (V): The question,
12 (V): interest, or rather, then there is no logic.
13 (M): 3 /
Intention. . . . 281e
WBTC078-85 30/05/2005 16:16 Page 563
Gegenstände, über die er denkt, sind ja garnicht in seinem Kopf (ebensowenig wie in seinen
Gedanken)!
Man muß nämlich die Gedanken, Intentionen (etc.) von außen betrachtet als solche
verstehen, ohne über die Bedeutung von etwas unterrichtet zu werden. Denn auch die
Relation des Bedeutens wird ja dann als ein Phänomen gesehen (und ich darf22
dann nicht
wieder auf eine Bedeutung des Phänomens hinweisen müssen, da ja dieses Bedeuten wieder
in dem Phänomen mit inbegriffen23
ist).
24
Wenn man den Gedanken betrachtet, so kann also von einem Verstehen keine Rede mehr
sein, denn, sieht man ihn, so muß man ihn als den Gedanken dieses Inhalts erkennen, es ist
nichts zu deuten. – Aber so ist es ja wirklich, wenn wir denken, da wird nicht gedeutet. –
25
Kann man Magenschmerzen von außen betrachtet als solche verstehen? Und was sind26
&Magenschmerzen von außen betrachtet“. Es sind Magenschmerzen gemeint die man hat nicht die
des Andern deren Wirkungen man sieht.
27
Freilich, sofern das Meinen eine spezifische28
Erfahrung ist kann man keine Andere das Meinen
nennen. Nur erklärt keine besondere Erfahrung die Richtung der Meinung. Und wenn wir sagten
&von außen betrachtet . . .“ so wollten wir auch gar nicht29
sagen die Meinung sei eine besondere
Erfahrung sondern sie sei nicht etwas was geschähe oder uns geschähe,30
(&denn das wäre ja tot“)
sondern etwas, was wir tun.
31
Das Subject falle hier nicht aus der Erfahrung heraus sondern sei so in ihr involviert daß sich die
Erfahrung nicht beschreiben ließe.
32
Es ist beinahe als sagte man wir können uns nicht dorthin gehen sehen da wir33
selbst gehen.
(Und also nicht stehen & zuschauen können.)34
Aber hier laborieren wir eben wie so sehr oft35
an
einer Ausdrucksweise die inadäquat ist die wir abschütteln wollen aber zugleich doch gebrauchen36
& kleiden den Protest gegen unsere eigene Ausdrucksweise in einen verneinenden Satz in dieser
Ausdrucksweise. Denn wenn man sagt &wir sehen uns dorthin gehen“ so meint man eben daß wir
sehen was man sieht wenn man selbst geht & nicht was man sieht wenn ein Andrer geht. Und man37
hat ja auch eine bestimmte Seherfahrung wenn man selbst geht.
38
Die kausale39
Erklärung des Bedeutens und Verstehens lautet im Wesentlichen so: einen
Befehl verstehen heißt, man würde ihn ausführen, wenn ein gewisser Riegel zurückgezogen
würde. – Es würde jemandem befohlen, einen Arm zu heben, und man sagt: den Befehl
verstehen heißt, den Arm zu heben. Das ist klar, wenn auch gegen unseren Sprachgebrauch
(wir nennen das ”den Befehl befolgen“). Nun sagt man [Frege] aber:40
den Befehl verstehen
heißt, entweder den Arm heben, oder, wenn das nicht, etwas bestimmtes Anderes tun – etwa
das Bein heben. Nun heißt das aber nicht ”verstehen“ im ersten Sinn, denn der Befehl war
nicht ”den Arm oder das Bein zu heben“. Der Befehl bezieht sich also (nach wie vor) auf
22 (V): kann
23 (V): in den Phänomenen inbegriffen
24 (M): 3
25 (M): 3
26 (V): was heißt es
27 (M): 3
28 (O): speziffische
29 (V): auch nicht
30 (V): geschähe, sondern
31 (M): 3
32 (M): 3
33 (V): wir nicht
34 (V): gehen (also nicht stehen & zuschauen können).
35 (V): wie so oft
36 (V): an einer Ausdrucksweise die wir abschütteln
wollen & doch gebrauchen
37 (V): man sieht ja
38 (M): ///
39 (V): kausale Bedeutung d
40 (V): man // Frege // aber:
381v
382
383
282 Intention. . . .
WBTC078-85 30/05/2005 16:16 Page 564
The things he’s thinking about aren’t in his head at all (any more than they’re in his
thoughts)!
For looking at them from the outside, one has to understand thoughts, intentions (etc.),
as such without being informed of the meaning of anything. For then the relation of
meaning is also seen as a phenomenon (and then I mustn’t be forced to14
point to a meaning
of the phenomenon in turn, because once again, this meaning is included in the
phenomenon15
).
16
So when one looks at a thought, there can no longer be any talk about understanding,
for if one sees the thought then one has to recognize it as the thought with this content;
there is nothing to interpret. – But that’s really the way it is when we think; there’s no interpreting
going on there. –
17
Can one understand a stomach-ache, viewed from outside, as a stomach-ache? And what is “a
stomach-ache viewed from outside”?18
What is meant is one’s own stomach-ache, not someone else’s,
the effects of which one sees.
19
To be sure, in so far as meaning something is a specific experience one can’t call any other experience
meaning. Except that no specific experience explains the direction of meaning. And when we
said “seen from outside . . .”, we didn’t in any way want to say that meaning something is a specific
experience; rather, we wanted to say that it isn’t something that happens, or happens to us (“for
that would be dead”), but something that we do.
20
We wanted to say that here the subject doesn’t drop out of experience; but is involved in it in
such a way that the experience can’t be described.
21
It’s almost as if we said that we can’t see ourselves walking towards that point over there, since
it is we who are doing the walking. (And so we can’t stand and watch.)22
But here, as so very often23
,
we labour under a form of expression that is inadequate. We want to shake it off, yet at the same
time we use it;24
and we clothe our protest against our own form of expression in a denial that
is couched in this very form of expression. For when we say “we see ourselves walking over in
that direction” we mean precisely that we see what we see when we ourselves are walking over in
that direction, and not what we see when someone else is walking. And we do have a particular visual
experience when we ourselves are walking.
25
The causal explanation of meaning and understanding goes essentially like this: Understanding
a command means that one would carry it out if a certain bolt were pulled back.
Let’s say someone were ordered to raise his arm and it is said: Understanding the command
means raising his arm. That is clear, although contrary to our use of language (we call that
“obeying the command”). However, one (Frege) does26
say: Understanding the command
means either to raise an arm, or if not that, to carry out some other specific action – for
instance to lift a leg. But this isn’t what “understanding” means in the first sense, for
the command was not “Lift an arm or a leg”. So the command refers to an action that has
14 (V): then I can’t
15 (V): phenomena
16 (M): 3
17 (M): 3
18 (V): what does “a stomach-ache viewed from outside”
mean?
19 (M): 3
20 (M): 3
21 (M): 3
22 (V): walking (and so we can’t stand and watch)^
23 (V): as so often
24 (V): labour under a way of expressing ourselves that
we want to shake off but still use;
25 (M): ///
26 (V): one // Frege // does
Intention. . . . 282e
WBTC078-85 30/05/2005 16:16 Page 565
eine Handlung, die nicht geschehen ist. Mit andern Worten, es bleibt der Unterschied
bestehen zwischen dem Verstehen und dem Befolgen des Befehls. Und weiter [Frege]:41
ein
unverstandener Befehl ist gar kein Befehl. – Dieses Verstehen des Befehls kann nicht irgend
eine Handlung sein, (etwa den Fuß heben), sondern42
sie muß das Wesen des Befehls selbst
enthalten.
43
”In dem Faktum des Verstehens muß das Verstehen (was immer es ist) seinen Ausdruck
finden.
In dem Vorgang des Verstehens (welcher immer der sei) muß das Verstehen ausgedrückt
sein.“
(Wenn ich Einem in die Seele sähe,44
müßte ich sehen, woran er denkt. Siehe Vorgang
des Denkens.)
45
In der Sprache wird alles ausgetragen.
Wenn ich in der Sprache denke so schweben mir nicht neben dem sprachlichen Ausdruck noch
Bedeutungen vor sondern die Sprache selbst ist das Vehikel der Gedanken.
46
Warum scheint mir mein Gedanke ein so exceptionelles Stück Wirklichkeit zu sein?
Doch nicht, weil ich ihn ”von innen“ kenne, das heißt nichts; sondern offenbar, weil ich
alles in Gedanken ausmache, und über das Denken auch nur wieder denke.
Alles47
wird auf den gemeinsamen Nenner der Sprache gebracht & dort verglichen.
41 (V): weiter // Frege // :
42 (O): heben) sondern
43 (M): ////
44 (V): sehe,
45 (M): 3
46 (M): ///
47 (V): Die Spra Alles
382v
383
382v
283 Intention. . . .
WBTC078-85 30/05/2005 16:16 Page 566
not taken place. In other words, the difference between understanding and carrying out the
command remains. And furthermore (Frege):27
A command that isn’t understood is no
command at all. – The understanding of a command can’t be some random action (say,
lifting one’s foot); rather, the action must contain the essence of the command itself.
28
“Understanding (whatever it is) must find its expression in the fact of understanding.
Understanding must be expressed in the process of understanding (whatever that might
be).”
(If I were to look29
into someone’s soul I’d have to see what he is thinking about. Cf. the
process of thinking.)
30
Everything is carried out in language.
When I’m thinking in a language I don’t have additional meanings in mind running alongside the
linguistic expression; rather, language itself is the vehicle of thought.
31
Why does my thought strike me as such an exceptional piece of reality? Certainly not
because I know it “from within”; that means nothing. Rather, obviously because I use thought
to find out everything; even concerning thinking, all I can do is to think.
Everything is brought to the common denominator of language and compared there.
27 (V): furthermore // Frege // :
28 (M): ////
29 (V): If I look
30 (M): 3
31 (M): ///
Intention. . . . 283e
WBTC078-85 30/05/2005 16:16 Page 567
82
Kein Gefühl der Befriedigung (kein
Drittes) kann das Kriterium dafür
sein, daß die Erwartung erfüllt ist.
1
Man könnte nämlich denken, wie ist es; der Gedanke und die Tatsache sind verschieden;
aber wir nennen den Gedanken: den, daß die Tatsache der Fall ist; oder die Tatsache: die,
welche den Gedanken wahr macht. Ist da das Eine eine Beschreibung mit Hilfe des
Anderen? Wird der Gedanke mittels der Tatsache, die ihn wahr macht, beschrieben, also einer
äußeren Eigenschaft nach beschrieben, wie wenn ich von jemandem sage, er sei mein Onkel?
2
Wenn man den Ausdruck ”der Gedanke, daß . . . der Fall ist“ als Beschreibung erklärt,
so ist damit wieder nichts erklärt, weil es sich fragt: wie ist eine solche Beschreibung möglich,
sie setzt3
selber wieder das Wesen des Gedankens voraus, denn sie enthält den Hinweis auf
eine Tatsache, die nicht geschehen ist, also gerade das, was problematisch war.
4
Die Erfüllung der Erwartung besteht nicht darin, daß ein Drittes geschieht, das man außer
eben als ”die Erfüllung der Erwartung“ auch noch anders beschreiben könnte, also z.B. als
ein Gefühl der Befriedigung, oder der Freude, oder wie immer.
Denn die Erwartung, daß p der Fall sein wird, muß das Gleiche sein, wie5
die Erwartung
der Erfüllung dieser Erwartung, dagegen wäre, wenn ich unrecht habe, die Erwartung,
daß p eintreffen wird, verschieden von der Erwartung, daß die Erfüllung dieser Erwartung
eintreffen wird.
6
Könnte denn die Rechtfertigung lauten: ”Du hast gesagt ,bring‘ etwas Rotes’ und dieses
hier hat mir daraufhin ein Gefühl der Befriedigung gegeben,7
darum habe ich es gebracht“?
8
Müßte man da nicht antworten: Ich habe Dir doch nicht geschafft, mir das zu bringen,
was Dir auf meine9
Worte hin ein solches Gefühl geben wird!
10
Ich gehe die gelbe Blume suchen. Auch wenn mir während des Gehens ein Bild
vorschwebt, brauche ich es denn, wenn ich die gelbe Blume – oder eine andere – sehe? –
Und wenn ich sage ”sobald ich eine gelbe Blume sehe, schnappt, gleichsam, etwas in dem
Gedächtnis11
ein“: kann ich denn dieses Einschnappen eher voraussehen, erwarten, als die
gelbe Blume? Ich wüßte nicht, warum. D.h., wenn es in einem bestimmten Fall wirklich so
1 (R): [Zu § 80 S. 375]
2 (M): ///
3 (O): selbst
4 (M): 3
5 (V): wie der
6 (M): 3
7 (V): erzeugt,
8 (M): 3
9 (V): Deine
10 (M): 3
11 (V): in der Erinnerung
384
385
284
WBTC078-85 30/05/2005 16:16 Page 568
82
No Feeling of Satisfaction (no
Third Thing) Can Be the Criterion
that Expectation has been Fulfilled.
1
For one might think, how can this be: Thoughts and facts are different; but we call the
thought that a fact is the case a thought; or we call the fact that verifies a thought a fact. Is
each of these a description in terms of the other? Is a thought described by way of the fact
that verifies it, i.e. is it described according to an external property, as if I say of someone
that he is my uncle?
2
If one declares that the expression “the thought that . . . is the case” is a description, then
once again nothing has been explained, for the question arises: How is such a description
possible? It itself presupposes the nature of thought, for it contains a reference to a fact that
hasn’t happened, which is just what was problematical.
3
The fulfilment of expectation doesn’t consist in the occurrence of some third thing that,
in addition to being described as “the fulfilment of the expectation”, could also be described
as something else, i.e. as a feeling of satisfaction, for instance, or of joy or whatever.
For the expectation that p will be the case has to be the same thing as the expectation of
the fulfilment of this expectation. On the other hand, if I’m wrong, the expectation that p
will happen would be different from the expectation that the fulfilment of this expectation
will happen.
4
Could my justification perhaps run like this: “You said ‘Bring something red’ and as a
result this thing gave me a feeling of satisfaction5
, and that’s why I brought it”?
6
Wouldn’t one have to answer: But I didn’t tell you to bring me whatever produces such
a feeling as a result of my7
words!
8
I go looking for a yellow flower. Even if I have a mental image while I’m walking, do
I really need it when I see a yellow flower – or some other one? – And if I say “As soon
as I see a yellow flower, something as it were clicks in my memory”, can I really foresee,
expect, this clicking more easily than the yellow flower? I don’t know why. That is, if in a
1 (R): [To § 80 p. 375]
2 (M): ///
3 (M): 3
4 (M): 3
5 (V): thing produced a feeling of satisfaction in me
6 (M): 3
7 (V): your
8 (M): 3
284e
WBTC078-85 30/05/2005 16:16 Page 569
ist, daß ich nicht die gelbe Blume, sondern ein anderes (indirektes) Kriterium erwarte, so
ist dies12
jedenfalls keine Erklärung des Erwartens.
13
Aber geht nicht mit dem Eintreffen des Erwarteten immer ein Phänomen der
Bejahung14
(oder Befriedigung)15
Hand in Hand? Dann frage ich: Ist dieses Phänomen ein
anderes, als das Eintreten des Erwarteten? Wenn ja, dann weiß ich nicht, ob so ein anderes
Phänomen die Erfüllung immer begleitet. – Oder ist es dasselbe, wie die Erfüllung?
Wenn ich sage: Der, dem die Erwartung erfüllt wird, muß doch nicht sagen ”ja, das ist es“
(oder dergleichen), so kann man mir antworten: ”gewiß, aber er muß doch wissen, daß
die Erwartung erfüllt ist“. – Ja, soweit das Wissen dazu gehört,16
daß sie17
erfüllt ist. In
diesem Sinne: wüßte er’s nicht, so wäre sie nicht erfüllt. – ”Wohl, aber, wenn einem
eine Erwartung erfüllt wird, so tritt doch immer eine Entspannung auf!“ – Woher weißt
Du das? –
18
Beim Versteckenspiel erwarte ich, den Fingerhut zu finden. Wenn ich ihn finde, gebe
ich ein Zeichen der Befriedigung von mir, oder ich fühle doch (eine) Befriedigung. Dieses
Phänomen mag ich auch erwartet haben (oder auch nicht), aber diese Erwartung ist nicht
die, den Fingerhut zu finden. Ich kann beide Erwartungen haben und die sind offenbar
ganz getrennt.
19
Es ist nicht so, daß wir das Phänomen einer Unbefriedigung20
bemerken,21
die dann
durch Finden22
des Fingerhutes vergeht,23
und nun sagen: ”also war jenes Phänomen die
Erwartung den Fingerhut zu finden“.24
Nein, das erste Phänomen ist die Erwartung den Fingerhut zu finden25
so sicher, wie26
das zweite das Finden des Fingerhutes ist. Der Ausdruck ”Finden27
des Fingerhuts“28
gehört
zu der Beschreibung des ersten so notwendig, wie zur Beschreibung des zweiten. Nur
verwechseln wir nicht ”die Bedeutung des Wortes ,Fingerhut‘ “ (den Ort dieses Worts im
grammatischen Raume) mit der Tatsache, daß ein Fingerhut hier ist.
12 (V): das
13 (M): 3
14 (V): Zustimmung
15 (V): Befriedigung)
16 (V): gehört, da
17 (V): die
18 (M): ///
19 (M): ////
20 (V): wir eine Unbefriedigung
21 (V): spüren // merken //,
22 (O): finden
23 (V): Fingerhutes aufgehoben wird
24 (V): die Erwartung des Fingerhutes“.
25 (V): die Erwartung des Fingerhutes
26 (V): als
27 (O): ”finden
28 (V): Fingerhutes ist. Das Wort ”Fingerhut“
386
285 Kein Gefühl der Befriedigung (kein Drittes) kann das Kriterium dafür sein . . .
WBTC078-85 30/05/2005 16:16 Page 570
particular instance it really is the case that I’m not expecting the yellow flower but another
(indirect) criterion, then in any case this9
isn’t an explanation of expecting.
10
But doesn’t a phenomenon of affirmation11
(or satisfaction) always go hand in hand with
the occurrence of what was expected? Then I ask: Is this phenomenon different from the
occurrence of what was expected? If it is then I don’t know whether such a different phenomenon
always accompanies fulfilment. – Or is it the same thing as fulfilment? If I say:
“After all, the person whose expectation is fulfilled doesn’t have to say ‘Yes, that’s it’ (or
some such thing)”, then someone can answer: “Certainly, but he must surely know that his
expectation is fulfilled”. – True, in so far as knowing is part of its being fulfilled. In this
sense: If he didn’t know that the expectation had been fulfilled then it wouldn’t have been
fulfilled. – “All right, but when someone’s expectation is fulfilled, it’s always accompanied
by a release of tension!” – How do you know that? –
12
When playing hide and seek, I expect to find the thimble. When I find it I signal satisfaction
or at least I feel (a) satisfaction. I may very well have expected this phenomenon (or
not), but this expectation isn’t that of finding the thimble. I can have both expectations, but
they are obviously quite separate.
13
It isn’t the case that we notice14
the phenomenon of dissatisfaction15
, that it disappears16
when we find the thimble, and then we say: “So that phenomenon was the expectation of
finding the thimble”.17
No, the first phenomenon is as surely the expectation of finding the thimble18
as the
second one is the finding of the thimble. The expression “finding the thimble”19
is just as
necessarily a part of the description of the first phenomenon as it is of the description of
the second phenomenon. Let’s just not confuse “the meaning of the word ‘thimble’” (the
location of this word in grammatical space) with the fact that there is a thimble here.
9 (V): that
10 (M): 3
11 (V): approval
12 (M): ///
13 (M): ////
14 (V): sense // feel //
15 (V): notice a dissatisfaction
16 (V): it is cancelled
17 (V): of the thimble”.
18 (V): of the thimble
19 (V): The word “thimble”
No Feeling of Satisfaction . . . 285e
WBTC078-85 30/05/2005 16:16 Page 571
83
Der Gedanke – Erwartung, Wunsch,
etc. – und die gegenwärtige Situation.
1
”Wenn der Gedanke ein Bild ist, so erscheint die Beschäftigung mit diesem Bild als Spielerei,2
wenn
sie sich nicht mit der uns interessierenden Wirklichkeit befaßt. Wenn ich hoffe, daß er zur
Tür hereinkommen wird, so beschäftige ich mich mit dieser Tür, etwa mit dem Boden, auf
den er treten wird. Und das Übrige, was die Phantasie tut, ist nicht Spiel, sondern eine Art
Vorbereitung, eine Tätigkeit (sozusagen eine Arbeit), die die Form des Bildes in sich trägt.
Etwa so (nur nicht unbedingt so explicit), wie wenn ich seinen Weg mit einem Teppich
belegen und an einer bestimmten Stelle einen Stuhl herrichten wollte.“
Denn warum sollen wir uns gerade für dieses Bild interessieren, wo wir uns doch sonst
mit Seelenzuständen, Magenschmerzen, etc. nicht befassen.
Der Kalkül des Denkens knüpft mit der Wirklichkeit an.
Die Erwartung ist eine vorbereitende Handlung. Eine vorbereitende Handlung innerhalb der
Sprache (Berechnung des Dampfkessels.)
3
Die Erwartg. ist eine Vorbereitung auf etwas eine Vorbereitung innerhalb d. Sprache.4
5
(Der Plan kann mich nur leiten, wenn ich auch auf dem Plan bin.)
6
Wenn ich mit verbundenen Augen die Richtung verloren habe und man mir nun sagt:
geh’ dort und dort hin, so hat dieser Befehl keinen Sinn für mich.
7
Ich erwarte mir, daß der Stab im selben Sinne 2 m hoch sein wird, in dem er jetzt 1 m
99 cm hoch ist.
In der Spr. wird alles ausgetr.
In demselben Sinne, in dem er jetzt 1m hoch ist, wird er später 1,5 m hoch sein.
8
Wäre der Gedanke sozusagen eine Privatbelustigung und hätte nichts mit der Außenwelt
zu tun, so wäre er für uns ohne jedes Interesse (wie etwa die Gefühle bei einer
Magenverstimmung). Was wir wissen wollen ist: Was hat der Gedanke mit dem zu tun,
was außer dem Gedanken vorfällt. Denn seine Bedeutung, ich meine seine Wichtigkeit,
bezieht er ja nur daher.
1 (M): 3
2 (V): “Die Beschäftigung mit dem Bild
erscheint als Spielerei,
3 (R): [gehört zur Erklärung des Wesens der Erwartg.
Erwartg als Hohlform eine Vollform fordernd.]
4 (V): Die Erwartung ist eine vorbereitende,
erwartende, Handlung // eine vorbereitende
Handlung. // Eine vorb. H. innerhalb der Sprache. In
der Sprache wird alles ausgetragen. (R): Siehe
S. 354
5 (M): ///
6 (M): ///
7 (R): [Zu S. 102]
8 (M): ///
387
386v
387
388
286
WBTC078-85 30/05/2005 16:16 Page 572
83
Thought – Expectation, Wish, etc. –
and the Present Situation.
1
“If a thought is a picture, then occupying oneself with this picture seems like idle play2
if it
doesn’t concern itself with the reality that we’re interested in. But when I’m hoping that
he’ll come in the door, I’m occupying myself with that door, perhaps with the floor that he’ll
step on. And everything else that imagination does is not a game, but a kind of preparation,
an activity (work, so to speak) that contains in it the form of a picture. More or less (just
not necessarily as explicitly) as if I wanted to lay a carpet in his path and set up a chair at a
certain spot.”
For why should it be precisely this picture that we’re interested in, when otherwise we’re
not concerned with psychological states, stomach-aches, etc.?
The calculus of thinking ties up with reality.
Expectation is a preparatory action. A preparatory action within language (calculation of the boiler).
3
Expectation is a preparation for something, a preparation within language.4
5
(The map can only guide me if I too am on it.)
6
If I’m blindfolded and have lost my orientation and am now told “Go here or there”,
that command makes no sense to me.
7
I expect that the bar will be 2 m high in the same sense in which it’s now 1 m 99 cm high.
Everything is carried out in language.
In the same sense in which it is now 1 m high, it will be 1.5 m high later.
8
If a thought were a private amusement, so to speak, and didn’t have anything to do with
the outside world, it would lack all interest for us (like, say, the feelings accompanying an
upset stomach). What we want to know is: What does a thought have to do with what
happens outside it? Because after all it is only from there that it derives its meaning, I
mean – its importance.
1 (M): 3
2 (V): “Occupying oneself with the picture seems
like idle play
3 (R): [belongs to the explanation of the nature of
expectation. Expectation as a concave shape requiring
one that is convex.]
4 (V): Expectation is a preparatory, expecting, action //
a preparatory action. // A preparatory action within
language. Everything is carried out in language.
(R): See p. 354
5 (M): ///
6 (M): ///
7 (R): [To p. 102]
8 (M): ///
286e
WBTC078-85 30/05/2005 16:16 Page 573
9
Was hat das, was ich denke, mit dem zu tun, was der Fall ist.
10
Das Denken als Ganzes mit seiner Anwendung11
geht sozusagen automatisch d.h. als Kalkül
vor sich. – Wieviele Zwischenstufen ich auch zwischen den Gedanken und die Anwendung
setze, immer folgt eine Zwischenstufe der nächsten – und die Anwendung der letzten – ohne
Zwischenglied. Und hier haben wir den gleichen Fall, wie wenn wir zwischen Entschluß
und Tat durch Zwischenglieder vermitteln wollen.
12
Wenn ich gehe, so enthält der einzelne Schritt nicht das Ziel, wohin mich das Gehen
bringen wird. Komme ich ans Ziel, so war jeder Schritt ein Schritt zu diesem Ziel.
13
”Worin besteht es, sich eine gelbe Blume zu wünschen? Wesentlich darin, daß man in
dem, was man sieht, eine gelbe Blume vermißt? Also auch darin, daß man erkennt, was in
dem Satz ausgedrückt ist ,ich sehe jetzt keine gelbe Blume‘.“
14
Könnte man auch sagen: Man kann die Erwartung nicht beschreiben, wenn man die gegenwärtige
Realität nicht beschreiben kann oder, man kann die Erwartung nicht beschreiben,
wenn man nicht eine vergleichende Beschreibung von Erwartung und Gegenwart geben kann
in der Form: Jetzt sehe ich hier einen roten Kreis und erwarte mir später dort ein blaues
Viereck.
D.h., der Sprachmaßstab muß an dem Punkt der Gegenwart angelegt werden und deutet
dann über ihn hinaus – etwa in der Richtung der Erwartung.
Ich will sagen um den Ort des Gewünschten zu bestimmen muß mein Satz wie ein Maßstab auf
die gegenwärtige Situation in gewisser Richtung aufgesetzt15
werden, denn wie sollte er sonst den
Punkt im Raum zeigen wo das Gewünschte sein soll?
Aber auch wenn so der Maßstab an der Wirklichkeit aufsitzt16
warum muß ich ihn dann als
gerade diesen Wunsch interpretieren? Die Schwierigkeit die man hier lösen will ist wieder: &wie
bestimmt der Wunsch das Gewünschte“. Und man trachtet wieder vergebens die Erfüllung des
Wunsches im Wunsche schon vorwegzunehmen.
17
Ich will sagen: wenn ich über eine gelbe Blume rede, muß ich zwar keine sehen, aber
ich muß etwas sehen und das Wort ”gelbe Blume“ hat quasi nur in Übereinstimmung mit
oder im Gegensatz zu dem Bedeutung, was ich sehe. Seine Bedeutung würde quasi nur von
dem aus bestimmt, was ich sehe, entweder als das, was ich sehe, oder als das, was davon in
der und der Richtung so und so weit liegt. Hier meine ich aber weder Richtung noch Distanz
räumlich im gewöhnlichen Sinn, sondern es kann die Richtung von Rot nach Blau und die
Farbendistanz von Rot auf ein bestimmtes Blaurot gemeint sein. – Aber auch so stimmt meine
Auffassung nicht. Es ist schon richtig, daß der Satz ”ich wünsche eine gelbe Blume“ den
Gesichtsraum voraussetzt, nämlich nur insofern, als er in unserer Sprache voraussetzt, daß
der Satz ”ich sehe jetzt eine gelbe Blume“ und sein Gegenteil Sinn hat.18
Ja, es muß auch
Sinn haben, oder vielmehr, es hat auch Sinn, zu sagen ”das Gelb, was ich mir wünsche, ist
grünlicher als das, welches ich sehe“. Aber anderseits wird der grammatische Ort des Wortes
”gelbe Blume“ nicht durch eine Maßangabe, bezogen auf das, was ich jetzt sehe, bestimmt.
Obwohl, soweit von einer solchen Entfernung und Richtung die Rede überhaupt sein kann,
9 (M): 3
10 (M): 3
11 (V): als Ganzes und seine Anwendung
12 (M): ///
13 (R): [Zu S. 102 § 29]
14 (R): [Zu S. 102]
15 (V): auf die gegenwärtige Situation aufgesetzt
16 (V): aufsteht
17 (R): [Zu S. 102]
18 (V): Sinn haben muß.
389
388v
389
287 Der Gedanke . . .
WBTC078-85 30/05/2005 16:16 Page 574
9
What does what I think have to do with what is the case?
10
Thinking as a whole, together with its11
application, proceeds automatically, so to speak,
i.e. as a calculus. – No matter how many intermediate steps I place between a thought and its
application, one intermediate step always follows the next – and the application follows the
last one – without an intermediate link. And it’s the same here as when we want to mediate
between decision and action with intermediate links.
12
When I walk, an individual step does not contain the goal that walking will get me to.
When I get to the goal, then each step was a step towards that goal.
13
“What does wishing for a yellow flower consist in? In essence, missing a yellow flower
in what one sees? And therefore also in realizing what is expressed in the sentence ‘Now I
don’t see a yellow flower’?”
14
Could one say as well: One can not describe expectation if one can’t describe present
reality, or one can’t describe expectation if one can’t give a comparative description of
expectation and the present in the form: I see a red circle here now, and I’m expecting
a blue quadrangle over there later.
That is, the linguistic measuring stick has to be applied at the point of the present, and
then it points beyond it – for instance in the direction of the expectation.
I want to say: To determine the location of what I’m wishing for, my proposition has to be superimposed
on the present situation like a measuring stick, but pointing in a certain direction.15
For how
else could it be expected to show that point in space where what I’m wishing for is supposed to be?
But even if the measuring stick sits16
alongside reality in that way, why do I then have to interpret
it as just that wish? Once again, the difficulty that one wants to solve here is: “How does the wish
specify what I’m wishing for?”. And once again one tries in vain to anticipate the fulfilment of the
wish in the wish.
17
I want to say: When I talk about a yellow flower I don’t have to see one, to be sure,
but I do have to see something, and the word “yellow flower” has meaning as it were only
in agreement with or in opposition to what I see. Its meaning can only be specified, so to
speak, in reference to what I see, either as what I see or as what is situated in such and such
a direction and at such and such a distance from it. But here I don’t mean either direction
or distance in the usual spatial sense; rather, for example, the direction from red to blue,
or the colour distance from red to a particular bluish red. – But even put this way, my understanding
isn’t right. It is true, to be sure, that the proposition “I want a yellow flower”
presupposes visual space, i.e. it does so only in so far as it presupposes that in our language
the proposition “Now I see a yellow flower” and its opposite make18
sense. Indeed to say
“The yellow that I’m wishing for has more green in it than the yellow I’m seeing” must also
make sense, or rather – it does make sense. But on the other hand the grammatical location
of the words “yellow flower” isn’t specified by a measurement that refers to what I am now
seeing. In so far as such distance and direction can be considered at all, they must already
9 (M): 3
10 (M): 3
11 (V): whole, and its
12 (M): ///
13 (R): [To p. 102 § 29]
14 (R): [To p. 102]
15 (V): like a measuring stick.
16 (V): stick has been placed
17 (R): [To p. 102]
18 (V): opposite must make
Thought – Expectation, Wish, etc. . . . 287e
WBTC078-85 30/05/2005 16:16 Page 575
durch die Beschreibung des gegenwärtigen Gesichtsbildes und des Gewünschten diese
Entfernung und Richtung im grammatischen Raum gegeben sein muß.
19
Ich habe das Gefühl, nur die Stellungnahme zu dem Bild kann es uns zur Wirklichkeit
machen, d.h., kann es mit der Wirklichkeit so verbinden, gleichsam wie eine Lasche,
20
die die Überleitung von dem Bild zur Wirklichkeit
herstellt, die beiden in der rechten Lage zueinander haltend, dadurch, daß beide für sie
dasselbe bedeuten.
Die Furcht verbindet das Bild mit dem21
Schrecken der Wirklichkeit.22
23
Ich könnte vielleicht auch fragen: Was ist es, was dem Bild seine Bedeutung gibt?
Die Kontinuität des Kalküls in mir. Ich benehme mich dem Bild gegenüber ähnlich wie der
Wirklichkeit gegenüber & der Kalkül das Nachdenken in mir vollzieht sich in einer Einstellung
oder einer kontinuierlichen Reihe von Einstellungen. D.h. ich erlebe das Bild in seiner Art, wie
die Wirklichkeit in ihrer Art.
24
Unsere Stellungnahme zu dem Bild, daß wir das Bild erleben25
, macht es uns26
zur Wirklichkeit.
27
D.h. verbindet es mit der Wirklichkeit, indem es eine Kontinuität herstellt.
28
Mit dem Bild ist der Satz gemeint. Und das Problem war: &Was hat mein Gedanke mit dem zu
tun, was der Fall ist?“ Das Problem der Abbildung, des Portraits29
etc. Oder (besser): &Dieses Kreuz
auf dem Plan bin ich“.
Ich bin empört, wenn ich von einem Mord lese,30
wie wenn ich einen Mord sehe.31
32
Das Problem läßt scheinbar zwei Lösungen zu: man kann sagen, daß der Kalkül den Gedanken
mit der Wirklichkeit verbindet,
33
Früher sagte ich, daß der Satz seine Bedeutung hat, indem er quasi in uns eingreift.
34
Dazu auch: Wir denken, sehen voraus, überlegen, weil wir nicht anders können.
35
Was macht uns die Erwartung zur Erwartung?
36
Man könnte fragen: Was macht uns das Bild, den Gedanken, zur Wirklichkeit?
37
Oder: Was macht uns den Glauben zur Wirklichkeit?
38
Nun das Glauben ist ein natürlicher Akt der Menschen.
Uns Bilder herzustellen ist Teil unseres Lebens.
19 (M): 3
20 (F): MS 108, S. 274.
21 (V): den
22 (V): mit der Wirklichkeit.
23 (M): 5555
24 (M): 555
25 (V): Daß wir das Bild erleben
26 (V): macht den Gedanken
27 (M): 555
28 (M): 3
29 (O): der Abbildung des Portraits
30 (V): wenn ich die Beschreibung eines Mordes lese,
31 (V): ich Zeuge des Mordes bin.
32 (M): ///
33 (M): ///
34 (M): ///
35 (M): ///
36 (M): ///
37 (M): 3 ||
38 (M): 3
390
389v
288 Der Gedanke . . .
WBTC078-85 30/05/2005 16:16 Page 576
have been given within grammatical space by a description of the present visual image and
of what I’m wishing for.
19
I have the feeling that only our attitude towards the picture can turn it into a reality for
us. That is, because picture and reality mean the same thing to it, it can link the former with
the latter – like a fishplate, as it were – 20
establishing the
transition from picture to reality, holding them in the right position to each other.
Fear connects the picture to the terror of reality.21
22
I might also ask: What is it that gives the picture its meaning?
The continuity of the calculus within me. I behave towards the picture in a similar way as towards
reality, and the calculus within me – the thought process – occurs as one take on it, or as a continuous
series of takes. That is to say, I experience the picture in its way as I do reality in its way.
23
Our attitude towards the picture, that we have an experience of the picture, turns it24
into
reality for us.
25
That is, it connects it to reality by establishing a continuity.
26
What is meant by the picture is a proposition. And the problem was: What does my thought
have to do with what is the case? The problem of depiction, of a portrait, etc. Or (better): “I am this
cross on the map”.
I am outraged when I read about a murder27
, just as I am when I see a murder.28
29
The problem seemingly admits of two solutions: one can say that the calculus connects thought
to reality.
30
I used to say that a proposition gets its meaning by engaging with us, as it were.
31
In this context as well: We think, foresee, consider, because we can’t do otherwise.
32
What turns expectation into expectation for us?
33
One could ask: What turns a picture, a thought, into reality for us?
34
Or: What turns belief into reality for us?
35
Well, believing is a natural act for humans.
Making pictures for ourselves is part of our lives.
19 (M): 3
20 (F): MS 108, p. 274.
21 (V): to reality.
22 (M): 5555
23 (M): 555
24 (M): That we experience the picture turns the
thought
25 (M): 555
26 (M): 3
27 (V): read the description of a murder
28 (V): when I am a witness to a murder.
29 (M): ///
30 (M): ///
31 (M): ///
32 (M): ///
33 (M): ///
34 (M): 3 ||
35 (M): 3
Thought – Expectation, Wish, etc. . . . 288e
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84
Glauben. Gründe des Glaubens.
1
Glauben. Hiermit verwandt ist: erwarten, hoffen, fürchten, wünschen. Aber auch:
zweifeln, suchen, etc.
Man sagt: ”Ich habe ihn von 5 bis 6 Uhr erwartet“, ”ich habe den ganzen Tag gehofft,
er werde kommen“, ”in meiner Jugend habe ich gewünscht . . .“, etc. Daher der falsche
Vergleich mit in der Zeit amorphen Zuständen (Zahnschmerz, das Hören eines Tones, etc.,
obwohl diese unter sich wieder verschieden sind).
2
Was heißt es nun: ”ich glaube, er wird um 5 Uhr kommen“? oder: ”er glaubt N werde
um 5 Uhr kommen“? Nun, woran erkenne ich, daß er das glaubt? Daran, daß er es sagt?
oder aus seinem übrigen Verhalten? oder aus beiden? Danach wird man dem Satz ”er glaubt
. . .“ verschiedenen Sinn geben können.
3
Man kann in Worten glauben.
4
Hat es einen Sinn zu fragen: ”Woher weißt Du, daß Du das glaubst“? Und ist etwa die
Antwort: ”ich erkenne es durch Introspection“?
In manchen Fällen wird man so etwas sagen können, in manchen aber nicht.
5
Es hat einen Sinn, zu fragen: ”liebe ich sie wirklich? mache ich mir das nicht nur vor?“
Und der Prozeß der Introspection ist hier das Aufrufen von Erinnerungen, das Vorstellen
möglicher Situationen und der Gefühle, die man hätte, etc.
6
Introspection nennt man einen Vorgang7
des Schauens, im Gegensatz zum Sehen.
8
”Wie9
weiß ich, daß ich das glaube?“, ”wie weiß ich, daß ich Zahnschmerzen habe?“:
in mancher Beziehung sind diese Beispiele10
ähnlich.
11
Man konstruiert hier nach dem Schema: ”Woher weißt Du, daß jemand im andern Zimmer
ist?“ – ”Ich habe ihn drin singen gehört“.
”Ich weiß, daß ich Zahnschmerzen habe, weil ich es fühle“ ist nach diesem Schema
konstruiert und heißt nichts.
1 (M): 5555
2 (M): ///
3 (M): 3
4 (M): 3
5 (M): ///
6 (M): 3 (E): Das Material von hier ab bis
einschließlich S. 398 hat Wittgenstein dem TS
211 entnommen. Ab ”Man könnte nun die
Sache so (falsch) auffassen . . .“ ist jede der darin
enthaltenen Bemerkungen mit der Zahl 36
überschrieben. TS 213 enthält zwei leicht
voneinander abweichende Versionen der S.
392, von denen wir die gewählt haben, die wir
für die spätere halten.
7 (V): Prozeß
8 (M): 3
9 (V): ”Woher
10 (V): Fälle
11 (M): 3
391
390v
392
391
289
WBTC078-85 30/05/2005 16:16 Page 578
84
Belief. Grounds for Belief.
1
Believing. Related to this is: expecting, hoping, fearing, wishing. But also: doubting,
searching, etc.
We say: “Between 5 and 6 o’clock I expected him”, “I hoped all day long that he would
come”, “When I was young I wished . . .”, etc. Hence the incorrect comparison to states
that are amorphous in time (a toothache, hearing a sound, etc., even though these differ
from each other in turn).
2
Now what does this mean: “I believe he’ll come at 5 o’clock”? or “He believes that N
will come at 5 o’clock”? Well, how do I tell that he believes that? By his saying so? Or by
the way he otherwise behaves? Or by both? Depending on this, one can give different senses
to the proposition “He believes . . .”.
3
One can believe with words.
4
Does it make sense to ask: “How do you know that you believe that”? And might the
answer be: “I know it by introspection”?
In some cases one will be able to say something like that, but not in others.
5
It does make sense to ask: “Do I really love her? Or am I only deceiving myself?” And
here the process of introspection consists in calling up memories, imagining possible situations
and feelings that one would have, etc.
6
Introspection is called a process of looking, as opposed to seeing.
7
“How8
do I know that I believe that?”, “How do I know that I have a toothache?”: in
some respects these examples9
are similar.
10
Here we are proceeding according to the schema: “How do you know that there is
someone in the other room?” – “I heard him singing in there”.
“I know that I have a toothache because I feel it” is constructed according to this schema,
and means nothing.
1 (M): 5555
2 (M): ///
3 (M): 3
4 (M): 3
5 (M): ///
6 (M): 3 (E): Beginning at this point and continuing
through Wittgenstein’s p. 398, the
remarks come from copies of pages from TS 211.
Beginning with “Now one could (mis)understand
the matter”, below, each remark is preceded by
the handwritten number 36. TS 213 contains two
versions of p. 392, which differ slightly. We have
chosen what we consider the second of these two
versions.
7 (M): 3
8 (V): “Whence
9 (V): cases
10 (M): 3
289e
WBTC078-85 30/05/2005 16:16 Page 579
Vielmehr: ich habe Zahnschmerzen = ich fühle Zahnschmerzen = ich fühle, daß ich
Zahnschmerzen habe (ungeschickter und irreführender Ausdruck). ”Ich weiß, daß ich
Zahnschmerzen habe“ sagt dasselbe, nur noch ungeschickter, es sei denn, daß unter
”ich habe Zahnschmerzen“ eine Hypothese verstanden wird. Wie in dem Fall: ”ich weiß,
daß die Schmerzen vom schlechten Zahn herrühren und nicht von einer Neuralgie“.
Denken wir12
an die Frage ”wie merkst Du, daß Du Zahnschmerzen hast?“, oder gar: ”wie
merkst Du, daß Du fürchterliche Zahnschmerzen hast?“ (Dagegen: ”wie merkst Du, daß
Du Zahnschmerzen bekommen wirst“.)
13
(Hierher gehört die Frage: welchen Sinn hat es, von der Verifikation des Satzes ”ich
habe Zahnschmerzen“ zu reden? Und hier sieht man deutlich, daß die Frage ”wie wird
dieser Satz verifiziert“ von einem Gebiet der Grammatik zum andern ihren Sinn ändert.)
14
In dem Sinn von &Z.“, in dem man geneigt ist zu sagen &ich kann nicht Z. haben ohne es zu
wissen“ heißt es eben darum nichts zu sagen: &ich weiß daß ich Z. habe“ es sei denn daß dies ein
ungeschickter Ausdruck ist statt des Satzes &ich habe Z.“.
15
Ist &Ich glaube . . .“ der Ausdruck des Glaubens oder die Beschreibung des Geisteszustandes?16
17
Ist der Satz &es regnet“ die Beschreibung meines Geisteszustandes? Da es doch die Wiedergabe
meines18
Gedankens ist daß es regnet. Wir werden nicht so leicht geneigt sein, den Satz die
Beschreibung des Geisteszustands zu nennen, wenn wir bedenken,19
daß das Denken im Reden
bestehen kann, keine Begleitung des Gedankenausdrucks ist.
20
Man kann in Worten glauben.
21
Anderseits warum sollen wir nicht sagen, daß die Aussage22
&Ich glaube . . .“ die Beschreibung
des Geisteszustandes ist? es ist ja damit nichts verredet. Denn &Geisteszustand“ & &Beschreibung
eines23
Geisteszustands“ heißt24
eben so Vieles.
25
Man könnte nun die Sache so (falsch) auffassen: Die Frage ”wie weißt Du, daß Du
Zahnschmerzen hast“ wird darum nicht gestellt, weil man dies von den Zahnschmerzen
(selbst) aus erster Hand erfährt, während man, daß ein Mensch im andern Zimmer ist, aus
zweiter Hand, etwa durch ein Geräusch, erfährt. Das eine weiß ich durch unmittelbare
Beobachtung, das andere erfahre ich indirekt. Also: ”Wie weißt Du, daß Du Zahnschmerzen
hast“ – ”Ich weiß es, weil ich sie habe“ – ”Du entnimmst es daraus, daß Du sie hast;
aber mußt Du dazu nicht schon wissen, daß Du sie hast?“. – Der Übergang von den
Zahnschmerzen zur Aussage ”ich habe Zahnschmerzen“ ist eben ein ganz anderer, als der
vom Geräusch zur Aussage ”in diesem Zimmer ist jemand“. Das heißt, die Übergänge gehören
zu ganz verschiedenen Sprachspielen.26
27
Ist, daß ich Zahnschmerzen habe ein Grund zur Annahme, daß ich Zahnschmerzen habe?
12 (V): wir auch
13 (M): 3
14 (M): ///
15 (M): 3
16 (V): Ist &Ich glaube . . .“ der Ausdruck oder die
Beschreibung des Geisteszustandes des Glaubens?
17 (M): 3
18 (V): des
19 (V): sehen,
20 (M): 3
21 (M): 3
22 (V): daß der Satz
23 (V): des
24 (O): Geisteszustands heißt
25 (M): 3
26 (V): gehören ganz andern Sprachspielen an.
27 (M): 1
391v
392v
392, 393
290 Glauben. . . .
WBTC078-85 30/05/2005 16:16 Page 580
Rather: I have a toothache = I’m feeling a pain in my tooth = I feel that I have a toothache
(a clumsy and misleading expression). “I know that I have a toothache” says the same thing,
only more clumsily still, unless we take “I have a toothache” to be a hypothesis. As in the
case: “I know that the pain stems from my bad tooth and not from neuralgia”.
Let’s think11
about the question “How do you notice that you have a toothache?” or even:
“How do you notice that you have a horrible toothache?” (On the other hand: “How do you
notice that you will get a toothache?”.)
12
(This question is relevant here: What sense is there in talking about the verification of
the proposition “I have a toothache”? And here one sees clearly that the question “How is
this proposition verified?” changes its sense from one area of grammar to another.)
13
From the very sense of “toothache” that we’re inclined to express by saying “I cannot have
a toothache without knowing it”, it follows that it means nothing to say “I know that I have a
toothache”, unless this is a clumsy replacement for the sentence “I have a toothache”.
14
Is “I believe . . .” an expression of belief, or a description of one’s mental state?15
16
Is the proposition “It’s raining” a description of my mental state? After all, it’s the expression of
my17
thought that it is raining. We will not be so readily inclined to call that proposition a description
of a mental state if we consider18
that thinking can consist in speaking, and that it isn’t an accompaniment
of an expression of thought.
19
One can believe with words.
20
On the other hand, why should we not say that the statement21
“I believe . . .” is the description
of a mental state? For nothing has been misspoken in saying that. For “mental state” and
“description of a mental state” mean so many things.
22
Now one could (mis)understand the matter in the following way: The question “How
do you know that you have a toothache?” doesn’t get asked because we get this information
first-hand from the toothache (itself), whereas one finds out second-hand that someone is
in the room next door, say because of a noise. I know the one thing by immediate observation,
the other I find out indirectly. Thus: “How do you know that you have a toothache?”
– “I know it because I have it” – “You deduce this from the fact that you have a toothache;
but to do this, don’t you already have to know that you have one?”. – The transition from
the toothache to the statement “I have a toothache” is completely different from that between
a noise and the statement “There’s someone in this room”. That means that the transitions
belong to completely different language-games.
23
Is the fact that I have a toothache a reason to assume that I have a toothache?
11 (V): think also
12 (M): 3
13 (M): ///
14 (M): 3
15 (V): Is “I believe . . .” an expression or a description
of a mental state of belief?
16 (M): 3
17 (V): of the
18 (V): see
19 (M): 3
20 (M): 3
21 (V): the proposition
22 (M): 3
23 (M): 1
Belief. . . . 290e
WBTC078-85 30/05/2005 16:16 Page 581
28
(Man kann die Philosophen dadurch verwirren (confound), daß man nicht bloß da Unsinn
spricht, wo auch sie es tun, sondern auch solchen, den zu sagen sie sich scheuen (würden).)
29
Erschließt man aus der Wirklichkeit einen Satz? Also etwa ”aus den wirklichen
Zahnschmerzen, darauf, daß man Zahnschmerzen hat“? Aber das ist doch nur eine unkorrekte
Ausdrucksweise; es müßte heißen: man schließt, daß man Zahnschmerzen hat daraus,
daß man Zahnschmerzen hat (offenbarer Unsinn).
30
”Warum glaubst Du, daß Du Dich an der Herdplatte verbrennen wirst?“ – Hast Du
Gründe für diesen Glauben, und brauchst Du Gründe?
Hast Du diese Gründe – gleichsam – immer bei Dir, wenn Du es glaubst?
Und glaubst Du es immer – ausdrücklich – wenn Du Dich etwa wehrst, die Herdplatte
anzurühren?
Meint man mit ”Gründen für den Glauben“31
dasselbe, wie mit ”Ursachen des Glaubens“
(Ursachen des Vorgangs des Glaubens)?
32
Was für einen Grund habe ich, anzunehmen, daß mein Finger, wenn er den Tisch
berühren, einen Widerstand spüren wird? Was für einen Grund, zu glauben, daß dieser Bleistift
sich nicht schmerzlos durch meine Hand stecken läßt? Wenn ich dies frage, melden sich
hundert Gründe, die einander gar nicht zu Wort kommen lassen wollen. ”Ich habe es doch
selbst ungezählte Male erfahren; und ebenso oft von ähnlichen Erfahrungen gehört; wenn
es nicht so wäre, würde . . . ; etc.“.
33
Glaube ich, wenn ich auf meine Türe zugehe, ausdrücklich, daß sie sich öffnen lassen
wird, – daß dahinter ein Zimmer und nicht ein Abgrund sein wird, etc.?
Setzen wir statt des Glaubens den Ausdruck des Glaubens. –
34
Was heißt es, etwas aus einem bestimmten Grunde glauben? Entspricht es, wenn wir
statt des Glaubens den Ausdruck des Glaubens setzen, dem, daß man35
den Grund sagt, ehe
man36
das Begründete sagt?
37
”Hast Du es aus diesen Gründen geglaubt?“ ist dann eine ähnliche Frage, wie: ”hast
Du, als Du mir sagtest, 25 × 25 sei 625, die Multiplikation wirklich ausgeführt?“
38
Die Frage ”aus welchen Gründen glaubst Du das“39
könnte bedeuten: ”aus welchen
Gründen leitest Du das jetzt ab (hast Du es jetzt abgeleitet)“; aber auch: ”welche Gründe
kannst Du mir nachträglich für diese Annahme angeben“.
40
Ich könnte also unter ”Gründen“ zu einer Meinung tatsächlich das allein41
verstehen,
was der Andere sich vorgesagt hat, ehe er zu der Meinung kam. Die Rechnung, die er
tatsächlich ausgeführt hat.
28 (M): 2
29 (M): 3
30 (M): 4
31 (V): ”Gründen des Glaubens“
32 (M): 1
33 (M): 2
34 (M): 3
35 (V): Einer
36 (V): er
37 (M): 4
38 (M): 1
39 (V): Frage ”warum glaubst Du das“
40 (M): 2
41 (V): tatsächlich nur das
394
395
291 Glauben. . . .
WBTC078-85 30/05/2005 16:16 Page 582
24
(One can confuse (confound) philosophers not only by talking nonsense where they do,
but also by saying the kinds of nonsensical things they (would) hesitate to say.)
25
Do we infer a proposition from reality? Do we, for instance, “infer that we have a toothache
from the real toothache”? But that’s nothing other than an incorrect way of expressing oneself;
we’d have to say: From having a toothache we conclude that we have a toothache (which
is obvious nonsense).
26
“Why do you believe that you’ll get burned on the hotplate?” – Do you have reasons
for this belief, and do you need reasons?
Do you have these reasons constantly on your person – as it were – when you believe this?
And do you always – explicitly – believe this, say when you refuse to touch the hotplate?
Does one mean the same thing by “reasons for believing”27
and “causes of believing”
(causes of the process of believing)?
28
What kind of a reason do I have for assuming that my finger will feel resistance when
it touches the table? What kind of a reason for believing that this pencil will not pierce my
hand without pain? When I ask this hundreds of reasons arise that try to shout each other
down. “I myself have experienced it countless times; and have heard of similar experiences
as often; if it weren’t like that then . . . would . . . ; etc.”.
29
When I approach my door, do I explicitly believe that it can be opened – that there will
be a room behind it and not an abyss, etc.?
Let’s substitute “expression of belief” for “belief”. –
30
What does it mean to believe something for a particular reason? If we substitute
“expression of belief” for “belief”, does that correspond to stating the reason before
stating31
what it is a reason for?
32
Then “Did you believe it for these reasons?” is a similar question to: “Did you really
carry out the multiplication when you told me that 25 × 25 was 625?”
33
The question “What reasons do you have for believing that?”34
could mean: “What
reasons do you now have for inferring that (did you just now have for having inferred it)?”;
but it could also mean: “What reasons can you subsequently give me for this assumption?”.
35
So by “reasons” for an opinion I could in fact understand solely what someone else said
to himself before he arrived at his opinion. The calculation that he actually carried out.
24 (M): 2
25 (M): 3
26 (M): 4
27 (V): “grounds of belief”
28 (M): 1
29 (M): 2
30 (M): 3
31 (V): to someone stating the reason before he
states
32 (M): 4
33 (M): 1
34 (V): question “Why do you believe that?”
35 (M): 2
Belief. . . . 291e
WBTC078-85 30/05/2005 16:16 Page 583
42
Frage ich jemand: ”warum glaubst Du, daß diese Armbewegung einen Schmerz mit sich
bringen wird?“, und er antwortet: ”weil sie ihn einmal hervorgebracht und einmal nicht
hervorgebracht hat“, so werde ich sagen: ”das ist doch kein Grund zu Deiner Annahme“.
Wie nun, wenn er mir darauf antwortet: ”oh doch! ich habe diese Annahme noch immer
gemacht, wenn ich diese Erfahrung gemacht hatte“? – Da würden wir doch sagen: ”Du
scheinst mir die Ursache (psychologische Ursache) Deiner Annahme anzugeben, aber nicht
den Grund“.
43
”Warum glaubst Du, daß das geschehen wird?“ – ”Weil ich es zweimal beobachtet habe.“
Oder: ”Warum glaubst Du, daß das geschehen wird?“ – ”Weil ich es mehrmals
beobachtet habe; und es geht offenbar so vor sich: . . .“ (es folgt eine Darlegung einer
umfassenden Hypothese). Aber diese Hypothese, dieses Gesamtbild, muß Dir einleuchten.
Hier geht die Kette der Gründe nicht weiter. – (Eher könnte man sagen, daß sie sich schließt.)
44
Man möchte sagen: Wir schließen nur dann aus der früheren Erfahrung auf die zukünftige,
wenn wir die Vorgänge verstehen (im Besitze der richtigen Hypothese sind). Wenn
wir den richtigen, tatsächlichen, Mechanismus zwischen den beiden beobachteten Rädern
annehmen. Aber denken wir doch nur: Was ist denn unser45
Kriterium dafür, daß unsere
Annahme die richtige ist? –
Das Bild und die Daten überzeugen uns und führen uns nicht wieder weiter – zu andern
Gründen.
46
Wir sagen: ”diese Gründe sind überzeugend“; und dabei handelt es sich nicht um
Prämissen, aus denen das folgt, wovon wir überzeugt wurden.
47
Wenn man sagt: ”die gegebenen Daten sind insofern Gründe, zu glauben, p werde
geschehen, als dies aus den Daten zusammen mit dem angenommenen Naturgesetz folgt“,
– dann kommt das eben darauf hinaus, zu sagen, das Geglaubte folge aus den Daten nicht,
sondern komme vielmehr einer neuen Annahme gleich.
48
Wenn man nun fragt: wie kann aber frühere Erfahrung ein Grund zur Annahme sein,
es werde später das und das eintreffen, – so ist die Antwort: welchen allgemeinen Begriff
vom Grund zu solch einer Annahme haben wir denn? Diese Art Angabe über die
Vergangenheit nennen wir eben Grund zur Annahme, es werde das in Zukunft geschehn.
– Und wenn man sich wundert, daß wir ein solches Sprachspiel49
spielen, dann berufe ich
mich auf die Wirkung einer vergangenen Erfahrung (daß ein gebranntes Kind das Feuer
fürchtet).
50
Wer sagt, er ist durch Angaben über Vergangenes nicht davon zu überzeugen, daß
in Zukunft etwas geschehen wird, der muß etwas anderes mit dem Wort ”überzeugen“
meinen, als wir es tun. – Man könnte ihn fragen: Was willst Du denn hören? Was für Angaben
nennst Du Gründe dafür, das zu glauben?51
Was nennst Du ”überzeugen“? Welche Art des
”Überzeugens“ erwartest Du Dir. – Wenn das keine Gründe sind, was sind denn Gründe?
– Wenn Du sagst, das seien52
keine Gründe, so mußt Du doch angeben können, was der
42 (M): 3
43 (M): 4
44 (M): 1
45 (V): das
46 (M): 2
47 (M): 3
48 (M): 4
49 (V): Spiel
50 (M): 1
51 (V): Gründe um, das zu glauben?
52 (V): sind
396
397
292 Glauben. . . .
WBTC078-85 30/05/2005 16:16 Page 584
36
If I ask somebody: “Why do you believe that this movement of your arm will be
accompanied by pain?” and he answers: “Because sometimes it produces it and sometimes
it doesn’t”, then I’ll say: “But that isn’t any reason for your assumption”.
Now what if he answers me: “Oh yes it is! Whenever I’ve had this experience, I’ve always
made that assumption”? – Then surely we’d say: “You seem to be giving the cause (the
psychological cause) for your assumption, but not the reason”.
37
“Why do you believe that that will happen?” – “Because I’ve observed it twice.”
Or: “Why do you believe that that will happen?” – “Because I’ve observed it several times;
and evidently this is how it happens: . . .” (and now an extensive hypothesis is laid out). But
this hypothesis, this total picture, must make sense to you. Here the chain of reasons does
not continue. – (It would be more correct to say that it comes to an end.)
38
One wants to say: We only infer a future experience from a past one when we
understand the processes (have the correct hypothesis). If we assume the correct, actual
mechanism between the two wheels we’re observing. But let’s just consider: What is our39
criterion for our assumption being the right one? –
The picture and the data convince us, but they don’t lead us further – towards other
reasons.
40
We say: “These reasons are convincing”; and here it isn’t a matter of premises, from
which what we were convinced of follows.
41
If one says: “The data we’re given are reasons for believing that p will happen, in so far
as this follows from the data, together with the assumption of a certain natural law” – then
this simply amounts to saying that what is believed does not follow from the data, but rather
is tantamount to a new assumption.
42
Now if one asks: But how can a previous experience be a reason for assuming that later
on this or that will happen? – the answer is: What general concept of a reason for such an
assumption do we really have? Well, this kind of a statement about the past is simply what
we call a reason for assuming that this will happen in the future. – And if you are surprised
that we are playing such a language-game43
, then I refer you to the effect of a past experience
(to the fact that a child who has been burned fears fire).
44
Whoever says that information about what happened in the past could not convince him
that something will happen in the future, must mean something else by the word “convince”
than we do. – One could ask him: What do you want to hear? What sort of information do
you call reasons for believing45
that? What do you call “convincing someone”? What kind of
“convincing” are you expecting? – If these aren’t reasons, what are? – If you are saying that
these are not reasons then you surely must be able to state what would have to be the case
36 (M): 3
37 (M): 4
38 (M): 1
39 (V): the
40 (M): 2
41 (M): 3
42 (M): 4
43 (V): game
44 (M): 1
45 (V): reasons to believe
Belief. . . . 292e
WBTC078-85 30/05/2005 16:16 Page 585
Fall sein müßte, damit wir mit Recht sagen könnten, es seien Gründe für unsern Glauben53
vorhanden. ”Keine Gründe“ – im54
Gegensatz wozu?
55
Denn, wohlgemerkt: Gründe sind hier nicht Sätze, aus denen das Geglaubte folgt.
56
Aber nicht, als ob wir sagen wollten:57
Für’s Glauben genügt eben weniger, als für das
Wissen. – Denn hier handelt es sich nicht um eine Annäherung an das logische Folgen.
58
Irregeführt werden wir durch die Redeweise:59
”Dieser Grund ist gut, denn er macht
das Eintreffen des Ereignisses wahrscheinlich“.60
Hier ist es, als ob wir nun etwas weiteres
über den Grund ausgesagt hätten, was ihn als (guten) Grund rechtfertigt;61
während mit
dem Satz, daß dieser Grund das Eintreffen wahrscheinlich macht, nichts gesagt ist, wenn
nicht, daß dieser Grund einem62
bestimmten Standard des guten Grundes entspricht, –
der Standard aber nicht begründet ist!
63
Ein guter Grund ist einer der so aussieht.
64
”Das ist ein guter Grund, denn er macht das Eintreffen wahrscheinlich“ erscheint uns
so wie: ”das ist ein guter Hieb, denn er macht den Gegner kampfunfähig“.
65
Man ist versucht zu66
sagen: ”ein guter Grund ist er nur darum, weil er das Eintreffen
wirklich wahrscheinlich macht“. Weil er sozusagen wirklich einen Einfluß auf das Ereignis
hat, also quasi einen erfahrungsmäßigen.
67
”Warum nimmst Du an, daß er besserer Stimmung sein wird, weil ich Dir sage, daß er
gegessen hat? ist denn das ein Grund?“ – ”Das ist ein guter Grund, denn das Essen hat
erfahrungsgemäß einen Einfluß auf seine Stimmung.“ Und das könnte man auch so sagen:
”Das Essen macht es wirklich wahrscheinlicher, daß er guter Stimmung sein wird“.
Wenn man aber fragen wollte: ”Und ist alles das, was Du von der früheren Erfahrung
vorbringst, ein guter Grund, anzunehmen, daß es sich auch diesmal so verhalten wird“, so
kann ich nun nicht sagen: ja, denn das macht das Eintreffen der Annahme wahrscheinlich.
Ich habe oben meinen Grund mit Hilfe des Standards für den guten Grund gerechtfertigt;
jetzt kann ich aber nicht den Standard rechtfertigen.
68
Wenn man sagt ”die Furcht ist begründet“, so ist nicht wieder begründet, daß wir das
als guten Grund zur Furcht ansehen. Oder vielmehr: es kann hier nicht wieder von einer
Begründung die Rede sein.
69
Wenn die Begründung eines Glaubens70
eine erfahrungsgemäße Beziehung wäre, so müßte
man weiter fragen ”und warum ist das ein Grund gerade für diesen Glauben“. Und so ginge
es weiter. (Z.B. ”warum nehmen wir das Gedächtnis als Grund für den Glauben, daß etwas
in der Vergangenheit geschehen ist“.)
53 (V): für unsere Annahme
54 (V): Gründe“ – : im
55 (M): 2
56 (M): 3
57 (V): als ob man sagen könnte (kön:
58 (M): 4
59 (V): Ausdrucksweise:
60 (V): ”Das ist ein guter // richtiger // Grund zu
unserer Annahme, denn er macht das
Eintreffen des Ereignisses wahrscheinlich“.
61 (V): was seine Zugrundelegung rechtfertigt;
62 (V): dem
63 (M): 1
64 (M): 2
65 (M): 3
66 (V): Man möchte
67 (M): 4
68 (M): 5
69 (M): 3
70 (V): Wenn der Grund, etwas zu glauben,
398
399
293 Glauben. . . .
WBTC078-85 30/05/2005 16:16 Page 586
for us to be justified in saying that reasons do exist for our belief.46
“Not reasons” – as opposed
to what?
47
For, mind you: Here reasons are not propositions from which what is believed logically
follows.
48
But it isn’t as if we wanted to say:49
Less will do for belief than for knowledge. – For
here it isn’t a matter of an approximation to logical inference.
50
We are misled by this way of speaking:51
“This reason is good because it makes the occurrence
of the event probable.”52
Now here it seems as if we had said something further about
the reason, which justifies it as a (good) reason;53
whereas saying that this reason makes the
occurrence probable says nothing except that this reason comes up to a54
particular standard
for a good reason – but that that standard has no rational basis!
55
A good reason is one that looks like this.
56
“That’s a good reason, since it makes the occurrence probable” seems to us like: “That’s
a good blow, since it disables the enemy”.
57
One is tempted58
to say: “It’s a good reason only because it makes the occurrence really
probable”. Because so to speak it really has an influence on the event, i.e. an empirical one,
as it were.
59
“Why do you assume that he’ll be in a better mood because I told you that he’s just
eaten? Is that any kind of reason?” – “That’s a good reason because, based on past experience,
eating influences his mood.” And that could also be put this way: “Eating really does
make it more probable that he will be in a good mood”.
But if one wanted to ask: “And is everything you put forth about past experience a good
reason to assume that this time too this is the way it will be?” then I can’t say: “Yes, because
that makes the occurrence of the assumption probable.” In what I said earlier I justified my
reason using a standard for a good reason; but now I can’t justify the standard.
60
If one says “That fear is justified” then that doesn’t in turn justify that we view this
justification as a good reason to be afraid. Or rather: Here there can be no further talk of
justification.
61
If the justification for a belief62
were an empirical relationship, one would have to go on to
ask “And why is that a reason for just this belief?”. And it would go on in this way. (For
example, “Why do we take memory as a reason for the belief that something happened in
the past?”.)
46 (V): assumption.
47 (M): 2
48 (M): 3
49 (V): if one could say:
50 (M): 4
51 (V): this mode of expression:
52 (V): “That is a correct // good // reason for our
assumption because it makes the occurrence of
the event probable.”
53 (V): justifies using it as a basis;
54 (V): to the
55 (M): 1
56 (M): 2
57 (M): 3
58 (V): One would like
59 (M): 4
60 (M): 5
61 (M): 3
62 (V): If the reason for believing something
Belief. . . . 293e
WBTC078-85 30/05/2005 16:16 Page 587
Wenn die Beziehung des Grundes zum Begründeten eine wäre, die die Erfahrung lehrt, so müßte
man weiter fragen: und mit welchem Recht nimmst Du das als Grund für diesen Glauben? Und so
ginge es weiter; & der Glaube wäre nie gerechtfertigt.
Vergleiche damit: &Wenn eine Verbindung zweier Dinge immer71
in einer Vermittlung durch ein
drittes Ding besteht, dann sind zwei Dinge nie miteinander verbunden.“ Das ist falsch; dagegen
könnte man sagen: &Wenn eine Verbindg. zweier Dinge – immer in einer Vermittlg. durch ein
drittes Ding besteht, das mit jedem der zwei verbunden ist, dann sind zwei Dinge nie mit einander
verbunden“. Das heißt72
aber nicht: eine Verbindung wird nie erreicht, sondern es hat keinen Sinn
zu sagen &die Verbindung wird erreicht“ (& also auch nicht das Gegenteil). D.h., es hat keinen
Sinn von73
&Verbindung“ zu reden; der Begriff der &Verbindung“ ist gar nicht erklärt worden.
Wir meinen, wir müssen74
den endlosen Regress ein paar Stufen weit75
durchlaufen & ihn dann
in Verzweiflung aufgeben. Während die Definitionsgleichung einfach keine Auflösung hat. Wir
haben keine solche Methode zu ihrer Auflösung festgelegt.
Die Erfahrung lehrt daß die Ursache von B A ist. Und also ist es ein guter Grund76
zur Annahme77
daß B geschehen wird78
daß A geschehen ist.
Aber man kann nicht sagen die Erfahrung lehre daß die wiederholte Erfahrung der Koinzidenz ein
guter Grund zur Annahme weiterer Koinzidenzen sei.79
71 (V): ”Wenn eine Verbindung immer
72 (O): verbunden”% & das heißt eigentlich:
73 (V): von einer
74 (V): müßten
75 (V): ein paarmal
76 (V): Grund d
77 (V): Grund anzunehmen
78 (V): wird wenn
79 (M):
294 Glauben. . . .
399v
WBTC078-85 30/05/2005 16:16 Page 588
If the relationship of a reason to what it is a reason for were taught by experience, one would have
to ask the next question: “And what is your justification for taking that as a reason for this belief?”
And it would go on in this way; and belief would never be justified.
Compare this to: “If a connection between two things always63
consists in a mediation by a third
thing, then two things are never connected to each other.” That is false; on the other hand one could
say: “If a connection between two things always consists in a mediation by a third thing that is connected
to each of the two, then two things are never connected to each other.” But that doesn’t
mean: A connection is never achieved; it’s just that it makes no sense to say “A connection is achieved”
(and therefore neither does its opposite). That is, it makes no sense to talk about “connection”; the
concept of “connection” has in no way been explained.
We believe that we have to proceed a few steps64
along an infinite regress, and then have to give
up in despair. Whereas the defining equation simply has no solution. We haven’t established any
method for its solution.
Experience teaches that A is the cause of B. And therefore that A has happened is a good reason
for the assumption65
that B will happen.
But we can’t say that experience teaches that a repeated experience of coincidence is a good
reason to assume further coincidences.66
63 (V): “If a connection always
64 (V): a few times
65 (V): reason to assume
66 (M):
Belief. . . . 294e
WBTC078-85 30/05/2005 16:16 Page 589
85
Grund, Motiv, Ursache.
1
Ich lege meine Hand auf die Herdplatte, fühle unerträgliche Hitze und ziehe die Hand
schnell zurück: War es nicht möglich, daß die Hitze der Platte im nächsten Augenblick aufgehört
hätte? konnte ich es wissen? Und war es nicht möglich, daß ich gerade durch meine
Bewegung mich weiterem2
Schmerz aussetzte?
Es müßte also kein guter Grund sein3
zu sagen: ”Ich mußte die Hand zurückziehen,4
weil die
Platte zu heiß war“?5
–
Wie kann man sicher sein, es darum getan zu haben?
Denken wir uns daß Hitze die Wirkung hätte uns zu zwingen die Hand gegen6
den heißen
Gegenstand zu pressen (etwa ähnlich wie man einen Leitungsdraht nicht auslassen kann)
7
Wenn man nun fragte: Bist Du sicher, daß Du es deswegen getan hast? Würde ich8
da
nicht schwören, daß ich9
es nur deswegen getan habe?10
Und ist es nicht doch Hypothese?11
Sollte man sagen: ich weiß daß ich es deshalb tun wollte;12
nicht, daß der Arm sich aus dieser
Ursache zurückgezogen hat? Man weiß13
das Motiv, nicht die Ursache.
”Ich hab’ es nicht mehr (länger) ausgehalten.“
”Ich halte es nicht mehr aus; ich muß die Hand zurückziehen.“ Aber worin besteht dieses
Zurückziehen, als in dem Wunsch, die Hand möchte sich zurückziehen, während sie
sich wirklich zurückzieht? Zieht sie sich nicht zurück, so können wir auch nichts machen.
Jedenfalls, möchte ich sagen, ist ”sie zurückziehen wollen“ eine Erfahrung, die wir zwar
wünschen können, aber nicht herbeiführen. &Wie was?“ muß ich fragen. Ich meinte mit
dem Herbeiführen14
nicht: Verursachen, sondern15
ein direktes (nichtkausales) Bewegen.16
(Denke
an die Erfahrung beim Zeichnen eines Quadrats mit seinen Diagonalen durch den Spiegel.)
17
Wenn ich sage, die Erfahrung des Wollens könne ich zwar wünschen, aber nicht herbeiführen,
so bin ich da wieder bei einem, für die Erkenntnistheorie so18
charakteristischen
Unsinn. Denn in dem Sinne, in welchem ich überhaupt etwas herbeiführen kann (etwa
Magenschmerzen durch Überessen), kann ich auch das Wollen herbeiführen. (In diesem Sinne
führe ich das Schwimmen-Wollen herbei, indem ich in’s tiefe Wasser springe.) Ich wollte
1 (M): 3
2 (V): einem
3 (V): Es ist also in gewissem Sinne keine gute
Begründung
4 (V): ”Ich zog die Hand zurück,
5 (V): war“!
6 (V): hätte daß uns zu zwingen gegen
7 (M): 3
8 (V): man
9 (V): man
10 (V): hat?
11 (V): Erfahrung?
12 (V): Müßte man nicht sagen: man würde
schwören, daß man es deshalb tun wollte;
13 (V): beschwört
14 (V): mit Herbeiführen
15 (V): nicht ein kausales sondern
16 (V): Herbeiführen^
17 (M): 3 zwangsläufiger Mechanismus
18 (V): sehr
400
401
400v
401
295
WBTC078-85 30/05/2005 16:16 Page 590
85
Reason, Motive, Cause.
1
I place my hand on a hotplate, feel unbearable heat and quickly pull my hand back: Mightn’t
the heat have stopped the very next moment? Could I know this? And mightn’t my very
motion have exposed me to further pain2
?
So, saying “I had to pull3
my hand back because the hotplate was too hot” wouldn’t
necessarily be a good reason?4
–
How can you be sure you’ve done it for that reason?
Let’s imagine that heat had the effect of forcing us to press our hand against the hot object
(in the same way you can’t let go of a live wire).
5
Now if someone asked: Are you sure you did it because of that? Wouldn’t I swear that I6
had done it only because of that? And yet, isn’t this a hypothesis7
? Should one say: I know that
I wanted to do it because of that8
; and not – that this was the cause of my arm’s drawing back?
One knows9
the motive, but not the cause.
“I couldn’t stand it any (longer).”
“I can’t stand it any more; I have to pull back my hand.” What else does this pulling back
consist in other than the wish that our hand should pull back while it’s actually pulling back?
If it doesn’t pull back, then there’s nothing we can do. In any case, I would like to say that
“wanting to pull it back” is an experience that we can wish for, to be sure, but can’t bring
about. “Like what?”, I have to ask. By “bringing about” I don’t mean “causing”, but10
“directly
(non-causally) moving”. (Think about the experience of drawing a square and its diagonals
using a mirror.)
11
If I say that I can wish for but not bring about the experience of wanting, then once
again I have arrived at a piece of nonsense that is so12
characteristic of epistemology. For
in the sense in which I can bring about anything at all (say a stomach-ache by overeating),
I can also bring about wanting. (In this sense I can bring about wanting to swim by jumping
into deep water.) No doubt I wanted to say that I couldn’t want wanting; that is, it makes
1 (M): 3
2 (V): to pain
3 (V): “I pulled
4 (V): hot” is, in a certain sense, not a good
justification!
5 (M): 3
6 (V): Wouldn’t one swear that one
7 (V): this an experience
8 (V): Mustn’t one say: One would swear that one
wanted to do it because of that
9 (V): invokes
10 (V): mean a causal but
11 (M): 3 ineluctable mechanism
12 (V): quite
295e
WBTC078-85 30/05/2005 16:16 Page 591
wohl sagen: ich könnte das Wollen nicht wollen; d.h., es hat keinen Sinn, vom Wollen-wollen
zu sprechen. Und hier denkt man sich das Wollen als ein direktes – d.h. nicht-kausales –
Herbeiführen.19
Und den kausalen Nexus20
als eine Verbindung zweier Maschinenteile durch einen
zwangsläufigen Mechanismus (etwa durch eine Reihe von Zahnrädern), die auslassen kann,21
wenn
der Mechanismus gestört wird), während der Nexus des Willens mit . . . ? etwa dem des Innern
mit dem22
Äußern entspricht, oder dem der Bewegung des physikalischen Körpers mit der23
Bewegung seines Gesichtsbildes.24
25
Man denkt nur an die Störungen, denen ein Mechanismus normalerweise ausgesetzt ist; nicht
daran, daß die Zahnräder plötzlich weich werden könnten, oder einander durchdringen, etc.
26
”Wie weißt Du, daß Du es aus diesem Motiv getan hast?“ – ”Ich erinnere mich daran,
es darum getan zu haben“. – ”Woran erinnerst Du Dich? – Hast Du es Dir damals gesagt;
oder erinnerst Du Dich an die Stimmung in der Du warst; oder daran, daß Du Mühe
hattest, einen Ausdruck Deines Gefühls zu unterdrücken?“ Daraus wird sich zeigen worin es
bestand es aus diesem Motiv getan zu haben.
Und wenn man etwa einen Ausdruck seines Gefühls nur mit Mühe unterdrückt hat, –
wie war das? Hatte man sich ihn damals leise vorgesagt? etc. etc.
27
Das Motiv ist nicht eine Ursache ”von innen gesehen“! Das Gleichnis von ”innen und
außen“ ist hier – wie so oft – gänzlich irreleitend. – Es ist von der Idee der Seele (eines
Lebewesens) im Kopfe (als Hohlraum vorgestellt) hergenommen.28
Aber diese Idee ist29
darin mit andern unverträglichen30
vermengt, wie die Metaphern in dem Satz: ”der Zahn der
Zeit, der alle Wunden heilt, etc.“.
31
Man nimmt an daß ein Mensch das Motiv seiner Tat weiß; – das zeigt uns, wie wir das Wort &Motiv“
gebrauchen.32
”Wie weißt Du, daß das wirklich der Grund ist, weswegen Du es glaubst? – (das) ist, als
fragte ich: ”wie weißt Du, daß es das ist, was Du glaubst“. Denn er gibt nicht die Ursache
eines Glaubens an, die er nur vermuten könnte, sondern beschreibt ein Operieren mit
Gedanken, das zu dem Geglaubten führt (und ihn etwa geführt hat).33
Einen Vorgang, der
seiner Art nach zu dem des Glaubens gehört. – Der Unterschied zwischen der Frage nach
19 (V): direktes (d.h. nicht-kausales) Herbeiführen.
20 (V): Herbeiführen. & hier denkt man sich den kaus.
Nexus . . .
21 (V): zu sprechen. Und mein falscher Ausdruck
kam daher, daß man sich das Wollen als ein
direktes, // unmittelbares, // nicht-kausales,
Herbeiführen // Bewegen // denken will. Dieser
Idee aber // Und dem wieder // liegt wieder eine
falsche Analogie zugrunde, etwa, daß der
kausale Nexus // Und dieser Idee liegt die
Vorstellung zu Grund daß der kausale Nexus //
durch eine Reihe von Zahnrädern gebildet
wird (die auslassen kann, // liegt die irreführende
Analogie zugrunde, in der der kausale Nexus als eine
Reihe von Zahnrädern erscheint // zugrunde, die den
kausalen Nexus als eine Reihe von Zahnrädern sieht //
zugrunde, der kaus. Nex. werde etwa durch eine
Reihe von Z. gebildet // zugrunde; der kausale Nexus
erscheint als ein Mechanismus der // erscheint durch
einen Mechan. hergestellt, der // zwei Maschinenteile
verbindet. Die Verbindung kann auslassen,
22 (V): Innern zum
23 (V): Körpers zur // & der
24 (V): Bewegung seiner Erscheinung.
25 (M): 3
26 (M): 3 ////
27 (M): 3
28 (V): hergeleitet.
29 (V): Es ist die Idee von der Seele, einem
Lebewesen, im Kopfe. Aber sie ist
30 (V): unvereinbaren
31 (M): 3
32 (V): – das zeigt sagt uns was etwas über die
Bedeutung des Wortes &Motiv“.
33 (V): beschreibt einen Vorgang von
Operationen, die zu dem Geglaubten führen
(und etwa geführt haben).
400v
401
402
401v
402
401
401
400v
401
402
296 Grund, Motiv, Ursache.
WBTC078-85 30/05/2005 16:16 Page 592
no sense to talk about wanting to want. And here one thinks of wanting as a direct – i.e.
non-causal – bringing13
about. And of the causal nexus14
as a connection of two parts of a machine
by an interlocking mechanism (say by a series of cogwheels). This is a connection that can fail if the
mechanism malfunctions,15
whereas the nexus of the will and . . . ? corresponds, say, to the
nexus of the inner with the16
external world, or to that of the movement of a physical body
with17
the movement of its visual image.18
19
One thinks only of the malfunctions that a mechanism is usually subject to; but not of the
possibility that the cogwheels could suddenly turn soft, or interpenetrate, etc.
20
“How do you know that you did it from this motive?” – “I remember having done it for
that reason.” – “What are you remembering? – Did you say that to yourself then; or do you
remember the mood you were in; or that you had trouble suppressing an expression of your
feeling?” These things will show what having done it from this motive consisted in.
And if perhaps you suppressed an expression of your feeling only with difficulty – how
did you do that? Did you at that time utter the expression softly to yourself? etc., etc.
21
A motive is not a cause “seen from within”! Here the simile of “inside and outside” is
totally misleading – as it so often is. – It is taken22
from the idea of the soul (of a living being)
in one’s head (imagined as a hollow space). But this idea has been mixed23
with other incompatible
ideas, like the mixed metaphors in the sentence: “The tooth of time that heals all
wounds, etc.”.
24
We assume that a person knows the motive for his action; – that shows us how we use the word
“motive”.25
“How do you know that this is really the reason you believe that?” – That’s like asking:
“How do you know that this is what you believe?”. For he isn’t stating the cause for a belief,
which cause he could only surmise; rather, he is describing an operation with thoughts
that leads26
to his belief (and may have led him there), a process that belongs inherently to
the process of believing. – The difference between asking for the cause of a belief and
13 (V): direct (i.e. non-causal) bringing
14 (V): about. And here one thinks of the causal nexus
. . .
15 (V): wanting to want. And my false expression
originates from wanting to think of wanting as
a direct, // immediate, // non-causal bringing about
// moving //. However, underlying this idea // And
underlying this in turn // is a false analogy, for
instance, // And underlying this idea is the mental
image // that the causal nexus is formed by a
series of cogwheels. // And underlying this idea is
the misleading analogy in which the causal nexus
appears as a series of cogwheels. // analogy that sees
the causal nexus as a series of cogwheels. // analogy,
that the causal nexus is formed by a series of cogwheels.
// However, underlying this idea is a false analogy;
the causal nexus appears as a mechanism that //
appears to be created through a mechanism that //
connects two parts of a machine. This connection
can fail
16 (V): inner to the
17 (V): body to the // body and the
18 (V): its appearance.
19 (M): 3
20 (M): 3 ////
21 (M): 3
22 (V): derived
23 (V): It is the idea of the soul, a living being, in one’s
head. But it is mixed
24 (M): 3
25 (V): – that shows says to us what something about
the meaning of the word “motive”.
26 (V): describing a process of operations that lead
Reason, Motive, Cause. 296e
WBTC078-85 30/05/2005 16:16 Page 593
der Ursache und der (Frage) nach dem Grund des Glaubens ist etwa34
der, zwischen den
Fragen: ”was ist die physikalische Ursache davon, daß Du von A nach B gekommen bist“ und:
”auf welchem Wege35
bist Du von A nach B gekommen“.36
– Und hier sieht man sehr klar,
wie auch die Angabe der Ursache als Angabe eines Weges aufgefaßt werden kann, aber in
ganz anderem Sinne.
37
”Man kann die Ursache einer Erscheinung nur vermuten“ (nicht wissen). – Das muß
eine Aussage die Grammatik betreffend sein. Sie38
sagt nicht,39
daß wir mit dem besten Willen40
die Ursache nicht wissen können. Der Satz ist insofern ähnlich dem: ”wir können in der
Zahlenreihe, soweit wir auch zählen, kein Ende erreichen“. Das heißt: von einem ”Ende der
Zahlenreihe“ kann keine Rede sein; und dies ist – irreführend – in das Gleichnis gekleidet
von Einem, der wegen der großen Länge des Weges das Ende nicht erreichen kann. – So
gibt es einen Sinn, in dem ich sagen kann: ”ich kann die Ursache dieser Erscheinung nur
vermuten“ d.h.: es ist mir noch nicht gelungen, sie (im gewöhnlichen Sinne) ”festzustellen“.
Also im Gegensatz zu dem Fall, in dem es mir gelungen ist, in dem41
ich also die Ursache
weiß. – Sage ich nun aber, im metaphysischen Sinn,42
”ich kann die43
Ursache immer nur
vermuten“, so sagt das eigentlich:44
ich will im Falle der Ursache immer nur von ”vermuten“
und nicht von ”wissen“ sprechen & so Fälle verschiedener Grammatik voneinander unter-
scheiden.45
(Das ist also so, als sagte ich:46
ich will in einer Gleichung das Zeichen ”=“ und
nicht das Wort ”ist“ gebrauchen.) Was an unserem Satz irreführend ist, ist das &nur“,47
aber freilich
gehört das eben ganz zu dem Gleichnis, das schon im Gebrauch des Wortes ”können“ liegt.
48
Nach den Gründen zu einer Annahme gefragt, besinnt man sich auf diese Gründe.
Geschieht hier dasselbe, wie, wenn man (darüber) nachdenkt, was die Ursachen eines
Ereignisses gewesen sein mögen?49
50
”Diese Gegend macht mich melancholisch.“ Woher weißt Du, daß es die Gegend ist?
Ist das eine Hypothese – wie Du auch nur glaubst, daß es jene Speise war, die die
Magenschmerzen verursachte, oder gehört es zur unmittelbaren Erfahrung. Wäre es also
widerlegt, wenn Du, in eine andere Gegend versetzt, melancholisch bliebest; oder ist es nicht
durch eine künftige Erfahrung zu widerlegen, da es die Beschreibung der gegenwärtigen ist?
Ja, wie bist Du auf den Gedanken gekommen, daß es die Gegend ist, die diese Stimmung
hervorruft? Oder handelt es sich eben gar nicht um einen durch sie hervorgerufenen
Zustand meiner Person, sondern, etwa, darum, daß das Bild der Gegend melancholisch ist?
(Dies hängt unmittelbar zusammen mit dem Problem: Motiv und Ursache.)
51
”Das ist ein furchtbarer Anblick“. – Das kannst Du nicht wissen. Vielleicht hättest Du
auch sonst gezittert.
34 (V): etwa so, wie
35 (O): gekommen bist und der Frage ”auf
welchem Wege
36 (V): der, zwischen der Frage: ”was ist die
physikalische Ursache davon, daß Du da bist“
und der Frage: ”auf welchem Wege bist Du
hergekommen“.
37 (M): 3
38 (O): sein. Er
39 (V): muß ein Satz der Grammatik sein. Es ist
nicht gemeint,
40 (V): wir @mit dem besten Willen!
41 (V): gelungen ist, wo
42 (V): aber, als metaphysischen Satz,
43 (V): eine
44 (V): so heißt das:
45 (V): sprechen, um so Fälle verschiedener
Grammatik voneinander (zu) unterscheiden.
46 (V): so, wie wenn ich sage:
47 (V): Was also an unserem ersten Beispiel falsch
ist, ist das Wort ”nur“,
48 (M): 3
49 (V): wie, wenn man über die Ursachen eines
Ereignisses nachdenkt?
50 (M): /// – Ursache.)
51 (M): 3
403
404
297 Grund, Motiv, Ursache.
WBTC078-85 30/05/2005 16:16 Page 594
(asking) for the reason for a belief is rather like the one between the questions:27
“What is
the physical cause for your having got from A to B28
?” and29
“Which path led you from A to
B30
?”. – And here one sees very clearly how specifying a cause can also be understood as
specifying a path, but in an entirely different sense.
31
“One can only surmise the cause of a phenomenon” (but not know it). – That is a
statement that refers to grammar. It doesn’t say32
that even with the firmest of intentions33
we
can’t know the cause. In this respect, the proposition is similar to this: “No matter how far
we count, we can’t get to an end of the numerical progression”. And that means: There can
be no talk of an “end to the numerical progression”; and this is – misleadingly – dressed up
in the simile of someone who can’t get to the end of the road because it’s so long. – So there
is a sense in which I can say: “I can only surmise the cause of this phenomenon”, i.e. I haven’t
yet succeeded in “ascertaining” it (in the usual sense). That is in contrast to the case in which
I have succeeded, in which34
I therefore know the cause. – But if I say, in a metaphysical
sense,35
“I can never do more than surmise the36
cause”, that really means:37
When it comes
to causes I want to talk only of “surmising” and not of “knowing”, in order to distinguish
cases with different grammars from each other. (As if I were to say:38
I want to use the sign
“=” in an equation and not the word “is”.) What is misleading about our proposition is “only”39
,
but of course that’s an integral part of the simile that’s already inherent in the use of the
word “can”.
40
If we’re asked about the reasons for an assumption, we give some thought to these reasons.
Does the same thing happen here as when we think (about) what the causes of an event might
have been?41
42
“This part of the country makes me melancholy.” How do you know that it is the region
that is doing that? Is that a hypothesis – just as you only believe that it was that food that
caused your stomach-ache, or is it a part of immediate experience? So would it be disproved
if you remained melancholy after moving to a different region; or is it not disprovable by a
future experience because it is the description of a present one?
How did you arrive at the thought that it is the region that evokes this mood? Or is it
simply not a matter of one’s mental state being brought about by the region but, say, of
the fact that a picture of the region is melancholy? (This is immediately connected with the
problem: motive and cause.)
43
“That’s a terrible sight.” – You can’t know that. Maybe you would have trembled
anyway.
27 (V): question:
28 (V): your being here
29 (V): and the question
30 (V): “Which path did you take to get here
31 (M): 3
32 (V): That is a proposition of grammar. It doesn’t
mean
33 (V): with !the firmest of intentionsE
34 (V): succeeded, where
35 (V): say, as a metaphysical proposition,
36 (V): a
37 (V): cause”, then that means:
38 (V): As if I say:
39 (V): So what is wrong with our first example is
the word “only”
40 (M): 3
41 (V): about the causes of an event?
42 (M): /// – and cause.)
43 (M): 3
Reason, Motive, Cause. 297e
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Wie hängt die Furcht mit dem Anblick zusammen? oder mit der furchtbaren Vorstellung?
Oder soll ich etwa sagen: ”sich vor dieser Vorstellung fürchten“ heißt, sie haben und sich
fürchten? Wenn man nun aber mehrere Vorstellungen hat, während man sich fürchtet (mehrere
sieht oder hört), ist da ein Zweifel darüber, was das Furchtbare ist? Oder weiß man es eben
aus früherer Erfahrung, wovor (von allen diesen Sachen) man sich fürchtet? Ich möchte auch
sagen ”das Fürchten ist eine Beschäftigung mit dem Anblick“.
Kann ich sagen; es sei ein sehr komplizierter Vorgang, in welchem die Vorstellung an charakteristischen
Stellen eintritt?
52
Denken wir an ein furchtbares Antlitz. Welche Rolle spielt der Anblick im Vorgang der
Furcht.
53
Ich will sagen: die Furcht begleitet nicht den Anblick. Sondern das Furchtbare und die
Furcht haben die Struktur des Gesichtes. Denken wir, daß wir den Zügen eines Gesichts
mit den Augen in Aufregung folgen. Sie gleichsam zitternd nachfahren. So daß die
Schwingungen der Furcht den Linien des Gesichts superponiert wären.
52 (M): /// 53 (M): 3
298 Grund, Motiv, Ursache.
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How is fear connected to what one sees? Or to a terrifying mental image? Or should I
perhaps say: “to be afraid of this mental image” means having it and being afraid? But now
what if you have (see or hear) several mental images while you’re afraid? Is there any doubt
about what it is that’s terrifying? Or does one simply know from previous experience what
(among all of these things) one is afraid of? I’m also inclined to say “Being afraid is a way
of occupying yourself with a sight”.
Can I say: It’s a very complicated process, into which a mental image enters at characteristic
points?
44
Let’s think about a terrifying face. What role does that sight play in the process of fear?
45
I want to say: Fear doesn’t accompany the sight. Rather, what is frightening as well as
fear have the structure of the face. Let’s imagine that we’re alarmed, and we’re following
the features of a face with our eyes. It’s as if we’re tracing them with a trembling hand. So
that the tremors of our fear are superimposed on the lines of the face.
44 (M): /// 45 (M): 3
Reason, Motive, Cause. 298e
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Philosophie.405
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Philosophy.
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86
Schwierigkeit der Philosophie, nicht
die intellektuelle Schwierigkeit der
Wissenschaften, sondern die
Schwierigkeit einer Umstellung.1
Widerstände des Willens sind zu
überwinden.
Wie ich oft gesagt habe, führt die Philosophie mich zu keinem Verzicht, da ich mich nicht
entbreche, etwas zu sagen, sondern eine gewisse Wortverbindung als sinnlos aufgebe. In
anderem Sinne aber erfordert die Philosophie dann eine Resignation, aber des Gefühls, nicht
des Verstandes. Und das ist es vielleicht, was sie Vielen so schwer macht. Es kann schwer
sein, einen Ausdruck nicht zu gebrauchen, wie es schwer ist, die Tränen zurückzuhalten,
oder einen Ausbruch der Wut.2
|(Tolstoi: die Bedeutung (Bedeutsamkeit) eines Gegenstandes liegt in seiner allgemeinen
Verständlichkeit. – Das ist wahr und falsch. Das, was den Gegenstand schwer verständlich
macht ist – wenn er bedeutend, wichtig, ist – nicht, daß irgendeine besondere Instruktion
über abstruse Dinge zu seinem Verständnis erforderlich wäre, sondern der Gegensatz
zwischen dem Verstehen des Gegenstandes und dem, was die meisten Menschen sehen
wollen. Dadurch kann gerade das Naheliegendste am allerschwersten verständlich werden.
Nicht eine Schwierigkeit des Verstandes, sondern des Willens ist zu überwinden.3
)|
Die Arbeit an der Philosophie ist – wie vielfach die Arbeit in der Architektur – eigentlich
mehr eine4
Arbeit an Einem selbst. An der eignen Auffassung. Daran, wie man die Dinge
sieht. (Und was man von ihnen verlangt.)
1 (E): Vgl. die folgende Bermerkung (MS 153b,
S. 30r):
Difficulty of our investigation: great length of
chain of thoughts. The difficulty is here essential
to the thought not as in the sciences com due
to its novelty. It is a difficulty which I cann’t
remove if I try to make you see the problems.
I cann’t give you a startling solution which
suddenly will remove all your difficulties. I cann’t
find one key which will unlock the door of our
safe. The unlocking must be done in you by
a difficult process of synoptizing certain facts.
2 (V): Ausbruch des Zorns.
3 (E): Vgl. die folgende Bermerkung (MS 158,
S. 34v):
Schopenhauer: ”If you find yourself stumped trying
to convince someone of something and not
being able to getting anywhere, tell yourself that
it’s the will & not the intellect you’re up against.“
4 (V): die
406
407
300
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86
Difficulty of Philosophy not
the Intellectual Difficulty of
the Sciences, but the Difficulty
of a Change of Attitude.1
Resistance of the Will Must
be Overcome.
As I have often said, philosophy does not lead me to any renunciation, since I do not abstain
from saying something, but rather abandon a certain combination of words as senseless.
In another sense, however, philosophy does require a resignation, but one of feeling, not of
intellect. And maybe that is what makes it so difficult for many. It can be difficult not to
use an expression, just as it is difficult to hold back tears, or an outburst of rage.2
|(Tolstoy: the meaning (meaningfulness) of a subject lies in its being generally understandable.
– That is true and false. What makes a subject difficult to understand – if it
is significant, important – is not that it would take some special instruction about abstruse
things to understand it. Rather it is the antithesis between understanding the subject and
what most people want to see. Because of this the very things that are most obvious can
become the most difficult to understand. What has to be overcome is not a difficulty of the
intellect, but of the will.3
)|
As is frequently the case with work in architecture, work on philosophy is actually closer
to working on oneself. On one’s own understanding. On the way one sees things. (And on
what one demands of them.)
1 (E): Cf. this remark (from MS 153b, p. 30r),
which Wittgenstein wrote in English:
Difficulty of our investigation: great length of
chain of thoughts. The difficulty is here essential
to the thought not as in the sciences com due
to its novelty. It is a difficulty which I cann’t
remove if I try to make you see the problems.
I cann’t give you a startling solution which
suddenly will remove all your difficulties. I cann’t
find one key which will unlock the door of our
safe. The unlocking must be done in you by
a difficult process of synoptizing certain facts.
2 (V): anger.
3 (E): Cf. this remark (from MS 158, p. 34v), which
Wittgenstein wrote in English:
Schopenhauer: “If you find yourself stumped
trying to convince someone of something and
not being able to getting anywhere, tell yourself
that it’s the will & not the intellect you’re up
against.”
300e
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Beiläufig gesprochen, hat es nach5
der alten Auffassung – etwa der, der (großen) westlichen
Philosophen – zweierlei6
Arten von Problemen im wissenschaftlichen Sinne gegeben:
wesentliche, große, universelle, und unwesentliche, quasi accidentelle Probleme. Und
dagegen ist unsere Auffassung, daß es kein großes, wesentliches Problem im Sinne der
Wissenschaft gibt.
5 (V): in 6 (V): zwei
301 Schwierigkeit der Philosophie . . .
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Roughly speaking, according to the4
old conception – for instance that of the (great) western
philosophers – there have been two kinds of5
intellectual problems: the essential, great,
universal ones, and the non-essential, quasi-accidental problems. We, on the other hand, hold
that there is no such thing as a great, essential problem in the intellectual sense.
4 (V): speaking, in the 5 (V): been two
Difficulty of Philosophy not the Intellectual Difficulty of the Sciences . . . 301e
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87
Die Philosophie zeigt die
irreführenden Analogien im
Gebrauch unsrer Sprache auf.
Ist die Grammatik, wie ich das Wort gebrauche, nur die Beschreibung der tatsächlichen
Handhabung der Sprache?1
So daß ihre Sätze eigentlich wie Sätze einer Naturwissenschaft
aufgefaßt werden könnten?
Das könnte man die descriptive Wissenschaft vom Sprechen nennen, im Gegensatz zu
der vom Denken.
Es könnten ja auch die Regeln des Schachspiels als Sätze aus der Naturgeschichte des
Menschen aufgefaßt werden. (Wie die Spiele der Tiere in naturgeschichtlichen Büchern
beschrieben werden.)
Wenn ich einen philosophischen Fehler rektifiziere und sage, man hat sich das immer so
vorgestellt, aber so ist es nicht, so muß ich immer eine Analogie aufzeigen, nach der man
gedacht hat, die man aber nicht als Analogie erkannt hat.2
Die Wirkung einer in die Sprache aufgenommenen falschen Analogie: Sie bedeutet einen
ständigen Kampf und Beunruhigung (quasi einen ständigen Reiz). Es ist, wie wenn ein Ding
aus der Entfernung ein Mensch zu sein scheint, weil wir dann Gewisses nicht wahrnehmen,
und in der Nähe sehen wir, daß es ein Baumstumpf ist. Kaum entfernen wir uns ein wenig
und verlieren die Erklärungen aus dem Auge, so erscheint uns eine Gestalt; sehen wir daraufhin3
näher zu, so sehen wir eine andere; nun entfernen wir uns wieder, etc. etc.
(Der aufregende Charakter der grammatischen Unklarheit.)
Philosophieren ist: falsche Argumente zurückweisen.
4
Der Philosoph trachtet, das erlösende Wort zu finden, das ist das Wort, das uns endlich
erlaubt, das zu fassen, was bis dahin5
immer, ungreifbar, unser Bewußtsein belastet hat.
(Es ist, wie wenn man ein Haar auf der Zunge liegen hat; man spürt es, aber kann es nicht
ergreifen6
und darum nicht loswerden.)
Der Philosoph liefert uns das Wort, womit man7
die Sache ausdrücken und unschädlich
machen kann.
1 (V): Sprachen?
2 (V): so zeige ich immer auf eine Analogie //
so muß ich immer . . . zeigen //, nach der man
sich gerichtet hat, und, daß diese Analogie
nicht stimmt.
3 (O): darauf-hin
4 (M): \
5 (V): jetzt
6 (V): erfassen
7 (V): ich
408
409
409
302
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87
Philosophy Points out the
Misleading Analogies in the Use
of our Language.
Is grammar, as I use the word, nothing but the description of the actual use of a language?1
So that its propositions could really be understood like the propositions of a natural science?
That could be called the descriptive science of speaking, in contrast to that of thinking.
Indeed, the rules of chess too could be taken as propositions that belong in the natural
history of man. (As the games animals play are described in books on natural history.)
If I rectify a philosophical mistake and say that this is the way it has always been conceived,
but this is not the way it is, I must always point out an analogy according to which
one had been thinking, but which one did not recognize as an analogy.2
The effect of a false analogy accepted into language: it means a constant battle and
uneasiness (a constant irritant, as it were). It is as if something seems to be a human being
from afar, because at that distance we don’t perceive certain things, but from close up we
see that it is a tree stump. The moment we move away a little and lose sight of the explanations,
one figure appears to us; if after that we look more closely, we see a different figure;
now we move away again, etc., etc.
(The upsetting character of grammatical unclarity.)
Philosophizing is: rejecting false arguments.
3
The philosopher strives to find the liberating word, and that is the word that finally
permits us to grasp what until then4
had constantly and intangibly weighed on our
consciousness.
(It’s like having a hair on one’s tongue; one feels it, but can’t get hold of5
it, and therefore
can’t get rid of it.)
The philosopher provides us with the word with which we6
can express the matter and
render it harmless.
1 (V): of languages?
2 (V): it is, I always point out an analogy // have
to point out an analogy // that one has taken as
a guideline and show // have to show // that this
analogy is incorrect.
3 (M): \
4 (V): now
5 (V): can’t grasp
6 (V): I
302e
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(Die Wahl unserer Worte ist so wichtig, weil es gilt, die Physiognomie der Sache genau
zu treffen, weil nur der genau gerichtete Gedanke auf die richtige Bahn führen kann. Der
Wagen muß haargenau auf die Schiene gesetzt werden, damit er richtig weiterrollen kann.)
8
Eine der wichtigsten Aufgaben ist es, alle falschen Gedankengänge so charakteristisch
auszudrücken, daß der Leser sagt ”ja, genau so habe ich es gemeint“. Die Physiognomie jedes
Irrtums nachzuzeichnen.
9
Wir können ja auch nur dann den Andern eines Fehlers überführen, wenn er diesen
Ausdruck (wirklich) als den richtigen Ausdruck seines Gefühls anerkennt.10
11
Nämlich, nur wenn er ihn als solchen anerkennt, ist er der richtige Ausdruck.
(Psychoanalyse.)
12
Was der Andre anerkennt, ist die Analogie die ich ihm darbiete, als Quelle seines
Gedankens.
8 (M): \
9 (M): \
10 (V): wenn er anerkennt, daß dies wirklich der
Ausdruck seines Gefühls ist.
11 (M): \
12 (M): \
410
303 Die Philosophie zeigt . . .
WBTC086-95 30/05/2005 16:15 Page 606
(The choice of our words is so important, because the point is to hit the physiognomy of
the matter exactly; because only the thought that is precisely targeted can lead the right way.
The railway carriage must be placed on the tracks exactly, so that it can keep on rolling as
it is supposed to.)
7
One of the most important tasks is to express all false thought processes so true to
character that the reader says, “Yes, that’s exactly the way I meant it”. To make a tracing
of the physiognomy of every error.
8
Indeed, we can only prove that someone made a mistake if he (really) acknowledges this
expression as the correct expression of his feeling.9
10
For only if he acknowledges it as such, is it the correct expression. (Psychoanalysis.)
11
What the other person acknowledges is the analogy I’m presenting to him as the source
of his thought.
7 (M): \
8 (M): \
9 (V): if he acknowledges that this really is the
expression of his feeling.
10 (M): \
11 (M): \
Philosophy Points out the Misleading Analogies . . . 303e
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88
Woher das Gefühl des
Fundamentalen unserer
grammatischen Untersuchungen?
1
(Es beschäftigen uns Fragen verschiedener Art, etwa ”wie groß ist das spezifische
Gewicht dieses Körpers“, ”wird es heute schön bleiben“, ”wer wird als nächster zur Tür
hereinkommen“, etc. Aber unter unseren Fragen finden sich solche von besonderer Art. Wir
haben hier ein anderes Erlebnis. Die Fragen scheinen fundamentaler zu sein als die anderen.
Und nun sage ich; wenn wir dieses Erlebnis haben, dann sind wir an der Grenze der Sprache
angelangt.)
2
Woher nimmt die Betrachtung ihre Wichtigkeit, da sie doch nur alles Interessante, d.h.
alles Große und Wichtige, zu zerstören scheint? (Gleichsam alle Bauwerke; indem sie nur
Steinbrocken und Schutt übrig läßt.)
3
Woher nimmt die Betrachtung ihre Wichtigkeit,4
die uns darauf aufmerksam macht, daß
man eine Tabelle auf mehr als eine Weise brauchen kann, daß man sich eine Tabelle als
Anleitung zum Gebrauch einer Tabelle ausdenken kann, daß man einen Pfeil auch als Zeiger
der Richtung von der Spitze zum Schwanzende auffassen kann, daß ich eine Vorlage auf
mancherlei Weise als Vorlage benützen kann?
Wir führen die Wörter von ihrer metaphysischen, wieder auf ihre normale5
Verwendung
in der Sprache zurück.
(Der Mann, der sagte, man könne nicht zweimal in den gleichen Fluß steigen, sagte etwas
Falsches; man kann zweimal in den gleichen Fluß steigen.)
Und so sieht die Lösung aller philosophischen Schwierigkeiten aus. Unsere6
Antworten
müssen, wenn sie richtig sind, gewöhnliche & triviale sein.7
Aber man muß sie im richtigen
Geist anschauen, dann macht das nichts.
[&Schlichter Unsinn“]
Woher nehmen8
die alten philosophischen Probleme ihre Bedeutung?
9
Der Satz der Identität z.B. schien eine fundamentale Bedeutung zu haben. Aber der Satz,
daß dieser ”Satz“ ein Unsinn ist, hat diese Bedeutung übernommen.
1 (M): \ (R): Gehört zu &müssen“, &können“
2 (M): \
3 (M): \
4 (V): Wichtigkeit:
5 (V): richtige
6 (V): Ihre
7 (V): sind, hausbacken und gewöhnlich sein.
8 (V): nahmen
9 (M): \
411
412
304
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88
Whence the Feeling that
our Grammatical Investigations
are Fundamental?
1
(Questions of different kinds occupy us. For instance, “What is the specific weight of
this body?”, “Will the weather stay nice today?”, “Who will come through the door next?”,
etc. But among our questions there are those of a special kind. Here we have a different
experience. These questions seem to be more fundamental than the others. And now I say:
When we have this experience, we have arrived at the limits of language.)
2
Where does our investigation get its importance from, since it seems only to destroy
everything interesting, that is, all that is great and important? (All the buildings, as it were,
leaving behind only bits of stone and rubble.)
3
Where does this observation get its importance from – the one4
that calls our attention
to the fact that a table can be used in more than one way, that we can think up a table that
instructs us how to use a table, that one can also understand an arrow as indicating the
direction from tip to tail, that I can use a model as a model in several different ways?
We’re bringing words back from their metaphysical to their normal5
use in language.
(The man who said that one cannot step into the same river twice was wrong; one can
step into the same river twice.)
And this is what the solution of all philosophical difficulties looks like. Our answers6
, if
they are correct, must be ordinary and trivial.7
But one must look at them in the proper spirit,
and then that doesn’t matter.
[“Plain nonsense”]
Where do8
the old philosophical problems get their importance from?
9
The Law of Identity, for example, seemed to be of fundamental importance. But now
the proposition that this “Law” is nonsense has taken over this importance.
1 (M): \ (R): Belongs to “must”, “can”
2 (M): \
3 (M): \
4 (V): from: the one
5 (V): correct
6 (V): Their answers
7 (V): be homespun and ordinary.
8 (V): did
9 (M): \
304e
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10
Ich könnte fragen: Warum empfinde ich einen grammatischen Witz in gewissem Sinne
als tief? (Und das ist natürlich die philosophische Tiefe.)
11
Warum empfinden wir die Untersuchung der Grammatik als fundamental?
12
(Das Wort ”fundamental“ kann auch nichts metalogisches, oder philosophisches
bedeuten, wo es überhaupt eine Bedeutung hat.)
13
Die Untersuchung der Grammatik ist im selben Sinne fundamental, wie wir die
Sprache fundamental – etwa ihr eigenes Fundament – nennen können.
14
Unsere grammatische Untersuchung unterscheidet sich ja von der eines Philologen
etc.: uns interessiert z.B. die Übersetzung von einer Sprache in andre, von uns erfundene
Sprachen. Überhaupt interessieren uns Regeln, die der Philologe gar nicht betrachtet.
Diesen Unterschied können wir also wohl hervorheben.
15
Anderseits wäre es irreführend zu sagen, daß wir das Wesentliche der Grammatik
behandeln (er, das Zufällige).
16
”Aber das ist ja nur eine äußere Unterscheidung“.17
Ich glaube, eine andere gibt es nicht.
18
Eher könnten wir sagen, daß wir doch etwas Anderes Grammatik nennen, als er. Wie
wir eben Wortarten unterscheiden, wo für ihn kein Unterschied (vorhanden) ist.
19
Die Wichtigkeit der Grammatik ist die Wichtigkeit der Sprache.
Man könnte auch ein Wort z.B. ”rot“20
wichtig nennen insofern, als es oft und zu Wichtigem
gebraucht wird, im Gegensatz etwa zu dem Wort ”Pfeifendeckel“. Und die Grammatik des
Wortes ”rot“ ist dann wichtig, weil sie die Bedeutung des Wortes ”rot“ beschreibt.
(Alles, was die Philosophie tun kann ist, Götzen zerstören. Und das heißt, keinen neuen
– etwa in der ”Abwesenheit eines Götzen“ – zu schaffen.)
10 (M): \
11 (M): \
12 (M): \
13 (M): \ |
14 (M): \
15 (M): \
16 (M): \
17 (V): nur ein äußerer Unterschied“.
18 (M): \
19 (M): \ ||
20 (V): @rot!
413
305 das Gefühl des Fundamentalen . . .
WBTC086-95 30/05/2005 16:15 Page 610
10
I could ask: Why do I feel that a grammatical joke is in a certain sense deep? (And of
course what I mean is philosophical depth.)
11
Why do we feel that the investigation of grammar is fundamental?
12
(When it has a meaning at all, the word “fundamental” cannot mean anything metalogical,
or philosophical.)
13
The investigation of grammar is fundamental in the same sense in which we can call
language fundamental – can call it its own foundation, for example.
14
After all, our grammatical investigation differs from that of a philologist, etc.: what
interests us, for instance, is the translation from one language into other languages we have
invented. In general the rules that the philologist totally ignores are the ones that interest
us. Thus we are justified in emphasizing this difference.
15
On the other hand it would be misleading to say that we deal with what is essential about
grammar (and that he deals with what is accidental).
16
“But that is only an external differentiation.”17
I believe there is no other.
18
We could come closer by saying that we are calling something else “grammar” than
he is. Just as we differentiate between kinds of words where for him there is no difference
(present).
19
The importance of grammar is the importance of language.
One could also call a word important – for instance “red” – in so far as it is used
frequently and for important things, in contrast, for instance, to the words “officer’s
orderly”. And then the grammar of the word “red” is important, because it describes the
meaning of the word “red”.
(All that philosophy can do is to destroy idols. And that means not creating a new one –
say in the “absence of an idol”.)
10 (M): \
11 (M): \
12 (M): \
13 (M): \ |
14 (M): \
15 (M): \
16 (M): \
17 (V): difference.”
18 (M): \
19 (M): \ ||
Whence the Feeling that our Grammatical . . . 305e
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89
Methode der Philosophie: die
übersichtliche Darstellung der
grammatischen1
Tatsachen.
Das Ziel: Durchsichtigkeit der
Argumente. Gerechtigkeit.
2
Es hat Einer gehört, daß der Anker eines Schiffes durch eine Dampfmaschine aufgezogen
werde. Er denkt nur an die, welche das Schiff treibt (und nach welcher es Dampfschiff heißt)
und kann sich, was er gehört hat, nicht erklären. (Vielleicht fällt ihm die Schwierigkeit auch
erst später ein.) Nun sagen wir ihm: Nein, es ist nicht diese Dampfmaschine, sondern außer
ihr gibt es noch eine Reihe anderer an Bord und eine von diesen hebt den Anker. – War
sein Problem ein philosophisches? War es ein philosophisches, wenn er von der Existenz
anderer Dampfmaschinen auf dem Schiff gehört hatte und nur daran erinnert werden mußte?
– Ich glaube, seine Unklarheit hat zwei Teile: Was der Erklärende ihm als Tatsache mitteilt,
hätte der Fragende sehr wohl als Möglichkeit sich selber ausdenken können, und seine
Frage in bestimmter Form, statt in der des bloßen Zugeständnisses der Unklarheit vorlegen
können. Diesen Teil des Zweifels hätte er selber beheben können, dagegen konnte ihn
Nachdenken nicht über die Tatsachen belehren. Oder: Die Beunruhigung, die davon
herkommt, daß er die Wahrheit nicht wußte, konnte ihm kein Ordnen seiner Begriffe nehmen.
Die andere Beunruhigung und Unklarheit wird durch die Worte ”hier stimmt mir etwas
nicht“ gekennzeichnet und die Lösung, durch (die Worte): ”Ach so, Du meinst nicht die
Dampfmaschine“ oder – für einen andern Fall – ” . . . Du meinst mit Dampfmaschine nicht
nur Kolbenmaschine“.
3
Die Arbeit des Philosophen ist ein Zusammentragen von Erinnerungen zu einem
bestimmten Zweck.
4
Eine philosophische Frage ist ähnlich der nach der Verfassung einer bestimmten
Gesellschaft. – Und es wäre etwa so, als ob eine Gesellschaft ohne klar geschriebene Regeln
zusammenkäme, aber mit einem Bedürfnis nach5
solchen: ja, auch mit einem Instinkt, durch
welchen sie gewisse Regeln in ihren Zusammenkünften einhalten;6
nur, daß dies dadurch
erschwert wird, daß nichts hierüber klar ausgesprochen ist und keine Einrichtung getroffen,
1 (V): sprachlichen
2 (R): ∀ S. 40/3 ?
3 (M): \
4 (M): \
5 (V): nach einem
6 (V): begut beobachten;
414
415
306
WBTC086-95 30/05/2005 16:15 Page 612
89
The Method of Philosophy: the
Clearly Surveyable Representation
of Grammatical1
Facts.
The Goal: the Transparency of
Arguments. Justice.
2
Someone has heard that a ship’s anchor is hauled up by a steam engine. He thinks only
of the one that powers the ship (and for which it is called a steamship) and cannot explain
to himself what he has heard. (Perhaps the difficulty doesn’t occur to him until later.) Now
we tell him: No, it is not that steam engine; besides it there are a number of other ones on
board, and one of these hoists the anchor. – Was his problem a philosophical one? Was it a
philosophical one if he had heard that there were other steam engines on the ship and only
had to be reminded of it? – I believe his confusion has two parts: what the explainer tells
him as fact the questioner could easily have conceived of as a possibility by himself, and he
could have put his question in a definite form instead of in the form of a simple admission
of confusion. He could have removed this part of his doubt by himself; however, reflection
couldn’t instruct him about the facts. Or: no ordering of his concepts could free him from
the uneasiness that comes from not having known the truth.
The other uneasiness and confusion is characterized by the words “Something’s wrong
here”, and the solution is characterized by (the words): “Oh, you don’t mean that
steam engine” or – in another case – “. . . By ‘steam engine’ you don’t mean just a piston
engine.”
3
The work of the philosopher consists in assembling reminders for a particular purpose.
4
A philosophical question is similar to one about the constitution of a particular society.
– And it’s as if a group of people came together without clearly written rules, but with a
need for them; indeed also with an instinct that caused them to observe certain rules at their
meetings; but this is made difficult by the fact that nothing has been clearly articulated about
1 (V): Linguistic
2 (R): ∀ p. 40/3 ?
3 (M): \
4 (M): \
306e
WBTC086-95 30/05/2005 16:15 Page 613
die die Regeln klar hervortreten läßt.7
So betrachten sie tatsächlich Einen von ihnen
als Präsidenten, aber er sitzt nicht oben an der Tafel, ist durch nichts kenntlich und das
erschwert die Verhandlung. Daher kommen wir und schaffen eine klare Ordnung: Wir
setzen den Präsidenten an einen leicht kenntlichen Platz und seinen Sekretär zu ihm an
ein eigenes Tischchen und die übrigen gleichberechtigten Mitglieder in zwei Reihen zu
beiden Seiten des Tisches etc. etc.
8
Wenn man die Philosophie fragt: ”was ist – z.B. – Substanz?“ so wird um eine Regel gebeten.
Eine allgemeine Regel, die für das Wort ”Substanz“ gilt, d.h.: nach welcher ich zu spielen
entschlossen bin. – Ich will sagen: die Frage ”was ist . . .“ bezieht sich nicht auf einen
besonderen – praktischen – Fall, sondern wir fragen sie von unserem Schreibtisch aus.
Erinnere Dich nur an den Fall des Gesetzes der Identität, um zu sehen, daß es sich bei
der Erledigung einer philosophischen Schwierigkeit nicht um das Aussprechen neuer
Wahrheiten über den Gegenstand der Untersuchung (der Identität) handelt.
Die Schwierigkeit besteht nun9
darin, zu verstehen, was uns die Festsetzung einer Regel
hilft. Warum die uns beruhigt, nachdem wir so tief10
beunruhigt waren. Was uns beruhigt
ist offenbar, daß wir ein System sehen, das diejenigen Gebilde (systematisch) ausschließt,
die uns immer beunruhigt haben, mit denen wir nichts anzufangen wußten und die wir doch
respektieren zu müssen glaubten. Ist die Festsetzung einer solchen grammatischen Regel
in dieser Beziehung nicht wie die Entdeckung einer Erklärung in der Physik? z.B., des
Copernicanischen Systems? Eine Ähnlichkeit ist vorhanden. – Das Seltsame an der
philosophischen Beunruhigung und ihrer Lösung möchte scheinen, daß sie ist, wie die Qual
des Asketen, der, eine schwere Kugel unter Stöhnen stemmend, da stand und den ein Mann
erlöste, indem er ihm sagte: ”lass’ sie fallen“. Man fragt sich: Wenn Dich diese Sätze
beunruhigten,11
Du nichts mit ihnen anzufangen wußtest, warum ließest Du sie nicht schon
früher fallen, was hat Dich daran gehindert? Nun, ich glaube, es war das falsche System,
dem er sich anbequemen zu müssen glaubte, etc.
Henne & Kreidestrich
12
(Die besondere Beruhigung, welche eintritt, wenn wir einem Fall, den wir für einzigartig
hielten, andere ähnliche Fälle an die Seite stellen können, tritt in unseren Untersuchungen
immer wieder ein, wenn wir zeigen, daß ein Wort nicht nur eine Bedeutung (oder, nicht nur
zwei) hat, sondern in fünf oder sechs verschiedenen (Bedeutungen) gebraucht wird.)
13
Die philosophischen Probleme kann man mit den Kassenschlössern vergleichen, die
durch Einstellen eines bestimmten Wortes oder einer bestimmten Zahl geöffnet werden,
sodaß keine Gewalt die Tür öffnen kann, ehe gerade dieses Wort getroffen ist, und ist es
getroffen, jedes Kind sie öffnen kann.14
15
Der Begriff der übersichtlichen Darstellung ist für uns von grundlegender Bedeutung.
Er bezeichnet unsere Darstellungsform, die Art, wie wir die Dinge sehen. (Eine Art der
”Weltanschauung“, wie sie scheinbar für unsere Zeit typisch ist. Spengler.)
7 (V): Regeln deutlich macht.
8 (M): \
9 (V): nur
10 (V): schwer
11 (O): beunruhigen,
12 (M): \
13 (M): \
14 (V): und ist es getroffen, keinerlei Anstrengung
nötig ist, die Tür // sie // zu öffnen.
15 (M): \
416
417
307 Methode der Philosophie . . .
WBTC086-95 30/05/2005 16:15 Page 614
this, and no arrangement has been made which brings the rules out clearly.5
Thus they in
fact view one of their own as president, but he doesn’t sit at the head of the table and has
no distinguishing marks, and that makes negotiations difficult. That is why we come along
and create a clear order: we seat the president at a clearly identifiable spot, seat his secretary
next to him at a little table of his own, and seat the other full members in two rows on
both sides of the table, etc., etc.
6
When one asks philosophy: “What is – for instance – substance?” then one is asking for
a rule. A general rule, which is valid for the word “substance”, i.e. a rule according to which
I have decided to play. – I want to say: The question “What is . . . ?” doesn’t refer to a
particular – practical – case, but is one we ask sitting at our desks. Just remember the case
of the Law of Identity in order to see that taking care of a philosophical problem is not a
matter of pronouncing new truths about the subject of the investigation (identity).
The difficulty lies in7
understanding how establishing a rule helps us. Why it calms us
after we have been so profoundly8
anxious. Obviously what calms us is that we see a system
that (systematically) excludes those structures that have always made us uneasy, those
we were unable to do anything with, and that we still thought we had to respect. Isn’t the
establishment of such a grammatical rule similar in this respect to the discovery of an explanation
in physics – for instance, of the Copernican system? There is a similarity. – The strange
thing about philosophical uneasiness and its resolution might seem to be that it is like the
suffering of an ascetic who stands there lifting a heavy ball above his head, amid groans, and
whom someone sets free by telling him: “Drop it”. One wonders: If these propositions made
you uneasy and you didn’t know what to do with them, why didn’t you drop them earlier?
What stopped you from doing this? Well, I believe it was the false system that he thought
he had to accommodate himself to, etc.
Hen and chalk-line
9
(The particular peace of mind that occurs when we can place other similar cases next to
one we thought was unique, occurs again and again in our investigations when we show that
a word doesn’t have just one meaning (or just two), but is used with five or six different
meanings.)
10
Philosophical problems can be compared to locks on safes, which are opened by dialling
a certain word or number, so that no force can open the door until just this word has been
found, and once it has been found, any child can open it.11
12
The concept of a surveyable representation is of fundamental significance for us. It designates
our form of representation, the way we look at things. (A kind of “Weltanschauung”,
as is apparently typical of our time. Spengler.)
5 (V): made that makes the rules clear.
6 (M): \
7 (V): lies only in
8 (V): very
9 (M): \
10 (M): \
11 (V): found, no effort at all is necessary to open
it. // open the door.
12 (M): \
The Method of Philosophy . . . 307e
WBTC086-95 30/05/2005 16:15 Page 615
16
Diese übersichtliche Darstellung vermittelt das Verständnis,17
welches eben darin
besteht, daß wir die ”Zusammenhänge sehen“. Daher die Wichtigkeit des Findens von
Zwischengliedern.18
Der Satz ist vollkommen logisch analysiert, dessen Grammatik vollkommen klargelegt ist.
Er mag in welcher Ausdrucksweise immer hingeschrieben oder ausgesprochen sein.
Unserer Grammatik fehlt es vor allem an Übersichtlichkeit.
19
Die Philosophie darf, was wirklich gesagt wird20
in keiner Weise antasten, sie kann es21
am Ende also nur beschreiben.
22
Denn sie kann es23
auch nicht begründen.
Sie läßt alles wie es ist.
Sie läßt auch die Mathematik wie sie ist (jetzt ist) und keine mathematische Entdeckung
kann sie weiter bringen.
Ein ”führendes Problem der mathematischen Logik“ (Ramsey) ist ein Problem der
Mathematik wie jedes andere.
24
(Ein Gleichnis gehört zu unserem Gebäude; aber wir können auch aus ihm keine Folgen
ziehen; es führt uns nicht über sich selbst hinaus, sondern muß als Gleichnis stehen bleiben.
Wir können keine Folgerungen daraus ziehen. So, wenn wir den Satz mit einem Bild
vergleichen (wobei ja, was wir unter ”Bild“ verstehen, schon früher25
in uns festliegen
muß) oder, wenn ich die Anwendung der Sprache mit der, etwa, des Multiplikationskalküls
vergleiche.
Die Philosophie stellt eben alles bloß hin und erklärt und folgert nichts.)
26
Da alles offen daliegt, ist auch nichts zu erklären, denn, was etwa verborgen ist,27
interessiert
uns nicht.
Die Antwort auf die Frage nach der Erklärung der Negation ist wirklich: verstehst Du
sie denn nicht? Nun, wenn Du sie verstehst, was gibt es da noch zu erklären, was hat eine
Erklärung da noch zu tun?
Wir müssen wissen, was Erklärung heißt. Es ist die ständige Gefahr, dieses Wort in der
Logik in einem Sinn verwenden zu wollen, der von der Physik hergenommen ist.
28
Methodologie, wenn sie von der Messung redet, sagt nicht, aus welchem Material etwa
wir den Maßstab am Vorteilhaftesten herstellen, um dies und dies Resultat zu erzielen; obwohl
doch das auch zur Methode des Messens gehört. Vielmehr interessiert diese Untersuchung
bloß, unter welchen Umständen wir sagen, eine Länge, eine Stromstärke, (u.s.w.) sei
gemessen. Sie will die, von uns bereits verwendeten, uns geläufigen, Methoden tabulieren,
um dadurch die Bedeutung der Worte ”Länge“, ”Stromstärke“, etc. festzulegen.)
16 (M): \
17 (V): Verstehen,
18 (V): Wichtigkeit der Zwischenglieder.
19 (M): \
20 (V): darf den wirklichen // tatsächlichen //
Gebrauch der Sprache
21 (V): ihn
22 (M): \
23 (O): ihn
24 (M): \
25 (V): vorher
26 (M): \
27 (V): erklären. Denn was etwa nicht offen
daliegt,
28 (M): VII 7
418
419
308 Methode der Philosophie . . .
WBTC086-95 30/05/2005 16:15 Page 616
13
This surveyable representation provides just that kind of understanding that consists
in our “seeing connections”. Hence the importance of finding connecting links.14
That proposition is completely logically analysed whose grammar has been completely
clarified. No matter how it’s expressed in writing or in speech.
Above all, our grammar is lacking in surveyability.
15
Philosophy may in no way infringe upon what is really said16
; in the end it can only
describe it.
17
Neither can it justify it.
It leaves everything as it is.
It also leaves mathematics as it is (is now), and no mathematical discovery can advance it.
A “leading problem of mathematical logic” (Ramsey) is a problem of mathematics like any
other.
18
(A simile is part of our edifice; but we cannot draw any conclusions from it either;
it doesn’t lead us beyond itself, but must remain standing as a simile. We can draw no
inferences from it. As when we compare a proposition to a picture (in which case, what we
understand by “picture” must have been established in us earlier19
), or when I compare the
application of language with that of the calculus of multiplication, for instance.
Philosophy simply sets everything out, and neither explains nor deduces anything.)
20
Since everything lies open to view there is nothing to explain, either. For anything that
might be hidden21
is of no interest to us.
The answer to a request for an explanation of negation is really: Don’t you understand
it? Well, if you understand it, what is there left to explain, what is there left for an explanation
to do?
We must know what “explanation” means. There is a constant danger of wanting to use
this word in logic in a sense that is derived from physics.
22
When methodology talks about measurement, it does not say out of which material, for
instance, it would be the most advantageous to make the measuring stick in order to achieve
this or that result: even though this too is, after all, part of the method of measuring. Rather,
this investigation is only interested in the circumstances under which we say that a length,
the strength of a current (etc.) has been measured. It wants to tabulate the methods we already
use and are familiar with, in order to establish the meaning of the words “length”, “strength
of current”, etc.)
13 (M): \
14 (V): of connecting links.
15 (M): \
16 (V): upon the real // actual // use of language
17 (M): \
18 (M): \
19 (V): in us before
20 (M): \
21 (V): anything that might not be open to view
22 (M): VII 7
The Method of Philosophy . . . 308e
WBTC086-95 30/05/2005 16:15 Page 617
29
Wollte man Thesen in der Philosophie aufstellen, es könnte nie über sie zur Diskussion
kommen, weil Alle mit ihnen einverstanden wären.
30
Das Lernen der Philosophie ist wirklich ein Rückerinnern. Wir erinnern uns, daß wir
die Worte wirklich auf diese Weise gebraucht haben.
31
Die philosophisch wichtigsten Aspekte der Sprache32
sind durch ihre Einfachheit und
Alltäglichkeit verborgen.
(Man kann es nicht bemerken, weil man es immer (offen) vor Augen hat.)
Die eigentlichen Grundlagen seiner Forschung fallen dem Menschen gar nicht auf. Es sei
denn, daß ihm dies einmal zum Bewußtsein gekommen33
ist. (Frazer etc. etc.)
Und das heißt, das Auffallendste (Stärkste) fällt ihm nicht auf.
34
(Eines der größten Hindernisse für die Philosophie ist die Erwartung neuer unerhörter35
Aufschlüsse.)
36
Philosophie könnte man auch das nennen, was vor allen neuen Entdeckungen und
Erfindungen da37
ist.
Das muß sich auch darauf beziehen, daß ich keine Erklärungen der Variablen ”Satz“ geben
kann. Es ist klar, daß dieser logische Begriff, diese Variable, von der Ordnung des Begriffs
”Realität“ oder ”Welt“ sein muß.
38
Wenn Einer die Lösung des ”Problems des Lebens“ gefunden zu haben glaubt, und sich
sagen wollte, jetzt ist alles ganz leicht, so brauchte er sich zu seiner Widerlegung nur erinnern,
daß es eine Zeit gegeben hat, wo diese ”Lösung“ nicht gefunden war; aber auch zu
der Zeit mußte man leben können und im Hinblick auf sie erscheint die gefundene Lösung
als39
ein Zufall. Und so geht es uns in der Logik. Wenn es eine ”Lösung“ der logischen
(philosophischen) Probleme gäbe, so müßten wir uns nur vorhalten, daß sie ja einmal nicht
gelöst waren (und auch da mußte man leben und denken können). –
Alle Überlegungen können viel hausbackener angestellt werden, als ich sie in früherer
Zeit angestellt habe. Und darum brauchen40
in der Philosophie auch keine neuen Wörter
angewendet werden, sondern die alten reichen aus.41
(Unsere Aufgabe ist es nur, gerecht zu sein. D.h., wir haben nur die Ungerechtigkeiten
der Philosophie aufzuzeigen und zu lösen, aber nicht neue Parteien – und
Glaubensbekenntnisse – aufzustellen.)
(Es ist schwer, in der Philosophie nicht zu übertreiben.)
(Der Philosoph übertreibt, schreit gleichsam in seiner Ohnmacht, so lange er den Kern
der Konfusion noch nicht entdeckt hat.)
Das philosophische Problem ist ein Bewußtsein der Unordnung in unsern Begriffen, und
durch Ordnen derselben zu heben.
29 (M): \
30 (M): \ VII 164
31 (M): \
32 (V): Dinge
33 (V): einmal aufgefallen
34 (M): \
35 (V): tiefer
36 (M): \
37 (V): möglich
38 (M): \
39 (V): wie
40 (V): brauchen wir
41 (V): sondern die alten, gewöhnlichen Wörter der
Sprache reichen aus.
420
421
309 Methode der Philosophie . . .
WBTC086-95 30/05/2005 16:15 Page 618
23
If one wanted to establish theses in philosophy, no debate about them could ever arise,
because everyone would be in agreement with them.
24
Learning philosophy is really recollecting. We remember that we really did use words
that way.
25
The aspects of language26
that are philosophically most important are hidden behind their
simplicity and ordinariness.
(One is unable to notice this importance because it is always (openly) before one’s eyes.)
The real foundations of their inquiry don’t strike people at all. Unless, at some point, they
have become aware of that fact.27
(Frazer, etc., etc.)
And this means that they are not struck by what is most striking (powerful).
28
(One of the greatest impediments for philosophy is the expectation of new, unheard of29
elucidations.)
30
One could also give the name “philosophy” to what is present31
before all new discoveries
and inventions.
This must also relate to the fact that I can’t give any explanations of the variable “proposition”.
It is clear that this logical concept, this variable, must belong to the same class as
the concept “reality” or “world”.
32
If someone believes he has found the solution to the “problem of life” and is inclined
to tell himself that now everything is simple, then to refute himself he would only have to
remember that there was a time when this “solution” had not been found; but at that time
too one had to be able to live, and in reference to this time the new solution seems like a
coincidence. And that’s what happens to us in logic. If there were a “solution” to logical
(philosophical) problems then we would only have to call to mind that, after all, at one time
they had not been solved (and then too one had to be able to live and think). –
All reflections can be carried out in a much more homespun way than I used to do. And
therefore no new words have to be used in philosophy – the old ones33
suffice.
(Our only task is to be just. That is, all we have to do is to point out and resolve the injustices
of philosophy; we must not set up new parties – and creeds.)
(It is difficult not to exaggerate in philosophy.)
(The philosopher exaggerates, screams, as it were, in his helplessness, so long as he
hasn’t yet discovered the core of the confusion.)
The philosophical problem is an awareness of the disorder in our concepts, and can be
solved by ordering them.
23 (M): \
24 (M): \ VII 164
25 (M): \
26 (V): of the things
27 (V): they have noticed that.
28 (M): \
29 (V): new, deep
30 (M): \
31 (V): possible
32 (M): \
33 (V): the old, ordinary words of language
The Method of Philosophy . . . 309e
WBTC086-95 30/05/2005 16:15 Page 619
42
Ein philosophisches Problem ist immer von der Form: ”Ich kenne mich einfach nicht
aus“.
Wie ich Philosophie betreibe, ist es ihre ganze Aufgabe, den Ausdruck so zu gestalten,
daß gewisse Beunruhigungen43
verschwinden. ( (Hertz.) )
Wenn ich Recht habe, so müssen sich philosophische Probleme wirklich restlos lösen lassen,
im Gegensatz zu allen andern.
Wenn ich sage: Hier sind wir an der Grenze der Sprache, so klingt44
das immer, als wäre
hier eine Resignation nötig, während im Gegenteil volle Befriedigung eintritt, da keine Frage
übrig bleibt.45
46
Die Probleme werden im eigentlichen Sinne aufgelöst – wie ein Stück Zucker im Wasser.
47
|Die Menschen, welche kein Bedürfnis nach Durchsichtigkeit ihrer Argumentation
haben, sind für die Philosophie verloren.|
42 (M): \
43 (V): Probleme
44 (V): scheint
45 (E) Vgl. MS 149, S. 6: ”(Funny that in
ordinary life we never feel that we have to
resign ourselves to something by using ordinary
language!)“
46 (M): \
47 (M): \
310 Methode der Philosophie . . .
WBTC086-95 30/05/2005 16:15 Page 620
34
A philosophical problem always has the form: “I simply don’t know my way about.”
As I do philosophy, its entire task is to shape expression in such a way that certain
worries35
disappear. ( (Hertz.) )
If I am right, then philosophical problems really must be solvable without remainder, in
contrast to all others.
When I say: Here we are at the limits of language, that always sounds36
as if resignation
were necessary at this point, whereas on the contrary complete satisfaction comes about,
since no question remains.37
38
The problems are solved in the literal sense of the word – dissolved like a lump of sugar
in water.
39
|People who have no need for transparency in their argumentation are lost to
philosophy.|
34 (M): \
35 (V): problems
36 (V): seems
37 (E): Cf. MS 149, p. 6: “(Funny that in ordinary
life we never feel that we have to resign ourselves
to something by using ordinary language!)”
38 (M): \
39 (M): \
The Method of Philosophy . . . 310e
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90
Philosophie.
Die Klärung des Sprachgebrauches.
Fallen der Sprache.
1
Wie kommt es, daß die Philosophie ein so komplizierter Aufbau2
ist. Sie sollte doch
gänzlich einfach sein, wenn sie jenes Letzte, von aller Erfahrung Unabhängige ist, wofür Du
sie ausgibst.3
– Die Philosophie löst die Knoten in unserem Denken auf; daher muß ihr Resultat
einfach sein, ihre Tätigkeit aber so kompliziert wie die Knoten, die sie auflöst.
Lichtenberg: ”Unsere ganze Philosophie ist Berichtigung des Sprachgebrauchs, also, die
Berichtigung einer Philosophie, und zwar der allgemeinsten.“4
(Die Veranlagung zur Philosophie liegt in der Empfänglichkeit,5
von einer Tatsache der
Grammatik einen starken und nachhaltigen Eindruck zu empfangen.) 6
Warum die grammatischen Probleme so hart und anscheinend unausrottbar sind – weil
sie mit den ältesten Denkgewohnheiten, d.h. mit den ältesten Bildern, die in unsere Sprache
selbst geprägt sind, zusammenhängen. ( (Lichtenberg.) )
|Das Lehren der Philosophie hat dieselbe ungeheure Schwierigkeit, welche der
Unterricht in der Geographie hätte, wenn der Schüler eine Menge falsche und falsch
vereinfachte7
Vorstellungen über den Lauf und Zusammenhang der Flüsse8
und Gebirge9
mitbrächte.|
|Die Menschen sind tief in den philosophischen, d.i. grammatischen Konfusionen
eingebettet. Und, sie daraus zu befreien, setzt voraus, daß man sie aus den ungeheuer
mannigfachen Verbindungen herausreißt, in denen sie gefangen sind. Man muß sozusagen
ihre ganze Sprache umgruppieren. – Aber diese Sprache ist ja so geworden,10
weil Menschen
die Neigung hatten – und haben – so zu denken. Darum geht das Herausreißen nur bei denen,
die in einer instinktiven Unbefriedigung mit der11
Sprache leben. Nicht bei denen, die ihrem
ganzen Instinkt nach in der Herde leben, die diese Sprache als ihren eigentlichen Ausdruck
geschaffen hat.|
1 (M): \
2 (V): Bau
3 (O): ausgibt.
4 (E): Georg Christoph Lichtenberg, Sudelbücher
H 146.
5 (V): (Die Fähigkeit zur Philosophie besteht in
der Fähigkeit,
6 (M): \ (R): Zu &Witz“ &Tiefe“
7 (V): falsche und viel zu einfache
8 (V): Flußläufe
9 (V): Gebirgsketten
10 (V): entstanden,
11 (V): instinktiven Auflehnung gegen die
422
423
311
WBTC086-95 30/05/2005 16:15 Page 622
90
Philosophy.
The Clarification of the Use of
Language. Traps of Language.
1
Why is philosophy such a complicated structure?2
After all, it should be completely
simple if it is that ultimate thing, independent of all experience, that you make it out to be.
– Philosophy unravels the knots in our thinking; hence its result must be simple, but its activity
as complicated as the knots it unravels.
Lichtenberg: “Our entire philosophy is correction of the use of language, and therefore
the correction of a philosophy – of the most general philosophy.”3
(A talent for philosophy consists in a receptiveness: in the ability4
to receive a strong and
lasting impression from a grammatical fact.)5
You ask why grammatical problems are so tough and seemingly ineradicable. – Because
they are connected with the oldest thought habits, i.e. with the oldest images that are engraved
into our language itself. ( (Lichtenberg.) )
|Teaching philosophy involves the same immense difficulty as instruction in geography
would have if a pupil brought with him a mass of false and falsely simplified6
ideas about
the courses and connections of rivers and mountains.7
|
|Human beings are deeply imbedded in philosophical, i.e. grammatical, confusions. And
freeing them from these presupposes extricating them from the immensely diverse associations
they are caught up in. One must, as it were, regroup their entire language. – But of
course this language developed8
as it did because human beings had – and have – the tendency
to think in this way. Therefore extricating them only works with those who live in an
instinctive state of dissatisfaction with9
language. Not with those who, following all of their
instincts, live within the very herd that has created this language as its proper expression.|
1 (M): \
2 (V): building?
3 (E): Georg Christoph Lichtenberg, Sudelbücher
H 146.
4 (V): (The ability to do philosophy consists in the
ability
5 (M): \ (R): To “joke” “deep”
6 (V): and much too simple
7 (V): of riverbeds and mountain chains.
8 (V): originated
9 (V): live in an instinctive state of rebellion
against
311e
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Die Sprache hat für Alle die gleichen Fallen bereit; das ungeheure Netz gut gangbarer12
Irrwege. Und so sehen wir also Einen nach dem Andern die gleichen Wege gehen und wissen
schon, wo er jetzt abbiegen wird, wo er geradaus fortgehen wird, ohne die Abzweigung
zu bemerken, etc. etc. Ich sollte also an allen den Stellen, wo falsche Wege abzweigen, Tafeln
aufstellen, die über die gefährlichen Punkte hinweghelfen.
Man hört immer wieder die Bemerkung, daß die Philosophie eigentlich keinen Fortschritt
mache, daß die gleichen philosophischen Probleme, die schon die Griechen beschäftigten,
uns noch beschäftigen. Die das aber sagen, verstehen nicht den Grund, warum es so sein
muß.13
Der ist aber, daß unsere Sprache sich gleich geblieben ist und uns immer wieder zu
denselben Fragen verführt. Solange es ein Verbum ”sein“ geben wird, das zu funktionieren
scheint wie ”essen“ und ”trinken“, solange es Adjektive ”identisch“, ”wahr“, ”falsch“,
”möglich“ geben wird, solange von einem Fluß der Zeit und von einer Ausdehnung des
Raumes die Rede sein wird, u.s.w., u.s.w., solange werden die Menschen immer wieder an
die gleichen rätselhaften Schwierigkeiten stoßen, und auf etwas starren, was keine Erklärung
scheint wegheben zu können.
Und dies befriedigt im Übrigen ein Verlangen nach dem Transcendenten,14
denn, indem
sie die ”Grenze des menschlichen Verstandes“ zu sehen glauben, glauben sie natürlich, über
ihn hinaus sehen zu können.
Ich lese ” . . . philosophers are no nearer to the meaning of ,Reality‘ than Plato got . . .“.
Welche seltsame Sachlage. Wie sonderbar, daß Plato dann überhaupt so weit kommen
konnte! Oder, daß wir dann nicht weiter kommen konnten! War es, weil Plato so gescheit war?
15
Der Konflikt, in welchem wir uns in logischen Betrachtungen immer wieder befinden,
ist wie der Konflikt zweier Personen, die miteinander einen Vertrag abgeschlossen haben,
dessen letzte Formulierungen in leicht mißdeutbaren Worten niedergelegt sind, wogegen
die Erläuterungen zu diesen Formulierungen alles in unmißverständlicher Weise erklären.
Die eine der beiden Personen nun hat ein kurzes Gedächtnis, vergißt die Erläuterungen
immer wieder, mißdeutet die Bestimmungen des Vertrages und gerät daher16
fortwährend
in Schwierigkeiten. Die andere muß immer von frischem an die Erläuterungen im Vertrag
erinnern und die Schwierigkeit wegräumen.
17
Erinnere Dich daran, wie schwer es Kindern fällt, zu glauben, (oder einzusehen) daß
ein Wort wirklich zwei ganz verschiedene Bedeutungen haben kann.18
19
Das Ziel der Philosophie ist es, eine Mauer dort zu errichten, wo die Sprache ohnehin
aufhört.
20
Die Ergebnisse der Philosophie sind die Entdeckung irgend eines schlichten Unsinns,
und Beulen, die sich der Verstand beim Anrennen an die Grenze21
der Sprache geholt hat.
Sie, die Beulen, lassen uns den Wert jener Entdeckung erkennen.22
23
Welcher Art ist unsere Untersuchung? Untersuche ich die Fälle, die ich als Beispiele
anführe, auf ihre Wahrscheinlichkeit? oder Tatsächlichkeit? Nein, ich führe nur an, was möglich
ist, gebe also grammatische Beispiele.
12 (V): erhaltener (O): ganzbarer
13 (V): so ist.
14 (V): Überirdischen,
15 (M): \
16 (V): und kommt
17 (M): \
18 (V): Bedeutungen hat.
19 (M): \
20 (M): \
21 (V): an das Ende
22 (V): verstehen.
23 (M): \
424
425
312 Philosophie. . . .
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Language has the same traps ready for everyone; the immense network of easily trodden10
false paths. And thus we see one person after another walking down the same paths and we
already know where he will make a turn, where he will keep going straight ahead without
noticing the turn, etc., etc. Therefore wherever false paths branch off I ought to put up signs
to help in getting past the dangerous spots.
One keeps hearing the remark that philosophy really doesn’t make any progress, that the
same philosophical problems that occupied the Greeks keep occupying us. But those who
say that don’t understand the reason it must be so.11
That reason is that our language has
remained constant and keeps seducing us into asking the same questions. So long as there
is a verb “be” that seems to function like “eat” and “drink”, so long as there are the adjectives
“identical”, “true”, “false”, “possible”, so long as there is talk about a flow of time and
an expanse of space, etc., etc., humans will continue to bump up against the same mysterious
difficulties, and stare at something that no explanation seems able to remove.
And this, by the way, satisfies a longing for the transcendental,12
for in believing that they
see the “limit of human understanding” they of course believe that they can see beyond it.
I read “. . . philosophers are no nearer to the meaning of ‘Reality’ than Plato got . . .”.
What a strange state of affairs. How strange in that case that Plato could get that far in the
first place! Or that after him we were not able to get further! Was it because Plato was so
clever?
13
The conflict in which we constantly find ourselves when we undertake logical
investigations is like the conflict between two people who have concluded a contract with
each other, the last formulations of which are set down in easily misunderstood words, whereas
the explanations of these formulations explain everything unambiguously. Now one of
the two people has a short memory, constantly forgets the explanations, misinterprets the
provisions of the contract, and therefore continually runs into difficulties. The other person
has to remind him over and over of the explanations in the contract and remove the difficulty.
14
Remember what a hard time children have believing (or realizing) that a word really can
have15
two completely different meanings.
16
The goal of philosophy is to erect a wall at the point where language ends anyway.
17
The results of philosophy are the discovery of one or another piece of plain nonsense,
and are the bumps that understanding got by running its head up against the limits18
of
language. These bumps allow us to recognize19
the value of that discovery.
20
What is the nature of our investigation? Am I investigating the cases that I give as
examples with a view toward their probability, or their actuality? No, I’m just presenting
what is possible, and am therefore giving grammatical examples.
10 (V): of well-kept
11 (V): it is so.
12 (V): supernatural,
13 (M): \
14 (M): \
15 (V): really has
16 (M): \
17 (M): \
18 (V): end
19 (V): understand
20 (M): \
Philosophy. . . . 312e
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Philosophie wird nicht in Sätzen, sondern in einer Sprache niedergelegt.
Wie Gesetze nur Interesse gewinnen, wenn sie übertreten werden24
so gewinnen gewisse
grammatische Regeln erst dann Interesse, wenn die Philosophen sie übertreten möchten.
Die Wilden haben Spiele (oder wir nennen es doch so), für die sie keine geschriebenen
Regeln, kein Regelverzeichnis besitzen. Denken wir uns nun die Tätigkeit eines Forschers,
die Länder dieser Völker zu bereisen und Regelverzeichnisse für ihre Spiele anzulegen. Das
ist das ganze Analogon zu dem, was der Philosoph tut. ( (Warum sage ich aber nicht: ”Die25
Wilden haben Sprachen (oder wir . . . ), . . . keine geschriebene Grammatik haben . . .“?) )26
24 (V): wenn die Neigung besteht, sie zu
übertreten,
25 (O): nicht: Die
26 (O): haben . . .“?
426
313 Philosophie. . . .
WBTC086-95 30/05/2005 16:15 Page 626
Philosophy is not laid down in propositions, but in a language.
Just as laws only become interesting when they are transgressed,21
certain grammatical rules
only get interesting when philosophers want to transgress them.
Savages have games (that’s what we call them, anyway) for which they have no written
rules, no inventory of rules. Now let’s imagine the activity of an explorer travelling throughout
the countries of these peoples and setting up lists of rules for their games. This is
completely analogous to what the philosopher does. ( (But why don’t I say: “Savages have
languages (that’s what we . . . ) . . . have no written grammar . . .”?) )
21 (V): when there is an inclination to transgress
them,
Philosophy. . . . 313e
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91
Die philosophischen Probleme treten
uns im praktischen Leben gar nicht
entgegen (wie etwa die der
Naturlehre), sondern erst, wenn wir
uns bei der Bildung unserer Sätze
nicht vom praktischen Zweck,
sondern von gewissen Analogien
in der Sprache leiten lassen.
Was zum Wesen der Welt gehört, kann die Sprache nicht ausdrücken. Daher kann sie
nicht sagen, daß Alles fließt. Nur was wir uns auch anders vorstellen könnten, kann die Sprache
sagen.
Daß Alles fließt, muß im Wesen der Berührung der Sprache mit der Wirklichkeit liegen.
Oder besser: daß Alles fließt, muß im Wesen der Sprache liegen. Und, erinnern wir uns:
im gewöhnlichen Leben fällt uns das nicht auf – sowenig, wie die verschwommenen Ränder
unseres Gesichtsfeldes (”weil wir so daran gewöhnt sind“, wird Mancher sagen). Wie, bei
welcher Gelegenheit, glauben wir denn darauf aufmerksam zu werden? Ist es nicht, wenn
wir Sätze gegen die Grammatik der Zeit bilden wollen?
Wenn man sagt, daß ”alles fließt“, so fühlen wir, daß wir gehindert sind, das Eigentliche,
die eigentliche Realität festzuhalten. Der Vorgang auf der Leinwand entschlüpft uns eben,
weil er ein Vorgang ist. Aber wir beschreiben doch etwas; und ist das ein anderer Vorgang?
Die Beschreibung steht doch offenbar gerade mit dem Bild auf der Leinwand in
Zusammenhang. Es muß dem Gefühl unserer Ohnmacht ein falsches Bild zugrunde liegen.
Denn was wir beschreiben wollen können, das können wir beschreiben.
Ist nicht dieses falsche Bild das eines Bilderstreifens, der so geschwind vorbeiläuft, daß
wir keine Zeit haben, ein Bild aufzufassen.
Wir würden nämlich in diesem Fall geneigt sein, dem Bilde nachzulaufen. Aber dazu
gibt es ja im Ablauf eines Vorgangs nichts analoges.
Es ist merkwürdig, daß wir das Gefühl, daß das Phänomen uns entschlüpft, den
ständigen Fluß der Erscheinung, im gewöhnlichen Leben nie spüren, sondern erst, wenn
wir philosophieren. Das deutet darauf hin, daß es sich hier um einen Gedanken handelt, der
uns durch eine falsche Verwendung unserer Sprache suggeriert wird.
427
428
314
WBTC086-95 30/05/2005 16:15 Page 628
91
We Don’t Encounter Philosophical
Problems at all in Practical Life
(as we do, for Example, Those of
Natural Science). We Encounter
them only When we are Guided
not by Practical Purpose in Forming
our Sentences, but by Certain
Analogies within Language.
Language cannot express what belongs to the essence of the world. Therefore it cannot
say that everything is in flux. Language can only say what we could also imagine
differently.
That everything is in flux must be inherent in the contact between language and reality.
Or better: That everything is in flux must be inherent to language. And, let’s remember: in
everyday life we don’t notice that – any more than we notice the blurred edges of our visual
field (“because we are so used to it”, some will say). How, on what occasion, is it that we
start noticing it? Isn’t it when we want to form sentences contrary to the grammar of our
present time?
When it is said that “everything is in flux” we feel that we are hindered in pinning
down the actual – actual reality. The event on the screen escapes us precisely because
it is an event. But we are describing something; and is that a different event? After all,
the description is obviously linked closely to the picture on the screen. There must be
a false image underlying our feeling of powerlessness. For what we can want to describe
we can describe.
Isn’t this false image that of a strip of pictures that runs by so quickly that we don’t have
any time to perceive a single picture?
For in that case we would be inclined to chase after the picture. But of course there is
nothing analogous to that in the course of an event.
It’s remarkable that in everyday life we never have the feeling that the phenomenon is
getting away from us, that we never sense the continual flow of appearance – not until we
philosophize. This points to the fact that here we are dealing with a thought that is suggested
to us by a misuse of our language.
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Das Gefühl ist nämlich, daß die Gegenwart in die Vergangenheit schwindet, ohne daß
wir es hindern können. Und hier bedienen wir uns doch offenbar des Bildes eines Streifens,
der sich unaufhörlich an uns vorbeibewegt und den wir nicht aufhalten können. Aber es ist
natürlich ebenso klar, daß das Bild mißbraucht ist. Daß man nicht sagen kann ”die Zeit fließt“
wenn man mit ”Zeit“ die Möglichkeit der Veränderung meint.
Daß uns nichts auffällt, wenn wir uns umsehen, im Raum herumsehen, unseren eigenen
Körper fühlen etc. etc., das zeigt, wie natürlich uns eben diese Dinge sind. Wir nehmen
nicht wahr, daß wir den Raum perspektivisch sehen oder daß das Gesichtsbild gegen den
Rand zu in irgendeinem Sinne verschwommen ist. Es fällt uns nie auf und kann uns nie
auffallen, weil es die Art der Wahrnehmung ist. Wir denken nie darüber nach, und es ist
unmöglich, weil es zu der Form unserer Welt keinen Gegensatz gibt.
Ich wollte sagen, es ist merkwürdig, daß die, die nur den Dingen, nicht unseren
Vorstellungen, Realität zuschreiben, sich in der Vorstellungswelt so selbstverständlich
bewegen und sich nie aus ihr heraussehnen.
D.h., wie selbstverständlich ist doch das Gegebene. Es müßte mit allen Teufeln zugehen,
wenn das das kleine, aus einem schiefen Winkel aufgenommene Bildchen wäre.
Dieses Selbstverständliche, das Leben, soll etwas Zufälliges, Nebensächliches sein;
dagegen etwas, worüber ich mir normalerweise nie den Kopf zerbreche, das Eigentliche!
D.h., das, worüber hinaus man nicht gehen kann, noch gehen will, wäre nicht die Welt.
Immer wieder ist es der Versuch, die Welt in der Sprache abzugrenzen und
hervorzuheben – was aber nicht geht. Die Selbstverständlichkeit der Welt drückt sich
eben darin aus, daß die Sprache nur sie bedeutet, und nur sie bedeuten kann.
Denn, da die Sprache die Art ihres Bedeutens erst von ihrer Bedeutung, von der Welt,
erhält, so ist keine Sprache denkbar, die nicht diese Welt darstellt.
In den Theorien und Streitigkeiten der Philosophie finden wir die Worte, deren
Bedeutungen uns vom alltäglichen Leben her wohlbekannt sind, in einem ultraphysischen
Sinne angewandt.
1
Wenn die Philosophen ein Wort gebrauchen und nach seiner Bedeutung forschen, muß
man sich immer fragen: wird denn dieses Wort in der Sprache, für die es geschaffen ist,2
je
tatsächlich so gebraucht?
Man wird dann meistens finden, daß es nicht so ist, und das Wort entgegen seiner
normalen3
Grammatik gebraucht wird. (”Wissen“, ”Sein“, ”Ding“.)
4
(Den Philosophen geht es oft wie den kleinen Kindern,5
die zuerst mit ihrem Bleistift beliebige6
Striche auf ein Papier kritzeln und dann7
den Erwachsenen fragen ”was ist das?“ – Das ging
so zu: Der Erwachsene hatte dem Kind öfters etwas vorgezeichnet und gesagt: ”das ist ein
Mann“, ”das ist ein Haus“, u.s.w. Und nun macht das Kind auch Striche und fragt: was ist
nun das?)
1 (M): \
2 (V): Sprache, die es geschaffen hat,
3 (V): Wort gegen seine normale
4 (M): \
5 (V): Die Philosophen sind oft wie kleine Kinder,
6 (V): Bleistift irgend welche
7 (V): nun
429
430
315 Die philosophischen Probleme . . .
WBTC086-95 30/05/2005 16:15 Page 630
For the feeling is that the present fades into the past without our being able to prevent
it. And here we are obviously using the image of a strip that constantly moves past us and
that we can’t stop. But of course it’s just as clear that the image has been misused. That one
cannot say “time flows” if by “time” one means the possibility of change.
That we don’t notice anything when we look around, look around in space, feel our own
bodies, etc., etc., shows how natural these very things are to us. We don’t perceive that we
see space perspectivally, or that our visual image is in some sense blurred towards its edge.
We never notice this, and can never notice it, because it is the mode of perception. We never
think about it, and it is impossible to do so, because there is no opposite to the form of our
world.
I wanted to say that it’s remarkable that those who ascribe reality only to things and
not to our ideas move about so comfortably in the world of ideas and never long to escape
from it.
In other words, how self-evident the given is! Things would have to have gone to blazes
for that to be just a tiny photograph taken from a skewed angle.
The self-evident, life, is supposed to be something accidental, unimportant; on the other
hand something that normally I never puzzle over is supposed to be what is real!
That is to say, what one neither can nor wants to go beyond would not be the world.
Again and again there is the attempt to delimit and to display the world in language – but
that doesn’t work. The self-evidence of the world is expressed in the very fact that language
signifies only it, and can only signify it.
Because language doesn’t have any way of signifying something until it gets it from what
it signifies, from the world, no language is conceivable that doesn’t represent this world.
In the theories and disputes of philosophy we find words whose meanings are well known
to us from everyday life being used in an ultraphysical sense.
1
When philosophers use a word and search for its meaning, one must always ask oneself:
Is this word ever really used this way in the language for which it has been created?2
Usually one will then find that it is not so, and that the word is being used contrary to3
its normal grammar. (“Knowing”, “Being”, “Thing”.)
4
(Philosophers often fare like little children,5
who first scribble random lines6
on a piece of
paper with their pencils, and then7
ask an adult “What is that?” – Here’s how this happened:
the adult had drawn something for the child several times and had said: “That’s a man”,
“That’s a house”, etc. And then the child draws lines too, and asks: “Now what’s that?”)
1 (M): \
2 (V): language that has created it?
3 (V): used against
4 (M): \
5 (V): Philosophers are often like little children,
6 (V): scribble some sort of lines
7 (V): now
We Don’t Encounter Philosophical Problems . . . 315e
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92
Methode in der Philosophie.
Möglichkeit des ruhigen
Fortschreitens.
Die eigentliche Entdeckung ist die, die mich fähig macht, mit dem Philosophieren
aufzuhören, wann ich will.
Die die Philosophie zur Ruhe bringt, so daß sie nicht mehr von Fragen gepeitscht wird,1
die sie selbst in Frage stellen.
Sondern es wird jetzt an Beispielen eine Methode gezeigt, und die Reihe dieser Beispiele
kann abgebrochen werden.2
Richtiger hieße es aber: Es werden Probleme gelöst (Beunruhigungen3
beseitigt), nicht ein
Problem.
Die Unruhe in der Philosophie kommt daher, daß die Philosophen die Philosophie falsch
ansehen, falsch sehen, nämlich gleichsam in (unendliche) Längsstreifen zerlegt, statt in (endliche)
Querstreifen. Diese Umstellung der Auffassung macht die größte Schwierigkeit. Sie wollen
also gleichsam den unendlichen Streifen erfassen, und klagen, daß dies4
nicht Stück für Stück
möglich ist. Freilich nicht, wenn man unter einem Stück einen endlosen Längsstreifen
versteht. Wohl aber, wenn man einen Querstreifen als ganzes, definitives Stück5
sieht. – Aber
dann kommen wir ja mit unserer Arbeit nie zu Ende! Gewiß6
nicht, denn sie hat ja keins.
(Statt der turbulenten Mutmaßungen und Erklärungen wollen wir die ruhige
Feststellung7
sprachlicher Tatsachen.)8
Wir müssen die ganze Sprache durchpflügen.
(Die meisten Menschen, wenn sie eine philosophische Untersuchung anstellen wollen,
machen es wie Einer, der äußerst nervös einen Gegenstand in einer Lade sucht. Er
wirft Papiere aus der Lade heraus – das Gesuchte mag darunter sein – blättert hastig und
ungenau unter den übrigen. Wirft wieder einige in die Lade zurück, bringt sie mit den
andern durcheinander, u.s.w. Man kann ihm dann nur sagen: Halt, wenn Du so suchst,
kann ich Dir nicht suchen helfen. Erst mußt Du anfangen, in vollster Ruhe methodisch
eins nach dem andern zu untersuchen; dann bin ich auch bereit, mit Dir zu suchen und
mich auch in der Methode nach Dir zu richten.)
1 (V): ist,
2 (V): kann man abbrechen.
3 (V): Schwierigkeiten
4 (V): es
5 (V): als Stück
6 (V): Freilich
7 (V): Festsetzung
8 (V): wollen wir ruhige Darlegungen // Erwägung
// Konstatierungen // sprachlicher Tatsachen
geben.) // von sprachlichen Tatsachen geben.)
431
432
316
WBTC086-95 30/05/2005 16:15 Page 632
92
Method in Philosophy.
The Possibility of
Quiet Progress.
The real discovery is the one that makes me able to stop doing philosophy when I want
to.
The one that quiets philosophy down, so that it is no longer lashed by questions that call
itself into question.
Instead, we now demonstrate a method with the help of examples; and the series of these
examples can be broken off.1
But more correctly, one should say: It’s not a single problem that is solved – rather, problems
are solved (worries2
removed).
Unrest in philosophy comes from philosophers looking at, seeing, philosophy all wrong,
namely, as cut up into (infinite) vertical strips, as it were, rather than into (finite) horizontal
strips. This change in understanding creates the greatest difficulty. They want to grasp
the infinite strip, as it were, and they complain that this3
is not possible piece by piece. Of
course it isn’t, if by “a piece” one understands an endless vertical strip. But it is, if one
sees a horizontal strip as a whole, definitive piece.4
– But then we’ll never get finished with
our work! Certainly5
not, because it doesn’t have an end.
(Instead of turbulent conjectures and explanations, we want the calm ascertaining of
linguistic facts.)6
We must plough though the whole of language.
(When most people want to engage in a philosophical investigation, they act like someone
who is quite nervously looking for an object in a drawer. He throws papers out of the
drawer – what he’s looking for may be among them – leafs around among the others hastily
and sloppily. Throws some back into the drawer, mixes them up with the others, and so on.
Then one can only tell him: Stop, if you search in that way, I can’t help you look. First you
have to start by examining one thing after another methodically, and in complete peace; then
I am willing to join you in your search and to follow you in terms of method as well.)
1 (V): and one can break off the series of these
examples.
2 (V): difficulties
3 (V): it
4 (V): as a piece.
5 (V): Of course
6 (V): we want to give a calm presentation //
consideration // statement // of linguistic facts.) //
of linguistic facts.)
316e
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93
Die Mythologie in den Formen
unserer Sprache. ((Paul Ernst.))
In den alten Riten haben wir den Gebrauch einer äußerst ausgebildeten Gebärdensprache.
Und wenn ich in Frazer lese, so möchte ich auf Schritt und Tritt sagen: Alle diese Prozesse,
diese Wandlungen der Bedeutung, haben wir noch in unserer Wortsprache vor uns. Wenn
das, was sich in der letzten Garbe verbirgt, der ”Kornwolf“ genannt wird, aber auch diese
Garbe selbst, und auch der Mann der sie bindet, so erkennen wir hierin einen uns
wohlbekannten sprachlichen Vorgang.
Der Sündenbock, auf den man seine Sünde legt und der damit in die Wüste hinausläuft,
– ein falsches Bild, ähnlich denen, die die philosophischen Irrtümer verursachen.
Ich möchte sagen: nichts zeigt unsere Verwandtschaft mit jenen Wilden besser, als daß
Frazer ein ihm und uns so geläufiges Wort wie ”ghost“ oder ”shade“ bei der Hand hat, um
die Ansichten dieser Leute zu beschreiben.
(Das ist ja doch etwas anderes, als wenn er etwa beschriebe, die Wilden bildeten1
sich ein,
daß ihnen ihr Kopf herunterfällt, wenn sie einen Feind erschlagen haben. Hier hätte unsere
Beschreibung nichts Abergläubisches oder Magisches an sich.)
Ja, diese Sonderbarkeit bezieht sich nicht nur auf die Ausdrücke ”ghost“ und ”shade“,
und es wird viel zu wenig Aufhebens davon gemacht, daß wir das Wort ”Seele“, ”Geist“
(”spirit“) zu unserem eigenen gebildeten Vokabular zählen. Dagegen ist es eine Kleinigkeit,
daß wir nicht glauben, daß unsere Seele ißt und trinkt.
In unserer Sprache ist eine ganze Mythologie niedergelegt.
Austreiben des Todes oder Umbringen des Todes; aber anderseits wird er als Gerippe
dargestellt, also selbst in gewissem Sinne tot. ”As dead as death.“ ”Nichts ist so tot wie der
Tod; nichts so schön wie die Schönheit selbst!“ Das Bild, worunter man sich hier die Realität
denkt ist, daß die Schönheit, der Tod, etc. die reinen (konzentrierten) Substanzen sind,2
während sie in einem schönen Gegenstand als Beimischung vorhanden sind.3
– Und
erkenne ich hier nicht meine eigenen Betrachtungen über ”Gegenstand“ und ”Komplex“?
(Plato.)
Die primitiven Formen unserer Sprache: Substantiv, Eigenschaftswort und Tätigkeitswort
zeigen das einfache Bild, auf dessen Form sie alles zu bringen sucht.
1 (V): bilden
2 (V): die reine (konzentrierte) Substanz ist,
3 (V): ist.
433
434
317
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93
The Mythology in the Forms of
our Language. ((Paul Ernst.))
In ancient rites we find the use of an extremely well-developed language of gestures.
And when I read Frazer, I would like to say every step of the way: All these processes,
these changes of meaning, we still have right in front of us in our word-language. If what
is hidden in the last sheaf, but also the sheaf itself, and also the man who binds it, is called
the “Cornwolf”, then we recognize in this a well-known linguistic process.
The scapegoat, on whom one lays one’s sin, and who runs out into the desert with it –
a false picture, similar to those that cause errors in philosophy.
I would like to say: Nothing shows our kinship with those savages better than that Frazer
has at hand a word like “ghost” or “shade” – which is so familiar to him and to us – to describe
the views of these people.
(Indeed, this is different than if he were to relate, for instance, that the savages imagined1
that their heads fall off when they have slain an enemy. Here there would be nothing superstitious
or magical about our description.)
Indeed, this oddity refers not only to the expressions “ghost” and “shade”, and much
too little is made of it that we include the words “soul” and “spirit” in our own educated
vocabulary. Compared to this it’s a trifle that we do not believe that our soul eats and drinks.
An entire mythology is laid down in our language.
Driving out death or killing death; but on the other hand it is portrayed as a skeleton, and
therefore as dead itself, in a certain sense. “As dead as death.” “Nothing is as dead as death;
nothing as beautiful as beauty itself!” The picture according to which reality is thought
of here is that beauty, death, etc., are the pure (concentrated) substances,2
whereas in a
beautiful object they are3
contained as an admixture. – And don’t I recognize my own
observations here about “object” and “complex”? (Plato.)
The primitive forms of our language – noun, adjective and verb – show the simple
picture to whose form language tries to reduce everything.
1 (V): imagine
2 (V): etc., is the pure (concentrated) substance,
3 (V): object it is
317e
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Solange man sich unter der Seele ein Ding, einen Körper vorstellt, der in unserem Kopfe
ist, solange ist diese Hypothese nicht gefährlich. Nicht in der Unvollkommenheit und Rohheit
unserer Modelle liegt die Gefahr, sondern in ihrer Unklarheit (Undeutlichkeit).
Die Gefahr beginnt, wenn wir merken, daß das alte Modell nicht genügt, es nun aber
nicht ändern, sondern nur gleichsam sublimieren. Solange ich sage, der Gedanke ist in meinem
Kopf, ist alles in Ordnung; gefährlich wird es, wenn wir sagen, der Gedanke ist nicht in
meinem Kopfe, aber in meinem Geist.
435
318 Die Mythologie in den Formen unserer Sprache. . . .
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As long as we imagine the soul as a thing, a body, in our heads, this hypothesis is not
dangerous. The danger doesn’t lie in the imperfection and crudity of our models, but in
their lack of clarity (vagueness).
The danger sets in when we notice that the old model is inadequate, but then we don’t
change it, but only sublimate it, as it were. So long as I say the thought is in my head, everything
is all right; things get dangerous when we say that the thought is not in my head, but
in my mind.
The Mythology in the Forms of our Language. . . . 318e
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Phänomenologie.436
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Phenomenology.
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94
Phänomenologie ist Grammatik.
Die Untersuchung der Regeln des Gebrauchs unserer Sprache, die Erkenntnis dieser
Regeln und übersichtliche Darstellung, läuft auf das hinaus, d.h. leistet dasselbe, was man
oft durch die Konstruktion einer phänomenologischen Sprache erzielen1
will.
Jedesmal, wenn wir erkennen, daß die und die Darstellungsweise auch durch eine andre
ersetzt werden kann, machen wir einen Schritt zu diesem Ziel.
»Angenommen, mein Gesichtsbild wären zwei gleichgroße rote Kreise auf blauem
Grund: was ist hier in zweifacher Zahl vorhanden, und was einmal? (Und was bedeutet
diese Frage überhaupt?) – Man könnte sagen: wir haben hier eine Farbe, aber zwei
Örtlichkeiten. Es wurde aber auch gesagt, rot und kreisförmig seien Eigenschaften von
zwei Gegenständen, die man Flecke nennen könnte, und die in gewissen räumlichen
Beziehungen zueinander stehen.« Die Erklärung ”es sind hier zwei Gegenstände – Flecke –,
die . . .“ klingt wie eine Erklärung der Physik. Wie wenn Einer fragt ”was sind das für
rote Kreise, die ich dort sehe“ und ich antworte ”das sind zwei rote Laternen, etc.“. Eine
Erklärung wird aber hier nicht gefordert (unsere Unbefriedigung durch eine Erklärung
lösen zu wollen ist der Fehler der Metaphysik). Was uns beunruhigt, ist die Unklarheit über
die Grammatik des Satzes ”ich sehe zwei rote Kreise auf blauem Grund“; insbesondere die
Beziehungen zur Grammatik der Sätze2
wie ”auf dem Tisch liegen zwei rote Kugeln“; und
wieder ”auf diesem Bild sehe ich zwei Farben“. Ich darf3
natürlich statt des ersten Satzes
sagen: ”ich sehe zwei Flecken von4
den Eigenschaften Rot und kreisförmig und in der
räumlichen Beziehung Nebeneinander“ – und ebensowohl: ”ich sehe die Farbe rot an zwei
kreisförmigen Örtlichkeiten nebeneinander“ – wenn ich bestimme, daß diese Ausdrücke
das gleiche bedeuten sollen, wie der obige Satz. Es wird sich dann einfach die Grammatik
der Wörter ”Fleck“, ”Örtlichkeit“, ”Farbe“, etc. nach der (Grammatik) der Wörter des
ersten Satzes richten müssen. Die Konfusion entsteht hier dadurch, daß wir glauben,
über das Vorhandensein oder Nichtvorhandensein eines Gegenstands (Dinges) – des Flecks
– entscheiden zu müssen: wie wenn man entscheidet, ob, was ich sehe (im physikalischen
Sinn) ein roter Anstrich oder ein Reflex ist.
Irrtümliche Anwendung unserer physikalischen Ausdrucksweise auf Sinnesdaten.
”Gegenstände“, d.h. Dinge, Körper im Raum des Zimmers – und ”Gegenstände“ im
Gesichtsfeld; der Schatten eines Körpers an der Wand als Gegenstand! Wenn man gefragt
wird: ”existiert der Kasten noch, wenn ich ihn nicht anschaue“, so ist die korrekte
Antwort: ”ich glaube nicht, daß ihn jemand gerade dann wegtragen wird, oder zerstören“.
Die Sprachform ”ich nehme x wahr“ bezieht sich ursprünglich auf ein Phänomen
(als Argument) im physikalischen Raum (ich meine hier: im ”Raum“ der alltäglichen
1 (V): leisten
2 (V): Grammatik eines Satzes
3 (V): kann
4 (V): mit
437
438
320
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94
Phenomenology is Grammar.
The investigation of the rules of the use of our language, the recognition of these rules,
and their clearly surveyable representation amounts to, i.e. accomplishes the same thing as,
what one often wants to achieve1
in constructing a phenomenological language.
Each time we recognize that such and such a mode of representation can be replaced by
another one, we take a step toward that goal.
»Say my visual image were of two red circles of equal size on a blue background: what is
there here in two’s and what once? (And what does this question mean, anyway?) – One
could say: Here we have one colour, but two locations. But it was also said that red and
circular were properties of two objects that one could call spots and that are spatially
related to each other in particular ways.« The explanation “here are two objects – spots, –
that . . .” sounds like an explanation from physics. As when someone asks “What sorts of
red circles are those that I see over there?”, and I answer “Those are two red lanterns,
etc.” But here no explanation is being demanded (wanting to remove our dissatisfaction
with an explanation is the mistake of metaphysics). What disturbs us is the lack of clarity
about the grammar of the sentence “I see two red circles on a blue background” – in particular
its relations to the grammar of sentences2
such as “Two red balls are lying on the
table”, and “I see two colours in this picture”. Of course instead of the first sentence I’m
allowed to3
say: “I see two spots with the properties of red and circular and in the spatial
relationship of being next to each other” – and equally well: “I see the colour red at two circular
locations next to each other” – if I stipulate that these expressions are to mean the same
thing as the sentence above. In that case the grammar of the words “spot”, “location”, “colour”,
etc. will just have to be guided by the grammar of the words of the first sentence. The confusion
arises from our believing that we have to decide about the presence or absence of an
object (a thing) – of the spot; as when I decide whether what I’m seeing (in a physical sense)
is a red coat of paint or a reflection.
Erroneous application of our physical mode of expression to sense data. “Objects”, i.e.
things, bodies in the space of a room – and “objects” in one’s visual field; the shadow of a
body on the wall as an object! If one is asked: “Does the wardrobe still exist when I don’t
look at it?” the correct answer is: “I don’t think that right then somebody is going to carry
it off or destroy it”. The linguistic form “I perceive x” originally refers to a phenomenon
(as an argument) in physical space (here I mean “in space” in the ordinary way of speaking).
Therefore I can’t automatically apply this form to what is called “sense data”, say to
1 (V): accomplish
2 (V): of a sentence
3 (V): sentence I can
320e
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Ausdrucksweise). Ich kann diese Form daher nicht unbedenklich auf das anwenden, was man
Sinnesdatum nennt, etwa auf ein optisches Nachbild. (Vergleiche auch, was wir über die
Identifizierung von Körpern, und anderseits von Farbflecken im Gesichtsfeld gesagt haben.)
Was es heißt: ich, das Subjekt, stehe dem Tisch, als Objekt, gegenüber, kann ich leicht
verstehen; in welchem Sinne aber stehe ich meinem optischen Nachbild des Tisches
gegenüber?
”Ich kann diese Glasscheibe nicht sehen, aber ich kann sie fühlen.“ Kann man sagen: ”ich
kann das Nachbild nicht sehen, aber . . .“?
Vergleiche:
”Ich sehe den Tisch deutlich“;
”Ich sehe das Nachbild deutlich“;
”Ich höre die Musik deutlich“;
”Ich höre das Ohrensausen deutlich“.
Ich sehe den Tisch nicht deutlich, heißt etwa: ich sehe nicht alle Einzelheiten des Tisches;
– was aber heißt es: ”ich sehe nicht alle Einzelheiten des Nachbildes“, oder: ”ich höre nicht
alle Einzelheiten des Ohrenklingens“?
Könnte man nicht sehr wohl statt ”ein Nachbild sehen“ sagen: ”ein Nachbild haben“?
Denn: ein Nachbild ”sehen“? im Gegensatz wozu? –
”Wenn Du mich auf den Kopf schlägst, sehe ich Kreise.“ – ”Sind es genaue Kreise, hast
Du sie gemessen?“ (Oder: ”sind es gewiß Kreise, oder täuscht Dich Dein Augenmaß?“) –
Was heißt es nun, wenn man sagt: ”wir können nie einen genauen Kreis sehen“? Soll das
eine Erfahrungstatsache sein, oder die Konstatierung einer logischen Unmöglichkeit? – Wenn
das letztere, so heißt es also, daß es keinen Sinn hat, vom Sehen eines genauen Kreises zu
reden. Nun, das kommt drauf an, wie man das Wort gebrauchen will. ”Genauer Kreis“ im
Gegensatz zu einem Gesichtsbild, das wir eine sehr kreisähnliche Ellipse5
nennen würden,
kann man doch gewiß sagen. Das Gesichtsbild ist dann ein genauer Kreis,6
welches uns wirklich,
wie wir sagen würden, kreisförmig erscheint und nicht vielleicht nur sehr ähnlich einem
Kreise.7
Ist anderseits von einem Gegenstand der Messung die Rede, so gibt es wieder verschiedene
Bedeutungen des Ausdrucks ”genauer Kreis“, je nach dem Erfahrungskriterium,
das ich für die genaue Kreisförmigkeit des Gegenstandes bestimme.8
Wenn wir nun sagen:9
”keine Messung ist absolut genau“, so erinnern wir hier an einen Zug in der Grammatik der
Angabe von Messungsresultaten. Denn sonst könnte uns Einer sehr wohl antworten: ”wie
weißt Du das, hast Du alle Messungen untersucht?“ – ”Man kann nie einen genauen Kreis
sehen“ kann die Hypothese sein, daß genauere Messung eines kreisförmig aussehenden
Gegenstandes immer zu dem Resultat führen wird, daß der Gegenstand von der Kreisform
abweicht. – Der Satz ”man kann ein 100-Eck nicht von einem Kreis unterscheiden“ hat
nur Sinn, wenn man die beiden auf irgend eine Weise unterscheiden kann, und sagen will,
man könne sie, etwa visuell, nicht unterscheiden. Wäre keine Methode der Unterscheidung
vorgesehen, so hätte es also keinen Sinn, zu sagen, daß diese zwei Figuren (zwar) gleich
aussehen, aber ”tatsächlich“10
verschieden sind. Und jener Satz wäre dann etwa die
Definition 100-Eck = Kreis.
5 (O): Elipse
6 (V): Das Gesichtsbild ist ein genauer Kreis,
7 (V): Kreis.
8 (V): je nach dem Erfahrungskriterium, welches
ich dafür bestimme, daß der Gegenstand genau
kreisförmig ist.
9 (V): Wenn ich nun sage:
10 (V): aber ”in Wirklichkeit“
439
440
321 Phänomenologie ist Grammatik.
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an optical after-image. (Compare also what we said about the identification of physical objects,
and on the other hand, of spots of colours within the visual field.) I can easily understand
what this means: “I, the subject, am standing across from the table, as an object”; but in
what sense am I standing across from my optical after-image of the table?
“I can’t see this pane of glass, but I can feel it.” Can one say: “I can’t see the after-image,
but . . .”?
Compare: “I see the table clearly”;
“I see the after-image clearly”;
“I hear the music clearly”;
“I hear the ringing in my ears clearly”.
“I don’t see the table clearly” might mean: I don’t see all of the details of the table. – But
what does this mean: “I don’t see all of the details of the after-image”? Or this: “I don’t hear
all of the details of the ringing in my ears”?
Instead of “see an after-image” couldn’t we perfectly well say: “have an after-image”? For:
“see” an after-image? In contrast to what? –
“If you hit me on the head, I see circles.” – “Are they true circles, did you measure them?”
(Or: “Are they circles for sure, or is your eye playing tricks on you?”) – Now what does it
mean when we say: “We can never see a true circle”? Is this supposed to be an empirical
fact, or the statement of a logical impossibility? – If it’s the latter, this means that it makes
no sense to talk about seeing a true circle. Well, that depends on how one wants to use the
word. Certainly one can say “true circle” in contrast to a visual image that we would call an
almost circular ellipse. Then that visual4
image is a true circle that really seems circular to
us, as we would say, and not perhaps only quite similar to a circle. If, on the other hand,
we are talking about an object of measurement, then once again there are different meanings
of the expression “true circle”, depending on the empirical criterion that I lay down for
the exact circularity of the object.5
Now if we6
say: “No measurement is absolutely accurate”,
we are calling to mind a grammatical feature of stating the results of measurements.
For otherwise someone could perfectly well answer us: “How do you know this, have you
checked every measurement?” – “One can never see a true circle” can be the hypothesis that
more precisely measuring an object that looks circular will result in the object deviating from
true circularity. – The sentence “One can’t distinguish a 100-sided figure from a circle” only
makes sense if we can distinguish the two in some way, and want to say we can’t distinguish
them visually, for example. So if no method of differentiation were provided for, it would
make no sense to say that these two figures (do) look alike, but “actually”7
are different. And
then that proposition would amount to the definition “100-sided figure = circle”.
4 (V): The visual
5 (V): that I determine for the circle’s being
exactly circular.
6 (V): I
7 (V): but “in reality”
Phenomenology is Grammar. 321e
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Ist in irgendeinem Sinne ein genauer Kreis im Gesichtsfeld undenkbar, dann muß der
Satz ”im Gesichtsfeld ist nie ein genauer Kreis“ von der Art des Satzes sein: ”im
Gesichtsfeld ist nie ein hohes C.“11
Der Farbenraum wird beiläufig dargestellt durch das Oktaeder,12
mit den reinen Farben
an den Eckpunkten und diese Darstellung ist eine grammatische, keine psychologische. Zu
sagen, daß unter den und den Umständen – etwa – ein rotes Nachbild sichtbar wird, ist
dagegen Psychologie (das kann sein, oder auch nicht, das andere ist a priori; das Eine kann
durch Experimente festgestellt werden, das Andere nicht.)
Was Mach ein Gedankenexperiment nennt, ist natürlich gar kein Experiment. Im
Grunde ist es eine grammatische Betrachtung.
Das Farbenoktaeder13
ist Grammatik, denn es sagt, daß wir von einem rötlichen Blau, aber
nicht von einem rötlichen Grün reden können, etc.
Die Oktaeder-Darstellung14
ist eine übersichtliche Darstellung der grammatischen Regeln.
Wenn Einer konstatieren wollte ”der Gesichtsraum ist farbig“, so wären wir versucht, ihm
zu antworten: ”Wir können ihn uns ja gar nicht anders vorstellen (denken)“. Oder: ”Wenn
er nicht färbig wäre, so wäre er in dem Sinne verschieden vom Gesichtsraum, wie ein
Klang von einer Farbe“. Richtiger aber könnte man sagen: er wäre dann eben nicht, was
wir ”Gesichtsraum“ nennen. In der Grammatik wird auch die Anwendung der Sprache
beschrieben; das, was man den Zusammenhang zwischen Sprache und Wirklichkeit nennen
möchte. Wäre er aber nicht beschrieben, so wäre einerseits die Grammatik unvollständig,
anderseits könnte sie aus dem Beschriebenen nicht vervollständigt werden. In dem Sinn, in
welchem wir ihn uns nicht anders denken können, ist die ”Färbigkeit“ in der Definition des
Begriffs ”Gesichtsraum“, d.h. in der Grammatik des Wortes ”Gesichtsraum“, enthalten.
Wenn manchmal gesagt wird: man könne das Helle nicht sehen, wenn man nicht das Dunkle
sähe; so ist das kein Satz der Physik oder Psychologie – denn hier stimmt es nicht und ich
kann sehr wohl eine ganz weiße Fläche sehen und nichts Dunkles daneben – sondern es muß
heißen: In unserer Sprache wird ”hell“ als ein Teil eines Gegensatzpaars hell – dunkel
gebraucht. Wie wenn man sagte: im Schachspiel wird die weiße Farbe von Figuren zur
Unterscheidung von der schwarzen Farbe andrer Figuren gebraucht.
Ist nicht die Harmonielehre wenigstens teilweise Phänomenologie, also Grammatik!
Die Harmonielehre ist nicht Geschmacksache.
Eine Kirchentonart verstehen, heißt nicht, sich an die Tonfolge gewöhnen, in dem Sinne,
in dem ich mich an einen Geruch gewöhnen kann und ihn nach einiger Zeit nicht mehr
unangenehm empfinde. Sondern es heißt, etwas Neues hören, was ich früher noch nicht gehört
habe, etwa in der Art – ja ganz analog – wie es wäre, 10 Striche ||||||||||, die ich früher
nur als 2 mal 5 Striche habe sehen können, plötzlich als ein charakteristisches Ganzes sehen
zu können. Oder die Zeichnung eines Würfels, die ich nur als flaches Ornament habe sehen
können, auf einmal räumlich zu sehen.
11 (V): dann muß der Satz ”ich sehe nie einen
genauen Kreis im Gesichtsfeld“ von der Art
des Satzes sein: ”ich sehe nie ein hohes C
im Gesichtsfeld“.
12 (O): Oktoeder,
13 (O): Farbenoktoeder
14 (O): Oktoeder-Darstellung
441
442
322 Phänomenologie ist Grammatik.
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If a true circle within one’s visual field is in some sense inconceivable, then the sentence
“There never is a true circle within a visual field” must be the same kind of sentence as
“There is never a high C within a visual field”.8
Colour space is roughly represented by an octahedron, the pure colours being at the
corners – and this representation is grammatical, not psychological. On the other hand, to
say that under such and such circumstances – say – a red after-image appears, is psychological
(it may or may not occur, the other is a priori; the one can be ascertained through
experiments, the other can’t).
What Mach calls a thought-experiment is of course not an experiment at all. At bottom
it is a grammatical examination.
The colour-octahedron is grammar because it tells us that we can talk about a reddish
blue, but not about a reddish green, etc.
The representation via the octahedron is a surveyable representation of the grammatical
rules.
If someone were to state: “Our visual space is in colour”, then we’d be tempted to answer:
“But we can’t even imagine (conceive of) it otherwise”. Or: “If it weren’t in colour then it
would differ from our visual space in the sense in which a sound differs from a colour”.
But one could say, more correctly: “Then it simply wouldn’t be what we call ‘visual space’”.
In grammar the application of language is also described – what we would like to call
the connection between language and reality. If it weren’t described then on the one hand
grammar would be incomplete, and on the other it couldn’t be completed from what
was described. In the sense in which we can’t think of it otherwise, “being coloured” is
contained in the definition of the concept “visual space”, i.e. in the grammar of the words
“visual space”.
If at times it’s said that one can’t see what is light if one doesn’t see what is dark, then
that isn’t a proposition of physics or psychology – for in these cases it is incorrect: I can
perfectly well see a totally white surface and nothing dark next to it. – Rather, this must be
phrased: In our language “light” is used as part of the oppositional pair light–dark. As if
one were to say: In chess the white colour of some pieces is used to distinguish them from
the black colour of other pieces.
Isn’t harmony, at least partially, phenomenology, i.e. grammar!
Harmony isn’t a matter of taste.
To understand an ecclesiastical mode doesn’t mean to get used to a sequence of tones in
the sense in which I can get used to an odour and after a while no longer find it unpleasant.
It means, rather, to hear something new, something that I haven’t heard before, say like –
indeed quite analogously to – suddenly being able to see 10 lines | | | | | | | | | | , which
earlier I was only able to see as 2 times 5 lines, as a characteristic whole. Or like suddenly
seeing spatially the drawing of a cube that I had previously been able to see only as a flat
decoration.
8 (V): then the sentence “I never see an exact
circle in my visual field” has to be the same kind
of sentence as: “I never see a high C in my visual
field”.
Phenomenology is Grammar. 322e
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95
Kann man in die Eigenschaften des
Gesichtsraumes tiefer eindringen?
etwa durch Experimente?
Die Tatsache, daß man ein physikalisches Hunderteck als Kreis sieht, es nicht von
einem physikalischen Kreis unterscheiden kann, sagt gar nichts über die Möglichkeit, ein
Hunderteck zu sehen.
Daß es mir nicht gelingt, einen physikalischen Körper zu finden, der das Gesichtsbild
eines Hundertecks gibt, ist nicht von logischer Bedeutung. Es frägt sich: Hat es Sinn von
einem Gesichts-Hunderteck zu reden? Oder: Hat es Sinn, von zugleich gesehenen 30
Strichen nebeneinander zu reden. Ich glaube, nein.
Der Vorgang ist gar nicht so, daß man zuerst ein Dreieck, dann ein Viereck, Fünfeck etc.
bis z.B. zum 50-Eck sieht und dann der Kreis kommt; sondern man sieht ein Dreieck, ein
Viereck etc. bis vielleicht zum Achteck, dann sieht man nur mehr Viel-Ecke mit mehr oder
weniger langen Seiten. Die Seiten werden kleiner, dann beginnt ein fluktuieren zum Kreis
hin und dann kommt der Kreis.
Daß eine physikalische Gerade als Tangente an einen Kreis gezogen das Gesichtsbild
einer geraden Linie gibt, die ein Stück weit mit der gekrümmten zusammenläuft, beweist
auch nicht, daß unser Sehraum nicht euklidisch ist, denn es könnte sehr wohl ein anderes
physikalisches Gebilde das der euklidischen Tangente entsprechende Bild erzeugen.
Tatsächlich aber ist ein solches Bild undenkbar.
Wenn man frägt, ob die Tonleiter eine unendliche Möglichkeit der Fortsetzung in sich
trägt, so ist die Antwort nicht dadurch gegeben, daß man Luftschwingungen, die eine gewisse
Schwingungszahl1
überschreiten, nicht mehr als Töne wahrnimmt, denn es könnte ja die
Möglichkeit bestehen, höhere Tonempfindungen auf andere Art und Weise hervorzurufen.
Die Geometrie unseres Gesichtsraumes ist uns gegeben, d.h., es bedarf keiner
Untersuchung bis jetzt verborgener Tatsachen, um sie zu finden. Die Untersuchung ist
keine, im Sinn einer physikalischen oder psychologischen Untersuchung. Und doch kann
man sagen, wir kennen diese Geometrie noch nicht. Diese Geometrie ist Grammatik und
die Untersuchung eine grammatische Untersuchung.
Man kann sagen, diese Geometrie liegt offen vor uns (wie alles Logische) – im Gegensatz
zur praktischen Geometrie des physikalischen Raumes.
1 (O): Schwingenzahl
443
444
323
WBTC086-95 30/05/2005 16:15 Page 646
95
Can one Penetrate More Deeply
into the Properties of Visual Space?
Say through Experiments?
The fact that one sees a physical hundred-sided figure as a circle, that one can’t distinguish
it from a physical circle, says nothing at all about the possibility of seeing a hundredsided
figure.
That I don’t succeed in finding a physical body that gives me the visual image of a
hundred-sided figure is of no logical importance. The question is: Does it make sense to
talk about a visual hundred-sided figure? Or does it make sense to talk about 30 lines next
to each other as being seen together simultaneously? I believe it doesn’t.
The process is not like this: at first one sees a triangle, then a square, a pentagon, etc., up
to say a fifty-sided figure, and then the circle follows; rather, one sees a triangle, a square,
etc., up to perhaps an octagon, and then one only sees polygons with longer or shorter sides.
The sides keep getting shorter, then a fluctuation around a circle sets in and then the circle
follows.
That a physical straight line, drawn as a tangent to a circle, results in the visual image of
a straight line that coincides for a stretch with the bent one, also doesn’t prove that our visual
space isn’t Euclidean, because a different physical structure could perfectly well produce the
image that corresponds to the Euclidean tangent. In actuality, however, such an image is
inconceivable.
If one asks whether a musical scale contains within it an infinite possibility of being continued,
the answer is not given by saying that one no longer perceives vibrations of the air
that exceed a particular frequency as notes; because the possibility could exist of eliciting
sensations of higher notes in a different way.
The geometry of our visual space is given to us, i.e. finding it doesn’t require an investigation
into hitherto hidden facts. In the sense of a physical or psychological investigation,
ours isn’t one at all. Nevertheless, one can say that we don’t yet know this geometry. This
geometry is grammar, and our investigation is a grammatical investigation.
One can say that this geometry lies before us in full view (as does everything that is logical)
– in contrast to the practical geometry of physical space.
323e
WBTC086-95 30/05/2005 16:15 Page 647
Niemand kann uns unseren2
Gesichtsraum näher kennen lehren. Aber wir können seine
sprachliche Darstellung übersehen lernen. Unterscheide die geometrische Untersuchung von
der Untersuchung der Vorgänge im Gesichtsraum.
Man könnte beinahe von einer externen und einer internen Geometrie reden. Das,
was im Gesichtsraum angeordnet ist, steht in dieser Art von Ordnung a priori, d.h. seiner
logischen Natur nach und die Geometrie ist hier einfach Grammatik. Was der Physiker in
der Geometrie des physikalischen Raumes in Beziehung zu einander setzt, sind Instrumentablesungen,
die ihrer internen Natur nach nicht anders sind, ob wir in einem geraden oder
sphärischen physikalischen Raum leben. D.h., nicht eine Untersuchung der logischen
Eigenschaften dieser Ablesungen führt den Physiker zu einer Annahme über die Art des
physikalischen Raumes, sondern die abgelesenen Tatsachen.
Die Geometrie der Physik hat es in diesem Sinn nicht mit der Möglichkeit, sondern mit
den Tatsachen zu tun. Sie wird von Tatsachen bestätigt; in dem Sinne nämlich, in dem ein
Teil einer Hypothese bestätigt wird.
Vergleich des Arbeitens an der Rechenmaschine mit dem Messen geometrischer Gebilde.
Machen wir bei dieser Messung ein Experiment, oder verhält es sich so, wie im Falle der
Rechenmaschine, daß wir nur interne Relationen feststellen und das physikalische Resultat
unserer Operationen nichts beweist?
Im Gesichtsraum gibt es natürlich kein geometrisches Experiment.
Ich glaube, daß hier der Hauptpunkt des Mißverständnisses über das a priori und a
posteriori der Geometrie liegt.
Jede Hypothese ist eine heuristische Methode. Und in dieser Lage ist, glaube ich, auch
die euklidische oder eine andere Geometrie auf den Raum der physikalischen Messungen
angewandt. Ganz anders verhält es sich mit dem, was man die Geometrie des Gesichtsraumes
nennen kann.
2 (V): den
445
324 die Eigenschaften des Gesichtsraumes . . .
WBTC086-95 30/05/2005 16:15 Page 648
Nobody can teach us to get to know our visual1
space better. But we can learn how to get
an overview of its linguistic representation. Distinguish a geometrical investigation from an
investigation of the processes in visual space.
One could almost talk about an external and internal geometry. What is arranged in
visual space is situated in this kind of order a priori, i.e. by virtue of its logical nature, and
in this case geometry is simply grammar. What the physicist puts in relation to each other
within the geometry of physical space are readings from instruments that, by virtue of their
internal nature, are no different whether we are living in a flat or a spherical physical space.
That is to say, it isn’t an investigation into the logical properties of these readings that leads
the physicist to an assumption about the nature of physical space, but the facts that he has
read off.
In this sense, the geometry of physics is not concerned with possibility, but with facts. It
is confirmed by facts; in the sense, that is, in which part of a hypothesis is confirmed.
Compare working with an adding machine to measuring geometric shapes. Are we conducting
an experiment when we perform such measurements, or are we only ascertaining
internal relationships, as in the case of the adding machine, and the physical result of our
operations doesn’t prove anything?
Of course there is no such thing as a geometric experiment within visual space.
I believe that herein lies the main point of the misunderstanding about the a priori and
the a posteriori in geometry.
Every hypothesis is a heuristic method. And I believe that Euclidean geometry too, or a
different geometry, is in that situation when it is applied to the space of physical measurements.
Things are quite different with what we might call the geometry of visual space.
1 (V): know visual
The Properties of Visual Space . . . 324e
WBTC086-95 30/05/2005 16:15 Page 649
96
Gesichtsraum im Gegensatz
zum euklidischen Raum.
Wenn die Aussage, daß wir nie einen genauen Kreis sehen, bedeuten soll, daß wir z.B.
keine Gerade sehen, die den Kreis in einem Punkt berührt (d.h., daß nichts1
in unserm
Sehraum die Multiplizität der einen Kreis berührenden Geraden hat) dann ist zu dieser
Ungenauigkeit nicht ein beliebig hoher Grad der Genauigkeit denkbar.
Das Wort ”Gleichheit“ hat eine andere Bedeutung, wenn wir es auf Strecken im
Sehraum anwenden, als die, die es, auf den physikalischen Raum angewendet, hat.2
Die
Gleichheit im Sehraum hat eine andere Multiplizität als die Gleichheit im physikalischen
1 (O): nicht
2 (O): als, die es auf den physikalischen Raum
angewendet hat.
3 (F): MS 108, S. 44.
4 (O): Schein nocht nicht mehr
5 (F): MS 108, S. 31.
446
447
Raum, darum können im Sehraum 3
g1 und g2 Gerade
(Sehgerade) sein und die Strecken a1 = a2, a2 = a3, etc.
aber nicht a1 = a4 sein. Ebenso hat der Kreis und die
Gerade im Gesichtsraum eine andere Multiplizität
als Kreis und Gerade im physikalischen Raum, denn ein kurzes Stück eines gesehenen Kreises
kann gerade sein; ”Kreis“ und ”Gerade“ eben im Sinne der Gesichtsgeometrie angewandt.
Die gewöhnliche Sprache hilft sich hier mit dem Wort ”scheint“ oder ”erscheint“. Sie
sagt a1 und a2 scheinen gleich zu sein, während zwischen a1 und a5 dieser Schein nicht
mehr4
besteht. Aber sie benutzt das Wort ”scheint“ zweideutig. Denn seine Bedeutung hängt
davon ab, was diesem Schein nun als das Sein entgegengestellt wird. In einem Fall ist es das
Resultat einer Messung, im anderen eine weitere Erscheinung. In diesen Fällen ist also die
Bedeutung des Wortes ”scheinen“ eine verschiedene.
Wenn ich sage 5
”die obere Strecke ist so lang wie die untere“ und mit
diesem Satz das meine, was sonst der Satz ”die obere Strecke erscheint mir so lang, wie die
untere“ sagt, dann hat in dem Satz das Wort ”gleich“ eine ganz andere Bedeutung, wie im
gleichlautenden Satz, für den die Verifikation die Übertragung der Länge mit dem Zirkel
ist. Darum kann ich z.B. im zweiten Fall von einem Verbessern der Vergleichsmethoden
reden, aber nicht im ersten Falle. Der Gebrauch desselben Wortes ”gleich“ in ganz verschiedenen
Bedeutungen ist sehr verwirrend. Er ist der typische Fall, daß Worte und
Redewendungen, die sich ursprünglich auf die ”Dinge“ der physikalischen Ausdrucksweise,
die ”Körper im Raum“ beziehen, auf die Teile unseres Gesichtsfeldes angewendet werden,
wobei sie ihre Bedeutung gänzlich wechseln müssen und die Aussagen ihren Sinn verlieren,
die früher einen hatten, und andere einen Sinn gewinnen, die in der ersten Ausdrucksart
keinen hatten. Wenn auch eine gewisse Analogie bestehen bleibt, eben die, die uns verführt,
den gleichen Ausdruck zu gebrauchen.
a1
a2
a3
a4
a5
g1
g2
325
WBTC096-100 30/05/2005 16:15 Page 650
96
Visual Space in Contrast
to Euclidean Space.
If the statement that we never see a true circle is supposed to mean that we don’t see a
straight line, for instance, that touches the circle at a point (i.e. that nothing in our visual
space has the multiplicity of a straight line touching a circle), then no degree of accuracy is
conceivable that matches this inaccuracy.
When we apply it to distances within visual space, the word “equality” has a different
meaning from when it is applied to physical space. Equality within visual space has a
different multiplicity from equality in physical space. Therefore within visual space g1
and g2
1
can be straight lines (visually straight lines) and
the lengths a1 = a2, a2 = a3, etc.; but a1 = a4 is not a
possibility. Likewise a circle and a straight line have a
different multiplicity in visual space than they do in
physical space, for a short piece of a visual circle can be
straight – “circle” and “straight line” being used here in the sense of visual geometry.
Here ordinary language helps itself out by using the word “seems” or “appears”. It says
a1 and a2 appear to be equal, whereas this appearance no longer exists between a1 and a5. But
it is using the word “appears” ambiguously. For its meaning depends on what is contrasted
to this appearance as reality. In the one case it is the result of a measurement, in the other
a further appearance. So in these cases the meaning of the word “appear” is
different.
If I say 2
“The upper distance is as long as the lower one”, and mean
by this what is otherwise expressed by the sentence “The upper distance seems to me to be
as long as the lower one”, then the word “equal” has a completely different meaning in this
proposition than in a proposition with the same wording, but whose verification is the transposition
of the length with a divider. Therefore I can speak, for instance, about improving
the methods of comparison in the second case but not in the first. The use of the same word
“equal” with completely different meanings is very confusing. It is typical of the practice
whereby words and phrases that originally refer to “things” in a physical mode of expression,
to “bodies in space”, are applied to parts of our field of vision. In this process they
have to change their meanings completely, and those statements that earlier had a sense lose
it, and others that had no sense in the first mode of expression gain one. Even though a
certain analogy remains – the very one that seduces us into using the same expression.
1 (F): MS 108, p. 44. 2 (F): MS 108, p. 31.
a1
a2
a3
a4
a5
g1
g2
325e
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Die visuelle Gerade berührt den visuellen Kreis nicht in einem Punkt, sondern in einer
visuellen Strecke. – Wenn ich die Zeichnung eines Kreises und einer Tangente ansehe,
so ist6
nicht das merkwürdig, daß7
ich etwa niemals einen vollkommenen Kreis und eine
vollkommene Gerade miteinander in Berührung sehe; interessant wird8
es erst, wenn ich sie
sehe, und dann die Tangente mit dem Kreis ein Stück zusammenläuft.
Die Verschwommenheit, Unbestimmtheit unserer Sinneseindrücke ist nicht etwas,
dem sich abhelfen läßt, eine Verschwommenheit, der auch völlige Schärfe entspricht (oder
entgegensteht). Vielmehr ist diese allgemeine Unbestimmtheit, Ungreifbarkeit, dieses
Schwimmen der Sinneseindrücke, das, was mit dem Worte ”alles fließt“ bezeichnet worden
ist. Wir sagen ”man sieht nie einen genauen Kreis“, und wollen sagen, daß, auch wenn wir
keine Abweichung von der Kreisform sehen, uns das keinen genauen Kreis gibt. (Es ist, als
wollten wir sagen: wir können dieses Werkzeug nie genau führen, denn wir halten nur den
Griff und das Werkzeug sitzt im Griff lose.) Was aber verstehen wir dann unter dem Begriff
”genauer Kreis“? Wie sind wir zu diesem Begriff überhaupt gekommen? Nun, wir denken
z.B. an eine genau gemessene Kreisscheibe aus einem sehr harten Stahl. Aha – also dorthin
zielen wir mit dem Begriff ”genauer Kreis“. Freilich, davon finden wir im Gesichtsbild nichts.
Wir haben eben die Darstellungsform gewählt, die die Stahlscheibe genauer nennt als die
Holzscheibe und die Holzscheibe genauer als die Papierscheibe. Wir haben den Begriff ”genau“
durch eine Reihe bestimmt und reden von den Sinneseindrücken als Bildern, ungenauen
Bildern, der physikalischen Gegenstände.
Zwingt mich etwas zu der Deutung, daß der Baum, den ich durch mein Fenster sehe,
größer ist als das Fenster? Das kommt darauf an, wie ich die Wörter ”größer“ und ”kleiner“
gebrauche. – Denken wir uns die alltägliche9
visuelle Erfährung wäre es für uns, Stäbe
in verschiedenen Lagen zu10
sehen, die durch Teilstriche in (visuell) gleiche Teile geteilt,
wären. Könnte sich da nicht ein doppelter Gebrauch der Worte ”länger“ und ”kürzer“
einbürgern. Wir würden nämlich manchmal den Stab den längeren nennen, der in mehr Teile
geteilt wäre; etc.
Messen einer Länge im Gesichtsfeld durch Anlegen11
eines visuellen Maßstabes. D.i., eines
Stabes, der durch Teilstriche in gleiche Teile geteilt ist. Es gibt hier eine Messung, die darin
besteht, daß der Maßstab an zwei Längen12
angelegt wird. Und zwar können 2 Maßstäbe,
je einer an eine Länge, angelegt13
werden und das Kriterium für die Gleichheit der
Maßeinheit ist, daß die Einheiten gleichlang aussehen. Es kann aber auch ein Maßstab von
einer Länge14
zur andern transportiert werden und das Kriterium der Konstanz der
Maßeinheit ist, daß wir keine Veränderung merken. Während das Kriterium dafür, daß
die gemessenen Längen sich nicht verändern etwa darin besteht, daß wir keine Bewegung
der Endpunkte wahrgenommen haben. Ich kann unzählige verschiedene Bestimmungen
darüber treffen, welches das Kriterium der Längengleichheit im Gesichtsbild sein soll und
danach werden sich wieder verschiedene Bedeutungen der Maßangaben ergeben.
Teilbarkeit. Unendliche Teilbarkeit.
Die unendliche Teilbarkeit der euklidischen Strecke besteht in der Regel (Festsetzung),
daß es Sinn hat, von einem n-ten Teil jedes Teils zu sprechen. Spricht man aber von der
6 (V): wäre
7 (V): wenn
8 (V): ist // wäre
9 (V): normale
10 (V): verschiedenen Längen zu
11 (O): Anlagen
12 (V): Strecken
13 (O): 2 Maßstäbe je einer an eine Länge
angelegt
14 (V): Strecke
448
449
326 Gesichtsraum im Gegensatz zum euklidischen Raum.
WBTC096-100 30/05/2005 16:15 Page 652
The visual straight line doesn’t touch the visual circle at a point, but along a visual span.
– When I look at the drawing of a circle with a tangent it isn’t remarkable that I never see3
a true circle and a true straight line touching each other; things don’t get4
interesting until
I do see them, and then the tangent coincides with the circle for a stretch.
The blurredness, indefiniteness, of our sense impressions is not something that can
be remedied; and it is not a blurredness to which absolute sharpness corresponds (or is
opposed). Rather this general indefiniteness, intangibility, this swimming of sense impressions
is what has been referred to by the expression “Everything is in flux”. We say “One
never sees a true circle”, and we want to say that even if we see no deviation from circularity,
this doesn’t give us a true circle. (It’s as if we wanted to say: We can never guide this
tool precisely, for all we’re holding is its handle, and the tool is loose at the handle.) But
what then do we understand by the concept “true circle”? How did we arrive at this concept,
anyhow? Well, for instance, we think of a precisely measured circular disc of very hard steel.
Aha – so that’s what we’re aiming for with the concept “true circle”. To be sure, we find
none of this in our visual image. We’ve simply chosen the form of representation that calls
the steel disc truer than the wooden disc and the wooden disc truer than the paper disc.
We’ve defined the concept “true” through a series, and we’re talking about sense impressions
as about pictures, inexact pictures of physical objects.
Does something force me into the interpretation that the tree that I see through my
window is bigger than the window? That depends on how I use the words “bigger” and
“smaller”. – Let’s imagine that seeing rods in different positions that were subdivided by
lines of graduation into (visually) equal parts were our everyday5
visual experience. Couldn’t
an ambiguous use of the words “longer” and “shorter” come into being here? For sometimes
we would call that rod longer that was divided into more parts; etc.
Measuring a length within the visual field by applying a visual yardstick. That is, a rod
that is subdivided into equal parts by lines of graduation. Here there is a measurement
that consists in applying the yardstick to two lengths.6
Specifically, 2 yardsticks can be laid
alongside the lengths, each alongside a separate one, and the criterion for the equality of the
measuring unit is that the units appear to be of the same length. But a yardstick can also be
moved from one length7
to another, and the criterion for the constancy of the measuring
unit is that we don’t notice any change. Whereas the criterion for the measured lengths not
changing might consist in our not having observed any movement of the end points. I can
set up countless different stipulations for what is to be the criterion for sameness of length
in a visual image, and accordingly different meanings of the measurements will result.
Divisibility. Infinite divisibility.
The infinite divisibility of a Euclidean section consists in the rule (stipulation) that it makes
sense to speak about an nth
part of each part. But if one is talking about the divisibility of a
3 (V): tangent then wouldn’t it be remarkable if I
never saw
4 (V): things aren’t // wouldn’t be
5 (V): normal
6 (V): stretches.
7 (V): stretch
Visual Space in Contrast to Euclidean Space. 326e
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Teilbarkeit einer Länge im Gesichtsraum und fragt, ob eine solche noch teilbar, oder
endlos teilbar ist, so suchen wir hier nach einer Regel, die einer gewissen Realität entspricht
(aber wie entspricht sie ihr?). Ich sehe einen schwarzen Streifen an der Wand vor mir, – ist
seine Breite teilbar? Was ist das Kriterium dafür? Hier gibt es nun unzählige Kriterien,
die wir alle als Kriterien der Teilbarkeit im Gesichtsfeld bezeichnen15
würden, und die
stufenweise in einander übergehen. Vor allem könnte die Bedeutung von ”Teilbarkeit“ so
festgelegt werden, daß ein Versuch sie erweist; dann ist es also nicht ”logische Möglichkeit“
der Teilung, sondern physische Möglichkeit, und die logische Möglichkeit, die hier in Frage
kommt, ist in der Beschreibung des Versuchs der Teilung gegeben – wie immer dieser Versuch
ausgehn mag.
Was würden wir nun einen ”Versuch der Teilung“ nennen? – Etwa den, einen Strich neben
den ersten zu malen, der gleichbreit aussieht und aus einem grünen und roten Längsstreifen
besteht, wobei die Erinnerung das Kriterium dafür gäbe, daß der schwarze Streifen die
gleiche Breite habe, die er hatte, als wir die Frage stellten. (D.h., daß wir als gleiche Breite
des schwarzen Streifens jetzt und früher das bezeichnen, was als gleichbreit erinnert wird.)
Anderseits könnte ich als Kriterium der Teilbarkeit des schwarzen Streifens festsetzen, daß
zugleich mit ihm ein gleich breit aussehender und geteilter Streifen gesehen wird. Und als
Vollzug der möglichen Teilung würde ich dann die Ersetzung des ungeteilten durch einen
16
geteilten bezeichnen, bei welcher der zuerst gesehene ungeteilte Streifen bestehen bleibt.
17
Ich würde also sagen ”a ist18
teilbar19
“ – weil ich b daneben sehe und ”a ist20
geteilt“,
wenn ich danach 2 Streifen von der Art b sehe. In der Aussage ”a ist geteilt“
bezeichnet ”a“ also einen Ort; das nämlich, was gleichbleibt, ob a geteilt oder ungeteilt
ist. Hier gibt es nun wieder Verschiedenes, was wir als ”Ort im Gesichtsfeld“ und
”Festlegung eines Ortes im Gesichtsfeld“ bezeichnen. – Wir könnten aber einen Streifen
nur dann teilbar nennen, wenn er sich in gleicher (gesehener) Breite in einen geteilten Streifen
fortsetzt, oder aber, wenn es uns 21
gelingt, einen geteilten Streifen zeitweilig
an ihn (im Gesichtsfeld) anzulegen, etc. etc. – Dann aber gibt es das Kriterium der
Vorstellbarkeit der Teilung. Wir sagen: ”oh ja, diesen Streifen kann ich mir noch ganz leicht
geteilt denken“ (oder ”vorstellen“). ”Wenn eine Teilung dieses Streifens a in ungleiche Teile
möglich ist, dann umsomehr in gleiche Teile.“ Und hier22
haben wir wieder die
Festsetzung eines neuen Kriteriums der Teilbarkeit in gleiche Teile. Und hier sagt man: ich
kann mir doch in diesem Fall gewiß denken, daß der Streifen halbiert wird.23
Aber worin
besteht diese Fähigkeit24
des Denkens? Kann ich es, wenn ich es versuche? Und wie, wenn
es mir nicht gelingt? Was hier mit dem ”ich kann mir . . . denken“ gemeint ist, erfährt man,
wenn man fragt ”wieso kannst Du Dir nun die Halbierung denken“. Darauf ist die Antwort:
”ich brauche mir doch nur den schwarzen Teil des Streifens etwas breiter zu denken“; und
es wird offenbar angenommen, daß, das zu denken, keine Schwierigkeit mehr hat. In
Wirklichkeit aber handelt es sich hier nicht um die Schwierigkeit,25
mir26
ein bestimmtes Bild
vor’s innere Auge zu rufen, und nicht um etwas, was ich versuchen und mir mißlingen kann;
sondern um die Anerkennung einer Regel der Ausdrucksweise. Diese Regel kann allerdings
} a
15 (V): anerkennen
16 (M): 59
17 (F): MS 113, S. 129r.
18 (V): sei
19 (O): geteilt (E): Auf Grund von MS 113 (S.
129v) haben wir hier ”geteilt“ durch ”teilbar“
ersetzt.
20 (V): sei
21 (F): MS 113, S. 129v.
22 (F): MS 113, S. 130r.
23 (V): wäre.
24 (V): Möglichkeit
25 (V): um Schwierigkeiten,
26 (V): sich
450
451
327 Gesichtsraum im Gegensatz zum euklidischen Raum.
a b
WBTC096-100 30/05/2005 16:15 Page 654
length within visual space, and asks whether this kind of a length is still divisible or infinitely
divisible, then in this case we’re looking for a rule that corresponds to a particular reality
(but how does it correspond to it?). I see a black strip on the wall in front of me – is its
breadth divisible? What is the criterion for this? Well, there are countless criteria here, all
of which we would designate8
as criteria of divisibility within the visual field, and which blend
into each other step by step. Above all, the meaning of “divisibility” could be stipulated in
such a way that a test would show it; so then it isn’t the “logical possibility” of division, but
a physical possibility; and the logical possibility that is in question here is given in the description
of the attempt at division – however this attempt may come out.
Now what would we call an “attempt at division”? – Perhaps painting another line next
to the first one, one that looks equally wide and consists of a green and red vertical strip,
with memory providing the criterion for the black strip being just as wide as when we asked
the question. (That is, for our designating what is remembered as being of the same width
as being the same width of the black strip, both now and before.) On the other hand I could
set down as a criterion for the divisibility of the black strip that a divided strip, appearing
to be of equal width, is seen simultaneously with it. And then what I would call the
carrying out of the possible division would be the replacement of the undivided strip by a
9
divided one, in the process of which the original undivided strip stays put. 10
Thus
I would say “a is divisible11
” – because I see b next to it, and “a is divided” when
I see the 2 strips in b coming after it. So in the statement “a is divided”, “a” refers
to a location, i.e. what remains the same whether a is divided or undivided. Here once
again there are various things that we call “a location in the visual field” and “establishing
a location in the visual field”. – But we can only call a strip divisible if it continues into a
divided strip of the same (visual) width or if we 12
manage temporarily to lay a
divided strip alongside it (in the visual field), etc., etc. – But then there is the criterion of
the imaginability of division. We say: “Oh yes, I can quite easily think of (or imagine) this
strip as divided.” “If this strip a can be divided into unequal parts, then all the more can it
be divided into equal parts.” And here 13
once again we have the establishment of
a new criterion of divisibility into equal parts. And here one says: Surely I can imagine in
this case that the strip is14
halved. But what does this mental ability15
consist in? Can I do
it if I try? And what if I don’t succeed in doing it? You can find out what is meant here by
“I can imagine . . .” by asking “How is it that you can now imagine the halving?” The answer
to that is: “All I have to do is imagine the black part of the strip as a little wider”; and
obviously it’s assumed that to imagine that is no longer difficult. But in this case it actually
isn’t a question of the difficulty16
of calling up a particular image before my17
mind’s eye,
nor is it a question of something that I can try but fail at; rather, it’s a question of acknowledging
a rule for a mode of expression. To be sure, this rule can be based on the ability to
imagine something; that is to say, in this case a mental image works like a pattern, that is,
like a sign, and 18
of course it can also be replaced by a painted pattern. For if I ask: “What
} a
8 (V): acknowledge
9 (M): 59
10 (F): MS 113, p. 129r.
11 (O): divided (E): TS 213 has “divided”
here, but we have replaced it with “divisible”,
from MS 113, p. 129v.
12 (F): MS 113, p. 129v.
13 (F): MS 113, p. 130r.
14 (V): were
15 (V): this possibility of thinking
16 (V): the difficulties
17 (V): one’s
18 (M): 59
Visual Space in Contrast to Euclidean Space. 327e
a b
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gegründet sein auf der27
Fähigkeit, sich etwas vorzustellen; d.h. die Vorstellung funktioniert
in diesem Fall als Muster, also als Zeichen, und kann 28
natürlich auch ersetzt werden
durch ein gemaltes Muster. Wenn ich nämlich frage: ”was versteht man unter dem
Wachsen der Breite eines Streifens“, so wird mir als Erklärung so etwas vorgeführt, es
wird mir ein Muster gegeben, das ich, oder dessen Erinnerung ich etwa meiner Sprache
einverleibe. Und so kann der, den ich frage ”wieso ist der breite Streifen a teilbar,
weil b teilbar ist“ 29
als Antwort den Streifen b verbreitern und mir vorführen,30
wie
aus b ein geteilter Streifen von der Breite des a werden kann.31
Aber bei dieser
Antwort hätte es nun sein Bewenden. Und was hat er zur Erklärung getan? Er hat
mir ein Zeichen, ein Muster, in mein Zeichensystem gegeben; das ist alles.
Gibt es nun für die Teilbarkeit des Streifens im Gesichtsraum eine Grenze? Nun – das
kann ich festsetzen, wie ich will. – Das heißt: ich kann ein Zeichensystem mit begrenzter
Teilbarkeit, oder eins mit unbegrenzter Teilbarkeit einführen – nur kann ich natürlich
die Tatsachen nicht kommandieren und muß sie dann mit dem von mir festgesetzten
Zeichensystem entsprechend beschreiben. Wenn also meine Vorstellung, bezw. das
Gesichtsbild eines geteilten Streifens, einen Teil meines Zeichensystems bildet, so endet
dieser Teil meines Symbolismus, wo ich, aus irgend welchen Gründen, unfähig32
bin, eine
weitere Verkleinerung der Teile herbeizuführen.33
Dann aber kann ich mich entscheiden:
entweder,34
zu sagen, es gäbe keine weitere Teilung mehr, d.h. von einer solchen zu reden
sei sinnlos – und in diesem Falle habe ich mich gebunden, ein eventuell auftretendes Phänomen,
das ich versucht wäre, eine weitere Teilung zu nennen, anders zu beschreiben; – oder aber:
die35
Teilbarkeit im Symbolismus weitergehen zu lassen, wodurch aber nichts geändert wird,
weil ja meine Reihe von Mustern, die auch zur Sprache gehört, ein Ende hat. Soweit diese
Reihe von Mustern eine Reihe von Zeichen ist, kommt durch jedes neue Muster ein 36
neues
Zeichen in die Sprache. Diese Betrachtung ist meist ohne Wichtigkeit; manchmal aber wird
sie wichtig. Wir haben einen dem Problem der Teilbarkeit analogen Fall,37
wenn gefragt wird:
ist es möglich, jede beliebige Anzahl 3n von Strichen |||||||||||| mit einem Blick als
Gruppe von Trippeln zu erfassen, oder jede beliebig lange Reihe solcher Striche als ein für
ihre Anzahl charakteristisches Bild zu sehen, wie wir es für38
| || ||| |||| können? Auch
hier können wir zur Beschreibung unserer Erfahrung ein endliches oder ein unendliches
Zahlensystem verwenden, – denn die Reihe der Muster übersehbarer Gruppen hat ein Ende
und sie determiniert den Sinn unsrer Sätze ebensosehr, wie das verwendete Zahlensystem.
Wenn ich also sagte ”wir suchen nach einer Regel, die einer gewissen Realität entspricht“,
so liegt die Entsprechung in der Einfachheit und leichten Verständlichkeit der Darstellung.
Die Regel wird durch die Tatsachen nur insofern gerechtfertigt, als die Wahl eines
Koordinatensystems durch ihre Anwendung auf eine Kurve gerechtfertigt wird, die sich in
dem System besonders einfach darstellen läßt.
39
Es ist möglich, im Gesichtsfeld zwei gleichlange (d.h. gleichlang gesehene40
) Strecken
zu sehen, deren jede durch Farbgrenzen in mehrere Teile, gleiche Teile, geteilt ist und beim
Zählen dieser Teile zu finden, daß ihre Anzahlen ungleich sind. Wie ist es nun mit einer
27 (V): die
28 (M): 59
29 (F): MS 113, S. 130v.
30 (V): zeigen,
31 (V): des a wird.
32 (O): Gründen unfähig
33 (V): Teile zu bewirken.
34 (V): entscheiden, entweder,
35 (V): aber, die
36 (M): 59
37 (V): der Teilung analogen Fall,
38 (O): wie es für
39 (M): 59 |
40 (O): gesehen
452
453
328 Gesichtsraum im Gegensatz zum euklidischen Raum.
a b
WBTC096-100 30/05/2005 16:15 Page 656
do we understand by a strip growing in width?”, then I might be presented with something
like that as an explanation, i.e. I might be given a pattern, and I might incorporate it or
the memory of it into my language. And in this way if I ask someone “How is it that
the broad strip a is divisible on account of b’s being divisible?”, 19
he can answer
by widening the strip b and thus demonstrating to me20
how b can turn into21
a
divided strip of the same width as a. But that answer would be as far as things could
go. And what has he done by way of explanation? He put a sign, a pattern, into my system
of signs; that’s all.
Now is there a limit to the divisibility of the strip in visual space? Well – I can establish
that any way I want. That is: I can introduce a sign system with limited divisibility or one
with unlimited divisibility – the only thing is, of course, that I can’t order the facts about,
and once I’ve established my sign system I have to use it to describe the facts accordingly.
So if my mental or visual image of a divided strip makes up a part of my system of signs,
then this part of my symbolism ends where, for whatever reasons, I am incapable of
narrowing the parts any further. But then I can decide: either22
to say that there is no
further division, i.e. that to speak about one is senseless – and in this case I have obliged
myself to give a different description of any phenomenon that might occur and that I might
be tempted to call a further division – or: to23
allow the divisibility to continue within my
symbolism. In this case, though, nothing is changed, because of course my series of patterns,
which also belongs to language, comes to an end. In so far as this series of patterns is a
series of signs, a 24
new sign enters language with each new pattern. Usually this consideration
is of no importance; but sometimes it becomes important. We have a case analogous
to the problem of divisibility when we ask: Is it possible at a glance to grasp any arbitrary
number 3n of lines |||||||||||| as a group of triplets, or to see any series of such lines,
of whatever length, as a picture that is expressive of their number, as we can do this for
| || ||| ||||? Here too we can use a finite or infinite number system to describe our
experience – for the series of patterns of surveyable groups has an end, and this series
determines the sense of our propositions just as much as the number system we use.
So when I said “We’re looking for a rule that corresponds to a particular reality”, the
correspondence lies in the simplicity and the ease of comprehending the representation of
that rule. The rule is justified by the facts only to the same extent as the choice of a system
of coordinates is justified by its application to a curve that can be represented with a particular
simplicity in that system.
25
It is possible to see in your visual field two straight lines of equal length (i.e. lines seen
to be of equal length), each of which is divided by different colours into several parts, equal
parts, and to find when counting those parts that their numbers are unequal. Now what about
19 (F): MS 113, p. 130v.
20 (V): thus showing me
21 (V): b turns into
22 (V): decide either
23 (V): or to
24 (M): 59
25 (M): 59 |
Visual Space in Contrast to Euclidean Space. 328e
a b
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Frage: ”Angenommen, ich könnte 30 und 31 Teile als Zahl übersehen, wäre es auch dann
möglich, zwei Strecken von 30 und 31 gesichtsgleichen Teilen als gleichlang zu sehen?“ –
Nun, wie ist diese Frage zu entscheiden? Vor allem: wie ist das, wenn man 30 Teile als Zahl
übersieht? Was kann man dafür als Erklärung geben? Wir können freilich niemandem einen
Centaur zeigen, weil es keinen gibt, aber es ist für die Bedeutung des Wortes ”Centaur“
wesentlich, daß wir einen malen, oder modellieren können. – So aber ist es auch für den
Sinn des Satzes ”ich kann 30 Teile als Zahl übersehen“ wesentlich, was ich etwa als Beispiel
dieses Überblickens zeigen kann, und daß ich keinen Fall eines Überblickens von 30
Strichen als Muster zeigen kann. Hier kann man sagen: ich kann mir das Überblicken von
30 Strichen als Zahlbild41
nicht vorstellen, ich weiß nicht, wie das wäre, und die Frage ”wie
wäre es, wenn . . .“ ist für mich unsinnig, denn es ist mir kein Kriterium zur Entscheidung
gegeben.
Wenn wir die Bedeutungen der Ausdrücke ”gleichlang“ und anderer im Gesichtsraum
mit den Bedeutungen derselben Wörter im euklidischen Raum verwechseln, dann kommen42
wir auf43
Widersprüche und fragen dann: ”Wie ist so eine Erfahrung möglich?! Wie ist
es möglich, daß 24 gleichlange Strecken zusammen die gleiche Länge ergeben, wie 25
ebensolange? Habe ich wirklich so eine Erfahrung gehabt?“
”Ist ein Feld eines Schachbretts einfacher, als das ganze Schachbrett?“ Das kommt darauf
an, wie Du das Wort ”einfacher“ gebrauchst. Meinst Du damit ”aus einer kleineren Anzahl
von Teilen bestehend“, so sage ich: Wenn diese Teile etwa die Atome des Schachbretts sind,
so ist also das Feld einfacher als das Schachbrett. – Wenn Du aber von dem sprichst, was
wir am Schachbrett sehen,44
so bestehen ja die Felder nicht aus Teilen, es sei denn, daß sie
wieder aus kleineren Flecken bestehen, und wenn Du dann den Fleck den einfacheren nennst,
der weniger Flecken enthält, so ist wieder das Feld einfacher als das Schachbrett. ”Ist aber
die gleichmäßig gefärbte Fläche einfach?“ – Wenn ”einfach“ bedeutet: nicht aus Flecken
mehrerer Farben zusammengesetzt, – ja!
Aber können wir nicht sagen: einfach ist, was sich nicht teilen läßt? – Wie teilen läßt? Mit
dem Messer? Und mit welchem Messer? Beschreibe mir erst die Methode der Teilung, die
Du erfolglos anwendest, dann werde ich wissen, was Du ”unteilbar“ nennst. Aber vielleicht
willst Du sagen:45
”unteilbar“ nenne46
ich nicht das, was man erfolglos zu teilen versucht,
sondern das, wovon es sinnlos (unerlaubt) ist zu sagen, es bestehe aus Teilen. – Dann ist
”unteilbar“ eine grammatische Bestimmung. Eine Bestimmung also, die Du selber machen
kannst und durch welche Du die Bedeutung, den Gebrauch andrer Wörter festlegst. Wenn
ich etwa sage: ein einfärbiger Fleck ist unteilbar (einfach), denn, wenn ich ihn – z.B. – durch
einen Strich teile, so ist er nicht mehr einfärbig, – so setze ich damit fest, in welcher Bedeutung
ich das Wort ”teilen“ gebrauchen will. Wenn nun gefragt wird: ”besteht das Gesichtsbild
aus minima visibilia“, so fragen wir zurück: wie verwendest Du das Wort ”aus . . .
bestehen“? Wenn in dem Sinn, in welchem ein Schachbrett aus schwarzen und weißen Feldern
besteht, – nein! – Denn Du wolltest doch nicht leugnen, daß wir einfärbige Flecke sehen
(ich meine Flecke, deren Erscheinung einfärbig ist). Wenn Du aber etwa sagen willst, daß
ein physikalischer Fleck (ein meßbarer Fleck im physikalischen Raum) verkleinert werden
kann, bis wir ihn aus einer bestimmten Entfernung nicht mehr sehen, daß er dann beim
Entschwinden gemessen und in dieser Ausdehnung der kleinst sichtbare Fleck genannt
werden kann, so stimmen wir bei.
41 (V): das Übersehen von 30 Strichen
42 (V): geraten
43 (V): in
44 (V): aber vom visuellen Schachbrett sprichst,
45 (V): sagen: @ist
46 (O): nene
454
455
329 Gesichtsraum im Gegensatz zum euklidischen Raum.
WBTC096-100 30/05/2005 16:15 Page 658
this question: “Assuming that at a glance I could take in 30 and 31 parts as numbers, then
would it also be possible to see two straight lines divided into 30 and 31 visually equal parts
as of equal length?” – Well how can this question be decided? Above all: What’s it like to
take in 30 parts as a number? What can be given as an explanation for that? To be sure,
we can’t show anyone a centaur because there is no such thing, but it is essential for the
meaning of the word “centaur” that we can paint or sculpt one. – And in the same way,
what I might be able to give as an example of “taking in at a glance” is essential to the sense
of the sentence “At a glance I can take in 30 parts as a number”, and it’s also essential
that I can’t produce any example of taking in 30 lines as a pattern. Here one can say: I can’t
imagine taking in 30 lines as a numerical image,26
I don’t know what that would be like, and
the question “What would it be like if . . .” makes no sense to me because I have been given
no criterion for deciding it.
If we confuse the meanings of “of the same length” and other expressions in visual space
with the meanings of the same words in Euclidean space, we encounter27
contradictions,
and then we ask: “How is this kind of an experience possible?! How is it possible that 24
and 25 segments of equal length add up to the same length? Have I really had this kind
of an experience?”
“Is one square of a chess board simpler than the whole chess board?” That depends
on how you are using the word “simpler”. If by this you mean “consisting of a smaller
number of parts” then I say: If these parts are, say, the atoms of the chess board, then the
square is simpler than the chess board. – But if you’re talking about what we see when we
look at the chess board,28
then of course the squares don’t consist of parts, unless they in
turn consist of smaller patches; and if you then call the patch simpler that contains fewer
patches, then once again the square is simpler than the chess board. “But is a uniformly
coloured surface simple?” – If “simple” means: not composed of patches of several colours
– yes!
But can’t we say: The simple is what can’t be divided? – Can’t be divided how? With a
knife? And with what kind of knife? First describe for me the method of partition that you
are applying unsuccessfully, and then I’ll know what you are calling “indivisible”. But maybe
you want to say: What I’m calling “indivisible” is not what one unsuccessfully tries to
partition but that of which it makes no sense (is forbidden) to say that it consists of parts.
– Then “indivisible” is a grammatical stipulation. And thus a stipulation that you can make
yourself and by means of which you can establish the meaning, the use, of other words. If
I say, for instance: A monochrome patch is indivisible (simple), because if I divide it – say
– with a line, then it’s no longer monochrome – I establish the meaning I want to give to
the word “divide”. Now if someone asks: “Does a visual image consist of minima visibilia?”,
then we respond: How are you using the words “consist of . . .”? If you’re using them in
the sense in which a chess board consists of black and white squares, then – no! – Because
surely you don’t want to deny that we see monochrome patches (I mean patches the
appearance of which is monochrome). But if perhaps you want to say that a physical patch
(a measurable patch in physical space) can be reduced until we can no longer see it from a
particular distance, and that it can then be measured at the vanishing point, and that at that
size it can be called the smallest visible patch, then we agree.
26 (V): 30 lines at a glance,
27 (V): we get into
28 (V): about a visual chess board,
Visual Space in Contrast to Euclidean Space. 329e
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Wenn wir in der Geometrie sagen, das regelmäßige Sechseck bestehe aus sechs gleichseitigen
Dreiecken, so heißt das, daß es Sinn hat, von einem regelmäßigen Sechseck zu reden,
das aus sechs gleichseitigen Dreiecken besteht. Wenn daraufhin gefragt würde ”ist also das
Sechseck einfach oder zusammengesetzt“, so müßte ich antworten: bestimme Du selbst, wie
Du die Wörter ”einfach“ und ”zusammengesetzt“ gebrauchen willst.
Es scheint, man kann einen einfärbigen Fleck nicht zusammengesetzt sehen, außer, wenn
man ihn sich nicht einfärbig vorstellt. Die Vorstellung einer Trennungslinie macht den Fleck
mehrfärbig,47
denn die Trennungslinie muß eine andere Farbe haben, als der übrige Fleck.48
Das würde heißen: Die einfachen Bestandteile des Gesichtsfeldes sind einfärbige Flecke.
Wie verhält es sich aber dann mit den kontinuierlichen Farbenübergängen!
Kann man sagen, daß der kleinere Fleck einfacher ist als der größere?
Nehmen wir an sie seien einfärbige Kreise, worin soll die größere Einfachheit des
kleineren Kreises bestehen?
Man könnte sagen, der größere kann zwar aus dem kleineren und noch einem Teil
bestehen, aber nicht vice versa. Aber warum soll ich nicht den kleineren als die Differenz
des größeren und des Ringes darstellen?
Es scheint mir also, der kleinere Fleck ist nicht einfacher als der größere.
Ob es einen Sinn hat zu sagen ”dieser Teil einer roten Fläche (der durch keine sichtbare
Grenze abgegrenzt ist) ist rot“ hängt davon ab, ob es einen absoluten Ort gibt. Denn, wenn
im Gesichtsraum von einem absoluten Ort die Rede sein kann, dann kann ich auch diesem
absoluten Ort eine Farbe zuschreiben, wenn seine Umgebung gleichfärbig ist.
Wir können in absolutem Sinne49
von einem Ort im Gesichtsfeld reden. Denken wir uns,
daß ein roter Fleck im Gesichtsfeld verschwindet und in gänzlich neuer Umgebung wieder
auftaucht, so hat es Sinn zu sagen, er tauche am gleichen Ort oder an einem andern Ort
wieder auf. (Wäre ein solcher Raum mit einer Fläche vergleichbar, die von Punkt zu Punkt
eine andere Krümmung hätte, so daß wir jeden Ort auf der Fläche als absolutes Merkmal
angeben könnten?)
Der Gesichtsraum ist ein gerichteter Raum, in dem es ein Oben und Unten, Rechts und
Links gibt. Und diese Bestimmungen 50
haben nichts mit der Richtung der Schwerkraft oder
der rechten und linken Hand zu tun. Sie würden auch dann ihren Sinn beibehalten, wenn
wir unser ganzes Leben lang durch ein Teleskop zu den Sternen sähen. – Dann wäre unser
Gesichtsfeld ein hellerer Kreis vom Dunkel begrenzt und im Kreis Lichtpunkte.51
Nehmen
wir an, wir hätten nie unsern Körper gesehen, sondern immer nur dieses Bild, wir könnten
also die Lage eines Sterns nicht mit der unseres Kopfes oder unserer Füße vergleichen: was
zeigt mir dann, daß mein Raum ein Oben und Unten etc. hat, oder einfach: daß er gerichtet
ist? Es hat Sinn, zu sagen, daß sich das ganze Sternbild im Kreis dreht, obwohl es dadurch
seine relative Lage zu nichts im Gesichtsraum ändert. Oder richtiger ausgedrückt: ich rede
47 (O): einfärbig, (E): Wir haben diese
Änderung auf Grund von MS 105 (S. 9), MS
111 (S. 31) und TS 208 (S. 1r), 209 (S. 113),
211 (S. 20) und 212 (S. 1248) durchgeführt.
48 (M): /Auslassung 1/. (E): Auf Grund von
Typoskript 208 (S. 1r–2) haben wir die
beiden nun folgenden Bemerkungen mit
einbezogen. Anscheinend wurden sie irrtümlich
ausgelassen.
49 (V): in einem absoluten Sinne
50 (M): 35
51 (V): wäre unser Gesichtsfeld dunkel mit einem
helleren Kreis und in diesem Lichtpunkte.
456
457
330 Gesichtsraum im Gegensatz zum euklidischen Raum.
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If in geometry we say that a regular hexagon consists of six equilateral triangles, then that
means that it makes sense to talk about a regular hexagon that consists of six equilateral
triangles. If the question were then raised: “Is the hexagon therefore simple or composite?”,
I’d have to answer: Decide for yourself how you want to use the words “simple” and
“composite”.
It seems that one cannot see a monochrome patch as composite unless one imagines it
as not monochrome. The mental image of a dividing line makes the patch polychrome,29
because the dividing line has to have a different colour from the rest of the patch.30
That would mean: The simple components of the visual field are monochrome patches.
But then what about continuous transitions of colour?
Can we say that a smaller patch is simpler than a larger one?
Let’s assume they are monochrome circles. What is the greater simplicity of the smaller
circle supposed to consist in?
We could say that the larger one can consist of the smaller plus another part, but not vice
versa. But why shouldn’t I represent the smaller as the difference between the larger and a
ring around the smaller?
So it seems to me that the smaller patch isn’t simpler than the larger.
Whether it makes sense to say “This part of a red surface (that is not delimited by any
visible boundary) is red” depends on whether an absolute location exists. Because if one
can speak about an absolute location within visual space, then I can also assign this absolute
location a colour – even if its surroundings are of the same colour.
We can speak about a location within the visual field in an absolute sense. If we imagine
that a red patch vanishes from the visual field and reappears in completely new surroundings,
then it makes sense to say that it either reappears in the same location or in another
one. (Would such a space be comparable to a surface that had different curvatures from one
point to the next, so that we could list each location on the surface as an absolute feature?)
Visual space is a directional space in which there is up and down, right and left. And these
determinations 31
have nothing to do with the direction of gravity or one’s right and left hand.
They would retain their sense even if we looked at the stars through a telescope for our entire
lives. – Then our visual field would be a bright circle bounded by darkness with points of
light in the circle.32
Let’s assume that we had never seen our bodies, but always just this
image, so that we couldn’t compare the position of a star with that of our head or our feet:
What would then show me that my space has an up and a down, etc., or simply: that it is
directional? It makes sense to say that an entire constellation rotates in a circle even though
in doing so it doesn’t change its position relative to anything in visual space. Or expressed
29 (O): monochrome, (E): We have made this
change based on previous versions of this
remark in MS 105 (p. 9), MS 111 (p. 31), and
TSS 208 (p. 1r), 209 (p. 113), 211 (p. 20), and
212 (p. 1248).
30 (R): /Omission 1/. (E): It appears from TS
208 (pp. 1r–2) that at this point Wittgenstein
had intended to include the next two remarks,
(“That would mean . . . than the larger.”),
which were mistakenly omitted from TS 213.
31 (M): 35
32 (V): would be dark, with a brighter circle and
with points of light in it.
Visual Space in Contrast to Euclidean Space. 330e
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auch dann von einer Drehung im Gesichtsraum, wenn keine relative Lageänderung in ihm
stattfindet.
Dieser Sachverhalt ist nicht vielleicht dadurch wegerklärt, daß man sagt: die Retina hat
eben ein Oben, Unten, etc., und so ist es leicht verständlich, daß es das Analoge im Gesichtsfeld
gibt. Vielmehr ist eben das nur eine Darstellung des Sachverhalts auf dem Umweg über die
Verhältnisse in der Retina.
52
Man könnte meinen: es verhält sich im Gesichtsfeld immer so, als sähen wir mit allem
Übrigen ein gerichtetes Koordinatenkreuz, wonach wir alle Richtungen fixieren können. –
Aber auch das ist keine richtige Darstellung; denn sähen wir wirklich ein solches Kreuz
(etwa mit Pfeilen), so wären wir im Stande, nicht nur die relativen Richtungen der Objekte
dagegen zu fixieren, sondern auch die Lage des Kreuzes selbst im Raum, gleichsam gegen
ein ungesehenes, im Wesen dieses Raums enthaltenes Koordinatensystem.
53
Ich kann die Figur 1 als Buchstaben, als Zeichen für ”kleiner“ oder für ”größer“ sehen,
auch ohne sie54
mit meinem Körper zusammen zu sehen. Vielleicht wird man sagen, daß ich
die Lage meines Körpers fühle, ohne ihn zu sehen. Gewiß, und ich sage eben, daß ”die gefühlte
Lage“ nicht ”die gesehene Lage“ ist; daher können sie auch nicht miteinander verglichen,
wohl aber einander zugeordnet werden.
Die Wörter ”oben“, ”unten“, ”rechts“, ”links“ haben andere Bedeutung im Gesichtsraum,
andere im Gefühlsraum. Aber auch das Wort ”Gefühlsraum“ ist mehrdeutig. (Definitionen
der55
Wörter ”oben“, ”unten“, etc. durch die Spitze des Buchstabens56
”V“, des Zeichens
”kleiner“ und ”größer“ einerseits, anderseits durch Kopf- und Fußschmerzen; oder durch
Gleichgewichtsgefühle.)
”Ist Distanz in der Struktur des Gesichtsraumes schon enthalten, oder scheint es uns nur
so,57
weil wir gewisse Erscheinungen des Gesichtsbildes mit gewissen Erfahrungen des
Tastsinnes assoziieren, welche letztere erst Distanzen betreffen?“ Woher nehmen wir diese
Vermutung? Wir scheinen dergleichen irgendwo angetroffen zu haben. Denken wir nicht an
folgenden Fall? diese Melodie mißfiele mir nicht, wenn ich sie nicht unter diesen unangenehmen
Umständen zum erstenmal gehört hätte. Aber hier gibt es zwei Möglichkeiten:
Entweder die Melodie mißfällt mir, wie manche andere, für deren Mißfallen ich jenen Grund
nicht angeben würde, und es ist bloß eine Vermutung, daß die Ursache meines Mißfallens
in jenem früheren Erlebnis liegt. Oder aber, wenn immer ich die Melodie höre, fällt mir
jenes Erlebnis ein und macht mir das Hören der Melodie unangenehm; dann ist meine Aussage
keine Hypothese über die Ursache meines Mißfallens, sondern eine Beschreibung dieses
Mißfallens selbst. – Wenn also gefragt wird: ”scheint es uns nur so, daß eine Strecke im
Gesichtsraum selbst länger ist, als eine andere und bezieht sich das ,länger‘ nicht bloß
auf eine Erfahrung des Tastsinns, die wir mit dem Gesehenen associieren“, – so ist zu
antworten: Weißt Du etwas von dieser Association? beschreibst Du mit ihr Dein Erlebnis,
oder vermutest Du sie nur als Ursache Deines Erlebnisses? – Wenn das letztere, so können
wir von Distanzen im Gesichtsraum reden, ohne auf die mögliche Ursache unserer
Erfahrung Rücksicht zu nehmen. Dabei muß man sich daran erinnern, daß die Aussagen
über Distanzen (daß diese Strecke gleichlang ist wie jene, oder länger als jene, etc.) einen
andern Sinn haben, wenn sie sich auf den Gesichtsraum, und einen andern, wenn sie sich
auf den euklidischen Raum beziehen.
52 (M): 35
53 (F): MS 112, S. 122r
54 (V): es
55 (V): (Definieren d
56 (O): Buchstaben
57 (V): nur, so
458
459
331 Gesichtsraum im Gegensatz zum euklidischen Raum.
WBTC096-100 30/05/2005 16:15 Page 662
more correctly: I also speak of rotation within visual space when no relative change of
position occurs within it.
Contrary to what we might believe, this state of affairs is not laid to rest by saying: It’s
simply that the retina has a top, bottom, etc., and so it’s easy to understand that there is
something analogous in the visual field. Rather, this explanation is only a representation of
the state of affairs – one that uses the detour of the properties of the retina.
33
One might think: In the visual field it is always as if, along with everything else, we were
seeing a N–S–E–W coordinate plane, according to which we can fix all directions. But
neither is that a correct representation; for if we really did see such a plane (say, with arrows)
then we would be able not only to fix the directions of objects relative to it, but also the
position of the plane itself in space, relative to an unseen system of coordinates that was
inherently contained in this space, as it were.
34
I can see the figure 1 as a letter, as a sign for “smaller” or “larger”, even without
seeing it35
together with my body. Perhaps it will be said that I feel the position of my
body without seeing it. Certainly. That’s exactly what I’m saying: that “the felt position”
is not “the seen position”; therefore they can’t be compared to each other, but they can be
coordinated with each other.
The words “up”, “down”, “right”, “left” have one meaning in visual space and another
in the space of feeling. But the phrase “space of feeling” is also ambiguous. (Definitions of
the words “up”, “down”, etc., that use the point of the letter “V” and the signs for “smaller”
and “larger”, on the one hand, and on the other hand, definitions that use a head- and
foot-ache, or sensations of equilibrium.)
“Is distance already contained in the structure of visual space, or does it only seem that
way to us because we associate certain visual images with certain experiences of our sense
of touch, and only the latter have to do with distances?” Where do we get this conjecture
from? We seem to have encountered something like it somewhere. Aren’t we thinking of the
following case? I would not dislike this melody if I hadn’t first heard it in these unpleasant
circumstances. But here there are two possibilities: Either I dislike the melody for other
reasons – as I do many another one – and it is merely a conjecture that the cause of my
dislike lies in that earlier experience. Or, whenever I hear the melody I remember that
experience, and that makes hearing the melody unpleasant for me; in that case my statement
isn’t a hypothesis about the cause of my dislike, but a description of the dislike itself. – So
if the question is: “Does it only seem to us that one line segment in visual space is longer
than another, and doesn’t this ‘longer’ merely refer to an experience of the sense of touch
that we associate with what we see?” – then the answer should be: Do you know anything
about this association? Are you using it to describe your experience, or do you merely
suspect that it is the cause of your experience? – If the latter, then we can speak of distances
within visual space without taking the possible cause of our experience into account. In this
context one has to remind oneself that statements about distances (that this line segment is
of the same length as that, or is longer than that, etc.) have one sense when they refer to
visual space, and another when they refer to Euclidean space.
33 (M): 35
34 (F): MS 112, p. 122r.
35 (V): seeing the sign
Visual Space in Contrast to Euclidean Space. 331e
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Zu sagen, der Punkt B ist nicht zwischen A und C58
(die Strecke a
nicht kürzer als c), sondern dies erscheine uns nur so wegen gewisser
Assoziationen, klingt und ist absurd, weil wir uns eben in unserer Aussage
gar nicht um eventuelle Ursachen der Erscheinung kümmern, sondern nur diese im
Gegensatz zu andern Erscheinungen beschreiben.
Wenn Du sagst, der Punkt B erscheint59
Dir nur zwischen A und C (zu liegen), so antworte
ich: das ist es ja, was ich sage, nur gebrauche ich dafür den Ausdruck ”er liegt zwischen A
und C“.
Und wenn Du fragst ”scheint es nicht nur so“, so antworte ich: Welche Methode würdest
Du denn anwenden, um die Antwort auf Deine Frage zu finden. Dann nämlich werde ich
verstehen, was Dein Verdacht eigentlich betrifft. Wenn Du sagst: ist auf diesem Tisch nicht
doch vielleicht etwas, was ich nicht sehe, so antworte ich: Wie könnten wir denn das Betreffende
finden? Versuche, mir doch eine Erfahrung zu beschreiben, die Dich veranlassen würde,
zu sagen:60
”es war doch noch etwas da“. Beschreibe mir die Erfahrung, die Dich davon
überzeugen würde, daß B doch nicht zwischen A und C liegt, und ich werde verstehen,
welcher Art dieser61
wirkliche Sachverhalt im Gegensatz zum scheinbaren ist. Aber Eines
ist klar: die Erfahrung, die Dich das lehrt, kann nicht diejenige ändern, die ich mit den
Worten beschreibe ”B liegt zwischen A und C“.
Dem Einwurf liegt aber eine falsche Auffassung der logischen Analyse zugrunde. Was
wir vermissen ist nicht ein genaueres Hinsehen (etwa auf A, B und C) und die Entdeckung
eines Vorgangs hinter dem oberflächlich62
beobachteten (dies wäre die Untersuchung eines
physikalischen oder psychologischen Phänomens), sondern die Klarheit in der Grammatik
der Beschreibung des alten Phänomens. Denn, sähen wir genauer hin, so sähen wir eben
etwas Anderes und hätten nichts für unser Problem gewonnen. Diese Erfahrung, nicht eine
andere, sollte beschrieben werden.
Hat das Gesichtsfeld einen Mittelpunkt? – Es hat Sinn, in einem Bild etwa ein
Kreuzchen anzubringen und zu sagen: schau’ auf das Kreuz; Du wirst dann auch das Übrige
sehen, aber das Kreuz ist dann im Mittelpunkt des Gesichtsfeldes.
Im Gesichtsraum gibt es absolute Lage. Wenn ich durch ein Aug’ schaue, sehe ich meine
Nasenspitze. Würde diese abgeschnitten und entfernt, mir aber dann in die Hand gegeben,
so könnte ich sie ohne Hilfe des Spiegels und bloß durch die Kontrolle des Sehens wieder
an ihre alte Stelle setzen; auch dann, wenn sich inzwischen alles in meinem Gesichtsbild
geändert hätte. Der Satz ”ich sehe das sehende Auge im Spiegel“ ist nur scheinbar von der
Form des Satzes ”ich sehe das Auge des Andern im Spiegel“, denn es hat keinen Sinn zu
sagen: ”ich sehe das sehende Auge“. Wenn ich ”visuelles Auge“ das Bild nenne, was mir
etwa das Auge eines Andern bietet, so kann ich sagen, daß das Wort ”das sehende Aug“
nicht einem visuellen Auge entspricht.
Im Gesichtsraum gibt es absolute Lage und daher auch absolute Bewegung. Man denke
sich das Bild zweier Sterne in stockfinsterer Nacht, in der ich nichts sehen kann als diese,
und diese bewegen sich im Kreise umeinander.
Mein Gesichtsfeld weist keine Unvollständigkeit auf, die mich dazu bringen könnte,
mich umzuwenden um63
zu sehen, was hinter mir liegt. Im Gesichtsraum gibt es kein
”hinter mir“; und wenn ich mich umwende, ändert sich ja bloß mein Gesichtsbild, wird
58 (F): MS 112, S. 125r.
59 (V): scheint
60 (V): Dich sagen lassen würde:
61 (V): der
62 (V): gewöhnlich
63 (V): und
460
461
332 Gesichtsraum im Gegensatz zum euklidischen Raum.
c
aA B C
WBTC096-100 30/05/2005 16:15 Page 664
To say that point B is not between A and C36
(or that distance a is
not shorter than c), but that this only appears to us to be the case because
of certain associations, sounds and is absurd, simply because in making
our statement we don’t even concern ourselves with possible causes of the phenomenon, but
merely describe it in contrast to other phenomena.
If you say, point B merely appears37
to you (to lie) between A and C, then I answer: But
that’s precisely what I’m saying, it’s just that to say this I use the expression “It’s between
A and C”.
And if you ask “Doesn’t it merely appear that way?”, I answer: What method would you
use to find the answer to your question? For then I’ll understand what your suspicion really
refers to. If you say: “Isn’t there possibly something on this table that, despite appearances,
I don’t see?”, then I answer: “How could we find that thing?” Try to describe an experience
for me that would cause38
you to say: “In spite of appearances there was something else
here”. Describe for me the experience that would convince you that B is not between A
and C after all, and I’ll understand the nature of this39
real state of affairs, in contrast to the
apparent one. But one thing is clear: whatever the experience is that teaches you that, it can’t
change the one that I describe with the words “B is between A and C”.
But the objection is based on a false understanding of logical analysis. What we’re missing
isn’t a more precise scrutiny (say of A, B and C), nor the discovery of a process behind
the one that is observed superficially40
(that would be the investigation of a physical or
psychological phenomenon), but clarity in the grammar of the description of the old
phenomenon. Because if we looked more closely we would simply see something else, and
would have made no advance on our problem. This experience, and not another, is what
needs to be described.
Does one’s visual field have a centre point? – It makes sense to make a little cross in a
picture and to say: Look at the cross; then you’ll also see the rest of the picture, but it is the
cross that will be at the centre point of your visual field.
There is absolute position in visual space. When I look through one eye I see the tip of
my nose. If it were cut off and removed, but then put into my hand, I could put it back in
its old location without the help of a mirror, merely by checking visually; and I could do
this even if everything in my visual image had changed in the meantime. The sentence “I
see the seeing eye in the mirror” only appears to have the same form as the sentence “I see
someone else’s eye in the mirror”, for it makes no sense to say: “I see the seeing eye”. If I
call the image that, for example, someone else’s eye presents to me a “visual eye” then I can
say that the expression “the seeing eye” doesn’t correspond to a visual eye.
There is absolute position in visual space, and hence there is also absolute movement.
Imagine the image of two stars in a pitch black night, when I can’t see anything but them
and they’re moving in circles around each other.
My visual field doesn’t exhibit any deficiency that could cause me to turn around to see41
what is behind me. There is no “behind me” in visual space; and if I turn around, my visual
36 (F): MS 112, p. 125r.
37 (V): seems
38 (V): get
39 (V): the
40 (V): that is usually observed
41 (V): around and to see
Visual Space in Contrast to Euclidean Space. 332e
c
aA B C
WBTC096-100 30/05/2005 16:15 Page 665
aber nicht vervollständigt. (Der ”Raum um mich herum“ ist eine Verbindung von Sehraum
und Muskelgefühlsraum.) Es hat64
keinen Sinn, im Gesichtsraum von der Bewegung eines
Gegenstandes zu reden, die um das sehende Auge hinten herum führt.
Beziehung zwischen physikalischem Raum und Gesichtsraum. Denke an das Sehen bei
geschlossenen Augen (Nachbilder, etc.) und an die Traumbilder.
64 (V): hat e
333 Gesichtsraum im Gegensatz zum euklidischen Raum.
WBTC096-100 30/05/2005 16:15 Page 666
image changes; it is not completed. (The “space around me” is a combination of visual space
and the space of muscle sensation.) It makes no sense to talk about the movement of an object
within visual space that goes behind and around the seeing eye.
The relation between physical and visual space. Think about seeing with your eyes closed
(after-images, etc.) and about images in dreams.
Visual Space in Contrast to Euclidean Space. 333e
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97
Das sehende Subjekt
und der Gesichtsraum.
Es ist unsinnig zu sagen ”ich sehe diesen Gegenstand1
im Gesichtsraum“. Im Gegensatz
wozu? Ist es denkbar, daß ich ihn2
höre, oder daß ein Anderer ihn3
sieht?
Darum kann ich auch nicht sagen, daß der Gegenstand in meinem Gesichtsraum die Ursache
davon4
ist, daß ich ihn sehe.
(Darum ist es auch Unsinn zu sagen: aus dem Urnebel haben sich die Sonnen, Planeten,
die einfachsten Lebewesen und endlich ein Wesen entwickelt, das so organisiert ist, daß es
all diese Dinge sehen und über sie Betrachtungen anstellen kann. Es sei denn, daß man unter
diesen Betrachtungen die (rein) physikalischen Äußerungen, im Sinne des Behaviourism
versteht. In diesem Sinne kann man auch von einer photographischen Kamera sagen, daß
sie etwas wahrnehme.)
Wenn man gefragt würde: was ist der Unterschied zwischen einem Ton und einer Farbe,
und die Antwort wäre ”Töne hören wir, dagegen sehen wir die Farben“; so ist das nur eine
durch Erfahrung gerechtfertigte Hypothese, wenn es überhaupt einen Sinn haben soll,
das zu sagen. Und in diesem Sinn ist es denkbar, daß ich einmal Töne mit den Augen
wahrnehmen, also sehen werde, und Farben hören. Das Wesentliche der Töne und Farben
ist offenbar in der Grammatik der Wörter für Töne und Farben gezeigt.
Wenn wir vom Gesichtsraum reden, so werden wir leicht zu der Vorstellung verführt, als
wäre er eine Art von Guckkasten, den jeder vor5
sich herumtrüge. D.h. wir verwenden dann
das Wort ”Raum“ ähnlich, wie wenn wir ein Zimmer einen Raum nennen. In Wirklichkeit
aber bezieht sich doch das Wort ”Gesichtsraum“ nur auf eine Geometrie, ich meine auf einen
Abschnitt der Grammatik unserer Sprache.
In diesem Sinne gibt es keine ”Gesichtsräume“, die etwa jeder seinen Besitzer hätten. (Und
etwa auch solche, vazierende, die gerade niemandem gehören?)
”Aber kann nicht ich in meinem Gesichtsraum eine Landschaft, und Du in dem Deinen
ein Zimmer sehen?“ – Nein, – ”ich sehe in meinem Gesichtsraum“ ist Unsinn. Es muß heißen
”ich sehe eine Landschaft und Du etc.“ – und das wird nicht bestritten. Was uns hier irreführt,
ist eben das Gleichnis vom Guckkasten, oder etwa von einer kreisrunden weißen Scheibe,
die wir gleichsam als Projektionsleinwand mit uns trügen, und die der Raum ist, in dem
das jeweilige Gesichtsbild erscheint. Aber der Fehler an diesem Gleichnis ist, daß es sich
die Gelegenheit – die Möglichkeit – zum Erscheinen eines visuellen Bildes selbst visuell
vorstellt; denn die weiße Leinwand ist ja selbst ein Bild.
1 (V): sehe die Dinge
2 (V): sie
3 (V): sie
4 (V): dessen
5 (V): mit
462
463
334
WBTC096-100 30/05/2005 16:15 Page 668
97
The Seeing Subject
and Visual Space.
It is nonsense to say “I see this object1
in visual space”. As opposed to what? Is it conceivable
that I hear it, or that someone else sees it?2
Therefore neither can I say that the object within my visual space is the cause of my
seeing it.
(Therefore it is also nonsense to say: From a primordial cloud there developed the suns,
the planets, the most primitive living things, and finally a being that is organized in such a
way that it can see all of these things and can contemplate them. Unless by “contemplate”
one understands (purely) physical utterances, in the sense of behaviourism. In this sense one
can also say of a camera that it perceives something.)
If one were asked: “What is the difference between a sound and a colour?”, and the answer
were “We hear sounds, but we see colours”, then this is only a hypothesis that is justified
through experience, if saying this makes any sense in the first place. And to that extent it is
conceivable that one day I shall perceive sounds with my eyes, i.e. see them, and shall hear
colours. What is essential to sounds and colours is obviously shown in the grammar of the
words for sounds and colours.
When we speak about visual space we are easily seduced into imagining that it is a kind
of peep-show box that everyone carries around in front of3
himself. That is to say, in doing
this we are using the word “space” in a way similar to when we call a room a space. But
in reality the word “visual space” only refers to a geometry, I mean to a section of the
grammar of our language.
In this sense there are no “visual spaces” each of which, say, would have its own owner.
(And are there maybe also such visual spaces that are itinerant, that don’t belong to anyone
at the moment?)
“But can’t I see a landscape in my visual space and you a room in yours?” – No – “I see
in my visual space” is nonsense. It must be “I see a landscape and you, etc.” – and that isn’t
being called into question. What is misleading us here is simply the simile of the peep-show
or, say, of a circular white disc that we carry with us as a projection screen, as it were, which
is the space where the respective visual image appears. But the flaw in this simile is that it
visually imagines the opportunity – the possibility – of a visual image itself appearing; for,
after all, the white screen is itself an image.
1 (V): see the things
2 (V): hear them, or that someone else sees them?
3 (V): carries along with
334e
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Es ist nun wichtig, daß der Satz ”das Auge, womit ich sehe, kann ich nicht unmittelbar
sehen“ ein verkappter Satz der Grammatik, oder Unsinn ist. Der Ausdruck ”näher am (oder,
weiter vom) sehenden Auge“ hat nämlich eine andere Grammatik, als der ”näher an dem
blauen Gegenstand, welchen ich sehe“. Die visuelle Erscheinung, die der Beschreibung
entspricht ”A setzt die Brille auf“, ist von der grundverschieden, die ich mit den Worten
beschreibe: ”ich setze die Brille auf“. Ich könnte nun sagen: ”mein Gesichtsraum hat
Ähnlichkeit mit einem Kegel“, aber dann muß es verstanden werden, daß ich hier den Kegel
als Raum, als Repräsentanten einer Geometrie, nicht als Teil eines Raumes (Zimmer) denke.
(Also ist es mit dieser Idee nicht verträglich, daß ein Mensch durch ein Loch in der Spitze
des Kegels in diesen hineinschaut.)6
6 (V): durch ein Loch an der Spitze in den Kegel
hineinschaut.)
464
335 Das sehende Subjekt . . .
WBTC096-100 30/05/2005 16:15 Page 670
It’s important that the proposition “I cannot directly see the eye with which I see” is a
grammatical proposition in disguise, or nonsense. For the expression “closer to (or further
from) the seeing eye” has a different grammar from “closer to the blue object that I see”.
The visual appearance that corresponds to the description “A is putting on his glasses” is
fundamentally different from the one that I describe in the words: “I am putting on my
glasses”. Now I could say: “My visual space is similar to a cone”, but then it has to be
understood that here I am imagining the cone as space, as representative of a geometry,
and not as part of a space (a room). (So that someone looking into the cone through a hole
in its tip is not compatible with this idea.)
The Seeing Subject . . . 335e
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98
Der Gesichtsraum mit einem
Bild (ebenen Bild) verglichen.
Wer aufgefordert würde, das Gesichtsbild zu malen und es im Ernst versuchte, würde
bald sehen, daß es unmöglich ist.
Verschiedene Bedeutungen der Wörter ”verschwommen“, ”unklar“.
Verschwommen, unklar, unscharf.
”Die Linien dieser Zeichnung sind unscharf“, ”meine Erinnerung an die Zeichnung
ist unklar, verschwommen“, ”die Gegenstände am Rand meines Gesichtsfeldes sehe ich
verschwommen“. – Wenn man von der Verschwommenheit der Bilder am Rande des
Gesichtsfeldes spricht, so schwebt einem1
oft ein Bild dieses Gesichtsfeldes vor, wie es
etwa Mach entworfen hat.2
Die Verschwommenheit aber der Ränder eines Bildes auf
der Papierfläche3
ist von gänzlich andrer Natur, als die, die man von den Rändern des
Gesichtsfeldes aussagt. So verschieden, wie die Blässe der Erinnerung an eine Zeichnung,
von der Blässe einer Zeichnung (selbst). Wenn im Film eine Erinnerung oder ein Traum
dargestellt werden sollte, so gab man den Bildern einen bläulichen Ton. Aber die Traumund
Erinnerungsbilder haben natürlich keinen bläulichen Ton – sowenig, wie unser
Gesichtsbild4
verwaschene Ränder hat; also sind die bläulichen Bilder auf der Leinwand5
nicht unmittelbar anschauliche Bilder der Träume, sondern ”Bilder“ in noch einem
andern Sinn. – Bemerken wir im gewöhnlichen Leben, wo wir doch unablässig schauen, die
Verschwommenheit an den Rändern des Gesichtsfeldes? Ja, welcher Erfahrung entspricht
sie eigentlich, denn im normalen Sehen kommt sie nicht vor! Nun, wenn wir den Kopf nicht
drehen und wir beobachten etwas, was wir durch Drehen der Augen gerade noch sehen
können, dann sehen wir etwa einen Menschen, können aber sein Gesicht nicht erkennen,
sondern sehen es in gewisser Weise verschwommen. Die Erfahrung hat nicht die geringste
Ähnlichkeit mit dem Sehen einer Scheibe, auf welcher6
Bilder gemalt sind, in der Mitte der
Scheibe mit scharfen Umrissen, nach dem Rand zu mehr und mehr verschwimmend, etwa
in ein allgemeines Grau unmerklich übergehend. Wir denken an so eine Scheibe, wenn
wir z.B. fragen: könnte man sich nicht ein Gesichtsfeld mit gleichbleibender Klarheit
der Umrisse etc. denken? Es gibt keine Erfahrung, die im Gesichtsfeld der entspräche,
wenn man den Blick einem Bild entlang gleiten läßt, das von scharfen Figuren zu immer
verschwommeneren übergeht.
1 (V): Einem
2 (E): Siehe: Ernst Mach, Die Analyse der
Empfindungen, Jena, 1922, S.15.
3 (O): Papierfläche haben können, (V): Die
Verschwommenheit aber, als die die Ränder
eines Bildes auf der Papierfläche haben
können,
4 (V): Gesichtsbild verschwach
5 (V): bläulichen Projektionen auf der Leinwand
6 (V): der
465
466
336
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98
Visual Space Compared to a Picture
(Two-Dimensional Picture).
Someone who was asked to paint his visual image and seriously tried to do this would
soon see that it is impossible.
Different meanings of the words “blurry”, “unclear”.
Blurry, unclear, out of focus.
“The lines of this drawing are out of focus”, “My memory of the drawing is unclear, blurry”,
“I see the objects at the edges of my visual field as blurred”. – When speaking of the blurredness
of images at the edge of one’s visual field, one frequently has in mind an image of this
visual field such as, say, sketched by Mach.1
But the blurredness of the edges of a picture
on2
a paper surface is inherently different from that ascribed to the edges of the visual field.
As different as the paleness of one’s memory of a drawing is from the paleness of the drawing
(itself). When a memory or a dream is represented in a film, the pictures are given a
bluish tint. But of course dream and memory images don’t have a bluish tint – any more
than our visual image has washed-out edges; so the bluish pictures3
on the screen are not
pictures that are taken of dreams, but “pictures” in another sense. – In everyday life, where
after all we are constantly looking at things, do we notice the blurredness at the edges of
our visual field? Indeed, what experience does this blurredness actually correspond to, for
it doesn’t occur in the normal process of seeing things! Well, if we observe something that
we can just barely see by turning our eyes but not our head, then we see, say, a person;
but we can’t make out his face, and see it in a certain way as blurry. This experience
doesn’t have the least similarity to seeing a disc on which pictures have been painted –
pictures with sharp outlines in the middle of the disc, blurring increasingly toward the
edge, imperceptibly transitioning, say, into a generic grey. We’re thinking of such a disc
when we ask, for instance: Couldn’t one imagine a visual field with consistent clarity of its
outlines, etc.? There is no experience in the visual field that might correspond to what
occurs when one allows one’s glance to glide over a picture whose figures transition from
being in focus to ever more blurred.
1 (E): Cf. Ernst Mach, Die Analyse der Empfindungen,
Jena, 1922, p. 15.
2 (V): blurredness that the edges of a picture can
have on
3 (V): projections
336e
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Es ist z.B. wichtig, daß in dem Satz ”ein roter Fleck befindet sich nahe an der Grenze
des Gesichtsfeldes“ das ”nahe an“ eine andere Bedeutung hat als in einem Satz ”der rote
Fleck im Gesichtsfeld befindet sich nahe an dem braunen Fleck“. Das Wort ”Grenze“ in
dem vorigen Satz hat ferner eine andere Bedeutung – und ist eine andere Wortart – als in
dem Satz ”die Grenze zwischen rot und blau im Gesichtsfeld ist ein Kreis“.
Welchen Sinn hat es, zu sagen: Unser Gesichtsbild ist an den Rändern undeutlicher als
gegen die Mitte? Wenn wir hier nämlich nicht davon reden, daß wir die physikalischen
Gegenstände in der Mitte des Gesichtsfeldes deutlicher sehen.
Eines der klarsten Beispiele der Verwechslung zwischen physikalischer und phänomenologischer
Sprache ist das Bild, welches Mach von seinem Gesichtsfeld entworfen hat und
worin die sogenannte Verschwommenheit der Gebilde gegen den Rand des Gesichtsfeldes
durch eine Verschwommenheit (in ganz anderem Sinne) der Zeichnung wiedergegeben wurde.
Nein, ein sichtbares Bild des Gesichtsbildes kann man nicht machen.
Kann ich also sagen, daß die Farbflecken in der Nähe des Randes des Gesichtsfeldes keine
scharfen Konturen mehr haben: Sind denn Konturen dort denkbar? Ich glaube es ist klar,
daß jene Undeutlichkeit eine interne Eigenschaft des Gesichtsraumes ist. Hat z.B. das Wort
”Farbe“ eine andere Bedeutung, wenn es sich auf Gebilde in der Randnähe bezieht?
Die Grenzenlosigkeit des Gesichtsraums ist ohne jene ”Verschwommenheit“ nicht
denkbar.
Die Gefahr, die darin liegt, Dinge einfacher sehen zu wollen, als sie in Wirklichkeit sind,
wird heute oft sehr überschätzt. Diese Gefahr besteht aber tatsächlich im höchsten Grade
in der phänomenologischen Untersuchung7
der Sinneseindrücke. Diese werden immer für
viel einfacher gehalten, als sie sind.
Es ist seltsam, daß ich geschrieben habe, der Gesichtsraum hat nicht die Form
8
und nicht, er habe nicht die Form9
; und daß ich das Erste geschrieben habe, ist
sehr bezeichnend.
Man bedenkt gar nicht, wie merkwürdig das dreidimensionale Sehen ist. Wie seltsam etwa
ein Bild, eine Photographie aussähe, wenn wir im Stande wären, sie als Verteilung grauer,
weißer und schwarzer Flecken in einer ebenen Fläche zu sehen. Was wir sehen, würde dann
ganz sinnlos wirken. Ebenso, wenn wir mit einem Aug’ flächenhaft sehen könnten. Es ist
z.B. gar nicht klar, was geschieht, wenn wir mit zwei Augen die Gegenstände plastischer
sehen, als mit einem. Denn sie wirken auch mit einem gesehen schon plastisch. Und der
Unterschied zwischen Relief und Rundplastik ist auch keine richtige Analogie.
7 (O): Unterschung
8 (F): MS 112, S. 28r.
9 (F): MS 112, S. 28v.
467
468
337 Der Gesichtsraum mit einem Bild verglichen.
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It’s important, for instance, that the word “close” has a different meaning in the sentence
“There is a red patch close to the border of my visual field” than in the sentence “The red
patch in my visual field is close to a brown patch”. Furthermore, the word “border” in the
previous sentence has a different meaning – and is a different type of word – than in the
sentence “In my visual field the border between red and blue is a circle”.
What sense is there in saying: Our visual image is less distinct at the edges than towards
the center? That is, if we’re not talking here about the fact that we see physical objects more
distinctly in the middle of our visual field.
One of the clearest examples of the confusion between physical and phenomenological language
is the picture Mach sketched of his field of vision, in which the so-called blurredness
of the shapes toward the edge of his visual field was reproduced by a blurredness (in a quite
different sense) in the drawing. No, you can’t make a visible picture of your visual image.
So can I say that colour patches close to the edge of one’s visual field no longer have sharply
defined contours: are contours even conceivable there? I believe it is clear that that indistinctness
is an internal property of visual space. Does the word “colour”, for instance, have a different
meaning when it refers to shapes close to the edge?
The limitlessness of visual space is inconceivable without that “blurredness”.
Nowadays too much is often made of the danger that lies in wanting to see things as
simpler than they really are. But this danger does actually exist to the highest degree in the
phenomenological investigation of sense impressions. They are always thought to be much
simpler than they are.
It’s strange that I wrote that visual space doesn’t have the form , 4
and not
that it doesn’t have the form5
, and it’s very significant that I wrote the
former.
We don’t even think about how remarkable three-dimensional seeing is. How strange, say,
a picture, a photograph would look if we could see it as a distribution of grey, white and
black spots on a flat surface. What we would see there would strike us as utterly senseless.
Likewise if we could see two-dimensionally with one eye. It isn’t at all clear, for example,
what happens when we see objects as more three-dimensional with two eyes than we do with
one. For even when seen with one, they already have a three-dimensional effect. And the
difference between relief and sculpture isn’t a proper analogy either.
4 (F): MS 112, p. 28r. 5 (F): MS 112, p. 28v.
Visual Space Compared to a Picture . . . 337e
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99
Minima visibilia.
Der einfärbige Fleck in der färbigen1
Ebene ist nicht aus kleineren Teilen zusammengesetzt,
außer so, wie die Zehn etwa aus tausend Hundertsteln.
Das kleinste sichtbare Stück ist ein Stück der physikalischen Fläche, nicht des
Gesichtsfeldes. Der Versuch, der das kleinste noch Sichtbare ermittelt, stellt eine Relation
fest zwischen zwei Erscheinungen.
Dieser2
Versuch untersucht nicht den Gesichtsraum und man kann den Gesichtsraum nicht
untersuchen. Nicht in ihn tiefer eindringen.
(Wenn man beschreiben wollte, was auf der Hand liegt, könnte man nicht ”untersuchen
wollen, was auf der Hand liegt“.)3
Man könnte glauben, das Gesichtsfeld sei aus den minima visibilia zusammengesetzt; etwa
aus lauter kleinen Quadraten, die man als unteilbare Flecke sieht. Unsinn.
Das Gesichtsfeld ist nicht zusammengesetzt, wenn wir die Zusammensetzung nicht
sehen. Denn bei dem Wort ”Zusammensetzung“ denken wir doch an die Zusammensetzung
eines größeren Flecks aus kleineren.
Von kleinsten sichtbaren Teilen des Gesichtsfeldes zu reden ist irreführend; gibt es
denn auch Teile des Gesichtsfeldes, die wir nicht mehr sehen? Und wenn wir etwa das
Gesichtsbild4
eines Fixsterns so nennen, so könnte das nur heißen, daß es keinen Sinn habe,
hier von ”kleiner“ zu reden, und nicht, daß tatsächlich kein Fleck im Gesichtsfeld kleiner
ist. Also ist der Superlativ ”das kleinste . . .“ falsch angewendet.
Der kleinste sichtbare Unterschied wäre einer, der in sich selbst das Kriterium des
Kleinsten trüge.
Denn im Fall des Flecks A zwischen B und C 5
unterscheiden wir eben
einige Lagen und andere unterscheiden wir nicht. Was wir aber brauchten, wäre sozusagen
ein infinitesimaler Unterschied, also ein Unterschied, der es in sich selbst trüge, der
Kleinste zu sein.
Der Gesichtsraum besteht offenbar nicht aus diskreten Teilen.
Denn sonst müßte man unmittelbar sagen können, aus welchen.
Oder er besteht nur sofern aus Teilen, als man sie angeben kann.
Gibt es einen kleinst sichtbaren Farbunterschied? – Welche Farben sind hier gemeint?
Nennen wir Farbe das Ergebnis der Mischung von Farbstoffen: dann kann ich das
Experiment machen, z.B. zu einer Menge eines roten Farbstoffes eine kleine Menge eines
B A C
1 (V): farbigen
2 (V): Der
3 (V): ”untersuchen, was auf der Hand liegt“.)
4 (V): Bild
5 (F): MS 108, S. 134.
469
470
338
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99
Minima Visibilia.
A monochrome patch on a coloured plane is not composed of smaller parts, except as,
say, ten is composed of a thousand hundredths.
The smallest visible piece is a piece of a physical surface, not a piece of the field of vision.
The experiment that discovers the smallest thing still visible ascertains a relation between
two phenomena.
This1
experiment doesn’t examine visual space, and one can’t examine visual space. Can’t
penetrate more deeply into it.
(One might want to describe what is obvious, but one can’t “want to examine2
what is
obvious”.)
One might think that one’s visual field is made up of minima visibilia; say, of nothing but
small squares that one sees as indivisible patches. Nonsense.
The visual field isn’t composite if we don’t see the composition. For the word “composition”
makes us think, after all, of a larger patch being composed of smaller ones.
It’s misleading to talk about the smallest visible parts of the visual field; are there really
such things as parts of the visual field that we no longer see? And if, say, that is what we
call the visual image3
of a fixed star, then all that could mean is that here it made no sense
to talk about “smaller”, and not that, in actuality, no patch in the visual field was smaller.
So the superlative “the smallest . . .” is used incorrectly.
The smallest visible difference would inherently contain the criterion for “smallest”.
For in the case of the patch A between B and C 4
we do distinguish
between some positions and not between others. But what we need is an infinitesimal difference,
so to speak, that is, a difference that inherently contains “being the smallest”.
Obviously visual space does not consist of discrete parts.
Otherwise one would have to be able straightaway to say which ones.
Or it consists of parts only in so far as one can list them.
Is there a smallest visible colour difference? – What colours are meant here? If colour
is what we call the result of a mixture of pigments, then I can perform the experiment of
mixing a small amount of yellow pigment with a certain amount of red pigment, for instance,
B A C
1 (V): The
2 (V): “examine
3 (V): the image
4 (F): MS 108, p. 134.
338e
WBTC096-100 30/05/2005 16:15 Page 677
gelben beizumischen und zu versuchen, ob ich einen Farbunterschied sehe; wenn ja, so
wiederhole ich den Versuch mit einem kleineren Zusatz des gelben Farbstoffes und immer
so fort, bis der Zusatz keinen sichtbaren Unterschied mehr hervorbringt; das kleinste Quantum,
welches noch einen sichtbaren Unterschied hervorbrachte, nenne ich, mit einem gewissen
Faktor von Ungenauigkeit, den kleinst sichtbaren Unterschied. Das Wesentliche ist (hier),
daß der Unterschied noch da war, also noch konstatiert wurde, als kein Unterschied mehr
gesehen wurde. Was ich so konstatiert habe, war der kleinst sichtbare Unterschied in den
Pigmenten. Und ähnlich könnte ich von einem kleinst sichtbaren Unterschied zwischen
farbigen Lichtern reden; wenn ich nur außer dem Gesicht ein anderes Mittel der
Unterscheidung habe. – Anders wird es, wenn man fragt: ”gibt es einen kleinst sichtbaren
Unterschied zwischen den gesehenen Farben“. Der müßte der kleinste in dem Sinne sein,
in dem die Null die kleinste Kardinalzahl ist. Es wäre also nicht ein Unterschied, den man
nicht mehr unterteilen könnte, weil das Experiment seiner Unterteilung immer mißlänge;
sondern die Unmöglichkeit der Unterteilung wäre eine logische, was so viel heißt, als daß
es keinen Sinn hätte, von einer Unterteilung zu reden. Der kleinst sichtbare Unterschied in
diesem Sinne wäre also ein Farbunterschied einer andern Art.
6
Wenn man einen schwarzen Streifen auf weißem Grund immer dünner und dünner
werden läßt, so kommt man endlich zu dem, was ich einen visuellen Strich (im Gegensatz
zu einer visuellen Linie, der Grenze zweier Farben) nennen möchte. Der Strich ist kein
Streifen, er hat keine Breite; d.h., wenn er von einem andern Strich
durchkreuzt wird,7
sehen wir nicht die 4 Eckpunkte, in denen sich die
Grenzlinien zweier Streifen schneiden. Es ist unsinnig, von der optischen
Unterteilung eines Strichs zu reden. Ihm entspricht die Erscheinung eines
Fixsterns, die sich zum visuellen Punkt, dem Schnitt zweier Farbgrenzen,
ebenso verhält, wie der Strich zur Farbgrenze. Den optischen Fixstern
könnte man also ein minimum visibile nennen. Aber man kann nun nicht etwa sagen,
das Gesichtsfeld bestehe aus solchen Teilen! Es bestünde nur aus ihnen,8
wenn wir sie
sähen. Das visuelle Bild9
eines Fixsternnebels im Fernrohr, besteht aus ihnen, soweit wir sie
unterscheiden können. Denn diese beiden Ausdrücke heißen eben dasselbe.
Wenn gefragt wird ”ist unser Gesichtsfeld kontinuierlich oder diskontinuierlich“, so
müßte man erst wissen, von welcher Kontinuität man redet. Einen Farbübergang nennen
wir kontinuierlich, wenn wir keine Diskontinuität in ihm sehen.
6 (M): 57 |
7 (F): MS 112, S. 130v.
8 (V): nur daraus,
9 (V): Das Bild
471
472
339 Minima visibilia.
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and checking whether I see a colour difference; if I do then I repeat the experiment with a
smaller addition of the yellow pigment, and so on in this way until the addition no longer
produces a visible difference; granted a certain margin of error, I call the smallest quantity
that still produces a visible difference the “smallest visible difference”. What is essential (here)
is that the difference was still there, and thus was still ascertained, when no more difference
was seen. What I ascertained in this manner was the smallest visible difference in the pigments.
And similarly I could talk about a smallest visible difference between coloured lights,
if only I have another means of differentiation in addition to sight. – The situation changes
when one asks: “Is there a smallest visible difference between colours we see?”. It would
have to be the smallest in the sense in which zero is the smallest cardinal number. Thus it
wouldn’t be a difference that one couldn’t subdivide any further because the experiment
to subdivide it would always fail; rather the impossibility of subdivision would be logical,
which is tantamount to saying that it would make no sense to talk about a subdivision. So
the smallest visible difference in this sense would be a colour difference of a different kind.
5
If one allows a black strip on a white background to become thinner and thinner, then
one finally arrives at what I’d like to call a visual streak (as opposed to a visual line, the
border between two colours). The streak is not a strip, for it doesn’t
have any width; thus, if it is crossed by another streak6
we do not see the
4 corner points where the borderlines of two strips intersect. It is nonsensical
to talk about an optical subdivision of a streak. What corresponds
to it is the appearance of a fixed star, an appearance that relates to a visual
point – the intersection of two colour-borders – precisely as does a streak
to a colour border. So one could call an optical fixed star a minimum visibile. But it doesn’t
follow that one can say, for instance, that one’s field of vision consists of such parts! It would
only consist of them if we saw them. The visual image7
of a fixed star nebula in a telescope
consists of them in so far as we can distinguish between them. For these two expressions
simply mean the same thing.
If someone asked “Is our field of vision continuous or discontinuous?”, then one
would first need to know what continuity he was speaking of. We call a colour transition
continuous if we see no discontinuity in it.
5 (M): 57 |
6 (F): MS 112, p. 130v.
7 (V): The image
Minima Visibilia. 339e
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100
Farben und Farbenmischung.
1
Zu sagen, daß diese Farbe jetzt an einem Ort ist, 2
heißt, diesen Ort vollständig beschreiben.
– Zwei Farben, zwei Dampfspannungen, zwei Geschwindigkeiten, zwei elektrische
Spannungen, haben nicht zugleich an einem Punkt3
Platz. – Eine merkwürdige Gesellschaft,
die sich da zusammenfindet. Und auch der ”Punkt“ von dem ich rede, hat verschiedene
Bedeutungen.
Wenn also ”f(x)“ sagt, x sei jetzt an einem bestimmten Ort, so ist also ”f(a) & f(b)“ ein
Widerspruch. Warum nenne ich aber ”f(a) & f(b)“4
einen Widerspruch; da doch p & ~p die
Form des Widerspruchs ist? Heißt5
es einfach, daß das Zeichen ”fa & fb“ kein Satz ist,
wie etwa ”ffaa“ keiner ist? Unsere Schwierigkeit ist nur, daß wir doch das Gefühl haben,
daß hier ein Sinn vorliegt, wenn auch ein degenerierter (Ramsey). Daß, wenn ich ”und“
zwischen zwei Aussagen setze, ein lebendes Wesen entstehen muß und nicht etwas
Totes, wie wenn ich etwa ”a & f“ geschrieben hätte. Das ist ein sehr merkwürdiges und
sehr tiefliegendes Gefühl. Man müßte sich darüber klar werden, was die Worte ”daß hier
ein Sinn vorliegt“ sagen wollen.
Die Entscheidung darüber, ob ”fa & fb“ Unsinn ist, wie ”a & f“, könnte man so fällen:
Ist p & ~(fa & fb) = p, oder ist die linke Seite dieser Gleichung (und also die Gleichung)
Unsinn? – Kann ich nicht entscheiden, wie ich will?
Kann ich die Regel, die dem allen zu Grunde liegt, so schreiben: fa = (fa & ~(fb))? d.i.:
aus fa folgt ~fb.
Ich glaubte, als ich die ”Abhandlung“ schrieb (und auch später noch) daß fa = fa & ~fb
nur möglich wäre, wenn fa das logische Produkt aus irgend einem andern Satz und ~fb –
also fa = p & ~fb – wäre, und war der Meinung, fa (z.B. eine Farbenangabe) werde sich in
ein solches Produkt zerlegen lassen. Dabei hatte ich keine klare Vorstellung davon, wie ich
mir die Auffindung einer solchen Zerlegung dachte. Oder vielmehr: ich dachte wohl an
die Konstruktion eines Zeichens, das die richtige grammatische 6
Verwendung in jedem
Zusammenhang durch seine Beschaffenheit zum Ausdruck brächte (d.h., seine Regeln
ganz einfach gestaltete und in gewissem Sinne schon in sich trüge, wie jede übersichtliche
Notation); aber ich übersah, daß, wenn diese Umgestaltung des Satzes f(a) in seiner
Ersetzung durch ein logisches Produkt bestehen sollte, dann die Faktoren dieses Produkts
einen unabhängigen und uns bereits bekannten Sinn haben müßten.7
Als ich dann eine solche Analyse einer Farbangabe durchführen wollte, zeigte sich,8
was
es war, was ich mir unter der Analyse vorgestellt hatte. Ich glaubte die Farbangabe als ein
logisches Produkt r & s & t . . . auffassen zu können, dessen einzelne Faktoren die
Ingredienzien angaben (wenn es mehrere waren), aus denen die Farbe (color, nicht pigmentum)
1 (R): ∀ S. 51/1
2 (M): 92
3 (V): Ort
4 (O): aber ”f(a) & f(b)‘
5 (V): Bedeutet
6 (M): 92
7 (V): mußten.
8 (V): wollte, kam zum Vorschein,
473
474
475
340
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100
Colours and the Mixing of Colours.
1
To say that this colour is now at a location 2
means describing that location completely. –
There is no room at a single point3
for two simultaneous colours, steam pressures, speeds,
voltages. – A strange company it is that meets there. And the “point” that I’m talking about
has different meanings, too.
So if “f(x)” says that x is now in a particular location, then for that reason “f(a) & f(b)”
is a contradiction. But why am I calling “f(a) & f(b)” a contradiction since, after all, p & ~p
is the form of contradiction? Does it simply mean that the sign “fa & fb” isn’t a proposition,
as, say, “ffaa” isn’t? Our only difficulty is that in spite of this we have the feeling that
there is a sense here, albeit a degenerate one (Ramsey). That a living being has to come
about if I place “and” between two statements, and not something dead, as if I had
written “a & f”, for example. That is a very remarkable and very deeply embedded feeling.
One has to clarify for oneself what the words “that there is a sense here” are trying to say.
One might decide whether “fa & fb” is nonsense, like “a & f”, in such a way: Is p &
~(fa & fb) = p, or is the left side of this equation (and therefore the equation) nonsense? –
Can’t I decide as I please?
Can I write the rule that underlies all of this like this: fa = (fa & ~fb)? That is, ~fb
follows from fa.
When I wrote the Tractatus (and later as well) I believed that fa = fa & ~fb would be
possible only if fa were the logical product of some other proposition and ~fb – and therefore
fa = p & ~fb – and I was of the opinion that fa (e.g., a colour-statement) could be
analysed into such a product. In this context I had no clear idea about how I was imagining
the discovery of such an analysis. Or rather: I was probably thinking of the construction
of a sign that, because of its make-up, would express the correct grammatical 4
use in any
context (i.e. that would fashion its rules quite simply and in a certain sense would contain
them inherently, like any surveyable notation); but what I overlooked was that if this transformation
of the proposition f(a) were to consist in its being replaced by a logical product,
then the factors of this product would have to have an independent sense that was already
known to us.
Then, when I wanted to carry out such an analysis of a colour statement, it became appar-
ent5
what I had imagined analysis to be. I believed I could understand a colour statement as
a logical product r & s & t . . . , the discrete factors of which indicated the ingredients (if
there were several) that the colour (“colour”, not “pigment”) consisted of. Then, of course,
it also had to be said that these are all the ingredients, and this requirement has the effect
1 (R): ∀ p. 51/1
2 (M): 92
3 (V): location
4 (M): 92
5 (V): manifest
340e
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besteht. Es muß dann natürlich auch gesagt werden, daß dies alle Ingredienzien sind und
diese abschließende Bemerkung bewirkt, daß r & s & t & S mit r & s & t & u & S in
Widerspruch steht. Die Farbangabe hieße dann: ”an diesem Ort sind jetzt diese Farben (oder:
ist jetzt diese Farbe) und sonst keine“. D.h.: die Farbangabe, die in unsrer gewöhnlichen
Ausdrucksweise lautet ”dies (oder: hier) ist rot“ würde nun ”hier ist rot und sonst keine
Farbe“ lauten müssen;9
während die Angabe ”hier ist rot und blau“ bedeuten sollte, daß die
Farbe dieses Orts eine Mischfarbe aus rot und blau sei. Die Sätze10
nähmen da folgende Form
an: ”in dieser Farbe ist rot enthalten“, ”in dieser Farbe ist nur rot enthalten“, ”in dieser Farbe
ist nur rot und blau enthalten“, etc. – Aber dies gibt nicht die rechte Grammatik: Es müßte
das Vorhandensein eines roten Stiches ohne irgend einen andern Stich die rein rote
Färbung dieses Orts bedeuten; das scheint uns unsinnig und der Fehler klärt sich so auf: Es
muß im Wesen (in der Grammatik) dieses roten Stiches liegen, daß ein Mehr oder Weniger
von ihm möglich ist; ein rötliches Blau kann dem reinen Rot näher und weniger nahe liegen,
also in diesem Sinne mehr oder weniger Rot enthalten. Der Satz, welcher angibt, daß Rot
als Ingrediens einer Farbe hier vorhanden ist, müßte also irgendwie eine Quantität von Rot
angeben;11
dann aber muß dieser Satz auch außerhalb des logischen Produkts Sinn haben,
und es müßte also Sinn haben zu sagen, daß dieser Ort rein rot gefärbt ist und die und
die Quantität von Rot enthalte; und das hat keinen Sinn. Und wie verhält es sich mit den
einzelnen Sätzen, die einem Ort verschiedene Quantitäten, oder Grade, von Rot zuschreiben?
Nennen wir zwei solche q1r und q2r: sollen sich diese widersprechen? Angenommen q2 sei
größer als q1, dann könnte zwar unsere Festsetzung sein, daß q2r & q1r kein Widerspruch
sein solle (wie die Sätze ”in diesem Korb sind 4 Äpfel“ und ”in diesem Korb sind 3
Äpfel“, wenn das ”nur“ fehlt), aber dann müssen q2r und ~q1r einander widersprechen; und
daher müßte nach meiner alten Auffassung q2r ein Produkt aus q1r und einem andern Satz
sein. Dieser andre Satz müßte die von q1 auf q2 fehlende Quantität angeben und für ihn
bestünde daher die selbe Schwierigkeit. – Das Schema der Ingredientien paßt nicht auf
den Fall der Farbenmischung, wenn man unter ”Farben“ nicht Farbstoffe versteht.12
Und
auch in diesem Schema sind verschiedene Angaben über das verwendete Quantum eines
Bestandteils widersprechende Angaben; oder, wenn ich festsetze, daß p (=ich habe 3kg
Salz verwendet) und q (= ich habe 5kg Salz verwendet) einander nicht widersprechen
sollen, dann widersprechen einander doch q und ~p.13
Und es läuft alles darauf hinaus,
daß der Satz ”ich habe 2kg Salz verwendet“ nicht heißt ”ich habe 1kg Salz verwendet und
ich habe 1kg Salz verwendet“, daß also f(1 + 1) nicht gleich ist f(1) & f(1).
Unsere Erkenntnis ist eben, daß wir es mit Maßstäben, und nicht quasi mit isolierten
Teilstrichen zu tun haben.
Der Satz ”an einem Ort hat zu einer Zeit nur eine Farbe Platz“ ist natürlich ein
verkappter Satz der Grammatik. Seine Verneinung ist kein Widerspruch, widerspricht aber
einer Regel unserer angenommenen Grammatik.
Die Regeln über ”und“, ”oder“, ”nicht“, etc., die ich durch die W-F-Notation dargestellt
habe, sind ein Teil der Grammatik über diese Wörter, aber nicht die ganze.
Wenn ich z.B. sage, ein Fleck ist zugleich hellrot und dunkelrot, so denke ich dabei, daß
der eine Ton den andern deckt.
9 (V): Farbe” zu lauten haben;
10 (V): Farbangaben
11 (V): nennen;
12 (O): Das Schema der Ingredientien paßt nicht
auf den Fall der Farbenmischung, wenn man
unter ”Farben“ nicht Farbstoffe versteht,
(nicht).
13 (V): dann doch q und ~p.
476
477
341 Farben und Farbenmischung.
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that r & s & t & S contradicts r & s & t & u & S. Then the colour statement would run as
follows: “Now these colours (or: this colour) and no others are (is) in this place”. That is, a
colour statement that we usually express as “This (or Here) is red” would now have to be
phrased: “Here is red and no other colour”; whereas the statement “Here is red and blue”
would have to mean that the colour at this location was a colour mixture of red and blue.
In this case the propositions6
would take on the following form: “Red is contained in this
colour”, “Only red is contained in this colour”, “Only red and blue are contained in this
colour”, etc. – But this doesn’t give us the proper grammar: the presence of a red tinge,
without any other tinge, would have to mean a purely red colour at this location; that seems
nonsensical to us, and the mistake is cleared up this way: the possibility of more or less of
the red tinge must be contained inherently in (in the grammar of) the red tinge; a reddish
blue can be situated closer to or further away from pure red, i.e. can in this sense contain
more or less red. So the proposition stating that red is present as an ingredient of a colour
would somehow have to indicate7
a quantity of red; but then this proposition must also
make sense outside the logical product, and therefore it ought to make sense to say that
this location is coloured pure red and contains such and such a quantity of red; and that
doesn’t make any sense. And how about the individual propositions that ascribe different
quantities or degrees of red to a location? Let’s call two such propositions q1r and q2r; will
they contradict each other? Assuming that q2 is larger than q1, we could stipulate, to be sure,
that q2r & q1r is not a contradiction (like the sentences “There are 4 apples in this basket”
and “There are 3 apples in this basket”, when there is no “only”). But then q2r and ~q1r
will contradict one another; and therefore, according to my former view, q2r would have to
be a product of q1r and another proposition. This other proposition would have to state the
difference in quantity between q1 and q2, and therefore would pose the same difficulty. –
The schema of ingredients does not fit the case of a colour mixture, if by “colours” one
doesn’t understand pigments. And in this schema too, different statements about the
quantity of an ingredient used are contradictory; or, even if I stipulate that p (= I’ve used
3 kg of salt) and q (= I’ve used 5 kg of salt) are not supposed to be contradictory, then
q and ~p still contradict one another.8
And it all amounts to this: that the sentence “I’ve
used 2 kg of salt” doesn’t mean “I’ve used 1 kg of salt and I’ve used 1 kg of salt”, i.e. that
f(1+1) isn’t equal to f(1) & f(1).
We’ve simply come to understand that we are dealing with rulers and not with isolated
graduation marks, as it were.
Of course the proposition “There is only room for one colour in one location at one time”
is a disguised grammatical proposition. Its negation isn’t a contradiction, but it contradicts a
rule of our normal grammar.
The rules for “and”, “or”, “not”, etc. that I have represented via the T–F notation are
part of, but not the entire, grammar of these words.
When I say, for instance, that a patch is bright and dark red at the same time, I am thinking
all the while that one shade is covering the other.
6 (V): the indications of colour
7 (V): name
8 (V): then q and ~p still are.
Colours and the Mixing of Colours. 341e
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Hat es dann aber noch einen Sinn zu sagen, der Fleck habe den unsichtbaren, verdeckten
Farbton?
Hat es gar einen Sinn, zu sagen, eine vollkommen schwarze Fläche sei weiß, man sehe
nur das Weiß nicht, weil es vom Schwarz gedeckt sei? Und warum deckt das Schwarz das
Weiß und nicht Weiß das Schwarz?
Wenn ein Fleck eine sichtbare und eine unsichtbare Farbe hat, so hat er diese zwei Farben14
jedenfalls in ganz verschiedenem Sinne.
”Rot und grün gehen nicht zugleich an denselben Ort“ heißt nicht, sie sind tatsächlich
nie beisammen, sondern, es ist Unsinn zu sagen, sie seien zugleich am selben Ort und also
auch Unsinn zu sagen, sie seien nie zugleich am selben Ort.
Eine Mischfarbe, oder besser Zwischenfarbe, von blau und rot ist dies durch eine
interne Relation zu den Strukturen von blau und rot. Richtiger ausgedrückt: was wir
”eine Zwischenfarbe von blau und rot“ (oder ”blaurot“) nennen, heißt so, wegen einer
Verwandtschaft, die sich in der Grammatik der Wörter15
”blau“, ”rot“ und ”blaurot“ zeigt.
(Der Satz, der von einer internen Relation der Strukturen redet, entspringt schon aus einer
unrichtigen Vorstellung; aus der, welche in den Begriffen ”rot“, ”blau“, etc. komplizierte
Strukturen16
sieht; deren innere17
Konstruktion die Analyse zeigen muß.) Die Verwandtschaft
aber der reinen Farben und ihrer Zwischenfarbe ist elementarer Art, d.h., sie besteht nicht
darin, daß der Satz, welcher einem Gegenstand die Farbe blaurot zuschreibt, aus den Sätzen
besteht, die ihm die Farben rot und blau zuschreiben. Und so ist auch die Verwandtschaft
verschiedener Grade eines rötlichen Blau, z.B., eine elementare Verwandtschaft.
Es hat Sinn von einer Färbung zu sagen, sie sei nicht rein rot, sondern enthalte einen
gelblichen, oder bläulichen, weißlichen, oder schwärzlichen Stich; und es hat Sinn zu sagen,
sie enthalte keinen dieser Stiche, sondern sei reines Rot. Man kann in diesem Sinne von
einem reinen Blau, Gelb, Grün, Weiß, Schwarz reden, aber nicht von einem reinen Orange,
Grau, oder Rötlichblau. (Von einem ”reinen Grau“ übrigens wohl, sofern man damit ein
nicht-grünliches, nicht-gelbliches u.s.w. Weiß-Schwarz meint: und ähnliches gilt für ”reines
Orange“, etc.) D.h. der Farbenkreis hat vier ausgezeichnete Punkte. Es hat nämlich Sinn zu
sagen ”dieses Orange liegt (nicht in der Ebene des Farbenkreises, sondern im Farbenraum)
näher dem Rot als jenes“; aber wir können nicht, um das gleiche auszudrücken sagen ”dieses
Orange liegt näher dem Blaurot als jenes“ oder ”dieses Orange liegt näher dem Blau als jenes“.
Die Farbenmischung, von der hier die Rede ist, bringt der Farbenkreisel hervor, aber auch
er nicht, wenn ich ihn nur ruhend und dann in rascher Drehung sehe. Denn es wäre ja denkbar,
daß der Kreisel im ruhenden Zustand halb rot und halb gelb ist und daß er in rascher Drehung
(aus welchen Ursachen18
immer) grün erscheint. Vielmehr bringt der Farbenkreisel die
Mischung nur insofern zustande, als wir sie optisch als solche wahrnehmen können.19
Wenn
er sich nämlich nach und nach schneller und schneller dreht und wir sehen, wie aus rot und
gelb orange wird. Wir sind aber darin nicht dem Farbkreisel ausgeliefert; sondern, wenn durch
irgend einen unbekannten Einfluß, während der Kreisel sich schneller und schneller dreht,
die Farbe seiner Scheibe ins Weißliche überginge, so würden wir nun nicht sagen, die
Zwischenfarbe zwischen Rot und Gelb sei ein weißliches Orange. So wenig, wie wir sagen
würden 3 + 4 sei 6, wenn beim Zusammenlegen von 3 und 4 Äpfeln einer auf unbekannte
14 (V): er diese Farben
15 (V): in den grammatischen Bestimmungen
über die Wörter
16 (V): Gebäude
17 (V): innere Strukture
18 (V): aus welcher Ursache
19 (V): optisch kontrollieren können.
478
479
342 Farben und Farbenmischung.
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But does it then still make sense to say that the patch has the invisible, covered-up shade
of colour?
Going further, does it make sense to say that a completely black surface is white – it’s just
that one doesn’t see the white because it is covered by the black? And why does the black
cover the white and not the white the black?
When a patch has a visible and an invisible colour then in any case it has these two colours9
in a completely different sense.
“Red and green don’t come together at the same location at the same time” does not mean
that in actuality they are never together but, rather, that it is nonsense to say that they are
at the same place at the same time, and therefore also nonsense to say that they are never at
the same place at the same time.
A mixed colour (or better, an intermediate colour) made up of blue and red is a
mixed (intermediate) colour via an internal relationship to the structures of blue and red.
Expressed more precisely: What we call “an intermediate colour between blue and red” (or
“bluish-red”) is so called because of a relationship that shows in the grammar of the10
words
“blue”, “red” and “bluish-red”. (The very proposition that talks about an internal relationship
between the structures originates in an incorrect idea – in that idea that sees complicated
structures11
in the concepts “red”, “blue”, etc., structures whose inner12
construction
must be shown by analysis.) But the relationship of pure colours to their intermediate colour
is of an elementary kind. That is to say, it doesn’t require that the proposition that ascribes
the colour bluish-red to an object is made up of the propositions that ascribe the colours
blue and red to it. And in a like manner the relationship among various shades of a reddishblue,
for instance, is an elementary relationship.
It makes sense to say of a colour that it is not pure red, but contains a yellowish, or
bluish, whitish or blackish tinge; and it makes sense to say that it contains none of these
tinges but is pure red. In this sense one can speak of a pure blue, yellow, green, white, black,
but not of a pure orange, grey or reddish-blue. (Incidentally, one can talk of a pure grey, in
so far as one means by this a non-greenish, non-yellowish, etc., whitish-black; and something
similar holds for “pure orange”, etc.) That is to say, the colour circle has four special
points. For it does make sense to say “This orange is situated closer to red than that one
(not on the plane of the colour circle, but within colour space)”; but it’s not an equivalent
expression to say “This orange is situated closer to bluish-red than that one” or “This orange
is situated closer to blue than that one”.
A colour wheel produces the mixture of colours that we’re talking about here, but it won’t
do it if I see it only in a still position and then turning rapidly. For it’s conceivable, after
all, that the wheel is half red and half yellow when it’s still, and that (for whatever reasons13
)
it appears to be green when it spins rapidly. Rather, the colour wheel produces the mixture
only in so far as we can visually perceive it as a mixture.14
That is, when little by little it
spins faster and faster and we see how red and yellow turn into orange. But here we’re not
at the mercy of the wheel; rather, if through some unknown influence its colour turned
whitish as the wheel spun faster and faster, then we wouldn’t say that the intermediate colour
between red and yellow was a whitish orange. No more than we’d say that 3 + 4 is 6 if,
when putting 3 and 4 apples together one were to disappear in some unknown way and 6
9 (V): these colours
10 (V): in the grammatical regulations for the
11 (V): buildings
12 (V): inner structure
13 (V): reason
14 (V): can have it under our visual control.
Colours and the Mixing of Colours. 342e
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Weise verschwände und 6 Äpfel vor uns lägen. Ich gebrauche hier den Farbenkreisel nicht
zu einem Experiment, sondern zu einer Rechnung.
Es scheint außer dem Übergang von Farbe zu Farbe auf dem Farbenkreis noch einen bestimmten
anderen zu geben, den wir vor uns haben, wenn wir kleine Flecke der einen Farbe
mit kleinen Flecken der andern untermischt sehen. Ich meine hier natürlich einen gesehenen
Übergang.
Und diese Art des Übergangs gibt dem Wort ”Mischung“ eine neue Bedeutung, die mit
der Relation Zwischen auf dem Farbenkreis nicht zusammenfällt.
Man könnte es so beschreiben: Einen orangefarbigen Fleck kann ich mir entstanden denken
durch Untermischen kleiner roter und gelber Flecke, dagegen einen roten nicht durch
Untermischen von violetten und orangefarbigen. – In diesem Sinne ist Grau eine Mischung
von Schwarz und Weiß, und Rosa eine von Rot und Weiß, aber Weiß nicht eine Mischung
von Rosa und einem weißlichen Grün.
Nun meine ich aber nicht, daß es durch ein Experiment der Mischung festgestellt wird,
daß gewisse Farben so aus anderen entstehen. Ich könnte das Experiment etwa mit einer
rotierenden Farbenscheibe anstellen. Es kann dann gelingen, oder nicht gelingen, aber das
zeigt nur, ob der betreffende visuelle Vorgang auf diese physikalische Weise hervorzurufen
ist, oder nicht; es zeigt aber nicht, ob er möglich ist. Genau so, wie die physikalische
Unterteilung einer Fläche nicht die visuelle Teilbarkeit beweisen oder widerlegen kann. Denn
angenommen, ich sehe eine physikalische Unterteilung nicht mehr als visuelle Unterteilung,
sehe aber die nicht geteilte Fläche im betrunkenen Zustande geteilt, war dann die visuelle
Fläche nicht teilbar?
Man könnte sagen, Violett und Orange löschen einander bei der Mischung teilweise aus,
nicht aber Rot und Gelb.
Orange ist jedenfalls ein Gemisch von Rot und Gelb in einem Sinne, in dem Gelb kein
Gemisch von Rot und Grün ist, obwohl ja Gelb im Kreis zwischen Rot und Grün liegt.
Und wenn das offenbar Unsinn wäre, so fragt es sich, an welcher Stelle es anfängt Sinn
zu werden; d.h., wenn ich nun im Kreis von Rot und Grün aus dem Gelb näherrücke und
Gelb ein Gemisch der betreffenden beiden Farben nenne.
Ich erkenne nämlich im Gelb wohl die Verwandtschaft zu Rot und Grün, nämlich die
Möglichkeit zum Rötlichgelb und Grünlichgelb – und dabei erkenne ich doch nicht Grün
und Rot als Bestandteile von Gelb in dem Sinne, in dem ich Rot und Gelb als Bestandteile
von Orange erkenne.
Ich will sagen, daß Rot nur in dem Sinn zwischen Violett und Orange ist, wie Weiß
zwischen Rosa und Grünlichweiß. Aber ist in diesem Sinn nicht jede Farbe zwischen jenen
zwei anderen, oder doch zwischen solchen zweien, zu denen man auf unabhängigen Wegen
von der dritten gelangen kann.
Kann man sagen, in diesem Sinne liegt eine Farbe nur in einem gegebenen kontinuierlichen
Übergang zwischen zwei andern. Also etwa Blau zwischen Rot und Schwarz.
Wenn man mir sagt, die Farbe eines Flecks liege zwischen Violett und Rot, so verstehe
ich das und kann mir ein rötlicheres Violett als das Gegebene denken. Sagt man mir nun,
die Farbe liege zwischen diesem Violett und einem Orange – wobei mir kein bestimmter
kontinuierlicher Übergang in Gestalt eines gemalten Farbenkreises vorliegt – so kann ich
mir höchstens denken, es sei auch hier ein rötlicheres Violett gemeint, es könnte aber auch
ein rötlicheres Orange gemeint sein, denn eine Farbe, die, abgesehen von einem gegebenen
Farbenkreis in der Mitte zwischen den beiden Farben liegt, gibt es nicht und aus eben diesem
Grunde kann ich auch nicht sagen, an welchem Punkt das Orange, welches die eine Grenze
480
481
343 Farben und Farbenmischung.
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apples now lay in front of us. Here I am using the colour wheel not as an experiment but
for a calculation.
Besides the transition from colour to colour on the colour wheel, there seems to be another
specific transition that we encounter when we see little patches of the one colour mixed in
with little patches of the other one. Of course what I mean here is a transition that’s seen.
And this kind of transition gives the word “mixture” a new meaning, which doesn’t coincide
with the relation “between” on the colour wheel.
One could describe it like this: I can imagine an orange-coloured patch as having arisen
through the intermixing of small red and yellow patches, whereas I can’t imagine a red patch
as having arisen through the intermixing of violet and orange-coloured patches. – In this
sense grey is a mixture of black and white, and pink of red and white, but white isn’t a
mixture of pink and a whitish green.
I don’t mean, though, that a mixing experiment establishes that certain colours arise
from others in this way. I could carry out the experiment, say, with a rotating colour wheel.
Then it might succeed or not succeed, but all that shows is whether or not the respective
visual process can be produced by this physical means; it doesn’t show whether it is possible.
Just as the physical dissection of a surface can neither prove nor disprove its visual
divisibility. For let’s assume that I no longer see a physical division as a visual division,
but that when I’m drunk I see the undivided surface as divided – then wasn’t the visual
surface divisible?
One could say that violet and orange partially cancel each other out when they are mixed,
but red and yellow don’t.
In any case, orange is a mixture of red and yellow in a sense in which yellow isn’t a
mixture of red and green, even though, as we know, yellow lies between red and green on
the colour circle.
And if that were obviously nonsense, then the question is, at what point it begins to make
sense. That is, when coming from red and green on the circle, I now move closer to yellow
and call yellow a mixture of the respective two colours.
For with yellow I do recognize its kinship to red and green, i.e. the possibility of its changing
into reddish-yellow and greenish-yellow – but still, in this process I don’t recognize
green and red as components of yellow in the sense in which I recognize red and yellow as
components of orange.
I want to say that red is between violet and orange only in the sense in which white is
between pink and greenish-white. But in that sense isn’t any colour between two others,
or in any case between two colours that can be reached by routes that are independent
of it?
Can one say that, in this sense, a colour always lies in a given continuum between two
others? Thus, say, that blue lies between red and black?
If I am told that the colour of a spot lies between violet and red, I understand that and
can imagine a more reddish violet than the one that is defined as such. Now if I’m told
that the colour is somewhere between this violet and an orange – and I don’t have a particular
continuous transition in front of me in the form of a painted colour circle – then
at best I can imagine that what is meant here as well is a more reddish violet; but a more
reddish orange could be meant as well, for there is no such thing as a colour that is situated
in the middle of those two colours, independently of a given colour circle. And for this very
reason neither can I say at what point the orange that constitutes the one border is too close
to yellow still to be mixable with the violet; I simply cannot tell which orange lies 45 degrees
Colours and the Mixing of Colours. 343e
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bildet, schon zu nahe dem Gelb liegt, um noch mit dem Violett gemischt werden zu
können; ich kann eben nicht erkennen, welches Orange in einem Farbenkreis 45 Grad vom
Violett entfernt liegt. Das Dazwischenliegen der Mischfarbe ist eben hier kein anderes, als
das des Rot zwischen Blau und Gelb.
Wenn ich im gewöhnlichen Sinn sage, Rot und Gelb geben Orange, so ist hier nicht von
einer Quantität der Bestandteile die Rede. Wenn daher ein Orange gegeben ist, so kann ich
nicht sagen, daß noch mehr Rot es zu einem röteren Orange gemacht hätte (ich rede ja nicht
von Pigmenten) obwohl es natürlich einen Sinn hat, von einem röteren Orange zu sprechen.
Es hat aber z.B. keinen Sinn zu sagen, dies Orange und dies Violett enthalten gleichviel Rot.
Und wieviel Rot enthielte Rot?
Der Vergleich, den man fälschlicherweise zu machen geneigt ist, ist der der Farbenreihe
mit einem System von 2 Gewichten an einem Maßstab, durch deren Vermehrung oder
Verschiebung ich den Schwerpunkt des Systems beliebig verschieben kann.
20
Es ist nun Unsinn, zu glauben, daß, wenn ich die Schale A auf Violett halte und B in
das Feld Rot-Gelb hineinverschiebe, S sich gegen Rot hin bewegen wird.
Und wie ist es mit den Gewichten, die ich auf die Schalen lege: Heißt es denn etwas, zu
sagen, ”mehr von diesem Rot“? Wenn ich nicht von Pigmenten spreche. Das kann nur dann
etwas heißen, wenn ich unter reinem Rot eine bestimmte, vorher angenommene Anzahl von
Einheiten verstehe. Dann aber bedeutet die volle Anzahl dieser Einheiten nichts, als, daß
die Waagschale21
auf Rot steht. Es ist also mit den Verhältniszahlen wieder nur ein Ort der
Waagschale,22
aber nicht ein Ort und ein Gewicht angegeben.
Solange ich nun im Farbenkreis mit meinen beiden Grenzfarben – z.B. – im Gebiete
Blau – Rot stehe und die rötere Farbe gegen Rot verschiebe, so kann ich sagen, daß die
Resultante auch gegen Rot wandert. Überschreite ich aber mit der einen Grenzfarbe das Rot
und bewege mich gegen Gelb, so wird die Resultierende nun nicht röter! Die Mischung eines
gelblichen Rot mit einem Violett macht Violett nicht röter, als die Mischung von reinem
Rot und dem Violett. Daß das eine Rot nun gelber geworden ist, nimmt ja vom Rot etwas
weg und gibt nicht Rot dazu.
Man könnte das auch so beschreiben: Habe ich einen Farbtopf mit violettem Pigment
und einen mit Orange und nun vergrößere ich die Menge des der Mischung zugesetzten
Oranges,23
so wird zwar die Farbe der Mischung nach und nach aus dem Violett ins Orange
übergehen, aber nicht über das reine Rot.
Ich kann von zwei verschiedenen Tönen von Orange sagen, daß ich von keinem Grund
habe zu sagen, er liege näher an Rot als an Gelb. – Ein ”in der Mitte“ gibt es hier nicht. –
Dagegen kann ich nicht zwei verschiedene Rot sehen und im Zweifel sein, ob eines, und
welches, von ihnen das reine Rot ist. Das reine Rot ist eben ein Punkt, das Mittel zwischen
Gelb und Rot aber nicht.
20 (F): MS 108, S. 84.
21 (O): Wagschale
22 (O): Wagschale,
23 (O): des der Mischung zugesetztem Orange,
482
483
483
344 Farben und Farbenmischung.
A B
S
Blau Violett Rot Gelb
WBTC096-100 30/05/2005 16:15 Page 688
away from violet in a colour circle. Here the intermediate position of the mixed colour is
simply no different from the intermediate position of red between blue and yellow.
If I say that red and yellow make orange in the ordinary sense of these words, then I’m
not talking about a quantity of the components. Thus, if an orange is given, I can’t say that
still more red would have turned it into a redder orange, (for I’m not talking about pigments)
even though it makes sense of course to talk about a redder orange. But it doesn’t make any
sense, for instance, to say that this orange and this violet contain equal amounts of red. For
how much red would red contain?
The comparison that one is erroneously inclined to draw is that between the colour-chart
and a system of two weights on a measuring stick, where by increasing or moving the weights
I can shift the system’s centre of gravity at will.
15
But it is nonsense to think that S will move toward red if I keep pan A on violet and
move B into the red-yellow section.
And what about the weights that I put into the pans: Does it mean anything to say “more
of this red”, if I’m not talking about pigments? That can have meaning only if by “pure red”
I understand a particular, previously agreed-upon number of units. But then the largest number
of these units means nothing other than that the pan of the scales is on red. So once
again the proportional numbers only indicate a position of the pan on the scales, not a place
and a weight.
Now so long as I’m standing within the colour circle with my two transitional colours –
for example, in the area blue-red – and am moving the redder colour towards red, I can say
that the resulting colour is also moving towards red. But if I cross over red with the one
transitional colour and move towards yellow, then the resulting colour doesn’t turn redder!
The mixture of a yellowish red with violet doesn’t make the violet redder than the mixture
of pure red and violet. That the one red has now become yellower subtracts something from
the red; it doesn’t add red.
One could also describe it this way: If I have one paint can with violet pigment and another
with orange, and I increase the amount of orange that I add to the mixture of the two, then
the colour of the mixture will gradually change from violet to orange, but not via pure red.
I can say of two different shades of orange that I have no reason to say of either that it
lies closer to red than to yellow. – Here there is no such thing as “in the middle”. – Conversely,
I can’t see two different reds and be in doubt whether one of them, and which one, is pure
red. Pure red is simply a point, but the middle area between yellow and red isn’t.
15 (F): MS 108, p. 84.
Colours and the Mixing of Colours. 344e
A B
S
Blue Violet Red Yellow
WBTC096-100 30/05/2005 16:15 Page 689
Es ist freilich wahr, daß man von einem Orange sagen kann, es sei beinahe Gelb, also es
liege ”näher am Gelb als am Rot“ und Analoges von einem beinahe roten Orange. Daraus
folgt aber nicht, daß es nun auch eine Mitte im Sinne eines Punktes zwischen Rot und Gelb
geben müsse. Es ist eben hier ganz wie in der Geometrie des Gesichtsraums, verglichen mit
der euklidischen. Es ist hier eine andere Art von Quantitäten als die, welche durch unsere
rationalen Zahlen dargestellt werden. Die Begriffe näher und weiter sind hier überhaupt nicht
zu brauchen, oder sind irreführend, wenn wir diese Worte anwenden.
Auch so: Von einer Farbe zu sagen, sie liege zwischen Rot und Blau, bestimmt sie nicht
scharf (eindeutig). Die reinen Farben aber müßte ich eindeutig durch die Angabe bestimmen,
sie liegen zwischen gewissen Mischfarben. Also bedeutet hier das Wort ”dazwischen
liegen“ etwas anderes als im ersten Fall. D.h.: Wenn der Ausdruck ”dazwischen liegen“
einmal die Mischung zweier einfacher Farben, ein andermal den gemeinsamen einfachen
Bestandteil zweier Mischfarben bezeichnet, so ist die Multiplizität seiner Anwendung in jedem
Falle eine andere. Und das ist kein Gradunterschied, sondern ein Ausdruck dafür, daß es
sich um 2 ganz verschiedene Kategorien handelt.
Wir sagen, eine Farbe kann nicht zwischen Grüngelb und Blaurot liegen, in demselben
Sinne, wie zwischen Rot und Gelb, aber das können wir nur sagen, weil wir in diesem Falle
den Winkel von 45 Grad unterscheiden können; weil wir Punkte Gelb, Rot sehen. Aber eben
diese Unterscheidung gibt es im andern Fall – wo die Mischfarben als primär angenommen
werden – nicht. Hier könnten wir also sozusagen nie sicher sein, ob die Mischung noch möglich
ist oder nicht.24
Freilich könnte ich beliebige Mischfarben wählen und bestimmen, daß sie
einen Winkel von 45 Graden einschließen, das wäre aber ganz willkürlich, wogegen es nicht
willkürlich ist, wenn wir sagen, daß es keine Mischung von Blaurot und Grüngelb im ersten
Sinne gibt.
In dem einen Falle gibt die Grammatik also den ”Winkel von 45 Grad“ und nun glaubt
man fälschlich, man brauche ihn nur zu halbieren und den nächsten Abschnitt ebenso um
einen andern Abschnitt von 45 Grad zu kriegen. Aber hier bricht eben das Gleichnis des Winkels
zusammen.
Man kann freilich auch alle Farbtöne in einer geraden Linie anordnen, etwa mit den Grenzen
Schwarz und Weiß, wie das geschehen ist, aber dann muß man eben durch Regeln gewisse
Übergänge ausschließen und endlich muß das Bild auf der Geraden die gleiche Art des
topologischen Zusammenhangs bekommen, wie auf dem Oktaeder.25
Es ist dies ganz analog,
wie das Verhältnis der gewöhnlichen Sprache zu einer ”logisch geklärten“ Ausdrucksweise.
Beide sind einander vollkommen äquivalent, nur drückt die eine die Regeln der Grammatik
schon durch die äußere Erscheinung aus.
Wenn mir 2 nahe aneinander liegende – etwa – rötliche Farbtöne gegeben sind, so ist es
unmöglich darüber zu zweifeln, ob beide zwischen Rot und Blau, beide zwischen Rot und
Gelb, oder der eine zwischen Rot und Blau, der andere zwischen Rot und Gelb gelegen ist.
Und mit dieser Entscheidung haben wir auch entschieden, ob beide sich mit Blau, mit Gelb,
oder der eine sich mit Blau, der andere mit Gelb mischen, und das gilt, wie nahe immer
man die Farbtöne aneinander bringt, solange wir die Pigmente überhaupt der Farbe nach
unterscheiden können.
24 (E): Im Typoskript wird dieser Satz hier
wiederholt.
25 (O): Oktoeder.
484
485
345 Farben und Farbenmischung.
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To be sure, it’s true that one can say of an orange that it is almost yellow, and that therefore
it’s situated “closer to yellow than to red”, and one can say something analogous of an
orange that’s almost red. But from this it doesn’t follow that there must also be a middle,
in the sense of a point, between red and yellow. Things here are simply the way they are in
the geometry of visual space, as compared to Euclidean geometry. Here the quantities are
of a different kind than those represented by our rational numbers. The concepts “closer”
and “further” can’t be used here at all, or are misleading when we apply these words.
One could also describe it like this: To say of a colour that it is situated between red and
blue doesn’t define it sharply (unambiguously). To define the pure colours unambiguously I
would have to say that they are situated between certain mixed colours. So here the words
“to be situated between” mean something different than in the first case. That is: If the expression
“to be situated between” sometimes refers to the mixture of two pure colours, but at
other times refers to the pure component that two mixed colours have in common, then
in each case the multiplicity of what it is applied to is different. And this is not a difference
of degree, but an expression of the fact that we are dealing with two completely different
categories.
We say that a colour can’t be situated between greenish-yellow and bluish-red in the
same sense as between red and yellow, but we can only say that because in this case we
can discern an angle of 45 degrees; because we see points of yellow, and red. But there’s no
discerning this in the other case – where mixed colours are taken as primary. So here we
could never be certain, so to speak, whether or not the mixture is still possible. To be sure,
I could pick out mixed colours at will, and stipulate that they form an angle of 45 degrees,
but that would be completely arbitrary, whereas it isn’t arbitrary when we say that there is
no such thing as a mixture of bluish-red and greenish-yellow in the first sense.
So in the one case grammar provides the “angle of 45 degrees”, and then one wrongly
believes that all one has to do is to halve it, and then do likewise to the next sector, in order
to get a new sector of 45 degrees. But here the simile of the angle simply breaks down.
To be sure, one can also arrange all the shades of colour along a straight line, say with
black and white as borders, as is often done. If one then applies some rules so as to exclude
certain transitions, the resulting picture of the line will then have the same kind of topological
nexus as on an octahedron. This is completely analogous to the relationship between
ordinary language and a mode of expression that is “logically clarified”. Both are completely
equivalent to each other; it’s just that in the one the rules of grammar are expressed simply
by its outward appearance.
If I am given two shades of colour that are close together – let’s say they’re reddish – then
I can’t possibly be in doubt whether both are situated between red and blue, both between
red and yellow, or the one between red and blue and the other between red and yellow. And
when we determine this we will also have determined whether both are mixed with blue, or
with yellow, or whether the one is mixed with blue and the other with yellow; and this holds
true no matter how close together one makes the shades of colour, so long as we’re able to
distinguish between the pigments in these colours at all.
Colours and the Mixing of Colours. 345e
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Idealismus, etc.486
WBTC101-107 30/05/2005 16:15 Page 692
Idealism, etc.
WBTC101-107 30/05/2005 16:15 Page 693
101
Die Darstellung des unmittelbar
Wahrgenommenen.
Es kommt uns vor, als wäre die Erinnerung eine etwas sekundäre Art der Erfahrung,
im Vergleich zur Erfahrung des Gegenwärtigen. Wir sagen ”daran können wir uns nur
erinnern“. Als wäre in einem primären Sinn die Erinnerung ein etwas schwaches und
unsicheres Bild dessen, was wir ursprünglich in voller Deutlichkeit vor uns hatten.
In der physikalischen Sprache stimmt das: Ich sage ”ich kann mich nur undeutlich an dieses
Haus erinnern“.
Und warum es nicht dabei sein Bewenden haben lassen? Denn diese Ausdrucksweise
sagt ja doch alles, was wir sagen wollen und was sich sagen läßt! Aber wir wollen sagen,
daß es sich auch noch anders sagen läßt; und das ist wichtig.
In dieser andern Ausdrucksweise wird der Nachdruck gleichsam auf etwas anderes gelegt.
Die Worte ”scheinen“, ”Irrtum“, etc. haben nämlich eine gewisse Gefühlsbetonung, die den1
Phänomenen nicht wesentlich ist. Sie hängt irgendwie mit dem Willen und nicht bloß mit
der Erkenntnis zusammen.
Wir reden z.B. von einer optischen Täuschung und verbinden mit diesem Ausdruck die
Idee eines Fehlers, obwohl ja nicht wesentlich ein Fehler vorliegt: und wäre im Leben für
gewöhnlich das Aussehen wichtiger, als die Resultate der Messung, so würde auch die Sprache
zu diesen Phänomenen eine andere Einstellung zeigen.
Es gibt nicht – wie ich früher glaubte – eine primäre Sprache im Gegensatz zu unserer
gewöhnlichen, der ”sekundären“. Aber insofern könnte man im Gegensatz zu unserer Sprache
von einer primären reden, als in dieser keine Bevorzugung gewisser Phänomene vor anderen
ausgedrückt sein dürfte; sie müßte sozusagen absolut sachlich sein.
Es ist jetzt an der Zeit, Kritik am Worte ”Sinnesdatum“ zu üben. Sinnesdatum ist die
Erscheinung dieses Baumes, ob nun ”wirklich ein Baum dasteht“ oder eine Attrappe, ein
Spiegelbild, eine Halluzination etc. Sinnesdatum ist die Erscheinung des Baumes, und was
wir sagen wollen ist, daß diese sprachliche Darstellung nur eine Beschreibung, aber nicht
die wesentliche ist. Genau so, wie man von dem Ausdruck ”mein Gesichtsbild“ sagen kann,
daß es nur eine Form der Beschreibung, aber nicht etwa die einzig mögliche und richtige ist.
Die Ausdrucksform ”die Erscheinung dieses Baumes“ enthält nämlich die Anschauung, als
bestünde ein notwendiger Zusammenhang dessen, was wir diese Erscheinung nennen, mit
der ”Existenz eines Baumes“ und zwar, entweder durch eine wahre Erkenntnis oder einen
Irrtum. D.h., wenn von der ”Erscheinung2
eines Baumes“ die Rede ist, so hielten wir entweder
etwas für einen Baum, was einer ist, oder etwas, was keiner ist. Dieser Zusammenhang aber
besteht nicht.
1 (O): dem 2 (O): der Erscheinung
487
488
347
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101
The Representation of what is
Immediately Perceived.
It seems to us that, compared to the experience of the present, memory is a somewhat
secondary kind of experience. We say “We can only remember something”. As if, in some
primary sense, memory were a rather weak and uncertain image of what was originally
before us with complete clarity.
That is correct in physical language, for I say “I can only vaguely remember this house”.
And why not leave it at that? For, after all, this mode of expression says everything we
want to say and that can be said! But we want to say that it can be said in yet another way;
and that is important.
In this other mode of expression, it’s as if the emphasis were placed on something else.
For the words “to seem”, “error”, etc. have a certain emotional emphasis that isn’t essential
to the phenomena. This emphasis is somehow connected to the will, and not merely to
knowledge.
We talk about an optical illusion, for instance, and associate the idea of a mistake with
this expression even though, really, this is not a case of a mistake; and if appearance were
usually more important in life than the results of measurement, language too would present
a different attitude towards such phenomena.
Contrary to my previous belief – there is no such thing as a primary language, as opposed
to our ordinary one, to “secondary” language. But one could speak about a primary language
as opposed to ours in so far as in the former, expressions that privilege certain phenomena
above others would be forbidden; it would have to be absolutely objective, so to speak.
The time has now come to subject the expression “sense datum” to criticism. A sense
datum is the appearance of this tree, whether there “really is a tree standing there”, or a
prop, a mirror image, a hallucination, etc. A sense datum is the appearance of the tree, and
what we want to say is that this linguistic representation is just one description, but not the
essential one. Just as one can say that the expression “my visual image” is only one form of
description, but by no means the only possible and correct one. For the form of expression
“the appearance of this tree” incorporates the idea that there is a necessary connection between
what we call this appearance and the “existence of a tree”, a connection made either by a
true perception or a mistake. That is, if we’re talking about the “appearance of a tree”, then
either we take something for a tree that is one, or something that isn’t one. But there is no
such connection.
347e
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Die Idealisten möchten der Sprache vorwerfen, daß sie das Sekundäre als primär und
das Primäre als sekundär darstellt. Aber das ist nur in diesen unwesentlichen, und mit
der Erkenntnis nicht zusammenhängenden Wertungen der Fall (”nur“ die Erscheinung).
Davon abgesehen enthält die gewöhnliche Sprache keine Entscheidung über primär und
sekundär. Es ist nicht einzusehen, inwiefern der Ausdruck ”die Erscheinung eines Baumes“
etwas dem Ausdruck ”Baum“ sekundäres darstellt. Der Ausdruck ”nur ein Bild“ geht auf
die Vorstellung zurück, daß wir das Bild eines Apfels nicht essen können.
Zur Frage nach der Existenz der Sinnesdaten. Man sagt, wenn etwas rot scheint, so muß
Etwas rot gewesen sein; wenn etwas kurze Zeit zu dauern schien, so muß Etwas kurze Zeit
gedauert haben; etc. Man könnte nämlich fragen: Wenn etwas rot schien, woher wissen wir
denn, daß es gerade rot schien. Handelt es sich da um eine erfahrungsmäßige Zuordnung
dieses Scheins mit3
dieser Wirklichkeit? Wenn etwas ”die Eigenschaft φ zu haben schien“,
woher wissen wir, daß es diese Eigenschaft zu haben schien –. Was für ein Zusammenhang
besteht zwischen ”es scheint so“ und ”es ist so“.
Vor allem kann der Schein recht haben,4
oder unrecht. – Er ist auch in einem Sinne
erfahrungsgemäß mit der Wirklichkeit verbunden. Man sagt ”das scheint Typhus zu
sein“ und das heißt, diese Symptome sind erfahrungsgemäß mit jenen Erscheinungen
verbunden. Wenn ich sage ”das scheint rot zu sein“ und dann ”ja, es ist wirklich rot“,
so habe ich für die zweite Entscheidung einen Test angewandt, der unabhängig von der
ersten Erscheinung war.
Die Hypothese kann so aufgefaßt werden, daß sie nicht über die Erfahrung hinausgeht,
d.h. nicht der Ausdruck der Erwartung künftiger Erfahrung ist. So kann der Satz ”es scheint
vor mir auf dem Tisch eine Lampe zu stehen“ nichts weiter tun, als meine Erfahrung (oder,
wie man sagt, unmittelbare Erfahrung) zu beschreiben.
Wie verhält es sich mit der Genauigkeit dieser Beschreibung. Ist es richtig zu sagen: Mein
Gesichtsbild ist so kompliziert, es ist unmöglich, es ganz zu beschreiben? Dies ist eine sehr
fundamentale Frage.
Das scheint nämlich zu sagen, daß man von Etwas sagen könnte, es könne nicht
beschrieben werden, oder nicht mit den jetzt vorhandenen Mitteln, oder (doch) man wisse
nicht, wie es beschreiben. (Die Frage, das Problem, in der Mathematik.)
Wie ist denn das Es gegeben, das ich nicht zu beschreiben weiß? – Mein Gesichtsbild
ist ja kein gemaltes Bild, oder der Ausschnitt der Natur den ich sehe, daß ich es näher
untersuchen könnte. – Ist dieses Es schon artikuliert, und die Schwierigkeit nur es in Worten
darzustellen, oder soll es noch auf seine Artikulation warten?
”Die Blume war von einem rötlichgelb, welches ich aber nicht genauer (oder, nicht genauer
mit Worten) beschreiben kann.“ Was heißt das?
”Ich sehe es vor mir und könnte es malen.“
Wenn man sagt, man könnte diese Farbe nicht mit Worten genauer beschreiben, so denkt
man (immer) an eine Möglichkeit einer solchen Beschreibung (freilich, denn sonst hätte der
Ausdruck5
”genaue Beschreibung“ keinen Sinn) und es schwebt einem dabei der Fall einer
Messung vor, die wegen unzureichender Mittel nicht ausgeführt wurde.
Es ist mir nichts zur Hand, was diese oder eine ähnliche Farbe hätte.
3 (V): und
4 (V): haben.
5 (V): hätte das Wort
489
490
348 Die Darstellung des unmittelbar Wahrgenommenen.
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Idealists like to reproach language by saying that it represents what is secondary as
primary and what is primary as secondary. But that is only the case with these insignificant
judgements that have nothing to do with knowledge (“only” an appearance). Aside from
that, ordinary language doesn’t differentiate between what is primary or secondary. There
is absolutely no good reason why the expression “the appearance of a tree” should represent
something that is secondary in relation to the expression “tree”. The expression “only an
image” goes back to the idea that we can’t eat the image of an apple.
Concerning the question of the existence of sense data. We say that if something seems
to be red then something must have been red; if something seemed to last for a short time,
then something must have lasted for a short time; etc. In this regard one could ask: If something
seemed to be red, how do we know that it seemed exactly red? Is it a matter of an
empirical link between this appearance and this reality? If something “seemed to have the
property φ” how do we know that it seemed to have this property? –. What kind of a
connection is there between “That’s the way it seems” and “That’s the way it is”?
Most importantly, appearance can be accurate or deceptive.1
– In one sense, it’s also
connected to reality empirically. We say “That seems to be typhoid”, and that means that
these symptoms are empirically connected to those phenomena. If I say “That seems to be
red”, and then “Yes, it really is red”, I’ve applied a test to reach the second determination
that was independent of the first phenomenon.
A hypothesis can be understood in such a way that it doesn’t exceed experience, i.e. that
it isn’t an expression of an expectation of future experience. Thus the proposition “There
seems to be a lamp standing in front of me on the table” often does nothing more than describe
my experience (or my immediate experience, as we say).
What about the accuracy of this description? Is it correct to say: My visual image is
so complicated that it’s impossible to describe it completely? This is a very fundamental
question.
For that seems to be saying that one could say of something that it can’t be described, or
not with the means that are now available, or (at least) that one doesn’t know how to describe
it. (The question, the problem, in mathematics.)
How has the “it” that I don’t know how to describe been given? – After all, my visual
image isn’t a painted picture, or a facet of nature, that I see, such that I could examine it
more closely. – Is this “it” already articulated, and the only difficulty is to represent it in
words, or is it supposed to await its articulation?
“The flower had a reddish-yellow colour, but one that I can’t describe more accurately
(or more accurately in words).” What does that mean?
“I see it in front of me and could paint it.”
If we say that we can’t describe this colour more accurately in words, then we’re
(always) thinking of a possibility of such a description (of course – otherwise the expres-
sion2
“accurate description” would make no sense), and here we have in mind the case
of a measurement that wasn’t carried out because of inadequate means.
There’s nothing at hand that has this or a similar colour.
1 (V): can be accurate. 2 (V): words
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Wenn man sagt, man könne das Gesichtsbild nicht ganz beschreiben, meint man, man
kann keine Beschreibung geben, nach der man sich dieses Gesichtsbild genau reproduzieren
könnte.
Aber was heißt hier ”genaue Reproduktion“? Hier liegt selbst wieder ein falsches Bild
zu Grunde.
Was ist das Kriterium der genauen Reproduktion?
Wir können von dem Gesichtsbild nicht weiter reden, als unsere Sprache jetzt reicht.
Und auch nicht weiter6
meinen (denken), als unsere Sprache reicht.7
(Nicht mehr meinen,
als wir sagen können.)
Einer der gefährlichsten Vergleiche ist der des Gesichtsfelds mit einer gemalten Fläche
(oder, was auf dasselbe hinauskommt, einem farbigen räumlichen Modell).
Hiermit hängt es zusammen: Könnte ich denn das Gesichtsbild ”mit allen Einzelheiten“
wiedererkennen? Oder vielmehr, hat diese Frage überhaupt einen Sinn?
Denn als einwandfreiste Darstellung des Gesichtsbildes erscheint uns immer noch
ein gemaltes Bild oder Modell. Aber, daß die Frage nach dem ”Wiedererkennen in allen
Einzelheiten“ sinnlos ist, zeigt schon, wie inadäquat Bild und Modell sind.
Phänomenologische Sprache: Die Beschreibung der unmittelbaren Sinneswahrnehmung,
ohne hypothetische Zutat. Wenn etwas, dann muß doch wohl die Abbildung durch ein gemaltes
Bild oder dergleichen eine solche Beschreibung der unmittelbaren Erfahrung sein. Wenn
wir also z.B. in ein Fernrohr sehen und die gesehene Konstellation aufzeichnen oder
malen. Denken wir uns sogar unsere Sinneswahrnehmung dadurch reproduziert, daß zu ihrer
Beschreibung ein Modell erzeugt wird, welches von einem bestimmten Punkt gesehen, diese
Wahrnehmungen erzeugt; das Modell könnte mit einem Kurbelantrieb in die richtige
Bewegung gesetzt werden und wir könnten durch Drehen der Kurbel die Beschreibung
herunterlesen. (Eine Annäherung hierzu wäre eine Darstellung im Film.)
Ist das keine Darstellung des Unmittelbaren – was sollte eine sein? – Was noch unmittelbarer
sein wollte, müßte es aufgeben, eine Beschreibung zu sein. Es kommt dann vielmehr
statt einer Beschreibung jener unartikulierte Laut heraus, mit dem manche Autoren die
Philosophie gerne anfangen möchten. (”Ich habe, um mein Wissen wissend, bewußt etwas“.
Driesch.8
)
”Was wir im physikalischen Raum denken, ist nicht das Primäre, das wir nur mehr oder
weniger anerkennen können; sondern, was vom physikalischen Raum wir erkennen können,
zeigt uns, wie weit das Primäre reicht und wie wir den physikalischen Raum zu deuten haben.“
Es scheint ein Einwand gegen die Beschreibung des unmittelbar Erfahrenen zu sein: ”für
wen beschreibe ich’s?“ Aber wie, wenn ich es abzeichne? Und die Beschreibung muß immer
ein Nachzeichnen sein.
Und soweit eine Person für das Verstehen in Betracht kommt, steht die meine und
die des Anderen auf einer Stufe. Es ist doch hier ebenso wie mit den Zahnschmerzen.
Beschreiben ist nachbilden, und ich muß nicht notwendigerweise für irgendjemand
nachbilden.
6 (V): mehr
7 (V): sagt.
8 (E): Vgl. Hans Driesch, Wirklichkeitslehre: ein
metaphysischer Versuch, Leipzig: Reinicke, 1917,
S. 9.
491
492
493
349 Die Darstellung des unmittelbar Wahrgenommenen.
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If we say that we can’t completely describe our visual image, we mean that we can’t give
any description that someone could use to reproduce this visual image accurately.
But what does “accurate reproduction” mean here? This expression itself is once again
based on a false image.
What is the criterion for accurate reproduction?
We can’t speak of our visual image beyond the current reach of our language. And neither
can we mean (think) beyond the reach of our language.3
(We can’t mean more than we can
say.)
One of the most dangerous comparisons is that between one’s field of vision and a painted
surface (or – what amounts to the same thing – a coloured three-dimensional model).
That is connected with this: Could I recognize my visual image “in all of its details”? Or
rather, does this question make any sense in the first place?
For it is still a painted picture or a model that seems to us to be the least objectionable
representation of a visual image. But the fact that the question about “recognizing in all of
its details” makes no sense already shows how inadequate the picture and the model are.
Phenomenological language: the description of immediate sense perception without
any hypothetical addition. If anything, then surely a portrayal in a painted picture or the
like must be such a description of immediate experience. Such as when we look through a
telescope, for instance, and draw or paint the constellation we see. Let’s even imagine that
our sense perception is reproduced by creating a model for describing it, a model that,
seen from a certain point, produces these perceptions; this machine could be set into proper
motion with a crank drive, and by turning the crank, we could read off the description. (An
approximation to this would be a representation in film.)
If that isn’t a representation of the immediate – then what can be? – Anything that
claimed to be even more immediate would have to forego being a description. Instead of a
description, what results in that case is that inarticulate sound with which some authors would
like to begin philosophy. (“Knowing of my knowing, I consciously possess something.” –
Driesch.4
)
“What we think of as being in physical space isn’t what is primary, which we can only
recognize more or less; rather the part of physical space we can perceive shows us how far
the primary extends, and how we are to interpret physical space.”
This seems to be an objection to describing immediate experience: “For whom am I
describing it?” But what if I copy it? And a description must always be a copying.
And in so far as a person is involved in understanding a description, I and anyone else
are on the same level. After all, it’s the same here as with a toothache. Describing is copying,
and I don’t necessarily have to copy for anyone.
3 (V): (think) more than our language says. 4 (E): Cf. Hans Driesch, Wirklichkeitslehre: ein
metaphysicher Versuch, Leipzig: Reinicke, 1917,
p. 9.
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Wenn ich mich mit der Sprache dem Andern verständlich mache, so muß es sich hier
um ein Verstehen im Sinne des Behaviourism handeln. Daß er mich verstanden hat, ist eine
Hypothese, wie, daß ich ihn verstanden habe.
”Für wen würde ich meine unmittelbare Erfahrung beschreiben? Nicht für mich, denn
ich habe sie ja: und nicht für jemand andern, denn der könnte sie nie aus der Beschreibung
entnehmen?“ – Er kann sie soviel und so wenig aus der Beschreibung entnehmen, wie
aus einem gemalten Bild. Die Vereinbarungen über die Sprache sind doch mit Hilfe von
gemalten Bildern (oder was diesen9
gleichkommt) getroffen worden. Und, unserer10
gewöhnlichen Ausdrucksweise nach, entnimmt er doch aus einem gemalten Bild etwas.
9 (O): diesem 10 (V): unserer Spr
350 Die Darstellung des unmittelbar Wahrgenommenen.
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If I use language to make myself understood to someone, then this has to be an instance
of understanding in the behaviourist sense. That he understood me is a hypothesis, as is my
having understood him.
“For whom would I describe my immediate experience? Not for myself, for I’m the one
who has it; and not for someone else, because he could never infer it from my description.”
– He can infer it as much and as little from my description as from a painted picture. After
all, our linguistic conventions have been arrived at with the help of painted pictures (or their
equivalents). And according to our usual way of speaking, he does infer something from a
painted picture.
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102
”Die Erfahrung im gegenwärtigen
Moment, die eigentliche Realität.“
Es ist nämlich die Anschauung aufzugeben, daß, um vom Unmittelbaren zu reden, wir von
dem Zustand in einem Zeitmoment reden müßten. Diese Anschauung ist darin ausgedrückt,
wenn man sagt: ”alles, was uns gegeben ist, ist das Gesichtsbild und die Daten der übrigen
Sinne, sowie die Erinnerung, in dem gegenwärtigen Augenblick“. Das ist Unsinn; denn was
meint man mit dem ”gegenwärtigen Augenblick“? Dieser Vorstellung liegt vielmehr schon
ein physikalisches Bild zu Grunde, nämlich das vom Strom der Erlebnisse, den ich nun in
einem Punkt1
quer durchschneide. Es liegt hier eine ähnliche Tendenz und ein ähnlicher
Fehler vor, wie beim Idealismus (oder Solipsismus).
Der Zeitmoment, von dem ich sage, er sei die Gegenwart, die alles enthält, was mir gegeben
ist, gehört selbst zur physikalischen Zeit.
Denn, wie ist so ein Moment bestimmt? Etwa durch einen Glockenschlag? Und kann ich
denn nun die ganze, mit diesem Schlag gleichzeitige Erfahrung wirklich beschreiben? Wenn
man daran denkt es zu versuchen, wird man sofort gewahr, daß es eine Fiktion ist, wovon
wir reden.
Wir stellen uns das Erleben wie einen Filmstreifen vor, so daß man sagen kann: dieses
Bild, und kein anderes, ist in diesem Augenblick vor der Linse.
Aber nur im Film kann man von einem in diesem Moment gegenwärtigen Bild reden;
nicht, wenn man aus dem physikalischen Raum und seiner Zeit in den Gesichtsraum und
seine Zeit übergeht.
Es ist eben irreführend, zu sagen ”das Gedächtnis sagt mir, daß dies dieselbe Farbe ist
etc.“. Sofern es mir etwas sagt, kann es mich auch täuschen (d.h. etwas falsches sagen).
Wenn ich die unmittelbar gegebene Vergangenheit beschreibe, so beschreibe ich mein
Gedächtnis, und nicht etwas, was dieses Gedächtnis anzeigt. (Wofür dieses Gedächtnis ein
Symptom wäre.)
Und ”Gedächtnis“ bezeichnet hier – wie früher ”Gesicht“ und ”Gehör“ – auch nicht ein
psychisches Vermögen, sondern einen bestimmten Teil der logischen Struktur unserer Welt.
Was wir die Zeit im Phänomen (specious present) nennen können, liegt nicht in der Zeit
(Vergangenheit, Gegenwart und Zukunft) der Geschichte, ist keine Strecke der Zeit.
Während der Vorgang der ”Sprache“2
in der homogenen geschichtlichen Zeit abläuft.
(Denke an den Mechanismus zur Beschreibung der unmittelbaren Wahrnehmung.)
1 (V): nun an einer Stelle 2 (V): Während, was wir unter ”Sprache“
verstehen,
494
495
351
WBTC101-107 30/05/2005 16:15 Page 702
102
“The Experience at the Present
Moment, Actual Reality.”
We have to give up the view that in order to speak about the immediate, we must speak
about a state at a moment in time. This view is expressed by saying: “All that is given to us
at the present moment is our visual image and the data of the other senses, as well as
memory”. That is nonsense; for what does one mean by the “present moment”? This idea
is already based on a physical image, that of the stream of experiences that I’m now bisecting
at a point.1
Here there’s a similar tendency and a similar mistake as with idealism
(or solipsism).
That moment in time – of which I say that it is the present, and which contains everything
that has been given to me – itself belongs to physical time.
For how is such a moment determined? By the stroke of a clock, perhaps? And can I really
describe the entire experience that is simultaneous with this stroke? If you think about
trying it you’ll notice immediately that what we’re talking about is fiction.
We imagine that experience is like a film strip, and that we can say: This picture and no
other is in front of the lens at this moment.
But only in a film can one talk about a picture that is present at this moment; not when
one moves from physical space and its time to visual space and its time.
It’s simply misleading to say “Memory tells me that this is the same colour, etc.” In so
far as it tells me something, it can also deceive me (i.e. say something false).
If I describe the immediately given past, then I’m describing my memory, and not
something that this memory signifies. (For which this memory would be a symptom.)
And – like “vision” and “hearing” earlier – neither does “memory” signify a psychological
ability, but a particular part of the logical structure of our world.
What we might call phenomenal time (the specious present) doesn’t occur in historical
time (past, present and future). It is not a stretch of time, whereas the process of2
“language” runs its course in homogeneous historical time. (Think of our earlier apparatus
for describing immediate perception.)
1 (V): spot. 2 (V): whereas what we understand by
351e
WBTC101-107 30/05/2005 16:15 Page 703
(Von welcher Wichtigkeit ist denn diese Beschreibung des gegenwärtigen Phänomens,
die für uns gleichsam zur fixen Idee werden kann. Daß wir darunter leiden, daß die
Beschreibung nicht das beschreiben kann, was beim Lesen der Beschreibung vor sich geht.
Es scheint, als wäre die Beschäftigung mit dieser Frage geradezu kindisch und wir in eine
Sackgasse hineingeraten. Und doch ist es eine bedeutungsvolle Sackgasse, denn in sie lockt
es Alle zu gehen; als wäre dort die letzte Lösung der philosophischen Probleme zu suchen.
– Es ist, als käme man mit dieser Darstellung des gegenwärtigen Phänomens in einen
verzauberten Sumpf, wo alles Erfaßbare verschwindet.)
Anderseits brauchen wir eine Ausdrucksweise, die Vorgänge3
des Gesichtsraums getrennt
von den Erfahrungen andrer Art darstellt.
(Wir befinden uns mit unserer Sprache (als physischer Erscheinung) sozusagen nicht im
Bereich des projizierten Bildes auf der Leinwand, sondern im Bereich des Films, der durch
die Laterne geht. Und wenn ich zu dem Vorgang auf der Leinwand Musik machen will,
muß das, was sie hervorruft, sich wieder im Gebiet des Films abspielen. Das gesprochene
Wort im Sprechfilm, das die Vorgänge auf der Leinwand begleitet, ist ebenso fliehend,4
wie
diese Vorgänge, und nicht das Gleiche wie der Tonstreifen. Der Tonstreifen begleitet nicht
das Spiel auf der Leinwand.)
Ein Gedanke über die Darstellbarkeit der unmittelbaren Realität durch die Sprache:
”Der Strom des Lebens, oder der Strom der Welt, fließt dahin, und unsere Sätze werden,
sozusagen, nur in Augenblicken verifiziert. Unsere Sätze werden nur von der Gegenwart
verifiziert. – Sie müssen also so gemacht sein, daß sie von ihr verifiziert werden können. Sie
müssen das Zeug haben, um von ihr verifiziert werden zu können. Dann sind sie also in
irgendeiner Weise mit der Gegenwart kommensurabel und dies können sie nicht sein5
trotz
ihrer raum-zeitlichen Natur, sondern diese muß sich zur Kommensurabilität verhalten,
wie die Körperlichkeit eines Maßstabes zu seiner Ausgedehntheit, mittels6
der er mißt. Im
Falle des Maßstabes kann man auch nicht sagen: ,Ja, der Maßstab mißt die Länge trotz
seiner Körperlichkeit; freilich, ein Maßstab, der nur Länge hätte, wäre das Ideal, wäre der
reine Maßstab‘. Nein, wenn ein Körper Länge hat, so kann es keine Länge ohne einen Körper
geben – und wenn ich auch verstehe, daß in einem bestimmten Sinn nur die Länge des
Maßstabs mißt, so bleibt doch, was ich in die Tasche stecke der Maßstab, – der Körper und
nicht die Länge.“
”Nur die Erfahrung des gegenwärtigen Augenblicks hat Realität“. – Soll das heißen, daß
ich heute früh nicht aufgestanden bin? Oder, daß ein Ereignis, dessen ich mich in diesem
Augenblick nicht entsinne,7
nicht stattgefunden hat? – Soll hier ”gegenwärtige Erfahrung“
im Gegensatz stehen zu zukünftiger und vergangener Erfahrung? Oder ist es ein Beiwort,
wie das Wort ”rational“ in ”rationale Zahl“, so daß man die beiden Wörter auch durch eines
ersetzen könnte und das Beiwort auf eine grammatische Eigentümlichkeit hinweist. Und
was wird in diesem Falle vom Subjekt ausgesagt, wenn ihm Realität zugesprochen wird?
Betonen wir hier nicht wieder eine grammatische Eigentümlichkeit, etwa, als wenn man sagte:8
”nur die Kardinalzahlen sind wirkliche Zahlen“. (Kronecker soll gesagt haben, nur die
Kardinalzahlen seien von Gott erschaffen, alle anderen seien Menschenwerk.) – Heißt es
3 (V): Phänomene
4 (V): fließend,
5 (V): Dann haben sie also in irgendeiner Weise
die Kommensurabilität mit der Gegenwart und
diese können sie nicht haben
6 (V): mit
7 (V): erinnere,
8 (V): Eigentümlichkeit, in derselben Weise, wie
wenn man sagt
496
497
498
352 Die Erfahrung im gegenwärtigen Moment . . .
WBTC101-107 30/05/2005 16:15 Page 704
(Anyway – what’s the importance of this description of the present phenomenon, which
can turn into an idée fixe for us, as it were? Such that we suffer because the description
cannot describe what goes on when we read the description. It seems that occupying oneself
with this question is nothing short of childish, and that we have got ourselves into a
blind alley. And yet it is a significant blind alley, for all are lured into entering it; as if the
ultimate solution of philosophical problems were to be sought there. – It is as if in representing
the present phenomenon this way one got into a bewitched swamp where everything
comprehensible vanishes.)
On the other hand we need a way of expressing ourselves that represents events3
in visual
space separately from other kinds of experiences.
(When we’re dealing with our language (as a physical phenomenon) we are not in the realm
of the picture projected on the screen, as it were, but rather in that of the film passing through
the projector. And if I want to add music to the event on the screen, then what produces
it must also take place in the realm of the film. The spoken word in a talking-picture that
accompanies the events on the screen is just as fleeting4
as these events, but it is not the same
thing as the sound-track. It’s not the sound-track that accompanies the action on the screen.)
A thought about the ability to represent immediate reality through language:
“The stream of life, or the stream of the world, flows along and our propositions are verified
only at an instant, so to speak. Our propositions are only verified by the present. – So they
must be constructed in such a way that they can be verified by it. They must have what it
takes to be verified by it. So in some way they are commensurable with the present, but they
cannot be so5
in spite of their spatio-temporal nature; rather, this nature must relate to their
commensurability as the corporality of a measuring stick does to its extendedness, by means
of which6
it measures. In the case of the measuring stick you can’t say: ‘Yes, the measuring
stick measures length in spite of its corporality; but a measuring stick that had only length
would be the ideal, would be the pure measuring stick’. No, if a body has length, then there
can’t be any length without a body – and even if I do understand that in a certain sense only
the length of the measuring stick does the measuring, yet what I put into my pocket is still
the measuring stick – the body and not the length.”
“Only the experience of the present moment has reality.” – Is that supposed to mean that
I didn’t get up this morning? Or that an event that I can’t think of7
right now didn’t take
place? – In this case is “present experience” supposed to stand in opposition to future and
past experience? Or is it an adjective like the word “rational” in “rational number”, such that
one could replace the two words by one, the adjective pointing to a grammatical idiosyncrasy?
And in that case what are we saying about the subject if we grant it reality? Aren’t we once
again emphasizing a grammatical idiosyncrasy here, as if for instance we said:8
“Only the
cardinal numbers are real numbers”. (Kronecker is supposed to have said that only the
cardinal numbers were created by God, all else is the work of man.) – If we say “present
3 (V): phenomena
4 (V): fluid
5 (V): they have commensurability with the
present and they cannot have it
6 (V): extendedness, with which
7 (V): can’t remember
8 (V): here, in the same as if we said:
The Experience at the Present Moment . . . 352e
WBTC101-107 30/05/2005 16:15 Page 705
”gegenwärtige Erfahrung“ im Gegensatz zu zukünftiger und vergangener, dann meint
man mit diesen Erfahrungen etwa physikalische Vorgänge; und wenn ich das Bild von der
Laterna magica gebrauche und die zeitlichen Beziehungen in räumliche übersetze, so ist
die gegenwärtige Erfahrung im physikalischen Sinn das Bild auf dem Filmstreifen, das
sich vor dem Objektiv der Laterne befindet. (Ich kann nicht sagen: ”das sich jetzt vor dem
Objektiv der Laterne befindet“.) Auf der einen Seite dieses Bildes liegen9
die vergangenen,
auf der andern die zukünftigen Bilder (die beiden Seiten sind durch Eigentümlichkeiten des
Apparates charakterisiert). Das Bild auf der Leinwand gehört der Zeit des Filmstreifens nicht
an; man kann von ihm nicht in dem eben beschriebenen Sinne sagen, es sei gegenwärtig.
(Im Gegensatz wozu? – Das Wort ”gegenwärtig“, wenn man es hier benützt, bezeichnet
nicht einen Teil10
eines Raumes im Gegensatz zu andern Teilen, sondern charakterisiert einen
Raum.) Der Satz, nur die gegenwärtige Erfahrung habe Realität, wäre nun hier der Satz,
daß nur das Bild vor dem Objektiv dem Bild auf der Leinwand entspricht. Und das könnte
allerdings ein Erfahrungssatz sein und das Gleichnis läßt uns hier im Stich, wenn wir die
Entsprechung zwischen Film und Leinwand (die Projektionsart) nicht so festlegen,11
daß sich
dadurch das Bild auf dem Film, welches dem Bild auf der Leinwand entspricht, als das Bild
vor dem Objektiv der Laterne ergibt.
9 (V): sind
10 (V): Teil d
11 (V): festsetzen,
353 Die Erfahrung im gegenwärtigen Moment . . .
WBTC101-107 30/05/2005 16:15 Page 706
experience” as opposed to future and past experience, then by these experiences we mean,
for example, physical events; and if I use the image of the laterna magica and translate the
temporal relationships into spatial ones, then in a physical sense the present experience is
the picture on the film strip that is in front of the projector’s lens. (I can’t say: “That is now
in front of the projector’s lens”.) On one side of this picture lie9
the past, on the other the
future pictures (the two sides are characterized by the peculiarities of the apparatus). The
picture on the screen is not part of the time of the film strip; one cannot say of it that it is
present in the sense just described. (In contrast to what? – The word “present”, when one uses
it here, doesn’t signify one part of a space in opposition to other parts; rather, it characterizes
a space.) So here the proposition that only present experience has reality would be the
proposition that only the picture in front of the lens corresponds to the picture on the screen.
And that could indeed be an empirical proposition, but here the simile leaves us in the lurch,
unless we set up the correspondence between film and screen (the mode of projection) so
that the picture on the film that corresponds to the picture on the screen turns out to be the
picture in front of the lens of the projector.
9 (V): are
The Experience at the Present Moment . . . 353e
WBTC101-107 30/05/2005 16:15 Page 707
103
Idealismus.
((Ich sehe undeutlich eine Verbindung zwischen dem Problem des Solipsismus oder
Idealismus und dem, der Bezeichnungsweise eines Satzes. Wird etwa das Ich in diesen Fällen
durch den Satz ersetzt und das Verhältnis des Ich zur Wirklichkeit durch das Verhältnis
von Satz und Wirklichkeit?))
Dem, der sagt ”aber es steht doch wirklich ein Tisch hier“ muß man antworten: ”Freilich
steht ein wirklicher Tisch hier, – im Gegensatz zu einem nachgemachten“.
Wenn er aber nun weiterginge und sagte: die Vorstellungen seien nur Bilder der Dinge,
so müßte ich (ihm) widersprechen und sagen, daß der Vergleich der Vorstellung mit einem
Bilde des Körpers gänzlich irreführend sei, da es für ein Bild wesentlich sei, daß es mit seinem
Gegenstand verglichen werden kann.
Wenn aber Einer sagt ”die Vorstellungen sind das einzig Wirkliche“, so muß ich
sagen, daß ich hier das Prädikat1
”wirklich“ nicht verstehe und nicht weiß, was für eine
Eigenschaft man damit eigentlich den Vorstellungen zuspricht und – etwa – den Körpern
abspricht. Ich kann ja nicht begreifen, wie man mit Sinn – ob wahr oder falsch – eine
Eigenschaft Vorstellungen und physischen Körpern zuschreiben kann.
(Der Mensch, der in den Spiegel sieht um sich zwinkern zu sehen; und was er nun
wirklich sieht. Ungeeignete physikalische Theorien.)
(Zeitdauer eines Tones und Zeitdauer einer akkustischen Schwingung.)
Das Wahre am Idealismus ist eigentlich, daß der Sinn des Satzes aus seiner Verifikation
ganz hervorgeht.
Wenn der Idealismus sagt, der Baum sei nur meine Vorstellung, so ist ihm vorzuhalten,
daß der Ausdruck ”dieser Baum“ nicht dieselbe Bedeutung hat wie ”meine Vorstellung von
diesem Baum“. Sagt der Idealismus, meine Vorstellung allein existiert (hat Realität) nicht
der Baum, so mißbraucht er das Wort ”existieren“ oder ”Realität haben“.
1.) Du scheinst ja hier zu sagen, daß die Vorstellung eine Eigenschaft hat, die der Baum
nicht hat. Aber wie weißt Du das? Hast Du alle Vorstellungen und Bäume daraufhin
untersucht. Oder ist das ein Satz a priori, dann muß er2
in eine grammatische Regel gefaßt
werden, die sagt, daß man von der Vorstellung etwas Bestimmtes mit Sinn aussagen darf,
nicht aber vom Baum. 2.) Was soll es aber heißen, von einer Vorstellung Realität auszusagen?
Dem Gebrauch3
entsprechend höchstens,4
daß diese Vorstellung vorhanden ist. In anderm
Sinne – freilich – sagen wir aber auch von einem Baum aus, er existiere (habe Realität) im
Gegensatz zu dem Fall etwa, daß er bereits umgehauen ist. Und es bleibt nur übrig, daß
1 (V): Wort
2 (O): dann er
3 (V): Sprachgebrauch
4 (V): nur,
499
500
501
354
WBTC101-107 30/05/2005 16:15 Page 708
103
Idealism.
((I see, indistinctly, a connection between the problem of solipsism or idealism and the
notational system of a proposition. In these cases is the “I” perhaps replaced by the proposition,
and the relationship of the “I” to reality by the relationship between the proposition
and reality?))
Someone who says “But there really is a table here” has to be answered: “Sure, there’s a
real table here – as opposed to an imitation”.
But if he now went further and said that mental images are just images of things, then
I’d have to contradict (him), and say that the comparison of a mental image with the image
of a physical object is completely misleading, since it’s essential to an image that it can be
compared with its object.
But if someone says “Ideas are the only entities that are real”, then I have to say that here
I don’t understand the predicate1
“real”, and that I don’t know what kind of property one
in fact grants to ideas and – perhaps – denies of objects, when one uses it. For I can’t grasp
how it can make any sense to ascribe (either truly or falsely) a single property to both ideas
and physical objects.
(The person who looks into the mirror in order to see himself wink; and what he then
actually sees. Physical theories that don’t fit.)
(Duration of a sound and duration of an acoustical oscillation.)
The truth in idealism is really that the sense of a proposition emerges completely from
its verification.
If idealism says that the tree is only my mental image, then one should point out to it that
the expression “this tree” doesn’t have the same meaning as “my mental image of this tree”.
If idealism says that only my mental image exists (has reality), not the tree, then it misuses
the words “exist” or “have reality”.
1. You seem to be saying here that the mental image has a property that the tree doesn’t.
But how do you know that? Did you examine all mental images and trees in that regard?
Or is that an a priori proposition? Then it should be subsumed under a grammatical rule
stating that one can sensibly say something specific about a mental image, but not about a
tree. 2. But what is stating that a mental image is real supposed to mean? According to our
use,2
the most3
it can mean is that this mental image is present. But in another sense – to
be sure – we also say of a tree that it exists (has reality), as opposed for example to the case
where it’s been cut down. And all that’s left is that the word “tree”, in the sense in which
1 (V): word
2 (V): our linguistic use,
3 (V): use, all
354e
WBTC101-107 30/05/2005 16:15 Page 709
das Wort ”Baum“ in der Bedeutung, in der man sagen kann ”der Baum wird umgehauen
und verbrannt“ einer anderen grammatischen Kategorie angehört, als der Ausdruck
”meine Vorstellung vom Baum“, etwa im Satz: ”Meine Vorstellung vom Baum wird immer
undeutlicher“. Sagt aber der Realismus, die Vorstellungen seien doch ”nur die subjektiven
Abbilder5
der Dinge“, so ist zu sagen, daß dem ein falscher Vergleich6
zwischen der
Vorstellung von einem Ding und dem Bild des Dinges zu Grunde liegt. Und zwar einfach,
weil es wohl möglich ist, ein Ding zu sehen und sein Bild (etwa nebeneinander), aber nicht
ein Ding und die Vorstellung davon.
Es handelt sich um die Grammatik des Wortes ”Vorstellung“ im Gegensatz zur
Grammatik der ”Dinge“.
|(Es könnte sich eine seltsame Analogie daraus ergeben, daß das Okular auch des
riesigsten Fernrohrs nicht größer sein darf,7
als unser Auge.)|
Wer den Satz, nur die gegenwärtige Erfahrung sei real, bestreiten will (was ebenso falsch
ist, wie ihn zu behaupten) wird etwa fragen, ob denn ein Satz wie ”Julius Cäsar ging über
die Alpen“ nur den gegenwärtigen Geisteszustand Desjenigen beschreibt, der sich mit dieser
Sache beschäftigt. Und die Antwort ist natürlich: Nein! er beschreibt ein Ereignis, das, wie
wir glauben, vor ca. 2000 Jahren stattgefunden hat. Wenn nämlich das Wort ”beschreibt“
so aufgefaßt wird, wie in dem Satz ”der Satz ,ich schreibe‘ beschreibt, was ich gegenwärtig
tue“. Der Name Julius Cäsar bezeichnet eine Person. – Aber was sagt denn das8
alles?
Ich scheine mich ja um die eigentliche philosophische Antwort drücken zu wollen! –
Aber Sätze, die von Personen handeln, d.h. Personennamen enthalten, können eben auf sehr
verschiedene Weise verifiziert werden. – Fragen wir uns nur, warum wir den Satz glauben.
– Daß es (z.B.) denkbar ist, die Leiche Cäsars noch zu finden, hängt unmittelbar mit dem
Sinn des Satzes über Julius Cäsar zusammen. Aber auch, daß es möglich9
ist, eine Schrift
zu finden, aus der hervorgeht, daß so ein Mann nie gelebt hat und seine Existenz zu
bestimmten Zwecken erdichtet worden sei.10
Solche11
Möglichkeiten gibt es (aber) für
einen Satz: ”ich sehe einen roten Fleck über einen grünen dahinziehen“ nicht. Und das
meinen wir, wenn wir sagen, dieser Satz habe in unmittelbarerer Art Sinn,12
als der13
über
Julius Cäsar.
5 (V): Bilder
6 (V): dem eine falsche Analogie
7 (V): nicht größer ist,
8 (V): das also
9 (V): denkbar
10 (V): ist.
11 (V): Diese
12 (V): grünen dahinziehen“ nicht; und das ist es,
was wir damit meinen, wenn wir sagen, daß
dieser Satz in unmittelbarerer Art Sinn hat,
13 (V): jener
502
355 Idealismus.
WBTC101-107 30/05/2005 16:15 Page 710
one can say “The tree is being cut down and burned”, belongs to a different grammatical
category than the expression “My mental image of the tree”, as in the sentence: “My
mental image of the tree is becoming less and less distinct”. But if realism says that mental
images are “only the subjective likenesses4
of things”, then it must be said that this is
based on a false comparison5
between the mental image and a picture of a thing. And it’s
false simply because it’s quite possible to see a thing and its picture (say, next to each
other), but not a thing and a mental image of it.
It’s a matter of the grammar of the words “mental image”, as opposed to the grammar of
“things”.
|(A strange analogy could arise from the fact that the eyepiece of even the most gigantic
telescope mustn’t be6
any bigger than our eye.)|
Anyone who wants to dispute the proposition that only present experience is real (which
is just as false as asserting it) might ask whether a sentence such as “Julius Caesar crossed
the Alps” really only describes the present mental state of the person who’s concerned
with the matter. And of course the answer is: No! It describes an event that took place – so
we believe – about 2,000 years ago. That is, if the word “describes” is understood as in the
sentence “The sentence ‘I’m writing’ describes what I’m presently doing”. The name Julius
Caesar designates a person. – But what does all of this say? I seem to be wanting to dodge
the actual philosophical answer! – But it’s simply that propositions about people, i.e. that
contain proper names, are verifiable in very different ways. – Let’s ask ourselves why we
believe the proposition. – That it is conceivable, (for instance), that we might still find Caesar’s
body, is immediately connected with the sense of the proposition about Julius Caesar.
But what’s also connected with this sense is that it’s possible to7
find a document indicating
that such a man never lived and that his existence was made up for certain reasons. (But)
there are no such possibilities8
for a proposition like: “I see a red patch eclipsing a green
one”. And that9
is what we mean when we say that this proposition makes sense in a more
immediate way than the one10
about Julius Caesar.
4 (V): the pictures
5 (V): a false analogy
6 (V): telescope isn’t
7 (V): it’s conceivable that we might
8 (V): (But) these possibilities don’t exist
9 (V): green one”; and that
10 (V): than that
Idealism. 355e
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104
”Schmerzen haben.“
Zur Erklärung des Satzes ”er hat Zahnschmerzen“ sagt man etwa: ”ganz einfach, ich weiß,
was es heißt, daß ich Zahnschmerzen habe, und wenn ich sage, daß er Zahnschmerzen hat,
so meine ich, daß er jetzt das hat, was ich damals hatte“. Aber was bedeutet ”er“ und was
bedeutet ”Zahnschmerzen haben“. Ist das eine Relation, die die Zahnschmerzen damals zu
mir hatten und jetzt zu ihm. Dann wäre ich mir also jetzt auch der Zahnschmerzen bewußt,
und dessen daß er sie jetzt hat, wie ich eine Geldbörse jetzt in seiner Hand sehen kann, die
ich früher in meiner gesehen habe.
Hat es einen Sinn zu sagen ”ich habe Schmerzen, ich merke sie aber nicht“? Denn in
diesem Satz könnte ich dann allerdings statt ”ich habe“ ”er hat“ einsetzen. Und umgekehrt,
wenn die Sätze ”er hat Schmerzen“ und ”ich habe Schmerzen“ auf der gleichen logischen
Stufe stehen, so muß ich im Satz ”er hat Schmerzen,1
die ich nicht fühle“ statt ”er hat“ ”ich
habe“ setzen können. – Ich könnte auch so sagen: Nur insofern ich Schmerzen haben kann,
die ich nicht fühle, kann er Schmerzen haben, die ich nicht fühle. Es könnte dann noch immer
der Fall sein, daß ich tatsächlich die Schmerzen, die ich habe, immer fühle, aber es muß
Sinn haben, das zu verneinen.
Der Begriff der Zahnschmerzen als eines Gefühlsdatums ist allerdings auf den Zahn des
Anderen ebenso anwendbar, wie auf den meinen, aber nur in dem Sinne, in dem es ganz
wohl möglich wäre, in dem Zahn in eines andern Menschen Mund Schmerzen zu
empfinden. Im Einklang mit der gegenwärtigen Ausdrucksweise würde man aber diese
Tatsache nicht durch die Worte ”ich fühle seinen Zahnschmerz“ ausdrücken, sondern durch
”ich habe in seinem Zahn Schmerzen“. – Man kann nun sagen: Freilich hast Du nicht
seinen Zahnschmerz, denn es ist auch dann sehr wohl möglich, daß er sagt ”ich fühle in
diesem Zahn nichts“. Und sollte ich in diesem Fall sagen ”Du lügst, ich fühle, wie Dein
Zahn schmerzt“?
Wenn ich jemand, der Zahnschmerzen hat, bemitleide, so setze ich mich in Gedanken an
seine Stelle. Aber ich setze mich an seine Stelle.
Die Frage ist, ob es Sinn hat zu sagen: ”Nur A kann den Satz ,A hat Schmerzen‘ verifizieren,
ich nicht“. Wie aber wäre es, wenn dieser Satz falsch wäre, wenn ich also den Satz
verifizieren könnte, 2
kann es etwas anderes heißen, als daß dann ich Schmerzen fühlen müßte!
Aber wäre das eine Verifikation? Vergessen wir nicht: es ist Unsinn, zu sagen, ich müßte
meine oder seine Schmerzen fühlen.
Man könnte auch so fragen: Was in meiner Erfahrung rechtfertigt das ”meine“ in ”ich
fühle meine Schmerzen“. Wo ist die Multiplizität des Gefühls, die dieses Wort rechtfertigt,
und es kann nur dann gerechtfertigt sein, wenn an seine Stelle auch ein anderes treten kann.
1 (V): Schmerzen!, 2 (V): könnte, daß
503
504
356
WBTC101-107 30/05/2005 16:15 Page 712
104
“Having Pain.”
To explain the proposition “He has a toothache” one might say: “It’s quite simple: I know
what it means that I have a toothache, and if I say that he has a toothache I mean that he
has now what I had then”. But what does “he” mean, and what “to have a toothache”? Is
this a relationship that the toothache had to me at that time and now has to him? So in that
case I too would be conscious of the toothache now, and of the fact that he has it now, just
as I can now see a wallet in his hand that I saw in mine before.
Does it make sense to say I have aches, but I don’t notice them? Because then I really
could insert “he has” for “I have” in this sentence. And, conversely, if the sentences “He
has aches” and “I have aches” are on the same logical level, then in the sentence “He has
aches that I don’t feel” I have to be able to replace “He has” with “I have”. – I could also
put it this way: Only in so far as I can have aches that I don’t feel can he have aches that I
don’t feel. Then it could still be the case that in fact I always feel the aches that I have, but
it must make sense to deny that.
To be sure, the concept of a toothache as a sense datum is just as applicable to someone
else’s tooth as to mine, but only in the sense in which it would be quite possible to feel pain
in a tooth in someone else’s mouth. In accordance with our present way of speaking, however,
one wouldn’t express this fact with the words “I feel his toothache”, but rather with
“I have an ache in his tooth”. – Now one can say: Of course you don’t have his toothache,
because in that case it might very well happen that he says “I don’t feel anything in this
tooth”. And should I say in this case “You’re lying, I feel that your tooth is aching”?
When I pity someone who has a toothache I mentally put myself in his place. But I put
myself in his place.
The question is whether it makes sense to say: “Only A can verify the proposition ‘A is
in pain’, not I”. But what if this proposition were false – i.e. if I could verify the proposition?
Can that mean anything other than that in that case, I would have to feel pain! But
would that be a verification? Let’s not forget: It’s nonsense to say that I had to feel my or
his pain.
One could also put the question this way: What in my experience justifies the “my” in
“I feel my pain”? Where is the multiplicity of feeling that justifies this word? – and it can
only be justified if it’s also possible for another to take its place.
356e
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”Ich habe Schmerzen“ ist, im Falle ich den Satz gebrauche, ein Zeichen ganz anderer Art,
als es für mich im Munde eines Anderen ist; und zwar darum, weil es im Munde eines Anderen
für mich so lange sinnlos ist, als ich nicht weiß, welcher Mund es ausgesprochen hat. Das
Satzzeichen besteht in diesem Falle nicht im Laut allein, sondern in der Tatsache, daß dieser
Mund den Laut hervorbringt. Während im Falle ich es sage, oder denke, das Zeichen der
Laut allein ist.
Angenommen, ich hätte stechende Schmerzen im rechten Knie und bei jedem Stich zuckt
mein rechtes Bein. Zugleich sehe ich einen anderen Menschen, dessen Bein in gleicher Weise
zuckt und der über stechende Schmerzen klagt; und zu gleicher Zeit fängt mein linkes Bein
ebenso an zu zucken, obwohl ich im linken Knie keine Schmerzen fühle. Nun sage ich: mein
Gegenüber hat offenbar in seinem Knie dieselben Schmerzen, wie ich in meinem rechten
Knie. Wie ist es aber mit meinem linken Knie, ist es nicht in genau dem gleichen Fall, wie
das Knie des Anderen?
Wenn ich sage ”A hat Zahnschmerzen“, so gebrauche ich die Vorstellung des
Schmerzgefühls in der selben Weise, wie etwa den Begriff des Fließens, wenn ich vom
Fließen des elektrischen Stromes rede.
Ich sammle gleichsam sinnvolle Sätze über Zahnschmerzen, das ist der charakteristische
Vorgang einer grammatischen Untersuchung. Ich sammle nicht wahre, sondern sinnvolle
Sätze und darum ist diese Betrachtung keine psychologische. (Man möchte sie oft eine
Metapsychologie nennen.)
Man könnte sagen: Die Philosophie sammle fortwährend ein Material von Sätzen, ohne
sich um ihre Wahr- oder Falschheit zu kümmern; nur im Falle der Logik und Mathematik
hat sie es nur mit den ”wahren“ Sätzen zu tun.
Die Erfahrung des Zahnschmerzgefühls ist nicht die, daß eine Person Ich etwas hat.
In den Schmerzen unterscheide ich eine Intensität, einen Ort, etc., aber keinen Besitzer.
Wie wären etwa Schmerzen, die gerade niemand hat? Schmerzen, die gerade niemandem
gehören?
Die Schmerzen werden als etwas dargestellt, das man wahrnehmen kann, im Sinne, in
dem man eine Zündholzschachtel wahrnimmt. – Das Unangenehme sind dann freilich nicht
die Schmerzen, sondern nur das Wahrnehmen der Schmerzen.
Wenn ich einen Anderen bedaure, weil er Schmerzen hat, so stelle ich mir wohl die
Schmerzen vor, aber ich stelle mir vor, daß ich sie habe.
Soll ich mir auch die Schmerzen eines auf dem Tisch liegenden Zahnes denken können,
oder die Schmerzen eines Teetopfs? Soll man etwa sagen: es ist nur nicht wahr, daß der
Teetopf Schmerzen hat, aber ich kann es mir denken?!
Die beiden Hypothesen, daß die Anderen Schmerzen haben, und die, daß sie keine haben,
und sich nur so benehmen wie ich, wenn ich welche habe, müssen ihrem Sinne nach
identisch sein, wenn alle mögliche Erfahrung, die die eine bestätigt, auch die andere bestätigt.
Wenn also keine Entscheidung zwischen beiden durch die Erfahrung denkbar ist.
Zu sagen, daß die Anderen keine Schmerzen haben, setzt aber voraus, daß es Sinn hat zu
sagen, daß sie Schmerzen haben.
Ich glaube, es ist klar, daß man in demselben Sinne sagt, daß andere Menschen
Schmerzen haben, in welchem man sagt, daß ein Stuhl keine hat.
505
506
507
357 ”Schmerzen haben.“
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When I use the sentence “I’m in pain” it’s a completely different kind of expression from
what it is for me when it comes from someone else’s mouth; and this is because when it is
in someone else’s mouth it makes no sense to me, so long as I don’t know which mouth
uttered it. In this case, the expression consists not only of the sound, but also in the fact
that this mouth is producing the sound. Whereas in the case where I say or think the
sentence, the expression is the sound alone.
Let’s assume that I have stabbing pains in my right knee and my right leg is twitching at
every stab. At the same time I see someone else who is complaining about stabbing pains,
and whose leg is twitching in the same way as mine; and then my left leg begins to twitch
in the same way, even though I don’t feel any pain in my left knee. Now I say: Evidently
the person across from me has the same pains in his knee as I do in my right knee. But what
about my left knee: isn’t it in exactly the same situation as the other person’s knee?
When I say “A has a toothache” I’m using the idea of feeling pain in the same way as I
use the concept of flow when I speak of the flow of an electric current.
I’m collecting meaningful propositions about a toothache, as it were; that is the process
characteristic of a grammatical investigation. I’m not collecting true propositions, but meaningful
ones, and therefore this is not a psychological investigation. (One is often inclined to
call it metapsychology.)
One could say: Philosophy is constantly collecting a stock of propositions without
worrying about their truth or falsity; only in the cases of logic and mathematics does it have
to do solely with “true” propositions.
The experience of feeling a toothache isn’t the experience that a person “I” has
something.
With regard to pain I discern an intensity, a location, etc., but no owner.
What would pains be like, for instance, that nobody has at the moment, that don’t belong
to anybody at the moment?
Pains are represented as something that one can perceive, in the sense in which one
perceives a box of matches. – Then, to be sure, the unpleasant thing is not the pains, but
only perceiving them.
If I feel sorry for someone else because he has pains, then I do imagine the pains, but I
imagine that I’m having them.
Should I also be able to imagine the pain of a tooth lying on the table, or the pain of a
teapot? Should one say, for instance: The only thing wrong with saying that a teapot is in
pain is that it’s not true, but I can imagine it?!
The two hypotheses, the one that others are in pain, the other that they are not in pain
but are only behaving the way I do when I am in pain, must be identical in sense if any
possible experience that confirms the one also confirms the other. In other words, if it is
inconceivable that experience could decide between the two.
But to say that others are not in pain presupposes that it makes sense to say that they
are in pain.
I think it is clear that one says that others are in pain in the same sense as one says that
a chair is not in pain.
“Having Pain.” 357e
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Wie wäre es, wenn ich zwei Körper hätte, d.h. wenn mein Körper aus zwei getrennten
Leibern bestünde?
Hier sieht man – glaube ich – wieder, wie das Ich nicht auf derselben Stufe mit den Andern
steht, denn wenn die Andern je zwei Körper hätten, so könnte ich es nicht erkennen.
Kann ich mir denn die Erfahrung mit zwei Leibern denken? Die Gesichtserfahrung gewiß
nicht.
Das Phänomen des Schmerzgefühls in einem Zahn, welches ich kenne, ist in der
Ausdrucksweise der gewöhnlichen Sprache dargestellt durch ,,ich habe in dem und dem
Zahn Schmerzen“. Nicht durch einen Ausdruck von der Art ”an diesem Ort ist ein
Schmerzgefühl“. Das ganze Feld dieser Erfahrung wird in dieser Sprache durch Ausdrücke
von der Form ”ich habe . . .“ beschrieben. Die Sätze von der Form ”N hat Zahnschmerzen“
sind für ein ganz anderes Feld reserviert. Wir können daher nicht überrascht sein, wenn
in den Sätzen ”N hat Zahnschmerzen“ nichts mehr auf jene Art mit der Erfahrung
Zusammenhängendes gefunden wird.
Wenn man sagt, die Sinnesdaten seien ”privat“,3
niemand anderer könne meine
Sinnesdaten sehen, hören, fühlen, und meint damit nicht eine Tatsache unserer Erfahrung,
so müßte das ein philosophischer Satz sein; und was gemeint ist, drückt sich darin aus, daß
eine Person in die Beschreibung von Sinnesdaten eintritt.
Denn, kann ein Anderer meine Zahnschmerzen nicht haben, so kann ich sie – in diesem
Sinne – auch nicht haben.
In dem Sinne, in welchem es nicht erlaubt ist zu sagen, der Andere habe diese
Schmerzen, ist es auch nicht erlaubt zu sagen, ich hätte4
sie.
Was wesentlich privat ist, oder scheint, hat keinen Besitzer.
Was soll es heißen: er hat diese Schmerzen? außer, er hat solche Schmerzen: d.h., von solcher
Stärke, Art, etc. Aber nur in dem5
Sinn kann auch ich ”diese Schmerzen“ haben.
Das heißt, die Subjekt-Objekt Form ist darauf nicht anwendbar.
Die Subjekt-Objekt Form bezieht sich auf den Leib und die Dinge um ihn, die auf ihn
wirken.
In der nicht-hypothetischen Beschreibung des Gesehenen, Gehörten – diese Wörter
bezeichnen hier grammatische Formen – tritt das Ich nicht auf, es ist hier von Subjekt und
Objekt nicht die Rede.
Der Solipsismus könnte durch die Tatsache widerlegt werden, daß das Wort ”ich“ in der
Grammatik keine zentrale Stellung hat, sondern ein Wort ist, wie jedes andre Wort.
Wie im Gesichtsraum, so gibt es in der Sprache kein metaphysisches Subjekt.
|Die Schwierigkeit, die uns das Sprechen über den Gesichtsraum ohne Subjekt macht
und über ”meine und seine Zahnschmerzen“, ist die, die Sprache einzurenken, daß sie richtig
in den Tatsachen sitzt.|
Behaviourism. ”Mir scheint, ich bin traurig, ich lasse den Kopf so hängen.“
Warum hat man kein Mitleid, wenn eine Tür ungeölt ist und beim Auf- und Zumachen
schreit? Haben wir mit dem Andern, der sich benimmt wie wir, wenn wir Schmerzen haben,
3 (V): ”privat“, sei
4 (V): habe
5 (V): dem eine
508
509
358 ”Schmerzen haben.“
WBTC101-107 30/05/2005 16:15 Page 716
What would it be like if I had two bodies, i.e. if my body consisted of two separate
bodies?
Here one sees once again – so I believe – how the “I” is not on the same level with other
people, for if others each had two bodies I couldn’t tell it.
Can I imagine what experience would be like with two bodies? Certainly not what visual
experience would be like.
In the way we express ourselves in everyday language, the phenomenon of feeling pain
in a tooth, with which I am acquainted, is represented by “I have a pain in this or that
tooth”. And not by an expression of the type “There is a feeling of pain at this location”.
In this language the entire field of the experience is described by expressions of the form
“I have . . .”. Propositions of the form “N has a toothache” are reserved for an entirely different
field. So we cannot be surprised if nothing that is connected to experience in the first
way is any longer found in the propositions “N has a toothache”.
If we say that sense data are “private”, that nobody else can see, hear, feel my sense data,
and if by this we don’t mean a fact that we’ve experienced, then this would have to be
a philosophical proposition; and what is meant is expressed by a person entering into the
description of sense data.
For if someone else cannot have my toothache, then – to that extent – I too cannot
have it.
In the sense in which it is not permissible to say that someone else has these pains it is
also not permissible to say that I have them.
What is essentially private, or seems that way, doesn’t have an owner.
What is this supposed to mean: He has these pains? Unless it is supposed to mean that he
has such pains: i.e. pains of such intensity, kind, etc. But only in that sense can I too have
“these pains”.
That means that the subject–object form is not applicable to this.
The subject–object form refers to the body and the things around it that have an effect
on it.
In a non-hypothetical description of what has been seen, heard – here these words designate
grammatical forms – the “I” does not appear; here there is no talk of subject and object.
Solipsism could be disproved by the fact that the word “I” doesn’t occupy a central position
in grammar, but is a word like any other.
As in visual space, there is no metaphysical subject in language.
|The difficulty created for us by talking about visual space without a subject, and about
“my and his toothache”, is one of setting language, as one does a bone, so that it is accurately
aligned with the facts.|
Behaviourism. “It seems to me that I’m sad, my head is hanging so low.”
Why doesn’t one feel pity when a door hasn’t been oiled and it screeches as it’s opened and
shut? Do we feel pity for someone else who behaves as we do when we’re in pain – feel it
“Having Pain.” 358e
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Mitleid – auf philosophische Erwägungen hin, die zu dem Ergebnis geführt haben, daß er
leidet, wie wir? Ebensogut können uns die Physiker damit Furcht einflößen, daß sie uns
versichern, der Fußboden sei gar nicht kompakt, wie er scheine, sondern bestehe aus losen
Partikeln, die regellos herumschwirren. ”Aber wir hätten doch mit dem Andern nicht Mitleid,
wenn wir wüßten, daß er nur eine Puppe ist, oder seine Schmerzen bloß heuchelt.“ Freilich,
– aber wir haben auch ganz bestimmte Kriterien dafür, daß etwas eine Puppe ist, oder daß
Einer seine Schmerzen heuchelt und diese Kriterien stehen eben im Gegensatz zu denen,
die wir Kriterien dafür nennen, daß etwas keine Puppe (sondern etwa ein Mensch) ist und
seine Schmerzen nicht heuchelt (sondern wirklich Schmerzen hat).
Hat es Sinn zu sagen, zwei Menschen hätten denselben Körper? Welches wären die
Erfahrungen, die wir mit diesem Satz beschrieben? Daß ich darauf käme, daß das, was ich
meine Hand nenne, und bewege, an dem Körper eines Andern sitzt, ist natürlich denkbar,
denn ich sehe, während ich jetzt schreibe, die Verbindung meiner Hand mit meinem übrigen
Körper nicht. Und ich könnte wohl darauf kommen, daß sich die frühere Verbindung
gelöst hat und also auch, daß meine Hand jetzt an dem Arm eines Andern sitzt.
Von Sinnesdaten in dem Sinne dieses Worts, in dem es undenkbar ist, daß der Andere
sie hat, kann man eben aus diesem Grunde auch nicht sagen, daß der Andere sie nicht hat.
Und eben darum ist es auch sinnlos zu sagen, daß ich, im Gegensatz zum Andern, sie habe.
– Wenn man sagt ”seine Zahnschmerzen kann ich nicht fühlen“, meint man damit, daß man
die Zahnschmerzen des Andern bis jetzt nie gefühlt hat? Wie unterscheiden sich seine
Zahnschmerzen von den meinen? Wenn das Wort ”Schmerzen“ in den Sätzen ”ich habe
Schmerzen“ und ”er hat Schmerzen“ die gleiche Bedeutung hat, – was heißt es dann zu
sagen, daß er nicht dieselben Schmerzen haben kann, wie ich? Wie können sich denn
verschiedene Schmerzen voneinander unterscheiden? Durch Stärke, durch den Charakter
des Schmerzes (stechend, bohrend, etc.) und durch die Lokalisation im Körper. Wenn
nun aber diese Charakteristika bei beiden dieselben sind? – Wenn man aber einwendet, der
Unterschied der Schmerzen6
sei eben der, daß in einem Falle ich sie habe, im andern Fall
er! – dann ist also die besitzende Person eine Charakteristik der Schmerzen selbst. Aber was
ist dann mit dem Satz ”ich habe Schmerzen“ oder ”er hat Schmerzen“ ausgesagt? – Wenn
das Wort ”Schmerzen“ in beiden Fällen die gleiche Bedeutung hat, dann muß man die
Schmerzen der Beiden miteinander vergleichen können; und wenn sie in Stärke etc., etc.
miteinander übereinstimmen, so sind sie die gleichen; wie zwei Anzüge die gleiche Farbe
besitzen, wenn sie in Bezug auf Helligkeit, Sättigung, etc. miteinander übereinstimmen.
Wenn man fragt ”ist es denkbar, daß ein Mensch die Schmerzen des Andern fühlt?“ so
schweben einem dabei die Schmerzen (etwa Zahnschmerzen) des Andern gleichsam als ein
Körper, ein Volumen, vor im Mund des Andern und die Frage scheint zu fragen, ob wir an
diesem Schmerzvolumen teilhaben können. Etwa dadurch, daß sich unser beider Wangen
durchdrängen. Aber auch das scheint dann nicht zu genügen und wir müßten uns ganz mit
ihm decken.7
1.) ”Ich habe Schmerzen“
”N hat Schmerzen“
dagegen 2.) ”Ich habe graue Haare“
”N hat graue Haare“8
6 (V): einwendet, ihr Unterschied
7 (V): und wir müßten ganz mit ihm
zusammenfallen.
8 (O): Haare
510
511
359 ”Schmerzen haben.“
WBTC101-107 30/05/2005 16:15 Page 718
as a result of philosophical deliberations that have led to the conclusion that he is suffering
like us? By the same token physicists could frighten us by assuring us that the floor wasn’t
nearly as firm as it seemed, but consisted of loose particles that were erratically whizzing
around. “But we wouldn’t feel pity for someone else if we knew that he were only a puppet
or were only simulating his pain.” Certainly – but we also have very specific criteria for something
being a puppet or for someone simulating pain, and these criteria are in contrast to
those we call criteria for something not being a puppet (but, say, a human being) and for
someone not simulating his pain (but really being in pain).
Does it make sense to say that two people have the same body? Which experiences would
we be describing with this sentence? It’s conceivable, of course, that I’d come to believe that
what I call my hand, and move, is attached to someone else’s body, for as I’m writing now
I don’t see the connection of my hand to the rest of my body. And I could easily come to
believe that the previous connection had come undone, and thus also that my hand is now
attached to someone else’s arm.
If we define the words “sense data” as something we can’t imagine anyone else having,
then that makes it impossible to say that someone else doesn’t have them. And for just that
reason it also makes no sense to say that I, as opposed to someone else, have them. – If one
says “I can’t feel his toothache” does one mean that, up until now, one has never felt someone
else’s toothache? How does his toothache differ from mine? If the word “pain” has the
same meaning in the sentences “I’m in pain” and “He’s in pain” – then what does it mean
to say that he can’t be in the same pain as I? How can various pains differ from each other?
With respect to intensity, the nature of the pains (stabbing, piercing, etc.), and their localization
within the body. And what if these characteristics are the same for both? – But if it’s
objected that the difference between the pains is1
precisely that I’m the one having pain in
the one case and he in the other! – then the owner of the pain is a characteristic of the pain
itself. But in that case: what has been stated by the proposition “I’m in pain”, or “He’s in
pain”? – If the word “pain” has the same meaning in both cases, then one has to be able to
compare the pains of both people; and if they coincide in intensity etc., etc., then they are
the same; just as two suits are the same colour if they correspond with respect to brightness,
saturation, etc.
If it’s asked “Is it conceivable that someone might feel someone else’s pain?” then one has
a mental image of someone else’s pain (say his toothache) as a body, as it were, a mass in
the other person’s mouth, and the question seems to be asking whether we can share in this
mass of pain. Say, by both of our cheeks interpenetrating. But even this doesn’t seem to be
enough, and we’d have to be completely congruent with him.
(1) “I’m in pain”
“N is in pain”
on the other hand (2) “I have grey hair”
“N has grey hair”
1 (V): that their difference is
“Having Pain.” 359e
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Die verschiedenen philosophischen Schwierigkeiten und Konfusionen in Verbindung mit
dem ersten Beispiel lassen sich zum größten Teil auf die Verwechslung der Grammatik der
Fälle 1) und 2) zurückführen.
Es hat Sinn zu sagen: ”ich sehe seine Haare, aber nicht die meinen“, oder ”ich sehe meine
Hände täglich, aber nicht die seinen“ und dieser Satz ist analog dem: ”ich sehe meine Wohnung
täglich, aber nicht die seine“. – Dagegen ist es Unsinn: ”ich fühle meine Schmerzen, aber
nicht die seinen“.
Die Ausdrucksweise unserer Sprache in den beiden Fällen 1) und 2) ist natürlich nicht
”falsch“, aber sie ist irreführend. ”Eine 9
herrenlose Wohnung“, ”herrenlose Zahnschmerzen“.
Es gibt Menschen, die Untersuchungen darüber anstellen, ”ob es ungesehene Gesichtsbilder
gibt“ und sie glauben, daß das eine Art wissenschaftlicher Untersuchung (über diese
Phänomene) ist.
”Wie ein Satz verifiziert wird, – das sagt er“: und nun sieh Dir daraufhin die Sätze an:
”Ich habe Schmerzen“, ”N hat Schmerzen“.
Wenn nun aber ich der N bin? – Dann haben dennoch die beiden Sätze verschiedenen
Sinn.
”Die Sache ist doch ganz einfach: ich spüre freilich seine Schmerzen nicht, aber er spürt
sie eben (und so sind alle Verhältnisse doch symmetrisch)“. Aber dieser Satz ist eben Unsinn.
– Um nun die Asymmetrie der Erfahrung mit Bezug auf mich und den Andern deutlich
zum Ausdruck zu bringen, könnte ich eine asymmetrische Ausdrucksweise vorschlagen:
Alte Ausdrucksweise: Neue Ausdrucksweise:
W. hat Schmerzen. Es sind Schmerzen vorhanden.
W. hat Schmerzen in seiner Es sind Schmerzen in der linken
linken Hand. Hand des W.
N. hat Schmerzen. N. benimmt sich wie W., wenn Schmerzen
vorhanden sind.
N. heuchelt Schmerzen in seiner Hand. N. heuchelt das Benehmen des W., wenn
Schmerzen in seiner Hand sind.
Ich bedauere N., weil er Schmerzen hat. Ich bedauere N., weil er sich benimmt, wie etc.
Da wir für jeden sinnvollen Ausdruck der alten Ausdrucksweise einen der neuen setzen
und für verschiedene alte,10
verschiedene neue, so muß, was Eindeutigkeit und Verständlichkeit
anlangt, die neue Ausdrucksweise der alten gleichwertig sein. – Aber könnte man denn nicht
eine solche asymmetrische Ausdrucksweise ebensogut für Sätze der Art ”ich habe graue
Haare“, ”N. hat graue Haare“ konstruieren? Nein. Man muß nämlich verstehen, daß der
Name ”W.“ in den Sätzen der rechten Seite sinnvoll durch andere Namen ersetzt werden
können muß. 11
Und ist das nicht der Fall, dann braucht weder ”W.“ noch ein anderer Name
in diesen Sätzen vorzukommen12
. Ersetzt man nämlich ”W.“ durch den Namen eines
andern Menschen, so wird etwa gesagt, daß ich in der Hand eines anderen Körpers als des
meinigen Schmerzen empfinde. Es wäre z.B. denkbar, daß ich mit einem Andern den Körper
wechsle;13
etwa aufwache, meinen alten Körper mir gegenüber auf einem Sessel sitzen sehe,
und mich im Spiegel sehend fände, daß ich das Gesicht und den Körper meines Freundes
angenommen habe. Ich betrachte nun den Personennamen als Namen eines Körpers. Und
es hat nun Sinn zu sagen: ”ich habe im Körper des N (oder im Körper N) Zahnschmerzen
(in der asymmetrischen Ausdrucksweise: ”in einem Zahn des N sind Schmerzen“); aber es
9 (M): 71
10 (V): alte, verschied
11 (M): 71
12 (V): vorkommen
13 (V): Andern Körper wechsle;
512
513
360 ”Schmerzen haben.“
WBTC101-107 30/05/2005 16:15 Page 720
The various philosophical difficulties and confusions connected with the first example can
be attributed, for the most part, to confusing the grammar of cases (1) and (2).
It makes sense to say: “I see his hair, but not mine”, or “Every day I see my hands, but
not his”, and this sentence is analogous to: “Every day I see my apartment, but not his”. –
On the other hand this is nonsense: “I feel my pain, but not his”.
Of course, the way of expressing this in our language in cases (1) and (2) isn’t “wrong”,
but it is misleading. “An apartment 2
without an owner”, “a toothache without an owner”.
There are people who conduct investigations into “whether unseen visual images exist”, and
they believe that doing this is a kind of scholarly investigation (into these phenomena).
“How a proposition is verified – that’s what it says”: and now, with this in mind, look at
the propositions: “I am in pain”, “N is in pain”.
But now what if I am N? – Then the two propositions still have different senses.
“Really the matter is quite simple: to be sure, I don’t feel his pain, but he does (and so,
after all, all the relationships are symmetrical).” But this proposition is simply nonsense. –
In order to express clearly the asymmetry of experience as between myself and someone else,
I could suggest an asymmetrical form of expression:
Old form of expression: New form of expression:
W is in pain. There is pain present.
W has a pain in his left hand. There is a pain in W’s left hand.
N is in pain. N is behaving like W when there is pain present.
N is simulating pain in his hand N is simulating W’s behaviour when there is
pain in his hand.
I feel sorry for N because I feel sorry for N because he behaves like, etc.
he is in pain.
Since we’re writing down an expression of the new form for each meaningful expression
of the old, and different new ones for different old ones, the new form of expression has to
be equivalent to the old in terms of clearness and the absence of ambiguity. – But couldn’t
we construct such an asymmetrical form of expression just as well for sentences of the kind
“I have grey hair”, “N has grey hair”? No. For it must be understood that the name “W”
in the sentences on the right must be meaningfully replaceable by other names. 3
And if
this is not the case, then neither “W” nor another name need appear in these sentences.
For if one replaces “W” by someone else’s name, then what is being said, for instance, is
that I feel pain in the hand belonging to a body other than my own. It’s conceivable, for
instance, that I might exchange bodies with someone else4
; that say I wake up, see my old
body sitting in a chair across from me and find, upon looking into the mirror, that I have
assumed the face and body of my friend. I look at a person’s name as the name of a body.
And then it makes sense to say: “I have a toothache in N’s body (or in the body N)”
(in the asymmetrical form of expression: “There is an ache in one of N’s teeth”); but it
2 (M): 71
3 (M): 71
4 (V): with others
“Having Pain.” 360e
WBTC101-107 30/05/2005 16:15 Page 721
hat keinen Sinn, zu sagen ”ich habe auf dem Kopf des N graue Haare“, außer, das soll heißen:
”N hat graue Haare“.
Aber ist (denn) die vorgeschlagene asymmetrische Ausdrucksweise richtig? Warum sage
ich ”N benimmt sich wie W, wenn er . . .“? Wodurch ist denn W charakterisiert? Doch
durch die Formen etc. seines Körpers und durch dessen kontinuierliche Existenz im Raum.
Sind aber diese Dinge für die Erfahrung der Schmerzen wesentlich? Könnte ich mir nicht
folgende Erfahrung denken: ich wache mit Schmerzen in der linken Hand auf und finde,
daß sie ihre Gestalt geändert hat und jetzt so aussieht, wie die Hand meines Freundes, während
er meine Hand erhalten hat. Und worin besteht die Kontinuität meiner Existenz im Raum?
Wenn mir jemand Verläßlicher erzählte, er sei, während ich geschlafen habe, bei mir gesessen,
plötzlich sei mein Körper verschwunden und sei plötzlich wieder erschienen – ist es unmöglich
das zu glauben? – Und worin besteht etwa die Kontinuität meines Gedächtnisses? In welcher
Zeit ist es kontinuierlich? Oder besteht die Kontinuität darin, daß im Gedächtnis keine Lücke
ist? Wie im Gesichtsfeld keine ist. (Denn überlege nur, wie wir den blinden Fleck merken!)
Und was hätte diese Kontinuität mit der zu tun, die für den Gebrauch des Personennamens
W. 14
wesentlich ist?15
Die Erfahrung der Schmerzen läßt sich in ganz anderer Umgebung
als der von uns gewohnten denken. (Denken wir doch nur, daß man tatsächlich Schmerzen
in der Hand haben kann, obwohl es diese im physikalischen Sinn gar nicht mehr gibt, weil
sie einem amputiert worden ist.) In diesem Sinne könnte man Zahnschmerzen ohne Zahn,
Kopfschmerzen ohne Kopf etc. haben. Wir machen eben hier einfach eine Unterscheidung,
wie die zwischen Gesichtsraum und physikalischem Raum, oder Gedächtniszeit und
physikalischer Zeit. – Danach nun ist es unrichtig, die Ausdrucksweise einzuführen ”N
benimmt sich wie W, wenn . . .“. Man könnte vielleicht sagen ”N benimmt sich, wie der
Mensch in dessen Hand Schmerzen sind“. Warum sollte man aber überhaupt die Erfahrung
der Schmerzen zur Beschreibung des bewußten Benehmens heranziehen? – Wir wollen doch
einfach zwei verschiedene Erfahrungsgebiete trennen; wie wenn wir Tasterfahrung und
Gesichtserfahrung an einem Körper trennen. Und verschiedener kann nichts sein, als die
Schmerzerfahrung und die Erfahrung, einen menschlichen Körper sich winden zu sehen,16
Laute ausstoßen zu hören, etc. Und zwar besteht hier kein Unterschied zwischen meinem
Körper und dem des Andern, denn es gibt auch die Erfahrung, die Bewegungen des
eigenen Körpers zu sehen und die von ihm ausgestoßenen Laute zu hören.
Denken wir uns, unser Körper würde aus unserem Gesichtsfeld entfernt, etwa, indem
man ihn gänzlich durchsichtig machte; er behielte aber die Fähigkeit, in einem geeigneten
Spiegel in der uns gewohnten Weise zu erscheinen, so daß wir etwa die sichtbaren
Äußerungen unserer Zahnschmerzen wesentlich wie die eines fremden Körpers wahrnähmen.
Dies ergäbe auch eine ganz andere Koordination zwischen sehendem Auge und
Gesichtsraum, als die uns selbstverständlich erscheinende alltägliche. (Denke an das
Zeichnen eines Vierecks mit seinen Diagonalen im Spiegel.) Wenn wir uns aber so die
Möglichkeit denken können, daß wir unsern sichtbaren Körper nur als Bild in einem Spiegel
kennten, so ist es nun auch denkbar, 17
daß dieser Spiegel wegfiele und wir ihn nicht anders
sähen, als irgend einen andern menschlichen Körper. – Wodurch wäre er dann aber als mein
Körper charakterisiert? Nun, nur dadurch, daß ich z.B. die Berührung dieses Körpers fühlen
würde, nicht aber die eines andern, etc. So ist es auch nicht mehr wesentlich, daß der Mund
unterhalb des sehenden Auges meine Worte spricht. (Und das ist von großer Wichtigkeit.)
Auch wenn ich meinen Körper sehe, wie ich ihn jetzt sehe, d.h. von seinen Augen aus, ist
14 (M): 71
15 (V): W. von Bedeutung ist?
16 (V): sich winden sehen,
17 (M): 71
514
515
361 ”Schmerzen haben.“
WBTC101-107 30/05/2005 16:15 Page 722
makes no sense to say “I have grey hair on N’s head”, unless that is supposed to mean: “N
has grey hair”.
But is the asymmetrical form of expression I’ve suggested (really) correct? Why do I say
“N is behaving like W when he . . .”? How is W characterized? Certainly by the shapes,
etc. of his body, and by its continuous existence in space. But are these things essential for
the experience of pain? Couldn’t I imagine the following experience: I wake up with a pain
in my left hand and find that it has changed shape and now looks like my friend’s hand,
whereas he has got my hand. And what does the continuity of my existence in space consist
in? If some reliable person were to tell me that he had been sitting next to me while I
was sleeping, and that suddenly my body had disappeared and then suddenly reappeared –
is it impossible to believe that? – And what does, say, the continuity of my memory consist
in? In which time is it continuous? Or does the continuity consist in there not being any gap
in memory? Just as there is none in the visual field. (For just think about how we notice our
blind spot!) And what would this continuity have to do with the one that is 5
essential to6
the
use of the proper name W? The experience of pain is imaginable in a completely different
context than the one we’re used to. (Let’s just remember that one can actually have pain in
one’s hand even though, in a physical sense, it doesn’t exist anymore, because it has been
amputated.) To that extent one could have a toothache without a tooth, a headache without
a head, etc. We’re simply making a distinction here like the one between visual space and
physical space, or memory-time and physical time. – Now according to this it’s incorrect
to introduce the form of expression “N is behaving like W, when . . .”. Possibly one could
say “N is behaving like the person in whose hand there is pain”. But why should one enlist
the experience of pain to describe conscious behaviour at all? – After all, we simply want
to separate two different areas of experience; as when we separate tactile and visual
experience in a body. And nothing can be more different than the experience of pain and
the experience of seeing a human body writhing, of hearing it utter sounds, etc. And,
what’s more, there’s no difference here between my body and someone else’s, for there’s
also the experience of seeing the movements of one’s own body and of hearing the sounds
uttered by it.
Let’s imagine that our body were removed from our field of vision, say by someone making
it completely diaphanous; but suppose it retained the ability to appear to us in the usual
way in an appropriate mirror, so that we would perceive, say, the visible manifestations of
our toothache essentially like those of a different body. This would also result in a completely
different coordination between the seeing eye and visual space than the everyday one that
seems to us a matter of course. (Think of drawing a quadrangle, with its diagonals, in the
mirror.) But if in this way we can imagine the possibility of recognizing our visible body
only as an image in a mirror, then it is also imaginable 7
that this mirror might be omitted
and that we wouldn’t see our body differently from any other human body. – But what would
then characterize it as my body? Well, only the fact that I would, for instance, feel this body
being touched, but not another one, etc. So it’s no longer essential that the mouth below the
seeing eye speaks my words. (And that is of great importance.) Even when I see my body,
as I now see it, i.e. through its eyes, it is imaginable that I might switch bodies with others.
5 (M): 71
6 (V): is important for
7 (M): 71
“Having Pain.” 361e
WBTC101-107 30/05/2005 16:15 Page 723
es denkbar, daß ich18
mit Andern den Körper tausche. Die Erfahrung bestünde einfach in
dem, was man als eine sprunghafte Änderung meines Körpers und seiner Umgebung
beschreiben würde. Ich würde einmal die Körper A, B, C, D 19
von E aus, und
E von den Augen dieses Körpers sehen, und plötzlich etwa C, D, E, A von
B aus und B aus dessen Augen; etc. Noch einfacher aber wird die Sache, wenn
ich alle Körper – meinen, sowie die fremden – überhaupt nicht aus Augen sehe, und sie also,
was ihre visuelle Erscheinung betrifft, alle auf gleicher Stufe stehen. Dann ist es klar, was
es heißt, daß ich im Zahn des Andern Schmerzen haben kann; – wenn ich dann überhaupt
noch bei der Bezeichnung bleiben will, die einen Körper ”meinen“ nennt und also einen anderen
den ”eines Andern“. Denn es ist nun vielleicht praktischer, die Körper einfach20
mit
Eigennamen zu bezeichnen. – Es gibt also jetzt eine Erfahrung:21
die, der Schmerzen in einem
Zahn eines der existierenden menschlichen Körper;22
das ist nicht die, die ich in der gewöhnlichen
Ausdrucksweise mit den Worten ”A hat Zahnschmerzen“ beschriebe, sondern mit
den Worten ”ich habe in einem Zahn des A Schmerzen“. Und es gibt die andere Erfahrung:
einen Körper, sei es meiner oder ein andrer, sich winden zu sehen. Denn, vergessen wir
nicht: Die Schmerzen haben zwar einen Ort im Raum, sofern man z.B. sagen kann, sie
wandern, oder seien an zwei Orten zugleich, etc.: aber ihr Raum ist nicht der visuelle oder
physikalische. – Und nun haben wir zwar eine neue Ausdrucksweise, sie ist aber nicht mehr
asymmetrisch. Sie bevorzugt nicht einen Körper, einen Menschen zum Nachteil des andern,
ist also nicht solipsistich. – So ist alle Erfahrung23
ohne Ansehen der Person verteilt. Aber
wir teilen anders. Es werden die Dinge in unsrer Betrachtungsweise anders zusammengefaßt.
Wie wenn man einmal die Zeit zum Raum rechnet und einmal nicht, oder wie wenn man
einen Wald als Holzblock mit Löchern ansähe. Oder die Bahn des Mondes um24
die Sonne
einmal als Kreisbahn um die Erde, die sich verschiebt, – ein andermal als Wellenlinie, die
um die Sonne läuft. (Wäre die Erde etwa nicht sichtbar, so könnte es eine merkwürdige neue
Betrachtungsweise sein, die Wellenbewegung des Mondes um die Sonne als Kreisbahn um
ein kreisendes Zentrum25
aufzufassen.) Man könnte auf diese Weise gewisse Vorurteile zerstören,
die auf die besondere uns geläufige Betrachtungsart aufgebaut wären. – Sehr klar wird
der Charakter der anderen Betrachtungsweise, wenn man an die analoge Veränderung26
der
Grenzen durch die Einführung des Begriffs der Gedächtniszeit denkt. Es ist ganz ähnlich
der veränderten Betrachtung der Mondbewegung. Eine Grenze, die früher mit anderen in
der Zeichnung zusammenlief, wird plötzlich stark ausgezogen und hervorgehoben. –
18 (V): daß ich mich
19 (F): MS 114, S. 26r.
20 (V): nur
21 (V): Erfahrung,
22 (V): Körpers;
23 (V): ist alles
24 (O): in
25 (V): um einen kreisenden Körper
26 (V): Verschiebung
516
362 ”Schmerzen haben.“
E
D
C
A B
WBTC101-107 30/05/2005 16:15 Page 724
The experience would simply consist in what one would describe as a sudden change of my
body and its surroundings. I would start out seeing the bodies A, B, C, D 8
from
E, and E through the eyes of E’s body, and then suddenly I would see, say C, D, E,
A from B and B through B’s eyes; etc. But the matter gets even simpler if I see all
bodies – my own as well as those of others – not through eyes at all, and if they are thus all
situated on the same level, so far as their visual appearance is concerned. Then it’s clear what
it means for me to be able to have an ache in someone else’s tooth; – if, in that case, I even
want to stick to a designation that calls one body “mine”, and a different one “someone else’s”.
Because now it might be more practical to designate the bodies simply9
with proper names.
– So now there is an experience: that of a pain in a tooth of one of the existing human
bodies; it isn’t the one that, using the usual form of expression, I would describe with the
words “A has a toothache”, but with the words “I have an ache in one of A’s teeth”. And
there is the other experience: of seeing a body writhing, be it mine or someone else’s. For
let’s not forget: it is true that pain has a location in space, in so far as one can say for instance
that it travels or is situated in two places at the same time, etc.: but its space isn’t visual or
physical space. – And now, to be sure, we do have a new form of expression, but it is no
longer asymmetrical. It doesn’t give preference to one body, one person to the disadvantage
of another, and therefore it’s not solipsistic. – In this way all experience10
is distributed without
respect to persons. But we divide differently. Things are combined differently in our way
of looking at things. As if one considers time as belonging to space at one time, and at another
as not, or as if one were to view a forest as a block of wood with holes in it. Or as if one saw
the orbit of the moon around the sun now as a shifting circular orbit around the earth, now
as a wavy line that runs around the sun. (If, say, the earth were invisible, it could be a strange
new way of looking at things to understand the wave-motion of the moon around the sun
as a circular orbit around a rotating centre.)11
In this way, one could destroy certain prejudices
that are based on our particular way of looking at things that we are used to. – What
is characteristic of looking at things in this other way becomes quite clear if one thinks about
the analogous change12
of boundaries that results from the introduction of the concept of
memory-time. This is quite similar to the change in view about the motion of the moon. A
boundary in a drawing that previously converged with others is suddenly drawn in bold and
emphasized. –
8 (F): MS 114, p. 26r.
9 (V): only
10 (V): way everything
11 (V): a body.)
12 (V): shift
“Having Pain.” 362e
E
D
C
A B
WBTC101-107 30/05/2005 16:15 Page 725
105
Gedächtniszeit.
”Ist die Zeit, in der die Erlebnisse des Gesichtsraums vor sich gehen, ohne Tonerlebnisse
denkbar? Es scheint, ja. Und doch, wie seltsam, daß etwas eine Form sollte haben können,
die auch ohne eben diesen Inhalt denkbar wäre. Oder lernt der, dem das Gehör geschenkt
würde, damit auch eine neue Zeit kennen?“
Die hergebrachten Fragen taugen zur logischen Untersuchung der Phänomene nicht. Diese
schaffen sich ihre eigenen Fragen, oder vielmehr, geben ihre eigenen Antworten.
Die Zeit ist ja nicht ein Zeitraum, sondern eine Ordnung.
Denn ”die Zeit“ hat eine andere Bedeutung, wenn wir das Gedächtnis als die Quelle der
Zeit auffassen und wenn wir es als ein aufbewahrtes Bild des vergangenen Ereignisses
auffassen.
Wenn wir das Gedächtnis als ein Bild auffassen, dann ist es ein Bild eines physikalischen
Ereignisses. Das Bild verblaßt und ich merke sein Verblassen, wenn ich es mit andern
Zeugnissen des Vergangenen vergleiche. Hier ist das Gedächtnis nicht die Quelle der Zeit,
sondern mehr oder weniger gute Aufbewahrerin dessen, was ”wirklich“ gewesen ist, und
dieses war eben etwas, wovon wir auch andere Kunde haben können, ein physikalisches
Ereignis. – Ganz anders ist es, wenn wir nun das Gedächtnis als Quelle der Zeit betrachten.
Es ist hier kein Bild und kann auch nicht verblassen – in dem Sinne, wie ein Bild verblaßt,
sodaß es seinen Gegenstand immer weniger getreu darstellt. Beide Ausdrucksweisen sind
in Ordnung und gleichberechtigt, aber nicht miteinander vermischbar. Es ist ja klar, daß
die Ausdrucksweise vom Gedächtnis als einem Bild, nur ein Bild ist; genau so, wie die
Ausdrucksweise, die die Vorstellungen ”Bilder der Gegenstände in unserem Geiste“ (oder
dergleichen) nennt. Was ein Bild ist, das wissen wir, aber die Vorstellungen sind doch gar
keine Bilder, denn sonst kann ich das Bild sehen und den Gegenstand, dessen Bild es ist,
aber hier ist es offenbar ganz anders. Wir haben eben ein Gleichnis gebraucht und nun tyrannisiert
uns das Gleichnis. In der Sprache dieses Gleichnisses kann ich mich nicht außerhalb
des Gleichnisses bewegen. Es muß zu Unsinn führen, wenn man mit der Sprache dieses
Gleichnisses1
über das Gedächtnis als Quelle unserer Erkenntnis, als Verifikation unserer
Sätze, reden will. Man kann von gegenwärtigen, vergangenen und zukünftigen Ereignissen
in der physikalischen Welt reden, aber nicht von gegenwärtigen, vergangenen und zukünf-
tigen2
Vorstellungen, wenn man als Vorstellung nicht doch wieder eine Art physikalischen
Gegenstand (etwa jetzt ein physikalisches Bild, statt des Körpers) bezeichnet; sondern
gerade eben das gegenwärtige. Man kann also den Zeitbegriff, d.h. die Regeln der Syntax,
wie sie von den physikalischen Substantiven gelten, nicht in der Welt der Vorstellung
anwenden, d.h. nicht dort, wo man sich einer radikal anderen Ausdrucksweise bedient.
1 (O): Gleichnis 2 (V): zukünftigen Ereignissen in
517
518
363
WBTC101-107 30/05/2005 16:15 Page 726
105
Memory-Time.
“Is the time in which the experiences of visual space take place conceivable without the
experiences of sound? It seems that it is. And yet, how strange that something should be
capable of having a form that’s also conceivable without this very content. Or does a person
who, say, is given the gift of hearing come to know a new time as a result?”
The traditional questions are of no use for the logical investigation of the phenomena.
The latter create their own questions, or rather, supply their own answers.
After all, time is not a temporal space, but an ordering.
For “time” has one meaning when we understand memory as the source of time and another
when we understand it as a preserved image of a past event.
If we understand memory as a picture, then it is a picture of a physical event. The picture
fades and I notice its fading when I compare it with other evidence of the past. Here
memory is not the source of time, but a good or not so good preserver of what “really”
happened, and this is simply something – a physical event – about which we can have heard
from other sources as well. – Now it’s completely different if we look at memory as the source
of time. In this case it isn’t a picture and neither can it fade – in the sense that a picture
fades so that it represents its object less and less accurately. Both forms of expression are in
order and are equally legitimate, but they can’t be intermixed with each other. It’s clear, of
course, that the form of expression of memory as a picture is only a picture; just like the
form of expression that calls mental images “pictures of objects in our mind” (or some such
thing). We know what a picture is, but mental images aren’t pictures at all, because I can
see a picture and the object of which it is a picture; but in the other case things are obviously
quite different. It’s just that we’ve used a simile, and now the simile is tyrannizing us.
In the language of this simile, I cannot move outside the simile. Wanting to use the language
of this simile to speak of memory as the source of our cognition, as the verification of our
propositions, has to lead to nonsense. One can talk about present, past and future events in
the physical world, but not about present, past and future mental images1
– that is, so long
as one doesn’t revert to calling a kind of physical object (say, a physical picture instead of
a body) a mental image, but calls just the present image by that name. So one can’t apply
the concept of time, i.e. the syntactical rules as they apply to physical nouns, to the world
of the mental image; that is, one can’t apply it where one uses a radically different kind
of expression.
1 (V): future events in
363e
WBTC101-107 30/05/2005 16:15 Page 727
Kann ich sagen, das Drama hat seine eigene Zeit, die nicht ein Abschnitt der historischen
Zeit ist. D.h., ich kann in ihm von früher und später reden, aber die Frage hat keinen Sinn,
ob die Ereignisse, etwa, vor oder nach Cäsars Tod geschehen sind.
Das Gleichnis vom Fließen3
der Zeit ist natürlich irreführend und muß uns, wenn wir
daran festhalten, in Verlegenheiten führen.4
Was Eddington5
über ”die Richtung der Zeit“ und den Entropiesatz6
sagt, läuft darauf
hinaus, daß die Zeit ihre Richtung umkehren würde, wenn die Menschen eines Tages
anfingen rückwärts zu gehen. Wenn man will, kann man das freilich so nennen: man muß
dann nur darüber klar sein, daß man damit nichts anderes sagt, als daß die Menschen ihre
Gehrichtung geändert haben.
Die meisten Rätsel, die uns das Wesen der Zeit aufzugeben scheint, kann man durch die
Betrachtung einer Analogie verstehen, die in einer oder der andern Form den verschiedenen
falschen Auffassungen zu Grunde liegt: Es ist der Vorgang, im Projektionsapparat, durch
welchen der Film läuft einerseits, und auf der Leinwand anderseits.
Wenn man sagt, die Zukunft sei bereits präformiert, so heißt das offenbar: die Bilder des
Filmstreifens, welche den zukünftigen Vorgängen auf der Leinwand entsprechen, sind bereits
vorhanden. Aber für das, was ich in einer Stunde tun werde, gibt es ja keine solchen
Bilder, und wenn es sie gibt, so dürfen wir wieder nicht die Bilder auf dem Zukunftsteil des
Filmstreifens mit den zukünftigen Ereignissen auf der Leinwand verwechseln. Nur von jenen
können wir sagen, daß sie präformiert sind, d.h. jetzt schon existieren. Und bedenken wir,
daß der Zusammenhang der Ereignisse auf der Leinwand mit dem, was die Filmbilder zeigen
ein empirischer ist; wir können aus ihnen kein Ereignis auf der Leinwand prophezeien,
sondern nur hypothetisch vorhersagen. Auch – und hier liegt eine andere Quelle des
Mißverständnisses – können wir nicht sagen ”es ist jetzt der Fall, daß dieses Ereignis in einer
Stunde eintreten wird“ oder ”es ist um 5 Uhr der Fall, daß ich um 7 Uhr spazierengehen
werde.“
»Wenn die Erinnerung kein Sehen in die Vergangenheit ist, wie wissen wir dann
überhaupt, daß sie mit Beziehung auf die Vergangenheit zu deuten ist? Wir könnten uns
dann einer Begebenheit erinnern und zweifeln, ob wir in unserm Erinnerungsbild ein Bild
der Vergangenheit oder der Zukunft haben.
Ich kann natürlich sagen: ich sehe nicht die Vergangenheit, sondern nur ein Bild der
Vergangenheit. Aber woher weiß ich, daß es ein Bild der Vergangenheit ist, wenn dies
nicht im Wesen des Erinnerungsbildes liegt. Haben wir etwa durch die Erfahrung gelernt,
diese Bilder als Bilder der Vergangenheit zu deuten? Aber was hieße hier überhaupt
”Vergangenheit“?«
Die Daten unseres Gedächtnisses sind geordnet; diese Ordnung nennen wir
Gedächtniszeit, im Gegensatz zur physikalischen Zeit, der Ordnung der Ereignisse in der
physikalischen Welt. Gegen den Ausdruck ”Sehen in die Vergangenheit“ sträubt sich unser
Gefühl mit Recht; denn es ruft das Bild hervor,7
daß Einer einen Vorgang in der physikalischen
Welt sieht, der jetzt gar nicht geschieht, sondern schon vorüber ist. Und die Vorgänge,
welche wir ”Vorgänge in der physikalischen Welt“, und die, welche wir ”Vorgänge in unserer
Erinnerung“ nennen, sind einander wirklich nur zugeordnet. Denn wir reden von einem
3 (V): Fluß
4 (V): landen.
5 (O): Edington
6 (O): Enthropiesatz
7 (V): es gibt uns ein Bild davon,
520
364 Gedächtniszeit.
519
WBTC101-107 30/05/2005 16:15 Page 728
Can I say that drama has its own time that is not a section of historical time? That is to
say, within a drama I can speak about before and after, but the question whether the events
took place, say, before or after Caesar’s death makes no sense.
Of course the simile of time flowing2
is misleading, and it necessarily leads us into
quandaries3
if we hold on to it.
What Eddington says about “the direction of time” and the principle of entropy amounts
to maintaining that time would reverse its direction if one day people started walking backwards.
If you want to, you can certainly state it this way; but then you have to realize that
in saying this you’re saying only that people have changed the direction in which they walk.
One can understand most of the riddles that the nature of time seems to present us with
by examining an analogy that, in one form or another, underlies the various false conceptions:
on the one hand it’s what goes on within the projector through which the film is
running, on the other it’s what goes on on the screen.
If one says that the future is already pre-formed, then this obviously means: the pictures
on the strip of film that correspond to the future events on the screen already exist. But
there just aren’t any such pictures for what I’ll be doing in an hour, and if there are, then
once again we mustn’t confuse the pictures on the future part of the film strip with the future
events on the screen. Only of the former can we say that they are pre-formed, i.e. that they
already exist now. And let’s bear in mind that the connection between the events on the screen
and what the film images show is an empirical one; we can’t prophesy any event on the screen
from those images, but can only make hypothetical predictions. Neither can we say – and
here is another source of misunderstanding – “It’s now the case that this event will occur
in an hour” or “It’s the case at 5 o’clock that I’ll take a walk at 7 o’clock.”
»If memory isn’t a looking into the past, then how do we know in the first place that it is
to be interpreted with reference to the past? If we didn’t know that, we might remember an
event and be in doubt whether in our memory image we have an image of the past or of the
future.
Of course I can say: I don’t see the past, but only an image of the past. But how do I know
that it is an image of the past if this isn’t contained in the essence of the memory-image?
Have we perhaps learned from experience to interpret these images as images of the past?
But what would “past” mean here, anyway?«
The data of our memory are ordered; we call this order memory-time, as opposed to
physical time, the order of events in the physical world. For good reason, we bristle at the
expression “looking into the past”; for it evokes4
the image of someone seeing an event in
the physical world that is not taking place now at all, but is already past. And the events
that we call “events in the physical world” and those that we call “events in our memory”
are really only attributed to each other. For we speak about remembering something
2 (V): of the flow of time
3 (V): necessarily lands us in quandaries
4 (V): it gives us
Memory-Time. 364e
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Fehlerinnern und das Gedächtnis ist nur eines von den Kriterien dafür, daß etwas in der
physikalischen Welt geschehen ist.
Die Erinnerungszeit unterscheidet sich unter anderem dadurch von der physikalischen,
daß sie ein Halbstrahl ist, dessen Anfangspunkt8
die Gegenwart ist. Der Unterschied zwischen
Erinnerungszeit und physikalischer Zeit ist natürlich ein logischer. D.h.: die beiden
Ordnungen könnten sehr wohl mit ganz verschiedenen Namen bezeichnet werden und man
nennt sie nur beide ”Zeit“, weil eine gewisse grammatische Verwandtschaft besteht, ganz
wie zwischen Kardinal- und Rationalzahlen; Gesichtsraum, Tastraum und physikalischem
Raum; Farbtönen und Klangfarben, etc., etc.
Gedächtniszeit. Sie ist (wie der Gesichtsraum) nicht ein Teil der großen Zeit, sondern
die spezifische Ordnung der Ereignisse oder Situationen in der Erinnerung.9
In dieser Zeit
gibt es z.B. keine Zukunft. Gesichtsraum10
und physikalischer Raum, Gedächtniszeit und
physikalische Zeit, verhalten sich zueinander nicht wie ein Stück der Kardinalzahlenreihe
zum Gesetz dieser Reihe (”der11
ganzen Zahlenreihe“), sondern, wie das System der
Kardinalzahlen zu dem, der rationalen Zahlen. Und dieses Verhältnis erklärt auch den Sinn
der Meinung, daß der eine Raum den andern einschließt, enthält.
Messung des Raumes und des räumlichen Gegenstandes. Das Seltsame am leeren Raum
und an der leeren Zeit. Die Zeit (und der Raum) ein ätherischer Stoff. Von Substantiven
verleitet, glauben wir an eine Substanz.12
Ja, wenn wir der Sprache die Zügel überlassen und
nicht dem Leben, dann entstehen die philosophischen Probleme.
”Was ist die Zeit?“ – schon in der Frage liegt der Irrtum: als wäre die Frage: woraus, aus
welchem Stoff, ist die Zeit gemacht. Wie man etwa sagt, woraus ist dieses feine Kleid gemacht.
Die alles gleichmachende Gewalt der Sprache, die sich am krassesten im Wörterbuch
zeigt, und die es möglich macht, daß die Zeit personifiziert werden konnte; was nicht weniger
merkwürdig ist, als es wäre, wenn wir Gottheiten der logischen Konstanten hätten.
8 (V): Endpunkt
9 (V): im Gedächtnis.
10 (O): Zukunft, Gesichtsraum
11 (V): ”zur
12 (V): verleitet, nehmen wir eine Substanz an.
522
365 Gedächtniszeit.
521
WBTC101-107 30/05/2005 16:15 Page 730
incorrectly, and memory is only one of the criteria for something having happened in the
physical world.
Among other things, memory-time is distinguished from physical time by being a ray,
the origin5
of which is the present. Of course, the difference between memory-time and
physical time is a logical difference. That is, the two orders could perfectly well be called
by completely different names, and one only calls both of them “time” because there’s a
certain grammatical relationship, just as between cardinal and rational numbers; as between
visual space, tactile space and physical space; as between shades of colour and tone colours,
etc., etc.
Memory-time. It (like visual space) is not a part of time in the larger sense, but is the
specific order of events or situations in memory. In this time there is no future, for instance;
visual and physical space, memory-time and physical time are not related to each other as a
section of the series of cardinal numbers is to the law of this series (“of6
the entire numberseries”),
but rather as the system of cardinal numbers is to that of the rational numbers.
And this relationship also makes sense of the idea that the one space encloses, contains, the
other one.
Measurement of space and an object in space. The strangeness of empty space and empty
time. Time (and space) an ethereal stuff. Seduced by substantives, we believe in7
Substance.
Indeed, when we hand over the reins to language and not to life, that’s when the philosophical
problems arise.
“What is time?” – the error is already contained in the question, as if the question were:
of what, of what material, is time made? As one might say – of what is this fine dress made?
The all-levelling power of language that appears in its crassest form in the dictionary, and
that makes it possible for time to be personified; which is no less remarkable than if we had
deities for the logical constants.
5 (V): the end-point
6 (V): (“to
7 (V): we assume
Memory-Time. 365e
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106
”Hier“ und ”Jetzt“.
In gewissem Sinne ist die Bedeutung der Wörter ”hier“, ”jetzt“ (etc.) die einzige, die ich
nicht von vornherein festlegen kann. Aber das ist natürlich irreführend ausgedrückt: Die
Bedeutung ist festzulegen und festgelegt, wenn die Regeln bezüglich dieser Worte festgelegt
sind, und das kann geschehen, ehe sie1
in einem bestimmten Fall angewandt werden; denn
wozu auch sonst ein Wort in verschiedenen Fällen gebrauchen.
Die Wörter ”hier“, ”jetzt“, etc. bezeichnen den Ursprung2
eines Koordinatensystems:
Wie der Buchstabe ”O“, aber sie stehen nicht für Beschreibungen der Lage des Punktes O
im Verhältnis zu räumlichen Gegenständen.3
Sie stehen nicht für die Beschreibung einer
räumlichen Situation.
Unterschied zwischen Sage und Märchen. Märchen4
(und andere Dichtungen) vom Jetzt
und Hier abgeschnitten.
Es ist aber ein wichtiger Satz in der Grammatik des Wortes ”hier“, daß es keinen Sinn
hat, ”hier“ zu schreiben, wo eine Ortsangabe stehen soll; daß ich also auf einen Gegenstand
kein Täfelchen befestigen soll, mit der Aufschrift ”Dieser Gegenstand ist immer nur hier
zu benützen“.
Ich kann natürlich in Bezug auf die Wörter ”jetzt“ und ”hier“ etc. nur tun, was ich sonst
tue, nämlich ihren Gebrauch beschreiben. Aber5
diese Beschreibung muß allgemein sein, d.h.
im Vorhinein, vor jedem Gebrauch.
Hier und Jetzt sind geometrische Begriffe, wie etwa der Mittelpunkt meines
Gesichtsfeldes.
Hier und Jetzt haben nicht eine größere Multiplizität, als sie zu haben scheinen. Das
anzunehmen ist die große Gefahr. Ersetze sie, durch welchen Ausdruck Du willst, immer
ist es nur ein Wort – und daher eins so gut wie das andere.
Das, was ”particular“ ist, ist das Ereignis. Das Ereignis, das durch die Worte beschrieben
wird, ”heute hat es geregnet“ und am nächsten Tag durch ”gestern hat es geregnet“.
Was ist denn die ”gegenwärtige Situation“? Nun, daß das und das der Fall ist. Nicht: ”daß
das und das jetzt der Fall ist“.
”Jetzt“ ist ein Wort. Wozu brauche ich dieses Wort? ”Jetzt“ – im Gegensatz wozu? – Im
Gegensatz zu ”in einer Stunde“, ”vor 5 Minuten“, etc. etc.
1 (V): die
2 (V): Anfangspunkt
3 (V): aber sie beschreiben nicht seine Lage
gegenüber den Gegenständen im Raum.
4 (O): Märchen, Märchen
5 (V): Und
523
524
366
WBTC101-107 30/05/2005 16:15 Page 732
106
“Here” and “Now”.
In a certain sense the meaning of the words “here”, “now” (etc.) is the only one that I can’t
specify in advance. But of course that’s putting it in a misleading way: The meaning can
be specified and is specified when the rules for these words have been specified, and that
can happen before they are applied in a particular case; otherwise why use one word for
different cases?
The words “here”, “now”, etc. refer to the origin of1
a system of coordinates, like the
letter “O”; but they don’t stand for descriptions of the location of point O relative to
objects in space. They don’t stand for the description of a spatial situation.2
The difference between myth and fairy tale. Fairy tales (and other works of fiction) are
cut off from the here and now.
But it’s an important proposition in the grammar of the word “here” that it makes no
sense to write “here” where a location still needs to be specified; that therefore I shouldn’t
attach a label to an object that reads “This object is to be used here exclusively”.
Of course, with respect to the words “now” and “here”, etc., I can only do what I
usually do, i.e. describe their use. But3
this description must be general, i.e. there from the
start, before any use.
“Here” and “now” are geometrical concepts, like, for instance, the centre point of my field
of vision.
“Here” and “now” have no greater multiplicity than they seem to have. There’s a huge
danger in assuming they do. Replace them with whatever expression you want, it’s always
just one word – and therefore one word is as good as the other.
What is “particular” is the event. The event described by the words “It rained today”,
and the next day by “It rained yesterday”.
What is the “present situation” anyway? Well, that this or that is the case. Not: “that this
or that is the case now”.
“Now” is a word. What do I need this word for? “Now” – as opposed to what? – As opposed
to “in an hour”, “5 minutes ago”, etc., etc.
1 (V): the beginning point for
2 (V): letter “O”; but they don’t describe its location
vis-à-vis the objects in space.
3 (V): And
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”Jetzt“ bezeichnet kein System, sondern gehört zu einem System. Es wirkt nicht
magisch; wie auch sonst kein Wort.
Wenn die Sprache sich mit dem Gelde vergleichen läßt, an dem an und für sich nichts
liegt, sondern das nur indirekt von Bedeutung ist, weil man damit6
Gegenstände kaufen kann,
die für uns Bedeutung haben; so möchte man vielleicht sagen,7
daß hier beim Gebrauch der
Wörter ”ich“, ”hier“, ”jetzt“ etc. der Tauschhandel in den Geldhandel eintritt.8
Wenn ich sage ”ich gehe jetzt dorthin“, so kommt in dem Symbol manches vor, was in
dem Zeichen allein nicht liegt. Der Satz, wenn ich ihn etwa von unbekannter Hand
geschrieben, irgendwo vorfinde, sagt gar nichts; das Wort ”ich“, das Wort ”jetzt“ und
”dorthin“ sind allein ohne die Gegenwart der sprechenden Person, der gegenwärtigen
Situation und der im Raum gezeigten Richtung bedeutungslos.
”Jetzt“, ”früher“, ”hier“, ”dort“, ”ich“, ”Du“, ”dieses“, sind solche Wörter zur
Anknüpfung an die Wirklichkeit.
”Aber die Wirklichkeit, die solcherart zum Symbol gehört, fällt unter die Herrschaft der
Grammatik.“
Nun könnte man fragen: Gehört die Windrose noch zum Plan? Oder vielmehr; gehört
die Regel, nach der die Windrose angewandt wird, noch zum Plan? Und es ist klar, daß ich
diese Regel durch eine andere Orientierungsregel ersetzen kann, in der von der Windrose
nicht die Rede ist, sondern statt deren9
etwa von einem Weg auf dem Plan und was ihm in
der Gegend entspricht.
Wenn (in einem Satz ”ich will, daß Du dorthin gehst“) der Sprechende, der
Angesprochene und der Pfeil der die Richtung weist, zum Symbolismus gehören, so
spielen sie in ihm jedenfalls eine ganz andere Rolle, als die Wörter.
Wenn aber die Grammatik den ganzen Symbolismus umfassen soll, wie zeigt sich in ihr
die Ergänzungsbedürftigkeit der Wörter ”ich“, ”Du“, ”dieses“, etc. durch Gegenstände der
Realität?
Denn, daß jener Satz ohne eine solche Ergänzung nichts sagt, muß die Grammatik sagen.
Wenn sie das vollständige Geschäftsbuch der Sprache sein soll (wie ich es meine).
Ich will immer zeigen, daß alles was in10
der Logik ”business“ ist, in der Grammatik gesagt
werden muß.
Wie etwa der Fortgang eines Geschäftes aus den Geschäftsbüchern muß vollständig
herausgelesen werden können. Sodaß man, auf die Geschäftsbücher deutend, muß sagen
können: Hier! hier muß sich alles zeigen; und was sich hier nicht zeigt, gilt nicht. Denn am
Ende muß sich hier alles Wesentliche abspielen.
Alles wirklich Geschäftliche – heißt das – muß sich in der Grammatik abwickeln.
Wie erklärt die Grammatik das Wort ”jetzt“? Doch wohl durch die Regeln, die sie für
seinen Gebrauch angibt. Das Gleiche 11
für das Wort ”ich“.
6 (V): man mit ihm
7 (V): so kann man sagen,
8 (M): (?)
9 (O): dessen
10 (V): an
11 (V): Gleiche gi
525
526
367 ”Hier“ und ”Jetzt“.
WBTC101-107 30/05/2005 16:15 Page 734
“Now” doesn’t designate a system; rather, it belongs to one. It works no magic – any more
than any other word.
If language can be compared to money – which means nothing in and of itself, but is only
of importance indirectly – because with it one can buy things that are important to us, then
one might be inclined to say4
that here, in the use of the words “I”, “here”, “now”, etc.,
barter enters into the commerce of money.5
If I say “Now I’m going there”, then some things occur in the symbol that aren’t contained
in the sign alone. If, say, I find the sentence somewhere, written by an unknown hand,
then it doesn’t mean anything at all; by themselves, in the absence of a speaker, a present
situation and an indication of a spatial direction, the word “I”, the word “now”, and “there”
are meaningless.
“Now”, “earlier”, “here”, “there”, “I”, “you”, “this” are words for ligatures to reality.
“But the reality that belongs to a symbol in this way falls under the domain of grammar.”
Now one could ask: Is the compass rose part of a map? Or rather: Is the rule according
to which the compass rose is applied part of a map? And it’s clear that I can replace this rule
with a different rule of orientation, in which there is no mention of a compass rose, but instead,
say, of a path on the map and of what corresponds to it on the ground.
If (in the sentence “I want you to go over there”) the speaker, the person being spoken
to, and the arrow indicating the direction all belong to the symbolism, then nevertheless they
play an entirely different role within it than do the words.
But if grammar is to encompass the entire symbolism, how is it shown in grammar that
there’s a need to supplement the words “I’’, “you’’, “this’’, etc., with real objects?
For it is grammar that has to say that a proposition with these words means nothing
without such a supplement. If it’s to be the complete ledger of language (as I believe).
I’m always wanting to show that everything that’s “business” in logic6
has to be stated in
grammar.
As, say, the progress of a business has to be completely discernible from its ledgers.
So that one has to be able to point to the ledgers and say: There! Everything has to show
up there; and what doesn’t show up there doesn’t count. For in the end, everything that is
essential must take place there.
And this means that everything actually pertaining to business has to be transacted in
grammar.
How does grammar explain the word “now’’? Surely via the rules it gives for its use. The
same goes for the word “I”.
4 (V): one can say
5 (M): (?)
6 (V): show that the “business” of logic
“Here” and “Now”. 367e
WBTC101-107 30/05/2005 16:15 Page 735
Ich könnte mir denken, daß Einer, um das Wort ”jetzt“ zu erklären, auf den gegenwärtigen
Zeigerstand einer Uhr zeigt.12
So wie13
er zur Erklärung des Ausdrucks ”in fünf Minuten“
auf die Ziffer der Uhr zeigen kann, wo der Zeiger sich in fünf Minuten befinden wird.
Es ist klar, daß dadurch nur die Uhr in unsere Zeichensprache einbezogen wird.
Das Wort ”jetzt“ wirkt gleichsam als Schlag eines Zeitmessers. Es gibt durch sein
Ertönen eine Zeit an. Man kann es ja auch wirklich durch ein anderes Zeitzeichen ersetzen.
Wenn man z.B. sagt: tu das, wenn ich in die Hände klatsche. Das Klatschen ist dann ein
Zeitzeichen, wie der Pfeil ein Richtungszeichen ist, wenn ich sage ”gehe dort → hin“.
Wenn mir z.B. die Rede, die ein Anderer gestern gesprochen hat, mitgeteilt wird: ”es14
geschieht heute das und das“, so muß ich verstehen, daß der Satz, wenn ich ihn höre, nicht
so verifiziert werden kann, wie er zu verifizieren war, als er ursprünglich ausgesprochen wurde.
Die Grammatik sagt mir: wenn ich gestern sagte ”heute geschieht es“, so heißt das soviel,
wie wenn ich heute sage ”gestern ist es geschehen“.
Wenn man nun sagt ”dieser Mensch heißt N“, so muß uns die Grammatik sagen, daß
diese Wortfolge keinen Sinn hat, wenn sie nicht durch ein Hinweisen ergänzt wird.
12 (V): gegenwärtigen Stand der Zeiger einer Uhr
zeigt.
13 (O): Sowie
14 (O): wird. ”es
527
368 ”Hier“ und ”Jetzt“.
WBTC101-107 30/05/2005 16:15 Page 736
I could well imagine that to explain the word “now’’, someone points to the present position
of the hands of a clock. Just as, in order to explain the expression “in five minutes’’, he
can point to the number on the clock where the hand will be in five minutes.
It is clear that in so doing he’s just including the clock in our language of signs.
The word “now’’ works like the ring of a timer, as it were. The sound gives the time.
Indeed, one can actually replace it with a different sign for time. If we say, for instance: “Do
this when I clap my hands”. There, clapping is a sign for time, as the arrow is a sign for
direction when I say “Go over there →”.
If, for instance, I’m told of a statement that someone made yesterday: “Such and such is
happening today”, then I have to understand that that proposition can’t be verified in the
same way when I hear it as it could be verified when it was originally uttered. Grammar
tells me: If yesterday I said “It’s happening today”, that means the same as if today I say
“It happened yesterday”.
Now if one says “This man’s name is N”, grammar has to tell us that this sequence of
words makes no sense unless it is supplemented with pointing.
“Here” and “Now”. 368e
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107
Farbe, Erfahrung, etc.
als formale Begriffe.
Man überlege: welchen Grund hat man, ein neues Phänomen Farbe zu nennen, wenn es
sich nicht in unser bisheriges Farbenschema einfügt.
Erfahrung ist nicht etwas, das man durch Bestimmungen von einem Andren abgrenzen
kann, was nicht Erfahrung ist; sondern eine logische Form.
Die Erfahrung (Der Begriff der Erfahrung) scheint (uns) von völligem Dunkel begrenzt.
Aber auch Schwarz ist1
eine Farbe, und wenn eine Farbe gegen Schwarz abgegrenzt ist,
so durch eine Farbgrenze, wie jede andre.
Unmittelbare Erfahrung (Sinnes-Datum) ist entweder ein Begriff von trivialer
Abgrenzung oder eine Form.
1 (V): wäre
528
369
WBTC101-107 30/05/2005 16:15 Page 738
107
Colour, Experience, etc.,
as Formal Concepts.
Consider: What reason does anyone have to call a new phenomenon a colour if it doesn’t
fit in with the colour scheme we’ve been using so far?
Experience isn’t something you can demarcate from something else that isn’t experience
by stipulation; rather, it’s a logical form.
Experience (the concept of experience) seems (to us) to be bordered by utter darkness.
But even black is1
a colour, and when a colour is demarcated from black that’s done with
a colour border, like any other colour border.
Immediate experience (a sense datum) is either a concept with trivial demarcations, or
a form.
1 (V): black would be
369e
WBTC101-107 30/05/2005 16:15 Page 739
Grundlagen der
Mathematik.
529
WBTC108-115 30/05/2005 16:20 Page 740
Foundations of
Mathematics.
WBTC108-115 30/05/2005 16:20 Page 741
108
Die Mathematik mit
einem Spiel verglichen.
Was spricht man der Mathematik ab wenn man sagt, sie sei nur ein Spiel (oder: sie sei
ein Spiel)?
Ein Spiel, im Gegensatz wozu? – Was spricht man ihr zu, wenn man sagt (sie sei kein
Spiel), ihre Sätze hätten Sinn?1
Der Sinn außerhalb des Satzes.
Und was geht uns der an? Wo zeigt er sich und was können wir mit ihm anfangen? (Auf
die Frage ”was ist der Sinn dieses Satzes?“ kommt ein Satz zur Antwort.)2
(”Aber der mathematische Satz drückt doch einen Gedanken aus“ – Welchen
Gedanken? – )
Kann er durch einen anderen Satz ausgedrückt werden? oder nur durch diesen Satz? –
Oder überhaupt nicht? In diesem Falle geht er uns nichts an.
Will man durch die mathematischen Sätze von andern Gebilden, den Hypothesen, etc.
etwa unterscheiden? Daran tut man Recht, und daß dieser Unterschied besteht, unterliegt
ja keinem Zweifel.
Will man sagen, die Mathematik werde gespielt, wie das Schach, oder eine Patience und
es laufe dabei auf ein Gewinnen oder Ausgehen hinaus,3
so ist das offenbar unrichtig.
Sagt man, daß die seelischen Vorgänge, die den Gebrauch der mathematischen Symbole
begleiten, andere sind, als die, die das Schachspiel4
begleiten, so weiß ich darüber nichts
zu sagen.
Es gibt auch beim Schach einige Konfigurationen, die unmöglich sind, obwohl jeder Stein
in einer ihm erlaubten Stellung steht. (Wenn z.B.5
die Anfangsstellung der Bauern intakt ist
und ein Läufer schon auf dem Feld.) Aber man könnte sich ein Spiel denken, in welchem6
die Anzahl der Züge vom Anfang der Partie notiert würde, und dann gäbe es den Fall, daß
nach n Zügen diese Konfiguration nicht eintreten könnte und man es der Konfiguration doch
nicht ohneweiters ansehen kann, ob sie als n-te möglich ist, oder nicht.
Die Handlungen im Spiel müssen den Handlungen im Rechnen entsprechen. (Ich meine:
darin muß die Entsprechung bestehen, oder, so müssen die beiden einander zugeordnet sein.)
1 (V): Was spricht man ihr zu, wenn man sagt, ihre
Sätze hätten Sinn?
2 (O): zur Antwort. (V): Satzes?“ antwortet ein
Satz.
3 (V): und es gebe dabei ein Gewinnen oder
Ausgehen,
4 (V): Schachspielen
5 (V): (Z.B. wenn
6 (V): denken, worin
530
531
371
WBTC108-115 30/05/2005 16:20 Page 742
108
Mathematics
Compared to a Game.
What do we deny mathematics when we say it is only a game (or: it is a game)?
A game, in contrast to what? – What do we award mathematics when we say (it isn’t a
game), its propositions make sense?1
The sense outside the proposition.
And what concern is it of ours? Where does it manifest itself and what can we do with
it? (The question “What is the sense of this proposition?” is answered by a proposition.)
(“But a mathematical proposition does express a thought!” – Which thought? – )
Can it be expressed by another proposition? Or only by this proposition? – Or not at all?
In that case it is no concern of ours.
Do you want to distinguish mathematical propositions from other constructions, such as
hypotheses? You are right to do so, and there is no doubt that this distinction exists.
If you want to say that mathematics is played like chess or patience, and the point of it
is2
winning or going out, that is obviously incorrect.
If you say that the mental processes accompanying the use of mathematical symbols are
different from those accompanying chess,3
I don’t know what to say about that.
In chess there are some configurations that are impossible even though each piece is in a
permissible position. (If, for example,4
all the pawns are still in their initial position, but a
bishop is already in play.) But one could imagine a game in which a record was kept of the
number of moves from the beginning of the game, and then there might be the situation
where a configuration couldn’t occur after n moves, and yet one couldn’t tell just by looking
at the configuration whether or not it was possible as the nth configuration.
Moves in games must correspond to moves in calculating. (I mean: that’s what the
correspondence must consist in, or, that’s the way that the two must be correlated with
each other.)
1 (V): when we say its propositions make sense?
2 (V): and in it there is
3 (V): accompanying the playing of chess,
4 (V): (For example, if
371e
WBTC108-115 30/05/2005 16:20 Page 743
Handelt die Mathematik von Schriftzeichen?7
Ebensowenig, wie das Schachspiel von
Holzfiguren handelt.
Wenn wir von dem Sinn mathematischer Sätze reden, oder; wovon sie handeln, so
gebrauchen wir ein falsches Bild. Es ist nämlich hier auch so, als ob unwesentliche,
willkürliche Zeichen das Wesentliche – eben den Sinn – miteinander gemein hätten.8
Weil die Mathematik9
ein Kalkül ist und daher wesentlich von nichts handelt, gibt es keine
Metamathematik.
Wie verhält sich die Schachaufgabe (das Schachproblem) zur Schachpartie? – Denn, daß
die Schachaufgabe der Rechenaufgabe entspricht, eine Rechenaufgabe ist, ist klar.
Ein arithmetisches Spiel wäre z.B. folgendes: Wir schreiben auf gut Glück eine vierstellige
Zahl hin, etwa 7368; dieser Zahl soll man sich dadurch nähern, daß man die Zahlen 7,
3, 6 und 8 in irgendeiner Reihenfolge miteinander multipliziert. Die Spielteilnehmer
rechnen mit Bleistift auf Papier, und wer in der geringsten Anzahl von Operationen der
Zahl 7368 am nächsten kommt, hat gewonnen. (Übrigens lassen sich eine Menge der
mathematischen Rätselfragen zu solchen Spielen umformen.)
Angenommen, einem Menschen wäre Arithmetik nur zum Gebrauch in einem arithmetischen
Spiel gelehrt worden. Hätte er etwas Anderes gelernt als der, welcher Arithmetik
zum gewöhnlichen10
Gebrauch lernt? Und wenn er nun im Spiel 21 mit 8 multipliziert und
168 erhält, tut er etwas Andres, als der, welcher herausfinden wollte, wieviel 21 × 8 ist?
Man wird sagen: Der Eine wollte doch eine Wahrheit finden, während der Andre nichts
dergleichen wollte.
Nun könnte man diesen Fall etwa mit dem des Tennisspiels11
vergleichen wollen, in welchem
der Spieler eine bestimmte Bewegung macht, der Ball darauf in bestimmter Weise fliegt und
man diesen Schlag nun als Experiment auffassen kann, durch welches man eine bestimmte
Wahrheit erfahren hat, oder aber auch als eine Spielhandlung, mit dem alleinigen Zweck,
das Spiel zu gewinnen.
Dieser Vergleich würde aber nicht stimmen, denn wir sehen im Schachzug kein
Experiment (was wir übrigens auch könnten), sondern eine Handlung einer Rechnung.
Es könnte Einer vielleicht sagen: In dem arithmetischen Spiel werden wir zwar multiplizieren
, aber die Gleichung 21 × 8 = 168 wird nicht im Spiel vorkommen. Aber ist
das nicht ein äußerlicher Unterschied? und warum sollen wir nicht auch so multiplizieren
(und gewiß dividieren), daß die Gleichung als solche angeschrieben wird?
Also kann man nur einwenden, daß in dem Spiel die Gleichung kein Satz ist. Aber was
heißt das? Wodurch wird sie dann zu einem Satz? Was muß noch dazu kommen, damit
sie ein Satz wird? – Handelt es sich nicht um die Verwendung12
der Gleichung (oder der
Multiplikation)? – Und Mathematik ist es wohl dann, wenn es zum Übergang von einem
Satz zu einem andern verwendet wird. Und so wäre das unterscheidende Merkmal
zwischen Mathematik und Spiel mit dem Begriff des Satzes (nicht ”mathematischen
Satzes“) gekuppelt, und verliert damit für uns seine Aktualität.
21 8
168
×
7 (V): Zeichen?
8 (V): miteinander gemeinsam haben.
9 (O): Grammatik (E): Sinngemäß und auf
Grund früherer Versionen des Typoskripts
[MS 110 (S. 10), TS 211 (S. 123), TS 212 (S.
1391)], haben wir hier statt ”Grammatik“
”Mathematik“ gesetzt.
10 (V): normalen
11 (O): Tennisspiel
12 (V): Anwendung
532
533
372 Die Mathematik mit einem Spiel verglichen.
WBTC108-115 30/05/2005 16:20 Page 744
Is mathematics about written signs5
? No more than chess is about wooden pieces.
When we talk about the sense of mathematical propositions, or what they are about, we
are using a false picture. For here too it’s as if there were inessential, arbitrary signs that
had6
something essential in common, namely their sense.
Because mathematics7
is a calculus and therefore is really about nothing, there isn’t any
metamathematics.
How is a chess exercise (a chess problem) related to a game of chess? – For it’s clear that
chess problems correspond to arithmetical problems, that they are arithmetical problems.
The following would be an example of an arithmetical game: We write down a four-digit
number at random, e.g. 7,368; we are to approach this number by multiplying the numbers
7, 3, 6, and 8 with each other in any order. The players calculate with pencil and paper, and
the person who comes closest to the number 7,368 in the smallest number of steps has won.
(Many mathematical puzzles, incidentally, can be turned into games of this kind.)
Suppose a human being had been taught arithmetic only for use in an arithmetical game.
Would he have learned something different from a person who learns arithmetic for its
ordinary8
use? And if he multiplies 21 by 8 in the game and gets 168, does he do something
different from a person who wanted to find out how much 21 × 8 is?
It will be said: But the one wanted to find out a truth, whereas the other wanted nothing
of the sort.
Well, we might want to compare this with a tennis game, where a player moves his racket
in a certain way and then the ball flies off in a particular way; and we can view this stroke
either as an experiment, by means of which we discovered a particular truth, or as a move
with the sole purpose of winning the game.
But this comparison wouldn’t be right, because we don’t regard a move in chess as an
experiment (which, incidentally we could do as well), but as a step in a calculation.
Someone might perhaps say: In the arithmetical game, we may do the multiplication
but the equation 21 × 8 = 168 needn’t occur in the game. But isn’t that a superficial
distinction? And why shouldn’t we multiply (and of course divide) in such a way that
the equation is written down as an equation?
So one can only object that in the game the equation is not a proposition. But what
does that mean? How does it become a proposition? What must be added to it to make it a
proposition? – Isn’t it a matter of the use9
of the equation (or of the multiplication)? – And
it is mathematics, I should think, when it is used for the transition from one proposition to
another. And thus the distinguishing mark between mathematics and a game becomes linked
to the concept of a proposition (not “mathematical proposition”), thereby causing the mark
to lose its actuality for us.
5 (V): about signs
6 (V): have
7 (O): grammar (E): On the basis of earlier
versions of this remark (in MS 110, p. 10; TS
211, p. 123; TS 212, p. 1391), we have replaced
“grammar” here with “mathematics”.
8 (V): normal
9 (V): application
Mathematics Compared to a Game. 372e
21
× 8
168,
WBTC108-115 30/05/2005 16:20 Page 745
Man könnte aber sagen, daß der eigentliche Unterschied darin bestehe, daß für Bejahung
und Verneinung im Spiel kein Platz sei. Es wird da z.B. multipliziert und 21 × 8 = 148 wäre
ein falscher Zug, aber ”~(21 × 8 = 148)“, welches ein richtiger arithmetischer Satz ist, hätte
in unserm Spiel nichts zu suchen.
(Da mag man sich daran erinnern, daß in der Volksschule nie mit Ungleichungen
gearbeitet wird, vom Kind nur die richtige Ausführung der Multiplikation verlangt wird
und nie – oder höchst selten – die Konstatierung einer Ungleichung.)
Wenn ich in unserm Spiel 21 × 8 ausrechne, und wenn ich es tue, um damit eine praktische
Aufgabe zu lösen, so ist jedenfalls die Handlung der Rechnung in beiden Fällen die
Gleiche (und auch für Ungleichungen könnte in einem Spiele Platz geschaffen werden).
Dagegen ist mein übriges Verhalten zu der Rechnung jedenfalls in den zwei Fällen verschieden.
Die Frage ist nun: kann man von dem Menschen, der im Spiel die Stellung ”21 × 8 =
168“ erhalten hat, sagen, er habe herausgefunden, daß 21 × 8 168 sei? Und was fehlt ihm
dazu? Ich glaube, es fehlt nichts, es sei denn eine Anwendung der Rechnung.
Die Arithmetik ein Spiel zu nennen, ist ebenso falsch, wie das Schieben von
Schachfiguren (den Schachregeln gemäß) ein Spiel zu nennen; denn es kann auch eine
Rechnung sein.
Man müßte also sagen: Nein, das Wort ”Arithmetik“ ist nicht der Name eines Spiels.
(Das ist natürlich wieder eine Trivialität.) – Aber die Bedeutung des Wortes ”Arithmetik“
kann erklärt werden durch die Beziehung der Arithmetik zu einem arithmetischen Spiel,
oder auch durch die Beziehung der Schachaufgabe zum Schachspiel.
Dabei aber ist es wesentlich, zu erkennen, daß dieses Verhältnis nicht das ist, einer
Tennisaufgabe zum Tennisspiel.
Mit ”Tennisaufgabe“ meine ich etwa die Aufgabe, einen Ball unter gegebenen
Umständen in bestimmter Richtung zurückzuwerfen. (Klarer wäre der Fall, vielleicht, einer
Billardaufgabe.) Die Billardaufgabe ist keine mathematische Aufgabe (obwohl zu ihrer
Lösung Mathematik angewendet werden kann). Die Billardaufgabe ist eine physikalische
Aufgabe und daher ”Aufgabe“ im Sinne der Physik; die Schachaufgabe ist eine mathematische
Aufgabe und daher ”Aufgabe“ in einem andern (im mathematischen) Sinn.
In dem Kampf zwischen dem ”Formalismus“ und der ”inhaltlichen Mathematik“, – was
behauptet denn jeder Teil? Dieser Streit ist so ähnlich dem, zwischen Realismus und
Idealismus! Darin z.B.,13
daß er bald obsolet (geworden) sein wird und daß beide Parteien,
entgegen ihrer täglichen Praxis, Ungerechtigkeiten behaupten.
Die Arithmetik ist kein Spiel, niemandem wäre es eingefallen, unter den Spielen der
Menschen die Arithmetik zu nennen.
Worin besteht denn das Gewinnen und Verlieren in einem Spiel (oder das Ausgehen der
Patience)? Natürlich nicht in der Konfiguration,14
die das Gewinnen – z.B. – hervorbringt.
Wer gewinnt, muß durch eine besondere15
Regel festgestellt werden. (”Dame“ und
”Schlagdame“ sind nur durch diese Regel unterschieden.)
13 (V): Idealismus! Auch darin
14 (V): der Situation des Spiels,
15 (V): eigene
534
535
373 Die Mathematik mit einem Spiel verglichen.
WBTC108-115 30/05/2005 16:20 Page 746
But one could say that the real distinction lay in the fact that in the game there is no
room for affirmation and negation. For instance, there is multiplication in the game, and
21 × 8 = 148 would be a false move, but “~(21 × 8 = 148)”, which is a correct arithmetical
proposition, would have no place in our game.
(Here we may remind ourselves that in elementary schools they never work with inequations.
The children are only asked to carry out multiplications correctly and never – or hardly
ever – asked to set up an inequation.)
When I work out 21 × 8 in our game, and when I do it in order to solve a practical problem,
the mechanics of the calculation are the same in both cases (and we could make room
in a game for inequations as well). But in all other respects my attitude to the calculation –
in these two cases, at any rate – is different.
Now the question is: Can we say of someone who has reached the position “21 × 8 = 168”
in the game that he has found out that 21 × 8 is 168? What is he lacking? I think the only
thing missing is an application for the calculation.
Calling arithmetic a game is just as wrong as calling the movement of chess pieces (according
to chess-rules) a game; for that can be a calculation too.
So one ought to say: No, the word “arithmetic” is not the name of a game. (Of course
once again this is trivial.) – But the meaning of the word “arithmetic” can be explained by
the relationship of arithmetic to an arithmetical game, or also by the relationship of a chess
problem to the game of chess.
But in doing so it is essential to recognize that this relationship is not that of a tennis
problem to the game of tennis.
By “tennis problem” I mean something like the problem of returning a ball in a particular
direction in given circumstances. (Perhaps a billiards problem would be a clearer case.)
A billiards problem isn’t a mathematical problem (although mathematics can be applied to
solve it). A billiards problem is a physical problem, and thus a “problem” in the sense of
physics; a chess problem is a mathematical problem, and thus a “problem” in a different
(the mathematical) sense.
In the battle between “formalism” and “content mathematics” – what is it that each side
is claiming? This dispute is similar to the one between realism and idealism! For instance,
in so far10
as it will soon be (have become) obsolete, and both parties make unjust claims
that conflict with their everyday practices.
Arithmetic is no game; nobody would have thought of listing arithmetic among the games
human beings play.
What constitutes winning and losing in a game (or going out in patience)? Not the con-
figuration11
that produces, say, the win. The winner must be established by a particular12
rule. (“Draughts” and “Losing Draughts” differ only because of this rule.)
10 (V): Among other reasons, in so far
11 (V): situation
12 (V): special
Mathematics Compared to a Game. 373e
WBTC108-115 30/05/2005 16:20 Page 747
Konstatiert nun die Regel etwas, die sagt, ”wer zuerst seine Steine im Feld des Andern
hat, hat gewonnen“? Wie ließe sich das verifizieren? Wie weiß ich, ob Einer gewonnen hat?
Etwa daraus, daß er sich freut?
Diese Regel sagt doch wohl: Du mußt versuchen, Deine Steine so rasch als möglich etc.
Die Regel in dieser Form bringt das Spiel schon mit dem Leben in Zusammenhang.
Und man könnte sich denken, daß in einer Volksschule, in der das Schachspielen ein
Lehrgegenstand16
wäre, die Reaktion des Lehrers auf das schlechte Spiel eines Schülers genau
dieselbe17
wäre, wie die auf eine falsch gerechnete Rechenaufgabe.
Ich möchte beinahe sagen: Im Spiel gibt es (zwar) kein ”wahr“ und ”falsch“, dafür gibt
es aber in der Arithmetik kein ”Gewinnen“ und ”Verlieren“.
Ich sagte einmal, es wäre denkbar, daß Kriege auf einer Art großem Schachbrett nach
den Regeln des Schachspiels ausgefochten würden. Aber: Wenn es wirklich bloß nach den
Regeln des Schachspiels ginge, dann brauchte man eben kein Schlachtfeld für diesen Krieg,
sondern er könnte auf einem gewöhnlichen Brett gespielt werden. Und dann wäre es (eben)
im gewöhnlichen18
Sinne kein Krieg. Aber man könnte sich ja auch eine Schlacht von
den Regeln des Schachspiels geleitet denken. Etwa so, daß der ”Läufer“ mit der ”Dame“
nur kämpfen dürfte, wenn seine Stellung zu ihr es ihm im Schachspiel erlaubte, sie zu
”nehmen“.
Könnte man sich eine Schachpartie gespielt denken, d.h., sämtliche Spielhandlungen
ausgeführt denken, aber in einer andern Umgebung, so daß dieser Vorgang von uns19
nicht
die Partie eines Spiels genannt werden könnte?20
Gewiß, es könnte sich ja um eine Aufgabe handeln, die die Beiden miteinander lösen. (Und
einen Fall für die Nützlichkeit einer solchen Aufgabe kann man sich ja nach dem Oberen
leicht konstruieren.)
Die Regel über das Gewinnen und Verlieren unterscheidet eigentlich nur zwei Pole. Welche
Bewandtnis es (dann) mit dem hat, der gewinnt (oder verliert), geht sie eigentlich nichts an.
Ob z.B. der Verlierende dann etwas zu zahlen hat.
(Und ähnlich, kommt es uns ja vor, verhält es sich mit dem ”richtig“ und ”falsch“ im
Rechnen.)
In der Logik geschieht immer wieder, was in dem Streit über das Wesen der Definition
geschehen ist. Wenn man sagt, die Definition habe es nur mit Zeichen zu tun und ersetze
bloß ein Zeichen durch ein anderes,21
so wehren sich die Menschen dagegen und sagen, die
Definition leiste nicht nur das, oder es gebe eben verschiedene Arten der Definition22
und
die interessante und wichtige sei nicht die (reine) ”Verbaldefinition“.
Sie glauben nämlich, man nehme der Definition ihre Bedeutung, Wichtigkeit, wenn man
sie als bloße Ersetzungsregel, die von Zeichen handelt, hinstellt. Während die Bedeutung der
Definition in ihrer Anwendung liegt, quasi in ihrer Lebenswichtigkeit. Und eben das geht
(heute) in dem Streit zwischen Formalismus, Intuitionismus, etc. vor sich. Es ist den Leuten
unmöglich, die Wichtigkeit einer Tatsache,23
ihre Konsequenzen, ihre Anwendung, von
16 (V): ein obligater Gegenstand
17 (V): Schülers dieselbe
18 (V): normalen
19 (O): Vorgang uns
20 (V): Spiels genannt würde?
21 (V): bloß ein kompliziertes Zeichen durch ein
einfacheres,
22 (V): Arten von Definitionen
23 (V): Sache // Handlung,
536
537
374 Die Mathematik mit einem Spiel verglichen.
WBTC108-115 30/05/2005 16:20 Page 748
Now does the rule that says “Whoever first gets his pieces on his opponent’s side has won”
state anything? How could this be verified? How do I know whether someone has won? Perhaps
by the fact that he is happy?
This rule really says: You have to try to get your pieces as quickly as possible, etc.
In this form the rule connects the game with life. And one could imagine that in an
elementary school in which chess were a subject,13
the teacher’s reaction to a pupil’s poor
playing would be exactly the same14
as to a calculation carried out incorrectly.
I’m almost inclined to say: In a game there is (to be sure) no “true” and “false”, but then
again in arithmetic there is no “winning” and “losing”.
I once said that is was conceivable that wars might be fought on a kind of huge chess board
according to the rules of chess. But if everything really went only according to the rules of
chess, then you wouldn’t need a battlefield for this war; it could be played on an ordinary
board. And then it wouldn’t be a war in the ordinary15
sense. But of course you could also
imagine a battle governed by the rules of chess. Such that, say, the “bishop” would be allowed
to fight with the “queen” only when his position relative to her would allow him to “take”
her in chess.
Could we imagine a game of chess being played (i.e. a complete set of chess moves being
carried out), but in such different surroundings that what went on wasn’t something we could16
call a game in a match?
Certainly – it could be a case of the two participants collaborating to solve a problem. (And
indeed, along these lines we can easily come up with a case where such a problem might be
useful.)
All the rule about winning and losing really does is to distinguish between two poles. What
happens later to the winner (or loser) is really none of its business. – Whether, for instance,
the loser has to pay something afterwards.
(And, so it seems, things are similar with “right” and “wrong” in calculations.)
What keeps happening in logic is the same thing that happened in the dispute about the
nature of definition. If someone says that a definition is concerned only with signs and does
no more than substitute one sign for another,17
people resist and say that that isn’t all a definition
does, or that there simply are different kinds of definition,18
and that the interesting and
important one isn’t the (purely) “verbal definition”.
For they think that if you make definition out to be a mere substitution rule for signs you
take away its significance and importance. Whereas the significance of a definition lies in its
application, in its importance for life, as it were. And precisely the same thing is happening
(today) in the dispute between formalism and intuitionism, etc. People can’t distinguish the
13 (V): a required subject,
14 (V): be the same
15 (V): normal
16 (V): would
17 (V): substitute a simpler for a complicated sign,
18 (V): definitions,
Mathematics Compared to a Game. 374e
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ihr selbst zu unterscheiden; die Beschreibung einer Sache von der Beschreibung ihrer
Wichtigkeit.
Immer wieder hören wir (so), daß der Mathematiker mit dem Instinkt arbeitet (oder etwa,
daß er nicht mechanisch nach der Art eines Schachspielers vorgehe), aber wir erfahren
nicht, was das mit dem Wesen der Mathematik zu tun haben soll. Und wenn ein solches
psychisches Phänomen in der Mathematik eine Rolle spielt, wie weit wir überhaupt exakt
über die Mathematik reden können, und wie weit nur mit der Art der Unbestimmtheit, mit
der wir über Instinkte, etc. reden müssen.
Immer wieder möchte ich sagen: Ich kontrolliere die Geschäftsbücher der Mathematiker;
die seelischen Vorgänge, Freuden, Depressionen, Instinkte, der Geschäftsleute, so wichtig
sie in andrer Beziehung sind, kümmern mich nicht.24
24 (V): die seelischen Vorgänge in den Inhabern,
so wichtig sie sind, kümmern mich nicht.
538
375 Die Mathematik mit einem Spiel verglichen.
WBTC108-115 30/05/2005 16:20 Page 750
importance, the consequences, the application, of a fact from the fact itself;19
they can’t
distinguish the description of a thing from the description of its importance.
Again and again we hear that a mathematician uses his instincts in his work (or, say, that
he doesn’t proceed mechanically like a chess player), but we aren’t told what that’s supposed
to have to do with the nature of mathematics. And if such a psychological phenomenon has
a role in mathematics, we are also not told how far we can speak about mathematics with
any exactitude, and how far we can speak about it only with the kind of indeterminacy we
have to use in speaking about instincts, etc.
Time and again I would like to say: I check the ledgers of mathematicians; as important
as they are in other respects, the mental processes, joys, depressions and instincts of these
business people are no concern of mine.20
19 (V): of a thing from the thing itself; // of an action
from the action itself;
20 (V): respects, the mental processes that take place
within the owners, as important as they are, are
no concern of mine.
Mathematics Compared to a Game. 375e
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109
Es gibt keine Metamathematik.
Kein Kalkül kann ein philosophisches Problem entscheiden.
Der Kalkül kann uns nicht prinzipielle Aufschlüsse über die Mathematik geben.
Es kann darum1
auch keine ”führenden Probleme“ der mathematischen Logik geben, denn
das wären solche, deren Lösung uns endlich das Recht geben würde2
Arithmetik zu treiben,
wie wir es tun.
Und dazu können wir nicht auf den Glücksfall der Lösung eines mathematischen
Problems warten.
Ich sagte oben ”Kalkül ist kein mathematischer Begriff“;3
das heißt, das Wort ”Kalkül“
ist kein Schachstein der Mathematik.
Es brauchte in der Mathematik nicht vorzukommen. – Und wenn es doch in einem Kalkül
gebraucht wird, so ist dieser nun kein Metakalkül. Vielmehr ist dann dieses Wort wieder
nur ein Schachstein wie alle andern.
Auch die Logik ist keine Metamathematik, d.h. auch das Arbeiten mit dem logischen Kalkül
kann4
keine wesentlichen Wahrheiten über die Mathematik zu Tage fördern. Siehe hierzu
das ”Entscheidungsproblem“ und ähnliches in der modernen mathematischen Logik.
|Durch Russell, aber besonders durch Whitehead, ist in die Philosophie eine
Pseudoexaktheit gekommen, die die schlimmste Feindin wirklicher Exaktheit ist. Am
Grunde liegt hier der Irrtum, ein Kalkül könne die metamathematische Grundlage der
Mathematik sein.|
Die Zahl ist durchaus kein ”grundlegender mathematischer Begriff“. Es gibt so viele
Rechnungen,5
in denen von Zahlen nicht die Rede ist.
Und was die Arithmetik betrifft, so ist es mehr oder weniger willkürlich, was wir noch
Zahlen nennen wollen. Und im Übrigen ist der Kalkül – z.B. – der Kardinalzahlen zu
beschreiben, d.h. seine Regeln sind anzugeben, und damit ist der Arithmetik der Grund gelegt.6
Lehre sie uns, dann hast Du sie begründet.
|Hilbert stellt Regeln eines bestimmten Kalküls als Regeln der7
Metamathematik auf.|
Es ist ein Unterschied, ob ein System auf ersten Prinzipien ruht, oder ob es bloß von
ihnen ausgehend entwickelt wird. Es ist ein Unterschied, ob es, wie ein Haus, auf seinen
1 (V): daher
2 (V): endlich berechtigen würde
3 (E): Siehe S. 231.
4 (V): auch Operationen des logischen Kalküls
können
5 (V): Kalküle,
6 (V): und damit sind die Grundlagen der
Arithmetik gegeben. // und damit ist die
Arithmetik begründet.
7 (V): einer
539
540
376
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109
There is no Metamathematics.
No calculus can decide a philosophical problem.
A calculus cannot give us fundamental insights into mathematics.
So neither can there be any “leading problems” of mathematical logic; for those would
be problems whose solution would at long last give us the right to1
do arithmetic as we do.
And to do that we can’t wait for the windfall of the solution of a mathematical problem.
Earlier I said “calculus is not a mathematical concept”;2
in other words, the word
“calculus” is not a chess piece in mathematics.
It needn’t occur in mathematics. – And if it is used in a calculus nonetheless, that
doesn’t make the calculus into a metacalculus; in such a case the word “calculus” is itself
just a chess piece, like all the others.
Neither is logic metamathematics; that is, work with the3
logical calculus can’t bring to
light any essential truths about mathematics either. In this connection look at the “decision
problem” and similar topics in modern mathematical logic.
|Through Russell, but especially through Whitehead, there entered into philosophy a
false exactitude that is the worst enemy of real exactitude. At the bottom of this lies the
erroneous belief that a calculus could be the metamathematical foundation of mathematics.|
Number is in no way a “fundamental mathematical concept”. There are plenty of calcu-
lations4
in which numbers aren’t mentioned.
And as for arithmetic, what we are willing to call numbers is more or less arbitrary. And
as for the rest, what has to be done is to describe the calculus – say – of the cardinal numbers.
That is, its rules must be given, and thereby the foundation is laid for arithmetic.5
Teach them to us, and then you have laid its foundation.
|Hilbert sets up rules of a particular calculus as rules of metamathematics.6
|
It makes a difference whether a system rests on first principles, or whether it is merely
developed from them. It makes a difference whether, like a house, it rests on its foundation
1 (V): last entitle us to
2 (E): Cf. p. 231e.
3 (V): that is, operations of the
4 (V): calculi
5 (V): thereby the foundations of arithmetic are
given. // thereby arithmetic is grounded.
6 (V): of a metamathematics.
376e
WBTC108-115 30/05/2005 16:20 Page 753
untersten Mauern ruht oder ob es, wie etwa ein Himmelskörper, im Raum frei schwebt und
wir bloß unten zu bauen angefangen haben, obwohl wir es auch irgendwo anders hätten
tun können.
Die Logik und die Mathematik ruht nicht auf Axiomen; so wenig eine Gruppe auf
den sie definierenden Elementen und Operationen ruht. Hierin liegt der Fehler, das
Einleuchten, die Evidenz, der Grundgesetze als Kriterium der Richtigkeit in der Logik zu
betrachten.
Ein Fundament, das auf nichts steht, ist ein schlechtes Fundament.
(p & q) 1 (p & ~q) 1 (~p & q) 1 (~p & ~q): Das wird meine Tautologie, und ich würde
dann nur sagen, daß sich jedes ”Gesetz der Logik“8
nach bestimmten Regeln auf diese Form
bringen läßt. Das heißt aber dasselbe, wie:9
sich von ihr ableiten läßt; und hier wären wir
bei der Russell’schen Art der Demonstration angelangt und alles, was wir dazusetzen ist nur,
daß diese Ausgangsform selber kein selbständiger Satz ist und daß dieses und alle anderen
”Gesetze der Logik“ die Eigenschaft haben p & Log = p, p 1 Log = Log.
Das Wesen des ”logischen Gesetzes“ ist es ja, daß es im Produkt mit irgendeinem Satz
diesen Satz ergibt. Und man könnte den Kalkül Russells auch mit Erklärungen beginnen
von der Art:
p ⊃ p .&. q = q
p .&. p 1 q = p etc.
8 (V): sich jeder ”Satz der Logik“ 9 (V): als:
541
377 Es gibt keine Metamathematik.
WBTC108-115 30/05/2005 16:20 Page 754
or whether, like a celestial body, it floats freely in space and we have just begun to build at
the bottom, although we could have begun building anywhere else.
Logic and mathematics are not based on axioms, any more than a group is based on the
elements and operations that define it. Herein lies the error: regarding the intuitiveness, the
self-evidence, of the fundamental rules as a criterion for correctness in logic.
A foundation that rests on nothing is a bad foundation.
(p & q) 1 (p & ~q) 1 (~p & q) 1 (~p & ~q): That will be my tautology, and all I would
say is that every “law7
of logic” can be reduced to this form in accordance with specific
rules. But that means the same as: can be derived from it. And now we’ve arrived at the
Russellian method of demonstration, and all we add to it is that this initial form is not
itself an independent proposition, and that this and all other “laws of logic” have the
property p & Log = p, p 1 Log = Log.
It is indeed the essence of a “logical law” that taken together with any proposition it yields
that proposition. And one could begin Russell’s calculus with definitions such as:
p ⊃ p .&. q = q
p .&. p 1 q = p, etc.
7 (V): “proposition
There is no Metamathematics. 377e
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110
Beweis der Relevanz.
Wenn man die Lösbarkeit beweist, so muß in diesem Beweis irgendwie der Begriff
”Lösung“ vorhanden sein. (In dem Mechanismus des Beweises muß irgend etwas diesem
Begriff entsprechen.) Aber dieser Begriff ist nicht durch eine äußere Beschreibung zu
repräsentieren, sondern nun wirklich darzustellen.
Der Beweis der Beweisbarkeit eines Satzes wäre der Beweis des Satzes selbst. Dagegen
gibt es etwas, was wir den Beweis der Relevanz nennen könnten. Das wäre z.B. der Beweis,
der mich davon überzeugt, daß ich die Gleichung 17 × 38 = 456 nachprüfen kann, noch ehe
ich es getan habe. Woran erkenne ich nun, daß ich 17 × 38 = 456 überprüfen kann, während
ich das beim Anblick eines Integralausdrucks vielleicht nicht weiß? Ich erkenne offenbar,
daß er nach einer bestimmten Regel gebaut ist und auch, wie die Regel1
zur Lösung der
Aufgabe an dieser Bauart des Satzes haftet. Der Beweis der Relevanz ist dann etwa eine
Darstellung der allgemeinen Form der Lösungsmethode, etwa der Multiplikationsaufgaben,
die die allgemeine Form der Sätze erkennen läßt, deren Kontrolle sie möglich macht. Ich
kann dann sagen, ich erkenne, daß diese Methode auch diese Gleichung nachprüft, obwohl
ich die Nachprüfung noch nicht vollzogen habe.
Wenn von Beweisen der Relevanz (und ähnlichen Dingen der Mathematik) geredet wird,
so geschieht es immer, als hätten wir, abgesehen von den einzelnen Operationsreihen, die
wir Beweise der Relevanz nennen, noch einen ganz scharfen umfassenden Begriff so eines
Beweises oder überhaupt eines mathematischen Beweises. Während in Wirklichkeit dieses
Wort wieder in vielen, mehr oder weniger verwandten Bedeutungen angewandt wird. (Wie
etwa die Wörter ”Volk“, ”König“, ”Religion“, etc.; siehe Spengler.) Denken wir nur an die
Rolle, die bei2
der Erklärung so eines Wortes ein Beispiel spielt. Denn, wenn ich erklären
will, was ich unter ”Beweis“ verstehe, werde ich auf Beispiele von Beweisen zeigen müssen,
wie ich bei der Erklärung des Wortes ”Apfel“ auf Äpfel zeigen werde. Mit der Erklärung
des Wortes ”Beweis“ verhält es sich nun wie mit der des Wortes ”Zahl“: ich kann das Wort
”Kardinalzahl“ erklären, indem ich auf Beispiele von Kardinalzahlen weise, ja, ich kann
geradezu für dieses Wort das Zeichen ”1, 2, 3, u.s.w. ad inf.“ gebrauchen; ich kann anderseits
das Wort ”Zahl“ erklären, indem ich auf verschiedene Zahlenarten hinweise; aber dadurch
werde ich den Begriff ”Zahl“ nun nicht so scharf fassen, wie früher den der Kardinalzahl,
es sein denn, daß ich sagen will, daß nur diejenigen Gebilde, die wir heute als Zahlen
bezeichnen, den Begriff ”Zahl“ konstituieren. Dann aber kann man von keiner neuen
Konstruktion sagen, sie sei die Konstruktion einer Zahlenart. Das Wort ”Beweis“ aber wollen
wir ja so gebrauchen, daß es nicht einfach durch eine Disjunktion gerade heute üblicher Beweise
definiert wird, sondern wir wollen es in Fällen gebrauchen,3
von denen wir uns heute ”noch
1 (V): Vorschrift
2 (V): in
3 (V): sondern in Fällen gebrauchen,
542
543
378
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110
Proof of Relevance.
If we prove that a problem can be solved, the concept “solution” must somehow occur in
the proof. (There must be something in the mechanism of the proof that corresponds to this
concept.) But the concept mustn’t be represented by an external description; it must really
be demonstrated.
The proof of the provability of a proposition is the proof of the proposition itself. On the
other hand there is something we could call a proof of relevance: an example would be a
proof convincing me that I can verify the equation 17 × 38 = 456 even before I have actually
done so. Now what lets me know that I can check 17 × 38 = 456, whereas I might not
know that I can do that when I look at an expression in the integral calculus? Obviously,
because I recognize that the equation is constructed in accordance with a particular rule,
and I also recognize how the rule1
for the solution of the problem is connected to the way
the proposition is constructed. In this case a proof of relevance might be something like a
presentation of the general form of the method for solving problems – say multiplication
problems – which enables us to recognize the general form of the propositions it makes it
possible to check. Then I can say I recognize that this method will verify this equation,
even though I haven’t yet carried out the verification.
When we speak of proofs of relevance (and other like things in mathematics) it always
looks as if we had, in addition to the particular series of operations called proofs of relevance,
another quite definite inclusive concept of such proofs, or of mathematical proof in
general. But in fact the word “proof” is used with many more or less related meanings. (Like
words such as “people”, “king”, “religion”, etc.; cf. Spengler.) Let’s think about the role of
an example in the explanation of such a word. For if I want to explain what I understand
by “proof”, I will have to point to examples of proofs, just as when explaining the word
“apple” I will point to apples. Now it’s the same with the definition of the word “proof” as
it is with that of “number”: I can explain the expression “cardinal number” by pointing to
examples of cardinal numbers; indeed, for this expression I can simply substitute the sign
“1, 2, 3, and so on ad inf.”. And I can define the word “number” by pointing to various
kinds of numbers; but in so doing, I won’t circumscribe the concept “number” as precisely
as I previously circumscribed the concept cardinal number, unless I want to say that only
those entities that we at present call numbers constitute the concept “number”. But in that
case we can’t say of any new construction that it is the construction of a kind of number.
And we want to use the word “proof” so that it isn’t simply defined by a disjunction of
proofs that happen to be currently in use; rather, we want to use it2
in cases that at present
1 (V): prescript 2 (V): we use it
378e
WBTC108-115 30/05/2005 16:20 Page 757
gar keine Vorstellung machen können“. Soweit der Begriff des Beweises scharf 4
gefaßt ist,
ist er es durch einzelne Beweise, oder durch Reihen von Beweisen (den Zahlenreihen
analog) und das müssen wir bedenken, wenn wir mit voller Exaktheit über Beweise der
Relevanz, der Widerspruchsfreiheit, etc. etc. reden wollen.5
Man kann sagen: Ein Beweis der Relevanz wird den Kalkül des Satzes, auf den er
sich bezieht, ändern. Einen Kalkül mit diesem Satz rechtfertigen kann er nicht; in dem Sinn,
in welchem die Ausführung der Multiplikation 17 × 23 das Anschreiben der Gleichung
17 × 23 = 391 rechtfertigt. Wir müßten nur dem Wort ”rechtfertigen“ ausdrücklich jene
Bedeutung geben. Dann darf man aber nicht glauben, daß die Mathematik, ohne diese
Rechtfertigung, in irgend einem allgemeineren und allgemein feststehenden Sinne unerlaubt,
oder mit einem Dolus behaftet sei. (Das wäre ähnlich, als wollte Einer sagen: ”der Gebrauch
des Wortes ‘Steinhaufen’ ist im Grunde unerlaubt, ehe wir nicht offiziell festgelegt haben,
wieviel Steine einen Haufen machen“. Durch so eine Festlegung würde der Gebrauch
des Wortes ”Haufen“ modifiziert, aber nicht in irgend einem allgemein anerkannten
Sinne ”gerechtfertigt“. Und wenn eine solche offizielle Definition gegeben würde,6
so
wäre dadurch nicht der Gebrauch, den man früher von dem Wort gemacht hat, als etwas
Unrichtiges7
gekennzeichnet.)
Der Beweis der Kontrollierbarkeit von 17 × 23 = 391 ist ”Beweis“ in einem andern Sinne
dieses Worts, als der, der Gleichung selbst. (Der Müller mahlt, der Maler malt: beide . . . )
Die Kontrollierbarkeit der Gleichung entnehmen8
wir aus ihrem Beweis in
analoger Weise, wie die Kontrollierbarkeit des Satzes ”die Punkte A und B sind
nicht durch eine Windung der Spirale getrennt“ aus der Figur.9
Und man sieht
auch schon, daß der Satz, der die Kontrollierbarkeit aussagt, ”Satz“ in einem andern
Sinne ist, als der, dessen Kontrollierbarkeit behauptet wird. Und hier kann man
wieder nur sagen: Sieh Dir den Beweis an, dann wirst Du sehen, was hier bewiesen
wird, was ”der bewiesene Satz“ genannt wird.
Kann man sagen, daß wir zu jedem Schritt eines Beweises eine frische Intuition brauchen?
(Individualität der Zahlen.) Es wäre etwa so: Ist mir eine allgemeine (variable) Regel
gegeben, so muß ich immer von neuem erkennen, daß diese Regel auch hier angewendet
werden kann (daß sie auch für diesen Fall gilt). Kein Akt der Voraussicht kann mir diesen
Akt der Einsicht ersparen. Denn tatsächlich ist die Form, auf die die Regel angewandt wird,
bei jedem neuen Schritte eine neue. – Es handelt sich aber hier nicht um einen Akt der Einsicht,
sondern um einen Akt der Entscheidung.
Der sogenannte Beweis der Relevanz steigt die Leiter zu seinem Satz nicht hinaus, denn
dazu muß man jede Stufe nehmen, sondern zeigt nur, daß die Leiter in der Richtung zu
jenem Satze führt. (In der Logik gibt es kein Surrogat.) Es ist auch der Pfeil, der die Richtung
weist, kein Surrogat für das Durchschreiten aller Stufen bis zum bestimmten Ziel.
4 (V): @scharf
5 (V): wenn wir uns anschicken, mit voller
Exaktheit über Beweise der Relevanz, der
Widerspruchsfreiheit, etc. etc. zu reden.
6 (V): wäre,
7 (V): als unrichtig
8 (V): ersehen
9 (F): MS 113, S. 103v.
544
545
379 Beweis der Relevanz.
A
B
WBTC108-115 30/05/2005 16:20 Page 758
we “can’t even conceive of”. To the extent that the concept of proof is sharply circumscribed,
it is so only by individual proofs, or by a series of proofs (analogous to the number series),
and we must keep that in mind if we want to3
speak absolutely precisely about proofs of
relevance, consistency proofs, etc., etc.
We can say: A proof of relevance will change the calculus of the proposition to which it
refers. It cannot justify a calculus containing this proposition, in the sense in which carrying
out the multiplication 17 × 23 justifies writing down the equation 17 × 23 = 391. Not,
that is, unless we expressly give the word “justify” that meaning. But in that case we
mustn’t believe that without this justification, mathematics is in some more general and widely
established sense illegitimate, or intentionally deceptive. (That’s like someone wanting to
say: “The use of the expression ‘pile of stones’ is fundamentally illegitimate until we have
laid down officially how many stones make a pile.” Such a stipulation would modify the
use of the word “pile”, but it wouldn’t “justify” it in any generally recognized sense; and if
such an official definition were given4
, it wouldn’t mean that the previous use of the word
would be labelled as something incorrect.) 5
The proof of the verifiability of 17 × 23 = 391 is a “proof” in a different sense of
this word than the proof of the equation itself. (A cobbler heels, a doctor heals: both . . . )
We infer6
the verifiability of the equation from its proof, analogously to our
inferring the verifiability of the proposition “The points A and B are not separated
by a turn of the spiral” from the figure.7
And we also see that the proposition that
expresses verifiability is a “proposition” in a different sense from the one whose
verifiability is asserted. And here again, one can only say: Take a look at the proof,
and you will see what is being proved here, what gets called “the proposition that
is proved”.
Can one say that for each step of a proof we need a fresh intuition? (The individuality
of numbers.) That might be argued something like this: If I have been given a general
(variable) rule, I must recognize each time anew that this rule can be applied here too (that
it holds for this case as well). No act of foresight can spare me this act of insight. For the
form to which the rule is applied is in fact a new one at each new step. – But here it’s not
a matter of an act of insight, but of an act of decision.
The so-called proof of relevance does not climb the ladder to its proposition – for that
requires you to climb rung by rung – but only shows that the ladder leads in the direction
of that proposition. (There is no surrogate in logic.) Neither is an arrow that indicates the
direction a surrogate for going through all the stages to a particular goal.
3 (V): we set out to
4 (V): definition had been given
5 (V): as incorrect.)
6 (V): see
7 (F): MS 113, p. 103v.
Proof of Relevance. 379e
A
B
WBTC108-115 30/05/2005 16:20 Page 759
111
Beweis der Widerspruchsfreiheit.
Irgendetwas sagt mir: eigentlich dürfte ein Widerspruch in den Axiomen eines Systems
nicht schaden, als bis er offenbar wird. Man denkt sich einen versteckten Widerspruch wie
eine versteckte Krankheit, die schadet, obwohl (und vielleicht gerade deshalb weil) sie sich
uns nicht deutlich zeigt. Zwei Spielregeln aber, die einander für einen bestimmten Fall widersprechen,
sind vollkommen in Ordnung, bis dieser Fall eintritt und dann erst wird1
es nötig,
durch eine weitere Regel zwischen ihnen zu entscheiden.
Der Beweis der Widerspruchsfreiheit der Axiome, von dem die Mathematiker heute
soviel Aufhebens machen. Ich habe das Gefühl: wenn in den Axiomen eines Systems
ein Widerspruch wäre, so wäre das gar nicht so ein großes Unglück. Nichts leichter, als ihn
zu beseitigen.
”Man darf ein System von Axiomen nicht benützen, ehe seine Widerspruchsfreiheit
nachgewiesen ist.“
”In den Spielregeln dürfen keine Widersprüche vorkommen.“
Warum nicht? ”Weil man dann nicht wüßte, wie man zu spielen hat“?
Aber wie kommt es, daß man auf den Widerspruch mit dem Zweifel reagiert?
Auf den Widerspruch reagiert man überhaupt nicht. Man könnte nur sagen: Wenn das
wirklich so gemeint ist (wenn der Widerspruch hier stehen soll), so versteh’ ich es nicht.
Oder: ich hab’ es nicht gelernt. Ich verstehe die Zeichen nicht. Ich habe nicht gelernt, was
ich daraufhin tun soll, ob es überhaupt ein Befehl ist; etc.
Wie wäre es etwa, wenn man in der Arithmetik zu den üblichen Axiomen die Gleichung
2 × 2 = 5 hinzunehmen wollte? Das hieße natürlich, daß das Gleichheitszeichen nun seine
Bedeutung geändert2
hätte, d.h., daß nun andere Regeln für das Gleichheitszeichen gälten.3
Wenn ich nun sagte: ”also kann ich es nicht als Ersetzungszeichen gebrauchen; so hieße
das, daß seine Grammatik nun nicht mehr mit der des Wortes ”ersetzen“ (”Ersetzungszeichen“,
etc.) übereinstimmt. Denn das Wort ”kann“ in diesem Satz deutet nicht auf eine
physische (physiologische, psychologische) Möglichkeit.4
”Die Regeln dürfen einander nicht widersprechen“, das ist wie: ”die Negation darf nicht
verdoppelt eine Negation ergeben“. Es liegt nämlich in der Grammatik des Wortes ”Regel“,
daß ”p & ~p“ keine Regel ist (wenn ”p“ eine Regel ist).5
1 (V): dann wird
2 (V): gewechselt
3 (O): gälte.
4 (O): (physiologische, psychologische Möglichkeit.
5 (V): daß ”p 1 ~p“ (wenn ”p“ eine Regel ist)
keine Regel ist.
546
547
380
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111
Consistency Proof.
Something tells me that a contradiction in the axioms of a system really can’t do any harm
until it is revealed. We think of a hidden contradiction as being like a hidden illness, which
does harm even though (and perhaps precisely because) it doesn’t clearly reveal itself to us.
But two rules of a game that contradict each other in a particular case are perfectly in order
until that case crops up, and it’s only then1
that it becomes necessary to decide between them
with a further rule.
The proof of the consistency of axioms that mathematicians make so much fuss about
nowadays. I have the feeling that if there were a contradiction in the axioms of a system it
wouldn’t be such a great misfortune. Nothing’s easier than removing it.
“We mustn’t use a system of axioms before its consistency has been proved.”
“There mustn’t be any contradictions in the rules of a game.”
Why not? “Because then one wouldn’t know how to play”?
But how does it happen that we react to a contradiction with doubt?
We don’t have any reaction to a contradiction. We can only say: If it’s really meant like
that (if the contradiction is supposed to be there), then I don’t understand it. Or: it isn’t
something I’ve learned. I don’t understand the signs. I haven’t learned what I’m supposed
to do with it, whether it’s a command; etc.
What would it be like, for instance, if one wanted to add the equation 2 × 2 = 5 to the
usual axioms of arithmetic? Of course that would mean that the equals-sign had now changed2
its meaning, i.e. that different rules would now apply to it.
Now if I said: “So I can’t use it as a substitution sign”, that would mean that its grammar
no longer matched the grammar of the word “substitute” (“substitution sign”, etc.). For
the word “can” in that proposition doesn’t indicate a physical (physiological, psychological)
possibility.
“Rules mustn’t contradict each other” is like: “Negation, when doubled, mustn’t yield
a negation”. For it is part of the grammar of the word “rule” that “p & ~p”3
is not a rule
(if “p” is a rule).
1 (V): it’s then
2 (V): switched
3 (V): “p 1 ~p”
380e
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Das heißt, man könnte also auch sagen: die Regeln dürfen6
einander widersprechen, wenn
andre Regeln für den Gebrauch des Wortes7
”Regel“ gelten – wenn das Wort ”Regel“ eine
andere Bedeutung hat.
Wir können eben auch hier nicht begründen (außer (etwa) biologisch oder historisch)
sondern nur die Übereinstimmung oder den Gegensatz der Regeln für gewisse Wörter
konstatieren, also sagen, daß diese Worte mit diesen Regeln gebraucht werden.8
Es läßt sich nicht zeigen, beweisen, daß man diese9
Regeln als Regeln dieser Handlung
gebrauchen kann.
Außer, indem man zeigt, daß die Grammatik der Beschreibung10
der Handlung mit der
jener Regeln übereinstimmt.
”In den Regeln darf kein Widerspruch sein“, das klingt so, wie eine Vorschrift: ”in einer
Uhr darf der Zeiger nicht locker auf seiner Welle sitzen“. Man erwartet sich dann eine
Begründung: weil sonst . . . Im ersten Falle könnte diese Begründung aber nur lauten: weil
es sonst kein Regelverzeichnis ist. Es ist eben wieder ein Fall der grammatischen Struktur,
die sich logisch nicht begründen läßt.
|Zum indirekten Beweis, daß eine Gerade über einen Punkt hinaus nur eine Fortsetzung
hat: Wir nahmen an, es könnte eine Gerade zwei Fortsetzungen haben. – Wenn wir das
annehmen, so muß diese Annahme einen Sinn haben –. Was heißt es aber: das annehmen?
Es heißt nicht, eine naturgeschichtlich falsche Annahme machen, wie etwa die, daß ein Löwe
zwei Schwänze hätte. – Es heißt nicht, etwas annehmen, was gegen die Konstatierung einer
Tatsache verstößt.11
Es heißt vielmehr, eine Regel annehmen; und gegen die ist weiter nichts
zu sagen, außer daß sie etwa einer anderen widerspricht und ich sie darum fallen lasse.
Wenn im Beweis nun gezeichnet wird , 12
und das eine Gerade darstellen soll,
die sich gabelt, so ist darin nichts Absurdes (Widersprechendes), es sei denn, daß wir eine
Festsetzung getroffen haben, der es widerspricht.13
Wenn nachträglich ein Widerspruch gefunden wird, so waren vorher die Regeln noch
nicht klar und eindeutig. Der Widerspruch macht also nichts, denn er ist dann durch das
Aussprechen einer Regel zu entfernen.
In einem grammatisch geklärten System14
gibt es keinen versteckten Widerspruch, denn
da muß die Regel gegeben sein, nach welcher ein Widerspruch zu finden ist. Versteckt kann
der Widerspruch nur in dem Sinn sein, daß er gleichsam in der Unordnung der Regeln,15
in dem16
ungeordneten Teil der Grammatik versteckt ist; dort aber macht er nichts,17
da er
durch ein Ordnen der Grammatik zu entfernen ist.
Warum dürfen sich Regeln nicht widersprechen? Weil es sonst keine Regeln wären.|
6 (V): können
7 (V): für das Wort
8 (V): historisch) und // sondern // (können) nur
beschreiben, wie das Wort ”Regel“ gebraucht
wird.
9 (V): gewisse
10 (V): Bezeichnung
11 (V): spricht.
12 (F): MS 112, S. 21r.
13 (V): Wenn im Beweis nun eine Gerade gezeichnet
wird, die sich gabelt, so darf das an und
für sich nicht absurd sein, und ich kann nur
sagen: so etwas // das // nenne ich keine
Gerade.
14 (V): In einem völlig geklärten System // mit
klarer Grammatik
15 (V): gleichsam im ”Kraut-und-Rüben“ der
Regeln,
16 (V): in dem g
17 (V): versteckt ist; das aber macht nichts,
548
549
381 Beweis der Widerspruchsfreiheit.
WBTC108-115 30/05/2005 16:20 Page 762
That means we could also say: Rules are allowed to4
contradict each other, if different
rules are valid for the use of the word5
“rule” – if the word “rule” has a different meaning.
Here too we simply can’t give any reasons (except, say, biological or historical ones); all
we can do is to note the agreement or disagreement between the rules for certain words,
and say that these words are used with these rules.6
It cannot be shown, proved, that one can use these7
rules as the rules for this activity.
Except by showing that the grammar of the description8
of the activity matches the grammar
of those rules.
“In the rules there must not be any contradiction.” This sounds like a directive: “In a clock
the hand mustn’t be loose on its shaft”. We then expect a reason: because otherwise. . . . But
in the first case this reason could only be: Because otherwise it won’t be a set of rules. Once
again it’s simply a case of a grammatical structure that can’t be given a logical foundation.
|In the indirect proof that a straight line has only one continuation beyond a point we
were assuming that a straight line could have two continuations. – If we assume that, then
that supposition must make sense. – But what does it mean to make that assumption? It
doesn’t mean making a false assumption of natural history, such as the assumption that a
lion has two tails. – It doesn’t mean making an assumption that contravenes9
a statement of
fact. Rather it means assuming a rule; and there’s nothing further to be said against that rule
except that perhaps it contradicts another rule, and that for that reason I drop it.
Now if in the proof there’s the following drawing, 10
and this is supposed to
represent a straight line that forks off, then there is nothing absurd (contradictory) in that
unless it contradicts some stipulation we have made.11
If a contradiction is found later on, that means that earlier the rules weren’t clear and
unambiguous. So the contradiction doesn’t matter, because we can get rid of it later by
enunciating a rule.
In a grammatically clear system12
there is no hidden contradiction, because such a system
must include the rule according to which a contradiction can be found. A contradiction can
only be hidden in the sense that it is hidden in a disarray13
of rules, in an unorganized part
of grammar, as it were; but there it doesn’t14
do any harm, since it can be removed by putting
the grammar in order.
Why are rules not allowed to contradict one another? Because otherwise they wouldn’t be
rules.|
4 (V): Rules can
5 (V): for the word
6 (V): ones); and // rather // we (can) only
describe how the word “rule” is used.
7 (V): certain
8 (V): designation
9 (V): contradicts
10 (F): MS 112, p. 21r.
11 (V): Now if in the proof a straight line that forks
off is drawn, then in and of itself that needn’t
be absurd, and all I can say is: I don’t call something
like that // call that // a straight line.
12 (V): In a completely clarified system // with a
clear grammar
13 (V): in the mish-mash
14 (V): but that doesn’t
Consistency Proof. 381e
WBTC108-115 30/05/2005 16:20 Page 763
112
Die Begründung der Arithmetik, in
der diese auf ihre Anwendungen
vorbereitet wird. (Russell, Ramsey.)
Man empfindet immer eine Scheu, die Arithmetik zu begründen, indem man etwas über
ihre Anwendung ausspricht. Sie scheint fest genug in sich selbst begründet zu sein. Und
das kommt natürlich daher, daß die Arithmetik ihre eigene Anwendung ist.
Man könnte sagen: Wozu die Anwendung der Arithmetik einschränken, sie sorgt für sich
selbst. (Ich kann ein Messer herstellen ohne Rücksicht darauf, welche Klasse von Stoffen
ich damit werde schneiden lassen; das wird sich dann schon zeigen.)
Gegen die Abgrenzung des Anwendungsgebiets spricht nämlich das Gefühl, daß wir die
Arithmetik verstehen können, ohne ein solches Gebiet im Auge zu haben. Oder sagen wir
so: Der Instinkt sträubt sich gegen alles, was nicht bloß eine Analyse der schon vorhandenen
Gedanken ist.
Man könnte sagen: Die Arithmetik ist eine Art Geometrie; d.h., was in der Geometrie
die Konstruktionen auf dem Papier sind, sind in der Arithmetik die Rechnungen (auf dem
Papier). – Man könnte sagen, sie ist eine allgemeinere Geometrie.
Es handelt sich immer darum, ob und wie es möglich ist, die allgemeinste Form der
Anwendung der Arithmetik darzustellen. Und hier ist eben das Seltsame, daß das in
gewissem Sinne nicht nötig zu sein scheint. Und wenn es wirklich nicht nötig ist, dann ist
es auch unmöglich.
Es scheint nämlich die allgemeine Form ihrer Anwendung dadurch dargestellt zu sein,
daß nichts über sie ausgesagt wird. (Und ist das eine mögliche Darstellung, so ist es auch
die einzig richtige.)
Der Sinn der Bemerkung, daß die Arithmetik eine Art Geometrie sei, ist eben, daß die
arithmetischen Konstruktionen autonom sind, wie die geometrischen, und daher sozusagen
ihre Anwendbarkeit selbst garantieren.
Denn auch von der Geometrie muß man sagen können, sie sei ihre eigene Anwendung.
(In dem Sinne von möglichen und wirklich gezogenen Geraden können1
wir auch von
möglichen und wirklich dargestellten Zahlen reden.)
2
Das ist eine arithmetische Konstruktion und in etwas erweitertem Sinn auch
eine geometrische.
1 (V): könnten 2 (F): TS 209, S. 47.
550
551
382
WBTC108-115 30/05/2005 16:20 Page 764
112
Laying the Foundations for
Arithmetic as Preparation for its
Applications. (Russell, Ramsey.)
One always shies away from giving arithmetic a foundation by uttering something about
its application. It appears firmly enough grounded in itself. And of course the reason for
that is that arithmetic is its own application.
You could say: Why limit the application of arithmetic? Its application takes care of itself.
(I can make a knife without bothering about what kinds of materials I will cut with it; that
will become apparent soon enough.)
What speaks against our demarcating an area of application is the feeling that we can understand
arithmetic without having such an area in mind. Or let’s put it like this: Our instinct
resists anything that isn’t merely an analysis of the thoughts already before us.
You could say arithmetic is a kind of geometry; that is to say, what are constructions on
paper in geometry are calculations (on paper) in arithmetic. – You could say it is a more
general kind of geometry.
It is always a question of whether and how it’s possible to represent the most general form
of the application of arithmetic. And here the strange thing is that in a certain sense this
doesn’t seem to be necessary. And if it really isn’t necessary, then it’s also impossible.
The general form of its application seems to be represented by the fact that nothing is said
about it. (And if that’s a possible representation, then it is also the only right one.)
The point of the remark that arithmetic is a kind of geometry is simply that arithmetical
constructions are autonomous, like geometrical ones, and that therefore they guarantee their
applicability themselves, so to speak.
For it must be possible to say of geometry too that it is its own application.
(In the sense in which we speak of possible versus actually drawn lines we can1
also speak
of possible versus actually represented numbers.)
2
This is an arithmetical construction, and in a somewhat extended sense also
a geometrical one.
1 (V): could 2 (F): TS 209, p. 47.
382e
WBTC108-115 30/05/2005 16:20 Page 765
Angenommen, mit dieser Rechnung wollte ich folgende Aufgabe lösen: Wenn ich 11 Äpfel
habe und Leute mit je 3 Äpfeln beteilen will, wieviele Leute kann ich beteilen? Die
Rechnung liefert mir die Lösung 3. Angenommen nun, ich vollzöge alle Handlungen des
Beteilens und am Ende hätten 4 Personen je 3 Äpfel in der Hand. Würde ich nun sagen,
die Ausrechnung hat ein falsches Resultat ergeben? Natürlich nicht. Und das heißt ja nur,
daß die Ausrechnung kein Experiment war.
Es könnte scheinen, als berechtigte uns die mathematische Ausrechnung zu einer
Vorhersagung, etwa, daß ich 3 Personen werde beteilen können und 2 Äpfel übrigbleiben
werden. So ist es aber nicht. Zu dieser Vorhersagung berechtigt uns eine physikalische
Hypothese, die außerhalb der Rechnung steht. Die Rechnung ist nur eine Betrachtung der
logischen Formen, der Strukturen, und kann an sich nichts Neues liefern.
Wenn 3 Striche auf dem Papier das Zeichen für die 3 sind, dann kann man sagen, die 3
ist in unserer Sprache so anzuwenden, wie sich 3 Striche anwenden lassen.
Ich sagte: ”Eine Schwierigkeit der Frege’schen Theorie ist die Allgemeinheit der Worte
’Begriff‘ und ’Gegenstand‘. Denn, da man Tische, Töne, Schwingungen und Gedanken zählen
kann, so ist es schwer, sie alle unter einen Hut zu bringen“. – Aber was heißt es: ”man kann
sie zählen“? Doch, daß es Sinn hat, auf sie die Kardinalzahlen anzuwenden.3
Wenn wir aber
das wissen, diese grammatische Regel wissen, was brauchen wir uns da den Kopf über die
andern grammatischen Regeln zu zerbrechen, wenn es sich uns nur um eine Rechtfertigung
der Anwendung der Kardinalarithmetik handelt? Es ist nicht schwer ”sie alle unter einen
Hut zu bringen“, sondern sie sind, soweit das für diesen Fall4
nötig ist, unter einen Hut
gebracht.
Die Arithmetik aber kümmert sich (wie wir alle sehr wohl wissen) überhaupt nicht um
diese Anwendung. Ihre Anwendbarkeit sorgt für sich selbst.
Daher ist alles ängstliche Suchen nach den Unterschieden zwischen Subjekt-PrädikatFormen,
aber auch die Konstruktion von Funktionen ”in extension“ (Ramsey), zur
Begründung der Arithmetik Zeitverschwendung.
Die Gleichung 4 Äpfel + 4 Äpfel = 8 Äpfel ist eine Ersetzungsregel, die ich verwende,
wenn ich nicht das Zeichen ”4 + 4“ durch ”8“, sondern das Zeichen ”4 Äpfel + 4 Äpfel“
durch ”8 Äpfel“ ersetze.
Man muß sich aber davor hüten zu glauben ”4 Äpfel + 4 Äpfel = 8 Äpfel“ ist die konkrete
Gleichung, dagegen 4 + 4 = 8 der abstrakte Satz, wovon die erste Gleichung nur eine spezielle
Anwendung sei.5
So daß zwar die Arithmetik der Äpfel viel weniger allgemein wäre,6
als die
eigentliche allgemeine, aber eben in ihrem beschränkten Bereich (für Äpfel) gälte. – Es gibt
aber keine ”Arithmetik der Äpfel“, denn die Gleichung 4 Äpfel + 4 Äpfel = 8 Äpfel7
ist nicht
ein Satz, der von Äpfeln handelt. Man kann sagen, daß in dieser Gleichung das Wort ”Äpfel“
keine Bedeutung hat. (Wie man es überhaupt von dem Zeichen in einer Zeichenregel sagen
kann, die seine Bedeutung bestimmen hilft.)
Wie kann man Vorbereitungen zum Empfang von etwas eventuell Existierendem treffen,
– in dem Sinn, in welchem Russell und Ramsey das (immer) tun wollten? Man bereitet etwa
die Logik für die Existenz von vielstelligen Relationen vor, oder für die Existenz einer
unendlichen Zahl von Gegenständen. –
3 (V): Sinn hat, sie zu zählen.
4 (V): Zweck
5 (V): ist.
6 (V): ist,
7 (V): Gleichung mit den benannten Zahlen
552
553
383 Die Begründung der Arithmetik . . .
WBTC108-115 30/05/2005 16:20 Page 766
Suppose I wished to use this calculation to solve the following problem: if I have
11 apples and want to share them among some people in such a way that each is given
3 apples, how many people can I share with? The calculation supplies me with the answer 3.
Now suppose I were to go through the whole process of sharing and at the end 4 people
each had 3 apples in their hands. Would I then say that the computation had come out
wrong? Of course not. And of course all that means is that the computation wasn’t an
experiment.
It might look as though the mathematical computation entitled us to make a prediction,
say, that I will be able to give three people their share and that two apples will be left over.
But that isn’t so. What justifies us in making this prediction is a hypothesis of physics,
which lies outside the calculation. The calculation is only an examination of logical forms,
of structures, and of itself can’t yield anything new.
If 3 lines on paper are the sign for the number 3, then you can say the number 3 is to be
applied in our language in the way in which the 3 lines can be applied.
I said: “One difficulty in the Fregean theory is the generality of the words ‘Concept’ and
‘Object’. For even though you can count tables, tones, vibrations and thoughts, it is difficult
to bring them all under one heading.” – But what does “you can count them” mean? What
it means is that it makes sense to apply the cardinal numbers to them.3
But if we know
that, if we know this grammatical rule, why do we need to rack our brains about the other
grammatical rules, when all we are concerned with is justifying the application of cardinal
arithmetic? It isn’t difficult “to bring them all under one heading”; rather, so far as is
necessary for the present case,4
they are already brought together.
But (as we all know perfectly well) arithmetic isn’t at all concerned with this application.
Its applicability takes care of itself.
Therefore all the anxious searching for distinctions between subject-predicate forms,
as well as the construction of functions “in extension” (Ramsey) in order to establish a
foundation of arithmetic, is a waste of time.
The equation 4 apples + 4 apples = 8 apples is a substitution rule that I use if I substitute
not “8” for the sign “4 + 4”, but “8 apples” for the sign “4 apples + 4 apples”.
But we must beware of thinking that “4 apples + 4 apples = 8 apples” is the concrete
equation whereas 4 + 4 = 8 is the abstract proposition, of which the former is only a special
application, so that despite the arithmetic of apples being much less general than the truly
general arithmetic, it is nevertheless valid in its restricted domain (for apples). – But there
isn’t any “arithmetic of apples”, because the equation 4 apples + 4 apples = 8 apples is5
not
a proposition about apples. We can say that in this equation the word “apples” has no meaning.
(As we can say in general about a sign in a rule that helps determine its meaning.)
How can we make preparations to receive something that possibly exists – in the sense
in which Russell and Ramsey (always) wanted to do this? We get logic ready for the
existence of many-placed relations, for instance, or for the existence of an infinite number
of objects. –
3 (V): sense to count them.
4 (V): purpose,
5 (V): equation with the given numbers is
Laying the Foundations for Arithmetic . . . 383e
WBTC108-115 30/05/2005 16:20 Page 767
Nun kann man doch für die Existenz eines Dinges vorsorgen: Ich mache z.B. ein
Kästchen, um den Schmuck hineinzulegen, der vielleicht einmal gemacht werden wird. –
Aber hier kann ich doch sagen, was der Fall sein muß, – welcher Fall es ist, für den ich
vorsorge. Dieser Fall läßt sich jetzt so gut beschreiben,8
wie, nachdem er schon eingetreten
ist; und auch dann, wenn er nie eintritt. (Lösung mathematischer Probleme.) Dagegen
sorgen Russell und Ramsey für eine eventuelle Grammatik vor.
Man denkt einerseits, daß es die Mathematik mit der Art der Funktionen zu tun hat und
ihren Argumenten,9
von deren Anzahlen sie handelt. Aber man will sich nicht durch die uns
jetzt bekannten Funktionen binden lassen und man weiß nicht, ob jemals eine gefunden
werden wird, die 100 Argumentstellen hat; also muß man vorsorgen und eine Funktion
konstruieren, die alles für die 100-stellige Relation vorbereitet, wenn sich eine finden sollte.
– Was heißt es aber überhaupt: ”es findet sich (oder: es gibt) eine 100-stellige Relation“?
Welchen Begriff haben wir von ihr? oder auch von einer 2-stelligen? – Als Beispiel einer
2-stelligen Relation gibt man etwa die zwischen Vater und Sohn. Aber welche Bedeutung
hat dieses Beispiel für die weitere logische Behandlung der 2-stelligen Relationen? Sollen
wir uns jetzt statt jedes ”aRb“ vorstellen ”a ist der Vater des b“? – Wenn aber nicht, ist
dann das Beispiel, oder irgend eins überhaupt, essentiell? Spielt dieses Beispiel nicht die
gleiche Rolle, wie eines in der Arithmetik, wenn ich jemandem 3 × 6 = 18 an 3 Reihen zu
je 6 Äpfeln erkläre?
Hier handelt es sich um unsern Begriff der Anwendung. – Man hat etwa die Vorstellung
von einem Motor, der erst leer geht, und dann eine Arbeitsmaschine treibt.
Aber was gibt die Anwendung der Rechnung?10
Fügt sie ihr einen neuen Kalkül
bei?11
Dann ist sie ja jetzt eine andere Rechnung. Oder gibt sie ihr in irgend einem, der
Mathematik (Logik) wesentlichen,12
Sinne Substanz? Wie kann man dann überhaupt, auch
nur zeitweise, von der Anwendung absehen?
Nein, die Rechnung mit Äpfeln ist wesentlich dieselbe, wie die mit Strichen oder Ziffern.
Die Arbeitsmaschine setzt den Motor fort, aber die Anwendung (in diesem Sinne) nicht die
Rechnung.
Wenn ich nun, um ein Beispiel zu geben, sage: ”die Liebe ist eine 2-stellige Relation“, –
13
sage ich hier etwas über die Liebe aus? Natürlich nicht. Ich gebe eine Regel für den Gebrauch
des Wortes ”Liebe“ und will etwa sagen, daß wir dieses Wort z.B. so gebrauchen.
Nun hat man aber doch das Gefühl, daß, mit dem Hinweis auf die 2-stellige Relation
”Liebe“, in die Hülse des Relationskalküls Sinn gesteckt wurde. – Denken wir uns eine
geometrische Demonstration statt an einer Zeichnung oder an analytischen Symbolen an einem
Lampenzylinder durchgeführt.14
In wiefern ist hier von der Geometrie eine Anwendung
gemacht? Tritt denn der Gebrauch des Glaszylinders als Lampenglas in die geometrische
Überlegung ein? Und tritt der Gebrauch des Wortes ”Liebe“ in einer Liebeserklärung in
meine Überlegungen15
über die 2-stelligen Relationen ein?
8 (V): vorsorge. Ich kann diesen Fall jetzt so gut
beschreiben,
9 (V): Gegenständen,
10 (V): Aber was erhält die Rechnung von ihrer
Anwendung?
11 (V): zu?
12 (O): wesentlichem,
13 (O): Relation, – (V): Wenn ich nun sage:
”die Liebe ist ein Beispiel einer 2-stelligen
Relation“, –
14 (V): vorgenommen.
15 (V): Überlegung
554
555
556
384 Die Begründung der Arithmetik . . .
WBTC108-115 30/05/2005 16:20 Page 768
We can make preparations for the existence of a thing: for example, I make a box for
jewellery that possibly will be crafted some day. – But here I can say what must be the
case – which case it is for which I am preparing. This case can be described6
just as well
now as after it has occurred; and even if it never occurs at all. (Solution of mathematical
problems.) On the other hand, what Russell and Ramsey are making preparations for is a
possible grammar.
On the one hand we think that mathematics has to do with the nature of functions and
their arguments,7
and that their number is its business. But we don’t want to let ourselves
be tied down to the functions now known to us, and we don’t know whether people will
ever discover one with 100 argument places; thus we have to make preparations and construct
a function that gets everything ready for a 100-place relation, just in case one is found.
– But what does “a 100-place relation is found (or exists)” mean, anyway? What concept
do we have of it? Or of a 2-place relation, for that matter? – As an example of a 2-place
relation we cite, say, the relation between father and son. But what is the significance of this
example for the further logical treatment of 2-place relations? Instead of every “aRb” are we
now to imagine “a is the father of b”? – If not, is this example or any example essential?
Doesn’t this example play the same role as an example in arithmetic, when I use 3 rows of
6 apples each to explain 3 × 6 = 18 to somebody?
Here it is a matter of our concept of application. – We have, say, a mental image of an
engine that first runs in neutral, and then powers a machine.
But what does the application of a calculation provide for?8
Does this application add
a new calculus to it?9
In that case it’s now a different calculation! Or does the application
give it substance in some sense that’s essential to mathematics (logic)? If so, how can we
disregard the application at all, even only temporarily?
No, a calculation with apples is essentially the same as a calculation with lines or numbers.
A machine is an extension of an engine, but an application is not (in the same sense)
an extension of a calculation.
Now if in order to give an example, I say “Love is a 2-place relation”10
– am I saying
something about love? Of course not. I am giving a rule for the use of the word “love”, and
I want to say something to the effect that we use this word in such and such a way, for instance.
Yet we do have the feeling that when we referred to the 2-place relation “love” we put
meaning into the husk of the calculus of relations. – Let’s imagine a geometrical demonstration
carried out11
using the cylinder of a lamp instead of a drawing or analytical symbols.
To what extent has geometry been applied here? Does the use of the glass cylinder as glass
for a lamp really enter into the geometrical considerations? And does the use of the word
“love” in a declaration of love enter into my considerations12
about 2-place relations?
6 (V): I can describe this case
7 (V): subjects,
8 (V): does the calculation receive from its
application?
9 (V): application inflict a new calculus on it?
10 (V): Now if I say: “Love is an example of a
2-place relation”
11 (V): demonstration done
12 (V): consideration
Laying the Foundations for Arithmetic . . . 384e
WBTC108-115 30/05/2005 16:20 Page 769
Wir haben es mit verschiedenen Verwendungen, Bedeutungen, des Wortes ”Anwendung“
zu tun. ”Die Multiplikation wird in der Division angewandt“; ”der Glaszylinder wird in der
Lampe angewandt“; ”die Rechnung ist auf diese Äpfel angewandt“.
Hier kann man nun sagen: Die Arithmetik ist ihre eigene Anwendung. Der Kalkül ist seine
eigene Anwendung.
Wir können nicht in der Arithmetik für eine grammatische Anwendung vorsorgen.
Denn, ist die Arithmetik nur ein Spiel, so ist für sie auch ihre Anwendung nur ein Spiel,
und entweder das gleiche Spiel (dann führt es uns nicht weiter), oder ein anderes – und dann
konnten wir das schon in der reinen Arithmetik betreiben.
Wenn also der Logiker sagt, er habe für eventuell existierende 6-stellige Relationen in der
Arithmetik vorgesorgt, so können wir fragen: Was wird denn nun zu dem, was Du vorbereitet
hast, hinzutreten,16
wenn es seine Anwendung finden wird?17
Ein neuer Kalkül? – aber den
hast Du ja eben nicht vorbereitet. Oder etwas, was den Kalkül nicht tangiert? – dann interessiert
uns das nicht, und der Kalkül, den Du uns gezeigt hast, ist uns Anwendung genug.
Die unrichtige Idee ist: die Anwendung eines Kalküls auf die wirkliche Sprache verleihe
ihm eine Realität, die er vorher18
nicht hatte.19
Aber, wie gewöhnlich in unserem Gebiet, liegt hier der Fehler nicht darin, daß man etwas
Falsches glaubt, sondern darin, daß man auf eine irreführende Analogie hinsieht.
Was geschieht denn, wenn die 6-stellige Relation gefunden wird? Wird quasi ein Metall
gefunden, das nun die gewünschten (vorher beschriebenen) Eigenschaften (das richtige
spezifische Gewicht, die Festigkeit, etc.) hat? Nein; ein Wort wird gefunden, das wir
tatsächlich in unsrer Sprache so verwenden, wie wir etwa den Buchstaben R verwendet
haben. ”Ja, aber dieses Wort hat doch Bedeutung und ’R‘ hatte keine! Wir sehen also jetzt,
daß dem ’R‘ etwas entsprechen kann.“ Aber die Bedeutung des Wortes besteht ja nicht
darin, daß ihm etwas entspricht. Außer etwa, wo es sich um Namen und benannten
Gegenstand handelt, aber da setzt der Träger des Namens nur den Kalkül fort, also die
Sprache. Und es ist nicht so, wie wenn man sagt: ”diese Geschichte hat sich tatsächlich
zugetragen, sie war nicht bloße Erfindung20
“.
Das alles hängt auch mit dem falschen Begriff der logischen Analyse zusammen, den Russell,
Ramsey und ich hatten. So daß man auf eine endliche logische Analyse der Tatsachen wartet,
wie auf eine chemische von Verbindungen. Eine Analyse, durch die man dann etwa eine
7-stellige Relation wirklich findet, wie ein Element, das tatsächlich das spezifische Gewicht
7 hat.
Die Grammatik ist für uns ein reiner Kalkül. (Nicht die Anwendung eines auf die Realität.)
”Wie kann man Vorbereitungen für etwas eventuell Existierendes treffen“ heißt: Wie
kann man die Arithmetik auf eine Logik aufbauen, in der man im Speziellen noch Resultate
einer Analyse unserer21
Sätze erwartet, und dabei für alle eventuellen Resultate durch eine
Konstruktion a priori aufkommen wollen? – Man will sagen: ”Wir wissen nicht, ob es sich
nicht herausstellen wird, daß es keine Funktionen mit 4 Argumentstellen gibt, oder, daß es
16 (V): hinzukommen,
17 (V): Anwendung findet?
18 (V): früher
19 (V): Die unrichtige Idee ist, daß die
Anwendung eines Kalküls in der Grammatik der
wirklichen Sprache, ihm eine Realität zuordnet,
eine Wirklichkeit gibt, die er früher nicht hatte.
20 (V): Fiktion
21 (V): der
557
558
559
385 Die Begründung der Arithmetik . . .
WBTC108-115 30/05/2005 16:20 Page 770
We are dealing with different uses, meanings, of the word “application”. “Multiplication
is applied in division”; “the glass cylinder is applied in the lamp”; “the calculation has been
applied to these apples”.
At this point we can say: arithmetic is its own application. The calculus is its own
application.
In arithmetic we cannot make preparations for a grammatical application. For if arithmetic
is only a game, its application too is only a game, and either the same game (in which case
it takes us no further) or a different game – and in that case we were already able to play it
in pure arithmetic.
So if the logician says that he has made preparations in arithmetic for the possible
existence of 6-place relations, we can ask him: When what you have prepared finds its
application, what will be added to it? A new calculus? – but in no way is that something
you have prepared for. Or something that doesn’t affect the calculus? – then that doesn’t
interest us, and the calculus you have shown us is application enough for us.
The incorrect idea is: the application of a calculus to real language endows it with a
reality that it didn’t have before.13
But here, as is usual in our field of study, the mistake lies not in believing something false,
but in turning one’s gaze toward a misleading analogy.
What happens, anyway, when the 6-place relation is found? Is it like a metal being found
that has the desired (previously described) properties (the right specific weight, strength, etc.)?
No; what is found is a word that we in fact use in our language as we used, say, the letter
R. “Yes, but this word has meaning, and ‘R’ had none! So now we see that something can
correspond to ‘R’.” But the meaning of the word doesn’t consist in something’s corresponding
to it; except, say, in the case of a name and the object it names; but in that case the
bearer of the name is merely a continuation of the calculus, i.e. language. And it is not like
saying “This story really happened, it wasn’t mere invention”.14
This is all connected as well with the false concept of logical analysis that Russell, Ramsey
and I used to have. Such that one awaits an ultimate logical analysis of facts, as one waits
for a chemical analysis of compounds. An analysis that actually enables one to find, say, a
7-place relation, like an element that actually has the specific weight 7.
Grammar is for us a pure calculus (not the application of a calculus to reality).
“How can we make preparations for something that may or may not exist?” means:
How can we base arithmetic upon a logic in which we are still awaiting the results of an
analysis of our15
propositions in particular cases, and at the same time hope to account for
all possible results through an a priori construction? – One wants to say: “We don’t know
whether it mightn’t turn out that there are no functions with 4 argument places, or that there
13 (V): The incorrect idea is that the application
of a calculus in the grammar of real language
assigns a reality to it, gives it a reality, that it
didn’t have before.
14 (V): fiction”.
15 (V): the
Laying the Foundations for Arithmetic . . . 385e
WBTC108-115 30/05/2005 16:20 Page 771
nur 100 Argumente gibt, die in Funktionen einer Variablen sinnvoll eingesetzt werden
können. Gibt es z.B. (die Annahme scheint immerhin möglich) nur eine solche Funktion F
und 4 Argumente a, b, c, d, und hat es in diesem Falle Sinn, zu sagen ’2 + 2 = 4‘, da es
keine Funktionen gibt, um die Teilung in 2 und 2 zu bewerkstelligen?“ Und nun, sagt man
sich, werden wir für alle eventuellen Fälle vorbauen. Aber das heißt natürlich nichts: Denn
einerseits baut der Kalkül nicht für eine eventuelle Existenz vor, sondern er konstruiert sich
die Existenz, die er überhaupt braucht. Anderseits sind die scheinbaren hypothetischen
Annahmen über die logischen Elemente (den logischen Aufbau) der Welt nichts andres, als
Angaben der Elemente eines Kalküls; und die können freilich auch so gemacht22
werden,
daß es darin ein 2 + 2 nicht gibt.
Treffen wir etwa Vorbereitungen für die Existenz von 100 Gegenständen, indem wir
100 Namen einführen und einen Kalkül mit ihnen. Und nehmen wir jetzt an, es werden
wirklich 100 Gegenstände gefunden. Aber wie ist das, wenn jetzt den Namen Gegenstände
zugeordnet werden, die ihnen früher nicht zugeordnet waren? ändert sich jetzt der
Kalkül? – was hat diese Zuordnung überhaupt mit ihm zu tun? Erhält er durch sie mehr
Wirklichkeit? Oder gehörte er früher bloß zur Mathematik, jetzt aber zur Logik? – Was ist
das für eine Frage: ”gibt es 3-stellige Relationen“, ”gibt es 1000 Gegenstände“? Wie ist
das zu entscheiden? – Aber es ist doch Tatsache, daß wir eine 2-stellige Relation angeben
können, etwa die Liebe, und eine 3-stellige, etwa die Eifersucht, aber, vielleicht, nicht eine
27-stellige! – Aber was heißt es ”eine 2-stellige Relation angeben“? Das klingt (ja) so, als
würden wir auf ein Ding hinweisen und sagen ”siehst Du, das ist so ein Ding“ (wie wir es
nämlich vorher beschrieben haben). Aber so etwas findet ja gar nicht statt (der Vergleich
von dem Hinweisen ist gänzlich falsch). ”Die Beziehung der Eifersucht kann nicht in
2-stellige Beziehungen aufgelöst werden“: das klingt ähnlich wie: ”Alkohol kann nicht in
Wasser und eine feste Substanz zerlegt werden“. Liegt das nun in der Natur der Eifersucht?
(Vergessen wir nicht: der Satz ”A ist wegen B auf C eifersüchtig“ kann ebenso wenig
zerlegt werden wie der: ”A ist wegen B auf C nicht eifersüchtig“.) Das, worauf man hinweist,
ist etwa die Gruppe der Leute A, B und C. – ”Aber wenn nun Lebewesen plötzlich den
3-dimensionalen Raum kennen lernten, nachdem sie bisher nur die Ebene kannten, aber in
ihr doch eine 3-dimensionale Geometrie entwickelt hätten?!“ Würde diese Geometrie damit23
geändert, würde sie inhaltsreicher? – ”Ja, aber ist es denn nicht so, als hätte ich mir z.B.
einmal beliebige Regeln gesetzt, die es mir verböten in meinem Zimmer bestimmte Wege
zu gehen, die ich, was die physikalischen Hindernisse betrifft, ohne weiteres gehen könnte,
– und als würden dann physikalische Bedingungen eintreten, etwa Möbel in das Zimmer
gestellt, die mich nun zwängen, mich nach den Regeln zu bewegen, die ich mir erst
willkürlich gegeben hatte? Wie also der 3-dimensionale Kalkül noch ein Spiel war, da gab
es eigentlich noch keine 3 Dimensionen; denn das x, y, z gehorchte24
nur den Regeln,
weil ich es so wollte; jetzt, wo wir sie mit den wirklichen 3 Dimensionen gekuppelt haben,
können sie sich nicht mehr anders bewegen.“ Aber das ist eine bloße Fiktion. Denn hier
handelt es sich nicht um eine Verbindung mit der Wirklichkeit, die nun die Grammatik
in ihrer Bahn hält! Die ”Verbindung der Sprache mit der Wirklichkeit“, etwa durch die
hinweisenden Definitionen, macht die Grammatik nicht zwangsläufig (rechtfertigt die
Grammatik nicht). Denn diese bleibt immer nur ein frei im Raume schwebender
Kalkül, der zwar25
erweitert, aber nicht gestützt werden kann. Die ”Verbindung mit der
Wirklichkeit“ erweitert nur die Sprache, aber zwingt sie zu nichts. Wir reden von der
22 (V): getroffen
23 (V): nun
24 (O): gehorchten
25 (V): nur
560
386 Die Begründung der Arithmetik . . .
WBTC108-115 30/05/2005 16:20 Page 772
are only 100 arguments that can be meaningfully inserted into functions of one variable. Is
there, for example (the supposition does appear possible) only one four-place function F and
4 arguments a, b, c, d, and does it then make sense to say ‘2 + 2 = 4’, since there aren’t
any functions to accomplish the division into 2 and 2?” So now, one says to oneself, we will
make provision for all possible cases. But of course that means nothing: for on the one hand
the calculus doesn’t provide for possible existence; it constructs all the existence it needs.
On the other hand what look like hypothetical assumptions about the logical elements
(the logical structure) of the world are nothing other than specifications of the elements
of a calculus; and of course you can make these in such a way that the calculus does not
contain any 2 + 2.
Let’s prepare, for example, for the existence of 100 objects by introducing 100 names and
a calculus to go along with them. And now let’s suppose 100 objects really are found. But
what happens if names are now correlated with objects they hadn’t been correlated with before?
Does the calculus now change? – What does this correlation have to do with it at all? Does
it acquire more reality because of this correlation? Or did the calculus previously belong
only to mathematics, but now to logic? – What sorts of questions are “Are there 3-place
relations?”, “Are there 1000 objects?”? How is this to be decided? – But surely it is a fact that
we can specify a 2-place relation, say love, and a 3-place one, say jealousy, but perhaps not
a 27-place one! – But what does “specify a 2-place relation” mean? That does sound as if
we were pointing to a thing and saying “See, that’s the kind of thing” (i.e. the kind of thing
we described earlier). But nothing like that takes place (the comparison with pointing is completely
wrong). “The relation of jealousy cannot be broken up into 2-place relationships”
sounds similar to “Alcohol cannot be broken down into water plus a solid substance”. Now
is that inherent in the nature of jealousy? (Let’s not forget: the proposition “A is jealous
of C because of B” is no more reducible than the proposition “A is not jealous of C because
of B”.) What is pointed to is, say, the group of people A, B and C. – “But what if living
beings who at first knew only plane surfaces, but had nonetheless developed a 3-dimensional
geometry, were to suddenly become acquainted with 3-dimensional space?!” Would this alter
their geometry,16
would it become richer in content? – “Yes, but isn’t this as if at some time,
for instance, I had made arbitrary rules for myself prohibiting me from taking certain routes
in my room, routes that, as far as physical obstacles were concerned, I could take without a
problem – and then physical conditions occurred (say furniture was put in the room) that
forced me to move in accordance with the rules I had originally imposed on myself arbitrarily?
Thus, when the 3-dimensional calculus was only a game, there actually weren’t three
dimensions; for the x, y, z dimensions only obeyed the rules because that’s the way I wanted
it; but now that we have linked them up to the real 3 dimensions, they are no longer able
to move otherwise.” But that is mere fiction. For here it’s not a matter of a connection
with reality that keeps grammar on track! The “connection of language with reality”, say by
means of ostensive definitions, doesn’t make grammar inevitable (doesn’t justify grammar).
For grammar always remains a calculus floating freely in space, a calculus that can be extended,
to be sure,17
but not supported. The “connection with reality” merely extends language; it
doesn’t force it to do anything. We speak of discovering a 27-place relation, but on the one
16 (V): Would their geometry now be altered, 17 (V): that can only be extended,
Laying the Foundations for Arithmetic . . . 386e
WBTC108-115 30/05/2005 16:20 Page 773
Auffindung einer 27-stelligen Relation: aber einerseits kann mich keine Entdeckung
zwingen, (das Zeichen und) den Kalkül der 27-stelligen Relation zu gebrauchen; andrerseits
kann ich die Handlungen dieses Kalküls26
selbst mittels dieser Notation beschreiben.
Wenn man in der Logik scheinbar mehrere verschiedene Universen betrachtet (wie Ramsey),
so betrachtet man in Wirklichkeit verschiedene Spiele. Die Erklärung eines ”Universums“
würde z.B. in Ramsey’s Fall einfach eine27
Definition (∃x).φx Def.
φa 1 φb 1 φc 1 φd sein.
26 (V): ich diesen Kalkül 27 (V): die
561
387 Die Begründung der Arithmetik . . .
==
WBTC108-115 30/05/2005 16:20 Page 774
hand no discovery can force me to use (the sign and) the calculus for a 27-place relation,
and on the other hand I can describe the operations of this calculus18
itself with this
notation.
When it seems in logic that we’re looking at several different universes (as does Ramsey),
in reality we are looking at different games. In Ramsey’s case, for instance, the definition of
a “universe” would simply be a definition like19
(∃x).φx Def.
φa 1 φb 1 φc 1 φd.
18 (V): describe this calculus 19 (V): be the definition
Laying the Foundations for Arithmetic . . . 387e
==
WBTC108-115 30/05/2005 16:20 Page 775
113
Ramsey’s Theorie der Identität.
Die Theorie der Identität bei Ramsey macht den Fehler, den man machen würde, wenn
man sagte, ein gemaltes Bild könne man auch als Spiegel benutzen, wenn auch nur für eine
einzige Stellung, wo dann übersehen wird, daß das Wesentliche am Spiegel gerade das ist,
daß man aus ihm die Stellung des Körpers vor dem Spiegel schließen kann, während man
im Fall des gemalten Bildes erst wissen muß, daß die Stellungen übereinstimmen, ehe man
das Bild als Spiegelbild auffassen kann.
Wenn die Dirichlet’sche Auffassung der Funktion einen strengen Sinn hat, so muß
sie sich in einer Definition ausdrücken, die das Funktionszeichen mit der Tabelle als
gleichbedeutend erklärt.
Ramsey erklärt ”x = x“ auf einem Umweg als die Aussage: ”jeder Satz ist sich selbst
äquivalent“ und ”x = y“ als die Aussage: ”jeder Satz ist jedem Satz äquivalent“.1
Er hat also mit seiner Erklärung nichts andres erreicht, als was die zwei Definitionen
x = x Def.
Tautologie
x = x Def.
Kontradiktion bestimmen. (Das Wort ”Tautologie“ kann hier durch jede
beliebige Tautologie ersetzt werden und das gleiche gilt für ”Kontradiktion“.)
Soweit ist nichts geschehn, als Erklärungen der zwei verschiedenen Zeichenformen
x = x und x = y zu geben. Diese Erklärungen können natürlich durch zwei Klassen von
Erklärungen ersetzt werden,2
z.B.:
a = a a = b
b = b = Taut. b = c = Cont.
c = c c = a
Nun aber schreibt Ramsey:
”(∃x, y).x ≠ y“, d.h. ”(∃x, y).~(x = y)“, –
dazu hat er aber gar kein Recht: denn, was bedeutet in diesem Zeichen das ”x = y“? Es
ist ja weder das Zeichen ”x = y“, welches ich in der Definition oben gebraucht habe, noch
natürlich das ”x = x“ in der vorhergehenden Definition. Also ist es ein noch unerklärtes
Zeichen. Um übrigens die Müßigkeit dieser3
Definitionen einzusehen, lese man sie (wie sie
der Unvoreingenommene lesen würde) so: Ich erlaube, statt des Zeichens ”Taut.“, dessen
1 (V): Ramsey definiert x = y als
(φe).φex ≡ φey.
Aber nach den Erklärungen, die er über seine
Funktionszeichen ”φe“ gibt,
ist (φe).φex ≡ φex die Aussage: ”jeder Satz ist sich
selbst äquivalent“
(φe).φex ≡ φey die Aussage: ”jeder Satz ist jedem
Satz äquivalent“.
2 (V): werden:
3 (V): jener
562
563
5
6
7
5
6
7
388
==
==
WBTC108-115 30/05/2005 16:20 Page 776
113
Ramsey’s Theory of Identity.
Ramsey’s theory of identity makes the same mistake that would be made if it were said
that you could also use a painting as a mirror, even if only for a single posture. What’s
overlooked here is that what is essential to a mirror is precisely that you can infer from it
the posture of a body in front of it, whereas in the case of a painting you have to know that
the postures are correlated before you can regard the picture as a mirror image.
If Dirichlet’s conception of function has a strict sense, it must be expressed in a definition
that declares the function sign to be equivalent to the table.
In a roundabout way, Ramsey defines “x = x” as the statement “Every proposition is
equivalent to itself” and “x = y” as the statement “Every proposition is equivalent to every
proposition”.1
So all he has achieved by his definition is what is laid down by the two definitions
x = x Def.
Tautology
x = x Def.
Contradiction. (Here the word “tautology” can be replaced by any arbitrary
tautology, and similarly for “contradiction”.)
So far all that has happened is that definitions have been given of the two different forms
of signs x = x and x = y. Of course these definitions can be replaced by two sets of definitions,
for example:
a = a a = b
b = b = Taut. b = c = Cont.
c = c c = a
And now Ramsey writes:
“(∃x, y).x ≠ y”, i.e. “(∃x, y).~(x = y)” –
but he has no right to do this: for what does “x = y” mean in this expression? It is neither
the sign “x = y” used in the definition above, nor of course the “x = x” in the preceding
definition. So it is a sign that is still unexplained. By the way, to understand the futility of
these2
definitions, you should read them (as an unbiased person would) as follows: I permit
the sign “Taut.”, whose use we know, to be replaced by the sign “a = a” or “b = b”, etc.;
1 (V): Ramsey defines x = y as (φe).φex ≡ φey. But
according to the definitions that he gives for his
function sign “φe”, (φe).φex ≡ φex is equivalent to
the statement “Every proposition is equivalent to
itself ”, and (φe).φex ≡ φey is equivalent to the
statement “Every proposition is equivalent to
every proposition”.
2 (V): those
5
6
7
5
6
7
388e
==
==
WBTC108-115 30/05/2005 16:20 Page 777
Gebrauch wir kennen, das Zeichen ”a = a“ oder ”b = b“, etc. zu setzen; und statt des Zeichens
”Cont.“ (”~Taut.“) die Zeichen ”a = b“, ”a = c“, etc. Woraus übrigens hervorgeht, daß
(a = b) = (c = d) = (a ≠ a) = etc.!
Es braucht wohl nicht gesagt zu werden, daß ein so definiertes Gleichheitszeichen nichts
mit demjenigen zu tun hat, welches wir zum Ausdruck einer Ersetzungsregel brauchen.
Ich kann nun ”(∃x, y).x ≠ y“ natürlich wieder erklären; etwa als a ≠ a .1. a ≠ b .1.
b ≠ c .1. a ≠ c; diese Erklärung aber ist eigentlich Humbug und ich sollte unmittelbar schreiben
(∃x, y).x ≠ y Def.
Taut. (D.h. das Zeichen auf der linken Seite würde mir als ein neues –
unnötiges – Zeichen für ”Taut.“ gegeben.) Denn wir dürfen nicht vergessen, daß nach
der Erklärung ”a = a“, ”a = b“, etc. unabhängige Zeichen sind und nur insofern
zusammenhängen, als eben die Zeichen ”Taut.“ und ”Cont.“.
Die Frage ist hier die nach der Nützlichkeit der ”extensiven“ Funktionen, denn die
Ramsey’sche Erklärung des Gleichheitszeichens ist ja so eine Bestimmung durch die
Extension. Worin besteht4
nun die extensive Bestimmung einer Funktion? Sie ist offenbar
eine Gruppe von Definitionen, z.B. die:
fa = p Def
fb = q Def
fc = r Def
Diese Definitionen erteilen uns die Erlaubnis, statt der uns bekannten Sätze ”p“, ”q“, ”r“
die Zeichen ”fa“, ”fb“, ”fc“ zu setzen. Zu sagen, durch diese drei Definitionen werde5
die
Funktion f(x) bestimmt, sagt gar nichts, oder dasselbe, was die drei Definitionen sagen.
Denn die Zeichen ”fa“, ”fb“, ”fc“ sind Funktion und Argument nur, sofern es auch die
Wörter ”Ko(rb)“, ”Ko(pf)“ und ”Ko(hl)“ sind. (Es macht dabei keinen Unterschied, ob die
”Argumente“ ”rb“, ”pf“, ”hl“ sonst noch als Wörter gebraucht werden, oder nicht.)
(Welchen Zweck also die Definitionen haben können, außer den, uns irrezuführen, ist schwer
einzusehn.)
Das Zeichen ”(∃x).fx“ heißt zunächst gar nichts; denn die Regeln für Funktionen im alten
Sinn des Wortes gelten ja hier nicht. Für diese wäre eine Definition fa = . . . . Unsinn. Das
Zeichen ”(∃x).fx“ ist, wenn keine ausdrückliche Erklärung dafür gegeben wird, nur wie ein
Rebus zu verstehen, in welchem auch die Zeichen eine Art uneigentliche Bedeutung haben.
Jedes der Zeichen ”a = a“, ”a = c“, etc. in den Definitionen (a = a) Def.
Taut., etc. ist
ein Wort.
Der Endzweck der Einführung der extensiven Funktionen war übrigens die Analyse
von Sätzen über unendliche Extensionen und dieser Zweck ist verfehlt, da eine extensive
Funktion durch eine Liste von Definitionen eingeführt wird.
Es besteht eine Versuchung, die Form der Gleichung für die Form von Tautologien und
Kontradiktionen zu halten, und zwar darum, weil es scheint, als könne man sagen,6
x = x
ist selbstverständlich wahr (und) x = y selbstverständlich falsch. Eher noch kann man
natürlich x = x mit einer Tautologie vergleichen, als x = y mit einer Kontradiktion,7
da
ja alle richtigen (und ”sinnvollen“) Gleichungen8
der Mathematik von der Form x = y
sind. Man könnte x = x eine degenerierte Gleichung nennen (Ramsey nannte sehr richtig
4 (V): Extension. Welcher Art ist
5 (V): sei
6 (V): sagen:
7 (V): kann man natürlich sagen, daß x = x die
Rolle einer Tautologie spielt, als x = y die der
Kontradiktion,
8 (O): (und ”sinnvollen“ Gleichungen
564
565
389 Ramsey’s Theorie der Identität.
==
==
WBTC108-115 30/05/2005 16:20 Page 778
and the sign “Cont.” (“~Taut.”) to be replaced by the signs “a = b”, “a = c” etc. From
which, incidentally, it follows that
(a = b) = (c = d) = (a ≠ a) = etc.!
It goes without saying that an equals-sign defined like that has nothing to do with the one
we use to express a substitution rule.
Now of course I can define “(∃x, y).x ≠ y” in turn; say as a ≠ a .1. a ≠ b .1. b ≠ c .1.
a ≠ c; but actually this definition is humbug, and I should straightaway write (∃x, y).x ≠ y
Def.
Taut. (That is, I would be given the sign on the left side as a new – unnecessary –
sign for “Taut.”) For we mustn’t forget that, according to the definition, “a = a”, “a = b”,
etc. are independent signs, and are connected with each other only in so far as are the signs
“Taut.” and “Cont.”.
The question that arises here is one about the usefulness of “extended” functions, for
Ramsey’s explanation of the equals-sign is just such a specification by extension. Now what
exactly does the specification of a function by its extension consist in?3
Obviously, it is a group
of definitions, for example:
fa = p Def.
fb = q Def.
fc = r Def.
These definitions allow us to substitute the signs “fa”, “fb”, “fc” for the known propositions
“p”, “q”, “r”. To say that the function f(ξ) is4
determined by these three definitions
says nothing at all, or the same thing that the three definitions say.
For the signs “fa”, “fb”, “fc” are function and argument only in so far as the words “Co(rn)”,
“Co(al)” and “Co(lt)” are as well. (Here it makes no difference whether or not the “arguments”
“rn”, “al”, “lt” are used as words in any other contexts.)
(So it is hard to see what purpose the definitions can have except to mislead us.)
To begin with, the sign “(∃x).fx” means nothing; for here the rules for functions in the
old sense of the word don’t hold. According to them a definition like fa = . . . would be nonsense.
If no explicit definition is given for it, the sign “(∃x).fx” can only be understood as
like a rebus, in which the signs have a kind of figurative meaning as well.
Each of the signs “a = a”, “a = c”, etc. in the definitions (a = a) Def.
Taut., etc. is a word.
Incidentally, the ultimate purpose for introducing extended functions was the analysis
of propositions about infinite extensions. But this purpose is misguided, since an extended
function is introduced by a list of definitions.
There is a temptation to regard the form of an equation as the form of tautologies and
contradictions, because it looks as if one could say that x = x is self-evidently true and
x = y self-evidently false. Of course one can more readily compare x = x to a tautology
than x = y to a contradiction,5
because all correct (and “meaningful”) equations of mathematics
are of the form x = y. We could call x = x a degenerate equation (Ramsey quite
correctly called tautologies and contradictions “degenerate propositions”), and indeed a
correct degenerate equation (the limiting case of an equation). For we use expressions of
3 (V): What is the nature of the specification of a
function by its extension?
4 (V): f(ξ) had been
5 (V): Of course one can more readily say that
x = x plays the role of a tautology than that
x = y plays the role of a contradiction,
Ramsey’s Theory of Identity. 389e
==
==
WBTC108-115 30/05/2005 16:20 Page 779
Tautologien und Kontradiktionen degenerierte Sätze) und zwar eine richtige degenerierte
Gleichung (den Grenzfall einer Gleichung). Denn wir gebrauchen Ausdrücke der Form
x = x wie richtige Gleichungen, wobei wir uns vollkommen bewußt sind, daß es sich um
degenerierte Gleichungen handelt. Im gleichen Fall sind Sätze in geometrischen Beweisen,
wie etwa: ”der Winkel α ist gleich dem Winkel β, der Winkel γ ist sich selbst gleich, . . .“.
Man könnte nun einwenden, daß richtige Gleichungen der Form x = y auch Tautologien,
dagegen falsche, Kontradiktionen sein müßten, weil man ja die richtige Gleichung muß
beweisen können und das, indem man die beiden Seiten der Gleichung transformiert, bis
eine Identität x = x herauskäme. Aber obwohl durch diesen Prozeß die erste Gleichung
als richtig erwiesen ist und insofern die Identität x = x das Endziel der Transformationen
war, so ist sie nicht das Endziel in dem Sinne, als hätte man durch die Transformationen
der Gleichung ihre richtige Form geben wollen, wie man einen krummen Gegenstand
zurechtbiegt, und als habe sie nun in der Identität diese vollkommene Form (endlich)
erreicht. Man kann also nicht sagen: die richtige Gleichung ist ja eigentlich eine Identität.
Sie ist eben keine Identität.
390 Ramsey’s Theorie der Identität.
WBTC108-115 30/05/2005 16:20 Page 780
the form x = x like correct equations, while being fully conscious that we are dealing
with degenerate equations. Propositions in geometrical proofs, such as “the angle α is equal
to the angle β, the angle γ is equal to itself . . .”, are in the same situation.
At this point the objection could be made that correct equations of the form x = y must
also be tautologies, whereas incorrect ones must be contradictions, because one has to be
able to prove a correct equation, and to do that by transforming each side of it until an
identity of the form x = x is reached. But although the original equation has been shown
by this process to be correct, and to that extent the identity x = x was the ultimate goal of
the transformations, still it is not the ultimate goal, in the sense that one wanted to give the
equation its correct form through the transformations – like bending a crooked object straight;
nor is it the ultimate goal in the sense that (at long last) the equation has now achieved this
perfect form in the identity. So we can’t say: a correct equation is actually an identity. The
point is precisely that it is not an identity.
Ramsey’s Theory of Identity. 390e
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114
Der Begriff der Anwendung der
Arithmetik (Mathematik).
Wenn man sagt: ”es muß der Mathematik wesentlich sein, daß sie angewandt werden kann“,
so1
meint man, daß diese Anwendbarkeit2
nicht die eines Stückes Holz ist, von dem ich sage
”das werde ich zu dem und dem anwenden können“.
Die Geometrie ist nicht die Wissenschaft (Naturwissenschaft) von den geometrischen
Ebenen, geometrischen Geraden und geometrischen Punkten, im Gegensatz etwa zu einer
anderen Wissenschaft, die von den groben, physischen Geraden, – Strichen, – Flächen etc.
handelt und deren Eigenschaften angibt. Der Zusammenhang der Geometrie mit Sätzen
des praktischen Lebens, die von Strichen, Farbgrenzen, Kanten und Ecken etc. handeln,
ist nicht der, daß sie über ähnliche Dinge spricht, wie diese Sätze, wenn auch über ideale
Kanten, Ecken, etc.; sondern der, zwischen diesen Sätzen und ihrer Grammatik. Die
angewandte Geometrie ist die Grammatik der Aussagen über die räumlichen Gegenstände.
Die sogenannte geometrische Gerade verhält sich zu einer Farbgrenze nicht wie etwas Feines
zu etwas Grobem, sondern wie Möglichkeit zur Wirklichkeit. (Denke an die Auffassung der
Möglichkeit als Schatten der Wirklichkeit.)
Man kann eine Kreisfläche beschreiben, die durch Durchmesser in 8 kongruente Teile
geteilt ist, aber es ist sinnlos, das von einer elliptischen3
Fläche zu sagen. Und darin liegt,
was die Geometrie in dieser Beziehung von der Kreis- und Ellipsenfläche4
aussagt.
|Ein Satz, der auf einer falschen Rechnung beruht (wie etwa ”er teilte das 3m lange Brett
in 4 Teile zu je 1m“) ist unsinnig5
und das beleuchtet, was es heißt ”Sinn zu haben“ und
”etwas mit dem Satz meinen“.6
|
Wie ist es mit dem Satz ”die Winkelsumme im Dreieck ist 180 Grad“? Dem sieht man
es jedenfalls nicht an, daß er ein Satz der Syntax ist.
Der Satz ”Gegenwinkel sind gleich“ heißt, ich werde, wenn sie sich bei der Messung nicht
als gleich erweisen, die Messung für falsch erklären und ”die Winkelsumme im Dreieck ist
180 Grad“ heißt, ich werde, wenn sie sich bei einer Messung nicht als 180 Grad erweist,
einen Messungsfehler annehmen. Der Satz ist also ein Postulat über die Art und Weise der
Beschreibung der Tatsachen. Also ein Satz der Syntax.
1 (V): so hei
2 (V): Anwendbarkeit
3 (O): eliptischen
4 (O): Elipsenfläche
5 (V): 1m“) hat keinen Sinn
6 (V): und das wirft ein Licht auf den Sinn der
Ausdrücke ”Sinn haben“ und ”etwas mit dem
Satz meinen“.
566
567
391
WBTC108-115 30/05/2005 16:20 Page 782
114
The Concept of the Application of
Arithmetic (Mathematics).
If we say: “It must be essential to mathematics that it can be applied” we mean that this
ability to be applied1
isn’t that of a piece of wood, of which I say “I’ll be able to use it for
this or that”.
Geometry isn’t the science (natural science) of geometric planes, lines and points, as opposed,
say, to some other science whose subject matter is gross physical lines, strips, surfaces, etc.
and that states their properties. The connection between geometry and propositions of
practical life, which are about strips, colour boundaries, edges and corners, etc. doesn’t consist
in geometry’s speaking of things similar to what these propositions speak of, although
geometry does speak about ideal edges and corners, etc.; rather, it consists in the connection
between these propositions and their grammar. Applied geometry is the grammar of
statements about spatial objects. The relation between what is called a geometrical line and
a boundary between two colours isn’t like the relation between something fine and something
coarse, but like the relation between possibility and actuality. (Think of understanding possibility
as the shadow of actuality.)
You can describe a circular surface divided by diameters into 8 congruent parts, but it
makes no sense to give such a description of an elliptical surface. And therein is contained
what geometry says in this respect about circular and elliptical surfaces.
|A proposition based on a false calculation (such as “He cut a board that was 3 metres
long into 4 one-metre parts”) is nonsensical,2
and that throws light on what is meant by
“making sense” and “meaning something by a proposition”.3
|
What about the proposition “The sum of the angles of a triangle is 180 degrees”? As
hard as you try, you can’t tell by looking at it that it is a proposition of syntax.
The proposition “corresponding angles are equal” means that if they aren’t found to be
equal when they are measured I will declare the measurement incorrect; and “the sum of
the angles of a triangle is 180 degrees” means that if it doesn’t turn out to be 180 degrees
when they are measured I will assume there has been a mistake in the measurement. So the
proposition is a postulate about the way of describing facts, and therefore a proposition
of syntax.
1 (V): this applicability
2 (V): parts”) makes no sense,
3 (V): and that throws light on the sense of
the expressions “making sense” and “meaning
something by a proposition”.
391e
WBTC108-115 30/05/2005 16:20 Page 783
Über
Kardinalzahlen.
568
WBTC108-115 30/05/2005 16:20 Page 784
On Cardinal
Numbers.
WBTC108-115 30/05/2005 16:20 Page 785
115
Kardinalzahlenarten.
Was die Zahlen sind? – Die Bedeutungen der Zahlzeichen; und die Untersuchung dieser
Bedeutung ist die Untersuchung der Grammatik der Zahlzeichen.
Wir suchen nicht nach einer Definition des Zahl-Begriffs, sondern versuchen eine
Darlegung der Grammatik des Wortes ”Zahl“ und der Zahlwörter.1
Es gibt unendlich viele Kardinalzahlen, weil wir dieses unendliche System konstruieren
und es das der Kardinalzahlen nennen. Es gibt auch ein Zahlensystem ”1, 2, 3, 4, 5, viele“
und auch eines: ”1, 2, 3, 4, 5“.2
Und3
warum sollte ich das nicht auch ein System von
Kardinalzahlen nennen? (und also ein endliches).
Daß das axiom of infinity nicht ist, wofür Russell es gehalten hat, daß es weder ein Satz
der Logik, noch auch – wie es da steht – ein Satz der Physik ist, ist klar. Ob der Kalkül
damit, in eine ganz andre Umgebung gebracht (in ganz anderer ”Interpretation“), irgendwo
eine praktische Anwendung finden könnte, weiß ich nicht.
Von den logischen Begriffen, z.B. von dem (oder: einem) der Unendlichkeit, könnte man
sagen: ihre Essenz beweise ihre Existenz.
(Frege hätte noch gesagt: ”es gibt vielleicht Menschen,4
die in der Kenntnis der
Kardinalzahlenreihe nicht über die 5 hinausgekommen sind (und etwa das Übrige der
Reihe nur in unbestimmter Form sehen), aber diese Reihe existiert unabhängig von uns“.
Existiert das Schachspiel unabhängig von uns, oder nicht? – )
Eine sehr interessante Erwägung über die Stellung des Zahlbegriffs in der Logik ist
die: Wie ist5
es mit dem Zahlbegriff, wenn ein Volk keine Zahlwörter besitzt, sondern
sich zum Zählen, Rechnen, etc. ausschließlich eines Abacus bedient, etwa der Russischen
Rechenmaschine?6
(Nichts wäre interessanter, als die Arithmetik dieser Menschen zu untersuchen und man
verstünde wirklich, daß es hier keinen Unterschied zwischen 20 und 21 gibt.)7
Könnte man auch eine Zahlenart den Kardinalzahlen entgegensetzen, deren Reihe der
der Kardinalzahlen ohne der 5 entspräche? Oh ja: nur wäre diese Zahlenart zu nichts zu
brauchen, wozu die Kardinalzahlen es sind. Und die 5 fehlt diesen Zahlen nicht, wie ein
Apfel, den man aus einer Kiste voller Äpfel herausgenommen8
hat und wieder hineinlegen
kann, sondern die 5 fehlt dem Wesen dieser Zahlen; sie kennen die 5 nicht (wie die
1 (V): sondern nach einer Klärung der Grammatik
des Wortes ”Zahl“ und der Zahlwörter.
2 (O): 4, 5,”.
3 (V): Und W
4 (V): Völker,
5 (V): steht
6 (V): sondern sich statt dieser immer eines
Abacus bedient, etwa einer Russischen
Rechenmaschine?
7 (V): daß hier kein Unterschied zwischen 20
und 21 existiert // besteht.)
8 (V): genommen
569
570
393
WBTC108-115 30/05/2005 16:20 Page 786
115
Kinds of Cardinal Numbers.
What are numbers? – The meanings of numerals. And an investigation of this meaning
is an investigation of the grammar of numerals.
We aren’t looking for a definition of the concept of number; rather, we’re attempting an
exposition1
of the grammar of the word “number” and of the numerals.
There are infinitely many cardinal numbers because we construct this infinite system and
call it the system of cardinal numbers. There is also a number system “1, 2, 3, 4, 5, many”
and even a system “1, 2, 3, 4, 5”. And why shouldn’t I call that too a system of cardinal
numbers (and thus a finite one)?
It is clear that the axiom of infinity is not what Russell took it for; that it is neither a proposition
of logic, nor – as it stands – a proposition of physics. Whether this means that the
calculus to which it belongs, if it were transplanted into quite different surroundings (with
an entirely different “interpretation”), could find a practical application somewhere, I do
not know.
One could say of logical concepts, for example, of the (or a) concept of infinity: Their
essence proves their existence.
(Frege would have insisted: “Perhaps there are people2
who have not got beyond the first
five in their knowledge of the series of cardinal numbers (and perhaps see the rest of the
series only in an indeterminate form), but this series exists independently of us”. Does the
game of chess exist independently of us, or not? – )
Here is a very interesting consideration about the position of the concept of number in
logic: What happens to the concept of number if a people have no numerals, but for counting,
calculating, etc. use an abacus, say a Russian abacus, exclusively?3
(Nothing would be more interesting than to investigate the arithmetic of these people;
and one would really understand that here there is no distinction4
between 20 and 21.)
Could one also – in contrast to the cardinal numbers – set up a kind of number whose
series corresponded to the series of cardinal numbers without the 5? Certainly; only this kind
of number could be used for none of the things we use the cardinal numbers for. And these
numbers are not missing the five like an apple that was taken out of5
a box full of apples
and can be put back again; rather they are essentially missing a 5; the numbers do not know
1 (V): we’re seeking a clarification
2 (V): peoples
3 (V): but instead of these always use an abacus,
say a Russian abacus?
4 (V): here no distinction exists
5 (V): taken from
393e
WBTC108-115 30/05/2005 16:20 Page 787
Kardinalzahlen die Zahl 1
/2 nicht kennen). Angewendet würden also diese Zahlen (wenn
man sie so nennen will) in einem Fall, in dem die Kardinalzahlen (mit der 5) nicht mit Sinn
angewendet werden könnten.
(Zeigt sich hier nicht die Unsinnigkeit des Geredes von der ”Grundintuition“?)
Wenn die Intuitionisten von der ”Grundintuition“ sprechen, – ist diese ein psychologischer
Prozeß? Und wie kommt er dann in die Mathematik? Oder ist, was sie meinen, nicht
doch nur ein Urzeichen (im Sinne Freges); ein Bestandteil eines Kalküls?
So seltsam es klingt, so ist es möglich, die Primzahlen bis – sagen wir – zur 7 zu kennen
und daher ein endliches System von Primzahlen zu besitzen. Und was wir die Erkenntnis
nennen, daß es unendlich viele Primzahlen gibt, ist in Wahrheit die Erkenntnis eines neuen,
und mit dem andern gleichberechtigten, Systems.9
Wenn man bei geschlossenen Augen ein Flimmern sieht, unzählige Lichtpünktchen, die
kommen und verschwinden, – wie man es etwa beschreiben würde – so hat es keinen Sinn,
hier von einer ”Anzahl“ der zugleich gesehenen Pünktchen zu reden. Und man kann nicht
sagen ”es sind immer eine bestimmte Anzahl von Lichtpünktchen da, wir wissen sie bloß
nicht“; dies entspräche einer Regel, die dort angewandt wird, wo von einer Kontrolle dieser
Anzahl gesprochen werden kann.
|Es hat Sinn zu sagen: Ich verteile viele unter viele. Aber der Satz ”ich konnte die
vielen Nüsse nicht unter die vielen Menschen verteilen“ kann nicht heißen, daß es logisch
unmöglich war. Man kann auch nicht sagen: ”in manchen Fällen ist es möglich, viele unter
viele zu verteilen und in manchen nicht“; denn darauf frage ich: in welchen Fällen ist
dies möglich und in welchen unmöglich? und darauf könnte nicht mehr im Viele-System
geantwortet werden.|
Von einem Teil meines Gesichtsfeldes zu sagen, er habe keine Farbe, ist Unsinn; ebenso
– natürlich auch – zu sagen, er habe Farbe (oder, eine Farbe). Anderseits10
hat es Sinn zu
sagen, er habe nur eine Farbe (sei einfärbig, oder gleichfärbig), er habe mindestens zwei Farben,
nur zwei Farben, u.s.w.
Ich kann also in dem Satz ”dieses Viereck in meinem Gesichtsfeld hat mindestens zwei
Farben“ statt ”zwei“ nicht ”eine“ substituieren. Oder auch: ”das Viereck hat nur eine Farbe“
heißt nicht – analog (∃x).φx & ~(∃x, y).φx & φy – ”das Viereck hat eine Farbe, aber nicht
zwei Farben“.
Ich rede hier von dem Fall, in welchem11
es sinnlos ist zu sagen,12
”der Teil des Raumes
hat13
keine Farbe“. Wenn ich die gleichfärbigen (einfärbigen) Flecke in dem Viereck zähle,
so hat es übrigens Sinn zu sagen, es seien keine solchen vorhanden, wenn die Farbe des Vierecks
sich kontinuierlich ändert. Es hat dann natürlich auch Sinn zu sagen, in dem Viereck sei
”ein gleichfärbiger Fleck oder mehrere“ und auch, das Viereck habe eine Farbe aber nicht
zwei Farben. – Von diesem Gebrauch aber des Satzes ”das Viereck hat keine Farbe“ sehe
ich jetzt ab und spreche von einem System, in welchem, daß eine Figur14
eine Farbe hat,
selbstverständlich genannt wird,15
also,16
richtig ausgedrückt, in welchem es diesen Satz nicht
9 (O): System.
10 (V): eine Farbe). Wohl aber
11 (V): dem
12 (V): sagen:
13 (V): habe
14 (V): daß eine Fläche // ein Viereck // ein
Flächenstück
15 (V): selbstverständlich ist
16 (O): wird also,
571
572
573
394 Kardinalzahlenarten.
WBTC108-115 30/05/2005 16:20 Page 788
the 5 (in the way that the cardinal numbers do not know the number 1
/2). So these numbers
(if you want to call them that) would be applied in cases where the cardinal numbers (with
the 5) couldn’t be applied sensibly.
(Doesn’t this show the absurdity of the talk about “basic intuition”?)
When the intuitionists speak of a “basic intuition” – is this a psychological process? And
if so, how does it get into mathematics? Or isn’t what they mean only a primitive sign (in
Frege’s sense); an element of a calculus?
Strange as it sounds, it is possible to know the prime numbers up to – let’s say – 7, and
thus to have a finite system of prime numbers. And what we call the realization that there
are infinitely many primes is in truth the realization of a new system that is as justified as
the other.
If you close your eyes and see a twinkling – countless little points of light coming and
vanishing, as we might describe it – it doesn’t make any sense to speak of a “number” of
simultaneously seen little points. And you can’t say “There is always a definite number of
little points of light here, we just don’t know what it is”; that would correspond to a rule
that is applied in a case where you can speak of checking this number.
|It makes sense to say: I distribute many among many. But the proposition “I couldn’t
distribute the many nuts among the many people” can’t mean that it was logically impossible.
Neither can you say: “In some cases it is possible to distribute many among many and
in some it isn’t”; for, in response to that, I ask: in which cases is this possible and in which
impossible? And to that no further answer could be given in the many-system.|
To say of a part of my visual field that it has no colour is nonsense; and of course it is
equally nonsensical to say that it has colour (or a colour). On the other hand it6
does make
sense to say that it has only one colour (is monochrome, or isochromatic), or that it has at
least two colours, only two colours, etc.
So in the proposition “This rectangle in my visual field has at least two colours” I
cannot substitute “one” for “two”. Or again: “The rectangle has only one colour” does not
mean – on the analogy of (∃x).φx & ~(∃x, y).φx & φy – “The rectangle has one colour
but not two colours”.
I am speaking here of the case in which it is senseless to say7
“That part of space has no
colour”. If I count the isochromatic (monochrome) spots in the rectangle, it does incidentally
make sense to say that there aren’t any there if the colour of the rectangle is continually
changing. In that case, of course, it also makes sense to say that there are “one or more
isochromatic spots” in the rectangle, and also that the rectangle has one colour but not two.
– But for the moment I am disregarding that use of the proposition “The square has no
colour”, and am speaking of a system in which we say it is self-evident8
that a figure9
has a
colour, and therefore, correctly expressed, of a system in which this proposition doesn’t exist.10
If you call the proposition self-evident you really mean what is expressed by a grammatical
6 (V): But it
7 (V): to say:
8 (V): in which it is self-evident
9 (V): a surface // a rectangle // a piece of a
surface
10 (V): in which this proposition is nonsense.
Kinds of Cardinal Numbers. 394e
WBTC108-115 30/05/2005 16:20 Page 789
gibt.17
Wenn man den Satz selbstverständlich nennt, so meint man eigentlich dasjenige,18
was eine grammatische Regel ausdrückt, die die Form der Sätze über den Gesichtsraum,
z.B., beschreibt. Wenn man nun die Zahlangabe der Farben im Viereck mit dem Satz ”in
dem Viereck ist eine Farbe“ beginnt, dann darf das natürlich nicht der Satz der Grammatik
über die ”Färbigkeit“ des Raumes sein.
Was meint man, wenn man sagt ”der Raum ist färbig“? (Und, eine sehr interessante Frage:
welcher Art ist diese Frage?) Nun, man sieht etwa zur Bestätigung herum und blickt auf die
verschiedenen Farben um sich her und möchte etwa sagen: wohin ich schaue, ist eine Farbe.
Oder: Es ist doch alles färbig, alles sozusagen angestrichen. Man denkt sich hier die Farben
im Gegensatz zu einer Art (von) Farblosigkeit, die aber bei näherem Zusehen wieder zur
Farbe wird. Wenn man übrigens zur Bestätigung sich umsieht, so schaut man vor allem auf
ruhige und einfärbige Teile des Raumes und lieber nicht auf unruhige,19
unklar gefärbte
(fließendes Wasser, Schatten, etc.). Muß man sich dann gestehen, daß man eben alles
Farbe nennt, was man sieht, so will man es nun als eine Eigenschaft des Raumes an und für
sich (nicht mehr der Raumteile) aussagen, daß er färbig sei. Das heißt aber, vom Schachspiel
zu sagen, daß es das Schachspiel sei und es kann nun nur auf eine Beschreibung des
Spiels hinauslaufen. Und nun kommen wir zu einer Beschreibung der räumlichen Sätze;
aber ohne (eine) Begründung, und als müßte man sie mit einer andern Wirklichkeit in
Übereinstimmung bringen.
Zur Bestätigung des Satzes ”der Gesichtsraum ist färbig“ sieht man sich (etwa) um und
sagt: das hier ist schwarz, und schwarz ist eine Farbe; das ist weiß, und weiß ist eine Farbe;
u.s.w. ”Schwarz ist eine Farbe“ aber faßt man so auf, wie ”Eisen ist ein Metall“20
(oder
vielleicht besser ”Gips ist eine Schwefelverbindung“).
Mache ich es sinnlos zu sagen, ein Teil des Gesichtsraumes habe keine Farbe, so
wird die (Frage nach der) Analyse der Angabe der Zahl der Farben in einem Teil des
Gesichtsraumes ganz ähnlich der, der Angabe der Zahl der Teile eines Vierecks, etwa, das
ich durch Striche in begrenzte Flächenteile teile.
Auch hier kann ich es als sinnlos ansehen, zu sagen, das Viereck ”bestehe aus 0 Teilen“.
Man kann daher nicht sagen, es bestehe ”aus einem oder mehreren Teilen“, oder es ”habe
mindestens einen Teil“. Denken wir uns den speziellen Fall eines Vierecks, das durch
parallele Striche geteilt ist. Daß dieser Fall sehr speziell ist, macht (uns) nichts, denn wir
halten ein Spiel nicht für weniger bemerkenswert, weil es nur eine sehr beschränkte
Anwendung hat. 21
Ich kann hier die Teile entweder so zählen, wie es
gewöhnlich geschieht, und dann heißt es nichts, zu sagen, es seien 0
Teile vorhanden. Ich könnte aber auch eine Zählung denken, die
den ersten Teil sozusagen als selbstverständlich ansieht und ihn nicht zählt oder als 0, und
die nur die Teile hinzuzählt, die hinzugeteilt wurden. Anderseits könnte man sich ein
Herkommen denken, nach dem etwa Soldaten in Reih und Glied immer mit der Anzahl
von Soldaten gezählt werden, welche über einen Soldaten angetreten sind (etwa, indem die
Anzahl der möglichen Kombinationen des Flügelmanns und eines andern Soldaten der
Reihe angegeben werden soll). Aber auch ein Herkommen könnte existieren, wonach die
Anzahl der Soldaten immer um 1 größer als die wirkliche angegeben wird. Das wäre etwa
ursprünglich geschehen, um einen bestimmten Vorgesetzten über die wirkliche Zahl zu
täuschen, dann aber habe es sich als Zählweise für Soldaten eingebürgert. (Akademisches
17 (V): in welchem dieser Satz Unsinn ist.
18 (V): das,
19 (V): bewegte,
20 (V): Metall“.
21 (F): MS 113, S. 10v.
574
575
395 Kardinalzahlenarten.
0 1 2 3 4
1 2 3 4 5
0
1
WBTC108-115 30/05/2005 16:20 Page 790
rule that describes the form of propositions about visual space, for instance. If you now begin
a statement of the number of colours in the rectangle with the proposition “There is one
colour in the rectangle”, then of course that can’t be a grammatical proposition about the
“colouredness” of space.
What do we mean when we say “space is coloured”? (And, a very interesting question:
what kind of a question is this?) Well, perhaps you look around for confirmation and look
at the different colours around you and feel the inclination to say, for instance: Wherever I
look there’s a colour. Or: Everything is coloured, everything painted, as it were. Here you
are imagining colours in contrast to a kind (of) colourlessness, which on closer inspection,
however, itself becomes a colour. Incidentally, when you look around for confirmation you
look first and foremost at static and monochromatic parts of space, and you prefer not to
look at busy,11
unclearly coloured parts (flowing water, shadows, etc.). If you then have to
admit to yourself that you simply call everything that you see colour, that’s when you want
to declare being coloured an inherent property of space (and no longer only of the parts of
space). But that amounts to the same as saying of chess that it is chess; and then it can only
boil down to a description of the game. And now we arrive at a description of spatial propositions;
but without (any) reasons for them, and as if we had to bring them into agreement
with another reality.
In order to confirm the proposition “visual space is coloured” one might look around and
say: This is black, and black is a colour; that is white, and white is a colour, etc. But “black
is a colour” is understood like “iron is a metal”12
(or perhaps better, “gypsum is a sulphur
compound”).
If I make it senseless to say that a part of visual space has no colour, then (asking for) the
analysis of a statement of the number of colours in a part of visual space becomes very much
like asking for the analysis of a statement of the number of parts of a rectangle, for instance,
that I mark off into areas with lines.
Here too I can regard it as senseless to say that the rectangle “consists of 0 parts”. Therefore
one cannot say that it consists “of one or more parts”, or that it “has at least one part”. Let’s
imagine the special case of a rectangle divided by parallel lines. It doesn’t matter to us that
this case is quite special, for we don’t regard a game as less remarkable just because it has
only a very limited application. 13
Here I can count the parts in the usual
way, and then it is meaningless to say there are 0 parts there. But I
could also imagine a way of counting which regards the first part as a
matter of course, so to speak, and doesn’t count it or counts it as 0, and which counts only
those parts that were added on. On the other hand, one could imagine a custom according
to which, say, soldiers in rank and file are always counted by giving the number of soldiers
who have fallen in over and above the first soldier (perhaps because we’re supposed to give
the number of possible combinations of the flank-man with another soldier of the rank).
But there could also be a custom of always giving the number of soldiers as 1 greater than
the real one. Perhaps this happened originally in order to deceive a particular officer about
the real number, but later came into general use as a way of counting soldiers. (The
11 (V): moving,
12 (V): metal”.
13 (F): MS 113, p. 10v.
Kinds of Cardinal Numbers. 395e
0 1 2 3 4
1 2 3 4 5
0
1
WBTC108-115 30/05/2005 16:20 Page 791
Viertel.) Die Anzahl der verschiedenen Farben in einer Fläche könnte22
auch23
durch die Anzahl
der möglichen Kombinationen zu zwei Gliedern angegeben werden. Und dann kämen für
diese Anzahl nur die Zahlen in Betracht und es wäre dann sinnlos, von 2 oder
4 Farben in einer Fläche zu reden, wie jetzt von oder i Farben. Ich will sagen, daß
nicht die Kardinalzahlen wesentlich primär und die – nennen wir’s – Kombinationszahlen
24
etc. sekundär sind. Man könnte auch eine Arithmetik der
Kombinationszahlen konstruieren und diese wäre in sich so
geschlossen, wie die Arithmetik der Kardinalzahlen. Aber ebenso natürlich kann es eine
Arithmetik der geraden Zahlen oder der Zahlen 1, 3, 4, 5, 6, 7 . . . geben. Es ist natürlich
das Dezimalsystem zur Schreibung dieser Zahlenarten ungeeignet.
Denken wir uns eine Rechenmaschine, die, anstatt mit Kugeln, mit Farben in einem Streifen
rechnet. Und während wir jetzt auf unserm Abacus25
mit Kugeln, oder den Fingern, die
Farben in einem Streifen zählen, so würden wir dann die Kugeln auf einer Stange, oder
die Finger an unserer Hand, mit Farben in einem Streifen zählen. Wie aber müßte diese
Farbenrechenmaschine konstruiert sein, um funktionieren zu können? Wir brauchten ein
Zeichen dafür, daß keine Kugeln an der Stange sitzen. Man muß sich den Abacus als ein
Gebrauchsinstrument denken und als Mittel der Sprache. Und, so, wie man etwa 5 durch
die fünf Finger einer Hand darstellen kann (man denke an eine Gebärdensprache), so würde
man es durch den Streifen mit 5 Farben darstellen. Aber für die 0 brauche ich ein Zeichen,
sonst habe ich die nötige Multiplizität nicht. Nun, da kann ich entweder die Bestimmung
treffen, daß die Fläche26
schwarz die 0 bezeichnen soll (dies ist natürlich willkürlich und die
einfärbige rote Fläche täte es ebensogut); oder aber die einfärbige Fläche soll 0 bezeichnen,
die zweifärbige 1, etc. Es ist ganz gleichgültig, welche Bezeichnungsweise ich wähle. Und
man sieht hier, wie sich die Mannigfaltigkeit der Kugeln auf die Mannigfaltigkeit der Farben
in einer Fläche projiziert.
27
Es hat keinen Sinn, von einem schwarzen Zweieck im weißen
Kreis zu reden; und dieser Fall ist analog dem: es ist sinnlos zu
sagen, das Viereck bestehe aus 0 Teilen (keinem Teil).
Hier haben wir etwas, wie eine untere Grenze des Zählens, noch ehe wir die Eins erreichen.
28
Ist Teile Zählen in I das Gleiche, wie
Punkte Zählen in IV? Und worin besteht der
Unterschied? Man kann das Zählen der Teile in
I auffassen als ein Zählen von Vierecken. Dann
kann man aber auch sagen ”in dieser Zeile ist kein
Viereck“; und dann zählt man nicht Teile. Es beunruhigt uns die Analogie zwischen dem
Zählen der Punkte und der Teile, und das Versagen29
dieser Analogie.
Darin, die ungeteilte Fläche als ”Eins“ zu zählen, ist etwas Seltsames; dagegen finden wir
keine Schwierigkeit darin, die einmal geteilte als Bild der 2 zu sehen. Man möchte hier viel
lieber zählen ”0, 2, 3, etc.“. Und dies entspricht der Satzreihe:30
”das Viereck ist ungeteilt“,
”das Viereck ist in 2 Teile geteilt“, etc.
2
n n( )⋅ − 1
2
22 (O): könne
23 (V): man
24 (F): MS 113, S. 11v.
25 (O): Abacuus
26 (V): Farbe
27 (F): MS 113, S. 37r.
28 (F): MS 113, S. 36v–37r.
29 (V): Auslassen
30 (V): Zahlenreihe:
576
577
396 Kardinalzahlenarten.
3 6 101
0
1
2
3
4
I II III IV V
WBTC108-115 30/05/2005 16:20 Page 792
academic quarter-hour). The number of different colours on a surface could also be given
by the number of their possible combinations in pairs. And in that case the only numbers
that would be considered would be numbers of the form , and then it would be
senseless to talk of 2 or 4 colours on a surface, as it now is to talk of or i colours.
I want to say that it is not the case that the cardinal numbers are essentially primary
and that what we might call the combination numbers –
etc. – 14
are secondary. We could also construct an arithmetic of
the combination numbers and it would be just as self-contained as the arithmetic of the
cardinal numbers. But just as naturally there can be an arithmetic of the even numbers,
or of the numbers 1, 3, 4, 5, 6, 7. . . . Of course the decimal system is unsuited for
writing these kinds of numbers.
Let’s imagine a calculating machine that calculates with colours on a strip of paper instead
of with beads. And whereas we now use the beads on our abacus or our fingers to count the
colours on a strip, we would then use the colours on a strip to count the beads on a bar or
the fingers on our hand. But how would this colour-calculating machine have to be constructed
in order to work? We would need a sign for there being no beads on the bar. We
must imagine the abacus as a practical tool and as an instrument of language. And just as
we can represent, say, 5 by the five fingers of a hand (think of a gesture-language) so we
would then represent it by a strip with five colours. But I need a sign for the 0. Otherwise
I do not have the necessary multiplicity. Well, I can either stipulate that a black surface15
is
to denote the 0 (this is of course arbitrary, and a monochromatic red surface would do just
as well); or that any one-coloured surface is to denote 0, a two-coloured surface 1, etc. It is
completely immaterial which method of denotation I choose. And we see here how the
multiplicity of the beads is projected onto the multiplicity of the colours on a surface.
16
It makes no sense to speak of a black figure with two angles
in a white circle; and this is analogous to its being senseless to
say that the rectangle consists of 0 parts (no part).
Here we have something like a lower limit of counting, even before we reach the number
one.
17
Is counting parts in I the same as counting dots
in IV? And what does the difference consist in?
We can regard counting the parts in I as counting
rectangles. But in that case one can also say:
“In this row there is no rectangle”; and then one
isn’t counting parts. We are disturbed both by the analogy between counting the dots and
counting the parts, and by the failure of this analogy.
There is something odd in counting the undivided surface as “one”; on the other hand
we find no difficulty in seeing a surface that has been divided once as a picture of 2. Here
we would much prefer to count “0, 2, 3” , etc. And this corresponds to the series of pro-
positions18
“The rectangle is undivided”, “The rectangle is divided into 2 parts”, etc.
2
n n( )⋅ − 1
2
14 (F): MS 113, p. 11v.
15 (V): colour
16 (F): MS 113, p. 37r.
17 (F): MS 113, pp. 36v–37r.
18 (V): numbers
Kinds of Cardinal Numbers. 396e
0
1
2
3
4
I II III IV V
3 6 101
WBTC108-115 30/05/2005 16:20 Page 793
Das Natürlichste ist, die Reihe der Schemata 1 aufzufassen als
A
A B
A B C
A B C D
etc.
Und hier kann man nun das31
erste Schema mit ”0“ bezeichnen, das zweite mit ”1“, das
dritte aber etwa mit ”3“, wenn man an alle möglichen Unterschiede denkt, und das vierte
mit ”6“. Oder man nennt das dritte Schema ”2“ (wenn man sich bloß um eine Anordnung
kümmert) und das vierte ”3“.
32
Man kann die Teiligkeit des Vierecks beschreiben, indem man sagt: es ist
in 533
Teile geteilt, oder: es sind 4 Teile davon abgetrennt worden, oder: es hat
das Teilungsschema ABCDE, oder: man kommt durch alle Teile, indem man
4 Grenzen passiert, oder: das Viereck ist geteilt (d.h. in 2 Teile), der eine Teil wieder geteilt
und beide Teile dieser Teilung geteilt, – etc.
Ich will zeigen, daß nicht nur eine Methode besteht, die Teiligkeit zu beschreiben.
Man wird sich aber vielleicht auch enthalten, den Unterschied überhaupt mit einer Zahl
zu bezeichnen, sondern sich ganz an die Schemata A, AB, ABC, etc. halten. Oder es auch
so beschreiben: 1, 12, 123, etc., oder, was auf das Gleiche hinauskommt: 0, 01, 012, etc.
Diese kann man sehr wohl auch Zahlzeichen nennen.
Die Schemata: A, AB, ABC, etc.; 1, 12, 123, etc.; |, ||, |||, etc.; 34
etc.;
0, 1, 2, 3, etc.; 1, 2, 3, etc.; 1, 12, 121323, etc.; etc. – sind alle gleich fundamental.
Man wundert sich nun darüber, daß das Zahlenschema, mit welchem man Soldaten in
einer Kaserne zählt, nicht auch für die Teile eines Vierecks gelten soll. Aber das Schema
der Soldaten in der Kaserne ist35
das der Teile des Vierecks36
.
Keines ist im Vergleich zum andern primär.
Ich kann die Reihe der Teilungsschemata sowohl mit der Reihe 1, 2, 3, etc. als auch mit
der Reihe 0, 1, 2, 3, etc. vergleichen.
Zähle ich die Teile, so gibt es in meiner Zahlenreihe keine 0, denn die Reihe
A
A B
A B C
etc.
fängt mit einem Buchstaben an, während die Reihe 37
etc. nicht mit einem Punkt
anfängt. Ich kann dagegen auch mit dieser Reihe alle Tatsachen der Teilung darstellen, nur
”zähle ich dann nicht die Teile“.
Unrichtig ausgedrückt, aber so, wie man es zunächst ausdrücken würde, lautet das Problem:
”warum kann man sagen ’es gibt 2 Farben auf dieser Fläche‘ und nicht ’es gibt eine Farbe
auf dieser Fläche‘?“ Oder: wie muß ich die grammatische Regel ausdrücken, daß ich nicht
mehr versucht bin Unsinniges zu sagen, und daß sie mir selbstverständlich ist? Wo liegt der
, , ,· ··
, ,· · ·, , ,
, , , ,
31 (V): die
32 (F): MS 113, S. 38v.
33 (V): fünf
34 (F): MS 113, S. 39r.
35 (F): MS 113, S. 39v.
36 (F): MS 113, S. 39v.
37 (F): MS 113, S. 40v.
578
579
397 Kardinalzahlenarten.
WBTC108-115 30/05/2005 16:20 Page 794
The most natural thing is to conceive of the series of schemata 1 as
A
A B
A B C
A B C D
etc.
And here we can call the first schema “0”, the second “1”, but the third say “3”, if we think
of all possible differences, and the fourth “6”. Or we call the third schema “2” (if we are
concerned simply with one arrangement) and the fourth “3”.
19
We can describe the way a rectangle is divided by saying: It is divided into
520
parts, or: Four parts have been cut out of it, or: Its division-schema is
ABCDE, or: You can traverse all the parts by crossing 4 boundaries or: The
rectangle is divided (i.e. into 2 parts), one part is subdivided, and both parts of this division
are subdivided – etc.
I want to show that there isn’t only one method of describing the way it’s divided.
But perhaps we’ll refrain altogether from using a number to denote the distinction and
keep instead just to the schemata A, AB, ABC, etc.; or we’ll describe it like this: 1, 12, 123,
etc., or, what amounts to the same, 0, 01, 012, etc.
We can perfectly well call these numerals too.
The schemata A, AB, ABC, etc.; 1, 12, 123, etc.; |, ||, |||, etc.; 21
etc.;
0, 1, 2, 3, etc.; 1, 2, 3, etc.; 1, 12, 121323, etc.; etc. – are all equally fundamental.
Now we are surprised that the number-schema by which we count soldiers in a barracks
doesn’t also hold for the parts of a rectangle. But the schema for the soldiers in the barracks
is 22
etc., the one for the parts of the rectangle is .23
Neither is
primary in comparison with the other.
I can compare the series of division-schemata with the series 1, 2, 3, etc. as well as with
the series 0, 1, 2, 3, etc.
If I count the parts, then there is no 0 in my number series because the series
A
A B
A B C
etc.
begins with one letter whereas the series 24
etc. doesn’t begin with one dot. On the
other hand, I can represent all the facts about the division by this series too, only “Then
I’m not counting the parts”.
Expressed incorrectly, but as one would express it at first, the problem is: “Why can one
say ‘There are 2 colours on this surface’ but not ‘There is one colour on this surface’?” Or:
How must I express the grammatical rule so that I’m no longer tempted to talk nonsense
and so that the rule is a matter of course for me? Where is the false thought, the false
, , ,· ··
, ,· · ·, , ,
, , , ,
19 (F): MS 113, p. 38v.
20 (V): five
21 (F): MS 113, p. 39r.
22 (F): MS 113, p. 39v.
23 (F): MS 113, p. 39v.
24 (F): MS 113, p. 40v.
Kinds of Cardinal Numbers. 397e
WBTC108-115 30/05/2005 16:20 Page 795
falsche Gedanke, die falsche Analogie, durch die ich verführt werde, die Sprache unrichtig
zu gebrauchen? Wie muß ich die Grammatik darstellen, daß diese Versuchung wegfällt? Ich
glaube, daß die Darstellung durch die Reihen
A
A B und
A B C
u.s.w. u.s.w.
38
die Unklarheit hebt.
Es kommt alles darauf an, ob ich mit einer Zahlenreihe zähle, die mit 0 anfängt, oder mit
einer, die mit 1 anfängt.
So ist es auch, wenn ich die Längen von Stäben, oder die Größen von Hüten zähle.
Wenn ich mit Zählstrichen zähle, so könnte ich sie dann so schreiben: , 39
um
zu zeigen, daß es auf den Richtungsunterschied ankommt und der einfache Strich der 0
entspricht (d.h. der Anfang ist).
Es hat hier übrigens mit den Zahlzeichen (1), ((1) + 1), etc. eine gewisse Schwierigkeit:
Nämlich die, daß wir sie nach einer gewissen Länge nicht mehr unterscheiden können,
ohne die Striche zu zählen, also ohne die Zeichen in andere zu übersetzen. ”||||||||||“
und ”|||||||||||“ kann man nicht in dem Sinne unterscheiden – sie sind also nicht in
demselben Sinn verschiedene Zeichen – wie ”10“ und ”11“. Übrigens würde dasselbe
natürlich auch im Dezimalsystem geschehen (denken wir an die Zahlen 1111111111 und
11111111111), aber das ist nicht ohne Bedeutung. –
Denken wir uns den Fall, es gäbe uns Einer eine Rechenaufgabe in der Strichnotation,
etwa: ||||||||||| + |||||||||| und während wir rechneten machte er sich den Spaß,
Striche, ohne daß wir es bemerkten, wegzuwischen und dazuzugeben. Er würde uns dann
immer sagen ”die Rechnung stimmt ja nicht“ und wir würden sie immer von Neuem
durchlaufen, stets zum Narren gehalten. – Ja, streng genommen, ohne den Begriff eines
Kriteriums der Richtigkeit der Rechnung. –
Hier könnte man nun Fragen aufwerfen, wie die: Ist es nun nur sehr wahrscheinlich,
daß 464 + 272 = 736 ist? Und ist also nicht auch 2 + 3 = 5 nur sehr wahrscheinlich? Und
wo40
ist denn die objektive Wahrheit, der sich diese Wahrscheinlichkeit nähert? D.h.,
wie bekommen wir denn einen Begriff davon, daß 2 + 3 eine gewisse Zahl wirklich ist,
abgesehen von dem, was sie uns zu sein scheint? –
Wenn man nämlich fragen würde: was ist das Kriterium in der Strichnotation, daß
wir zweimal das gleiche Zahlzeichen vor uns haben? – Die Antwort könnte sein: ”wenn es
beidemale gleich aussieht“, oder ”wenn es beidemale die gleiche Anzahl von Strichen
enthält“. Oder soll es heißen: wenn eine eins-zu-eins Zuordnung etc. möglich ist?
Wie kann ich wissen, daß |||||||| und |||||||| dasselbe Zeichen sind? Es genügt doch
nicht, daß sie ähnlich ausschauen. Denn es ist nicht die ungefähre Gleichheit der Gestalt,
was die Identität der Zeichen ausmachen darf, sondern gerade eben die Zahlengleichheit.
|Das Problem der Unterscheidung von 1 + 1 + 1 + 1 + 1 + 1 + 1 und 1 + 1 + 1 + 1 + 1
+ 1 + 1 + 1 ist viel fundamentaler,41
als es auf den ersten Blick scheint. Es handelt sich um
den Unterschied zwischen physikalischer und visueller Zahl.|
, , ,
38 (F): MS 113, S. 41v.
39 (F): MS 113, S. 42r.
40 (V): was
41 (V): wichtiger,
580
398 Kardinalzahlenarten.
·
··
WBTC108-115 30/05/2005 16:20 Page 796
analogy by which I am misled into using language incorrectly? How must I represent the
grammar so that this temptation falls by the wayside? I believe that representing it with the
series
A
A B and
A B C
and so on and so on
25
removes the unclarity.
Everything depends on whether I count using a number series that begins with 0, or one
that begins with 1.
That’s also the way it is if I’m counting the lengths of sticks or the sizes of hats.
If I counted with tally lines, I could write them like this: 26
in order to show
that what matters is the difference in direction and that a simple line corresponds to 0 (i.e.
is the beginning).
Here, incidentally, there is a certain difficulty with the numerals (1), ( (1) + 1), etc.: beyond
a certain length we can no longer distinguish them without counting the lines, i.e. without
translating the signs into different ones. “||||||||||” and “|||||||||||” can’t be distinguished
in the same sense as “10” and “11”, and so they aren’t different signs in the same
sense. Incidentally, the same thing could of course also happen in the decimal system (think
of the numbers 1111111111 and 11111111111), and that is not without significance. –
Let’s imagine that someone gave us a calculation problem to do in a stroke-notation, say
|||||||||| + |||||||||||, and while we were calculating he’d amuse himself by removing
and adding strokes without our noticing. He’d keep on telling us: “But your calculation
isn’t right”, and we’d keep going through it again, fooled every time. – Indeed, strictly speaking,
without any concept of a criterion for the correctness of the calculation. –
Here one could raise questions like: Is it only very probable that 464 + 272 = 736? And
therefore isn’t 2 + 3 = 5 also only very probable? And where27
is the objective truth which
this probability approaches? That is, how do we get a concept of 2 + 3 really being a certain
number, apart from what it seems to us to be? –
For if it were asked: What is the criterion in the stroke-notation for our having the
same numeral in front of us twice? – the answer could be: “If it looks the same both times”
or “If it contains the same number of strokes both times”. Or should it be: If a one-to-one
correlation, etc. is possible?
How can I know that |||||||| and |||||||| are the same sign? After all, it is not enough
that they look similar. For having roughly the same gestalt mustn’t be what constitutes the
identity of the signs; rather it has to be precisely the sameness of their number.
|The problem of the distinction between 1 + 1 + 1 + 1 + 1 + 1 + 1 and 1 + 1 + 1 + 1 +
1 + 1 + 1 + 1 is much more fundamental28
than appears at first sight. It is a matter of the
distinction between physical and visual number.|
, , ,
25 (F): MS 113, p. 41v.
26 (F): MS 113, p. 42r.
27 (V): what
28 (V): important
Kinds of Cardinal Numbers. 398e
·
··
WBTC108-115 30/05/2005 16:20 Page 797
116
2 + 2 = 4.
Die Kardinalzahl ist eine interne Eigenschaft einer Liste.
Hat die Anzahl wesentlich etwas mit einem Begriff zu tun? Ich glaube, das kommt darauf
hinaus, zu fragen, ob es einen Sinn hat, von einer Anzahl von Gegenständen zu reden, die
nicht unter einen Begriff gebracht sind. Hat es z.B. Sinn zu sagen ”a, b und c sind drei
Gegenstände“? – Es ist allerdings ein Gefühl vorhanden, das uns sagt: Wozu von Begriffen
reden, die Zahl hängt ja nur vom Umfang des Begriffes ab, und wenn der einmal bestimmt
ist, so kann der Begriff sozusagen abtreten. Der Begriff ist1
nur ein Hilfsmittel,2
um einen
Umfang zu bestimmen, der Umfang aber ist selbständig und in seinem Wesen unabhängig
vom Begriff; denn es kommt ja auch nicht darauf an, durch welchen Begriff wir den Umfang
bestimmt haben. Das ist das Argument für die extensive Auffassung. Dagegen kann man
zuerst sagen: Wenn der Begriff wirklich nur ein Hilfsmittel ist, um zum Umfang zu gelangen,
dann hat der Begriff in der Arithmetik nichts zu suchen; dann muß man eben die Klasse
gänzlich von dem zufällig mit ihr verknüpften Begriff scheiden. Im3
entgegengesetzten Fall
aber ist der vom Begriff unabhängige Umfang nur eine Chimaire und dann ist es besser,
von ihm überhaupt nicht zu reden, sondern nur vom Begriff.
Das Zeichen für den Umfang eines Begriffes ist eine Liste. Man könnte – beiläufig – sagen:
die Zahl4
ist die externe Eigenschaft eines Begriffs und die interne seines Umfangs (der Liste
der Gegenstände, die unter ihn fallen). Die Anzahl ist das Schema eines Begriffsumfangs.
D.h.: die Zahlangabe ist, wie Frege sagte, die Aussage über einen Begriff (ein Prädikat). Sie
bezieht sich nicht auf einen Begriffsumfang, d.i. auf eine Liste, die etwa der Umfang eines
Begriffes sein kann. Aber die Zahlangabe über einen Begriff ist ähnlich dem Satz, welcher
aussagt, daß eine bestimmte Liste der Umfang dieses Begriffs sei. Von so einer Liste wird
Gebrauch gemacht, wenn ich sage: ”a, b, c, d fallen unter den Begriff F(x)“. ”a, b, c, d“ ist
die Liste. Natürlich sagt der Satz nichts anderes, als Fa & Fb & Fc & Fd; aber er zeigt, mit
Hilfe der Liste geschrieben, seine Verwandtschaft mit ”(∃x, y, z, u).Fx & Fy & Fz & Fu“,
welches wir kurz ”(∃||||x).F(x)“ schreiben können.
Die Arithmetik hat es mit dem Schema |||| zu tun. – Aber redet denn die Arithmetik
von Strichen, die ich mit Bleistift auf Papier mache? – Die Arithmetik redet nicht von den
Strichen, sie operiert mit ihnen.
Die Zahlangabe enthält nicht immer eine Verallgemeinerung oder Unbestimmtheit:
”Die Strecke AB ist in zwei (3, 4, etc.5
) gleiche Teile geteilt“.
Wenn man wissen will, was ”2 + 2 = 4“ heißt, muß man fragen, wie wir es (erhalten),
es ausrechnen. Wir betrachten dann den Vorgang der Berechnung als das Wesentliche, und
1 (V): ist ein
2 (V): nur eine Methode,
3 (V): Im g
4 (V): Anzahl
5 (O): et.
581
582
583
399
WBTC116-120 30/05/2005 16:14 Page 798
116
2 + 2 = 4.
A cardinal number is an internal property of a list.
Is quantity inherently linked to concepts? I believe this amounts to asking whether it makes
sense to speak about a quantity of things that haven’t been brought under a concept. Does
it, for example, make sense to say “a, b and c are three objects”? – Admittedly we have a
feeling that says to us: Why talk about concepts, the number depends only on the extension
of the concept, and once that has been determined the concept can make its exit, as it were.
The concept is only an aid1
for determining an extension, but the extension is autonomous
and is essentially independent of the concept; for it’s quite immaterial which concept we
have used to determine the extension. That is the argument for the extensional viewpoint.
The first objection to it is: if a concept really is only an aid for getting to an extension, then
there’s no place for concepts in arithmetic; in that case we must simply divorce a class
completely from the concept that happens to be linked to it. But if the opposite is true,
then an extension independent of the concept is just a chimera, and in that case it’s better
not to speak of it at all, but only of the concept.
The sign for the extension of a concept is a list. We could say, as an approximation, that
number2
is an external property of a concept and an internal property of its extension (of
the list of objects that fall under it). A number is a schema for the extension of a concept.
That is: as Frege said, a statement of number is a statement about a concept (a predicate).
It doesn’t refer to an extension of a concept, i.e. to a list that may be something like the
extension of a concept. But a number-statement about a concept is similar to a proposition
saying that a determinate list is the extension of this concept. I use such a list when I say
“a, b, c, d fall under the concept F(x)”: “a, b, c, d” is the list. Of course this proposition
says nothing more than Fa & Fb & Fc & Fd; but when it’s written with the help of the list,
the proposition shows its relationship to “(∃x, y, z, u).Fx & Fy & Fz & Fu”, which we can
abbreviate as “(∃||||x).F(x)”.
Arithmetic is concerned with the schema ||||. – But does arithmetic talk about strokes
that I draw with a pencil on paper? – Arithmetic doesn’t talk about the strokes, it operates
with them.
A statement of number doesn’t always contain a generalization or indeterminacy: “The
line AB is divided into two (3, 4, etc.) equal parts”.
If you want to know what 2 + 2 = 4 means, you have to ask how we arrive at it, how we
calculate it. In that case we view the process of calculation as what’s essential; and this is
1 (V): is only a method 2 (V): quantity
399e
WBTC116-120 30/05/2005 16:14 Page 799
diese Betrachtungsweise ist die des gewöhnlichen Lebens, wenigstens, was die Zahlen
anbelangt, für die wir einer6
Ausrechnung bedürfen. Wir dürfen uns ja nicht schämen, die
Zahlen7
und Rechnungen so aufzufassen, wie sie die alltägliche Arithmetik jedes Kaufmanns
auffaßt. Wir rechnen dann 2 + 2 = 4 und überhaupt die Regeln des kleinen Einmaleins gar
nicht aus, sondern nehmen sie – sozusagen als Axiome – an und rechnen nur mit ihrer Hilfe.
Wir könnten aber natürlich auch 2 + 2 = 4 ausrechnen und die Kinder tun es auch durch
Abzählen. Gegeben die Ziffernfolge 1 2 3 4 5 6, ist die Ausrechnung:
1 2 1 2
1 2 3 4 .
Definitionen zur Abkürzung:8
6 (O): eine
7 (V): Ziffern
8 (F): MS 113, S. 5r.
9 (O): (L|| + ||| x).φx & ψx. (E): Wir
haben die Formel auf Grund von MS 153b,
S. 44v korrigiert.
10 (V): noch
11 (O): machen,
12 (V): Keineswegs!
13 (V): hat mir
584
585
400 2 + 2 = 4.
Hat man damit den arithmetischen Satz 2 + 3 = 5 demonstriert? Natürlich nicht. Man
hat auch nicht gezeigt, daß (L||x)φx & (L|||x )ψx & Ind. :⊃: (L|| + |||x).φx 1 ψx9
tautologisch ist, denn von einer Summe ”|| + |||“ war in unsern Definitionen ja10
gar keine
Rede. (Ich werde die Tautologie zur Abkürzung in der Form ”L|| & L||| ⊃ L|||||“
schreiben.) Wenn nun die Frage ist, welche Anzahl von Strichen rechts von ”⊃“ bei gegebener
linker Seite das Ganze zu einer Tautologie macht,11
so kann man diese Zahl finden, man
kann auch finden, daß sie im vorigen Fall || + ||| ist, aber genau so gut, daß sie | + ||||
oder | + ||| + | ist, denn sie ist dies alles. Man kann aber auch eine Induktion finden,
die zeigt, daß – algebraisch ausgedrückt – Ln & Lm .⊃. L(n + m) tautologisch wird. Dann
habe ich z.B. ein Recht, L17 & L28 .⊃. L(17 + 28) als Tautologie anzusehen. Aber ist nun
dadurch die Gleichung 17 + 28 = 45 gegeben? Durchaus nicht!12
Dies muß ich mir vielmehr
nun erst ausrechnen. Es hat nun auch Sinn, nach dieser allgemeinen Regel L2 & L3 ⊃ L5
als Tautologie hinzuschreiben; wenn ich, (sozusagen), noch nicht weiß, was 2 + 3 ergeben
wird; denn 2 + 3 hat13
nur sofern Sinn, als es noch ausgerechnet werden muß.
Daher hat die Gleichung || + ||| = ||||| nur dann einen Witz, wenn das Zeichen
”|||||“ so wiedererkannt wird, wie das Zeichen ”5“; nämlich unabhängig von der
Gleichung.
Mein Standpunkt unterscheidet sich dadurch von dem der Leute, die heute über die
Grundlagen der Arithmetik schreiben, daß ich es nicht nötig habe, einen bestimmten Kalkül,
z.B. den des Dezimalsystems, zu verachten. Einer ist für mich so gut wie der andere. Einen
besondern Kalkül gering zu achten ist so, als wollte man Schach spielen ohne wirkliche Figuren,
weil das zu wenig abstrakt, zu speziell sei. Soweit es auf die Figuren nicht ankommt,
(∃x).φx .&. ~(∃x, y).φx & φy Def. (εx).φx
(∃x, y).φx & φy .&. ~(∃x, y, z)φx & φy & φz Def. (εx, y).φx & φy, u.s.w.
(εx).φx Def. (ε|x)φx
(εx, y).φx & φy) .=. (ε||x)φx .=. (ε2x)φx, u.s.w.
Man kann zeigen daß
(ε||x)φx & (ε|||x)ψx & ~(∃x).φx & ψx .⊃. (ε|||||x)φx 1 ψx eine Tautologie ist.
1
4
2
4
3
Ind.
WBTC116-120 30/05/2005 16:14 Page 800
4
Does that demonstrate the arithmetical proposition 2 + 3 = 5? Of course not. It also
doesn’t show that (L||x)φx & (L|||x)ψx & Ind. :⊃: (L|| + |||x).φx 1 ψx)5
is tautologous,
because nothing at all was said6
in our definitions about a sum “|| + |||”. (In
order to abbreviate I will write the tautology in the form “L||& L||| ⊃ L|||||”.) Now
if the question is – given the left-hand side – what number of strokes to the right of “⊃”
makes the whole thing a tautology, then we can find this number, and can also discover
that in the case above it is || + |||; but we can equally well discover that it is |+ ||||,
or | + ||| + |, for it is all of these. We can also find an inductive proof, however, that
shows that the algebraic expression Ln & Lm .⊃. L(n + m) is tautologous. In that case I
have a right to regard a proposition like L17 & L28 .⊃. L(17 + 28) as a tautology. But does
that give us the equation 17 + 28 = 45? Certainly not!7
On the contrary, I still have to
work this out. In accordance with this general rule, it now also makes sense to write
L2 & L3 .⊃. L5 as a tautology if, (as it were), I don’t yet know what 2 + 3 comes out as; for
2 + 3 has sense only in so far as it has still to be worked out.
Therefore the equation || + ||| = ||||| only has a point if the sign “|||||” is
recognized in the same way as the sign “5”, that is, independently of the equation.
The difference between my point of view and that of contemporary writers on the
foundations of arithmetic is that I don’t need to scoff at a particular calculus like that of
the decimal system. For me one calculus is as good as another. To disparage a particular
calculus is like wanting to play chess without real pieces because playing with pieces isn’t
abstract enough – is too particularized. To the extent that the pieces really don’t matter, one
how we look at the matter in ordinary life, at least as far as the numbers we need to figure
out are concerned. After all, we mustn’t be ashamed of regarding numbers3
and calculations
in the same way as every merchant does in his everyday arithmetic. For there we don’t figure
out 2 + 2 = 4, nor the rules of the multiplication tables in general; rather, we accept them
– as axioms, as it were – and only calculate with their help. But of course we could also work
out 2 + 2 = 4, and children in fact do so by counting off. Given the sequence of numbers
1 2 3 4 5 6 the calculation is:
1 2 1 2
1 2 3 4
Abbreviative Definitions:
3 (V): numerals
4 (F): MS 113, p. 5r.
5 (O): (L|| + ||| x).φx & ψx. (E): We have
taken the correct version of this formula from
MS 153b, p. 44v.
6 (V): was yet said
7 (V): By no means!
2 + 2 = 4. 400e
(∃x).φx .&. ~(∃x, y).φx & φy Def. (εx).φx
(∃x, y).φx & φy .&. ~(∃x, y, z).φx & φy & φz Def. (εx, y).φx & φy, etc.
(εx).φx Def. (ε|x).φx
(εx, y).φx & φy) .=. (ε||x).φx .=. (ε2x).φx, etc.
It can be shown that
(ε||x).φx & (ε|||x).ψx & ~(∃x).φx & ψx .⊃. (ε|||||x).φx 1 ψx is a tautology.
1
4
2
4
3
Ind.
WBTC116-120 30/05/2005 16:14 Page 801
sind eben die einen so gut wie die andern. Und soweit die Spiele sich doch voneinander
unterscheiden,14
ist eben ein Spiel so gut, d.h. so interessant, wie das andere. Keines aber
ist sublimer als das andre.
Welches ist der Beweis von L|| & L||| ⊃ L|||||, der der Ausdruck unseres Wissens
ist, daß dies ein richtiger logischer Satz ist?
Er macht offenbar davon Gebrauch, daß man (∃x) . . . als logische Summe behandeln kann.
Wir übersetzen etwa von dem Symbolismus 15
(”wenn in jedem Quadrat ein Stern ist,
so sind zwei im ganzen Rechteck“) in den Russell’schen. Und es ist nicht, als gäben wir mit
der Tautologie in dieser Schreibweise einer Meinung Ausdruck, die uns plausibel erscheint
und (die) der Beweis dann bestätigt; sondern, was uns plausibel erscheint ist, daß dieser
Ausdruck eine Tautologie (ein Gesetz der Logik) ist.
Die Reihe von Sätzen
(∃x): aRx & xRb
(∃x, y): aRx & xRy & yRb
(∃x, y, z): aRx & xRy & yRz & zRb u.s.f.
kann man sehr wohl so ausdrücken:
”es gibt ein Glied zwischen a und b“
”es gibt zwei Glieder zwischen a und b“ u.s.w.
und kann das etwa schreiben16
(∃1x).aRxRb,
(∃2x).aRxRb, etc.
Es ist aber klar, daß zum Verständnis dieser Ausdrücke die obere Erklärung nötig ist, weil
man sonst nach Analogie von (∃2x).fx = (∃x, y).φx & φy glauben könnte, (∃2x).aRxRb sei
gleichbedeutend einem Ausdruck (∃x, y).aRxRb & aRyRb.
Ich könnte natürlich auch statt ”(∃x, y).F(x, y)“ schreiben ”(∃2x, y).F(x, y)“. Aber die
Frage wäre nun: was habe ich dann unter ”(∃3x, y).F(x, y)“ zu verstehen? Aber hier läßt
sich eine Regel geben; und zwar brauchen wir eine, die uns in der Zahlenreihe beliebig
weiterführt. Z.B. die:
(∃3x, y).F(x, y) = (∃x, y, z): F(x, y) & F(x, z) & F(y, z)
(∃4x, y).F(x, y) = (∃x, y, z, u): F(x, y) & F(x, z) & . . .
es folgen die Kombinationen zu zwei Elementen. U.s.f. Es könnte aber auch definiert
werden:
(∃3x, y).F(x, y) = (∃x, y, z): (F(x, y) & F(y, x) & F(x, z) & F(z, x) & F(y, z) & F(z, y)
u.s.f.
”(∃3x).F(x, y)“ entspräche etwa dem Satz der Wortsprache ”F(x, y) wird von 3 Dingen
befriedigt“ und auch dieser Satz bedürfte einer Erklärung um eindeutig zu werden.
Soll ich nun sagen, daß in diesen17
verschiedenen Fällen das Zeichen ”3“ eine verschiedene18
Bedeutung hat? Drückt nicht vielmehr das Zeichen ”3“ das aus, was den verschiedenen
14 (V): Und soweit ein Spiel sich von dem andern
doch unterscheidet,
15 (F): MS 113, S. 6v.
16 (O): Schreiben
17 (V): den
18 (V): andere
586
401 2 + 2 = 4.
WBTC116-120 30/05/2005 16:14 Page 802
set is as good as another. And to the extent that the games really are distinct from each other,8
one game is as good, i.e. as interesting, as the other. But none of them is more sublime than
any other.
Which proof of L|| & L||| ⊃ L||||| expresses our knowledge that this is a correct
logical proposition?
Obviously, one that makes use of the fact that one can treat (∃x) . . . as a logical sum. For
example, we translate from this symbolism 9
(“If there is a star in each square, then
there are two in the whole rectangle”) into the Russellian one. And it isn’t as if we were
using the tautology in this notation to express an idea that appears plausible, and (that) is
then confirmed by the proof; rather, what appears plausible to us is that this expression is
a tautology (a law of logic).
The series of propositions
(∃x): aRx & xRb
(∃x, y): aRx & xRy & yRb
(∃x, y, z): aRx & xRy & yRz & zRb, etc.
can perfectly well be expressed as follows:
“There is one term between a and b.”
“There are two terms between a and b.”, etc.,
and can, for example, be written as
(∃1x).aRxRb,
(∃2x).aRxRb, etc.
But it is clear that in order to understand these expressions we need the explanation above,
because otherwise by analogy with (∃2x).φx = (∃x, y).φx & φy, one might believe that
(∃2x).aRxRb was equivalent to an expression like (∃x, y).aRxRb & aRyRb.
Of course I could also write “(∃2x, y).F(x, y)” instead of “(∃x, y).F(x, y)”. But then the
question would be: What am I to take “(∃3x, y).F(x, y)” as meaning? But here a rule can be
given; specifically, we need one that takes us further in the number series – to wherever we
want to go. For example, this one:
(∃3x, y).F(x, y) = (∃x, y, z):F(x, y) & F(x, z) & F(y, z)
(∃4x, y).F(x, y) = (∃x, y, z, u):F(x, y) & F(x, z) & . . .
followed by the combinations of two elements. And so on. But we could also give the following
definition:
(∃3x, y).F(x, y) = (∃x, y, z):(F(x, y) & F(y, x) & F(x, z) & F(z, x) & F(y, z) & F(z, y)
and so on.
For example, “(∃3x, y).F(x, y)” would correspond to the proposition in word-language
“F(x, y) is satisfied by 3 things”; and this proposition too would need an explanation to become
unambiguous.
Am I now to say that in these10
different cases the sign “3” has a different11
meaning?
Isn’t it rather that the sign “3” expresses what is common to the different interpretations?
8 (V): And to the extent that one game really
differs from another,
9 (F): MS 113, p. 6v.
10 (V): the
11 (V): another
2 + 2 = 4. 401e
WBTC116-120 30/05/2005 16:14 Page 803
Interpretationen gemeinsam ist? Warum hätte ich es sonst gewählt. Es gelten ja auch die
gleichen Regeln von dem Zeichen ”3“ in19
jedem dieser Zusammenhänge.20
Es ist nach wie
vor durch 2 + 1 zu ersetzen; etc. Allerdings aber ist ein Satz nach dem Vorbild von L|| &
L||| ⊃ L||||| nun keine Tautologie. Zwei Menschen, die miteinander in Frieden leben
und drei weitere Menschen, die miteinander in Frieden leben geben nicht fünf Menschen,
die miteinander in Frieden leben. Aber das heißt nicht, daß nun 2 + 3 nicht 5 ist. Vielmehr
läßt sich die Addition nur nicht so anwenden. Denn man könnte sagen: 2 Menschen, die
. . . und 3 Menschen, die . . . und von denen jeder mit jedem der ersten Gruppe in Frieden
lebt = 5 Menschen die. . . .
Mit andern Worten die Zeichen von der Form (∃1x, y).F(x, y), (∃2x, y).F(x, y), etc. haben
die Multiplizität der Kardinalzahlen, wie die Zeichen (∃lx).φx, (∃2x).φx, etc. und wie auch
die Zeichen (L1x).φx, (L2x).φx, etc.
”Es gibt nur 4 rote Dinge, aber die bestehen nicht aus 2 und 2, weil es keine Funktion
gibt, die sie zu je zweien unter einen Hut bringt“. Das hieße, den Satz 2 + 2 = 4 so
auffassen: Wenn auf einer Fläche 4 Kreise zu sehen sind, so haben je 2 von ihnen immer
eine bestimmte Eigentümlichkeit miteinander gemein; sagen wir etwa ein
Zeichen innerhalb des Kreises. 21
(Dann sollen natürlich auch je 322
der Kreise
ein Zeichen gemeinsam haben,23
etc.) Denn, wenn ich überhaupt etwas über
die Wirklichkeit annehme, warum nicht das? Das ”axiom of reducibility“
ist wesentlich von keiner andern Art. In diesem Sinne könnte man sagen, daß
zwar 2 und 2 immer 4 ergeben, aber 4 nicht immer aus 2 und 2 besteht. (Nur durch die
gänzliche Vagueheit und Allgemeinheit des Reduktionsaxioms werden wir zu dem Glauben
verleitet, es handle sich hier24
– wenn überhaupt um einen sinnvollen Satz – um mehr, als
eine willkürliche Annahme, zu der kein Grund vorhanden ist. Drum ist es hier und in allen
ähnlichen Fällen äußerst klärend, diese Allgemeinheit, die die Sache ja doch nicht mathematischer
macht, ganz fallen zu lassen und statt ihrer ganz spezialisierte Annahmen zu machen).
Man möchte sagen: 4 muß nicht immer aus 2 und 2 bestehen, aber es kann, wenn es
wirklich aus Gruppen besteht, aus 2 und 2 wie aus 3 und 1, etc., bestehen; aber nicht aus 2
und 1, oder 3 und 2, etc.; und so bereiten wir eben alles für den Fall vor, daß 4 in Gruppen
zerlegbar ist. Aber dann hat es eben die Arithmetik gar nicht mit der wirklichen Zerlegung
zu tun, sondern nur mit jener Möglichkeit der Zerlegung. Die Behauptung könnte ja auch
die sein, daß, wenn immer ich eine Gruppe von 4 Punkten auf einem Papier sehe, je 2 von
ihnen durch eine Klammer verbunden sind.25
Oder:26
um je 2 solche Gruppen von 2 Punkten sei in der Welt immer ein Kreis gezogen.
Dazu kommt nun, daß, z.B., die Aussage, daß in einem weißen Viereck 2 schwarze Kreise
zu sehen sind, nicht die Form ”(∃x, y).etc.“ hat. Denn, gebe ich den Kreisen Namen, dann
beziehen sich diese Namen gerade auf die Orte der Kreise und ich kann nicht von ihnen
sagen, sie seien entweder in dem einen oder dem andern Viereck. Ich kann wohl sagen: ”in
beiden Vierecken zusammen sind 4 Kreise“, aber das heißt nicht, daß ich von jedem einzeln
19 (V): in di
20 (V): ”3“ in dieser wie // und // in jener
Verwendung.
21 (F): MS 113, S. 62r.
22 (V): 3 Krei
23 (V): Zeichen gemein haben,
24 (V): verleitet, als handle es sich hier
25 (V): Die Behauptung könnte ja auch die sein,
daß von einer Gruppe von 4 Punkten auf
dem Papier immer je 2 durch einen Strich
verbunden sind.
26 (V): sind.
Denken wir gar an die Annahme$
587
588
402 2 + 2 = 4.
1 26
5
3
2 6
57
41
6
7
3
54
7
WBTC116-120 30/05/2005 16:14 Page 804
Why else would I have chosen it? Certainly, in each of these contexts,12
the same rules hold
for the sign “3”. It continues to be replaceable by 2 + 1, and so on. However, a proposition
on the pattern of L||& L||| ⊃ L||||| isn’t a tautology. Two people who live at peace
with each other and three other people who live at peace with each other do not make five
people who live at peace with each other. But that does not mean that 2 + 3 aren’t 5; it is
just that addition cannot be applied in that way. For one could say: 2 people who . . .
and 3 people who . . . , each of whom lives at peace with each of the first group, = 5 people
who . . . .
In other words, the signs of the form (∃1x, y).F(x, y), (∃2x, y).F(x, y), etc. have the
multiplicity of the cardinal numbers, as do the signs (∃1x).φx, (∃2x).φx, etc., and also the
signs (L1x).φx, (L2x).φx, etc.
“There are only 4 red things, but they don’t consist of 2 and 2, for there is no function
that brings them under one heading in pairs.” That would mean taking the proposition
2 + 2 = 4 this way: if you can see 4 circles on a surface, any two of them
always have a particular characteristic in common; say a sign inside the circle.
13
(Then of course any three of the circles will also have a sign in common, and
so on.) For if I assume anything at all about reality, why not that? The “axiom
of reducibility” is essentially no different. In this sense one could say, that, to
be sure, 2 and 2 always make 4, but 4 doesn’t always consist of 2 and 2. (It is only the
complete vagueness and generality of the axiom of reducibility that seduces us into believing
that – if it’s a matter of a meaningful proposition at all – here it’s a matter of more than
an arbitrary assumption for which there is no reason. Therefore, in this and all similar cases
it clarifies things immensely to drop this generality altogether – which after all doesn’t make
the matter any more mathematical – and in its place to make very specific assumptions.)
We feel like saying: 4 doesn’t always have to consist of 2 and 2, but if it really does
consist of groups, it can consist of 2 and 2, as well as of 3 and 1 etc.; but not of 2 and 1 or
3 and 2, etc.; and this is how we get everything prepared for the case that 4 is divisible into
groups. But then arithmetic doesn’t have anything at all to do with actual division, but only
with the possibility of division. After all, the assertion could also be that whenever I see a
group of 4 dots on a piece of paper every two of them are connected with a parenthesis.14
Or:15
that in the real world there is always a circle drawn around every 2 such groups of
2 dots.
Now add to this that a statement like “you can see two black circles in a white rectangle”
doesn’t have the form “(∃x, y).etc.”. For if I give the circles names, the names refer to the
precise locations of the circles, and I can’t say of them that they are either in the one or
in the other rectangle. To be sure, I can say “There are 4 circles in both rectangles taken
together”, but that doesn’t mean that I can say of each individual circle that it is in one
12 (V): in this use as well as // and // in that one,
13 (F): MS 113, p. 62r.
14 (V): After all, the assertion could also be that,
of a group of 4 dots on a piece of paper, every
2 are connected by a line.
15 (V): parenthesis.
Let’s even consider the assumption$
2 + 2 = 4. 402e
1 26
5
3
2 6
57
41
6
7
3
54
7
WBTC116-120 30/05/2005 16:14 Page 805
sagen kann, daß er im einen oder andern Viereck sei. Denn der Satz ”dieser Kreis ist in
diesem Viereck“ ist im angenommenen Fall sinnlos.
Was bedeutet nun der Satz ”in den 2 Vierecken zusammen sind 4 Kreise“? Wie konstatiere
ich das? Indem ich die Zahlen in beiden addiere? Die Zahl der Kreise in beiden Vierecken
zusammen bedeutet also dann das Resultat der Addition der beiden Zahlen. – Oder ist es
etwa das Resultat einer eigenen27
Zählung, die durch beide Vierecke geht; oder die Zahl von
Strichen, die ich erhalte, wenn ich einen Strich einem Kreis zuordne, ob er nun in einem
oder im andern Viereck ist. Man kann nämlich sagen: ”jeder Strich
28
ist entweder einem Kreis zugeordnet, der in dem einen, oder einem
Kreis, der in dem andern Viereck steht“; aber nicht: ”dieser Kreis steht
entweder in diesem oder im andern Viereck“, wenn ”dieser Kreis“ eben
durch seine Lage charakterisiert ist. Dies kann nur dann hier sein, wenn
”dies“ und ”hier“ nicht dasselbe bedeuten. Dagegen kann dieser Strich einem Kreis in diesem
Viereck zugeordnet sein, denn er bleibt dieser Strich, auch wenn er einem Kreis im andern
Viereck zugeordnet ist.
29
Sind in diesen beiden Kreisen zusammen 9 Punkte oder 7? Wie man
es gewöhnlich versteht, 7. Aber muß ich es so verstehen? Warum soll ich
nicht die Punkte, die beiden Kreisen gemeinsam angehören, doppelt
zählen:30
Anders ist es, wenn man fragt: ”wieviel Punkte sind innerhalb
der stark ausgezogenen Grenze?“ 31
Denn hier kann ich sagen: es
sind 7, in dem Sinne, in welchem in den Kreisen 5 und 4 sind.
Man könnte nun sagen: die Summe von 4 und 5 nenne ich die Zahl, welche die unter
den Begriff φx 1 ψx fallenden Gegenstände haben, wenn (L4x).φx & (L5x).ψx & Ind.
der Fall ist. Und zwar heißt das (nun) nicht, daß die Summe von 4 und 5 nur in der
Verbindung mit Sätzen von der Art (∃4x).φx etc. verwendet werden darf, sondern es heißt:
Wenn Du die Summe von n und m bilden willst, setze die Zahlen links von ”.⊃.“ in die
Form (∃nx).φx & (∃mx).ψx etc. ein, und die Zahl, die rechts stehen muß, um aus dem
ganzen Satz32
eine Tautologie zu machen, ist die Summe von m und n. Dies ist also eine
Additionsmethode, und zwar eine äußerst umständliche.
Vergleiche: ”Wasserstoff und Sauerstoff geben zusammen Wasser“ – ”2 Punkte und 3
Punkte geben zusammen 5 Punkte“.
Bestehen denn z.B. 4 Punkte in meinem Gesichtsfeld, die ich ”als 4“, nicht ”als 2 und 2“
sehe, aus 2 und 2? Ja, was heißt das? Soll es heißen, ob sie in irgendeinem Sinne in Gruppen
von je 2 Punkten geteilt waren? Gewiß nicht. (Denn dann müßten sie ja wohl auch in
allen andern denkbaren Weisen geteilt sein.) Heißt es, daß sie sich in Gruppen von 2 und
2 teilen lassen? also, daß es Sinn hat, von solchen Gruppen in den vieren zu reden? –
Jedenfalls entspricht doch das dem Satz ”2 + 2 = 4“, daß ich nicht sagen kann, die Gruppe
der 4 Punkte, die ich gesehen habe, habe aus getrennten Gruppen von 2 und 3 Punkten
bestanden. Jeder wird sagen: das ist unmöglich, denn 3 + 2 = 5. (Und ”unmöglich“ heißt
hier ”unsinnig“.)
27 (V): besondern
28 (F): MS 113, S. 63v.
29 (F): MS 113, S. 63v.
30 (F): MS 113, S. 63v.
31 (F): MS 113, S. 63v.
32 (V): Ausdruck
589
590
403 2 + 2 = 4.
1·
2·
3·
· 6
· 7
5·9
4·8
WBTC116-120 30/05/2005 16:14 Page 806
rectangle or the other. For in the case supposed, the proposition “This circle is in this rectangle”
makes no sense.
But now what does the proposition “There are 4 circles in the 2 rectangles taken together”
mean? How do I ascertain that? By adding the numbers in each? In that case the number
of the circles in the two rectangles together means the result of the addition of the two
numbers. – Or is it perhaps the result of a separate16
count through both rectangles? Or is
it the number of lines I get if I correlate a line with a circle, no matter whether it is in
one rectangle or in the other? For if “this circle” is characterized precisely by its position
we can say: “Every line 17
is correlated either with a circle in the one
rectangle or with a circle in the other”, but not: “This circle is either
in this rectangle or in the other”. This can only be here if “this”
and “here” don’t mean the same thing. By contrast this line can be
correlated with a circle in this rectangle because it remains this line,
even if it is correlated with a circle in the other rectangle.
18
Are there 9 dots or 7 in these two circles? The way this is normally understood,
there are 7. But do I have to understand it this way? Why shouldn’t
I count the dots that jointly belong to both circles twice:
19
It’s different if we ask: “How many dots are there in the borders outlined
16 (V): special
17 (F): MS 113, p. 63v.
18 (F): MS 113, p. 63v.
19 (F): MS 113, p. 63v.
20 (F): MS 113, p. 63v.
21 (V): expression
2 + 2 = 4. 403e
1·
2·
3·
· 6
· 7
5·9
4·8
in bold?” 20
For here I can say: There are 7, in the same sense
in which there are 5 and 4 in the circles.
Now we could say: “I call that number ‘The sum of 4 and 5’ that objects falling under the
concept φx 1 ψx have, if (L4x).φx & (L5x).ψx & Ind. is the case”. (Now) that doesn’t
mean that the sum of 4 and 5 can only be used in connection with propositions of the type
(∃4x).φx etc.; it means, rather: “If you want to construct the sum of n and m, insert the
numbers on the left-hand side of ‘.⊃.’ into the form (∃nx).φx & (∃mx).ψx, etc., and the
sum of m and n will be the number which has to go on the right-hand side in order to
turn the whole proposition21
into a tautology”. So that is a method of addition, and a very
long-winded one at that.
Compare: “Hydrogen and oxygen together yield water” – “2 dots and 3 dots together yield
5 dots”.
So do, for example, 4 dots in my visual field, that I “see as 4” and not “as 2 and 2”, consist
of 2 and 2? Well, what does that mean? Is it asking whether in some sense they had been
divided into groups of 2 dots each? Certainly not (for in that case they would presumably
have to have been divided in all other conceivable ways as well). Does it mean that they can
be divided into groups of 2 and 2, i.e. that it makes sense to speak of such groups in the four?
– At any rate it does correspond to the proposition 2 + 2 = 4 that I can’t say that the group
of 4 dots I saw consisted of separate groups of 2 and 3. Everyone will say: That’s impossible,
because 3 + 2 = 5. (And “impossible” here means “nonsensical”.)
WBTC116-120 30/05/2005 16:14 Page 807
”Bestehen 4 Punkte aus 2 und 2“ kann eine Frage nach einer physikalischen oder
visuellen33
Tatsache sein; dann ist es nicht die Frage der Arithmetik. Die arithmetische Frage
könnte aber allerdings in der Form gestellt werden: ”Kann eine Gruppe von 4 Punkten aus
getrennten Gruppen von je 2 Punkten bestehen“.
”Angenommen, ich glaubte, es gäbe überhaupt nur eine Funktion und die 4
Gegenstände, die sie befriedigen. Später komme ich darauf, daß sie noch von einem fünften
Ding befriedigt wird; ist jetzt das Zeichen ”4“ sinnlos geworden?“ – Ja, wenn es im Kalkül
die 4 nicht gibt, dann ist ”4“34
sinnlos.35
Wenn man sagt, es wäre möglich, mit Hilfe der Tautologie
(∃n2x).φx & (∃n3x).ψx & Ind. .⊃. (∃n5x).φx 1 ψx . . . A
zu addieren, so wäre das folgendermaßen zu verstehen: Zuerst ist es möglich, nach gewissen
Regeln herauszufinden, daß
(∃nx).φx & (∃nx).ψx & Ind. .⊃. (∃nx, y) : φx 1 ψx .&. φy 1 ψy
tautologisch ist. (∃nx).φx ist eine Abkürzung für
(∃x).φx & ~(∃x, y).φx & φy. Ich werde ferner Tautologien der Art A zur Abkürzung so
schreiben:
(L ) & (L ) ⊃ (L ).
So geht also aus den Regeln hervor, daß (Lx) & (Lx) ⊃ (Lx, y), (Lx, y) & (Lx) ⊃ (Lx, y, z)
und andere Tautologien. Ich schreibe ”und andere“ und nicht ”u.s.w. ad inf.“),36
weil man
mit diesem Begriff noch nicht operieren muß.37
38
Als die Zahlen im Dezimalsystem hingeschrieben waren, gab es Regeln, nämlich die der
Addition für je zwei Zahlen von 0 bis 9, und die reichten mir, entsprechend angewandt,
für Additionen aller Zahlen aus. Welche Regel entspricht nun diesen Elementarregeln? Es
ist offenbar, daß wir uns in einer Rechnung wie σ weniger Regeln merken brauchen als in
17 + 28. Ja, wohl nur eine allgemeine und gar keine der Art 3 + 2 = 5. Im Gegenteil, wieviel
3 + 2 ist, scheinen wir jetzt ableiten, ausrechnen zu können.
Die Aufgabe ist 2 + 3 = ? und man schreibt
1,2,3,4,5,6,7
1,2;1,2,3
33 (V): optischen
34 (V): ist sie
35 (V): Ja, wenn im Kalkül die 4 nicht existiert,
dann ist ”4“ sinnlos.
36 (O): ”u.s.w. ad inf.),
37 (E): Hier fehlen die letzten drei Worte im TS.
Wir haben sie nach MS 113, S. 67r eingesetzt.
38 (E): In MS 111 (S. 156) und TS 211 (S. 98)
steht eine weitere Bemerkung vor der vorliegenden.
Sie wurde anscheinend irrtümlich
ausgelassen, ist aber zur Erklärung von σ nötig.
Sie lautet:
17 + 28 kann ich mir nach Regeln ausrechnen,
ich brauche 17 + 28 = 45 (α) nicht als Regel
zu geben. Kommt also in einem Beweis der
Übergang von f(17 + 28) auf f(45) vor, so
brauche ich nicht sagen, er geschähe nach der
Regel α sondern nach andern Regeln des 1 + 1.
Wie ist es hiermit aber in der (((1) + 1) + 1)
Notation? Kann ich sagen ich könne mir in
ihr z.B. 2 + 3 ausrechnen? Und nach welcher
Regel? Es geschähe so: σ . . . [(1) + 1] +
[((1) + 1) + 1] = (([(1) + 1] + 1) + 1) + 1 =
.[((((1) + 1) + 1) + 1) + 1]
591
591
404 2 + 2 = 4.
WBTC116-120 30/05/2005 16:14 Page 808
“Do 4 dots consist of 2 and 2?” can be a question about a physical or a visual22
fact; then
it isn’t an arithmetical question. The arithmetical question could be put in this form, though:
“Can a group of 4 dots consist of separate groups of 2 each?”
“Suppose that I used to believe that there wasn’t anything at all except one function and
the 4 objects that satisfy it. Later I discover that it is satisfied by a fifth thing too: has the
sign ‘4’ now become senseless?” – Indeed, if 4 doesn’t exist in the calculus then “4”23
makes
no sense.
If you say it would be possible to do addition using the tautology
(∃n2x).φx & (∃n3x).ψx & Ind. .⊃. (∃n5x).φx 1 ψx . . . A
that would have to be understood this way: first it is possible to establish according to certain
rules that
(∃nx).φx & (∃nx).ψx & Ind. .⊃. (∃nx, y):φx 1 ψx .&. φy 1 ψy
is tautological. (∃nx).φx is an abbreviation for
(∃x).φx & ~(∃x, y).φx & φy. Furthermore, for the sake of abbreviation, I will write
tautologies of type A thus:
(L ) & (L ) ⊃ (L ).
Therefore (Lx) & (Lx) ⊃ (Lx, y), (Lx, y) & (Lx) ⊃ (Lx, y, z) and other tautologies follow
from the rules. I write “and other tautologies” and not “and so on ad inf.” since I don’t yet
need to work with that concept.24
25
When the numbers were written down in the decimal system there were rules, namely
the addition rules for every pair of numbers from 0 to 9, and, used appropriately, these sufficed
for the addition of all numbers. Now which rule corresponds to these elementary rules? It’s
obvious that in a calculation like σ we don’t have to remember as many rules as in 17 + 28.
Indeed we need to remember only one general rule, and none of the kind 3 + 2 = 5; on the
contrary, we now seem to be able to deduce, work out, how much 3 + 2 is.
The problem is 2 + 3 = ? and we write
1, 2, 3, 4, 5, 6, 7
1, 2; 1, 2, 3
22 (V): or an optical
23 (V): it
24 (E): This sentence is incomplete in the TS.
We have supplied the missing words from the
corresponding sentence in MS 113, p. 67r.
25 (E): In MS 111 (p. 156) and TS 211 (p. 98) this
remark is preceded by another remark that
appears to have been left out of TS 213 by mistake.
It is required to give sense to the reference
to σ in this remark. It reads:
I can calculate 17 + 28 according to rules; I
don’t need to give 17 + 28 = 45 (α) as a rule.
So if the step from f(17 + 28) to f(45) occurs
in a proof, I don’t need to say it took place
according to rule (α), but according to other
rules for 1 + 1.
But what is this like in the (((1) + 1) + 1)
notation? Can I say that in it I could calculate,
for example, 2 + 3? And according to which
rule? It would go like this: σ . . . [(1) + 1] +
[((1) + 1) + 1] = (([(1) + 1] + 1) + 1) + 1 =
.[((((1) + 1) + 1) + 1) + 1]
2 + 2 = 4. 404e
WBTC116-120 30/05/2005 16:14 Page 809
So rechnen Kinder tatsächlich, wenn sie ”abzählen“. (Und dieser Kalkül muß so gut sein
wie ein anderer.)
Es ist übrigens klar, daß das Problem, ob 5 + (4 + 3) = (5 + 4) + 3 ist, sich so lösen läßt:
39
denn diese Konstruktion hat genau die Multiplizität jedes andern
Beweises dieses Satzes.
Wenn ich die Zahl nach ihrem
letzten Buchstaben nenne, so beweist
das, daß (E + D) + C = E + (D + C) =
L. Diese Form des Beweises ist gut,
weil sie deutlich zeigt, daß das Ergebnis
wirklich errechnet ist und weil man aus
ihr doch auch wieder den allgemeinen
Beweis herauslesen kann.
Es ist hier eine gute Mahnung – so seltsam sie klingt –: treibe hier nicht Philosophie,
sondern Mathematik.
Unser Kalkül braucht überhaupt noch nichts von der Bildung einer Reihe ”(Lx)“,
”(Lx, y)“, ”(Lx, y, z)“, etc. zu wissen, sondern kann einfach einige, etwa 3, dieser Zeichen
einführen, ohne das ”u.s.w.“. Wir können nun einen Kalkül mit einer endlichen Reihe von
Zeichen einführen, indem wir eine Reihenfolge gewisser Zeichen festsetzen, etwa die der
Buchstaben des Alphabets, und schreiben:
(La) & (La) ⊃ (La, b)
(La, b) & (La) ⊃ (La, b, c)
(La, b) & (La, b) ⊃ (La, b, c, d)
u.s.w. bis zum z.
Die rechte Seite (rechts vom ”⊃“) kann man dann aus der linken durch einen Kalkül der
Art finden:
a b c d e f . . . z
a b - - -
- a b c B
a b c d e
Dieser Kalkül ergäbe sich aus den Regeln zur Bildung der Tautologien als eine
Vereinfachung. – Dieses Gesetz der Bildung eines Reihenstückes aus zwei andern vorausgesetzt,
kann ich für das erste nun die Bezeichnung ”Summe der beiden andern“ einführen
und also definieren:
a + a Def.
ab
a + ab Def.
abc
u.s.w. bis z.
Hätte man an einigen Beispielen die Regel des Kalküls B erklärt, so könnte man auch diese
Definitionen als Spezialfälle einer allgemeinen Regel betrachten und nun Aufgaben stellen
von der Art: ”abc + ab = ?“. Es liegt nun nahe, die Tautologie
39 (F): MS 111, S. 159.
592
593
405 2 + 2 = 4.
5 4 3
A
A
A
A
A
A
B
B
C
C
D
D
E
E, A
A
A
E, A
G
B
B
H
C
C
I
D
I,
D,
J
A
A
K
B
B
L
C
L
C
G
G
L
M N OF
==
==
WBTC116-120 30/05/2005 16:14 Page 810
This is in fact how children calculate when they “count off”. (And that calculus must be as
good as any other.)
It’s clear, incidentally, that the problem whether 5 + (4 + 3) = (5 + 4) + 3 can be solved
this way: 26
for this construction has precisely the same multiplicity
as every other proof of this proposition.
If I assign a number to each letter’s
position in the alphabet, that is a proof
that (E + D) + C = E + (D + C) = L.
This form of proof is good, because
it shows clearly that the result has
really been worked out and because
from it you can read off the general
proof as well.
It may sound odd, but it is a good admonition at this point: Don’t do philosophy here,
do mathematics.
Our calculus needn’t yet know anything at all about the construction of a series “(Lx)”,
“(Lx, y)”, “(Lx, y, z)”, etc.; it can simply introduce a few, say three, of these signs without
the “etc.”. We can then introduce a calculus with a finite series of signs by setting down
a sequence of certain signs, say that of the letters of the alphabet, and write:
(La) & (La) ⊃ (La, b)
(La, b) & (La) ⊃ (La, b, c)
(La, b) & (La, b) ⊃ (La, b, c, d)
etc., up to z.
The right-hand side (the side to the right of “⊃”) can then be found from the left-hand side
by a calculus of the type:
a b c d e f . . . z
a b - - -
- a b c B
a b c d e
This calculus would result from the rules for the construction of tautologies as a
simplification. – If I presuppose this law for constructing a fragment of the series out of
two others, I can then introduce the designation “sum of the two others” for the first
composite fragment, and thus give the definition:
a + a Def.
ab
a + ab Def.
abc
and so on up to z.
If the rule for the calculus B had been explained by a few examples, we could regard those
definitions too as particular cases of a general rule, and then pose problems of the type:
“abc + ab = ?”. Now what is tempting is to confuse the tautology
26 (F): MS 111, p. 159.
2 + 2 = 4. 405e
5 4 3
A
A
A
A
A
A
B
B
C
C
D
D
E
E, A
A
A
E, A
G
B
B
H
C
C
I
D
I,
D,
J
A
A
K
B
B
L
C
L
C
G
G
L
M N OF
==
==
WBTC116-120 30/05/2005 16:14 Page 811
α) (La, b) & (La, b) ⊃ (La, b, c, d) mit der Gleichung
β) (ab + ab) = abcd zu verwechseln.
– Aber diese ist eine Ersetzungsregel, jene ist keine Regel, sondern eben eine Tautologie.
Das Zeichen ”⊃“ in α entspricht in keiner Weise dem ”=“ in β.
Man vergißt, daß das Zeichen ”⊃“ in α ja nicht sagt, daß die beiden Zeichen rechts und
links von ihm eine Tautologie ergeben.
Dagegen könnte man einen Kalkül konstruieren, in welchem die Gleichung ξ + η = ζ als
eine Transformation erhalten wird (aus) der Gleichung:
γ) (Lξ) & (Eη ) ⊃ (Lζ ) = Taut.
So, daß ich also sozusagen ζ = ξ + η erhalte, wenn ich ζ aus der Gleichung γ herausrechne.
Wie tritt der Begriff der Summe in diese Überlegungen ein? – Im ursprünglichen Kalkül,
der (etwa) feststellt, daß die Form
δ) Lξ) & (Lη) ⊃ (Lζ)
(z.B.) tautologisch wird für ξ = xy, η = x und ζ = xyz, ist von Summierung nicht die Rede.
– Dann bringen wir ein Zahlensystem in den Kalkül (etwa das System a b c d . . . z). Und
endlich definieren wir die Summe zweier Zahlen als diejenige Zahl ζ, welche die Gleichung
γ löst.
(Wenn wir statt ”(Lx) & (Lx) ⊃ (Lx, y)“ schrieben: ”(Lx) & (Lx) ⊃ (Lx + x)“, so hätte
das keinen Sinn; es sei denn, daß die Notation von vornherein nicht
ι) ”(Lx) etc.“, ”(Lx, y) etc.“, ”(Lx, y, z) etc.“ lautet, sondern:
κ) ”(Lx) etc.“, ”(Lx + x) etc.“, ”(Lx + x + x) etc.“.
Denn warum sollten wir plötzlich statt
”(Lx, y) & (Lx) ⊃ (Lx, y, z)“ schreiben:
”(Lx, y) & (Lx) ⊃ (Lxy + x)“?
das wäre nur eine Verwirrung der Notation. – Nun sagt man: Es vereinfacht doch das
Hinschreiben der Tautologie sehr, wenn man in der rechten Klammer gleich die Ausdrücke
der beiden linken hinschreiben kann. Aber diese Schreibweise ist ja noch gar nicht erklärt;
ich weiß ja nicht, was (Lxy + x) bedeutet, daß nämlich (Lxy + x) = (Lx, y, z) ist.
Wenn man aber von vornherein die Notation ”(Lx)“, ”(Lx + x)“, ”(Lx + x + x)“
gebrauchte, so40
hätte vorerst nur der Ausdruck ”(Lx + x + x + x)“ Sinn, aber nicht
”(L(x + x) + (x + x))“.
Die Notation κ ist auf einer Stufe mit41
ι. Ob42
sich in der Form δ eine Tautologie ergibt,
kann man etwa kurz durch das Ziehen von Verbindungslinien kalkulieren, also
40 (O): ”(Lx + x + x)“ , so
41 (V): ist im gleichen Fall wie
42 (V): Daß
594
595
406 2 + 2 = 4.
(Lx, y) & (Lx, y) ⊃ (Lx, y, z, u) und analog
(Lx + x) & (Lx + x) ⊃ (Lx + x + x + x).
WBTC116-120 30/05/2005 16:14 Page 812
α) (La, b) & (La, b) ⊃ (La, b, c, d) with the equation
β) (ab + ab) = abcd.
– But the latter is a replacement rule, and the former isn’t a rule but – as indicated – a
tautology. The sign “⊃” in α in no way corresponds to the “=” in β.
We forget that the sign “⊃” in α does not say that the two signs to the right and left of
it yield a tautology.
On the other hand we could construct a calculus in which the equation ξ + η = ζ is obtained
as a transformation from the equation
γ) (Lξ) & (Lη) ⊃ (Lζ) = Taut.
So that as it were I get ζ = ξ + η if I figure out ζ from the equation γ.
How does the concept of sum enter into these considerations? – There is no mention of
summation in the original calculus that, say, stipulates that the form
δ) (Lξ) & (Lη) ⊃ (Lζ)
becomes tautologous, (for instance), where ξ = xy, η = x, and ζ = xyz. – Later we introduce
a number system (say the system a, b, c, d, . . . z) into the calculus, and finally we define
the sum of two numbers as the number ζ that solves the equation γ.
(If instead of “(Lx) & (Lx) ⊃ (Lx + x)” we wrote: “(Lx) & (Lx) ⊃ (Lx, y)” it would make
no sense; unless, from the very start, the notation read, not
ι) “(Lx) etc.”, “(Lx, y) etc.”, “(Lx, y, z) etc.”
but κ) “(Lx) etc.”, “(Lx + x) etc.”, “(Lx + x + x) etc.”
For why should we suddenly write
“(Lx, y) & (Lx) ⊃ (Lxy + x)” instead of
“(Lx, y) & (Lx) ⊃ (Lx, y, z)”?
That would just be a confusion in the notation. – Now we say: But it will greatly simplify
writing the tautology down if we can immediately write the expressions in the two left
parentheses in the right parenthesis. But that notation hasn’t yet been explained: I don’t
know what (Lxy + x) means, i.e. that (Lxy + x) = (Lx, y, z).
But if we used the notation “(Lx)”, “(Lx + x)”, “(Lx + x + x)” right from the start,
then at that point only the expression “(Lx + x + x + x)” would make sense, but not
“(L(x + x) + (x + x))”.
The notation κ is on the same level as27
ι. A quick way of calculating whether28
you get
a tautology of the form δ is, say, to draw connecting lines, i.e.
27 (V): is in the same situation as 28 (V): that
2 + 2 = 4. 406e
(Lx, y) & (Lx, y) ⊃ (Lx, y, z, u) and analogously
(Lx + x) & (Lx + x) ⊃ (Lx + x + x + x).
WBTC116-120 30/05/2005 16:14 Page 813
43
Die Verbindungslinien44
entsprechen nur der Regel, die in jedem Fall für die Kontrolle
der Tautologie gegeben sein muß. Von einer Addition ist hier noch keine Rede. Die tritt
erst ein, wenn ich mich entschließe – z.B. – statt ”xyzu“ ”xy + xy“ zu schreiben, und zwar
in Verbindung mit einem Kalkül, der nach Regeln die Ableitung einer Ersetzungsregel
”xy + xy = xyzu“ erlaubt. Addition liegt auch dann nicht vor, wenn ich in der Notation κ
schreibe ”(Lx) & (Lx) ⊃ (Lx + x)“, sondern erst, wenn ich zwischen ”x + x“ und ”(x) + (x)“
unterscheide und schreibe: (x) + (x) = (x + x).)
Ich kann ”die Summe von ξ und η“ (”ξ + η“) als die Zahl ζ definieren (oder: ”den
Ausdruck“ – wenn wir uns scheuen, das Wort Zahl zu gebrauchen) – ich kann ”ξ + η“
als die Zahl ζ definieren, die den Ausdruck δ tautologisch macht; – man kann aber auch
”ξ + η“, z.B., durch den Kalkül B definieren (unabhängig von dem der Tautologien) und
nun die Gleichung (Lξ) & (Lη) ⊃ (Lξ + η) = Taut. ableiten.45
Eine Frage, die sich leicht einstellt, ist die: müssen wir die Kardinalzahlen in Verbindung
mit der Notation (∃x, y, . . . ).φx & φy . . . einführen? Ist der Kalkül der Kardinalzahlen
irgendwie an den mit den Zeichen ”(∃x, y . . . ).φx & φy . . .“ gebunden? Ist etwa der
letztere die einzige, und vielleicht wesentlich einzige, Anwendung des ersteren?46
Was
die ”Anwendung der Kardinalarithmetik auf die47
Grammatik“ betrifft, so kann man auf
das verweisen, was wir über den Begriff der Anwendung eines Kalküls gesagt haben. – Man
könnte nun unsere Frage auch so stellen: Kommen die Kardinalzahlen in den Sätzen unserer
Sprache immer hinter dem Zeichen ”∃“ vor: wenn wir uns nämlich die Sprache in die
Russell’sche Notation übersetzt denken? Diese Frage hängt unmittelbar mit der zusammen:
Wird das Zahlzeichen in der Sprache immer als Charakterisierung eines Begriffes – einer
Funktion – gebraucht? Die Antwort darauf ist, daß unsere Sprache die Zahlzeichen immer
als Attribute von48
Begriffswörtern gebraucht – daß aber diese Begriffswörter unter sich
gänzlich verschiedenen grammatischen Systemen angehören (was man daraus sieht, daß
das eine in Verbindungen Bedeutung hat, in denen das andre sinnlos ist), so daß die Norm,
die sie zu Begriffswörtern macht, für uns uninteressant wird. Eine ebensolche Norm aber
ist die Schreibweise ”(∃x, y, . . . ) etc.“; sie ist die direkte Übersetzung einer Norm unserer
Wortsprachen, nämlich des Ausdruckes ”es gibt . . .“, eines Sprachschemas,49
in das
unzählige grammatische50
Formen gepreßt sind.
Übrigens ist das Zahlzeichen, jetzt in einem andern Sinne, nicht mit ”∃“ verbunden: da51
nämlich ”(∃3)x . . .“ nicht in ”(∃2 + 3)x . . .“ enthalten ist.52
Wenn wir von den mittels ”=“ konstruierten Funktionen (x = a 1 x = b etc.) absehen,
so wird nach Russells Theorie 5 = 1, wenn es keine Funktion gibt, die nur von einem
Argument, oder nur von 5 Argumenten befriedigt wird. Dieser Satz scheint natürlich auf
den ersten Blick unsinnig; denn, wie kann man dann sinnvoll sagen, daß es keine solchen
Funktionen gibt. Russell müßte sagen, daß man die beiden Aussagen, daß es Fünfer- und
Einserfunktionen gibt, nur dann getrennt machen kann, wenn wir in unserem Symbolismus
eine Fünfer- und eine Einserklasse haben. Er könnte etwa sagen, daß seine Auffassung richtig
sei, weil ich, ohne das Paradigma der Klasse 5 im Symbolismus, gar nicht sagen könne, eine
43 (F): MS 113, S. 68v.
44 (V): Bögen
45 (V): beweisen.
46 (V): Anwendung der Kardinalzahlen // des
ersten?
47 (V): Kardinalarithmetik in der
48 (V): immer in Verbindung mit
49 (V): Ausdrucksschemas,
50 (V): logische
51 (V): insofern
52 (V): verbunden: insofern nämlich ”(∃3x) . . .“
nicht in ”(∃2 + 3x) . . .“ enthalten ist.
596
597
407 2 + 2 = 4.
WBTC116-120 30/05/2005 16:14 Page 814
29
The connecting lines30
correspond only to the rule which we have to give in any case for
checking the tautology. Here addition doesn’t come up yet. It doesn’t enter the picture until
I decide – for example – to write “xy + xy” instead of “xyzu”, and to do this in connection
with a calculus whose rules allow the derivation of the replacement rule “xy + xy = xyzu”.
Neither is it a case of addition when I write in the notation κ: “(Lx) & (Lx) ⊃ (Lx + x)”;
it only becomes one when I distinguish between “x + x” and “(x) + (x)”, and write:
(x) + (x) = (x + x).)
I can define “the sum of ξ and η” (“ξ + η”) as the number ζ (or “the expression” if
we are reluctant to use the word “number”) – I can define “ξ + η” as the number ζ that
makes the expression δ tautologous; but we can also define “ξ + η” (independently of the
calculus of tautologies) by the calculus B, for instance, and then derive31
the equation
(Lξ) & (Lη) ⊃ (Lξ + η) = Taut.
A question that quickly arises is this: Must we introduce the cardinal numbers in
connection with the notation (∃x, y, . . . ).φx & φy . . . ? Is the calculus of the cardinal
numbers somehow tied to the calculus of the signs “(∃x, y, . . . ).φx & φy . . .”? Might the
latter calculus be the only, indeed maybe essentially the only, application of the former?32
As far as the “application of cardinal arithmetic to33
grammar” is concerned, we can refer
to what we said about the concept of the application of a calculus. Now we could also
put our question this way: Do the cardinal numbers always occur after the “∃” sign in
the propositions of our language, if we imagine this language translated into Russellian
notation? This question is directly connected with another: Is a numeral, within language,
always used to characterize a concept – a function? The answer to this is that our language
does always use the numerals as attributes of34
concept-words – but that, within their group,
these concept-words belong to such totally different grammatical systems (which you can
see from the fact that some of them have meaning in contexts in which others make no
sense) that a norm that makes them concept-words holds no interest for us. But the
notation “(∃x, y, . . . ) etc.” is just such a norm. It is a direct translation of a norm of our
word-languages, i.e. of the expression “there is . . .”, of a schema of language35
into which
countless grammatical36
forms have been squeezed.
Incidentally – and now in a different sense – numerals are not connected with “∃”: that
is, because37
“(∃3)x . . .” is not contained in “(∃2 + 3)x . . .”.38
If we disregard the functions (x = a 1 x = b, etc.) that contain “=”, then, according to
Russell’s theory, 5 = 1 if there is no function that is satisfied by only one argument, or by
only 5 arguments. Of course at first glance this proposition seems nonsensical; for in that
case how can one sensibly say that there are no such functions? Russell would have to say
that the two statements that there are five-functions and that there are one-functions can
only be made separately if we have a five-class and a one-class in our symbolism. He could
say, for example, that his view was correct because without the paradigm of the class 5 in
the symbolism, there would be no way for me to say that a function was satisfied by five
29 (F): MS 113, p. 68v.
30 (V): The arcs
31 (V): prove
32 (V): application of the cardinal numbers? //
application of the first?
33 (V): in
34 (V): numerals in connection with
35 (V): expression
36 (V): logical
37 (V): with “∃”: in so far as
38 (V): with “∃”: in so far as “(∃3x) . . .” is not
contained in “(∃2 + 3x) . . .”.
2 + 2 = 4. 407e
WBTC116-120 30/05/2005 16:14 Page 815
Funktion werde von 5 Argumenten befriedigt. – D.h., daß aus der Existenz des Satzes
”(∃φ): (L1x).φx“ seine Wahrheit schon hervorgeht. – Man scheint also sagen zu können:
schau’ auf diesen Satz, dann wirst Du sehen, daß er wahr ist. Und in einem, für uns
irrelevanten, Sinn ist das auch möglich: Denken wir uns etwa auf die Wand eines Zimmers
mit roter Farbe geschrieben: ”in diesem Zimmer befindet sich etwas Rotes“. –
Dieses Problem hängt damit zusammen, daß ich in der hinweisenden Definition von
dem Paradigma (Muster) nichts aussage, sondern nur mit seiner Hilfe Aussagen mache;
daß es zum Symbolismus gehört und nicht einer der Gegenstände ist, auf den ich den
Symbolismus53
anwende.
Ist z.B. ”1 Fuß“ definiert als die Länge eines bestimmten Stabes in meinem Zimmer, und
ich würde etwa statt ”diese Tür ist 6 Fuß hoch“ sagen: ”diese Tür hat sechsmal ,diese
54
Länge (wobei ich auf den Einheitsstab zeige)“, – dann könnte man nicht (etwa) sagen: ”der
Satz ,es gibt einen Gegenstand von 1 Fuß Länge’ beweist sich selbst, denn ich könnte diesen
Satz gar nicht aussprechen, wenn es keinen Gegenstand von dieser Länge gäbe“; denn
vom Einheitsstab kann ich nicht aussagen, daß er 1 Fuß lang sei. (Wenn ich nämlich statt
”1 Fuß“ das Zeichen ”diese 55
Länge“ einführe, so hieße die Aussage, daß der Einheitsstab
die Länge 1 Fuß hat: ”dieser Stab hat diese Länge“ (wobei ich beide Male auf den gleichen
Stab zeige).) So kann man von der Gruppe der Striche, welche etwa als Paradigma der
3 steht nicht sagen, sie56
bestehe aus 3 Strichen.
”Wenn jener Satz nicht wahr ist, so gibt es diesen Satz gar nicht“ – das heißt: ”wenn es
diesen Satz nicht gibt, so gibt es ihn nicht“. Und ein Satz kann das Paradigma im andern
nie57
beschreiben, sonst ist es eben nicht Paradigma. Wenn die Länge des Einheitsstabes durch
die Längenangabe ”1Fuß“ beschrieben werden kann, dann ist er nicht das Paradigma der
Längeneinheit, denn sonst müßte jede Längenangabe mit seiner Hilfe gemacht werden.
Ein Satz, ”~(∃φ): (Lx).φx“ muß, wenn wir ihm überhaupt einen Sinn geben, von der Art
des Satzes58
sein: ”es gibt keinen Kreis auf dieser Fläche, der nur einen schwarzen Fleck
enthält“. (Ich meine: er muß einen ähnlich bestimmten Sinn haben; und nicht vague bleiben,
wie er in der Russell’schen Logik und in meiner der Abhandlung wäre.)
Wenn nun aus den Sätzen ”~(∃φ):(Lx).φx“59
. . . ρ
und ”~(∃φ):(Lx, y).φx & φy“60
. . . σ
folgt, daß 1 = 2 ist, so ist hier mit ”1“ und ”2“ nicht das gemeint, was wir sonst damit meinen,
denn die Sätze ρ und σ würden in der Wortsprache lauten: ”es gibt keine Funktion, die nur
von einem Ding befriedigt wird“ und ”es gibt keine Funktion, die nur von 2 Dingen befriedigt
wird“. Und dies sind nach den Regeln unserer Sprache Sätze mit verschiedenem Sinn.
Man ist versucht zu sagen: ”Um ,(∃x, y).φx & φy‘ auszudrücken,61
brauchen wir 2 Zeichen
,x‘ und ,y‘.“ Aber das heißt nichts. Was wir dazu brauchen, sind, etwa, die Schreibutensilien,
nicht die Bestandteile des Satzes. Ebensowenig hieße es, zu sagen: ”Um ,(∃x, y).φx & φy‘
auszudrücken, brauchen wir das Zeichen ,(∃x, y).φx & φy‘.“62
Wenn man63
fragt: ”was heißt denn dann ,5 + 7 = 12‘ – was für ein Sinn oder Zweck
bleibt denn noch für diesen Ausdruck, nachdem man die Tautologien etc. aus dem
53 (V): ich ihn
54 (F): MS 113, S. 71r.
55 (F): MS 113, S. 71r.
56 (O): es
57 (V): niemals
58 (V): dessen
59 (V): ”~(∃φ): (Lx).φx“ und
60 (O): und ~(∃φ): (Lx, y).φx & φy
61 (V): ausdrücken zu können,
62 (V): Was wir dazu brauchen, ist vielleicht
Papier und Feder; und der Satz heißt so wenig,
wie: ”um ,p‘ auszudrücken, brauchen wir ,p‘“.
63 (V): man s
598
599
408 2 + 2 = 4.
WBTC116-120 30/05/2005 16:14 Page 816
arguments – i.e. that from the existence of the proposition “(∃φ): (L1x).φx” its truth already
follows. – So you seem to be able to say: Look at this proposition, and you will see that
it is true. And in a sense that’s irrelevant to us, that is indeed possible: Let’s imagine, for
example, “In this room there is something red” written in red on the wall of a room. –
This problem is connected with the fact that in an ostensive definition I do not state
anything about the paradigm (sample), but only use it to make statements; that it belongs
to the symbolism and is not one of the objects to which I apply the symbolism.39
For example, if “1 foot” is defined as the length of a particular rod in my room, and, instead
of saying, e.g., “This door is 6 ft. high” I were to say “This door measures six times this
40
length” (pointing to the unit rod) – then we couldn’t say (for instance): “The proposition
‘There is an object that is 1 ft. long’ proves itself, because I couldn’t even utter this proposition
if there were no object of that length”; for I cannot say of the unit rod that it is
one foot long. (For if I introduced the sign “this 41
length” instead of “1 foot”, then the
statement that the unit rod is 1 foot long would mean “This rod has this length” (with
me pointing both times to the same rod).) Similarly one cannot say of a group of strokes
serving as a paradigm of 3, that it consists of 3 strokes.
“If that proposition isn’t true, then this proposition doesn’t even exist” means: “If this
proposition doesn’t exist, then it doesn’t exist”. And one proposition can never describe
the paradigm in another; otherwise it simply isn’t a paradigm. If the length of the unit rod
can be described by the statement of length “1 foot”, then this unit rod isn’t the paradigm
of the unit of length; otherwise every statement of length would have to be made by means
of it.
If we give any sense at all to a proposition of the form “~(∃φ):(Lx).φx” it must be a proposition
like42
: “There is no circle on this surface containing only one black speck” (I mean:
the proposition must have a similarly determined sense, and not remain vague as it would be
in Russellian logic and in my logic in the Tractatus.)
Now if it follows from the propositions
“~(∃φ):(Lx).φx” . . . . . ρ
and “~(∃φ):(Lx, y).φx & φy” . . . . . σ
that 1 = 2, then here “1” and “2” don’t mean what we usually mean by them, because in
word-language the proposition ρ would be “There is no function that is satisfied by only
one thing” and σ would be “There is no function that is satisfied by only 2 things”. And
according to the rules of our language these are propositions with different senses.
One is tempted to say: “In order to express43
‘(∃x, y).φx & φy’ we need 2 signs, ‘x’ and
‘y’.” But that means nothing. What we need for this are, for instance, writing utensils, not
the components of the proposition. It would be equally meaningless to say: “In order to
express ‘(∃x, y).φx & φy’ we need the sign ‘(∃x, y).φx & φy’.”44
If we ask: But what then does “5 + 7 = 12” mean – what kind of sense or purpose is left
for this expression after the tautologies, etc. have been eliminated45
from the arithmetical
39 (V): apply it.
40 (F): MS 113, p. 71r.
41 (F): MS 113, p. 71r.
42 (V): must be like
43 (V): say: “To be able to express
44 (V): nothing. What we need for this, are,
perhaps, paper and pen; and the proposition
means no more than: “In order to express ‘p’ we
need ‘p’.”
45 (V): excluded
2 + 2 = 4. 408e
WBTC116-120 30/05/2005 16:14 Page 817
arithmetischen Kalkül ausgeschaltet64
hat, – so ist die Antwort: Diese Gleichung ist
eine Ersetzungsregel, die sich auf bestimmte allgemeine Ersetzungsregeln, die Regeln der
Addition, stützt. Der Inhalt von 5 + 7 = 12 ist (wenn einer es nicht wüßte) genau das, was
den Kindern Schwierigkeiten macht, wenn sie diesen Satz im Rechenunterricht lernen.
Keine Untersuchung der Begriffe, nur die Einsicht in den Zahlenkalkül kann vermitteln,
daß 3 + 2 = 5 ist. Das ist es, was sich in uns auflehnt gegen den Gedanken, daß
”(L3x).φx & (L2x).ψx & Ind. .⊃. (L5x).φx 1 ψx“65
der Satz 3 + 2 = 5 sein könnte. Denn dasjenige,66
wodurch wir jenen67
Ausdruck als Tautologie
erkennen, kann sich selbst nicht aus einer Betrachtung von Begriffen ergeben, sondern
muß aus dem Kalkül zu ersehen sein. Denn die Grammatik ist ein Kalkül. D.h., was im
Tautologien-Kalkül noch außer dem Zahlenkalkül da ist, rechtfertigt diesen nicht und ist,
wenn wir uns für ihn interessieren, nur Beiwerk.
Die Kinder lernen in der Schule wohl 2 × 2 = 4, aber nicht 2 = 2.
64 (V): ausgeschlossen
65 (E): Im Manuskript steht: ”(∃3x)φx · (∃2x)ψx ·
Ind. ⊃ (∃5x)φx 1 ψx“.
66 (V): das,
67 (V): diesen
409 2 + 2 = 4.
WBTC116-120 30/05/2005 16:14 Page 818
calculus? – the answer is: This equation is a replacement rule which rests on certain general
replacement rules, the rules of addition. The content of 5 + 7 = 12 (if someone didn’t
know it) is precisely what children find difficult when they are learning this proposition in
maths class.
No investigation of concepts, only insight into the number calculus, can get across that
3 + 2 = 5. That is what makes us rebel against the idea that
“(L3x).φx & (L2x).ψx & Ind. .⊃. (L5x).φx 1 ψx”46
could be the proposition 3 + 2 = 5. For what enables us to tell that that47
expression is a
tautology cannot itself be the result of an examination of concepts, but must be ascertainable
from the calculus. For grammar is a calculus. That is, whatever else the tautology
calculus contains apart from the number calculus does not justify the latter; and if the
number calculus is what we are interested in, it is mere decoration.
Children do learn 2 × 2 = 4 in school, but not 2 = 2.
46 (E): The manuscript source has: “(∃3x)φx ·
(∃2x)ψx · Ind. ⊃ (∃5x)φx 1 ψx”.
47 (V): this
2 + 2 = 4. 409e
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117
Zahlangaben innerhalb
der Mathematik.
Worin liegt der Unterschied zwischen der Zahlangabe, die sich auf einen Begriff1
und
der Zahlangabe, die sich auf eine Variable bezieht? Die Erste ist ein Satz, der von dem Begriff
handelt, die zweite eine grammatische Regel die Variable betreffend.
Kann ich aber nicht eine Variable dadurch bestimmen, daß ich sage, ihre Werte sollen
alle Gegenstände sein, die eine bestimmte Funktion befriedigen? – Dadurch bestimme ich ja
die Variable nicht, außer wenn ich weiß, welche Gegenstände die Funktion befriedigen, d.h.,
wenn mir diese Gegenstände auch auf andre Weise (etwa durch eine Liste) gegeben sind;
und dann wird die Angabe der Funktion überflüssig. Wissen wir nicht, ob ein Gegenstand
die Funktion befriedigt, so wissen wir nicht, ob er ein Wert der Variablen sein soll und die
Grammatik der Variablen ist dann in dieser Beziehung einfach nicht bestimmt.2
Zahlangaben in der Mathematik (z.B. ”die Gleichung x2
= 1 hat 2 Wurzeln“) sind daher
von ganz anderer Art, als Zahlangaben außerhalb der Mathematik (”auf dem Tisch liegen
2 Äpfel“.)
Wenn man sagt, A B lasse 2 Permutationen zu, so klingt das, als mache man eine allgemeine
Aussage, analog der ”in dem Zimmer sind 2 Menschen“, wobei über die Menschen
noch nichts weiter gesagt ist und bekannt sein braucht. Das ist aber im Falle A B nicht so.
Ich kann A B, B A nicht allgemeiner beschreiben und daher kann der Satz, es seien
2 Permutationen möglich, nicht weniger sagen, als, es sind die Permutationen A B und
B A möglich. Zu sagen, es sind 6 Permutationen von 3 Elementen möglich kann nicht
weniger, d.h. etwas allgemeineres sagen, als das Schema zeigt:
A B C
A C B
B A C
B C A
C A B
C B A
Denn es ist unmöglich, die Zahl der möglichen Permutationen zu kennen, ohne diese selbst
zu kennen. Und wäre das nicht so, so könnte die Kombinatorik nicht zu ihren allgemeinen
Formeln kommen. Das Gesetz, welches wir in der Bildung der Permutationen erkennen, ist
durch die Gleichung p = n! dargestellt. Ich glaube, in demselben Sinn, wie der Kreis durch
die Kreisgleichung. – Ich kann freilich die Zahl 2 den Permutationen A B, B A zuordnen,
sowie die 6 den ausgeführten Permutationen von A, B, C, aber das gibt mir nicht den Satz
1 (V): Zahlangabe über einen Begriff 2 (V): ausgesprochen.
600
601
410
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117
Statements of Number
within Mathematics.
What distinguishes a statement of number that refers to a concept1
from one that refers
to a variable? The first is a proposition about the concept, the second a grammatical rule
concerning the variable.
But can’t I specify a variable by saying that its values are to be all objects satisfying a
certain function? – If I do that, I’m really not specifying the variable unless I know which
objects satisfy the function, that is, if these objects are also given to me in another way (say
by a list); and then giving the function becomes superfluous. If we do not know whether an
object satisfies the function, then we do not know whether it is to be a value of the variable,
and then the grammar of the variable is in this respect simply not determined.2
Statements of number in mathematics (e.g. “The equation x2
= 1 has 2 roots”) are therefore
quite different in kind from statements of number outside mathematics (“There are
2 apples lying on the table”).
If we say that A B admits of 2 permutations, it sounds as if we are making a general
assertion, analogous to “There are 2 people in the room”, in which nothing further has yet
been said or need be known about the people. But this isn’t so in the case of A B. I cannot
give a more general description of A B, B A, and so the proposition that 2 permutations
are possible cannot say less than that the permutations A B and B A are possible. To say
that 6 permutations of 3 elements are possible cannot say less, i.e. anything more general,
than is shown by the schema:
A B C
A C B
B A C
B C A
C A B
C B A
For it’s impossible to know the number of possible permutations without knowing which they
are. And if this weren’t so, the theory of combinations couldn’t arrive at its general
formulae. The law that we discern in the formation of the permutations is represented by
the equation p = n!. In the same sense, I believe, as a circle is represented by the equation
for a circle. – To be sure, I can correlate the number 2 with the permutations A B, B A,
just as I can the number 6 with the development of the permutations of A, B, C, but that
doesn’t give me the theorem of combination theory. What I see in A B, B A is an internal
1 (V): number about a concept 2 (V): expressed.
410e
WBTC116-120 30/05/2005 16:14 Page 821
der Kombinationslehre. – Das was ich in A B, B A sehe, ist eine interne Relation, die sich
daher nicht beschreiben läßt. D.h. das läßt sich nicht beschreiben, was diese Klasse von
Permutationen komplett macht. – Zählen kann ich nur, was tatsächlich da ist, nicht die
Möglichkeiten. Ich kann aber z.B. berechnen, wieviele Zeilen ein Mensch schreiben muß,
wenn er in jede Zeile eine Permutation von 3 Elementen setzt und solange permutiert,
bis er ohne Wiederholung nicht weiter kann. Und das heißt, er braucht 6 Zeilen, um
auf diese Weise die Permutationen A B C, A C B, etc. hinzuschreiben, denn dies sind
eben ”die Permutationen von A, B, C“. Es hat aber keinen Sinn zu sagen, dies seien
alle Permutationen von A B C.
Eine Kombinationsrechenmaschine ist denkbar ganz analog der Russischen.
Es ist klar, daß es eine mathematische Frage gibt:3
”wieviele Permutationen von – z.B. –
4 Elementen gibt es“, eine Frage von genau derselben Art, wie die ”wieviel ist 25 × 18“.
Denn es gibt eine allgemeine Methode zur Lösung beider.
Aber die Frage gibt es auch nur mit Bezug auf diese Methode.
Der Satz, es gibt 6 Permutationen von 3 Elementen, ist identisch mit dem Permutationsschema
und darum gibt es hier keinen Satz ”es gibt 7 Permutationen von 3 Elementen“,
denn dem entspricht kein solches Schema.
Man könnte die Zahl 6 in diesem Falle auch als eine andere Art von Anzahl, die
Permutationszahl von A, B, C auffassen. Das Permutieren als eine andere Art des Zählens.
Wenn man wissen will, was ein Satz bedeutet, so kann man immer fragen ”wie weiß ich
das“. Weiß ich, daß es 6 Permutationen von 3 Elementen gibt, auf die gleiche Weise wie,
daß 6 Personen im Zimmer sind? Nein. Darum ist jener Satz von anderer Art als dieser.
Man kann auch sagen, der Satz ”es gibt 6 Permutationen von 3 Elementen“ verhält
sich genau so zum Satz ”es sind 6 Leute im Zimmer“, wie der Satz 3 + 3 = 6, den man
auch in der Form ”es gibt 6 Einheiten in 3 + 3“ aussprechen könnte. Und wie ich in
dem einen Fall die Reihen im Permutationsschema zähle, so kann ich im andern die
Striche in
|||
||| zählen.
Wie ich 4 × 3 = 12 durch das Schema beweisen kann: , 4
so kann ich 3! = 6 durch
das Permutationsschema beweisen.
Der Satz ”die Relation R verbindet zwei Gegenstände miteinander“, wenn das soviel heißen
soll, wie ”R ist eine zweistellige Relation“ ist ein Satz der Grammatik.
3 (O): gibt; 4 (F): MS 108, S. 75.
602
603
411 Zahlangaben innerhalb der Mathematik.
WBTC116-120 30/05/2005 16:14 Page 822
relation, which therefore cannot be described. That is to say, what makes this class of
permutations complete cannot be described. – I can only count what is actually there, not
possibilities. But I can figure out, for example, how many lines someone must write if he
puts a permutation of 3 elements in each line and continues his permutations until he can’t
go any further without repetition. And this means he needs 6 lines to write down the
permutations A B C, A C B, etc. in this way, for these just are “the permutations of A, B,
C”. But it makes no sense to say that these are all permutations of A B C.
We could imagine a calculating-machine for combinations that is completely analogous
to the Russian one.
It’s clear that there is a mathematical question: “How many permutations of – say –
4 elements are there?”. This is a question of precisely the same kind as “How much is
25 × 18?”. For there is a general method for solving both problems.
But it is only with respect to this method that this question exists.
The proposition that there are 6 permutations of 3 elements is identical with the permutation
schema, and therefore there is no proposition here that says “There are 7 permutations
of 3 elements”, for no such schema corresponds to that.
You could also conceive of the number 6 in this case as another kind of number, the
permutation-number of A, B, C. Permuting as another kind of counting.
If you want to know what a proposition means, you can always ask “How do I know that?”.
Do I know that there are 6 permutations of 3 elements in the same way in which I know
that there are 6 people in a room? No. Therefore the former proposition is of a different
kind from the latter.
You can also say that the proposition “There are 6 permutations of 3 elements” is related
to the proposition “There are 6 people in the room” in precisely the same way as is the
proposition “3 + 3 = 6”, which you could also express in the form “There are 6 units
in 3 + 3”. And just as in the one case I count the rows in the permutation schema, so in
the other I can count the strokes in
| | |
| | | .
Just as I can prove that 4 × 3 = 12 by means of the schema , 3
I can also prove that
3! = 6 by means of the permutation schema.
The proposition “The relation R links two objects”, if it is to amount to the same thing
as “R is a two-place relation”, is a proposition of grammar.
3 (F): MS 108, p. 75.
Statements of Number within Mathematics. 411e
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118
Zahlengleichheit.
Längengleichheit.
Wie soll man nun den Satz auffassen ”diese Hüte haben die gleiche Größe“, oder ”diese
Stäbe haben die gleiche Länge“, oder ”diese Flecke1
haben die gleiche Farbe“? Soll man sie
in der Form schreiben: ”(∃L).La & Lb“? Aber wenn das in der gewöhnlichen Weise gemeint
wird, also mit den gewöhnlichen Regeln gebraucht wird, so müßte es ja dann Sinn haben
zu schreiben ”(∃L).La“ also ”der Fleck a hat eine Farbe“, ”der Stab hat eine Länge“. Ich
kann freilich ”(∃L).La & Lb“ für ”a und b sind gleichlang“ schreiben, wenn ich nur weiß
und berücksichtige, daß ”(∃L).La“ sinnlos ist; aber dann wird die Notation irreführend und
verwirrend. (”eine Länge haben“, ”einen Vater haben“). – Wir haben hier den Fall, den wir
in der gewöhnlichen Sprache2
oft so ausdrücken: ”Wenn a die Länge L hat, so hat b auch
L“; aber hier hätte der Satz ”a hat die Länge L“ gar keinen Sinn, oder doch nicht als Aussage
über a; und der Satz lautet richtiger ”nennen wir die Länge von a ,L‘, so ist die Länge von
b auch ,L‘“3
und „L“ ist eben hier wesentlich eine Variable. Der Satz hat übrigens die Form
eines Beispiels, eines Satzes, der als Beispiel zum allgemeinen Satz dienen kann und man
würde etwa auch fortfahren:4
”wenn z.B. a die Länge 5m hat,5
so hat b auch 5m, u.s.w.“. –
Zu sagen ”die Stäbe a und b haben die gleiche Länge“ sagt nämlich gar nichts über die Länge
jedes Stabes; denn es sagt auch nicht, ”daß jeder der beiden eine Länge hat“. Der Fall hat
also gar keine Ähnlichkeit mit dem: ”A und B haben den gleichen Vater“ und ”der Name
des Vaters von A und B ist ,N‘“, wo ich einfach für die allgemeine Bezeichnung den
Eigennamen einsetze. „5m“ ist aber6
nicht der Name der betreffenden Länge, von der zuerst
nur gesagt wurde, daß a und b sie beide besäßen. Wenn es sich um Längen im Gesichtsfeld
handelt, können wir zwar sagen, die beiden Längen seien gleich, aber wir können sie im
allgemeinen nicht mit einer Zahl ”benennen“. – Der Satz ”ist L die Länge von a, so hat
auch b die Länge L“ schreibt seine Form nur als eine von der eines Beispiels7
derivierte
(Form) hin. Und man könnte den allgemeinen Satz auch wirklich durch eine Aufzählung8
von Beispielen mit einem ”u.s.w.“ ausdrücken. Und es ist eine Wiederholung desselben
Satzes, wenn ich sage: ”a und b sind gleichlang; ist die Länge von a L, so ist die Länge von
b auch L; ist a 5m lang, so ist auch b 5m lang, ist a 7m, so ist b 7m, u.s.w.“. Die dritte
Fassung zeigt schon, daß in dem Satz nicht das ”und“ zwischen zwei Formen steht, wie in
”(∃x).φx & ψx“, so daß man auch ”(∃x).φx“ und ”(∃x).ψx“ schreiben dürfte.
Nehmen wir als Beispiel auch den Satz ”in den beiden Kisten sind gleichviel Äpfel“.
Wenn man diesen Satz in der Form schreibt ”es gibt eine Zahl, die die9
Zahl der Äpfel in
1 (V): Flecken
2 (V): Sprache so
3 (O): auch L‘“
4 (V): fortsetzen:
5 (V): a 5m lang ist,
6 (V): aber d
7 (V): als eine von der Form eines // des //
Beispiels
8 (V): Anführung
9 (V): Zahl, die die Äp
604
605
606
412
WBTC116-120 30/05/2005 16:14 Page 824
118
Sameness of Number.
Sameness of Length.
How should we understand the proposition “These hats are of the same size”, or “These
rods have the same length”, or “These patches have the same colour”? Should we write them
in the form “(∃L).La & Lb”? But if that is meant in the usual way, i.e. is used with the
usual rules, then it ought to make sense to write “(∃L).La”, i.e. “The patch a has a colour”,
“The rod has a length”. To be sure, I can write “(∃L).La & Lb” for “a and b have the same
length”, provided that I know and take into account that “(∃L).La” is senseless; but then
the notation becomes misleading and confusing (“to have a length”, “to have a father”). –
Here we have the case that we often express in ordinary language as follows: “If a has the
length L, so does b”; but here the proposition “a has the length L” has no sense at all, or
at least not as a statement about a; and more correctly phrased, the proposition reads “If
we call the length of a ‘L’, then the length of b is also L”, and here “L” is essentially a
variable. That proposition, incidentally, has the form of an example, of a proposition that
can serve as an example of a general proposition; and we might continue: “For example, if
a has the length of 5 metres,1
then b also has the length of 5 metres, etc.” – For saying “The
rods a and b have the same length” says nothing at all about the length of each rod; neither
does it say “that each of the two has a length”. So this case is quite unlike “A and B have
the same father” and “The name of A’s and B’s father is ‘N’”, where I simply substitute
the proper name for the general description. But “5 m” is not the name of the length in
question, of which it was first said merely that a and b both had it. If it is a matter of lengths
in the visual field we can say, to be sure, that the two lengths are equal, but in general we
cannot “name” them with a number. – The written form of the proposition “If L is the length
of a, then b has that length as well” is derived solely from the (form) of an example.2
And
indeed, we could express the universal proposition by enumerating3
examples, followed by
“etc.”. And if I say, “a and b are of equal length; if the length of a is L, then the length of
b is also L; if a is 5 m long, then b is also 5 m long, if a is 7 m long, then b is 7 m long, etc.”,
I am repeating the same proposition. The third formulation shows that in the proposition
the “and” doesn’t stand between two forms, as it does in “(∃x).φx & ψx”, so that one can
also write “(∃x).φx” and “(∃x).ψx”.
Let us take as a further example the proposition “There are the same number of apples
in each of the two boxes”. If we write this in the form “There is a number that is the
1 (V): a is 5 metres long,
2 (V): from one of the forms of an // the // example.
3 (V): listing
412e
WBTC116-120 30/05/2005 16:14 Page 825
beiden Kisten ist“, so kann man auch hier nicht die Form bilden: ”es gibt eine Zahl, die
die Zahl der Äpfel in dieser Kiste ist“, oder ”die Äpfel in dieser Kiste haben eine Zahl“.
Schreibe ich:
(∃x).φx .&. ~(∃x, y).φx & ψx .=. (∃n1x).φx .=. φ1 etc.,
so könnte man den Satz ”die Anzahl der Äpfel in den beiden Kisten ist die gleiche“ schreiben:
”(∃n).φn & ψn“. ”(∃n).φn“ aber wäre kein Satz.
Will man den Satz ”unter φ und ψ fallen gleichviele Gegenstände“ in übersichtlicher
Notation schreiben, so ist man vor allem versucht, ihn in der Form ”φn & ψn“ zu schreiben.
Und ferner empfindet man das nicht als logisches Produkt von φn und ψn, so daß es also
auch Sinn hätte zu schreiben φn & ψ5 – sondern es ist wesentlich, daß nach ”φ“ und ”ψ“
der gleiche Buchstabe folgt und φn & ψn ist eine Abstraktion aus logischen Produkten φ4
& ψ4, φ5 & ψ5 etc., nicht selbst ein logisches Produkt.
(Es würde also auch nicht aus φn & ψn φn folgen. ”φn & ψn“ verhält sich vielmehr zu
einem logischen Produkt ähnlich wie der Differenzialquotient zu einem Quotienten.) Es ist
so wenig ein logisches Produkt, wie die Photographie einer Familiengruppe eine Gruppe
von Photographien ist. Darum kann uns also die Form ”φn & ψn“ irreführen und es wäre
vielleicht eine Schreibweise der Art ” “ 10
vorzuziehen; aber auch ”(∃n).φn & ψn“,
wenn die Grammatik dieses Zeichens festgelegt ist. Man kann dann festlegen:
(∃n).φn = Taut., was soviel heißt wie (∃n).φn & p = p.
Also (∃n).φn 1 ψn = Taut., (∃n)φn ⊃ ψn = Taut., (∃n).φn|ψn = Cont., etc.
φ1 & ψ1 & (∃n).φn & ψn = φ1 & (∃n).φn & ψn
φ2 & ψ2 & (∃n).φn & ψn = φ2 & (∃n).φn & ψn
etc. ad inf.
Und überhaupt sind die Rechnungsregeln für (∃n).φn & ψn daraus abzuleiten, daß man
schreiben kann:
(∃n).φn & ψn = φ0 & ψ0 .1. φ1 & ψ1 .1. φ2 & ψ2 u.s.w. ad inf.).
Es ist klar, daß dies keine logische Summe ist, da ”u.s.w. ad inf.“ kein Satz ist. Die Notation
(∃n).φn & ψn ist aber auch nicht unmißverständlich; denn man könnte sich wundern, warum
man hier statt φn & ψn nicht Φn sollte setzen können und dann sollte ja ”(∃n).Φn“
nichtssagend werden. Das klärt sich natürlich auf, wenn man auf die Notation ~(∃x).φx für
φ0, (∃x).φx & ~(∃x, y).φx & φy für φ1, etc. zurückgeht, beziehungsweise auf (∃n0x).φx für
φ0, (∃n1x).φx für φ1, etc. Denn dann ist zu unterscheiden zwischen
(∃n1x).φx & (∃n1x).ψx und
(∃n1x).φx & ψx.
Und geht man auf11
(∃n).φn & ψn über, so bedeutet das (∃n):(∃nnx).φx & (∃nnx).ψx (welches
nicht nichtssagend ist) und nicht (∃n):(∃nnx).φx & ψx, welches nichtssagend ist.
Die Worte ”gleichzahlig“, ”längengleich“, ”gleichfärbig“, etc. haben ähnliche aber nicht
die gleiche Grammatik.12
– In allen Fällen liegt die Auffassung des Satzes als eine endlose
logische Summe nahe, deren Glieder die Form φn & ψn haben. Außerdem hat jedes dieser
φn & ψn
10 (F): MS 113, S. 16r.
11 (V): auf (En
12 (V): aber verschiedene Grammatik.
607
608
413 Zahlengleichheit. Längengleichheit.
WBTC116-120 30/05/2005 16:14 Page 826
number of the apples in each of the two boxes”, then here too we cannot construct the
form “There is a number that is the number of apples in this box” or “The apples in this
box have a number”. If I write:
(∃x).φx .&. ~(∃x, y).φx & ψx .=. (∃n1x).φx .=. φ1, etc.,
then we could write the proposition “The number of apples in both boxes is the same” as:
“(∃n).φn & ψn”. But “(∃n).φn” wouldn’t be a proposition.
If you want to write the proposition “The same number of objects falls under φ and ψ”
in a surveyable notation, the strongest temptation is to write it in the form “φn & ψn”. And
that doesn’t feel as if it were a logical product of φn and ψn, so that it would also make sense
to write φn & ψ5; rather, it is essential that the same letter should follow φ and ψ; and φn
& ψn is an abstraction from the logical products φ4 & ψ4, φ5 & ψ5, etc., and not a logical
product itself.
(So neither would φn follow from φn & ψn. Rather, the relation of “φn & ψn” to a
logical product is similar to that of a differential quotient to a quotient.) φn & ψn is no
more a logical product than the photograph of a family group is a group of photographs.
Therefore the form “φn & ψn” can mislead us, and perhaps a notation of the form
“ ”4
would be preferable; or even “(∃n).φn & ψn”, provided that the grammar of
this sign has been fixed. We can then stipulate:
(∃n).φn = Taut., which amounts to the same thing as (∃n).φn & p = p.
Therefore (∃n).φn 1 ψn = Taut., (∃n).φn ⊃ ψn = Taut., (∃n).φn|ψn = Cont., etc.
φ1 & ψ1 & (∃n).φn & ψn = φ1 & (∃n).φn & ψn
φ2 & ψ2 & (∃n).φn & ψn = φ2 & (∃n).φn & ψn
etc. ad inf.
And in general the calculation rules for (∃n).φn & ψn can be derived from the fact that we
can write:
(∃n).φn & ψn = φ0 & ψ0 .1. φ1 & ψ1 .1. φ2 & ψ2 and so on ad inf.
It’s clear that this is not a logical sum, because “and so on ad inf.” is not a proposition. But
the notation (∃n).φn & ψn isn’t exempt from misunderstanding either, because you could
wonder why you can’t put Φn instead of φn & ψn here, though if you did (∃n)Φn would
of course be meaningless. Of course this gets cleared up if we go back to the notation
~(∃x).φx for φ0, (∃x).φx & ~(∃x, y).φx & φy for φ1, etc., and to (∃n0x).φx for φ0,
(∃n1x).φx for φ1 respectively, and so on. For then we can distinguish between
(∃n1x).φx & (∃n1x).ψx and
(∃n1x).φx & ψx
And if we make the transition to (∃n).φn & ψn, that means (∃n):(∃n nx).φx & (∃n nx).ψx
(which is not meaningless), and not (∃n):(∃n nx).φx & ψx, which is meaningless.
The expressions “of the same number”, “of the same length”, “of the same colour”, etc.
have grammars that are similar but not the same.5
– In each case it seems natural to regard
the proposition as an endless logical sum whose terms have the form φn & ψn. Moreover,
φn & ψn
4 (F): MS 113, p. 16r. 5 (V): similar, but different.
Sameness of Number. Sameness of Length. 413e
WBTC116-120 30/05/2005 16:14 Page 827
Worte mehrere verschiedene Bedeutungen, d.h., könnte selbst wieder durch mehrere
Wörter mit verschiedener Grammatik ersetzt werden. Denn ”gleichzahlig“ heißt etwas anderes,
wenn es auf Striche angewandt wird, die gleichzeitig im Gesichtsraum sind, als wenn es sich
auf die Äpfel in zwei Kisten bezieht; und ”gleichlang“ im13
Gesichtsraum angewandt ist
verschieden von ”gleichlang“ im euklidischen Raum; und die Bedeutung von ”gleichfärbig“
hängt von dem Kriterium ab, das wir für die Gleichfärbigkeit annehmen.
Wenn es sich um Flecke im Gesichtsraum handelt, die wir zu gleicher Zeit sehen, so
hat das Wort ”gleichlang“ verschiedene Bedeutung, je nachdem die Strecken unmittelbar
angrenzend oder von einander entfernt sind. In der Wortsprache hilft man sich da häufig14
mit dem Wort ”es scheint“.
Die Gleichzahligkeit, wenn es sich um eine Anzahl von Strichen handelt, ”die man übersehen
kann“, ist eine andere als die, welche nur durch Zählen der Striche festgestellt werden
kann.
15
Verschiedene Kriterien der Gleichzahligkeit: I und II die Zahl, die
man unmittelbar erkennt; III das Kriterium der Zuordnung; IV hier
muß man beide Klassen zählen; V man erkennt das gleiche Muster.
(Das sind natürlich nicht die einzigen Fälle.)
Im Fall der Längengleichheit im euklidischen Raum mag man
sagen, sie bestehe darin, daß beide Strecken die gleiche Zahl von cm
messen, beide 5cm, beide 10cm, etc. Wenn es sich aber um die
Längengleichheit zweier Strecken im Gesichtsraum handelt, so gibt es
hier nicht eine Länge L die beide haben.
Man möchte sagen: zwei Stäbe müssen immer entweder gleichlang
oder verschieden lang sein. Aber was heißt das? Es ist natürlich eine
Regel der Ausdrucksweise. ”In den zwei Kisten müssen entweder gleichviel
Äpfel oder verschiedene Anzahlen sein.“ Das Anlegen zweier Maßstäbe an je eine
Strecke soll die Methode sein, wie ich herausfinde, ob die beiden Strecken gleichlang sind:
sind sie aber gleichlang, wenn die beiden Maßstäbe gerade nicht angelegt16
sind? Wir würden
in diesem Fall sagen, wir wissen nicht, ob die beiden während dieser Zeit gleich oder
verschieden lang sind. Aber man könnte auch sagen, sie haben während dieser Zeit keine
Längen, oder etwa keine numerischen Längen.
Ähnliches, wenn auch nicht das Gleiche, gilt von der Zahlengleichheit.
Es gibt hier die Erfahrung, daß wir eine Anzahl Punkte sehen, deren Anzahl wir nicht
unmittelbar sehen können, die wir aber während des Zählens überblicken können, so daß es
Sinn hat zu sagen, sie haben sich während des Zählens nicht verändert. Anderseits aber gibt
es auch den Fall einer Gruppe von Gegenständen17
oder Flecken, die wir nicht übersehen
können, während wir sie zählen, so daß es hier das frühere Kriterium dafür, daß18
die Gruppe
sich während des Zählens nicht verändert, nicht gibt.
Russells Erklärung der Gleichzahligkeit ist aus verschiedenen Gründen ungenügend. Aber
die Wahrheit ist, daß man in der Mathematik keine solche Erklärung der Gleichzahligkeit
braucht. Hier ist überhaupt alles falsch aufgezäumt.
13 (V): ”gleichlang“ auf den
14 (V): oft
15 (F): MS 113, S. 17v.
16 (O): gerade angelegt
17 (V): Körpern
18 (O): Kriterium, daß
609
610
414 Zahlengleichheit. Längengleichheit.
I
II
III
IV
V
A B
WBTC116-120 30/05/2005 16:14 Page 828
each of these words has several different meanings, i.e. each can itself be replaced by several
words with different grammars. For “of the same number” means something different
when applied to lines simultaneously present in visual space from when it refers to the
apples in two crates; and “of the same length” applied in6
visual space is different from
“of the same length” in Euclidean space; and the meaning of “of the same colour” depends
on the criterion we adopt for sameness of colour.
If it’s a matter of patches in visual space seen simultaneously, the expression “of the same
length” varies in meaning depending on whether the line segments are immediately adjacent
to or at a distance from each other. In word-language we often7
get out of this difficulty
by using the expression “it seems”.
Sameness of number, when it is a matter of a quantity of lines “that one can take in at a
glance”, is a different sameness from that which can only be ascertained by counting the
lines.
8
Different criteria for sameness of number: in I and II the number
that one immediately recognizes; in III the criterion of correlation; in
IV we have to count both groups; in V we recognize the same pattern.
(Of course these are not the only cases.)
We might say that equality of length in Euclidean space consists in
both line segments measuring the same number of cm – both 5 cm, both
10 cm, etc. – but that where it is a case of two line segments in visual
space being of equal length, there is no one length L that both line
segments have.
One would like to say: Two rods must always be either the same length
or different lengths. But what does that mean? Of course, that’s a rule
for the way we speak. “The number of apples in the two crates must
be either the same or different.” The method whereby I discover whether two line segments
are of the same length is supposed to be the laying of a ruler against each segment: but do
they have the same length when the two rulers happen not to be laid next to them? In that
case we would say that we don’t know whether during that time the two line segments are
of the same or different lengths. But we could also say that during that time they have no
lengths, or, say, no numerical lengths.
Something similar, although not the same thing, holds for sameness of number.
Related to this is the experience where we see a multitude of dots whose number we can’t
take in immediately, but which we can keep in view while we count, so that it makes sense
to say that they haven’t changed during the counting. On the other hand, there’s also the
case of a group of objects9
or patches that we can’t keep in view while we’re counting them,
so that here we don’t have the same criterion as before for the group’s not changing while
it is being counted.
Russell’s definition of sameness of number is inadequate for various reasons. The truth
is, however, that in mathematics we don’t need any such definition of sameness of number.
Here in general everything is set up wrong.
6 (V): to
7 (V): frequently
8 (F): MS 113, p. 17v.
9 (V): bodies
Sameness of Number. Sameness of Length. 414e
I
II
III
IV
V
A B
WBTC116-120 30/05/2005 16:14 Page 829
Was uns verführt die Russell’sche oder Frege’sche, Erklärung anzunehmen, ist der
Gedanke, zwei Klassen von Gegenständen (Äpfeln in zwei Kisten) seien gleichzahlig, wenn
man sie einander 1 zu 1 zuordnen könne. Man denkt sich die Zuordnung als eine Kontrolle
der Gleichzahligkeit. Und hier macht man in Gedanken wohl noch eine Unterscheidung
zwischen Zuordnung und Verbindung durch eine Relation; und zwar wird die Zuordnung
zur Verbindung, was die ”geometrische Gerade“ zu einer wirklichen ist, eine Art idealer
Verbindung; einer Verbindung, die quasi von der Logik vorgezeichnet ist und durch die
Wirklichkeit nun nachgezogen werden kann. Es ist die Möglichkeit, aufgefaßt als eine
schattenhafte Wirklichkeit. Dies hängt dann wieder mit der Auffassung von ”(∃x)φx“ als
Ausdruck der Möglichkeit von φx zusammen.
”φ und ψ sind gleichzahlig“ (ich werde dies schreiben ”S(φ, ψ)“, oder auch einfach ”S“)
soll ja aus ”φ5 & ψ5“ folgen; aber aus φ5 & ψ5 folgt nicht, daß φ und ψ durch eine 1–1
Relation R verbunden sind (dies werde ich ”π(φ, ψ)“ oder ”π“ schreiben). Man hilft sich,
indem man sagt, es bestehe dann eine Relation der Art
”x = a & y = b .1. x = c & y = d .1. u.s.w.“.
Aber, erstens, warum definiert man dann nicht gleich S als das Bestehen einer solchen
Relation. Und wenn man darauf antwortet, diese Definition19
würde die Gleichzahligkeit
bei unendlichen Anzahlen nicht einschließen, so ist zu sagen, daß dies nur auf eine Frage
der ”Eleganz“ hinausläuft, da ich letzten Endes für endliche Zahlen meine Zuflucht doch
zu den ”extensiven“ Beziehungen nehmen müßte. Aber diese führen uns auch zu nichts:
denn, zu sagen, zwischen φ und ψ bestehe eine Beziehung – z.B. – der Form x = a &
y = b .1. x = c & y = d sagt nichts andres, als
(∃x, y).φx & φy .&. ~(∃x, y, z).φx & φy & φz :&: (∃x, y).ψx & ψy .&. ~(∃x, y, z).ψx &
ψy & ψz. (Was ich in der Form schreibe (∃n2x).φx & (∃n2x).ψx.) Und, zu sagen, zwischen
φ und ψ bestehe eine der Beziehungen x = a & y = b; x = a & y = b .1. x = c & y = d; etc.
etc., heißt nichts andres als, es bestehe eine der Tatsachen φ1 & ψ1 ; φ2 & ψ2; etc. etc. Nun
hilft man sich mit der größeren Allgemeinheit, indem man sagt, zwischen φ und ψ bestehe
irgend eine 1–1 Relation und vergißt, daß man dann doch für die Bezeichnung20
dieser
Allgemeinheit die Regel festlegen muß, nach welcher ”irgend eine Relation“ auch die Relationen
der Form x = a & y = b etc. einschließt. Dadurch, daß man mehr sagt, kommt man nicht
drum herum, das Engere zu sagen, das in dem Mehr vorhanden sein soll. (Die Logik läßt
sich nicht betrügen.)
In dem Sinne von S also, in welchem S aus φ5 & ψ5 folgt, wird es durch die Russell’sche
Erklärung nicht erklärt. Vielmehr braucht man da eine Reihe von Erklärungen
φ0 & S = φ0 & ψ0 = ψ0 & S
φ1 & S = φ1 & ψ1 = ψ1 & S
. . . α
etc. ad inf.
Dagegen wird π als Kriterium der Gleichzahligkeit gebraucht und kann natürlich in einem
andern Sinne von S auch S gleichgesetzt werden. (Und man kann dann nur sagen: Wenn in
einer21
Notation S = π ist, dann bedeutet S nichts andres als π.)
Es folgt zwar nicht π aus φ5 & ψ5, wohl aber φ5 & ψ5 aus π & φ5.
π & φ5 = π & φ5 & ψ5 = π & ψ5
u.s.w.
19 (V): Erklärung
20 (V): Beziehung
21 (V): Deiner
611
612
415 Zahlengleichheit. Längengleichheit.
5
6
7
WBTC116-120 30/05/2005 16:14 Page 830
What seduces us into accepting the Russellian or Fregean definition is the thought that
two classes of objects (apples in two crates) have the same number if they can be correlated
1 to 1. We think of correlation as a way of checking sameness of number. And here we distinguish
in thought between being correlated and being connected by a relation; and to be
precise, in relation to connection, correlation becomes what the “geometrical straight line”
is in relation to a real line, namely a kind of ideal connection; a connection that has been
sketched out by logic, as it were, and can now be drawn in bold by reality. It is possibility
conceived of as a shadowy actuality. This in turn is connected with the idea of “(∃x).φx” as
an expression of the possibility of φx.
“φ and ψ have the same number” (I will write this as “S(φ, ψ)”, or simply “S”) is
supposed to follow from “φ5 & ψ5”; but it doesn’t follow from φ5 & ψ5 that φ and ψ are
connected by a 1–1 relation R (this I will write as “π(φ, ψ)”, or “π”). We get out of the
difficulty by saying that in that case there is a relation of the type
“x = a & y = b .1. x = c & y = d .1. etc.”.
But then why don’t we define S as the existence of such a relation in the first place? And if
you reply that this definition10
wouldn’t include sameness of number in the case of infinite
numbers, we shall have to say that this boils down only to a question of “elegance”, because
for finite numbers, I would still in the end have to take refuge in “extensional” relations.
But these too get us nowhere, because saying that between φ and ψ there is a relation, e.g.
of the form x = a .&. y = b .1. x = c .&. y = d says only that
(∃x, y).φx & ψy .&. ~(∃x, y, z).φx & φy & φz :&: (∃x, y).ψx & ψy .&. ~(∃x, y, z).ψx &
ψy & ψz. (Which I write in the form (∃n 2x).φx & (∃n 2x).ψx.) And saying that between φ
and ψ there is one of the relations x = a & y = b or x = a & y = b .1. x = c & y = d, etc.,
etc., means only that one of the facts φ1 & ψ1 or φ2 & ψ2, etc., etc., obtains. Then we try
to help ourselves out by becoming even more general, saying that between φ and ψ there is
some 1–1 relation, forgetting that in order to specify this generality we still have to construct
the rule according to which “some relation” also includes relations of the form x = a &
y = b, etc. One can’t, by saying more, avoid saying the less that is supposed to occur in the
more. (Logic cannot be duped.)
So in the sense of S in which S follows from φ5 & ψ5, it is not defined by Russell’s definition.
Rather, what we need here is a series of definitions
φ0 & S = φ0 & ψ0 = ψ0 & S
φ1 & S = φ1 & ψ1 = ψ1 & S
. . . α
etc. ad inf.
On the other hand, π is used as a criterion for sameness of number and can, of course, in
another sense of S, also be equated with S. (And then we can only say: if in a given notation11
S = π, then S means nothing other than π.)
Though π doesn’t follow from φ5 & ψ5, φ5 & ψ5 does follow from π & φ5.
π & φ5 = π & φ5 & ψ5 = π & ψ5
etc.
10 (V): explanation 11 (V): in your notation
Sameness of Number. Sameness of Length. 415e
5
6
7
WBTC116-120 30/05/2005 16:14 Page 831
Also kann man schreiben:
π & φ0 = π & φ0 & ψ0 = π & φ0 & S
π & φ1 = π & φ1 & ψ1 = π & φ1 & S . . . . . β
π & φ2 = π & φ2 & ψ2 = π & φ2 & S
u.s.w. ad inf.
Und dies kann man dadurch ausdrücken, daß man sagt, die Gleichzahligkeit folge aus π.
Und man kann auch die Regel geben π & S = π, die mit den Regeln, oder der Regel, β und
der Regel α übereinstimmt.
Die Regel ”aus π folgt S“ also π & S = π könnte man auch ganz gut weglassen; die Regel
β tut denselben Dienst.
Schreibt man S in der Form
φ0 & ψ0 .1. φ1 & ψ1 .1. φ2 & ψ2 .1. . . . ad inf.,
so kann man mit grammatischen Regeln, die der gewohnten Sprache entsprechen, leicht
π & S = π ableiten. Denn
(φ0 & ψ0 .1. φ1 & ψ1 etc. ad inf.) & π = φ0 & ψ0 & π .1. φ1 & ψ1 & π .1. etc. ad
inf. = φ0 & π .1. φ1 & π .1. φ2 & π .1. etc. ad inf. = π & (φ0 1 φ1 1 φ2 1 etc.
ad inf.) = π.
Der Satz ”φ0 1 φ1 1 φ2 1 etc. ad inf.“ muß als Tautologie behandelt werden.
22
Man kann den Begriff der Gleichzahligkeit so auffassen, daß es keinen Sinn hat,
von zwei Gruppen von Punkten Gleichzahligkeit oder das Gegenteil auszusagen,
wenn es sich nicht um zwei Reihen handelt, deren eine zum mindesten einem Teil der
andern 1–1 zugeordnet ist. Zwischen solchen Reihen kann 23
dann nur von einseitiger
oder gegenseitiger Inklusion24
die Rede sein. Und diese hat eigentlich mit besondern
Zahlen so wenig zu tun, wie die Längengleichheit oder Ungleichheit im Gesichtsraum
mit Maßzahlen. Die Verbindung mit den Zahlen kann gemacht werden, muß aber
nicht gemacht werden. Wird die Verbindung mit der Zahlenreihe gemacht, so wird die
Beziehung der gegenseitigen Inklusion oder Längengleichheit der Reihen zur Beziehung der
Zahlengleichheit. Aber nun folgt nicht nur ψ5 aus π & φ5 sondern auch π aus φ5 & ψ5.
Das heißt, hier ist S = π.
22 (F): MS 113, S. 21v.
23 (V): kann n
24 (V): Einschließung
613
416 Zahlengleichheit. Längengleichheit.
5
6
7
.
.
.
. .
. .
. .
. .
. .
. .
. .
WBTC116-120 30/05/2005 16:14 Page 832
We can therefore write:
π & φ0 = π & φ0 & ψ0 = π & φ0 & S
π & φ1 = π & φ1 & ψ1 = π & φ1 & S . . . . . β
π & φ2 = π & φ2 & ψ2 = π & φ2 & S
and so on ad inf.
And we can express this by saying that the sameness of number follows from π. And we can
also give the rule (π & S) = π, which agrees with the rules, or the rule, β and the rule α.
We could perfectly well drop the rule “S follows from π”, that is, π & S = π; the rule β
does the same job.
If we write S in the form
(φ0 & ψ0) 1 (φ1 & ψ1) 1 (φ2 & ψ2) 1 . . . ad inf.,
we can easily derive (π & S) = π by using grammatical rules that correspond to ordinary
language. For
(φ0 & ψ0 .1. φ1 & ψ1 etc. ad inf.) & π = φ0 & ψ0 & π .1. φ1 & ψ1 & π .1. etc.
ad inf. = φ0 & π .1. φ1 & π .1. φ2 & π .1. etc. ad inf. = π & (φ0 1 φ1 1 φ2 1 etc.
ad inf.) = π.
The proposition “φ0 1 φ1 1 φ2 1 etc. ad inf.” must be treated as a tautology.
12
We can understand the concept of sameness of number in such a way that it
makes no sense to attribute sameness of number or its opposite to two groups of dots
except in the case of two series of which one is correlated 1–1 to at least a part of the
other. Then all we can talk about is unilateral or mutual inclusion between such series.
And this inclusion really doesn’t have any more to do with particular numbers than
equality or inequality of length in the visual field has to do with figures that we
measure. We can, but don’t have to, make a connection with numbers. If we connect
it with the number series, then the relation of mutual inclusion or of equality of length between
the series becomes the relation of sameness of number. But then not only does ψ5 follow
from π & φ5, but π also follows from φ5 & ψ5. That means that here S = π.
12 (F): MS 113, p. 21v.
Sameness of Number. Sameness of Length. 416e
.
.
.
. .
. .
. .
. .
. .
. .
. .
5
6
7
WBTC116-120 30/05/2005 16:14 Page 833
Mathematischer
Beweis.
614
WBTC116-120 30/05/2005 16:14 Page 834
Mathematical
Proof.
WBTC116-120 30/05/2005 16:14 Page 835
119
Wenn ich sonst etwas suche, so kann
ich das Finden beschreiben, auch
wenn es nicht eingetreten ist; anders,
wenn ich die Lösung eines
mathematischen Problems suche.
Mathematische Expedition
und Polarexpedition.
Wie kann es in der Mathematik Vermutungen geben? Oder vielmehr: Welcher Natur
ist das, was in der Mathematik wie eine Vermutung aussieht? Wenn ich also etwa
Vermutungen über die Verteilung der Primzahlen anstelle.
Ich könnte mir z.B. denken, daß jemand in meiner Gegenwart Primzahlen der Reihe nach
hinschriebe, ich wüßte nicht, daß es die Primzahlen sind – ich könnte etwa glauben, es seien
Zahlen, wie sie ihm eben einfielen – und nun versuchte ich irgendein Gesetz in ihnen zu
finden. Ich könnte nun geradezu eine Hypothese über diese Zahlenfolge aufstellen, wie über
jede andere, die ein physikalisches Experiment ergibt.
In welchem Sinne habe ich nun hiedurch eine Hypothese über die Verteilung der
Primzahlen aufgestellt?
Man könnte sagen, eine Hypothese in der Mathematik hat den Wert, daß sie die
Gedanken an einen bestimmten Gegenstand – ich meine ein bestimmtes Gebiet – heftet und
man könnte sagen ”wir werden gewiß etwas Interessantes über diese Dinge herausfinden“.
Das Unglück ist, daß unsere Sprache so grundverschiedene Dinge mit jedem der Worte
”Frage“, ”Problem“, ”Untersuchung“, ”Entdeckung“ bezeichnet. Ebenso mit den Worten
”Schluß“, ”Satz“, ”Beweis“.
Es frägt sich wieder, welche Art der Verifikation lasse ich für meine Hypothese gelten?
Oder kann ich vorläufig – faute de mieux – die empirische gelten lassen, solange ich noch
keinen ”strengen Beweis“ habe? Nein. Solange ein solcher Beweis nicht besteht, besteht gar
keine Verbindung zwischen meiner Hypothese und dem ”Begriff“ der Primzahl.
Erst der sogenannte Beweis verbindet die Hypothese überhaupt mit den Primzahlen als
solchen. Und das zeigt sich daran, daß – wie gesagt – bis dahin die Hypothese als eine rein
physikalische aufgefaßt werden kann. – Ist andererseits der Beweis geliefert, so beweist
er gar nicht, was vermutet worden war, denn in die Unendlichkeit hinein kann ich nicht
vermuten. Ich kann nur vermuten, was bestätigt werden kann, aber durch die Erfahrung
615
616
418
WBTC116-120 30/05/2005 16:14 Page 836
119
If I am Looking for Something in
Other Cases I Can Describe Finding
it, Even if it Hasn’t Happened; it is
Different if I am Looking for the
Solution to a Mathematical Problem.
Mathematical Expeditions and
Polar Expeditions.
How can there be conjectures in mathematics? Or rather, what sort of thing is it that looks
like a conjecture in mathematics? If, for instance, I make conjectures about the distribution
of the primes.
I could imagine, for example, that someone were writing primes one after the other in my
presence, but that I didn’t know they were the primes – I might think that they were the
numbers that just happened to occur to him – and now I might try to detect some sort of
law in them. I might form a hypothesis about this number sequence straightaway, as I might
about any other sequence of numbers generated by an experiment in physics.
Now in what sense have I, by so doing, made a hypothesis about the distribution of the
primes?
We could say that a hypothesis in mathematics is of value in so far as it fastens our thoughts
onto a particular object – I mean a particular area – and we could say “We shall surely find
out something interesting about these things”.
Unfortunately, our language uses each of the words “question”, “problem”, “investigation”,
“discovery” to refer to such fundamentally different things. As it does with the words
“conclusion”, “proposition”, “proof”.
Again, the question arises: what kind of verification do I count as valid for my
hypothesis? Or can I – for lack of something better – allow an empirical one to count for
the time being, until I have “strict proof”? No. So long as there is no such proof, there
is no connection at all between my hypothesis and the “concept” of a prime number.
Only the so-called proof establishes any connection between the hypothesis and the
primes as such. And that is shown by the fact that – as I’ve said – until then the hypothesis
can be understood as one belonging purely to physics. – On the other hand, once a proof
has been supplied, it in no way proves what had been conjectured, for I can’t conjecture on
into infinity. I can only conjecture what can be confirmed, but experience can only confirm
418e
WBTC116-120 30/05/2005 16:14 Page 837
kann nur eine endliche Zahl von Vermutungen bestätigt werden, und den Beweis kann
man nicht vermuten, solange man ihn nicht hat, und dann auch nicht.1
Angenommen, es hätte Einer den pythagoräischen Lehrsatz zwar nicht bewiesen, wäre
aber durch Messungen der Katheten und Hypotenusen2
zur ”Vermutung“ dieses Satzes
geführt worden. Und nun fände er den Beweis und sagt, er habe nun bewiesen, was er
früher vermutet hatte: so ist doch wenigstens das eine merkwürdige Frage: An welchem
Punkt des Beweises kommt denn nun das heraus, was er früher durch die einzelnen
Versuche bestätigt fand? denn der Beweis ist doch wesensverschieden von der früheren
Methode. – Wo berühren sich diese Methoden, da sie angeblich in irgendeinem Sinne das
Gleiche ergeben? D.h.: Wenn der Beweis und die Versuche nur verschiedene Ansichten
Desselben (derselben Allgemeinheit) sind.
(Ich sagte ”aus der gleichen Quelle fließt nur Eines“ und man könnte sagen, es wäre
doch zu sonderbar, wenn aus so verschiedenen Quellen dasselbe fließen sollte. Der Gedanke,
daß aus verschiedenen Quellen dasselbe fließen kann ist uns von der Physik, d.h. von den
Hypothesen so vertraut.3
Dort schließen wir immer von Symptomen4
auf die Krankheiten
und wissen, daß die verschiedensten Symptome, Symptome Desselben sein können.)
Wie konnte man nach der Statistik das vermuten, was dann der Beweis zeigte?
Wo soll aus dem Beweis dieselbe Allgemeinheit hervorspringen, die die früheren
Versuche wahrscheinlich machten?
Ich hatte die Allgemeinheit vermutet, ohne den Beweis zu vermuten (nehme ich an) und
nun beweist der Beweis gerade die Allgemeinheit, die ich vermutete!?
Angenommen, jemand untersuchte gerade Zahlen auf das Stimmen des Goldbach’schen
Satzes hin. Er würde nun die Vermutung aussprechen – und die läßt sich aussprechen
– daß, wenn er mit dieser Untersuchung fortfährt, er, solange er lebt, keinen widersprechenden
Fall antreffen werde. Angenommen, es werde nun ein Beweis des Satzes
gefunden, – beweist der dann auch die Vermutung des Mannes? Wie ist das möglich?
Nichts ist verhängnisvoller für das philosophische Verständnis, als die Auffassung von Beweis
und Erfahrung als zweier verschiedener, also doch vergleichbarer Verifikationsmethoden.
Welcher Art war Sheffers5
Entdeckung, daß p .1. q und ~p sich durch p|q ausdrücken
lassen? – Man hatte keine Methode, nach p|q zu suchen und wenn man heute eine fände,
so könnte das keinen Unterschied machen.
Was war es, was wir vor der Entdeckung nicht wußten? (Es war nichts, was wir nicht
wußten, sondern etwas, was wir nicht kannten.)
Das sieht man sehr deutlich, wenn man sich den Einspruch erhoben denkt, p|p sei gar
nicht das, was ~p sagt. Die Antwort ist natürlich, daß es sich nur darum handelt, daß
das System p|q etc. die nötige Multiplizität hat. Sheffer6
hat also ein symbolisches System
gefunden, das die nötige Multiplizität hat.
Ist es ein Suchen, wenn ich das System Sheffers7
nicht kenne und sage, ich möchte ein
System mit nur einer logischen Konstanten konstruieren. Nein!
1 (V): und auch nicht.
2 (O): Hypothenusen
3 (V): geläufig.
4 (O): Symtomen
5 (O): Scheffers
6 (O): Scheffers
7 (O): Scheffers
617
618
419 Wenn ich sonst etwas suche . . .
WBTC116-120 30/05/2005 16:14 Page 838
a finite number of conjectures, and you can’t conjecture the proof until you’ve got it, and
not then, either.
Suppose that someone hadn’t proved Pythagoras’ theorem, but had been led to “conjecture”
it by measuring the sides and hypotenuses of right-angled triangles. And suppose he
now discovered the proof, and said that now he had proved what he had earlier conjectured.
In these circumstances this is at the least a remarkable question: At what point of the
proof does what he had earlier found confirmed by individual trials emerge? For the proof,
after all, is essentially different from the earlier method. – Where do these methods make
contact, since supposedly in some sense they yield the same thing? That is, where do they
make contact if the proof and the trials are only different aspects of the same thing (the same
generality)?
(I have said: “Only one thing flows from the same spring”, and one could say that it would
certainly be odd beyond belief if the same thing were to flow from such different springs.
The idea that the same thing can come from different sources is quite familiar to us from
the physical sciences, i.e. from hypotheses. There we are always inferring illnesses from symptoms
and we know that the most varied symptoms can be symptoms of the same thing.)
How were we able to surmise from statistics the very thing the proof later showed?
From where in a proof is the same generality to spring forth that the earlier trials made
probable?
(I’m assuming that) I had conjectured the generality without conjecturing the proof, and
now the proof proves exactly the generality that I conjectured!?
Suppose someone were investigating even numbers to see if they confirmed Goldbach’s
conjecture. He might express the surmisal – and it can be expressed – that if he continued
with this investigation, he would never meet a counterexample as long as he lived. Now let’s
assume that a proof of the conjecture is discovered – will it also prove his surmisal? How is
that possible?
Nothing is more disastrous to philosophical understanding than the notion of proof and
experience as two different – yet still comparable – methods of verification.
What kind of discovery did Sheffer make when he found that p .1. q and ~p can be
expressed by p|q? No one had a method for looking for p|q, and if someone were to find
one today, that wouldn’t make any difference.
What was it that we didn’t know before the discovery? (It wasn’t anything that we didn’t
know; it was something with which we weren’t acquainted.)
You can see this very clearly if you imagine someone objecting that p|p isn’t at all
what is said by ~p. The reply of course is that it’s only a question of the system p|q, etc.,
having the requisite multiplicity. So Sheffer found a symbolic system with the requisite
multiplicity.
Is it a search if I’m unaware of Sheffer’s system and say that I’d like to construct a
system with only one logical constant. No!
If I am Looking for Something in . . . 419e
WBTC116-120 30/05/2005 16:14 Page 839
Die Systeme sind ja nicht in einem Raum, so daß ich sagen könnte: Es gibt Systeme mit
3 und 2 logischen Konstanten und nun suche ich die Zahl der Konstanten in der selben Weise
zu vermindern. Es gibt hier keine selbe Weise.
|Wenn auf die Lösung – etwa – des Fermat’schen Problems Preise ausgesetzt sind,
so könnte man mir vorhalten: Wie kannst Du sagen,8
daß es dieses Problem nicht gebe;
wenn Preise auf die Lösung ausgesetzt sind, so muß es das Problem wohl geben. Ich
müßte sagen: Gewiß, nur mißverstehen die, die darüber reden, die Grammatik des Wortes
”mathematisches Problem“ und des Wortes ”Lösung“. Der Preis ist eigentlich auf die Lösung
einer naturwissenschaftlichen Aufgabe gesetzt; (gleichsam) auf das Äußere der Lösung
(darum spricht man z.B. auch von einer Riemann’schen Hypothese). Die Bedingungen der
Aufgabe sind äußerliche; und wenn die Aufgabe gelöst ist, so entspricht, was geschehen ist,
der gestellten Aufgabe,9
wie die Lösung einer physikalischen Aufgabe dieser Aufgabe.
Wäre die Aufgabe, eine Konstruktion des regelmäßigen Fünfecks zu finden, so ist die
Konstruktion in dieser Aufgabenstellung durch das physikalische Merkmal charakterisiert,
daß sie tatsächlich ein durch Messung definiertes regelmäßiges Fünfeck liefern soll. Denn
den Begriff der konstruktiven Fünfteilung (oder des konstruktiven Fünfecks) erhalten wir ja erst
durch die Konstruktion.10
Ebenso im Fermat’schen Satz haben wir ein empirisches Gebilde, das wir als Hypothese
deuten, also – natürlich – nicht als Ende einer Konstruktion. Die Aufgabe fragt also, in
gewissem Sinne, nach etwas Anderem, als was die Lösung gibt.|
|Natürlich steht auch der Beweis des Gegenteils des Fermat’schen Satzes, z.B., – im
gleichen Verhältnis zur Aufgabe, wie der Beweis des Satzes. (Beweis der Unmöglichkeit
einer Konstruktion.)|
Sofern man die Unmöglichkeit der 3-Teilung als eine physische Unmöglichkeit darstellen
kann, indem man z.B. sagt: ”versuch‘ nicht, den Winkel in 3 gleiche Teile zu teilen, es
ist hoffnungslos!“, insofern beweist der ”Beweis der Unmöglichkeit“ diese nicht. Daß es
hoffnungslos ist, die Teilung zu versuchen, das hängt mit physikalischen Tatsachen
zusammen.
Denken wir uns, jemand stellte sich dieses11
Problem: Es ist ein Spiel zu erfinden: das
Spiel soll auf einem Schachbrett gespielt werden; jeder Spieler soll 8 Steine haben; von den
weißen Steinen sollen 2 (die ”Konsulen“), die an den Enden der Anfangsposition stehen,
durch die Regeln irgendwie ausgezeichnet sein; sie sollen eine größere Bewegungsfreiheit
haben als die andern; von den schwarzen Steinen soll einer (der ”Feldherr“) ein12
ausgezeichneter
sein; ein weißer Stein nimmt einen schwarzen (und umgekehrt), indem er sich
an dessen Stelle setzt; das ganze Spiel soll eine gewisse Analogie mit den Punischen
Kriegen haben. Das sind die Bedingungen, denen das Spiel zu genügen hat. – Das ist gewiß
eine Aufgabe, und eine Aufgabe ganz andrer Art, als die, herauszufinden, wie Weiß im
Schachspiel unter gewissen Bedingungen gewinnen könne. – Denken wir uns nun aber das
Problem:13
”Wie kann Weiß in dem14
Kriegsspiel, dessen Regeln wir noch nicht genau
kennen, in 20 Zügen gewinnen?“ – Dieses Problem wäre ganz analog den Problemen der
Mathematik (nicht ihren Rechenaufgaben).
8 (V): behaupten,
9 (V): der Stellung der Aufgabe,
10 (V): Fünfecks) haben wir ja noch gar nicht.
11 (V): folgendes
12 (O): ”Feldherr“ ein
13 (V): aber die Frage:
14 (V): unserm
619
620
420 Wenn ich sonst etwas suche . . .
WBTC116-120 30/05/2005 16:14 Page 840
For the systems aren’t in one space, so that I could say: There are systems with 3 and 2
logical constants and now I’ll try to reduce the number of constants in the same way. There
is no same way here.
|If prizes have been offered for the solution – say – of Fermat’s theorem, someone might
reproach me: How can you say1
that this problem doesn’t exist? If prizes have been offered
for its solution, then surely the problem must exist. I would have to say: Certainly, it’s just
that the people who talk about it misunderstand the grammar of the expression “mathematical
problem” and of the word “solution”. The prize has actually been offered for the solution
of a scientific problem; for the exterior of the solution (as it were) (that’s why we also talk
about a Riemannian hypothesis, for instance). The conditions of the problem are external
conditions; and when the problem has been solved, what has happened corresponds to the
problem that was set up2
in the same way in which the solution to a problem in physics
corresponds to that problem.|
If our assignment were to find a construction for a regular pentagon, then in the settingup
of the problem this construction is characterized by the physical criterion that it really
is supposed to yield a regular pentagon, as defined by measurement. For we don’t get the
concept of constructive division into five (or of a constructive pentagon) until we get it from the
construction.3
Likewise in Fermat’s theorem we have an empirical structure that we interpret as a
hypothesis, and so not – of course – as the end of a construction. So in a certain sense the
problem asks for something other than what the solution gives.
|Of course a proof of the contradictory of Fermat’s theorem (for instance) stands in the
same relation to the problem as a proof of the theorem itself. (Proof of the impossibility of
a construction.)|
In so far as we can represent the impossibility of the trisection of an angle as a physical
impossibility (by saying things like “Don’t try to divide an angle into 3 equal parts, it’s
hopeless!”) the “proof of impossibility” does not prove this impossibility. That it is hopeless
to attempt the trisection is connected with physical facts.
Imagine someone gave himself this4
problem. He is to invent a game; the game is to
be played on a chessboard; each player is to have eight pieces; two of the white ones (the
“consuls”) that are at the ends of the beginning position of the game are to be given some
special status by the rules; they are to have a greater freedom of movement than the other
pieces; one of the black pieces (the “general”) is to have a special status; a white piece takes
a black one (and vice versa) by being put in its place; the whole game is to have a certain
analogy with the Punic wars. Those are the conditions that the game has to satisfy. – There
is no doubt that that is a problem, a problem of a completely different kind from that of
finding out how under certain conditions white can win in chess. – But now let’s imagine
the problem:5
“How can white win in 20 moves in the6
war game whose rules we don’t yet
know precisely?” – That problem would be quite analogous to the problems of mathematics
(not to its problems of calculation).
1 (V): claim
2 (V): to the setting-up of the problem
3 (V): For we don’t even have the concept of
constructive division into five (or of a constructive
pentagon) yet.
4 (V): himself the following
5 (V): question:
6 (V): our
If I am Looking for Something in . . . 420e
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Was versteckt ist, muß gefunden werden können. (Versteckter Widerspruch.)
Was versteckt ist, muß sich auch, ehe es gefunden wurde, ganz beschreiben lassen, als
wäre es (schon) gefunden.
Wenn man sagt, der Gegenstand ist so versteckt, daß es unmöglich ist, ihn zu finden, so
hat das guten Sinn und die Unmöglichkeit ist hier natürlich keine logische; d.h., es hat Sinn,
von dem Finden des Gegenstandes zu reden und auch, es zu beschreiben; und wir leugnen
nur, daß es15
geschehen wird.
16
Man könnte so sagen: Wenn ich etwas suche – ich meine, den Nordpol, oder ein Haus
in London – so kann ich das, was ich suche, vollständig beschreiben, ehe ich es gefunden
habe (oder gefunden habe, daß es nicht da ist) und diese Beschreibung wird in jedem
Fall logisch einwandfrei sein. Während ich im Falle des ”Suchens“ in der Mathematik, wo
es nicht in einem System geschieht, das was ich suche, nicht beschreiben kann, oder nur
scheinbar; denn, könnte ich es in allen Einzelheiten beschreiben, so hätte ich es eben
schon, und ehe es vollständig beschrieben ist, kann ich nicht sicher sein, ob das was ich suche,
logisch einwandfrei ist, sich also überhaupt beschreiben läßt; d.h. diese unvollkommene
Beschreibung läßt gerade das aus, was notwendig wäre, damit etwas gesucht werden könnte.
Sie ist also nur eine Scheinbeschreibung des ”Gesuchten“.
Irregeführt wird man hier leicht durch die Rechtmäßigkeit einer unvollkommenen
Beschreibung im Falle des Suchens eines wirklichen Gegenstandes, und hier spielt wieder
eine Unklarheit über die Begriffe ”Beschreibung“ und ”Gegenstand“ hinein. Wenn man sagt,
ich gehe auf den Nordpol und erwarte mir dort eine Flagge zu finden, so hieße das in
der Russell’schen Auffassung: ich erwarte mir Etwas (ein X) zu finden, das eine Flagge –
etwa von dieser und dieser Farbe und Größe – ist. Und es scheint dann, als bezöge sich
die Erwartung (das Suchen) auch hier nur auf eine indirekte Kenntnis17
und nicht auf den
Gegenstand selbst, den ich erst dann direkt18
kenne (knowledge by acquaintance), wenn
ich ihn vor mir habe (während ich vorher19
nur indirekt mit ihm bekannt bin). Aber das ist
Unsinn. Was immer ich dort wahrnehmen kann – soweit es eine Bestätigung meiner
Erwartung ist – kann ich auch schon vorher beschreiben. Und ”beschreiben“ heißt hier
nicht, etwas darüber aussagen, sondern es aussprechen, d.h.: Was ich suche, muß ich
vollständig beschreiben können.
Die Frage ist: Kann man sagen, daß die Mathematik heute gleichsam ausgezackt – oder
ausgefranst – ist und daß man sie deshalb wird abrunden können. Ich glaube, man kann das
erstere nicht sagen, ebensowenig wie man sagen kann, die Realität sei struppig, weil es
4 primäre Farben, sieben Töne in einer Oktave,20
drei Dimensionen im Sehraum etc. gäbe.
Die Mathematik ”abrunden“21
kann man so wenig, wie man sagen kann ”runden wir die
vier primären Farben auf fünf oder zehn ab“, oder ”runden wir die acht Töne einer Oktave22
auf zehn ab“.
Vergleich zwischen einer mathematischen Expedition und einer Polarexpedition. Diesen
Vergleich anzustellen hat Sinn und ist sehr nützlich.
15 (V): das
16 (M): ///
17 (V): eine Beschreibung
18 (V): eigentlich
19 (V): früher
20 (O): Oktav,
21 (V): Mathematik abrunden
22 (O): Oktav
621
622
421 Wenn ich sonst etwas suche . . .
WBTC116-120 30/05/2005 16:14 Page 842
It must be possible to find what has been hidden. (Hidden contradiction.)
What has been hidden must also be completely describable before it is found, as if it had
(already) been found.
It makes good sense to say that an object is hidden in such a way that it is impossible to
find it; and of course the impossibility here is not a logical one; that is, it makes sense to
talk of finding the object and also to describe the finding; we are merely denying that it7
will happen.
8
We could put it like this: If I’m searching for something – I mean, the North Pole, or a
house in London – I can completely describe what I am looking for before I’ve found it (or
have found that it isn’t there) and either way this description will be logically unobjectionable.
Whereas if I’m “searching” for something in mathematics, unless I’m doing so within
a system, I cannot describe what I am looking for, or can do so only apparently; for if I could
describe it in every particular, then I would already have it; and before it has been completely
described I can’t be sure whether what I’m searching for is logically unobjectionable, and
therefore describable at all. That is to say, this incomplete description leaves out just what
would enable me to search for something. So it is only an apparent description of what is
being “searched for”.
Here we are easily misled by the legitimacy of an incomplete description when we are
searching for a real object, and here again a lack of clarity about the concepts “description”
and “object” comes into play. If someone says, I am going to the North Pole and I expect
to find a flag there, that would mean, along Russellian lines: I expect to find something
(an x) that is a flag – say of such and such a colour and size. And then it seems as if in
this case as well, the expectation (the searching) referred only to indirect knowledge,9
and
not to the object itself – which object I don’t know directly10
(knowledge by acquaintance)
until I have it in front of me (whereas previously I was only indirectly acquainted with it).
But that is nonsense. Whatever I can perceive there – to the extent that it is a fulfilment of
my expectation – I can also describe in advance. And here “describe” doesn’t mean saying
something about it, but rather expressing it. That is, I must be able to describe completely
what I am looking for.
The question is: Can one say that at present mathematics is as it were jagged – or frayed
– and that for that reason we shall be able to round it off? I think you can’t say the former,
any more than you can say that reality is unkempt because there are 4 primary colours,
7 notes in an octave, 3 dimensions in visual space, etc.
You can’t “round off”11
mathematics any more than you can say “Let’s round off the
4 primary colours to 5 or 10” or “Let’s round off the 8 tones in an octave to 10”.
The comparison between a mathematical expedition and a polar expedition. It makes
sense and is very useful to draw this comparison.
7 (V): that that
8 (M): ///
9 (V): to a description,
10 (V): don’t really know
11 (V): round off
If I am Looking for Something in . . . 421e
WBTC116-120 30/05/2005 16:14 Page 843
Wie seltsam wäre es, wenn eine geographische Expedition nicht sicher wüßte, ob sie
ein Ziel, also auch ob sie überhaupt einen Weg hat. Das können wir uns nicht denken, es
gibt Unsinn. Aber in der mathematischen Expedition verhält es sich gerade so. Also wird
es vielleicht am besten sein, den Vergleich ganz fallen zu lassen.
Es wäre wie eine Expedition, die des Raumes nicht sicher wäre!
Könnte man sagen, daß die arithmetischen oder geometrischen Probleme immer so ausschauen,
oder fälschlich so aufgefaßt werden können, als bezögen sie sich auf Gegenstände
im Raum, während sie sich auf den Raum selbst beziehen?
Raum nenne ich das, dessen man beim Suchen gewiß sein kann.
623
422 Wenn ich sonst etwas suche . . .
WBTC116-120 30/05/2005 16:14 Page 844
How strange it would be if a geographical expedition didn’t know for sure whether it had
a goal, and so also whether it even had a route. We can’t imagine such a thing; it’s nonsense.
But this is precisely the way things are in a mathematical expedition. And so perhaps it will
be best to drop the comparison altogether.
It would be like an expedition that wasn’t sure of space!
Can one say that arithmetical or geometrical problems always seem, or can be falsely
conceived, to refer to objects in space, whereas they refer to space itself?
I call “space” what one can be certain of while searching.
If I am Looking for Something in . . . 422e
WBTC116-120 30/05/2005 16:14 Page 845
120
Beweis, und Wahrheit und Falschheit
eines mathematischen Satzes.
Der bewiesene mathematische Satz hat in seiner Grammatik zur Wahrheit hin ein
Übergewicht. Ich kann, um den Satz von 25 × 25 = 625 zu verstehen, fragen: wie wird dieser
Satz bewiesen. Aber ich kann nicht fragen: wie wird – oder würde – sein Gegenteil
bewiesen; denn es hat keinen Sinn, vom Beweis des Gegenteils von 25 × 25 = 625 zu reden.
Will ich also eine Frage stellen, die von der Wahrheit des Satzes unabhängig ist, so muß
ich von der Kontrolle seiner Wahrheit, nicht von ihrem Beweis, oder Gegenbeweis, reden.
Die Methode der Kontrolle entspricht dem, was man den Sinn des mathematischen Satzes
nennen kann. Die Beschreibung dieser Methode ist allgemein und bezieht sich auf ein System
von Sätzen, etwa den Sätzen der Form a × b = c.
Man kann nicht sagen: ”ich werde ausrechnen, daß es so ist“, sondern ”ob es so ist“. Also,
ob so, oder anders.
Die Methode der Kontrolle der Wahrheit entspricht dem Sinn des mathematischen
Satzes. Kann von so einer Kontrolle nicht die Rede sein, dann bricht die Analogie der
”mathematischen Sätze“ mit dem, was wir sonst Satz nennen, zusammen. So gibt es eine
Kontrolle für die Sätze der Form ”(∃k)n
m . . .“ und ”~(∃k)n
m . . .“, die sich auf Intervalle beziehen.
Denken wir1
nun an die Frage: ”hat die Gleichung x2
+ ax + b = 0 eine reelle Lösung“.
Hier gibt es wieder eine Kontrolle und die Kontrolle scheidet zwischen den Fällen (∃ . . . )
etc. und ~(∃ . . . ) etc. Kann ich aber in demselben Sinne auch fragen und kontrollieren,
”ob die Gleichung eine Lösung hat“? es sei denn, daß ich diesen Fall wieder mit andern in
ein System bringe.
(In Wirklichkeit konstruiert der ”Beweis des Hauptsatzes der Algebra“ eine neue Art von
Zahlen.)
Gleichungen sind eine Art von Zahlen. (D.h. sie können den Zahlen ähnlich behandelt
werden.)
Der ”Satz der Mathematik“, welcher durch eine Induktion bewiesen ist, – so2
aber, daß
man nach dieser Induktion nicht in einem System von Kontrollen fragen3
kann, – ist nicht
”Satz“ in dem Sinne, in welchem es4
die Antwort auf eine mathematische Frage ist.
”Jede Gleichung G hat eine Wurzel.“ Und wie, wenn sie keine hat? können wir diesen
Fall beschreiben, wie den, daß sie keine rationale Lösung hat? Was ist das Kriterium dafür,
daß eine Gleichung keine Lösung hat? Denn dieses Kriterium muß gegeben werden,5
wenn
1 (V): Wir an
2 (O): ist –, so
3 (V): suchen
4 (V): er
5 (V): sein,
624
625
423
WBTC116-120 30/05/2005 16:14 Page 846
120
Proof, and the Truth and Falsity of
Mathematical Propositions.
A mathematical proposition that has been proved tips the scale towards truth in its grammar.
In order to understand the proposition 25 × 25 = 625 I can ask: How is this proposition
proved? But I can’t ask how its contradictory is or would be proved, because it makes
no sense to speak of a proof of the contradictory of 25 × 25 = 625. So if I want to ask a
question that’s independent of the truth of the proposition, I have to speak of checking its
truth, not of proving or disproving it. The method of checking corresponds to what one
can call the sense of a mathematical proposition. The description of this method is general,
and refers to a system of propositions, for instance of propositions of the form a × b = c.
We can’t say “I am going to figure out that it is so”; rather, we have to say “whether it is
so”, i.e. whether it is so or otherwise.
The method of checking the truth corresponds to the sense of a mathematical proposition.
If it’s impossible to speak of such a check, then the analogy between “mathematical
propositions” and what we otherwise call propositions collapses. Thus there is a check for
propositions of the form “(∃k)n
m . . .” and “~(∃k)n
m . . .”, which refer to intervals.
Now let’s consider the question “Does the equation x2
+ ax + b = 0 have a solution in real
numbers?” Here again there is a check, and the check makes a distinction between (∃ . . . ),
etc. and ~(∃ . . . ), etc. But can I in the same sense also ask and check “whether the equation
has a solution”? Not unless I include this case too in a system with others.
(In reality the “proof of the fundamental theorem of algebra” constructs a new kind of
number.)
Equations are a kind of number. (That is, they can be treated similarly to the numbers.)
A “proposition of mathematics” that has been proved by induction is not a “proposition”
in the same sense as the answer to a mathematical question, unless one can ask1
for this
induction in a system of checks.
“Every equation G has a root.” And what if it doesn’t have one? Can we describe that
case as we can describe the one where the equation doesn’t have a rational solution? What
is the criterion for an equation not having a solution? For this criterion must be given2
if
1 (V): search 2 (V): must have been given
423e
WBTC116-120 30/05/2005 16:14 Page 847
die mathematische Frage einen Sinn haben soll und wenn der Existenzsatz6
Antwort auf eine
Frage sein soll.7
(Worin besteht die Beschreibung des Gegenteils; worauf stützt sie sich; auf welche Beispiele,
und wie sind diese Beispiele mit einem besonderen Fall des bewiesenen Gegenteils verwandt?
Diese Fragen sind nicht etwa nebensächlich, sondern absolut wesentlich.)
(Die Philosophie der Mathematik besteht in einer genauen Untersuchung der mathematischen
Beweise – nicht darin, daß man die Mathematik mit einem Dunst umgibt.)
Wenn in den Diskussionen über die Beweisbarkeit der mathematischen Sätze gesagt wird,
es gäbe wesentlich Sätze der Mathematik, deren Wahrheit oder Falschheit unentschieden
bleiben müsse, so bedenken8
die, die es sagen,9
nicht, daß solche Sätze, wenn wir sie gebrauchen
können und ”Sätze“ nennen wollen, ganz andere Gebilde sind, als was sonst ”Satz“ genannt
wird: denn der Beweis ändert die Grammatik des Satzes. Man kann wohl ein und dasselbe
Brett einmal als Windfahne, ein andermal als Wegweiser verwenden; aber das feststehende
nicht als Windfahne und das bewegliche nicht als Wegweiser. Wollte jemand sagen ”es
gibt auch bewegliche Wegweiser“, so würde ich ihm antworten: ”Du willst wohl sagen,
’es gibt auch bewegliche Bretter‘; und ich sage nicht, daß das bewegliche Brett unmöglich
irgendwie verwendet werden kann, – nur nicht als Wegweiser“.
Das Wort ”Satz“, wenn es hier überhaupt Bedeutung haben soll, ist äquivalent einem
Kalkül und zwar jedenfalls dem,10
in welchem p .1. ~p = Taut. ist (das ”Gesetz des
ausgeschlossenen Dritten“ gilt). Soll es nicht gelten, so haben wir den Begriff des Satzes
geändert. Aber wir haben damit keine Entdeckung gemacht (etwas gefunden, das ein Satz
ist, und dem und dem Gesetz nicht gehorcht); sondern eine neue Festsetzung getroffen, ein
neues Spiel angegeben.
6 (O): Existenzssatz
7 (V): und wenn das, was die Form eines
Existenzsatzes hat, ”Satz“ im Sinne der
Antwort auf eine Frage sein soll.
8 (V): wissen
9 (O): bedenken // wissen // ,die es sagen,
10 (O): den,
627
626
626
424 Beweis . . .
WBTC116-120 30/05/2005 16:14 Page 848
the mathematical question is to have sense, and if the existence proposition is to be an
answer to a question.3
(What does the description of the contradictory consist in? On what is it based? On
what examples, and how are these examples related to particular cases of the proved
contradictory? Far from being side issues, these questions are absolutely essential.)
(The philosophy of mathematics consists in an exact investigation of mathematical proofs
– not in surrounding mathematics with a mist.)
If it’s said in discussions about the provability of mathematical propositions that in
essence there are propositions of mathematics whose truth or falsehood must remain
undecided, those who say that don’t consider4
that such propositions, if we can use them
and want to call them “propositions”, are completely different structures from what is
otherwise called “propositions”; for a proof alters the grammar of a proposition. You can
use one and the same board now as a weathervane and now as a signpost; but you can’t
use it as a weathervane when it’s fixed nor as a signpost when it’s movable. If someone were
to say “There are also movable signposts” I would answer him “You want to say, don’t you,
‘There are also movable boards’; and I’m not saying that a movable board can’t possibly be
used in some way – just not as a signpost”.
The word “proposition”, if it is to have any meaning here at all, is equivalent to a calculus:
to a calculus, in any case, in which p .1. ~p = Taut. (in which the “law of the excluded
middle” is valid). If it is not supposed to be valid, then we have altered the concept of
a proposition. But in so doing we haven’t made a discovery (have not found something
that is a proposition and doesn’t obey such and such a law); rather we have made a new
stipulation, set up a new game.
3 (V): and if what has the form of an existence
proposition is to be a “proposition” in the sense
of an answer to a question.
4 (V): know
Proof . . . 424e
WBTC116-120 30/05/2005 16:14 Page 849
121
Wenn Du wissen willst, was bewiesen
wurde, schau den Beweis an.
Die Mathematiker verirren sich nur dann, wenn sie über Kalküle im Allgemeinen reden
wollen; und zwar darum, weil sie dann die besondern Bestimmungen vergessen, die jedem
besonderen Kalkül zu Grunde liegen.1
Der Grund, warum alle Philosophen der Mathematik fehlgehen, ist der, daß man in der
Logik nicht allgemeine Dicta durch Beispiele begründen kann, wie in der Naturgeschichte.
Sondern jeder besondere Fall hat die volle2
Bedeutung, aber alles ist mit ihm erschöpft. Und3
man kann keinen allgemeinen Schluß aus ihm ziehen (also keinen Schluß).
Eine logische Fiktion gibt es nicht und darum kann man nicht mit logischen Fiktionen
arbeiten; und muß jedes Beispiel ganz ausführen.
In der Mathematik kann es nur mathematische troubles4
geben, nicht philosophische.
Der Philosoph notiert eigentlich nur das, was der Mathematiker so gelegentlich über seine
Tätigkeit hinwirft.
Der Philosoph kommt leicht in die Lage eines ungeschickten Direktors, der, statt seine
Arbeit zu tun und nur darauf zu schauen, daß seine Angestellten ihre Arbeit richtig machen,
ihnen ihre Arbeit abnimmt und sich so eines Tages mit fremder Arbeit überladen sieht,
während die Angestellten zuschaun und ihn kritisieren.
Besonders ist er geneigt, sich die Arbeit des Mathematikers aufzuhalsen.
Wenn Du wissen willst, was der Ausdruck ”Stetigkeit einer Funktion“ bedeutet, schau’
den Beweis der Stetigkeit an; der wird ja zeigen, was er beweist. Aber sieh nicht das Resultat
an, wie es in Prosa ausgedrückt5
ist und auch nicht, wie es in der Russell’schen Notation
lautet, die ja bloß eine Übersetzung des Prosaausdrucks ist; sondern richte Deinen Blick
dorthin, wo im Beweis noch gerechnet wird. Denn der Wortausdruck des angeblich
bewiesenen Satzes ist meist irreführend, denn er verschleiert das eigentliche Ziel des
Beweises, das in diesem mit voller Klarheit zu sehen ist.
”Wird die Gleichung von irgend welchen Zahlen befriedigt?“; ”sie wird von Zahlen
befriedigt“; ”sie wird von allen Zahlen (von keiner Zahl) befriedigt“. Hat Dein Kalkül Beweise?
und welche? daraus erst wird man den Sinn dieser Sätze und Fragen entnehmen können.
Sage mir wie Du suchst und ich werde Dir sagen was Du suchst.
1 (V): Kalkül als Grundlage dienen.
2 (V): größtmögliche
3 (V): Bedeutung, und anderseits // wieder // ist
mit ihm alles erschöpft, und
4 (V): Schwierigkeiten
5 (V): hingeschrieben
628
629
630
425
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121
If you Want to Know What was
Proved, Look at the Proof.
Mathematicians always get lost when they start to talk about calculi in general; they do so
because they forget the particular stipulations that are the foundations1
of each particular
calculus.
The reason every philosopher of mathematics goes astray is that in logic, as opposed to
natural history, one cannot justify generalizations with examples. Rather, meaning2
is fully
contained in the particular case, and it is exhaustive. You3
can’t draw any general conclusion
from it (and that means any conclusion).
There is no such thing as a logical fiction, and therefore you can’t work with logical
fictions; you have to work out each example fully.
In mathematics there can only be mathematical troubles4
, not philosophical ones.
Really, all the philosopher ever jots down is what the mathematician occasionally dashes
off about his activities.
The philosopher easily gets into the position of a clumsy manager, who, instead of doing
his own work and merely making sure that his employees are doing their work well, relieves
them of their work and then one day finds himself overburdened with other people’s work,
while his employees watch and criticize him.
He is particularly inclined to saddle himself with the work of the mathematician.
If you want to know what the expression “continuity of a function” means, look at the
proof of continuity; that will show what it proves. But don’t look at the result as it is expressed5
in prose, or as it reads in Russellian notation, which is simply a translation of the prose expression;
rather, direct your attention to the part of the proof where calculation is still going on.
For the verbal expression of the allegedly proved proposition is in most cases misleading,
because it veils the real goal of the proof, which can be seen completely clearly in the
proof itself.
“Is the equation satisfied by any numbers?”; “It is satisfied by numbers”; “It is satisfied
by all numbers (no number)”. Does your calculus have proofs? And what proofs? It is only
from them that we will be able to gather the sense of these propositions and questions.
Tell me how you are searching and I will tell you what you are searching for.
1 (V): that serve as the foundation
2 (V): Rather, the greatest possible meaning
3 (V): case, and on the other hand you // and then
again you
4 (V): difficulties
5 (V): as it has been written down
425e
WBTC121-130 30/05/2005 16:14 Page 851
Wir werden uns zuerst fragen müssen: Ist der mathematische Satz bewiesen? und wie?
Denn der Beweis gehört zur Grammatik des Satzes! – Daß das so oft nicht eingesehen wird,
kommt daher, daß wir hier wieder auf der Bahn einer uns irreführenden Analogie denken.
Es ist, wie gewöhnlich in diesen Fällen, eine Analogie aus unserm naturwissenschaftlichen
Denken. Wir sagen z.B. ”dieser Mann ist vor 2 Stunden gestorben“, und wenn man uns
fragt ”wie läßt sich das feststellen“, so können wir eine Reihe von Anzeigen (Symptomen)6
dafür angeben. Wir lassen aber auch die Möglichkeit dafür offen, daß etwa die Medizin
bis jetzt unbekannte Methoden entdeckt, die Zeit des Todes festzustellen und das heißt:
Wir können solche mögliche Methoden auch jetzt schon beschreiben, denn nicht
ihre Beschreibung wird entdeckt, sondern, es wird nur experimentell festgestellt, ob die
Beschreibung den Tatsachen entspricht. So kann ich z.B. sagen: eine Methode besteht darin,
die Quantität des Hämoglobins im Blut zu finden, denn diese nehme mit der Zeit nach dem
Tode, nach dem und dem Gesetz, ab. Das stimmt natürlich nicht, aber, wenn es stimmte,
so würde sich dadurch an der von mir erdichteten Beschreibung nichts ändern. Nennt man
nun die medizinische Entdeckung ”die Entdeckung eines Beweises dafür, daß der Mann vor
2 Stunden gestorben ist“, so muß man sagen, daß diese Entdeckung an der Grammatik des
Satzes ”der Mann ist vor 2 Stunden gestorben“ nichts ändert. Die Entdeckung ist die
Entdeckung, daß eine bestimmte Hypothese wahr ist (oder: mit den Tatsachen übereinstimmt).
Diese Denkweise sind wir nun so gewöhnt, daß wir den Fall der Entdeckung eines
Beweises in der Mathematik unbesehen für den gleichen oder einen ähnlichen halten. Mit
Unrecht: denn, kurz gesagt, den mathematischen Beweis konnte man nicht beschreiben, ehe
er gefunden war.
Der ”medizinische Beweis“ hat die Hypothese, die er bewiesen hat, nicht in einen neuen
Kalkül eingegliedert und ihm also keinen neuen Sinn gegeben; der mathematische Beweis
gliedert den mathematischen Satz in einen neuen Kalkül ein, er verändert seine Stellung in
der Mathematik. Der Satz mit seinem Beweis gehört einer andern Kategorie an, als der Satz
ohne den Beweis. (Der unbewiesene mathematische Satz – Wegweiser der mathematischen
Forschung, Anregung zu mathematischen Konstruktionen.)
Sind die Variablen von derselben Art in den Gleichungen:
x2
+ y2
+ 2xy = (x + y)2
x2
+ 3x + 2 = 0
x2
+ ax + b = 0
x2
+ xy + z = 0 ?
Das kommt auf die Verwendung dieser Gleichungen an. – Aber der Unterschied zwischen
No
1 und No
27
(wie sie gewöhnlich gebraucht werden) ist nicht einer der Extension der Werte,
die sie8
befriedigen. Wie beweist Du den Satz ”Nr. 1 gilt für alle Werte von x und y“ und
wie den Satz ”es gibt Werte von x, die Nr. 2 befriedigen“? So viel Analogie in diesen Beweisen
ist, soviel Analogie ist im Sinn der beiden Sätze.
Aber kann ich nicht von einer Gleichung sagen: ”Ich weiß, sie stimmt für einige Substitutionen
nicht – ich erinnere mich nicht, für welche – ; ob sie aber allgemein nicht stimmt,
das weiß ich nicht“? – Aber was meinst Du damit, wenn Du sagst, Du weißt das? Wie weißt
Du es? Hinter den Worten ”ich weiß . . .“ ist ja nicht ein bestimmter Geisteszustand, der
der Sinn dieser Worte wäre. Was kannst Du mit diesem Wissen anfangen? denn das wird
6 (O): (”Symptomen)
7 (V): zwischen Nr. 1 und Nr. 2
8 (V): sich
631
632
426 Wenn Du wissen willst . . .
WBTC121-130 30/05/2005 16:14 Page 852
We shall first have to ask ourselves: Has the mathematical proposition been proved? And
how? For the proof is part of the grammar of the proposition! – The fact that this is so often
not understood arises from our thinking once again along the lines of a misleading analogy.
As usual in these cases, it is an analogy from the way we think in the natural sciences. We
say, for example, “This man died two hours ago” and if someone asks us “How can you
tell that?” we can give a series of indications (symptoms) for it. But we also leave open the
possibility that, say, medicine will discover hitherto unknown methods for ascertaining the
time of death. And that means that we can already describe such possible methods; for it
isn’t their description that’s discovered. Rather, the only thing anyone ascertains experimentally
is whether the description corresponds to the facts. Thus I can say, for example: One method
consists in discovering the quantity of haemoglobin in the blood, because this diminishes
according to such and such a law in proportion to the time after death. Of course that isn’t
correct, but if it were correct, nothing in the description I’ve made up would change because
of it. Now if you call the medical discovery “the discovery of a proof that the man died two
hours ago” you must also say that this discovery doesn’t change anything in the grammar
of the proposition “The man died two hours ago”. The discovery is the discovery that a
particular hypothesis is true (or: agrees with the facts). Now we are so accustomed to this
way of thinking that we take the case of the discovery of a proof in mathematics, sight
unseen, as being the same, or similar. We are wrong to do so because, to put it concisely,
the mathematical proof couldn’t be described before it had been discovered.
The “medical proof” didn’t incorporate the hypothesis it proved into a new calculus,
and so didn’t give it a new sense; a mathematical proof incorporates the mathematical
proposition into a new calculus, and alters its position in mathematics. A proposition with
its proof belongs to a different category than a proposition without a proof. (Unproved
mathematical propositions – signposts for mathematical research, stimuli for mathematical
constructions.)
Are the variables in the following equations all of the same kind?
x2
+ y2
+ 2xy = (x + y)2
x2
+ 3x + 2 = 0
x2
+ ax + b = 0
x2
+ xy + z = 0
That depends on the use of these equations. – But the distinction between No. 1 and
No. 26
(as they are ordinarily used) is not between the extension of the values that satisfy
them. How do you prove the proposition “No. 1 holds for all values of x and y” and how
do you prove the proposition “There are values of x that satisfy No. 2?” There’s as much
similarity in these proofs as there is in the senses of the two propositions.
But can’t I say of an equation “I know it doesn’t hold for some substitutions. I’ve
forgotten for which; but whether it fails to hold in general, I don’t know”? But what do you
mean when you say you know that? How do you know it? It’s not as if behind the words
“I know . . .” there’s a certain state of mind that’s the sense of those words. What can you
do with that knowledge? For that’s what will show what that knowledge consists in. Do
6 (V): Nr. 1 and Nr. 2
If you Want to Know What was Proved . . . 426e
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zeigen, worin dieses Wissen besteht. Kennst Du eine Methode, um festzustellen, daß die
Gleichung allgemein ungiltig ist? Erinnerst Du Dich daran, daß die Gleichung für einige
Werte von x zwischen 0 und 1000 nicht stimmt? Hat Dir jemand bloß die Gleichung gezeigt
und gesagt, er habe Werte für x gefunden, die die Gleichung nicht befriedigen, und weißt
Du vielleicht selbst nicht, wie man dies für einen gegebenen Wert konstatiert? etc. etc.
”Ich habe ausgerechnet, daß es keine Zahl gibt, welche . . .“. – In welchem
Rechnungssystem kommt diese Rechnung vor? – Dies wird uns zeigen, in welchem
Satzsystem der errechnete Satz ist. (Man fragt auch: ”wie rechnet man so etwas aus?“)
”Ich habe gefunden, daß es eine solche9
Zahl gibt“.
”Ich habe ausgerechnet, daß es keine solche Zahl gibt“.
Im ersten Satz darf ich nicht ”keine“ statt ”eine“ einsetzen. – Und wie, wenn ich im zweiten
statt ”keine“ ”eine“ setze? Nehmen wir an, eine10
Rechnung ergibt nicht den Satz ”~(∃n)
etc.“, sondern ”(∃n) etc.“. Hat es dann etwa Sinn zu sagen: ”nur Mut! jetzt mußt Du
einmal auf eine solche Zahl kommen, wenn Du nur lang genug probierst“? Das hat nur
Sinn, wenn der Beweis nicht ”(∃n) etc.“ ergeben, sondern dem Probieren Grenzen gesteckt
hat, also etwas ganz anderes geleistet hat. D.h., das, was wir den Existenzsatz nennen, der
uns eine Zahl suchen lehrt, hat zum Gegenteil nicht den Satz ”(n) etc.“, sondern einen Satz,
der sagt, daß in dem und dem Intervall keine Zahl ist, die . . . . Was ist das Gegenteil des
Bewiesenen? – Dazu muß man auf den Beweis schauen. Man kann sagen: das Gegenteil des
bewiesenen Satzes ist das, was statt seiner durch einen bestimmten Rechnungsfehler im
Beweis bewiesen worden wäre. Wenn nun z.B. der Beweis, daß ~(∃n) etc. der Fall ist, eine
Induktion ist die zeigt, daß, soweit ich auch gehe, eine solche Zahl nicht vorkommen kann,
so ist das Gegenteil dieses Beweises (ich will einmal diesen Ausdruck gebrauchen) nicht
der Existenzbeweis in unserem Sinne. – Es ist hier nicht, wie im Fall des Beweises, daß
keine oder eine der Zahlen a, b, c, d die Eigenschaft L hat; und diesen Fall hat man immer
als Vorbild vor Augen. Hier könnte ein Irrtum darin bestehen, daß ich glaube c hätte die
Eigenschaft und, nachdem ich den Irrtum eingesehen hätte, wüßte ich, daß keine der Zahlen
die Eigenschaft hat. Die Analogie bricht eben hier zusammen.
(Das hängt damit zusammen, daß ich nicht in jedem Kalkül, in dem ich Gleichungen
gebrauchen, eo ipso auch die Verneinungen von Gleichungen gebrauchen darf. Denn
2 × 3 ≠ 7 heißt nicht, daß die Gleichung ”2 × 3 = 7“ nicht vorkommen soll, wie etwa die
Gleichung ”2 × 3 = sinus“, sondern die Verneinung ist eine Ausschließung innerhalb eines
von vornherein bestimmten Systems. Eine Definition kann ich nicht verneinen, wie eine
nach Regeln abgeleitete Gleichung.)
Sagt man, das Intervall im Existenzbeweis sei nicht wesentlich, da ein andres Intervall es
auch getan hätte, so heißt das natürlich nicht, daß das Fehlen einer Intervallangabe es
auch getan hätte. – Der Beweis der Nichtexistenz hat zum Beweis der Existenz nicht das
Verhältnis eines Beweises von p zum Beweis des Gegenteils.
Man sollte glauben, in den11
Beweis des Gegenteils von ”(∃n) etc.“ müßte sich eine
Negation einschleichen12
können, durch die irrtümlicherweise ”~(∃n) etc.“ bewiesen wird.
Gehen wir doch einmal, umgekehrt, von den Beweisen aus und nehmen wir an, sie wären
uns ursprünglich gezeigt worden und man hätte uns dann gefragt: was beweisen diese
Rechnungen? Sieh auf die Beweise und entscheide dann, was sie beweisen.
9 (V): es so eine
10 (V): die
11 (O): dem
12 (V): verirren
633
634
427 Wenn Du wissen willst . . .
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you know a method for ascertaining that the equation doesn’t hold in general? Do you
remember that the equation doesn’t hold for some values of x between 0 and 1,000? Did
someone merely show you the equation and say he had found values of x that didn’t satisfy
the equation, and do you yourself perhaps not know how to establish this for a given value?
Etc., etc.
“I have figured out that there is no number that. . . .” In what system of calculation does
that calculation occur? That will show us which propositional system the proposition that
has been figured out belongs to. (One also asks: “How does one calculate something like that?”)
“I have discovered that there is such a number.”
“I have figured out that there is no such number.”
In the first sentence I’m not allowed to substitute “no such” for “such a”. – And what if
in the second I put “such a” for “no such”? Let’s suppose the result of a7
calculation isn’t
the proposition “~(∃n) etc.”, but “(∃n) etc.” Does it then maybe make sense to say
“Courage! You have to get to such a number at some point, if only you try long enough”?
That only makes sense if the proof hasn’t yielded “(∃n) etc.”, but has set limits to trying,
i.e. has accomplished something altogether different. That is, the contradictory of what we
call an existence theorem, a theorem that teaches us to look for a number, is not the proposition
“(n) etc.”, but a proposition that says that in such and such an interval there is no
number that. . . . What is the contradictory of what has been proved? – For that you must
look at the proof. We can say that the contradictory of a proved proposition is what would
have been proved instead of it if a certain miscalculation had been made in the proof. Now
if, for instance, the proof that ~(∃n) etc. is the case is an induction that shows that however
far I go such a number cannot occur, then the contradictory of this proof (I’m going to go
ahead and use this expression) is not an existence proof in our sense. – Here it isn’t the case,
as it is in that of the proof, that none or one of the numbers a, b, c, d has the property ε;
and this is the case one always has in mind as a paradigm. Here it would be a mistake for
me to believe that c had the property, and after I had recognized my mistake I would know
that none of the numbers had the property. At this point the analogy just collapses.
(This is connected with the fact that not in every calculus in which I am allowed to use
equations am I eo ipso also allowed to use the negations of equations as well. For 2 × 3 ≠ 7
doesn’t mean that the equation “2 × 3 = 7” is not supposed to occur, like, say, the equation
2 × 3 = sine; rather the negation is an exclusion within a predetermined system. I can’t
negate a definition as I can an equation derived according to rules.)
If you say that the interval in an existence proof isn’t essential because another interval
might also have served the purpose, then of course that doesn’t mean that not specifying an
interval would also have served the purpose. – The relation of a proof of non-existence to a
proof of existence is not the same as that of a proof of p to a proof of its contradictory.
One should suppose that in a proof of the contradictory of “(∃n) etc.” it ought to be
possible for a negation to creep in,8
because of which “~(∃n) etc.” is proved erroneously.
Let’s for once start at the other end with the proofs, and suppose that they had been shown
to us at the outset, and that then we had been asked: What do these calculations prove? Look
at the proofs and then decide what they prove.
7 (V): the 8 (V): to stray into it,
If you Want to Know What was Proved . . . 427e
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Ich brauche nicht zu behaupten, man müsse die n Wurzeln der Gleichung n-ten Grades
konstruieren können, sondern ich sage nur, daß der Satz ”diese Gleichung hat n Wurzeln“
etwas anderes heißt, wenn ich ihn durch Abzählen der konstruierten Wurzeln, und wenn ich
ihn anderswie bewiesen habe. Finde ich aber eine Formel für die Wurzeln einer Gleichung,
so habe ich einen neuen Kalkül konstruiert und keine Lücke eines alten ausgefüllt.
Es ist daher Unsinn zu sagen, der Satz ist erst bewiesen, wenn man eine solche
Konstruktion aufzeigt. Denn dann haben wir eben etwas Neues konstruiert, und was wir
jetzt unter dem Hauptsatz der Algebra verstehen, ist eben, was der gegenwärtige ”Beweis“
uns zeigt.
”Jeder Existenzbeweis muß eine Konstruktion dessen enthalten, dessen Existenz er
beweist.“ Man kann nur sagen ”ich nenne ,Existenzbeweis‘ nur einen, der eine solche
Konstruktion enthält“. Der Fehler liegt darin,13
daß man vorgibt14
einen klaren Begriff der
Existenz15
zu besitzen.
Man glaubt ein Etwas, die Existenz, beweisen zu können, sodaß man nun unabhängig
vom Beweis von ihr überzeugt ist. (Die Idee der, voneinander – und daher wohl auch vom
Bewiesenen – unabhängigen Beweise!) In Wirklichkeit ist Existenz das, was man mit dem
beweist, was man ”Existenzbeweis“ nennt. Wenn die Intuitionisten und Andere darüber reden,
so sagen sie: ”Diesen16
Sachverhalt, die Existenz, kann man nur so, und nicht so, beweisen“.
Und sehen nicht, daß sie damit einfach das definiert haben, was sie Existenz nennen. Denn
die Sache verhält sich eben nicht so, wie wenn man sagt: ”daß ein Mann in dem Zimmer
ist, kann man nur dadurch beweisen, daß man hineinschaut, aber nicht, indem man an der
Türe horcht“.
Wir haben keinen Begriff der Existenz unabhängig von unserm Begriff des
Existenzbeweises.
Warum ich sage, daß wir einen Satz, wie den Hauptsatz der Algebra, nicht finden,
sondern konstruieren? – Weil wir ihm beim Beweis einen neuen Sinn geben, den er früher
gar nicht gehabt hat. Für diesen Sinn gab es vor dem sogenannten Beweis nur eine
beiläufige Vorlage in der Wortsprache.
Denken wir, Einer würde sagen: das Schachspiel mußte nur entdeckt werden, es war immer
da! Oder das reine Schachspiel war immer da, nur das materielle, von Materie verunreinigte,
haben wir gemacht.
Wenn durch Entdeckungen ein Kalkül der Mathematik geändert wird, – können wir den
alten Kalkül nicht behalten (aufheben)? (D.h., müssen wir ihn wegwerfen?) Das ist ein sehr
interessanter Aspekt. Wir haben nach der Entdeckung des Nordpols nicht zwei Erden: eine
mit, und eine ohne den Nordpol. Aber nach der Entdeckung des Gesetzes der Verteilung
der Primzahlen, zwei Arten von Primzahlen.
Die mathematische Frage muß so exakt sein, wie der mathematische Satz. Wie
irreführend die Ausdrucksweise der Wortsprache den Sinn der mathematischen Sätze
darstellt, sieht man, wenn man sich die Multiplizität eines mathematischen Beweises vor
Augen führt17
und bedenkt, daß der Beweis zum Sinn des bewiesenen Satzes gehört, d.h.
den Sinn bestimmt. Also nicht etwas ist, was bewirkt, daß wir einen bestimmten Satz glauben,
13 (V): Fehler ist,
14 (V): glaubt
15 (V): Begriff des Existenzbeweises
16 (O): ”Dieser
17 (V): stellt
635
636
428 Wenn Du wissen willst . . .
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I don’t need to assert that it must be possible to construct the n roots of equations of the
nth degree; rather, I’m only saying that the proposition “This equation has n roots” means
something different if I proved it by enumerating the constructed roots than if I proved it
some other way. But if I find a formula for the roots of an equation then I have constructed
a new calculus, and have not filled in a gap in an old one.
Therefore it’s nonsense to say that the proposition hasn’t been proved until we produce
such a construction. For in that case we simply have constructed something new, and what
we now understand by the fundamental theorem of algebra is simply what the present “proof”
shows us.
“Every existence proof must contain a construction of what it proves the existence of.”
All you can say is: “I won’t call anything an ‘existence proof’ that doesn’t contain such a
construction”. The mistake lies in pretending to possess9
a clear concept of existence.10
We think we can prove something, existence, in such a way that we are then convinced
of it independently of the proof. (The idea of proofs independent of each other – and so
presumably also independent of what has been proved!) In reality, existence is what is
proved by what we call “existence proofs”. When the intuitionists and others talk about this
they say: “This state of affairs, existence, can be proved only thus and not so.” And they
don’t see that in saying that they have simply defined what they call existence. For it isn’t
at all like saying “That someone is in the room can only be proved by looking inside, not
by listening at the door”.
We have no concept of existence independent of our concept of an existence proof.
Why do I say that we don’t “discover” a proposition like the fundamental theorem of
algebra, but instead “construct” it? – Because in proving it we give it a new sense that it
didn’t have before. Before the so-called proof there was only a rough model of that sense in
word-language.
Suppose someone were to say: Chess only had to be discovered, it was always there! Or:
the pure game of chess was always there; what we’ve made is only the material game, tainted
by matter.
If a calculus in mathematics is altered by discoveries, can’t we keep (hold on to) the old
calculus? (That is, do we have to throw it away?) That is a very interesting way of looking
at the matter. After the discovery of the North Pole we didn’t have two earths, one with and
one without the North Pole. But after the discovery of the law of the distribution of the
primes, we do have two kinds of primes.
A mathematical question must be as exact as a mathematical proposition. You can see how
misleadingly our way of speaking in word-language represents the sense of mathematical
propositions if you call to mind the multiplicity of a mathematical proof, and consider that
the proof belongs to the sense of the proved proposition, i.e. determines that sense. So it isn’t
something that causes us to believe a particular proposition, but something that shows us
what we believe – if we can talk of believing here at all. Concept words in mathematics: prime
9 (V): mistake is in believing that one possesses 10 (V): of an existence proof.
If you Want to Know What was Proved . . . 428e
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sondern etwas, was uns zeigt, was wir glauben, – wenn hier von Glauben18
eine Rede sein
kann. Begriffswörter in der Mathematik: Primzahl, Kardinalzahl, etc. Es scheint darum unmittelbar
Sinn zu haben, wenn gefragt wird: ”Wieviel Primzahlen gibt es?“ (”Es glaubt der
Mensch, wenn er nur Worte hört, . . .“.)19
In Wirklichkeit ist diese Wortzusammenstellung
einstweilen Unsinn; bis für sie eine besondere Syntax gegeben wurde. Sieh’ den Beweis dafür
an, ”daß es unendlich viele Primzahlen gibt“ und dann die Frage, die er zu beantworten
scheint. Das Resultat eines intrikaten Beweises kann nur insofern einen einfachen
Wortausdruck haben, als das System von Ausdrücken, dem dieser Ausdruck angehört,
in seiner Multiplizität einem System solcher Beweise entspricht. – Die Konfusionen in
diesen Dingen sind ganz darauf zurückzuführen, daß man die Mathematik als eine
Art Naturwissenschaft behandelt. Und das wieder hängt damit zusammen, daß sich die
Mathematik von der Naturwissenschaft abgelöst hat. Denn, solange sie in unmittelbarer
Verbindung mit der Physik betrieben wird, ist es klar, daß sie keine Naturwissenschaft ist.
(Etwa, wie man einen Besen nicht für ein Einrichtungsstück des Zimmers halten kann, solange
man ihn dazu benützt, die Einrichtungsgegenstände zu säubern.)
Ist nicht die Hauptgefahr die, daß uns der Prosa-Ausdruck des Ergebnisses einer
mathematischen Operation einen Kalkül vortäuscht, der gar nicht vorhanden ist. Indem er
seiner äußern Form nach einem System anzugehören scheint, das es hier gar nicht gibt.
Ein Beweis ist Beweis eines (bestimmten) Satzes, wenn er es nach einer Regel ist, nach
der dieser Satz diesem Beweis zugeordnet ist. D.h., der Satz muß einem System von Sätzen
angehören und der Beweis einem System von Beweisen. Und jeder Satz der Mathematik
muß einem Kalkül der Mathematik angehören. (Und kann nicht in Einsamkeit thronen20
und sich sozusagen nicht unter andere Sätze mischen.)
Also ist auch der Satz ”jede Gleichung n-ten Grades hat n Lösungen“ nur ein Satz der
Mathematik, sofern er einem System von Sätzen, und sein Beweis einem korrespondierenden
System von Beweisen, entspricht. Denn welchen guten Grund habe ich, dieser Kette von
Gleichungen etc. (dem sogenannten Beweis) diesen Prosasatz zuzuordnen. Es muß doch aus
dem Beweis – nach einer Regel – hervorgehen, von welchem Satz er der Beweis ist.
Nun liegt es aber im Wesen dessen, was wir als Satz bezeichnen, daß es sich verneinen
lassen muß. Und auch die Verneinung des bewiesenen Satzes muß mit dem Beweis zusammenhängen;
so nämlich, daß sich zeigen läßt, unter welchen andern, entgegengesetzten,
Bedingungen sie herausgekommen wäre.
18 (O): glauben
19 (E): Vgl. Goethe, Faust I, 2565–2566.
20 (O): tronen
637
429 Wenn Du wissen willst . . .
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number, cardinal number, etc. That is why it seems to make sense to ask without hesitation,
“How many prime numbers are there?” (“Men always believe, if only they hear words
/ . . .”)11
In reality this combination of words is nonsense for the time being; until it’s given
a special syntax. Look at the proof for the claim “that there are infinitely many primes”, and
then at the question that it appears to answer. The result of an intricate proof can have a
simple verbal expression only in so far as the system of expressions to which this expression
belongs has a multiplicity corresponding to a system of such proofs. – The confusions in
these matters are entirely attributable to treating mathematics as a kind of natural science.
And this is connected in turn with the fact that mathematics has detached itself from
natural science; for as long as it is done in immediate connection with physics, it is clear
that it isn’t a natural science. (Somewhat in the same way that you can’t take a broom for
part of the furnishings of a room so long as you use it to clean the furniture.)
Isn’t this the main danger – that expressing the result of a mathematical operation in
prose will give us the illusion of a calculus that doesn’t even exist, by outwardly appearing
to belong to a system that isn’t even there?
A proof is a proof of a (particular) proposition if it performs its proof following a rule that
correlates the proposition with the proof. That is, the proposition must belong to a system
of propositions, and the proof to a system of proofs. And every proposition in mathematics
must belong to a calculus of mathematics. (It cannot sit in splendid isolation and refuse to
mingle with the other propositions, so to speak.)
So the proposition “Every equation of nth degree has n roots” is a proposition of mathematics
only in so far as it corresponds to a system of propositions and its proof corresponds
to an appropriate system of proofs. For what good reason do I have to correlate this prose
sentence to that chain of equations, etc. (to the so-called proof)? Surely it must emerge from
the proof – according to a rule – which proposition it is a proof of.
Now it is a part of the nature of what we call propositions that we must able to negate them.
And the negation of a proved proposition must also be connected with the proof; in such a
way, that is, that we can show under what different, contrasting, conditions the negation
would have been the result.
11 (E): Cf. Goethe, Faust I, 2565–6.
If you Want to Know What was Proved . . . 429e
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122
Das mathematische Problem.
Arten der Probleme.
Suchen.
”Aufgaben“ in der Mathematik.
Wo man fragen kann, kann man auch suchen, und wo man nicht suchen kann, kann man
auch nicht fragen. Und auch nicht antworten.
Wo es keine Methode des Suchens gibt, da kann auch die Frage keinen Sinn haben. –
Nur wo eine Methode der Lösung ist, ist eine Frage (d.h. natürlich nicht: ”nur wo die Lösung
gefunden ist, ist eine Frage“). – D.h.: dort wo die Lösung des Problems nur von einer Art
Offenbarung erwartet werden kann, ist auch keine Frage. Einer Offenbarung entspricht keine
Frage. –
Die Annahme der Unentscheidbarkeit setzt voraus, daß zwischen den beiden Seiten einer
Gleichung, sozusagen, eine unterirdische Verbindung besteht; daß die Brücke nicht in
Symbolen geschlagen werden kann. Aber dennoch besteht; denn sonst wäre die Gleichung
sinnlos. – Aber die Verbindung besteht nur, wenn wir sie durch einen Kalkül1
gemacht haben.
Der Übergang ist nicht durch eine dunkle Spekulation hergestellt, von andrer Art als das
was er verbindet. (Wie ein dunkler Gang zwischen zwei lichten Orten.)
Ich kann den Ausdruck ”die Gleichung G ergibt die Lösung L“ nicht eindeutig
anwenden, solange ich keine Methode der Lösung besitze; weil ”ergibt“ eine Struktur
bedeutet, die ich, ohne sie zu kennen, nicht bezeichnen kann. Denn das heißt das Wort
”ergibt“ zu verwenden, ohne seine Grammatik zu kennen. Ich könnte aber auch sagen:
Das Wort ”ergibt“ hat andere Bedeutung, wenn ich es so verwende, daß es sich auf eine
Methode der Lösung bezieht, und eine andere, wenn dies nicht der Fall ist. Es verhält
sich hier mit ”ergibt“ ähnlich, wie mit dem Wort ”gewinnen“ (oder ”verlieren“), wenn
das Kriterium des ”Gewinnens“ einmal ein bestimmter Verlauf der Partie ist (hier muß
ich die Spielregeln kennen, um sagen zu können, ob Einer gewonnen hat), oder ob ich mit
”gewinnen“ etwas meine, was sich beiläufig2
durch ”zahlen müssen“ ausdrücken ließe.
Wenn wir ”ergibt“ im ersten Sinne3
anwenden, so heißt ”die Gleichung ergibt L“:
wenn ich die Gleichung nach gewissen Regeln transformiere, so erhalte ich L. So wie die
Gleichung 25 × 25 = 620 besagt, daß ich 620 erhalte, wenn ich auf 25 × 25 die Multiplikationsregeln
anwende. Aber diese Regeln müssen mir hier4
schon gegeben sein, ehe das Wort
”ergibt“ Bedeutung hat, und ehe die Frage einen Sinn hat, ob die Gleichung L ergibt.
1 (V): durch Symbole
2 (V): etwa
3 (V): ”ergibt“ in der ersten Bedeutung
4 (V): nun
638
639
430
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122
Mathematical Problems.
Kinds of Problems.
Searching.
“Tasks” in Mathematics.
Where you can ask you can also search, and where you cannot search you cannot ask either.
Nor can you answer.
Where there is no method of searching, a question cannot have any sense. – Only where
there is a method for a solution is there a question (of course that doesn’t mean: “only where
the solution has been found is there a question”). – That is: where we can only expect the
solution of the problem from some sort of revelation, there is no question. To a revelation
no question corresponds. –
The supposition of undecidability presupposes that there is, so to speak, an underground
connection between the two sides of an equation; that the bridge cannot be built in symbols,
but does exist; because otherwise the equation would be senseless. – But the connection
only exists if we have made it via a calculus;1
the transition isn’t produced by some dark
speculation that is different in kind from what it connects (like a dark passage between two
sunlit places).
I can’t use the expression “the equation G yields the solution L” unambiguously so long
as I don’t have a method for solving the equation; because “yields” denotes a structure that
I can’t designate unless I am acquainted with it. If I could, that would mean using the word
“yields” without knowing its grammar. But I could also say: The word “yields” has one
meaning when I use it in such a way that it refers to a method for solving the equation, and
it has another meaning otherwise. Here the word “yields” is similar to the word “win” (or
“lose”), when at one time the criterion for “winning” is a particular sequence of events in
the game (in which case I must know the rules of the game in order to be able to say whether
someone has won) and at another time by “winning” I mean something that I could express
roughly2
by “having to pay”.
If we use “yields” in the first sense,3
then “the equation yields L” means: if I transform
the equation in accordance with certain rules, I get L. Just as the equation 25 × 25 = 620
says that I get 620 if I apply the rules of multiplication to 25 × 25. But in this case4
these
rules must already have been given to me before the word “yields” has a meaning, and before
the question whether the equation yields L has a sense.
1 (V): via symbols;
2 (V): could perhaps express
3 (V): “yields” with the first meaning,
4 (V): But now
430e
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Es genügt also nicht zu sagen ”p ist beweisbar“, sondern es muß heißen: beweisbar nach
einem bestimmten System.
Und zwar behauptet der Satz nicht, p sei beweisbar nach dem System S, sondern nach
seinem System, dem System von p. Daß p dem System S angehört, das läßt sich nicht
behaupten (das muß sich zeigen). – Man kann nicht sagen, p gehört zum System S; man
kann nicht fragen, zu welchem System p gehört; man kann nicht das System von p suchen.
”p verstehen“ heißt, sein System kennen. Tritt p scheinbar von einem System in das andere
über, so hat in Wirklichkeit p seinen Sinn gewechselt.
Es ist unmöglich, Entdeckungen neuartiger Regeln zu machen, die von einer uns
bekannten Form (etwa dem sinus eines Winkels) gelten. Sind es neue Regeln, so ist es nicht
die alte Form.
Kenne ich die Regeln der elementaren Trigonometrie, so kann ich den Satz sin 2x =
2 sin xcos x kontrollieren, aber nicht den Satz sin x = x − – . . . . Das heißt aber,
daß der sinus der elementaren Trigonometrie und der sinus der höheren Trigonometrie
verschiedene Begriffe sind.
Die beiden Sätze stehen gleichsam auf zwei verschiedenen Ebenen. In der ersten kann
ich mich bewegen, soweit ich will, ich werde nie zu dem Satz auf der höheren Ebene
kommen.
Der Schüler, dem das Rüstzeug der elementaren Trigonometrie zur Verfügung stünde
und von dem die Überprüfung der Gleichung sin x = x − . . . verlangt würde, fände das,
was er zur Bewältigung dieser Aufgabe braucht, eben nicht vor. Er kann die Frage nicht nur
nicht beantworten, sondern er kann sie auch nicht verstehen. (Sie wäre wie die Aufgabe, die
der Fürst im Märchen dem Schmied stellt: ihm einen ”Klamank“ zu bringen. Busch,
Volksmärchen.)
Man nennt es eine Aufgabe, wenn gefragt wird ”wieviel ist 25 × 16“, aber auch eine Aufgabe:
was ist das ∫sin2
xdx? Die erste hält man zwar für viel leichter als die zweite, sieht aber
nicht, daß sie in verschiedenem Sinn ”Aufgaben“ sind. Der Unterschied ist natürlich kein
psychologischer; denn5
es handelt sich nicht drum, ob der Schüler die Aufgabe lösen kann,
sondern ob der Kalkül sie lösen kann, oder, welcher Kalkül sie lösen kann.
Die Unterschiede, auf die ich aufmerksam machen kann, sind solche, wie sie jeder Bub
in der Schule wohl kennt. Aber man verachtet diese Unterschiede später, wie die Russische
Rechenmaschine (und den zeichnerischen Beweis in der Geometrie) und sieht sie als
unwesentlich an, statt als wesentlich und fundamental.
Es ist uninteressant, ob der Schüler6
eine Regel weiß, nach der er7
∫sin2
xdx gewiß lösen
kann, aber nicht, ob der Kalkül, den wir vor uns haben (und den er zufälligerweise benützt)
eine solche Regel enthält.
Nicht, ob der Schüler es kann, sondern ob der Kalkül es kann und wie er es tut,
interessiert uns.
Im Falle 25 × 16 = 370 nun, schreibt der Kalkül, den wir benützen, jeden Schritt zur
Prüfung dieser Gleichung vor.
Ein merkwürdiges Wort: ”Es ist mir gelungen, das zu beweisen“.
(Das ist es, was im Falle 25 × 16 = 400 niemand sagen würde.)
x3
3!
x x3 5
3 5! !
+
5 (V): und
6 (V): ob man
7 (V): man
640
641
431 Das mathematische Problem. . . .
WBTC121-130 30/05/2005 16:14 Page 862
So it is not enough to say “p is provable”; we must say: provable according to a particular
system.
And, to be more specific, the proposition doesn’t assert that p is provable according to
the system S, but according to its own system, the system of p. That p belongs to the system
S cannot be asserted (that has to show itself). – We can’t say, p belongs to the system
S; we can’t ask, to which system p belongs; we can’t search for p’s system. “Understanding
p” means – knowing its system. If p appears to cross over from one system to another, then
really it has changed its sense.
It is impossible to discover novel rules that are valid for a form familiar to us (say for the
sine of an angle). If they are new rules, then it is not the old form.
If I know the rules of elementary trigonometry, I can check the proposition sin 2x =
2 sin xcos x, but not the proposition sin x = x − −. . . . But that means that the sine
function of elementary trigonometry and that of higher trigonometry are different concepts.
The two propositions stand on two different planes, as it were. However far I range on
the first plane I will never come to the proposition on the higher plane.
A schoolboy who had at his disposal the toolbox of elementary trigonometry and was told
to verify the equation sin x = x − just wouldn’t find in it what he needs to come to grips
with this problem. Not only couldn’t he answer the question – he couldn’t understand it,
either. (It would be like the task the prince sets the smith in the fairy tale: to bring him a
“hubbub”. Busch, Volksmärchen.)
We call it a problem when we are asked “How much is 25 × 16?”, but also when we are
asked: what is ∫sin2
xdx? We regard the first as much easier than the second, but we don’t
see that they are “problems” in different senses. Of course the difference isn’t a psychological
one; for5
it isn’t a matter of whether the pupil can solve the problem, but whether the
calculus can solve it, or which calculus can solve it.
The distinctions to which I can draw attention are ones of the kind that are quite familiar
to every schoolboy. But later on we disdain those distinctions, as we do the Russian abacus
(and geometrical proofs that use drawings); and we regard them as inessential, rather than
as essential and fundamental.
Whether a pupil6
knows a rule for ensuring a solution to the ∫sin2
xdx is of no interest; what
does interest us is whether the calculus we have before us (and that he happens to be using)
contains such a rule.
What interests us is not whether the pupil can do it, but whether the calculus can do it,
and how it does it.
In the case of 25 × 16 = 370, the calculus we use prescribes every step for checking the
equation.
A remarkable expression: “I succeeded in proving this”.
(That’s something no one would say in the case of 25 × 16 = 400.)
x3
3!
x x3 5
3 5! !
+
5 (V): and 6 (V): Whether one
Mathematical Problems. . . . 431e
WBTC121-130 30/05/2005 16:14 Page 863
Man könnte festlegen:8
”Was man anfassen kann, ist ein Problem. – Nur wo ein Problem
sein kann, kann etwas behauptet werden.“
Würde denn aus dem Allen nicht das Paradox folgen: daß es in der Mathematik keine
schweren Probleme gibt; weil, was schwer ist, kein Problem ist? Was folgt, ist, daß das ”schwere
mathematische Problem“, d.h. das Problem der mathematischen Forschung, zur Aufgabe
”25 × 25 = ?“ nicht in dem Verhältnis steht, wie etwa ein akrobatisches Kunststück zu einem
einfachen Purzelbaum (also einfach in dem Verhältnis: sehr leicht zu sehr schwer), sondern
daß es ”Probleme“ in verschiedenen Bedeutungen des Wortes sind.
”Du sagst ,wo eine Frage ist, da ist auch ein Weg zu ihrer Beantwortung‘, aber in der
Mathematik gibt es doch Fragen, zu deren Beantwortung wir keinen Weg sehen.“ – Ganz
richtig, und daraus folgt nur, daß wir in diesem Fall das Wort ”Frage“ in anderem Sinn
gebrauchen, als im oberen Fall. Und ich hätte vielleicht sagen sollen ”es sind hier zwei
verschiedene Formen und nur für die erste möchte ich das Wort ,Frage‘ gebrauchen“. Aber
dieses Letztere ist nebensächlich. Wichtig ist, daß wir es hier mit zwei verschiedenen Formen
zu tun haben. (Und daß Du Dich in der Grammatik des Wortes ”Art“ nicht auskennst, wenn
Du nun sagen willst, es seien eben nur9
zwei verschiedene Arten von Fragen.)
”Ich weiß, daß es für diese Aufgabe eine Lösung gibt, obwohl ich die Lösung10
noch nicht
habe“. – In welchem Symbolismus weiß ich es?11
12
”Ich weiß, daß es da ein Gesetz geben muß.“ Ist dieses Wissen ein
amorphes, das Aussprechen des Satzes begleitendes Gefühl? Dann interessiert
es uns nicht. Und ist es ein symbolischer Prozeß – nun, dann ist die Aufgabe,
ihn in einem klaren13
Symbolismus darzustellen.14
Was heißt es: den Goldbach’schen Satz glauben? Worin besteht dieser Glaube? In einem
Gefühl der Sicherheit, wenn wir den Satz aussprechen, oder hören? Das interessiert uns nicht.
Ich weiß ja auch nicht, wie weit dieses Gefühl durch den Satz selbst hervorgerufen sein mag.
Wie greift der Glaube in diesen Satz ein? Sehen wir nach, welche Konsequenzen er hat, wozu
er uns bringt. ”Er bringt mich zum Suchen nach einem Beweis dieses Satzes.“ – Gut,
jetzt sehen wir noch nach, worin Dein Suchen eigentlich besteht; dann werden wir wissen,
was es mit dem Glauben an den Satz auf sich hat.15
Man darf nicht an einem Unterschied der Formen vorbeigehen – wie man wohl an einem
Unterschied zwischen Anzügen vorbeigehen kann, wenn er etwa sehr gering ist.
In gewissem Sinne gibt es für uns – nämlich in der Grammatik – nicht ”geringe
Unterschiede“. Und überhaupt bedeutet ja das Wort Unterschied etwas ganz anderes, als
dort wo es sich um einen Unterschied zweier Sachen16
handelt.
Der Philosoph spürt Wechsel im Stil seiner Ableitung, an denen der Mathematiker von
heute, mit seinem stumpfen Gesicht ruhig vorübergeht. – Eine höhere Sensitivität ist es
eigentlich, was den Mathematiker der Zukunft von dem heutigen unterscheiden wird; und
die wird die Mathematik – gleichsam – stutzen; weil man dann mehr auf die absolute Klarheit,
als auf das17
Erfinden neuer Spiele bedacht sein wird.
8 (V): erklären:
9 (V): nur ver
10 (V): die Art der Lösung
11 (V): Symbolismus weißt Du es?
12 (F): MS 112, S. 70v.
13 (V): offenbaren
14 (V): auszudrücken.
15 (V): wissen, wie es sich mit Deinem Glauben an
den Satz verhält.
16 (V): Dinge
17 (V): ein
642
643
432 Das mathematische Problem. . . .
WBTC121-130 30/05/2005 16:14 Page 864
One could stipulate:7
“Whatever one can get a grip on is a problem. – Only where there
can be a problem can something be asserted.”
Wouldn’t a paradox follow from all of this: namely, there are no difficult problems in
mathematics, since what is difficult isn’t a problem? What follows is, that the “difficult
mathematical problems”, i.e. the problems of mathematical research, aren’t related to the
problem “25 × 25 = ?” as, say, a feat of acrobatics is to a simple somersault (i.e. the relationship
isn’t simply: very easy to very difficult). Rather, they are “problems” in different
senses of the word.
“You say ‘Where there is a question, there is also a way to answer it’, but in mathematics
there are questions that we don’t see any way to answer.” – Quite right, but all that
follows from that is that in this case we are using the word “question” in a different sense
than in the case above. And perhaps I should have said “Here there are two different forms,
and I am going to restrict the word ‘question’ to the first”. But this latter point is a side
issue. What is important is that we are dealing here with two different forms. (And that you
do not know your way about in the grammar of the word “kind” if you now want to say that
these forms are just two different kinds of questions.)
“I know that there is a solution for this problem, although I don’t yet have the solution.”8
– In what symbolism do I know9
it?
10
“I know there must be a rule here.” Is this knowledge an amorphous
feeling accompanying the utterance of the sentence? Then it doesn’t interest us.
And if it is a symbolic process – well, then the task is to represent it in a clear
symbolism.11
What does it mean to believe Goldbach’s conjecture? What does this belief consist in? In
a feeling of certainty when we utter or hear the conjecture? That doesn’t interest us. After
all, I don’t know to what extent this feeling may have been caused by the proposition itself.
How does my belief engage with this proposition? Let’s look and see what its consequences
are, what it gets us to do. “It makes me search for a proof of this proposition.” – Very
well; and now we’ll look and see what your searching really consists in. Then we’ll know
what belief in the proposition amounts to.12
We mustn’t overlook a difference between forms – as we may well overlook a difference
between suits, if for example it is very slight.
In a certain sense – in grammar, that is – there are for us no such things as “negligible
distinctions”. And in general the word “distinction” means something completely different
from when it is a question of a distinction between two objects.13
A philosopher notices changes in the style of a derivation which today’s mathematician
passes over calmly, with a blank expression on his face. – What will distinguish the mathematicians
of the future from those of today will really be a greater sensitivity; and that will
– as it were – prune mathematics; for then everyone will be more concerned with absolute
clarity than with the14
invention of new games.
7 (V): explain:
8 (V): the type of solution.”
9 (V): do you know
10 (F): MS 112, p. 70v.
11 (V): to express it in an obvious symbolism.
12 (V): know how things stand with your belief in
the proposition.
13 (V): things.
14 (V): an
Mathematical Problems. . . . 432e
WBTC121-130 30/05/2005 16:14 Page 865
Die philosophische Klarheit wird auf das Wachstum der Mathematik den gleichen
Einfluß haben, wie das Sonnenlicht auf das Wachsen der Kartoffeltriebe. (Im dunklen18
Keller
wachsen sie meterlang.)
Den Mathematiker muß es bei meinen mathematischen Ausführungen grausen, denn seine
Schulung hat ihn immer davon abgelenkt, sich Gedanken und Zweifeln, wie ich sie aufrolle,
hinzugeben. Er hat sie als etwas Verächtliches ansehen lernen und hat, um eine Analogie
aus der Psychoanalyse (dieser Absatz erinnert an Freud) zu gebrauchen, einen Ekel vor
diesen Dingen erhalten, wie vor etwas Infantilem. D.h., ich rolle alle jene Probleme auf, die
etwa ein Kind19
beim Lernen der Arithmetik, etc. als Schwierigkeiten empfindet und die der
Unterricht unterdrückt, ohne sie zu lösen. Ich sage also zu diesen unterdrückten Zweifeln:
ihr habt ganz recht, fragt nur, und verlangt nach Aufklärung!
18 (V): dunkeln 19 (V): Knabe
644
433 Das mathematische Problem. . . .
WBTC121-130 30/05/2005 16:14 Page 866
Philosophical clarity will have the same effect on the growth of mathematics as sunlight
has on the growth of potato shoots. (In a dark cellar they grow several metres long.)
A mathematician is bound to be horrified when faced with my mathematical remarks, since
his schooling has always diverted him from giving himself over to thoughts and doubts of
the kind that I am bringing up. He has learned to regard them as something contemptible
and, to use an analogy from psychoanalysis (this paragraph is reminiscent of Freud), he has
acquired a revulsion against these things as against something infantile. That is to say, I’m
bringing up all of those problems that a child15
learning arithmetic, etc., finds difficult,
the problems that classroom instruction suppresses without solving. So I’m saying to those
suppressed doubts: You are quite right, go ahead and ask – and demand clarification!
15 (V): boy
Mathematical Problems. . . . 433e
WBTC121-130 30/05/2005 16:14 Page 867
123
Eulerscher Beweis.
Kann man aus der Ungleichung: 1 + + . . . ≠ (1 + + . . . ) · (1 +
+ . . . ) eine Zahl ν konstruieren,1
die jedenfalls in den Kombinationen der rechten
Seite noch fehlt? Der Euler’sche Beweis dafür, daß es ”unendlich viele Primzahlen gibt“ soll
ja ein Existenzbeweis sein, und wie ist der ohne Konstruktion möglich?
~ 1 + + . . . = (1 + + . . . ) · (1 + + . . . ) das Argument läuft so:
Das rechte Produkt ist eine Reihe von Brüchen , in deren Nenner alle Kombinationen 2ν
3µ
vorkommen; wären das alle Zahlen, so müßte diese Reihe die gleiche sein, wie die 1 +
+ . . . und dann müßten auch die Summen gleich sein. Die linke ist aber ∞ und die rechte
nur eine endliche Zahl × = 3, also fehlen in der rechten Reihe unendlich viele Brüche,
d.h. es gibt in der linken2
Reihe Brüche, die in der rechten3
nicht vorkommen. Und nun
handelt es sich darum: ist dieses Argument richtig? Wenn es sich hier um endliche Reihen
handelte, so wäre alles durchsichtig.4
Denn dann könnte man aus der Methode der Summation
eben herausfinden, welche Glieder der linken Reihe auf die rechte Reihe fehlen. Man
könnte nun fragen: wie kommt es, daß die linke5
Reihe ∞ gibt, was muß sie außer den Gliedern
der rechten6
enthalten, daß es so wird? Ja es frägt sich: hat eine Gleichung, wie die obere
1 + + + . . . = 3 überhaupt einen Sinn? Ich kann ja aus ihr nicht herausfinden, welche
Glieder links zuviel sind. Wie wissen wir, daß alle Glieder der rechten auch in der linken
Seite vorkommen? Im Fall endlicher Reihen kann ich es erst sagen, wenn ich mich Glied
für Glied davon überzeugt habe; – und dann sehe ich zugleich, welche übrigbleiben. – Es fehlt
uns hier die Verbindung zwischen dem Resultat der Summe und den Gliedern, die einzige,
die den Beweis erbringen könnte. – Am klarsten wird alles, wenn man sich die Sache mit
einer endlichen Gleichung ausgeführt denkt: 1 + + + + + ≠ (1 + ) · (1 + ) =
1 + + + . Wir haben hier wieder das Merkwürdige, was man etwa einen Indizienbeweis
in der Mathematik nennen könnte – der ewig unerlaubt ist. Oder, einen Beweis durch
Symptome. Das Ergebnis der Summation ist ein Symptom dessen (oder wird als eines
aufgefaßt), daß links Glieder sind, die rechts fehlen.7
Die Verbindung des Symptoms, mit
dem, was man bewiesen haben möchte,8
ist lose. D.h. es ist eine Brücke nicht geschlagen,
aber man gibt sich damit zufrieden, daß man das andere Ufer sieht.
Alle Glieder der rechten Seite kommen in der linken Seite vor, aber die Summe links gibt
∞ und die rechte nur einen endlichen Wert – also müssen. . . . aber in der Mathematik muß
garnichts, außer was ist.
Die Brücke muß geschlagen werden.
1
6
1
3
1
2
1
3
1
2
1
6
1
5
1
4
1
3
1
2
1
3
1
2
3
2
2
1
1
3
1
2
1
n
1
3
1
32+1
2
1
22+1
2
1
3
+
1
3
1
32+
1
2
1
2
1
22 3+ +1
2
1
3
1
4
+ +
1 (V): ableiten,
2 (O): rechten
3 (O): linken
4 (V): klar.
5 (O): rechte
6 (O): linken
7 (O): daß rechts Glieder sind, die links fehlen.
8 (V): man beweisen möchte,
645
646
647
434
WBTC121-130 30/05/2005 18:48 Page 868
123
Euler’s Proof.
From the inequality 1 + + . . . ≠ (1 + + . . . ) · (1 + + . . . )
can we construct1
a number ν which is still missing – at least from the combinations on the
right side? After all, Euler’s proof that “there are infinitely many prime numbers” is supposed
to be an existence proof, and how is such a proof possible without a construction?
~ 1 + + . . . = (1 + + . . . ) · (1 + + . . . ) The argument goes like
this: The product on the right is a series of fractions in whose denominators all combinations
of the form 2ν
3µ
occur; if there were no numbers besides these, then this series
would have to be the same as the series 1 + + . . . , and in that case the sums also would
have to be the same. But the sum on the left side is ∞ and the one on the right side is only
a finite number × = 3, so there are infinitely many fractions missing in the series on the
right; that is, there are fractions in the series on the left that do not occur on the right.2
And
now the question is: Is this argument correct? If it were a question of a finite series here,
everything would be transparent.3
For then the method of summation would enable us to
find out which terms occurring in the series on the left were missing from the right. Now
we could ask: How does it come about that the series on the left4
gives us ∞? What must it
contain in addition to the terms on the right5
to make it infinite? Indeed the question arises:
Does an equation like 1 + + + . . . = 3 above have any sense at all? After all, I can’t find
out from it which terms on the left are in excess. How do we know that all the terms on
the right side also occur on the left? In the case of a finite series I can’t say that until I
have ascertained it term by term; – and then I see at the same time which are the extra
ones. – Here we are lacking the connection between the result of the sum and the terms,
the only connection that could furnish a proof. Everything becomes clearest if we imagine
the matter carried out with a finite equation: 1 + + + + + ≠ (1 + ) · (1 + ) =
1 + + + . Here once again we have that remarkable phenomenon that we might
call circumstantial proof in mathematics – something that is absolutely never permitted.
Or perhaps we could call it a proof by symptoms. The result of the summation is (or is
understood as) a symptom that there are terms on the left that are missing on the right.6
The connection of the symptom to what we would like to have proved7
is loose. That is,
no bridge has been built, so we settle for seeing the other bank.
All the terms on the right side occur on the left, but the sum on the left side yields ∞
and the one on the right only a finite value – so . . . must; but in mathematics the only thing
that must be is what is.
The bridge has to be built.
1
6
1
3
1
2
1
3
1
2
1
6
1
5
1
4
1
3
1
2
1
3
1
2
3
2
2
1
1
3
1
2
1
n
1
3
1
32+1
2
1
22+1
2
1
3
+
1
3
1
32+1
2
1
2
1
22 3+ +1
2
1
3
1
4
+ +
1 (V): derive
2 (O): on the right that do not occur on the left.
3 (V): clear.
4 (O): right
5 (O): left
6 (O): terms on the right that are missing on the
left.
7 (V): like to prove
434e
WBTC121-130 30/05/2005 16:14 Page 869
In der Mathematik gibt es kein Symptom, das kann es nur im psychologischen Sinne für
den Mathematiker geben.
Man könnte auch so sagen: Es kann sich in der Mathematik nicht auf etwas schließen
lassen, was sich nicht sehen läßt.
9
Das ganze lose Wesen jener Beweisführung beruht wohl auf der Verwechslung der Summe
und des Grenzwerts der Summe.
Das sieht man klar: wie weit immer man die rechte Reihe fortsetzt, immer kann man die
linke auch so weit bringen, daß sie alle Glieder der rechten einschließt. (Dabei bleibt noch
offen, ob die dann auch noch andre Glieder enthält.)
10
Man könnte auch so fragen: Wenn man11
nur diesen Beweis hätte,12
was könnte man13
nun daraufhin wagen? Wenn wir etwa die Primzahlen bis N gefunden hätten, könnten
wir nun daraufhin ins Unendliche auf die Suche nach einer weiteren Primzahl gehen – da
uns der Beweis verbürgt, daß wir eine finden werden? Das ist doch Unsinn. – Denn das
”wenn wir nur lange genug suchen“ heißt garnichts. (Bezieht sich auf Existenzbeweise im
Allgemeinen.)
14
Könnte ich auf diesen Beweis hin weitere Primzahlen links hinzufügen? Gewiß nicht,
denn ich weiß ja garnicht, wie ich welche finden kann und das heißt: ich habe15
ja gar keinen
Begriff der Primzahl, der Beweis hat mir keinen gegeben. Ich könnte nur beliebige Zahlen
(bezw. Reihen) hinzufügen.
(Die Mathematik ist angezogen mit falschen Deutungen.)
(”Es muß noch eine Primzahl16
kommen“ heißt in der Mathematik nichts. Das hängt unmittelbar
damit zusammen, daß es ”in der Logik nichts Allgemeineres und Spezielleres gibt“.)
Wenn die Zahlen alle Kombinationen von 2 und 3 wären, so müßte
17
ergeben, – sie ergibt ihn aber nicht. . . .
Was folgt daraus? (Satz des ausgeschlossenen
Dritten.) Daraus folgt nichts, als daß die
Grenzwerte der Summen verschieden sind;
also nichts (Neues). Nun könnte man aber untersuchen, woran das liegt. Dabei wird man
vielleicht auf Zahlen stoßen, die durch 2ν
× 3µ
nicht darstellbar sind, also auf größere
Primzahlen, nie aber wird man sehen, daß keine Anzahl solcher ursprünglicher Zahlen zur
Darstellung aller Zahlen genügt.
1 + + + . . . ≠ 1 + + + + . . .
Wieviel Glieder der Form 1
–2v ich auch zusammennehmen mag, nie ergibt es mehr als 2,
während die ersten vier Glieder der linken Reihe schon mehr als 2 ergeben. (Hierin muß
also schon der Beweis liegen.) Und hierin liegt er auch und zugleich die Konstruktion einer
Zahl, die keine Potenz von 2 ist, denn die Regel heißt nun: finde den Abschnitt der Reihe,
der jedenfalls 2 übertrifft, dieser muß eine Zahl enthalten, die keine Potenz von 2 ist.
1
23
1
22
1
2
1
3
1
2
9 (M): 14 \1
10 (M): 14 \2
11 (V): du
12 (V): hättest,
13 (V): was könntest du
14 (M): 14 \3
15 (V): habe g
16 (V): eine solche Zahl
17 (F): MS 108, S. 284.
435 Eulerscher Beweis.
648
lim lim lim
n
n
n
n
m
n
n m
1
2
1
3
den
1
n→ ν
ν
ν
ν
ν
ν
∞
=
=
→∞
=
=
→∞
=
=
∑ ∑ ∑
⎛
⎝
⎜
⎞
⎠
⎟ ⋅
⎛
⎝
⎜
⎞
⎠
⎟
0 0 1
WBTC121-130 30/05/2005 16:14 Page 870
In mathematics there are no symptoms: only in a psychological sense can that sort of thing
exist for mathematicians.
We could also put it like this: In mathematics nothing can be inferred that cannot be seen.
8
All of the looseness of that line of reasoning most likely rests on the confusion between
a sum and the limiting value of a sum.
We see this clearly: however far we continue the series on the right we can always get the
one on the left to the point that it includes all the terms of the one on the right. (And this
leaves it open whether the one on the left then contains other terms as well.)
9
We could also put the question this way: If one had only this proof, what would one stake
on it?10
If, say, we had found the primes up to N, could we then go on infinitely in search
of a further prime number – since the proof gives us the guarantee that we will find one?
Surely that is nonsense. – For “if we only search long enough” means nothing at all. (And
this refers to existence proofs in general).
11
Could I, based on this proof, add further prime numbers to the left side? Certainly not,
because I have no idea how to find any, and that means: I have absolutely no concept of
prime numbers; the proof hasn’t given me any. I could only add arbitrary numbers (or series).
(Mathematics is dressed in false interpretations).
(“Another prime12
number has to turn up” means nothing in mathematics. That is directly
connected with the fact that “in logic there is nothing ‘more general’ or ‘more particular’”).
If the numbers were all combinations of 2 and 3 then –
13
but it doesn’t . . . What follows from
that? (The law of the excluded middle.)
Nothing follows, except that the limiting
values of the sums are different; that
is, nothing (new). But now we could investigate why this is so. In so doing we may hit upon
numbers that are not representable by 2ν
× 3µ
. That is to say, we may hit upon larger prime
numbers, but we will never see that no quantity of such original numbers will suffice for the
representation of all numbers.
1 + + + . . . ≠ 1 + + + + . . .
However many terms of the form 1
–2v I put might together, they never add up to more than
2, whereas the first four terms of the series on the left already add up to more than 2. (So
the proof must already be contained in this.) And this is what it is contained in, and so is –
at the same time – the construction of a number that is not a power of 2, for the rule now
says: find the segment of the series that adds up to more than 2 – this must contain a number
that is not a power of 2.
1
23
1
22
1
2
1
3
1
2
8 (M): 14 \1
9 (M): 14 \2
10 (V): way: If you had only this proof, what
would you stake on it?
11 (M): 14 \3
12 (V): such
13 (F): MS 108, p. 284.
Euler’s Proof. 435e
lim lim lim
n
n
n
n
m
n
n m
1
2
1
3
would yield
1
n→ ν
ν
ν
ν
ν
ν
∞
=
=
→∞
=
=
→∞
=
=
∑ ∑ ∑
⎛
⎝
⎜
⎞
⎠
⎟ ⋅
⎛
⎝
⎜
⎞
⎠
⎟
0 0 1
WBTC121-130 30/05/2005 16:14 Page 871
(1 + + + . . . ) · (1 + + + . . . ) · . . . (1 + + + . . . ) = n.18
Wenn ich nun die Summe 1 + + + . . . so weit ausdehne, bis sie n überschreitet, dann
muß dieser Teil ein Glied enthalten, das in der rechten Reihe nicht gefunden werden kann,
denn enthielte die rechte Reihe alle diese Glieder, dann müßte sie eine größere und keine
kleinere Summe ergeben.
Die Bedingung, unter der ein Teil der Reihe 1 + + + . . . , etwa + + +
. . . , gleich oder größer als 1 wird, ist folgende:
Es soll also werden:
+ + + . . . => 1
Formen wir die linke Seite um in:
=
=
= => 1
∴ 2nν + 2ν − 2n2
− 2n + 2n + 2 − n2
− nν + n + ν => 0
nν + 3ν − 3n2
+ 2 + n => 0
ν => < 3n − 1.193n n
n
2
( )− +
+
2
3
1
1
2 2
1
−
−
+
+
− +
+
n
n
n
n
ν
ν
n n n n
n
n
n
n +( ) ( )− ⋅ − ⋅ + − ++
1
2
1
11 1ν ν
1 1 1 1 1
( ) ( ) . . . ( ) . . .+ − + − + − + + + + +−
−
1
n + 1
2
n + 2
n
n + (n 1)
n
2n
n
2n + 1
n
2n + 2
n
n +
n
ν
1 . . .+ + +n
n + 1
n
n + 2
n
n +
n
ν
1
n + ν
1
n + 2
1
n + 1
1
n
1
n + ν
1
n + 2
1
n + 1
1
n
1
3
1
2
1
3
1
2
1
n2
1
n
1
32
1
3
1
22
1
2
18 (F): MS 108, S. 285.
19 (F): MS 108, S. 286.
436 Eulerscher Beweis.
649
WBTC121-130 30/05/2005 16:14 Page 872
(1 + + + . . . ) · (1 + + + . . . ) · . . . (1 + + + . . . ) = n.14
Now if I extend the sum 1 + + + . . . until it is greater than n, this part must contain
a term that can’t be found in the series on the right, for if this series contained all those
terms, it would have to yield a larger, not a smaller, sum.
The condition for a segment of the series 1 + + + . . . , say + + + . . .
, being equal to or greater than 1 is as follows:
To make:
+ + + . . . => 1
transform the left side into:
=
=
= => 1
∴ 2nν + 2ν − 2n2
− 2n + 2n + 2 − n2
− nν + n + ν => 0
nν + 3ν − 3n2
+ 2 + n => 0
ν => < 3n − 1.153n n
n
2
( )− +
+
2
3
1
1
2 2
1
−
−
+
+
− +
+
n
n
n
n
ν
ν
n n n n
n
n
n
n +( ) ( )− ⋅ − ⋅ + − ++
1
2
1
11 1ν ν
1 1 1 1 1
( ) ( ) . . . ( ) . . .+ − + − + − + + + + +−
−
1
n + 1
2
n + 2
n
n + (n 1)
n
2n
n
2n + 1
n
2n + 2
n
n +
n
ν
1 . . .+ + +n
n + 1
n
n + 2
n
n +
n
ν
1
n + ν
1
n + 2
1
n + 1
1
n
1
n + ν
1
n + 2
1
n + 1
1
n
1
3
1
2
1
3
1
2
1
n2
1
n
1
32
1
3
1
22
1
2
14 (F): MS 108, p. 285.
15 (F): MS 108, p. 286.
Euler’s Proof. 436e
WBTC121-130 30/05/2005 16:14 Page 873
124
Dreiteilung des Winkels, etc.
Man könnte sagen: In der Geometrie der euklidischen Ebene kann man nach der 3-Teilung
des Winkels nicht suchen, weil es sie nicht gibt – und nach der 2-Teilung nicht, weil es
sie gibt.
In der Welt der euklidischen1
Elemente kann ich ebensowenig nach der 3-Teilung des
Winkels fragen, wie ich nach ihr suchen kann. Es ist von ihr einfach nicht die Rede.
(Ich kann der Aufgabe der 3-Teilung des Winkels in einem größern System ihren
Platz bestimmen, aber nicht im System der euklidischen2
Geometrie danach fragen, ob sie
lösbar ist.3
In welcher Sprache sollte ich denn danach fragen? in der euklidischen? – Und
ebensowenig kann ich in der euklidischen Sprache nach der Möglichkeit der 2-Teilung des
Winkels im euklidischen System fragen. Denn das würde in dieser Sprache auf eine Frage
nach der Möglichkeit schlechthin hinauslaufen, welche immer Unsinn ist.)
Wir müssen übrigens hier eine Unterscheidung zwischen gewissen Arten von Fragen
machen, eine Unterscheidung, die wieder zeigt, daß, was wir in der Mathematik ”Frage“
nennen, von dem verschieden ist, was wir im alltäglichen Leben so nennen. Wir müssen
unterscheiden zwischen einer Frage ”wie teilt man den Winkel in 2 gleiche Teile“ und der
Frage ”ist diese Konstruktion die Halbierung des Winkels“. Die Frage hat nur Sinn in einem
Kalkül, der uns eine Methode zu ihrer Lösung gibt; nun kann uns ein Kalkül sehr wohl
eine Methode zur Beantwortung der einen Frage geben, aber nicht zur Beantwortung der
andern. Euklid z.B. lehrt uns nicht nach der Lösung seiner Probleme suchen, sondern gibt
sie uns und beweist, daß es die Lösungen sind. Das ist aber keine psychologische oder
pädagogische Angelegenheit, sondern eine mathematische. D.h.4
der Kalkül (den er uns
gibt) ermöglicht es uns nicht, nach der Konstruktion zu suchen. Und ein Kalkül, der es
ermöglicht, ist eben ein anderer. (Vergleiche auch Methoden des Integrierens mit denen des
Differenzierens; etc.)
Es gibt eben in der Mathematik sehr Verschiedenes, was alles Beweis genannt wird und
diese Verschiedenheiten sind logische. Was also ”Beweis“ genannt wird, hat nicht mehr
miteinander zu tun, als was5
”Zahl“ genannt wird.
Welcher Art ist der Satz ”die 3-Teilung des Winkels mit Zirkel und Lineal ist
unmöglich“? Doch wohl von derselben, wie: ”in der Reihe der Winkelteilungen F(n) kommt
keine F(3) vor, wie in der Reihe der Kombinationszahlen keine 4“. Aber welcher
Art ist dieser Satz? Von der des Satzes: ”in der Reihe der Kardinalzahlen kommt nicht1
2
n n
2
( )⋅ − 1
1 (O): Euklidischen
2 (O): Euklidischen
3 (V): Geometrie nach der Möglichkeit der 3Teilung
fragen // nach ihrer Lösbarkeit fragen.
4 (O): D.H.
5 (V): was $Zah
650
651
652
437
WBTC121-130 30/05/2005 16:14 Page 874
124
The Trisection of an Angle, etc.
We could say: In Euclidean plane geometry we can’t search for the trisection of an angle,
because there is no such thing – and we can’t search for the bisection of an angle, because
there is such a thing.
In the world of Euclid’s Elements I can no more ask about the trisection of an angle than
I can search for it. It just isn’t mentioned.
(I can assign the problem of the trisection of an angle its place within a larger system, but
I can’t ask within the system of Euclidean geometry whether it’s solvable.1
In what language
should I ask this? In the Euclidean? – And it is equally impossible for me to ask in Euclidean
language about the possibility of bisecting an angle within the Euclidean system. For in
that language that would amount to a question about absolute possibility, which is always
nonsense.)
Incidentally, here we must make a distinction between certain types of questions, a
distinction that will show once again that what we call a “question” in mathematics is
different from what we call by that word in everyday life. We must distinguish between the
question “How does one divide an angle into two equal parts?” and the question “Is this
construction the bisection of an angle?” A question makes sense only in a calculus which
gives us a method for its solution; and a calculus may well give us a method for answering
the one question, but not the other. For instance, Euclid doesn’t teach us to search for the
solutions to his problems; rather, he gives them to us and then proves that they are the
solutions. And this isn’t a psychological or pedagogical matter, but a mathematical one. That
is, the calculus (that he gives us) doesn’t enable us to search for the construction. And a
calculus that does enable us to do that is simply a different one. (Compare also the methods
of integration with those of differentiation, etc.)
In mathematics there are just very different things that are all called proofs, and these
disparities are logical disparities. So the things that are called “proofs” have no more to do
with each other than the things called “numbers”.
What kind of proposition is “The trisection of an angle with ruler and compass is impossible”?
The same kind, no doubt, as “There is no F(3) in the series of angle-divisions
F(n), just as there is no 4 in the series of combination-numbers ”. But what kind of
proposition is that? The same kind as “ doesn’t occur in the series of cardinal numbers”.
That is evidently a (superfluous) rule of the game, something like: in draughts there is
no piece called “the queen”. And the question whether trisection is possible is then the
question whether there is such a thing in the game as trisection, whether there is a piece in
1
2
n n
2
( )⋅ − 1
1 (V): possible.
437e
WBTC121-130 30/05/2005 16:14 Page 875
vor“. Das ist offenbar eine (überflüssige) Spielregel, etwa wie die: im Damespiel kommt
keine Figur vor, die ”König“ genannt wird. Und die Frage, ob eine 3-Teilung möglich ist,
ist dann die, ob es eine 3-Teilung im Spiel gibt, ob es eine Figur im Damespiel gibt, die
”König“ genannt wird, und etwa eine ähnliche Rolle spielt, wie der Schachkönig. Diese
Frage wäre natürlich einfach durch eine Bestimmung zu beantworten, aber sie würde
kein Problem, keine Rechenaufgabe stellen. Hätte also einen andern Sinn, als eine, deren
Antwort lautete: ich werde ausrechnen, ob es so etwas gibt. (Etwa: ”ich werde ausrechnen,
ob es unter den Zahlen 5, 7, 18, 25, eine gibt, die durch 3 teilbar ist“.) Ist nun die Frage
nach der Möglichkeit der 3-Teilung des Winkels von dieser Art? Ja, – wenn man im Kalkül
ein allgemeines System hat, um, etwa, die Möglichkeit der n-Teilung zu berechnen.
Warum nennt man diesen Beweis den Beweis dieses Satzes? Der Satz ist ja kein Name,
sondern gehört (als Satz) einem Sprachsystem an: Wenn ich sagen kann ”es gibt keine
3-Teilung“, so hat es Sinn zu sagen ”es gibt keine 4-Teilung“ etc. etc. Und ist dies ein
Beweis des ersten Satzes (ein Teil seiner Syntax), so muß es also entsprechende Beweise
(oder Gegenbeweise) für die andern Sätze des Satzsystems geben, denn sonst gehören sie
nicht zu demselben System.
Ich kann nicht fragen, ob die 4 unter den Kombinationszahlen vorkommt, wenn das6
mein
Zahlensystem ist. Und nicht, ob unter den Kardinalzahlen vorkommt, oder zeigen, daß
es nicht eine von ihnen ist, außer, wenn ich ”Kardinalzahlen“ einen Teil eines Systems nenne,
welches auch enthält. (Ebensowenig kann ich aber auch sagen oder beweisen, daß 3 eine
der Kardinalzahlen ist.) Die Frage heißt vielmehr etwa so: ”Geht die Division 1 : 2 in ganzen
Zahlen aus“, und das läßt sich nur fragen in einem System, worin das Ausgehen und das
Nichtausgehen bekannt ist.7
(Die Ausrechnung muß Sinn haben.)
Bezeichnen wir mit ”Kardinalzahlen“ nicht einen Teil der rationalen Zahlen, so können
wir nicht ausrechnen, ob 81/3 eine Kardinalzahl ist, sondern, ob die Division 81 : 3 ausgeht
oder nicht.
Statt des Problems der 3-Teilung des Winkels mit Lineal und Zirkel können wir nun ein
ganz entsprechendes, aber viel übersichtlicheres, untersuchen. Es steht uns ja frei, die Möglichkeiten
der Konstruktion mit Lineal und Zirkel weiter einzuschränken. So können wir z.B.
die Bedingung setzen, daß sich die Öffnung des Zirkels nicht verändern läßt. Und wir können
festsetzen, daß die einzige Konstruktion, die wir kennen – oder besser: die unser Kalkül
kennt8
– diejenige ist, die man zur Halbierung einer Strecke AB benützt, nämlich:9
(Das könnte z.B. tatsächlich die primitive Geometrie eines Volkes sein. Und für sie gälte
das, was ich über die Gleichberechtigung der Zahlenreihe ”1, 2, 3, 4, 5, viele“ mit der Reihe
der Kardinalzahlen gesagt habe. Überhaupt ist es für unsere Untersuchungen ein guter Trick,
sich die Arithmetik oder Geometrie eines primitiven Volks vorzustellen.)10
Ich will diese Geometrie das System α nennen und fragen: ”ist die 3-Teilung der Strecke
im System α möglich?“
Welche 3-Teilung ist in dieser Frage gemeint? – denn davon hängt offenbar der Sinn der
Frage ab. Ist z.B. die physikalische 3-Teilung gemeint? D.h. die 3-Teilung durch Probieren
und Nachmessen. In diesem Falle ist die Frage vielleicht zu bejahen. Oder die optische
3-Teilung? d.h. die Teilung, deren Resultat drei gleichlang aussehende Teile sind?
1
2
1
2
6 (V): dieses
7 (V): Nichtausgehen vorkommt.
8 (V): nennt
9 (F): MS 113, S. 114v.
10 (V): auszumalen.)
653
654
438 Dreiteilung des Winkels . . .
A B
WBTC121-130 30/05/2005 16:14 Page 876
draughts called “the queen” which perhaps has a role like that of the queen in chess. Of
course this question could be answered simply by a stipulation; but it wouldn’t pose any
problem, any task of calculation; and so it would have a different sense from one whose answer
was: I will figure out whether there is such a thing. (For example: “I will figure out whether
there is one among the numbers 5, 7, 18, 25 that is divisible by 3”.) Now is the question
about the possibility of trisecting an angle of that kind? It is – if you have a general system
in the calculus for calculating, for instance, the possibility of division into n equal parts.
Why does one call this proof the proof of this proposition? A proposition isn’t a name,
after all; it belongs (as a proposition) to a system of language. If I can say “There is no such
thing as trisection” then it makes sense to say “There is no such thing as quadrisection”,
etc., etc. And if this is a proof of the first proposition (a part of its syntax), then there
must be corresponding proofs (or disproofs) for the other propositions of the propositional
system, for otherwise they don’t belong to the same system.
I can’t ask whether 4 occurs among the combination-numbers if that2
is my number
system. And I can’t ask whether occurs among the cardinal numbers, or show that it isn’t
one of them, unless I mean by “cardinal numbers” part of a system that contains as well.
(But neither can I say or prove that 3 is one of the cardinal numbers.) Rather, the question
goes something like this: “Does the division 1 ÷ 2 come out in whole numbers?”, and
that can only be asked in a system in which coming out even and not coming out even are
familiar.3
(The calculation must make sense.)
If by “cardinal numbers” we aren’t referring to a subset of the rational numbers, then
we can’t work out whether 81/3 is a cardinal number; what we can work out is whether the
division 81 ÷ 3 comes out even or not.
Instead of the problem of trisecting an angle with ruler and compass we can now investigate
one that corresponds to it completely, but can be surveyed much more easily. After
all, there is nothing to prevent us from further restricting the possibilities of construction
with ruler and compass. For instance, we might lay down the condition that the opening of
the compass can’t be changed. And we can stipulate that the only construction we know – or
better: that our calculus knows – is the one used to bisect a line-segment AB, namely4
(That might actually be the primitive geometry of a tribe, for instance. And what I said
earlier about the number series “1, 2, 3, 4, 5, many” having equal rights with the series of
cardinal numbers would be valid for this geometry too. In general it is a good trick in our
investigations to imagine5
the arithmetic or geometry of a primitive people.)
I want to call this geometry system α and ask: “Is the trisection of a line possible in
system α?”
What kind of trisection is meant in this question? – for that’s obviously what the sense
of the question depends on. For instance, is what is meant physical trisection? Trisection,
that is, by trial and error and measurement? In that case the answer is perhaps yes. Or is
optical trisection meant? A division, that is, that yields three parts that seem to have the
1
2
1
2
2 (V): this
3 (V): coming out even occurs.
4 (F): MS 113, p. 114v.
5 (V): picture
The Trisection of an Angle . . . 438e
A B
WBTC121-130 30/05/2005 16:14 Page 877
Wenn wir z.B. durch ein verzerrendes Medium sehen, so ist es ganz leicht
vorstellbar, daß uns die Teile a, b, und c gleichlang erscheinen.11
Nun könnte man die Resultate der Teilungen im System α nach der
Zahl der erzeugten Teile durch die Zahlen 2, 22
, 23
, u.s.w. darstellen; und
die Frage, ob die 3-Teilung möglich ist, könnte bedeuten: ist eine der Zahlen
in dieser Reihe = 3. Diese Frage kann freilich nur gestellt werden, wenn
die 2, 22
, 23
, etc. in einem andern System (etwa den Kardinalzahlen)
eingebettet sind; nicht, wenn sie selbst unser Zahlensystem sind; denn dann
kennen wir – oder unser System – eben die 3 nicht. – Aber wenn unsere Frage lautet:
ist eine der Zahlen 2, 22
, etc. gleich 3, so ist hier eigentlich von einer 3-Teilung der Strecke
nicht die Rede. Immerhin könnte12
die Frage nach der Möglichkeit der 3-Teilung so
aufgefaßt werden. – Eine andere Auffassung erhalten wir nun, wenn wir dem System α ein
System β hinzufügen, worin es die Streckenteilung nach Art dieser Figur
13
gibt. Es kann nun gefragt werden: ist die Teilung β in 108 Teile
eine Teilung der Art α? Und diese Frage könnte wieder auf die hinauslaufen:
ist 108 eine Potenz von 2? aber sie könnte auch auf eine andere
Entscheidungsart hinweisen (einen andern Sinn haben), wenn wir die
Systeme α und β zu einem geometrischen Konstruktionssystem
verbinden; so zwar, daß es sich nun in diesem System beweisen läßt,
daß die beiden Konstruktionen die gleichen Teilungspunkte B, C, D
”liefern müssen“.14
Denken wir nun, es hätte Einer im System α eine Strecke AB in 8
Teile geteilt, nähme15
diese nun zu den Strecken a, b, c zusammen und
fragte: ist das eine Teilung in 3 gleiche Teile.16
(Wir könnten uns den Fall übrigens leichter mit
einer größeren Anzahl ursprünglicher Teile vorstellen, die es möglich
macht, 3 gleichlang aussehende Gruppen von Teilen zu bilden.) Die Antwort auf diese Frage
wäre der Beweis, daß 23
nicht durch 3 teilbar ist; oder der Hinweis darauf, daß sich die Teile
a, b, c wie 1 : 3 : 4 verhalten. Und nun könnte man fragen: habe ich also im System α nicht
doch einen Begriff von der 3-Teilung, nämlich der Teilung, die die Teile a, b, c im Verhältnis
1 : 1 : 1 hervorbringt? Gewiß, ich habe nun einen neuen Begriff ”3-Teilung einer Strecke“
eingeführt; wir könnten ja sehr wohl sagen, daß wir durch die 8-Teilung der Strecke AB
die Strecke CB in 3 gleiche Teile geteilt haben, wenn das eben heißen soll: wir haben eine
Strecke erzeugt, die aus 3 gleichen Teilen besteht.17
Die Perplexität, in der wir uns bezüglich des
Problems der 3-Teilung befanden, war etwa die: Wenn die 3-Teilung des Winkels
unmöglich ist – logisch unmöglich – wie kann man dann überhaupt nach ihr fragen? Wie
kann man das logisch Unmögliche beschreiben und nach seiner Möglichkeit sinnvoll fragen?
D.h., wie kann man logisch nicht zusammenpassende Begriffe zusammenstellen (gegen
die Grammatik, also unsinnig) und sinnvoll nach der Möglichkeit dieser Zusammenstellung
fragen? – Aber dieses Paradox fände sich ja wieder, wenn man fragt: ”ist 25 × 25 = 620?“ –
da es doch logisch unmöglich ist, daß diese Gleichung stimmt; ich kann ja nicht beschreiben,
wie es wäre, wenn –. Ja, der Zweifel ob 25 × 25 = 620 (oder der, ob es = 625 ist) hat eben
den Sinn, den die Methode der Prüfung ihm gibt. Und die Frage nach der Möglichkeit der
11 (F): MS 113, S. 115r.
12 (V): kann
13 (F): MS 113, S. 115v.
14 (F): MS 113, S. 115v.
15 (O): nehme
16 (V): eine 3-Teilung. (F): MS 113, S. 115v.
17 (F): MS 113, S. 116r.
655
656
439 Dreiteilung des Winkels . . .
a b c
α β
A B C D E
A
a b c
B
A C B
WBTC121-130 30/05/2005 16:14 Page 878
same length? If, for instance, we look through some distorting medium, it
is quite easily imaginable that the parts a, b, and c might appear to us to
be of the same length.6
Now we could represent the results of the divisions in system α by the
numbers 2, 22
, 23
, etc., corresponding to the number of segments produced;
and the question whether trisection is possible could mean: Does any of
the numbers in this series = 3? To be sure, this question can only be asked
if 2, 22
, 23
, etc. are embedded in another system (say in the system of cardinal numbers);
if these numbers are themselves our number system it can’t be asked, for in that case we –
or our system – simply aren’t acquainted with the number 3. – But if our question is: Is one
of the numbers 2, 22
, etc. equal to 3?, then we’re not really talking about a trisection of the
line-segment here. Nonetheless, we might7
understand the question about the possibility of
trisection in that way. – Now we get a different understanding if we
adjoin to the system α a system β in which line-segments are divided
in the manner of this figure. 8
Now we can ask: Is a division β into
108 sections a division of type α? And this question could again boil
down to: Is 108 a power of 2? But it could also point to a different
decision procedure (have a different sense) if we connected the
systems α and β to form a system of geometrical constructions in
such a way that now it could be proved in this system that the two
constructions “must yield” the same division points B, C, D.9
Now suppose that someone, having divided a line-segment AB
into 8 sections in system α, were now to group these into the linesegments
a, b, c, and were to ask: Is this a division into 3 equal
parts?10
(Incidentally, we could imagine the case more easily with a larger number of original sections,
which would make it possible to form 3 groups of sections which seemed to be of equal
length.) The answer to this question would be a proof that 23
is not divisible by 3; or an
indication that the sections a, b, c are in the ratio 1 : 3 : 4. And now you could ask: So don’t
I have a concept of trisection in system α after all, i.e. a concept of a division which yields
the parts a, b, c, in the ratio 1 : 1 : 1? Of course, I have now introduced a new concept of
“trisection of a line-segment”; we could perfectly well say that by dividing the line-segment
AB into eight parts we have divided the line-segment CB into 3 equal parts, if that is to
mean: we have produced a line that consists of 3 equal parts.11
The perplexity in which we found ourselves in relation to the problem of trisection was
roughly this: if the trisection of an angle is impossible – logically impossible – then how can
we ask about it at all? How can we describe what is logically impossible and meaningfully
ask about its possibility? That is, how can one (in violation of grammar, and thus nonsensically)
put together concepts that don’t fit together logically, and meaningfully ask about the
possibility of this combination? But this paradox would recur if we asked “Is 25 × 25 = 620?”,
since it’s logically impossible that this equation is correct; after all, I can’t describe what
it would be like if . . . . – Indeed, the doubt whether 25 × 25 = 620 (or whether it = 625)
has precisely the sense that the method of checking gives it. And the question about the
6 (F): MS 113, p. 115r.
7 (V): can
8 (F): MS 113, p. 115v.
9 (F): MS 113, p. 115v.
10 (V): a tripartite division? (F): MS 113,
p. 115v.
11 (F): MS 113, p. 116r.
The Trisection of an Angle . . . 439e
a b c
α β
A B C D E
A
a b c
B
A C B
WBTC121-130 30/05/2005 16:14 Page 879
3-Teilung hat den Sinn, den die Methode der Prüfung ihr gibt. Es ist ganz richtig: wir stellen
uns hier nicht vor, oder beschreiben, wie es ist, wenn 25 × 25 = 620 ist, und das heißt eben,
daß wir es hier mit einer andern (logischen) Art von Frage zu tun haben, als etwa der: ”ist
diese Straße 620 oder 625m lang?“
(Wir sprechen von einer ”Teilung des Kreises in 7 Teile“ und von einer Teilung des Kuchens
in 7 Teile.)
440 Dreiteilung des Winkels . . .
WBTC121-130 30/05/2005 16:14 Page 880
possibility of trisection has the sense that the method of checking gives it. It is quite correct
that: Here we don’t imagine, or describe, what it is like if 25 × 25 = 620; and what that means
is simply that here we are dealing with a type of question that is (logically) different from,
say, “Is this street 620 or 625 metres long”?
(We talk about a “division of a circle into 7 segments” and of a division of a cake into
7 segments.)
The Trisection of an Angle . . . 440e
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125
Suchen und Versuchen.
Wenn man jemandem,1
der es noch nicht versucht hat, sagt ”versuche die Ohren zu
bewegen“, so wird er zuerst etwas in der Nähe der Ohren bewegen, was er schon früher
bewegt hat, und dann werden sich entweder auf einmal seine Ohren bewegen oder nicht.
Man könnte nun von diesem Vorgang sagen: er versucht die Ohren zu bewegen. Aber wenn
das ein Versuch genannt werden kann, so ist es einer in einem ganz anderen Sinn als der,
die Ohren (oder die Hände) zu bewegen, wenn wir zwar ”wohl wissen, wie es zu machen
ist“, aber sie jemand hält, sodaß wir sie schwer oder nicht bewegen können. Der Versuch
im ersten Sinne entspricht einem Versuch, ”ein mathematisches Problem zu lösen“, zu dessen
Lösung es keine2
Methode gibt. Man kann sich immer um das scheinbare Problem
bemühen. Wenn man mir sagt ”versuche, durch den bloßen Willen den Krug dort am anderen
Ende des Zimmers zu bewegen“ so werde ich ihn anschauen und vielleicht irgendwelche
seltsame Bewegungen mit meinen Gesichtsmuskeln machen; also selbst in diesem Falle
scheint es einen Versuch zu geben.
Denken wir daran, was es heißt, etwas im Gedächtnis zu suchen.
Hier liegt gewiß etwas wie ein Suchen im eigentlichen Sinn vor.
Versuchen, eine Erscheinung hervorzurufen, aber heißt nicht, sie suchen.
Angenommen, ich taste meine Hand nach einer schmerzhaften Stelle ab, so suche ich wohl
im Tastraum, aber nicht im Schmerzraum. D.h., was ich eventuell finde, ist eigentlich eine
Stelle und nicht der Schmerz. D.h., wenn die Erfahrung auch ergeben hat, daß Drücken3
einen Schmerz hervorruft, so ist doch das Drücken kein Suchen nach einem Schmerz. So
wenig, wie das Drehen einer Elektrisiermaschine das Suchen nach einem Funken ist.
|Kann man versuchen, zu einer Melodie den falschen Takt zu schlagen? Oder: Wie
verhält sich dieser Versuch4
zu dem, ein Gewicht zu heben, das uns zu schwer ist? |
|Es ist nicht nur höchst bedeutsam, daß man die Gruppe ||||| auf vielerlei Arten
sehen kann (in vielerlei Gruppierungen), sondern (noch) viel bemerkenswerter,5
daß man es
willkürlich tun kann. D.h., daß es einen ganz bestimmten Vorgang gibt, eine bestimmte
”Auffassung“ auf Befehl zu bekommen; und daß es – dem entsprechend – auch einen ganz
bestimmten Vorgang des vergeblichen Versuchens gibt. So kann man auf Befehl die Figur
6
so sehen, daß der eine oder der andere Vertikalstrich die Nase, dieser oder jener
Strich der Mund wird, und kann unter Umständen das eine oder das andere
vergeblich versuchen.|
1 (O): jemanden,
2 (O): eine
3 (O): drücken
4 (V): sich dieses Versuchen
5 (O): viel mehr bemerkenswerter,
6 (F): MS 112, S. 50r.
657
658
441
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125
Trying to Find and Trying.
If you say to someone who hasn’t yet tried it, “Try to move your ears”, he will first move
some part of his body near his ears that he has moved before, and then either his ears will
suddenly move or they won’t. Now you could say of this process: He is trying to move his
ears. But if that can be called trying, it’s trying in a completely different sense from trying
to move our ears (or our hands) when we already “know how to do it” but someone is holding
them, so that we can move them only with difficulty or not at all. Trying in the first
sense corresponds to trying “to solve a mathematical problem” when there is no method for
its solution. One can always strive to solve the apparent problem. If someone says to me
“Try to move that jug at the other end of the room using only your will power”, I will look
at it and perhaps make some strange movements with my facial muscles; so even in that
case there seems to be such a thing as trying.
Let’s think of what it means to try to find something in one’s memory.
Here there is certainly something like trying to find in the strict sense.
But trying to evoke a phenomenon does not mean trying to find it.
Suppose I am feeling for a painful place on my hand; in that case I am trying to find it
in the space of touch, but not in the space of pain. That means: what I may find is actually
a location and not the pain. That means that even if experience has shown that pressing
produces a pain, pressing still isn’t trying to find a pain, any more than turning the handle
of an electrostatic machine is trying to find a spark.
|Can one try to beat the wrong time to a melody? Or: How is trying to do this like
trying to lift a weight that’s too heavy for us?|
|It’s not only highly significant that one can see the group ||||| in many different ways
(in many different groupings); what is (still) more noteworthy is that one can do it at will.
That is, that there is a very definite process of taking a particular “view” on command; and
that correspondingly there is also a very definite process of vainly trying to do so. Thus,
when ordered to do so you can see this figure 1
in such a way that one or the other
vertical line becomes the nose, this or that line the mouth; and in certain circumstances
you can try in vain to do the one or the other.|
1 (F): MS 112, p. 50r.
441e
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|Das Wesentliche ist hier, daß dieser Versuch den Charakter desjenigen hat, ein Gewicht
mit der Hand zu heben; nicht den Charakter des Versuchs, in welchem man Verschiedenes
tut, verschiedene Mittel ausprobiert, um (z.B.) ein Gewicht zu heben. In den zwei
Fällen hat das Wort ”Versuch“ ganz verschiedene Bedeutungen. (Eine außerordentlich
folgenreiche grammatische Tatsache.)|
442 Suchen und Versuchen.
659
WBTC121-130 30/05/2005 16:14 Page 884
|The essential thing here is that this kind of trying is like trying to lift a weight with one’s
hand; not like the kind of trying where one does different things, tries out different means,
in order (for example) to lift a weight. In the two cases the word “trying” has completely
different meanings. (A grammatical fact that is extraordinarily rich in consequences.)|
Trying to Find and Trying. 442e
WBTC121-130 30/05/2005 16:14 Page 885
Induktionsbeweis.
Periodizität.
660
WBTC121-130 30/05/2005 16:14 Page 886
Inductive Proofs.
Periodicity.
WBTC121-130 30/05/2005 16:14 Page 887
126
Inwiefern beweist der
Induktionsbeweis einen Satz?
Ist der Induktionsbeweis ein Beweis von a + (b + c) = (a + b) + c, so muß man sagen
können: die Rechnung liefert, daß a + (b + c) = (a + b) + c ist (und kein anderes Resultat).
Denn dann muß erst die Methode der Berechnung (allgemein) bekannt sein und, wie wir
darauf 25 × 16 ausrechnen können, so auch a + (b + c). Es wird also erst eine allgemeine
Regel zur Ausrechnung aller solcher Aufgaben gelehrt und danach die besondere gerechnet.
– Welches ist aber hier die allgemeine Methode der Ausrechnung? Sie muß auf allgemeinen
Zeichenregeln beruhen ( – etwa, wie dem associativen Gesetz – ).
Wenn ich a + (b + c) = (a + b) + c negiere, so hat das nur Sinn, wenn ich etwa sagen will:
es ist nicht a + (b + c) = (a + b) + c, sondern = (a + 2b) + c. Denn es fragt sich: was ist der
Raum, in welchem ich den Satz negiere? wenn ich ihn abgrenze, ausschließe, – wovon?
Die Kontrolle von 25 × 25 = 625 ist die Ausrechnung von 25 × 25, die Berechnung der
rechten Seite; – kann ich nun a + (b + c) = (a + b) + c errechnen, das Resultat (a + b) + c
ausrechnen? Je nachdem man es als berechenbar oder unberechenbar betrachtet, ist es
beweisbar oder nicht. Denn ist der Satz eine Regel, der jede Ausrechnung folgen muß, ein
Paradigma, dann hat es keinen Sinn, von einer Ausrechnung der Gleichung zu reden; sowenig,
wie von der einer Definition.
Das, was die Ausrechnung möglich macht, ist das System, dem der Satz angehört und
das auch die Rechenfehler bestimmt, die sich bei der Ausrechnung machen lassen. Z.B.
ist (a + b)2
= a2
+ 2ab + b2
und nicht = a2
+ ab + b2
; aber (a + b)2
= −4 ist kein möglicher
Rechenfehler in diesem System.
Ich könnte ja auch ganz beiläufig (siehe andere Bemerkungen1
) sagen: ”25 × 64 = 160,
64 × 25 = 160; das beweist, daß a × b = b × a ist“ (und diese Redeweise ist nicht vielleicht
lächerlich und falsch; sondern man muß sie nur recht deuten). Und man kann richtig daraus
schließen; also läßt sich ”a · b = b · a“ in einem Sinne beweisen.2
Und ich will sagen: Nur in dem Sinne, in welchem die Ausrechnung so eines Beispiels
Beweis des algebraischen Satzes genannt werden kann, ist der Induktionsbeweis ein Beweis
dieses Satzes. Nur insofern kontrolliert er den algebraischen Satz. (Er kontrolliert seinen Bau,3
nicht seine Allgemeinheit.)
(Die Philosophie prüft nicht die Kalküle der Mathematik, sondern nur, was die
Mathematiker über diese Kalküle sagen.)
1 (O): Bemerungen
2 (V): berechnen.
3 (V): kontrolliert seine Struktur,
661
662
444
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126
To what Extent does a Proof by
Induction Prove a Proposition?
If a proof by induction is a proof of a + (b + c) = (a + b) + c, we must be able to say: the
calculation gives the result that a + (b + c) = (a + b) + c (and no other result).
For in that case the method of calculation must first be known (in general), and as easily
as we can then calculate 25 × 16, so can we calculate a + (b + c). So first a general rule is
taught for calculating all such problems, and afterwards the particular problem is calculated.
– But what is the general method of calculation here? It must be based on general rules for
signs (– such as, say, the associative law – ).
If I negate a + (b + c) = (a + b) + c, this only makes sense if I want to say, for instance:
a + (b + c) isn’t (a + b) + c, but is (a + 2b) + c. For the question is: What is the space in
which I negate the proposition? If I mark it off and exclude it – from what?
To check 25 × 25 = 625 I calculate 25 × 25, computing the right side; – can I now calculate
a + (b + c) = (a + b) + c, computing the result (a + b) + c? Depending on whether we
look at it as calculable or not, it is or isn’t provable. For if the proposition is a rule which
every calculation has to follow, a paradigm, then it makes no more sense to talk about
calculating that equation than to talk about calculating a definition.
What makes the calculation possible is the system to which the proposition belongs; and
that also determines the mistakes that can be made in the calculation. For example, (a + b)2
is a2
+ 2ab + b2
, and not a2
+ ab + b2
; but (a + b)2
= −4 is not a possible miscalculation in
this system.
I could also say, quite off the cuff (see other remarks): “25 × 64 = 160, 64 × 25 = 160;
that proves that a × b = b × a” (and this way of speaking is not ridiculous and incorrect; you
just have to interpret it correctly). And you can infer correctly from that; so in one sense
“a · b = b · a” can be proved.1
And I want to say: It is only in the sense in which you can call calculating such an example
a proof of the algebraic proposition that the proof by induction is a proof of this
proposition. Only to that extent is it a check of the algebraic proposition. (It is a check of
its construction2
, not its generality).
(Philosophy doesn’t examine the calculi of mathematics, but only what mathematicians
say about these calculi.)
1 (V): calculated. 2 (V): structure
444e
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127
Der rekursive Beweis und der
Begriff des Satzes. Hat der Beweis
einen Satz als wahr erwiesen und
sein Gegenteil1
als falsch?
Hat der rekursive Beweis von a + (b + c) = (a + b) + c . . . . . A
eine Frage beantwortet? und welche? Hat er eine Behauptung als wahr erwiesen und also
ihr Gegenteil als falsch?
Das, was Skolem2
den rekursiven Beweis von A nennt, kann man so schreiben:
a + (b + 1) = (a + b) + 1
a + (b + (c + 1)) = a + ((b + c) + 1) = (a + (b + c)) + 1 . . . . . B
(a + b) + (c + 1) = ((a + b) + c) + 1
In diesem Beweis kommt offenbar der bewiesene Satz gar nicht vor. – Man müßte nur
eine allgemeine Bestimmung machen,3
die den Übergang zu ihm erlaubt. Diese Bestimmung
könnte man so ausdrücken:
α φ(1) = ψ(1)
β φ(c + 1) = F (φ(c)) φ(c) = ψ(c) . . . . . ∆
γ ψ(c + 1) = F (ψ(c))
Wenn 3 Gleichungen von der Form α, β, γ bewiesen sind, so sagen wir, es sei ”die Gleichung
∆ für alle Kardinalzahlen bewiesen“. Das ist eine Erklärung dieser Ausdrucksform durch
die erste. Sie zeigt, daß wir das Wort ”beweisen“ im zweiten Fall anders gebrauchen als im
ersten. Es ist jedenfalls irreführend zu sagen, wir hätten die Gleichung ∆ oder A bewiesen,
und vielleicht besser zu sagen, wir hätten ihre Allgemeingültigkeit bewiesen, obwohl das
wieder in anderer Hinsicht irreführend ist.
Hat nun der Beweis B eine Frage beantwortet, eine Behauptung als wahr erwiesen?
Ja, welches ist denn der Beweis B: ist4
es die Gruppe der 3 Gleichungen von der Form
α, β, γ, oder die Klasse der Beweise dieser Gleichungen? Diese Gleichungen behaupten
ja etwas (und beweisen nichts in dem Sinne, in dem sie bewiesen werden). Die Beweise
von α, β, γ aber beantworten die Frage, ob diese 3 Gleichungen stimmen, und erweisen
die Behauptung als wahr, daß sie stimmen. Ich kann nun erklären: die Frage, ob A für alle
Kardinalzahlen gilt, solle bedeuten: ”gelten für die Funktionen
1 (V): und einen andern
2 (V): man
3 (V): treffen,
4 (V): B. Ist
663
664
5
6
7
5
6
7
445
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127
Recursive Proof and the Concept of a
Proposition. Did the Proof Prove
a Proposition True and its
Contradictory1
False?
Did the recursive proof of a + (b + c) = (a + b) + c . . . . . A
answer a question? And if so, what question? Did it show an assertion to be true and thus
its contradictory to be false?
What Skolem2
calls the recursive proof of B can be written thus:
a + (b + 1) = (a + b) + 1
a + (b + (c + 1)) = a + ((b + c) + 1) = (a + (b + c)) + 1 . . . . . B
(a + b) + (c + 1) = ( (a + b) + c) + 1
In this proof the proposition proved obviously doesn’t even occur. – All we have to do
is to make3
a general stipulation that allows the transition to it. This stipulation could be
expressed thus:
α φ(1) = ψ(1)
β φ(c + 1) = F (φ(c)) φ(c) = ψ(c) . . . . . ∆
γ ψ(c + 1) = F (ψ(c))
If 3 equations of the form α, β, γ have been proved, we say “The equation ∆ has been proved
for all cardinal numbers”. This is a definition of this latter form of expression in terms of
the first. It shows that the word “prove” is used differently in the second case than in the
first. In any case it is misleading to say that we have proved the equation ∆ or A. Perhaps
it is better to say that we have proved its general validity, though that too is misleading in
other respects.
Now has proof B answered a question, proved an assertion true? Well, which is proof B,
anyway? Is it the group of three equations of the form α, β, γ, or the class of proofs of these
equations? After all, these equations assert something (and don’t prove anything in the sense
in which they are proved). But the proofs of α, β, γ answer the question whether these three
equations are correct, and show the truth of the assertion that they are correct. So I can
produce a definition: the question whether A is valid for all cardinal numbers is to mean:
“Are the equations α, β, and γ valid for the functions
1 (V): and Another Proposition
2 (V): one
3 (V): produce
5
6
7
5
6
7
445e
WBTC121-130 30/05/2005 16:14 Page 891
φ(ξ) = a + (b + ξ), ψ(ξ) = (a + b) + ξ
Gleichungen α, β und γ?“ Und dann ist diese Frage durch den rekursiven Beweis von
A beantwortet, wenn hierunter die Beweise von α, β, γ verstanden werden (bezw. die
Festsetzung von α und die Beweise von β und γ mittels α).
Ich kann also sagen, daß der rekursive Beweis ausrechnet, daß die Gleichung A einer
gewissen Bedingung genügt; aber es ist nicht eine Bedingung der Art, wie sie etwa die
Gleichung (a + b)2
= a2
+ 2ab + b2
erfüllen muß, um ”richtig“ genannt zu werden. Nenne
ich A ”richtig“, weil sich Gleichungen von der Form α, β, γ dafür beweisen lassen, so
verwende ich jetzt das Wort ”richtig“ anders, als im Falle der Gleichungen α, β, γ oder
(a + b)2
= a2
+ 2ab + b2
.
Was heißt ”1 : 3 = 0,3“? heißt es dasselbe wie ”11: 3 = 0,3“? – Oder ist diese Division
1
der Beweis des ersten Satzes? D.h.: steht sie zu ihm im Verhältnis der Ausrechnung zum
Bewiesenen?
”1 : 3 = 0,3“ ist ja nicht von der Art, wie
”1 : 2 = 0,5“; vielmehr entspricht
”1 : 2 = 0,5“ dem ”1 : 3 = 0,3“ (aber nicht dem ”1 : 3 = 0,3“).
0 1 1
Ich will einmal statt der Schreibweise ”1 : 4 = 0,25“ die annehmen:5
”1 : 4 = 0,25“ also z.B. ”3 : 8 = 0,375“
0 0
dann kann ich sagen, diesem Satz entspricht nicht der: 1 : 3 = 0,3, sondern z.B der:
”1 : 3 = 0,333“. 0,3 ist nicht in dem Sinne Resultat (Quotient) der Division, wie 0,375.
1
Denn die Ziffer ”0,375“6
war uns vor der Division 3 : 8 bekannt; was aber bedeutet ”0,3“
losgelöst von der periodischen Division? – Die Behauptung, daß die Division a : b als
Quotienten 0, 5 ergibt, ist dieselbe wie die: die erste Stelle des Quotienten sei c und der
erste Rest gleich dem Dividenden.
Nun steht B zur Behauptung, A gelte für alle Kardinalzahlen, im selben Verhältnis, wie
1 : 3 = 0,3 zu 1 : 3 = 0,3.
1
Der Gegensatz zu der Behauptung ”A gilt für alle Kardinalzahlen“ ist nun: eine der
Gleichungen α, β, γ sei falsch. Und die entsprechende Frage sucht keine Entscheidung
zwischen einem (x).fx und einem (∃x).~fx.
Die Konstruktion der Induktion ist nicht ein Beweis, sondern eine bestimmte
Zusammenstellung (ein Muster im Sinne von Ornament) von Beweisen. Man kann ja
auch nicht sagen: ich beweise eine Gleichung, wenn ich drei beweise. Wie die Sätze einer
Suite nicht einen Satz ergeben.
5 (V): gebrauchen: 6 (V): die Zahl 0,375
665
666
446 Der rekursive Beweis . . .
WBTC121-130 30/05/2005 16:14 Page 892
φ(ξ) = a + (b + ξ), ψ(ξ) = (a + b) + ξ?”
And then that question will have been answered by the recursive proof of A, if by that
the proofs of α, β, γ are understood (or the laying down of α and the proofs of β and γ
using it).
So I can say that the recursive proof shows that the equation A satisfies a certain condition;
but it isn’t the kind of condition that, say, the equation (a + b)2
= a2
+ 2ab + b2
has to
fulfill in order to be called “correct”. If I call A “correct” because equations of the form
α, β, γ can be proved for it, I am now using the word “correct” differently than with the
equations α, β, γ, or (a + b)2
= a2
+ 2ab + b2
.
What does “1 ÷ 3 = 0·3”? mean? The same as “1 ÷ 3 = 0·3”? – Or is that division the
1
proof of the first proposition? That is, does it have the same relationship to it as a
calculation has to what is proved?
“1 ÷ 3 = 0·3” is not the same kind of thing as
“1 ÷ 2 = 0·5”; rather, what
“1 ÷ 2 = 0·5” corresponds to is “1 ÷ 3 = 0·3” (but not “1 ÷ 3 = 0·3”).
0 1 1
Instead of the notation “1 ÷ 4 = 0·25” let me adopt4
for this occasion the following:
“1 ÷ 4 = 0·25”. So, for example “3 ÷ 8 = 0·375”.
0 0
Then I can say that what corresponds to this proposition is not 1 ÷ 3 = 0·3, but, for
example:
“1 ÷ 3 = 0·333”. 0·3 is not a result of division (quotient) in the same sense as 0·375.
1
For we were acquainted with the numeral “0·375”5
before the division 3 ÷ 8; but what does
“0·3” mean when detached from the periodic division? – The assertion that the division
a ÷ b gives 0·5 as a quotient is the same as the assertion that the first place of the quotient
is c and the first remainder is the same as the dividend.
The relation of B to the assertion that A holds for all cardinal numbers is the same as
that of
1 ÷ 3 = 0·3 to 1 ÷ 3 = 0·3.
1
The contradictory of the assertion “A is valid for all cardinal numbers” is that one of the
equations α, β, γ is false. And the corresponding question doesn’t ask for a decision between
a (x).fx and a (∃x).~fx.
The construction of the induction is not one proof, but a particular combination of
proofs (a pattern of proofs in the sense of a decoration). For neither can one say: If I
prove three equations, then I prove one. Just as the movements of a suite don’t amount
to a single movement.
4 (V): use 5 (V): the number 0·375
Recursive Proof and the Concept of a Proposition. . . . 446e
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Man kann auch so sagen: Sofern man die Regel, in irgendeinem Spiel Dezimalbrüche zu
bilden, die nur aus der Ziffer 3 bestehen, sofern man diese Regel als eine Art Zahl auffaßt,
kann eine Division sie nicht zum Resultat haben, sondern nur das, was man periodische
Division nennen kann und was die Form a : b = c hat.
a
447 Der rekursive Beweis . . .
WBTC121-130 30/05/2005 16:14 Page 894
We can also put it this way: In so far as one understands this rule – the rule in a certain
game of forming decimal fractions consisting only of the numeral 3 – as a kind of number,
this number cannot be the result of a division, but only of what we can call a periodic
division, of the form a ÷ b = c.
a
Recursive Proof and the Concept of a Proposition. . . . 447e
WBTC121-130 30/05/2005 16:14 Page 895
128
Induktion, (x)φx und (∃x)φx.
Inwiefern erweist die Induktion den
allgemeinen Satz als wahr und einen
Existentialsatz als falsch?
3 × 2 = 5 + 1
3 × (a + 1) = 3 + (3 × a) = (5 + b) + 3 = 5 + (b + 3)
Warum nennst Du denn diese Induktion den Beweis dafür, daß (n): n > 2 .⊃. 3 × n ≠ 5?!
– Nun, siehst Du denn nicht, daß der Satz, wenn er für n = 2 gilt, auch für n = 3 gilt, und
dann auch für n = 4, und daß es immer so weiter geht? (Was erkläre ich denn, wenn ich das
Funktionieren des induktiven Beweises erkläre?) Du nennst ihn also einen Beweis für ”f(2)
& f(3) & f(4) & u.s.w.“, ist er aber nicht vielmehr die Form der Beweise für ”f(2)“ und ”f(3)“
und ”f(4)“ u.s.w.? Oder kommt das auf eins hinaus? Nun, wenn ich die Induktion den Beweis
eines Satzes nenne, dann darf ich es nur, wenn das nichts anderes heißen soll, als daß sie
jeden Satz einer gewissen Form beweist. (Und mein Ausdruck bedient sich der Analogie
vom Verhältnis der Sätze ”alle Säuren färben Lackmuspapier1
rot“, ”Schwefelsäure färbt
Lackmuspapier2
rot“.)
Denken wir nun, jemand sagte ”prüfen wir nach, ob f(n) für alle n gilt“ und nun fängt
er an, die Reihe zu schreiben:
3 × 2 = 5 + 1
3 × (2 + 1) = (3 × 2) + 3 = (5 + 1) + 3 = 5 + (1 + 3)
3 × (2 + 2) = (3 × (2 + 1)) + 3 = (5 + (1 + 3)) + 3 = 5 + (1 + 3 + 3)
und nun bricht er ab und sagt: ”ich sehe schon, daß es für alle n gilt“. – So hat er also eine
Induktion gesehen! Aber hatte er denn nach einer Induktion gesucht? Er hatte ja gar keine
Methode, um nach einer3
zu suchen. Und hätte er nun keine entdeckt, hätte er damit eine
Zahl gefunden, die der Bedingung nicht entspricht? – Die Regel der Kontrolle kann ja nicht
lauten: sehen wir nach, ob sich eine Induktion findet, oder ein Fall, für den das Gesetz nicht
gilt. – Wenn das Gesetz vom ausgeschlossenen Dritten nicht gilt, so heißt das nur, daß unser
Ausdruck nicht mit einem Satz zu vergleichen ist.
Wenn wir sagen, die Induktion beweise den allgemeinen Satz, so wollen wir natürlich zur
Ausdrucksform übergehen, sie beweise, daß dies, und nicht sein Gegenteil der Fall ist.4
Welches
wäre aber das Gegenteil des Bewiesenen? Nun, daß (∃n).~fn der Fall ist. Damit verbinden
1 (O): Lakmuspapier
2 (O): Lakmuspapier
3 (V): ihr
4 (V): so denken wir: sie beweist, daß dieser Satz
und nicht sein Gegenteil wahr ist.
667
668
448
WBTC121-130 30/05/2005 16:14 Page 896
128
Induction, (x).φx and (∃x).φx. To
What Extent does Induction Prove a
Universal Proposition True and an
Existential Proposition False?
3 × 2 = 5 + 1
3 × (a + 1) = 3 + (3 × a) = (5 + b) + 3 = 5 + (b + 3)
Why do you call this induction the proof that (n): n > 2 .⊃. 3 × n ≠ 5?! Well, don’t you see
that if the proposition is valid for n = 2, it’s also valid for n = 3, and then also for n = 4, and
that it goes on like that for ever? (What am I explaining, anyway, when I explain the way a
proof by induction works?) So you call it a proof of “f(2) & f(3) & f(4), etc.”, but isn’t it
rather the form of the proofs of “f(2)” and “f(3)” and “f(4)”, etc.? Or does that amount to
the same thing? Well, if I call the induction the proof of a single proposition, I am allowed to
do so only if that is supposed to mean no more than that it proves every proposition of a
certain form. (And my expression uses the analogy of the relationship between the propositions
“All acids turn litmus paper red” and “Sulphuric acid turns litmus paper red”).
Now suppose someone were to say “Let’s check whether f(n) is valid for all n”, and then
he begins to write the series
3 × 2 = 5 + 1
3 × (2 + 1) = (3 × 2) + 3 = (5 + 1) + 3 = 5 + (1 + 3)
3 × (2 + 2) = (3 × (2 + 1)) + 3 = (5 + (1 + 3)) + 3 = 5 + (1 + 3 + 3)
and then he breaks off and says “I already see that it’s valid for all n”. – So he has seen
an induction! But was he trying to find an induction? After all, he didn’t have any method
for trying to find one.1
And if he hadn’t discovered one, would he thereby have found a
number which did not satisfy the condition? – The rule for checking clearly can’t be: let’s
see whether an induction turns up, or a case for which the law doesn’t hold. – If the law of
the excluded middle doesn’t hold, that only means that our expression isn’t comparable to
a proposition.
When we say that induction proves the universal proposition, of course we want to switch
to the mode of expression that it proves that this and not its contradictory is the case.2
But
what would be the contradictory of what was proved? Well, that (∃n).~fn is the case. Here
we are combining two concepts: one derived from my current concept of the proof of (n).f(n),
1 (V): it.
2 (V): When we say that induction proves the
universal proposition, we think: it proves that
this proposition and not its contradictory is
true.
448e
WBTC121-130 30/05/2005 16:14 Page 897
wir zwei Begriffe: den einen, den ich aus meinem gegenwärtigen Begriff des Beweises von
(n).f(n) herleite, und einen andern, der von der Analogie mit (∃x).φx hergenommen ist. (Wir
müssen ja bedenken, daß ”(n).fn“ kein Satz ist, solange ich kein Kriterium seiner Wahrheit
habe; und dann nur den Sinn hat, den ihm dieses Kriterium gibt. Ich konnte freilich, schon
ehe ich das Kriterium besaß,5
etwa nach einer Analogie zu (x).fx ausschauen.) Was ist nun
das Gegenteil von dem, was die Induktion beweist? Der Beweis von (a + b)2
= a2
+ 2ab + b2
rechnet diese Gleichung aus im Gegensatz etwa zu (a + b)2
= a2
+ 3ab + b2
. Was rechnet der
Induktionsbeweis aus?
Die Gleichungen: 3 + 2 = 5 + 1, 3 × (a + 1) = (3 × a) + 3, (5 + b) + 3 = 5 + (b + 3) im
Gegensatz also etwa zu 3 + 2 = 5 + 6, 3 × (a + 1) = (4 × a) + 2, etc. Aber dieses Gegenteil
entspricht ja nicht dem Satz (∃x).φx. – Ferner ist nun6
mit jener Induktion im Gegensatz
jeder Satz von der Form ~ f(n), d.h.7
der Satz ”~f(2)“, ”~f(3)“, u.s.w.; d.h. die Induktion
ist das Gemeinsame in der Ausrechnung8
von f(2), f(3), u.s.w.; aber sie ist nicht die
Ausrechnung ”aller Sätze der Form f(n)“, da ja nicht eine Klasse von Sätzen in dem
Beweis vorkommt, die ich ”alle Sätze der Form f(n)“ nenne. Jede einzelne nun von diesen
Ausrechnungen ist die Kontrolle eines Satzes von der Form f(n). Ich konnte nach der
Richtigkeit dieses Satzes fragen und eine Methode zu ihrer Kontrolle anwenden, die durch
die Induktion nur auf eine einfache Form gebracht war. Nenne ich aber die Induktion ”den
Beweis eines allgemeinen Satzes“, so kann ich nach der Richtigkeit dieses Satzes nicht
fragen (sowenig, wie nach der Richtigkeit der Form der Kardinalzahlen). Denn, was ich
Induktionsbeweis nenne, gibt mir keine Methode zur Prüfung, ob der allgemeine Satz richtig
oder falsch ist; diese Methode müßte mich vielmehr lehren, auszurechnen (zu prüfen), ob
sich für einen bestimmten Fall eines Systems von Sätzen eine Induktion bilden läßt, oder
nicht. (Was so geprüft wird, ist, ob alle n die oder jene Eigenschaft haben, wenn ich so sagen
darf; aber nicht, ob alle sie haben, oder ob es einige gibt, die sie nicht haben. Wir rechnen
z.B. aus, daß9
die Gleichung x2
+ 3x + 1 = 0 keine rationalen Lösungen hat (daß es keine
rationale Zahl gibt, die . . . ) und nicht die Gleichung x2
+ 2x + , – 10
dagegen die Gleichung
x2
+ 2x + 1 = 0, etc.)
Daher wir es seltsam empfinden, wenn uns gesagt wird, die Induktion beweise den allgemeinen
Satz; da wir das richtige Gefühl haben, daß wir ja in der Sprache der Induktion
die allgemeine Frage gar nicht hätten stellen können. Da uns ja nicht zuerst eine Alternative
gestellt war (sondern nur zu sein schien, solange uns ein Kalkül mit endlichen Klassen
vorschwebte).
Die Frage nach der Allgemeinheit hatte vor dem Beweis noch gar keinen Sinn, also war
sie auch keine Frage, denn die hätte nur Sinn gehabt, wenn eine allgemeine Methode der
Entscheidung bekannt war, ehe der besondere Beweis bekannt war.11
Denn der Induktionsbeweis entscheidet nicht in einer Streitfrage.12
Wenn gesagt wird: ”der Satz ,(n).fn‘ folgt aus der Induktion“ heiße nur: jeder Satz der
Form f(n) folge aus der Induktion; – ”der Satz ,(∃n). ~f(n)‘ widerspricht13
der Induktion“
1
2
5 (V): hatte,
6 (V): nun nicht
7 (V): ~f(n), nämlich
8 (V): in den Ausrechnungen
9 (V): der
10 (O): x2
+ 2x + – ,
11 (V): Die Frage nach der Allgemeinheit hätte
// hatte // von // vor dem Beweis noch gar
1
2
keinen Sinn, also ist sie auch keine Frage, denn
die Frage hätte nur Sinn gehabt, wenn eine
allgemeine Methode zur Entscheidung bekannt
war, ehe der besondere Beweis bekannt war.
12 (V): entscheidet nichts. // entscheidet keine
Streitfrage.
13 (V): widerspreche
669
670
670
449 Induktion, (x)φx und (∃x)φx. . . .
WBTC121-130 30/05/2005 16:14 Page 898
and another taken from the analogy with (∃x).φx. (For we have to keep in mind that “(n).fn”
isn’t a proposition until I have a criterion for its truth; and that then it only has the sense
this criterion gives it. To be sure, even before I had the criterion I could be on the lookout,
say, for an analogy to (x).fx.) Now what is the opposite of what the induction proves?
The proof of (a + b)2
= a2
+ 2ab + b2
solves this equation, in contrast to, say, (a + b)2
=
a2
+ 3ab + b2
. What does the inductive proof solve?
The equations: 3 + 2 = 5 + 1, 3 × (a + 1) = (3 × a) + 3, (5 + b) + 3 = 5 + (b + 3), as
opposed to, say, 3 + 2 = 5 + 6, 3 × (a + 1) = (4 × a) + 2, etc. But of course this opposite
doesn’t correspond to the proposition (∃x).φx. – Furthermore, every proposition of the form
~f(n), i.e.3
the propositions “~f(2)”, “~f(3)”, etc., now conflicts with that induction; that is
to say, the induction is the common element in the calculation4
of f(2), f(3), etc.; but it isn’t
the calculation “of all propositions of the form f(n)”, since of course no class of propositions
occurs in the proof that I call “all propositions of the form f(n)”. Now each one of
these calculations is a check on a proposition of the form f(n). I was able to investigate the
correctness of this proposition and employ a method to check it, which the induction just
brought into a simple form. But if I call the induction “the proof of a universal proposition”,
I can’t ask whether that proposition is correct (any more than whether the form of
the cardinal numbers is correct). Because what I call inductive proof gives me no method of
checking whether the universal proposition is correct or incorrect; rather, what this method
teaches me is how to calculate (check) whether or not an induction can be constructed for
a particular case within a system of propositions. (What is checked in this way is whether
all n have this or that property, if I may put it like that; not whether all of them have it,
or whether there are some that don’t have it. For example, we calculate that the equation
x2
+ 3x + 1 = 0 has no rational roots (that there is no rational number that . . . ), and
neither does the equation x2
+ 2x + = 0 – but the equation x2
+ 2x + 1 = 0 does, etc.)
Thus we find it odd when we are told that the induction proves the universal proposition;
for we feel rightly that in the language of the induction we couldn’t even have posed
the universal question. After all, we weren’t first given an alternative (there only seemed to
be one so long as we had a calculus with finite classes in mind).
Prior to the proof, the question about universality made no sense at all, and so it wasn’t
even a question, because a question would only have made sense if a general method for
making a decision had been known before the particular proof was known.5
Proof by induction isn’t decisive for a disputed question.6
If it is said: “The proposition ‘(n).fn’ follows from the induction” only means: every proposition
of the form f(n) follows from the induction, and “The proposition (∃n).~ f(n)
contradicts the induction” only means: every proposition of the form ~f(n) is disproved by
1
2
3 (V): namely,
4 (V): calculations
5 (V): Prior to the proof, the question about the
universal proposition would make // made // no
sense at all, and so it isn’t even a question,
because the question would only have made sense
if a general method of decision were known before
the particular proof was discovered.
6 (V): induction decides nothing. // induction
doesn’t decide a disputed question.
Induction, (x).φx and (∃x).φx. . . . 449e
WBTC121-130 30/05/2005 16:14 Page 899
heiße nur: jeder Satz der Form ~f(n) werde durch die Induktion widerlegt, – so kann man
damit einverstanden sein,14
aber wird15
jetzt fragen: Wie gebrauchen wir den Ausdruck ”der
Satz (n).f(n)“ richtig? Was ist seine Grammatik. (Denn daraus, daß ich ihn in gewissen
Verbindungen gebrauche, folgt nicht, daß ich ihn überall dem Ausdruck ”der Satz (x).φx“
analog gebrauche.)
Denken wir, es stritten sich Leute darüber, ob in der Division 1 : 3 lauter Dreier im
Quotienten herauskommen müßten; sie hätten aber keine Methode, um dies zu entscheiden.16
Nun bemerkt Einer von ihnen die induktive Eigenschaft von 1,0 : 3 = 0,3 und sagt: jetzt
1
weiß ich’s, es müssen lauter 3 im Quotienten stehen. Die Andern hatten an diese Art
der Entscheidung nicht gedacht. Ich nehme an, es habe ihnen unklar etwas von einer
Entscheidung durch stufenweise Kontrolle vorgeschwebt, und daß sie diese Entscheidung
freilich nicht herbeiführen könnten. Halten sie nun an ihrer extensiven Auffassung fest,
so ist allerdings durch die Induktion eine Entscheidung herbeigeführt, denn die Induktion
zeigt für jede Extension des Quotienten, daß sie aus lauter 3 besteht. Lassen sie aber die
extensive Auffassung fallen, so entscheidet die Induktion nichts. Oder nur das, was die
Ausrechnung von 1,0 : 3 = 0,3 entscheidet: nämlich, daß ein Rest bleibt, der gleich dem
1
Dividenden ist. Aber mehr nicht. Und nun kann es allerdings eine richtige Frage geben,
nämlich: ist der Rest, der bei dieser Division bleibt, gleich dem Dividenden? und diese
Frage ist jetzt an die Stelle der alten extensiven getreten und ich kann natürlich den alten
Wortlaut beibehalten, aber er ist jetzt außerordentlich irreleitend, denn er17
läßt es immer
so erscheinen, als wäre die Erkenntnis der Induktion nur ein Vehikel, das uns in die
Unendlichkeit tragen kann. (Das hängt auch damit zusammen, daß das Zeichen ”u.s.w.“ sich
auf eine interne Eigenschaft des Reihenstückes, das ihm vorhergeht, bezieht und nicht auf
seine Extension.)
Die Frage ”gibt es eine rationale Zahl, die die Wurzel von x2
+ 3x + 1 = 0 ist“ ist freilich
durch eine Induktion entschieden:18
– aber hier habe ich eben eine Methode konstruiert,
um Induktionen zu bilden; und die Frage hat ihren Wortlaut nur, weil es sich um
eine Konstruktion von Induktionen handelt. D.h. die Frage wird durch eine Induktion
entschieden, wenn ich nach dieser Induktion fragen konnte. Wenn mir also ihr Zeichen
von vornherein auf ja und nein bestimmt war, so daß ich rechnerisch zwischen ihnen
entscheiden konnte, wie z.B., ob der Rest in 5 : 7 gleich oder ungleich dem Dividenden sein
wird. (Die Verwendung der Ausdrücke ”alle . . .“ und ”es gibt . . .“ für diese Fälle hat eine
gewisse Ähnlichkeit mit der Verwendung des Wortes ”unendlich“ im Satz ”heute habe ich
ein Lineal mit unendlichem Krümmungsradius gekauft“.)
1 : 3 = 0,3 entscheidet durch ihre Periodizität nichts, was früher offen gelassen war. Wenn
1
vor der Entdeckung der Periodizität Einer vergebens nach einer 4 in der Entwicklung
von 1 : 3 gesucht hätte, so hätte er doch die Frage ”gibt es eine 4 in der Entwicklung von
1 : 3“ nicht sinnvoll stellen können; d.h., abgesehen davon, daß er tatsächlich zu keiner 4
gekommen war, können wir ihn davon überzeugen, daß er keine Methode besitzt, seine Frage
zu entscheiden. Oder wir könnten auch sagen: abgesehen von dem Resultat seiner Tätigkeit
könnten wir ihn über die Grammatik seiner Frage und die Natur seines Suchens aufklären
14 (V): so kann man sich damit zufrieden geben,
15 (V): wirdn
16 (V): Methode, wie dies zu entscheiden sei.
17 (V): sie
18 (V): entschieden,
671
672
450 Induktion, (x)φx und (∃x)φx. . . .
WBTC121-130 30/05/2005 16:14 Page 900
the induction, then we can agree;7
but then we shall ask: How do we use the expression “the
proposition (n).f(n)” correctly? What is its grammar? (For from the fact that I use it in
certain contexts it doesn’t follow that I use it everywhere analogously to the expression
“the proposition (x).φx”.)
Let’s imagine that people argued about whether the quotient of the division 1 ÷ 3 had to
contain only threes, but had no method of deciding this.8
Now one of them notices the inductiveness
of 1·0 ÷ 3 = 0·3 and says: Now I have it: there must be only threes in the quotient.
1
The others hadn’t thought of that kind of decision. I suppose that they had vaguely
imagined some kind of decision by checking step by step, realizing that, to be sure, they
wouldn’t be able to reach this decision. Now if they hold on to their extensional viewpoint,
the induction has indeed produced a decision, because in the case of each extension of the
quotient it shows that the extension consists of nothing but threes. But if they drop their
extensional viewpoint the induction decides nothing; or only what is decided by figuring out
1·0 ÷ 3 = 0·3, namely that there is a remainder that’s the same as the dividend. But nothing
1
more than that. And now a real question can be asked, namely, is the remainder left after
this division equal to the dividend? And this question has now taken the place of the old
extensional question, and of course I can keep the old wording, but it is now extremely misleading,
for it9
always makes it look as if the knowledge of induction were merely a vehicle
– a vehicle that could carry us into infinity. (This is also connected with the fact that the
sign “etc.” refers to an internal property of the part of the series that precedes it, and not
to its extension.)
To be sure, the question “Is there a rational number that is a root of x2
+ 3x + 1 = 0?” is
decided by an induction: – but here I’ve just constructed a method for forming inductions;
and the question is only phrased as it is because it is a matter of constructing inductions.
That is, a question is decided by an induction, if I was able to ask about that induction;
i.e. if its sign was settled for me from the start, right down to yes and no, so that by calculating
I could decide between the two; as I can decide, for instance, whether the remainder
in 5 ÷ 7 will be equal to the dividend or not. (The use in these cases of the expressions
“all . . .” and “there is . . .” has a certain similarity to the use of the word “infinite” in the
sentence “Today I bought a ruler with an infinite radius of curvature”.)
The periodicity of 1 ÷ 3 = 0·3 decides nothing that had earlier been left open. Even if
1
before the discovery of its periodicity, someone had been looking in vain for a 4 in the
development of 1 ÷ 3, he still couldn’t have meaningfully asked the question “Is there a
4 in the development of 1 ÷ 3?”. That is, independently of the fact that he hadn’t actually
come upon a 4, we can convince him that he doesn’t have any method of deciding his
question. Or we could also say: apart from the result of his activity we could enlighten him
as to the grammar of his question and the nature of his search (as we could enlighten a
7 (V): can rest content with this;
8 (V): method of finding out how this was to be
decided.
9 (V): for the question
Induction, (x).φx and (∃x).φx. . . . 450e
WBTC121-130 30/05/2005 16:14 Page 901
(wie einen heutigen Mathematiker über analoge Probleme). ”Aber als Folge der Entdeckung
der Periodizität hört er nun doch gewiß auf, nach einer 4 zu suchen! Sie überzeugt ihn also,
daß er nie eine finden wird“. – Nein. Die Entdeckung der Periodizität bringt ihn vom Suchen
ab, wenn er sich nun neu einstellt. Man könnte ihn fragen: ”Wie ist es nun, willst Du noch
immer nach einer 4 suchen?“ (Oder hat Dich, sozusagen, die Periodizität auf andere
Gedanken gebracht.)
Und die Entdeckung der Periodizität ist in Wirklichkeit die Konstruktion eines neuen
Zeichens und Kalküls. Denn es ist irreführend ausgedrückt, wenn wir sagen, sie bestehe darin,
daß es uns aufgefallen sei, daß der erste Rest gleich dem Dividenden ist. Denn hätte man
Einen, der die periodische Division nicht kannte, gefragt:19
ist in dieser Division der erste
Rest gleich dem Dividenden, so hätte er natürlich ”ja“ gesagt; es wäre ihm also aufgefallen.
Aber damit hätte ihm nicht die Periodizität auffallen müssen;20
d.h.: er hätte damit nicht
den Kalkül mit den Zeichen a : b = c gefunden.
a
Ist nicht, was ich hier sage, das,21
was Kant damit22
meinte, daß 5 + 7 = 12 nicht
analytisch, sondern synthetisch a priori sei?
19 (V): gefragt,
20 (V): brauchen;
21 (V): sage, immer dasselbe,
22 (V): damit g
451 Induktion, (x)φx und (∃x)φx. . . .
WBTC121-130 30/05/2005 16:14 Page 902
contemporary mathematician about analogous problems). “But as a result of discovering the
periodicity surely he’ll stop looking for a 4! So it convinces him that he’ll never find one.”
– No. The discovery of the periodicity will dissuade him from looking if he now adopts
a new frame of mind. We could ask him: “Well, how about it, do you still want to look for
a 4?” (Or has the periodicity, so to say, changed your mind?)
And the discovery of the periodicity is really the construction of a new sign and a new
calculus. For it is misleading to say that this discovery consists in our having noticed that
the first remainder is equal to the dividend. For if we had asked someone unacquainted
with periodic division: “Is the first remainder in this division equal to the dividend?”, of
course he would have answered “Yes”; so he would have noticed this. But in noticing it he
wouldn’t have had to notice the periodicity; that is, in noticing it he would not have discovered
the calculus with the signs a ÷ b = c.
a
Isn’t what I’m saying here what10
Kant meant by saying that 5 + 7 = 12 is not analytic
but synthetic a priori?
10 (V): here the same as what
Induction, (x).φx and (∃x).φx. . . . 451e
WBTC121-130 30/05/2005 16:14 Page 903
129
Wird aus der Anschreibung des
Rekursionsbeweises noch ein weiterer
Schluß auf die Allgemeinheit gezogen,
sagt das Rekursionsschema nicht
schon alles was zu sagen war?
Man sagt für gewöhnlich, die rekursiven Beweise zeigen,1
daß die algebraischen
Gleichungen für alle Kardinalzahlen gelten; aber es kommt hier momentan nicht darauf an,
ob dieser Ausdruck glücklich oder schlecht gewählt ist, sondern nur darauf, ob er in allen
Fällen die gleiche, klarbestimmte, Bedeutung hat.2
Und ist es da nicht klar, daß die rekursiven Beweise tatsächlich dasselbe für alle
”bewiesenen“ Gleichungen zeigen?
Und das heißt doch, daß zwischen dem rekursiven Beweis und dem von ihm bewiesenen
Satz immer die gleiche (interne) Beziehung besteht?
Es ist ja übrigens ganz klar, daß es so einen rekursiven, oder richtiger, iterativen ”Beweis“
geben muß. (Der uns die Einsicht vermittelt, daß es ”mit allen Zahlen so gehen muß“.)
|D.h. es scheint mir klar, und daß ich einem Anderen die Richtigkeit dieser Sätze für die
Kardinalzahlen durch einen Prozeß der Iteration begreiflich machen könnte.|
Wie aber weiß ich 28 + (45 + 17) = (28 + 45) + 17 ohne es bewiesen zu haben? Wie kann
mir ein allgemeiner Beweis einen besonderen Beweis schenken? Denn ich könnte doch den
besondern Beweis führen, und wie treffen sich da die beiden Beweise, und wie, wenn sie
nicht übereinstimmen?
D.h.: Ich möchte Einem zeigen, daß das distributive Gesetz wirklich im Wesen der Anzahl
liegt und nicht etwa nur in diesem bestimmten Fall zufällig gilt; werde ich da nicht durch
einen Prozeß der Iteration zu zeigen versuchen, daß das Gesetz gilt und immer weiter
gelten muß? Ja, – daraus ersehen wir, was wir hier darunter verstehen, daß ein Gesetz für
alle Zahlen gelten muß.
Und inwiefern kann man diesen Vorgang nicht einen3
Beweis des (distributiven) Gesetzes
nennen?
Und dieser Begriff des ”begreiflich-Machens“ ist hier ein Segen.4
1 (V): beweisen,
2 (V): gleiche Bedeutung
3 (V): den
4 (V): ”begreiflich-Machens“ kann uns hier wirklich
helfen // kann uns hier helfen.
673
674
452
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129
Is a Further Inference to Generality
Drawn from Writing Down the
Recursive Proof? Doesn’t the
Recursion Schema Already Say all
that Needed to be Said?
We commonly say that recursive proofs show1
that the algebraic equations hold for all
cardinal numbers; but here, for the moment, it doesn’t matter whether this expression is
well or ill chosen; the only thing that matters is whether it has the same clearly defined
meaning2
in all cases.
And isn’t it clear here that recursive proofs really do show the same thing for all “proved”
equations?
And doesn’t that mean that between the recursive proof and the proposition it proves,
there is always the same (internal) relation?
It is quite clear, by the way, that there must be a recursive, or more correctly, iterative
“proof” of this kind. (A proof conveying the insight that “that’s the way it must be with all
numbers”.)
|That is, it seems clear to me; and it also seems clear to me that by a process of iteration
I could make the correctness of these theorems for the cardinal numbers comprehensible
to someone else.|
But how do I know that 28 + (45 + 17) = (28 + 45) + 17 without having proved it? How
can a general proof give me a particular proof? For after all I could carry out the particular
proof, and then how do the two proofs meet? And what happens if they don’t agree?
That is: suppose I would like to show someone that the distributive law really is part of
the nature of number, and doesn’t only happen to hold in this particular case; won’t I use
a process of iteration to try to show that the law holds and must go on holding? Yes – that
process shows us what we understand here by a law having to be valid for all numbers.
And to what extent can we not call this process a3
proof of the (distributive) law?
And here this concept of “making something comprehensible” is a blessing.4
1 (V): prove
2 (V): the same meaning
3 (V): the
4 (V): comprehensible” can really help us here. //
can help us here.
452e
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Denn man könnte sagen: das Kriterium dafür, ob etwas ein Beweis eines Satzes ist, ist,
ob man ihn dadurch begreiflich machen kann. (Natürlich handelt es sich da wieder nur um
eine Erweiterung unserer grammatischen Betrachtungen des Wortes5
”Beweis“; nicht um
ein psychologisches Interesse an dem Vorgang des Begreiflich-machens.)
|”Dieser Satz ist für alle Zahlen durch das rekursive Verfahren bewiesen.“ Das ist der
Ausdruck, der so ganz irreführend ist. Es klingt so, als würde hier ein Satz, der konstatiert,
daß das und das für alle Kardinalzahlen gilt, auf einem Wege als wahr erwiesen, und als sei
dieser Weg ein Weg in einem Raum denkbarer Wege.
Während die Rekursion in Wahrheit nur sich selber zeigt, wie auch die Periodizität nur
sich selbst zeigt.6
|
Wir sagen nicht, daß der Satz f(x), wenn f(l) gilt und aus f(c) f(c + 1) folgt, darum für alle
Kardinalzahlen wahr ist; sondern: ”der Satz f(x) gilt für alle Kardinalzahlen“ heißt ”er gilt
für x = 1 und f(c + 1) folgt aus f(c)“.
Und hier ist ja der Zusammenhang mit der Allgemeinheit in endlichen Bereichen ganz
klar, denn eben das wäre in einem endlichen Bereich allerdings der Beweis dafür, daß f(x)
für alle Werte von x gilt und eben das ist der Grund, warum wir auch im arithmetischen
Falle sagen, f(x) gelte für alle Zahlen.
Zum mindesten muß ich sagen, daß, welcher Einwand gegen den Beweis B7
gilt, auch
z.B. gegen den der Formel (a + b)n
= etc. gilt.
Auch hier, müßte ich dann sagen, nehme ich nur eine algebraische Regel in Übereinstimmung
mit den Induktionen der Arithmetik an.
f(n) × (a + b) = f(n + 1)
f(1) = a + b
also: f(1) × (a + b) = (a + b)2
= f(2)
also: f(2) × (a + b) = (a + b)3
= f(3) u.s.w.
Soweit ist es klar. Aber nun: ”also (a + b)n
= f(n)“!
Ist denn hier ein weiterer Schluß gezogen? Ist denn hier noch etwas zu konstatieren?
Ich würde aber doch fragen, wenn mir Einer die Formel (a + b)n
= f(n) zeigt: wie ist
man denn dazugekommen? Und als Antwort käme doch die Gruppe
f(n) × (a + b) = f(n + 1)
f(1) = a + b.
Ist sie also nicht ein Beweis des algebraischen Satzes? – Oder antwortet sie nicht eher auf
die Frage ”was bedeutet der algebraische Satz“?
Ich will sagen: hier ist doch mit der Induktion alles erledigt.
Der Satz, daß A für alle Kardinalzahlen gilt, ist eigentlich der Komplex B. Und sein Beweis
der Beweis von β und γ. Aber das zeigt auch, daß dieser Satz in einem andern Sinne Satz
ist, als eine Gleichung, und dieser8
Beweis in anderm Sinne Beweis eines Satzes.
Vergiß hier nicht, daß wir nicht erst den Begriff des Satzes haben, dann wissen, daß die
Gleichungen mathematische Sätze sind, und dann erkennen, daß es noch andere Arten von
mathematischen Sätzen gibt!
5 (V): Betrachtungen über das Wort
6 (V): selber zeigt, wie auch die Periodizität.
7 (E): Vgl. den Anfang von Abschnitt 127.
8 (V): sein
675
676
453 Wird aus der Anschreibung des Rekursionsbeweises . . .
WBTC121-130 30/05/2005 16:14 Page 906
For we could say: the criterion for whether something is a proof of a proposition is whether
we can use it for making the proposition comprehensible. (Of course here again it is only a
matter of extending our grammatical examinations of5
the word “proof”, and not a matter
of our taking any psychological interest in the process of making things comprehensible.)
|“This proposition is proved for all numbers by the recursive procedure.” That is the
expression that is so very misleading. It sounds as if here a proposition stating that such
and such holds for all cardinal numbers were shown to be true by a particular route, and
as if this route were a route within a space of conceivable routes.
Whereas in reality recursion shows nothing but itself, just as periodicity too shows
nothing but itself.6
|
We are not saying that when f(1) holds and when f(c + 1) follows from f(c), the proposition
f(x) is therefore true of all cardinal numbers; but rather: “The proposition f(x) holds for
all cardinal numbers” means “It holds for x = 1, and f(c + 1) follows from f(c)”.
And here the connection with generality in finite domains is quite clear, for in a finite
domain that very thing would indeed be a proof that f(x) holds for all values of x, and precisely
that is the reason we say in the arithmetical case too that f(x) holds for all numbers.
At least I have to say that whatever objection holds against the proof B7
holds also, e.g.,
against the proof of the formula (a + b)n
= etc.
Here too, I would then have to say, I am merely assuming an algebraic rule that agrees
with the inductions of arithmetic.
f(n) × (a + b) = f(n + 1)
f(1) = a + b
therefore f(1) × (a + b) = (a + b)2
= f(2)
therefore f(2) × (a + b) = (a + b)3
= f(3), etc.
So far it’s clear. But now: “therefore (a + b)n
= f(n)”!
Has a further inference been drawn here? Is there something else to be ascertained here?
For if someone shows me the formula (a + b)n
= f(n) I would still ask: How was it arrived
at? And surely the answer would be: the group
f(n) × (a + b) = f(n + 1)
f(1) = a + b.
So isn’t it a proof of the algebraic proposition? – Or isn’t it rather an answer to the question
“What does the algebraic proposition mean?”
I want to say: Once you’ve got the induction, it’s all been taken care of.
The proposition that A holds for all cardinal numbers is really the complex B. And the
proof of that proposition is the proof of β and γ. But that also shows that this proposition
is a proposition in a different sense than an equation is, and that this8
proof is a proof of a
proposition in a different sense.
Don’t forget in this context that it isn’t that we first of all have the concept of a proposition,
and then come to know that equations are mathematical propositions, and still later
come to know that there are other kinds of mathematical propositions as well!
5 (V): concerning
6 (V): just as does periodicity.
7 (E): Cf. the beginning of Chapter 127.
8 (V): its
Is a Further Inference to Generality Drawn . . . 453e
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130
Inwiefern verdient der
Rekursionsbeweis den Namen
eines ”Beweises“?1
Inwiefern ist der Übergang nach
dem Paradigma A durch den Beweis
von B gerechtfertigt?2
Man kann nicht eine Rechnung zum Beweis eines Satzes ernennen.3
Ich möchte sagen: Muß man die Induktionsrechnung4
den Beweis des Satzes I nennen?5
D.h., tut’s keine andere Beziehung?
(Die unendliche Schwierigkeit ist die ”allseitige Betrachtung“ des Kalküls.)
”Der Übergang ist gerechtfertigt“ heißt in einem Falle, daß er nach bestimmten
gegebenen Formen vollzogen werden kann. Im andern Fall wäre die Rechtfertigung, daß
der Übergang nach Paradigmen geschieht, die selbst eine bestimmte Bedingung befriedigen.
Man denke sich, daß für ein Brettspiel solche Regeln gegeben würden, die aus lauter Wörtern
ohne ”r“ bestünden, und daß ich eine Regel gerechtfertigt nenne, wenn sie kein ”r“ enthält.
Wenn nun jemand sagte, er habe für das und das Spiel nur eine Regel aufgestellt, nämlich,
daß die Züge Regeln entsprechen müßten, die kein ”r“ enthalten. – Ist denn das eine Spielregel
(im ersten Sinn)? Geht das Spiel nicht doch nach der Klasse von Regeln6
vor sich, die nur
alle jener ersten Regel entsprechen sollen?
Es macht mir jemand die Konstruktion von B vor und sagt nun, A ist bewiesen. Ich frage:
”Wieso? – ich sehe nur, daß Du um A eine Konstruktion mit Hilfe von ρ7
gemacht hast“.
Nun sagt er: ”Ja, aber wenn das möglich ist, so sage ich eben, A sei bewiesen“. Darauf antworte
ich: ”Damit hast Du mir nur gezeigt, welchen neuen Sinn Du mit dem Wort ’beweisen‘
verbindest“.
1 (O): ”Beweises“.
2 (E): Zum Verständnis von ”A“ und ”B“ siehe
die ersten beiden Bemerkungen in Kapitel 127.
”A“ steht dort für a + (b + c) = (a + b) + c und
”B“ für a + (b + 1) = (a + b) + 1.
3 (V): Rechnung als den Beweis eines Satzes
bestimmen.
4 (V): man diese Rechnung
5 (E): Aus einer früheren Version dieser
Bemerkung (MS 111, S. 161) geht hervor, daß
sich ”I“ auf ”a + (b + c) = (a + b) + c“ bezieht.
6 (V): nach den Regeln
7 (V): α (E): Bezieht sich wahrscheinlich auf
die mit ”ρ“ bezeichneten Zeilen in MS 111,
S. 148 (siehe Anhang I).
677
678
454
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130
To what Extent does a
Recursive Proof Deserve
the Name “Proof”?
To what Extent is a Step in
Accordance with the Paradigm A
Justified by the Proof of B?1
We cannot appoint a calculation to be a2
proof of a proposition.
I would like to say: Do we have to call the inductive calculation3
the proof of the proposition
I?4
That is, won’t another relationship do?
(What is infinitely difficult is a “comprehensive examination” of the calculus.)
In the one case “This step is justified” means that it can be carried out in accordance with
specific forms that have been given. In the other case the justification might be that the step
takes place in accordance with paradigms that themselves satisfy a specific condition.
Suppose that for a certain board game rules were given consisting only of words with no
“r” in them, and that I called a rule “justified” if it contained no “r”. Now if someone said
he had laid down only one rule for a certain game, namely, that its moves had to obey rules
containing no “r”’s – would that be a rule of the game (in the first sense)? Isn’t the game
played in accordance with the class that the rules belong to5
, the only proviso being that all
of them are supposed to satisfy that first rule?
Someone demonstrates the construction of B for me and then says that A has been proved.
I ask “How come? All I see is that you have used ρ6
to carry out a construction around A”.
Then he says “Right, but when that’s possible, that’s when I say that A has been proved”.
To that I answer: “That only shows me what new sense you are attaching to the word ‘prove’.”
1 (E): For the likely referents of “A” and “B”,
cf. the first two remarks of Chapter 127. “A”
presumably refers to a + (b + c) = (a + b) + c,
whereas “B” seems to refer to a + (b + 1) =
(a + b) + 1.
2 (V): cannot stipulate a calculation as being the
3 (V): call this calculation
4 (E): On the basis of an earlier version of this
remark in MS 111 (p. 161), it is clear that “I”
refers to “a + (b + c) = (a + b) + c”.
5 (V): with the rules
6 (V): α (E): Presumably this is a reference
to the line(s) labelled “ρ” in MS 111, p. 148.
They are included here in Appendix I.
454e
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In einem Sinn heißt es, daß Du das Paradigma mittels ρ8
so und so konstruiert hast, in
dem andern, nach wie vor, daß eine Gleichung dem Paradigma entspricht.
Wenn wir fragen ”ist das ein Beweis oder nicht?“, so bewegen wir uns in der
Wortsprache.9
Nun ist natürlich nichts dagegen einzuwenden, wenn Einer sagt: Wenn die Glieder des
Übergangs in einer Konstruktion der und der Art stehen, so sage ich, die Rechtmäßigkeit
des Übergangs ist bewiesen.
Was wehrt sich in mir gegen die Auffassung von B als einem Beweis von A? Zuerst
entdecke ich, daß ich den Satz von ”allen Kardinalzahlen“ in meiner Rechnung nirgends
brauche. Ich habe den Komplex B mit Hilfe von ρ konstruiert und bin dann auf die Gleichung
A übergegangen; von ”allen Kardinalzahlen“ war dabei keine Rede. (Dieser Satz ist eine
Begleitung der Rechnung in der Wortsprache, die mich hier nur10
verwirren kann.) Aber
nicht nur fällt dieser allgemeine Satz überhaupt fort, sondern kein anderer tritt an seine Stelle.
Der Satz, der die Allgemeinheit behauptet, fällt also weg, ”es ist nichts bewiesen“, ”es folgt
nichts“.
”Ja, aber die Gleichung A folgt, sie steht nun an Stelle des allgemeinen Satzes“. – Ja
in wiefern folgt sie denn? Offenbar verwende ich hier ”folgt“ in einem ganz andern Sinn,
als dem normalen, da das, woraus A folgt, kein Satz ist. Das ist es auch, warum wir fühlen,
daß das Wort ”folgen“ nicht richtig angewandt ist.
Wenn man sagt ”aus dem Komplex B folgt, daß a + (b + c) = (a + b) + c“, so schwindelt
Einem. Man fühlt, daß man da auf irgend eine Weise einen Unsinn geredet hat, obwohl es
äußerlich richtig klingt.
Daß eine Gleichung folgt, heißt eben schon etwas (hat seine bestimmte Grammatik).
Aber wenn ich höre ”aus B folgt A“, so möchte ich fragen: ”was folgt?“ Daß a + (b + c)
gleich (a + b) + c ist, ist ja eine Festsetzung, wenn es nicht auf normale Weise aus einer
Gleichung folgt.
Wir können unsern Begriff des Folgens dem A und B nicht aufsetzen,11
er paßt hier nicht.
”Ich werde Dir beweisen, daß a + (b + n) = (a + b) + n.“ Niemand erwartet sich nun den
Komplex B zu sehen. Man erwartet eine andere Regel über a, b und n zu hören, die den
Übergang von der einen auf die andere Seite vermittelt. Wenn mir statt dessen B und das
Schema R12
gegeben wird, so kann ich das keinen Beweis nennen, eben weil ich unter Beweis
etwas anderes verstehe.
Ja ich werde dann etwa sagen: ”Ach so, das nennst Du ,Beweis‘, ich habe mir
vorgestellt . . .“.
Der Beweis von 17 + (18 + 5) = (17 + 18) + 5 wird allerdings nach dem Schema B geführt
und dieser Zahlensatz ist von der Form A. Oder auch: B ist der Beweis des Zahlensatzes;
aber eben deshalb nicht von A.
8 (O): α (E): Das ρ, das wir hier eingesetzt
haben, findet sich in der handschriftlichen
Version dieser Bemerkung (MS 112, S. 35v).
9 (V): in den Formen der Wortsprache.
10 (V): die mich nur
11 (V): des Folgens mit A und B nicht zur
Deckung bringen. // des Folgens dem A und B
nicht aufpassen.
12 (E): Siehe die in Anhang I zitierte Stelle aus MS
111.
679
680
455 Inwiefern verdient der Rekursionsbeweis den Namen eines ”Beweises“ . . .
WBTC121-130 30/05/2005 16:14 Page 910
In one sense this means that you have used ρ7
to construct the paradigm in such and such
a way; in another, it means, as before, that an equation is in accordance with the paradigm.
If we ask “Is that a proof or not?” we are operating within word-language.8
Now of course there can’t be any objection if someone says: When the terms of a step are
in a construction of such and such a kind, I say that the legitimacy of the step has been proved.
What is it in me that resists understanding B as a proof of A? First, I discover that nowhere
in my calculation do I use the proposition about “all cardinal numbers”. I used ρ to construct
the complex B and then I made the step to the equation A; there was no mention
of “all cardinal numbers” in that process. (This proposition is an accompaniment of the
calculation in word-language, and all it can do here is9
confuse me.) But it isn’t just that
this universal proposition completely drops out; it’s that no other takes its place.
So the proposition asserting the generalization drops out; “nothing is proved”, “nothing
follows”.
“Right, but the equation A follows; it now takes the place of the general proposition.” –
Well, to what extent does it follow? Obviously, I’m using “follows” here in a sense quite
different from the normal one, because what A follows from isn’t a proposition. And that’s
why we feel that the word “follows” hasn’t been applied correctly.
If you say “It follows from the complex B that a + (b + c) = (a + b) + c”, your head reels.
You feel that somehow or other you’ve said something nonsensical although outwardly it
sounds right.
That an equation follows, already has a meaning (has its own particular grammar).
But when I hear “A follows from B”, I would like to ask: “What follows?” If it doesn’t
follow from an equation in the normal way, that a + (b + c) is equal to (a + b) + c is no more
than a stipulation.
We can’t impose our concept of following on A and B; it doesn’t fit there.10
“I will prove to you that a + (b + n) = (a + b) + n.” No one then expects to see the
complex B. You expect to hear a different rule for a, b, and n that mediates the transition
from one side to the other. If instead of that I am given B and the schema R11
I can’t call
that a proof, because I understand something different by “proof”.
In that case, I shall say something like “Oh, so that’s what you call a ‘proof’; I had
imagined . . .”.
To be sure, the proof of 17 + (18 + 5) = (17 + 18) + 5 is carried out in accordance with
the schema B, and this numerical proposition is of the form A. Or alternatively: B is the
proof of the numerical proposition: but for that very reason, it isn’t a proof of A.
7 (O): α (E): The ρ we have included here
occurs in the handwritten version of this remark
in MS 112, p. 35v.
8 (V): within the forms of word-language.
9 (V): do is
10 (V): We cannot superimpose our concept of
following on A and B. // We cannot fit our
concept of following onto A and B.
11 (E): Cf. the passage from MS 111 that is
included in Appendix I of this book.
Does a Recursive Proof Deserve the Name “Proof” . . . 455e
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”Ich werde Dir AI,13
AII, AIII
14
aus einem15
Satz ableiten“. – Man denkt dabei natürlich
an eine Ableitung, wie sie mit Hilfe dieser Sätze gemacht wird. – Man denkt, es wird eine
Art von kleineren Kettengliedern gegeben werden, durch die wir alle diese großen ersetzen
können.
Und da haben wir doch ein bestimmtes Bild; und es wird uns etwas ganz Anderes geboten.
Die Gleichung wird durch den induktiven Beweis, quasi, der Quere, statt der Länge nach
zusammengesetzt.
Wenn wir nun die Ableitung rechnen,16
so kommen wir endlich zu dem Punkt, wo die
Konstruktion von B vollendet ist. Aber hier heißt es nun ”also gilt diese Gleichung“. Aber
diese Worte heißen ja nun etwas anderes als, wo wir sonst eine Gleichung aus Gleichungen
folgern. Die Worte ”die Gleichung folgt daraus“ haben ja schon eine Bedeutung. Und hier
wird eine Gleichung allerdings konstruiert, aber nach einem andern Prinzip.
Wenn ich sage ”aus dem Komplex folgt die Gleichung“, so ”folgt“ hier eine Gleichung
aus etwas, was gar keine Gleichung ist.
Man kann nicht sagen: die Gleichung, wenn sie aus B folgt, folge doch aus einem Satz,
nämlich aus α & β & γ; denn es kommt eben darauf an, wie ich17
aus diesem Satz A erhalte;
ob nach einer Regel des Folgens. Welches die Verwandtschaft der Gleichung zum Satz α
& β & γ ist. (Die Regel, die in diesem Falle zu A führt macht gleichsam einen Querschnitt
durch α & β & γ, sie faßt den Satz anders auf, als eine Regel des Folgens.)
Wenn uns die Ableitung von A aus α versprochen war und wir sehen nun den Übergang
von B auf A, so möchten wir sagen: ”ach, so war es nicht gemeint“. So, als hätte jemand
mir versprochen, er werde mir etwas schenken und nun sagt er: so, jetzt schenke ich Dir
mein Vertrauen.18
Darin, daß der Übergang von B auf A kein Folgen ist, liegt auch, was ich damit meinte,
daß nicht das logische Produkt α & β & γ die Allgemeinheit ausdrückt.
Ich sage, (a + b)2
= etc. ist mit Hilfe von AI, AII, etc. bewiesen, weil die Übergänge von
(a + b)2
zu a2
+ 2ab + b2
alle von der Form AI, oder AII, etc., sind. In diesem Sinne ist
in III auch der Übergang von (b + 1) + a auf (b + a) + 1 nach AI gemacht, aber nicht der
Übergang von a + n auf n + a!
Daß man sagt ”die Richtigkeit der Gleichung ist bewiesen“, zeigt schon, daß Beweis
nicht jede Konstruktion der Gleichung ist.19
Es zeigt mir jemand die Komplexe B und ich sage ”das sind keine Beweise der
Gleichungen A“. Nun sagt er: ”Du siehst aber noch nicht das System, nach dem diese
Komplexe gebildet sind“, und macht mich darauf aufmerksam.20
Wie konnte das die B zu
Beweisen21
machen? –
Durch diese Einsicht steige ich in eine andere, sozusagen höhere, Ebene; während der
Beweis auf der tieferen geführt werden müßte.22
13 (V): AI, au d
14 (E): Siehe die in Anhang I zitierte Stelle aus MS
111.
15 (V): aus dem einen
16 (V): ausführen,
17 (V): ich sie
18 (V): Dir meine Zeit.
19 (V): nicht jede Ableitung // Konstruktion // ist.
20 (V): und zeigt es mir.
21 (O): beweisen
22 (V): tieferen hätte geführt werden müssen.
681
682
456 Inwiefern verdient der Rekursionsbeweis den Namen eines ”Beweises“ . . .
WBTC121-130 30/05/2005 16:14 Page 912
“I will derive AI, AII, AIII
12
from a single13
proposition”. – In this context one is, of course,
thinking of a derivation of the kind that is made using these propositions. – We think we
shall be given smaller links of some kind that we can use to replace all these large ones in
the chain.
Here we have a particular picture in mind; and we are offered something quite different.
The inductive proof puts the equation together crosswise, as it were, instead of
lengthwise.
Now when we figure out14
the derivation, we finally come to the point at which the
construction of B has been completed. And at this point we say “Therefore this equation
holds”. But now these words mean something different from when we otherwise deduce
an equation from equations. The words “The equation follows from it” already have a
meaning. And an equation is constructed here, but according to a different principle.
If I say “The equation follows from the complex”, then here an equation “follows” from
something that is not an equation at all.
We can’t say: if the equation follows from B, then it follows from a proposition, namely
from α & β & γ; for what matters is how I get A from that proposition; whether I do so in
accordance with a rule of inference; and what the relationship of the equation is to the proposition
α & β & γ. (The rule leading to A in this case makes a cross-section, as it were, through
α & β & γ; it views the proposition differently from the way a rule of inference does.)
If we had been promised a derivation of A from α and now we see the step from B to A,
we feel like saying “Oh, that isn’t the way it was meant”. As if someone had promised to
give me a gift, and then he says: Here, now I’m giving you my trust.15
The fact that the step from B to A is not an inference also indicates what I meant when
I said that it isn’t the logical product α & β & γ that expresses the generalization.
I say that (a + b)2
= etc. has been proved using AI, AII, etc., because the steps from
(a + b)2
to a2
+ 2ab + b2
are all of the form AI or AII, etc. In this sense the step in III from
(b + 1) + a to (b + a) + 1 has also been made in accordance with A, but the step from a + n
to n + a hasn’t!
The fact that we say “the correctness of the equation has been proved” already shows that
not every construction of the equation is16
a proof.
Someone shows me the complexes B and I say “They are not proofs of the equations A”.
Then he says: “But you haven’t yet seen the system according to which these complexes
have been constructed”, and he points it out to me.17
How could that turn the B’s into
proofs? –
Through this insight I ascend to another, as it were higher, level; whereas a proof would
have to be18
carried out on a lower level.
12 (E): Cf. the passage from MS 111 that is
included in Appendix I of this book.
13 (V): from the one
14 (V): we carry out
15 (V): time.
16 (V): not every derivation // construction // is
17 (V): and he shows it to me.
18 (V): have to have been
Does a Recursive Proof Deserve the Name “Proof” . . . 456e
WBTC121-130 30/05/2005 16:14 Page 913
Nur ein bestimmter Übergang von Gleichungen zu einer Gleichung ist ein Beweis dieser
letzteren. Dieser findet hier nicht statt23
und alles Andere kann B nicht mehr zum Beweis
von A machen.24
Aber kann ich eben nicht sagen, daß, wenn ich dies über A bewiesen habe, ich damit A
bewiesen habe? Und woher kam dann überhaupt die Täuschung, daß ich es dadurch bewiesen
hätte? denn diese muß doch einen tieferen Grund haben.
Nun, wenn es eine Täuschung ist, so kam sie jedenfalls von unserer Ausdrucksweise in
der Wortsprache her ”dieser Satz gilt für alle Zahlen“; denn der algebraische Satz war ja
nach dieser Auffassung nur eine andere Schreibweise dieses Satzes (der Wortsprache). Und
diese Ausdrucksweise ließ den Fall aller Zahlen mit dem Fall ”aller Menschen in diesem
Zimmer“ verwechseln. (Während wir, um die Fälle zu unterscheiden, fragen: Wie verifiziert
man den einen und wie den andern.)
Wenn ich mir die Funktionen φ, ψ, F exakt definiert25
denke und nun das Schema des
Induktionsbeweises schreibe, -
26
auch dann kann ich nicht sagen, der Übergang von φr auf ψr sei auf Grund von ρ gemacht
worden (wenn der Übergang in α, β, γ nach ρ gemacht wurde – in speziellen Fällen ρ = α).
Er bleibt der Gleichung A entsprechend gemacht und ich könnte nur sagen, er entspreche
dem Komplex B, wenn ich nämlich diesen als ein anderes Zeichen statt der Gleichung A
auffasse.
Denn das Schema des Übergangs mußte ja α, β und γ enthalten.
Tatsächlich ist R nicht das Schema des Induktionsbeweises BIII; dieses ist viel komplizierter,
da es das Schema BI enthalten muß.
Es ist nur dann nicht ratsam, etwas ”Beweis“ zu nennen, wenn die übliche Grammatik
des Wortes ”Beweis“ mit der Grammatik des betrachteten Gegenstandes nicht übereinstimmt.
Die tiefgehende Beunruhigung rührt am Schluß von einem kleinen, aber offen zu Tage
liegenden Zug des überkommenen Ausdrucks her.
Was heißt es, daß R den Übergang von der Form A27
rechtfertigt? Es heißt wohl, daß
ich mich entschieden habe, nur solche Übergänge in meinem Kalkül zuzulassen, denen ein
Schema B entspricht, dessen Sätze α, β, γ wieder aus28
ρ ableitbar sein sollen. (Und das
hieße natürlich nichts anderes, als daß ich nur die Übergänge AI, AII, etc. zuließe und diesen
Schemata B entsprächen.) Richtiger wäre es, zu schreiben ”und diesen Schemata der
Form R entsprechen“. Ich wollte mit dem Nachsatz in der Klammer sagen, der Schein
der Allgemeinheit – ich meine, der Allgemeinheit des Begriffs der Induktionsmethode –
ist unnötig, denn es kommt am Schluß doch nur darauf hinaus, daß die speziellen
Konstruktionen BI, BII, etc. um die Seiten der Gleichungen AI, AII, etc. konstruiert wurden.
23 (V): Dieser ist hier nicht gemacht
24 (V): kann auf die Sprache keinen Einfluß
(mehr) haben.
25 (V): bestimmt
26 (F): MS 112, S. 57r.
27 (V): den Übergang A
28 (V): nach
683
684
685
457 Inwiefern verdient der Rekursionsbeweis den Namen eines ”Beweises“ . . .
B
α
β
γ
1
2
3
φ(1) = ψ(1)
φ(c + 1) = F(φ(c) )
ψ(c + 1) = F(ψ(c) )
R
5
6
7
. . . φn = ψn
A
WBTC121-130 30/05/2005 16:14 Page 914
Only a definite transition to one equation from multiple equations is a proof of that
equation. This definite transition doesn’t take place here,19
and nothing else suffices to turn
B into a proof of A.20
But can’t I say that if I have proved this about A, I have thereby proved A? And if not,
then where then did the illusion come from that in doing this I had proved it? For surely
there must be some deeper reason for this illusion.
Well, if it is an illusion, in any case it arose from our mode of expression in wordlanguage,
“This proposition holds for all numbers”; for according to this view the algebraic
proposition was only another way of writing the proposition (of word-language). And this
form of expression allowed the case of all the numbers to be confused with the case of “all
the people in this room”. (Whereas to distinguish the cases, we ask: How does one verify
the one, and how the other?)
If I imagine the functions φ, ψ, F as defined21
exactly and then write the schema for the
inductive proof:
22
even then I can’t say that the step from φr to ψr was taken on the basis of ρ (i.e. if the
step in α, β, γ was made in accordance with ρ – in particular cases ρ = α). The step is still
made in accordance with the equation A, and I can only say that the step corresponds to the
complex B if I regard that complex as another sign in place of the equation A.
For of course the schema for the step had to contain α, β and γ.
In fact R isn’t the schema for the inductive proof BIII; that is much more complicated,
since it has to contain the schema BI.
The only time it is inadvisable to call something a “proof” is when the usual grammar of
the word “proof” doesn’t accord with the grammar of the object under consideration.
In the last analysis our profound uneasiness stems from a small but obvious feature of the
conventional expression.
What does it mean, that R justifies a step of the form A?23
No doubt it means that I have
decided to allow in my calculus only those steps to which there corresponds a schema B,
and whose propositions α, β, γ are supposed to be derivable from24
ρ. (And of course
that would only mean that I was allowing just the steps AI, AII, etc., and that schemata B
corresponded to them). It would be more accurate to write “and that schemata of the form
R correspond to them”. The sentence added in parentheses was intended to say that the
appearance of generality – I mean of the generality of the concept of the inductive method
– is unnecessary, for in the end all of this only amounts to the fact that the particular
constructions BI, BII, etc. were constructed flanking the equations AI, AII, etc. Or that it
is superfluous to pick out the common feature of these constructions at that point; all that
19 (V): This definite transition isn’t made here,
20 (V): nothing else can have any (more) influence
on language.
21 (V): determined
22 (F): MS 112, p. 57r.
23 (V): justifies a transition A?
24 (V): derivable according to
Does a Recursive Proof Deserve the Name “Proof” . . . 457e
B
α
β
γ
1
2
3
φ(1) = ψ(1)
φ(c + 1) = F(φ(c) )
ψ(c + 1) = F(ψ(c) )
R
5
6
7
. . . φn = ψn
A
WBTC121-130 30/05/2005 16:14 Page 915
Oder: es ist ein Luxus, dann noch das Gemeinsame dieser Konstruktionen zu erkennen; alles
was maßgebend ist, sind diese Konstruktionen (selber). Denn alles, was da steht, sind diese
Beweise. Und der Begriff, unter den die Beweise fallen, ist überflüssig, denn wir haben nie
etwas mit ihm gemacht. Wie der Begriff Sessel überflüssig ist, wenn ich nur – auf die
Gegenstände weisend – sagen will ”stelle dies und dies und dies in mein Zimmer“ (obwohl
die drei Gegenstände Sessel sind). (Und eignen sich diese Geräte nicht, um darauf zu
sitzen, so wird das dadurch nicht anders, daß man auf eine Ähnlichkeit zwischen ihnen
aufmerksam macht.) Das heißt aber nichts anderes, als daß der einzelne Beweis unsere
Anerkennung als solchen braucht (wenn ”Beweis“ bedeuten soll, was es bedeutet); hat er
die nicht, so kann keine Entdeckung einer Analogie mit anderen solchen Gebilden sie
ihm verschaffen.29
Und der Schein des Beweises entsteht dadurch, daß α, β, γ und A
Gleichungen sind, und daß eine allgemeine Regel gegeben werden kann, nach der man aus
B A bilden (und es in diesem Sinne ableiten) kann.
Auf diese allgemeine Regel kann man nachträglich aufmerksam werden. (Wird man nun
dadurch aber darauf aufmerksam, daß die B doch in Wirklichkeit Beweise der A sind?)
Man wird da auf eine Regel aufmerksam, mit der man hätte beginnen können und mittels
der und α man AI, AII etc. hätte konstruieren30
können. Niemand aber würde sie in diesem
Spiel einen Beweis genannt haben.
Woher dieser Konflikt: ”Das ist doch kein Beweis!“ – ”das ist doch ein Beweis!“?
Man könnte sagen: Es ist wohl wahr, ich zeichne im Beweis von B mittels α die Konturen
der Gleichung A nach,31
aber nicht auf die Weise, die ich nenne ”A32
mittels α beweisen“.
Die Schwierigkeit, die in dieser33
Betrachtung zu überwinden ist34
ist, den
Induktionsbeweis als etwas Neues, sozusagen, naiv zu betrachten.
Wenn wir also oben sagten, wir können mit R beginnen, so ist dieses Beginnen mit R in
gewisser Weise Humbug. Es ist nicht so, wie wenn ich eine Rechnung mit der Ausrechnung
von 526 × 718 beginne. Denn hier ist diese Problemstellung der Anfangspunkt eines Weges.
Während ich dort das R sofort wieder verlasse und wo anders beginnen muß. Und wenn es
geschehen ist, daß ich einen Komplex von der Form R konstruiert habe, dann ist es wieder
gleichgültig, ob ich mir das früher äußerlich vorgesetzt habe, weil mir dieser Vorsatz,
mathematisch (gesprochen), d.h. im Kalkül, doch nichts geholfen hat. Es bleibt also bei der
Tatsache, daß ich jetzt einen Komplex von der Form R vor mir habe.
Wir könnten uns denken, wir kennten nur den Beweis BI
35
und würden nun sagen:
Alles, was wir haben, ist diese Konstruktion. Von einer Analogie dieser mit anderen
Konstruktionen, von einem allgemeinen Prinzip bei der Ausführung dieser Konstruktionen,
ist gar keine Rede. – Wenn ich nun so B und A sehe, muß ich fragen: warum nennst
Du das aber einen Beweis gerade von AI? (ich frage noch nicht: warum nennst Du es einen
Beweis von A). Was hat dieser Komplex mit AI zu tun? Als Antwort muß er mich auf die
Beziehung zwischen A und B aufmerksam machen, die in V ausgedrückt ist.36
29 (V): geben.
30 (V): bauen
31 (V): von B die Konturen der Gleichung A
mittels α nach,
32 (V): nenne$ ”A
33 (V): die durch diese
34 (V): Betrachtung überwunden werden soll
35 (V): BI mit d
36 (E): Im Manuskript (MS 112, S. 45r) erscheint
”V“, einige Seiten vor der vorliegenden
Bemerkung, als Kurzbezeichnung für eine
längere Formel. Wir haben sie in Anhang I
wiedergegeben.
686
687
458 Inwiefern verdient der Rekursionsbeweis den Namen eines ”Beweises“ . . .
WBTC121-130 30/05/2005 16:14 Page 916
is relevant are those constructions (themselves). For these proofs are all that is there. And the
concept under which the proofs fall is superfluous, because we never did anything with it.
Just as the concept “chair” is superfluous if – pointing to the objects – all I want to say is
“Put that and that and that in my room” (even though the three objects are chairs). (And if
those devices aren’t suitable for sitting on, then that doesn’t change by someone’s drawing
attention to a similarity between them.) But that only means that the individual proof
needs our acknowledgment of it as such (if “proof” is to mean what it means); and if it
doesn’t have it, no discovery of an analogy with other such constructions can provide
such an acknowledgment for it.25
The reason it looks like a proof is that α, β, γ and A are
equations, and that a general rule can be given, according to which we can construct (and
in that sense derive) A from B.
After the event we might become aware of this general rule. (But does that then make us
aware that the B’s are really proofs of the A’s?) What we become aware of is a rule we might
have started with, and using which (in conjunction with α) we could have constructed26
AI, AII, etc. But no one would have called it a proof in this game.
Whence this conflict: “But that isn’t any proof!” – “But that is a proof.”?
We could say: It’s no doubt true that in proving B, I use α to trace the contours of the
equation A, but not in the way I call “proving A by α”.27
The difficulty that has to be28
overcome in this29
investigation is the difficulty of looking
at the proof by induction as something new, naively, as it were.
So when we said above that we could begin with R, this beginning with R is, in a way,
humbug. It isn’t like beginning a calculation by calculating 526 × 718. For in the latter case
this setting out of the problem is the starting point of a journey, whereas in the former case
I immediately abandon the R and have to begin somewhere else. And when it turns out that
I’ve constructed a complex of the form R, it is again immaterial whether I explicitly set out
to do this earlier, since mathematically speaking, i.e. in the calculus, this intention didn’t
help me. So what remains is the fact that I now have a complex of the form R in front
of me.
We could imagine we were acquainted only with the proof BI, and then we would say:
All we have is this construction. There is no mention whatever of an analogy between this
and other constructions, or of a general principle in carrying out these constructions. – Now
if I see B and A like this I have to ask: But why do you call that a proof specifically of AI?
– (I am not yet asking: Why do you call it a proof of A?) What does this complex have to do
with AI? In answering he’ll have to point out to me the relation between A and B which is
expressed in V.30
25 (V): can give it to it.
26 (V): built
27 (V): say: It’s no doubt true that in proving B, I
trace the contours of the equation A, but not in
the way I call “proving A by α”.
28 (V): that should be
29 (V): overcome through this
30 (E): In the manuscript source (MS 112, p. 45r)
“V” appears a few pages before this remark,
as a designation for a lengthy formula. We have
included this formula here in Appendix I.
Does a Recursive Proof Deserve the Name “Proof” . . . 458e
WBTC121-130 30/05/2005 16:14 Page 917
Es zeigt uns jemand BI und erklärt uns den Zusammenhang mit AI, d.i., daß die rechte
Seite von A so und so erhalten wurde, etc. etc. Wir verstehen ihn; und er fragt uns (nun):
ist nun das ein Beweis von A? Wir werden37
antworten: gewiß nicht!
Hatten wir nun alles verstanden, was über diesen Beweis zu verstehen war? Ja. Hatten
wir auch die allgemeine Form des Zusammenhangs von B und A gesehen? Ja!
Und wir könnten auch daraus schließen, daß man so aus jedem A ein B konstruieren kann
und also auch umgekehrt A aus B.
Dieser Beweis ist nach einem bestimmten Plan gebaut (nach dem noch andere Beweise
gebaut sind). Aber dieser Plan kann den Beweis nicht zum Beweis machen. Denn wir
haben jetzt hier nur die eine Verkörperung dieses Planes, und können von dem Plan als
allgemeinem Begriff (ganz) absehen. Der Beweis muß für sich sprechen und der Plan ist
nur in ihm verkörpert, aber selbst kein Instrument38
des Beweises. (Das wollte ich immer
sagen.) Daher nützt es mir39
nichts, wenn man mich auf die Ähnlichkeiten zwischen
Beweisen aufmerksam macht, um mich davon zu überzeugen, daß sie Beweise sind.
Ist nicht unser Prinzip: kein Begriffswort40
zu verwenden, wo keines41
nötig ist? – D.h. die
Fälle, in denen das Begriffswort in Wirklichkeit für eine Aufzählung42
steht, als solche zu
erklären.43
Wenn ich nun früher sagte ”das ist doch kein Beweis“, so meinte ich ”Beweis“ in einem
bereits festgelegten44
Sinne, in welchem es aus A und B allein zu ersehen ist. Denn in diesem
Sinne kann ich sagen: Ich verstehe doch ganz genau, was B tut und in welchem Verhältnis
es zu A steht. Jede weitere Belehrung ist überflüssig und das, was da ist, ist kein Beweis.45
In diesem Sinne habe ich es nur mit B und A allein zu tun; ich sehe außer ihnen nichts und
nichts anderes geht mich an.
Dabei sehe ich das Verhältnis nach der Regel V sehr wohl,46
aber es kommt für mich
als Konstruktionsbehelf gar nicht in Frage. Sagte mir jemand, während meiner Betrachtung
von B und A, daß man auch hätte B aus A (oder umgekehrt) nach einer Regel konstruieren
können, so könnte ich ihm nur sagen ”komm’ mir nicht mit unwesentlichen Sachen“. Denn
das ist ja selbstverständlich, und ich sehe sofort, daß es B nicht zu einem Beweis von A macht.
Denn diese allgemeine Regel könnte nur zeigen,47
daß B der Beweis gerade von A48
ist, wenn
es überhaupt ein Beweis wäre. D.h., daß der Zusammenhang zwischen B und A einer Regel
gemäß ist, kann nicht zeigen, daß B ein Beweis von A ist. Und jeder solche Zusammenhang
könnte zur Konstruktion von B aus A (und umgekehrt) benützt werden.
Wenn ich also sagte ”V49
wird ja gar nicht zur Konstruktion benützt, also haben wir mit
ihm nichts zu tun“, so hätte es heißen müssen: Ich habe es doch nur mit A und B allein zu
tun. Es genügt doch, wenn ich A und B miteinander konfrontiere und nun frage ”ist B
ein Beweis von A“; und also brauche ich A nicht aus B nach einer vorher festgelegten
Regel zu konstruieren, sondern es genügt, daß ich die einzelnen A – wie viele es sind – den
37 (V): würden
38 (V): kein Bestandteil
39 (O): mich
40 (V): Prinzip: keinen Begriff
41 (V): keiner
42 (V): Liste
43 (V): – D.h. die Fälle zu zeigen, in denen das
Begriffswort in Wirklichkeit für eine Liste //
Aufzählung // steht. // – D.h. in den Fällen, in
denen das Begriffswort für eine Liste steht,
dies klar zu machen.
44 (O): festgelegtem
45 (V): und das ist kein Beweis.
46 (V): gut,
47 (V): Denn, daß es so eine allgemeine Regel
gibt, könnte nur zeigen,
48 (V): der Beweis von A und keinem andern Satz
49 (V): ”R
688
689
688
459 Inwiefern verdient der Rekursionsbeweis den Namen eines ”Beweises“ . . .
WBTC121-130 30/05/2005 16:14 Page 918
Someone shows us BI and explains the connection with AI, that is, that the right side of
A was obtained in such and such a way, etc., etc. We understand him; and (then) he asks
us: Now, is that a proof of A? We will31
answer: Certainly not!
Now had we understood everything there was to understand about this proof? Yes. Had
we also seen the general form of the connection between B and A? Yes!
And we could also infer from this that in this way we can construct a B from every A,
and therefore conversely an A from every B.
This proof has been constructed according to a certain plan (a plan according to which
further proofs have also been constructed). But this plan cannot make the proof a proof. For
all we have here is the one embodiment of this plan, and we can (completely) disregard the
plan as a general concept. The proof has to speak for itself and the plan is only embodied
in it, but isn’t itself an instrument32
of the proof. (That’s what I’ve been wanting to say all
along). Therefore it’s of no use to me if someone draws my attention to the similarities between
proofs in order to convince me that they are proofs.
Isn’t our principle: not to use a concept-word33
where none is necessary? – That is, in cases
where the concept-word really stands for an enumeration,34
to say as much.35
Now when I said earlier “But that isn’t a proof”, I meant “proof” in an already established
sense in which its being a proof can be gathered from A and B alone. For in this sense
I can say: But I understand precisely what B does and what relationship it has to A. All further
instruction is superfluous, and what is there36
is no proof. In this sense I am concerned
only with B and A; I don’t see anything other than them, and nothing else concerns me.
In this context, I certainly do see the relationship in accordance with the rule V, but it is
completely out of the question for me as an aid in construction. If someone told me while I
was examining B and A that we also could have constructed B from A (or conversely) according
to a rule, I could only say to him “Don’t bother me with irrelevancies”. For of course
that goes without saying, and I see immediately that it doesn’t make B a proof of A. For this
general rule37
could only show that B is a proof specifically of A,38
if it were a proof at all.
That means that the fact that the connection between B and A is in accordance with a rule
can’t show that B is a proof of A. And every such connection could be used to construct B
from A (and conversely).
So when I said “V39
isn’t even used for the construction, so we have nothing to do with
it” I should have said: After all, I’m only dealing with A and B. It is surely enough if I
confront A and B with each other and then ask: “Is B a proof of A?” And so I don’t need
to construct A from B according to a previously established rule; it’s sufficient for me to
place the particular A’s – however many there are – in confrontation with particular B’s.
31 (V): would
32 (V): itself a component
33 (V): concept
34 (V): for a list,
35 (V): – That is, to point out the cases in which
the concept-word really stands for a list // an
enumeration. // – That is, in cases where the
concept-word stands for a list, to make this clear.
36 (V): and that
37 (V): For that there is such a general rule
38 (V): proof of A and no other proposition,
39 (V): “R
Does a Recursive Proof Deserve the Name “Proof” . . . 459e
WBTC121-130 30/05/2005 16:14 Page 919
einzelnen B gegenüberstelle. Ich brauche eine Konstruktionsregel nicht; und das ist wahr.
Ich brauche eine vorher aufgestellte Konstruktionsregel nicht (aus der ich dann erst die A
gewonnen hätte).
Ich meine: Im Skolem’schen Kalkül brauchen wir keinen solchen Begriff,50
es51
genügt die
Liste.
Es geht uns nichts verloren, wenn wir nicht sagen ”wir haben die Grundgesetze A auf
diese Weise bewiesen“,52
sondern bloß zeigen, daß sich ihnen – in gewisser Beziehung
analoge – Konstruktionen zuordnen lassen.
Der Begriff der Allgemeinheit (und der Rekursion), der in diesen Beweisen gebraucht wird,
ist nicht allgemeiner, als er aus diesen Beweisen unmittelbar herauszulesen ist.
Die Klammer } in R, welche α, β, und γ zusammenhält, kann weiter nichts bedeuten, als
daß wir den Übergang in A (oder einem von der Form A) als berechtigt ansehen, wenn
die Glieder (Seiten) des Übergangs in einer, durch das Schema B charakterisierten
Beziehung, zu einander stehen. Es nimmt dann B den Platz von A. Und wie es früher
hieß: der Übergang ist in meinem Kalkül erlaubt, wenn er einem der A entspricht, so heißt
es jetzt:53
er ist erlaubt, wenn er einem der B entspricht.
Damit aber hätten wir noch keine Vereinfachung, keine Reduktion gewonnen.
Der Gleichungskalkül ist gegeben. In diesem Kalkül hat ”Beweis“ eine fixe54
Bedeutung.
Nenne ich nun auch die induktive Rechnung einen Beweis, so erspart mir dieser Beweis
doch nicht die Kontrolle, ob die Übergänge der Gleichungskette, nach diesen bestimmten
Regeln (oder Paradigmen) gemacht sind. Ist das der Fall, so sage ich, die letzte Gleichung
der Kette sei bewiesen; oder auch, die Gleichungskette stimme.
Denken wir uns, wir kontrollieren die Rechnung (a + b)3
= . . . auf die erste55
Weise und
beim ersten Übergang sagt er: ”ja, dieser Übergang geschieht zwar56
nach a · (b + c) =
ab + ac, aber stimmt das auch?“ Und nun zeigten wir ihm die Ableitung dieser Gleichung
im induktiven Sinne. –
In einer Bedeutung heißt die Frage ”stimmt die Gleichung“:57
läßt sie sich nach den
Paradigmen herleiten? – Im andern Fall heißt es: lassen sich die Gleichungen α, β, γ nach
dem Paradigma (oder den Paradigmen) herleiten? – Und hier haben wir die beiden
Bedeutungen der Frage (oder des Wortes ”Beweis“) auf eine Ebene gestellt (in einem System
ausgedrückt) und können sie nun vergleichen (und sehen, daß sie nicht Eines sind).
Und zwar leistet dieser neue Beweis nicht, was man annehmen könnte, daß er nämlich
den Kalkül auf eine engere58
Grundlage setzte – wie es etwa geschieht, wenn wir durch
p | q p 1 q und ~p ersetzen, oder die Zahl der Axiome vermindern. Denn, wenn man
nun sagt, man habe alle die Grundgleichungen A aus ρ allein abgeleitet, so heißt hier das
Wort ”abgeleitet“ etwas (ganz) andres. (Was man sich bei dieser Versprechung erwartet,
ist die Ersetzung der großen Kettenglieder durch kleinere, nicht durch zwei halbe
Kettenglieder.) Und in einem Sinne hat man durch diese Ableitungen alles beim alten
gelassen. Denn es bleibt im neuen Kalkül ein Kettenglied des alten wesentlich als ein
50 (V): brauchen wir diesen Begriff nicht,
51 (V): es
52 (V): ”wir haben die Grundgesetze A bewiesen“,
53 (V): so kann es jetzt heißen:
54 (V): festgelegte
55 (V): = . . . in der ersten
56 (V): (wohl)
57 (O): stimmt die Gleichung G“:
58 (V): kleinere
690
691
460 Inwiefern verdient der Rekursionsbeweis den Namen eines ”Beweises“ . . .
WBTC121-130 30/05/2005 16:14 Page 920
I don’t need a construction rule; and that’s the truth. I don’t need a previously established
construction rule (from which only then could I have obtained the A’s).
What I mean is: in Skolem’s calculus we don’t need any such concept;40
the list is sufficient.
We lose nothing if instead of saying “We have proved the fundamental laws A in this
fashion”,41
we merely show that constructions that are analogous to them in certain respects
can be coordinated with them.
The concept of generality (and of recursion) used in these proofs has no greater generality
than can be read immediately from these proofs.
The brace } in R that unites α, β, and γ can’t mean any more than that we regard the
step in A (or a step of the form A) as justified if the terms (sides) of the step are in a
relation to each other characterized by the schema B. B then takes the place of A. And just
as before we said: the step is permitted in my calculus if it corresponds to one of the A’s,
so we now say:42
it is permitted if it corresponds to one of the B’s.
But that still wouldn’t have gained us any simplification or reduction.
The calculus of equations has been given. In that calculus “proof” has a fixed meaning.43
Even if I now call the inductive calculation a proof, this proof still doesn’t spare me from
checking whether the steps in the chain of equations have been taken in accordance with
these particular rules (or paradigms). If they have been, I say that the last equation of the
chain has been proved, or alternatively, that the chain of equations is correct.
Suppose that we were using the first method to check the calculation (a + b)3
= . . . , and
at the first step someone said: “To be sure, that step is being taken in accordance with
a · (b + c) = ab + ac, but is that really correct?” And now we were to show him the inductive
derivation of that equation. –
One meaning of the question “Is the equation correct?”44
is: Can it be derived in accordance
with the paradigms? – Another is: Can the equations α, β, and γ be derived in accordance
with the paradigm (or the paradigms?) – And here we have put the two meanings of
the question (or of the word “proof”) on the same level (expressed them in a single system)
and can now compare them (and see that they are not the same).
Specifically, this new proof doesn’t produce what you might expect, i.e. it doesn’t set the
calculus on a narrower45
foundation – as happens, for example, if we replace p 1 q and ~p
by p | q, or reduce the number of axioms. For if we now say that all the basic equations A
have been derived from ρ alone, then the word “derived” here means something (entirely)
different. (What we expect, given this promise, is for the big links in the chain to be replaced
by smaller ones, not by two half links.) And in one sense these derivations have left everything
as it was. For in the new calculus a link of the old chain continues in essence to
40 (V): need this concept;
41 (V): fundamental laws A”,
42 (V): now can say:
43 (V): has an established meaning.
44 (O): equation G correct?”
45 (V): smaller
Does a Recursive Proof Deserve the Name “Proof” . . . 460e
WBTC121-130 30/05/2005 16:14 Page 921
solches bestehn. Die alte Struktur wird nicht aufgelöst. So daß man sagen muß, der alte Gang
des Beweises bleibt bestehen. Und es bleibt im alten Sinne auch die Unreduzierbarkeit.
Man kann daher auch nicht sagen, Skolem habe das algebraische System auf eine kleinere
Grundlage gesetzt, denn er hat es in einem andern Sinne als dem der Algebra ”begründet“.59
Wird ein Zusammenhang der A durch die Induktionsbeweise mittels α gezeigt und ist
dies nicht das Zeichen dafür, daß wir es hier doch mit Beweisen zu tun haben? – Es
wird nicht der Zusammenhang gezeigt, den ein Zerlegen der Übergänge A in Übergänge ρ
herstellen würde. Und ein Zusammenhang der A ist ja schon vor jedem Beweis zu sehen.
Ich kann die Regel R
. . . R
auch60
so schreiben:
. . . S
61
oder auch so:
a + (b + 1) = (a + b) + 1,
wenn ich R oder S als Erklärung oder Ersatz für diese Form nehme.
Wenn ich nun sage, in
α a + (b + 1) = (a + b) + 1 5
β a + (b + (c + 1)) = a + ((b + c) + 1) = (a + (b + c)) + 1 6 . . . B
γ (a + b) + (c + 1) = ((a + b) + c) + 1 7
62
seien die Übergänge durch die Regel R gerechtfertigt, – so kann man mir drauf antworten:
”Wenn Du das eine Rechtfertigung nennst, so hast Du die Übergänge gerechtfertigt. Du
hättest uns aber ebensoviel gesagt, wenn Du uns nur auf die Regel R und ihre formale
Beziehung zu α (oder zu α, β und γ) aufmerksam gemacht hättest.“
Ich hätte also auch sagen können: Ich nehme die Regel R in der und der Weise als
Paradigma meiner Übergänge.
Wenn nun Skolem etwa nach seinem Beweis für das assoziative Gesetz übergeht zu:
a + 1 = 1 + a 5
a + (b + 1) = (a + b) + 1 6 . . . C
(b + 1) + a = b + (1 + a) = b + (a + 1) = (b + a) + 1 7
63
und sagt, der erste und dritte Übergang in der dritten Zeile seien nach dem bewiesenen
assoziativen Gesetz gerechtfertigt, – so erfahren wir damit nicht mehr,64
als wenn er sagte,
die Übergänge seien nach dem Paradigma a + (b + c) = (a + b) + c gemacht (d.h., sie
entsprechen dem Paradigma) und es sei ein Schema α, β, γ mit Übergängen nach dem
a
a
a
(a
(a
a
( )
( )
(( ) )
)
)
( ( ))
+ +
+ +
+ + +
=
+ +
+ +
+ + +
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
1 1
1
1 1
1 1
1
1 1
ξ
ξ
ξ
ξ
a
a
a
(a
(a
a
( )
( )
(( ) )
)
)
(( ) )
+ +
+ +
+ + +
=
+ +
+ +
+ + +
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
1 1
1
1 1
1 1
1
1 1
ξ
ξ
ξ
ξ
59 (V): denn er hat es in einem andern Sinne als
dem algebraischen ”begründet“.
60 (O): Ich kann die Regel R auch
61 (F): MS 114, S. 1v.
62 (F): MS 114, S. 1v.
63 (F): MS 114, S. 1v.
64 (V): so sagt er uns damit nicht mehr,
692
693
461 Inwiefern verdient der Rekursionsbeweis den Namen eines ”Beweises“ . . .
WBTC121-130 30/05/2005 16:14 Page 922
exist as a link. The old structure is not broken up. So we have to say that the old proofprocedure
continues to exist. And in the old sense the irreducibility remains as well.
Neither can we say that Skolem has set the algebraic system onto a smaller foundation,
for he has “given it foundations” in a different sense from that of algebra.46
Do the inductive proofs – by means of α – show a connection between the A’s? And isn’t
this a sign that here we’re dealing with proofs after all? – The connection shown is not the
one that breaking up the A steps into ρ steps would establish. And one connection between
the A’s is already visible before any proof.
I can also write the rule R
. . . R
like47
this:
. . . S
48
or also like this:
a + (b + 1) = (a + b) + 1,
if I take R or S as a definition or substitute for that form.
Now if I say that the steps in:
α a + (b + 1) = (a + b) + 1 5
β a + (b + (c + 1)) = a + ((b + c) + 1) = (a + (b + c)) + 1 6 . . . B
γ (a + b) + (c + 1) = ((a + b) + c) + 1 7
49
are justified by the rule R, you can reply: “If that’s what you call a justification, then you
have justified the steps. But you’d have told us as much if you had just drawn our attention
to the rule R and its formal relationship to α (or to α, β, and γ ).”
So I could also have said: I take the rule R in such and such a way as a paradigm for my
steps.
So if Skolem, after his proof of the associative law, now takes the step to:
a + 1 = 1 + a 5
a + (b + 1) = (a + b) + 1 6 . . . C
(b + 1) + a = b + (1 + a) = b + (a + 1) = (b + a) + 1 7
50
and says that the first and third steps in the third line are justified according to the already
proved associative law – that tells us no more51
than if he had said that the steps were taken
in accordance with the paradigm a + (b + c) = (a + b) + c (i.e. they correspond to the paradigm)
and a schema α, β, γ was derived in steps according to the paradigm α. – “But does B
justify these steps, or not?” – “What do you mean by the word ‘justify’?” – “Well, the step
a
a
a
(a
(a
a
( )
( )
(( ) )
)
)
( ( ))
+ +
+ +
+ + +
=
+ +
+ +
+ + +
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
1 1
1
1 1
1 1
1
1 1
ξ
ξ
ξ
ξ
a
a
a
(a
(a
a
( )
( )
(( ) )
)
)
(( ) )
+ +
+ +
+ + +
=
+ +
+ +
+ + +
⎡
⎣
⎢
⎢
⎤
⎦
⎥
⎥
1 1
1
1 1
1 1
1
1 1
ξ
ξ
ξ
ξ
46 (V): sense than in the algebraic one.
47 (O): I can also write the rule R like
48 (F): MS 114, p. 1v.
49 (F): MS 114, p. 1v.
50 (F): MS 114, p. 1v.
51 (V): law – then we don’t find out anything
more from this
Does a Recursive Proof Deserve the Name “Proof” . . . 461e
WBTC121-130 30/05/2005 16:14 Page 923
Paradigma α abgeleitet. – ”Aber rechtfertigt B nun diese Übergänge, oder nicht?“ – Was
meinst Du mit dem Wort ”rechtfertigen“? – ”Nun, der Übergang ist gerechtfertigt, wenn
wirklich ein Satz, der für alle Zahlen gilt, bewiesen ist.“ – Aber in welchem Falle wäre das
geschehen? Was nennst Du einen Beweis davon, daß ein Satz für alle Kardinalzahlen gültig
ist? Wie weißt Du, ob der Satz (wirklich) für alle Kardinalzahlen gültig65
ist, da Du es nicht
ausprobieren kannst. Dein einziges Kriterium ist ja der Beweis. Du bestimmst also wohl eine
Form und nennst sie die des Beweises, daß ein Satz für alle Kardinalzahlen gilt. Dann haben
wir eigentlich gar nichts davon, daß uns zuerst die allgemeine Form dieser Beweise gezeigt
wird; da ja dadurch nicht gezeigt wird, daß nun der besondere Beweis wirklich das leistet,
was wir von ihm verlangen; ich meine: da hiedurch der besondere Beweis nicht als einer
gerechtfertigt, erwiesen ist, der einen Satz für alle Kardinalzahlen beweist. Der rekursive
Beweis muß vielmehr seine eigene Rechtfertigung sein. Wenn wir unsern Beweisvorgang
wirklich als den Beweis einer solchen Allgemeinheit rechtfertigen wollen, tun wir vielmehr
etwas anderes: wir gehen Beispiele einer Reihe durch, und diese Beispiele und das Gesetz,
was wir in ihnen erkennen, befriedigt uns nun, und wir sagen: ja, unser Beweis leistet
wirklich, was wir wollten. Aber wir müssen nun bedenken, daß wir mit der Angabe dieser
Beispielreihe die Schreibweise B und C nur in eine andere (Schreibweise) übersetzt haben.
(Denn die Beispielreihe ist nicht die unvollständige Anwendung der allgemeinen Form,
sondern ein anderer Ausdruck des Gesetzes.)66
Und weil die Wortsprache, wenn sie den
Beweis erklärt, erklärt was er beweist, den Beweis nur in eine andere Ausdrucksform
übersetzt, so können wir diese Erklärung auch ganz weglassen. Und wenn wir das tun,
so werden die mathematischen Verhältnisse viel klarer, nicht verwischt durch die Vieles
bedeutenden67
Ausdrücke der Wortsprache. Wenn ich z.B. B unmittelbar neben A setze,
ohne Vermittlung durch den Ausdruck der Wortsprache ”für alle Kardinalzahlen etc.“,68
so
kann kein falscher Schein eines Beweises von A durch B entstehen. Wir sehen dann die
nüchternen, (nackten) Beziehungen zwischen A und B, und wie weit sie reichen.69
Man
lernt so erst, unbeirrt von der alles gleichmachenden Form der Wortsprache, die eigentliche
Struktur dieser Beziehung kennen, und was es mit ihr auf sich hat.
Man sieht hier vor allem, daß wir an70
dem Baum der Strukturen B, C, etc. interessiert
sind, und daß an ihm zwar allenthalben die Form
φ(1) = ψ(1)
φ(n + 1) = F(φn)
ψ(n + 1) = F(ψn)
zu sehen ist, gleichsam eine bestimmte Astgabelung, – daß aber diese Gebilde in verschiedenen
Anordnungen, und Verbindungen untereinander, auftreten, und daß sie nicht in dem Sinne
Konstruktionselemente sind,71
wie die Paradigmen im Beweis von a + (b + (c + 1)) =
(a + (b + c)) +1 oder (a + b)2
= a2
+ 2ab + b2
. Der Zweck der ”rekursiven Beweise“ ist ja,
den algebraischen Kalkül mit dem der Zahlen in Verbindung zu setzen. Und der Baum der
rekursiven Beweise ”rechtfertigt“ den algebraischen Kalkül nur, wenn das heißen soll, daß
er ihn mit dem arithmetischen in Verbindung bringt. Nicht aber in dem Sinn, in welchem
die Liste der Paradigmen den algebraischen Kalkül, d.h. die Übergänge in ihm, rechtfertigt.
65 (V): giltig
66 (V): Ausdruck dieser Form.)
67 (V): die mehrdeutigen
68 (V): ohne Dazwischenkunft des Wortes ”alle“,
69 (V): Wir sehen dann ganz nüchtern, wie weit die
Beziehungen von B zu A und zu a + b = b + a
reichen und wo sie aufhören.
70 (V): in
71 (V): bilden,
694
695
462 Inwiefern verdient der Rekursionsbeweis den Namen eines ”Beweises“ . . .
WBTC121-130 30/05/2005 16:14 Page 924
is justified if a theorem that holds for all numbers really has been proved.” – But in what
case would that have happened? What do you call a proof that a theorem holds for all
cardinal numbers? How do you know whether a theorem is (really) valid for all cardinal
numbers, since you can’t test it? After all, your only criterion is the proof itself. So most
likely you stipulate a form and call it the form of the proof that a proposition holds for all
cardinal numbers. In that case we really gain nothing by being first shown the general form
of these proofs, because that doesn’t show that the individual proof really accomplishes what
we demand of it; I mean, because it doesn’t justify, doesn’t show the individual proof to be
a proof of a theorem for all cardinal numbers. Rather, the recursive proof has to be its
own justification. If we really want to justify our proof procedure as a proof of a generality
of this kind, we do something different: we go through examples of a series, and then these
examples and the law we recognize in them satisfy us, and we say: Yes, our proof really
does accomplish what we wanted. But now we must recognize that by giving this series of
examples we have only translated the notations B and C into another (notation). (For the
series of examples is not an incomplete application of the general form, but another expression
of the law.)52
And because word-language, in explaining the proof (i.e. what the proof
proves), only translates it into another form of expression, we can just as well drop this
explanation altogether. And if we do so, the mathematical relationships become much clearer,
no longer blurred by the expressions of word-language with their multiple53
meanings. For
example, if I put B right beside A, without using the expression of word-language “for all
cardinal numbers, etc.”,54
then no false appearance of a proof of A by B can arise. We then
see the sober (bare) relationships between A and B, and how far they extend.55
Only in this
way, unflustered by the all-levelling form of word-language, do we come to know the actual
structure of that relationship and what it is all about.
Here we see first and foremost that we are interested in the tree of the structures B, C,
etc., and that in it there is visible everywhere – a particular forking of the branches, as it
were – the following form:
φ(1) = ψ(1)
φ(n + 1) = F(φn)
ψ(n + 1) = F(ψn)
But we see that these patterns turn up in different arrangements and interconnections, and
that they are not56
elements of the construction in the same sense as are the paradigms in
the proof of a + (b + (c + 1)) = (a + (b + c)) + 1, or (a + b)2
= a2
+ 2ab + b2
. The purpose
of the “recursive proofs” is of course to connect the algebraic calculus with the calculus of
numbers. And the tree of the recursive proofs only “justifies” the algebraic calculus if that
is supposed to mean that it connects it with the arithmetical one. It doesn’t justify it in the
sense in which the list of paradigms justifies the algebraic calculus, i.e. the steps in it.
52 (V): of this form.)
53 (V): ambiguous
54 (V): without the word “all” coming in
between,
55 (V): We then see quite soberly how far the
relationships of B to A and to a + b = b + a
extend, and where they end.
56 (V): they do not form
Does a Recursive Proof Deserve the Name “Proof” . . . 462e
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Wenn man also die Paradigmen der Übergänge tabuliert, so hat das dort Sinn, wo das
Interesse darin liegt, zu zeigen, daß die und die Transformationen alle bloß mit Hilfe jener
– im übrigen willkürlich gewählten – Übergangsformen zustande gebracht sind. Nicht
aber dort, wo sich die Rechnung in einem andern Sinne rechtfertigen soll, wo also das
Anschauen der Rechnung – ganz abgesehen von dem Vergleich mit einer Tabelle vorher
festgelegter Normen – uns lehren muß, ob wir sie zulassen sollen oder nicht. Skolem hätte
uns also keinen Beweis des assoziativen und kommutativen Gesetzes versprechen brauchen,72
sondern einfach sagen können, er werde uns einen Zusammenhang der Paradigmen der Algebra
mit den Rechnungsregeln der Arithmetik zeigen. Aber ist das nicht Wortklauberei? hat er
denn nicht die Zahl der Paradigmen reduziert und uns z.B. statt jener beiden Gesetze eines,
nämlich a + (b + 1) = (a + b) + 1 gegeben? Nein. Wenn wir z.B. (a + b)4
= etc. (r) beweisen,
so könnten wir dabei von dem vorher bewiesenen Satz (a + b)2
= etc. (s) Gebrauch machen.
Aber in diesem Fall lassen sich die Übergänge in r, die durch s gerechtfertigt wurden, auch
durch jene Regeln rechtfertigen, mit denen s bewiesen wurde. Und es verhält sich dann s
zu jenen ersten Regeln, wie ein durch Definition eingeführtes Zeichen zu den primären
Zeichen, mit deren Hilfe es definiert wurde. Man kann die Definition immer auch eliminieren
und auf die primären Zeichen übergehen. Wenn wir aber in C einen Übergang machen, der
durch B gerechtfertigt ist, so können wir diesen Übergang nun nicht auch mit α allein machen.
Wir haben eben mit dem, was hier Beweis genannt wird, nicht einen Übergang73
in Stufen
zerlegt, sondern etwas ganz andres getan.
72 (V): sollen, 73 (V): Schritt
463 Inwiefern verdient der Rekursionsbeweis den Namen eines ”Beweises“ . . .
WBTC121-130 30/05/2005 16:14 Page 926
So tabulating the paradigms for the steps makes sense in the cases where we are interested
in showing that such and such transformations are all brought about only by means of
those transitional forms – ones that are, incidentally, arbitrarily chosen. But it doesn’t make
sense where the calculation is to justify itself in another sense, i.e. where merely looking at
the calculation – leaving aside any comparison with a table of previously established norms
– must show us whether we are to allow it or not. So Skolem wouldn’t have had to promise
us57
a proof of the associative and commutative laws; he could simply have said he would
show us a connection between the paradigms of algebra and the calculation rules of arithmetic.
But isn’t this hairsplitting? Didn’t he, after all, reduce the number of paradigms and
give us a single law, namely, a + (b + 1) = (a + b) + 1, instead of those two laws? No. In
proving, for example, (a + b)4
= etc. (r) we can make use of the previously proved proposition
(a + b)2
= etc. (s). But in that case the steps in (r) which were justified by (s) can also
be justified by the rules used to prove (s). And then the relation of (s) to those first rules is
the same as that of a sign introduced by definition to the primary signs used to define it: we
can always eliminate the definitions and step back to the primary signs. But when we take
a step in C that is justified by B, we can’t take the same step with (α) alone. And this means:
In what we’re here calling “proof”, we haven’t simply broken a transition58
into smaller steps;
we’ve done something quite different.
57 (V): Skolem shouldn’t have promised us 58 (V): step
Does a Recursive Proof Deserve the Name “Proof” . . . 463e
WBTC121-130 30/05/2005 16:14 Page 927
131
Der rekursive Beweis reduziert die
Anzahl der Grundgesetze nicht.
Wir haben also hier nicht den Fall, in welchem eine Gruppe von Grundgesetzen durch
eine mit weniger Gliedern bewiesen wird, aber nun weiter in den Beweisen alles im
Gleichen bleibt. (Wie auch in einem System von Grundbegriffen an der späteren
Entwicklung dadurch nichts geändert wird, daß man die Anzahl der Grundbegriffe durch
Definitionen reduziert.)
(Übrigens, welche verdächtige Analogie, zwischen ”Grundgesetzen“ und
”Grundbegriffen“!)
Es ist etwa1
so: der Beweis eines alten Grundgesetzes setzt sonst das System der Beweise
(einfach) nach rückwärts fort. Die Rekursionsbeweise aber setzen das System von algebraischen
Beweisen (mit den alten Grundgesetzen) nicht nach rückwärts fort, sondern sind ein neues
System, das mit dem ersten nur parallel zu laufen scheint.
Das ist eine seltsame Bemerkung, daß in den Induktionsbeweisen der Grundregeln
nach wie vor ihre Unreduzierbarkeit (Unabhängigkeit) sich zeigen muß.2
Was, wenn man
das für den Fall von gewöhnlichen Beweisen (oder Definitionen) sagte, also für den Fall, wo
die Grundregeln eben weiter reduziert werden, eine neue Verwandtschaft zwischen ihnen
gefunden (oder konstruiert) wird.
Wenn ich darin Recht habe, daß durch die Rekursionsbeweise die Unreduzierbarkeit3
intakt
bleibt, dann ist damit (wohl) alles gesagt, was ich gegen den Begriff vom Rekursions-”Beweis“
sagen4
wollte.5
Der induktive Beweis zerlegt den Übergang in A nicht. Ist es nicht das, was macht, daß
ich mich dagegen sträube, ihn Beweis zu nennen? Warum ich versucht bin zu sagen, er kann
auf keinen Fall – nämlich auch, wenn man A durch R und α konstruiert – mehr tun, als
etwas über den Übergang zu zeigen.
Wenn man sich einen Mechanismus aus Zahnrädern und diese aus lauter gleichen keilförmigen
Stücken und je einem Ring, der sie zu einem Rad zusammenhält, zusammengesetzt
denkt, so blieben in einem gewissen Sinne die Einheiten des Mechanismus doch die
Zahnräder.
Es ist so: Wenn ein Faß aus Dauben und Böden besteht, so halten doch nur alle diese in
dieser (bestimmten) Verbindung (als Komplex) die Flüssigkeit und bilden als Behälter neue
Einheiten.
1 (V): gleichsam
2 (V): (Unabhängigkeit) zu Tage treten muß.
3 (V): Unabhängigkeit
4 (V): vorbringen
5 (V): kann.
696
697
464
WBTC131-135 30/05/2005 16:14 Page 928
131
The Recursive Proof Doesn’t Reduce
the Number of Fundamental Laws.
So here we don’t have a case where a group of fundamental laws is proved by a group
with fewer terms while everything in the proofs remains the same. (Just as in a system of
fundamental concepts, nothing is altered in the later development by using definitions to
reduce the number of fundamental concepts.)
(And by the way, what a fishy analogy that, between “fundamental laws” and “fundamental
concepts”!)
It is something like this:1
Normally the proof of an old fundamental law (simply) continues
the system of proofs backwards. But the recursive proofs don’t continue the system
of algebraic proofs (with their old fundamental laws) backwards; rather they are a new
system, which only seems to run parallel to the first one.
It’s a strange remark that in the inductive proofs of the fundamental rules their irreducibility
(independence) must show itself,2
the same as before. What if we were to say the same thing
about normal proofs (or definitions), i.e. about the case where the fundamental rules are
further reduced, and a new relationship between them is discovered (or constructed)?
If I am right in saying that the irreducibility3
remains untouched by the recursive proofs,
then that (pretty well) sums up everything I wanted to say4
against the concept of recursive
“proof”.
The inductive proof doesn’t break up the step in A. Isn’t that what makes me balk at
calling it a proof? Isn’t that why I’m tempted to say that it can’t possibly do more than
show something about the step – not even if A is constructed by R and α?
If we imagine a mechanism made of cogwheels, and each of these cogwheels were made
of a number of identical wedge-shaped pieces plus a ring that holds them together to form
a wheel, it would still be the cogwheels that remained, in a certain sense, the units of the
mechanism.
It’s like this: if a barrel consists of staves and bottoms, it is still only when they are taken
as a whole, linked in this (particular) way (as a complex), that they hold liquid and that they
serve as individual containers.
1 (V): It is like this, as it were:
2 (V): must become apparent,
3 (V): independence
4 (V): I can proffer
464e
WBTC131-135 30/05/2005 16:14 Page 929
Denken wir uns eine Kette, sie besteht aus Gliedern und es ist möglich, (je) ein solches
Glied durch zwei kleinere zu ersetzen. Die Verbindung, die die Kette macht, kann dann,
statt durch die großen, ganz durch die kleinen6
Glieder gemacht werden. Man könnte sich
aber auch denken, daß jedes Glied der Kette aus – etwa – zwei halbringförmigen Teilen
bestünde, die zusammen das Glied bildeten, einzeln aber nicht als Glieder verwendet
werden könnten.
Es hätte nun ganz verschiedenen Sinn, einerseits, zu sagen: die Verbindung, die die großen
Glieder machen, kann durch lauter kleine Glieder gemacht werden; – und anderseits: diese
Verbindung kann durch lauter halbe große Glieder gemacht werden. Was ist der
Unterschied?
Der eine Beweis ersetzt eine großgliedrige Kette durch eine kleingliedrige, der andere zeigt,
wie man die (alten) großen Glieder aus mehreren Bestandteilen zusammensetzen kann.
Ähnlichkeit, sowie7
Verschiedenheit der beiden Fälle sind klar zu Tage liegend.8
Der Vergleich des Beweises mit der Kette ist natürlich ein logischer Vergleich und also
ein vollkommen exakter Ausdruck dessen, was er illustriert.
6 (V): kleineren
7 (V): und
8 (V): sind augenfällig.
698
465 Der rekursive Beweis reduziert die Anzahl der Grundgesetze nicht.
WBTC131-135 30/05/2005 16:14 Page 930
Let’s imagine a chain: it consists of links, and it’s possible to replace each such link by
two smaller ones. Then the connection that the chain makes can be made entirely by the
small5
links instead of by the large ones. But we could also imagine every link in the chain
consisting of, say, two parts, each shaped like half a ring, which together formed the link,
but could not be used individually as links.
Now it would mean completely different things to say, on the one hand: The connection
made by the large links can be made entirely by small links – and on the other: This connection
can be made entirely by large half-links. What’s the difference?
The one proof replaces a chain with large links by a chain with small links, the other shows
how one can fabricate the (old) large links from more than one part.
The similarity, as well as the difference, between6
the two cases is clear to see.7
The comparison between the proof and the chain is, of course, a logical comparison, and
therefore a completely exact expression of what it illustrates.
5 (V): smaller
6 (V): similarity and the difference between
7 (V): is obvious.
The Recursive Proof Doesn’t Reduce . . . 465e
WBTC131-135 30/05/2005 16:14 Page 931
132
Periodizität.
1 : 3 = 0,3.1
Man faßt die Periodizität eines Bruches, z.B. 1
/3, so auf, als bestünde2
sie darin, daß etwas,
was man die Extension des unendlichen Dezimalbruchs nennt, nur aus3
Dreien besteht, und
daß die Gleichheit des Restes dieser Division mit dem Dividenden nur das Anzeichen für
diese Eigenschaft der unendlichen Extension sei. Oder aber man korrigiert diese Meinung
dahin, daß nicht eine unendliche Extension diese Eigenschaft habe, sondern eine unendliche
Reihe endlicher Extensionen; und hierfür sei wieder die Eigenschaft der Division ein
Anzeichen.4
Man kann nun sagen: die Extension mit einem Glied sei 0,3, die mit 2 Gliedern
0,33, die mit dreien 0,333, u.s.w. Das ist eine Regel und das ”u.s.w.“ bezieht sich auf die
Regelmäßigkeit, und die Regel könnte auch geschrieben werden ”[0,3, 0,ξ, 0,ξ3]“. Das, was
aber durch die Division 1 : 3 = 0,3 bewiesen ist, ist diese Regelmäßigkeit im Gegensatz zu
1
einer andern, nicht die Regelmäßigkeit im Gegensatz zur Unregelmäßigkeit. Die periodische
Division, also 1 : 3 = 0,3 (im Gegensatz zu 1 : 3 = 0,3) beweist eine Periodizität der Quotienten,
1 1
d.h. sie bestimmt die Regel (die Periode), legt sie fest, aber ist nicht ein Anzeichen dafür, daß
eine Regelmäßigkeit ”vorhanden ist“. Wo ist sie denn vorhanden? Etwa in den bestimmten
Entwicklungen, die ich auf diesem Papier gebildet habe. Aber das sind doch nicht ”die
Entwicklungen“. (Hier werden wir irregeführt von der Idee der nicht aufgeschriebenen,
idealen Extensionen, die ein ähnliches Unding sind, wie die idealen, nicht gezogenen
geometrischen Geraden, die wir gleichsam nur in der Wirklichkeit nachziehen, wenn wir sie
zeichnen.) Wenn ich sagte ”das ,u.s.w.‘ bezieht sich auf die Regelmäßigkeit“, so unterschied
ich es von dem ”u.s.w.“ in ”er las alle Buchstaben: a, b, c, u.s.w.“. Wenn ich sage:
”die Extensionen von 1 : 3 sind 0,3, 0,33, 0,333, u.s.w.“, so gebe ich drei Extensionen und
– eine Regel. Unendlich ist nur diese, und zwar in keiner andern Weise, als die Division
1 : 3 = 0,3.
1
Von dem Zeichen ”0,3“ kann man sagen: es ist keine Abkürzung.
Und das Zeichen ”[0,3, 0,ξ, 0,ξ3]“ ist kein Ersatz für eine Extension, sondern das vollwertige
Zeichen selbst; und ebensogut ist ”0,3“. Es sollte uns doch zu denken geben, daß
ein Zeichen der Art ”0,3“ genügt, um damit zu machen, was wir brauchen. Es ist kein Ersatz,
und im Kalkül gibt es keinen Ersatz.
1 (O): 0,3. (E): geändert nach TS 212, S.
1705.
2 (V): bestehe
3 (V): nennt, aus lauter
4 (O): Anzeigen.
699
700
466
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132
Periodicity.
1 ÷ 3 = 0·3.1
We conceive of the periodicity of a fraction, for instance, of 1/3, as if it consisted in the
fact that something called the extension of the infinite decimal consists only of 2
threes; and
we hold that the sameness of remainder and dividend in this division is merely a symptom
of this property of infinite extension. Or we correct this view by saying that it isn’t an infinite
extension that has this property, but an infinite series of finite extensions; and it is of this
that the property of the division is a symptom. Now we can say: the extension taken to one
term is 0·3, to two terms 0·33, to three terms 0·333, and so on. That is a rule and the “and
so on” refers to the regularity; and the rule could also be written “|0·3, 0·ξ, 0·ξ3|” But what
is proved by the division 1 ÷ 3 = 0·3 is this regularity in contrast to another, not regularity
1
in contrast to irregularity. The periodic division 1 ÷ 3 = 0·3 (in contrast to 1 ÷ 3 = 0·3)
1 1
proves one periodicity of the quotients, i.e. it determines the rule (the period), it lays it down;
but it isn’t a symptom that a regularity “is present”. Where is it present anyway? Perhaps in
the particular expansions that I have developed on this piece of paper. But they certainly
aren’t “the expansions”. (Here we are misled by the idea of unwritten ideal extensions, which
are a kind of absurdity similar to those ideal, undrawn, geometric straight lines that we are
merely tracing, as it were, when we draw them in reality.) When I said “the ‘and so on’ refers
to the regularity” I was distinguishing it from the “and so on” in “He read all the letters of
the alphabet: a, b, c and so on”. When I say “the extensions of 1 ÷ 3 are 0·3, 0·33, 0·333 and
so on” I’m giving three extensions and – a rule. Only the latter is infinite, and it is infinite
in exactly the same way as the division 1 ÷ 3 = 0·3.
1
One can say of the sign “0·3”: it is not an abbreviation.
And the sign “|0·3, 0·ξ, 0·ξ3|” isn’t a substitute for an extension, but the full-valued
sign itself; and “0·3” is just as good. It should give us food for thought that a sign like
“0·3” suffices to do what we need. It isn’t a mere substitute, and in the calculus there are no
substitutes.
1 (O): 0·3. (E) Changed following TS 212,
p. 1705.
2 (V): consists of nothing but
466e
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Wenn man meint, die besondere Eigenschaft der Division 1 : 3 = 0,3 sei ein Anzeichen
1
für die Periodizität des unendlichen Dezimalbruchs, oder der Dezimalbrüche der Entwicklung,
so ist das ein Anzeichen dafür,5
daß etwas regelmäßig ist; aber was? Die Extensionen, die
ich gebildet habe? Aber andere gibt es ja nicht. Am absurdesten würde die Redeweise, wenn
man sagte: die Eigenschaft der Division sei ein Anzeichen dafür, daß das Resultat die Form
[0,a, 0,ξ, 0,ξa] habe; das wäre so, als wollte man sagen: eine Division ist das Anzeichen dafür,
daß eine Zahl herauskommt. Das Zeichen ”0,3“ drückt seine Bedeutung nicht von einer
größeren Entfernung aus, als ”0,333 . . .“, denn dieses Zeichen gibt eine Extension von drei
Gliedern und eine Regel; die Extension 0,333 ist für unsere Zwecke nebensächlich und so
bleibt nur die Regel, die ”[0,3, 0,ξ, 0,ξ3]“ ebensogut gibt. Der Satz ”die Division wird nach
der ersten Stelle periodisch“ heißt soviel wie: ”der erste Rest ist gleich dem Dividenden“.
Oder auch: der Satz ”die Division wird von der ersten Stelle an ins Unendliche die gleiche
Ziffer erzeugen“ heißt ”der erste Rest ist gleich dem Dividenden“; so wie der Satz ”dieses
Lineal hat einen unendlichen Radius“ heißt, es sei gerade.
Man könnte nun sagen: die Stellen eines6
Quotienten von 1 : 3 sind notwendig alle 3, und
das würde wieder nur heißen, daß der erste Rest gleich dem Dividenden ist und die erste
Stelle des Quotienten 3. Die Verneinung des ersten Satzes ist daher gleich der Verneinung
des zweiten. Es ist also dem ”notwendig alle“ nichts entgegengesetzt, was man ”zufällig alle“
nennen könnte; ”notwendig alle“ ist sozusagen ein Wort. Ich brauche nur fragen: Was ist
das Kriterium der notwendigen Allgemeinheit, und was wäre das, der zufälligen (das
Kriterium dafür also, daß zufällig alle Zahlen die Eigenschaft ε haben)?
5 (V): so heißt das, 6 (V): des
701
467 Periodizität. . . .
WBTC131-135 30/05/2005 16:14 Page 934
If you think that the special property of the division 1 ÷ 3 = 0·3 is a symptom of the
1
periodicity of the infinite decimal fraction, or of the decimal fractions of the expansion, then
that is indeed a sign3
that something is regular; but what? The extensions I’ve constructed?
But there are no others. Our way of speaking would become most absurd if we said that the
property of the division was an indication that the result has the form “|0·a, 0·ξ, 0·ξa|”;
that would be like saying that a division was an indication that the result was a number. The
sign “0·3” doesn’t express its meaning from a greater distance than “0·333 . . .”, because this
sign gives an extension of three terms and a rule; the extension 0·333 is irrelevant to our
purposes, and so all that remains is the rule, which is given just as well by “|0·3, 0·ξ, 0·ξ3|”.
The proposition “After the first place the division becomes periodic” means “The first
remainder is the same as the dividend”. Or again: the proposition “Starting with the first
place the division will produce the same number to infinity” means “The first remainder is
the same as the dividend”, just as the proposition “This ruler has an infinite radius” means
it is straight.
Now we could say: the places of a4
quotient of 1 ÷ 3 are necessarily all threes, and all that
would mean in turn would be that the first remainder is equal to the dividend and that the
first place of the quotient is 3. The negation of the first proposition is therefore equivalent
to the negation of the second. So there’s nothing in contrast to “necessarily all” that
could be called “accidentally all”; “necessarily all” is as it were one word. All I have to ask
is: What is the criterion of the necessary generality, and what would be the criterion of the
accidental generality (i.e. the criterion for all numbers accidentally having the property ε)?
3 (V): then that means 4 (V): the
Periodicity. . . . 467e
WBTC131-135 30/05/2005 16:14 Page 935
133
Der rekursive Beweis
als Reihe von Beweisen.
Der ”rekursive Beweis“ ist das allgemeine Glied einer Reihe von Beweisen. Er ist also
ein Gesetz, nach dem man Beweise konstruieren kann. Wenn gefragt wird, wie es möglich
ist, daß mir diese allgemeine Form den Beweis eines speziellen Satzes, z.B. 7 + (8 + 9) =
(7 + 8) + 9 ersparen kann, so ist die Antwort, daß sie nur alles zum Beweis dieses Satzes
vorbereitet hat, ihn aber nicht beweist (er kommt ja in ihr nicht vor). Der Beweis besteht
vielmehr aus der allgemeinen Form zusammen mit dem Satz.
Unsere gewöhnliche Ausdrucksweise trägt den Keim der Verwirrung in ihre
Fundamente, indem sie das Wort ”Reihe“ einerseits im Sinne von ”Extension“, anderseits
im Sinne von ”Gesetz“ gebraucht. Das Verhältnis der beiden kann
man sich an der Maschine klarmachen, die Schraubenfedern erzeugt.
1
Hier wird durch einen schraubenförmig gewundenen Gang ein Draht
geschoben, der nun so viele Schraubenwindungen erzeugt, als man
erzeugen will. Das, was man die unendliche Schraube nennt, ist nicht vielleicht etwas von
der Art der endlichen Drahtstücke, oder etwas, dem sich diese nähern je länger sie werden,
sondern das Gesetz der Schraube, wie es in dem kurzen Gangstück verkörpert ist. Der
Ausdruck ”unendliche Schraube“ oder ”unendliche Reihe“ ist daher irreführend.
Wir können also den rekurrierenden Beweis immer auch als Reihenstück mit dem
”u.s.w.“ anschreiben und er verliert dadurch nicht seine Strenge. Und zugleich zeigt diese
Schreibweise klarer sein Verhältnis zur Gleichung A. Denn nun verliert der rekursive Beweis
jeden Schein einer Rechtfertigung von A im Sinne eines algebraischen Beweises – etwa
von (a + b)2
= a2
+ 2ab + b2
. Dieser Beweis mit Hilfe der algebraischen Rechnungsregeln ist
vielmehr ganz analog einer Ziffernrechnung.
5 + (4 + 3) = 5 + (4 + (2 + 1))
= 5 + ((4 + 2) + 1)
= (5 + (4 + 2)) + 1
= (5 + (4 + (1 + 1))) + 1
= ( (5 + 4) + 2) + 1
= (5 + 4) + 3 . . . . . L
Das ist einerseits der Beweis von 5 + (4 + 3) = (5 + 4) + 3, anderseits kann man es als
Beweis von 5 + (4 + 4) = (5 + 4) + 4 etc. etc. gelten lassen, d.h. benützen.
Wenn ich nun sage: L ist der Beweis des Satzes a + (b + c) = (a + b) + c, so würde das
Eigentümliche am2
Übergang vom Beweis zum Satz viel auffälliger.
1 (F): MS 113, S. 121v. 2 (V): am
702
703
468
WBTC131-135 30/05/2005 16:14 Page 936
133
The Recursive Proof
as a Series of Proofs.
A “recursive proof” is the general term of a series of proofs. So it is a law for the construction
of proofs. To the question how this general form can save me the proof of a
particular proposition, e.g. of 7 + (8 + 9) = (7 + 8) + 9, the answer is that it merely has got
everything ready for the proof of this proposition, but it doesn’t prove it (indeed the proposition
doesn’t occur in it). The proof consists rather of the general form together with the
proposition.
Our usual way of speaking carries the seeds of confusion right into its foundations, by
using the word “series” in the sense of “extension” on the one hand, and in the sense of
“law” on the other. The relationship between the two can be clarified by considering a machine
for making coiled springs. 1
Here a wire is pushed through a helically shaped
tube to make as many coils as desired. What is called an infinite helix is
nothing like the finite pieces of wire, or something that they approach the
longer they become; it is the law of the helix, as it is embodied in the short
piece of tube. Therefore the expression “infinite helix” or “infinite series” is misleading.
So we can always write out the recursive proof as a limited series with “and so on” without
its thereby losing its rigour. And at the same time this notation shows its relation to the
equation A more clearly. For now the recursive proof loses all appearance of a justification
of A in the sense of an algebraic proof – like the proof of (a + b)2
= a2
+ 2ab + b2
. Rather,
this latter proof, with the help of algebraic calculation rules, is completely analogous to a
calculation with numbers.
5 + (4 + 3) = 5 + (4 + (2 + 1) )
= 5 + ((4 + 2) + 1)
= (5 + (4 + 2)) + 1
= (5 + (4 + (1 + 1))) + 1
= ((5 + 4) + 2) + 1
= (5 + 4) + 3 . . . . . L
On the one hand that is a proof of 5 + (4 + 3) = (5 + 4) + 3, but on the other hand we
can let it count, i.e. use it, as a proof of 5 + (4 + 4) = (5 + 4) + 4, etc., etc.
If I now say that L is the proof of the proposition a + (b + c) = (a + b) + c, the peculiarity
of the step from the proof to the proposition becomes much more noticeable.
1 (F): MS 113, p. 121v.
468e
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Definitionen führen nur praktische Abkürzungen ein, aber wir könnten auch ohne sie
auskommen. Aber wie ist es mit den rekursiven Definitionen?
Anwendung der Regel a + (b + 1) = (a + b) + 1 kann man zweierlei nennen: 4 + (2 + 1)
= (4 + 2) + 1 ist eine Anwendung in einem Sinne, im andern: 4 + (2 + 1) = ((4 + 1) + 1)
+ 1 = (4 + 2) + 1.
Die rekursive Definition ist eine Regel zur Bildung von Ersetzungsregeln. Oder auch
das allgemeine Glied einer Reihe von Definitionsreihen. Sie ist ein Wegweiser, der alle
Ausdrücke einer bestimmten Form auf einem Wege heimweist.
Man könnte – wie gesagt – den Induktionsbeweis ganz ohne die Benützung von
Buchstaben (mit voller Strenge) anschreiben. Die rekursive Definition a + (b + 1) = (a + b)
+ 1 müßte dann als Definitionsreihe geschrieben werden. Diese Reihe verbirgt sich nämlich
in der Erklärung ihres Gebrauchs. Man kann natürlich auch der Bequemlichkeit halber
die Buchstaben in der Definition beibehalten, muß sich aber dann in der Erklärung auf
ein Zeichen der Art ”1, (1) + 1, ((1) + 1) + 1, u.s.w.“ beziehen; oder, was auf dasselbe
hinausläuft, ”[1, ξ, ξ + 1]“. Hier darf man aber nicht etwa glauben, daß dieses Zeichen
eigentlich lauten sollte ”(ξ).[1, ξ, ξ + 1]“! –
Der Witz unserer Darstellung ist ja, daß der Begriff ”alle Zahlen“ nur durch eine Struktur
der Art ”[1, ξ, ξ + 1]“ gegeben ist. Die Allgemeinheit ist durch diese Struktur im
Symbolismus dargestellt und kann nicht durch ein (x).fx beschrieben werden.
Natürlich ist die sogenannte ”rekursive Definition“ keine Definition im hergebrachten
Sinne des Worts, weil keine Gleichung. Denn die Gleichung ”a + (b + 1) = (a + b) + 1“
ist nur ein Bestandteil von ihr. Noch ist sie das logische Produkt von Gleichungen. Sie ist
vielmehr ein Gesetz, wonach Gleichungen gebildet werden; wie [1, ξ, ξ + 1] keine Zahl
ist, sondern ein Gesetz etc. (Das Überraschende3
am Beweis von a + (b + c) = (a + b) + c
ist ja, daß er aus einer Definition allein hervorgehen soll. Aber α ist keine Definition,
sondern eine allgemeine Additionsregel.)
Anderseits ist die Allgemeinheit dieser Regel keine andere, als die der periodischen
Division 1 : 3 = 0,3. D.h. es ist in der Regel nichts offen gelassen, ergänzungsbedürftig
1
oder dergleichen.
Und vergessen wir nicht: Das Zeichen
”[1, ξ, ξ + 1]“ . . . N
interessiert uns nicht als ein suggestiver Ausdruck des allgemeinen Gliedes der
Kardinalzahlenreihe, sondern nur, sofern es mit analog gebauten Zeichen in Gegensatz
tritt: N im Gegensatz zu, etwa, [2, ξ, ξ + 3]; kurz als Zeichen, als Instrument, in einem
Kalkül. Und das Gleiche gilt natürlich von 1 : 3 = 0,3. (Offen gelassen wird in der Regel
1
nur ihre Anwendung.)
1 + (1 + 1) = (1 + 1) + 1, 2 + (1 + 1) = (2 + 1) + 1, 3 + (1 + 1) = (3 + 1) + 1 . . . u.s.w.
1 + (2 + 1) = (1 + 2) + 1, 2 + (2 + 1) = (2 + 2) + 1, 3 + (2 + 1) = (3 + 2) + 1 . . . u.s.w.
1 + (3 + 1) = (1 + 3) + 1, 2 + (3 + 1) = (2 + 3) + 1, 3 + (3 + 1) = (3 + 3) + 1 . . . u.s.w.
u.s.w.
So könnte man die Regel ”a + (b + 1) = (a + b) + 1“ anschreiben.
3 (V): Verblüffende
704
705
469 Der rekursive Beweis als Reihe von Beweisen.
WBTC131-135 30/05/2005 16:14 Page 938
Definitions merely introduce practical abbreviations, and so we could get along without
them. But how about recursive definitions?
Two different things can be called applications of the rule a + (b + 1) = (a + b) + 1:
in one sense 4 + (2 + 1) = (4 + 2) + 1 is an application, in another sense 4 + (2 + 1) =
((4 + 1) + 1) + 1 = (4 + 2) + 1 is.
A recursive definition is a rule for constructing replacement rules. But it’s also the general
term of a number of series of definitions. It’s a signpost that shows all expressions of a
certain form the same way home.
As we said, we could write the inductive proof without using letters at all (with no loss
of rigour). Then the recursive definition a + (b + 1) = (a + b) + 1 would have to be written
as a series of definitions. For this series is concealed in the explanation of its use. Of course
we can also keep the letters in the definition for the sake of convenience, but in that case we
have to refer in the explanation to a sign like “1, (1) + 1, ((1) + 1) + 1 and so on”, or – what
boils down to the same thing – “|1, ξ, ξ + 1|”. But here we mustn’t believe perchance that
this sign should really be “(ξ).|1, ξ, ξ + 1|”! –
The point of our formulation is of course that the concept “all numbers” has been given
only by a structure like “|1, ξ, ξ + 1|”. The generality has been represented by this structure
in the symbolism and cannot be described by an (x).fx.
Of course the so-called “recursive definition” isn’t a definition in the customary sense of
the word, because it isn’t an equation. For the equation “a + (b + 1) = (a + b) + 1” is only
a part of it. Nor is it a logical product of equations. Rather, it is a law for the construction
of equations; just as |1, ξ, ξ + 1| isn’t a number, but a law etc. (The amazing2
thing about
the proof of a + (b + c) = (a + b) + c is that it’s supposed to emerge from a definition alone.
But α isn’t a definition, but a general rule for addition.)
On the other hand the generality of this rule is none other than that of the periodic
division 1 ÷ 3 = 0·3. That means that nothing in the rule is left open or is in need of
1
completion or the like.
And let’s not forget: the sign
“|1, ξ, ξ + 1|” . . . . . N
doesn’t interest us as a suggestive expression of the general term of the series of cardinal
numbers, but only in so far as it is contrasted with signs that are similarly constructed:
N as opposed to, say, |2, ξ, ξ + 3|; in short, it interests us as a sign, an instrument, in a
calculus. And of course the same holds for 1 ÷ 3 = 0·3. (The only thing left open in the
1
rule is its application.)
1 + (1 + 1) = (1 + 1) + 1, 2 + (1 + 1) = (2 + 1) + 1, 3 + (1 + 1) = (3 + 1) + 1 . . . and so on
1 + (2 + 1) = (1 + 2) + 1, 2 + (2 + 1) = (2 + 2) + 1, 3 + (2 + 1) = (3 + 2) + 1 . . . and so on
1 + (3 + 1) = (1 + 3) + 1, 2 + (3 + 1) = (2 + 3) + 1, 3 + (3 + 1) = (3 + 3) + 1 . . . and so on
and so on.
That’s the way we could write the rule “a + (b + 1) = (a + b) + 1”.
2 (V): astounding
The Recursive Proof as a Series of Proofs. 469e
WBTC131-135 30/05/2005 16:14 Page 939
. . . R
4
In der Anwendung der Regel R, deren Beschreibung ja zu der Regel selbst als ein Teil ihres
Zeichens gehört, läuft a der Reihe [1, ξ, ξ + 1] entlang und das könnte natürlich durch ein
beigefügtes Zeichen, etwa ”a → N“ angegeben werden. (Die zweite und dritte Zeile der Regel
R könnte man zusammen die Operation nennen, wie das zweite und dritte Glied des Zeichens
N.) So ist auch die Erläuterung zum Gebrauch der rekursiven Definition α ein Teil dieser
Regel selber; oder auch eine Wiederholung ebenderselben5
Regel in andrer Form: sowie
”1, 1 + 1, 1 + 1 + 1, u.s.w.“ das gleiche bedeutet, wie (d.h. übersetzbar ist in) ”[1, ξ, ξ + 1]“.
Die Übersetzung in die Wortsprache erklärt den Kalkül mit den neuen Zeichen, da wir den
Kalkül mit den Zeichen der Wortsprache schon beherrschen.
Das Zeichen einer Regel ist ein Zeichen eines Kalküls wie jedes andere; seine Aufgabe ist
nicht, suggestiv (auf eine Anwendung hin) zu wirken, sondern, im Kalkül nach einem System6
gebraucht zu werden. Daher ist die äußere Form, wie die eines Pfeiles 6 nebensächlich,
wesentlich aber das System, worin das Regelzeichen verwendet wird. Das System von
Gegensätzen – sozusagen – wovon7
das Zeichen sich unterscheidet, etc.
Das, was ich hier die Beschreibung der Anwendung nenne, enthält ja selbst ein ”u.s.w.“,
kann also nur eine Ergänzung oder ein Ersatz des Regelzeichens selbst sein.
Was ist nun der Gegensatz eines allgemeinen Satzes, wie a + (b + (1 + 1)) = a + ((b + 1)
+ 1)? Welches ist das System von Sätzen, innerhalb dessen dieser Satz8
verneint wird? Oder
auch: wie, in welcher Form, kann dieser Satz mit andern in Widerspruch geraten? Oder:
Welche Frage beantwortet er? Gewiß nicht9
die, ob (n).fn oder (∃n).~fn der Fall ist, etc.
Die Allgemeinheit einer Regel kann eo ipso nicht in Frage gestellt werden.10
Denken wir uns nun den allgemeinen Satz als Reihe geschrieben
p11, p12, p13, . . . .
p21, p22, p23, . . . .
p31, p32, p33, . . . .
. . . . . . . . . . . . . .
und verneint. Wenn wir ihn als (x).f(x) auffassen, so betrachten wir ihn als logisches Produkt11
und sein Gegenteil ist die logische Summe der Verneinungen von p11, p12, etc. Diese
Disjunktion (nun) ist mit jedem beliebigen Produkt p11 & p21 & p22 & p12 . . . pmn vereinbar.
(Gewiß, wenn man den Satz mit einem logischen Produkt vergleicht, so wird er unendlich
vielsagend und sein Gegenteil nichtssagend.) (Bedenke aber: das ”u.s.w.“ steht im Satz nach
einem Beistrich, nicht nach einem ”und“ (”&“). Das ”u.s.w.“ ist kein Zeichen ihrer
Unvollständigkeit.)
Ist denn die Regel R unendlich vielsagend? wie ein ungeheuer langes logisches Produkt?
Daß man die Zahlenreihe durch die Regel laufen läßt, ist eine gegebene Form; darüber
wird nichts behauptet und kann nichts verneint werden.
4 (F): MS 113, S. 139r.
5 (V): der
6 (V): nach Gesetzen
7 (V): – sozusagen – von denen // worin
8 (V): dessen diese Regel
9 (V): er? Nicht
10 (V): Oder: welche Frage kann er beantworten,
zwischen welchen Alternativen entscheiden? –
Nicht zwischen einer ”(n)fn“ und einer
”(∃n)~fn“; denn die Allgemeinheit ist dem
Satz von der Regel R zugebracht. Sie kann
ebensowenig in Frage gestellt // gezogen //
werden, wie das System der Kardinalzahlen.
11 (V): so ist er ein logisches Produkt
706
707
706
707
470 Der rekursive Beweis als Reihe von Beweisen.
a
a
a
(a
(a
a
( )
( )
(( ) )
)
)
(( ) )
+ +
↓
+ +
+ + +
= + +
↓
+ +
+ + +
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
1 1
1
1 1
1 1
1
1 1
ξ
ξ
ξ
ξ
WBTC131-135 30/05/2005 16:14 Page 940
. . . R
3
In the application of the rule R (whose description is, of course, an inherent part of the
rule itself, since it is a part of its sign), a ranges over the series |1, ξ, ξ + 1|; and of course
that could be expressly indicated with an additional sign, say “a → N”. (We might call the
second and third lines of the rule R taken together “the operation”, like the second and third
terms of the sign N.) Thus the elucidation of the use of the recursive definition α is also
a part of that rule itself; or, if you like, a repetition of the very same rule4
in another
form; just as “1, 1 + 1, 1 + 1 + 1 and so on” means the same thing as (i.e. is translatable into)
“|1, ξ, ξ + 1|”. The translation into word-language explains the calculus with the new signs,
because we have already mastered the calculus with the signs of word-language.
The sign of a rule is a sign of a calculus, like any other sign; its job isn’t to suggest
(a particular application), but to be used in the calculus according to a system.5
Therefore
the exterior form, like that of an arrow 6, is unimportant. What is essential, though, is the
system in which the sign for the rule is employed. The system of contraries – so to speak –
from6
which the sign is distinguished, etc.
What I am here calling the description of the application contains an “and so on” itself,
and so it can be no more than a supplement to, or substitute for, the sign for the rule itself.
So what is the contradictory of a general proposition like a + (b + (1 + 1)) = a + ((b + 1)
+ 1)? What is the system of propositions within which this proposition7
is negated? Or
again, how, in what form, can this proposition come into contradiction with others? Or:
What question does it answer? Certainly not8
the question whether (n).fn or (∃n).~fn is
the case, etc. The generality of a rule is eo ipso incapable of being called into question.9
Now let’s imagine the general proposition written as a series:
p11, p12, p13, . . . .
p21, p22, p23, . . . .
p31, p32, p33, . . . .
. . . . . . . . . . . . . .
and then negated. If we understand it as (x).fx, then we are treating it as a10
logical product,
and its opposite is the logical sum of the negations of p11, p12 etc. (Now) this disjunction is
consistent with any random product p11 & p21 & p22 & p12 . . . pmn. (Certainly if you compare
the proposition with a logical product, it becomes infinitely significant and its opposite void
of significance.) (But consider this: the “and so on” in the proposition comes after a comma,
not after an “and” (“&”). The “and so on” is not a sign of their incompleteness.)
Is the rule R really infinitely significant? Like an enormously long logical product?
That you can let the number series run through the rule is a form that is given; nothing
is affirmed about this and nothing can be denied.
3 (F): MS 113, p. 139r.
4 (V): the rule
5 (V): calculus according to laws.
6 (V): within
7 (V): rule
8 (V): answer? Not
9 (V): others? Or: which question can it answer?
Between which alternatives can it decide? –
Not between “(n)fn” or “(∃n)~fn”; for generality
is given to the proposition by the rule R.
It can no more be called into question than the
system of cardinal numbers.
10 (V): then it is a
The Recursive Proof as a Series of Proofs. 470e
a
a
a
(a
(a
a
( )
( )
(( ) )
)
)
(( ) )
+ +
↓
+ +
+ + +
= + +
↓
+ +
+ + +
⎡
⎣
⎢
⎢
⎢
⎢
⎤
⎦
⎥
⎥
⎥
⎥
1 1
1
1 1
1 1
1
1 1
ξ
ξ
ξ
ξ
WBTC131-135 30/05/2005 16:14 Page 941
Das Durchleiten des Zahlenstromes ist ja nichts, wovon ich sagen kann, ich könne es
beweisen. Beweisen kann ich nur etwas über die Form, den Model, durch den ich den
Zahlenstrom leite.
Kann man nun nicht sagen, daß die allgemeine Zahlenregel
a + (b + c) = (a + b) + c . . . A
eben die Allgemeinheit hat wie a + (1 + 1) = (a + 1) + 1 (indem diese für jede Kardinalzahl,
jene für jedes Kardinalzahlentrippel gilt); und daß der Induktionsbeweis12
von A die Regel
A rechtfertigt? Daß wir also die Regel A geben dürfen, weil der Beweis zeigt, daß sie immer
stimmt?
Rechtfertigt 1 : 3 = 0,3 die Regel
1
. . . P
A ist eine vollkommen verständliche Regel; so wie die Ersetzungsregel P. Eine solche Regel
kann ich aber darum nicht geben, weil ich die einzelnen Fälle von A schon durch eine andere
Regel berechnen kann, wie ich P nicht als Regel geben kann, wenn ich eine Regel gegeben
habe, mit der ich , etc. berechnen kann.
Wie wäre es, wenn man außer den Multiplikationsregeln noch ”25 × 25 = 625“ als Regel
festsetzen wollte? (Ich sage nicht ”25 × 25 = 624“!) – 25 × 25 = 625 hat nur Sinn, wenn die
Art der Rechnung13
bekannt ist, die zu dieser Gleichung gehört, und hat nur Sinn in Bezug
auf diese Rechnung. A hat nur Sinn mit Bezug auf die Art der Ausrechnung von A. Denn
die erste Frage wäre hier eben: ist das eine Festsetzung,14
oder ein errechneter Satz? Denn
ist 25 × 25 = 625 eine Festsetzung (Grundregel), dann bedeutet das Multiplikationszeichen
etwas anderes, als es z.B. in Wirklichkeit bedeutet. (D.h. wir haben es mit einer andern
Rechnungsart zu tun.) Und ist A eine Festsetzung, dann definiert das die Addition anders,
als wenn es ein errechneter Satz ist. Denn die Festsetzung ist ja dann eine Erklärung des
Additionszeichens und die Rechenregeln,15
die A auszurechnen erlauben, eine andere
Erklärung desselben Zeichens. Ich darf hier nicht vergessen, daß α, β, γ nicht der Beweis
von A ist, sondern nur die Form des Beweises, oder des Bewiesenen ist; α, β, γ definiert
also A.
Darum kann ich nur sagen ”25 × 25 = 625 wird bewiesen“, wenn die Beweismethode
fixiert ist, unabhängig von dem speziellen Beweis. Denn diese Methode bestimmt erst die
Bedeutung von ”ξ · η“, also, was bewiesen wird. Insofern gehört also die Form a
a: b = c
zur Beweismethode, die den Sinn von 5 erklärt. Etwas anderes ist dann die Frage, ob ich
richtig gerechnet habe. – Und so gehört α, β, γ zur Beweismethode, die den Sinn des Satzes
A erklärt.
Die Arithmetik ist ohne eine Regel A vollständig, es fehlt ihr nichts. Der Satz A wird
(nun) mit Entdeckung einer Periodizität, mit der Konstruktion eines neuen Kalküls, in
die Arithmetik eingeführt. Die Frage nach der Richtigkeit dieses Satzes hätte vor dieser
Entdeckung (oder Konstruktion) so wenig Sinn, wie die Frage nach der Richtigkeit von
. . . ad inf.“.”1 3 0 3 1 3 0 33
1 2
: , , : , ,= =
”1 3 0 3 1 3 0 33 1 3 0 333
1 2 3
: , , : , , : , , “?= = = u.s.w.
12 (V): der rekursive Beweis
13 (V): Ausrechnung
14 (V): Bestimmung,
15 (V): Rechenregel,
708
709
471 Der rekursive Beweis als Reihe von Beweisen.
1 3 0 3
1
: ,=
WBTC131-135 30/05/2005 16:14 Page 942
After all, running the stream of numbers through is not something I can say I can prove.
I can only prove something about the form, the pattern, through which I run that stream.
Now can’t we say that the general number rule
a + (b + c) = (a + b) + c . . . A
has the very same generality as a + (1 + 1) = (a + 1) + 1 (in that the latter holds for every
cardinal number and the former for every triple of cardinal numbers); and that the induc-
tive11
proof of A justifies the rule A? Can’t we say that we are thus allowed to give the rule
A, because the proof shows that it is always right?
Does 1 ÷ 3 = 0·3 justify the rule
1
. . . P
A is a completely intelligible rule; just like the replacement rule P. But I can’t give
such a rule, because I can already calculate the particular instances of A using another rule;
just as I cannot give P as a rule if I have given a rule with which I can calculate
= 0·3, etc.
What would it be like if someone wanted to lay down “25 × 25 = 625” as a rule in
addition to the multiplication rules? (I’m not saying “25 × 25 = 624”!) – 25 × 25 = 625
only makes sense if the kind of calculation that belongs to this equation is already known,
and it only makes sense with reference to that calculation. A only makes sense with reference
to how A is figured out. For the first question here would be: Is that a stipulation, or
a proposition that’s been calculated? For if 25 × 25 = 625 is a stipulation (foundational rule),
then the multiplication sign means something different than it does, for example, in reality.
(That is, we are dealing with a different kind of calculation.) And if A is a stipulation, then
that defines addition differently than if it is a proposition that has been calculated. For
in that case the stipulation is of course a definition of the addition sign, and the rules of
calculation that allow12
A to be figured out are a different definition of the same sign. Here
I mustn’t forget that α, β, γ isn’t the proof of A, but only the form of the proof, or of what
has been proved; so α, β, γ is a definition of A.
Therefore I can only say “25 × 25 = 625 is proved” if the method of proof is fixed
independently of the specific proof. For it is this method that determines the meaning of
“ξ · η”, i.e. determines what is proved. So to that extent the form a
a÷ b = c belongs to the
method of proof that explains the sense of 5. It’s another question whether I’ve calculated
correctly. – And thus α, β, γ belong to the method of proof that explains the sense of the
proposition A.
Arithmetic is complete without a rule like A; without it it doesn’t lack anything. The
proposition A is introduced into arithmetic with the discovery of periodicity and the
construction of a new calculus. Before this discovery (or construction) a question about
the correctness of that proposition would have had as little sense as a question about the
correctness of . . . ad inf.”.“ , ,1 3 0 3 1 3 0 33
1 2
÷ = ⋅ ÷ = ⋅
“1 3 0 3 1 3 0 33 1 3 0 333
1 2 3
, , , ”?÷ = ⋅ ÷ = ⋅ ÷ = ⋅ and so on
11 (V): recursive 12 (V): the rule of calculation that allows
The Recursive Proof as a Series of Proofs. 471e
1 3
1
÷
WBTC131-135 30/05/2005 16:14 Page 943
Nun ist die Festsetzung P verschieden vom Satz ”1 : 3 = 0,3“ und in diesem Sinne ist
”a + (b + 5) = (a + b) + 5“ verschieden von einer Regel (Festsetzung) A. Die beiden gehören
andern Kalkülen an. Der Beweis, die Rechtfertigung, einer Regel A ist der Beweis von α,
β, γ16
nur insofern,17
als er die allgemeine Form der Beweise arithmetischer Sätze von der
Form A ist.
Die Periodizität ist nicht das Anzeichen (Symptom) dafür, daß es so weitergeht, aber der
Ausdruck ”so geht es immer weiter“ ist nur eine Übersetzung in eine andere Ausdrucksweise
des periodischen Zeichens.18
(Gäbe es außer dem periodischen Zeichen noch etwas, wofür
die Periodizität nur ein Symptom ist, so müßte dieses Etwas einen spezifischen Ausdruck
haben, der nichts anderes wäre, als der vollständige Ausdruck dieses Etwas.)
16 (V): γ nu
17 (V): Der Beweis, die Rechtfertigung, einer
Ersetzungsregel A ist der rekursive Beweis nur
insofern,
18 (V): Ausdrucksweise der Periodizität des
Zeichens.
472 Der rekursive Beweis als Reihe von Beweisen.
WBTC131-135 30/05/2005 16:14 Page 944
The stipulation of P is different from the proposition 1 ÷ 3 = 0·3, and in that sense
“a + (b + 5) = (a + b) + 5” is different from a rule (stipulation) such as A. The two belong
to different calculi. The proof, the justification of a rule like A, is the proof of α, β, γ13
only
in so far as it is the general form of the proofs of arithmetical propositions of the form A.
Periodicity is not an indication (symptom) that it goes on and on like that; rather, the
expression “it goes on like that forever” is only a translation of the sign for periodicity14
into
another form of expression. (If there were something other than the periodic sign of which
periodicity was only a symptom, that something would have to have a specific expression,
which would be nothing less than the complete expression of that something.)
13 (V): justification of a substitution proof A is a
recursive proof
14 (V): translation of the periodicity of the sign
The Recursive Proof as a Series of Proofs. 472e
WBTC131-135 30/05/2005 16:14 Page 945
134
Ein Zeichen auf bestimmte
Weise sehen, auffassen.
Entdecken eines Aspekts eines
mathematischen Ausdrucks.
”Den Ausdruck in bestimmter
Weise sehen“.
Hervorhebungen.
Ich sprach früher von Verbindungsstrichen, Unterstreichungen, etc. um die korrespondierenden,
homologen, Teile der Gleichungen eines Rekursionsbeweises zu zeigen.
Im Beweis
1
entspricht z.B. die Eins α nicht der β sondern dem c der nächsten Gleichung; β aber entspricht
nicht δ, sondern dem ε; und γ nicht dem δ sondern dem c + δ, etc.
Oder in:
2
entspricht nicht ι dem κ und ε dem λ, sondern ι dem α und ε dem β; und nicht β dem ζ,
aber ζ dem θ und α dem δ und β dem γ und γ dem µ, aber nicht dem θ, u.s.w.
Wie verhält es sich mit einer Rechnung wie:
(5+3)2
= (5+3)(5+3) = 5(5+3) + 3(5+3) = 5×5 + 5×3 + 3×5 + 3×3 = 52
+ 2×5×3 + 32
. . . R
1 (F): MS 112, S. 10r. 2 (F): MS 112, S. 10r.
710
711
a + (b + 1) = (a + b) + 1
γ α
a + (b + (c + 1) ) = (a + (b + c) ) + 1
δ β
(a + b) + (c + 1) = ( (a + b) + c) + 1
ξ ε
1
4
2
4
3
(a + 1) + 1 = (a + 1) + 1
ι
1 + (a + 1) = (1 + a) + 1
µ
δ
η θ
α
β ε ζ
γ
χ λ
473
WBTC131-135 30/05/2005 16:14 Page 946
134
Seeing and Understanding a Sign
in a Particular Way.
Discovering an Aspect of
a Mathematical Expression.
“Seeing an Expression
in a Particular Way.”
Marks of Emphasis.
Earlier I spoke of the use of connecting lines, underlinings, etc. to show the corresponding,
homologous, parts of the equations of a recursive proof. In the proof
1
the 1 marked α, for example, doesn’t correspond to the 1 marked β but to c in the next
equation; and β corresponds not to δ but to ε; and γ not to δ but to c + δ, etc.
Or in
ι 2
doesn’t correspond to κ and ε doesn’t correspond to λ; rather, ι corresponds to α and ε
to β; and β doesn’t correspond to ζ, but ζ corresponds to θ, and α to δ and β to γ and γ to
µ, not to θ, and so on.
What about a calculation like:
(5+3)2
= (5+3)(5+3) = 5(5+3) + 3(5+3) = 5×5 + 5×3 + 3×5 + 3×3 = 52
+ 2×5×3 + 32
. . . R
1 (F): MS 112, p. 10r. 2 (F): MS 112, p. 10r.
a + (b + 1) = (a + b) + 1
γ α
a + (b + (c + 1) ) = (a + (b + c) ) + 1
δ β
(a + b) + (c + 1) = ( (a + b) + c) + 1
ξ ε
1
4
2
4
3
(a + 1) + 1 = (a + 1) + 1
ι
1 + (a + 1) = (1 + a) + 1
µ
δ
η θ
α
β ε ζ
γ
χ λ
473e
WBTC131-135 30/05/2005 16:14 Page 947
aus welcher wir auch eine allgemeine Regel des Quadrierens eines Binoms herauslesen
können?
Wir können diese Rechnung sozusagen arithmetisch und algebraisch auffassen.3
Und dieser Unterschied in der Auffassung träte4
z.B. zu Tage, wenn das Beispiel gelautet
hätte 5
und wir nun in der algebraischen Auffassung die 2 an den
Stellen β einerseits, und an der Stelle α anderseits unterscheiden mußten, während sie in
der arithmetischen Auffassung nicht zu unterscheiden wären. Wir betreiben eben – glaube
ich – beide Male einen andern Kalkül.
Nach der einen Auffassung wäre z.B. die vorige6
Rechnung ein Beweis von
(7+8)2
= 72
+ 2×7×8 + 82
, nach der anderen nicht.
Wir könnten ein Beispiel rechnen, um uns zu vergewissern, daß (a + b)2
gleich
a2
+ b2
+ 2ab und nicht a2
+ b2
+ 3ab ist – wenn wir es etwa vergessen hätten; aber wir
könnten nicht in diesem Sinn kontrollieren, ob die Formel allgemein gilt. Auch diese
Kontrolle gibt es natürlich und ich könnte in der Rechnung
(5 + 3) 2
= . . . = 52
+ 2×5×3 + 32
nachsehen, ob die 2 im zweiten Glied ein allgemeiner Zug der Gleichung ist oder einer, der
von den speziellen Zahlen des Beispiels abhängt.
Ich mache (5 + 2) 2
= 52
+ 2×2×5 + 22
zu einem andern Zeichen, indem ich schreibe:
und dadurch ”andeute, welche Züge der rechten Seite von den besonderen Zahlen der linken
herrühren“, etc.
(Ich erkenne jetzt die Wichtigkeit dieses Prozesses der Zuordnung. Er ist der Ausdruck
einer neuen Betrachtung der Rechnung und daher der7
Betrachtung einer neuen Rechnung.)
Ich muß, um ”A zu beweisen“, erst – wie man sagen würde – die Aufmerksamkeit auf
ganz bestimmte Züge von8
B lenken.9
(Wie in der Division 1,0 : 3 = 0,3.)
1
(Und von dem, was ich dann sehe, hatte das α sozusagen noch gar keine Ahnung.)
Es verhält sich hier zwischen Allgemeinheit und Beweis der Allgemeinheit, wie zwischen
Existenz und Existenzbeweis.
Wenn α, β, γ bewiesen sind, muß der allgemeine Kalkül erst erfunden werden.
Es kommt uns ganz selbstverständlich vor, auf die Induktionsreihe hin ”a + (b + c) =
(a + b) + c“ zu schreiben; weil wir nicht sehen, daß wir damit einen ganz neuen Kalkül
beginnen. (Ein Kind, das gerade rechnen lernt, würde in dieser Beziehung klarer sehen
als wir.)
(5 + 2)2
= 52
+ 2·2·5 + 22
α β α β α β
( )5 2 5 2 25 2+ = + ⋅ ⋅ +2 2 2
α β β
3 (V): ansehen.
4 (O): träge
5 (F): MS 112, S. 25r.
6 (V): obige
7 (V): die
8 (V): in
9 (V): auf etwas ganz Bestimmtes richten.
712
713
474 Ein Zeichen auf bestimmte Weise sehen . . .
WBTC131-135 30/05/2005 16:14 Page 948
Can we also infer a general rule for the squaring of a binomial from this?
We can understand3
this calculation as it were arithmetically or algebraically.
And this difference between the two ways of looking at it would come out, for example,
if the example had been . 4
And in the algebraic way of looking
at it we’d have to distinguish the 2’s in the positions marked β, on the one hand, from the
2 in the position marked α, on the other, whereas in the arithmetical way they needn’t be
distinguished. We are – I believe – simply employing a different calculus in each case.
According to the one, but not the other, way of looking at it, the previous calculation5
would for instance be a proof of (7+8)2
= 72
+ 2×7×8 + 82
.
We could work out an example to make sure that (a + b)2
is equal to a2
+ b2
+ 2ab,
not to a2
+ b2
+ 3ab – if we had forgotten it, for instance; but we couldn’t check in that
sense whether the formula holds generally. Of course there is that sort of check too, and in
the calculation
(5 + 3)2
= . . . = 52
+ 2×5×3 + 32
I could check whether the 2 in the second term is a general feature of the equation or
something that depends on the particular numbers occurring in this example.
I turn (5 + 2)2
= 52
+ 2×2×5 + 22
into a different sign by writing
and thus “indicating which features of the right-hand side originate from the particular
numbers on the left”, etc.
(Now I recognize the importance of this process of coordination. It expresses a new way
of looking at the calculation and therefore a way of looking6
at a new calculation.)
In order to “prove A” I must first of all – as we would say – direct attention to very
particular features of7
B.8
(As in the division 1·0 ÷ 3 = 0·3.)
1
(And α had absolutely no inkling, so to speak, of what I see when I do.)
Here the relationship between generality and proof of generality is like the relationship
between existence and proof of existence.
When α, β, γ have been proved, the general calculus still has to be invented.
Writing “a + (b + c) = (a + b) + c” in the induction series seems to us a completely
obvious thing to do, because we fail to see that in doing so we are starting a totally new
calculus. (A child just learning to calculate would see more clearly than we do in this
respect.)
(5 + 2)2
= 52
+ 2·2·5 + 22
α β α β α β
( )5 2 5 2 25 2+ = + ⋅ ⋅ +2 2 2
α β β
3 (V): can look at
4 (F): MS 112, p. 25r.
5 (V): the calculation above
6 (V): therefore is the looking
7 (V): in
8 (V): attention to something quite specific.
Seeing and Understanding a Sign in a Particular Way. . . . 474e
WBTC131-135 30/05/2005 16:14 Page 949
Die Hervorhebungen geschehen durch das Schema R und könnten so ausschauen:
10
Es hätte aber natürlich auch genügt (d.h. wäre ein Symbol derselben Multiplizität gewesen)
B anzuschreiben und dazu:
f1ξ = a + (b + ξ), f2ξ = (a + b) + ξ.
(Und dabei ist wieder zu bedenken,11
daß jedes Symbol – wie explizit auch immer –
mißverstanden werden kann. – )
Wer etwa zuerst darauf aufmerksam macht, daß B so gesehen werden kann, der führt ein
neues Zeichen ein; ob er nun die Hervorhebungen mit B verbindet oder auch das Schema
R daneben schreibt. Denn dann ist eben R das neue Zeichen. Oder, wenn man will, auch B
zusammen mit R. Die Weise, wie er darauf aufmerksam gemacht hat, gibt das neue Zeichen.
Man könnte etwa sagen: Hier wurde die untere Gleichung als a + b = b + a gebraucht;
und analog: hier wurde B als A gebraucht, wobei B aber gleichsam der Quere nach
gelesen wurde. Oder: B wurde als A gebraucht, aber das neue Zeichen12
wird aus α & β &
γ so zusammengestellt, daß, indem man nun A aus B herausliest, α & β & γ nicht in jener
Art von Verkürzung erscheint,13
in der man die Prämisse im Folgesatz vor sich hat.14
Was heißt es nun: ”ich mache Dich drauf aufmerksam, daß hier in beiden Funktionszeichen
das gleiche Zeichen15
steht (vielleicht hast Du es nicht bemerkt)“? Heißt das, daß
er den Satz nicht verstanden hatte? – Und doch hat er etwas nicht bemerkt, was wesentlich
zum Satz gehörte; nicht etwa (so), als hätte er eine externe Eigenschaft des Satzes nicht bemerkt.
(Hier sieht man wieder, welcher Art das ist, was man ”verstehen eines Satzes“ nennt.)
Der Vergleich vom längs und quer Durchlaufen ist wieder ein logisches Bild und darum
nicht als unverbindliches Gleichnis über die Achsel anzusehen,16
sondern ein korrekter
Ausdruck einer grammatischen Tatsache.17
Wenn ich sagte, das neue Zeichen mit den Hervorhebungen müsse ja doch aus dem
alten ohne die Hervorhebungen abgeleitet sein,18
so heißt das nichts, weil ich ja das Zeichen
mit den Hervorhebungen abgesehen von seiner Entstehung betrachten kann. Es stellt
sich mir dann (Frege) dar, als drei Gleichungen, d.h., als die Figur dreier Gleichungen mit
gewissen Unterstreichungen etc.
10 (F): MS 112, S. 41r.
11 (V): anzumerken,
12 (V): aber die neue Gleichung // der neue Satz
13 (V): so zusammengestellt, daß, indem man nun
A aus B herausliest, man nicht α & β & γ in jener
Art von Verkürzung liest,
14 (V): im Folgesatz liest.
15 (V): Argument
16 (V): und darum nicht ein unverbindliches
Gleichnis,
17 (V1): Ausdruck eines grammatischen
Verhältnisses. (V2): Das Bild vom längs
und quer Durchlaufen ist natürlich wieder ein
logisches Bild und darum ein ganz exakter
Ausdruck eines grammatischen Verhältnisses.
Es ist also nicht davon zu sagen: ”das ist ein
bloßes Gleichnis, wer weiß, wie es sich in der
Wirklichkeit verhält“.
18 (V): Hervorhebungen entstehen,
714
715
714
475 Ein Zeichen auf bestimmte Weise sehen . . .
a + (b + 1) = (a + b) + 1
f1 (1)
a + (b + (c + 1) ) = |a + (b + c)| + 1
(a + b) + (c + 1) = |(a + b) + c| + 1
f2 (1)
f1 (c + 1) f1 (c) + 1
f2 (c + 1) f2 (c) + 1
5
4
4
6
4
4
7
WBTC131-135 30/05/2005 16:14 Page 950
Emphases are brought out by the schema R, and they might look like this:
9
Of course it would also have been enough (i.e. it would have been a symbol of the same
multiplicity) if we had written B and added
f1ξ = a + (b + ξ), f2ξ = (a + b) + ξ.
(And here again we must bear in mind10
that every symbol – however explicit – can be
misunderstood. – )
Whoever first draws attention to the fact that B can be seen in that way introduces a new
sign, whether he attaches the marks of emphasis to B or writes the schema R beside it. For
then it is simply R that is the new sign. Or, if you prefer, B together with R. It is the way
in which he has drawn attention to this that produces the new sign.
We could say, for instance, that here the lower equation was used as a + b = b + a;
and analogously that here B – being read crosswise as it were – was used as A. Or that B
was used as A, but the new sign11
was put together from α & β & γ, in such a way that in
now reading A out of B, α & β & γ doesn’t appear in the sort of abbreviation in which the
premiss turns up in the conclusion.12
What does it mean to say: “I am drawing your attention to the fact that the same sign13
occurs here in both function signs (perhaps you didn’t notice it)”? Does that mean that
he hadn’t understood the proposition? – For what he failed to notice was an essential
part of the proposition; (so) it’s not as if he just hadn’t noticed some external property of
the proposition. (Here again we see what kind of thing it is that is called “understanding a
proposition”.)
The simile of reading a sign lengthways and sideways is once again a logical image, and
for that reason it must not be looked down upon as an arbitrary comparison; rather14
, it’s a
correct expression of a grammatical fact.15
When I said that the new sign with the marks of emphasis had to be derived16
from the
old one without the marks, that means nothing, because of course I can think about the
sign with the marks without regard to its origin. In that case it presents itself to me as three
equations (Frege), that is, in the form of three equations with certain underlinings, etc.
9 (F): MS 112, p. 41r.
10 (V): must note
11 (V): the new equation // the new proposition
12 (V): in now reading A out of B, one doesn’t
read α & β & γ in the sort of abbreviation in
which one finds // reads // the premiss in the
conclusion.
13 (V): argument
14 (V): image, and therefore not an arbitrary
comparison, but
15 (V1): of a grammatical relationship. (V2):
The simile of reading a sign lengthways and
sideways is once again a logical image, and for
that reason is a completely exact expression of
a grammatical relationship. Therefore one cannot
say of it: “That is a mere simile, who knows
how things are in reality?”
16 (V): had to arise
Seeing and Understanding a Sign in a Particular Way. . . . 475e
a + (b + 1) = (a + b) + 1
f1 (1)
a + (b + (c + 1) ) = |a + (b + c)| + 1
(a + b) + (c + 1) = |(a + b) + c| + 1
f2 (1)
f1 (c + 1) f1 (c) + 1
f2 (c + 1) f2 (c) + 1
5
4
4
6
4
4
7
WBTC131-135 30/05/2005 16:14 Page 951
Daß diese Figur ganz analog der der drei Gleichungen ohne den Unterstreichungen
ist, ist allerdings bedeutsam, wie es ja auch bedeutsam ist, daß die Kardinalzahl 1 und die
Rationalzahl 1 analogen Regeln unterworfen sind, aber es hindert nicht, daß wir hier ein neues19
Zeichen haben.
Ich treibe jetzt etwas ganz Neues mit diesem Zeichen.
Verhält es sich hier nicht so, wie in dem Fall, den ich einmal annahm, daß der Kalkül
der Wahrheitsfunktionen von Frege und Russell mit der Kombination ~p & ~q der
Zeichen ”~“ und ”&“ betrieben worden wäre, ohne daß man das gemerkt hätte, und daß
nun Sheffer,20
statt eine neue Definition zu geben, nur auf eine Eigentümlichkeit der
bereits benützten Zeichen aufmerksam gemacht hätte.
Man hätte immer dividieren21
können, ohne je auf die Periodizität aufmerksam zu
werden. Hat man sie gesehen, so hat man etwas Neues gesehn.
Könnte man das aber dann nicht ausdehnen und sagen: ich hätte Zahlen miteinander multiplizieren
können, ohne je auf den Spezialfall aufmerksam zu werden, in dem ich eine Zahl
mit sich selbst multipliziere, und also ist x2
nicht einfach x · x“. Die Schaffung des Zeichens
”x2
“ könnte man den Ausdruck dafür nennen, daß man auf diesen Spezialfall aufmerksam
geworden ist. Oder, man hätte (immer) a mit b multiplizieren und durch c dividieren
können, ohne darauf aufmerksam zu werden, daß man ” “ auch ” “ schreiben kann
und daß das analog a · b ist. Und weiter: das ist doch der Fall des Wilden, der die Analogie
zwischen ||||| und |||||| noch nicht sieht,22
oder die, zwischen || und |||||.
und allgemein:
23
Man könnte die Definition U sehen, ohne zu wissen, warum ich so abkürze.24
Man könnte die Definition sehen, ohne ihren Witz zu verstehen. – Aber dieser Witz ist
eben etwas Neues, das in ihr als spezieller Ersetzungsregel noch nicht liegt.
Auch ist ”7“ natürlich kein Gleichheitszeichen, in dem Sinn wie sie in α, β und γ
stehen.
Aber man kann leicht zeigen, daß 7 gewisse formale Eigenschaften mit = gemeinsam hat.
Es wäre – nach den angenommenen Regeln – falsch, das Gleichheitszeichen so zu gebrauchen:
[(a + b)2
= a · (a + b) + b · (a + b) = . . .
= a2
+ 2ab + b2
] .=. [(a + b)2
= a2
+ 2ab + b2
] . . . . . ∆
wenn damit gemeint sein soll, daß die linke Seite der Beweis der rechten ist.
Könnte man sich aber nicht diese Gleichung als Definition aufgefaßt denken? Wenn es
z.B. immer Gebrauch gewesen wäre, statt der rechten Seite die ganze Kette hinzuschreiben,25
und man nun die Abkürzung einführte.
a b
c
⋅a b
c
⋅
19 (V): anderes
20 (O): Scheffer,
21 (O): Dividieren
22 (V): sieht, aber
23 (F): MS 112, S. 45r.
24 (V): definiere.
25 (V): anzuschreiben,
716
717
476 Ein Zeichen auf bestimmte Weise sehen . . .
[(a + (b + 1) = (a + b) + 1] & [a + (b + (c + 1)) = (a + (b + c)) + 1] & [(a + b)
+ (c + 1) = ((a + b) + c) + 1] .Def.. (a + (b + c)) .7. ((a + b) + c) . . . U
α
γ
β
[f1(1) = f2(1) ] & [f1(c + 1) = f1(c) + 1] & [f2(c + 1) = f2(c) + 1] .Def..
f1(c) .7. f2(c) . . . V.
ρ β γ
WBTC131-135 30/05/2005 16:14 Page 952
It is indeed significant that this form is completely analogous to that of the three equations
without the underlinings; as it is also significant that the cardinal number 1 and the
rational number 1 are governed by analogous rules; but that doesn’t stand in the way of the
fact that here we have a new17
sign.
I am now doing something completely new with this sign.
Isn’t this like the supposition I once made that the Frege–Russell calculus of truthfunctions
might have been operated with the signs “~” and “&” combined into “~p & ~q”
without anyone noticing, and that then Sheffer, instead of giving a new definition, would
merely have drawn attention to a peculiarity of the signs already in use?
We could have gone on dividing without ever becoming aware of periodicity. Once we’ve
seen it, we’ve seen something new.
But couldn’t we extend that and say “I could have multiplied numbers by each other without
ever noticing the special case in which I multiply a number by itself; and that means
that x2
is not simply x · x”? We could call the creation of the sign “x2
” the expression of our
having become aware of that special case. Or, we could have gone on multiplying a by b and
dividing it by c without noticing that we could also write “ ” as “a · ”, and that the
latter is analogous to a · b. Or again: this is what happens to a savage who doesn’t yet see
the analogy between ||||| and ||||||, or between || and |||||.
and in general:
18
You could see the definition U without knowing why I’m abbreviating19
things in this
way.
You could see the definition without understanding its point. – But its point is something
new, something that is not yet contained in the definition as a specific replacement rule.
Of course, “7” isn’t an equals sign either, in the same sense as the ones occurring in α,
β, and γ.
But we can easily show that 7 has certain formal properties in common with =.
It would be incorrect – according to the usual rules – to use the equals sign like this:
[(a+b)2
= a · (a+b) + b · (a+b) = . . . = a2
+ 2ab + b2
] .=. [(a+b)2
= a2
+ 2ab + b2
] . . . . . ∆
if that is supposed to mean that the left-hand side is the proof of the right.
But couldn’t we imagine that this equation were understood as a definition? For instance,
if it had always been the custom to write down the whole chain instead of the right-hand
side, and we now introduced the abbreviation.
b
c
a b
c
⋅
17 (V): different
18 (F): MS 112, p. 45r.
19 (V): defining
Seeing and Understanding a Sign in a Particular Way. . . . 476e
[(a + (b + 1) = (a + b) + 1] & [a + (b + (c + 1)) = (a + (b + c)) + 1] & [(a + b)
+ (c + 1) = ((a + b) + c) + 1] .Def.. (a + (b + c)) .7. ((a + b) + c) . . . U
α
γ
β
[f1(1) = f2(1) ] & [f1(c + 1) = f1(c) + 1] & [f2(c + 1) = f2(c) + 1] .Def..
f1(c) .7. f2(c) . . . V.
ρ β γ
WBTC131-135 30/05/2005 16:14 Page 953
Freilich kann26
∆ als Definition aufgefaßt werden! Denn das linke Zeichen wird tatsächlich
gebraucht, und warum sollte man es dann nicht27
nach dieser Übereinkunft abkürzen.28
Nur gebraucht man dann das rechte oder linke Zeichen anders, als es jetzt üblich ist.29
Es ist nie genügend hervorgehoben worden, daß ganz verschiedene Arten von Zeichenregeln
in der Form der Gleichung geschrieben werden.
Die ”Definition“ x · x = x2
könnte30
so aufgefaßt werden, daß sie nur erlaubt, statt des
Zeichens ”x · x“ das Zeichen ”x2
“ zu setzen, also analog der Definition 1 + 1 = 2; aber auch
so (und so wird sie tatsächlich aufgefaßt), daß sie erlaubt, a2
statt a · a, und (a + b)2
statt
(a + b) · (a + b) zu setzen; auch so, daß für das x jede beliebige Zahl eintreten kann.
Wer entdeckt, daß ein Satz p aus einem von der Form q ⊃ p & q folgt, der konstruiert
ein neues Zeichen, das Zeichen dieser Regel. (Ich nehme dabei an, ein Kalkül mit p, q, ⊃,
&, sei schon früher gebraucht worden, und nun träte diese Regel hinzu und schaffe damit
einen neuen Kalkül.)
In der Notation ”x2
“ verschwindet ja wirklich die Möglichkeit, den einen der Faktoren
x31
durch eine andere Zahl zu ersetzen. Ja, es wären zwei Stadien der Entdeckung (oder
Konstruktion) von x2
denkbar. Daß man etwa zuerst statt ”x2
“ ”x=
“ setzt, ehe es Einem
nämlich auffällt, daß es das System x · x, x · x · x, etc. gibt, und daß man dann erst
hierauf kommt. Ähnliches ist in der Mathematik unzählige Male vorgekommen. (Liebig
bezeichnete ein Oxyd noch nicht so, daß der Sauerstoff in der Notation32
als Element
wie das oxydierte33
auftrat. Und, so seltsam das klingt, man könnte auch mit allen uns
heute bekannten Daten dem Sauerstoff durch eine ungeheuer34
künstliche Interpretation –
d.h. grammatische Konstruktion – eine solche Ausnahmestellung verschaffen; natürlich nur
in der Form der Darstellung.)
Mit den Definitionen x · x = x2
, x · x · x = x3
kommen nur die Zeichen ”x2
“ und ”x3
“ zur
Welt (und so weit war es noch nicht nötig, Ziffern als Exponenten zu schreiben).
|Der Prozeß der Verallgemeinerung35
schafft ein neues Zeichensystem.|
Sheffers36
Entdeckung ist natürlich nicht die der Definition ~p & ~q = p|q. Diese Definition
hätte Russell sehr wohl haben können, ohne doch damit das Sheffer’sche37
System zu
besitzen, und anderseits hätte Sheffer38
auch ohne diese Definition sein System begründen
können. Sein System ist ganz in dem Zeichen ”~p & ~p“ für ”~p“ und ”~(~p & ~q )
& ~(~p & ~q)“ für ”p 1 q“ enthalten und ”p|q“ gestattet nur eine Abkürzung. Ja, man
kann sagen, daß einer sehr wohl hätte das Zeichen ”~(~p & ~q) & ~(~p & ~q)“ für
”p 1 q“ kennen können, ohne das System (p|q) | (p|q) in ihm zu erkennen.
Machen wir die Sache noch klarer durch die Annahme der beiden Frege’schen Urzeichen
”~“ und ”&“, so bleibt hier die Entdeckung bestehen, wenn auch die Definitionen
26 (V): kann
27 (V): es nicht
28 (V): nicht durch das rechte ersetzen.
29 (V1): als wir es jetzt gebrauchen. (V2): Nur
gebraucht man dann dieses oder jenes anders,
als es jetzt üblich ist.
30 (V): kann
31 (V): Möglichkeit, das eine der x
32 (V): Sauerstoff darin
33 (V): als gleichwertes Element mit dem
oxydierten
34 (O): ungeheur
35 (V): Generalisation
36 (O): Scheffers
37 (O): Scheffer’sche
38 (O): Scheffer
718
719
477 Ein Zeichen auf bestimmte Weise sehen . . .
WBTC131-135 30/05/2005 16:14 Page 954
Of course ∆ can20
be regarded as a definition! For the sign on the left-hand side is in fact
used, and why shouldn’t we then abbreviate21
it according to this convention?22
It’s just that
in that case either the sign on the right or the sign on the left23
is used in a different way
from the usual one.24
It has never been sufficiently emphasized that totally different kinds of rules for signs get
written in the form of an equation.
The “definition” x · x = x2
could25
be understood as merely allowing us to replace the sign
“x · x” by the sign “x2
”, i.e. in analogy to the definition “1 + 1 = 2”; but it could also be
understood (and in fact is understood) as allowing us to replace a · a with a2
, and (a + b) ·
(a + b) with (a + b)2
; and it could also be understood in such a way that any arbitrary
number could be substituted for the x.
Whoever discovers that a proposition p follows from one of the form q ⊃ p & q constructs
a new sign, the sign for that rule. (Here I’m assuming that a calculus with p, q, ⊃, &, has
already been in use, and that this rule is now added to create a new calculus.)
In the notation “x2
” the possibility of replacing one of the factors x26
by another number
really does vanish. Indeed, we could imagine two stages in the discovery (or construction)
of x2
. Perhaps at first people write “x=
” instead of “x2
”, i.e. before they notice that there is
the system x · x, x · x · x, etc.; and only then do they hit upon the latter notation. Similar
things have occurred in mathematics countless times. (When Liebig designated an oxide,
oxygen did not appear in his notation as an element in the same way as27
what was oxidized.
And, odd as it sounds, even with all the data known to us today, we could – via an incredibly
artificial interpretation, that is to say, grammatical construction – give oxygen such a
privileged position; only, of course, in the form of its representation.)
All that the definitions x · x = x2
, x · x · x = x3
bring into the world are the signs “x2
” and
“x3
” (and until they did it wasn’t necessary to write numbers as exponents).
|The process of generalization creates a new system of signs.|
Of course Sheffer’s discovery is not the discovery of the definition ~p & ~q = p|q. Russell
could very well have had that definition without thereby being in possession of Sheffer’s
system, and on the other hand Sheffer could have established his system even without this
definition. His system is completely contained in the use of the signs “~p & ~p” for “~p”
and “~(~p & ~q) & ~(~p & ~q)” for “p 1 q”, and all “p|q” does is to permit an abbreviation.
Indeed, we can say that someone could very well have been acquainted with the use
of the sign “~(~p & ~q) & ~(~p & ~q)” for “p 1 q” without recognizing the system
(p|q)|(p|q) in it.
If we make matters clearer still by adopting Frege’s two primitive signs “~” and “&”,
then the discovery still stands, even if the definitions are written ~p & ~p = ~p and
20 (V): can
21 (V): we abbreviate
22 (V): we then replace it with the one on the right?
23 (V): either the latter or the former
24 (V): from the way we use it now.
25 (V): can
26 (V): one of the x’s
27 (V): appear there as an element equivalent to
Seeing and Understanding a Sign in a Particular Way. . . . 477e
WBTC131-135 30/05/2005 16:14 Page 955
geschrieben werden, ~p & ~p = ~p und ~(~p & ~p ) & ~(~q & ~q ) = p & q. Hier hat
sich an den Urzeichen scheinbar gar nichts geändert.
Man könnte sich auch denken, daß jemand die ganze Frege’sche oder Russell’sche Logik
schon in diesem System hingeschrieben hätte und doch, wie Frege, ”~“ und ”&“ seine
Urzeichen nennte, weil er das andere System in seinen Sätzen nicht sähe.
Es ist klar, daß die Entdeckung des Sheffer’schen39
Systems in ~p & ~p = ~p und
~(~p & ~p) & ~(~q & ~q) = p & q der Entdeckung entspricht, daß x2
+ ax + ein
Spezialfall von a2
+ 2ab + b2
ist.
Daß etwas so angesehen werden kann, sieht man erst, wenn es so angesehen ist.
Daß ein Aspekt möglich ist, sieht man erst, wenn er da ist.
Das klingt, als könnte die Sheffer’sche40
Entdeckung gar nicht in Zeichen dargestellt
werden (periodische Division).41
Aber das liegt daran, daß man die Anwendung42
des
Zeichens in seiner Einführung nicht vorausnehmen43
kann (die Regel ist und bleibt ein Zeichen
und von ihrer Anwendung getrennt).
Die allgemeine Regel für den Induktionsbeweis kann ich natürlich nur dann anwenden,
wenn ich die Substitution entdecke, durch die sie anwendbar wird. So wäre es möglich, daß
einer die Gleichungen
(a + 1) + 1 = (a + 1) + 1
1 + (a + 1) = (1 + a) + 1
sähe, ohne auf die Substitution
44
zu kommen.
Wenn ich übrigens sage, ich verstehe die Gleichungen als besondern Fall jener Regel, so
muß doch das Verständnis das sein, was sich in der Erklärung der Beziehung zwischen der
Regel und den Gleichungen zeigt, also, was wir durch die Substitutionen ausdrücken. Sehe
ich diese nicht als einen Ausdruck dessen an, was ich verstehe, dann gibt es keinen; aber
dann hat es auch keinen Sinn, von einem Verständnis zu reden, zu sagen, ich verstehe etwas
Bestimmtes. Denn nur dort hat es Sinn, vom Verstehen zu reden, wo wir eines verstehen,
im Gegensatz zu etwas anderem. Und diesen Gegensatz45
drücken eben Zeichen aus.
Ja, das Sehen der internen Beziehung kann nur wieder das Sehen von etwas sein, das sich
beschreiben läßt, wovon man sagen kann, ”ich sehe, daß es sich so verhält“, also wirklich
etwas von der Natur der Zuordnungszeichen46
(wie Verbindungsstriche, Klammern,
Substitutionen, etc.). Und alles andere kann nur in der Anwendung des Zeichens der
allgemeinen Regel in einem besonderen Fall liegen.
Es ist, als entdeckten wir an gewissen Körpern, die vor uns liegen, Flächen, mit denen
sie aneinandergereiht werden können. Oder vielmehr, als entdeckten wir, daß sie mit den
a2
4
39 (O): Scheffer’schen
40 (O): Scheffer’sche
41 (O): werden. (periodische Division)
42 (V): Verwendung
43 (O): voraus nehmen
44 (F): MS 111, S. 148.
45 (V): Und dies
46 (V): von der Natur der Zeichen der Zuordnung
720
721
478 Ein Zeichen auf bestimmte Weise sehen . . .
a = x, F1(x) = x + 1, F1(x + 1) = (x + 1) + 1, F2(x + 1) = 1 + (x + 1), F2(x) = 1 + x
WBTC131-135 30/05/2005 16:14 Page 956
~(~p & ~p) & ~(~q & ~q) = p & q. Here apparently nothing at all has changed in the
primitive signs.
We could also imagine that someone had already written the whole Fregean or Russellian
logic in this system, and yet, like Frege, called “~” and “&” his primitive signs, because he
didn’t see the other system in his propositions.
It’s clear that the discovery of Sheffer’s system in ~p & ~p = ~p and ~(~p & ~p) &
~(~q & ~q ) = p & q corresponds to the discovery that x2
+ ax + is a special case of
a2
+ 2ab + b2
.
We don’t see that something can be looked at in a certain way until it has been so
looked at.
We don’t see that an aspect is possible until it is there.
That sounds as if Sheffer’s discovery couldn’t even be represented in signs (periodic
division). But that’s because we can’t anticipate the use28
of the sign in its introduction (the
rule is and remains a sign, separated from its application).
Of course I can only apply the general rule for the induction proof if I discover the substitution
that makes it applicable. So it would be possible for someone to see the equations
(a + 1) + 1 = (a + 1) + 1
1 + (a + 1) = (1 + a) + 1
without arriving at the substitution29
Incidentally, when I say that I understand the equations as particular cases of that rule,
my understanding has to be an understanding that shows itself in the explanation of the
relation between the rule and the equations, i.e. what we express by the substitutions. If I
don’t regard that as an expression of what I understand, then there is no such expression;
but then neither does it make any sense to speak of understanding, to say that I understand
something specific. For it only makes sense to speak of understanding in cases
where we understand one thing, as opposed to something else. And it is this contrast that30
signs express.
Indeed, seeing the internal relation can in its turn only be seeing something that can be
described, something of which one can say: “I see that this is the way things are”; so really
it can only be something like the signs for correlations (such as connecting lines, brackets,
substitutions, etc.). And everything else has to be contained in the application of the sign
for the general rule in a particular case.
It’s as if there were certain solid objects lying in front of us, and we discovered surfaces
on them which enabled us to place them in a continuous row. Or rather, as if we discovered
a = x, F1(x) = x + 1, F1(x + 1) = (x + 1) + 1, F2(x + 1) = 1 + (x + 1), F2(x) = 1 + x
a2
4
28 (V): application
29 (F): MS 111, p. 148.
30 (V): this that
Seeing and Understanding a Sign in a Particular Way. . . . 478e
WBTC131-135 30/05/2005 16:14 Page 957
und den Flächen, die wir auch schon früher gesehen47
hatten, aneinandergereiht werden
können. Es ist das die Art der Lösung vieler Spiele oder Rätselfragen.
Der, welcher48
die Periodizität entdeckt, erfindet einen neuen Kalkül. Die Frage ist, wie
unterscheidet sich der Kalkül mit der periodischen Division von dem Kalkül, der die
Periodizität nicht kennt?
(Wir hätten einen Kalkül mit Würfeln betreiben können, ohne je auf die Idee zu
kommen, sie zu Prismen aneinanderzureihen.)
47 (V): gekannt 48 (V): der
479 Ein Zeichen auf bestimmte Weise sehen . . .
WBTC131-135 30/05/2005 16:14 Page 958
that by using certain surfaces, which we had seen31
before, we could place them in a continuous
row. That’s the way many games or puzzles are solved.
The person who discovers periodicity invents a new calculus. The question is, how does
the calculus with periodic division differ from the calculus in which periodicity is unknown?
(We could have operated a calculus with cubes without ever hitting upon the idea of combining
them to make prisms.)
31 (V): known
Seeing and Understanding a Sign in a Particular Way. . . . 479e
WBTC131-135 30/05/2005 16:14 Page 959
135
Der Induktionsbeweis,
Arithmetik und Algebra.
Wozu brauchen wir denn das kommutative Gesetz? Doch nicht, um die Gleichung
4 + 6 = 6 + 4 anschreiben zu können, denn diese Gleichung wird durch ihren besonderen
Beweis gerechtfertigt. Und es kann freilich auch der Beweis des kommutativen Gesetzes als
ihr Beweis verwendet werden, aber dann ist er eben jetzt1
ein spezieller (arithmetischer)
Beweis. Ich brauche das Gesetz also, um danach mit Buchstaben zu operieren.
Und diese Berechtigung kann mir der Induktionsbeweis nicht geben.
Aber eines ist klar: Wenn uns der Rekursionsbeweis das Recht gibt, algebraisch zu
rechnen, dann gibt uns auch der arithmetische Beweis L dieses Recht.2
Auch so: Der Rekursionsbeweis hat es – natürlich3
– wesentlich mit Zahlen zu tun. Aber
was gehen mich die an, wenn ich rein algebraisch operieren will. Oder: Der Rekursionsbeweis
ist nur dann zu benützen,4
wenn ich durch ihn einen5
Übergang in einer Zahlenrechnung
rechtfertigen will.
Man könnte nun aber fragen: Also brauchen wir (beide:) sowohl den Induktionsbeweis als
auch das assoziative Gesetz, da ja dieses Übergänge der Zahlenrechnung nicht begründen
kann, und jener nicht Transformationen in der Algebra?
Ja, hat man (denn) vor den Skolem’schen Beweisen das assoziative Gesetz – z.B. –
hingenommen, ohne den entsprechenden Übergang in einer Zahlenrechnung durch
Rechnung ausführen6
zu können? D.h.: konnte man vorher 5 + (4 + 3) = (5 + 4) + 3 nicht
ausrechnen, sondern hat es als Axiom betrachtet?
Wenn ich sage, die periodische Zahlenrechnung beweist den Satz, der mich zu jenen
Übergängen berechtigt, wie hätte dieser Satz gelautet, wenn man ihn als Axiom angenommen
und nicht bewiesen hätte?
Wie hätte der Satz gelautet, nach welchem ich 5 + (7 + 9) = (5 + 7) + 9 gesetzt hätte,
ohne es beweisen zu können? Es ist doch offenbar, daß es so einen Satz nie gegeben hat.
Könnte man auch so sagen: In der Arithmetik wird das assoziative Gesetz überhaupt nicht
gebraucht, sondern da arbeiten wir (nur) mit besonderen Zahlenrechnungen.
Und die Algebra, auch wenn sie sich der arithmetischen Notation bedient, ist ein ganz
anderer Kalkül, und nicht aus dem arithmetischen abzuleiten.
1 (V): (hier)
2 (V): dann auch der arithmetische Beweis L.
3 (V): offenbar
4 (V): gebrauchen,
5 (V): ich mit ihm den
6 (V): begründen
722
723
480
WBTC131-135 30/05/2005 16:14 Page 960
135
Proof by Induction,
Arithmetic and Algebra.
What do we need the commutative law for, anyway? Surely not so as to be able to write
the equation 4 + 6 = 6 + 4, because that equation is justified by its own particular proof.
And to be sure, the proof of the commutative law can also be used to prove it, but in that
case it1
is simply a particular (arithmetical) proof. So the reason I need the law is to apply
it to operations with letters.
And it is this justification that the inductive proof cannot give me.
But one thing is clear: if the recursive proof gives us the right to calculate algebraically,
then the arithmetical proof L gives us this right too.2
Again: Of course3
the recursive proof deals essentially with numbers. But of what concern
are they to me if I want to operate purely algebraically? Or again: The recursive proof
is only to be used if I want to justify a step in a number-calculation with it.
But now someone could ask: So we need both, don’t we? The inductive proof as well as
the associative law, since the latter can’t justify steps in numerical calculations, and the
former can’t justify transformations in algebra.
But before Skolem developed his proofs, did we really just accept the associative law, for
example, without anyone’s being able to figure out4
the corresponding step in a numerical
calculation? That is, were we previously unable to figure out 5 + (4 + 3) = (5 + 4) + 3, and
did we view it instead as an axiom?
If I say that the periodic calculation proves the proposition that justifies me in those steps,
then how would this proposition have read if it had been accepted as an axiom, and hadn’t
been proved?
How would the proposition have read, following which I would have postulated
5 + (7 + 9) = (5 + 7) + 9) without being able to prove it? It’s quite obvious that there
has never been such a proposition.
Could we also put it like this: In arithmetic the associative law isn’t used at all; there we
work (only) with particular number calculations?
And even when algebra uses arithmetical notation, it’s a totally different calculus, and it
can’t be derived from the arithmetical one.
1 (V): case (here) it
2 (V): then so does the arithmetical proof L.
3 (V): Again: obviously
4 (V): to justify
480e
WBTC131-135 30/05/2005 16:14 Page 961
Auf die Frage ”ist 5 × 4 = 20?“ könnte man antworten: ”sehen wir nach, ob es mit den
Grundregeln der Arithmetik übereinstimmt“; und entsprechend könnte ich sagen: sehen wir
nach, ob A mit den Grundregeln übereinstimmt. Aber mit welchen? Nun, wohl mit α.
Aber zwischen α und A liegt eben die Notwendigkeit einer Festsetzung darüber, was wir
hier ”Übereinstimmung“ nennen wollen.
D.h. zwischen α und A liegt die Kluft von7
Arithmetik und8
Algebra, und wenn B als
Beweis von A gelten soll, so muß diese (Kluft) durch eine Bestimmung überbrückt werden.
Nun ist ganz klar, daß wir Gebrauch von so einer Idee der Übereinstimmung machen,
wenn wir uns nur z.B. rasch ein Zahlenbeispiel ausrechnen, um dadurch die Richtigkeit eines
algebraischen Satzes zu kontrollieren.
Und in diesem Sinne könnte ich z.B. rechnen 25 × 16 16 × 25
25 32
150 80
400 400
und sagen: ”ja, ja, es stimmt, a × b ist gleich b × a“ – wenn ich mir vorstelle, daß ich das
vergessen hätte.
A, als Regel für das algebraische Rechnen, kann nicht rekursiv bewiesen werden; das
würde man besonders klar sehen, wenn man den ”rekursiven Beweis“ als eine Reihe arithmetischer
Ausdrücke hinschriebe. Denkt man sie sich hingeschrieben (d.h. ein Reihenstück
mit dem ”u.s.w.“), aber ohne die Absicht irgend etwas zu ”beweisen“, und nun fragte Einer:
”beweist dies a + (b + c) = (a + b) + c?“, so würden wir erstaunt zurückfragen: ”wie kann
es denn so was beweisen? in der Reihe kommen doch nur Ziffern und keine Buchstaben
vor!“ – Wohl aber könnte man nun sagen: Wenn ich für das Buchstabenrechnen die Regel
A einführe, so kommt dieser Kalkül dadurch in einem bestimmten Sinn in Einklang mit dem
Kalkül der Kardinalzahlen, wie ich ihn durch das Gesetz der Additionsregeln (rekursive
Definition a + (b + 1) = (a + b) + 1) festgelegt habe.
7 (V): von der 8 (V): zur
724
725
481 Der Induktionsbeweis, Arithmetik und Algebra.
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If you were asked “Is 5 × 4 = 20”?, you could answer: “Let’s check whether this agrees
with the basic rules of arithmetic”, and in the same way I could say: “Let’s check whether
A agrees with the basic rules”. But with which basic rules? Well, presumably with α.
But in the case of the relation between α and A we do need to define what we want to
call “agreement”.
That means that between α and A there lies the gulf between arithmetic and algebra, and
if B is to count as a proof of A, this (gulf) has to be bridged by a definition.
Now it’s quite clear that we do use such an idea of agreement when, for instance, we proceed
quickly to work out a numerical example in order to check the correctness of an algebraic
proposition.
And in this sense I could calculate, for example 25 × 16 16 × 25
25 32
150 80
400 400
and say: “Oh yes, that’s right, a × b is equal to b × a” – supposing I had forgotten that.
As a rule for algebraic calculation, A cannot be proved recursively. We would see
that especially clearly if we wrote down the “recursive proof” as a series of arithmetical
expressions. Imagine them written down (i.e. a fragment of the series, plus “and so on”) but
without the intention of “proving” anything, and now suppose someone asks: “Does this
prove a + (b + c) = (a + b) + c?”. We would respond in astonishment: “How can it prove
anything of the kind? After all, the series contains only numbers, and no letters!”. But no
doubt we would then say: If I introduce A as a rule for calculation with letters, in a certain
sense that brings this calculus into unison with the calculus of the cardinal numbers,
the calculus I established by the law for the rules of addition (the recursive definition
a + (b + 1) = (a + b) + 1).
Proof by Induction, Arithmetic and Algebra. 481e
WBTC131-135 30/05/2005 16:14 Page 963
Das Unendliche
in der Mathematik.
Extensive Auffassung.
726
WBTC136-140 30/05/2005 16:14 Page 964
The Infinite in
Mathematics. The
Extensional Viewpoint.
WBTC136-140 30/05/2005 16:14 Page 965
136
Allgemeinheit in der Arithmetik.
”Welchen Sinn hat ein Satz der Art ,(∃n).3 + n = 7‘?“ Man ist hier in einer seltsamen
Schwierigkeit: einerseits empfindet man es als Problem, daß der Satz die Wahl zwischen
unendlich vielen Werten von n hat, andrerseits scheint uns der Sinn des Satzes in sich gesichert
und nur für uns (etwa) noch zu erforschen, da wir doch ”wissen, was ,(∃x).φx‘ bedeutet“.
Wenn Einer sagte, er wisse nicht, welchen Sinn ”(∃n).3 + n = 7“ habe,1
so würde man ihm
antworten: ”aber Du weißt doch, was dieser Satz sagt: 3 + 0 = 7 .1. 3 + 1 = 7 .1. 3 + 2 = 7
und so weiter!“ Aber darauf kann man antworten: ”Ganz richtig – der Satz ist also keine
logische Summe, denn die2
endet nicht mit ’und so weiter‘ und das, worüber ich nicht klar
bin, ist eben diese Satzform ,φ(0) 1 φ(1) 1 φ(2) 1 u.s.w.‘ – und Du hast mir nur statt der
ersten unverständlichen Satzform3
eine zweite gegeben und zwar mit dem Schein, als gäbest
Du mir etwas altbekanntes, nämlich eine Disjunktion.“
Wenn wir nämlich meinen, daß wir doch unbedingt ”(∃n) etc.“ verstehen, so denken
wir zur Rechtfertigung an andre Fälle des Gebrauchs der Notation ”(∃ . . . ) . . .“,
beziehungsweise der Ausdrucksform ”es gibt . . .“ unserer Wortsprache. Darauf kann man
aber nur sagen: Du vergleichst also den Satz ”(∃n) . . .“ mit jenem Satz ”es gibt ein Haus in
dieser Stadt, welches . . .“, oder ”es gibt zwei Fremdwörter auf dieser Seite“. Aber mit dem
Vorkommen der Worte ”es gibt“ in diesen Sätzen ist ja die Grammatik dieser Allgemeinheit
noch nicht bestimmt. Und dieses Vorkommen weist auf nichts andres hin, als eine gewisse
Analogie in den Regeln.4
Wir werden also die Grammatik der Allgemeinheit ”(∃n)etc.“
ohne vorgefaßtes Urteil untersuchen können,5
d.h., ohne6
uns von der Bedeutung, die
”(∃ . . .) . . .“ in andern Fällen hat,7
stören zu lassen.
”Alle Zahlen haben vielleicht die Eigenschaft ε.“ Wieder ist die Frage: was ist die Grammatik
dieses allgemeinen Satzes? Denn damit ist uns nicht gedient, daß wir die Verwendung des
Ausdrucks ”alle . . .“ in andern grammatischen Systemen kennen. Sagt man: ”Du weißt doch,
was es heißt! es heißt: ε(0) & ε(1) & ε(2) u.s.w.“, so ist damit wieder nichts erklärt; außer,
daß der Satz kein logisches Produkt ist. Und man wird, um die Grammatik des Satzes
verstehen zu lernen, fragen: Wie gebraucht8
man diesen Satz? Was sieht man als Kriterium
seiner Wahrheit an? Was ist seine Verifikation? – Wenn keine Methode vorgesehen ist, um
zu entscheiden, ob der Satz wahr oder falsch ist, ist er ja zwecklos und d.h. sinnlos. Aber
hier kommen wir nun zur Illusion, daß allerdings eine solche Methode der Verifikation
vorgesehen ist, die sich nur einer menschlichen Schwäche wegen nicht durchführen läßt.
1 (V): nicht, was ”(∃n).3 + n = 7“ bedeute,
2 (V): sie
3 (V): Satzart
4 (V): in Regeln.
5 (V): Wir werden also ruhig diese Regeln von
vorne untersuchen können,
6 (V): können, ohne
7 (V): Bedeutung von ”(∃ . . .) . . .“ in andern
Fällen
8 (O): gebrauchst
727
728
483
WBTC136-140 30/05/2005 16:14 Page 966
136
Generality in Arithmetic.
“What is the sense of such a proposition as ‘(∃n).3 + n = 7’?” Here we are in a strange
quandary: on the one hand we feel it to be a problem that the proposition has the choice
between infinitely many values of n, and on the other hand the sense of the proposition seems
inherently guaranteed, something that only remains for us to explore, because after all we
“know what ‘(∃x).φx’ means”. If someone said he didn’t know what the sense of “(∃n).3 +
n = 7” was,1
he would get the answer “But you do know what this proposition says: 3 + 0
= 7 .1. 3 + 1 = 7 .1. 3 + 2 = 7, and so on!” But to that one can reply “Quite right – so the
proposition isn’t a logical sum, because a logical sum doesn’t end with ‘and so on’; and what
I am not clear about is precisely this propositional form ‘φ(0) 1 φ(1) 1 φ(2) 1 and so on’
– and all you have done is to substitute a second unintelligible form2
of proposition for
the first one, all the while seeming to give me something familiar, namely a disjunction.”
For if we believe that we definitely do understand “(∃n) etc.”, we justify this by thinking
of other cases where we use the notation “(∃ . . . ) . . .”, or of the expression “There is
. . .” in our word-language. But to that one can only say: So you’re comparing the proposition
“(∃n) . . .” with the proposition “There is a house in this city which . . .” or
“There are two foreign words on this page”. But the occurrence of the words “there is”
in those sentences doesn’t suffice to determine the grammar of this generalization; all it
does is to indicate a certain analogy in the rules. And so we can investigate the grammar
of the generalization “(∃n) etc.” without a preconceived judgement, that is, without letting
the meaning of “(∃ . . . ) . . .” in other cases bother us.3
“Perhaps all numbers have the property ε.” Again the question is: What is the grammar
of this general proposition? For our being acquainted with the use of the expression
“all . . .” in other grammatical systems doesn’t help us. If we say “But you know what it
means: it means ε(0) & ε(1) & ε(2) and so on”, again this explains nothing; except that
the proposition is not a logical product. And in order to understand the grammar of the
proposition, we should ask: How is this proposition used? What is regarded as the criterion
of its truth? What is its verification? – If no method is provided for deciding whether the
proposition is true or false, then it is pointless, i.e. senseless. But it is here that we arrive at
the illusion that such a method of verification has indeed been provided, a method that is
1 (V): what “(∃n).3 + n = 7” meant,
2 (V): kind
3 (V): Therefore we can go ahead and examine
these rules from the beginning, without letting
the meaning of “(∃ . . . ) . . .” // the meaning
that “(∃ . . . ) . . .” has // in other cases
bother us.
483e
WBTC136-140 30/05/2005 16:14 Page 967
Diese Verifikation besteht darin, daß man alle (unendlich vielen) Glieder des Produktes
ε(O) & ε(1) & ε(2) . . . auf ihre Richtigkeit prüft. Hier wird das, was man ”logische
Unmöglichkeit“ nennt, mit physischer Unmöglichkeit verwechselt.9
Denn dem Ausdruck
”alle Glieder des unendlichen Produktes auf ihre Richtigkeit prüfen“ glaubt man Sinn gegeben
zu haben, weil man das Wort ”unendlich viele“ für die Bezeichnung einer riesig großen Zahl
hält. Und bei der ”Unmöglichkeit, die unendliche Zahl von Sätzen zu prüfen“ schwebt uns
die Unmöglichkeit vor, eine sehr große Anzahl von Sätzen zu prüfen, wenn wir etwa nicht
die nötige Zeit haben.
Erinnere Dich daran, daß, in dem Sinn, in welchem es unmöglich ist, eine unendliche
Anzahl von Sätzen zu prüfen, es auch unmöglich ist, das10
zu versuchen. – Wenn wir uns
mit den Worten ”Du weißt doch, was ,alle . . .‘ heißt“ auf die Fälle berufen, in welchen diese
Redeweise gebraucht wird, so kann es uns doch nicht gleichgültig sein, wenn wir einen
Unterschied zwischen diesen Fällen und dem Fall sehen, für welchen der Gebrauch der
Worte erklärt11
werden sollte. – (Gewiß), wir wissen, was es heißt, ”eine Anzahl von Sätzen
auf ihre Richtigkeit prüfen“ und gerade auf dieses Verständnis berufen wir uns ja, wenn wir
verlangen, man solle nun auch den Ausdruck ”unendlich viele Sätze . . .“ verstehen. Aber
hängt denn der Sinn des ersten Ausdrucks nicht von den spezifischen Erfahrungen ab, die
ihm entsprechen?12
Und gerade diese Erfahrungen fehlen ja in der Verwendung (dem Kalkül)
des zweiten Ausdrucks; es sei denn, daß ihm solche Erfahrungen zugeordnet werden, die
von den ersten grundverschieden sind.
Ramsey schlug einst vor, den Satz, daß unendlich viele Gegenstände eine Funktion f(ξ)
befriedigen, durch die Verneinung sämtlicher Sätze
~(∃x).fx
(∃x).fx & ~(∃x, y).fx & fy
(∃x, y),fx & fy .&. ~(∃x, y, z).fx & fy & fz
u.s.w.
auszudrücken. – Aber diese Verneinung ergäbe die Reihe
(∃x).fx
(∃x, y).fx & fy
(∃x, y, z) etc. etc.
Aber diese Reihe ist wieder ganz überflüssig: denn erstens enthält ja der zuletzt
angeschriebene Satz alle vorhergehenden und zweitens nützt uns dieser auch nichts, da
er ja nicht von einer unendlichen Anzahl von Gegenständen handelt. Die Reihe kommt
also in Wirklichkeit auf einen Satz hinaus:
”(∃x, y, z . . . ad inf.).fx & fy & fz . . . ad inf.“.
Und mit diesem Zeichen können wir gar nichts anfangen, wenn wir nicht seine Grammatik
kennen. Eines aber ist klar: wir haben es nicht mit einem Zeichen von der Form
”(∃x, y, z).fx & fy & fz“ zu tun; wohl aber mit einem Zeichen, dessen Ähnlichkeit mit
diesem dazu gemacht scheint, uns irrezuführen.
9 (V): Hier wird logische mit physischer
Möglichkeit verwechselt.
10 (V): es
11 (V): gerechtfertigt
12 (V): Aber ist denn der Sinn des ersten
Ausdrucks von der Erfahrung // den
Erfahrungen //, die mit ihm verknüpft ist //
sind //, unabhängig?
730
728
484 Allgemeinheit in der Arithmetik.
729
WBTC136-140 30/05/2005 16:14 Page 968
kept from being carried out only because of a human weakness. This verification consists
in checking the correctness of all the (infinitely many) terms of the product ε(0) & ε(1)
& ε(2) . . . . Here what is called “logical impossibility” is being confused with physical
impossibility.4
For we think we have given a sense to the expression “checking the correctness
of all the terms of the infinite product” because we take the expression “infinitely
many” as designating an enormously large number. And when we hear of “the impossibility
of checking an infinite number of propositions” we have in mind the impossibility of
checking a very large number of propositions, say when we don’t have enough time.
Remember that in the sense in which it is impossible to check an infinite number of propositions
it is also impossible to try to do so. – If we say “But you do know what ‘all . . .’
means”, and cite as our authority the cases in which this mode of speech is used, then it
certainly can’t be a matter of indifference to us whether we see a distinction between these
cases and the case for which the use of the words needs to be explained.5
– (To be sure),
we know what is meant by “checking a number of propositions for correctness”, and it is
precisely this understanding that we are appealing to when we demand that one should
also understand the expression “infinitely many propositions . . .”. But doesn’t the sense
of the first expression depend on the specific experiences that correspond to it?6
And
it’s precisely these experiences that are lacking in the use (in the calculus) of the second
expression; unless experiences are correlated with it that are fundamentally different in
kind from the first ones.
Ramsey once proposed to express the proposition that infinitely many objects satisfy a
function f(ξ) by negating all propositions like:
~(∃x).fx
(∃x).fx & ~(∃x, y).fx & fy
(∃x, y).fx & fy .&. ~(∃x, y, z).fx & fy & fz
and so on.
– But such a negation would yield the series
(∃x).fx
(∃x, y).fx & fy
(∃x, y, z) etc., etc.
But this series too is completely superfluous: for in the first place the last proposition at any
point surely contains all the previous ones, and secondly even it is of no use to us, because
of course it isn’t about an infinite number of objects. So in reality the series boils down to
one proposition:
“(∃x, y, z . . . ad inf.).fx & fy & fz . . . ad inf.”.
And we can’t do anything with that sign unless we know its grammar. But one thing is clear:
we are not dealing with a sign of the form “(∃x, y, z).fx & fy & fz”, but with a sign whose
similarity to that sign seems to be made to mislead us.
4 (V): Here logical possibility is being confused
with physical possibility.
5 (V): justified.
6 (V): But is the sense of the first expression
independent of the experience // experiences //
that is // are // connected with it?
Generality in Arithmetic. 484e
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”m > n“ kann ich allerdings definieren als (∃x).m − n = x, aber dadurch habe ich es in
keiner Weise analysiert. Man denkt nämlich, daß durch die Verwendung des Symbolismus
”(∃ . . . ) . . .“ eine Verbindung hergestellt sei13
zwischen ”m > n“ und andern Sätzen von
der Form ”es gibt . . .“, vergißt aber, daß damit zwar eine gewisse Analogie betont ist,
aber nicht mehr; da das Zeichen ”(∃ . . . ) . . .“ in unzählig vielen verschiedenen ”Spielen“
gebraucht wird. (Wie es eine ”Dame“ im Schach- und im Damespiel gibt.) Wir müssen also
erst die Regeln wissen, nach denen14
es hier verwendet wird. Und da wird sofort klar, daß
diese Regeln hier mit den Regeln für die Subtraktion zusammenhängen. Denn, wenn wir
– wie gewöhnlich – fragen: ”wie weiß ich – d.h. woraus geht es hervor –, daß es eine Zahl
x gibt, die der Bedingung m − n = x genügt“, so kommen darauf die Regeln für die Subtraktion
zur Antwort. Und nun sehen wir, daß wir mit unserer Definition nicht viel gewonnen haben.
Ja, wir hätten gleich als Erklärung von ”m > n“ die Regeln angeben können, nach welchen
man so einen Satz – z.B. im Falle ”32 > 17“ – überprüft.
Wenn ich sage: ”für jedes n gibt es ein δ, das die Funktion kleiner macht als n“, so
muß ich mich auf ein allgemeines arithmetisches Kriterium beziehen, das anzeigt, wann
F(δ) < n.
Wenn ich wesentlich keine Zahl hinschreiben kann, ohne ein Zahlensystem, so muß sich
das auch in der allgemeinen Behandlung der Zahl wiederspiegeln. Das Zahlensystem ist nicht
etwas Minderwertiges – wie eine Russische Rechenmaschine – das nur für Volksschüler
Interesse hat, während die höhere, allgemeine Betrachtung davon absehen kann.
Es geht auch nichts von der Allgemeinheit der Betrachtung verloren, wenn ich die
Regeln, die die Richtigkeit und Falschheit von ”m > n“ (also seinen Sinn) bestimmen, etwa
für das15
Dezimalsystem gebe. Ein System brauche ich ja doch und die Allgemeinheit ist
dadurch gewahrt, daß man die Regeln gibt, nach denen von einem System in ein anderes
übersetzt wird.
Ein Beweis in der Mathematik ist allgemein, wenn er allgemein anwendbar ist. Eine
andere Allgemeinheit kann nicht im Namen der Strenge gefordert werden. Jeder Beweis
stützt sich auf bestimmte Zeichen, auf eine bestimmte Zeichengebung. Es kann nur die
eine Art der Allgemeinheit eleganter erscheinen, als die andere. ((Dazu die Verwendung
des Dezimalsystems in Beweisen über δ und η.))
”Streng“ heißt: klar.16
”Den mathematischen Satz kann man sich vorstellen als ein Lebewesen, das selbst weiß,
ob es wahr oder falsch ist. (Zum Unterschied von den Sätzen der Empirie.)17
Der mathematische Satz weiß selbst, daß er wahr, oder daß er falsch ist. Wenn er von
allen Zahlen handelt, so muß er auch schon alle Zahlen übersehen. Wie der Sinn, so muß
auch seine Wahrheit oder Falschheit in ihm liegen.“
”Es ist, als wäre die Allgemeinheit eines Satzes ,(n) · ε(n)‘ nur eine Anweisung auf
die eigentliche, wirkliche, mathematische Allgemeinheit eines Satzes. Gleichsam nur eine
13 (V): ist
14 (V): wissen, wie
15 (V): etwa im
16 (E): In TS 212, S. 1776 notiert sich
Wittgenstein zu dieser Bemerkung: ”[Gegen
Hardy + zu der Verteidigung des Dezimalsystems
in Beweisen, etc.]“
17 (V): den empirischen Sätzen.)
731
732
733
485 Allgemeinheit in der Arithmetik.
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I certainly can define “m > n” as (∃x).m − n = x, but in doing so I haven’t in any way
analysed it. For you think that by using the symbolism “(∃ . . . ) . . .” you’ve established
a connection between “m > n” and other propositions of the form “there is . . .”; but you
forget that in doing this you’ve stressed a certain analogy – and nothing more – because
the sign “(∃ . . . ) . . .” is used in countless different “games”. (Just as there is a “king” in
chess and draughts.) So first we have to know the rules governing its use here; and then it
immediately becomes clear that these rules are connected with the rules for subtraction.
For if we ask the usual question “How do I know – i.e. from what does it follow – that
there is a number x that satisfies the condition m − n = x?”, then it is the rules for subtraction
that come as the answer. And then we see that we haven’t gained much by our
definition. Indeed, as an explanation of “m > n” we could have straightaway given the
rules for checking such a proposition – for example, in the case of “32 >17”.
If I say: “For any n there is a δ that makes the function smaller than n”, I have to be
referring to a general arithmetical criterion that indicates when F(δ) < n.
If in the nature of the case I cannot write down a number independently of a number
system, this must also be reflected in the general treatment of number. A number system is
not something inferior – like a Russian abacus – that is only of interest to elementary school
pupils, whereas a higher-level general consideration can afford to disregard it.
Neither do I lose anything of the generality of my examination if I give the rules that
determine the correctness and incorrectness (and thus the sense) of “m > n” for7
, say, the
decimal system. After all I do need a system, and the generality is preserved by giving the
rules according to which translation takes place from one system into another.
A proof in mathematics is general if it is generally applicable. You can’t demand some
other kind of generality in the name of rigour. Every proof rests on particular signs, on a
particular assignment of signs. All that can happen is that one type of generality may appear
more elegant than another. ((Cf. the use of the decimal system in proofs concerning δ
and η.))
“Rigorous” means: clear.8
“We can imagine a mathematical proposition as a living being that knows on its own
whether it is true or false. (As opposed to empirical propositions.)
A mathematical proposition knows on its own that it is true or that it is false. If it is
about all numbers, it must also have an overview of all the numbers. As its sense must be
contained in it, so too must its truth or falsity.”
“It’s as though the generality of a proposition ‘(n) · ε(n)’ only gave directions to the
actual, real mathematical generality of a proposition. As though it were only a description
7 (V): within
8 (E): At this point in the predecessor typescript
(TS 212, p. 1776), Wittgenstein has written next
to the remark: “[Contra Hardy, and in defence
of the decimal system in proofs, etc.]”.
Generality in Arithmetic. 485e
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Beschreibung der Allgemeinheit, nicht diese selbst. Als bilde der Satz nur auf rein äußerliche
Weise ein Zeichen, dem erst von innen Sinn gegeben werden muß.“
”Wir fühlen: Die Allgemeinheit, die die mathematische Behauptung hat, ist anders als die
Allgemeinheit des Satzes, der bewiesen ist.“
”Man könnte sagen: ein mathematischer Satz ist der Hinweis auf einen Beweis.“
Wie wäre es, wenn ein Satz seinen Sinn selber nicht ganz erfaßte. Wenn er sich quasi
selber zu hoch wäre? – Und das nehmen eigentlich die Logiker an.
Den Satz, der von allen Zahlen handelt, kann man sich nicht durch ein endloses
Schreiten verifiziert denken, denn, wenn das Schreiten endlos ist, so führt es ja eben nicht
zu einem Ziel.
Denken wir uns eine unendlich lange Baumreihe, und ihr entlang, damit wir sie
inspizieren können, einen Weg. Sehr gut, so muß dieser Weg endlos sein. Aber wenn er
endlos ist, so heißt das, daß man ihn nicht zu Ende gehen kann. D.h., er bringt mich nicht
dazu, die Reihe zu übersehen. Der endlose Weg hat nämlich nicht ein ”unendlich fernes“
Ende, sondern kein Ende.
Man kann auch nicht sagen: ”Der Satz kann alle Zahlen nicht successive erfassen, so
muß er sie durch den Begriff fassen“, – als ob das faute de mieux so wäre: ”Weil er es
so nicht kann, muß er es auf andre Weise tun“. Aber ein successives Erfassen ist schon
möglich, nur führt es eben nicht zur Gesamtheit. Diese liegt: nicht auf dem Weg, den wir
schrittweise gehen, – und nicht: am unendlich fernen Ende dieses Weges. (Das alles heißt
nur ”ε(0) & ε(1) & ε(2) & u.s.w.“ ist nicht das Zeichen eines logischen Produkts.)
”Alle Zahlen können nicht zufällig eine Eigenschaft ε besitzen; sondern nur ihrem Wesen
nach.“18
– Der Satz ”die Menschen, welche rote Nasen haben, sind gutmütig“ hat auch dann
nicht denselben Sinn wie der Satz ”die Menschen, welche Wein trinken, sind gutmütig“,
wenn die Menschen, welche rote Nasen haben, eben die sind, die Wein trinken. Dagegen:
wenn die Zahlen m, n, o der Umfang eines mathematischen Begriffs sind, so daß also fm &
fn & fo der Fall ist, dann hat19
der Satz, welcher sagt, daß die Zahlen, die f befriedigen, die
Eigenschaft ε haben, den gleichen Sinn wie ”ε(m) & ε(n) & ε(o)“. Denn die beiden Sätze
”f(m) & f(n) & f(o)“ und ”ε(m) & ε(n) & ε(o)“ lassen sich, ohne daß wir dabei den Bereich
der Grammatik verlassen, in einander umformen.
Sehen wir uns nun den Satz an: ”alle n Zahlen, welche der Bedingung F(ξ) genügen, haben
zufälligerweise die Eigenschaft ε“. Da kommt es drauf an, ob die Bedingung F(ξ) eine
mathematische ist. Ist sie das, nun dann kann ich ja aus F(x) ε(x) ableiten, wenn auch über
die Disjunktion der n Werte von F(ξ). (Denn hier gibt es eben eine Disjunktion.) Hier werde
ich also nicht von einem Zufall reden. – Ist die Bedingung eine nicht-mathematische, so wird
man dagegen vom Zufall reden können. Z.B. wenn ich sage: alle Zahlen, die ich heute auf
den Omnibussen gelesen habe, waren zufällig Primzahlen. (Dagegen kann man natürlich
nicht sagen: ”die Zahlen 17, 3, 5, 31, sind zufällig Primzahlen“, ebensowenig wie: ”die Zahl
3 ist zufällig eine Primzahl“.) ”Zufällig“ ist wohl der Gegensatz von ”allgemein ableitbar“;
aber man kann sagen: der Satz ”17, 3, 5, 31 sind Primzahlen“ ist allgemein ableitbar – so
sonderbar das klingt –, wie auch der Satz 2 + 3 = 5.
Sehen wir nun zu unserm ersten Satz zurück, so fragen wir wieder: Wie soll denn der
Satz ”alle Zahlen haben die Eigenschaft ε“ gemeint sein? wie soll man ihn denn wissen
können? denn diese Festsetzung gehört ja zur Festsetzung seines Sinnes! Das Wort
18 (V): Wesen (als Zahlen) nach.“ 19 (V): sagt
734
735
486 Allgemeinheit in der Arithmetik.
WBTC136-140 30/05/2005 16:14 Page 972
of the generality, as it were, and not the generality itself. As if the proposition formed a sign
only in a purely external way, a sign that still needed to be given sense from within.”
“We feel: the generality of a mathematical assertion is different from the generality of the
proposition proved.”
“We could say: a mathematical proposition is a pointer to a proof.”
What would it be like if a proposition didn’t quite grasp its own sense? If its sense were,
so to speak, over its head? – And that really is what logicians assume.
A proposition about all numbers can’t be thought of as verified by an endless succession
of steps, for if a succession is endless, it doesn’t lead to any goal.
Let’s imagine an infinitely long row of trees, and a path alongside it so that we can inspect
them. Excellent: so this path must be endless. But if it’s endless that means you can’t walk
to the end of it. That is, it does not put me in a position to survey the row. For the endless
path does not have an end that’s “infinitely distant” – it has no end.
Nor can you say: “A proposition can’t grasp all the numbers one by one, so it has to grasp
them through a concept”, as if this were so for lack of a better alternative: “Because it can’t
do it like this, it has to do it another way.” But grasping the numbers one by one is possible;
it’s just that it doesn’t lead to the totality. That lies: not on the path on which we go
step by step – nor at the infinitely distant end of that path. (All of this only means that
“ε(0) & ε(1) & ε(2) and so on” is not the sign for a logical product.)
“No number can possess a property ε accidentally; it can do so only essentially.”9
– The
proposition “People who have red noses are good-natured” does not have the same sense as
the proposition “People who drink wine are good-natured”, even if the people who have red
noses are the very same people who drink wine. On the other hand, if the numbers m, n, o
are the extension of a mathematical concept, such that it is the case that fm & fn & fo, then
the proposition that the numbers that satisfy f have the property ε has the same sense as
“ε(m) & ε(n) & ε(o)”. For the two propositions “f(m) & f(n) & f(o)” and “ε(m) & ε(n) &
ε(o)” can be transformed into each other without thereby leaving the realm of grammar.
Now let’s look at the proposition: “All the n numbers that satisfy the condition F(ξ)
just happen to have the property ε”. What matters here is whether the condition F(ξ) is a
mathematical one. If it is, then I can derive ε(x) from F(x), if only through the disjunction
of the n values of F(ξ). (For here there is in fact a disjunction.) So here I won’t talk about
coincidence. – On the other hand, if the condition is a non-mathematical one we can speak
of coincidence. For example, if I say: “All the numbers I saw today on buses happened
to be prime numbers”. (But of course we can’t say: “The numbers 17, 3, 5, 31 happen to
be prime numbers”, any more than “The number 3 happens to be a prime number”.)
“Coincidental” is the opposite of “derivable through a general rule”; but one can say that
the proposition “17, 3, 5, 31 are prime numbers” is derivable through a general rule –
however odd this sounds – just like the proposition 2 + 3 = 5.
If we now look back at our first proposition, we ask again: How is the proposition “All
numbers have the property ε” supposed to be meant? How is one supposed to be able to
know it? For to establish this is of course part of establishing its sense! After all, the word
9 (V): essentially (as a number)”.
Generality in Arithmetic. 486e
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”zufällig“ deutet doch auf eine Verifikation durch successive Versuche und dem widerspricht,
daß wir nicht von einer endlichen Zahlenreihe reden.
In der Mathematik sind Beschreibung und Gegenstand äquivalent. ”Die20
fünfte Zahl
der Zahlenreihe hat diese Eigenschaften“ sagt dasselbe wie ”5 hat diese Eigenschaften“. Die
Eigenschaften eines Hauses folgen nicht aus seiner Stellung in einer Häuserreihe; dagegen
sind die Eigenschaften einer Zahl die Eigenschaften einer Stellung.
Man kann sagen, daß die Eigenschaften einer bestimmten Zahl nicht vorauszusehen sind.
Man sieht sie erst, wenn man zu ihr kommt.
Das Allgemeine ist die Wiederholung einer Operation. Jedes Stadium dieser
Wiederholung hat seine Individualität. Nun ist es nicht etwa so, daß ich durch die
Operation von einer Individualität zur andern fortschreite. So daß die Operation das
Mittel wäre, um von einer zur andern zu kommen. Gleichsam das Vehikel, das bei jeder
Zahl anhält, die man nun betrachten kann. Sondern die dreimal iterierte21
Operation + 1
erzeugt und ist die Zahl drei.
(Im Kalkül sind Prozeß und Resultat einander äquivalent.)
Ehe ich aber nun von ”allen diesen Individualitäten“, oder ”der Gesamtheit dieser
Individualitäten“ sprechen wollte, müßte ich mir gut überlegen, welche Bestimmungen ich
in diesem Falle für den Gebrauch der Worte ”alle“ und ”Gesamtheit“ gelten lassen will.
Es ist schwer, sich von der extensiven Auffassung ganz frei zu machen: So denkt man:
”Ja, aber es muß doch eine innere Beziehung zwischen x3
+ y3
und z3
bestehen, da doch
(zum mindesten) die Extensionen dieser Ausdrücke, wenn ich sie nur kennte, das Resultat
einer solchen Beziehung darstellen müßten“. Etwa: ”Es müssen doch entweder wesentlich
alle Zahlen die Eigenschaft ε haben, oder nicht; da doch alle Zahlen die Eigenschaften haben,
oder nicht; wenn ich auch nicht wissen kann, welches der Fall ist.“22
”Wenn ich die Zahlenreihe durchlaufe, so komme ich entweder einmal zu einer Zahl von
der Eigenschaft ε, oder niemals.“ Der Ausdruck ”die Zahlenreihe durchlaufen“ ist Unsinn;
außer es wird ihm ein Sinn gegeben, der aber die vermutete Analogie mit dem ”durchlaufen
der Zahlen von 1 bis 100“ aufhebt.
Wenn Brouwer die Anwendung des Satzes vom ausgeschlossenen Dritten in der
Mathematik bekämpft, so hat er Recht, soweit er sich gegen ein Vorgehen richtet, das
den Beweisen empirischer Sätze analog ist. Man kann in der Mathematik nie etwas auf
die Art beweisen: Ich habe 2 Äpfel auf dem Tisch liegen gesehen; jetzt ist nur einer da;
also hat A einen Apfel gegessen. – Man kann nämlich nicht durch Ausschließung gewisser
Möglichkeiten eine neue beweisen, die nicht, durch die von uns gegebenen Regeln, schon
in jener Ausschließung liegt. Insofern gibt es in der Mathematik keine echten Alternativen.
Wäre die Mathematik die Untersuchung von erfahrungsmäßig gegebenen Aggregaten, so
könnte man durch die Ausschließung eines Teils das Nichtausgeschlossene beschreiben, und
hier wäre der nicht ausgeschlossene Teil der Ausschließung des andern nicht äquivalent.
Die Betrachtungsweise: daß ein logisches Gesetz, weil es für ein Gebiet der Mathematik
gilt, nicht notwendig auch für ein anderes gelten müsse, ist in der Mathematik gar nicht
am Platz, ihrem Wesen ganz entgegen. Obwohl23
manche Autoren gerade das für besonders
subtil halten, und entgegen den Vorurteilen.
20 (O): ”die
21 (V): die dreimalige
22 (V): wenn ich das auch nicht wissen kann.“
23 (V): Obwohl ei
736
737
487 Allgemeinheit in der Arithmetik.
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“coincidental” suggests a verification by successive tests, and that is contradicted by the fact
that we are not speaking of a finite series of numbers.
In mathematics description and object are equivalent. “The fifth number of the number
series has these properties” says the same as “5 has these properties”. The properties of a
house do not follow from its position in a row of houses; but the properties of a number are
the properties of a position.
You might say that the properties of a particular number can’t be foreseen. You don’t see
them until you’ve got there.
What is general is the repetition of an operation. Each stage of this repetition has its own
individuality. But it isn’t as if I use the operation to move from one individual to the next,
so that the operation would be the means for getting from one to the other – the vehicle, as
it were, that stops at every number so we can study it. Rather, the operation +1 repeated
three times produces10
and is the number 3.
(In the calculus, process and result are equivalent to each other.)
But before deciding to speak of “all these individualities” or “the totality of these
individualities” I would have to consider carefully what stipulations I wanted to allow for
the use of the words “all” and “totality”.
It’s difficult to extricate yourself completely from the extensional viewpoint: And so you
keep thinking “Yes, but still, there must be an internal relation between x3
+ y3
and z3
, since
(at least) the extensions of these expressions, if only I knew them, would have to show the
result of such a relation”. Or perhaps: “It must surely be essential to all numbers either to
have the property ε or not to have it; since after all, all numbers have their properties or
don’t have them; even if I can’t know which is the case.”11
“If I run through the number series, either I eventually come to a number with the
property ε, or I never do.” The expression “to run through the number series” is nonsense;
unless a sense is given to it which eliminates the assumed analogy with “running through
the numbers from 1 to 100”.
When Brouwer battles against the application of the law of the excluded middle in mathematics,
he is right in so far as he is directing his attack against a process that is analogous
to the proofs of empirical propositions. In mathematics you can never prove anything this
way: I saw two apples lying on the table, and now there is only one there, so A has eaten an
apple. – For you can’t by excluding certain possibilities prove a new one which isn’t already
contained in that exclusion by virtue of the rules we have laid down. To that extent there
are no genuine alternatives in mathematics. If mathematics were the investigation of empirically
given aggregates, one could use the exclusion of one part to describe what was not
excluded, and in that case the non-excluded part wouldn’t be equivalent to the exclusion of
the other.
This way of looking at things: that a logical law, just because it is valid for one area of
mathematics, doesn’t necessarily have to be valid for another area as well – is completely
out of place in mathematics, completely contrary to its essence. Although some authors
hold just this approach to be particularly subtle and to be an antidote to prejudices.
10 (V): the triple operation + 1 produces 11 (V): know that.”
Generality in Arithmetic. 487e
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Wie es sich nun mit derjenigen Allgemeinheit, mit den Sätzen der Mathematik verhält,
die nicht von ”allen Kardinalzahlen“, sondern z.B. von ”allen reellen Zahlen“ handeln,24
kann
man nur erkennen, indem25
man diese Sätze und ihre Beweise untersucht.
Wie ein Satz verifiziert wird,26
das sagt er. Vergleiche die Allgemeinheit in der Arithmetik
mit der Allgemeinheit von nicht arithmetischen Sätzen. Sie wird anders verifiziert und ist
darum eine andere. Die Verifikation ist nicht bloß ein27
Anzeichen der Wahrheit, sondern
sie bestimmt den Sinn des Satzes. (Einstein: wie eine Größe gemessen wird, das ist sie.)
24 (V): Wie es sich nun mit derjenigen
Allgemeinheit in der Mathematik verhält,
deren Sätze nicht //, die nicht // von ”allen
Kardinalzahlen“, sondern z.B. von ”allen
reellen Zahlen“ handeln, // spricht
25 (V): wenn
26 (V): ist,
27 (V): nicht ein bloßes
488 Allgemeinheit in der Arithmetik.
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We can only come to know about that generality, about those propositions of mathematics
that are not about “all cardinal numbers”,12
but, say, about “all real numbers”, by
investigating13
those propositions and their proofs.
How a proposition is verified14
is what it says. Compare generality in arithmetic with the
generality of non-arithmetical propositions. It is verified differently and so is of a different
kind. The verification is not merely an indication15
of the truth, but determines the sense of
the proposition. (Einstein: how a magnitude is measured is what it is.)
12 (V): We can only come to know about that
generality in mathematics whose propositions are
// which is // not about // does not speak about
// “all cardinal numbers”,
13 (V): numbers”, if one investigates
14 (V): proposition has been verified
15 (V): not a mere indication
Generality in Arithmetic. 488e
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137
Zur Mengenlehre.
|”Die rationalen Punkte liegen auf der Zahlengeraden nahe beisammen1
“: irreführendes
Bild.|
Ist ein Raum denkbar, der nur alle rationalen Punkte, aber nicht die irrationalen enthält?
Wäre etwa diese Struktur für unsern Raum zu grob?2
Weil wir die irrationalen Punkte dann
nur annäherungsweise erreichen könnten?3
Unser Netz wäre also nicht fein genug? Nein.
Die Gesetze gingen uns ab, nicht die Extensionen.
Ist ein Raum denkbar, der nur alle rationalen aber nicht die irrationalen Punkte enthält?
Und das heißt nur: Sind die irrationalen Zahlen nicht in den rationalen präjudiziert?
So wenig, wie das Schachspiel im Damespiel.
Die irrationalen Zahlen füllen keine Lücke aus, die die rationalen offen lassen.
Man wundert sich darüber, daß ”zwischen den überall dicht liegenden rationalen
Punkten“ noch die irrationalen Platz haben. (Welche Verdummung!) Was zeigt eine
Konstruktion, wie die des Punktes ? Zeigt sie diesen Punkt, wie er doch noch zwischen
den rationalen Punkten Platz hat? Sie zeigt, daß der durch die Konstruktion erzeugte Punkt,
nämlich als Punkt dieser Konstruktion, nicht rational ist. – Und was entspricht dieser
Konstruktion in der Arithmetik? Etwa eine Zahl, die sich doch noch zwischen die rationalen
Zahlen hineinzwängt? Ein Gesetz, das nicht vom Wesen der rationalen Zahl ist.
Die Erklärung des Dedekind’schen Schnittes gibt vor, anschaulich zu sein,4
wenn sie
sagt:5
Es gibt 3 Fälle: entweder hat die Klasse R ein erstes Glied und L kein letztes,
etc. In Wahrheit lassen sich 2 dieser 3 Fälle gar nicht vorstellen. Außer, wenn die
Wörter ”Klasse“, ”erstes Glied“, ”letztes Glied“ gänzlich ihre vorgeblich6
beibehaltenen
alltäglichen Bedeutungen wechseln. Wenn man nämlich – starr darüber, daß Einer von einer
Klasse von Punkten redet, die rechts von einem gegebenen Punkt liegt und keinen Anfang
hat – sagt: gib uns doch ein Beispiel so einer Klasse, – so zieht er das von den rationalen
Zahlen hervor! Aber hier ist ja gar keine Klasse von Punkten im ursprünglichen7
Sinn!
Der Schnittpunkt zweier Kurven ist nicht das gemeinsame Glied zweier Klassen von
Punkten, sondern der Durchschnitt zweier Gesetze. Es sei denn, daß man die erste
Ausdrucksweise, sehr irreführend, durch die zweite definiert.
Es mag nach dem Vielen, was ich schon darüber gesagt habe, trivial klingen, wenn ich
jetzt sage, daß der Fehler in der mengentheoretischen Betrachtungsweise immer wieder
2
1 (V): nahe bei einander
2 (V): ungenau?
3 (V): Weil wir zu den irrationalen Punkten
dann (immer) nur annäherungsweise gelangen
könnten?
4 (V): gibt vor, sie wäre anschaulich,
5 (V): wenn gesagt wird:
6 (V): anscheinend
7 (V): alltäglichen
738
739
740
489
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137
On Set Theory.
|“The rational points lie close together1
on the number line”: a misleading picture.|
Is a space conceivable that contains only all the rational points, but not the irrational ones?
Would this structure perhaps be too coarse2
for our space? Since then we could only reach3
the irrational points approximately? Would this mean that our net wasn’t fine enough? No.
What we would lack would be the laws, not the extensions.
Is a space conceivable that contains only all rational points but not the irrational points?
And that only means: Aren’t the irrational numbers prejudged in the rational ones?
No more than chess is prejudged in draughts.
The irrational numbers do not fill a gap that the rational ones leave open.
We are surprised that “between the ubiquitously and densely distributed rational points”,
there is still room for the irrationals. (What stupidity!) What does a construction like that
of the point show? Does it show how this point, in spite of everything, still finds room
between the rational points? It shows that the point produced by the construction, that is
produced as a point by this construction, is not rational. – And what corresponds to this
construction in arithmetic? Perhaps a number that manages after all to squeeze in between
the rational numbers? A law that does not have the nature of a rational number.
The explanation of the Dedekind cut pretends to be clear when it says4
: There are three
cases: either the class R has a first member and L no last member, etc. In fact two of these
three cases cannot even be imagined, unless the words “class”, “first member”, “last
member”, completely change the everyday meanings they supposedly5
have retained. For
if we’re dumbfounded by someone’s talk of a class of points that lies to the right of a
given point and has no beginning, and we say: Do give us an example of such a class – then
he trots out the example of the rational numbers! But here there is no class of points, in
the original6
sense!
The point of intersection of two curves isn’t the common member of two classes of
points; it’s the intersection of two laws. Unless, quite misleadingly, we use the second form
of expression to define the first.
After the many things I have already said about this, it may sound trivial if I now say
that the mistake in the set-theoretical approach consists time and again in viewing laws and
2
1 (V): close to each other
2 (V): inaccurate
3 (V): only (ever) get to
4 (V): it is said
5 (V): apparently
6 (V): everyday
489e
WBTC136-140 30/05/2005 16:14 Page 979
darin liegt, Gesetze und Aufzählungen (Listen) als wesentlich Eins zu betrachten und sie
aneinander zu reihen; da, wo das eine nicht ausreicht, das Andere seinen Platz ausfüllt.
Das Symbol für eine Klasse ist eine Liste.
Die Schwierigkeit liegt auch hier wieder in der Bildung mathematischer Scheinbegriffe.
Wenn man z.B. sagt: Man kann die Kardinalzahlen ihrer Größe nach in eine Folge ordnen,
aber nicht die rationalen Zahlen, so ist darin unbewußt die Voraussetzung enthalten, als hätte
der Begriff des Ordnens der Größe nach für die rationalen Zahlen doch einen Sinn, und als
erwiese sich dieses Ordnen nun beim Versuch als unmöglich (was voraussetzt, daß der Versuch
denkbar ist). – So denkt man, ist es möglich zu versuchen, die reellen Zahlen (als wäre es ein
Begriff wie etwa ”Äpfel auf diesem Tisch“) in eine Reihe zu ordnen, und es erwiese sich
nun als undurchführbar.
Wenn der Mengenkalkül sich in seiner Ausdrucksweise soviel als möglich an die
Ausdrucksweise des Kalküls der Kardinalzahlen anlehnt, so ist das wohl in mancher
Hinsicht belehrend, weil es auf gewisse formale Ähnlichkeiten hinweist, aber auch
irreführend, wenn er gleichsam noch etwas ein Messer nennt, das weder Griff noch Klinge
mehr hat. (Lichtenberg.)
(Die Eleganz eines mathematischen Beweises kann nur den einen Sinn haben, gewisse
Analogien besonders stark zu Tage treten zu lassen, wenn das gerade erwünscht ist, sonst
entspringt sie dem Stumpfsinn und hat nur die eine Wirkung, das zu verhüllen, was klar
und offenbar sein sollte. Das stumpfsinnige Streben nach Eleganz ist eine Hauptursache,
warum die Mathematiker ihre eigenen Operationen nicht verstehen, oder es entspringt die
Verständnislosigkeit und jenes Streben einer gemeinsamen Quelle.)
Die Menschen sind im Netz der Sprache verstrickt8
und wissen es nicht.
”Es gibt einen Punkt, in dem die beiden Kurven einander schneiden.“ Wie weißt Du das?
Wenn Du es mir sagst, werde ich wissen, was der Satz ”es gibt . . .“ für einen Sinn hat.
Wenn man wissen will, was der Ausdruck ”das Maximum einer Kurve“ bedeutet, so frage
man sich: wie findet man es? – Was anders gefunden wird, ist etwas anderes. Man definiert
es als den Punkt der Kurve, der höher liegt als alle andern, und hat dabei wieder die Idee,
daß es nur unsere menschliche Schwäche ist, die uns verhindert, alle Punkte der Kurve einzeln
durchzugehen und den höchsten unter ihnen auszuwählen. Und dies führt zu der Meinung,
daß der höchste Punkt unter einer endlichen Anzahl von Punkten wesentlich dasselbe ist,
wie der höchste Punkt einer Kurve, und daß man hier eben auf zwei verschiedene Methoden
das Gleiche findet, wie man auf verschiedene Weise feststellt, daß jemand im Nebenzimmer
ist: anders etwa, wenn die Tür geschlossen ist und wir zu schwach sind, sie zu öffnen,
und anders, wenn wir hinein können. Aber, wie gesagt, menschliche Schwäche liegt dort
nicht vor, wo die scheinbare Beschreibung der Handlung ”die wir nicht ausführen können“
sinnlos ist. Es würde freilich nichts schaden, ja sehr interessant sein, die Analogie zwischen
dem Maximum einer Kurve und dem Maximum (in anderm Sinne) einer Klasse von Punkten
zu sehen, so lange uns die Analogie nicht das Vorurteil eingibt, es liege im Grunde beide
Male dasselbe vor.
Es ist der gleiche Fehler unserer Syntax, der den geometrischen Satz ”die Strecke
läßt sich durch einen Punkt in zwei Teile teilen“ als die gleiche Form darstellt, wie den
8 (V): gefangen
741
742
490 Zur Mengenlehre.
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enumerations (lists) as essentially the same kind of thing and placing them on a line next
to each other, the one filling in the gaps left by the other.
The symbol for a class is a list.
Here again, the difficulty lies in the formation of mathematical pseudo-concepts. For
instance, if we say that we can arrange the cardinal numbers, but not the rational numbers,
in a sequence according to their size, we are unconsciously presupposing that the concept
of an ordering by size does make sense for the rational numbers, but that when we try it, it
turns out that this ordering is impossible (which presupposes that the attempt is conceivable).
– Thus we think that it is possible to attempt to arrange the real numbers (as if that
were a concept like “apples on this table”) in a series, and that now it turns out to be
impracticable.
When the calculus of sets has its form of expression follow as much as possible the form
of expression of the calculus of cardinal numbers, this is in some ways instructive. For it
points to certain formal similarities. But it’s also misleading when it still, so to speak, calls
something a knife that no longer has either handle or blade. (Lichtenberg.)
(The only sense there can be to elegance in a mathematical proof is to allow certain
analogies to come out in particularly clear profile, when that is what’s wanted; otherwise
elegance is a product of mindlessness, and all it does is to obscure what ought to be clear
and manifest. The mindless pursuit of elegance is a principal reason for mathematicians not
understanding their own operations. Or perhaps the lack of understanding and that pursuit
of elegance have a common origin.)
Humans are entangled7
in the net of language without knowing it.
“There is a point at which the two curves intersect.” How do you know that? When you
tell me I will know what sort of sense the proposition “There is . . .” has.
If you want to know what the expression “the maximum of a curve” means, ask yourself:
How does one find it? – What is found in a different way is a different thing. We define
the maximum as the point on the curve that’s higher than all the others, and in so doing
we again have the idea that it is only our human weakness that prevents us from sifting
through all the points of the curve one by one and selecting the highest among them. And
this leads to the idea that the highest point among a finite number of points is essentially
the same thing as the highest point of a curve, and that here we are simply finding out the
same thing by two different methods, just as we ascertain in different ways that someone is
in the next room – in one way if the door is shut and we aren’t strong enough to open it,
and in another way if we can get inside. But as I said, it isn’t a matter of human weakness,
where what appears to be the description of an action “we cannot carry out” is senseless.
To be sure it wouldn’t hurt, indeed it would be very interesting, to see the analogy between
the maximum of a curve and the maximum (in another sense) of a class of points, so long
as the analogy doesn’t instil in us the prejudice that in both cases we have fundamentally
the same thing.
It’s the same defect as the one in our syntax that represents the proposition in geometry
“a line can be divided by a point into two parts”, as a proposition of the same form as “a
7 (V): caught
On Set Theory. 490e
WBTC136-140 30/05/2005 16:14 Page 981
Satz: ”die Strecke ist unbegrenzt teilbar“; so daß man scheinbar in beiden Fällen sagen
kann: ”nehmen wir an, die mögliche Teilung sei vollzogen9
“. ”In zwei Teile teilbar“ und
”unbegrenzt teilbar“ haben eine gänzlich verschiedene Grammatik. Man operiert fälschlich
mit dem Worte ”unendlich“, wie mit einem Zahlwort; weil beide in der Umgangssprache
auf die Frage ”wieviele . . .“ zur Antwort kommen.
”Das Maximum ist doch aber höher, als jeder beliebige andre Punkt der Kurve.“ Aber
die Kurve besteht ja nicht aus Punkten, sondern ist ein Gesetz, dem Punkte gehorchen.
Oder auch: ein Gesetz, nach dem Punkte konstruiert werden können. Wenn man nun fragt:
”welche Punkte“, – so kann ich nur sagen: ”nun, z.B., die Punkte P, Q, R, etc.“. Und es
ist einerseits so, daß keine Anzahl von Punkten gegeben werden kann, von denen man sagen
könnte, sie seien alle Punkte, die auf der Kurve liegen, daß man anderseits auch nicht von
einer solchen Gesamtheit von Punkten reden kann, die nur wir Menschen nicht aufzählen
können, die sich aber beschreiben läßt und die man die Gesamtheit aller Punkte der Kurve
nennen könnte, – eine Gesamtheit, die für uns Menschen zu groß wäre. Es gibt ein Gesetz
einerseits und Punkte auf der Kurve anderseits – aber nicht ”alle Punkte der Kurve“. Das
Maximum liegt höher als irgend welche Punkte der Kurve, die man etwa konstruiert, aber
nicht höher als eine Gesamtheit von Punkten; es sei denn, daß das Kriterium hiervon, und
also der Sinn dieser Aussage, wieder nur die Konstruktion aus dem Gesetz der Kurve ist.
Das Gewebe der Irrtümer auf diesem Gebiet ist natürlich ein sehr kompliziertes. Es tritt
z.B. noch die Verwechslung zweier verschiedener Bedeutungen des Wortes ”Art“ hinzu.
Man gibt nämlich zu, daß die unendlichen Zahlen eine andre Art Zahlen sind, als die
endlichen, aber man mißversteht nun, worin hier der Unterschied verschiedener Arten
besteht. Daß es sich nämlich hier nicht um die Unterscheidung von Gegenständen nach
ihren Eigenschaften handelt, wie wenn man rote Äpfel von gelben unterscheidet,10
sondern
um verschiedene logische Formen. – So versucht Dedekind eine unendliche Klasse zu
beschreiben; indem er sagt, es sei eine, die einer echten Teilklasse ihrer selbst ähnlich ist.
Hierdurch hat er scheinbar eine Eigenschaft angegeben, die die Klasse haben muß, um unter
den Begriff ”unendliche Klasse“ zu fallen. (Frege.) Denken wir uns nun die Anwendung
dieser11
Definition. Ich soll also in einem bestimmten Fall untersuchen, ob eine Klasse endlich
ist oder nicht, etwa ob eine bestimmte Baumreihe endlich oder endlos ist. Ich nehme also,
der Definition folgend, eine Teilklasse dieser Baumreihe und untersuche, ob sie der ganzen
Klasse ähnlich (d.h. 1–1 koordinierbar) ist! (Hier fängt gleichsam schon Alles an zu lachen.)
Das heißt ja gar nichts: denn, nehme ich eine ”endliche Klasse“ als Teilklasse, so muß ja
der Versuch, sie der ganzen Klasse 1 zu 1 zuzuordnen eo ipso mißlingen; und mache ich
den Versuch an einer unendlichen Teilklasse, – aber das heißt ja schon erst recht nichts,
denn, wenn sie unendlich ist, kann ich den Versuch dieser Zuordnung gar nicht machen. –
Das, was man im Fall einer endlichen Klasse ”Zuordnung aller ihrer Glieder mit andern“
nennt, ist etwas ganz anderes, als das, was man z.B. eine Zuordnung aller Kardinalzahlen
mit allen Rationalzahlen nennt. Die beiden Zuordnungen, oder, was man in den zwei Fällen
mit diesem Wort bezeichnet, gehören verschiedenen logischen Typen12
an. Und es ist nicht
die ”unendliche Klasse“ eine Klasse, die mehr Glieder im gewöhnlichen Sinn des Wortes
”mehr“ enthält, als die endlichen. Und wenn man sagt, daß eine unendliche Zahl größer ist,
als eine endliche, so macht das die beiden nicht vergleichbar, weil in dieser Aussage das Wort
”größer“ eine andere Bedeutung hat, als etwa im Satz ”5 größer als 4“.
9 (V): ausgeführt
10 (O): unterscheitet,
11 (V): der
12 (V): Kathegorien
743
491 Zur Mengenlehre.
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line can be divided infinitely”; so that it looks as if in both cases we can say “Let’s suppose
we’ve completed8
the possible division”. “Divisible into two parts” and “infinitely divisible”
have completely different grammars. We wrongly use the word “infinite” like a number word
because in everyday speech both are given as answers to the question “How many . . . ?”.
“But after all, the maximum is higher than any other arbitrary point of the curve.” But
the curve doesn’t consist of points; it’s a law that points obey, or again, a law according to
which points can be constructed. If you now ask: “Which points?” – I can only say, “Well,
for instance, the points P, Q, R, etc.” On the one hand we can’t give a number of points
that could be said to be all the points that lie on the curve, and on the other hand we can’t
speak of a totality of points that are just not countable by us humans, but can be described
and could be called the totality of all points of the curve – a totality too large for us human
beings. On the one hand there is a law, and on the other there are points on the curve – but
there is no such thing as “all the points of the curve”. The maximum is higher than any
points of the curve that we happen to construct, but it isn’t higher than a totality of points,
unless the criterion for that, and thus the sense of that statement, is once again simply the
construction according to the law of the curve.
Of course the web of errors in this area is very complicated. For example, the confusion
between two different meanings of the word “kind” adds to the complexity. For we admit
that the infinite numbers are a different kind of number from the finite ones, but then here
we misunderstand what the difference between different kinds consists of. We don’t realize,
that is, that here it’s not a matter of distinguishing between objects according to their
properties, as when we distinguish between red and yellow apples, but a matter of different
logical forms. – Thus Dedekind tries to describe an infinite class by saying that it is a class
that is similar to a proper subclass of itself. In this way he appears to have stated a property
that a class must have in order to fall under the concept “infinite class” (Frege). Now let’s
imagine applying this9
definition. Suppose I am to investigate in a particular case whether a
class is finite or not, say whether a certain row of trees is finite or infinite. So, in accordance
with the definition, I take a subclass of this row of trees and investigate whether it is
similar (i.e. can be coordinated one-to-one) to the whole class! (Here everybody is already
beginning to laugh, as it were.) This clearly doesn’t mean anything at all; for if I take a “finite
class” as a subclass, the attempt to correlate it one-to-one with the whole class must eo ipso
fail; and if I make the attempt with an infinite subclass – but that is even more of a piece
of nonsense, for if it is infinite, then I can’t even attempt this correlation. – What we call
a “correlation of all the members of a class with others” in the case of a finite class is
something quite different from what we call a correlation of all cardinal numbers with all
rational numbers, for instance. The two correlations, or what one refers to with this word
in the two cases, belong to different logical types.10
And an “infinite class” is not a class that
contains more members than a finite one, in the ordinary sense of the word “more”. And
if we say that an infinite number is greater than a finite one, that doesn’t make the two
comparable, because in that statement the word “greater” has a different meaning than it has,
say, in the proposition “5 is greater than 4”.
8 (V): we’ve carried out
9 (V): the
10 (V): categories.
On Set Theory. 491e
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Die Definition gibt nämlich vor, daß aus dem Gelingen oder Mißlingen des Versuchs,
eine wirkliche Teilklasse der ganzen Klasse zuzuordnen, hervorgeht, daß sie unendlich
bezw. endlich ist. Während es einen solchen entscheidenden Versuch gar nicht gibt. –
”Unendliche Klasse“ und ”endliche Klasse“ sind verschiedene logische Kategorien;13
was
von der einen Kategorie14
sinnvoll ausgesagt werden kann, kann es nicht von der andern.
Der Satz, daß eine Klasse einer ihrer Subklassen nicht ähnlich ist, ist für endliche Klassen
nicht wahr, sondern eine Tautologie. Die grammatischen Regeln über die Allgemeinheit
jener15
generellen Implikation im Satz ”k ist eine Subklasse von K“16
enthalten das, was
der Satz, K sei eine unendliche17
Klasse, sagt.
|Ein Satz (wie) ”es gibt keine letzte Kardinalzahl“ verletzt den naiven – und rechten –
Sinn. Wenn ich frage ”wer war der letzte Mann der Prozession“ und die Antwort lautet ”es
gibt keinen letzten“, so verwirrt sich mir das Denken; was heißt das ”es gibt keinen letzten“?
ja, wenn die Frage geheißen hätte ”wer war der Fahnenträger“, so hätte ich die Antwort
verstanden ”es gibt keinen Fahnenträger“. Und nach einer solchen Antwort ist ja jene
verwirrende18
gebildet. Wir fühlen nämlich mit Recht: wo von einem Letzten die Rede sein
kann, da kann nicht ”kein Letzter“ sein. Das heißt aber natürlich: Der Satz ”es gibt keine
letzte“ müßte richtig lauten: es hat keinen Sinn, von einer ”letzten Kardinalzahl“ zu reden,
dieser Ausdruck ist unrechtmäßig gebildet.|
”Hat die Prozession ein Ende“ könnte auch heißen: ist sie eine in sich geschlossene
Prozession. Und nun höre ich die Mathematiker sagen19
”da siehst Du ja, daß Du Dir sehr
wohl einen solchen Fall vorstellen kannst, daß etwas kein Ende hat; warum soll es dann nicht
auch andere solche Fälle20
geben können?“ – Aber die Antwort ist: Die ”Fälle“ in diesem
Sinn des Wortes sind grammatische Fälle und sie bestimmen erst den Sinn der Frage. Die
Frage ”warum soll es nicht auch andere Fälle geben können“ ist der analog gebildet: ”Warum
soll es nicht noch andere Fälle von Mineralien21
geben können, die im Dunkeln leuchten“,
aber hier handelt es sich um Fälle der Wahrheit einer Aussage, dort um Fälle, die den Sinn
bestimmen.22
Die Ausdrucksweise: m = 2n ordne eine Klasse einer ihrer echten Subklassen23
zu, kleidet
einen trivialen24
Sinn durch Heranziehung einer irreführenden Analogie in eine paradoxe
Form. (Und statt sich dieser paradoxen Form als etwas Lächerlichem zu schämen, brüstet
man sich eines Sieges über alle Vorurteile des Verstandes.) Es ist genau so, als stieße man
die Regeln des Schach um und sagte, es habe sich gezeigt, daß man Schach auch ganz anders
spielen könne. So verwechselt man erst das Wort ”Zahl“ mit einem Begriffswort wie ”Äpfel“,
spricht dann von einer ”Anzahl der Anzahlen“ und sieht nicht, daß man in diesem
Ausdruck nicht beidemal das gleiche Wort ”Anzahl“ gebrauchen sollte; und endlich hält
man es für eine Entdeckung, daß die Anzahl der geraden Zahlen die gleiche ist wie die der
geraden und ungeraden.
13 (O): Kathegorien;
14 (O): Kathegorie
15 (V): der
16 (V): Die grammatischen Regeln über die
Allgemeinheit der generellen Implikation in
dem Satz ”k ist eine Subklasse von K“
17 (V): endliche
18 (V): sinnlose
19 (V): Und nun könnte man sagen
20 (V): nicht auch einen andern solchen Fall
21 (V): nicht noch andere Mineralien
22 (V): dort um Fälle, die den Sinn eines Satzes
bestimmen.
23 (V): Teilklassen
24 (V): einfachen
744
745
744
492 Zur Mengenlehre.
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That is to say, the definition pretends that what follows from the success or failure of the
attempt to correlate a proper subclass with the whole class is that the class is infinite or finite,
respectively. Whereas there just isn’t any such decisive test. – “Infinite class” and “finite
class” are different logical categories; what can be meaningfully asserted of the one category
cannot be meaningfully asserted of the other.
With regard to finite classes the proposition that a class is not similar to one of its
subclasses is not a truth, but a tautology. The grammatical rules for the generality of that11
general implication in the proposition “k is a subclass of K” contain what is said by the proposition
that K is an infinite12
class.
|A proposition (like) “There is no last cardinal number” is an affront to naive – and
correct – common sense. If I ask “Who was the last person in the procession?” and am told
“There isn’t any last person”, my thinking goes topsy-turvy; what does “There isn’t any last
person” mean? To be sure, if the question had been “Who was the standard bearer?” I
would have understood the answer “There isn’t any standard bearer”; and of course that
bewildering13
answer above is modelled on an answer of the latter kind. For we feel, correctly,
that where we can speak of a last one at all, it’s impossible for there to be “no last one”.
But of course that means: The proposition “There is no last cardinal number” should read,
correctly: “It makes no sense to speak of a ‘last cardinal number’ – that expression has been
unlawfully formed”.|
“Does the procession have an end?” could also mean: Is it a self-contained procession?
And now I hear the mathematicians saying14
: “See, you can perfectly well imagine such a
case of something not having an end; so why shouldn’t it be possible that there are other
such cases15
as well?” – But the answer is: The “cases” in this sense of the word are
grammatical cases, and it is they that determine the meaning of the question. The question
“Why shouldn’t other such cases be possible as well?” is formed in analogy to this:
“Why shouldn’t other such cases of minerals16
that shine in the dark be possible?”; but
the latter is about instances of the truth of a statement, the former about cases that determine
the sense.17
The form of expression “m = 2n correlates a class with one of its proper subclasses”
recruits a misleading analogy to clothe a trivial18
sense in a paradoxical form. (And instead
of being ashamed of this paradoxical form as something ridiculous, people boast about a
victory over all prejudices of the intellect.) It’s exactly as if one overturned the rules of
chess and said it had been shown that chess could also be played quite differently. Thus
one first confuses the word “number” with a concept word like “apples”, then one talks
about a “number of numbers” and doesn’t see that in this expression one shouldn’t use
the same word “number” both times; and finally one deems it a discovery that the number
of the even numbers is equal to the number of the even and odd numbers.
11 (V): the
12 (V): is a finite
13 (V): senseless
14 (V): now one could say
15 (V): that there’s another such case
16 (V): other minerals
17 (V): cases that determine the sense of a
proposition.
18 (V): simple
On Set Theory. 492e
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Weniger irreführend ist es, zu sagen, ”m = 2n gibt die Möglichkeit der Zuordnung jeder
Zahl mit einer andern“, als ”m = 2n ordnet alle Zahlen anderen zu“. Aber auch hier muß
erst die Grammatik die Bedeutung des Ausdrucks ”Möglichkeit der Zuordnung“ lehren.
(Es ist beinahe unglaublich, wie ein Problem durch die irreführenden Ausdrucksweisen,
die Generation auf Generation rundherum stellt, gänzlich, auf Meilen, blockiert wird, so
daß es beinahe unmöglich wird, dazuzukommen.)
Wenn zwei25
Pfeile in derselben Richtung zeigen, ist es dann nicht absurd, diese
Richtungen ”gleich lang“ zu nennen, weil, was in der Richtung des einen Pfeiles liegt,
auch in der des andern liegt? – Die Allgemeinheit von m = 2n ist ein Pfeil, der der
Operationsreihe entlang weist. Und zwar kann man sagen, der Pfeil weist in’s Unendliche;
aber heißt das, daß es ein Etwas, das Unendliche, gibt, auf das er – wie auf ein Ding –
hinweist? – Der Pfeil bezeichnet gleichsam die Möglichkeit der Lage von Dingen in seiner
Richtung. Das Wort ”Möglichkeit“ ist aber irreführend, denn, was möglich ist, wird man
sagen, soll eben nun wirklich werden. Auch denkt man dabei immer an zeitliche Prozesse
und schließt daraus, daß die Mathematik nichts mit der Zeit zu tun hat, daß die Möglichkeit
in ihr bereits Wirklichkeit ist.
Die ”unendliche Reihe der Kardinalzahlen“ oder ”der Begriff der Kardinalzahl“ ist
nur so eine Möglichkeit, – wie aus dem Symbol ”[0, ξ, ξ + 1]“ klar hervorgeht. Dieses
Symbol selbst ist ein Pfeil, dessen Feder die ”0“, dessen Spitze ”ξ + 1“ ist. Es ist möglich,
von Dingen zu reden, die in der Richtung des Pfeils liegen, aber irreführend oder absurd,
von allen möglichen Lagen der Dinge in der Pfeilrichtung als einem Äquivalent dieser
Richtung selbst zu reden. Wenn ein Scheinwerfer Licht in den unendlichen Raum wirft,
so beleuchtet er allerdings alles, was in der Richtung seiner Strahlen liegt, aber man soll
nicht sagen, er beleuchtet die Unendlichkeit.
Es ist immer mit Recht höchst verdächtig,26
wenn Beweise in der Mathematik allgemeiner
geführt werden, als es der bekannten Anwendung des Beweises entspricht. Es liegt hier
immer der Fehler vor, der in der Mathematik allgemeine Begriffe und besondere Fälle sieht.
In der Mengenlehre treffen wir auf Schritt und Tritt diese verdächtige Allgemeinheit.
Man möchte immer sagen: ”Kommen wir zur Sache!“
Jene allgemeinen Betrachtungen haben stets nur Sinn, wenn man einen bestimmten
Anwendungsbereich im Auge hat.
Es gibt eben in der Mathematik keine Allgemeinheit, deren Anwendung auf spezielle
Fälle sich noch nicht voraussehen ließe.
Man empfindet darum die allgemeinen Betrachtungen der Mengenlehre (wenn man sie
nicht als Kalkül ansieht) immer als Geschwätz und ist ganz erstaunt, wenn einem eine27
Anwendung dieser Betrachtungen gezeigt wird. Man empfindet, es geht da etwas nicht ganz
mit rechten Dingen zu.
Der Unterschied zwischen etwas Allgemeinem, das man wissen könne und dem
Besonderen, das man aber nicht wisse; oder zwischen der Beschreibung des Gegenstandes,
die man kenne, und dem Gegenstand, den man nicht gesehen hat, ist auch ein Stück, das
man von der physikalischen Beschreibung der Welt in die Logik hinüber genommen hat.
Daß unsere Vernunft Fragen erkennen kann, aber deren Antworten nicht, gehört auch
hierher.
25 (V): 2
26 (O): verdächtlich,
27 (V): die
747
493 Zur Mengenlehre.
746
WBTC136-140 30/05/2005 16:14 Page 986
It is less misleading to say “m = 2n allows the possibility of correlating every number
with another” than to say “m = 2n correlates all numbers with others”. But here again we
don’t know the meaning of the expression “possibility of correlation” until grammar has
taught it to us.
(It’s almost unbelievable how a problem gets completely barricaded for miles by the
misleading formulations of it that generation upon generation throw up around it, so that it
becomes virtually impossible to get to it.)
When two arrows are pointing in the same direction, isn’t it absurd to call these directions
“equally long”, because whatever lies in the direction of the one arrow, also lies in that
of the other? – The generality of m = 2n is an arrow that points along the series generated
by the operation. And in fact you can say that the arrow points into the infinite; but does
that mean that there is a something, the infinite, that it points to – as to a thing? – The arrow
designates the possibility of things lying in its direction, as it were. But the word “possibility”
is misleading for, as someone will say, what is possible might now become actual.
Furthermore, we always think of temporal processes in this context, and infer from the fact
that mathematics has nothing to do with time, that in it possibility is already actuality.
The “infinite series of cardinal numbers” or “the concept of cardinal number” is a
possibility only in this way – as emerges clearly from the symbol “|0, ξ, ξ + 1|”. This
symbol is itself an arrow, with the “0” as its tail and the “ξ + 1” as its tip. It’s possible to
speak of things which lie in the direction of the arrow, but it’s misleading or absurd to speak
of all possible positions of things lying in the direction of the arrow as an equivalent for this
direction itself. If a searchlight sends light out into infinite space it does indeed illuminate
everything lying in the direction of its rays, but you shouldn’t say it illuminates infinity.
It is always right to be extremely suspicious when proofs in mathematics are carried
out in a more general fashion than is warranted by the known application of the proof.
These are always instances of the mistake that sees general concepts and particular cases in
mathematics. In set theory we meet this suspect generality at every step.
One always feels like saying “Let’s get down to the matter at hand”.
In all cases these general considerations only make sense when we have a particular area
of application in mind.
In mathematics there just isn’t any such thing as a generalization whose application to
particular cases is still unforeseeable.
That’s why the general discussions of set theory (if they aren’t viewed as calculi) always
strike us as blather, and why we are greatly astounded when we are shown an application
for them. We feel that something is going on here that isn’t quite right.
The distinction between something general, which one can know, and the particular, which
one doesn’t know, or between the “known” description of the object, and the object that one
hasn’t seen, is another piece that has been carried over from the physical description of the
world into logic. That our reason can discover questions but not their answers also belongs
in this context.
On Set Theory. 493e
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Die Mengenlehre sucht das Unendliche auf eine allgemeinere Art zu fassen, als es
die Untersuchung der Gesetze der reellen Zahlen kann. Sie sagt, daß das wirklich
Unendliche mit dem mathematischen Symbolismus überhaupt nicht zu fassen ist, und
daß es also nur beschrieben und nicht dargestellt werden kann. Die Beschreibung würde es
etwa so erfassen, wie man eine Menge von Dingen, die man nicht alle in der Hand
halten kann, in einer Kiste verpackt trägt. Sie sind dann unsichtbar, und doch wissen
wir, daß wir sie tragen (gleichsam indirekt). Man könnte von dieser Theorie sagen, sie
kaufe die Katze im Sack. Soll sich’s das Unendliche in seiner Kiste einrichten, wie es will.
Darauf beruht auch die Idee, daß man logische Formen beschreiben kann. In so einer
Beschreibung werden uns die Strukturen in einer Verpackung gezeigt, die ihre Form
unkenntlich macht28
und so sieht es aus, als könne man von einer Struktur reden, ohne
sie in der Sprache selber wiederzugeben. So verpackte Begriffe dürfen wir allerdings
verwenden, aber unsere Zeichen haben ihre Bedeutung dann über Definitionen, die eben
die Strukturen29
so verhüllt haben; und gehen wir diesen Definitionen nach, so werden
die Strukturen wieder enthüllt. (Vergl. Russells Definition von ”R*“.)
Es geht, sozusagen, die Logik nichts an, wieviele Äpfel vorhanden sind, wenn von
”allen Äpfeln“ geredet wird; dagegen ist es anders mit den Zahlen: für die ist sie einzeln
verantwortlich.
Die Mathematik besteht ganz aus30
Rechnungen.
In der Mathematik ist alles Algorithmus,31
nichts Bedeutung; auch dort, wo es so scheint,
32
weil wir mit Worten über die mathematischen Dinge zu sprechen scheinen. Vielmehr bilden
wir dann eben mit diesen Worten einen Algorithmus.33
In der Mengenlehre müßte man das, was Kalkül ist, trennen von dem, was Lehre sein
will (und natürlich nicht sein kann). Man muß also die Spielregeln von unwesentlichen
Aussagen über die Schachfiguren trennen.
Wie Frege in Cantor’s angebliche Definition von ”größer“, ”kleiner“, ”+“, ”−“, etc. statt
dieser Zeichen neue Wörter einsetzte, um zu zeigen, daß keine wirkliche Definition vorliege,
ebenso könnte man in der ganzen Mathematik statt der geläufigen Wörter, insbesondere
statt des Wortes ”unendlich“ und seiner Verwandten ganz neue, bisher bedeutungslose
Ausdrücke setzen, um zu sehen, was der Kalkül mit diesen Zeichen wirklich leistet und
was er nicht leistet. Wenn die Meinung verbreitet wäre, daß das Schachspiel uns einen
Aufschluß über Könige und Türme gäbe, so würde ich vorschlagen, den Figuren neue
Formen und andere Namen zu geben, um zu demonstrieren,34
daß alles zum Schachspiel
Gehörige in seinen35
Regeln liegen muß.
Was ein geometrischer Satz bedeutet, was für eine Art der Allgemeinheit36
er hat, das muß
sich alles zeigen, wenn wir sehen, wie er angewendet wird. Denn, wenn Einer auch etwas
28 (V): werden die Strukturen und etwa zuordnende
Relationen in verpacktem Zustand
präsentiert // gezeigt //
29 (V): Begriffe
30 (V): besteht aus
31 (O): Algoritmus,
32 (V): scheint, als
33 (O): Algorismus. // Algoritmus.
34 (V): um die Einsicht zu erleichtern,
35 (V): den
36 (V): bedeutet, welche Allgemeinheit
748
749
494 Zur Mengenlehre.
WBTC136-140 30/05/2005 16:14 Page 988
Set theory attempts to grasp the infinite in a more general way than the investigation of
the laws of the real numbers can. It says that you can’t grasp the actual infinite through mathematical
symbolism at all, and that therefore it can only be described and not represented.
The description would encompass it in something like the way in which you carry with you
a quantity of things too numerous to be held in your hand by packing them in a box. They’re
then invisible, but still we know that we’re carrying them (so to speak, indirectly). One
could say of this theory that it buys a pig in a poke. Let the infinite accommodate itself in
its box as it likes.
This is also the basis for the idea that we can describe logical forms. In a description of
this sort the structures are shown to us in wrapping that makes their shape unrecognizable,19
and so it looks as if one could speak of a structure without reproducing it in language itself.
Concepts that are packed up like this may, to be sure, be used, but then our signs will derive
their meaning from the very definitions that have covered up the structures20
in this way;
and if we pursue these definitions, the structures are uncovered again. (Cf. Russell’s
definition of “R*”.)
It’s none of logic’s business, so to speak, how many apples there are when we speak of
“all apples”. With numbers it is different, though; logic is responsible for each and every
one of them.
Mathematics consists entirely of21
calculations.
In mathematics everything is algorithm, nothing meaning; even when it seems there’s
meaning, because we appear to be speaking about mathematical things in words. What we’re
really doing in that case is simply constructing an algorithm with those words.
In set theory what is calculus ought to be separated from what claims to be (and of course
cannot be) theory. The rules of the game have thus to be separated from inessential statements
about the chessmen.
Frege replaced those signs in Cantor’s alleged definitions of “greater”, “smaller”, “+”,
“−”, etc., with new words, to show that here there wasn’t any real definition. In the same
way, in all of mathematics one could replace the usual words, especially the word “infinite”
and its cognates, with entirely new and hitherto meaningless expressions, in order to see
what the calculus with these signs really achieves and what it doesn’t achieve. If the idea
were widespread that chess gave us information about kings and castles, I would propose
giving the pieces new shapes and different names, in order to demonstrate22
that everything
belonging to chess has to be contained in its23
rules.
What a geometrical proposition means, what kind of generality24
it has – all of this must
show itself when we see how it is applied. For even if someone were to mean something
19 (V): structures and possible coordinate relationships
are presented // shown // in a
wrapped-up state,
20 (V): concepts
21 (V): consists of
22 (V): to facilitate the insight
23 (V): the
24 (V): means, which generality
On Set Theory. 494e
WBTC136-140 30/05/2005 16:14 Page 989
Unerreichbares37
mit ihm meinte,38
so hilft ihm das nicht, da er ihn ja doch nur ganz offen,39
und jedem verständlich, anwenden40
kann.
Wenn sich etwa jemand unter dem Schachkönig auch etwas Mystisches vorstellt,
so kümmert uns das nicht, weil er ja doch mit ihm nur auf den 8 × 8 Feldern des
Schachbretts ziehen kann.
Es gibt ein Gefühl: ”In der Mathematik kann es nicht Wirklichkeit und Möglichkeit
geben. Alles ist auf einer Stufe. Und zwar in gewissem Sinne wirklich“. – Und das ist
richtig. Denn Mathematik ist ein Kalkül; und der Kalkül sagt von keinem Zeichen, daß
es nur möglich wäre, sondern er hat es nur mit den Zeichen zu tun, mit denen er wirklich
operiert. (Vergleiche die Begründung der Mengenlehre mit der Annahme eines möglichen
Kalküls mit unendlichen Zeichen.)
Die Mengenlehre, wenn sie sich auf die menschliche Unmöglichkeit eines direkten
Symbolismus des Unendlichen beruft, führt dadurch die denkbar krasseste Mißdeutung
ihres eigenen Kalküls ein. Es ist freilich eben diese Mißdeutung, die für die Erfindung dieses
Kalküls verantwortlich ist. Aber der Kalkül an sich ist natürlich dadurch nicht als etwas
Falsches erwiesen (höchstens als etwas Uninteressantes), und es ist sonderbar, zu glauben,
daß dieser Teil der Mathematik durch irgendwelche philosophische (oder mathematische)
Untersuchungen gefährdet ist. (Ebenso könnte das Schachspiel durch die Entdeckung
gefährdet werden, daß sich Kriege zwischen zwei Armeen nicht so abspielen, wie der
Kampf auf dem Schachbrett.) Was der Mengenlehre verloren gehen muß, ist vielmehr die
Atmosphäre von Gedankennebeln, die den bloßen Kalkül umgibt. Also die Hinweise auf
einen, der Mengenlehre zugrunde liegenden, fiktiven Symbolismus, der nicht zu ihrem
Kalkül verwendet wird, und dessen scheinbare Beschreibung in Wirklichkeit Unsinn ist.
(In der Mathematik dürfen41
wir alles fingieren, nur nicht einen Teil unseres Kalküls.)
37 (V): Unfaßbares
38 (V): mit ihm meinen könnte,
39 (V): offenbar,
40 (O): anweden
41 (V): können
750
495 Zur Mengenlehre.
WBTC136-140 30/05/2005 16:14 Page 990
inaccessible by it25
it wouldn’t help him, because he can only apply it in a way that’s
completely open26
and intelligible to every one.
Even if someone imagines the chess king as something mystical, that doesn’t concern us,
because all he can do is make moves with it on the 8 × 8 squares of the chess board.
There is a feeling: “There can’t be actuality and possibility in mathematics. Everything
is on one level. And in fact, is in a certain sense actual”. – And that’s correct. For mathematics
is a calculus; and a calculus does not say of any sign that it is merely possible; rather,
a calculus is concerned only with the signs with which it actually operates. (Cf. justifying
set theory by assuming a possible calculus that uses infinite signs.)
When set theory appeals to the human impossibility of a direct symbolization of the infinite
it thereby introduces the crudest imaginable misinterpretation of its own calculus. To be sure,
it is this very misinterpretation that is responsible for the invention of that calculus. But
of course that doesn’t show the calculus to be something inherently incorrect (at most it
shows it to be something uninteresting), and it’s odd to believe that this part of mathematics
is imperilled by any kind of philosophical (or mathematical) investigations. (With
equal justification chess might be imperilled by the discovery that wars between two armies
do not follow the same course as the battle on the chess board.) What set theory has to lose
is rather the atmosphere of thought-fog surrounding the bare calculus, that is to say, the
references to a fictional symbolism underlying set theory, a symbolism that isn’t employed
in its calculus, and the apparent description of which is really nonsense. (In mathematics
we’re allowed27
to make up everything, except for a part of our calculus.)
25 (V): someone succeeded in meaning something
incomprehensible by it
26 (V): obvious
27 (V): able
On Set Theory. 495e
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138
Extensive Auffassung
der reellen Zahlen.
|Das Rätselhafte am Kontinuum ist, wie das Rätselhafte der Zeit für Augustinus,
dadurch bedingt, daß wir durch die Sprache verleitet werden, ein Bild auf es1
anzuwenden,
das nicht paßt. Die Mengenlehre behält das unpassende Bild des Diskontinuierlichen bei,
aber sagt diesem Bilde Widersprechendes von ihm aus, mit der Idee, mit Vorurteilen zu
brechen. Während in Wirklichkeit darauf hingewiesen werden sollte, daß dieses Bild eben
nicht paßt und daß man es allerdings nicht strecken kann, ohne es zu zerreißen,2
aber ein
neues und in 3
gewissem Sinne dem alten ähnliches brauchen kann.|
|Der Wirrwarr in der Auffassung des ”wirklich Unendlichen“ kommt von dem unklaren
Begriff der irrationalen Zahl her. D.h. davon, daß die logisch verschiedensten Gebilde, ohne
klare Begrenzung des Begriffs, ”irrationale Zahl“ genannt werden. Die Täuschung, als hätte
man einen festen Begriff, beruht darauf,4
daß man in Zeichen von der Art ”0,abc5
. . . ad
inf.“ einen Standard6
zu haben glaubt, dem sie (die Irrationalzahlen) jedenfalls entsprechen
müssen.|
”Angenommen, ich schneide eine Strecke dort, wo kein rationaler Punkt (keine rationale
Zahl) ist.“ Aber kann man denn das? von was für Strecken sprichst Du? – ”Aber, wenn
meine Meßinstrumente fein genug wären, so könnte ich mich doch durch fortgesetzte
Bisektionen einem gewissen Punkt unbegrenzt nähern.“ – Nein, denn ich könnte ja eben
niemals erfahren, ob mein Punkt ein solcher ist. Meine Erfahrung wird immer nur sein, daß
ich ihn bis jetzt nicht erreicht habe. ”Aber wenn ich nun mit einem absolut genauen Reißzeug
die Konstruktion der durchgeführt hätte und mich nun dem erhaltenen Punkt durch
Bisektion nähere, dann weiß ich doch, daß dieser Prozeß den konstruierten Punkt niemals
erreichen wird.“ – Aber das wäre doch sonderbar, wenn so die eine Konstruktion der andern
sozusagen etwas vorschreiben könnte! Und so ist es ja auch nicht. Es ist sehr leicht möglich,
daß ich bei der ”genauen“ Konstruktion der zu einem Punkt komme, den die Bisektion,
sagen wir nach 100 Stufen, erreicht; – aber dann werden wir sagen: unser Raum ist nicht
euklidisch. –
Der ”Schnitt in einem irrationalen Punkt“ ist ein Bild, und ein irreführendes Bild.
Ein Schnitt ist ein Prinzip der Teilung in größer und kleiner.
Sind durch den Schnitt einer Strecke die Resultate aller Bisektionen, die sich dem
Schnittpunkt nähern sollen, vorausbestimmt? Nein.
2
2
1 (O): sie
2 (V): zerbrechen,
3 (V): neues in
4 (V): Begriff, rührt daher,
5 (V): ”O,abcd
6 (V): Begriff // Bild
751
752
496
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138
The Extensional Conception of
the Real Numbers.
|Like the enigma of time for Augustine, what is enigmatic about the continuum arises
because language misleads us into applying a picture to it that doesn’t fit. Set theory preserves
the inappropriate picture of something discontinuous, but makes statements about it
that contradict this picture, with the idea of breaking with prejudices; whereas what really
ought to be done is to point out that the picture just doesn’t fit, and that although one can’t
stretch it without tearing1
it, one can use a new picture that is in a certain sense similar to
the old one.|
|The confusion in the conception of the “actual infinite” arises from the unclear concept
of “irrational number”, that is, from the fact that constructs that are logically quite different
are called “irrational numbers” without any clear limits being given to the concept.
The illusion that we have a firm concept rests on2
our belief that in signs of the form
“0·abc3
. . . ad infinitum” we have a standard4
to which they (the irrational numbers) have
to conform whatever happens.|
“Suppose I cut a line-segment at a place where there is no rational point (no rational
number).” But can you do that? What sort of line-segment are you speaking of? – “But
if my measuring instruments were fine enough, then by continued bisection I could get
infinitely close to a certain point.” – No, for I could never find out whether my point was a
point of this kind. All my experience will ever be is that I haven’t reached it up to now. “But
if I had carried out the construction of with absolutely precise drawing instruments,
and now by bisection draw nearer to the point I had produced, then I know that this process
will never reach the constructed point.” – But it would be odd if the one construction
could, as it were, prescribe something to the others in this way! And indeed that’s not the
way it is. It’s quite possible that in the process of the “exact” construction of I get to a
point that is reached by the bisection after say 100 steps; – but in that case we’ll say: our
space is not Euclidean. –
The “cut at an irrational point” is a picture, and a misleading picture.
A cut is a principle of division into greater and smaller.
Does a cut through a length determine in advance the results of all bisections that are
supposed to approach the point of the cut? No.
2
2
1 (V): breaking
2 (V): concept comes from
3 (V): “0·abcd
4 (V): concept // picture
496e
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In dem vorigen Beispiel,7
in dem ich mich bei der successiven Einschränkung eines
Intervalls durch Bisektionen einer Strecke von den Ergebnissen des Würfelns leiten ließ,
hätte ich ebensowohl das Anschreiben eines Dezimalbruches von Würfeln leiten lassen
können. So bestimmt auch die Beschreibung ”endloser Vorgang des Wählens zwischen 1
und 0“ beim Anschreiben eines Dezimalbruches kein Gesetz. Man möchte etwa sagen:
Die Vorschrift des endlosen Wählens zwischen 0 und 1 in diesem Falle könnte durch ein
Symbol ” “ wiedergegeben werden. Wenn ich aber ein Gesetz so andeute:
”0,001001001 . . . ad inf.“, so ist es nicht das endliche Reihenstück als Specimen der
unendlichen Reihe, was ich zeigen will, sondern die aus ihm entnehmbare Gesetzmäßigkeit.
Aus ” “ entnehme ich eben kein Gesetz, sondern gerade den Mangel eines
Gesetzes.
(Welches8
Kriterium gibt es dafür, daß die irrationalen Zahlen komplett sind? Sehen wir
uns eine irrationale Zahl an: Sie läuft entlang einer Reihe rationaler Näherungswerte. Wann
verläßt sie diese Reihe? Niemals. Aber sie kommt allerdings auch niemals zu einem Ende.
Angenommen, wir hätten die Gesamtheit aller irrationalen Zahlen mit Ausnahme einer
einzigen. Wie würde uns diese abgehen? Und wie würde sie nun – wenn sie dazukäme – die
Lücke füllen? – Angenommen, es wäre π. Wenn die irrationale Zahl durch die Gesamtheit
ihrer Näherungswerte gegeben ist, so gäbe es bis zu jedem beliebigen Punkt eine Reihe, die
mit der von π übereinstimmt. Allerdings kommt für jede solche Reihe ein Punkt der Trennung.
Aber dieser Punkt kann beliebig weit ”draußen“ liegen, so daß ich zu jeder Reihe, die
π begleitet, eine finden kann, die es weiter begleitet. Wenn ich also die Gesamtheit der
irrationalen Zahlen habe, außer π, und nun π einsetze, so kann ich keinen Punkt angeben,
an dem π nun wirklich nötig wird, es hat an jedem Punkt einen Begleiter, der es vom Anfang
an begleitet.
Auf die Frage ”wie würde uns π abgehen“, müßte man antworten: π, wenn es eine Extension
wäre, würde uns niemals abgehen. D.h., wir könnten niemals eine Lücke bemerken, die es
füllt. Wenn man uns fragte: ”aber hast Du auch einen unendlichen Dezimalbruch, der die
Ziffer m an der r-ten Stelle hat und n an der s-ten, etc.?“ – wir könnten ihm immer dienen.)
”Die gesetzmäßig fortschreitenden unendlichen Dezimalbrüche sind noch ergänzungsbedürftig
durch eine unendliche Menge regelloser9
unendlicher Dezimalbrüche, die ,unter
den Tisch fielen‘, wenn wir uns auf die gesetzmäßig erzeugten beschränkten.“ Wo ist so
ein nicht gesetzmäßig erzeugter unendlicher Dezimalbruch? Und wie können wir ihn
vermissen? Wo ist die Lücke, die er auszufüllen hätte?
Wie ist es, wenn man die verschiedenen Gesetze der Bildung von Dualbrüchen durch
die Menge der endlichen Kombinationen der Ziffern 0 und 1 sozusagen kontrolliert? –
Die Resultate eines Gesetzes durchlaufen die endlichen Kombinationen und die Gesetze
sind daher, was ihre Extensionen anlangt, komplett, wenn alle endlichen Kombinationen
durchlaufen werden.
Wenn man sagt: zwei Gesetze sind identisch, wenn sie auf jeder Stufe das gleiche Resultat
ergeben, so erscheint uns das wie eine ganz allgemeine Regel. In Wirklichkeit aber hat dieser
Satz verschiedenen Sinn, je nachdem was das Kriterium dafür ist, daß sie auf jeder Stufe
das gleiche Resultat liefern. (Denn die supponierte allgemein anwendbare Methode des
endlosen Probierens gibt es ja nicht!) Wir decken also die verschiedensten Bedeutungen
7 (E): Das Beispiel findet sich auf S. 504 unten.
8 (O): ”Welches
9 (V): ungeordneter
753
754
497 Extensive Auffassung der reellen Zahlen.
0, 111 . . . ad inf.000
0, 111 . . . ad inf.000
WBTC136-140 30/05/2005 16:14 Page 994
In my previous example,5
where I threw dice to guide me in the successive reduction of
an interval by the bisection of a length, I could just as well have thrown dice to guide me
in the writing of a decimal fraction. In the same way, the description “endless process of
choosing between 1 and 0” does not determine a law in the writing of a decimal fraction.
You feel like saying, for instance: The instruction for the endless choice between 0 and 1 in
this case could be rendered by a symbol like “ ”. But if I refer to a
law in this way: “0·001001001 . . . ad inf.”, what I want to show is not the finite section
of the series as a specimen of the infinite series, but rather the regularity to be gathered
from the specimen. What I infer from “ ” is not a law, but precisely the
lack of a law.
(What criterion is there for the irrational numbers being complete? Let’s look at an
irrational number: it runs alongside a series of rational approximations. When does it leave
this series? Never. But on the other hand it also never comes to an end.
Suppose we had the totality of all irrational numbers with one exception. How would we
miss that one? And how would it – if it were added, fill the gap? – Suppose that it were π.
If an irrational number is given through the totality of its approximations, then up to any
point taken at random there would be a series coinciding with that of π. To be sure, for each
such series there comes a point where they diverge. But this point can lie arbitrarily far
“outside”, so that for any series agreeing with π I can find one agreeing with it further.
So if I have the totality of all irrational numbers except π, and now insert π, I can’t cite a
point at which π now really becomes necessary. At every point it has a companion agreeing
with it from the beginning.
To the question “How would we miss π?” our answer would have to be “If π were an
extension, we would never miss it”, i.e. we would never notice a gap that it fills. If someone
asked us “But do you have an infinite decimal expansion with the number m in the rth
place and n in the sth place, etc.?” we could always oblige him.)
“The infinite decimal fractions developed in accordance with a law still need supplementing
by an infinite set of irregular6
infinite decimal fractions that would ‘fall by the
wayside’ if we were to restrict ourselves to those generated by a law.” Where is there such
an infinite decimal fraction that isn’t generated according to a law? And how can we miss
it? Where is the gap that it would have to fill?
What is it like if someone so to speak “inspects” the various laws for the construction of
binary fractions using the set of finite combinations of the numerals 0 and 1? – The results
of a law run through the finite combinations, and therefore the laws are complete as far as
their extensions are concerned, if all the finite combinations are gone through.
If one says “Two laws are identical if they yield the same result at every stage”, this
looks to us like a completely general rule. But in reality this proposition has different senses,
depending on what the criterion is for their yielding the same result at every stage. (For
of course there’s no such thing as the supposed generally applicable method of infinite
5 (E): The example is on p. 504e below. 6 (V): random
The Extensional Conception of the Real Numbers. 497e
0, 111 . . . ad inf.000
0, 111 . . . ad inf.000
WBTC136-140 30/05/2005 16:14 Page 995
mit einer, von einer Analogie hergenommenen, Redeweise und glauben nun, wir hätten
die verschiedensten Fälle in einem System vereinigt.
(Die Gesetze,10
die den irrationalen Zahlen entsprechen, gehören insofern alle der gleichen
Type an, als sie alle schließlich Vorschriften zur successiven Erzeugung von Dezimalbrüchen
sein müssen. Die gemeinsame Dezimalnotation bedingt in gewissem Sinne, eine gemeinsame
Type.)
Man könnte das auch so sagen: Beim Approximieren durch fortgesetzte Zweiteilung
kann man sich jedem Punkt der Strecke durch rationale Zahlen nähern. Es gibt keinen Punkt,
dem man sich nur durch irrationale Schritte einer bestimmten Type nähern könnte. Dies
ist natürlich nur, in andere Worte gekleidet, die Erklärung, daß wir unter irrationaler Zahl
einen unendlichen Dezimalbruch verstehen. Und diese Erklärung wieder ist weiter nichts,
als eine beiläufige Erklärung der Dezimalnotation, etwa mit einer Andeutung, daß wir Gesetze
unterscheiden, die periodische Dezimalbrüche liefern und andere.
Durch die falsche Auffassung des Wortes ”unendlich“ und der Rolle der ”unendlichen
Entwicklung“ in der Arithmetik der reellen Zahlen, wird man zu der Meinung verführt, es
gäbe eine einheitliche Notation der irrationalen Zahlen (nämlich eben die der unendlichen
Extension, z.B. der unendlichen Dezimalbrüche).
Dadurch, daß man bewiesen hat, daß für jedes Paar von Kardinalzahlen x und y ( )2
≠ 2
ist, ist doch nicht einer Zahlenart – genannt ”die irrationalen Zahlen“ – eingeordnet. Diese
Zahlenart müßte ich doch erst aufbauen; oder: von der neuen Zahlenart ist mir doch nicht
mehr bekannt, als ich bekannt mache.
2
x
y
10 (V): Vorschriften,
755
498 Extensive Auffassung der reellen Zahlen.
WBTC136-140 30/05/2005 16:14 Page 996
checking!) By adopting a mode of speaking from an analogy, we conceal the most various
meanings, and then believe that we have united the most disparate cases within a single
system.
(The laws7
corresponding to the irrational numbers all belong to the same type, to the
extent that they must all ultimately be recipes for the successive construction of decimal
fractions. In a certain sense the common decimal notation gives rise to a common type.)
We could also put it this way: In the process of approximation by continued bisection one
can draw nearer to every point in a length by way of rational numbers. There is no point
that could only be approached by way of irrational steps of a specified type. Of course, that
is nothing but the explanation – clothed in different words – that by irrational numbers we
mean endless decimal fractions; and that explanation in turn is only a rough explanation of
the decimal notation, including, say, a suggestion that we distinguish between laws that
yield recurring decimals and laws that don’t.
The incorrect conception of the word “infinite”, and of the role of “infinite expansion”
in the arithmetic of the real numbers, seduces us into thinking that there is a uniform
notation for irrational numbers (namely the notation of the infinite extension, e.g. of infinite
decimal fractions).
In proving that for every pair of cardinal numbers x and y, ( )2
≠ 2, we have not
correlated with a single kind of number – called “the irrational numbers”. This type
of number is something that I still have to construct; or: I don’t know any more about the
new type of number than I make known.
2
x
y
7 (V): prescripts
The Extensional Conception of the Real Numbers. 498e
WBTC136-140 30/05/2005 16:14 Page 997
139
Arten irrationaler Zahlen.
(π′, P, F)
π′ ist eine Regel zur Erzeugung von Dezimalbrüchen, und zwar ist die Entwicklung
von π′ dieselbe, wie die von π, außer wenn in der Entwicklung von π eine Gruppe 777
vorkommt; in diesem Falle tritt statt dieser Gruppe die Gruppe 000. Unser Kalkül kennt
keine Methode, um zu finden, wo wir in der Entwicklung von π auf so eine Gruppe stoßen.
P ist eine Regel zur Erzeugung von Dualbrüchen. In der Entwicklung steht an der n-ten
Stelle eine 1 oder eine 0, je nachdem n prim ist oder nicht.
F ist eine Regel zur Erzeugung von Dualbrüchen. An der n-ten Stelle steht eine 0, außer
dann, wenn ein Zahlentrippel x, y, z aus den ersten 100 Kardinalzahlen die Gleichung
xn
+ yn
= zn
löst.
Man möchte sagen, die einzelnen Ziffern der Entwicklung (von π z.B.) sind immer nur
die Resultate, die Rinde des fertigen Baumes. Das, worauf es ankommt, oder woraus noch
etwas Neues wachsen kann, ist im Innern des Stammes, wo die Triebkräfte sind. Eine
Änderung des Äußeren ändert den Baum überhaupt nicht. Um ihn zu ändern, muß man in
den noch lebenden Stamm gehen.
Ich nenne ”πn“ die Entwicklung von π bis zur n-ten Stelle. Dann kann ich sagen: Welche
Zahl π′100 bedeutet,1
verstehe ich; nicht aber, (welche) π′, weil2
π ja gar keine Stellen hat,
ich also auch keine durch andere ersetzen kann. Anders wäre es, wenn ich z.B. die Division
3
als eine Regel zur Erzeugung von Dezimalbrüchen erkläre, durch Division und
Ersetzung jeder 5 im Quotienten durch eine 3. Hier kenne ich z.B. die Zahl4
– Und wenn unser Kalkül eine Methode enthält, ein Gesetz der Lagen von 777 in der
Entwicklung von π zu berechnen, dann ist nun im Gesetz von π von 777 die Rede, und
das Gesetz kann durch die Substitution von 000 für 777 geändert werden. Dann aber ist π′
etwas anderes, als das, was ich oben definiert habe; es hat eine andere Grammatik, als die
von mir angenommene. In unserm Kalkül gibt es keine Frage, ob π gleich oder größer ist
als π′5
und keine solche Gleichung oder Ungleichung. π′ ist mit π unvergleichbar. Und zwar
kann man nun nicht sagen ”noch unvergleichbar“, denn, sollte ich einmal etwas π′ Ähnliches
konstruieren, das mit π vergleichbar ist, dann wird das eben darum nicht mehr π′ sein. Denn
π′ sowie π sind ja Bezeichnungen für ein Spiel, und ich kann nicht sagen, das Damespiel
werde noch mit weniger Steinen gespielt als das Schach, da es sich ja einmal zu einem Spiel
mit 16 Steinen entwickeln könne. Dann wird es nicht mehr das sein, was wir ”Damespiel“
nennen. (Es sei denn, daß ich mit diesem Wort gar nicht ein Spiel bezeichne, sondern
1 (V): ist,
2 (V): nicht aber π′, weil
3 (F): MS 113, S. 133r.
4 (F): MS 113, S. 133r.
5 (O): ob π π″ ist oder nicht (V): ob π π′ ist
oder nicht
756
757
499
a : b
5 3→
1 : 7
5 3→
.
WBTC136-140 30/05/2005 16:14 Page 998
139
Kinds of Irrational Numbers.
(π′, p, f)
π′ is a rule for the formation of decimal fractions; specifically, the expansion of π′ is the
same as the expansion of π except where the sequence 777 occurs in the expansion of π;
in that case the sequence 000 replaces the sequence 777. There is no method known to our
calculus for discovering where we will encounter such a sequence in the expansion of π.
P is a rule for the construction of binary fractions. At the nth place of the expansion there
occurs a 1 or a 0, depending on whether n is prime or not.
F is a rule for the construction of binary fractions. At the nth place there is a 0, except
when a triple x, y, z from the first 100 cardinal numbers satisfies the equation xn
+ yn
= zn
.
I’m tempted to say that the individual numbers of the expansion (of π, for example) are
always only the results, the bark of the fully grown tree. What counts, or what something
new can still grow from, is in the inside of the trunk, where the tree’s vital energy is. Altering
the exterior doesn’t change the tree at all. To change it, you have to enter into the living
part of the trunk.
I call “πn” the expansion of π up to the nth place. Then I can say: I understand what
number π′100 means,1
but not (what number) π′ means,2
since π has no places at all, and
therefore I can’t replace any of its places with others. It would be different if for example I
defined the division3
as a rule for the formation of decimal fractions by division and
the replacement of every 5 in the quotient by a 3. In this case I am acquainted, for instance,
with the number4
– And if our calculus contains a method to calculate a law of the
positions of 777 in the expansion of π, then 777 is mentioned in the law of π, and the law
can be altered by the substitution of 000 for 777. But in that case π′ is something different
from what I defined above; it has a different grammar from the one I supposed. In our
calculus there is no question whether π is equal to or greater than π′,5
and there is no such
equation or inequality. π′ is not comparable to π. And, furthermore, one can’t say “not yet
comparable”, because if at some time I should construct something similar to π′ that is
comparable to π, then for that very reason it will no longer be π′. For π′, like π, is a way of
denoting a game, and I cannot say that draughts is still played with fewer pieces than chess,
since one day it might very well develop into a game with 16 pieces. In that case it would
no longer be what we call “draughts” (unless I’m not using this word to designate a game
1 (V): is,
2 (V): but not π′,
3 (F): MS 113, p. 133r.
4 (F): MS 113, p. 133r.
5 (V): whether π is or is not π′,
499e
a b÷
→5 3
1 7÷
→5 3
.
WBTC136-140 30/05/2005 16:14 Page 999
etwa eine Charakteristik mehrerer Spiele; und auch diesen Nachsatz kann man auf π′ und
π anwenden.) Da es nun ein Hauptcharakteristikum einer Zahl ist, mit andern Zahlen
vergleichbar zu sein, so ist die Frage, ob man π′ eine Zahl nennen soll und ob eine reelle
Zahl; wie immer man es aber nennt, so ist das Wesentliche, daß π′ in einem andern Sinne
Zahl ist, als π. – Ich kann ja auch ein Intervall einen Punkt nennen; ja es kann einmal
praktisch sein, das zu tun; aber wird es nun einem Punkt ähnlicher, wenn ich vergesse, daß
ich hier das Wort ”Punkt“ in doppelter Bedeutung gebraucht habe?
Es zeigt sich hier klar, daß die Möglichkeit der Dezimalentwicklung π′ nicht zu einer Zahl
im Sinne von π macht. Die Regel für diese Entwicklung ist natürlich eindeutig, so eindeutig
wie die für π oder , aber das ist kein Argument dafür, daß π′ eine reelle Zahl ist; wenn
man die Vergleichbarkeit mit rationalen Zahlen6
für ein wesentliches Merkmal der reellen
Zahl nimmt. Man kann ja auch von dem Unterschied zwischen den rationalen und den irrationalen
Zahlen abstrahieren, aber der Unterschied verschwindet doch dadurch nicht. Daß
π′ eine eindeutige Regel zur Entwicklung von Dezimalbrüchen ist, konstituiert7
natürlich
eine Ähnlichkeit zwischen π′ und π oder ; aber auch ein Intervall8
hat Ähnlichkeit mit
einem Punkt, etc. Allen Irrtümern, die in diesem Kapitel der Philosophie der Mathematik
gemacht werden, liegt immer wieder die Verwechslung zu Grunde zwischen internen
Eigenschaften einer Form (der Regel als Bestandteil des Regelverzeichnisses) und dem, was
man im gewöhnlichen Leben ”Eigenschaft“ nennt (rot als Eigenschaft dieses Buches). Man
könnte auch sagen; die Widersprüche und Unklarheiten werden dadurch hervorgerufen,
daß die Menschen9
einmal unter einem Wort, z.B. ”Zahl“, ein bestimmtes Regelverzeichnis
verstehen, ein andermal ein variables Regelverzeichnis; so als nennte ich ”Schach“ einmal
das bestimmte Spiel, wie wir es heute spielen, ein andermal das Substrat einer bestimmten
historischen Entwicklung.
”Wie weit muß ich π entwickeln, um es einigermaßen zu kennen?10
“ – Das heißt
natürlich nichts. Wir kennen es also schon, ohne es überhaupt zu entwickeln. Und, in
diesem Sinne, könnte man sagen, kenne ich π′ gar nicht. Hier zeigt sich nur ganz deutlich,
daß π′ einem anderen System angehört als π, und das erkennt man, wenn man, statt ”die
Entwicklungen“ der beiden zu vergleichen, die Art der Gesetze allein ins Auge faßt.
Zwei mathematische Gebilde, deren eines ich in meinem Kalkül mit jeder rationalen
Zahl vergleichen kann, das andere nicht, – sind nicht Zahlen im gleichen Sinne des Wortes.
Der Vergleich der Zahl mit einem Punkt auf der Zahlengeraden11
ist nur stichhältig, wenn
man für je zwei Zahlen a und b sagen kann, ob a rechts von b, oder b rechts von a liegt.
Es genügt nicht, daß man den Punkt durch Verkleinerung seines Aufenthaltsortes –
angeblich – mehr und mehr bestimmt, sondern man muß ihn konstruieren. Fortgesetztes
Würfeln schränkt zwar den möglichen Aufenthalt des Punktes unbeschränkt ein, aber es
bestimmt keinen Punkt. Der Punkt ist nach jedem Wurf (oder jeder Wahl) noch unendlich
unbestimmt – oder richtiger: er ist nach jedem Wurf unendlich unbestimmt. Ich glaube, hier
werden wir von der absoluten Größe der Gegenstände in unserem Gesichtsraum irregeführt;
und andrerseits von der Zweideutigkeit des Ausdrucks ”sich einem Punkte12
nähern“.
Von einer Strecke im Gesichtsfeld kann man sagen, sie nähere sich durch Einschrumpfen
immer mehr einem Punkt; d.h. sie werde einem Punkt immer ähnlicher. Dagegen wird
die euklidische Strecke durch Einschrumpfen einem Punkt nicht ähnlicher, sie bleibt ihm
2
2
6 (V): mit andern reellen Zahlen
7 (V): bedeutet
8 (O): Interval
9 (V): Mathematiker
10 (V): erkennen?
11 (V): Zahlgeraden
12 (V): Gegenstand
758
759
500 Arten irrationaler Zahlen. . . .
WBTC136-140 30/05/2005 16:14 Page 1000
at all, but, say, a characteristic of several games; and this qualification can be applied to
π′ and π as well). Now since being comparable with other numbers is a fundamental characteristic
of a number, the question arises whether one is to call π′ a number, and whether
one is to call it a real number; but whatever it is called, the essential thing is that π′ is a
number in a different sense than π. I can also call an interval a point; and indeed on occasion
it can be practical to do so; but does it become more like a point if I forget that here
I’ve used the word “point” with two different meanings?
Here it’s clear that the possibility of the decimal expansion doesn’t turn π′ into a number
in the same sense as π. Of course the rule for this expansion is unambiguous, as unambiguous
as that for π or ; but that doesn’t mean that π′ is a real number, if one takes comparability
with rational6
numbers to be an essential feature of real numbers. Indeed, one can
also abstract from the distinction between the rational and irrational numbers, but that does
not make the distinction disappear. Of course the fact that π′ is an unambiguous rule for
the development of decimal fractions does constitute7
a similarity between π′ and π or ;
but then too an interval is similar to a point, etc. All the errors that are made in this
chapter of the philosophy of mathematics are based, again and again, on the confusion
between the internal properties of a form (of a rule as part of a list of rules) and what we
call “properties” in everyday life (red as a property of this book). We could also say: The
contradictions and unclarities are brought about by the fact that by a single word, e.g.
“number”, people8
understand at one time a definite set of rules, and at another time a
variable set: as if what I call “chess” was on one occasion the specific game as we play it
today, and on another the substratum of a particular historical development.
“How far must I expand π in order to know it to some extent?” – Of course that means
nothing. We already know it without expanding it at all. And in this sense it could be said
that I don’t know π′ at all. Here it becomes quite clear that π′ belongs to a different system
from π, and that is something we recognize if we look solely at the nature of the laws instead
of comparing “the expansions” of the two numbers.
Two mathematical structures, of which one but not the other can be compared in my
calculus with every rational number, are not numbers in the same sense of the word. The
comparison of a number to a point on the number line is valid only if we can say for every
pair of numbers a and b whether a is to the right of b or b to the right of a.
It’s not enough that we – supposedly – determine a point ever more closely by narrowing
down its whereabouts; rather, we must construct it. To be sure, continued throwing of
a die indefinitely restricts the possible location of a point, but it doesn’t determine a point.
After every throw (or every choice) the point is still infinitely indeterminate – or more
correctly, after every throw it is infinitely indeterminate. I think that we are misled here by
the absolute size of the objects in our visual space; and on the other hand, by the ambiguity
of the expression “to approach a point”.9
We can say of a line in the visual field that in
shrinking, it is getting closer and closer to a point; that is, that it is becoming more and
more like a point. On the other hand a Euclidean line does not become any more like a
point by shrinking; it always remains equally dissimilar to it, because its length, so to speak,
2
2
6 (V): with other real
7 (V): does mean
8 (V): “number”, the mathematicians
9 (V): approach an object”.
Kinds of Irrational Numbers. . . . 500e
WBTC136-140 02/06/2005 14:40 Page 1001
vielmehr immer gleich unähnlich, weil ihre Länge den Punkt, sozusagen, gar nichts angeht.
Wenn man von der euklidischen Strecke sagt, sie nähere sich durch Einschrumpfen einem
Punkt, so hat das nur Sinn, sofern schon ein Punkt bezeichnet ist, dem sich ihre Enden
nähern, und kann nicht heißen, sie erzeuge durch Einschrumpfen einen Punkt. Sich einem
Punkt nähern hat eben zwei Bedeutungen: es heißt einmal, ihm räumlich näher kommen,
dann muß er schon da sein, denn ich kann mich in diesem Sinne einem Menschen nicht
nähern, der nicht vorhanden ist. Anderseits heißt es ”einem Punkt ähnlicher werden“, wie
man etwa sagt, die Affen haben sich dem Stadium des Menschen in ihrer Entwicklung
genähert, die Entwicklung habe den Menschen erzeugt.
Zu sagen ”zwei13
reelle Zahlen sind identisch, wenn sie in allen Stellen ihrer Entwicklung
übereinstimmen“, hat nur dann Sinn, wenn ich dem Ausdruck ”in allen Stellen übereinstimmen“,
durch eine Methode diese Übereinstimmung festzustellen, einen Sinn gegeben
habe. Und das Gleiche gilt natürlich für den Satz ”sie stimmen nicht überein, wenn sie an
irgend einer Stelle nicht übereinstimmen“.
Könnte man aber nicht auch umgekehrt π′ als das Ursprüngliche, und also als den zuerst
angenommenen Punkt, betrachten und14
dann über die Berechtigung von π im Zweifel
sein? – Was ihre Extensionen betrifft, sind sie natürlich gleichberechtigt; was uns aber dazu
veranlaßt, π einen Punkt auf der Zahlengeraden zu nennen, ist seine Vergleichbarkeit mit
den Rationalzahlen.
Wenn ich π, oder sagen wir , als Regel zur Erzeugung von Dezimalbrüchen auffasse,
so kann ich natürlich eine Modifikation dieser Regel erzeugen, indem ich sage, es solle jede
7 in der Entwicklung von durch eine 5 ersetzt werden; aber diese Modifikation ist von
ganz andrer Art15
als die, welche, etwa, durch eine Änderung des Radikanten, oder des
Wurzelexponenten erzeugt wird. Ich nehme z.B. in das modifizierte Gesetz eine Beziehung
zum Zahlensystem der Entwicklung auf, die in dem ursprünglichen Gesetz nicht
vorhanden war. Die Änderung des Gesetzes ist von viel fundamentalerer Art, als es zuerst
den Anschein haben könnte. Ja, wenn wir das falsche Bild von der unendlichen Extension
vor uns haben, dann kann es allerdings scheinen, als ob ich durch die Hinzufügung der
Ersetzungsregel 7 → 5 zur diese viel weniger verändert hätte, als etwa durch Änderung
der in , denn die Entwicklungen16
von lauten denen von sehr ähnlich,
während die Entwicklung der schon nach der zweiten Stelle gänzlich von der der
abweicht.
Gebe ich eine Regel ρ zur Bildung von Extensionen an, aber so, daß mein Kalkül kein
Mittel kennt, vorherzusagen, wie oft höchstens sich eine scheinbare Periode der Extension
wiederholen kann, dann ist ρ von einer reellen Zahl insofern verschieden, als ich ρ − a
in gewissen Fällen nicht mit einer Rationalzahl vergleichen kann, so daß der Ausdruck
ρ − a = b unsinnig wird. Wäre z.B. die mir bekannte Entwicklung von ρ bis auf weiteres
3,141111 . . . , so ließe es sich von der Differenz ρ – 3,14M nicht sagen, sie sei größer, oder
sie sei kleiner, als 0; sie läßt sich also in diesem Sinne nicht mit 0 vergleichen, also nicht
mit einem Punkt der Zahlenachse, und man kann sie17
und ρ nicht in demselben Sinne
Zahl nennen wie einen dieser Punkte.
22 1,
221,2
2
2
2
2
13 (V): sagen: “zwei
14 (V): betrachten; und
15 (V): Natur
16 (V): Entwicklung
17 (O): und sie
760
761
501 Arten irrationaler Zahlen. . . .
2
7 5→
WBTC136-140 30/05/2005 16:14 Page 1002
10 (V): nature 11 (V): expansion
Kinds of Irrational Numbers. . . . 501e
isn’t any of a point’s business. If we say of a Euclidean line that in shrinking it’s getting
closer to a point, that only makes sense in so far as a point has already been designated which
its ends are approaching; it can’t mean that in shrinking it produces a point. To approach
a point simply has two meanings: on the one hand it means to come spatially closer to it,
and in that case the point must already be there, because in this sense I cannot approach
a person who isn’t there; on the other hand, it means “to become more like a point”,
as we say for instance that the apes have approached the stage of human beings in their
development, the development that produced human beings.
To say “Two real numbers are identical if they coincide in all places of their expansion”
only makes sense if, by producing a method for establishing this coincidence, I have given
a sense to the expression “coincide in all places”. And the same naturally holds for the proposition
“They do not coincide if they don’t coincide in any one place”.
But conversely couldn’t one see π′ as the original, and therefore as the point that was
assumed first, and then be in doubt about the justification of π? – As far as their extensions
are concerned, they are of course equally legitimate; but what causes us to call π a point
on the number line is its comparability with the rational numbers.
If I understand π, or let’s say , as a rule for generating decimal fractions, I can of course
produce a modification of this rule by saying that every 7 in the expansion of is to be
replaced by a 5; but this modification is of a completely different kind10
from the one that’s
produced, say, by an alteration of the radicand, or of the exponent of the radical sign. For
instance, in the modified law I am including a reference to the number system of the expansion
which wasn’t present in the original rule for . This change in the law is of a much
more fundamental kind than might at first appear. Of course, if we have the incorrect
picture of the infinite extension before our minds, it can indeed appear as if by appending
the substitution rule 7 → 5 to I had altered that much less than by altering into
; for the expansions11
of are very similar to those of , whereas the expansion
of deviates completely from that of from the second place onward.
If I state a rule ρ for the formation of extensions, but in such a way that my calculus knows
no way of predicting what is the maximum number of times an apparently recurring period
of the extension can be repeated, then ρ differs from a real number in so far as in certain
cases I can’t compare ρ − a with a rational number, so that the expression ρ − a = b becomes
nonsensical. If for instance the expansion of ρ so far known to me were 3·14 followed by an
unlimited series of ones (3·1411111 . . . ), it wouldn’t be possible to say of the difference
ρ − 3·14M that it was greater or less than 0; so in this sense it can’t be compared with 0,
i.e. with a point on the number axis; and it and ρ can’t be called numbers in the same
sense as one of these points.
22 1⋅
22 1⋅
22
2
2
2
2
7 5→
WBTC136-140 30/05/2005 16:14 Page 1003
|Die Ausdehnung eines Begriffes der Zahl, des Begriffs ”alle“, etc. erscheint uns (ganz)
harmlos; aber sie ist es nicht, sobald18
wir vergessen, daß wir unsern Begriff tatsächlich
geändert haben.|
|Was die irrationalen Zahlen betrifft, so sagt meine Untersuchung nur, daß es falsch
(oder irreführend) ist, von Irrationalzahlen zu sprechen, indem man sie als Zahlenart
den Kardinalzahlen und Rationalzahlen gegenüberstellt, weil man ”Irrationalzahlen“ in
Wirklichkeit verschiedene Zahlenarten nennt, – voneinander so verschieden, wie die
Rationalzahlen von jeder dieser Arten.|
Es wäre eine gute Frage für die Scholastiker gewesen: ”Kann Gott alle Stellen von π
kennen“.
Es tritt uns bei diesen Überlegungen immer wieder etwas entgegen, was man ”arithmetisches
Experiment“ nennen möchte. Was herauskommt ist zwar durch das Gegebene bestimmt,
aber ich kann nicht erkennen, wie es dadurch bestimmt ist. So geht es mit dem Auftreten
der 7 in der Entwicklung von π; so ergeben sich auch die Primzahlen als Resultate eines
Experiments. Ich kann mich davon überzeugen, daß 31 eine Primzahl ist, aber ich sehe den
Zusammenhang nicht zwischen ihr (ihrer Lage in der Reihe der Kardinalzahlen) und der
Bedingung, der sie entspricht. – Aber diese Perplexität ist nur die Folge eines falschen
Ausdrucks. Der Zusammenhang, den ich nicht zu sehen glaube, existiert gar nicht. Ein –
sozusagen unregelmäßiges – Auftreten der 7 in der Entwicklung von π gibt es gar nicht, denn
es gibt ja keine Reihe, die ”die Entwicklung von π“ hieße. Es gibt Entwicklungen von π,
nämlich die, die man entwickelt hat (vielleicht 1000) und in diesen kommt die 7 nicht
”regellos“ vor, denn ihr Auftreten in ihnen läßt sich beschreiben. – (Dasselbe für die
”Verteilung der Primzahlen“. Wer uns ein Gesetz dieser Verteilung gibt, gibt uns eine
neue Zahlenreihe, neue Zahlen.) (Ein Gesetz des Kalküls, das ich nicht kenne, ist kein
Gesetz.) (Nur was ich sehe, ist ein Gesetz; nicht, was ich beschreibe. Nur das hindert mich,
mehr in meinen Zeichen auszudrücken, als ich verstehen kann.)
Hat es keinen Sinn, – auch dann, wenn der Fermat’sche Satz bewiesen ist, – zu sagen
F = 0,11? (Wenn ich etwa in der Zeitung davon läse.) Ja, ich werde dann sagen: ”nun
können wir also schreiben ,F = 0,11‘“. D.h. es liegt nahe, das Zeichen ”F“ aus dem früheren
Kalkül, in dem es keine Rationalzahl bezeichnete, in den neuen hinüberzunehmen und nun
0,11 damit zu bezeichnen.
F wäre ja eine Zahl, von der wir nicht wüßten ob sie rational oder irrational ist. Denken
wir uns eine Zahl, von der wir nicht wüßten, ob sie eine Kardinalzahl oder eine Rationalzahl
ist. – Eine Beschreibung im Kalkül gilt eben nur als dieser bestimmte Wortlaut und hat
nichts mit einem Gegenstand der Beschreibung zu tun, der vielleicht einmal gefunden
werden wird.
Man könnte was ich meine auch in den Worten ausdrücken: Man kann keine Verbindung
von Teilen der Mathematik oder Logik herausfinden, die schon vorhanden war, ohne daß
man es wußte.
In der Mathematik gibt es kein ”noch nicht“ und kein ”bis auf weiteres“ (außer in dem
Sinne, in welchem man sagen kann, man habe noch nicht 1000-stellige Zahlen miteinander
multipliziert).
18 (V): wenn
762
763
502 Arten irrationaler Zahlen. . . .
WBTC136-140 30/05/2005 16:14 Page 1004
|The extension of a concept of number, of the concept “all”, etc., seems (quite) harmless
to us; but it stops being harmless as soon as we12
forget that we have in fact changed
our concept.|
|As far as the irrational numbers are concerned, my investigation says only that it is
incorrect (or misleading) to speak of irrational numbers by contrasting them as a type of
number with cardinal numbers and rational numbers; because what are called “irrational
numbers” are actually different types of numbers – as different from each other as the
rational numbers are different from each of these types.|
“Can God know all the places of π?” would have been a good question for the scholastics.
In these considerations we encounter again and again something that one is inclined
to call an “arithmetical experiment”. The result, to be sure, is determined by the data, but
I can’t make out how they determine it. That is how it goes with the occurrences of the
7s in the expansion of π; in the same way, the primes are produced as the results of an
experiment. I can ascertain that 31 is a prime number, but I don’t see the connection between
it (its position in the series of cardinal numbers) and the condition it satisfies. – But this
perplexity is nothing but the consequence of an incorrect expression. The connection that
I think I don’t see doesn’t even exist. There is no such thing as an – as it were irregular –
occurrence of 7s in the expansion of π, because there isn’t any series that might be called
the expansion of π. There are expansions of π, namely those that have been calculated
(perhaps 1,000) and in those the 7s don’t occur “irregularly”, because their occurrence there
can be described. – (The same goes for the “distribution of the primes”. Whoever gives
us a law for this distribution gives us a new number series, new numbers.) (A law of the
calculus that I do not know is not a law). (Only what I see is a law; not what I describe.
That is the only thing standing in the way of my expressing more in my signs than I can
understand.)
Does it make no sense to say – even after Fermat’s theorem has been proved – that
F = 0·11? (If, say, I were to read about it in the papers.) Indeed, I will then say, “So now
we can write ‘F = 0·11’.” That is, it is tempting to transfer the sign “F” from the earlier
calculus, in which it didn’t denote a rational number, into the new one, and now to denote
0·11 with it.
F, after all, would be a number of which we wouldn’t know whether it was rational
or irrational. Let’s imagine a number of which we didn’t know whether it was a cardinal
number or a rational number. – A description in a calculus is valid only with a particular
wording, and it has nothing to do with an object that is given by description and that may
someday be found.
What I mean could also be expressed by the words: We can’t discover any connection
between parts of mathematics or logic that was already there without our knowing it.
In mathematics there is no “not yet” and no “until further notice” (except in the sense
in which we can say that we have not yet multiplied 1,000-digit numbers by each other.).
12 (V): harmless when we
Kinds of Irrational Numbers. . . . 502e
WBTC136-140 30/05/2005 16:14 Page 1005
”Ergibt die Operation, z.B., eine rationale Zahl?“ – wie kann das gefragt werden, wenn
man keine Methode zur Entscheidung der Frage hat? denn die Operation ergibt doch nur
im festgesetzten Kalkül. Ich meine: ”ergibt“ ist doch wesentlich zeitlos.19
Es heißt doch nicht:
”ergibt mit der Zeit“! – sondern: ergibt nach der jetzt bekannten, festgesetzten Regel.20
”Die Lage aller Primzahlen muß doch irgendwie vorausbestimmt sein. Wir rechnen
sie nur successive aus, aber sie sind alle schon bestimmt. Gott kennt sie sozusagen alle. Und
dabei scheint es doch möglich, daß sie nicht durch ein Gesetz bestimmt sind. –“ – Immer
wieder das Bild von der Bedeutung eines Wortes, als einer vollen Kiste, deren Inhalt uns
mit ihr und in ihr verpackt gebracht wird, und den wir nur zu untersuchen haben. – Was
wissen wir denn von den Primzahlen? Wie ist uns denn dieser Begriff überhaupt gegeben?
Treffen wir nicht selbst die Bestimmungen über ihn? Und wie seltsam, daß wir dann
annehmen, es müssen Bestimmungen über ihn getroffen sein, die wir nicht getroffen haben.
Aber der Fehler ist begreiflich. Denn wir gebrauchen das Wort ”Primzahlen“ und es lautet
ähnlich wie ”Kardinalzahlen“, ”Quadratzahlen“, ”gerade Zahlen“, etc. So denken wir, es wird
sich ähnlich gebrauchen lassen, vergessen aber, daß wir ganz andere –andersartige– Regeln
für das Wort ”Primzahl“ gegeben haben, und kommen nun mit uns selbst in einen seltsamen
Konflikt. – Aber wie ist das möglich? die Primzahlen sind doch die uns wohlbekannten
Kardinalzahlen, – wie kann man dann sagen, der Begriff der Primzahl sei in anderem Sinne
ein Zahlbegriff, als der der Kardinalzahl? Aber hier spielt uns wieder die Vorstellung einer
”unendlichen Extension“ als eines Analogons21
zu den uns bekannten ”endlichen“ Extensionen
einen Streich. Der Begriff ”Primzahl“ ist freilich mit Hilfe des Begriffes ”Kardinalzahl“
erklärt, aber nicht ”die Primzahlen“ mit Hilfe der ”Kardinalzahlen“; und den Begriff
”Primzahl“ haben wir in wesentlich anderer Weise aus dem Begriff ”Kardinalzahl“
abgeleitet, als, etwa, den Begriff ”Quadratzahl“. (Wir können uns also nicht wundern, wenn
er sich anders benimmt.) Man könnte sich sehr wohl eine Arithmetik denken, die –
sozusagen – beim Begriff ”Kardinalzahl“ sich nicht aufhält, sondern gleich zu dem der
Quadratzahl übergeht (diese Arithmetik wäre natürlich nicht so anzuwenden, wie die unsere).
Aber der Begriff ”Quadratzahl“ hätte dann nicht den Charakter, den er in unserer Arithmetik
hat; daß er nämlich wesentlich ein Teilbegriff sei, daß die Quadratzahlen wesentlich ein Teil
der Kardinalzahlen seien; sondern sie wären eine komplette Reihe mit einer kompletten
Arithmetik. Und nun denken wir uns dasselbe für die Primzahlen gemacht! Da würde es
klar, daß diese nun in einem andern Sinne ”Zahlen“ seien, als z.B. die Quadratzahlen; und
als die Kardinalzahlen.
Könnten die Berechnungen eines Ingenieurs ergeben, daß die Stärken eines
Maschinenteils bei gleichmäßig wachsender Belastung in der Reihe der Primzahlen
fortschreiten müssen?22
19 (V): präsens.
20 (V): nach der gegenwärtigen Regel.
21 (V): als einem Analogon
22 (V): daß die Stärke // daß eine Dimension //
eines Maschinenteils bei gleichmäßig wachsender
Belastung in der Reihe der Primzahlen
fortschreiten müsse?
764
765
503 Arten irrationaler Zahlen. . . .
WBTC136-140 30/05/2005 16:14 Page 1006
“Does the operation yield a rational number, for instance?” – How can that be asked, if
we have no method for deciding the question? For it is only in an established calculus that
the operation yields results. I mean: “yields” is essentially timeless.13
It doesn’t mean “yields,
given time!” – but: yields in accordance with the rule currently known and established.14
“The position of all primes must somehow be predetermined. We just figure them out
successively, but they are all already determined. God, as it were, knows them all. And yet
for all that it seems possible that they have not been determined by a law.” – Time and again
there’s this picture of the meaning of a word as a full box, whose contents are brought to us
along with the box and packed up in it, and now all we have to do is examine them. – What
do we know about the prime numbers, anyway? How has this concept been given to us at
all? Don’t we ourselves specify it? And how odd that we then assume that specifications must
have been made about it that we ourselves didn’t make! But the mistake is understandable.
For we use the expression “prime numbers”, and it sounds similar to “cardinal numbers”,
“square numbers”, “even numbers”, etc. So we think it can be used in a similar way, but
forget that for the expression “prime number” we have given quite different rules – rules
different in kind – and now we get into a strange conflict with ourselves. – But how is that
possible? After all, the prime numbers are the familiar cardinal numbers – how can one then
say that the concept of prime number is a number concept in a different sense from the
concept of cardinal number? But here again we are tricked by the mental image of an “infinite
extension” as an analogue to the familiar “finite” extensions. To be sure, the concept “prime
number” is defined by means of the concept “cardinal number”, but “the prime numbers”
aren’t defined by means of “the cardinal numbers”; and we did derive the concept of “prime
number” from the concept “cardinal number” in an essentially different way from the way
we derived, say, the concept “square number”. (So we cannot be surprised if it behaves
differently.) One could easily imagine an arithmetic which – as it were – didn’t waste time
with the concept “cardinal number”, but went straight on to that of square numbers. (Of
course that arithmetic couldn’t be applied in the same way as ours.) But then the concept
“square number” wouldn’t have the characteristic it has in our arithmetic, namely that
of being essentially a part-concept, with the square numbers essentially a subclass of the
cardinal numbers; in that case the square numbers would be a complete series with a
complete arithmetic. And now let’s imagine the same thing done for the prime numbers!
That would make it clear that they are “numbers” in a different sense than, for example,
the square numbers; and the cardinal numbers.
Could the calculations of an engineer yield the result that the thicknesses of a machine
part must increase in accordance with the series of primes, given an increase of load at a
uniform rate?15
13 (V): essentially present tense.
14 (V): with the current rule.
15 (V): that the thickness // one dimension // of a
machine part must increase in accordance with
the series of primes, given an increase of load
at a uniform rate?
Kinds of Irrational Numbers. . . . 503e
WBTC136-140 30/05/2005 16:14 Page 1007
140
Regellose unendliche Dezimalzahl.
”Regellose unendliche Dezimalzahl“. Die Auffassung ist immer die, als ob wir nur Wörter
unserer Umgangssprache zusammenstellen brauchten, und die Zusammenstellung hätte
damit einen Sinn, den wir jetzt eben erforschen müßten – wenn er uns nicht gleich ganz
klar sein sollte. Es ist, als wären die Wörter Ingredientien einer chemischen Verbindung,
die wir zusammenschütten, sich miteinander verbinden lassen, und nun müßten wir eben
die Eigenschaften der (betreffenden) Verbindung untersuchen. Wer sagte, er verstünde den
Ausdruck ”regellose unendliche Dezimalzahl“ nicht, dem würde geantwortet: ”das ist nicht
wahr, Du verstehst ihn sehr gut! weißt Du nicht, was die Worte ”regellos“, ”unendlich“
und ”Dezimalzahl“ bedeuten?! – Nun, dann verstehst Du auch ihre Verbindung“. Und mit
dem ”Verständnis“ ist hier gemeint, daß er diese Wörter in gewissen Fällen anzuwenden
weiß und etwa eine Vorstellung mit ihnen verbindet. In Wirklichkeit tut der, welcher diese
Worte zusammenstellt und fragt ”was bedeutet das“ etwas ähnliches, wie die kleinen
Kinder, die ein Papier mit regellosen Strichen bekritzeln, es dem Erwachsenen zeigen und
fragen: ”was ist das?“
”Unendlich kompliziertes Gesetz“, ”unendlich komplizierte Konstruktion“. (”Es glaubt
der Mensch, wenn er nur Worte hört, es müsse sich dabei auch etwas denken lassen.“1
)
Wie unterscheidet sich ein unendlich kompliziertes Gesetz vom Fehlen eines Gesetzes?
(Vergessen wir nicht: Die Überlegungen der Mathematiker über das Unendliche sind
doch lauter endliche Überlegungen. Womit ich nur meine, daß sie ein Ende haben.)
”Eine regellose unendliche Dezimalzahl kann man sich z.B. dadurch erzeugt denken,
daß endlos gewürfelt wird und die Zahl der Augen jedesmal eine Dezimalstelle ist.“ Aber,
wenn endlos gewürfelt wird, kommt ja eben kein endgültiges Resultat heraus.
”Nur der menschliche Intellekt kann das nicht erfassen, ein höherer könnte es!“ Gut, dann
beschreibe mir die Grammatik des Ausdrucks ”höherer Intellekt“; was kann ein solcher
erfassen und was nicht, und in welchem Falle (der Erfahrung)2
sage ich, daß ein Intellekt
etwas erfaßt? Du wirst dann sehen, daß die Beschreibung des Erfassens das Erfassen
selbst ist. (Vergleiche: Lösung eines mathematischen Problems.)
Nehmen wir an, wir würfen mit einer Münze ”Kopf und Adler“ und teilen nun eine
Strecke AB nach folgender Regel: ”Kopf“ sagt: 3
nimm die linke Hälfte
und teile sie, wie der nächste Wurf vorschreibt. ”Adler“ sagt: nimm die
rechte Hälfte etc. Durch fortgesetztes Würfeln erzeuge ich dann Schnittpunkte, die sich
in einem immer kleineren Intervall4
bewegen. Beschreibt es nun die Lage eines Punktes,
1 (E): Vgl. Goethe, Faust I, 2565-2566.
2 (V): und unter welchen Umständen
3 (F): MS 113, S. 81r.
4 (O): Interval
766
767
768
A B
504
WBTC136-140 30/05/2005 16:14 Page 1008
140
Irregular Infinite Decimals.
“Irregular infinite decimal number.” We always have the idea that all we have to do is to
put together the words of our everyday language, and that in so doing the combination has
a sense that we now just have to explore – if it happens not to be completely clear to us right
away. It’s as if words were ingredients of a chemical compound, which we pour together
and allow to combine with each other; and now we simply have to investigate the properties
of the (respective) compound. Someone who said that he didn’t understand the expression
“irregular infinite decimal number” would get the answer “That’s not true, you
understand it perfectly well: don’t you know what the words ‘irregular’, ‘infinite’, and
‘decimal number’ mean? – Well, then, you understand their combination as well.” And
what is meant by “understanding” here is that he knows how to apply these words in
certain cases, and say connects a mental image with them. In fact, someone who puts these
words together and asks “What does that mean?” is doing something similar to what small
children do when they cover a piece of paper with random scribbles, show it to a grown-up,
and ask “What’s that?”
“Infinitely complicated law”, “infinitely complicated construction”. (“Men always
believe, if only they hear words / That some sort of meaning must also be there.”1
)
How does an infinitely complicated law differ from the absence of a law?
(Let’s not forget: mathematicians’ deliberations about the infinite are all finite deliberations,
after all. By which I mean only that they come to an end.)
“One can imagine an irregular infinite decimal being generated, for example, by an endless
throwing of dice, with the number of spots in each case being a decimal place.” But if
the throwing goes on for ever, what comes out is precisely not a final result.
“It’s only the human intellect that can’t grasp that. A higher intellect could do it!” Fine,
then describe to me the grammar of the expression “higher intellect”; what such an intellect
can grasp and what it can’t grasp, and in what (empirical) situation do2
I say that an
intellect grasps something? You will then see that describing grasping is itself grasping.
(Compare: the solution of a mathematical problem.)
Suppose we throw heads and tails with a coin and now divide a length AB in accordance
with the following rule: “Heads” means: 3
take the left half and divide
it in the way the next throw prescribes. “Tails” says “take the right half,
etc.” By repeated throws I then create dividing-points that move in an ever smaller
interval. Now does it describe the position of a point if I say that it is to be the one infinitely
1 (E): Goethe, Faust I, 2565–2566.
2 (V): in what circumstances do
3 (F): MS 113, p. 81r.
A B
504e
WBTC136-140 30/05/2005 16:14 Page 1009
wenn ich sage, es solle der sein, dem sich bei fortgesetztem Würfeln die Schnitte unendlich
nähern? Hier glaubt man etwa einen Punkt bestimmt zu haben, der einer regellosen
unendlichen Dezimalzahl entspricht. Aber die Beschreibung bestimmt doch ausdrücklich:
keinen Punkt; es sei denn, daß man sagt, daß die Worte ”Punkt auf dieser Strecke“ auch
”einen Punkt bestimmen“. Wir verwechseln hier die Vorschrift des Würfelns mit der
mathematischen Vorschrift, etwa Dezimalstellen der zu erzeugen. Diese mathematischen
Vorschriften sind die Punkte. D.h., es lassen sich zwischen diesen Vorschriften Beziehungen
finden, die in ihrer Grammatik den Beziehungen ”größer“ und ”kleiner“ zwischen zwei
Strecken analog sind und daher mit diesen Worten bezeichnet werden. Die Vorschrift, Stellen
der auszurechnen, ist das Zahlzeichen der irrationalen Zahl selbst; und ich rede hier
von einer ”Zahl“, weil ich mit diesen Zeichen (gewissen Vorschriften zur Bildung von
Rationalzahlen) ähnlich rechnen kann, wie mit den Rationalzahlen selbst. Will ich also
analog sagen, die Vorschrift des endlosen Halbierens nach Kopf und Adler bestimme einen
Punkt, eine Zahl, so müßte das heißen, daß diese Vorschrift als Zahlzeichen, d.h. analog andern
Zahlzeichen, gebraucht werden kann. Das ist aber natürlich nicht der Fall. Sollte diese
Vorschrift einem Zahlzeichen entsprechen, so höchstens (sehr entfernt) dem unbestimmten
Zahlwort ”einige“, denn sie tut nichts, als eine Zahl offen zu lassen. Mit einem Wort, ihr
entspricht nichts anderes, als das ursprüngliche Intervall5
AB.
2
2
5 (O): Interval
505 Regellose unendliche Dezimalzahl.
WBTC136-140 30/05/2005 16:14 Page 1010
approached by the cuts, given continued throwing of the coin? Here one might believe
he’s determined a point corresponding to an irregular infinite decimal. But the description,
after all, determines: no point; unless one says that the words “point on this line” also “determine
a point”! Here we’re confusing the instruction to throw the coin with a mathematical
instruction, say to generate the decimal places of . Those mathematical instructions are
the points. That is, you can find relations between those instructions that are analogous
in their grammar to the relations “larger” and “smaller” between two lengths, and are therefore
designated by these words. The instruction to work out the places of is the numeral
for the irrational number itself; and the reason I speak of a “number” here is that I can
calculate with these signs (certain rules for the construction of rational numbers) just as I
can with the rational numbers themselves. So if, in an analogous fashion, I want to say that
the instruction to bisect endlessly using heads and tails determines a point, a number, that
would have to mean that this instruction could be used as a numeral, i.e. analogously to other
numerals. But of course that is not the case. Should this instruction correspond to a numeral
at all, it would at best (very remotely) correspond to the indeterminate numeral “some”,
for all it does is to leave a number open. In a word, nothing corresponds to it except the
original interval AB.
2
2
Irregular Infinite Decimals. 505e
WBTC136-140 30/05/2005 16:14 Page 1011
Anhang I
(Hg.): Das hier Folgende haben wir dem MS 111, S. 147–8 entnommen. Wittgenstein
scheint sich in Kap. 130 darauf zu beziehen.
a + (b + 1) = (a + b) + 1 . . . (R)
a · 1 = a . . . (D)
a · (b + 1) = a · b + a . . . (M)
(Eine Untersuchung Schritt für Schritt dieses Beweises wäre sehr lehrreich.) Der erste
Übergang in I a + (b + (c + 1)) = a + ((b + c) + 1) wenn er nach R vor sich gehn soll
zeigt, daß die Variablen in R anders gemeint sind als die in den Gleichungen von I denn
sonst erlaubte R nur a + (b + 1) durch (a + b) + 1 zu ersetzen aber nicht b + (c + 1)
durch (b + c) + 1. Dasselbe zeigen auch die andern Übergänge dieses Beweises.
Wenn ich nun sagte, der Vergleich der beiden Zeilen des Beweises berechtigt mich die
Regel a + (b + c) = (a + b) + c zu folgern, so hieße das gar nichts, es sei denn ich hätte nach
einer vorher aufgestellten Regel so geschlossen. Diese Regel aber könnte wohl nur
F1(1) = F2(1), F1(x + 1) = f(F1(x))
F1(x) = F2(x) . . . (ρ)
F2(x + 1) = f(F2(x)) sein.
Aber diese Regel ist vag in Bezug auf F1, F2 + f.
a + (b + (c + 1)) = a + ((b + c) + 1) = a + ((b + c) + 1
R R
(a + b) + (c + 1)
R
= ((a + b) + c) + 1
5
6
7
a + (b + c)
= (a + b) + c
(I)
(a + 1) + 1 = (a + 1) + 1
7
1 + (a + 1) = (1 + a) + 1
5
6
7
(II)R a + 1 = 1 + a
a + (b + 1) = (a + b) + 1
R
(b + 1) + a = b + (1 + a) = b + (a + 1) = (b + a) + 1
5
6
7
(III)R II R a + b = b + a
a · (b + (c + 1)) = a · ((b + c) +1) = a · (b + c) + a
R
a · b + (a · (c + 1)) = a · b + (a · c + a) = (a · b + a · c) + a
5
6
7
(IV)M I
a · (b + c)
= a · b + a · c
M
5
6
7
506
WBTC136-140 30/05/2005 16:14 Page 1012
Appendix I
(Eds.): We have taken this section from MS 111, pp. 147–8. Parts of it appear to be
referred to in Chapter 130 of the present work.
a + (b + 1) = (a + b) + 1 . . . (R)
a · 1 = a . . . (D)
a · (b + 1) = a · b + a . . . (M)
(A step-by-step investigation of this proof would be very instructive.) The first step in I,
a + (b + (c + 1)) = a + ((b + c) + 1), if it is made in accordance with R, shows that the meanings
of the variables in R are different from those in the equations in I. For otherwise R
would only allow the replacement of a + (b + 1) by (a + b) + 1, and not the replacement of
b + (c + 1) by (b + c) + 1. The same thing appears in the other steps in this proof.
If I said that a comparison of the two lines of the proofs justifies me in inferring the rule
a + (b + c) = (a + b) + c, that would mean nothing, unless I had deduced that in accordance
with a previously established rule. But this rule could only be:
F1(1) = F2(1), F1(x + 1) = f(F1(x))
F1(x) = F2(x) . . . (ρ)
F2(x + 1) = f(F2(x))
But this rule is vague with regard to F1, F2, and f.
a + (b + (c + 1)) = a + ((b + c) + 1) = a + ((b + c) + 1
R R
(a + b) + (c + 1)
R
= ((a + b) + c) + 1
5
6
7
a + (b + c)
= (a + b) + c
(I)
(a + 1) + 1 = (a + 1) + 1
7
1 + (a + 1) = (1 + a) + 1
5
6
7
(II)R a + 1 = 1 + a
a + (b + 1) = (a + b) + 1
R
(b + 1) + a = b + (1 + a) = b + (a + 1) = (b + a) + 1
5
6
7
(III)R II R a + b = b + a
a · (b + (c + 1)) = a · ((b + c) +1) = a · (b + c) + a
R
a · b + (a · (c + 1)) = a · b + (a · c + a) = (a · b + a · c) + a
5
6
7
(IV)M I
a · (b + c)
= a · b + a · c
M
5
6
7
506e
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378–81, 383–5, 389, 393, 398, 401, 407,
414, 424–5, 430, 432, 442, 452, 456, 460,
467, 471, 483, 489, 491, 493–4, 497,
500–1, 503
Bedeutungskörper, 37, 125
Befehl, befehlen(d), 4, 6, 8, 10, 12–17, 19, 20,
34, 38–43, 71–5, 77, 108, 110, 112, 130,
132, 136–7, 138, 142, 146, 148, 150, 151–2,
162–3, 189, 193–4, 216, 219–20, 226, 252,
254–6, 266, 268, 275–8, 282–3, 286, 380,
441
behaviourism, behaviouristisch, 133, 221–2, 273,
334, 350, 358
Beweis der Relevanz, 378–9
Broad, C. D., 62
Brouwer, L. E. J., 107, 487
Busch, Wilhelm, 431
Cantor, Georg, 494
Carnap, Rudolf, 82
Carroll, Lewis, 59
Dedekind, Richard, 489, 491
Deduktion siehe: Widerspruch
definition, 10, 13, 28, 30–1, 34, 38–42, 44–6,
54, 56–7, 102, 119, 129, 134–5, 189,
194–8, 201, 207, 230–1, 253, 255–7, 261,
294, 321–2, 331, 374, 379, 386–9, 393, 400,
405, 408, 415, 427, 444, 463–4, 469–70,
476–7, 481, 485, 491–2, 494
hinweisende definition, 30, 39–41, 45–6,
134–5, 386, 408
denken (sich denken), 7–8, 16, 24–5, 30–2, 35,
42–3, 57, 59, 62–3, 78, 80, 82, 84–5, 97,
99, 101, 103, 106–8, 112, 118–19, 125,
129–30, 137, 139, 146–7, 149–51, 154,
157, 162–3, 166, 169, 172, 178–9, 181,
186, 190, 192–3, 195–6, 198–9, 201–3,
206, 208, 211, 216, 220, 227, 235–6, 242,
247, 249, 251, 253, 256, 261, 271, 295–6,
304, 313, 317, 322, 326–7, 330, 332, 336,
340, 343, 349, 357–8, 361, 368, 371, 374,
380, 384, 395–6, 398, 407–8, 415, 418–20,
Ableitung, ableiten, Deduktion, 127, 152, 210,
220, 222, 224, 233, 278, 377, 404, 407, 416,
432, 434, 456–8, 460, 486
Absicht, 117–18, 145, 216–18, 280, 481;
siehe auch: Intention, Vorsatz
Allgemeinheit, Allgemeinheits-, 44, 50, 57,
86, 91–2, 204, 208, 238, 242–3, 246–51,
253–8, 271, 383, 402, 415, 419, 444, 449,
452–3, 455–7, 460, 462, 467, 469–71, 474,
483, 485–6, 488, 492–4
Assoziatives Gesetz siehe: Gesetz
auffassen, 6, 8, 14, 32, 88, 129, 144, 146, 166,
192, 199, 237, 251, 256, 289–90, 304, 340,
363, 372, 388, 396, 402, 411–12, 416, 457,
470, 474, 501
Auffassung, 3, 10–11, 21, 23–4, 35, 82, 103–4,
158, 176–7, 202–3, 221, 230, 234, 249,
259, 266, 281, 287, 300–1, 316, 332, 341,
364, 388, 391, 399, 407, 413, 415, 419, 421,
439, 441, 450, 455, 457, 474, 487, 496, 498,
504
Augustinus, Aurelius, 23–4, 114, 496
ausgeschlossenes Drittes siehe unter: Gesetz
axiom of infinity, 393
bedeuten, 5, 11, 26–7, 30–2, 34, 41–2, 44,
52–3, 68, 89, 106, 115, 124–6, 134–5, 145,
151, 154, 168, 180, 185–6, 197, 201, 209,
211–12, 248, 258, 264, 267, 269, 276, 288,
291, 302, 305, 315, 320, 325, 329, 341,
344–5, 356, 388, 403, 406, 411, 413, 415,
425, 430, 432, 439, 445–6, 453, 458, 460,
470–1, 483, 490, 494, 499, 500, 504
Bedeutung, 2, 6, 10, 15, 22–38, 40–2, 44–6, 48,
50–2, 54, 56–8, 66, 72, 74–6, 84–5, 88,
90, 96–8, 107–8, 110, 115–16, 118–19,
121–5, 130–1, 134–7, 142, 145, 148,
152–3, 155, 157–8, 160, 163, 184–5, 191,
198–9, 201, 203, 206–7, 210–11, 223, 228,
245, 253–4, 258, 272–3, 282–3, 285–8,
295, 300, 304–5, 307–8, 312, 315, 317,
321, 323, 325–7, 329, 331, 336–7, 340, 343,
352, 354–5, 359, 361, 363, 366–7, 373–4,
Register
507
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422, 428, 434, 439, 448, 450, 454, 456–8,
460, 464–5, 470, 476, 478, 481, 485–6,
490–1, 502–4
denken an, über, 6, 16–18, 23–4, 27, 40, 42, 44,
52–3, 58, 62–4, 67, 78, 84, 87–8, 98, 107,
117–19, 135, 138–9, 142, 148–9, 153–4,
160, 162–3, 165–6, 172–3, 176–81, 192,
195–6, 200, 208, 223, 225, 230, 233–4,
242, 247, 255, 264–6, 271–2, 280–4,
287–8, 290, 292, 295–6, 298, 302, 306,
309, 311, 326–7, 331, 333, 335–8, 340–1,
348–9, 351, 357, 361–2, 378, 384, 391,
395–8, 402, 423, 426, 441, 448, 450, 456,
466, 483, 487, 493
Denken, das, 43, 52, 67, 76, 80, 87, 102, 120,
156, 160, 165, 167–9, 172–3, 175–80, 228,
234, 264, 283, 286–7, 290, 302, 311, 327,
426, 492
Deutung, deuten, 3, 14–16, 38, 43, 48, 56, 120,
123, 127, 137, 152, 176, 278, 282, 287,
326, 349, 364, 393, 420, 425, 444, 477;
vgl.: Auffassung
Dirichlet, Lejeune, 388
Disjunktion, disjungieren, 86, 91–2, 242, 244–5,
254–6, 279, 378, 470, 483, 486
distributives Gesetz siehe: Gesetz
Driesch, Hans, 349
Drury, Maurice O’Connor, 115
Eddington, Arthur, 96, 364
Einstein, Albert, 208, 488
Elementarsatz, 82, 203
Erinnerung, Erinnerungs-, 7–8, 33, 35, 67, 86,
132, 136–8, 140, 151, 171, 225, 279, 284,
289, 306, 327–8, 336, 347, 351, 364–5;
siehe auch: Gedächtnis
Erklärung, 3, 8–11, 15, 23–4, 27–41, 44, 46–8,
50, 54, 61, 65, 68, 88–9, 103, 116, 118–19,
129–30, 132–7, 139, 141–2, 145, 152–3,
168, 189, 191, 195, 198, 200, 206, 209, 211,
219, 221, 249, 251, 256, 258, 262, 272–4,
282, 285–6, 302, 307–9, 312, 316, 320,
328–9, 356, 368, 377–8, 384, 387–9, 401,
404, 414–15, 445, 461–2, 469, 471, 478,
485, 489, 498
hinweisende Erklärung, 38–9, 41, 44, 46,
132–4, 142, 184–5, 276
Ernst, Paul, 317
Erwartung, erwarten, 4, 14, 18–19, 34, 38, 40,
70–1, 73–4, 87, 94, 102, 118, 165, 171,
178, 180, 223, 237, 259, 263–71, 273,
275–80, 284–9, 292, 309, 348, 381, 385,
421, 430, 455, 460
Euklid, euklidisch, nicht-euklidisch, 105–6, 184,
255, 323–6, 329, 331, 345, 414, 437, 496,
500–1
Euler, Leonhard, 434
Existenzbeweis, 427–8, 434–5, 474
Farbe, Farb(en)-, -farbe, (-)färbig, (-)farbig, 7,
19, 23–5, 27–8, 31, 33–5, 37, 40–5, 56, 63,
78–9, 83, 85, 89, 91–2, 95, 99, 123, 135–9,
148, 152–3, 158, 163, 187–8, 196–7, 202,
208, 232, 235, 238, 242, 244, 248–51, 253,
267–8, 270–1, 274, 279, 287, 320–2,
328–30, 334, 337–45, 348–9, 351, 359,
365, 369, 391, 394–7, 408, 412, 421
Fermat’scher Satz, Fermat’sches Problem, 211,
420, 502
Folgen, das siehe: Schluß
Frazer, Sir James George, 309, 317
Frege, Gottlob, 3, 34, 39, 55, 81, 160–1, 206,
210, 247–8, 258, 282–3, 383, 393–4, 399,
415, 475–8, 491, 494
Freud, Sigmund, 433
Galton, Sir Francis, 101
Gebärde, Gebärden-, 8, 13, 24, 37–40, 45–6,
71–2, 130, 138–9, 154, 184, 317, 396;
siehe auch: Geste
Gedächtnis, Gedächtnis-, 33–4, 111, 136–7,
139, 166–7, 284, 293, 312, 351, 361–5, 491;
siehe auch: Erinnerung
Gedanke, 11, 14, 18, 26, 52, 54, 71–2, 88–9,
117–19, 124, 126, 135, 141, 148, 164–5,
167–8, 172, 175–80, 207, 210, 214, 218,
222–3, 225–6, 233, 246, 251, 257–8, 262,
264, 268, 273–4, 280–4, 286–8, 290,
296–7, 303, 314, 318, 322, 352, 356, 371,
382–3, 398, 409, 415, 418–19, 433, 451, 495
Geometrie, -geometrie, geometrisch, 44, 50, 56,
100, 103, 105, 122–4, 126–7, 129, 150,
184, 202, 206, 237, 243, 253, 323–5, 330,
334–5, 345, 366, 382, 384, 386, 390–1,
415, 422, 431, 437–9, 466, 490, 494
Gesetz, -gesetz, Satz (Gesetz), 45, 62, 94,
99–102, 104, 107, 115, 126–7, 144, 147,
190–1, 197, 203, 210, 216, 249, 257, 261,
292, 304, 307, 313, 365, 377, 401, 405, 410,
418, 424, 426, 428, 432, 435, 444, 448, 452,
460–4, 468–9, 480–1, 487, 489, 491, 494,
497–504
assoziatives Gesetz, 444, 461, 463, 480
distributives Gesetz, 452
Gesetz, Satz, des ausgeschlossenen Dritten,
80, 107, 424, 435, 448, 487
kommutatives Gesetz, 463, 480
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denken (sich denken) (cont’d)
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Gesichtsfeld, 255, 314, 320–2, 325–8, 330–2,
336–9, 349, 361, 366, 394, 403, 412, 500;
siehe auch: Gesichtsraum, Sehraum
Gesichtsraum, 64, 191, 237, 242, 245, 255,
287, 322–5, 327–35, 337–8, 345, 351–2,
358, 361, 363, 365, 395, 414, 416, 500;
siehe auch: Sehraum, Gesichtsfeld
Geste, 6, 8, 13, 27, 37, 40, 45–6, 87, 130,
154–5, 251; siehe auch: Gebärde
Goethe, Johann Wolfgang von, 429, 504
Goldbach’scher Satz, 419, 432
Grammatik, grammati(kali)sch, 10, 14, 18, 26,
28–9, 31–2, 35–6, 38, 44–5, 48, 50–4, 59,
61–4, 67, 73, 79–81, 83, 85–6, 91–2, 95,
100, 105–7, 115–18, 120–5, 127, 137,
141–9, 150, 152–3, 156, 158, 161, 163,
165, 178, 181, 183–9, 191, 195–6, 202–3,
206–9, 238–9, 242, 246–7, 249–50, 253,
255–6, 259, 270, 276, 285, 287–8, 290,
297, 302, 304–8, 311–15, 320–4, 329, 332,
334–5, 340–2, 345, 352, 354–5, 357–8,
360, 365–8, 372, 380–1, 383–6, 391, 393,
395, 397–8, 407, 409–11, 413, 416, 420,
423–4, 426, 430, 432, 439, 442, 450, 453,
455, 457, 475, 477, 483–4, 486, 491–3,
499, 504–5
Hardy, G. H., 485
heißen (bedeuten), 2–3, 7–9, 11–15, 17, 19–20,
27–8, 30–2, 37–8, 40–3, 45–6, 48, 50–3,
55–6, 58, 60–1, 64–5, 67–70, 73, 76–80,
83–5, 88–9, 95–6, 101, 103, 106, 108,
110–12, 114, 116–18, 120–1, 123–5, 130,
132, 136, 138, 142, 147–8, 151–3, 156–8,
163, 168, 173, 175, 178, 180, 184, 186, 188,
191, 193–5, 197–8, 201, 211, 214, 217,
219–20, 223, 225–6, 230–2, 234, 237, 243,
245–6, 248, 250, 252–3, 255–7, 260–1,
270–1, 273–4, 276, 279, 282–3, 289–91,
294, 297–8, 305, 308–9, 321–2, 328, 330,
338–40, 342, 344, 348–9, 352, 354, 356,
358–9, 361–2, 364, 367–8, 372, 376–7,
381, 383–6, 389, 391, 394–5, 399, 402–3,
408, 411, 413–16, 421, 426–8, 430–2,
435, 439–41, 446, 448–50, 452–8, 460,
462, 467, 475, 483–6, 489, 491–3, 500–1,
505
Hertz, Heinrich, 310
Hilbert, David, 376
hinweisende Definition siehe unter: Definition
hinweisende Erklärung siehe unter: Erklärung
Hypothese, hypothetisch, 15, 19, 30, 69, 92,
94–6, 98, 103, 111, 115–17, 133–4, 162,
176, 180, 199, 205, 214, 290, 292, 295, 297,
318, 321, 324, 331, 334, 348–50, 357–8,
364, 371, 383, 386, 418–20, 426
Idealismus, Idealist, 161, 346, 348, 351, 354, 373
indirekter Beweis, 381
Induktion, Induktions-, induktiv, 98, 400, 423,
427, 446, 448–50, 453–4, 456–8, 460–1,
464, 469, 471, 474, 478, 480
Induktionsbeweis, 234, 443–4, 449, 457–8, 461,
464, 469, 471, 478, 480; siehe auch:
induktiver Beweis
induktiver Beweis, 448, 456, 464; siehe auch:
Induktionsbeweis
Intention, 10, 118–19, 143, 170, 213, 218, 221,
223, 225, 280–2; siehe auch: Absicht,
Vorsatz
Interpretation, interpretieren siehe: Deutung
James, William, 29, 37
Kalkül, Kalkül-, -kalkül, 3, 23, 33, 35, 39, 52–4,
65, 79, 81–2, 90, 110, 119–20, 134, 139,
146, 153, 157, 159, 161, 165, 181–2, 186,
188, 191, 196–8, 203–4, 209–11, 228, 231,
241–2, 253, 258–62, 286–8, 308, 372,
376–7, 379, 384–7, 393–4, 400, 404–7,
409, 424–31, 437–8, 444, 449, 451, 454,
457–8, 460, 462, 466, 469, 470–2, 474,
476–7, 479–81, 484, 487, 490, 493–5,
499–503
Kalkulation siehe: Rechnung
Kant, Immanuel, 451
Kardinalzahl, 53, 248, 260–2, 339, 352, 365,
376, 378, 383, 392–4, 396, 399, 402, 407,
429, 437–9, 445–6, 449, 452–3, 455, 462,
469–71, 476, 481, 488, 490–3, 498–9,
502–3
Kepler, Johannes, 210
kommutatives Gesetz siehe: Gesetz
Kontradiktion siehe: Widerspruch
Kopfschmerz siehe: Schmerz
Kronecker, Leopold, 352
Lagrange, Joseph-Louis, 210
Lichtenberg, Georg Christoph, 311, 490
Liebig, Justus von, 477
Logik, logisch, 3, 12–14, 24, 35, 44, 46, 50, 53,
55–7, 61–2, 71–2, 78, 80, 82–3, 90–3, 95,
99, 101–2, 111, 122–3, 130–1, 160, 189,
195, 198, 201, 204–7, 209, 223, 229–31,
233, 237–9, 242–4, 248–50, 252–3, 257–8,
281, 293, 308–9, 312, 321, 323–4, 327,
332, 339–41, 345, 351, 356–7, 363, 365,
367, 369, 374, 376–7, 379, 381, 383–7, 393,
Register 509
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401, 407–8, 413, 415, 419–21, 425, 435,
437, 439–40, 456, 465, 469–70, 475, 478,
483–4, 486–7, 491–4, 496, 502
Logisch-philosophische Abhandlung, 82, 141, 158,
203, 340, 408
Mach, Ernst, 322, 336–7
Magenschmerzen siehe: Schmerz
Mathematik, mathematisch, 2, 31, 48, 52, 54,
63, 82, 90, 92, 139, 162, 209–10, 231, 308,
348, 357, 370–3, 375–9, 383–4, 386, 389,
391, 394, 402, 405, 410–11, 414, 417–18,
420–6, 428–30, 432–5, 437, 441, 444,
453, 458, 462, 473, 477, 482, 485–8, 490,
493–5, 500, 502, 504–5
Mechanik, 82, 101
meinen (intendieren), 3–6, 9–10, 13–15, 21, 23,
27–8, 33, 36–7, 40–3, 50, 52, 56, 58, 60–3,
65, 74–5, 80, 83–4, 87–9, 91–2, 94–6,
98–9, 113, 116–17, 119–20, 122–4, 126–7,
133, 142, 146, 148, 151–4, 156–8, 160–1,
163, 170, 177, 184, 192–3, 195–201, 204,
207, 217, 226–8, 230, 232–4, 243, 251–6,
258, 260, 267, 275–6, 278, 281–2, 286–8,
292, 294–5, 297, 303, 306, 315, 320, 325,
327, 329, 334, 338, 342–3, 349, 351, 353,
355–6, 358–9, 367, 371, 373, 380, 391,
394–5, 402, 408, 412, 418, 421, 426, 430,
438, 451, 456–7, 459–60, 462, 476, 483,
486, 495, 502–4
Meinung, 2–3, 10–11, 21, 74, 101–2, 116, 127,
153, 157, 170, 227–8, 231, 248, 281–2, 291,
340, 365, 401, 466, 490, 494, 498
Mengenlehre, 48, 489, 493–6
Metalogik, 2–3, 13, 158, 220, 223, 305
Metamathematik, 372, 376
Metaphilosophie, 54
Metaphysik, 2, 320
Metapsychologie, 357
minima visibilia, 329, 338
Mißverständnis siehe Verständnis
Nachprüfung, 378; siehe auch: Verifikation
Naturwissenschaft, Naturlehre,
naturwissenschaftlich, 199, 201, 302, 314,
391, 420, 426, 429
Negation, Negations-, 88–91, 122–6, 130, 146,
154, 176–7, 187, 191, 235, 308, 380, 427;
siehe auch: Verneinung
Nicod, Jean, 156
Periodizität, 443, 450–1, 453, 466–7, 471–2,
476, 479
Phänomenologie, 91, 319–20, 322, 337
Phantasiebild, 14, 136, 157, 275
Philosoph, 35, 273, 291, 301–2, 306, 309,
312–13, 315–16, 425, 432
Philosophie, philosophisch, 23, 48, 50–1, 54,
56–8, 111, 148–50, 160–1, 191, 202, 206,
258–9, 299–300, 302, 304–17, 349, 352,
355, 357–60, 365, 376, 405, 419, 424–5,
433, 444, 495, 500
philosophieren, 178, 203, 302, 314, 316
Physik, -physik, 92, 95–7, 101, 307–8, 320, 322,
324, 373, 393, 419, 429
physikalisch (zur Physik gehörend), 103, 383,
418, 420
Physiker, 96, 324, 359
Plato(n), 23, 35, 54, 170, 176, 195, 270, 312, 317
psychisch, 3, 13, 65, 130, 134, 160, 172, 176,
218, 221–2, 351, 375; siehe auch: seelisch
Psychologie, psychologisch, 34, 46, 106, 111,
124, 126, 130–1, 134, 148, 160, 165, 172,
176, 180, 209, 218, 221–3, 238, 251, 286,
292, 322–3, 332, 351, 357, 380, 394, 431,
435, 437, 453
Ramsey, Frank P., 34, 62, 198, 308, 340, 382–5,
387–9, 484
Realismus, 161, 355, 373
Realität, 70, 93–6, 142, 186, 226, 266, 268,
277–8, 281, 287, 309, 314–15, 317, 327–8,
351–4, 367, 385, 421
Rechnung, Rechnungs-, -rechnung,
Kalkulation, 104, 162, 168, 172, 176, 179,
181–2, 185, 222, 258, 278, 286, 291, 343,
373–4, 376, 382–5, 391, 398–400, 404,
413, 427, 438, 444, 446, 449–50, 454–5,
458, 460, 463, 468, 471, 473–4, 480, 494,
503
Regel, Regel-, -regel, 3, 18, 24–5, 28–9, 32,
43–5, 47, 50, 52–5, 57, 59, 61–4, 74, 80–1,
88, 90–2, 95, 97, 101–3, 106–7, 110–11,
115–19, 121–6, 132–4, 140, 142, 144–53,
158, 181, 184–203, 206, 214, 216–20, 222,
224, 231–2, 234–5, 238, 241–2, 245, 249,
251–62, 270, 274, 276, 278–9, 302, 305–7,
313, 320, 322, 326–8, 340–1, 345, 354,
363, 366–7, 373–4, 376–81, 383–4, 386,
389, 394–5, 397, 400–2, 404–10, 412–16,
420, 427, 429–31, 435, 438, 444, 447–8,
453–6, 458–61, 463–4, 466–72, 474,
476–8, 481, 483, 485, 487, 492, 494, 497,
499–501, 503–4; vgl.: Gesetz
Rekursionsbeweis, rekursiver Beweis, 445–6,
452, 454, 462, 464, 468, 471–3, 480–1
Riemann, Bernhard, 420
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Logik, logisch (cont’d)
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Russell, Bertrand, 59, 156, 197–8, 203, 208,
231, 247–9, 376–7, 382–5, 393, 401,
407–8, 414–15, 421, 425, 476–8, 494
Satz siehe: Gesetz
Schluß, Schluß-, 229–33, 236–8, 256, 293, 418,
425, 452–3, 455–6
Schmerz, Schmerz-, -schmerz, schmerz-, 2, 17,
29, 41, 78, 83–6, 94, 163, 166–7, 174–6,
265, 280, 290–2, 295, 356–62, 441
Kopfschmerz, 111, 361
Magenschmerzen, 2, 174, 280–2, 286, 295,
297
Zahnschmerz, 18, 93, 95, 111, 114, 120, 156,
163, 166, 279, 281, 289–91, 349, 356–62
Schopenhauer, Arthur, 281, 300
Seelenzustände, 286
seelisch, 3, 11, 124, 160, 223, 251, 371, 375;
siehe auch: psychisch
Sehraum, 323, 325, 333, 421; siehe auch:
Gesichtsfeld; Gesichtsraum
Shakespeare, William, 194
Sheffer, Henry M., 419, 476–8
Sinn, 2–3, 6–10, 12–13, 15–17, 19, 21, 24–6,
28–30, 32, 34–9, 41–6, 48, 51–4, 56–61,
63–5, 67, 69, 71, 76, 78–80, 82–5, 89,
92–4, 96, 98–101, 105–8, 110, 112, 116,
118–21, 123–7, 129–30, 137–9, 144–5,
147, 149, 151–2, 154–8, 161–3, 166,
173–4, 176–7, 181, 184, 186–93, 195,
197–8, 201, 204, 206–12, 214, 217–20,
222, 225–7, 230–7, 239, 241–2, 248, 251,
254–6, 258, 260, 262, 264, 267–8, 270,
272, 274–82, 285–7, 289–90, 294–7,
300–1, 305, 308, 310, 315, 317, 320–6,
328–34, 336–45, 347–50, 352–61, 363–6,
368–9, 371–4, 379, 381–4, 386, 388–91,
394, 396, 398–400, 402–3, 406–8, 410–16,
418–21, 423–32, 434–5, 437–41, 444–6,
449, 453–6, 458–65, 468–9, 471–2, 474,
476, 478, 481, 483–93, 495–8, 500–4;
siehe auch: Bedeutung
Sinnesdatum, 320–1, 347–8, 351, 358–9
sinnvoll, 52, 59, 63, 81, 83, 148, 219, 357, 360,
386, 389, 402, 407, 439, 450, 492
Skolem, Thoralf, 445, 460–1, 463, 480
Sokrates, 35, 54, 56, 170, 270
Solipsismus, 166, 351, 354, 358
Spengler, Oswald, 204, 307, 378
Spiel, Spiel-, -spiel, 4–5, 7, 23–5, 34–5, 45–7,
53–8, 61, 63–4, 79, 81, 90–2, 97, 101, 104,
106–7, 110–11, 113–14, 116–19, 124, 126,
133, 134–6, 139–40, 145–7, 149, 150–2,
156–9, 161–2, 177, 181, 184, 185–9,
192–6, 198–200, 202, 206, 211, 238, 253,
260–1, 285–6, 290, 292, 302, 313, 371–4,
380, 385–7, 393, 395, 401, 420, 424, 428,
430, 432, 438, 447, 454, 458, 479, 485, 494,
499–500
Sprachspiel, 4, 7, 63, 156, 159, 162, 290, 292
Sraffa, Piero, 190
Syllogismus, 204
Tautologie, 79, 126, 249, 377, 388–90, 400–9,
416, 492
Tolstoi, Leo, 300
Verifikation, 41, 69, 81, 94, 112, 137, 187,
207–9, 211, 242, 290, 325, 354, 356,
363, 418–19, 483–4, 487–8; siehe:
Nachprüfung
Verneinung, Verneinungs-, 37, 79, 86–8, 90,
96, 117, 122–4, 161, 191, 341, 373, 427,
429, 467, 470, 484; siehe: Negation
Verstand, 300, 312, 492
Verständnis, 3, 6–10, 12–13, 27, 29, 35, 38,
40–2, 46, 69, 85, 112–14, 116, 120, 123,
127, 133–5, 176, 190, 200, 216, 230, 252,
266, 272, 278, 300, 308, 401, 419, 454, 478,
484, 490, 504
Mißverständnis, 3, 9, 30–2, 46, 54, 56, 58,
90, 149, 202, 267, 274, 324, 364
unmißverständlich, 42, 45, 312, 413
verstehen, 2–3, 5–17, 19–20, 23, 27, 29–31,
35, 37, 39–41, 43, 46, 48, 56, 59–60, 62,
76, 88, 92, 100–1, 108, 110–11, 113–14,
116, 119–20, 122–4, 129–30, 132, 139,
142, 146, 148, 152, 158, 163, 166, 173,
176–7, 180, 191, 198, 200, 208, 210–11,
230, 235, 238, 251–2, 256, 261–2, 281–2,
291–2, 307–8, 312, 316, 321–2, 326, 332,
334, 341, 343–4, 351–2, 354, 360, 364,
368, 378, 380, 382, 389, 401, 403–4, 420,
423, 428, 431, 452, 455, 459, 475–6, 478,
483–4, 490, 494, 498–500, 502, 504
Verstehen, das, 1–3, 6–7, 9, 12–15, 18–19, 27,
37, 40, 57, 60, 109, 113–14, 120, 122, 148,
165, 176, 197, 230, 251–2, 262, 282–3,
300, 308, 349–50, 475, 478
Vorsatz, 118, 458; siehe: Absicht, Intention
vorschweben, 15, 119, 283, 359
Vorstellung, Vorstellungs-, 13–14, 19, 26,
33–4, 40, 52, 65, 69, 78–9, 84, 111, 118,
127, 132, 134–5, 138, 155, 157, 170–1,
173, 190, 210, 225–7, 266–7, 270, 272–3,
281, 296, 298, 311, 315, 328, 330, 334,
340, 342, 348, 351, 354–5, 357, 363, 384,
503–4
Register 511
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Wahrheitsfunktion, 61, 82, 91–3, 230–2, 476
Wahrscheinlichkeit, Wahrscheinlichkeits-,
98–104, 312, 398
Whitehead, Alfred North, 376
Widerspruch, Kontradiktion, 34, 79, 105, 126,
181, 329, 340–1, 380–1, 388–90, 421, 470,
500
Widerspruchsfreiheit, Beweis der, 379–80
Wirklichkeit, 14, 19, 23, 28, 32, 38, 45, 50,
52, 56, 59, 67, 70–1, 96, 104, 122, 129,
135, 137–8, 141–3, 148, 151, 156,
158–9, 170–1, 181, 184, 187, 190, 198,
202, 217, 226–8, 232, 255, 258–9, 266–7,
270, 272, 274, 277, 283, 286–8, 291, 314,
321–2, 327, 334, 337, 348, 354, 367, 378,
385–7, 391, 395, 402, 415, 423, 428–9,
431, 451, 458–9, 466, 471, 475, 484, 497,
502, 504
Wissenschaft, wissenschaftlich, 2, 57, 95, 198,
202, 300–2, 360, 391
Wortsprache, 38–9, 45, 78, 87, 114, 129–30,
139, 155, 184, 191–2, 247, 317, 401, 407–8,
414, 428, 455, 457, 462, 470, 483
Zahnschmerz siehe: Schmerz
Zeit, zeit-, -zeit, 18–19, 21, 57, 74, 84, 91–3,
100, 102, 104, 109, 113–16, 118, 124, 136,
156, 158, 179–80 186, 193, 197–9, 201,
208, 216–17, 227, 253, 264–5, 279, 289,
309, 312, 314–15, 322, 341, 348, 351–4,
361–5, 368, 384, 414, 426, 484, 493, 496,
503
512 Register
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136–7, 146, 148, 150–2, 162–3, 189,
193–4, 219–20, 226, 252, 254–6, 266,
268, 275–8, 282–3, 286, 380, 441; see also
order
commutative law see law – commutative law
consistency proof, 379–80
contradiction, 79, 105, 126, 181, 193, 329,
340–1, 380–1, 388–90, 421, 470, 500
Dedekind, Richard, 489, 491
deduction (deduce), 102, 127, 152, 207,
210, 233, 290, 308, 404, 456; see also
derivation
definition, 10, 13, 24, 28, 30–1, 34, 38–42,
44–6, 54, 56–7, 102, 108, 119, 129, 134–5,
137, 157, 189, 194–8, 201, 207, 230–1,
253, 255–7, 261, 321, 322, 331, 374, 377–9,
386–9, 393, 400–1, 405, 408, 414–15, 427,
444–5, 461, 463–4, 469–71, 476–7, 481,
485, 491–2, 494
derivation (derive), 24, 40, 127, 203–4, 220,
222, 224, 231, 255, 278, 308, 377, 407,
412–13, 427, 432, 456–8, 460–1, 475, 480,
486, 494, 503; see also deduction
Dirichlet, Lejeune, 388
disjunction, 86, 91–2, 242, 244–5, 254–6, 279,
378, 470, 483, 486
distributive law see law – distributive law
Driesch, Hans, 349
Drury, Maurice O’Connor, 115
Eddington, Arthur, 96, 364
Einstein, Albert, 208, 488
elementary proposition, 82, 203
entailment, 230–2, 234, 256; see also inference
Ernst, Paul, 317
Euclid (Euclidean), 105–6, 184, 255, 323–6,
329, 331, 345, 414, 437, 496, 500–1
Euler, Leonhard, 434
excluded middle see law – law of the excluded
middle
existence proof (proof of existence), 427–8,
434–5, 474
associative law see law – associative law
Augustine, Aurelius, 23–4, 114, 496
axiom of infinity, 393
behaviourism (behaviourist), 133, 221–2, 273,
334, 350, 358
Broad, C. D., 62
Brouwer, L. E. J., 107, 487
Busch, Wilhelm, 431
calculation, 54, 168, 172, 176, 179, 181–2, 185,
222, 249, 253, 258, 278, 286, 291, 343,
372–4, 376, 382–5, 391, 398–400, 404,
413, 420, 425, 427, 438, 444, 446, 449, 452,
454–5, 458, 460, 463, 468, 471, 473–4,
480–1, 494, 503
calculus, 3, 23, 33, 35, 39, 52–4, 65, 79, 81–2,
90, 104, 110, 119–20, 134, 139, 146, 153,
157–8, 161, 165, 181, 185, 188, 191,
196–8, 203–4, 209–11, 228, 231, 241–2,
253, 258–62, 286–8, 308, 372, 376–9,
384–7, 393–4, 400, 404–7, 409, 424–31,
437–8, 444, 449, 451, 454, 457–8, 460, 462,
466, 469, 470–2, 474, 476–7, 479–81, 484,
487, 490, 493–5, 499–503
Cantor, Georg, 494
cardinal numbers, 53, 248, 260–2, 339, 352,
365, 376, 378, 383, 392–4, 396, 399, 402,
407, 429, 437–9, 445–6, 449, 452–3, 455,
462, 469–71, 476, 481, 488, 490–3, 498–9,
502–3
Carnap, Rudolf, 82
Carroll, Lewis, 59
colour (coloured), 7, 19, 23–5, 27–8, 31, 33–5,
37, 40–5, 56, 63, 78–9, 83, 85, 89, 91–2,
95, 99, 123, 125, 134–9, 148, 152–3, 158,
163, 184, 187–8, 196–7, 202, 208, 232,
235, 238, 242, 244, 248, 250–1, 253, 268,
270–1, 274–5, 279, 287, 320–2, 328–30,
334, 337–45, 348–9, 351, 359, 365, 369,
391, 394–7, 412–14, 421
command, 6, 8, 10, 12–15, 17, 19, 20, 34,
38–40, 71–5, 77, 108, 110, 112, 132,
Index
513
WBTD02-Index(E) 02/06/2005 15:52 Page 513
expectation (expect), 3, 14, 18–19, 34, 38, 40,
70–1, 73–4, 87, 94, 102, 118, 151, 156,
165, 171, 178, 180, 202, 223, 237, 259,
263–71, 273, 275–80, 284–9, 292, 309,
348, 381, 421, 430, 455, 460
explanation (explain), 3–4, 8–11, 13, 15, 23,
26–41, 44–8, 50, 54, 58, 65, 88–9, 92–6,
103–4, 106, 116, 118–19, 129, 130, 132–6,
138–9, 141–2, 145, 152–4, 156, 158, 163,
168, 181, 184–5, 189, 191, 195, 198–200,
206–7, 209, 211, 219, 221, 243, 245, 249,
251, 253, 256, 258, 261–2, 272–4, 276,
282–6, 294, 302, 306–9, 312, 316, 320,
328–9, 331, 356, 367–8, 373, 378, 384,
388–9, 401, 405–6, 415, 432, 448, 459,
462, 469–71, 478, 483–5, 489, 498
Fermat’s theorem, 211, 420, 502
Frazer, Sir James George, 309, 317
Frege, Gottlob, 3, 34, 39, 55, 81, 160–1, 206,
210, 247–8, 258, 282–3, 383, 393–4, 399,
415, 475–8, 491, 494
Freud, Sigmund, 433
Galton, Sir Francis, 101
game, 3, 5, 7, 23–5, 34, 35, 45–7, 53–8, 61,
63–4, 79, 81, 90–2, 97, 101, 104, 106–7,
110–11, 113–14, 116–20, 124, 126, 130,
133, 134–6, 139–40, 145–7, 149, 150–2,
156–62, 177, 181, 184, 185–9, 192–200,
202, 204, 206, 211, 238, 253, 260–1, 286,
290, 292, 302, 313, 371–4, 380, 385–7,
393, 395, 401, 420, 424, 428, 430, 432, 437,
447, 454, 458, 479, 485, 494, 499–500;
see also language-game
generality, 44, 50, 57, 86, 91–2, 204, 208, 238,
242–3, 246–51, 253, 254–8, 271, 383, 402,
415, 419, 444, 452–3, 455–7, 460, 462,
467, 469–71, 474, 483, 485–6, 488, 492–4
geometry (geometrical), 44, 50, 56, 100, 103,
105, 122–4, 126–7, 129, 150, 184, 202,
206, 237, 243, 253, 323–5, 330, 334–5,
345, 366, 382, 384, 386, 390–1, 415, 422,
431, 437–9, 466, 490, 494
gesture, 6, 8, 13, 24, 27, 37, 38–40, 45–6, 71–2,
87, 130, 138–9, 154–5, 184, 251, 317, 396
Goethe, Johann Wolfgang von, 429, 504
Goldbach’s conjecture, 419, 432
grammar (grammatical), 10, 14, 18, 26, 28–9,
31–2, 35–6, 38, 44, 45, 48, 50–4, 59, 61–4,
67, 73, 79–81, 86, 91–2, 95, 100, 105–6,
107, 115–18, 120–5, 127, 137, 141–50,
152–3, 156, 158, 161, 163, 165, 178, 181,
183–9, 191, 195–6, 202–3, 206–9, 238–9,
242, 246–7, 249–50, 253, 255–6, 259, 270,
276, 285, 287–8, 290, 297, 302, 304–8,
311–15, 320–4, 329, 332, 334–5, 340–2,
345, 352, 354–5, 357–8, 360, 365–8, 372,
380–1, 383–6, 391, 393–5, 397–8, 407,
409–11, 413, 416, 420, 423–4, 426, 430,
432, 439, 442, 450, 453, 455, 457, 475, 477,
483–4, 486, 491–3, 499, 504–5
Hardy, G. H., 485
Hertz, Heinrich, 310
Hilbert, David, 376
hypothesis (hypothetical), 15, 19, 30, 69, 92,
94–6, 98, 103, 111, 115–17, 133, 162, 176,
180, 199, 205, 214, 290, 292, 295, 297, 318,
321, 324, 331, 334, 348–50, 357–8, 364,
371, 383, 386, 418–20, 426
idealism (idealists), 161, 346, 351, 348, 354,
373
indirect proof, 381
induction (inductive), 98, 234, 400, 423, 427,
443–4, 446, 448–50, 453–4, 456–8,
460–1, 464, 469, 471, 474, 478, 480
inference, 229, 231, 233, 236–8, 293, 452–3,
456; see also entailment
intention, 10, 21, 118–19, 143, 145, 153, 170,
213, 216–18, 221, 223, 225, 227, 280–2,
458, 481
interpretation (interpret), 3, 14–16, 38, 43, 48,
56, 120, 123, 127, 137, 152, 176, 266, 278,
282, 287, 326, 349, 364, 393, 420, 444,
477
James, William, 29, 37
Kant, Immanuel, 451
Kepler, Johannes, 210
Kronecker, Leopold, 352
Lagrange, Joseph-Louis, 210
language-game, 3, 7, 63–4, 156, 159, 162, 290,
292
law, 26, 62, 94, 99–102, 104, 107, 115, 126–7,
144, 147, 190–1, 197, 203, 210, 216, 249,
257, 261, 292, 304, 307, 313, 365, 377, 401,
405, 410, 418, 424–6, 428, 435, 444, 448,
452, 460–4, 468–9, 480–1, 487, 489, 491,
494, 497–504
associative law, 444, 461, 463, 480
commutative law, 463, 480
distributive law, 452
law of the excluded middle, 80, 107, 424,
435, 448, 487
Lichtenberg, Georg Christoph, 311, 490
514 Index
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Liebig, Justus von, 477
logic (logical), 3, 12–14, 24, 35, 44, 46, 50, 53,
55–7, 61–2, 71–2, 78, 80, 82–3, 90–3, 95,
99, 101–12, 111, 122–3, 130–1, 160, 189,
195, 198, 201, 203–7, 209, 223, 229–31,
233, 237–9, 242–4, 248–50, 252–3, 257–8,
281, 293, 308–9, 312, 321, 323–4, 327,
332, 339–41, 345, 351, 356–7, 363, 365,
367, 369, 374, 376–7, 379, 381, 383–7, 393,
401, 407–8, 413, 415, 419–21, 425, 435,
437, 439–40, 456, 465, 469–70, 475, 478,
483–4, 486–7, 491–4, 496, 502
Mach, Ernst, 322, 336–7
mathematics (mathematical), 2, 31, 48, 52, 54,
63, 82, 90, 92, 139, 162, 209–10, 231, 308,
348, 357, 370–3, 375–9, 383–4, 386, 389,
391, 394, 402, 405, 410–11, 414, 417–18,
420–6, 428–9, 430, 432–5, 437, 441, 444,
453, 458, 462, 473, 477, 482, 485–8, 490,
493–5, 500, 502, 504–5
meaning (mean), 2–7, 9–10, 15, 17, 21, 23–38,
40–2, 44–6, 48, 50–2, 54, 56–61, 63–6,
72, 74–6, 78–85, 88–90, 95–6, 98, 107–8,
110, 113–25, 127, 130–2, 134–8, 142, 145,
147–8, 151–8, 160, 163, 168, 170, 177, 180,
184–5, 191, 195–6, 198–9, 200–1, 206–7,
210–11, 219, 223, 226–8, 231, 233–4, 243,
245, 253–4, 258, 269–70, 272–4, 276, 278,
281–3, 285–8, 290, 292, 296, 300, 303,
305–8, 312, 315, 317, 321, 325–7, 329,
331, 336–7, 340, 343–4, 349, 354, 357–60,
363, 366–7, 373, 378–81, 383–6, 389, 391,
393, 395, 401–3, 407–8, 413–14, 424–5,
428, 430, 432, 439, 441–2, 450, 452–3, 455,
456–7, 460, 462, 467, 471, 483, 486, 489,
491–5, 498, 500–1, 503–4
meaning-body, 37, 125
mechanics, 82, 101, 373
memory, 8, 33–5, 67, 86, 111, 132, 136–9,
151, 166, 167, 171, 225, 279, 284, 289,
293, 312, 327–8, 336, 347, 351, 361–5,
441
mental image, 14, 19, 26, 33–4, 40, 52, 69,
78–9, 111, 118, 127, 132, 135, 138, 157,
170–1, 173, 179, 190, 210, 225–7, 266–7,
270, 272–3, 275, 281, 284, 296, 298, 327,
330, 354–5, 359, 363, 384, 503–4
metalogic, 2, 3, 13, 158, 220, 223, 305
metamathematics, 372, 376
metaphilosophy, 54
metaphysics, 2, 320
metapsychology, 357
minima visibilia, 329, 338
negation, 37, 85–90, 117, 122–6, 130, 146, 154,
176–7, 187, 191, 235, 308, 341, 373, 380,
427, 429, 444, 467, 470, 484
Nicod, Jean, 156
order, 3, 10, 15–16, 40–3, 75, 130, 132, 136–8,
142, 162, 216, 275, 278, 282, 441; see also
command
ostensive definition, 30, 38–41, 45–6, 134–5,
386, 408
ostensive explanation, 27, 38–9, 41, 44, 46,
132–4, 142, 184–5, 276
pain, 2, 17, 29, 41, 78, 83–6, 94, 163, 166–7,
174–6, 265, 280, 290–2, 295, 356–62, 441
headache, 111, 361
stomach ache, 2, 174, 280–2, 286, 295, 297
toothache, 18, 93, 95, 111, 114, 120, 156, 163,
166, 279, 281, 289–91, 349, 356–62
periodicity, 443, 450–1, 453, 466–7, 471–2, 476,
479
phenomenology, 91, 319–20, 322, 337
philosopher (philosophers), 35, 273, 291, 301–2,
306, 309, 312–13, 315–16, 425, 432
philosophy (philosophical, philosophize), 23, 48,
50–1, 54, 56–8, 111, 148–50, 160–1, 178,
191, 202–3, 206, 258–9, 299–300, 302,
304–17, 349, 352, 355, 357–60, 365, 376,
405, 419, 424–5, 433, 444, 495, 500
physics, 92, 95–7, 101, 103, 307–8, 320, 322,
324, 373, 383, 393, 418, 420, 429
physicists, 96, 324, 359
Plato, 23, 35, 54, 170, 176, 195, 270, 312, 317
probability, 98–104, 312, 398
proof by induction, 448–9
proof of relevance, 378–9
psychology (psychological), 3, 13, 34, 46, 65,
106, 111, 124, 126, 130–1, 134, 148, 160,
165, 172, 176, 180, 209, 218, 221–3, 238,
251, 286, 292, 322–3, 332, 351, 357, 375,
380, 394, 431, 435, 437, 453
Ramsey, Frank P., 34, 62, 198, 308, 340, 382–5,
387–9, 484
realism, 161, 355, 373
reality, 14, 19, 23, 32, 38, 45, 50, 52, 56, 59, 67,
70–1, 93–6, 122, 129, 135, 138, 141–3,
148, 156, 158–9, 170–1, 181, 184, 186–7,
190, 198, 202, 217, 226, 228, 232, 258–9,
266–8, 270, 272, 274, 277–8, 281, 283,
286–98, 291, 309, 314–15, 317, 321–2, 325,
327, 328, 334, 348, 351–4, 367, 385–7, 395,
402, 415, 421, 423, 428–9, 453, 466, 471,
475, 484, 497
Index 515
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recursive proof, 445–6, 452, 454, 462, 464, 468,
471–3, 480–1
Riemann, Bernhard, 420
rule (rules), 3, 18, 24, 25, 28, 29, 32, 43–5,
47, 50, 52–5, 57, 59, 61–4, 74, 80–1, 88,
90–2, 95, 97, 101–2, 106–7, 110–11,
115–19, 121–6, 132–4, 140, 142, 144–53,
158, 181, 184–203, 206, 214, 216–20,
222, 224, 231–2, 234–5, 238, 241–2,
245, 249, 251–62, 270, 274, 276, 278–9,
302, 305–7, 313, 320, 322, 326–8, 340–1,
345, 354, 363, 366–7, 373–4, 376–81,
383–4, 386, 389, 394–5, 397, 400–2,
404–10, 412–16, 420, 427, 429–32, 435,
437–8, 444, 447–8, 453–6, 458–61,
463–4, 466–72, 474, 476–8, 481,
483, 485–7, 492, 494, 497, 499–501,
503–5
Russell, Bertrand, 59, 156, 197–8, 203, 208,
231, 247–9, 376–7, 382–5, 393, 401,
407–8, 414–15, 421, 425, 476–8, 494
Schopenhauer, Arthur, 281, 300
science, 2, 57, 95, 199, 201–2, 300, 302, 314,
391, 419–20, 426, 429
sense, 2, 3, 6–10, 12–13, 15–17, 19, 21, 24–6,
28–30, 32, 34–9, 41–6, 48, 51–4, 56–61,
63–5, 67, 69, 71, 76, 78–80, 82–5, 89,
92, 94, 96, 98–101, 105–8, 110, 112, 116,
118–21, 123–7, 129–30, 133, 137–9,
144–5, 147–9, 151–2, 154–8, 161–3, 166,
173–4, 176–7, 180, 184, 186–93, 195,
197–8, 201, 203–4, 206–11, 214, 217–20,
222, 225–7, 230–1, 233–7, 239, 241–2,
248, 251, 254–6, 258, 260, 262, 264,
267–8, 270, 272, 275–82, 285–7, 289–90,
292, 294–7, 300–1, 305, 308, 310, 314–15,
317, 320–6, 328–45, 347–50, 352–61,
363–6, 368–9, 371–4, 379, 381–4, 386,
388–391, 394, 396, 398–400, 402–4,
406–8, 410–16, 418–21, 423–32, 434–5,
437–41, 444–6, 449, 453–6, 458–64,
468–9, 471–2, 474, 476, 478, 481,
483–93, 495–8, 500–4; see also
meaning
sense data, 320, 347–8, 351, 358–9
set theory, 48, 489, 493–6
Shakespeare, William, 194
Sheffer, Henry M., 419, 476–8
Skolem, Thoralf, 445, 460–1, 463, 480
Socrates, 35, 54, 56, 170, 270
solipsism, 166, 351, 354, 358
Spengler, Oswald, 204, 307, 378
Sraffa, Piero, 190
syllogism, 204
tautology, 79, 126, 249, 377, 388–90, 400–9,
416, 492
thought (thinking), 5, 11, 14, 18, 23–4, 31–2,
43, 52, 54, 57, 62, 67, 71–3, 75–6, 80, 82,
87–90, 98, 102, 117–20, 124–6, 135, 141,
148, 156, 160, 164–9, 172–82, 196, 200,
207, 210, 214, 218, 222–3, 225–8, 233–4,
246–7, 249, 257–8, 262, 264–5, 268, 274,
280–4, 286–8, 290, 296–7, 300, 302–3,
307, 311, 314, 317–18, 322, 327, 331,
336–7, 340–1, 348, 352, 371, 373, 382–3,
397, 415, 418, 426, 433, 450, 456, 466, 483,
486–7, 492, 495, 498
time, 16, 21, 74, 84, 91–3, 102, 104, 109,
113–16, 118, 124–7, 156, 158, 166, 180,
186, 193, 197–201, 204, 208, 217, 227, 236,
239, 253, 279, 289, 296, 307, 309, 312,
314–15, 341–2, 347–8, 351, 353, 356–7,
361–5, 368, 379, 386, 398, 414, 426, 430,
434–5, 441, 484, 493, 496, 503
Tolstoy, Leo, 300
Tractatus Logico-Philosophicus, 82, 141, 158, 203,
340, 408
truth-function, 61, 82, 91–3, 230–2, 476
understanding (understand), 1–3, 5–10, 12–17,
18–21, 23–4, 27, 29–32, 35, 37–9, 40–3,
46, 48, 55–7, 59–60, 69, 76, 82, 85, 88–9,
92, 100–1, 108–14, 116–17, 119–27,
129–30, 132–5, 139, 142, 146, 148, 152,
163, 165–6, 176–7, 180–1, 190–1, 197–8,
200, 207–8, 210–11, 216, 230, 235, 247,
249, 251–2, 256, 261–2, 266, 272, 278,
281–3, 287, 290–2, 300, 304, 307–8, 312,
316, 321–2, 324, 326, 328, 331–2, 334,
340–1, 343, 349–50, 352, 354, 362–4, 368,
378, 380–2, 388, 391, 393, 401, 403, 412,
416, 419, 423, 428, 431, 439, 447, 452,
455, 459, 470, 473–6, 478, 483–4, 490,
499–504
misunderstanding, 3, 9, 30–2, 46, 54, 56, 58,
90, 149, 202, 267, 270, 274, 290, 324, 364,
413, 420, 491
verification, 41, 69, 81, 94, 112, 137, 187,
207–9, 211, 242, 290, 325, 354, 356, 363,
378, 418–19, 483–4, 487–8
visual space, 237, 242, 245, 255, 287, 322–5,
327–35, 337–8, 345, 351–2, 358, 361, 363,
365, 395, 414, 421, 500
Whitehead, Alfred North, 376
word-language, 38–9, 45, 78, 87, 114, 129–30,
139, 155, 184, 191–2, 247, 317, 401, 407–8,
414, 428, 455, 457, 462, 470, 483
516 Index
WBTD02-Index(E) 30/05/2005 16:13 Page 516
THE BLUE AND THE BROWN BOOKS
Page i
Page Break ii
Page ii
Ludwig Wittgenstein from Blackwell Publishers
The Wittgenstein Reader
Edited by Anthony Kenny
Zettel
Second Edition
Edited by G.E.M. Anscombe and G.H. von Wright
Remarks on Colour
Edited by G.E.M. Anscombe
Remarks on the Foundations of Mathematics
Third Edition
Edited by G.H. von Wright, Rush Rhees and G.E.M. Anscombe
Philosophical Remarks
Edited by Rush Rhees
Philosophical Grammar
Edited by Rush Rhees
On Certainty
Edited by G.E.M. Anscombe and G.H. von Wright
Lectures and Conversations
Edited by Cyril Barrett
Culture and Value
Edited by G.H. von Wright
The Blue and Brown Books
Preliminary Studies for the Philosophical Investigations
Remarks on the Philosophy of Psychology, Vol. I
Edited by G.E.M. Anscombe and G.H. von Wright
Remarks on the Philosophy of Psychology, Vol. II
Edited by G.E.M. Anscombe and Heikki Nyman
Last Writings on the Philosophy of Psychology, Vol. I
Edited by G.H. von Wright and Heikki Nyman
Last Writings on the Philosophy of Psychology, Vol. II
Edited by G.H. von Wright and Heikki Nyman
Philosophical Investigations (German-English edition)
Translated by G.E.M. Anscombe
Page Break iii
Titlepage
Page iii
PRELIMINARY STUDIES FOR THE "PHILOSOPHICAL INVESTIGATIONS"
Generally known as
THE BLUE AND BROWN BOOKS
By
LUDWIG WITTGENSTEIN
BLACKWELL
Oxford UK & Cambridge USA
Page Break iv
Copyright page
Page iv
Copyright © Blackwell Publishers Ltd, 1958
Second Edition 1969
Reprinted 1972, 1975, 1978, 1980, 1984, 1987, 1989, 1992, 1993, 1994, 1997, 1998
Blackwell Publishers Ltd
108 Cowley Road
Oxford OX4 1JF, UK
ISBN 0-631-14660-1
Note on the Second Impression
There are a few alterations, taken from the text of the Blue Book in the possession of Mr. P. Sraffa. With the
exception of changes on pp. 1 and 17 they make no difference to the sense, being mostly improvements in
punctuation or grammar.
We rejected such changes as were no improvement.
The text of the Second Edition remains unaltered, but an index has been added.
All rights reserved. Except for the quotation of short passages for the purposes of criticism and review, no part of
this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher.
Except in the United States of America, this book is sold subject to the condition that it shall not, by way of trade or
otherwise, be lent, re-sold, hired out, or otherwise circulated without the publisher's prior consent in any form of
binding or cover other than that in which it is published and without a similar condition including this condition
being imposed on the subsequent purchaser.
Printed and bound in Great Britain by Athenæum Press Ltd, Gateshead, Tyne & Wear
This book is printed on acid-free paper
Page Break v
PREFACE
Page v
WITTGENSTEIN dictated the "Blue Book" (though he did not call it that) to his class in Cambridge during the
session 1933-34, and he had stencilled copies made. He dictated the "Brown Book" to two of his pupils (Francis
Skinner and Alice Ambrose) during 1934-35. He had only three typed copies made of this, and he showed them
only to very close friends and pupils. But people who borrowed them made their own copies, and there was a trade
in them. If Wittgenstein had named these dictations, he might have called them "Philosophical Remarks" or
"Philosophical Investigations". But the first lot was bound in blue wrappers and the second in brown, and they were
always spoken of that way.
Page v
He sent a copy of the Blue Book to Lord Russell later on, with a covering note.
DEAR RUSSELL,
Two years ago, or so, I promised to send you a manuscript of mine. Now the one I am sending you to-day isn't that
manuscript. I'm still pottering about with it, and God knows whether I will ever publish it, or any of it. But two years
ago I held some lectures in Cambridge and dictated some notes to my pupils so that they might have something to carry
home with them, in their hands if not in their brains. And I had these notes duplicated. I have just been correcting
misprints and other mistakes in some of the copies and the idea came into my mind whether you might not like to have a
copy. So I'm sending you one. I don't wish to suggest that you should read the lectures; but if you should have nothing
better to do and if you should get some mild enjoyment out of them I should be very pleased indeed. (I think it's very
difficult to understand them, as so many points are just hinted at. They are meant only for the people who heard the
lectures.) As I say, if you don't read them it doesn't matter at all.
Yours ever,
LUDWIG WITTGENSTEIN.
Page v
That was all the Blue Book was, though: a set of notes. The Brown Book was rather different, and for a time
he thought of it as a draft of something he might publish. He started more than once to make revisions of a German
version of it. The last was in August, 1936. He brought this, with some minor changes and insertions, to the
beginning of the discussion of voluntary action-about page 154 in
Page Break vi
our text. Then he wrote, in heavy strokes, "Dieser ganze 'Versuch einer Umarbeitung' vom (Anfang) bis hierher ist
nichts wert". ("This whole attempt at a revision, from the start right up to this point, is worthless.") That was when
he began what we now have (with minor revisions) as the first part of the Philosophical Investigations.
Page vi
I doubt if he would have published the Brown Book in English, whatever happened. And anyone who can
read his German will see why. His English style is often clumsy and full of Germanisms. But we have left it that
way, except in a very few cases where it marred the sense and the correction was obvious. What we are printing here
are notes he gave to his pupils, and a draft for his own use; that is all.
Page vi
Philosophy was a method of investigation, for Wittgenstein, but his conception of the method was changing.
We can see this in the way he uses the notion of "language games", for instance. He used to introduce them in order
to shake off the idea of a necessary form of language. At least that was one use he made of them, and one of the
earliest. It is often useful to imagine different language games. At first he would sometimes write "different forms of
language"--as though that were the same thing; though he corrected it in later versions, sometimes. In the Blue Book
he speaks sometimes of imagining different language games, and sometimes of imagining different notations--as
though that were what it amounted to. And it looks as though he had not distinguished clearly between being able to
speak and understanding a notation.
Page vi
He speaks of coming to understand what people mean by having someone explain the meanings of the
words, for instance. As though "understanding" and "explaining" were somehow correlative. But in the Brown Book
he emphasizes that learning a language game is something prior to that. And what is needed is not explanation but
training--comparable with the training you would give an animal. This goes with the point he emphasizes in the
Investigations, that being able to speak and understand what is said--knowing what it means--does not mean that
you can say what it means; nor is that what you have learned. He says there too (Investigations, par. 32) that
"Augustine describes the learning of human language as if the child came into a strange country and did not
understand the language of the country; that is, as if it already had a language, only not this one". You might see
whether the child knows French by asking him what the expressions mean. But that is not how you tell whether a
child can speak. And it is not what he learns when he learns to speak.
Page Break vii
Page vii
When the Brown Book speaks of different language games as "systems of communication" (Systeme
menschlicher Verständigung), these are not just different notations. And this introduces a notion of understanding,
and of the relation of understanding and language, which does not come to the front in the Blue Book at all. In the
Brown Book he is insisting, for example, that "understanding" is not one thing; it is as various as the language games
themselves are. Which would be one reason for saying that when we do imagine different language games, we are
not imagining parts or possible parts of any general system of language.
Page vii
The Blue Book is less clear about that. On page 17 he says that "the study of language games is the study of
primitive forms of language or primitive languages". But then he goes on, "If we want to study the problems of truth
and falsehood, of the agreement and disagreement of propositions with reality, of the nature of assertion,
assumption and question, we shall with great advantage look at primitive forms of language in which these forms of
thinking appear without the confusing background of highly complicated processes of thought. When we look at
such simple forms of language, the mental mist which seems to enshroud our ordinary use of language disappears.
We see activities, reactions, which are clear-cut and transparent. On the other hand we recognize in these simple
processes forms of language not separated by a break from our more complicated ones. We see that we can build up
the complicated forms from the primitive ones by gradually adding new forms."
Page vii
That almost makes it look as though we were trying to give something like an analysis of our ordinary
language. As though we wanted to discover something that goes on in our language as we speak it, but which we
cannot see until we take this method of getting through the mist that enshrouds it. And as if "the nature of assertion,
assumption and question" were the same there; we have just found a way of making it transparent. Whereas the
Brown Book is denying that. That is why he insists in the Brown Book (p. 81) that he is "not regarding the language
games which we describe as incomplete parts of a language, but as languages complete in themselves". So that, for
instance, certain grammatical functions in one language would not have any counterpart in another at all. And
"agreement or disagreement with reality" would be something different in the different languages-so that the study
of it in that language might not show you much about what it is in this one. That is why he asks in the
Page Break viii
Brown Book whether "Brick" means the same in the primitive language as it does in ours; this goes with his
contention that the simpler language is not an incomplete form of the more complicated one. That discussion of
whether we have to do here with an elliptical sentence is an important part of his account of what different language
games are. But there is not even an anticipation of it in the Blue Book.
Page viii
In one of Wittgenstein's note-books there is a remark about language games, which he must have written at
the beginning of 1934. I suspect it is later than the one I have quoted from page 17; anyway, it is different. "Wenn
ich bestimmte einfache Sprachspiele beschreibe, so geschieht es nicht, um mit ihnen nach und nach die Vorgänge
der ausgebildeten Sprache--oder des Denkens--aufzubauen, was nur zu Ungerechtigkeiten führt (Nicod und
Russell),--sondern ich stelle die Spiele als solche hin, und lasse sie ihre aufklärende Wirkung auf die besonderen
Probleme ausstrahlen." ("When I describe certain simple language games, this is not in order to construct from them
gradually the processes of our developed language--or of thinking--which only leads to injustices (Nicod and
Russell). I simply set forth the games as what they are, and let them shed their light on the particular problems.")
Page viii
I think that would be a good description of the method in the first part of the Brown Book. But it also points
to the big difference between the Brown Book and the Investigations.
Page viii
In the Brown Book the account of the different language games is not directly a discussion of particular
philosophical problems, although it is intended to throw light on them. It throws light on various aspects of
language, especially--aspects to which we are often blinded just by the tendencies that find their sharpest expression
in the problems of philosophy. And in this way the discussion does suggest where it is that the difficulties arise
which give birth to those problems.
Page viii
For instance, in what he says about "can", and the connection between this and "seeing what is common", he
is raising the question of what it is that you learn when you learn the language; or of what you know when you
know what something means. But he is also raising the question of what it would mean to ask how the language can
be developed--"Is that still something that has sense? Are you still speaking now, or is it gibberish?" And this may
lead on to the question of "What can be said", or again of "How we should know it was a proposition"; or what a
proposition is, or what language is. The way in which he describes the language games here is intended
Page Break ix
to show that one need not be led into asking those questions, and that it would be a misunderstanding if one were.
But the trouble is that we are left wondering why people constantly are. And in this the Investigations is different.
Page ix
The language games there (in the Investigations) are not stages in the exposition of a more complicated
language, any more than they are in the Brown Book; less so, if anything. But they are stages in a discussion leading
up to the "big question" of what language is (in par. 65).
Page ix
He brings them in--in the Investigations and in the Brown Book too--to throw light on the question about
the relation of words and what they stand for. But in the Investigations he is concerned with "the philosophical
conception of meaning" which we find in Augustine, and he shows that this is the expression of a tendency which
comes out most plainly in that theory of logically proper names which holds that the only real names are the
demonstratives this and that. He calls this "a tendency to sublime the logic of our language" (die Logik unserer
Sprache zu sublimieren) (par. 38)--partly because, in comparison with the logically proper names, "anything else we
might call a name was one only in an inexact, approximate sense". It is this tendency which leads people to talk
about the ultimate nature of language, or the logically correct grammar. But why should people fall into it? There is
no simple answer, but Wittgenstein begins an answer here by going on to discuss the notions of "simple" and
"complex" and the idea of logical analysis. (He does not do this in the Brown Book at all; and if all he wanted were
to throw light on the functioning of language, there would be no need to.)
Page ix
The whole idea of a logical analysis of language, or the logical analysis of propositions, is a queer and
confused one. And in setting forth his language games Wittgenstein was not trying to give any analysis at all. If we
call them "more primitive" or "simpler" languages, that does not mean that they reveal anything like the elements
which a more complicated language must have. (Cf. Investigations, par. 64.) They are different languages--not
elements or aspects of "Language". But then we may want to ask what there is about them that makes us say that
they are all languages. What makes anything a language, anyway? And that is the "big question" (par. 65) about the
nature of language or the nature of the proposition, which has lain behind the whole discussion up to this point.
Page ix
We might even say that the discussion up to this point in the Investigations has been an attempt to bring out
the sense of treating philosophical
Page Break x
problems by reference to language games at all. Or perhaps better: to show how the use of language games can make
clear what a philosophical problem is.
Page x
In the Brown Book, on the other hand, he passes from examples of different sorts of naming to a discussion
of various ways of "comparing with reality". This is still a discussion of the relations of words and what they stand
for, no doubt. But he is not trying here to bring out the tendency behind that way of looking at words which has
given trouble in philosophy.
Page x
In the Investigations he goes on then to a discussion of the relations of logic and language, but he does not
do that in the Brown Book--although it is closely connected with what he says there. I mean especially what he says
there about "can", and the connection of that with the idea of what can be said. ("When do we say that this is still
language? When do we say it is a proposition?") For the temptation then is to think of a calculus, and of what can be
said in it. But Wittgenstein would call that a misunderstanding of what a rule of language is, and of what using
language is. When we speak as we generally do, we are not using precisely definable concepts, nor precise rules
either. And the intelligibility is something different from intelligibility in a calculus.
Page x
It was because people thought of "what can be said" as "what is allowed in a calculus" ("For what other
sense of 'allowed' is there?")--it was for that reason that logic was supposed to govern the unity of language: what
belongs to language and what does not; what is intelligible and what is not; what is a proposition and what is not. In
the Brown Book Wittgenstein is insisting that language does not have that kind of unity. Nor that kind of
intelligibility. But he does not really discuss why people have wanted to suppose that it has.
Page x
You might think he had done that earlier, in the Blue Book, but I do not think he did. I do not think he sees
the question about logic and language there which the Brown Book is certainly bringing out, even if it does not make
quite clear what kind of difficulty it is. On page 25 of the Blue Book he says that "in general we don't use language
according to strict rules--it hasn't been taught us by means of strict rules, either. We, in our discussions on the other
hand, constantly compare language with a calculus proceeding according to exact rules." When he asks (at the
bottom of the page) why we do this, he replies simply, "The answer is that the puzzles which we try to remove
always spring from just this attitude towards language". And
Page Break xi
you might wonder whether that is an answer. His point, as he puts it on page 27, for instance, is that "the man who
is philosophically puzzled sees a law in the way a word is used, and, trying to apply this law consistently, comes up
against.... paradoxical results". And at first that looks something like what he said later, in the Investigations, about
a tendency to sublime the logic of our language. But here in the Blue Book he does not bring out what there is about
the use of language or the understanding of language that leads people to think of words in that way. Suppose we
say that it is because philosophers look on language metaphysically. All right; but when we ask what makes them do
that, Wittgenstein answers in the Blue Book that it is because of a craving for generality, and because "philosophers
constantly see the method of science before their eyes, and are irresistibly tempted to ask and answer questions in
the way science does" (p. 18). In other words, he does not find the source of metaphysics in anything specially
connected with language. That is one very important point here, and it means that he was not anything like as clear
about the character of philosophical puzzlement as he was when he wrote the Investigations. But in any case, it is
not that tendency--to ask and answer questions in the way science does--or it is not primarily that, which leads
philosophers to think of an ideal language or of a logically correct grammar when they are puzzled about language or
about understanding. That comes in a different way.
Page xi
Wittgenstein is quite clear in the Blue Book that we do not use language according to strict rules, and that we
do not use words according to laws like the laws that science speaks of. But he is not quite clear about the notions of
"knowing the meaning" or of "understanding"; and that means that he is still unclear about a great deal in the notion
of "following a rule" too. And for that reason he does not altogether recognize the kind of confusions that there may
be when people say that knowing the language is knowing what can be said.
Page xi
"What does the possibility of the meanings of our words depend on?" That is what is behind the idea of
meaning that we find in the theory of logically proper names and of logical analysis. And it goes with a question of
what you learn when you learn the language; or of what learning the language is. Wittgenstein makes it plain in the
Blue Book that words have the meanings we give them, and that it would be a confusion to think of an investigation
into their real meanings. But he has not yet seen clearly the difference between learning a
Page Break xii
language game and learning a notation. And for that reason he cannot quite make out the character of the confusion
he is opposing.
Page xii
In other words, in the Blue Book Wittgenstein had not seen clearly what the question about the requirements
of language or the intelligibility of language is. That is why he can say, on page 28, that "ordinary language is all
right". Which is like saying "it is a language, all right". And that seems to mean that it satisfies the requirements. But
when he speaks like that, he is himself in the sort of confusion that he later brought out. And it seems to me to
obscure the point of ideal languages--to obscure what those who spoke of them were trying to do--if one speaks, as
Wittgenstein does here, as though "making up ideal languages" were what he was doing when he made up language
games. He would not have spoken that way later.
Page xii
It may be this same unclarity, or something akin to it, that leads Wittgenstein to speak more than once in the
Blue Book of "the calculus of language" (e.g. p. 42, top paragraph; or, better, p. 65, middle paragraph and last
line)--although he has also said that it is only in very rare cases that we use language as a calculus. If you have not
distinguished between a language and a notation, you may hardly see any difference between following a language
and following a notation. But in that case you may well be unclear about the difficulties in connection with the
relation between language and logic.
Page xii
Those difficulties become much clearer in the Brown Book, even though he does not explicitly refer to them
there. We might say that they are the principal theme of the Investigations.
Page xii
For that is the theme that underlies the discussions of "seeing something as something" as well as the earlier
parts. And once again we find that Wittgenstein in the Investigations is making these discussions into an exposition
of the philosophical difficulties, in a way that he has not done in the Brown Book.
Page xii
At one time Wittgenstein was interested in the question of what it is to "recognize it as a proposition" (even
though it may be entirely unfamiliar), or to recognize something as language--to recognize that that is something
written there, for instance--independently of your recognition of what it says. The second part of the Brown Book
bears on this. And it shows that when such "recognitions" are rightly seen, they should not lead to the kinds of
questions that philosophers have asked. The analogies that he draws between understanding a sentence and
understanding a musical theme, for instance; or between wanting to say that this sentence means something and
wanting to
Page Break xiii
say that this colour pattern says something--show clearly that it is not as though you were recognizing any general
character (of intelligibility, perhaps) and that you ought to be able to tell us what that is, any more than it would
make sense for you to ask me what the colour pattern does say.
Page xiii
But why have people wanted to speak of "meta-logic" in this connection, for instance? The Brown Book
does something to explain that, and hints at more. But there is something about the way in which we use language,
and in the connection of language and thinking-the force of an argument, and the force of expressions
generally--which makes it seem as though recognizing it as a language were very different even from recognizing it
as a move in a game. (As though understanding were something outside the signs; and as though to be a language it
needs something that does not appear in the system of signs themselves.) And in the last sections of the
Investigations he is trying to take account of this.
Page xiii
He had spoken of "operating with signs". And someone might say, "You make it look just like operating a
mechanism; like any other mechanism. And if that is all there is to it--just the mechanism--then it is not a language."
Well, there is no short answer to that. But it is an important question. So is the question of what we mean by
"thinking with signs", for instance. What is that? And is the reference to making pencil strokes on paper really
helpful?
Page xiii
Much of all this can be answered by emphasizing that speaking and writing belong to intercourse with other
people. The signs get their life there, and that is why the language is not just a mechanism.
Page xiii
But the objection is that someone might do all that, make the signs correctly in the "game" with other people
and get along all right, even if he were "meaning-blind". Wittgenstein used that expression in analogy with
"colour-blind" and "tone-deaf". If I say an ambiguous word to you, like "board", for instance, I may ask you what
meaning you think of when you hear it, and you may say that you think of a committee like the Coal Board, or
perhaps you do not but think of a plank. Well, could we not imagine someone who could make no sense of such a
question? If you just said a word to him like that, it gave him no meaning. And yet he could "react with words" to
the sentences and other utterances he encountered, and to situations too, and react correctly. Or can we not imagine
that? Wittgenstein was not sure, I think. If a man were "meaning-blind", would that make
Page Break xiv
any difference to his use of language? Or does the perception of meaning fall outside the use of language?
Page xiv
There is something wrong about that last question; something wrong about asking it. But it seems to show
that there is still something unclear in our notion of "the use of language".
Page xiv
Or again, if we simply emphasize that signs belong to intercourse with people, what are we going to say
about the role of "insight" in connection with mathematics and the discovery of proofs, for instance?
Page xiv
So long as there are such difficulties, people will still think that there must be something like an
interpretation. They will still think that if it is language then it must mean something to me. And so on. And for this
reason--in order to try to understand the kinds of difficulties these are--it was necessary for Wittgenstein to go into
the whole complicated matter of "seeing something as something" in the way he was doing.
Page xiv
And the method has to be somewhat different there. One cannot do so much with language games.
R. R.
March, 1958
Page Break xv
THE BLUE BOOK
Page xv
Page Break xvi
Page Break 1
Page 1
THE BLUE BOOK
Page 1
WHAT is the meaning of a word?
Page 1
Let us attack this question by asking, first, what is an explanation of the meaning of a word; what does the
explanation of a word look like?
Page 1
The way this question helps us is analogous to the way the question "how do we measure a length?" helps us
to understand the problem "what is length?"
Page 1
The questions "What is length?", "What is meaning?", "What is the number one?" etc., produce in us a
mental cramp. We feel that we can't point to anything in reply to them and yet ought to point to something. (We are
up against one of the great sources of philosophical bewilderment: a substantive makes us look for a thing that
corresponds to it.)
Page 1
Asking first "What's an explanation of meaning?" has two advantages. You in a sense bring the question
"what is meaning?" down to earth. For, surely, to understand the meaning of "meaning" you ought also to
understand the meaning of "explanation of meaning". Roughly: "let's ask what the explanation of meaning is, for
whatever that explains will be the meaning." Studying the grammar of the expression "explanation of meaning" will
teach you something about the grammar of the word "meaning" and will cure you of the temptation to look about
you for some object which you might call "the meaning".
Page 1
What one generally calls "explanations of the meaning of a word" can, very roughly, be divided into verbal
and ostensive definitions. It will be seen later in what sense this division is only rough and provisional (and that it is,
is an important point). The verbal definition, as it takes us from one verbal expression to another, in a sense gets us
no further. In the ostensive definition however we seem to make a much more real step towards learning the
meaning.
Page 1
One difficulty which strikes us is that for many words in our language there do not seem to be ostensive
definitions; e.g. for such words as "one", "number", "not", etc.
Page 1
Question: Need the ostensive definition itself be understood?--Can't the ostensive definition be
misunderstood?
Page Break 2
Page 2
If the definition explains the meaning of a word, surely it can't be essential that you should have heard the
word before. It is the ostensive definition's business to give it a meaning. Let us then explain the word "tove" by
pointing to a pencil and saying "this is tove". (Instead of "this is tove" I could here have said "this is called 'tove' ". I
point this out to remove, once and for all, the idea that the words of the ostensive definition predicate something of
the defined; the confusion between the sentence "this is red", attributing the colour red to something, and the
ostensive definition "this is called 'red'".) Now the ostensive definition "this is tove" can be interpreted in all sorts of
ways. I will give a few such interpretations and use English words with well established usage. The definition then
can be interpreted to mean:
"This is a pencil",
"This is round",
"This is wood",
"This is one",
"This is hard", etc. etc.
Page 2
One might object to this argument that all these interpretations presuppose another word-language. And this
objection is significant if by "interpretation" we only mean "translation into a word-language".--Let me give some
hints which might make this clearer. Let us ask ourselves what is our criterion when we say that someone has
interpreted the ostensive definition in a particular way. Suppose I give to an Englishman the ostensive definition
"this is what the Germans call 'Buch'". Then, in the great majority of cases at any rate, the English word "book" will
come into the Englishman's mind. We may say he has interpreted "Buch" to mean "book". The case will be different
if e.g. we point to a thing which he has never seen before and say: "This is a banjo". Possibly the word "guitar" will
then come into his mind, possibly no word at all but the image of a similar instrument, possibly nothing at all.
Supposing then I give him the order "now pick a banjo from amongst these things." If he picks what we call a
"banjo" we might say "he has given the word 'banjo' the correct interpretation"; if he picks some other
instrument--"he has interpreted 'banjo' to mean 'string instrument'".
Page 2
We say "he has given the word 'banjo' this or that interpretation", and are inclined to assume a definite act of
interpretation besides the act of choosing.
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Page 3
Our problem is analogous to the following:
Page 3
If I give someone the order "fetch me a red flower from that meadow", how is he to know what sort of flower
to bring, as I have only given him a word?
Page 3
Now the answer one might suggest first is that he went to look for a red flower carrying a red image in his
mind, and comparing it with the flowers to see which of them had the colour of the image. Now there is such a way
of searching, and it is not at all essential that the image we use should be a mental one. In fact the process may be
this: I carry a chart co-ordinating names and coloured squares. When I hear the order "fetch me etc." I draw my
finger across the chart from the word "red" to a certain square, and I go and look for a flower which has the same
colour as the square. But this is not the only way of searching and it isn't the usual way. We go, look about us, walk
up to a flower and pick it, without comparing it to anything. To see that the process of obeying the order can be of
this kind, consider the order "imagine a red patch". You are not tempted in this case to think that before obeying
you must have imagined a red patch to serve you as a pattern for the red patch which you were ordered to imagine.
Page 3
Now you might ask: do we interpret the words before we obey the order? And in some cases you will find
that you do something which might be called interpreting before obeying, in some cases not.
Page 3
It seems that there are certain definite mental processes bound up with the working of language, processes
through which alone language can function. I mean the processes of understanding and meaning. The signs of our
language seem dead without these mental processes; and it might seem that the only function of the signs is to
induce such processes, and that these are the things we ought really to be interested in. Thus, if you are asked what
is the relation between a name and the thing it names, you will be inclined to answer that the relation is a
psychological one, and perhaps when you say this you think in particular of the mechanism of association.--We are
tempted to think that the action of language consists of two parts; an inorganic part, the handling of signs, and an
organic part, which we may call understanding these signs, meaning them, interpreting them, thinking. These latter
activities seem to take place in a queer kind of medium, the mind; and the mechanism of the mind, the nature of
which, it seems, we don't quite understand, can bring about effects which no material mechanism could. Thus e.g. a
thought (which is such a mental
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process) can agree or disagree with reality; I am able to think of a man who isn't present; I am able to imagine him,
'mean him' in a remark which I make about him, even if he is thousands of miles away or dead. "What a queer
mechanism," one might say, "the mechanism of wishing must be if I can wish that which will never happen".
Page 4
There is one way of avoiding at least partly the occult appearance of the processes of thinking, and it is, to
replace in these processes any working of the imagination by acts of looking at real objects. Thus it may seem
essential that, at least in certain cases, when I hear the word "red" with understanding, a red image should be before
my mind's eye. But why should I not substitute seeing a red bit of paper for imagining a red patch? The visual image
will only be the more vivid. Imagine a man always carrying a sheet of paper in his pocket on which the names of
colours are co-ordinated with coloured patches. You may say that it would be a nuisance to carry such a table of
samples about with you, and that the mechanism of association is what we always use instead of it. But this is
irrelevant; and in many cases it is not even true. If, for instance, you were ordered to paint a particular shade of blue
called "Prussian Blue", you might have to use a table to lead you from the word "Prussian Blue" to a sample of the
colour, which would serve you as your copy.
Page 4
We could perfectly well, for our purposes, replace every process of imagining by a process of looking at an
object or by painting, drawing or modelling; and every process of speaking to oneself by speaking aloud or by
writing.
Page 4
Frege ridiculed the formalist conception of mathematics by saying that the formalists confused the
unimportant thing, the sign, with the important, the meaning. Surely, one wishes to say, mathematics does not treat
of dashes on a bit of paper. Frege's idea could be expressed thus: the propositions of mathematics, if they were just
complexes of dashes, would be dead and utterly uninteresting, whereas they obviously have a kind of life. And the
same, of course, could be said of any proposition: Without a sense, or without the thought, a proposition would be
an utterly dead and trivial thing. And further it seems clear that no adding of inorganic signs can make the
proposition live. And the conclusion which one draws from this is that what must be added to the dead signs in
order to make a live proposition is something immaterial, with properties different from all mere signs.
Page 4
But if we had to name anything which is the life of the sign, we should have to say that it was its use.
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Page 5
If the meaning of the sign (roughly, that which is of importance about the sign) is an image built up in our
minds when we see or hear the sign, then first let us adopt the method we just described of replacing this mental
image by some outward object seen, e.g. a painted or modelled image. Then why should the written sign plus this
painted image be alive if the written sign alone was dead?-In fact, as soon as you think of replacing the mental image
by, say, a painted one, and as soon as the image thereby loses its occult character, it ceases to seem to impart any life
to the sentence at all. (It was in fact just the occult character of the mental process which you needed for your
purposes.)
Page 5
The mistake we are liable to make could be expressed thus: We are looking for the use of a sign, but we look
for it as though it were an object co-existing with the sign. (One of the reasons for this mistake is again that we are
looking for a "thing corresponding to a substantive.")
Page 5
The sign (the sentence) gets its significance from the system of signs, from the language to which it belongs.
Roughly: understanding a sentence means understanding a language.
Page 5
As a part of the system of language, one may say, the sentence has life. But one is tempted to imagine that
which gives the sentence life as something in an occult sphere, accompanying the sentence. But whatever
accompanied it would for us just be another sign.
Page 5
It seems at first sight that that which gives to thinking its peculiar character is that it is a train of mental states,
and it seems that what is queer and difficult to understand about thinking is the processes which happen in the
medium of the mind, processes possible only in this medium. The comparison which forces itself upon us is that of
the mental medium with the protoplasm of a cell, say, of an amoeba. We observe certain actions of the amoeba, its
taking food by extending arms, its splitting up into similar cells, each of which grows and behaves like the original
one. We say "of what a queer nature the protoplasm must be to act in such a way", and perhaps we say that no
physical mechanism could behave in this way, and that the mechanism of the amoeba must be of a totally different
kind. In the same way we are tempted to say "the mechanism of the mind must be of a most peculiar kind to be able
to do what the mind does". But here we are making two mistakes. For what struck us as being queer about thought
and thinking was not at all that it had curious effects which
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we were not yet able to explain (causally). Our problem, in other words, was not a scientific one; but a muddle felt
as a problem.
Page 6
Supposing we tried to construct a mind-model as a result of psychological investigations, a model which, as
we should say, would explain the action of the mind. This model would be part of a psychological theory in the way
in which a mechanical model of the ether can be part of a theory of electricity. (Such a model, by the way, is always
part of the symbolism of a theory. Its advantage may be that it can be taken in at a glance and easily held in the mind.
It has been said that a model, in a sense, dresses up the pure theory; that the naked theory is sentences or equations.
This must be examined more closely later on.)
Page 6
We may find that such a mind-model would have to be very complicated and intricate in order to explain the
observed mental activities; and on this ground we might call the mind a queer kind of medium. But this aspect of the
mind does not interest us. The problems which it may set are psychological problems, and the method of their
solution is that of natural science.
Page 6
Now if it is not the causal connections which we are concerned with, then the activities of the mind lie open
before us. And when we are worried about the nature of thinking, the puzzlement which we wrongly interpret to be
one about the nature of a medium is a puzzlement caused by the mystifying use of our language. This kind of
mistake recurs again and again in philosophy; e.g. when we are puzzled about the nature of time, when time seems
to us a queer thing. We are most strongly tempted to think that here are things hidden, something we can see from
the outside but which we can't look into. And yet nothing of the sort is the case. It is not new facts about time which
we want to know. All the facts that concern us lie open before us. But it is the use of the substantive "time" which
mystifies us. If we look into the grammar of that word, we shall feel that it is no less astounding that man should
have conceived of a deity of time than it would be to conceive of a deity of negation or disjunction.
Page 6
It is misleading then to talk of thinking as of a "mental activity". We may say that thinking is essentially the
activity of operating with signs. This activity is performed by the hand, when we think by writing; by the mouth and
larynx, when we think by speaking; and if we think by imagining signs or pictures, I can give you no agent that
thinks. If then you say that in such cases the mind thinks, I would only draw your attention to the fact that you are
using a metaphor, that here
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the mind is an agent in a different sense from that in which the hand can be said to be the agent in writing.
Page 7
If again we talk about the locality where thinking takes place we have a right to say that this locality is the
paper on which we write or the mouth which speaks. And if we talk of the head or the brain as the locality of
thought, this is using the expression "locality of thinking" in a different sense. Let us examine what are the reasons
for calling the head the place of thinking. It is not our intention to criticize this form of expression, or to show that it
is not appropriate. What we must do is: understand its working, its grammar, e.g. see what relation this grammar has
to that of the expression "we think with our mouth", or "we think with a pencil on a piece of paper".
Page 7
Perhaps the main reason why we are so strongly inclined to talk of the head as the locality of our thoughts is
this: the existence of the words "thinking" and "thought" alongside of the words denoting (bodily) activities, such as
writing, speaking, etc., makes us look for an activity, different from these but analogous to them, corresponding to
the word "thinking". When words in our ordinary language have prima facie analogous grammars we are inclined to
try to interpret them analogously; i.e. we try to make the analogy hold throughout.--We say, "The thought is not the
same as the sentence; for an English and a French sentence, which are utterly different, can express the same
thought". And now, as the sentences are somewhere, we look for a place for the thought. (It is as though we looked
for the place of the king of which the rules of chess treat, as opposed to the places of the various bits of wood, the
kings of the various sets.)--We say, "surely the thought is something; it is not nothing"; and all one can answer to
this is, that the word "thought" has its use, which is of a totally different kind from the use of the word "sentence".
Page 7
Now does this mean that it is nonsensical to talk of a locality where thought takes place? Certainly not. This
phrase has sense' if we give it sense. Now if we say "thought takes place in our heads", what is the sense of this
phrase soberly understood? I suppose it is that certain physiological processes correspond to our thoughts in such a
way that if we know the correspondence we can, by observing these processes, find the thoughts. But in what sense
can the physiological processes be said to correspond to thoughts, and in what sense can we be said to get the
thoughts from the observation of the brain?
Page 7
I suppose we imagine the correspondence to have been verified experimentally. Let us imagine such an
experiment crudely. It
Page Break 8
consists in looking at the brain while the subject thinks. And now you may think that the reason why my
explanation is going to go wrong is that of course the experimenter gets the thoughts of the subject only indirectly
by being told them, the subject expressing them in some way or other. But I will remove this difficulty by assuming
that the subject is at the same time the experimenter, who is looking at his own brain, say by means of a mirror. (The
crudity of this description in no way reduces the force of the argument.)
Page 8
Then I ask you, is the subject-experimenter observing one thing or two things? (Don't say that he is
observing one thing both from the inside and from the outside; for this does not remove the difficulty. We will talk
of inside and outside later.†1) The subject-experimenter is observing a correlation of two phenomena. One of them
he, perhaps, calls the thought. This may consist of a train of images, organic sensations, or on the other hand of a
train of the various visual, tactual and muscular experiences which he has in writing or speaking a sentence.--The
other experience is one of seeing his brain work. Both these phenomena could correctly be called "expressions of
thought"; and the question "where is the thought itself?" had better, in order to prevent confusion, be rejected as
nonsensical. If however we do use the expression "the thought takes place in the head", we have given this
expression its meaning by describing the experience which would justify the hypothesis that the thought takes
places in our heads, by describing the experience which we wish to call "observing thought in our brain".
Page 8
We easily forget that the word "locality" is used in many different senses and that there are many different
kinds of statements about a thing which in a particular case, in accordance with general usage, we may call
specifications of the locality of the thing. Thus it has been said of visual space that its place is in our head; and I
think one has been tempted to say this, partly, by a grammatical misunderstanding.
Page 8
I can say: "in my visual field I see the image of the tree to the right of the image of the tower" or "I see the
image of the tree in the middle of the visual field". And now we are inclined to ask "and where do you see the visual
field?" Now if the "where" is meant to ask for a locality in the sense in which we have specified the locality of the
image of the tree, then I would draw your attention to the fact that you have not yet given this question sense; that
is, that you have been
Page Break 9
proceeding by a grammatical analogy without having worked out the analogy in detail.
Page 9
In saying that the idea of our visual field being located in our brain arose from a grammatical
misunderstanding, I did not mean to say that we could not give sense to such a specification of locality. We could,
e.g., easily imagine an experience which we should describe by such a statement. Imagine that we looked at a group
of things in this room, and, while we looked, a probe was stuck into our brain and it was found that if the point of
the probe reached a particular point in our brain, then a particular small part of our visual field was thereby
obliterated. In this way we might co-ordinate points of our brain to points of the visual image, and this might make
us say that the visual field was seated in such and such a place in our brain. And if now we asked the question
"Where do you see the image of this book?" the answer could be (as above) "To the right of that pencil", or "In the
left hand part of my visual field", or again: "Three inches behind my left eye".
Page 9
But what if someone said "I can assure you I feel the visual image to be two inches behind the bridge of my
nose"; what are we to answer him? Should we say that he is not speaking the truth, or that there cannot be such a
feeling? What if he asks us "do you know all the feelings there are? How do you know there isn't such a feeling?"
Page 9
What if the diviner tells us that when he holds the rod he feels that the water is five feet under the ground? or
that he feels that a mixture of copper and gold is five feet under the ground? Suppose that to our doubts he
answered: "You can estimate a length when you see it. Why shouldn't I have a different way of estimating it?"
Page 9
If we understand the idea of such an estimation, we shall get clear about the nature of our doubts about the
statements of the diviner, and of the man who said he felt the visual image behind the bridge of his nose.
Page 9
There is the statement: "this pencil is five inches long", and the statement, "I feel that this pencil is five inches
long", and we must get clear about the relation of the grammar of the first statement to the grammar of the second.
To the statement "I feel in my hand that the water is three feet under the ground" we should like to answer: "I don't
know what this means". But the diviner would say: "Surely you know what it means. You know what 'three feet
under the ground' means, and you know what 'I feel' means!" But I should answer him: I know what a word means
in certain contexts. Thus I
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understand the phrase, "three feet under the ground", say, in the connections "The measurement has shown that the
water runs three feet under the ground", "If we dig three feet deep we are going to strike water", "The depth of the
water is three feet by the eye". But the use of the expression "a feeling in my hands of water being three feet under
the ground" has yet to be explained to me.
Page 10
We could ask the diviner "how did you learn the meaning of the word 'three feet'? We suppose by being
shown such lengths, by having measured them and such like. Were you also taught to talk of a feeling of water being
three feet under the ground, a feeling, say, in your hands? For if not, what made you connect the word 'three feet'
with a feeling in your hand?" Supposing we had been estimating lengths by the eye, but had never spanned a length.
How could we estimate a length in inches by spanning it? I.e., how could we interpret the experience of spanning in
inches? The question is: what connection is there between, say, a tactual sensation and the experience of measuring
a thing by means of a yard rod? This connection will show us what it means to 'feel that a thing is six inches long'.
Supposing the diviner said "I have never learnt to correlate depth of water under the ground with feelings in my
hand, but when I have a certain feeling of tension in my hands, the words 'three feet' spring up in my mind." We
should answer "This is a perfectly good explanation of what you mean by 'feeling the depth to be three feet', and the
statement that you feel this will have neither more, nor less, meaning than your explanation has given it. And if
experience shows that the actual depth of the water always agrees with the words 'n feet' which come into your
mind, your experience will be very useful for determining the depth of water".--But you see that the meaning of the
words "I feel the depth of the water to be n feet" had to be explained; it was not known when the meaning of the
words "n feet" in the ordinary sense (i.e. in the ordinary contexts) was known.--We don't say that the man who tells
us he feels the visual image two inches behind the bridge of his nose is telling a lie or talking nonsense. But we say
that we don't understand the meaning of such a phrase. It combines well-known words, but combines them in a way
we don't yet understand. The grammar of this phrase has yet to be explained to us.
Page 10
The importance of investigating the diviner's answer lies in the fact that we often think we have given a
meaning to a statement P if only we assert "I feel (or I believe) that P is the case." (We shall talk at a later
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occasion†1 of Prof. Hardy saying that Goldbach's theorem is a proposition because he can believe that it is true.) We
have already said that by merely explaining the meaning of the words "three feet" in the usual way we have not yet
explained the sense of the phrase "feeling that water is three feet etc." Now we should not have felt these difficulties
had the diviner said that he had learnt to estimate the depth of the water, say, by digging for water whenever he had
a particular feeling and in this way correlating such feelings with measurements of depth. Now we must examine the
relation of the process of learning to estimate with the act of estimating. The importance of this examination lies in
this, that it applies to the relation between learning the meaning of a word and making use of the word. Or, more
generally, that it shows the different possible relations between a rule given and its application.
Page 11
Let us consider the process of estimating a length by the eye: It is extremely important that you should
realise that there are a great many different processes which we call "estimating by the eye".
Page 11
Consider these cases:--
Page 11
(1) Someone asks "How did you estimate the height of this building?" I answer: "It has four storeys; I suppose each
storey is about fifteen feet high; so it must be about sixty feet."
(2) In another case: "I roughly know what a yard at that distance looks like; so it must be about four yards long."
(3) Or again: "I can imagine a tall man reaching to about this point; so it must be about six feet above the ground."
(4) Or: "I don't know; it just looks like a yard."
Page 11
This last case is likely to puzzle us. If you ask "what happened in this case when the man estimated the
length?" the correct answer may be: "he looked at the thing and said 'it looks one yard long'." This may be all that
has happened.
Page 11
We said before that we should not have been puzzled about the diviner's answer if he had told us that he had
learnt how to estimate depth. Now learning to estimate may, broadly speaking, be seen in two different relations to
the act of estimating; either as a cause of the phenomenon of estimating, or as supplying us with a rule (a table, a
chart, or some such thing) which we make use of when we estimate.
Page 11
Supposing I teach someone the use of the word "yellow" by repeatedly pointing to a yellow patch and
pronouncing the word.
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On another occasion I make him apply what he has learnt by giving him the order, "choose a yellow ball out of this
bag". What was it that happened when he obeyed my order? I say "possibly just this: he heard my words and took a
yellow ball from the bag". Now you may be inclined to think that this couldn't possibly have been all; and the kind
of thing that you would suggest is that he imagined something yellow when he understood the order, and then
chose a ball according to his image. To see that this is not necessary remember that I could have given him the order,
"Imagine a yellow patch". Would you still be inclined to assume that he first imagines a yellow patch, just
understanding my order, and then imagines a yellow patch to match the first? (Now I don't say that this is not
possible. Only, putting it in this way immediately shows you that it need not happen. This, by the way, illustrates the
method of philosophy.)
Page 12
If we are taught the meaning of the word "yellow" by being given some sort of ostensive definition (a rule of
the usage of the word) this teaching can be looked at in two different ways.
Page 12
A. The teaching is a drill. This drill causes us to associate a yellow image, yellow things, with the word
"yellow". Thus when I gave the order "Choose a yellow ball from this bag" the word "yellow" might have brought
up a yellow image, or a feeling of recognition when the person's eye fell on the yellow ball. The drill of teaching
could in this case be said to have built up a psychical mechanism. This, however, would only be a hypothesis or else
a metaphor. We could compare teaching with installing an electric connection between a switch and a bulb. The
parallel to the connection going wrong or breaking down would then be what we call forgetting the explanation, or
the meaning, of the word. (We ought to talk further on about the meaning of "forgetting the meaning of a word"†1).
Page 12
In so far as the teaching brings about the association, feeling of recognition, etc. etc., it is the cause of the
phenomena of understanding, obeying, etc.; and it is a hypothesis that the process of teaching should be needed in
order to bring about these effects. It is conceivable, in this sense, that all the processes of understanding, obeying,
etc., should have happened without the person ever having been taught the language. (This, just now, seems
extremely paradoxical.)
Page 12
B. The teaching may have supplied us with a rule which is itself involved in the processes of understanding,
obeying, etc.; "involved",
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however, meaning that the expression of this rule forms part of these processes.
Page 13
We must distinguish between what one might call "a process being in accordance with a rule", and, "a
process involving a rule" (in the above sense).
Page 13
Take an example. Some one teaches me to square cardinal numbers; he writes down the row
1 2 3 4,
and asks me to square them. (I will, in this case again, replace any processes happening 'in the mind' by processes of
calculation on the paper.) Suppose, underneath the first row of numbers, I then write:
1 4 9 16.
What I wrote is in accordance with the general rule of squaring; but it obviously is also in accordance with any
number of other rules; and amongst these it is not more in accordance with one than with another. In the sense in
which before we talked about a rule being involved in a process, no rule was involved in this. Supposing that in order
to get to my results I calculated 1 × 1, 2 × 2, 3 × 3, 4 × 4 (that is, in this case wrote down the calculations); these
would again be in accordance with any number of rules. Supposing, on the other hand, in order to get to my results I
had written down what you may call "the rule of squaring", say algebraically. In this case this rule was involved in a
sense in which no other rule was.
Page 13
We shall say that the rule is involved in the understanding, obeying, etc., if, as I should like to express it, the
symbol of the rule forms part of the calculation. (As we are not interested in where the processes of thinking,
calculating, take place, we can for our purpose imagine the calculations being done entirely on paper. We are not
concerned with the difference: internal, external.)
Page 13
A characteristic example of the case B would be one in which the teaching supplied us with a table which we
actually make use of in understanding, obeying, etc. If we are taught to play chess, we may be taught rules. If then
we play chess, these rules need not be involved in the act of playing. But they may be. Imagine, e.g., that the rules
were expressed in the form of a table; in one column the shapes of the chessmen are drawn, and in a parallel column
we find diagrams showing the 'freedom' (the legitimate moves) of the pieces. Suppose now that the way the game is
played involves making the transition from
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the shape to the possible moves by running one's finger across the table, and then making one of these moves.
Page 14
Teaching as the hypothetical history of our subsequent actions (understanding, obeying, estimating a length,
etc.) drops out of our considerations. The rule which has been taught and is subsequently applied interests us only
so far as it is involved in the application. A rule, so far as it interests us, does not act at a distance.
Page 14
Suppose I pointed to a piece of paper and said to someone: "this colour I call 'red'". Afterwards I give him the
order: "now paint me a red patch". I then ask him: "why, in carrying out my order, did you paint just this colour?"
His answer could then be: "This colour (pointing to the sample which I have given him) was called red; and the
patch I have painted has, as you see, the colour of the sample". He has now given me a reason for carrying out the
order in the way he did. Giving a reason for something one did or said means showing a way which leads to this
action. In some cases it means telling the way which one has gone oneself; in others it means describing a way
which leads there and is in accordance with certain accepted rules. Thus when asked, "why did you carry out my
order by painting just this colour?" the person could have described the way he had actually taken to arrive at this
particular shade of colour. This would have been so if, hearing the word "red", he had taken up the sample I had
given him, labelled "red", and had copied that sample when painting the patch. On the other hand he might have
painted it 'automatically' or from a memory image, but when asked to give the reason he might still point to the
sample and show that it matched the patch he had painted. In this latter case the reason given would have been of
the second kind; i.e. a justification post hoc.
Page 14
Now if one thinks that there could be no understanding and obeying the order without a previous teaching,
one thinks of the teaching as supplying a reason for doing what one did; as supplying the road one walks. Now
there is the idea that if an order is understood and obeyed there must be a reason for our obeying it as we do; and, in
fact, a chain of reasons reaching back to infinity. This is as if one said: "Wherever you are, you must have got there
from somewhere else, and to that previous place from another place; and so on ad infinitum". (If, on the other hand,
you had said, "wherever you are, you could have got there from another place ten yards away; and to that other
place from a third, ten yards further away, and so on ad infinitum", if you had said this you would have stressed the
infinite possibility of making a step.
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Thus the idea of an infinite chain of reasons arises out of a confusion similar to this: that a line of a certain length
consists of an infinite number of parts because it is indefinitely divisible; i.e., because there is no end to the
possibility of dividing it.)
Page 15
If on the other hand you realize that the chain of actual reasons has a beginning, you will no longer be
revolted by the idea of a case in which there is no reason for the way you obey the order. At this point, however,
another confusion sets in, that between reason and cause. One is led into this confusion by the ambiguous use of the
word "why". Thus when the chain of reasons has come to an end and still the question "why?" is asked, one is
inclined to give a cause instead of a reason. If, e.g., to the question, "why did you paint just this colour when I told
you to paint a red patch?" you give the answer: "I have been shown a sample of this colour and the word 'red' was
pronounced to me at the same time; and therefore this colour now always comes to my mind when I hear the word
'red' ", then you have given a cause for your action and not a reason.
Page 15
The proposition that your action has such and such a cause, is a hypothesis. The hypothesis is well-founded
if one has had a number of experiences which, roughly speaking, agree in showing that your action is the regular
sequel of certain conditions which we then call causes of the action. In order to know the reason which you had for
making a certain statement, for acting in a particular way, etc., no number of agreeing experiences is necessary, and
the statement of your reason is not a hypothesis. The difference between the grammars of "reason" and "cause" is
quite similar to that between the grammars of "motive" and "cause". Of the cause one can say that one can't know it
but can only conjecture it. On the other hand one often says: "Surely I must know why I did it" talking of the
motive. When I say: "we can only conjecture the cause but we know the motive" this statement will be seen later on
to be a grammatical one. The "can" refers to a logical possibility.
Page 15
The double use of the word "why", asking for the cause and asking for the motive, together with the idea that
we can know, and not only conjecture, our motives, gives rise to the confusion that a motive is a cause of which we
are immediately aware, a cause 'seen from the inside', or a cause experienced.--Giving a reason is like giving a
calculation by which you have arrived at a certain result.
Page 15
Let us go back to the statement that thinking essentially consists in operating with signs. My point was that it
is liable to mislead us
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if we say 'thinking is a mental activity'. The question what kind of an activity thinking is is analogous to this: "Where
does thinking take place?" We can answer: on paper, in our head, in the mind. None of these statements of locality
gives the locality of thinking. The use of all these specifications is correct, but we must not be misled by the
similarity of their linguistic form into a false conception of their grammar. As, e.g., when you say: "Surely, the real
place of thought is in our head". The same applies to the idea of thinking as an activity. It is correct to say that
thinking is an activity of our writing hand, of our larynx, of our head, and of our mind, so long as we understand the
grammar of these statements. And it is, furthermore, extremely important to realize how, by misunderstanding the
grammar of our expressions, we are led to think of one in particular of these statements as giving the real seat of the
activity of thinking.
Page 16
There is an objection to saying that thinking is some such thing as an activity of the hand. Thinking, one
wants to say, is part of our 'private experience'. It is not material, but an event in private consciousness. This
objection is expressed in the question: "Could a machine think?" I shall talk about this at a later point,†1 and now
only refer you to an analogous question: "Can a machine have toothache?" You will certainly be inclined to say: "A
machine can't have toothache". All I will do now is to draw your attention to the use which you have made of the
word "can" and to ask you: "Did you mean to say that all our past experience has shown that a machine never had
toothache?" The impossibility of which you speak is a logical one. The question is: What is the relation between
thinking (or toothache) and the subject which thinks, has toothache, etc.? I shall say no more about this now.
Page 16
If we say thinking is essentially operating with signs, the first question you might ask is: "What are
signs?"--Instead of giving any kind of general answer to this question, I shall propose to you to look closely at
particular cases which we should call "operating with signs". Let us look at a simple example of operating with
words. I give someone the order: "fetch me six apples from the grocer", and I will describe a way of making use of
such an order: The words "six apples" are written on a bit of paper, the paper is handed to the grocer, the grocer
compares the word "apple" with labels on different shelves. He finds it to agree with one of the labels, counts from 1
to the number written on the slip of paper, and for every number counted
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takes a fruit off the shelf and puts it in a bag.-And here you have a case of the use of words. I shall in the future again
and again draw your attention to what I shall call language games. These are ways of using signs simpler than those
in which we use the signs of our highly complicated everyday language. Language games are the forms of language
with which a child begins to make use of words. The study of language games is the study of primitive forms of
language or primitive languages. If we want to study the problems of truth and falsehood, of the agreement and
disagreement of propositions with reality, of the nature of assertion, assumption, and question, we shall with great
advantage look at primitive forms of language in which these forms of thinking appear without the confusing
background of highly complicated processes of thought. When we look at such simple forms of language the mental
mist which seems to enshroud our ordinary use of language disappears. We see activities, reactions, which are
clear-cut and transparent. On the other hand we recognize in these simple processes forms of language not separated
by a break from our more complicated ones. We see that we can build up the complicated forms from the primitive
ones by gradually adding new forms.
Page 17
Now what makes it difficult for us to take this line of investigation is our craving for generality.
Page 17
This craving for generality is the resultant of a number of tendencies connected with particular philosophical
confusions. There is
Page 17
(a) The tendency to look for something in common to all the entities which we commonly subsume under a
general term.--We are inclined to think that there must be something in common to all games, say, and that this
common property is the justification for applying the general term "game" to the various games; whereas games
form a family the members of which have family likenesses. Some of them have the same nose, others the same
eyebrows and others again the same way of walking; and these likenesses overlap. The idea of a general concept
being a common property of its particular instances connects up with other primitive, too simple, ideas of the
structure of language. It is comparable to the idea that properties are ingredients of the things which have the
properties; e.g. that beauty is an ingredient of all beautiful things as alcohol is of beer and wine, and that we therefore
could have pure beauty, unadulterated by anything that is beautiful.
Page 17
(b) There is a tendency rooted in our usual forms of expression,
Page Break 18
to think that the man who has learnt to understand a general term, say, the term "leaf", has thereby come to possess
a kind of general picture of a leaf, as opposed to pictures of particular leaves. He was shown different leaves when he
learnt the meaning of the word "leaf"; and showing him the particular leaves was only a means to the end of
producing 'in him' an idea which we imagine to be some kind of general image. We say that he sees what is in
common to all these leaves; and this is true if we mean that he can on being asked tell us certain features or
properties which they have in common. But we are inclined to think that the general idea of a leaf is something like a
visual image, but one which only contains what is common to all leaves. (Galtonian composite photograph.) This
again is connected with the idea that the meaning of a word is an image, or a thing correlated to the word. (This
roughly means, we are looking at words as though they all were proper names, and we then confuse the bearer of a
name with the meaning of the name.)
Page 18
(c) Again, the idea we have of what happens when we get hold of the general idea 'leaf', 'plant', etc. etc., is
connected with the confusion between a mental state, meaning a state of a hypothetical mental mechanism, and a
mental state meaning a state of consciousness (toothache, etc.).
Page 18
(d) Our craving for generality has another main source: our preoccupation with the method of science. I
mean the method of reducing the explanation of natural phenomena to the smallest possible number of primitive
natural laws; and, in mathematics, of unifying the treatment of different topics by using a generalization.
Philosophers constantly see the method of science before their eyes, and are irresistibly tempted to ask and answer
questions in the way science does. This tendency is the real source of metaphysics, and leads the philosopher into
complete darkness. I want to say here that it can never be our job to reduce anything to anything, or to explain
anything. Philosophy really is 'purely descriptive'. (Think of such questions as "Are there sense data?" and ask: What
method is there of determining this? Introspection?)
Page 18
Instead of "craving for generality" I could also have said "the contemptuous attitude towards the particular
case". If, e.g., someone tries to explain the concept of number and tells us that such and such a definition will not do
or is clumsy because it only applies to, say, finite cardinals I should answer that the mere fact that he could have
given such a limited definition makes this definition extremely important to
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us. (Elegance is not what we are trying for.) For why should what finite and transfinite numbers have in common be
more interesting to us than what distinguishes them? Or rather, I should not have said "why should it be more
interesting to us?"--it isn't; and this characterizes our way of thinking.
Page 19
The attitude towards the more general and the more special in logic is connected with the usage of the word
"kind" which is liable to cause confusion. We talk of kinds of numbers, kinds of propositions, kinds of proofs; and,
also, of kinds of apples, kinds of paper, etc. In one sense what defines the kind are properties, like sweetness,
hardness, etc. In the other the different kinds are different grammatical structures. A treatise on pomology may be
called incomplete if there exist kinds of apples which it doesn't mention. Here we have a standard of completeness in
nature. Supposing on the other hand there was a game resembling that of chess but simpler, no pawns being used in
it. Should we call this game incomplete? Or should we call a game more complete than chess if it in some way
contained chess but added new elements? The contempt for what seems the less general case in logic springs from
the idea that it is incomplete. It is in fact confusing to talk of cardinal arithmetic as something special as opposed to
something more general. Cardinal arithmetic bears no mark of incompleteness; nor does an arithmetic which is
cardinal and finite. (There are no subtle distinctions between logical forms as there are between the tastes of different
kinds of apples.)
Page 19
If we study the grammar, say, of the words "wishing", "thinking", "understanding", "meaning", we shall not
be dissatisfied when we have described various cases of wishing, thinking, etc. If someone said, "surely this is not all
that one calls 'wishing'", we should answer, "certainly not, but you can build up more complicated cases if you like."
And after all, there is not one definite class of features which characterize all cases of wishing (at least not as the
word is commonly used). If on the other hand you wish to give a definition of wishing, i.e., to draw a sharp
boundary, then you are free to draw it as you like; and this boundary will never entirely coincide with the actual
usage, as this usage has no sharp boundary.
Page 19
The idea that in order to get clear about the meaning of a general term one had to find the common element
in all its applications has shackled philosophical investigation; for it has not only led to no result, but also made the
philosopher dismiss as irrelevant the concrete cases, which alone could have helped him to understand the usage of
Page Break 20
the general term. When Socrates asks the question, "what is knowledge?" he does not even regard it as a preliminary
answer to enumerate cases of knowledge.†1 If I wished to find out what sort of thing arithmetic is, I should be very
content indeed to have investigated the case of a finite cardinal arithmetic. For
Page 20
(a) this would lead me on to all the more complicated cases,
Page 20
(b) a finite cardinal arithmetic is not incomplete, it has no gaps which are then filled in by the rest of
arithmetic.
Page 20
What happens if from 4 till 4.30 A expects B to come to his room? In one sense in which the phrase "to
expect something from 4 to 4.30" is used it certainly does not refer to one process or state of mind going on
throughout that interval, but to a great many different activities and states of mind. If for instance I expect B to come
to tea, what happens may be this: At four o'clock I look at my diary and see the name "B" against today's date; I
prepare tea for two; I think for a moment "does B smoke?" and put out cigarettes; towards 4.30 I begin to feel
impatient; I imagine B as he will look when he comes into my room. All this is called "expecting B from 4 to 4.30".
And there are endless variations to this process which we all describe by the same expression. If one asks what the
different processes of expecting someone to tea have in common, the answer is that there is no single feature in
common to all of them, though there are many common features overlapping. These cases of expectation form a
family; they have family likenesses which are not clearly defined.
Page 20
There is a totally different use of the word "expectation" if we use it to mean a particular sensation. This use
of the words like "wish", "expectation", etc., readily suggests itself. There is an obvious connection between this use
and the one described above. There is no doubt that in many cases if we expect some one, in the first sense, some,
or all, of the activities described are accompanied by a peculiar feeling, a tension; and it is natural to use the word
"expectation" to mean this experience of tension.
Page 20
There arises now the question: is this sensation to be called "the sensation of expectation", or "the sensation
of expectation that B will come"? In the first case to say that you are in a state of expectation admittedly does not
fully describe the situation of expecting that so-and-so will happen. The second case is often rashly suggested as an
explanation of the use of the phrase "expecting that so-and-so will happen", and you may even think that with this
explanation you are
Page Break 21
on safe ground, as every further question is dealt with by saying that the sensation of expectation is indefinable.
Page 21
Now there is no objection to calling a particular sensation "the expectation that B will come". There may
even be good practical reasons for using such an expression. Only mark:--if we have explained the meaning of the
phrase "expecting that B will come" in this way no phrase which is derived from this by substituting a different
name for "B" is thereby explained. One might say that the phrase "expecting that B will come" is not a value of a
function "expecting that x will come". To understand this compare our case with that of the function "I eat x". We
understand the proposition "I eat a chair" although we weren't specifically taught the meaning of the expression
"eating a chair".
Page 21
The role which in our present case the name "B" plays in the expression "I expect B" can be compared with
that which the name "Bright" plays in the expression "Bright's disease".†1 Compare the grammar of this word, when
it denotes a particular kind of disease, with that of the expression "Bright's disease" when it means the disease which
Bright has. I will characterize the difference by saying that the word "Bright" in the first case is an index in the
complex name "Bright's disease"; in the second case I shall call it an argument of the function "x's disease". One may
say that an index alludes to something, and such an allusion may be justified in all sorts of ways. Thus calling a
sensation "the expectation that B will come" is giving it a complex name and "B" possibly alludes to the man whose
coming had regularly been preceded by the sensation.
Page 21
Again we may use the phrase "expectation that B will come" not as a name but as a characteristic of certain
sensations. We might, e.g., explain that a certain tension is said to be an expectation that B will come if it is relieved
by B's coming. If this is how we use the phrase then it is true to say that we don't know what we expect until our
expectation has been fulfilled (cf. Russell). But no one can believe that this is the only way or even the most
common way of using the word "expect". If I ask someone "whom do you expect?" and after receiving the answer
ask again "Are you sure that you don't expect someone else?" then, in most cases, this question would be regarded
as absurd, and the answer will be something like "Surely, I must know whom I expect".
Page 21
One may characterize the meaning which Russell gives to the word
Page Break 22
"wishing" by saying that it means to him a kind of hunger.--It is a hypothesis that a particular feeling of hunger will
be relieved by eating a particular thing. In Russell's way of using the word "wishing" it makes no sense to say "I
wished for an apple but a pear has satisfied me".†1 But we do sometimes say this, using the word "wishing" in a way
different from Russell's. In this sense we can say that the tension of wishing was relieved without the wish being
fulfilled; and also that the wish was fulfilled without the tension being relieved. That is, I may, in this sense, become
satisfied without my wish having been satisfied.
Page 22
Now one might be tempted to say that the difference which we are talking about simply comes to this, that in
some cases we know what we wish and in others we don't. There are certainly cases in which we say, "I feel a
longing, though I don't know what I'm longing for" or, "I feel a fear, but I don't know what I'm afraid of", or again: "I
feel fear, but I'm not afraid of anything in particular".
Page 22
Now we may describe these cases by saying that we have certain sensations not referring to objects. The
phrase "not referring to objects" introduces a grammatical distinction. If in characterizing such sensations we use
verbs like "fearing", "longing", etc., these verbs will be intransitive; "I fear" will be analogous to "I cry". We may cry
about something, but what we cry about is not a constituent of the process of crying; that is to say, we could
describe all that happens when we cry without mentioning what we are crying about.
Page 22
Suppose now that I suggested we should use the expression "I feel fear", and similar ones, in a transitive way
only. Whenever before we said "I have a sensation of fear" (intransitively) we will now say "I am afraid of
something, but I don't know of what". Is there an objection to this terminology?
Page 22
We may say: "There isn't, except that we are then using the word 'to know' in a queer way". Consider this
case:--we have a general undirected feeling of fear. Later on, we have an experience which makes us say, "Now I
know what I was afraid of. I was afraid of so-and-so happening". Is it correct to describe my first feeling by an
intransitive verb, or should I say that my fear had an object although I did not know that it had one? Both these
forms of description can be used. To understand this examine the following example: It might be found practical to
call a certain state of decay in a tooth, not accompanied by what we commonly call toothache, "unconscious
Page Break 23
toothache" and to use in such a case the expression that we have toothache, but don't know it. It is in just this sense
that psychoanalysis talks of unconscious thoughts, acts of volition, etc. Now is it wrong in this sense to say that I
have toothache but don't know it? There is nothing wrong about it, as it is just a new terminology and can at any
time be retranslated into ordinary language. On the other hand it obviously makes use of the word "to know" in a
new way. If you wish to examine how this expression is used it is helpful to ask yourself "what in this case is the
process of getting to know like?" "What do we call 'getting to know' or, 'finding out'?"
Page 23
It isn't wrong, according to our new convention, to say "I have unconscious toothache". For what more can
you ask of your notation than that it should distinguish between a bad tooth which doesn't give you toothache and
one which does? But the new expression misleads us by calling up pictures and analogies which make it difficult for
us to go through with our convention. And it is extremely difficult to discard these pictures unless we are constantly
watchful; particularly difficult when, in philosophizing, we contemplate what we say about things. Thus, by the
expression "unconscious toothache" you may either be misled into thinking that a stupendous discovery has been
made, a discovery which in a sense altogether bewilders our understanding; or else you may be extremely puzzled
by the expression (the puzzlement of philosophy) and perhaps ask such a question as "How is unconscious
toothache possible?" You may then be tempted to deny the possibility of unconscious toothache; but the scientist
will tell you that it is a proved fact that there is such a thing, and he will say it like a man who is destroying a
common prejudice. He will say: "Surely it's quite simple; there are other things which you don't know of, and there
can also be toothache which you don't know of. It is just a new discovery". You won't be satisfied, but you won't
know what to answer. This situation constantly arises between the scientist and the philosopher.
Page 23
In such a case we may clear the matter up by saying: "Let's see how the word 'unconscious', 'to know', etc.
etc., is used in this case, and how it's used in others". How far does the analogy between these uses go? We shall
also try to construct new notations, in order to break the spell of those which we are accustomed to.
Page 23
We said that it was a way of examining the grammar (the use) of the word "to know", to ask ourselves what,
in the particular case we are examining, we should call "getting to know". There is a temptation
Page Break 24
to think that this question is only vaguely relevant, if relevant at all, to the question: "what is the meaning of the word
'to know'?" We seem to be on a side-track when we ask the question "What is it like in this case 'to get to know'?"
But this question really is a question concerning the grammar of the word "to know", and this becomes clearer if we
put it in the form: "What do we call 'getting to know'?" It is part of the grammar of the word "chair" that this is what
we call "to sit on a chair", and it is part of the grammar of the word "meaning" that this is what we call "explanation
of a meaning"; in the same way to explain my criterion for another person's having toothache is to give a
grammatical explanation about the word "toothache" and, in this sense, an explanation concerning the meaning of
the word "toothache".
Page 24
When we learnt the use of the phrase "so-and-so has toothache" we were pointed out certain kinds of
behaviour of those who were said to have toothache. As an instance of these kinds of behaviour let us take holding
your cheek. Suppose that by observation I found that in certain cases whenever these first criteria told me a person
had toothache, a red patch appeared on the person's cheek. Supposing I now said to someone "I see A has
toothache, he's got a red patch on his cheek". He may ask me "How do you know A has toothache when you see a
red patch?" I should then point out that certain phenomena had always coincided with the appearance of the red
patch.
Page 24
Now one may go on and ask: "How do you know that he has got toothache when he holds his cheek?" The
answer to this might be, "I say, he has toothache when he holds his cheek because I hold my cheek when I have
toothache". But what if we went on asking:--"And why do you suppose that toothache corresponds to his holding
his cheek just because your toothache corresponds to your holding your cheek?" You will be at a loss to answer this
question, and find that here we strike rock bottom, that is we have come down to conventions. (If you suggest as an
answer to the last question that, whenever we've seen people holding their cheeks and asked them what's the matter,
they have answered, "I have toothache",-remember that this experience only co-ordinates holding your cheek with
saying certain words.)
Page 24
Let us introduce two antithetical terms in order to avoid certain elementary confusions: To the question
"How do you know that so-and-so is the case?", we sometimes answer by giving 'criteria' and sometimes
Page Break 25
by giving 'symptoms'. If medical science calls angina an inflammation caused by a particular bacillus, and we ask in a
particular case "why do you say this man has got angina?" then the answer "I have found the bacillus so-and-so in
his blood" gives us the criterion, or what we may call the defining criterion of angina. If on the other hand the answer
was, "His throat is inflamed", this might give us a symptom of angina. I call "symptom" a phenomenon of which
experience has taught us that it coincided, in some way or other, with the phenomenon which is our defining
criterion. Then to say "A man has angina if this bacillus is found in him" is a tautology or it is a loose way of stating
the definition of "angina". But to say, "A man has angina whenever he has an inflamed throat" is to make a
hypothesis.
Page 25
In practice, if you were asked which phenomenon is the defining criterion and which is a symptom, you
would in most cases be unable to answer this question except by making an arbitrary decision ad hoc. It may be
practical to define a word by taking one phenomenon as the defining criterion, but we shall easily be persuaded to
define the word by means of what, according to our first use, was a symptom. Doctors will use names of diseases
without ever deciding which phenomena are to be taken as criteria and which as symptoms; and this need not be a
deplorable lack of clarity. For remember that in general we don't use language according to strict rules-it hasn't been
taught us by means of strict rules, either. We, in our discussions on the other hand, constantly compare language
with a calculus proceeding according to exact rules.
Page 25
This is a very one-sided way of looking at language. In practice we very rarely use language as such a
calculus. For not only do we not think of the rules of usage--of definitions, etc.--while using language, but when we
are asked to give such rules, in most cases we aren't able to do so. We are unable clearly to circumscribe the
concepts we use; not because we don't know their real definition, but because there is no real 'definition' to them. To
suppose that there must be would be like supposing that whenever children play with a ball they play a game
according to strict rules.
Page 25
When we talk of language as a symbolism used in an exact calculus, that which is in our mind can be found
in the sciences and in mathematics. Our ordinary use of language conforms to this standard of exactness only in rare
cases. Why then do we in philosophizing constantly compare our use of words with one following exact rules?
Page Break 26
The answer is that the puzzles which we try to remove always spring from just this attitude towards language.
Page 26
Consider as an example the question "What is time?" as Saint Augustine and others have asked it. At first
sight what this question asks for is a definition, but then immediately the question arises: "What should we gain by a
definition, as it can only lead us to other undefined terms?" And why should one be puzzled just by the lack of a
definition of time, and not by the lack of a definition of "chair"? Why shouldn't we be puzzled in all cases where we
haven't got a definition? Now a definition often clears up the grammar of a word. And in fact it is the grammar of
the word "time" which puzzles us. We are only expressing this puzzlement by asking a slightly misleading question,
the question: "What is...?" This question is an utterance of unclarity, of mental discomfort, and it is comparable with
the question "Why?" as children so often ask it. This too is an expression of a mental discomfort, and doesn't
necessarily ask for either a cause or a reason. (Hertz, Principles of Mechanics.) Now the puzzlement about the
grammar of the word "time" arises from what one might call apparent contradictions in that grammar.
Page 26
It was such a "contradiction" which puzzled Saint Augustine when he argued: How is it possible that one
should measure time? For the past can't be measured, as it is gone by; and the future can't be measured because it
has not yet come. And the present can't be measured for it has no extension.
Page 26
The contradiction which here seems to arise could be called a conflict between two different usages of a
word, in this case the word "measure". Augustine, we might say, thinks of the process of measuring a length: say,
the distance between two marks on a travelling band which passes us, and of which we can only see a tiny bit (the
present) in front of us. Solving this puzzle will consist in comparing what we mean by "measurement" (the grammar
of the word "measurement") when applied to a distance on a travelling band with the grammar of that word when
applied to time. The problem may seem simple, but its extreme difficulty is due to the fascination which the analogy
between two similar structures in our language can exert on us. (It is helpful here to remember that it is sometimes
almost impossible for a child to believe that one word can have two meanings.)
Page 26
Now it is clear that this problem about the concept of time asks for an answer given in the form of strict rules.
The puzzle is about rules.--Take another example: Socrates' question "What is knowledge?"
Page Break 27
Here the case is even clearer, as the discussion begins with the pupil giving an example of an exact definition, and
then analogous to this a definition of the word "knowledge" is asked for. As the problem is put, it seems that there is
something wrong with the ordinary use of the word "knowledge". It appears we don't know what it means, and that
therefore, perhaps, we have no right to use it. We should reply: "There is no one exact usage of the word
'knowledge'; but we can make up several such usages, which will more or less agree with the ways the word is
actually used".
Page 27
The man who is philosophically puzzled sees a law in the way a word is used, and, trying to apply this law
consistently, comes up against cases where it leads to paradoxical results. Very often the way the discussion of such
a puzzle runs is this: First the question is asked "What is time?" This question makes it appear that what we want is a
definition. We mistakenly think that a definition is what will remove the trouble (as in certain states of indigestion we
feel a kind of hunger which cannot be removed by eating). The question is then answered by a wrong definition; say:
"Time is the motion of the celestial bodies". The next step is to see that this definition is unsatisfactory. But this only
means that we don't use the word "time" synonymously with "motion of the celestial bodies". However in saying
that the first definition is wrong, we are now tempted to think that we must replace it by a different one, the correct
one.
Page 27
Compare with this the case of the definition of number. Here the explanation that a number is the same thing
as a numeral satisfies that first craving for a definition. And it is very difficult not to ask: "Well, if it isn't the numeral,
what is it?"
Page 27
Philosophy, as we use the word, is a fight against the fascination which forms of expression exert upon us.
Page 27
I want you to remember that words have those meanings which we have given them; and we give them
meanings by explanations. I may have given a definition of a word and used the word accordingly, or those who
taught me the use of the word may have given me the explanation. Or else we might, by the explanation of a word,
mean the explanation which, on being asked, we are ready to give. That is, if we are ready to give any explanation;
in most cases we aren't. Many words in this sense then don't have a strict meaning. But this is not a defect. To think
it is would be like saying that the light of my reading lamp is no real light at all because it has no sharp boundary.
Page 27
Philosophers very often talk about investigating, analysing, the
Page Break 28
meaning of words. But let's not forget that a word hasn't got a meaning given to it, as it were, by a power
independent of us, so that there could be a kind of scientific investigation into what the word really means. A word
has the meaning someone has given to it.
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There are words with several clearly defined meanings. It is easy to tabulate these meanings. And there are
words of which one might say: They are used in a thousand different ways which gradually merge into one another.
No wonder that we can't tabulate strict rules for their use.
Page 28
It is wrong to say that in philosophy we consider an ideal language as opposed to our ordinary one. For this
makes it appear as though we thought we could improve on ordinary language. But ordinary language is all right.
Whenever we make up 'ideal languages' it is not in order to replace our ordinary language by them; but just to
remove some trouble caused in someone's mind by thinking that he has got hold of the exact use of a common
word. That is also why our method is not merely to enumerate actual usages of words, but rather deliberately to
invent new ones, some of them because of their absurd appearance.
Page 28
When we say that by our method we try to counteract the misleading effect of certain analogies, it is
important that you should understand that the idea of an analogy being misleading is nothing sharply defined. No
sharp boundary can be drawn round the cases in which we should say that a man was misled by an analogy. The use
of expressions constructed on analogical patterns stresses analogies between cases often far apart. And by doing this
these expressions may be extremely useful. It is, in most cases, impossible to show an exact point where an analogy
begins to mislead us. Every particular notation stresses some particular point of view. If, e.g., we call our
investigations "philosophy", this title, on the one hand, seems appropriate, on the other hand it certainly has misled
people. (One might say that the subject we are dealing with is one of the heirs of the subject which used to be called
"philosophy".) The cases in which particularly we wish to say that someone is misled by a form of expression are
those in which we would say: "he wouldn't talk as he does if he were aware of this difference in the grammar of
such-and-such words, or if he were aware of this other possibility of expression" and so on. Thus we may say of
some philosophizing mathematicians that they are obviously not aware of the difference between the many different
usages of the word "proof"; and that they are not clear about
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the difference between the uses of the word "kind", when they talk of kinds of numbers, kinds of proofs, as though
the word "kind" here meant the same thing as in the context "kinds of apples". Or, we may say, they are not aware
of the different meanings of the word "discovery", when in one case we talk of the discovery of the construction of
the pentagon and in the other case of the discovery of the South Pole.
Page 29
Now when we distinguished a transitive and an intransitive use of such words as "longing", "fearing",
"expecting", etc., we said that some one might try to smooth over our difficulties by saying: "The difference between
the two cases is simply that in one case we know what we are longing for and in the other we don't". Now who says
this, I think, obviously doesn't see that the difference which he tried to explain away reappears when we carefully
consider the use of the word "to know" in the two cases. The expression "the difference is simply..." makes it appear
as though we had analysed the case and found a simple analysis; as when we point out that two substances with
very different names hardly differ in composition.
Page 29
We said in this case that we might use both expressions: "we feel a longing" (where "longing" is used
intransitively) and "we feel a longing and don't know what we are longing for". It may seem queer to say that we
may correctly use either of two forms of expression which seem to contradict each other; but such cases are very
frequent.
Page 29
Let us use the following example to clear this up. We say that the equation x2 = - 1 has the solution ±
. There was a time when one said that this equation had no solution. Now this statement, whether
agreeing or disagreeing, with the one which told us the solutions, certainly hasn't its multiplicity. But we can easily
give it that multiplicity by saying that an equation x2 + ax + b = 0 hasn't got a solution but comes α near to the
nearest solution which is β. Analogously we can say either "A straight line always intersects a circle; sometimes in
real, sometimes in complex points", or, "A straight line either intersects a circle, or it doesn't and is α far from doing
so". These two statements mean exactly the same. They will be more or less satisfactory according to the way a man
wishes to look at it. He may wish to make the difference between intersecting and not intersecting as inconspicuous
as possible. Or on the other hand he may wish to stress it; and either tendency may be justified, say, by his particular
practical purposes. But this may not be the reason at all why he prefers one form of expression to the other. Which
form he
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prefers, and whether he has a preference at all, often depends on general, deeply rooted, tendencies of his thinking.
Page 30
(Should we say that there are cases when a man despises another man and doesn't know it; or should we
describe such cases by saying that he doesn't despise him but unintentionally behaves towards him in a way-speaks
to him in a tone of voice, etc.--which in general would go together with despising him? Either form of expression is
correct; but they may betray different tendencies of the mind.)
Page 30
Let us revert to examining the grammar of the expressions "to wish", "to expect", "to long for", etc., and
consider that most important case in which the expression, "I wish so and so to happen" is the direct description of a
conscious process. That is to say, the case in which we should be inclined to answer the question "Are you sure that
it is this you wish?" by saying: "Surely I must know what I wish". Now compare this answer to the one which most
of us would give to the question: "Do you know the ABC?" Has the emphatic assertion that you know it a sense
analogous to that of the former assertion? Both assertions in a way brush aside the question. But the former doesn't
wish to say "Surely I know such a simple thing as this" but rather: "The question which you asked me makes
nosense". We might say: We adopt in this case a wrong method of brushing aside the question. "Of course I know"
could here be replaced by "Of course, there is no doubt" and this interpreted to mean "It makes, in this case, no
sense of talk of a doubt". In this way the answer "Of course I know what I wish" can be interpreted to be a
grammatical statement.
Page 30
It is similar when we ask, "Has this room a length?", and someone answers: "Of course it has". He might
have answered, "Don't ask nonsense".. On the other hand "The room has length" can be used as a grammatical
statement. It then says that a sentence of the form "The room is feet long" makes sense.
Page 30
A great many philosophical difficulties are connected with that sense of the expressions "to wish", "to think",
etc., which we are now considering. These can all be summed up in the question: "How can one think what is not the
case?"
Page 30
This is a beautiful example of a philosophical question. It asks "How can one...?" and while this puzzles us
we must admit that nothing is easier than to think what is not the case. I mean, this shows us again that the difficulty
which we are in does not arise through our inability to imagine how thinking something is done; just as the
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philosophical difficulty about the measurement of time did not arise through our inability to imagine how time was
actually measured. I say this because sometimes it almost seems as though our difficulty were one of remembering
exactly what happened when we thought something, a difficulty of introspection, or something of the sort; whereas
in fact it arises when we look at the facts through the medium of a misleading form of expression.
Page 31
"How can one think what is not the case? If I think that King's College is on fire when it is not on fire, the
fact of its being on fire does not exist. Then how can I think it? How can we hang a thief who doesn't exist?" Our
answer could be put in this form: "I can't hang him when he doesn't exist; but I can look for him when he doesn't
exist".
Page 31
We are here misled by the substantives "object of thought" and "fact", and by the different meanings of the
word "exist".
Page 31
Talking of the fact as a "complex of objects" springs from this confusion (cf. Tractatus
Logico-philosophicus). Supposing we asked: "How can one imagine what does not exist?" The answer seems to be:
"If we do, we imagine non-existent combinations of existing elements". A centaur doesn't exist, but a man's head
and torso and arms and a horse's legs do exist. "But can't we imagine an object utterly different from any one which
exists?"--We should be inclined to answer: "No; the elements, individuals, must exist. If redness, roundness and
sweetness did not exist, we could not imagine them".
Page 31
But what do you mean by "redness exists"? My watch exists, if it hasn't been pulled to pieces, if it hasn't
been destroyed. What would we call "destroying redness"? We might of course mean destroying all red objects; but
would this make it impossible to imagine a red object? Supposing to this one answered: "But surely, red objects
must have existed and you must have seen them if you are able to imagine them"?--But how do you know that this
is so? Suppose I said "Exerting a pressure on your eye-ball produces a red image". Couldn't the way by which you
first became acquainted with red have been this? And why shouldn't it have been just imagining a red patch? (The
difficulty which you may feel here will have to be discussed at a later occasion.†1)
Page 31
We may now be inclined to say: "As the fact which would make our thought true if it existed does not
always exist, it is not the fact which we think". But this just depends upon how I wish to use the word
Page Break 32
"fact". Why shouldn't I say: "I believe the fact that the college is on fire"? It is just a clumsy expression for saying: "I
believe that the college is on fire". To say "It is not the fact which we believe", is itself the result of a confusion. We
think we are saying something like: "It isn't the sugar-cane which we eat but the sugar", "It isn't Mr. Smith who
hangs in the gallery, but his picture".
Page 32
The next step we are inclined to take is to think that as the object of our thought isn't the fact it is a shadow of
the fact. There are different names for this shadow, e.g. "proposition", "sense of the sentence".
Page 32
But this doesn't remove our difficulty. For the question now is: "How can something be the shadow of a fact
which doesn't exist?"
Page 32
I can express our trouble in a different form by saying: "How can we know what the shadow is a shadow
of?"--The shadow would be some sort of portrait; and therefore I can restate our problem by asking: "What makes a
portrait a portrait of Mr. N?" The answer which might first suggest itself is: "The similarity between the portrait and
Mr. N". This answer in fact shows what we had in mind when we talked of the shadow of a fact. It is quite clear,
however, that similarity does not constitute our idea of a portrait; for it is in the essence of this idea that it should
make sense to talk of a good or a bad portrait. In other words, it is essential that the shadow should be capable of
representing things as in fact they are not.
Page 32
An obvious, and correct, answer to the question "What makes a portrait the portrait of so-and-so?" is that it
is the intention. But if we wish to know what it means "intending this to be a portrait of so-and-so" let's see what
actually happens when we intend this. Remember the occasion when we talked of what happened when we expect
some one from four to four-thirty. To intend a picture to be the portrait of so-and-so (on the part of the painter, e.g.)
is neither a particular state of mind nor a particular mental process. But there are a great many combinations of
actions and states of mind which we should call "intending..." It might have been that he was told to paint a portrait
of N, and sat down before N, going through certain actions which we call "copying N's face". One might object to
this by saying that the essence of copying is the intention to copy. I should answer that there are a great many
different processes which we call "copying something". Take an instance. I draw an ellipse on a sheet of paper and
ask you to copy it. What characterizes the process of copying? For it is clear that it isn't the fact that you draw a
similar
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ellipse. You might have tried to copy it and not succeeded; or you might have drawn an ellipse with a totally
different intention, and it happened to be like the one you should have copied. So what do you do when you try to
copy the ellipse? Well, you look at it, draw something on a piece of paper, perhaps measure what you have drawn,
perhaps you curse if you find it doesn't agree with the model; or perhaps you say "I am going to copy this ellipse"
and just draw an ellipse like it. There are an endless variety of actions and words, having a family likeness to each
other, which we call "trying to copy".
Page 33
Suppose we said "that a picture is a portrait of a particular object consists in its being derived from that
object in a particular way". Now it is easy to describe what we should call processes of deriving a picture from an
object (roughly speaking, processes of projection). But there is a peculiar difficulty about admitting that any such
process is what we call "intentional representation". For describe whatever process (activity) of projection we may,
there is a way of reinterpreting this projection. Therefore--one is tempted to say--such a process can never be the
intention itself. For we could always have intended the opposite by reinterpreting the process of projection. Imagine
this case: We give someone an order to walk in a certain direction by pointing or by drawing an arrow which points
in the direction. Suppose drawing arrows is the language in which generally we give such an order. Couldn't such an
order be interpreted to mean that the man who gets it is to walk in the direction opposite to that of the arrow? This
could obviously be done by adding to our arrow some symbols which we might call "an interpretation". It is easy
to imagine a case in which, say to deceive someone, we might make an arrangement that an order should be carried
out in the sense opposite to its normal one. The symbol which adds the interpretation to our original arrow could, for
instance, be another arrow. Whenever we interpret a symbol in one way or another, the interpretation is a new
symbol added to the old one.
Page 33
Now we might say that whenever we give someone an order by showing him an arrow, and don't do it
'mechanically' (without thinking), we mean the arrow in one way or another. And this process of meaning, of
whatever kind it may be, can be represented by another arrow (pointing in the same or the opposite sense to the
first). In this picture which we make of 'meaning and saying' it is essential that we should imagine the processes of
saying and meaning to take place in two different spheres.
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Page 34
Is it then correct to say that no arrow could be the meaning, as every arrow could be meant the opposite
way?--Suppose we write down the scheme of saying and meaning by a column of arrows one below the other.
Then if this scheme is to serve our purpose at all, it must show us which of the three levels is the level of meaning. I
can, e.g., make a scheme with three levels, the bottom level always being the level of meaning. But adopt whatever
model or scheme you may, it will have a bottom level, and there will be no such thing as an interpretation of that. To
say in this case that every arrow can still be interpreted would only mean that I could always make a different model
of saying and meaning which had one more level than the one I am using.
Page 34
Let us put it in this way:--What one wishes to say is: "Every sign is capable of interpretation; but the
meaning mustn't be capable of interpretation. It is the last interpretation." Now I assume that you take the meaning
to be a process accompanying the saying, and that it is translatable into, and so far equivalent to, a further sign. You
have therefore further to tell me what you take to be the distinguishing mark between a sign and the meaning. If
you do so, e.g., by saying that the meaning is the arrow which you imagine as opposed to any which you may draw
or produce in any other way, you thereby say that you will call no further arrow an interpretation of the one which
you have imagined.
Page 34
All this will become clearer if we consider what it is that really happens when we say a thing and mean what
we say.--Let us ask ourselves: If we say to someone "I should be delighted to see you" and mean it, does a
conscious process run alongside these words, a process which could itself be translated into spoken words? This will
hardly ever be the case.
Page 34
But let us imagine an instance in which it does happen. Supposing I had a habit of accompanying every
English sentence which I said aloud by a German sentence spoken to myself inwardly. If then, for some reason or
other, you call the silent sentence the meaning of the one spoken aloud, the process of meaning accompanying the
process of saying would be one which could itself be translated into outward signs. Or, before any sentence which
we say aloud we say its meaning
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(whatever it may be) to ourselves in a kind of aside. An example at least similar to the case we want would be saying
one thing and at the same time seeing a picture before our mind's eye which is the meaning and agrees or disagrees
with what we say. Such cases and similar ones exist, but they are not at all what happens as a rule when we say
something and mean it, or mean something else. There are, of course, real cases in which what we call meaning is a
definite conscious process accompanying, preceding, or following the verbal expression and itself a verbal
expression of some sort or translatable into one. A typical example of this is the 'aside' on the stage.
Page 35
But what tempts us to think of the meaning of what we say as a process essentially of the kind which we
have described is the analogy between the forms of expression:
"to say something"
"to mean something",
which seem to refer to two parallel processes.
Page 35
A process accompanying our words which one might call the "process of meaning them" is the modulation
of the voice in which we speak the words; or one of the processes similar to this, like the play of facial expression.
These accompany the spoken words not in the way a German sentence might accompany an English sentence, or
writing a sentence accompany speaking a sentence; but in the sense in which the tune of a song accompanies its
words. This tune corresponds to the 'feeling' with which we say the sentence. And I wish to point out that this
feeling is the expression with which the sentence is said, or something similar to this expression.
Page 35
Let us revert to our question: "What is the object of a thought?" (e.g. when we say, "I think that King's
College is on fire").
Page 35
The question as we put it is already the expression of several confusions. This is shown by the mere fact that
it almost sounds like a question of physics; like asking: "What are the ultimate constituents of matter?" (It is a
typically metaphysical question; the characteristic of a metaphysical question being that we express an unclarity
about the grammar of words in the form of a scientific question.)
Page 35
One of the origins of our question is the two-fold use of the propositional function "I think x". We say, "I
think that so-and-so will happen" or "that so-and-so is the case", and also "I think just the same thing as he"; and we
say "I expect him", and also "I expect that he will come". Compare "I expect him" and "I shoot him". We can't
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shoot him if he isn't there. This is how the question arises: "How can we expect something that is not the case?",
"How can we expect a fact which does not exist?"
Page 36
The way out of this difficulty seems to be: what we expect is not the fact, but a shadow of the fact; as it were,
the next thing to the fact. We have said that this is only pushing the question one step further back. There are several
origins to this idea of a shadow. One of them is this: we say "Surely two sentences of different languages can have
the same sense"; and we argue, "therefore the sense is not the same as the sentence", and ask the question "What is
the sense?" And we make of 'it' a shadowy being, one of the many which we create when we wish to give meaning
to substantives to which no material objects correspond.
Page 36
Another source of the idea of a shadow being the object of our thought is this: We imagine the shadow to be
a picture the intention of which cannot be questioned, that is, a picture which we don't interpret in order to
understand it, but which we understand without interpreting it. Now there are pictures of which we should say that
we interpret them, that is, translate them into a different kind of picture, in order to understand them; and pictures of
which we should say that we understand them immediately, without any further interpretation. If you see a telegram
written in cipher, and you know the key to this cipher, you will, in general, not say that you understand the telegram
before you have translated it into ordinary language. Of course you have only replaced one kind of symbols by
another; and yet if now you read the telegram in your language no further process of interpretation will take
place.--Or rather, you may now, in certain cases, again translate this telegram, say into a picture; but then too you
have only replaced one set of symbols by another.
Page 36
The shadow, as we think of it, is some sort of a picture; in fact, something very much like an image which
comes before our mind's eye; and this again is something not unlike a painted representation in the ordinary sense.
A source of the idea of the shadow certainly is the fact that in some cases saying, hearing, or reading a sentence
brings images before our mind's eye, images which more or less strictly correspond to the sentence, and which are
therefore, in a sense, translations of this sentence into a pictorial language.--But it is absolutely essential for the
picture which we imagine the shadow to be that it is what I shall call a "picture by similarity". I don't mean by this
that it is a picture similar to what it is intended to
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represent, but that it is a picture which is correct only when it is similar to what it represents. One might use for this
kind of picture the word "copy". Roughly speaking, copies are good pictures when they can easily be mistaken for
what they represent.
Page 37
A plane projection of one hemisphere of our terrestrial globe is not a picture by similarity or a copy in this
sense. It would be conceivable that I portrayed some one's face by projecting it in some queer way, though correctly
according to the adopted rule of projection, on a piece of paper, in such a way that no one would normally call the
projection "a good portrait of so-and-so" because it would not look a bit like him.
Page 37
If we keep in mind the possibility of a picture which, though correct, has no similarity with its object, the
interpolation of a shadow between the sentence and reality loses all point. For now the sentence itself can serve as
such a shadow. The sentence is just such a picture, which hasn't the slightest similarity with what it represents. If we
were doubtful about how the sentence "King's College is on fire" can be a picture of King's College on fire, we need
only ask ourselves: "How should we explain what the sentence means?" Such an explanation might consist of
ostensive definitions. We should say, e.g., "this is King's College" (pointing to the building), "this is a fire" (pointing
to a fire). This shews you the way in which words and things may be connected.
Page 37
The idea that that which we wish to happen must be present as a shadow in our wish is deeply rooted in our
forms of expression. But, in fact, we might say that it is only the next best absurdity to the one which we should
really like to say. If it weren't too absurd we should say that the fact which we wish for must be present in our wish.
For how can we wish just this to happen if just this isn't present in our wish? It is quite true to say: The mere
shadow won't do; for it stops short before the object; and we want the wish to contain the object itself.--We want
that the wish that Mr. Smith should come into this room should wish that just Mr. Smith, and no substitute, should
do the coming, and no substitute for that, into my room, and no substitute for that. But this is exactly what we said.
Page 37
Our confusion could be described in this way: Quite in accordance with our usual form of expression we
think of the fact which we wish for as a thing which is not yet here, and to which, therefore, we cannot point. Now in
order to understand the grammar of our expression "object of our wish" let's just consider the answer which we give
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to the question: "What is the object of your wish?" The answer to this question of course is "I wish that so-and-so
should happen". Now what would the answer be if we went on asking: "And what is the object of this wish?" It
could only consist in a repetition of our previous expression of the wish, or else in a translation into some other form
of expression. We might, e.g., state what we wished in other words or illustrate it by a picture, etc., etc. Now when
we are under the impression that what we call the object of our wish is, as it were, a man who has not yet entered
our room, and therefore can't yet be seen, we imagine that any explanation of what it is we wish is only the next best
thing to the explanation which would show the actual fact--which, we are afraid, can't yet be shown as it has not yet
entered.--It is as though I said to some one "I am expecting Mr. Smith", and he asked me "Who is Mr. Smith?", and
I answered, "I can't show him to you now, as he isn't there. All I can show you is a picture of him". It then seems as
though I could never entirely explain what I wished until it had actually happened. But of course this is a delusion.
The truth is that I needn't be able to give a better explanation of what I wished after the wish was fulfilled than
before; for I might perfectly well have shown Mr. Smith to my friend, and have shown him what "coming in"
means, and have shown him what my room is, before Mr. Smith came into my room.
Page 38
Our difficulty could be put this way: We think about things,--but how do these things enter into our
thoughts? We think about Mr. Smith; but Mr. Smith need not be present. A picture of him won't do; for how are we
to know whom it represents? In fact no substitute for hint will do. Then how can he himself be an object of our
thoughts? (I am here using the expression "object of our thought" in a way different from that in which I have used
it before. I mean now a thing I am thinking about, not 'that which I am thinking'.)
Page 38
We said the connection between our thinking, or speaking, about a man and the man himself was made
when, in order to explain the meaning of the word "Mr. Smith" we pointed to him, saying "this is Mr. Smith". And
there is nothing mysterious about this connection. I mean, there is no queer mental act which somehow conjures up
Mr. Smith in our minds when he really isn't here. What makes it difficult to see that this is the connection is a
peculiar form of expression of ordinary language, which makes it appear that the connection between our thought
(or the expression of our thought) and the thing we think about must have subsisted during the act of thinking.
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Page 39
"Isn't it queer that in Europe we should be able to mean someone who is in America?"--If someone had said
"Napoleon was crowned in 1804", and we asked him "Did you mean the man who won the battle of Austerlitz?" he
might say "Yes, I meant him". And the use of the past tense "meant" might make it appear as though the idea of
Napoleon having won the battle of Austerlitz must have been present in the man's mind when he said that Napoleon
was crowned in 1804.
Page 39
Someone says, "Mr. N. will come to see me this afternoon"; I ask "Do you mean him?" pointing to someone
present, and he answers "Yes". In this conversation a connection was established between the word "Mr. N." and
Mr. N. But we are tempted to think that while my friend said, "Mr. N. will come to see me", and meant what he said,
his mind must have made the connection.
Page 39
This is partly what makes us think of meaning or thinking as a peculiar mental activity; the word "mental"
indicating that we mustn't expect to understand how these things work.
Page 39
What we said of thinking can also be applied to imagining. Someone says, he imagines King's College on
fire. We ask him: "How do you know that it's King's College you imagine on fire? Couldn't it be a different building,
very much like it? In fact, is your imagination so absolutely exact that there might not be a dozen buildings whose
representation your image could be?"--And still you say: "There's no doubt I imagine King's College and no other
building". But can't saying this be making the very connection we want? For saying it is like writing the words
"Portrait of Mr. So-and-so" under a picture. It might have been that while you imagined King's College on fire you
said the words "King's College is on fire". But in very many cases you certainly don't speak explanatory words in
your mind while you have the image. And consider, even if you do, you are not going the whole way from your
image to King's College, but only to the words "King's College". The connection between these words and King's
College was, perhaps, made at another time.
Page 39
The fault which in all our reasoning about these matters we are inclined to make is to think that images and
experiences of all sorts, which are in some sense closely connected with each other, must be present in our mind at
the same time. If we sing a tune we know by heart, or say the alphabet, the notes or letters seem to hang together,
and each seems to draw the next after it, as though they were a string of
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pearls in a box, and by pulling out one pearl I pulled out the one following it.
Page 40
Now there is no doubt that, having the visual image of a string of beads being pulled out of a box through a
hole in the lid, we should be inclined to say: "These beads must all have been together in the box before". But it is
easy to see that this is making a hypothesis. I should have had the same image if the beads had gradually come into
existence in the hole of the lid. We easily overlook the distinction between stating a conscious mental event, and
making a hypothesis about what one might call the mechanism of the mind. All the more as such hypotheses or
pictures of the working of our mind are embodied in many of the forms of expression of our everyday language. The
past tense "meant" in the sentence "I meant the man who won the battle of Austerlitz" is part of such a picture, the
mind being conceived as a place in which what we remember is kept, stored, before we express it. If I whistle a tune
I know well and am interrupted in the middle, if then someone asks me "did you know how to go on?" I should
answer "yes, I did". What sort of process is this knowing how to go on? It might appear as though the whole
continuation of the tune had to be present while I knew how to go on.
Page 40
Ask yourself such a question as: "How long does it take to know how to go on?" Or is it an instantaneous
process? Aren't we making a mistake like mixing up the existence of a gramophone record of a tune with the
existence of the tune? And aren't we assuming that whenever a tune passes through existence there must be some
sort of a gramophone record of it from which it is played?
Page 40
Consider the following example: A gun is fired in my presence and I say: "This crash wasn't as loud as I had
expected". Someone asks me: "How is this possible? Was there a crash, louder than that of a gun, in your
imagination?" I must confess that there was nothing of the sort. Now he says: "Then you didn't really expect a
louder crash-but perhaps the shadow of one.--And how did you know that it was the shadow of a louder
crash?"--Let's see what, in such a case, might really have happened. Perhaps in waiting for the report I opened my
mouth, held on to something to steady myself, and perhaps I said: "This is going to be terrible". Then, when the
explosion was over: "It wasn't so loud after all".--Certain tensions in my body relax. But what is the connection
between these tensions, opening my mouth, etc., and a real louder crash? Perhaps this connection
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was made by having heard such a crash and having had the experiences mentioned.
Page 41
Examine expressions like "having an idea in one's mind", "analysing the idea before one's mind". In order not
to be misled by them see what really happens when, say, in writing a letter you are looking for the words which
correctly express the idea which is "before your mind". To say that we are trying to express the idea which is before
our mind is to use a metaphor, one which very naturally suggests itself; and which is all right so long as it doesn't
mislead us when we are philosophizing. For when we recall what really happens in such cases we find a great variety
of processes more or less akin to each other.--We might be inclined to say that in all such cases, at any rate, we are
guided by something before our mind. But then the words "guided" and "thing before our mind" are used in as
many senses as the words "idea" and "expression of an idea".
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The phrase "to express an idea which is before our mind" suggests that what we are trying to express in
words is already expressed, only in a different language; that this expression is before our mind's eye; and that what
we do is to translate from the mental into the verbal language. In most cases which we call "expressing an idea, etc."
something very different happens. Imagine what it is that happens in cases such as this: I am groping for a word.
Several words are suggested and I reject them. Finally one is proposed and I say: "That is what I meant!"
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(We should be inclined to say that the proof of the impossibility of trisecting the angle with ruler and
compasses analyses our idea of the trisection of an angle. But the proof gives us a new idea of trisection, one which
we didn't have before the proof constructed it. The proof led us a road which we were inclined to go; but it led us
away from where we were, and didn't just show us clearly the place where we had been all the time.)
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Let us now revert to the point where we said that we gained nothing by assuming that a shadow must
intervene between the expression of our thought and the reality with which our thought is concerned. We said that if
we wanted a picture of reality the sentence itself is such a picture (though not a picture by similarity).
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I have been trying in all this to remove the temptation to think that there 'must be' what is called a mental
process of thinking, hoping, wishing, believing, etc., independent of the process of expressing a thought, a hope, a
wish, etc. And I want to give you the following
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rule of thumb: If you are puzzled about the nature of thought, belief, knowledge, and the like, substitute for the
thought the expression of the thought, etc. The difficulty which lies in this substitution, and at the same time the
whole point of it, is this: the expression of belief, thought, etc., is just a sentence;--and the sentence has sense only
as a member of a system of language; as one expression within a calculus. Now we are tempted to imagine this
calculus, as it were, as a permanent background to every sentence which we say, and to think that, although the
sentence as written on a piece of paper or spoken stands isolated, in the mental act of thinking the calculus is
there--all in a lump. The mental act seems to perform in a miraculous way what could not be performed by any act
of manipulating symbols. Now when the temptation to think that in some sense the whole calculus must be present
at the same time vanishes, there is no more point in postulating the existence of a peculiar kind of mental act
alongside of our expression. This, of course, doesn't mean that we have shown that peculiar acts of consciousness
do not accompany the expressions of our thoughts! Only we no longer say that they must accompany them.
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"But the expression of our thoughts can always lie, for we may say one thing and mean another". Imagine
the many different things which happen when we say one thing and mean another!--Make the following experiment:
say the sentence "It is hot in this room", and mean: "it is cold". Observe closely what you are doing.
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We could easily imagine beings who do their private thinking by means of 'asides' and who manage their lies
by saying one thing aloud, following it up by an aside which says the opposite.
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"But meaning, thinking, etc., are private experiences. They are not activities like writing, speaking, etc."--But
why shouldn't they be the specific private experiences of writing--the muscular, visual, tactile sensations of writing
or speaking?
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Make the following experiment: say and mean a sentence, e.g.: "It will probably rain tomorrow". Now think
the same thought again, mean what you just meant, but without saying anything (either aloud or to yourself). If
thinking that it will rain tomorrow accompanied saying that it will rain tomorrow, then just do the first activity and
leave out the second.--If thinking and speaking stood in the relation of the words and the melody of a song, we
could leave out the speaking and do the thinking just as we can sing the tune without the words.
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Page 43
But can't one at any rate speak and leave out the thinking? Certainly--but observe what sort of thing you are
doing if you speak without thinking. Observe first of all that the process which we might call "speaking and meaning
what you speak" is not necessarily distinguished from that of speaking thoughtlessly by what happens at the time
when you speak. What distinguishes the two may very well be what happens before or after you speak.
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Suppose I tried, deliberately, to speak without thinking;--what in fact would I do? I might read out a sentence
from a book, trying to read it automatically, that is, trying to prevent myself from following the sentence with images
and sensations which otherwise it would produce. A way of doing this would be to concentrate my attention on
something else while I was speaking the sentence, e.g., by pinching my skin hard while I was speaking.--Put it this
way: Speaking a sentence without thinking consists in switching on speech and switching off certain
accompaniments of speech. Now ask yourself: Does thinking the sentence without speaking it consist in turning
over the switch (switching on what we previously switched off and vice versa); that is: does thinking the sentence
without speaking it now simply consist in keeping on what accompanied the words but leaving out the words? Try
to think the thoughts of a sentence without the sentence and see whether this is what happens.
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Let us sum up: If we scrutinize the usages which we make of such words as "thinking", "meaning",
"wishing", etc., going through this process rids us of the temptation to look for a peculiar act of thinking,
independent of the act of expressing our thoughts, and stowed away in some peculiar medium. We are no longer
prevented by the established forms of expression from recognizing that the experience of thinking may be just the
experience of saying, or may consist of this experience plus others which accompany it. (It is useful also to examine
the following case: Suppose a multiplication is part of a sentence; ask yourself what it is like to say the multiplication
7 × 5 = 35, thinking it, and, on the other hand, saying it without thinking.) The scrutiny of the grammar of a word
weakens the position of certain fixed standards of our expression which had prevented us from seeing facts with
unbiassed eyes. Our investigation tried to remove this bias, which forces us to think that the facts must conform to
certain pictures embedded in our language.
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"Meaning" is one of the words of which one may say that they have odd jobs in our language. It is these
words which cause most
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philosophical troubles. Imagine some institution: most of its members have certain regular functions, functions
which can easily be described, say, in the statutes of the institution. There are, on the other hand, some members
who are employed for odd jobs, which nevertheless may be extremely important.--What causes most trouble in
philosophy is that we are tempted to describe the use of important 'odd-job' words as though they were words with
regular functions.
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The reason I postponed talking about personal experience was that thinking about this topic raises a host of
philosophical difficulties which threaten to break up all our commonsense notions about what we should commonly
call the objects of our experience. And if we were struck by these problems it might seem to us that all we have said
about signs and about the various objects we mentioned in our examples may have to go into the melting-pot.
Page 44
The situation in a way is typical in the study of philosophy; and one sometimes has described it by saying
that no philosophical problem can be solved until all philosophical problems are solved; which means that as long as
they aren't all solved every new difficulty renders all our previous results questionable. To this statement we can only
give a rough answer if we are to speak about philosophy in such general terms. It is, that every new problem which
arises may put in question the position which our previous partial results are to occupy in the final picture. One then
speaks of having to reinterpret these previous results; and we should say: they have to be placed in a different
surrounding.
Page 44
Imagine we had to arrange the books of a library. When we begin the books lie higgledy-piggledy on the
floor. Now there would be many ways of sorting them and putting them in their places. One would be to take the
books one by one and put each on the shelf in its right place. On the other hand we might take up several books
from the floor and put them in a row on a shelf, merely in order to indicate that these books ought to go together in
this order. In the course of arranging the library this whole row of books will have to change its place. But it would
be wrong to say that therefore putting them together on a shelf was no step towards the final result. In this case, in
fact, it is pretty obvious that having put together books which belong together was a definite achievement, even
though the whole row of them had to be shifted. But some of the greatest achievements in philosophy could only be
compared with taking up some books which seemed to belong together, and putting them on different
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shelves; nothing more being final about their positions than that they no longer lie side by side. The onlooker who
doesn't know the difficulty of the task might well think in such a case that nothing at all had been achieved.--The
difficulty in philosophy is to say no more than we know. E.g., to see that when we have put two books together in
their right order we have not thereby put them in their final places.
Page 45
When we think about the relation of the objects surrounding us to our personal experiences of them, we are
sometimes tempted to say that these personal experiences are the material of which reality consists. How this
temptation arises will become clearer later on.
Page 45
When we think in this way we seem to lose our firm hold on the objects surrounding us. And instead we are
left with a lot of separate personal experiences of different individuals. These personal experiences again seem vague
and seem to be in constant flux. Our language seems not to have been made to describe them. We are tempted to
think that in order to clear up such matters philosophically our ordinary language is too coarse, that we need a more
subtle one.
Page 45
We seem to have made a discovery--which I could describe by saying that the ground on which we stood
and which appeared to be firm and reliable was found to be boggy and unsafe.--That is, this happens when we
philosophize; for as soon as we revert to the standpoint of common sense this general uncertainty disappears.
Page 45
This queer situation can be cleared up somewhat by looking at an example; in fact a kind of parable
illustrating the difficulty we are in, and also showing the way out of this sort of difficulty: We have been told by
popular scientists that the floor on which we stand is not solid, as it appears to common sense, as it has been
discovered that the wood consists of particles filling space so thinly that it can almost be called empty. This is liable
to perplex us, for in a way of course we know that the floor is solid, or that, if it isn't solid, this may be due to the
wood being rotten but not to its being composed of electrons. To say, on this latter ground, that the floor is not solid
is to misuse language. For even if the particles were as big as grains of sand, and as close together as these are in a
sandheap, the floor would not be solid if it were composed of them in the sense in which a sandheap is composed of
grains. Our perplexity was based on a misunderstanding; the picture of the thinly filled space had been wrongly
applied. For this picture of the structure of matter was meant to explain the very phenomenon of solidity.
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As in this example the word "solidity" was used wrongly and it
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seemed that we had shown that nothing really was solid, just in this way, in stating our puzzles about the general
vagueness of sense-experience, and about the flux of all phenomena, we are using the words "flux" and "vagueness"
wrongly, in a typically metaphysical way, namely without an antithesis; whereas in their correct and everyday use
vagueness is opposed to clearness, flux to stability, inaccuracy to accuracy, and problem to solution. The very word
"problem", one might say, is misapplied when used for our philosophical troubles. These difficulties, as long as they
are seen as problems, are tantalizing, and appear insoluble.
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There is a temptation for me to say that only my own experience is real: "I know that I see, hear, feel pains,
etc., but not that anyone else does. I can't know this, because I am I and they are they."
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On the other hand I feel ashamed to say to anyone that my experience is the only real one; and I know that
he will reply that he could say exactly the same thing about his experience. This seems to lead to a silly quibble.
Also I am told: "If you pity someone for having pains, surely you must at least believe that he has pains". But how
can I even believe this? How can these words make sense to me? How could I even have come by the idea of
another's experience if there is no possibility of any evidence for it?
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But wasn't this a queer question to ask? Can't I believe that someone else has pains? Is it not quite easy to
believe this?--Is it an answer to say that things are as they appear to common sense?--Again, needless to say, we
don't feel these difficulties in ordinary life. Nor is it true to say that we feel them when we scrutinize our experiences
by introspection, or make scientific investigations about them. But somehow, when we look at them in a certain
way, our expression is liable to get into a tangle. It seems to us as though we had either the wrong pieces, or not
enough of them, to put together our jigsaw puzzle. But they are all there, only all mixed up; and there is a further
analogy between the jig-saw puzzle and our case: It's no use trying to apply force in fitting pieces together. All we
should do is to look at them carefully and arrange them.
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There are propositions of which we may say that they describe facts in the material world (external world).
Roughly speaking, they treat of physical objects: bodies, fluids, etc. I am not thinking in particular of the laws of the
natural sciences, but of any such proposition as "the tulips in our garden are in full bloom", or "Smith will come in
any moment". There are on the other hand propositions
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describing personal experiences, as when the subject in a psychological experiment describes his sense-experiences;
say his visual experience, independent of what bodies are actually before his eyes and, n.b., independent also of any
processes which might be observed to take place in his retina, his nerves, his brain, or other parts of his body. (That
is, independent of both physical and physiological facts.)
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At first sight it may appear (but why it should can only become clear later) that here we have two kinds of
worlds, worlds built of different materials; a mental world and a physical world. The mental world in fact is liable to
be imagined as gaseous, or rather, aethereal. But let me remind you here of the queer role which the gaseous and the
aethereal play in philosophy,--when we perceive that a substantive is not used as what in general we should call the
name of an object, and when therefore we can't help saying to ourselves that it is the name of an aethereal object. I
mean, we already know the idea of 'aethereal objects' as a subterfuge, when we are embarrassed about the grammar
of certain words, and when all we know is that they are not used as names for material objects. This is a hint as to
how the problem of the two materials, mind and matter, is going to dissolve.
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It seems to us sometimes as though the phenomena of personal experience were in a way phenomena in the
upper strata of the atmosphere as opposed to the material phenomena which happen on the ground. There are views
according to which these phenomena in the upper strata arise when the material phenomena reach a certain degree
of complexity. E.g., that the mental phenomena, sense experience, volition, etc., emerge when a type of animal body
of a certain complexity has been evolved. There seems to be some obvious truth in this, for the amoeba certainly
doesn't speak or write or discuss, whereas we do. On the other hand the problem here arises which could be
expressed by the question: "Is it possible for a machine to think?" (whether the action of this machine can be
described and predicted by the laws of physics or, possibly, only by laws of a different kind applying to the
behaviour of organisms). And the trouble which is expressed in this question is not really that we don't yet know a
machine which could do the job. The question is not analogous to that which someone might have asked a hundred
years ago: "Can a machine liquefy a gas?" The trouble is rather that the sentence, "A machine thinks (perceives,
wishes)": seems somehow nonsensical. It is as though we had asked "Has the number 3 a colour?" ("What colour
could it be, as it obviously has none of the colours known to
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us?") For in one aspect of the matter, personal experience, far from being the product of physical, chemical,
physiological processes, seems to be the very basis of all that we say with any sense about such processes. Looking
at it in this way we are inclined to use our idea of a building-material in yet another misleading way, and to say that
the whole world, mental and physical, is made of one material only.
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When we look at everything that we know and can say about the world as resting upon personal experience,
then what we know seems to lose a good deal of its value, reliability, and solidity. We are then inclined to say that it
is all "subjective"; and "subjective" is used derogatorily, as when we say that an opinion is merely subjective, a
matter of taste. Now, that this aspect should seem to shake the authority of experience and knowledge points to the
fact that here our language is tempting us to draw some misleading analogy. This should remind us of the case when
the popular scientist appeared to have shown us that the floor which we stand on is not really solid because it is
made up of electrons.
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We are up against trouble caused by our way of expression.
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Another such trouble, closely akin, is expressed in the sentence: "I can only know that I have personal
experiences, not that anyone else has".--Shall we then call it an unnecessary hypothesis that anyone else has
personal experiences?--But is it an hypothesis at all? For how can I even make the hypothesis if it transcends all
possible experience? How could such a hypothesis be backed by meaning? (Is it not like paper money, not backed
by gold?)--It doesn't help if anyone tells us that, though we don't know whether the other person has pains, we
certainly believe it when, for instance, we pity him. Certainly we shouldn't pity him if we didn't believe that he had
pains; but is this a philosophical, a metaphysical belief? Does a realist pity me more than an idealist or a
solipsist?--In fact the solipsist asks: "How can we believe that the other has pain; what does it mean to believe this?
How can the expression of such a supposition make sense?"
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Now the answer of the common-sense philosopher--and that, n.b., is not the commonsense man, who is as
far from realism as from idealism--the answer of the common-sense philosopher is that surely there is no difficulty
in the idea of supposing, thinking, imagining that someone else has what I have. But the trouble with the realist is
always that he does not solve but skip the difficulties which his adversaries see, though they too don't succeed in
solving them. The
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realist answer, for us, just brings out the difficulty; for who argues like this overlooks the difference between
different usages of the words "to have", "to imagine". "A has a gold tooth" means that the tooth is in A's mouth. This
may account for the fact that I am not able to see it. Now the case of his toothache, of which I say that I am not able
to feel it because it is in his mouth, is not analogous to the case of the gold tooth. It is the apparent analogy, and
again the lack of analogy, between these cases which causes our trouble. And it is this troublesome feature in our
grammar which the realist does not notice. It is conceivable that I feel pain in a tooth in another man's mouth; and
the man who says that he cannot feel the other's toothache is not denying this. The grammatical difficulty which we
are in we shall only see clearly if we get familiar with the idea of feeling pain in another person's body. For
otherwise, in puzzling about this problem, we shall be liable to confuse our metaphysical proposition "I can't feel his
pain" with the experiential proposition, "We can't have (haven't as a rule) pains in another person's tooth". In this
proposition the word "can't" is used in the same way as in the proposition "An iron nail can't scratch glass". (We
could write this in the form "experience teaches that an iron nail doesn't scratch glass", thus doing away with the
"can't".) In order to see that it is conceivable that one person should have pain in another person's body, one must
examine what sort of facts we call criteria for a pain being in a certain place. It is easy to imagine the following case:
When I see my hands I am not always aware of their connection with the rest of my body. That is to say, I often see
my hand moving but don't see the arm which connects it to my torso. Nor do I necessarily, at the time, check up on
the arm's existence in any other way. Therefore the hand may, for all I know, be connected to the body of a man
standing beside me (or, of course, not to a human body at all). Suppose I feel a pain which on the evidence of the
pain alone, e.g., with closed eyes, I should call a pain in my left hand. Someone asks me to touch the painful spot
with my right hand. I do so and looking round perceive that I am touching my neighbour's hand (meaning the hand
connected to my neighbour's torso).
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Ask yourself: How do we know where to point to when we are asked to point to the painful spot? Can this
sort of pointing be compared with pointing to a black spot on a sheet of paper when someone says: "Point to the
black spot on this sheet"? Suppose someone said "You point to this spot because you know before you point that
the
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pains are there"; ask yourself "What does it mean to know that the pains are there?" The word "there" refers to a
locality;--but in what space, i.e., a 'locality' in what sense? Do we know the place of pain in Euclidean space, so that
when we know where we have pains we know how far away from two of the walls of this room, and from the floor?
When I have pain in the tip of my finger and touch my tooth with it, is my pain now both a toothache and a pain in
my finger? Certainly, in one sense the pain can be said to be located on the tooth. Is the reason why in this case it is
wrong to say I have toothache, that in order to be in the tooth the pain should be one sixteenth of an inch away from
the tip of my finger? Remember that the word "where" can refer to localities in many different senses. (Many
different grammatical games, resembling each other more or less, are played with this word. Think of the different
uses of the numeral "1".) I may know where a thing is and then point to it by virtue of that knowledge. The
knowledge tells me where to point to. We here conceived this knowledge as the condition for deliberately pointing to
the object. Thus one can say: "I can point to the spot you mean because I see it", "I can direct you to the place
because I know where it is; first turning to the right, etc." Now one is inclined to say "I must know where a thing is
before I can point to it". Perhaps you will feel less happy about saying: "I must know where a thing is before I can
look at it". Sometimes of course it is correct to say this. But we are tempted to think that there is one particular
psychical state or event, the knowledge of the place, which must precede every deliberate act of pointing, moving
towards, etc. Think of the analogous case: "One can only obey an order after having understood it".
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If I point to the painful spot on my arm, in what sense can I be said to have known where the pain was
before I pointed to the place? Before I pointed I could have said "The pain is in my left arm". Supposing my arm had
been covered with a meshwork of lines numbered in such a way that I could refer to any place on its surface. Was it
necessary that I should have been able to describe the painful spot by means of these coordinates before I could
point to it? What I wish to say is that the act of pointing determines a place of pain. This act of pointing, by the way,
is not to be confused with that of finding the painful spot by probing. In fact the two may lead to different results.
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An innumerable variety of cases can be thought of in which we should say that someone has pains in
another person's body; or, say,
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in a piece of furniture, or in any empty spot. Of course we mustn't forget that a pain in a particular part of our body,
e.g., in an upper tooth, has a peculiar tactile and kinaesthetic neighbourhood. Moving our hand upward a little
distance we touch our eye; and the word "little distance" here refers to tactile distance or kinaesthetic distance, or
both. (It is easy to imagine tactile and kinaesthetic distances correlated in ways different from the usual. The distance
from our mouth to our eye might seem very great 'to the muscles of our arm' when we move our finger from the
mouth to the eye. Think how large you imagine the cavity in your tooth when the dentist is drilling and probing it.)
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When I said that if we moved our hand upward a little, we touch our eye, I was referring to tactile evidence
only. That is, the criterion for my finger touching my eye was to be only that I had the particular feeling which
would have made me say that I was touching my eye, even if I had no visual evidence for it, and even if, on looking
into a mirror, I saw my finger not touching my eye, but, say, my forehead. Just as the 'little distance' I referred to was
a tactile or kinaesthetic one, so also the places of which I said, "they lie a little distance apart" were tactile places. To
say that my finger in tactile and kinaesthetic space moves from my tooth to my eye then means that I have those
tactile and kinaesthetic experiences which we normally have when we say "my finger moves from my tooth to my
eye". But what we regard as evidence for this latter proposition is, as we all know, by no means only tactile and
kinaesthetic. In fact if I had the tactile and kinaesthetic sensations referred to, I might still deny the proposition "my
finger moves etc...." because of what I saw. That proposition is a proposition about physical objects. (And now don't
think that the expression "physical objects" is meant to distinguish one kind of object from another.) The grammar
of propositions which we call propositions about physical objects admits of a variety of evidences for every such
proposition. It characterizes the grammar of the proposition "my finger moves, etc." that I regard the propositions "I
see it move", "I feel it move", "He sees it move", "He tells me that it moves", etc. as evidences for it. Now if I say "I
see my hand move", this at first sight seems to presuppose that I agree with the proposition "my hand moves". But if
I regard the proposition "I see my hand move" as one of the evidences for the proposition "my hand moves", the
truth of the latter is, of course, not presupposed in the truth of the former. One might therefore suggest the
expression "It looks as
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though my hand were moving" instead of "I see my hand moving". But this expression, although it indicates that my
hand may appear to be moving without really moving, might still suggest that after all there must be a hand in order
that it should appear to be moving; whereas we could easily imagine cases in which the proposition describing the
visual evidence is true and at the same time other evidences make us say that I have no hand. Our ordinary way of
expression obscures this. We are handicapped in ordinary language by having to describe, say, a tactile sensation by
means of terms for physical objects such as the word "eye", "finger", etc., when what we want to say does not entail
the existence of an eye or finger, etc.. We have to use a roundabout description of our sensations. This of course
does not mean that ordinary language is insufficient for our special purposes, but that it is slightly cumbrous and
sometimes misleading. The reason for this peculiarity of our language is of course the regular coincidence of certain
sense experiences. Thus when I feel my arm moving I mostly also can see it moving. And if I touch it with my hand,
also that hand feels the motion, etc. (The man whose foot has been amputated will describe a particular pain as pain
in his foot.) We feel in such cases a strong need for such an expression as: "a sensation travels from my tactual
cheek to my tactual eye". I said all this because, if you are aware of the tactual and kinaesthetic environment of a
pain, you may find a difficulty in imagining that one could have toothache anywhere else than in one's own teeth.
But if we imagine such a case, this simply means that we imagine a correlation between visual, tactual, kinaesthetic,
etc., experiences different from the ordinary correlation. Thus we can imagine a person having the sensation of
toothache plus those tactual and kinaesthetic experiences which are normally bound up with seeing his hand
travelling from his tooth to his nose, to his eyes, etc., but correlated to the visual experience of his hand moving to
those places in another person's face. Or again, we can imagine a person having the kinaesthetic sensation of moving
his hand, and the tactual sensation, in his fingers and face, of his fingers moving over his face, whereas his
kinaesthetic and visual sensations should have to be described as those of his fingers moving over his knee. If we
had a sensation of toothache plus certain tactual and kinaesthetic sensations usually characteristic of touching the
painful tooth and neighbouring parts of our face, and if these sensations were accompanied by seeing my hand
touch, and move about on, the edge of my table, we should feel doubtful whether to call this
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experience an experience of toothache in the table or not. If, on the other hand, the tactual and kinaesthetic
sensations described were correlated to the visual experience of seeing my hand touch a tooth and other parts of the
face of another person, there is no doubt that I would call this experience "toothache in another person's tooth".
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I said that the man who contended that it was impossible to feel the other person's pain did not thereby wish
to deny that one person could feel pain in another person's body. In fact, he would have said: "I may have toothache
in another man's tooth, but not his toothache".
Page 53
Thus the propositions "A has a gold tooth" and "A has toothache" are not used analogously. They differ in
their grammar where at first sight they might not seem to differ.
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As to the use of the word "imagine"-one might say: "Surely there is quite a definite act of imagining the other
person to have pain". Of course we don't deny this, or any other statement about facts. But let us see: If we make an
image of the other person's pain, do we apply it in the same way in which we apply the image, say, of a black eye,
when we imagine the other person having one? Let us again replace imagining, in the ordinary sense, by making a
painted image. (This could quite well be the way certain beings did their imagining.) Then let a man imagine in this
way that A has a black eye. A very important application of this picture will be comparing it with the real eye to see
if the picture is correct. When we vividly imagine that someone suffers pain, there often enters in our image what
one might call a shadow of a pain felt in the locality corresponding to that in which we say his pain is felt. But the
sense in which an image is an image is determined by the way in which it is compared with reality. This we might
call the method of projection. Now think of comparing an image of A's toothache with his toothache. How would
you compare them? If you say, you compare them 'indirectly' via his bodily behaviour, I answer that this means you
don't compare them as you compare the picture of his behaviour with his behaviour.
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Again, when you say, "I grant you that you can't know when A has pain, you can only conjecture it", you
don't see the difficulty which lies in the different uses of the words "conjecturing" and "knowing". What sort of
impossibility were you referring to when you said you couldn't know? Weren't you thinking of a case analogous to
that when one couldn't know whether the other man had a gold tooth in his mouth because he had his mouth shut?
Here what you didn't know you could nevertheless imagine knowing; it made sense to say that you
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saw that tooth although you didn't see it; or rather, it makes sense to say that you don't see his tooth and therefore it
also makes sense to say that you do. When on the other hand, you granted me that a man can't know whether the
other person has pain, you do not wish to say that as a matter of fact people didn't know, but that it made no sense
to say they knew (and therefore no sense to say they don't know). If therefore in this case you use the term
"conjecture" or "believe", you don't use it as opposed to "know". That is, you did not state that knowing was a goal
which you could not reach, and that you have to be contented with conjecturing; rather, there is no goal in this
game. Just as when one says "You can't count through the whole series of cardinal numbers", one doesn't state a fact
about human frailty but about a convention which we have made. Our statement is not comparable, though always
falsely compared, with such a one as "it is impossible for a human being to swim across the Atlantic"; but it is
analogous to a statement like "there is no goal in an endurance race". And this is one of the things which the person
feels dimly who is not satisfied with the explanation that though you can't know... you can conjecture....
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If we are angry with someone for going out on a cold day with a cold in his head, we sometimes say: "I won't
feel your cold". And this can mean: "I don't suffer when you catch a cold". This is a proposition taught by
experience. For we could imagine a, so to speak, wireless connection between the two bodies which made one
person feel pain in his head when the other had exposed his to the cold air. One might in this case argue that the
pains are mine because they are felt in my head; but suppose I and someone else had a part of our bodies in
common, say a hand. Imagine the nerves and tendons of my arm and A's connected to this hand by an operation.
Now imagine the hand stung by a wasp. Both of us cry, contort our faces, give the same description of the pain, etc.
Now are we to say we have the same pain or different ones? If in such a case you say: "We feel pain in the same
place, in the same body, our descriptions tally, but still my pain can't be his", I suppose as a reason you will be
inclined to say: "because my pain is my pain and his pain is his pain". And here you are making a grammatical
statement about the use of such a phrase as "the same pain". You say that you don't wish to apply the phrase, "he
has got my pain" or "we both have the same pain", and instead, perhaps, you will apply such a phrase as "his pain is
exactly like mine". (It would be no argument to say that the two couldn't have the same pain
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because one might anaesthetize or kill one of them while the other still felt pain.) Of course, if we exclude the phrase
"I have his toothache" from our language, we thereby also exclude "I have (or feel) my toothache". Another form of
our metaphysical statement is this: "A man's sense data are private to himself". And this way of expressing it is even
more misleading because it looks still more like an experiential proposition; the philosopher who says this may well
think that he is expressing a kind of scientific truth.
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We use the phrase "two books have the same colour", but we could perfectly well say: "They can't have the
same colour, because, after all, this book has its own colour, and the other book has its own colour too". This also
would be stating a grammatical rule-a rule, incidentally, not in accordance with our ordinary usage. The reason why
one should think of these two different usages at all is this: We compare the case of sense data with that of physical
bodies, in which case we make a distinction between: "this is the same chair that I saw an hour ago" and "this is not
the same chair, but one exactly like the other". Here it makes sense to say, and it is an experiential proposition: "A
and B couldn't have seen the same chair, for A was in London and B in Cambridge; they saw two chairs exactly
alike". (Here it will be useful if you consider the different criteria for what we call the "identity of these objects".
How do we apply the statements: "This is the same day...", "This is the same word...", "This is the same occasion...",
etc.?)
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What we did in these discussions was what we always do when we meet the word "can" in a metaphysical
proposition. We show that this proposition hides a grammatical rule. That is to say, we destroy the outward
similarity between a metaphysical proposition and an experiential one, and we try to find the form of expression
which fulfils a certain craving of the metaphysician which our ordinary language does not fulfil and which, as long as
it isn't fulfilled, produces the metaphysical puzzlement. Again, when in a metaphysical sense I say "I must always
know when I have pain", this simply makes the word "know" redundant; and instead of "I know that I have pain", I
can simply say "I have pain". The matter is different, of course, if we give the phrase "unconscious pain" sense by
fixing experiential criteria for the case in which a man has pain and doesn't know it, and if then we say (rightly or
wrongly) that as a matter of fact nobody has ever had pains which he didn't know of.
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When we say "I can't feel his pain", the idea of an insurmountable
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barrier suggests itself to us. Let us think straight away of a similar case: "The colours green and blue can't be in the
same place simultaneously". Here the picture of physical impossibility which suggests itself is, perhaps, not that of a
barrier; rather we feel that the two colours are in each other's way. What is the origin of this idea?--We say three
people can't sit side by side on this bench; they have no room. Now the case of the colours is not analogous to this;
but it is somewhat analogous to saying: "3 × 18 inches won't go into 3 feet". This is a grammatical rule and states a
logical impossibility. The proposition "three men can't sit side by side on a bench a yard long" states a physical
impossibility; and this example shows clearly why the two impossibilities are confused. (Compare the proposition
"He is 6 inches taller than I" with "6 foot is 6 inches longer than 5 foot 6". These propositions are of utterly different
kinds, but look exactly alike.) The reason why in these cases the idea of physical impossibility suggests itself to us is
that on the one hand we decide against using a particular form of expression, on the other hand we are strongly
tempted to use it, since (a) it sounds English, or German, etc., all right, and (b) there are closely similar forms of
expression used in other departments of our language. We have decided against using the phrase "They are in the
same place"; on the other hand this phrase strongly recommends itself to us through the analogy with other phrases,
so that, in a sense, we have to turn this form of expression out by force. And this is why we seem to ourselves to be
rejecting a universally false proposition. We make a picture like that of the two colours being in each other's way, or
that of a barrier which doesn't allow one person to come closer to another's experience than to the point of observing
his behaviour; but on looking closer we find that we can't apply the picture which we have made.
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Our wavering between logical and physical impossibility makes us make such statements as this: "If what I
feel is always my pain only, what can the supposition mean that someone else has pain?" The thing to do in such
cases is always to look how the words in question are actually used in our language. We are in all such cases
thinking of a use different from that which our ordinary language makes of the words. Of a use, on the other hand,
which just then for some reason strongly recommends itself to us. When something seems queer about the grammar
of our words, it is because we are alternately tempted to use a word in several different ways. And it is particularly
difficult to discover that an assertion which the metaphysician makes expresses
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discontentment with our grammar when the words of this assertion can also be used to state a fact of experience.
Thus when he says "only my pain is real pain", this sentence might mean that the other people are only pretending.
And when he says "this tree doesn't exist when nobody sees it", this might mean: "this tree vanishes when we turn
our backs to it". The man who says "only my pain is real", doesn't mean to say that he has found out by the
common criteria--the criteria, i.e., which give our words their common meanings--that the others who said they had
pains were cheating. But what he rebels against is the use of this expression in connection with these criteria. That is,
he objects to using this word in the particular way in which it is commonly used. On the other hand, he is not aware
that he is objecting to a convention. He sees a way of dividing the country different from the one used on the
ordinary map. He feels tempted, say, to use the name "Devonshire" not for the county with its conventional
boundary, but for a region differently bounded. He could express this by saying: "Isn't it absurd to make this a
county, to draw the boundaries here?" But what he says is: "The real Devonshire is this". We could answer: "What
you want is only a new notation, and by a new notation no facts of geography are changed". It is true, however, that
we may be irresistibly attracted or repelled by a notation. (We easily forget how much a notation, a form of
expression, may mean to us, and that changing it isn't always as easy as it often is in mathematics or in the sciences.
A change of clothes or of names may mean very little and it may mean a great deal.)
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I shall try to elucidate the problem discussed by realists, idealists, and solipsists by showing you a problem
closely related to it. It is this: "Can we have unconscious thoughts, unconscious feelings, etc.?" The idea of there
being unconscious thoughts has revolted many people. Others again have said that these were wrong in supposing
that there could only be conscious thoughts, and that psychoanalysis had discovered unconscious ones. The
objectors to unconscious thought did not see that they were not objecting to the newly discovered psychological
reactions, but to the way in which they were described. The psychoanalysts on the other hand were misled by their
own way of expression into thinking that they had done more than discover new psychological reactions; that they
had, in a sense, discovered conscious thoughts which were unconscious. The first could have stated their objection
by saying "We don't wish to use the phrase 'unconscious thoughts'; we wish to reserve the word 'thought' for
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what you call 'conscious thoughts'". They state their case wrongly when they say: "There can only be conscious
thoughts and no unconscious ones". For if they don't wish to talk of "unconscious thought" they should not use the
phrase "conscious thought", either.
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But is it not right to say that in any case the person who talks both of conscious and unconscious thoughts
thereby uses the word "thoughts" in two different ways?--Do we use a hammer in two different ways when we hit a
nail with it and, on the other hand, drive a peg into a hole? And do we use it in two different ways or in the same
way when we drive this peg into this hole and, on the other hand, another peg into another hole? Or should we only
call it different uses when in one case we drive something into something and in the other, say, we smash
something? Or is this all using the hammer in one way and is it to be called a different way only when we use the
hammer as a paper weight?--In which cases are we to say that a word is used in two different ways and in which that
it is used in one way? To say that a word is used in two (or more) different ways does in itself not yet give us any
idea about its use. It only specifies a way of looking at this usage by providing a schema for its description with two
(or more) subdivisions. It is all right to say: "I do two things with this hammer: I drive a nail into this board and one
into that board". But I could also have said: "I am doing only one thing with this hammer; I am driving a nail into
this board and one into that board". There can be two kinds of discussions as to whether a word is used in one way
or in two ways: (a) Two people may discuss whether the English word "cleave" is only used for chopping up
something or also for joining things together. This is a discussion about the facts of a certain actual usage. (b) They
may discuss whether the word "altus", standing for both "deep" and "high", is thereby used in two different ways.
This question is analogous to the question whether the word "thought" is used in two ways or in one when we talk
of conscious and unconscious thought. The man who says "surely, these are two different usages" has already
decided to use a two-way schema, and what he said expressed this decision.
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Now when the solipsist says that only his own experiences are real, it is no use answering him: "Why do you
tell us this if you don't believe that we really hear it?" Or anyhow, if we give him this answer, we mustn't believe that
we have answered his difficulty. There is no common sense answer to a philosophical problem. One can defend
common sense against the attacks of philosophers only by solving
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their puzzles, i.e., by curing them of the temptation to attack common sense; not by restating the views of common
sense. A philosopher is not a man out of his senses, a man who doesn't see what everybody sees; nor on the other
hand is his disagreement with common sense that of the scientist disagreeing with the coarse views of the man in the
street. That is, his disagreement is not founded on a more subtle knowledge of fact. We therefore have to look round
for the source of his puzzlement. And we find that there is puzzlement and mental discomfort, not only when our
curiosity about certain facts is not satisfied or when we can't find a law of nature fitting in with all our experience,
but also when a notation dissatisfies us--perhaps because of various associations which it calls up. Our ordinary
language, which of all possible notations is the one which pervades all our life, holds our mind rigidly in one
position, as it were, and in this position sometimes it feels cramped, having a desire for other positions as well. Thus
we sometimes wish for a notation which stresses a difference more strongly, makes it more obvious, than ordinary
language does, or one which in a particular case uses more closely similar forms of expression than our ordinary
language. Our mental cramp is loosened when we are shown the notations which fulfil these needs. These needs can
be of the greatest variety.
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Now the man whom we call a solipsist and who says that only his own experiences are real, does not thereby
disagree with us about any practical question of fact, he does not say that we are simulating when we complain of
pains, he pities us as much as anyone else, and at the same time he wishes to restrict the use of the epithet "real" to
what we should call his experiences; and perhaps he doesn't want to call our experiences "experiences" at all (again
without disagreeing with us about any question of fact). For he would say that it was inconceivable that experiences
other than his own were real. He ought therefore to use a notation in which such a phrase as "A has real toothache"
(where A is not he) is meaningless, a notation whose rules exclude this phrase as the rules of chess exclude a pawn's
making a knight's move. The solipsist's suggestion comes to using such a phrase as "there is real toothache" instead
of "Smith (the solipsist) has toothache". And why shouldn't we grant him this notation? I needn't say that in order to
avoid confusion he had in this case better not use the word "real" as opposed to "simulated" at all; which just means
that we shall have to provide for the distinction "real"/"simulated" in some other way. The solipsist who says
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"only I feel real pain", "only I really see (or hear)" is not stating an opinion; and that's why he is so sure of what he
says. He is irresistibly tempted to use a certain form of expression; but we must yet find why he is.
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The phrase "only I really see" is closely connected with the idea expressed in the assertion "we never know
what the other man really sees when he looks at a thing" or this, "we can never know whether he calls the same thing
'blue' which we call 'blue' ". In fact we might argue: "I can never know what he sees or that he sees at all, for all I
have is signs of various sorts which he gives me; therefore it is an unnecessary hypothesis altogether to say that he
sees; what seeing is I only know from seeing myself; I have only learnt the word 'seeing' to mean what I do". Of
course this is just not true, for I have definitely learned a different and much more complicated use of the word "to
see" than I here profess. Let us make clear the tendency which guided me when I did so, by an example from a
slightly different sphere: Consider this argument: "How can we wish that this paper were red if it isn't red? Doesn't
this mean that I wish that which doesn't exist at all? Therefore my wish can only contain something similar to the
paper's being red. Oughtn't we therefore to use a different word instead of 'red' when we talk of wishing that
something were red? The imagery of the wish surely shows us something less definite, something hazier, than the
reality of the paper being red. I should therefore say, instead of 'I wish this paper were red', something like 'I wish a
pale red for this paper' ". But if in the usual way of speaking he had said, "I wish a pale red for this paper," we
should, in order to fulfil his wish, have painted it a pale red--and this wasn't what he wished. On the other hand there
is no objection to adopting the form of expression which he suggests as long as we know that he uses the phrase "I
wish a pale x for this paper", always to mean what ordinarily we express by "I wish this paper had the colour x".
What he said really recommended his notation, in the sense in which a notation can be recommended. But he did
not tell us a new truth and did not show us that what we said before was false. (All this connects our present
problem with the problem of negation. I will only give you a hint, by saying that a notation would be possible in
which, to put it roughly, a quality had always two names, one for the case when something is said to have it, the
other for the case when something is said not to have it. The negation of "This paper is red" could then be, say, "This
paper is not rode". Such a notation would actually fulfil some of the
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wishes which are denied us by our ordinary language and which sometimes produce a cramp of philosophical
puzzlement about the idea of negation.)
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The difficulty which we express by saying "I can't know what he sees when he (truthfully) says that he sees a
blue patch" arises from the idea that "knowing what he sees" means: "seeing that which he also sees"; not, however,
in the sense in which we do so when we both have the same object before our eyes: but in the sense in which the
object seen would be an object, say, in his head, or in him. The idea is that the same object may be before his eyes
and mine, but that I can't stick my head into his (or my mind into his, which comes to the same) so that the real and
immediate object of his vision becomes the real and immediate object of my vision too. By "I don't know what he
sees" we really mean "I don't know what he looks at", where 'what he looks at' is hidden and he can't show it to me;
it is before his mind's eye. Therefore, in order to get rid of this puzzle, examine the grammatical difference between
the statements "I don't know what he sees" and "I don't know what he looks at", as they are actually used in our
language.
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Sometimes the most satisfying expression of our solipsism seems to be this: "When anything is seen (really
seen), it is always I who see it".
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What should strike us about this expression is the phrase "always I". Always who?--For, queer enough, I
don't mean: "always L. W." This leads us to considering the criteria for the identity of a person. Under what
circumstances do we say: "This is the same person whom I saw an hour ago"? Our actual use of the phrase "the
same person" and of the name of a person is based on the fact that many characteristics which we use as the criteria
for identity coincide in the vast majority of cases. I am as a rule recognized by the appearance of my body. My body
changes its appearance only gradually and comparatively little, and likewise my voice, characteristic habits, etc. only
change slowly and within a narrow range. We are inclined to use personal names in the way we do, only as a
consequence of these facts. This can best be seen by imagining unreal cases which show us what different
'geometries' we would be inclined to use if facts were different. Imagine, e.g., that all human bodies which exist
looked alike, that on the other hand, different sets of characteristics seemed, as it were, to change their habitation
among these bodies. Such a set of characteristics might be, say, mildness, together with a high pitched voice, and
slow movements, or a choleric temperament, a deep voice, and jerky movements, and such like. Under such
circumstances, although
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it would be possible to give the bodies names, we should perhaps be as little inclined to do so as we are to give
names to the chairs of our dining-room set. On the other hand, it might be useful to give names to the sets of
characteristics, and the use of these names would now roughly correspond to the personal names in our present
language.
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Or imagine that it were usual for human beings to have two characters, in this way: People's shape, size and
characteristics of behaviour periodically undergo a complete change. It is the usual thing for a man to have two such
states, and he lapses suddenly from one into the other. It is very likely that in such a society we should be inclined to
christen every man with two names, and perhaps to talk of the pair of persons in his body. Now were Dr. Jekyll and
Mr. Hyde two persons or were they the same person who merely changed? We can say whichever we like. We are
not forced to talk of a double personality.
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There are many uses of the word "personality" which we may feel inclined to adopt, all more or less akin.
The same applies when we define the identity of a person by means of his memories. Imagine a man whose
memories on the even days of his life comprise the events of all these days, skipping entirely what happened on the
odd days. On the other hand, he remembers on an odd day what happened on previous odd days, but his memory
then skips the even days without a feeling of discontinuity. If we like we can also assume that he has alternating
appearances and characteristics on odd and even days. Are we bound to say that here two persons are inhabiting the
same body? That is, is it right to say that there are, and wrong to say that there aren't, or vice versa? Neither. For the
ordinary use of the word "person" is what one might call a composite use suitable under the ordinary
circumstances. If I assume, as I do, that these circumstances are changed, the application of the term "person" or
"personality" has thereby changed; and if I wish to preserve this term and give it a use analogous to its former use, I
am at liberty to choose between many uses, that is, between many different kinds of analogy. One might say in such
a case that the term "personality" hasn't got one legitimate heir only. (This kind of consideration is of importance in
the philosophy of mathematics. Consider the use of the words "proof", "formula", and others. Consider the
question: "Why should what we do here be called 'philosophy'? Why should it be regarded as the only legitimate
heir of the different activities which had this name in former times?")
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Now let us ask ourselves what sort of identity of personality it is we are referring to when we say "when
anything is seen, it is always I who see". What is it I want all these cases of seeing to have in common? As an answer
I have to confess to myself that it is not my bodily appearance. I don't always see part of my body when I see. And
it isn't essential that my body, if seen amongst the things I see, should always look the same. In fact I don't mind
how much it changes. And I feel the same way about all the properties of my body, the characteristics of my
behaviour, and even about my memories.--When I think about it a little longer I see that what I wished to say was:
"Always when anything is seen, something is seen". I.e., that of which I said it continued during all the experiences
of seeing was not any particular entity "I", but the experience of seeing itself. This may become clearer if we imagine
the man who makes our solipsistic statement to point to his eyes while he says "I". (Perhaps because he wishes to be
exact and wants to say expressly which eyes belong to the mouth which says "I" and to the hands pointing to his
own body). But what is he pointing to? These particular eyes with the identity of physical objects? (To understand
this sentence, you must remember that the grammar of words of which we say that they stand for physical objects is
characterized by the way in which we use the phrase "the same so-and-so", or "the identical so-and-so", where
"so-and-so" designates the physical object.) We said before that he did not wish to point to a particular physical
object at all. The idea that he had made a significant statement arose from a confusion corresponding to the
confusion between what we shall call "the geometrical eye" and "the physical eye". I will indicate the use of these
terms: If a man tries to obey the order "Point to your eye", he may do many different things, and there are many
different criteria which he will accept for having pointed to his eye. If these criteria, as they usually do, coincide, I
may use them alternately and in different combinations to show me that I have touched my eye. If they don't
coincide, I shall have to distinguish between different senses of the phrase "I touch my eye" or "I move my finger
towards my eye". If, e.g., my eyes are shut, I can still have the characteristic kinaesthetic experience in my arm
which I should call the kinaesthetic experience of raising my hand to my eye. That I had succeeded in doing so, I
shall recognize by the peculiar tactile sensation of touching my eye. But if my eye were behind a glass plate fastened
in such a way that it prevented me from exerting a pressure on my eye with my
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finger, there would still be a criterion of muscular sensation which would make me say that now my finger was in
front of my eye. As to visual criteria, there are two I can adopt. There is the ordinary experience of seeing my hand
rise and come towards my eye, and this experience, of course, is different from seeing two things meet, say, two
finger tips. On the other hand, I can use as a criterion for my finger moving towards my eye, what I see when I look
into a mirror and see my finger nearing my eye. If that place on my body which, we say, 'sees' is to be determined
by moving my finger towards my eye, according to the second criterion, then it is conceivable that I may see with
what according to other criteria is the tip of my nose, or places on my forehead; or I might in this way point to a
place lying outside my body. If I wish a person to point to his eye (or his eyes) according to the second criterion
alone, I shall express my wish by saying: "Point to your geometrical eye (or eyes)". The grammar of the word
"geometrical eye" stands in the same relation to the grammar of the word "physical eye" as the grammar of the
expression "the visual sense datum of a tree" to the grammar of the expression "the physical tree". In either case it
confuses everything to say "the one is a different kind of object from the other"; for those who say that a sense
datum is a different kind of object from a physical object misunderstand the grammar of the word "kind", just as
those who say that a number is a different kind of object from a numeral. They think they are making such a
statement as "A railway train, a railway station, and a railway car are different kinds of objects", whereas their
statement is analogous to "A railway train, a railway accident, and a railway law are different kinds of objects".
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What tempted me to say "it is always I who see when anything is seen", I could also have yielded to by
saying: "whenever anything is seen, it is this which is seen", accompanying the word "this" by a gesture embracing
my visual field (but not meaning by "this" the particular objects which I happen to see at the moment). One might
say, "I am pointing at the visual field as such, not at anything in it". And this only serves to bring out the
senselessness of the former expression.
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Let us then discard the "always" in our expression. Then I can still express my solipsism by saying, "Only
what I see (or: see now) is really seen". And here I am tempted to say: "Although by the word 'I' I don't mean L. W.,
it will do if the others understand 'I' to mean L. W., if just now I am in fact L. W." I could also express my claim by
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saying: "I am the vessel of life"; but mark, it is essential that everyone to whom I say this should be unable to
understand me. It is essential that the other should not be able to understand 'what I really mean', though in practice
he might do what I wish by conceding to me an exceptional position in his notation. But I wish it to be logically
impossible that he should understand me, that is to say, it should be meaningless, not false, to say that he
understands me. Thus my expression is one of the many which is used on various occasions by philosophers and
supposed to convey something to the person who says it, though essentially incapable of conveying anything to
anyone else. Now if for an expression to convey a meaning means to be accompanied by or to produce certain
experiences, our expression may have all sorts of meanings, and I don't wish to say anything about them. But we
are, as a matter of fact, misled into thinking that our expression has a meaning in the sense in which a
non-metaphysical expression has; for we wrongly compare our case with one in which the other person can't
understand what we say because he lacks a certain information. (This remark can only become dear if we understand
the connection between grammar and sense and nonsense.)
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The meaning of a phrase for us is characterized by the use we make of it. The meaning is not a mental
accompaniment to the expression. Therefore the phrase "I think I mean something by it", or "I'm sure I mean
something by it", which we so often hear in philosophical discussions to justify the use of an expression is for us no
justification at all. We ask: "What do you mean?", i.e., "How do you use this expression?" If someone taught me the
word "bench" and said that he sometimes or always put a stroke over it thus: " ", and that this meant
something to him, I should say: "I don't know what sort of idea you associate with this stroke, but it doesn't interest
me unless you show me that there is a use for the stroke in the kind of calculus in which you wish to use the word
'bench'".--I want to play chess, and a man gives the white king a paper crown, leaving the use of the piece unaltered,
but telling me that the crown has a meaning to him in the game, which he can't express by rules. I say: "as long as it
doesn't alter the use of the piece, it hasn't what I call a meaning".
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One sometimes hears that such a phrase as "This is here", when while I say it I point to a part of my visual
field, has a kind of primitive meaning to me, although it can't impart information to anybody else.
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When I say "Only this is seen", I forget that a sentence may come ever so natural to us without having any
use in our calculus of language.
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Think of the law of identity, "a = a", and of how we sometimes try hard to get hold of its sense, to visualize it, by
looking at an object and repeating to ourselves such a sentence as "This tree is the same thing as this tree". The
gestures and images by which I apparently give this sentence sense are very similar to those which I use in the case
of "Only this is really seen". (To get clear about philosophical problems, it is useful to become conscious of the
apparently unimportant details of the particular situation in which we are inclined to make a certain metaphysical
assertion. Thus we may be tempted to say "Only this is really seen" when we stare at unchanging surroundings,
whereas we may not at all be tempted to say this when we look about us while walking.)
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There is, as we have said, no objection to adopting a symbolism in which a certain person always or
temporarily holds an exceptional place. And therefore, if I utter the sentence "Only I really see", it is conceivable that
my fellow creatures thereupon will arrange their notation so as to fall in with me by saying "so-and-so is really seen"
instead of "L. W. sees so-and-so", etc., etc. What, however, is wrong, is to think that I can justify this choice of
notation. When I said, from my heart, that only I see, I was also inclined to say that by "I" I didn't really mean L. W.,
although for the benefit of my fellow men I might say "It is now L. W. who really sees" though this is not what I
really mean. I could almost say that by "I" I mean something which just now inhabits L. W., something which the
others can't see. (I meant my mind, but could only point to it via my body.) There is nothing wrong in suggesting
that the others should give me an exceptional place in their notation; but the justification which I wish to give for it:
that this body is now the seat of that which really lives--is senseless. For admittedly this is not to state anything
which in the ordinary sense is a matter of experience. (And don't think that it is an experiential proposition which
only I can know because only I am in the position to have the particular experience.) Now the idea that the real I
lives in my body is connected with the peculiar grammar of the word "I", and the misunderstandings this grammar is
liable to give rise to. There are two different cases in the use of the word "I" (or "my") which I might call "the use as
object" and "the use as subject". Examples of the first kind of use are these: "My arm is broken", "I have grown six
inches", "I have a bump on my forehead", "The wind blows my hair about". Examples of the second kind are: "I see
so-and-so", "I hear so-and-so", "I try to lift my arm", "I think it will
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rain", "I have toothache". One can point to the difference between these two categories by saying: The cases of the
first category involve the recognition of a particular person, and there is in these cases the possibility of an error, or
as I should rather put it: The possibility of an error has been provided for. The possibility of failing to score has been
provided for in a pin game. On the other hand, it is not one of the hazards of the game that the balls should fail to
come up if I have put a penny in the slot. It is possible that, say in an accident, I should feel a pain in my arm, see a
broken arm at my side, and think it is mine, when really it is my neighbour's. And I could, looking into a mirror,
mistake a bump on his forehead for one on mine. On the other hand, there is no question of recognizing a person
when I say I have toothache. To ask "are you sure that it's you who have pains?" would be nonsensical. Now, when
in this case no error is possible, it is because the move which we might be inclined to think of as an error, a 'bad
move', is no move of the game at all. (We distinguish in chess between good and bad moves, and we call it a mistake
if we expose the queen to a bishop. But it is no mistake to promote a pawn to a king.) And now this way of stating
our idea suggests itself: that it is as impossible that in making the statement "I have toothache" I should have
mistaken another person for myself, as it is to moan with pain by mistake, having mistaken someone else for me. To
say, "I have pain" is no more a statement about a particular person than moaning is. "But surely the word 'I' in the
mouth of a man refers to the man who says it; it points to himself; and very often a man who says it actually points
to himself with his finger". But it was quite superfluous to point to himself. He might just as well only have raised
his hand. It would be wrong to say that when someone points to the sun with his hand, he is pointing both to the
sun and himself because it is he who points; on the other hand, he may by pointing attract attention both to the sun
and to himself.
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The word "I" does not mean the same as "L. W." even if I am L. W., nor does it mean the same as the
expression "the person who is now speaking". But that doesn't mean: that "L. W." and "I" mean different things. All
it means is that these words are different instruments in our language.
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Think of words as instruments characterized by their use, and then think of the use of a hammer, the use of a
chisel, the use of a square, of a glue pot, and of the glue. (Also, all that we say here can be understood only if one
understands that a great variety of games is played
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with the sentences of our language: Giving and obeying orders; asking questions and answering them; describing an
event; telling a fictitious story; telling a joke; describing an immediate experience; making conjectures about events
in the physical world; making scientific hypotheses and theories; greeting someone, etc., etc.) The mouth which says
"I" or the hand which is raised to indicate that it is I who wish to speak, or I who have toothache, does not thereby
point to anything. If, on the other hand, I wish to indicate the place of my pain, I point. And here again remember
the difference between pointing to the painful spot without being led by the eye and on the other hand pointing to a
scar on my body after looking for it. ("That's where I was vaccinated".)--The man who cries out with pain, or says
that he has pain, doesn't choose the mouth which says it.
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All this comes to saying that the person of whom we say "he has pain" is, by the rules of the game, the
person who cries, contorts his face, etc. The place of the pain--as we have said--may be in another person's body. If,
in saying "I", I point to my own body, I model the use of the word "I" on that of the demonstrative "this person" or
"he". (This way of making the two expressions similar is somewhat analogous to that which one sometimes adopts
in mathematics, say in the proof that the sum of the three angles of a triangle is 180°.
We say "α = α', β = β', and γ = γ". The first two equalities are of an entirely different kind from the third.) In "I have
pain", "I" is not a demonstrative pronoun.
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Compare the two cases: 1. "How do you know that he has pains?"--"Because I hear him moan". 2. "How do
you know that you have pains?"--"Because I feel them". But "I feel them" means the same as "I have them".
Therefore this was no explanation at all. That, however, in my answer I am inclined to stress the word "feel" and not
the word "I" indicates that by "I" I don't wish to pick out one person (from amongst different persons).
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The difference between the propositions "I have pain" and "he has pain" is not that of "L. W. has pain" and
"Smith has pain". Rather, it corresponds to the difference between moaning and saying that someone moans.--"But
surely the word 'I' in 'I have pain' serves to
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distinguish me from other people, because it is by the sign 'I' that I distinguish saying that I have pain from saying
that one of the others has". Imagine a language in which, instead of "I found nobody in the room", one said "I found
Mr. Nobody in the room". Imagine the philosophical problems which would arise out of such a convention. Some
philosophers brought up in this language would probably feel that they didn't like the similarity of the expressions
"Mr. Nobody" and "Mr. Smith". When we feel that we wish to abolish the "I" in "I have pain", one may say that we
tend to make the verbal expression of pain similar to the expression by moaning.--We are inclined to forget that it is
the particular use of a word only which gives the word its meaning. Let us think of our old example for the use of
words: Someone is sent to the grocer with a slip of paper with the words "five apples" written on it. The use of the
word in practice is its meaning. Imagine it were the usual thing that the objects around us carried labels with words
on them by means of which our speech referred to the objects. Some of these words would be proper names of the
objects, others generic names (like table, chair, etc.), others again, names of colours, names of shapes, etc. That is to
say, a label would only have a meaning to us in so far as we made a particular use of it. Now we could easily imagine
ourselves to be impressed by merely seeing a label on a thing, and to forget that what makes these labels important is
their use. In this way we sometimes believe that we have named something when we make the gesture of pointing
and utter words like "This is..." (the formula of the ostensive definition). We say we call something "toothache", and
think that the word has received a definite function in the dealings we carry out with language when, under certain
circumstances, we have pointed to our cheek and said: "This is toothache". (Our idea is that when we point and the
other "only knows what we are pointing to" he knows the use of the word. And here we have in mind the special
case when 'what we point to' is, say, a person and "to know that I point to" means to see which of the persons
present I point to.)
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We feel then that in the cases in which "I" is used as subject, we don't use it because we recognize a
particular person by his bodily characteristics; and this creates the illusion that we use this word to refer to
something bodiless, which, however, has its seat in our body. In fact this seems to be the real ego, the one of which
it was said, "Cogito, ergo sum".--"Is there then no mind, but only a body?" Answer: The word "mind" has meaning,
i.e., it has a use in our
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language; but saying this doesn't yet say what kind of use we make of it.
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In fact one may say that what in these investigations we were concerned with was the grammar of those
words which describe what are called "mental activities": seeing, hearing, feeling, etc. And this comes to the same as
saying that we are concerned with the grammar of 'phrases describing sense data'.
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Philosophers say it as a philosophical opinion or conviction that there are sense data. But to say that I believe
that there are sense data comes to saying that I believe that an object may appear to be before our eyes even when it
isn't. Now when one uses the word "sense datum", one should be clear about the peculiarity of its grammar. For the
idea in introducing this expression was to model expressions referring to 'appearance' after expressions referring to
'reality'. It was said, e.g., that if two things seem to be equal, there must be two somethings which are equal. Which
of course means nothing else but that we have decided to use such an expression as "the appearances of these two
things are equal" synonymously with "these two things seem to be equal". Queerly enough, the introduction of this
new phraseology has deluded people into thinking that they had discovered new entities, new elements of the
structure of the world, as though to say "I believe that there are sense data" were similar to saying "I believe that
matter consists of electrons". When we talk of the equality of appearances or sense data, we introduce a new usage
of the word "equal". It is possible that the lengths A and B should appear to us to be equal, that B and C should
appear to be equal, but that A and C do not appear to be equal. And in the new notation we shall have to say that
though the appearance (sense datum) of A is equal to that of B and the appearance of B equal to that of C, the
appearance of A is not equal to the appearance of C; which is all right if you don't mind using "equal" intransitively.
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Now the danger we are in when we adopt the sense datum notation is to forget the difference between the
grammar of a statement about sense data and the grammar of an outwardly similar statement about physical objects.
(From this point one might go on talking about the misunderstandings which find their expression in such sentences
as: "We can never see an accurate circle", "All our sense data are vague". Also, this leads to the comparison of the
grammar of "position", "motion", and "size" in Euclidean and in visual space.
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There is, e.g., absolute position, absolute motion and size, in visual space.)
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Now we can make use of such an expression as "pointing to the appearance of a body" or "pointing to a
visual sense datum". Roughly speaking, this sort of pointing comes to the same as sighting, say, along the barrel of a
gun. Thus we may point and say: "This is the direction in which I see my image in the mirror". One can also use
such an expression as "the appearance, or sense datum, of my finger points to the sense datum of the tree" and
similar ones. From these cases of pointing, however, we must distinguish those of pointing in the direction a sound
seems to come from, or of pointing to my forehead with dosed eyes, etc.
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Now when in the solipsistic way I say "This is what's really seen", I point before me and it is essential that I
point visually. If I pointed sideways or behind me--as it were, to things which I don't see--the pointing would in this
case be meaningless to me; it would not be pointing in the sense in which I wish to point. But this means that when I
point before me saying "this is what's really seen", although I make the gesture of pointing, I don't point to one thing
as opposed to another. This is as when travelling in a car and feeling in a hurry, I instinctively press against
something in front of me as though I could push the car from inside.
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When it makes sense to say "I see this", or "this is seen", pointing to what I see, it also makes sense to say "I
see this", or "this is seen", pointing to something I don't see. When I made my solipsist statement, I pointed, but I
robbed the pointing of its sense by inseparably connecting that which points and that to which it points. I
constructed a clock with all its wheels, etc., and in the end fastened the dial to the pointer and made it go round with
it. And in this way the solipsist's "Only this is really seen" reminds us of a tautology.
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Of course one of the reasons why we are tempted to make our pseudo-statement is its similarity with the
statement "I only see this", or "this is the region which I see", where I point to certain objects around me, as opposed
to others, or in a certain direction in physical space (not in visual space), as opposed to other directions in physical
space. And if, pointing in this sense, I say "this is what is really seen", one may answer me: "This is what you, L. W.,
see; but there is no objection to adopting a notation in which what we used to call 'things which L. W. sees' is called
'things really seen'". If, however, I believe that by pointing to that which in my grammar has no
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neighbour I can convey something to myself (if not to others), I make a mistake similar to that of thinking that the
sentence "I am here" makes sense to me (and, by the way, is always true) under conditions different from those very
special conditions under which it does make sense. E.g., when my voice and the direction from which I speak is
recognized by another person. Again an important case where you can learn that a word has meaning by the
particular use we make of it.--We are like people who think that pieces of wood shaped more or less like chess or
draught pieces and standing on a chess board make a game, even if nothing has been said as to how they are to be
used.
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To say "it approaches me" has sense, even when, physically speaking, nothing approaches my body; and in
the same way it makes sense to say, "it is here" or "it has reached me" when nothing has reached my body. And, on
the other hand, "I am here" makes sense if my voice is recognized and heard to come from a particular place of
common space. In the sentence "it is here" the 'here' was a here in visual space. Roughly speaking, it is the
geometrical eye. The sentence "I am here", to make sense, must attract attention to a place in common space. (And
there are several ways in which this sentence might be used.) The philosopher who thinks it makes sense to say to
himself "I am here" takes the verbal expression from the sentence in which "here" is a place in common space and
thinks of "here" as the here in visual space. He therefore really says something like "Here is here".
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I could, however, try to express my solipsism in a different way: I imagine that I and others draw pictures or
write descriptions of what each of us sees. These descriptions are put before me. I point to the one which I have
made and say: "Only this is (or was) really seen". That is, I am tempted to say: "Only this description has reality
(visual reality) behind it". The others I might call--"blank descriptions". I could also express myself by saying: "This
description only was derived from reality; only this was compared with reality". Now it has a clear meaning when
we say that this picture or description is a projection, say, of this group of objects--the trees I look at--or that it has
been derived from these objects. But we must look into the grammar of such a phrase as "this description is derived
from my sense datum". What we are talking about is connected with that peculiar temptation to say: "I never know
what the other really means by 'brown', or what he really sees when he (truthfully) says that he sees a brown
object".--We could propose to one who says this to use
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two different words instead of the one word "brown"; one word for his particular impression, the other word with
that meaning which other people besides himself can understand as well. If he thinks about this proposal he will see
that there is something wrong in his conception of the meaning, function, of the word "brown" and others. He looks
for a justification of his description where there is none. (Just as in the case when a man believes that the chain of
reasons must be endless. Think of the justification by a general formula for performing mathematical operations; and
of the question: Does this formula compel us to make use of it in this particular case as we do?) To say "I derive a
description from visual reality" can't mean anything analogous to: "I derive a description from what I see here". I
may, e.g., see a chart in which a coloured square is correlated to the word "brown", and also a patch of the same
colour elsewhere; and I may say: "This chart shows me that I must use the word 'brown' for the description of this
patch". This is how I may derive the word which is needed in my description. But it would be meaningless to say
that I derive the word "brown" from the particular colour-impression which I receive.
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Let us now ask: "Can a human body have pain?" One is inclined to say: "How can the body have pain? The
body in itself is something dead; a body isn't conscious!" And here again it is as though we looked into the nature of
pain and saw that it lies in its nature that a material object can't have it. And it is as though we saw that what has pain
must be an entity of a different nature from that of a material object; that, in fact, it must be of a mental nature. But
to say that the ego is mental is like saying that the number 3 is of a mental or an immaterial nature, when we
recognize that the numeral "3" isn't used as a sign for a physical object.
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On the other hand we can perfectly well adopt the expression "this body feels pain", and we shall then, just
as usual, tell it to go to the doctor, to lie down, and even to remember that when the last time it had pains they were
over in a day. "But wouldn't this form of expression at least be an indirect one?"--Is it using an indirect expression
when we say "Write '3' for 'x' in this formula" instead of "Substitute 3 for x"? (Or on the other hand, is the first of
these two expressions the only direct one, as some philosophers think?) One expression is no more direct than the
other. The meaning of the expression depends entirely on how we go on using it. Let's not imagine the meaning as
an occult connection the mind makes between
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a word and a thing, and that this connection contains the whole usage of a word as the seed might be said to contain
the tree.
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The kernel of our proposition that that which has pains or sees or thinks is of a mental nature is only, that the
word "I" in "I have pains" does not denote a particular body, for we can't substitute for "I" a description of a body.
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THE BROWN BOOK
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Page Break 76
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THE BROWN BOOK
I
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AUGUSTINE, in describing his learning of language, says that he was taught to speak by learning the names of
things. It is clear that whoever says this has in mind the way in which a child learns such words as "man", "sugar",
"table", etc. He does not primarily think of such words as "today", "not", "but", "perhaps".
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Suppose a man described a game of chess, without mentioning the existence and operations of the pawns.
His description of the game as a natural phenomenon will be incomplete. On the other hand we may say that he has
completely described a simpler game. In this sense we can say that Augustine's description of learning the language
was correct for a simpler language than ours. Imagine this language:--
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1). Its function is the communication between a builder A and his man B. B has to reach A building stones.
There are cubes, bricks, slabs, beams, columns. The language consists of the words "cube", "brick", "slab",
"column". A calls out one of these words, upon which B brings a stone of a certain shape. Let us imagine a society
in which this is the only system of language. The child learns this language from the grown-ups by being trained to
its use. I am using the word "trained" in a way strictly analogous to that in which we talk of an animal being trained
to do certain things. It is done by means of example, reward, punishment, and suchlike. Part of this training is that
we point to a building stone, direct the attention of the child towards it, and pronounce a word. I will call this
procedure demonstrative teaching of words. In the actual use of this language, one man calls out the words as
orders, the other acts according to them. But learning and teaching this language will contain this procedure: The
child just 'names' things, that is, he pronounces the words of the language when the teacher points to the things. In
fact, there will be a still simpler exercise: The child repeats words which the teacher pronounces.
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(Note. Objection: The word "brick" in language 1) has not the meaning which it has in our language.--This is
true if it means that
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in our language there are usages of the word "brick" different from our usages of this word in language 1). But don't
we sometimes use the word "brick!" in just this way? Or should we say that when we use it, it is an elliptical
sentence, a shorthand for "Bring me a brick"? Is it right to say that if we say "brick!" we mean "Bring me a brick"?
Why should I translate the expression "brick!" into the expression "Bring me a brick"? And if they are synonymous,
why shouldn't I say: If he says "brick!" he means "brick!"...? Or: Why shouldn't he be able to mean just "brick!" if he
is able to mean "Bring me a brick", unless you wish to assert that while he says aloud "brick!" he as a matter of fact
always says in his mind, to himself, "Bring me a brick"? But what reason could we have to assert this? Suppose
someone asked: If a man gives the order, "Bring me a brick", must he mean it as four words, or can't he mean it as
one composite word synonymous with the one word "brick!"? One is tempted to answer: He means all four words if
in his language he uses that sentence in contrast with other sentences in which these words are used, such as, for
instance, "Take these two bricks away". But what if I asked "But how is his sentence contrasted with these others?
Must he have thought them simultaneously, or shortly before or after, or is it sufficient that he should have one time
learnt them, etc.?" When we have asked ourselves this question, it appears that it is irrelevant which of these
alternatives is the case. And we are inclined to say that all that is really relevant is that these contrasts should exist in
the system of language which he is using, and that they need not in any sense be present in his mind when he utters
his sentence. Now compare this conclusion with our original question. When we asked it, we seemed to ask a
question about the state of mind of the man who says the sentence, whereas the idea of meaning which we arrived at
in the end was not that of a state of mind. We think of the meaning of signs sometimes as states of mind of the man
using them, sometimes as the role which these signs are playing in a system of language. The connection between
these two ideas is that the mental experiences which accompany the use of a sign undoubtedly are caused by our
usage of the sign in a particular system of language. William James speaks of specific feelings accompanying the use
of such words as "and", "if", "or". And there is no doubt that at least certain gestures are often connected with such
words, as a collecting gesture with "and", and a dismissing gesture with "not". And there obviously are visual and
muscular sensations connected with these gestures. On the other hand it is clear enough that these sensations do not
accompany every
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use of the word "not" and "and". If in some language the word "but" meant what "not" means in English, it is clear
that we should not compare the meanings of these two words by comparing the sensations which they produce. Ask
yourself what means we have of finding out the feelings which they produce in different people and on different
occasions. Ask yourself: "When I said, 'Give me an apple and a pear, and leave the room', had I the same feeling
when I pronounced the two words 'and'?" But we do not deny that the people who use the word "but" as "not" is
used in English will, broadly speaking, have similar sensations accompanying the word "but" to those the English
have when they use "not". And the word "but" in the two languages will on the whole be accompanied by different
sets of experiences.)
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2). Let us now look at an extension of language 1). The builder's man knows by heart the series of words
from one to ten. On being given the order, "Five slabs!", he goes to where the slabs are kept, says the words from
one to five, takes up a slab for each word, and carries them to the builder. Here both the parties use the language by
speaking the words. Learning the numerals by heart will be one of the essential features of learning this language.
The use of the numerals will again be taught demonstratively. But now the same word, e.g., "three", will be taught by
pointing either to slabs, or to bricks, or to columns, etc. And on the other hand, different numerals will be taught by
pointing to groups of stones of the same shape.
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(Remark: We stressed the importance of learning the series of numerals by heart because there was no
feature comparable to this in the learning of language 1). And this shows us that by introducing numerals we have
introduced an entirely different kind of instrument into our language. The difference of kind is much more obvious
when we contemplate such a simple example than when we look at our ordinary language with innumerable kinds of
words all looking more or less alike when they stand in the dictionary.--
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What have the demonstrative explanations of the numerals in common with those of the words "slab",
"column", etc., except a gesture and pronouncing the words? The way such a gesture is used in the two cases is
different. This difference is blurred if one says, "In one case we point to a shape, in the other we point to a number".
The difference becomes obvious and clear only when we contemplate a
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complete example (i.e., the example of a language completely worked out in detail).)
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3). Let us introduce a new instrument of communication,-a proper name. This is given to a particular object
(a particular building stone) by pointing to it and pronouncing the name. If A calls the name, B brings the object.
The demonstrative teaching of a proper name is different again from the demonstrative teaching in the cases 1) and
2).
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(Remark: This difference does not lie, however, in the act of pointing and pronouncing the word or in any
mental act (meaning?) accompanying it, but in the role which the demonstration (pointing and pronouncing) plays in
the whole training and in the use which is made of it in the practice of communication by means of this language.
One might think that the difference could be described by saying that in the different cases we point to different
kinds of objects. But suppose I point with my hand to a blue jersey. How does pointing to its colour differ from
pointing to its shape?--We are inclined to say the difference is that we mean something different in the two cases.
And 'meaning' here is to be some sort of process taking place while we point. What particularly tempts us to this
view is that a man on being asked whether he pointed to the colour or the shape is, at least in most cases, able to
answer this and to be certain that his answer is correct. If on the other hand, we look for two such characteristic
mental acts as meaning the colour and meaning the shape, etc., we aren't able to find any, or at least none which
must always accompany pointing to colour, pointing to shape, respectively. We have only a rough idea of what it
means to concentrate one's attention on the colour as opposed to the shape, or vice versa. The difference, one might
say, does not lie in the act of demonstration, but rather in the surrounding of that act in the use of the language.)
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4). On being ordered, "This slab!", B brings the slab to which A points. On being ordered, "Slab, there!", he
carries a slab to the place indicated. Is the word "there" taught demonstratively? Yes and no! When a person is
trained in the use of the word "there", the teacher will in training him make the pointing gesture and pronounce the
word "there". But should we say that thereby he gives a place the name "there"? Remember that the pointing gesture
in this case is part of the practice of communication itself.
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(Remark: It has been suggested that such words as "there", "here", "now", "this" are the 'real proper names'
as opposed to what in ordinary
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life we call proper names, and, in the view I am referring to, can only be called so crudely. There is a widespread
tendency to regard what in ordinary life is called a proper name only as a rough approximation of what ideally could
be called so. Compare Russell's idea of the 'individual'. He talks of individuals as the ultimate constituents of reality,
but says that it is difficult to say which things are individuals. The idea is that further analysis has to reveal this. We,
on the other hand, introduced the idea of a proper name in a language in which it was applied to what in ordinary life
we call "objects", "things" ("building stones").
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---"What does the word 'exactness' mean? Is it real exactness if you are supposed to come to tea at 4.30 and
come when a good clock strikes 4.30? Or would it only be exactness if you began to open the door at the moment
the clock began to strike? But how is this moment to be defined and how is 'beginning to open the door' to be
defined? Would it be correct to say, 'It is difficult to say what real exactness is, for all we know is only rough
approximations'?")
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5). Questions and answers: A asks, "How many slabs?" B counts them and answers with the numeral.
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Systems of communication as for instance 1), 2), 3), 4), 5) we shall call "language games". They are more or
less akin to what in ordinary language we call games. Children are taught their native language by means of such
games, and here they even have the entertaining character of games. We are not, however, regarding the language
games which we describe as incomplete parts of a language, but as languages complete in themselves, as complete
systems of human communication. To keep this point of view in mind, it very often is useful to imagine such a
simple language to be the entire system of communication of a tribe in a primitive state of society. Think of primitive
arithmetics of such tribes.
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When the boy or grown-up learns what one might call special technical languages, e.g., the use of charts and
diagrams, descriptive geometry, chemical symbolism, etc., he learns more language games. (Remark: The picture we
have of the language of the grown-up is that of a nebulous mass of language, his mother tongue, surrounded by
discrete and more or less clear-cut language games, the technical languages.)
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6). Asking for the name: we introduce new forms of building stones. B points to one of them and asks,
"What is this?"; A answers,
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"This is a...". Later on A calls out this new word, say "arch", and B brings the stone. The words, "This is..." together
with the pointing gesture we shall call ostensive explanation or ostensive definition. In case 6) a generic name was
explained, in actual fact, as the name of a shape. But we can ask analogously for the proper name of a particular
object, for the name of a colour, of a numeral, of a direction.
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(Remark: Our use of expressions like "names of numbers", "names of colours", "names of materials",
"names of nations" may spring from two different sources. One is that we might imagine the functions of proper
names, numerals, words for colours, etc., to be much more alike than they actually are. If we do so we are tempted
to think that the function of every word is more or less like the function of a proper name of a person, or such
generic names as "table", "chair", "door", etc. The second source is this, that if we see how fundamentally different
the functions of such words as "table", "chair", etc., are from those of proper names, and how different from either
the functions of, say, the names of colours, we see no reason why we shouldn't speak of names of numbers or
names of directions either, not by way of saying some such thing as "numbers and directions are just different forms
of objects", but rather by way of stressing the analogy which lies in the lack of analogy between the functions of the
words "chair" and "Jack" on the one hand, and "east" and "Jack" on the other hand.)
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7). B has a table in which written signs are placed opposite to pictures of objects (say, a table, a chair, a
tea-cup, etc.). A writes one of the signs, B looks for it in the table, looks or points with his finger from the written
sign to the picture opposite, and fetches the object which the picture represents.
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Let us now look at the different kinds of signs which we have introduced. First let us distinguish between
sentences and words. A sentence†1 I will call every complete sign in a language game, its constituent signs are
words. (This is merely a rough and general remark about the way I will use the words "proposition"†1 and "word".)
A proposition may consist of only one word. In 1) the signs "brick!", "column!" are the sentences. In 2) a sentence
consists of two words. According to the role which propositions play in a language game, we distinguish between
orders, questions, explanations, descriptions, and so on.
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8). If in a language game similar to 1) A calls out an order: "slab, column, brick!" which is obeyed by B by
bringing a slab, a column and a brick, we might here talk of three propositions, or of one only. If, on the other hand,
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9). the order of words shows B the order in which to bring the building stones, we shall say that A calls out a
proposition consisting of three words. If the command in this case took the form, "Slab, then column, then brick!"
we should say that it consisted of four words (not of five). Amongst the words we see groups of words with similar
functions. We can easily see a similarity in the use of the words "one," "two", "three", etc. and again one in the use
of "slab", "column" and "brick", etc., and thus we distinguish parts of speech. In 8) all the words of the proposition
belonged to the same part of speech.
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10). The order in which B had to bring the stones in 9) could have been indicated by the use of the ordinals
thus: "Second, column; first, slab; third, brick!". Here we have a case in which what was the function of the order of
words in one language game is the function of particular words in another.
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Reflections such as the preceding will show us the infinite variety of the functions of words in propositions,
and it is curious to compare what we see in our examples with the simple and rigid rules which logicians give for the
construction of propositions. If we group words together according to the similarity of their functions, thus
distinguishing parts of speech, it is easy to see that many different ways of classification can be adopted. We could
indeed easily imagine a reason for not classing the word "one" together with "two", "three", etc., as follows:
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11). Consider this variation of our language game 2). Instead of calling out, "One slab!", "One cube!", etc., A
just calls "Slab!", "Cube!", etc., the use of the other numerals being as described in 2). Suppose that a man
accustomed to this form 11) of communication was introduced to the use of the word "one" as described in 2). We
can easily imagine that he would refuse to classify "one" with the numerals "2" "3" etc.
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(Remark: Think of the reasons for and against classifying 'o' with the other cardinals. "Are black and white
colours?" In which cases would you be inclined to say so and which not?--Words can in
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many ways be compared to chess men. Think of the several ways of distinguishing different kinds of pieces in the
game of chess (e.g., pawns and 'officers').
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Remember the phrase, "two or more".)
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It is natural for us to call gestures, as those employed in 4), or pictures as in 7), elements or instruments of
language. (We talk sometimes of a language of gestures.) The pictures in 7) and other instruments of language which
have a similar function I shall call patterns. (This explanation, as others which we have given, is vague, and meant to
be vague.) We may say that words and patterns have different kinds of functions. When we make use of a pattern
we compare something with it, e.g., a chair with the picture of a chair. We did not compare a slab with the word
"slab". In introducing the distinction, 'word/pattern', the idea was not to set up a final logical duality. We have only
singled out two characteristic kinds of instruments from the variety of instruments in our language. We shall call
"one", "two", "three", etc., words. If instead of these signs we used "--", "-- -- ", "-- -- --", "-- -- -- --", we might call
these patterns. Suppose in a language the numerals were "one", "one one", "one one one", etc., should we call "one"
a word or a pattern? The same element may in one place be used as word and in another as pattern. A circle might
be the name for an ellipse, or on the other hand a pattern with which the ellipse is to be compared by a particular
method of projection. Consider also these two systems of expression:
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12). A gives B an order consisting of two written symbols, the first an irregularly shaped patch of a certain
colour, say green, the second the drawn outline of a geometrical figure, say a circle. B brings an object of this outline
and that colour, say a circular green object.
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13). A gives B an order consisting of one symbol, a geometrical figure painted a particular colour, say a green
circle. B brings him a green circular object. In 12) patterns correspond to our names of colours and other patterns to
our names of shape. The symbols in 13) cannot be regarded as combinations of two such elements. A word in
inverted commas can be called a pattern. Thus in the sentence "He said 'Go to hell'", "Go to hell" is a pattern of what
he said. Compare these cases: a) Someone says "I whistled..." (whistling a tune);
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b) Someone writes, "I whistled ". An onomatopoeic
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word like "rustling" may be called a pattern. We call a very great variety of processes "comparing an object with a
pattern". We comprise many kinds of symbols under the name "pattern". In 7) B compares a picture in the table with
the objects he has before him. But what does comparing a picture with the object consist in? Suppose the table
showed: a) a picture of a hammer, of pincers, of a saw, of a chisel; b) on the other hand, pictures of twenty different
kinds of butterflies. Imagine what the comparison in these two cases would consist in, and note the difference.
Compare with these cases a third case c) where the pictures in the table represent building stones drawn to scale, and
the comparing has to be done with ruler and compasses. Suppose that B's task is to bring a piece of cloth of the
colour of the sample. How are the colours of sample and cloth to be compared? Imagine a series of different cases:
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14). A shows the sample to B, upon which B goes and fetches the material 'from memory'.
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15). A gives B the sample, B looks from the sample to the materials on the shelves from which he has to
choose.
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16). B lays the sample on each bolt of material and chooses that one which he can't distinguish from the
sample, for which the difference between the sample and the material seems to vanish.
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17). Imagine on the other hand that the order has been, "Bring a material slightly darker than this sample". In
14) I said that B fetches the material 'from memory', which is using a common form of expression. But what might
happen in such a case of comparing 'from memory' is of the greatest variety. Imagine a few instances:
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14a). B has a memory image before his mind's eye when he goes for the material. He alternately looks at
materials and recalls his image. He goes through this process with, say, five of the bolts, in some instances saying to
himself, "Too dark", in some instances saying to himself, "Too light". At the fifth bolt he stops, says, "That's it" and
takes it from the shelf.
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14b). No memory image is before B's eye. He looks at four bolts, shaking his head each time, feeling some
sort of mental tension. On reaching the fifth bolt, this tension relaxes, he nods his head, and takes the bolt down.
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14c). B goes to the shelf without a memory image, looks at five bolts one after the other, takes the fifth bolt
from the shelf.
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'But this can't be all comparing consists in.'
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When we call these three preceding cases, cases of comparing from memory, we feel that their description is
in a sense unsatisfactory, or incomplete. We are inclined to say that the description has left out the essential feature
of such a process and given us accessory features only. The essential feature it seems would be what one might call a
specific experience of comparing and of recognizing. Now it is queer that on closely looking at cases of comparing,
it is very easy to see a great number of activities and states of mind, all more or less characteristic of the act of
comparing. This in fact is so, whether we speak of comparing from memory or of comparing by means of a sample
before our eyes. We know a vast number of such processes, processes similar to each other in a vast number of
different ways. We hold pieces whose colours we want to compare together or near each other for a longer or
shorter period, look at them alternately or simultaneously, place them under different lights, say different things
while we do so, have memory images, feelings of tension and relaxation, satisfaction and dissatisfaction, the various
feelings of strain in and around our eyes accompanying prolonged gazing at the same object, and all possible
combinations of these and many other experiences. The more such cases we observe and the closer we look at them,
the more doubtful we feel about finding one particular mental experience characteristic of comparing. In fact, if after
you had scrutinized a number of such closely, I admitted that there existed a peculiar mental experience which you
might call the experience of comparing, and that if you insisted, I should be willing to adopt the word "comparing"
only for cases in which this peculiar feeling had occurred, you would now feel that the assumption of such a peculiar
experience had lost its point, because this experience was placed side by side with a vast number of other
experiences which after we have scrutinized the cases seems to be that which really constitutes what connects all the
cases of comparing. For the 'specific experience' we had been looking for was meant to have played the role which
has been assumed by the mass of experiences revealed to us by our scrutiny: We never wanted the specific
experience to be just one among a number of more or less characteristic experiences. (One might say that there are
two ways of looking at this matter, one as it were, at dose quarters, the other as though from a distance and through
the medium of a peculiar atmosphere.) In fact we have found that the use which we really make of the word
"comparing" is different from that which looking at it
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from far away we were led to expect. We find that what connects all the cases of comparing is a vast number of
overlapping similarities, and as soon as we see this, we feel no longer compelled to say that there must be some one
feature common to them all. What ties the ship to the wharf is a rope, and the rope consists of fibres, but it does not
get its strength from any fibre which runs through it from one end to the other, but from the fact that there is a vast
number of fibres overlapping.
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'But surely in case 14c) B acted entirely automatically. If all that happened was really what was described
there, he did not know why he chose the bolt he did choose. He had no reason for choosing it. If he chose the right
one, he did it as a machine might have done it'. Our first answer is that we did not deny that B in case 14c) had what
we should call a personal experience, for we did not say that he didn't see the materials from which he chose or that
which he chose, nor that he didn't have muscular and tactile sensations and suchlike while he did it. Now what
would such a reason which justified his choice and made it non-automatic be like? (i.e.: What do we imagine it to be
like?) I suppose we should say that the opposite of automatic comparing, as it were, the ideal case of conscious
comparing, was that of having a clear memory image before our mind's eye or of seeing a real sample and of having
a specific feeling of not being able to distinguish in a particular way between these samples and the material chosen.
I suppose that this peculiar sensation is the reason, the justification, for the choice. This specific feeling, one might
say, connects the two experiences of seeing the sample, on the one hand, and the material on the other. But if so,
what connects this specific experience with either? We don't deny that such an experience might intervene. But
looking at it as we did just now, the distinction between automatic and non-automatic appears no longer clear-cut
and final as it did at first. We don't mean that this distinction loses its practical value in particular cases, e.g., if asked
under particular circumstances "Did you take this bolt from the shelf automatically, or did you think about it?", we
may be justified in saying that we did not act automatically and give as an explanation that we had looked at the
material carefully, had tried to recall the memory image of the pattern, and had uttered to ourselves doubts and
decisions. This may in the particular case be taken to distinguish automatic from non-automatic. In another case,
however, we may distinguish between an automatic
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and a non-automatic way of the appearance of a memory image, and so on.
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If our case 14c) troubles you, you may be inclined to say: "But why did he bring just this bolt of material?
How has he recognized it as the right one? What by?"--If you ask 'why', do you ask for the cause or for the reason?
If for the cause, it is easy enough to think up a physiological or psychological hypothesis which explains this choice
under the given conditions. It is the task of the experimental sciences to test such hypotheses. If on the other hand
you ask for a reason the answer is, "There need not have been a reason for the choice. A reason is a step preceding
the step of the choice. But why should every step be preceded by another one?"
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'But then B didn't really recognize the material as the right one.'You needn't reckon 14c) among the cases of
recognizing, but if you have become aware of the fact that the processes which we call processes of recognition form
a vast family with overlapping similarities, you will probably feel not disinclined to include 14c) in this family,
too.--'But doesn't B in this case lack the criterion by which he can recognize the material? In 14a), e.g., he had the
memory image and he recognized the material he looked for by its agreement with the image.'--But had he also a
picture of this agreement before him, a picture with which he could compare the agreement between the pattern and
the bolt to see whether it was the right one? And, on the other hand, couldn't he have been given such a picture?
Suppose, e.g., that A wished B to remember that what was wanted was a bolt exactly like the sample, not, as
perhaps in other cases, a material slightly darker than the pattern. Couldn't A in this case have given to B an example
of the agreement required by giving him two pieces of the same colour (e.g., as a kind of reminder)? Is any such link
between the order and its execution necessarily the last one?--And if you say that in 14b) at least he had the relaxing
of the tension by which to recognize the right material, had he to have an image of this relaxation about him to
recognize it as that by which the right material was to be recognized?--
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'But supposing B brings the bolt, as in 14c), and on comparing it with the pattern it turns out to be the wrong
one?'--But couldn't that have happened in all the other cases as well? Suppose in 14a) the bolt which B brought back
was found not to match with the pattern. Wouldn't we in some such cases say that his memory image had changed,
in others that the pattern or the material had changed, in
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others again that the light had changed? It is not difficult to invent cases, imagine circumstances, in which each of
these judgments would be made.--'But isn't there after all an essential difference between the cases 14a) and
14c)?'--Certainly! Just that pointed out in the description of these cases.--
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In 1) B learnt to bring a building stone on hearing the word "column!" called out. We could imagine what
happened in such a case to be this: In B's mind the word called out brought up an image of a column, say; the
training had, as we should say, established this association. B takes up that building stone which conforms to his
image.--But was this necessarily what happened? If the training could bring it about that the idea or
image-automatically--arose in B's mind, why shouldn't it bring about B's actions without the intervention of an
image? This would only come to a slight variation of the associative mechanism. Bear in mind that the image which
is brought up by the word is not arrived at by a rational process (but if it is, this only pushes our argument further
back), but that this case is strictly comparable with that of a mechanism in which a button is pressed and an indicator
plate appears. In fact this sort of mechanism can be used instead of that of association.
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Mental images of colours, shapes, sounds, etc., etc., which play a role in communication by means of
language we put in the same category with patches of colour actually seen, sounds heard.
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18). The object of the training in the use of tables (as in 7)) may be not only to teach the use of one particular
table, but it may be to enable the pupil to use or construct himself tables with new co-ordinations of written signs
and pictures. Suppose the first table a person was trained to use contained the four words "hammer", "pincers",
"saw", "chisel" and the corresponding pictures. We might now add the picture of another object which the pupil had
before him, say of a plane, and correlate with it the word "plane". We shall make the correlation between this new
picture and word as similar as possible to the correlations in the previous table. Thus we might add the new word
and picture on the same sheet, and place the new word under the previous words and the new picture under the
previous pictures. The pupil will now be encouraged to make use of the new picture and word without the special
training which we gave him when we taught him to use the first table. These acts of encouragement will be of
various kinds, and many such acts will only be possible if
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the pupil responds, and responds in a particular way. Imagine the gestures, sounds, etc., of encouragement you use
when you teach a dog to retrieve. Imagine on the other hand, that you tried to teach a cat to retrieve. As the cat will
not respond to your encouragement, most of the acts of encouragement which you performed when you trained the
dog are here out of the question.
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19). The pupil could also be trained to give things names of his own invention and to bring the objects when
the names are called. He is, e.g., presented with a table on which he finds pictures of objects around him on one side
and blank spaces on the other, and he plays the game by writing signs of his own invention opposite the pictures and
reacting in the previous way when these signs are used as orders. Or else
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20). the game may consist in B's constructing a table and obeying orders given in terms of this table. When
the use of a table is taught, and the table consists, say, of two vertical columns, the left hand one containing the
names, the right hand one the pictures, a name and a picture being correlated by standing on a horizontal line, an
important feature of the training may be that which makes the pupil slide his finger from left to right, as it were the
training to draw a series of horizontal lines, one below the other. Such training may help to make the transition from
the first table to the new item.
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Tables, ostensive definitions, and similar instruments I shall call rules, in accordance with ordinary usage.
The use of a rule can be explained by a further rule.
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21). Consider this example: We introduce different ways of reading tables. Each table consists of two
columns of words and pictures, as above. In some cases they are to be read horizontally from left to right, i.e.,
according to the scheme:
In others according to such schemes as:
or:
etc.
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Schemes of this kind can be adjoined to our tables, as rules for reading them. Could not these rules again be
explained by further rules? Certainly. On the other hand, is a rule incompletely explained if no rule for its usage has
been given?
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We introduce into our language games the endless series of numerals. But how is this done? Obviously the
analogy between this process and that of introducing a series of twenty numerals is not the same as that between
introducing a series of twenty numerals and introducing a series of ten numerals. Suppose that our game was like 2)
but played with the endless series of numerals. The difference between it and 2) would not be just that more
numerals were used. That is to say, suppose that as a matter of fact in playing the game we had actually made use of,
say, 155 numerals, the game we play would not be that which could be described by saying that we played the game
2), only with 155 instead of 10 numerals. But what does the difference consist in? (The difference would seem to be
almost one of the spirit in which the games are played.) The difference between games can lie, say, in the number of
the counters used, in the number of squares of the playing board, or in the fact that we use squares in one case and
hexagons in the other, and suchlike. Now the difference between the finite and infinite game does not seem to lie in
the material tools of the game; for we should be inclined to say that infinity can't be expressed in them, that is, that
we can only conceive of it in our thoughts, and hence that it is in these thoughts that the finite and infinite game
must be distinguished. (It is queer though that these thoughts should be capable of being expressed in signs.)
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Let us consider two games. They are both played with cards carrying numbers, and the highest number takes
the trick.
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22). One game is played with a fixed number of such cards, say 32. In the other game we are under certain
circumstances allowed to increase the number of cards to as many as we like, by cutting pieces of paper and writing
numbers on them. We will call the first of these games bounded, the second unbounded. Suppose a hand of the
second game was played and the number of cards actually used was 32. What is the difference in this case between
playing a hand a) of the unbounded game and playing a hand b) of the bounded game?
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The difference will not be that between a hand of a bounded game with 32 cards and a hand of a bounded
game with a greater number of cards. The number of cards used was, we said, the same. But there
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will be differences of another kind, e.g., the bounded game is played with a normal pack of cards, the unbounded
game with a large supply of blank cards and pencils. The unbounded game is opened with the question, "How high
shall we go?" If the players look up the rules of this game in a book of rules, they will find the phrase "and so on" or
"and so on ad inf." at the end of certain series of rules. So the difference between the two hands a) and b) lies in the
tools we use, though admittedly not in the cards they are played with. But this difference seems trivial and not the
essential difference between the games. We feel that there must be a big and essential difference somewhere. But if
you look closely at what happens when the hands are played, you find that you can only detect a number of
differences in details, each of which would seem inessential. The acts, e.g., of dealing and playing the cards may in
both cases be identical. In the course of playing the hand a), the players may have considered making up more cards,
and again discarded the idea. But what was it like to consider this? It could be some such process as saying to
themselves or aloud "I wonder whether I should make up another card". Again, no such consideration may have
entered the minds of the players. It is possible that the whole difference in the events of a hand of the bounded, and
a hand of the unbounded, game lay in what was said before the game started, e.g., "Let's play the bounded game".
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'But isn't it correct to say that hands of the two different games belong to two different systems?' Certainly.
Only the facts which we are referring to by saying that they belong to different systems are much more complex
than we might expect them to be.
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Let us now compare language games of which we should say that they are played with a limited set of
numerals, with language games of which we should say that they are played with the endless series of numerals.
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23). Like 2) A orders B to bring him a number of building stones. The numerals are the signs "1", "2",.......
"9", each written on a card. A has a set of these cards and gives B the order by showing him one of the set and
calling out one of the words, "slab", "column", etc.
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24). Like 23), only there is no set of indexed cards. The series of numerals 1 ... 9 is learned by heart. The
numerals are called out in the orders, and the child learns them by word of mouth.
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25). An abacus is used. A sets the abacus, gives it to B, B goes with it to where the slabs lie, etc.
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26). B is to count the slabs in a heap. He does it with an abacus, the abacus has twenty beads. There are never
more than 20 slabs in a heap. B sets the abacus for the heap in question and shows A the abacus thus set.
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27). Like 26). The abacus has 20 small beads and one large one. if the heap contains more than 20 slabs, the
large bead is moved. (So the large bead in some way corresponds to the word "many".)
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28). Like 26). If the heap contains n slabs, n being more than 20 but less than 40, B moves n-20 beads, shows
A the abacus thus set and claps his hands once.
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29). A and B use the numerals of the decimal system (written or spoken) up to 20. The child learning this
language learns these numerals by heart, as in 2).
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30). A certain tribe has a language of the kind 2). The numerals used are those of our decimal system. No one
numeral used can be observed to play the predominant role of the last numeral in some of the above games (27),
28)). (One is tempted to continue this sentence by saying, "although there is of course a highest numeral actually
used".) The children of the tribe learn the numerals in this way: They are taught the signs from 1 to 20 as in 2) and to
count rows of beads of no more than 20 on being ordered, "Count these". When in counting the pupil arrives at the
numeral 20, one makes a gesture suggestive of "Go on", upon which the child says (in most cases at any rate) "21".
Analogously, the children are made to count to 22 and to higher numbers, no particular number playing in these
exercises the predominant role of a last one. The last stage of the training is that the child is ordered to count a group
of objects, well above 20, without the suggestive gesture being used to help the child over the numeral 20. If a child
does not respond to the suggestive gesture, it is separated from the others and treated as a lunatic.
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31). Another tribe. Its language is like that in 30). The highest numeral observed in use is 159. In the life of
this tribe the numeral 159 plays a peculiar role. Supposing I said, "They treat this number as their highest",--but what
does this mean? Could we answer: "They just say that it is the highest"?--They say certain words, but
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how do we know what they mean by them? A criterion for what they mean would be the occasions on which the
word we are inclined to translate into our word "highest" is used, the role, we might say, which we observe this word
to play in the life of the tribe. In fact we could easily imagine the numeral 159 to be used on such occasions, in
connection with such gestures and forms of behaviour as would make us say that this numeral plays the role of an
unsurmountable limit, even if the tribe had no word corresponding to our "highest", and the criteria for numeral 159
being the highest numeral did not consist of anything that was said about the numeral.
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32). A tribe has two systems of counting. People learned to count with the alphabet from A to Z and also
with the decimal system as in 30.) If a man is to count objects with the first system, he is ordered to count "in the
closed way", in the second case, "in the open way"; and the tribe uses the words "closed" and "open" also for a
closed and open door.
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(Remarks: 23) is limited in an obvious way by the set of cards. 24): Note analogy and lack of analogy
between the limited supply of cards in 23) and of words in our memory in 24). Observe that the limitation in 26) on
the one hand lies in the tool (the abacus of 20 beads) and its usage in our game, on the other hand (in a totally
different way) in the fact that in the actual practice of playing the game no more than 20 objects are ever to be
counted. In 27) that latter kind of limitation was absent, but the large bead rather stressed the limitation of our
means. Is 28) a limited or an unlimited game? The practice we have described gives the limit 40. We are inclined to
say this game 'has it in it' to be continued indefinitely, but remember that we could also have construed the
preceding games as beginnings of a system. In 29) the systematic aspect of the numerals used is even more
conspicuous than in 28). One might say that there was no limitation imposed by the tools of this game, if it were not
for the remark that the numerals up to 20 are learnt by heart. This suggests the idea that the child is not taught to
'understand' the system which we see in the decimal notation. Of the tribe in 30) we should certainly say that they
are trained to construct numerals indefinitely, that the arithmetic of their language is not a finite one, that their series
of numbers has no end. (It is just in such a case when numerals are constructed 'indefinitely' that we say that people
have the infinite series of numbers.) 31) might show you what a vast variety of cases can be imagined in which we
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should be inclined to say that the arithmetic of the tribe deals with a finite series of numbers, even in spite of the fact
that the way in which the children are trained in the use of numerals suggests no upper limit. In 32) the terms
"closed" and "open" (which could by a slight variation of the example be replaced by "limited" and "unlimited") are
introduced into the language of the tribe itself. Introduced in that simple and clearly circumscribed game, there is of
course nothing mysterious about the use of the word "open". But this word corresponds to our "infinite", and the
games we play with the latter differ from 31) only by being vastly more complicated. In other words, our use of the
word "infinite" is just as straightforward as that of "open" in 31), and our idea that its meaning is 'transcendent' rests
on a misunderstanding.)
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We might say roughly that the unlimited cases are characterized by this: that they are not played with a
definite supply of numerals, but instead with a system for constructing numerals (indefinitely). When we say that
someone has been supplied with a system for constructing numerals, we generally think of one of three things: a) of
giving him a training similar to that described in 30), which, experience teaches us, will make him pass tests of the
kind mentioned there; b) of creating a disposition in the same man's mind, or brain, to react in that way; c) of
supplying him with a general rule for the construction of numerals.
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What do we call a rule? Consider this example:
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33). B moves about according to rules which A gives him. B is supplied with the following table:
A gives an order made up of the letters in the table, say: "aacaddd". B looks up the arrow corresponding to each
letter of the order and moves accordingly; in our example thus:
The table 33) we should call a rule (or else "the expression of a rule". Why I give these synonymous expressions will
appear later.) We shan't be inclined to call the sentence "aacaddd" itself a rule. It is of course the description of the
way B has to take. On the other hand,
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such a description would under certain circumstances be called a rule, e.g., in the following case:
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34). B is to draw various ornamental linear designs. Each design is a repetition of one element which A gives
him. Thus if A gives the order "cada", B draws a line thus:
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In this case I think we should say that "cada" is the rule for drawing the design. Roughly speaking, it
characterizes what we call a rule to be applied repeatedly, in an indefinite number of instances. Cf., e.g., the
following case with 34):
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35). A game played with pieces of various shapes on a chess board. The way each piece is allowed to move is
laid down by a rule. Thus the rule for a particular piece is "ac", for another piece "acaa", and so on. The first piece
then can make a move like this: , the second, like this: . Both a formula like "ac" or a diagram
like that corresponding to such a formula might here be called a rule.
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36). Suppose that after playing the game 33) several times as described above, it was played with this
variation: that B no longer looked at the table, but reading A's order the letters call up the images of the arrows (by
association), and B acts according to these imagined arrows.
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37). After playing it like this for several times, B moves about according to the written order as he would
have done had he looked up or imagined the arrows, but actually without any such picture intervening. Imagine even
this variation:
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38). B in being trained to follow a written order, is shown the table of 33) once, upon which he obeys A's
orders without further intervention of the table in the same way in which B in 33) does with the help of the table on
each occasion.
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In each of these cases, we might say that the table 33) is a rule of the game. But in each one this rule plays a
different role. In 33) the table is an instrument used in what we should call the practice of the game. It is replaced in
36) by the working of association. In 37) even this shadow of the table has dropped out of the practice of the game,
and in 38) the table is admittedly an instrument for the training of B only.
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But imagine this further case:
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39). A certain system of communication is used by a tribe. I will describe it by saying that it is similar to our
game 38) except that no table is used in the training. The training might have consisted in several times leading the
pupil by the hand along the path one wanted him to go. But we could also imagine a case:
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40). where even this training is not necessary, where, as we should say, the look of the letters abcd naturally
produced an urge to move in the way described. This case at first sight looks puzzling. We seem to be assuming a
most unusual working of the mind. Or we may ask, "How on earth is he to know which way to move if the letter a is
shown him?". But isn't B's reaction in this case the very reaction described in 37) and 38), and in fact our usual
reaction when for instance we hear and obey an order? For, the fact that the training in 38) and 39) preceded the
carrying out of the order does not change the process of carrying out. In other words the 'curious mental mechanism'
assumed in 40) is no other than that which we assumed to be created by the training in 37) and 38). 'But could such a
mechanism be born with you?' But did you find any difficulty in assuming that that mechanism was born with B,
which enabled him to respond to the training in the way he did? And remember that the rule or explanation given in
table 33) of the signs abcd was not essentially the last one, and that we might have given a table for the use of such
tables, and so on. (Cf. 21).)
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How does one explain to a man how he should carry out the order, "Go this way!" (pointing with an arrow
the way he should go)? Couldn't this mean going the direction which we should call the opposite of that of the
arrow? Isn't every explanation of how he should follow the arrow in the position of another arrow? What would you
say to this explanation: A man says, "If I point this way (pointing with his right hand) I mean you to go like this"
(pointing with his left hand the same way)? This just shows you the extremes between which the uses of signs vary.
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Let us return to 39). Someone visits the tribe and observes the use of the signs in their language. He describes
the language by saying that its sentences consist of the letters abcd used according to the table (of 33)). We see that
the expression, "A game is played according to the rule so and so" is used not only in the variety of cases
exemplified by 36), 37), and 38), but even in cases where the rule is neither an instrument of the training nor of the
practice of the game, but stands
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in the relation to it in which our table stands to the practice of our game 39). One might in this case call the table a
natural law describing the behaviour of the people of this tribe. Or we might say that the table is a record belonging
to the natural history of the tribe.
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Note that in the game 33) I distinguished sharply between the order to be carried out and the rule employed.
In 34), on the other hand, we called the sentence "cada" a rule, and it was the order. Imagine also this variation:
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41). The game is similar to 33), but the pupil is not just trained to use a single table; but the training aims at
making the pupil use any table correlating letters with arrows. Now by this I mean no more than that the training is
of a peculiar kind, roughly speaking one analogous to that described in 30). I will refer to a training more or less
similar to that in 30) as a "general training". General trainings form a family whose members differ greatly from one
another. The kind of thing I'm thinking of now mainly consists: a) of a training in a limited range of actions, b) of
giving the pupil a lead to extend this range, and c) of random exercises and tests. After the general training the order
is now to consist in giving him a sign of this kind:
He carries out the order by moving thus:
Here I suppose we should say the table, the rule, is part of the order.
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Note, we are not saying 'what a rule is' but just giving different applications of the word "rule": and we
certainly do this by giving applications of the words "expression of a rule".
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Note also that in 41) there is no clear case against calling the whole symbol given the sentence, though we
might distinguish in it between the sentence and the table. What in this case more particularly tempts us to this
distinction is the linear writing of the part outside the table. Though from certain points of view we should call the
linear character of the sentence merely external and inessential, this character and similar ones play a great role in
what as logicians we are inclined to say about sentences and propositions. And therefore if we conceive
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of the symbol in 41) as a unit, this may make us realize what a sentence can look like.
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Let us now consider these two games:
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42). A gives orders to B: They are written signs consisting of dots and dashes, and B executes them by doing
a figure in dancing with a particular step. Thus the order "-- •" is to be carried out by taking a step and a hop
alternately; the order "• • -- -- --" by alternately taking two hops and three steps, etc. The training in this game is
'general' in the sense explained in 41); and I should like to say, "The orders given don't move in a limited range. They
comprise combinations of any number of dots and dashes".--But what does it mean to say that the orders don't
move in a limited range? Isn't this nonsense? Whatever orders are given in the practice of the game constitute the
limited range.--Well, what I meant to say by "The orders don't move in a limited range" was that neither in the
teaching of the game nor in the practice of it a limitation of the range plays a 'predominant' role (see 30)), or, as we
might say, the range of the game (it is superfluous to say limited) is just the extent of its actual ('accidental') practice.
(Our game is in this way like 30).) Cf. with this game the following:
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43). The orders and their execution as in 42); but only these three signs are used: "--", "-- • •", "• -- --". We
say that in 42) B, in executing the order, is guided by the sign given to him. But if we ask ourselves whether the
three signs in 43) guide B in executing the orders, it seems that we can say both yes and no according to the way we
look at the execution of the orders.
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If we try to decide whether B in 43) is guided by the signs or not, we are inclined to give such answers as the
following: a) B is guided if he doesn't just look at an order, say "• -- --" as a whole and then act, but if he reads it
'word by word' (the words used in our language being "•" and "--") and acts according to the words he has read.
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We could make these cases clearer if we imagine that the 'reading word by word' consisted in pointing to
each word of the sentence in turn with one's finger as opposed to pointing at the whole sentence at once, say by
pointing to the beginning of the sentence. And the 'acting according to the words' we shall for the sake of simplicity
imagine to consist in acting (stepping or hopping) after each word of the sentence in turn.--b) B is guided if he goes
through a conscious process which makes a connection between the pointing to a word and the act of hopping and
stepping. Such a connection could be
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imagined in many different ways. E.g., B has a table in which a dash is correlated to the picture of a man making a
step and a dot to a picture of a man hopping. Then the conscious acts connecting reading the order and carrying it
out might consist in consulting the table, or in consulting a memory image of it 'with one's mind's eye'. c) B is guided
if he does not just react to looking at each word of the order, but experiences the peculiar strain of 'trying to
remember what the sign means', and further, the relaxing of this strain when the meaning, the right action, comes
before his mind.
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All these explanations seem in a peculiar way unsatisfactory, and it is the limitation of our game which
makes them unsatisfactory. This is expressed by the explanation that B is guided by the particular combination of
words in one of our three sentences if he could also have carried out orders consisting in other combinations of dots
and dashes. And if we say this, it seems to us that the 'ability' to carry out other orders is a particular state of the
person carrying out the orders of 42). And at the same time we can't in this case find anything which we should call
such a state.
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Let us see what role the words "can" or "to be able to" play in our language. Consider these examples:
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44). Imagine that for some purpose or other people use a kind of instrument or tool; this consists of a board
with a slot in it guiding the movement of a peg. The man using the tool slides the peg along the slot. There are such
boards with straight slots, circular slots, elliptic slots, etc. The language of the people using this instrument has
expressions for describing the activity of moving the peg in the slot. They talk of moving it in a circle, in a straight
line, etc. They also have a means of describing the board used. They do it in this form: "This is a board in which the
peg can be moved in a circle". One could in this case call the word "can" an operator by means of which the form of
expression describing an action is transformed into a description of the instrument.
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45). Imagine a people in whose language there is no such form of sentence as "the book is in the drawer" or
"water is in the glass", but wherever we should use these forms they say, "The book can be taken out of the drawer",
"The water can be taken out of the glass".
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46). An activity of the men of a certain tribe is to test sticks as to their hardness. They do it by trying to bend
the sticks with their
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hands. In their language they have expressions of the form, "This stick can be bent easily", or "This stick can be bent
with difficulty". They use these expressions as we use "This stick is soft", or "This stick is hard". I mean to say that
they don't use the expression, "This stick can be bent easily" as we should use the sentence, "I am bending the stick
with ease". Rather they use their expression in a way which would make us say that they are describing a state of the
stick. I.e., they use such sentences as, "This hut is built of sticks that can be bent easily". (Think of the way in which
we form adjectives out of verbs by means of the ending "able", e.g., "deformable".)
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Now we might say that in the last three cases the sentences of the form "so and so can happen" described the
state of objects, but there are great differences between these examples. In 44) we saw the state described before our
eyes. We saw that the board had a circular or a straight slot, etc. In 45), in some instances at least this was the case,
we could see the objects in the box, the water in the glass, etc. In such cases we use the expression "state of an
object" in such a way that there corresponds to it what one might call a stationary sense experience.
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When, on the other hand, we talk of the state of a stick in 46), observe that to this 'state' there does not
correspond a particular sense experience which lasts while the state lasts. Instead of that, the defining criterion for
something being in this state consists in certain tests.
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We may say that a car travels 20 miles an hour even if it only travels for half an hour. We can explain our
form of expression by saying that the car travels with a speed which enables it to make 20 miles an hour. And here
also we are inclined to talk of the velocity of the car as of a state of its motion. I think we should not use this
expression if we had no other 'experiences of motion' than those of a body being in a particular place at a certain
time and in another place at another time; if, e.g., our experiences of motion were of the kind which we have when
we see that the hour hand of the clock has moved from one point of the dial to the other.
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47). A tribe has in its language commands for the execution of certain actions of men in warfare, something
like "Shoot!", "Run!", "Crawl!", etc. They also have a way of describing a man's build. Such a description has the
form "He can run fast", "He can throw the spear far". What justifies me in saying that these sentences are
descriptions of the man's build is the use which they make of sentences of this
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form. Thus if they see a man with bulging leg muscles but who as we should say has not the use of his legs for some
reason or other, they say he is a man who can run fast. The drawn image of a man which shows large biceps they
describe as representing a man "who can throw a spear far".
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48). The men of a tribe are subjected to a kind of medical examination before going into war. The examiner
puts the men through a set of standardized tests. He lets them lift certain weights, swing their arms, skip, etc. The
examiner then gives his verdict in the form "So and so can throw a spear" or "can throw a boomerang" or "is fit to
pursue the enemy", etc. There are no special expressions in the language of this tribe for the activities performed in
the tests; but these are referred to only as the tests for certain activities in warfare.
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It is an important remark concerning this example and others which we give that one may object to the
description which we give of the language of a tribe, that in the specimens we give of their language we let†1 them
speak English, thereby already presupposing the whole background of the English language, that is, our usual
meanings of the words. Thus if I say that in a certain language there is no special verb for "skipping", but that this
language uses instead the form "making the test for throwing the boomerang", one may ask how I have
characterized the use of the expressions, "make a test for" and "throwing the boomerang", to be justified in
substituting these English expressions for whatever their actual words may be. To this we must answer that we have
only given a very sketchy description of the practices of our fictitious languages, in some cases only hints, but that
one can easily make these descriptions more complete. Thus in 48) I could have said that the examiner uses orders
for making the men go through the tests. These orders all begin with one particular expression which I could
translate into the English words, "Go through the test". And this expression is followed by one which in actual
warfare is used for certain actions. Thus there is a command upon which men throw their boomerangs and which
therefore I should translate into, "Throw the boomerangs". Further, if a man gives an account of the battle to his
chief, he again uses the expression I have translated into "throw a boomerang", this time in a description. Now what
characterizes an order as such, or a description as such,
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or a question as such, etc., is--as we have said--the role which the utterance of these signs plays in the whole practice
of the language. That is to say, whether a word of the language of our tribe is rightly translated into a word of the
English language depends upon the role this word plays in the whole life of the tribe; the occasions on which it is
used, the expressions of emotion by which it is generally accompanied, the ideas which it generally awakens or
which prompt its saying, etc., etc. As an exercise ask yourself: in which cases would you say that a certain word
uttered by the people of the tribe was a greeting? In which cases should we say it corresponded to our "Goodbye",
in which to our "Hello"? In which cases would you say that a word of a foreign language corresponded to our
"perhaps"?--to our expressions of doubt, trust, certainty? You will find that the justifications for calling something
an expression of doubt, conviction, etc., largely, though of course not wholly, consist in descriptions of gestures, the
play of facial expressions, and even the tone of voice. Remember at this point that the personal experiences of an
emotion must in part be strictly localized experiences; for if I frown in anger I feel the muscular tension of the frown
in my forehead, and if I weep, the sensations around my eyes are obviously part, and an important part, of what I
feel. This is, I think, what William James meant when he said that a man doesn't cry because he is sad but that he is
sad because he cries. The reason why this point is often not understood, is that we think of the utterance of an
emotion as though it were some artificial device to let others know that we have it. Now there is no sharp line
between such 'artificial devices' and what one might call the natural expressions of emotion. Cf. in this respect: a)
weeping, b) raising one's voice when one is angry, c) writing an angry letter, d) ringing the bell for a servant you wish
to scold.
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49) Imagine a tribe in whose language there is an expression corresponding to our "He has done so and so",
and another expression corresponding to our "He can do so and so", this latter expression, however, being only used
where its use is justified by the same fact which would also justify the former expression. Now what can make me
say this? They have a form of communication which we should call narration of past events because of the
circumstances under which it is employed. There are also circumstances under which we should ask and answer
such questions as "Can so and so do this?" Such circumstances can be described, e.g., by saying that a chief picks
men
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suitable for a certain action, say crossing a river, climbing a mountain, etc. As the defining criteria of "the chief
picking men suitable for this action", I will not take what he says but only the other features of the situation. The
chief under these circumstances asks a question which, as far as its practical consequences go, would have to be
translated by our "Can so and so swim across this river?" This question however, is only answered affirmatively by
those who actually have swum across this river. This answer is not given in the same words in which under the
circumstances characterizing narration he would say that he has swum across this river, but it is given in the terms of
the question asked by the chief. On the other hand, this answer is not given in cases in which we should certainly
give the answer, "I can swim across this river", if, e.g., I had performed more difficult feats of swimming though not
just that of swimming across this particular river.
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By the way, have the two phrases "He has done so and so" and "He can do so and so" the same meaning in
this language or have they different meanings? If you think about it, something will tempt you to say the one,
something to say the other. This only shows that the question has here no clearly defined meaning. All I can say is:
If the fact that they only say "He can..." if he has done... is your criterion for the same meaning, then the two
expressions have the same meaning. If the circumstances under which an expression is used make its meaning, the
meanings are different. The use which is made of the word "can"--the expression of possibility in 49)--can throw a
light upon the idea that what can happen must have happened before (Nietzsche). It will also be interesting to look,
in the light of our examples, on the statement that what happens can happen.
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Before we go on with our consideration of the use of 'the expression of possibility', let us get clearer about
that department of our language in which things are said about past and future, that is, about the use of sentences
containing such expressions as "yesterday", "a year ago", "in five minutes", "before I did this", etc. Consider this
example:
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50). Imagine how a child might be trained in the practice of 'narration of past events'. He was first trained in
asking for certain things (as it were, in giving orders. See 1)). Part of this training was the exercise of 'naming the
things'. He has thus learnt to name (and ask for) a dozen of his toys. Say now that he has played with three of them
(e.g., a ball, a stick, and a rattle), then they are taken away from
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him, and now the grown up says such a phrase as, "He's had a ball, a stick, and a rattle". On a similar occasion he
stops short in the enumeration and induces the child to complete it. On another occasion, perhaps, he only says,
"He's had..." and leaves the child to give the whole enumeration. Now the way of 'inducing the child to go on' can be
this: He stops short in his enumeration with a facial expression and a raised tone of voice which we should call one
of expectancy. All then depends on whether the child will react to this 'inducement' or not. Now there is a queer
misunderstanding we are most liable to fall into, which consists in regarding the 'outward means' the teacher uses to
induce the child to go on as what we might call an indirect means of making himself understood to the child. We
regard the case as though the child already possessed a language in which it thought and that the teacher's job is to
induce it to guess his meaning in the realm of meanings before the child's mind, as though the child could in his own
private language ask himself such a question as, "Does he want me to continue, or repeat what he said, or something
else?" (Cf. with 30).)
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51). Another example of a primitive kind of narration of past events: we live in a landscape with characteristic
natural landmarks against the horizon. It is therefore easy to remember the place at which the sun rises at a particular
season, or the place above which it stands when at its highest point, or the place at which it sets. We have some
characteristic pictures of the sun in different positions in our landscape. Let us call this series of pictures the sun
series. We have also some characteristic pictures of the activities of a child, lying in bed, getting up, dressing,
lunching, etc. This set I'll call the life pictures. I imagine that the child can frequently see the position of the sun while
about the day's activities. We draw the child's attention to the sun's standing in a certain place while the child is
occupied in a particular way. We then let it look both at a picture representing its occupation and at a picture
showing the sun in its position at that time. We can thus roughly tell the story of the child's day by laying out a row
of the life pictures, and above it what I called the sun series, the two rows in the proper correlation. We shall then
proceed to let the child supplement such a picture story, which we leave incomplete. And I wish to say at this point
that this form of training (see 50) and 30)) is one of the big characteristic features in the use of language, or in
thinking.
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52). A variation of 51). There is a big clock in the nursery, for simplicity's sake imagine it with an hour hand
only. The story of the child's day is narrated as above, but there is no sun series; instead we write one of the numbers
of the dial against each life picture.
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53). Note that there would have been a similar game in which also, as we might say, time was involved, that
of just laying out a series of life pictures. We might play this game with the help of words which would correspond
to our "before" and "after". In this sense we may say that 53) involves the ideas of before and after, but not the idea
of a measurement of time. I needn't say that an easy step would lead us from the narrations in 51), 52), and 53) to
narrations in words. Possibly someone considering such forms of narration might think that in them the real idea of
time isn't yet involved at all, but only some crude substitute for it, the positions of a clock hand and suchlike. Now if
a man claimed that there is an idea of five o'clock which does not bring in a clock, that the clock is only the coarse
instrument indicating when it is five o'clock or that there is an idea of an hour which does not bring in an instrument
for measuring the time, I will not contradict him, but I will ask him to explain to me what his use of the term "an
hour" or "five o'clock" is. And if it is not that involving a clock, it is a different one; and then I will ask him why he
uses the terms "five o'clock", "an hour", "a long time", "a short time", etc., in one case in connection with a clock, in
the other independent of one; it will be because of certain analogies holding between the two uses, but we have now
two uses of these terms, and no reason to say that one of them is less real and pure than the other. This might get
dearer by considering the following example:
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54). If we give a person the order "Say a number, any one which comes into your mind", he can generally
comply with it at once. Suppose it were found that the numbers thus said on request increased--with every normal
person--as the day went on; a man starts out with some small number every morning and reaches the highest
number before falling asleep at night. Consider what could tempt one to call the reactions described "a means of
measuring time" or even to say that they are the real milestones in the passage of time, the sun clocks, etc., being
only indirect markers. (Examine the statement that the human heart is the real clock behind all the other clocks.)
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Let us now consider further language games into which temporal expressions enter.
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55). This arises out of 1). If an order like "slab!", "column!", etc. is called out, B is trained to carry it out
immediately. We now introduce a clock into this game, an order is given, and we train the child not to carry it out
until the hand of our clock reaches a point indicated before with the finger. (This might, e.g., be done in this way:
You first trained the child to carry out the order immediately. You then give the order, but hold the child back,
releasing it only when the hand of the clock has reached the point of the dial to which we point with our fingers.)
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We could at this stage introduce such a word as "now". We have two kinds of orders in this game, the orders
used in 1), and orders consisting of these, together with a gesture indicating a point of the clock dial. In order to
make the distinction between these two kinds more explicit, we may affix a particular sign to the orders of the first
kind and, e.g., say: "slab, now!".
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It would be easy now to describe language games with such expressions as "in five minutes", "half an hour
ago".
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56). Let us now have the case of a description of the future, a forecast. One might, e.g., awaken the tension
of expectation in a child by keeping his attention for a considerable time on some traffic lights changing their colour
periodically. We also have a red, a green, and a yellow disc before us and alternately point to one of these discs by
way of forecasting the colour which will appear next. It is easy to imagine further developments of this game.
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Looking at these language games, we don't come across the ideas of the past, the future and the present in
their problematic and almost mysterious aspect. What this aspect is and how it comes about that it appears can be
almost characteristically exemplified if we look at the question "Where does the present go when it becomes past,
and where is the past?"--Under what circumstances has this question an allurement for us? For under certain
circumstances it hasn't, and we should wave it away as nonsense.
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It is clear that this question most easily arises if we are preoccupied with cases in which there are things
flowing by us,--as logs of wood float down a river. In such a case we can say the logs which have passed us are all
down towards the left and the logs which will pass us are all up towards the right. We then use this situation as a
simile for all happening in time and even embody the simile in our language, as when we say that 'the present event
passes by' (a log passes by),
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'the future event is to come' (a log is to come). We talk about the flow of events; but also about the flow of time--the
river on which the logs travel.
Page 108
Here is one of the most fertile sources of philosophic puzzlement: we talk of the future event of something
coming into my room, and also of the future coming of this event.
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We say, "Something will happen", and also, "Something comes towards me"; we refer to the log as
"something", but also the log's coming towards me.
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Thus it can come about that we aren't able to rid ourselves of the implications of our symbolism, which
seems to admit of a question like "Where does the flame of a candle go to when it's blown out?" "Where does the
light go to?", "Where does the past go to"? We have become obsessed with our symbolism.--We may say that we
are led into puzzlement by an analogy which irresistibly drags us on.--And this also happens when the meaning of
the word "now" appears to us in a mysterious light. In our example 55) it appears that the function of "now" is in no
way comparable to the function of an expression like "five o'clock", "midday", "the time when the sun sets", etc.
This latter group of expressions I might call "specifications of times". But our ordinary language uses the word
"now" and specifications of time in similar contexts. Thus we say
"The sun sets at six o'clock".
"The sun is setting now".
We are inclined to say that both "now" and "six o'clock" 'refer to points of time'. This use of words produces a
puzzlement which one might express in the question "What is the 'now'?--for it is a moment of time and yet it can't
be said to be either the 'moment at which I speak' or 'the moment at which the clock strikes', etc., etc."--Our answer
is: The function of the word "now" is entirely different from that of a specification of time.--This can easily be seen if
we look at the role this word really plays in our usage of language, but it is obscured when instead of looking at the
whole language game, we only look at the contexts, the phrases of language in which the word is used. (The word
"today" is not a date, but it isn't anything like it either. It doesn't differ from a date as a hammer differs from a mallet,
but as a hammer differs from a nail; and surely we may say there is both a connection between a hammer and a
mallet and between a hammer and a nail.)
Page 108
One has been tempted to say that "now" is the name of an instant of
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time, and this, of course, would be like saying that "here" is the name of a place, "this" the name of a thing, and "I"
the name of a man. (One could, of course, also have said "a year ago" was the name of a time, "over there" the name
of a place, and "you" the name of a person.) But nothing is more unlike than the use of the word "this" and the use
of a proper name--I mean the games played with these words, not the phrases in which they are used. For we do say
"This is short" and "Jack is short"; but remember that "This is short" without the pointing gesture and without the
thing we are pointing to would be meaningless.--What can be compared with a name is not the word "this" but, if
you like, the symbol consisting of this word, the gesture, and the sample. We might say: Nothing is more
characteristic of a proper name A than that we can use it in such a phrase as "This is A"; and it makes no sense to
say "This is this" or "Now is now" or "Here is here".
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The idea of a proposition saying something about what will happen in the future is even more liable to puzzle
us than the idea of a proposition about the past. For comparing future events with past events, one may almost be
inclined to say that though the past events do not really exist in the full light of day, they exist in an underworld into
which they have passed out of the real life; whereas the future events do not even have this shadowy existence. We
could, of course, imagine a realm of the unborn, future events, whence they come into reality and pass into the realm
of the past; and, if we think in terms of this metaphor, we may be surprised that the future should appear less
existent than the past. Remember, however, that the grammar of our temporal expressions is not symmetrical with
respect to an origin corresponding with the present moment. Thus the grammar of the expressions relating to
memory does not reappear 'with opposite sign' in the grammar of the future tense. This is the reason why it has been
said that propositions concerning future events are not really propositions. And to say this is all right as long as it
isn't meant to be more than a decision about the use of the term "proposition"; a decision which, though not
agreeing with the common usage of the word "proposition", may come natural to human beings under certain
circumstances. If a philosopher says that propositions about the future are not real propositions, it is because he has
been struck by the asymmetry in the grammar of temporal expressions. The danger is, however, that he imagines he
has made a kind of scientific statement about 'the nature of the future'.
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Page 110
57). A game is played in this way: A man throws a die, and before throwing he draws on a piece of paper
some one of the six faces of the die. If, after having thrown, the face of the die turning up is the one he has drawn, he
feels (expresses) satisfaction. If a different face turns up, he is dissatisfied. Or, let there be two partners and every
time one guesses correctly what he will throw his partner pays him a penny, and if incorrectly, he pays his partner.
Drawing the face of the die will under the circumstances of this game be called "making a guess" or "a conjecture".
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58). In a certain tribe contests are held in running, putting the weight, etc., and the spectators stake
possessions on the competitors. The pictures of all the competitors are placed in a row, and what I called the
spectator's staking property on one of the competitors consists in laying this property (pieces of gold) under one of
the pictures. If a man has placed his gold under the picture of the winner in the competition he gets back his stake
doubled. Otherwise he loses his stake. Such a custom we should undoubtedly call betting, even if we observed it in a
society whose language held no scheme for stating 'degrees of probability', 'chances' and the like. I assume that the
behaviour of the spectators expresses great keenness and excitement before and after the outcome of the bet is
known. I further imagine that on examining the placing of the bets I can understand 'why' they were thus placed. I
mean: In a competition between two wrestlers, mostly the bigger man is the favourite; or if the smaller, I find that he
has shown greater strength on previous occasions, or that the bigger had recently been ill, or had neglected his
training, etc. Now this may be so although the language of the tribe does not express reasons for the placing of the
bets. That is to say, nothing in their language corresponds to our saying, e.g., "I bet on this man because he has kept
fit, whereas the other has neglected his training", and such like. I might describe this state of affairs by saying that
my observation has taught me certain causes for their placing their bets as they do, but that the bettors used no
reasons for acting as they did.
Page 110
The tribe may, on the other hand, have a language which comprises 'giving reasons'. Now this game of giving
the reason why one acts in a particular way does not involve finding the causes of one's actions (by frequent
observations of the conditions under which they arise). Let us imagine this:
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59). If a man of our tribe has lost his bet and upon this is chaffed or scolded, he points out, possibly
exaggerating, certain features of the man on whom he has laid his bet. One can imagine a discussion of pros and
cons going on in this way: two people pointing out alternately certain features of the two competitors whose
chances, as we should say, they are discussing; A pointing with a gesture to the great height of the one, B in answer
to this shrugging his shoulders and pointing to the size of the other's biceps, and so on. I could easily add more
details which would make us say that A and B are giving reasons for laying a bet on one person rather than on the
other.
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Now one might say that giving reasons in this way for laying their bets certainly presupposes that they have
observed causal connections between the result of a fight, say, and certain features of the bodies of the fighters, or of
their training. But this is an assumption which, whether reasonable or not, I certainly have not made in the
description of our case. (Nor have I made the assumption that the bettors give reasons for their reasons.) We should
in a case like that just described not be surprised if the language of the tribe contained what we should call
expressions of degrees of belief, conviction, certainty. These expressions we could imagine to consist in the use of a
particular word spoken with different intonations, or a series of words. (I am not thinking, however, of the use of a
scale of probabilities.)--It is also easy to imagine that the people of our tribe accompany their betting by verbal
expressions which we translate into "I believe that so and so can beat so and so in wrestling," etc.
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60). Imagine in a similar way conjectures being made as to whether a certain load of gunpowder will be
sufficient to blast a certain rock, and the conjecture to be expressed in a phrase of the form "This quantity of
gunpowder can blast this rock".
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61). Compare with 60) the case in which the expression "I shall be able to lift this weight", is used as an
abbreviation for the conjecture "My hand holding this weight will rise if I go through the process (experience) of
'making an effort to lift it'". In the last two cases the word "can" characterized what we should call the expression of
a conjecture. (Of course I don't mean that we call the sentence a conjecture because it contains the word "can"; but
in calling a sentence a conjecture we referred to the role which the sentence played in the language game; and we
translate a word our tribe uses by "can" if
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"can" is the word we should use under the circumstances described.) Now it is clear that the use of "can" in 59), 60),
61) is closely related to the use of "can" in 46) to 49); differing, however in this, that in 46) to 49) the sentences
saying that something could happen were not expressions of conjecture. Now one might object to this by saying:
Surely we are only willing to use the word "can" in such cases as 46) to 49) because it is reasonable to conjecture in
these cases what a man will do in the future from the tests he has passed or from the state he is in.
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Now it is true that I have deliberately made up the cases 46) to 49) so as to make a conjecture of this kind
seem reasonable. But I have also deliberately made them up so as not to contain a conjecture. We can, if we like,
make the hypothesis that the tribe would never use such a form of expression as that used in 49), etc. if experience
had not shown them that... etc. But this is an assumption which, though possibly correct, is in no way presupposed
in the games 46) to 49) as I have actually described them.
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62). Let the game be this: A writes down a row of numbers. B watches him and tries to find a system in the
sequence of these numbers. When he has done so he says: "Now I can go on". This example is particularly
instructive because 'being able to go on' here seems to be something setting in suddenly in the form of a clearly
outlined event.--Suppose then that A had written down the row 1, 5, 11, 19, 29. At that point B shouts "Now I can
go on". What was it that happened when suddenly he saw how to go on? A great many different things might have
happened. Let us assume then that in the present case, while A wrote one number after the other, B busied himself
with trying out several algebraic formulae to see whether they fitted. When A had written "19" B had been led to try
the formula an = n2 + n - 1. A's writing 29 confirms his guess.
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63). Or, no formula came into B's mind. After looking at the growing row of numbers A was writing,
possibly with a feeling of tension and with hazy ideas floating in his mind, he said to himself the words "He's
squaring and always adding one more"; then he made up the next number of the sequence and found it to agree with
the numbers A then wrote down.--
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64). Or, the row A wrote down was 2, 4, 6, 8. B looks at it, and says "Of course I can go on", and continues
the series of even
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numbers. Or he says nothing, and just goes on. Perhaps when looking at the row 2, 4, 6, 8 which A had written
down, he had some sensation, or sensations, often accompanying such words as "That's easy!" A sensation of this
kind is, for instance, the experience of a slight, quick intake of breath, what one might call a slight start.
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Now, should we say that the proposition "B can continue the series", means that one of the occurrences just
described takes place? Isn't it dear that the statement "B can continue..." is not the same as the statement that the
formula an = n2 + n - 1 comes into B's mind? This occurrence might have been all that actually took place. (It is
dear, by the way, that it can make no difference to us here whether B has the experience of this formula appearing
before his mind's eye, or the experience of writing or speaking the formula, or of picking it out with his eyes from
amongst several formulae written down beforehand.) If a parrot had uttered the formula, we should not have said
that he could continue the series.--Therefore, we are inclined to say "to be able to..." must mean more than just
uttering the formula--and in fact more than any one of the occurrences we have described. And this, we go on,
shows that saying the formula was only a symptom of B's being able to go on, and that it was not the ability of going
on itself. Now what is misleading in this is that we seem to intimate that there is one peculiar activity, process, or
state called "being able to go on" which somehow is hidden from our eyes but manifests itself in those occurrents
which we call symptoms (as an inflammation of the mucous membranes of the nose produces the symptom of
sneezing). This is the way talking of symptoms, in this case, misleads us. When we say "Surely there must be
something else behind the mere uttering of the formula, as this alone we should not call 'being able to...' ", the word
"behind" here is certainly used metaphorically, and 'behind' the utterance of the formula may be the circumstances
under which it is uttered. It is true, "B can continue..." is not the same as to say "B says the formula...", but it doesn't
follow from this that the expression "B can continue..." refers to an activity other than that of saying the formula, in
the way in which "B says the formula" refers to the well-known activity. The error we are in is analogous to this:
Someone is told the word "chair" does not mean this particular chair I am pointing to, upon which he looks round
the room for the object which the word "chair" does denote. (The case would be even more a striking illustration if
he tried to look inside the chair in order to find the real meaning of the word "chair".) It is clear that when with
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reference to the act of writing or speaking the formula etc., we use the sentence "He can continue the series", this
must be because of some connection between writing down a formula and actually continuing the series. And the
connection in experience of these two processes or activities is clear enough. But this connection tempts us to
suggest that the sentence "B can continue..." means something like "B does something which, experience has shown
us, generally leads to his continuing the series". But does B, when he says "Now I can go on" really mean "Now I
am doing something which, as experience has shown us, etc., etc."? Do you mean that he had this phrase in his
mind or that he would have been prepared to give it as an explanation of what he had said? To say the phrase "B can
continue..." is correctly used when prompted by such occurrences as described in 62), 63), 64) but that these
occurrences justify its use only under certain circumstances (e.g. when experience has shown certain connections) is
not to say that the sentence "B can continue..." is short for the sentence which describes all these circumstances, i.e.
the whole situation which is the background of our game.
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On the other hand we should under certain circumstances be ready to substitute "B knows the formula", "B
has said the formula" for "B can continue the series". As when we ask a doctor "Can the patient walk?", we shall
sometimes be ready to substitute for this "Is his leg healed?"--"Can he speak?" under circumstances means "Is his
throat all right?", under others (e.g., if he is a small child) it means "Has he learned to speak?"-To the question "Can
the patient walk?", the doctor's answer may be "His leg is all right".--We use the phrase "He can walk, as far as the
state of his leg is concerned", especially when we wish to oppose this condition for his walking to some other
condition, say the state of his spine. Here we must beware of thinking that there is in the nature of the case
something which we might call the complete set of conditions, e.g., for his walking; so that the patient, as it were,
can't help walking, must walk, if all these conditions are fulfilled.
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We can say: The expression "B can continue the series" is used under different circumstances to make
different distinctions. Thus it may distinguish a) between the case when a man knows the formula and the case
when he doesn't; or b) between the case when a man knows the formula and hasn't forgotten how to write the
numerals of the decimal system, and the case when he knows the formula and has forgotten how to write the
numerals; or c) (as perhaps in 64)) between the case when a man is feeling his normal self and the case when he is
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still in a condition of shell-shock; or d) between the case of a man who has done this kind of exercise before and the
case of a man who is new at it. These are only a few of a large family of cases.
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The question whether "He can continue..." means the same as "He knows the formula" can be answered in
several different ways: We can say "They don't mean the same, i.e., they are not in general used as synonyms as,
e.g., the phrases 'I am well' and 'I am in good health'"; or we may say "Under certain circumstances 'He can
continue...' means he knows the formula". Imagine the case of a language (somewhat analogous to 49)) in which two
forms of expression, two different sentences, are used to say that a person's legs are in working order. The one form
of expression is exclusively used under circumstances when preparations are going on for an expedition, a walking
tour, or the like; the other is used in cases when there is no question of such preparations. We shall here be doubtful
whether to say the two sentences have the same meaning or different meanings. In any case the true state of affairs
can only be seen when we look into the detail of the usage of our expressions.--And it is clear that if in our present
case we should decide to say that the two expressions have different meanings, we shall certainly not be able to say
that the difference is that the fact which makes the second sentence true is a different one from the fact which makes
the first sentence true.
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We are justified in saying that the sentence "He can continue..." has a different meaning from this: "He
knows the formula". But we mustn't imagine that we can find a particular state of affairs 'which the first sentence
refers to', as it were on a plane above that on which the special occurrences (like knowing the formula, imagining
certain further terms, etc.) take place.
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Let us ask the following question: Suppose that, on one ground or another, B has said "I can continue the
series", but on being asked to continue it he had shown himself unable to do so should we say that this proved that
his statement, that he could continue, was wrong, or should we say that he was able to continue when he said he
was? Would B himself say "I see I was wrong", or "What I said was true, I could do it then but I can't now"?--There
are cases in which he would correctly say the one and cases in which he would correctly say the other. Suppose a)
when he said he could continue he saw the formula before his mind, but when he was asked to continue he found he
had forgotten it;--or, b) when he said he could continue he had said to himself the next five terms of the series, but
now finds
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that they don't come into his mind; or, c) before, he had continued the series calculating five more places, now he
still remembers these five numbers but has forgotten how he had calculated them;--or, d) he says "Then I felt I could
continue, now I can't";--or, e) "When I said I could lift the weight my arm didn't hurt, now it does"; etc.
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On the other hand we say "I thought I could lift this weight, but I see I can't", "I thought I could say this
piece by heart, but I see I was mistaken".
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These illustrations of the use of the word "can" should be supplemented by illustrations showing the variety
of uses we make of the terms "forgetting" and "trying", for these uses are closely connected with those of the word
"can". Consider these cases: a) Before, B had said to himself the formula, now, "he finds a complete blank there". b)
Before, he had said to himself the formula, now, for a moment he isn't sure 'whether it was 2n or 3n'. c) He has
forgotten a name and it is 'on the tip of his tongue'. Or, d) he is not certain whether he has ever known the name or
has forgotten it.
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Now look at the way in which we use the word "trying": a) A man is trying to open a door by pulling as hard
as he can. b) He is trying to open the door of a safe by trying to find the combination. c) He is trying to find the
combination by trying to remember it, or d) by turning the knobs and listening with a stethoscope. Consider the
various processes we call "trying to remember". Compare e) trying to move your finger against a resistance (e.g.,
when someone is holding it), and f) when you have intertwined the fingers of both hands in a particular way and feel
'you don't know what to do in order to make a particular finger move'.
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(Consider also the class of cases in which we say "I can do so and so but I won't": "I could if I tried"--e.g., lift
100 pounds; "I could if I wished"--e.g., say the alphabet.)
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One might perhaps suggest that the only case in which it is correct to say, without restriction, that I can do a
certain thing, is that in which while saying that I can do it, I actually do it, and that otherwise I ought to say "I can do
it as far as... is concerned". One may be inclined to think that only in the above case has a person given a real proof
of being able to do a thing.
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65). But if we look at a language game in which the phrase "I can..." is used in this way (i.e., a game in which
doing a thing is taken as the only justification for saying that one is able to do it),
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we see that there is not the metaphysical difference between this game and one in which other justifications are
accepted for saying "I can do so and so". A game of the kind 65), by the way, shows us the real use of the phrase "If
something happens it certainly can happen"; an almost useless phrase in our language. It sounds as though it had
some very clear and deep meaning, but like most of the general philosophical propositions it is meaningless except
in very special cases.
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66). Make this clear to yourself by imagining a language (similar to 49)) which has two expressions for such
sentences as "I am lifting a fifty pound weight"; one expression is used whenever the action is performed as a test
(say, before an athletic competition), the other expression is used when the action is not performed as a test.
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We see that a vast net of family likenesses connects the cases in which the expressions of possibility, "can",
"to be able to", etc. are used. Certain characteristic features, we may say, appear in these cases in different
combinations: there is, e.g., the element of conjecture (that something will behave in a certain way in the future); the
description of the state of something (as a condition for its behaving in a certain way in the future); the account of
certain tests someone or something has passed.--
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There are, on the other hand, various reasons which incline us to look at the fact of something being possible,
someone being able to do something, etc., as the fact that he or it is in a particular state. Roughly speaking, this
comes to saying that "A is in the state of being able to do something" is the form of representation we are most
strongly tempted to adopt; or, as one could also put it, we are strongly inclined to use the metaphor of something
being in a peculiar state for saying that something can behave in a particular way. And this way of representation, or
this metaphor, is embodied in the expressions "He is capable of...", "He is able to multiply large numbers in his
head", "He can play chess": in these sentences the verb is used in the present tense, suggesting that the phrases are
descriptions of states which exist at the moment when we speak.
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The same tendency shows itself in our calling the ability to solve a mathematical problem, the ability to enjoy
a piece of music, etc., certain states of the mind; we don't mean by this expression 'conscious mental phenomena'.
Rather, a state of the mind in this sense is the state of a hypothetical mechanism, a mind model meant to explain
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the conscious mental phenomena. (Such things as unconscious or subconscious mental states are features of the
mind model.) In this way also we can hardly help conceiving of memory as of a kind of storehouse. Note also how
sure people are that to the ability to add or to multiply or to say a poem by heart, etc., there must correspond a
peculiar state of the person's brain, although on the other hand they know next to nothing about such
psycho-physiological correspondences. We regard these phenomena as manifestations of this mechanism, and their
possibility is the particular construction of the mechanism itself.
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Now looking back to our discussion of 43), we see that it was no real explanation of B's being guided by the
signs when we said that B was guided if he could also have carried out orders consisting in other combinations of
dots and dashes than those of 43). In fact, when we considered the question whether B in 43) was guided by the
signs, we were all the time inclined to say some such thing as that we could only decide this question with certainty
if we could look into the actual mechanism connecting seeing the signs with acting according to them. For we have a
definite picture of what in a mechanism we should call certain parts being guided by others. In fact, the mechanism
which immediately suggests itself when we wish to show what in such a case as 43) we should call "being guided by
the signs" is a mechanism of the type of a pianola. Here, in the working of the pianola we have a clear case of certain
actions, those of the hammers of the piano, being guided by the pattern of the holes in the pianola roll. We could use
the expression "The pianola is reading off the record made by the perforations in the roll", and we might call
patterns of such perforations complex signs or sentences, opposing their function in a pianola to the function which
similar devices have in mechanisms of a different type, e.g., the combination of notches and teeth which form a key
bit. The bolt of a lock is caused to slide by this particular combination, but we should not say that the movement of
the bolt was guided by the way in which we combined teeth and notches, i.e., we should not say that the bolt moved
according to the pattern of the key bit. You see here the connection between the idea of being guided and the idea
of being able to read new combinations of signs; for we should say that the pianola can read any pattern of
perforations, of a particular kind, it is not built for one particular tune or set of tunes (like a musical box),-whereas
the bolt of the lock reacts to that pattern of the key bit only which is predetermined in the construction of the lock.
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We could say that the notches and teeth forming a key bit are not comparable to the words making up a sentence
but to the letters making up a word, and that the pattern of the key bit in this sense did not correspond to a complex
sign, to a sentence, but to a word.
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It is clear that although we might use the ideas of such mechanisms as similes for describing the way in
which B acts in the games 42) and 43), no such mechanisms are actually involved in these games. We shall have to
say that the use which we made of the expression "to be guided" in our examples of the pianola and of the lock is
only one use within a family of usages, though these examples may serve as metaphors, ways of representation, for
other usages.
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Let us study the use of the expression "to be guided" by studying the use of the word "reading". By
"reading" I here mean the activity of translating script into sounds, also of writing according to dictation or of
copying in writing a page of print, and suchlike; reading in this sense does not involve any such thing as
understanding what you read. The use of the word "reading" is, of course, extremely familiar to us in the
circumstances of our ordinary life (it would be extremely difficult to describe these circumstances even roughly). A
person, say an Englishman, has as a child gone through one of the normal ways of training in school or at home, he
has learned to read his language, later on he reads books, newspapers, letters, etc. What happens when he reads the
newspaper?--His eyes glide along the printed words, he pronounces them aloud or to himself, but he pronounces
certain words just taking their pattern in as a whole, other words he pronounces after having seen their first few
letters only, others again he reads out letter by letter. We should also say that he had read a sentence if while letting
his eyes glide along it he had said nothing aloud or to himself, but on being asked afterwards what he had read he
was able to reproduce the sentence verbatim or in slightly different words. He may also act as what we might call a
mere reading machine, I mean, paying no attention to what he spoke, perhaps concentrating his attention on
something totally different. We should in this case say that he read if he acted faultlessly like a reliable
machine.--Compare with this case the case of a beginner. He reads the words by spelling them out painfully. Some
of the words, however, he just guesses from their contexts, or possibly he knows the piece by heart. The teacher
then says that he is pretending to read the words, or just that he is not really reading them. If, looking at this
example, we asked ourselves what reading was, we should be
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inclined to say that it was a particular conscious mental act. This is the case in which we say "Only he knows
whether he is reading; nobody else can really know it". Yet we must admit that as far as the reading of a particular
word goes, exactly the same thing might have happened in the beginner's mind when he 'pretended' to read as what
happened in the mind of the fluent reader when he read the word. We are using the word "reading" in a different
way when we talk about the accomplished reader on the one hand and the beginner on the other hand. What in the
one case we call an instance of reading we don't call an instance of reading in the other.--Of course we are inclined to
say that what happened in the accomplished reader and in the beginner when they pronounced the word could not
have been the same. The difference lying, if not in their conscious states, then in the unconscious regions of their
minds, or in their brains. We here imagine two mechanisms, the internal working of which we can see, and this
internal working is the real criterion for a person's reading or not reading. But in fact no such mechanisms are known
to us in these cases. Look at it in this way:
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67). Imagine that human beings or animals were used as reading machines; assume that in order to become
reading machines they need a particular training. The man who trains them says of some of them that they already
can read, of others that they can't. Take a case of one who has so far not responded to the training. If you put before
him a printed word he will sometimes make sounds, and every now and then it happens 'accidentally' that these
sounds more or less correspond to the printed word. A third person hears the creature under training uttering the
right sound on looking at the word "table". The third person says "He's reading", but the teacher answers "No, he
isn't, it is mere accident". But supposing now that the pupil on being shown other words and sentences goes on
reading them correctly. After a time the teacher says "Now he can read".--But what about the first word "table"?
Should the teacher say "I was wrong. He read that, too"?, or should he say "No, he only started reading later"?
When did he really begin to read, or: Which was the first word, or the first letter, which he read? It is clear that this
question here makes no sense unless I give an 'artificial' explanation such as: "The first word which he reads = the
first word of the first hundred consecutive words he reads correctly."--Suppose on the other hand that we used the
word "reading" to distinguish between the case when a particular conscious
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process of spelling out the words takes place in a person's mind from the case in which this does not happen.--Then,
at least the person who is reading could say that such and such a word was the first which he actually read.--Also, in
the different case of a reading machine which is a mechanism connecting signs with the reactions to these signs, e.g.,
a pianola, we could say "Only after such and such a thing had been done to the machine, e.g., certain parts had been
connected by wires, the machine actually read; the first letter which it read was a d".--
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In the case 67), by calling certain creatures "reading machines" we meant only that they react in a particular
way to seeing printed signs. No connection between seeing and reacting, no internal mechanism enters into this case.
It would be absurd if the trainer had answered to the question whether he read the word "table" or not, "Perhaps he
read it", for there is no doubt in this case about what he actually did. The change which took place was one which
we might call a change in the general behaviour of the pupil, and we have in this case not given a meaning to the
expression "the first word in the new era". (Compare with this the following case:
................ . . . . . . . .
In our figure a row of dots with large intervals succeeds a row of dots with small intervals. Which is the last dot in
the first sequence and which the first dot in the second? Imagine our dots were holes in the revolving disc of a siren.
Then we should hear a tone of low pitch following a tone of high pitch (or vice versa). Ask yourself: At which
moment does the tone of low pitch begin and the other end?)
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There is a great temptation on the other hand to regard the conscious mental act as the only real criterion
distinguishing reading from not reading. For we are inclined to say "Surely a man always knows whether he is
reading or pretending to read", or "Surely a man always knows when he is really reading". If A tries to make B
believe that he is able to read Cyrillic script, cheating him by learning a Russian sentence by heart and then saying it
while looking at the printed sentence, we may certainly say that A knows that he is pretending and that his not
reading in this case is characterized by a particular personal experience, namely, that of saying the sentence by heart.
Also, if A makes a slip in saying it by heart, this experience will be different from that which a person has who
makes a slip in reading.
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68). But supposing now that a man who could read fluently and who was made to read sentences which he
had never read before read these sentences, but all the time with the peculiar feeling of knowing the sequence of
words by heart. Should we in this case say that he was not reading, i.e., should we regard his personal experience as
the criterion distinguishing between reading and not reading?
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69). Or imagine this case: A man under the influence of a certain drug is shown a group of five signs, not
letters of an existing alphabet; and looking at them with all the outward signs and personal experiences of spelling
out a word, pronounces the word "ABOVE". (This sort of thing happens in dreams. After waking up we then say, "It
seemed to me that I was reading these signs though they weren't really signs at all".) In such a case some people
might be inclined to say that he is reading, others that he isn't. We could imagine that after he had spelt out the word
"above" we showed him other combinations of the five signs and that he read them consistently with his reading of
the first permutation of signs shown to him. By a series of similar tests we might find that he used what we might
call an imaginary alphabet. If this was so, we should be more ready to say "He is reading" than "He imagines that he
reads, but he doesn't really".
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Note also that there is a continuous series of intermediary cases between the case when a person knows by
heart what is in print before him, and the case in which he spells out the letters of every word without any such help
as guessing from the context, knowing by heart, and such like.
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Do this: Say by heart the series of cardinals from one to twelve.--Now look at the dial of your watch and
read this sequence of numbers. Ask yourself what in this case you called reading, that is, what did you do to make it
reading?
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Let us try this explanation: A person reads if he derives the copy which he is producing from the model
which he is copying. (I will use the word "model" to mean that which he is reading off, e.g., the printed sentences
which he is reading or copying in writing, or such signs as "-- -- • • --" in 42) and 43) which he is "reading" by his
movements, or the scores which a pianist plays off, etc. The word "copy" I use for the sentence spoken or written
from the printed one, for the movements made according to such signs as "-- -- • • --", for the movements of the
pianist's fingers or the tune which he plays from the scores, etc.) Thus if we had taught a person the Cyrillic alphabet
and had
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taught him how each letter was pronounced, if then we gave him a piece printed in the Cyrillic script and he spelt it
out according to the pronunciation of each letter as we had taught it, we should undoubtedly say that he was
deriving the sound of every word from the written and spoken alphabet taught him. And this also would be a clear
case of reading. (We might use the expression, "We have taught him the rule of the alphabet".)
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But let us see; what made us say that he derived the spoken words from the printed by means of the rule of
the alphabet? Isn't all we know that we told him that this letter was pronounced this way, that letter that way, etc.,
and that he afterwards read out words in the Cyrillic script? What suggests itself to us as an answer is that he must
have shown somehow that he did actually make the transition from the printed to the spoken words by means of the
rule of the alphabet which we had given him. And what we mean by his showing this will certainly get clearer if we
alter our example and:
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70) assume that he reads off a text by transcribing it, say, from block letters into cursive script. For in this
case we can assume the rule of the alphabet to have been given in the form of a table which shows the block
alphabet and the cursive alphabet in parallel columns. Then the deriving the copy from the text we should imagine
this way: The person who copies looks up the table for each letter at frequent intervals, or he says to himself such
things as, "Now what's a small a like?", or he tries to visualize the table, refraining from actually looking at it.--
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71). But what if, doing all this, he then transcribed an "A" into a "b", a "B" into a "c", and so on? Should we
not call this "reading", "deriving", too? We might in this case describe his procedure by saying that he used the table
as we should have used it had we not looked straight from left to right like this:
but like this:
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though he actually when looking up the table passed with his eyes or finger horizontally from left to right.--But let us
suppose now
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72) that going through the normal process of looking up, he transcribed an "A" into an "n", a "B" into an "x",
in short, acted, as we might say, according to a scheme of arrows which showed no simple regularity. Couldn't we
call this "deriving" too?--But suppose that
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73) he didn't stick to this way of transcribing. In fact he changed it, but according to a simple rule: After
having transcribed "A" into "n", he transcribes the next "A" into "o", and the next "A" into "p", and so on. But where
is the sharp line between this procedure and that of producing a transcription without any system at all? Now you
might object to this by saying "In the case 71), you obviously assumed that he understood the table differently; he
didn't understand it in the normal way". But what do we call "understanding the table in a particular way?" But
whatever process you imagine this 'understanding' to be, it is only another link interposed between the outward and
inward processes of deriving I have described and the actual transcription. In fact this process of understanding
could obviously be described by means of a schema of the kind used in 71), and we could then say that in a
particular case he looked up the table like this:
understood the table like this:
and transcribed it like this:
But does this mean that the word "deriving" (or "understanding") has really no meaning, as by following up its
meaning this seems to trail off into nothing? In case 70) the meaning of "deriving" stood out quite clearly, but we
told ourselves that this was only one special case of deriving. It seemed to us that the essence of the
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process of deriving was here presented in a particular dress and that by stripping it of this we should get at the
essence. Now in 71), 72), 73) we tried to strip our case of what had seemed but its peculiar costume only to find that
what had seemed mere costumes were the essential features of the case. (We acted as though we had tried to find
the real artichoke by stripping it of its leaves.) The use of the word "deriving" is indeed exhibited in 70), i.e., this
example showed us one of the family of cases in which this word is used. And the explanation of the use of this
word, as that of the use of the word "reading" or "being guided by symbols", essentially consists in describing a
selection of examples exhibiting characteristic features, some examples showing these features in exaggeration,
others showing transitions, certain series of examples showing the trailing off of such features. Imagine that
someone wished to give you an idea of the facial characteristics of a certain family, the So and so's, he would do it
by showing you a set of family portraits and by drawing your attention to certain characteristic features, and his
main task would consist in the proper arrangement of these pictures, which, e.g., would enable you to see how
certain influences gradually changed the features, in what characteristic ways the members of the family aged, what
features appeared more strongly as they did so.
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It was not the function of our examples to show us the essence of 'deriving', 'reading', and so forth through a
veil of inessential features; the examples were not descriptions of an outside letting us guess at an inside which for
some reason or other could not be shown in its nakedness. We are tempted to think that our examples are indirect
means for producing a certain image or idea in a person's mind,--that they hint at something which they cannot
show. This would be so in some such case as this: Suppose I wish to produce in someone a mental image of the
inside of a particular eighteenth-century room which he is prevented from entering. I therefore adopt this method: I
show him the house from the outside, pointing out the windows of the room in question, I further lead him into
other rooms of the same period.--
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Our method is purely descriptive; the descriptions we give are not hints of explanations.
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II
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1. Do we have a feeling of familiarity whenever we look at familiar objects? Or do we have it usually?
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When do we actually have it?
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It helps us to ask: What do we contrast the feeling of familiarity with?
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One thing we contrast it with is surprise.
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One could say: "Unfamiliarity is much more of an experience than familiarity".
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We say: A shows B a series of objects. B is to tell A whether the object is familiar to him or not. a) The
question may be "Does B know what the objects are?" or b) "Does he recognize the particular object?"
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1). Take the case that B is shown a series of apparatus--a balance, a thermometer, a spectroscope, etc.
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2). B is shown a pencil, a pen, an inkpot, and a pebble. Or:
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3). Besides familiar objects he is shown an object of which he says "That looks as though it served some
purpose, but I don't know what purpose".
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What happens when B recognizes something as a pencil?
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Suppose A had shown him an object looking like a stick. B handles this object, suddenly it comes apart, one
of the parts being a cap, the other a pencil. B says "Oh, this is a pencil". He has recognized the object as a pencil.
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4). We could say "B always knew what a pencil looked like; he could, e.g., have drawn one on being asked
to. He didn't know that the object he was given contained a pencil which he could have drawn any time". Compare
with this case 5):
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5). B is shown a word written on a piece of paper held upside down. He does not recognize the word. The
paper is gradually turned round until B says "Now I see what it is. It is 'pencil'".
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We might say "He always knew what the word 'pencil' looked like. He did not know that the word he was
shown would, when turned round, look like 'pencil'".
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In both cases 4) and 5) you might say something was hidden. But note the different application of "hidden".
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6). Compare with this: You read a letter and can't read one of its words. You guess what it must be from the
context, and now can read it. You recognize this scratch as an e, the second as an a, the third as a t. This is different
from the case where the word "eat" was covered by a blotch of ink, and you only guessed that the word "eat" must
have been in this place.
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7). Compare: You see a word and can't read it. Someone alters it slightly by adding a dash, lengthening a
stroke, or such like. Now you can read it. Compare this alteration with the turning in 5), and note that there is a sense
in which while the word was turned round you saw that it was not altered. I.e., there is a case in which you say "I
looked at the word while it was turned, and I know that it is the same now as it was when I didn't recognize it".
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8). Suppose the game between A and B just consisted in this, that B should say whether he knows the object
or not but does not say what it is. Suppose he was shown an ordinary pencil, after having been shown a hygrometer
which he had never seen before. On being shown the hygrometer he said that he was not familiar with it, on being
shown the pencil, that he knew it. What happened when he recognized it? Must he have told himself, though he
didn't tell A, that what he saw was a pencil? Why should we assume this?
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Then, when he recognized the pencil, what did he recognize it as? 9). Suppose even that he had said to
himself "Oh, this is a pencil", could you compare this case with 4) or 5)? In these cases one might have said "He
recognized this as that" (pointing, e.g., for "this" to the covered up pencil and for "that" to an ordinary pencil, and
similarly in 5)).
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In 8) the pencil underwent no change and the words "Oh, this is a pencil" did not refer to a paradigm, the
similarity of which with the pencil shown B had recognized.
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Asked "What is a pencil?", B would not have pointed to another object as the paradigm or sample, but could
straight away have pointed to the pencil shown to him.
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"But when he said 'Oh, this is a pencil', how did he know that it was if he didn't recognize it as
something?"--This really comes to saying "How did he recognize 'pencil' as the name of this sort of thing?" Well,
how did he recognize it? He just reacted in this particular way by saying this word.
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10). Suppose someone shows you colours and asks you to name them.
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Pointing to a certain object you say "This is red". What would you answer if you were asked "How do you know
that this is red?"?
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Of course there is the case in which a general explanation was given to B, say, "We shall call 'pencil' anything
that one can easily write with on a wax tablet." Then A shows B amongst other objects a small pointed object, and B
says "Oh, this is a pencil", after having thought "One could write with this quite easily". In this case, we may say, a
derivation takes place. In 8), 9), 10) there is no derivation. In 4) we might say that B derived that the object shown to
him was a pencil by means of a paradigm, or else no such derivation might have taken place.
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Now should we say that B on seeing the pencil after seeing instruments which he didn't know had a feeling
of familiarity? Let us imagine what really might have happened. He saw a pencil, smiled, felt relieved, and the name
of the object he saw came into his mind or mouth.
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Now isn't the feeling of relief just that which characterizes the experience of passing from unfamiliar to
familiar things?
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2. We say we experience tension and relaxation, relief, strain and rest in cases as different as these: A man
holds a weight with outstretched arm; his arm, his whole body is in a state of tension. We let him put down the
weight, the tension relaxes. A man runs, then rests. He thinks hard about the solution of a problem in Euclid, then
finds it, and relaxes. He tries to remember a name, and relaxes on finding it.
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What if we asked "What do all these cases have in common that makes us say that they are cases of strain
and relaxation?"?
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What makes us use the expression "seeking in our memory", when we try to remember a word?
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Let us ask the question "What is the similarity between looking for a word in your memory and looking for
my friend in the park?" What would be the answer to such a question?
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One kind of answer certainly would consist in describing a series of intermediate cases. One might say that
the case which looking in your memory for something is most similar to is not that of looking for my friend in the
park, but, say, that of looking up the spelling of a word in a dictionary. And one might go on interpolating cases.
Another way of pointing out the similarity would be to say, e.g., "In
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both these cases at first we can't write down the word and then we can". This is what we call pointing out a common
feature.
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Now it is important to note that we needn't be aware of such similarities thus pointed out when we are
prompted to use the words "seeking", "looking for", etc., in the case of trying to remember.
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One might be inclined to say "Surely a similarity must strike us, or we shouldn't be moved to use the same
word".--Compare that statement with this: "A similarity between these cases must strike us in order that we should
be inclined to use the same picture to represent both". This says that some act must precede the act of using this
picture. But why shouldn't what we call "the similarity striking us" consist partially or wholly in our using the same
picture? And why shouldn't it consist partially or wholly in our being prompted to use the same phrase?
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We say: "This picture (or this phrase) suggests itself to us irresistibly". Well, isn't this an experience?
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We are treating here of cases in which, as one might roughly put it, the grammar of a word seems to suggest
the 'necessity' of a certain intermediary step, although in fact the word is used in cases in which there is no such
intermediary step. Thus we are inclined to say: "A man must understand an order before he obeys it", "He must
know where his pain is before he can point to it", "He must know the tune before he can sing it", and suchlike.
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Let us ask the question: Suppose I had explained to someone the word "red" (or the meaning of the word
"red") by having pointed to various red objects and given the ostensive explanation.--What does it mean to say
"Now if he has understood the meaning, he will bring me a red object if I ask him to"? This seems to say: If he has
really got hold of what is in common between all the objects I have shown him, he will be in the position to follow
my order. But what is it that is in common to these objects?
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Could you tell me what is in common between a light red and a dark red? Compare with this the following
case: I show you two pictures of two different landscapes. In both pictures, amongst many other objects, there is the
picture of a bush, and it is exactly alike in both. I ask you "Point to what these two pictures have in common", and as
an answer you point to this bush.
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Now consider this explanation: I give someone two boxes containing various things, and say "The object
which both boxes have in common is called a toasting fork". The person I give this explanation has to
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sort out the objects in the two boxes until he finds the one they have in common, and thereby, we may say, he
arrives at the ostensive explanation. Or, this explanation: "In these two pictures you see patches of many colours;
the one colour which you find in both is called 'mauve'".--In this case it makes a dear sense to say "If he has seen (or
found) what is in common between these two pictures, he can now bring me a mauve object".
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There is also this case: I say to someone "I shall explain to you the word 'w' by showing you various objects.
What's in common to them all is what 'w' means." I first show him two books, and he asks himself "Does 'w' mean
'book'?" I then point to a brick, and he says to himself "Perhaps 'w' means 'parallelepiped'". Finally I point to glowing
coal, and he says to himself "Oh, it's 'red' he means, for all these objects had something red about them". It would be
interesting to consider another form of this game where the person has at each stage to draw or paint what he thinks
I mean. The interest of this version lies in this, that in some cases it would be quite obvious what he has got to draw,
say, when he sees that all the objects I have shown him so far bear a certain trademark (he'd draw the trade
mark).--What, on the other hand, should be paint if he recognizes that there is something red about each object? A
red patch? And of what shape and shade? Here a convention would have to be laid down, say, that painting a red
patch with ragged edges does not mean that the objects have that red patch with ragged edges in common, but
something red.
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If, pointing to patches of various shades of red, you asked a man "What have these in common that makes
you call them red?", he'd be inclined to answer "Don't you see?" And this of course would not be pointing out a
common element.
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There are cases where experience teaches us that a person is not able to carry out an order, say, of the form
"Bring me x" if he did not see what was in common between the various objects to which I pointed as an
explanation of "x". And 'seeing what they have in common' in some cases consisted in pointing to it, in letting one's
glance rest on a coloured patch after a process of scrutiny and comparing, in saying to oneself "Oh, it's red he
means", and perhaps at the same time glancing at all the red patches on the various objects, and so on.--There are
cases, on the other hand, in which no process takes place comparable with this intermediary 'seeing what's in
common', and where we still use this phrase, though this time we ought to say
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"If after showing him these things he brings me another red object, then I shall say that he has seen the common
feature of the objects I showed him". Carrying out the order is now the criterion for his having understood.
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3. 'Why do you call "strain" all these different experiences?'--'Because they have some element in
common.'-'What is it that bodily and mental strain have in common?'--'I don't know, but obviously there is some
similarity.'
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Then why did you say the experiences had something in common? Didn't this expression just compare the
present case with those cases in which we primarily say that two experiences have something in common? (Thus we
might say that some experiences of joy and of fear have the feeling of heart-beat in common.) But when you said
that the two experiences of strain had something in common, these were only different words for saying that they
were similar. It was then no explanation to say that the similarity consisted in the occurrence of a common element.
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Also, shall we say that you had a feeling of similarity when you compared the two experiences, and that this
made you use the same word for both? If you say you have a feeling of similarity, let us ask a few questions about it:
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Could you say the feeling was located here or there?
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When did you actually have this feeling? For, what we call comparing the two experiences is quite a
complicated activity: perhaps you called the two experiences before your mind, and imagining a bodily strain, and
imagining a mental strain, was each in itself imagining a process and not a state uniform through time. Then ask
yourself at what time during all this you had the feeling of similarity.
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'But surely I wouldn't say they are similar if I had no experience of their similarity.'--But must this experience
be anything you should call a feeling? Suppose for a moment it were the experience that the word "similar"
suggested itself to you. Would you call this a feeling?
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'But is there no feeling of similarity?'--I think there are feelings which one might call feelings of similarity. But
you don't always have any such feeling if you 'notice similarity'. Consider some of the different experiences which
you have if you do so.
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a) There is a kind of experience which one might call being hardly able to distinguish. You see, e.g., two
lengths, two colours, almost exactly alike. But if I ask myself "Does this experience consist in
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having a peculiar feeling?", I should have to say that it certainly isn't characterized by any such feeling alone, that a
most important part of the experience is that of letting my glance oscillate between the two objects, fixing it intently
now on the one, now on the other, perhaps saying words expressive of doubt, shaking my head, etc., etc. There is,
one might say, hardly any room left for a feeling of similarity between these manifold experiences.
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b) Compare with this the case in which it is impossible to have any difficulty in distinguishing the two
objects. Supposing I say "I like to have the two kinds of flowers in this bed of similar colours to avoid a strong
contrast". The experience here might be one which one may describe as an easy sliding of the glance from one to the
other.
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c) I listen to a variation on a theme and say "I don't see yet how this is a variation of the theme, but I see a
certain similarity". What happened was that at certain points of the variation, at certain turning points of the key, I
had an experience of 'knowing where I was in the theme'. And this experience might again have consisted in
imagining certain figures of the theme, or in seeing them written before my mind or in actually pointing to them in
the score, etc.
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'But when two colours are similar, the experience of similarity should surely consist in noticing the similarity
which there is between them'.--But is a bluish green similar to a yellowish green or not? In certain cases we should
say they are similar and in others that they are most dissimilar. Would it be correct to say that in the two cases we
noticed different relations between them? Suppose I observed a process in which a bluish green gradually changed
into a pure green, into a yellowish green, into yellow, and into orange. I say "It only takes a short time from bluish
green to yellowish green, because these colours are similar".--But mustn't you have had some experience of
similarity to be able to say this?--The experience may be this, of seeing the two colours and saying that they are both
green. Or it may be this, of seeing a band whose colour changes from one end to the other in the way described, and
having some one of the experiences which one may call noticing how close to each other bluish green and yellowish
green are, compared to bluish green and orange.
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We use the word "similar" in a huge family of cases.
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There is something remarkable about saying that we use the word "strain" for both mental and physical strain
because there is a similarity between them. Should you say we use the word "blue" both for light blue and dark blue
because there is a similarity between them? If
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you were asked "Why do you call this 'blue' also?", you would say "Because this is blue, too".
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One might suggest that the explanation is that in this case you call 'blue" what is in common between the two
colours, and that, if you called 'strain" what was in common between the two experiences of strain, it would have
been wrong to say "I called them both 'strain' because they had a certain similarity", but that you would have had to
say "I used the word 'strain' in both cases because there is a strain present in both".
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Now what should we answer to the question "What do light blue and dark blue have in common?"? At first
sight the answer seems obvious: "They are both shades of blue". But this is really a tautology. So let us ask "What
do these colours I am pointing to have in common?" (Suppose one is light blue, the other dark blue.) The answer to
this really ought to be "I don't know what game you are playing". And it depends upon this game whether I should
say they had anything in common, and what I should say they had in common.
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Imagine this game: A shows B different patches of colours and asks him what they have in common. B is to
answer by pointing to a particular primary colour. Thus if A points to pink and orange, B is to point to pure red. If A
points to two shades of greenish blue, B is to point to pure green and pure blue, etc. If in this game A showed B a
light blue and a dark blue and asked what they had in common, there would be no doubt about the answer. If then
he pointed to pure red and pure green, the answer would be that these have nothing in common. But I could easily
imagine circumstances under which we should say that they had something in common and would not hesitate to
say what it was: Imagine a use of language (a culture) in which there was a common name for green and red on the
one hand and yellow and blue on the other. Suppose, e.g., that there were two castes, one the patrician caste,
wearing red and green garments the other, the plebeian, wearing blue and yellow garments. Both yellow and blue
would always be referred to as plebeian colours, green and red as patrician colours. Asked what a red patch and a
green patch have in common, a man of our tribe would not hesitate to say they were both patrician.
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We could also easily imagine a language (and that means again a culture) in which there existed no common
expression for light blue and dark blue, in which the former, say, was called "Cambridge",
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the latter "Oxford". If you ask a man of this tribe what Cambridge and Oxford have in common, he'd be inclined to
say "Nothing".
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Compare this game with the one above: B is shown certain pictures, combinations of coloured patches. On
being asked what these pictures have in common, he is to point to a sample of red, say, if there is a red patch in both,
to green if there is a green patch in both, etc. This shows you in what different ways this same answer may be used.
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Consider such an explanation as "I mean by 'blue' what these two colours have in common".--Now isn't it
possible that someone should understand this explanation? He would, e.g., on being ordered to bring another blue
object, carry out this order satisfactorily. But perhaps he will bring a red object and we shall be inclined to say: "He
seems to notice some sort of similarity between samples we showed him and that red thing".
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Note: Some people when asked to sing a note which we strike for them on the piano, regularly sing the fifth
of that note. That makes it easy to imagine that a language might have one name only for a certain note and its fifth.
On the other hand we should be embarrassed to answer the question: What do a note and its fifth have in common?
For of course it is no answer to say: "They have a certain affinity".
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It is one of our tasks here to give a picture of the grammar (the use) of the word "a certain".
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To say that we use the word "blue" to mean 'what all these shades of colour have in common' by itself says
nothing more than that we use the word "blue" in all these cases.
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And the phrase "He sees what all these shades have in common", may refer to all sorts of different
phenomena, i.e., all sorts of phenomena are used as criteria for 'his seeing that...'. Or all that happens may be that on
being asked to bring another shade of blue he carries out our order satisfactorily. Or a patch of pure blue may appear
before his mind's eye when we show him the different samples of blue: or he may instinctively turn his head towards
some other shade of blue which we haven't shown him for sample, etc., etc.
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Now should we say that a mental strain and a bodily strain were 'strains' in the same sense of the word or in
different (or 'slightly different') senses of the word?--There are cases of this sort in which we should not be doubtful
about the answer.
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4. Consider this case: We have taught someone the use of the words "darker" and "lighter". He could, e.g.,
carry out such an order
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as "Paint me a patch of colour darker than the one I am showing you". Suppose now I said to him: "Listen to the five
vowels a, e, i, o, u and arrange them in order of their darkness". He may just look puzzled and do nothing, but he
may (and some people will) now arrange the vowels in a certain order (mostly i, e, a, o, u). Now one might imagine
that arranging the vowels in order of darkness presupposed that when a vowel was sounded a certain colour came
before a man's mind, that he then arranged these colours in their order of darkness and told you the corresponding
arrangement of the vowels. But this actually need not happen. A person will comply with the order: "Arrange the
vowels in their order of darkness", without seeing any colours before his mind's eye.
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Now if such a person was asked whether u was 'really' darker than e, he would almost certainly answer some
such thing as "It isn't really darker, but it somehow gives me a darker impression".
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But what if we asked him "What made you use the word 'darker' in this case at all?"?
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Again we might be inclined to say "He must have seen something that was in common both to the relation
between two colours and to the relation between two vowels". But if he isn't capable of specifying what this
common element was, this leaves us just with the fact that he was prompted to use the words "darker", "lighter" in
both these cases.
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For, note the word "must" in "He must have seen something...". When you said that, you didn't mean that
from past experience you conclude that he probably did see something, and that's just why this sentence adds
nothing to what we know, and in fact only suggests a different form of words to describe it.
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If someone said: "I do see a certain similarity, only I can't describe it", I should say: "This itself characterizes
your experience".
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Suppose you look at two faces and say "They are similar, but I don't know what it is that's similar about
them". And suppose that after a while you said: "Now I know; their eyes have the same shape", I should say "Now
your experience of their similarity is different from what it was when you saw similarity and didn't know what it
consisted in". Now to the question "What made you use the word 'darker'...?" the answer may be "Nothing made me
use the word 'darker',--that is, if you ask me for a reason why I use it. I just used it, and what is more, I used it with
the same intonation of voice, and perhaps with the same facial expression and gesture, which I should in certain
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cases be inclined to use when applying the word to colours".--It is easier to see this when we speak of a deep sorrow,
a deep sound, a deep well. Some people are able to distinguish between fat and lean days of the week. And their
experience when they conceive a day as a fat one consists in applying this word together perhaps with a gesture
expressive of fatness and a certain comfort.
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But you may be tempted to say: This use of the word and gesture is not their primary experience. First of all
they have to conceive the day as fat and then they express this conception by word or gesture.
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But why do you use the expression "They have to"? Do you know of an experience in this case which you
call "the conception, etc."? For if you don't, isn't it just what one might call a linguistic prejudice that made you say
"He had to have a conception before... etc."?
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Rather, you can learn from this example and from others that there are cases in which we may call a
particular experience "noticing, seeing, conceiving that so and so is the case", before expressing it by word or
gestures, and that there are other cases in which if we talk of an experience of conceiving at all, we have to apply this
word to the experience of using certain words, gestures, etc.
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When the man said "u isn't really darker than e... ", it was essential that he meant to say that the word
"darker" was used in different senses when one talked of one colour being darker than another and, on the other
hand, of one vowel being darker than another.
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Consider this example: Suppose we had taught a man to use the words "green", "red", "blue" by pointing to
patches of these colours. We had taught him to fetch us objects of a certain colour on being ordered "Bring me
something red!", to sort out objects of various colours from a heap, and such like. Suppose we now show him a
heap of leaves, some of which are a slightly reddish brown, others a slightly greenish yellow, and give him the order
"Put the red leaves and the green leaves on separate heaps". It is quite likely that he will upon this separate the
greenish yellow leaves from the reddish brown ones. Now should we say that we had here used the words "red" and
"green" in the same sense as in the previous cases, or did we use them in different but similar senses? What reasons
would one give for adopting the latter view? One could point out that on being asked to paint a red patch, one
should certainly not have painted a slightly reddish brown one, and therefore one might say "red" means something
different in the two cases. But why shouldn't I say that it had one meaning only but was, of course, used according
to the circumstances?
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The question is: Do we supplement our statement that the word has two meanings by a statement saying that
in one case it had this, in the other that meaning? As the criterion for a word's having two meanings, we may use the
fact of there being two explanations given for a word. Thus we say the word "bank" has two meanings; for in one
case it means this sort of thing (pointing, say, to a river bank), in the other case that sort of thing (pointing to the
Bank of England). Now what I point to here are paradigms for the use of the words. One could not say: "The word
'red' has two meanings because in one case it means this (pointing to a light red), in the other that (pointing to a dark
red)", if, that is to say, there had been only one ostensive definition for the word "red" used in our game. One could,
on the other hand, imagine a language game in which two words, say "red" and "reddish", were explained by two
ostensive definitions, the first showing a dark red object, the second a light red one. Whether two such explanations
were given or only one might depend on the natural reactions of the people using the language. We might find that a
person to whom we give the ostensive definition, "This is called 'red'" (pointing to one red object) thereupon fetches
any red object of whatever shade of red on being ordered: "Bring me something red!" Another person might not do
so, but bring objects of a certain range of shades only in the neighbourhood of the shade pointed out to him in the
explanation. We might say that this person 'does not see what is in common between all the different shades of red'.
But remember please that our only criterion for that is the behaviour we have described.
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Consider the following case: B has been taught a use of the words "lighter" and "darker". He has been shown
objects of various colours and has been taught that one calls this a darker colour than that, trained to bring an object
on being ordered "Bring something darker than this", and to describe the colour of an object by saying that it is
darker or lighter than a certain sample, etc., etc. Now he is given the order to put down a series of objects, arranging
them in the order of their darkness. He does this by laying out a row of books, writing down a series of names of
animals, and by writing down the five vowels in the order u, o, a, e, i. We ask him why he put down that latter series,
and he says, "Well, o is lighter than u, and e lighter than o".--We shall be astonished at his attitude, and at the same
time admit that there is something in what he says. Perhaps we shall say: "But look, surely e isn't lighter than o in the
way this book is lighter than
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that".--But he may shrug his shoulders and say, "I don't know, but e is lighter than o, isn't it?"
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We may be inclined to treat this case as some kind of abnormality, and to say, "B must have a different
sense, with the help of which he arranges both coloured objects and vowels". And if we tried to make this idea of
ours (quite) explicit, it would come to this: The normal person registers lightness and darkness of visual objects on
one instrument, and, what one might call the lightness and darkness of sounds (vowels) on another, in the sense in
which one might say that we record rays of a certain wave length with the eyes, and rays of another range of wave
length with our sense of temperature. B on the other hand, we wish to say, arranges both sounds and colours by the
readings of one instrument (sense organ) only (in the sense in which a photographic plate might record rays of a
range which we could only cover with two of our senses).
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This roughly is the picture standing behind our idea that B must have 'understood' the word "darker"
differently from the normal person. On the other hand let us put side by side with this picture the fact that there is in
our case no evidence for 'another sense'.--And in fact the use of the word "must" when we say "B must have
understood the word differently" already shows us that this sentence (really) expresses our determination to look at
the phenomena we have observed after†1 the picture outlined in this sentence.
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'But surely he used "lighter" in a different sense when he said e was lighter than u'.--What does this mean?
Are you distinguishing between the sense in which he used the word and his usage of the word? That is, do you
wish to say that if someone uses the word as B does, some other difference, say in his mind, must go along with the
difference in usage? Or is all you want to say that surely the usage of "lighter" was a different one when he applied it
to vowels?
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Now is the fact that the usages differ anything over and above what you describe when you point out the
particular differences?
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What if somebody said, pointing to two patches which I had called red, "Surely you are using the word 'red'
in two different ways"?--I should say "This is light red and the other dark red,--but why should I have to talk of two
different usages?"--
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It certainly is easy to point out differences between that part of the game in which we applied "lighter" and
"darker" to coloured objects and that part in which we applied these words to vowels. In the first
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part there was comparison of two objects by laying them side by side and looking from one to the other, there was
painting a darker or lighter shade than a certain sample given; in the second there was no comparison by the eye, no
painting, etc. But when these differences are pointed out, we are still free to speak of two parts of the same game (as
we have done just now) or of two different games.
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'But don't I perceive that the relation between a lighter and a darker bit of material is a different one than that
between the vowels e and u,--as on the other hand I perceive that the relation between u and e is the same as that
between e and i?'--Under certain circumstances we shall in these cases be inclined to talk of different relations, under
certain others to talk of the same relation. One might say, "It depends how one compares them".
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Let us ask the question "Should we say that the arrows → and ← point in the same direction or in different
directions?"--At first sight you might be inclined to say "Of course, in different directions". But look at it this way: If
I look into a looking glass and see the reflection of my face, I can take this as a criterion for seeing my own head. If
on the other hand I saw in it the back of a head I might say "It can't be my own head I am seeing, but a head looking
in the opposite direction". Now this could lead me on to say that an arrow and the reflection of an arrow in a glass
have the same direction when they point towards each other, and opposite directions when the head of the one
points to the tail end of the other. Imagine the case that a man had been taught the ordinary use of the word "the
same" in the cases of "the same colour", "the same shape", "the same length". He had also been taught the use of the
word "to point to" in such contexts as "The arrow points to the tree". Now we show him two arrows facing each
other, and two arrows one following the other, and ask him in which of these two cases he'd apply the phrase "The
arrows point the same way". Isn't it easy to imagine that if certain applications were uppermost in his mind, he
would be inclined to say that the arrows → ← point 'the same way'?
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When we hear the diatonic scale we are inclined to say that after every seven notes the same note recurs, and,
asked why we call it the same note again one might answer "Well, it's a c again". But this isn't the explanation I want,
for I should ask "What made one call it a c again?" And the answer to this would seem to be "Well, don't you hear
that it's the same note only an octave higher?"--Here, too, we could imagine that a man had been taught our use of
the word
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"the same" when applied to colours, lengths, directions, etc., and that we now played the diatonic scale for him and
asked him whether he'd say that he heard the same notes again and again at certain intervals, and we could easily
imagine several answers, in particular for instance, this, that he heard the same note alternately after every four or
three notes (he calls the tonic, the dominant, and the octave the same note).
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If we had made this experiment with two people A and B, and A had applied the expression "the same note"
to the octave only, B to the dominant and octave, should we have a right to say that the two hear different things
when we play to them the diatonic scale?-If we say they do, let us be dear whether we wish to assert that there must
be some other difference between the two cases besides the one we have observed, or whether we wish to make no
such statement.
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5. All the questions considered here link up with this problem: Suppose you had taught someone to write
down series of numbers according to rules of the form: Always write down a number n greater than the preceding.
(This rule is abbreviated to "Add n".) The numerals in this game are to be groups of dashes |, ||, |||, etc. What I call
teaching this game, of course, consisted in giving general explanations and doing examples.--These examples are
taken from the range, say, between 1 and 85. We now give the pupil the order "Add 1". After some time we observe
that after passing 100 he did what we should call adding 2; after passing 300 he does what we should call adding 3.
We have him up for this: "Didn't I tell you always to add 1? Look what you have done before you got to
100!"--Suppose the pupil said, pointing to the numbers 102, 104, etc., "Well, didn't I do the same here? I thought this
was what you wanted me to do."--You see that it would get us no further here again to say "But don't you see...?",
pointing out to him again the rules and examples we had given to him. We might, in such a case, say that this person
naturally understands (interprets) the rule (and examples) we have given as we should understand the rule (and
examples) telling us: "Add 1 up to 100, then 2 up to 200, etc."
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(This would be similar to the case of a man who did not naturally follow an order given by a pointing gesture
by moving in the direction shoulder to hand, but in the opposite direction. And understanding here means the same
as reacting.)
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'I suppose what you say comes to this, that in order to follow the rule "Add 1" correctly a new insight,
intuition, is needed at every step.'--
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But what does it mean to follow the rule correctly? How and when is it to be decided which at a particular point is
the correct step to take?'The correct step at every point is that which is in accordance with the rule as it was meant,
intended.'--I suppose the idea is this: When you gave the rule "Add 1" and meant it, you meant him to write 101 after
100, 199 after 198, 1041 after 1040, and so on. But how did you do all these acts of meaning (I suppose an infinite
number of them) when you gave him the rule? Or is this misrepresenting it? And would you say that there was only
one act of meaning, from which, however, all these others, or any one of them, followed in turn? But isn't the point
just: 'What does follow from the general rule?' You might say "Surely I knew when I gave him the rule that I meant
him to follow up 100 by 101". But here you are misled by the grammar of the word "to know". Was knowing this
some mental act by which you at the time made the transition from 100 to 101, i.e., some act like saying to yourself
"I want him to write 101 after 100"? In this case ask yourself how many such acts you performed when you gave
him the rule. Or do you mean by knowing some kind of disposition?--then only experience can teach us what it was
a disposition for.'But surely if one had asked me which number he should write after 1568, I should have answered
"1569".'--I dare say you would, but how can you be sure of it? Your idea really is that somehow in the mysterious
act of meaning the rule you made the transitions without really making them. You crossed all the bridges before you
were there.--This queer idea is connected with a peculiar use of the word "to mean". Suppose our man got to the
number 100 and followed it up by 102. We should then say "I meant you to write 101". Now the past tense in the
word "to mean" suggests that a particular act of meaning had been performed when the rule was given, though as a
matter of fact this expression alludes to no such act. The past tense could be explained by putting the sentence into
the form "Had you asked me before what I wanted you to do at this stage, I should have said...". But it is a
hypothesis that you would have said that.
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To get this clearer, think of this example: Someone says "Napoleon was crowned in 1804". I ask him "Did
you mean the man who won the battle of Austerlitz?" He says "Yes, I meant him".--Does this mean that when he
'meant him', he in some way thought of Napoleon's winning the battle of Austerlitz?
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The expression "The rule meant him to follow up 100 by 101" makes it appear that this rule, as it was meant,
foreshadowed all the
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transitions which were to be made according to it. But the assumption of a shadow of a transition does not get us
any further, because it does not bridge the gulf between it and the real transition. If the mere words of the rule could
not anticipate a future transition, no more could any mental act accompanying these words.
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We meet again and again with this curious superstition, as one might be inclined to call it, that the mental act
is capable of crossing a bridge before we've got to it. This trouble crops up whenever we try to think about the ideas
of thinking, wishing, expecting, believing, knowing, trying to solve a mathematical problem, mathematical induction,
and so forth.
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It is no act of insight, intuition, which makes us use the rule as we do at the particular point of the series. It
would be less confusing to call it an act of decision, though this too is misleading, for nothing like an act of decision
must take place, but possibly just an act of writing or speaking. And the mistake which we here and in a thousand
similar cases are inclined to make is labelled by the word "to make" as we have used it in the sentence "It is no act of
insight which makes us use the rule as we do", because there is an idea that 'something must make us' do what we
do. And this again joins on to the confusion between cause and reason. We need have no reason to follow the rule
as we do. The chain of reasons has an end.
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Now compare these sentences: "Surely it is using the rule 'Add 1' in a different way if after 100 you go on to
102, 104, etc." and "Surely it is using the word 'darker' in a different way if after applying it to coloured patches we
apply it to the vowels".--I should say: "That depends on what you call a 'different way'".--
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But I should certainly say that I should call the application of "lighter" and "darker" to vowels 'another usage
of the words'; and I also should carry on the series 'Add 1' in the way 101, 102, etc., but not--or not
necessarily--because of some other justifying mental act.
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6. There is a kind of general disease of thinking which always looks for (and finds) what would be called a
mental state from which all our acts spring as from a reservoir. Thus one says, "The fashion changes because the
taste of people changes". The taste is the mental reservoir. But if a tailor to-day designs a cut of dress different from
that which he designed a year ago, can't what is called his change of taste have consisted, partly or wholly, in doing
just this?
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And here we say "But surely designing a new shape isn't in itself
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changing one's taste,--and saying a word isn't meaning it,-and saying that I believe isn't believing; there must be
feelings, mental acts, going along with these lines and these words".--And the reason we give for saying this is that a
man certainly could design a new shape without having changed his taste, say that he believes something without
believing it, etc. And this obviously is true. But it doesn't follow that what distinguishes a case of having changed
one's taste from a case of not having done so isn't under certain circumstances just designing what one hasn't
designed before. Nor does it follow that in cases in which designing a new shape is not the criterion for a change of
taste, the criterion must be a change in some particular region of our mind.
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That is to say, we don't use the word "taste" as the name of a feeling. To think that we do is to represent the
practice of our language in undue simplification. This, of course, is the way in which philosophical puzzles generally
arise; and our case is quite analogous to that of thinking that wherever we make a predicative statement we state that
the subject has a certain ingredient (as we really do in the case, "Beer is alcoholic").
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It is advantageous in treating our problem to consider parallel with the feeling or feelings characteristic for
having a certain taste, changing one's taste, meaning what one says, etc., etc., the facial expression (gestures or tone
of voice) characteristic for the same states or events. If someone should object, saying that feelings and facial
expressions can't be compared, as the former are experiences and the latter aren't, let him consider the muscular,
kinaesthetic and tactile experiences bound up with gestures and facial expressions.
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7. Let us then consider the proposition "Believing something cannot merely consist in saying that you believe
it, you must say it with a particular facial expression, gesture, and tone of voice". Now it cannot be doubted that we
regard certain facial expressions, gestures, etc. as characteristic for the expression of belief. We speak of a 'tone of
conviction'. And yet it is clear that this tone of conviction isn't always present whenever we rightly speak of
conviction. "Just so", you might say, "this shows that there is something else, something behind these gestures, etc.
which is the real belief as opposed to mere expressions of belief".--"Not at all", I should say, "many different criteria
distinguish, under different circumstances, cases of believing what you say from those of not believing what you
say". There may be
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cases where the presence of a sensation other than those bound up with gestures, tone of voice, etc. distinguishes
meaning what you say from not meaning it. But sometimes what distinguishes these two is nothing that happens
while we speak, but a variety of actions and experiences of different kinds before and after.
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To understand this family of cases it will again be helpful to consider an analogous case drawn from facial
expressions. There is a family of friendly facial expressions. Suppose we had asked "What feature is it that
characterizes a friendly face?" At first one might think that there are certain traits which one might call friendly traits,
each of which makes the face look friendly to a certain degree, and which when present in a large number constitute
the friendly expression. This idea would seem to be borne out by our common speech, talking of 'friendly eyes',
'friendly mouth', etc. But it is easy to see that the same eyes of which we say they make a face look friendly do not
look friendly, or even look unfriendly, with certain other wrinkles of the forehead, lines round the mouth, etc. Why
then do we ever say that it is these eyes which look friendly? Isn't it wrong to say that they characterize the face as
friendly, for if we say they do so 'under certain circumstances' (these circumstances being the other features of the
face) why did we single out the one feature from amongst the others? The answer is that in the wide family of
friendly faces there is what one might call a main branch characterized by a certain kind of eyes, another by a certain
kind of mouth, etc.; although in the large family of unfriendly faces we meet these same eyes when they don't
mitigate the unfriendliness of the expression.--There is further the fact that when we notice the friendly expression of
a face, our attention, our gaze, is drawn to a particular feature in the face, the 'friendly eyes', or the 'friendly mouth',
etc., and that it does not rest on other features although these too are responsible for the friendly expression.
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'But is there no difference between saying something and meaning it, and saying it without meaning
it?'--There needn't be a difference while he says it, and if there is, this difference may be of all sorts of different kinds
according to the surrounding circumstances. It does not follow from the fact that there is what we call a friendly and
an unfriendly expression of the eye that there must be a difference between the eye of a friendly and the eye of an
unfriendly face.
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One might be tempted to say "This trait can't be said to make the face look friendly, as it may be belied by
another trait". And this is like saying "Saying something with the tone of conviction can't be the
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characteristic of conviction, as it may be belied by experiences going along with it". But neither of these sentences is
correct. It is true that other traits in this face could take away the friendly character of this eye, and yet in this face it
is the eye which is the outstanding friendly feature.
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It is such phrases as "He said it and meant it" which are most liable to mislead us.--Compare meaning "I shall
be delighted to see you" with meaning "The train leaves at 3.30". Suppose you had said the first sentence to
someone and were asked afterwards "Did you mean it?", you would then probably think of the feelings, the
experiences, which you had while you said it. And accordingly you would in this case be inclined to say "Didn't you
see that I meant it?" Suppose that on the other hand, after having given someone the information "The train leaves at
3.30", he asked you "Did you mean it?", you might be inclined to answer "Certainly. Why shouldn't I have meant
it?"
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In the first case we shall be inclined to speak about a feeling characteristic of meaning what we said, but not
in the second. Compare also lying in both these cases. In the first case we should be inclined to say that lying
consisted in saying what we did but without the appropriate feelings or even with the opposite feelings. If we lied in
giving the information about the train, we would be likely to have different experiences while we gave it than those
which we have in giving truthful information, but the difference here would not consist in the absence of a
characteristic feeling, but perhaps just in the presence of a feeling of discomfort.
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It is even possible while lying to have quite strong experience of what might be called the characteristic for
meaning what one says--and yet under certain circumstances, and perhaps under the ordinary circumstances, one
refers to just this experience in saying, "I meant what I said", because the cases in which something might give the
lie to these experiences do not come into the question. In many cases therefore we are inclined to say: "Meaning
what I say" means having such and such experiences while I say it.
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If by "believing" we mean an activity, a process, taking place while we say that we believe, we may say that
believing is something similar to or the same as expressing a belief.
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8. It is interesting to consider an objection to this: What if I said "I believe it will rain" (meaning what I say)
and someone wanted to
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explain to a Frenchman who doesn't understand English what it was I believed. Then, you might say, if all that
happened when I believed what I did was that I said the sentence, the Frenchman ought to know what I believe if
you tell him the exact words I used, or say "Il croit 'It will rain' ". Now it is clear that this will not tell him what I
believe and consequently, you might say, we failed to convey just that to him which was essential, my real mental
act of believing.--But the answer is that even if my words had been accompanied by all sorts of experiences, and if
we could have transmitted these experiences to the Frenchman, he would still not have known what I believed. For
"knowing what I believe" just doesn't mean: feeling what I do while I say it; just as knowing what I intend with this
move in our game of chess doesn't mean knowing my exact state of mind while I'm making the move. Though, at
the same time, in certain cases, knowing this state of mind might furnish you with very exact information about my
intention.
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We should say that we had told the Frenchman what I believed if we translated my words for him into
French. And it might be that thereby we told him nothing--even indirectly--about what happened 'in me' when I
uttered my belief. Rather, we pointed out to him a sentence which in his language holds a similar position to my
sentence in the English language.-Again one might say that, at least in certain cases, we could have told him much
more exactly what I believed if he had been at home in the English language, because then, he would have known
exactly what happened within me when I spoke.
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We use the words "meaning", "believing", "intending" in such a way that they refer to certain acts, states of
mind given certain circumstances; as by the expression "checkmating somebody" we refer to the act of taking his
king. If on the other hand, someone, say a child, playing about with chessmen, placed a few of them on a chess
board and went through the motions of taking a king, we should not say the child had checkmated anyone.--And
here, too, one might think that what distinguished this case from real checkmating was what happened in the child's
mind.
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Suppose I had made a move in chess and someone asked me "Did you intend to mate him?", I answer "I
did", and he now asks me "How could you know you did, as all you knew was what happened within you when you
made the move?", I might answer "Under these circumstances this was intending to mate him".
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9. What holds for 'meaning' holds for 'thinking'.--We very often find it impossible to think without speaking
to ourselves half aloud,--and nobody asked to describe what happened in this case would ever say that
something--the thinking--accompanied the speaking, were he not led into doing so by the pair of verbs
"speaking"/"thinking", and by many of our common phrases in which their uses run parallel. Consider these
examples: "Think before you speak!" "He speaks without thinking", "What I said didn't quite express my thought",
"He says one thing and thinks just the opposite", "I didn't mean a word of what I said", "The French language uses
its words in that order in which we think them".
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If anything in such a case can be said to go with the speaking, it would be something like the modulation of
voice, the changes in timbre, accentuation, and the like, all of which one might call means of expressiveness. Some
of these, like the tone of voice and the accent, nobody for obvious reasons would call the accompaniments of the
speech; and such means of expressiveness as the play of facial expression or gestures which can be said to
accompany speech, nobody would dream of calling thinking.
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10. Let us revert to our example of the use of "lighter" and "darker" for coloured objects and the vowels. A
reason which we should like to give for saying that here we have two different uses and not one is this: 'We don't
think that the words "darker", "lighter" actually fit the relation between the vowels, we only feel a resemblance
between the relation of the sounds and the darker and lighter colours'. Now if you wish to see what sort of feeling
this is, try to imagine that without previous introduction you asked someone "Say the vowels a, e, i, o, u, in the order
of their darkness". If I did this, I should certainly say it in a different tone from that in which I should say "Arrange
these books in the order of their darkness"; that is, I should say it haltingly, in a tone similar to that of "I wonder if
you understand me", perhaps smiling slyly as I say it. And this, if anything, describes my feeling.
And this brings me to the following point: When someone asks me "What colour is the book over there?", and I say
"Red", and then he asks "What made you call this colour 'red'?", I shall in most cases have to say: "Nothing makes
me call it red; that is, no reason. I just looked at it and said 'It's red'". One is then inclined to say: "Surely this isn't all
that happened; for I could look at a colour and say a
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word and still not name the colour". And then one is inclined to go on to say: "The word 'red' when we pronounce it,
naming the colour we look at, comes in a particular way". But, at the same time, asked "Can you describe the way
you mean?"--one wouldn't feel prepared to give any description. Suppose now we asked: "Do you, at any rate,
remember that the name of the colour came to you in that particular way whenever you named colours on former
occasions?"--he would have to admit that he didn't remember a particular way in which this always happened. In
fact one could easily make him see that naming a colour could go along with all sorts of different experiences.
Compare such cases as these: a) I put an iron in the fire to heat it to light red heat. I am asking you to watch the iron
and want you to tell me from time to time what stage of heat it has reached. You look and say: "It is beginning to get
light red". b) We stand at a street crossing and I say: "Watch out for the green light. When it comes on, tell me and
I'll run across." Ask yourself this question: If in one such case you shout "Green!" and in another "Run!", do these
words come in the same way or in different ways? Can one say anything about this in a general way? c) I ask you:
"What's the colour of the bit of material you have in your hand?" (and I can't see). You think: "Now what does one
call this? Is this 'Prussian blue' or 'indigo'?"
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Now it is very remarkable that when in a philosophical conversation we say: "The name of a colour comes in
a particular way", we don't trouble to think of the many different cases and ways in which such a name comes.--And
our chief argument is really that naming the colour is different from just pronouncing a word on some different
occasion while looking at a colour. Thus one might say: "Suppose we counted some objects lying on our table, a
blue one, a red one, a white one, and a black one,--looking at each in turn we say: 'One, two, three, four'. Isn't it easy
to see that something different happens in this case when we pronounce the words than what would happen if we
had to tell someone the colours of the objects? And couldn't we, with the same right as before, have said 'Nothing
else happens when we say the numerals than just saying them while looking at the objects'? "Now two answers can
be given to this: First, undoubtedly, at least in the great majority of cases, counting the objects will be accompanied
by different experiences from naming their colours. And it is easy to describe roughly what the difference will be. In
counting we know a certain gesture, as it were, beating the number out with one's finger or by nodding one's head.
There is, on the other hand, an
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experience which one might call "concentrating one's attention on the colour", getting the full impression of it. And
these are the sort of things one recalls when one says "It is easy to see that something different happens when we
count the objects and when we name their colours". But it is in no way necessary that certain peculiar experiences
more or less characteristic for counting take place while we are counting, nor that the peculiar phenomenon of
gazing at the colour takes place when we look at the object and name its colour. It is true that the processes of
counting four objects and of naming their colours will, in most cases at any rate, be different taken as a whole, and
this is what strikes us; but that doesn't mean at all that we know that something different happens every time in
these two cases when we pronounce a numeral on the one hand and a name of a colour on the other.
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When we philosophize about this sort of thing we almost invariably do something of this sort: We repeat to
ourselves a certain experience, say, by looking fixedly at a certain object and trying to 'read off' as it were the name
of its colour. And it is quite natural that doing so again and again we should be inclined to say, "Something particular
happens while we say the word 'blue'". For we are aware of going again and again through the same process. But ask
yourself: Is this also the process which we usually go through when on various occasions--not philosophizing--we
name the colour of an object?
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11. The problem which we are concerned with we also encounter in thinking about volition, deliberate and
involuntary action. Think, say, of these examples: I deliberate whether to lift a certain heavyish weight, decide to do
it, I then apply my force to it and lift it. Here, you might say, you have a full-fledged case of willing and intentional
action. Compare with this such a case as reaching a man a lighted match after having lit with it one's own cigarette
and seeing that he wishes to light his; or again the case of moving your hand while writing a letter, or moving your
mouth, larynx, etc. while speaking.--Now when I called the first example a full-fledged case of willing, I deliberately
used this misleading expression. For this expression indicates that one is inclined in thinking about volition to regard
this sort of example as one exhibiting most clearly the typical characteristic of willing. One takes one's ideas, and
one's language, about volition from this kind of example and thinks that they must apply--if not in such an obvious
way-to all cases which one can properly call cases of willing.-It is the same case that we have met over and over
again:
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The forms of expression of our ordinary language fit most obviously certain very special applications of the words
"willing", "thinking", "meaning", "reading", etc., etc. And thus we might have called the case in which a man 'first
thinks and then speaks' the full-fledged case of thinking and the case in which a man spells out the words he is
reading the full-fledged case of reading. We speak of an 'act of volition' as different from the action which is willed,
and in our first example there are lots of different acts clearly distinguishing this case from one in which all that
happens is that the hand and the weight lift: there are the preparations of deliberation and decision, there is the effort
of lifting. But where do we find the analogues to these processes in our other examples and in innumerable ones we
might have given?
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Now on the other hand it has been said that when a man, say, gets out of bed in the morning, all that
happens may be this: he deliberates, "Is it time to get up?", he tries to make up his mind, and then suddenly he finds
himself getting up. Describing it this way emphasizes the absence of an act of volition. Now first: where do we find
the prototype of such a thing, i.e., how did we come by the idea of such an act? I think the prototype of the act of
volition is the experience of muscular effort.--Now there is something in the above description which tempts us to
contradict it; we say: "We don't just 'find', observe, ourselves getting up, as though we were observing someone else!
It isn't like, say, watching certain reflex actions. If, e.g., I place myself sideways close to a wall, my wall-side arm
hanging down outstretched, the back of the hand touching the wall, and if now keeping the arm rigid I press the back
of the hand hard against the wall, doing it all by means of the deltoid muscle, if then I quickly step away from the
wall, letting my arm hang down loosely, my arm without any action of mine, of its own accord begins to rise; this is
the sort of case in which it would be proper to say, 'I find my arm rising'."
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Now here again it is clear that there are many striking differences between the case of observing my arm
rising in this experiment or watching someone else getting out of bed and the case of finding myself getting up.
There is, e.g., in this case a perfect absence of what one might call surprise, also I don't look at my own movements
as I might look at someone turning about in bed, e.g., saying to myself "Is he going to get up?" There is a difference
between the voluntary act of getting out of bed and the involuntary rising of my arm. But there is not one common
difference between so-called
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voluntary acts and involuntary ones, viz, the presence or absence of one element, the 'act of volition'.
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The description of getting up in which a man says "I just find myself getting up" suggests that he wishes to
say that he observes himself getting up. And we may certainly say that an attitude of observing is absent in this case.
But the observing attitude again is not one continuous state of mind or otherwise which we are in the whole time
while, as we should say, we are observing. Rather, there is a family of groups of activities and experiences which we
call observing attitudes. Roughly speaking one might say there are observation-elements of curiosity, observant
expectation, surprise, and there are, we should say, facial expressions and gestures of curiosity, of observant
expectation, and of surprise; and if you agree that there is more than one facial expression characteristic for each of
these cases, and that there can be these cases without any characteristic facial expression, you will admit that to each
of these three words a family of phenomena corresponds.
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12. If I had said "When I told him that the train was leaving at 3.30, believing that it did, nothing happened
than that I just uttered the sentence", and if someone contradicted me, saying "Surely this couldn't have been all, as
you might 'just say a sentence' without believing it",--my answer should be "I didn't wish to say that there was no
difference between speaking, believing what you say, and speaking, not believing what you say; but the pair
'believing'/'not believing' refers to various differences in different cases (differences forming a family), not to one
difference, that between the presence and the absence of a certain mental state."
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13. Let us consider various characteristics of voluntary and involuntary acts. In the case of lifting the heavy
weight, the various experiences of effort are obviously most characteristic for lifting the weight voluntarily. On the
other hand, compare with this the case of writing, voluntarily, where in most of the ordinary cases there will be no
effort; and even if we feel that the writing tires our hands and strains their muscles, this is not the experience of
'pulling' and 'pushing' which we would call typical voluntary actions. Further compare the lifting of your hand when
you lift a weight with it with lifting your hand when, e.g., you point to some object above you. This will certainly be
regarded as a voluntary act, though the element of effort will most likely be entirely absent; in fact this raising of the
arm
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to point at an object is very much like raising the eye to look at it, and here we can hardly conceive of an
effort.--Now let us describe an act of involuntarily raising your arm. There is the case of our experiment, and this
was characterized by the utter absence of muscular strain and also by our observant attitude towards the lifting of
the arm. But we have just seen a case in which muscular strain was absent, and there are cases in which we should
call an action voluntary although we take an observant attitude towards it. But in a large class of cases it is the
peculiar impossibility of taking an observant attitude towards a certain action which characterizes it as a- voluntary
one. Try, e.g., to observe your hand rising when you voluntarily raise it. Of course you see it rising as you do, say, in
the experiment; but you can't somehow follow it in the same way with your eye. This might get clearer if you
compare two different cases of following lines on a piece of paper with your eye; a) some irregular line like this:
b) a written sentence. You will find that in a) the eye, as it were, alternately slips and gets stuck, whereas in reading a
sentence it glides along smoothly.
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Now consider a case in which we do take up an observant attitude towards a voluntary action, I mean the
very instructive case of trying to draw a square with its diagonals by placing a mirror on your drawing paper and
directing your hand by what you see by looking at it in the mirror. And here one is inclined to say that our real
actions, the ones to which volition immediately applies, are not the movements of our hand but something further
back, say, the actions of our muscles. We are inclined to compare the case with this: Imagine we had a series of
levers before us, through which, by a hidden mechanism, we could direct a pencil drawing on a sheet of paper. We
might then be in doubt which levers to pull in order to get the desired movement of the pencil; and we could say that
we deliberately pulled this particular lever, although we didn't deliberately produce the wrong result that we thereby
produced. But this comparison, though it easily suggests itself, is very misleading. For in the case of the levers which
we saw before us, there was such a thing as deciding which one we were going to pull before pulling it. But does our
volition, as it were, play on a keyboard of muscles, choosing which one it was going to use next?--For some actions
which we call deliberate it is characteristic
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that we, in some sense, 'know what we are going to do' before we do it. In this sense we say that we know what
object we are going to point to, and what we might call 'the act of knowing' might consist in looking at the object
before we point to it or in describing its position by words or pictures. Now we could describe our drawing the
square through the mirror by saying that our acts were deliberate as far as their motor aspect is concerned, but not as
far as their visual aspect is concerned. This would, e.g., be demonstrated by our ability to repeat a movement of the
hand which had produced a wrong result, on being told to do so. But it would obviously be absurd to say that this
motor character of voluntary motion consisted in our knowing beforehand what we were going to do, as though we
had had a picture of the kinaesthetic sensation before our mind and decided to bring about this sensation.
Remember the experiment where the subject has his fingers intertwined; if here, instead of pointing from a distance
to the finger which you order him to move, you touch that finger, he will always move it without the slightest
difficulty. And here it is tempting to say "Of course I can move it now, because now I know which finger it is I'm
asked to move." This makes it appear as though I had now shown you which muscle to contract in order to bring
about the desired result. The word "of course" makes it appear as though by touching your finger I had given you an
item of information telling you what to do. (As though normally when you tell a man to move such and such a
finger he could follow your order because he knew how to bring the movement about.)
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(It is interesting here to think of the case of sucking a liquid through a tube; if asked what part of your body
you sucked with, you would be inclined to say your mouth, although the work was done by the muscles by which
you draw your breath.)
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Let us now ask ourselves what we should call "speaking involuntarily". First note that when normally you
speak, voluntarily, you could hardly describe what happened by saying that by an act of volition you move your
mouth, tongue, larynx, etc. as a means to producing certain sounds. Whatever happens in your mouth, larynx, etc.
and whatever sensations you have in these parts while speaking would almost seem secondary phenomena
accompanying the production of sounds, and volition, one wishes to say, operates on the sounds themselves
without intermediary mechanism. This shows how loose our idea of this agent 'volition' is.
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Now to involuntary speaking. Imagine you had to describe a
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case,--what would you do? There is of course the case of speaking in one's sleep; this is characterized by our doing it
without being aware of it and not remembering having done it. But this obviously you wouldn't call the characteristic
of an involuntary action.
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A better example of involuntary speaking would, I suppose, be that of involuntary exclamations: "Oh!",
"Help!", and such like, and these utterances are akin to shrieking with pain. (This, by the way, could set us thinking
about 'words as expressions of feelings'.) One might say: "Surely these are good examples of involuntary speech,
because there is in these cases not only no act of volition by which we speak, but in many cases we utter these
words against our will". I should say: I certainly should call this involuntary speaking; and I agree that an act of
volition preparatory to or accompanying these words is absent,--if by "act of volition" you refer to certain acts of
intention, premeditation, or effort. But then in many cases of voluntary speech I don't feel an effort, much that I say
voluntarily is not premeditated, and I don't know of any acts of intention preceding it.
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Crying out with pain against our will could be compared with raising our arm against our will when someone
forces it up while we are struggling against him. But it is important to notice that the will--or should we say
'wish'--not to cry out is overcome in a different way from that in which our resistance is overcome by the strength of
the opponent. When we cry out against our will, we are as it were taken by surprise; as though someone forced up
our hands by unexpectedly sticking a gun into our ribs, commanding "Hands up!"
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14. Consider now the following example, which is of great help in all these considerations: In order to see
what happens when one understands a word, we play this game: You have a list of words, partly these words are
words of my native language, partly words of foreign languages more or less familiar to me, partly words of
languages entirely unknown to me (or, which comes to the same, nonsensical words invented for the occasion).
Some of the words of my native tongue, again, are words of ordinary everyday usage: and some of these, like
"house", "table", "man", are what we might call primitive words, being among the first words a child learns, and
some of these again, words of baby talk like "Mamma", "Papa". Again, there are more or less common technical
terms such as "carburettor", "dynamo", "fuse"; etc., etc. All these words are read out
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to me, and after each one I have to say "Yes" or "No" according to whether I understand the word or not. I then try
to remember what happened in my mind when I understood the words I did understand, and when I didn't
understand the others. And here again it will be useful to consider the particular tone of voice and facial expression
with which I say "Yes" and "No", alongside of the so-called mental events.--Now it may surprise us to find that
although this experiment will show us a multitude of different characteristic experiences, it will not show us any one
experience which we should be inclined to call the experience of understanding. There will be such experiences as
these: I hear the word "tree" and say "Yes" with the tone of voice and sensation of "Of course". Or I hear
"corroboration" I say to myself, "Let me see", vaguely remember a case of helping, and say "Yes". I hear "gadget", I
imagine the man who always used this word, and say "Yes". I hear "Mamma", this strikes me as funny and
childish--"Yes". A foreign word I shall very often translate in my mind into English before answering. I hear
"spinthariscope", and say to myself, "Must be some sort of scientific instrument", perhaps try to think up its
meaning from its derivation and fail, and say "No". In another case I might say to myself, "Sounds like
Chinese"--"No". Etc. There will, on the other hand, be a large class of cases in which I am not aware of anything
happening except hearing the word and saying the answer. And there will also be cases in which I remember
experiences (sensations, thoughts) which, as I should say, had nothing to do with the word at all. Thus amongst the
experiences which I can describe there will be a class which I might call typical experiences of understanding and
some typical experiences of not understanding. But opposed to these there will be a large class of cases in which I
should have to say "I know of no particular experience at all, I just said 'Yes', or 'No'".
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Now if someone said "But surely something did happen when you understood the word 'tree', unless you
were utterly absentminded when you said 'Yes'", I might be inclined to reflect and say to myself, "Didn't I have a
sort of homely feeling when I took in the word 'tree'?" But then, do I always have this feeling which now I referred to
when I hear that word used or use it myself, do I remember having had it, do I even remember a set of, say, five
sensations some one of which I had on every occasion when I could be said to have understood the word? Further,
isn't that 'homely feeling' I referred
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to an experience rather characteristic for the particular situation I'm in at present, i.e., that of philosophizing about
'understanding'?
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Of course in our experiment we might call saying "Yes" or "No" characteristic experiences of understanding
or not understanding, but what if we just hear a word in a sentence, where there isn't even a question of this reaction
to it?--We are here in a curious difficulty: on the one hand it seems we have no reason to say that in all cases in
which we understand a word one particular experience--or even one of a set--is present. On the other hand we may
feel it's plainly wrong to say that in such a case all that happens may be that I hear or say the word. For that seems to
be saying that part of the time we act as mere automatons. And the answer is that in a sense we do and in a sense we
don't.
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If someone talked to me with a kindly play of facial expressions, is it necessary that in any short interval his
face should have looked such that seeing it under any other circumstances I should have called its expression
distinctly kindly? And if not, does this mean that his 'kindly play of expression' was interrupted by periods of
inexpressiveness?--We certainly should not say this under the circumstances which I am assuming, and we don't
feel that the look at this moment interrupts the expressiveness, although taken alone we should call it inexpressive.
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Just in this way we refer by the phrase "understanding a word" not necessarily to that which happens while
we are saying or hearing it, but to the whole environment of the event of saying it. And this also applies to our
saying that someone speaks like an automaton or like a parrot. Speaking with understanding certainly differs from
speaking like an automaton, but this doesn't mean that the speaking in the first case is all the time accompanied by
something which is lacking in the second case. Just as when we say that two people move in different circles this
doesn't mean that they mayn't walk the street in identical surroundings.
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Thus also, acting voluntarily (or involuntarily) is, in many cases, characterized as such by a multitude of
circumstances under which the action takes place rather than by an experience which we should call characteristic of
voluntary action. And in this sense it is true to say that what happened when I got out of bed--when I should
certainly not call it involuntary--was that I found myself getting up. Or rather, this is a possible case; for of course
every day something different happens.
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15. The troubles which we have been turning over since §7 were all closely connected with the use of the
word "particular". We have been inclined to say that seeing familiar objects we have a particular feeling, that the
word "red" came in a particular way when we recognized the colour as red, that we had a particular experience when
we acted voluntarily.
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Now the use of the word "particular" is apt to produce a kind of delusion and roughly speaking this delusion
is produced by the double usage of this word. On the one hand, we may say, it is used preliminary to a specification,
description, comparison; on the other hand, as what one might describe as an emphasis. The first usage I shall call
the transitive one, the second the intransitive one. Thus, on the one hand I say "This face gives me a particular
impression which I can't describe". The latter sentence may mean something like: "This face gives me a strong
impression". These examples would perhaps be more striking if we substituted the word "peculiar" for "particular",
for the same comments apply to "peculiar". If I say "This soap has a peculiar smell: it is the kind we used as
children", the word "peculiar" may be used merely as an introduction to the comparison which follows it, as though
I said "I'll tell you what this soap smells like:...". If, on the other hand, I say "This soap has a peculiar smell!" or "It
has a most peculiar smell", "peculiar" here stands for some such expression as "out of the ordinary", "uncommon",
"striking".
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We might ask "Did you say it had a peculiar smell, as opposed to no peculiar smell, or that it had this smell,
as opposed to some other smell, or did you wish to say both the first and the second?"-Now what was it like when,
philosophizing, I said that the word "red" came in a particular way when I described something I saw as red? Was it
that I was going to describe the way in which the word "red" came, like saying "It always comes quicker than the
word 'two' when I'm counting coloured objects", or "It always comes with a shock", etc.?--Or was it that I wished to
say that "red" comes in a striking way?--Not exactly that either. But certainly rather the second than the first. To see
this more dearly, consider another example: You are, of course, constantly changing the position of your body
throughout the day; arrest yourself in any such attitude (while writing, reading, talking, etc., etc.) and say to yourself
in the way in which you say "'Red' comes in a particular way...", "I am now in a particular attitude". You will find
that you can quite naturally say
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this. But aren't you always in a particular attitude? And of course you didn't mean that you were just then in a
particularly striking attitude. What was it that happened? You concentrated on, as it were stared at, your sensations.
And this is exactly what you did when you said that "red" came in a particular way.
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"But didn't I mean that 'red' came in a different way from 'two'?"--You may have meant this, but the phrase,
"They come in different ways", is itself liable to cause confusion. Suppose I said "Smith and Jones always enter my
room in different ways": I might go on and say "Smith enters quickly, Jones slowly", I am specifying the ways. I
might on the other hand say "I don't know what the difference is", intimating that I'm trying to specify the
difference, and perhaps later on I shall say "Now I know what it is; it is...".--I could on the other hand tell you that
they came in different ways, and you wouldn't know what to make of this statement, and perhaps answer "Of course
they come in different ways; they just are different".--We could describe our trouble by saying that we feel as
though we could give an experience a name without at the same time committing ourselves about its use, and in fact
without any intention to use it at all. Thus when I say "red" comes in a particular way..., I feel that I might now give
this way a name if it hasn't already got one, say "A". But at the same time I am not at all prepared to say that I
recognize this to be the way "red" has always come on such occasions, nor even to say that there are, say, four
ways, A, B, C, D, in one of which it always comes. You might say that the two ways in which "red" and "two" come
can be identified by, say, exchanging the meaning of the two words, using "red" as the second cardinal numeral,
"two" as the name of a colour. Thus, on being asked how many eyes I had, I should answer "red", and to the
question "What is the colour of blood?", "two". But the question now arises whether you can identify the 'way in
which these words come' independently of the ways in which they are used,--I mean the ways just described. Did
you wish to say that as a matter of experience, the word when used in this way always comes in the way A, but
may, the next time, come in the way "two" usually comes? You will see then that you meant nothing of the sort.
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What is particular about the way "red" comes is that it comes while you're philosophizing about it, as what
is particular about the position of your body when you concentrated on it was concentration. We appear to
ourselves to be on the verge of describing the way, whereas we aren't really opposing it to any other way. We are
emphasizing,
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not comparing, but we express ourselves as though this emphasis was really a comparison of the object with itself;
there seems to be a reflexive comparison. Let me explain myself in this way: suppose I speak of the way in which A
enters the room, I may say "I have noticed the way in which A enters the room", and on being asked "What is it?", I
may answer "He always sticks his head into the room before coming in". Here I'm referring to a definite feature, and
I could say that B had the same way, or that A no longer had it. Consider on the other hand the statement "I've now
been observing the way A sits and smokes". I want to draw him like this. In this case I needn't be ready to give any
description of a particular feature of his attitude, and my statement may just mean "I've been observing A as he sat
and smoked".-'The way' can't in this case be separated from him. Now if I wished to draw him as he sat there, and
was contemplating, studying, his attitude, I should while doing so be inclined to say and repeat to myself "He has a
particular way of sitting". But the answer to the question "What way?" would be "Well, this way", and perhaps one
would give it by drawing the characteristic outlines of his attitude. On the other hand, my phrase "He has a particular
way...", might just have to be translated into "I'm contemplating his attitude". Putting it in this form we have, as it
were, straightened out the proposition; whereas in its first form its meaning seems to describe a loop, that is to say,
the word "particular" here seems to be used transitively and, more particularly, reflexively, i.e., we are regarding its
use as a special case of the transitive use. We are inclined to answer the question "What way do you mean?" by
"This way", instead of answering: "I didn't refer to any particular feature; I was just contemplating his position". My
expression made it appear as though I was pointing out something about his way of sitting, or, in our previous case,
about the way the word "red" came, whereas what makes me use the word "particular" here is that by my attitude
towards the phenomenon I am laying an emphasis on it: I am concentrating on it, or retracing it in my mind, or
drawing it, etc.
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Now this is a characteristic situation to find ourselves in when thinking about philosophical problems. There
are many troubles which arise in this way, that a word has a transitive and an intransitive use, and that we regard the
latter as a particular case of the former, explaining the word when it is used intransitively by a reflexive construction.
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Thus we say, "By 'kilogram' I mean the weight of one litre of water", "By 'A' I mean. 'B', where B is an
explanation of A".
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But there is also the intransitive use: "I said that I was sick of it and meant it". Here again, meaning what you said
could be called "retracing it", "laying an emphasis on it". But using the word "meaning" in this sentence makes it
appear that it must have sense to ask "What did you mean?" and to answer "By what I said I meant what I said,"
treating the case of "I mean what I say" as a special case of "By saying 'A' I mean 'B' ". In fact one uses the
expression "I mean what I mean" to say, "I have no explanation for it". The question, "What does this sentence p
mean?", if it doesn't ask for a translation of p into other symbols, has no more sense than "What sentence is formed
by this sequence of words?"
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Suppose to the question, "What's a kilogram?" I answered, "It is what a litre of water weighs", and someone
asked, "Well, what does a litre of water weigh?"--
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We often use the reflexive form of speech as a means of emphasizing something. And in all such cases our
reflexive expressions can be 'straightened out'. Thus we use the expression "If I can't, I can't", "I am as I am", "It is
just what it is", also "That's that". This latter phrase means as much as "That's settled", but why should we express
"That's settled" by "That's that"? The answer can be given by laying before ourselves a series of interpretations which
make a transition between the two expressions. Thus for "That's settled", I will say "The matter is closed". And this
expression, as it were, files the matter and shelves it. And filing it is like drawing a line around it, as one sometimes
draws a line around the results of a calculation, thereby marking it as final. But this also makes it stand out; it is a
way of emphasizing it. And what the expression "That's that" does is to emphasize the 'that'.
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Another expression akin to those we have just considered is this: "Here it is; take it or leave it!" And this
again is akin to a kind of introductory statement which we sometimes make before remarking on certain alternatives,
as when we say: "It either rains or it doesn't rain; if it rains we'll stay in my room, if it doesn't...". The first part of this
sentence is no piece of information (just as "Take it or leave it" is no order). Instead of, "It either rains or it doesn't
rain" we could have said, "Consider the two cases...". Our expression underlines these cases, presents them to your
attention.
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It is closely connected with this that in describing a case like 30)†1 we are tempted to use the phrase, "There
is, of course, a number beyond
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which no one of the tribe has ever counted; let this number be...". Straightened out this reads: "Let the number
beyond which no one of the tribe has ever counted be...". Why we tend to prefer the first expression to the one
straightened out is that it more strongly directs our attention to the upper end of the range of numerals used by our
tribe in their actual practice.
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16. Let us now consider a very instructive case of that use of the word "particular" in which it does not point
to a comparison and yet seems most strongly to do so,--the case when we contemplate the expression of a face
primitively drawn in this way:
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Let this face produce an impression on you. You may then feel inclined to say: "Surely I don't see mere
dashes. I see a face with a particular expression". But you don't mean that it has an outstanding expression nor is it
said as an introduction to a description of the expression, though we might give such a description and say, e.g., "It
looks like a complacent business man, stupidly supercilious, who though fat, imagines he's a lady killer". But this
would only be meant as an approximate description of the expression. "Words can't exactly describe it", one
sometimes says. And yet one feels that what one calls the expression of the face is something that can be detached
from the drawing of the face. It is as though we could say: "This face has a particular expression: namely this"
(pointing to something). But if I had to point to anything in this place it would have to be the drawing I am looking
at. (We are, as it were, under an optical delusion which by some sort of reflection makes us think that there are two
objects where there is only one. The delusion is assisted by our using the verb "to have", saying "The face has a
particular expression". Things look different when, instead of this, we say: "This is a peculiar face". What a thing is,
we mean, is bound up with it; what it has can be separated from it.)
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'This face has a particular expression.'--I am inclined to say this when I am trying to let it make its full
impression upon me.
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What goes on here is an act, as it were, of digesting it, getting hold of it, and the phrase "getting hold of the
expression of this face" suggests that we are getting hold of a thing which is in the face and different from it. It
seems we are looking for something, but we don't
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do so in the sense of looking for a model of the expression outside the face we see, but in the sense of sounding the
thing without attention. It is, when I let the face make an impression on me, as though there existed a double of its
expression, as though the double was the prototype of the expression and as though seeing the expression of the
face was finding the prototype to which it corresponded--as though in our mind there had been a mould and the
picture we see had fallen into that mould, fitting it. But it is rather that we let the picture sink into our mind and make
a mould there.
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When we say, "This is a face, and not mere strokes", we are, of course, distinguishing such a drawing
from such a one
And it is true: If you ask anyone: "What is this?" (pointing to the first drawing) he will certainly say: "It's a face", and
he will be able straight away to reply to such questions as, "Is it male or female?", "Smiling or sad?", etc. If on the
other hand you ask him: "What is this?" (pointing to the second drawing), he will most likely say, "This is nothing at
all", or "These are just dashes". Now think of looking for a man in a picture puzzle; there it often happens that what
at first sight appears as 'mere dashes' later appears as a face. We say in such cases: "Now I see it is a face". It must be
quite clear to you that this doesn't mean that we recognize it as the face of a friend or that we are under the delusion
of seeing a 'real' face: rather, this 'seeing it as a face' must be compared with seeing this drawing
either as a cube or as a plane figure consisting of a square and two rhombuses; or with seeing this
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'as a square with diagonals', or 'as a swastika', that is, as a limiting case of this;
or again with seeing these four dots .... as two pairs of dots side by side with each other, or as two interlocking pairs,
or as one pair inside the other, etc.
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The case of 'seeing
as a swastika' is of special interest because this expression might mean being, somehow, under the optical delusion
that the square is not quite closed, that there are the gaps which distinguish the swastika from our drawing. On the
other hand it is quite clear that this was not what we meant by "seeing our drawing as a swastika". We saw it in a
way which suggested the description, "I see it as a swastika". One might suggest that we ought to have said "I see it
as a closed swastika";--but then, what is the difference between a closed swastika and a square with diagonals? I
think that in this case it is easy to recognize 'what happens when we see our figure as a swastika'. I believe it is that
we retrace the figure with our eyes in a particular way, viz., by staring at the centre, looking along a radius, and along
a side adjacent to it, starting at the centre again, taking the next radius and the next side, say in a right-handed sense
of rotation, etc. But this explanation of the phenomenon of seeing the figure as a swastika is of no fundamental
interest to us. It is of interest to us only in so far as it helps one to see that the expression "seeing the figure as a
swastika" did not mean seeing this or that, seeing one thing as something else, when, essentially, two visual objects
entered the process of doing so.-Thus also seeing the first figure as a cube did not mean 'taking it to be a cube'. (For
we might never have seen a cube and still have this experience of 'seeing it as a cube'.)
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And in this way 'seeing dashes as a face' does not involve a comparison between a group of dashes and a real
human face; and, on the other hand, this form of expression most strongly suggests that we are alluding to a
comparison.
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Consider also this example: Look at W once as a capital double-U,
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and another time as a capital M upside down. Observe what doing the one and doing the other consists in.
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We distinguish seeing a drawing as a face and seeing it as something else or as 'mere dashes'. And we also
distinguish between superficially glancing at a drawing (seeing it as a face), and letting the face make its full
impression on us. But it would be queer to say: "I am letting the face make a particular impression on me" (except
in such cases in which you can say that you can let the same face make different impressions on you). And in letting
the face impress itself on me and contemplating its 'particular impression', no two things of the multiplicity of a face
are compared with each other; there is only one which is laden with emphasis. Absorbing its expression, I don't find
a prototype of this expression in my mind; rather, I, as it were, cut a seal from the impression.
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And this also describes what happens when in 15)†1 we say to ourselves "The word 'red' comes in a
particular way...". The reply could be: "I see, you're repeating to yourself some experience and again and again
gazing at it."
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17. We may shed light on all these considerations if we compare what happens when we remember the face
of someone who enters our room, when we recognize him as Mr. So and So,--when we compare what really
happens in such cases with the representation we are sometimes inclined to make of the events. For here we are
often obsessed by a primitive conception, viz., that we are comparing the man we see with a memory image in our
mind and we find the two to agree. I.e., we are representing 'recognizing someone' as a process of identification by
means of a picture (as a criminal is identified by his photo). I needn't say that in most cases in which we recognize
someone no comparison between him and a mental picture takes place. We are, of course, tempted to give this
description by the fact that there are memory images. Very often, for instance, such an image comes before our
mind immediately after having recognized someone. I see him as he stood when we last saw each other ten years
ago.
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I will here again describe the kind of thing that happens in your mind and otherwise when you recognize a
person coming into your room by means of what you might say when you recognize him. Now this may just be:
"Hello!" And thus we may say that one kind of event of recognizing a thing we see consists in saying "Hello!" to it in
words,
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gestures, facial expressions, etc.--And thus also we may think that when we look at our drawing and see it as a face,
we compare it with some paradigm, and it agrees with it, or it fits into a mould ready for it in our mind. But no such
mould or comparison enters into our experience, there is only this shape, not any other to compare it with, and as it
were, say "Of course" to. As when in putting together a jig-saw puzzle, somewhere a small space is left unfilled and I
see a piece obviously fitting it and put it in the place saying to myself "Of course". But here we say, "Of course"
because the piece fits the mould, whereas in our case of seeing the drawing as a face, we have the same attitude for
no reason.
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The same strange illusion which we are under when we seem to seek the something which a face expresses
whereas, in reality, we are giving ourselves up to the features before us--that same illusion possesses us even more
strongly if repeating a tune to ourselves and letting it make its full impression on us, we say "This tune says
something", and it is as though I had to find what it says. And yet I know that it doesn't say anything such that I
might express in words or pictures what it says. And if, recognizing this, I resign myself to saying "It just expresses a
musical thought", this would mean no more than saying "It expresses itself".--"But surely when you play it you
don't play it anyhow, you play it in this particular way, making a crescendo here, a diminuendo there, a caesura in
this place, etc."--Precisely, and that's all I can say about it, or may be all that I can say about it. For in certain cases I
can justify, explain the particular expression with which I play it by a comparison, as when I say "At this point of the
theme, there is, as it were, a colon", or "This is, as it were, the answer to what came before", etc. (This, by the way,
shows what a 'justification' and an 'explanation' in aesthetics is like.) It is true I may hear a tune played and say "This
is not how it ought to be played, it goes like this"; and I whistle it in a different tempo. Here one is inclined to ask
"What is it like to know the tempo in which a piece of music should be played?" And the idea suggests itself that
there must be a paradigm somewhere in our mind, and that we have adjusted the tempo to conform to that
paradigm. But in most cases if someone asked me "How do you think this melody should be played?", I will, as an
answer, just whistle it in a particular way, and nothing will have been present to my mind but the tune actually
whistled (not an image of that).
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This doesn't mean that suddenly understanding a musical theme
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may not consist in finding a form of verbal expression which I conceive as the verbal counterpoint of the theme.
And in the same way I may say "Now I understand the expression of this face", and what happened when the
understanding came was that I found the word which seemed to sum it up.
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Consider also this expression: "Tell yourself that it's a waltz, and you will play it correctly".
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What we call "understanding a sentence" has, in many cases, a much greater similarity to understanding a
musical theme than we might be inclined to think. But I don't mean that understanding a musical theme is more like
the picture which one tends to make oneself of understanding a sentence; but rather that this picture is wrong, and
that understanding a sentence is much more like what really happens when we understand a tune than at first sight
appears. For understanding a sentence, we say, points to a reality outside the sentence. Whereas one might say
"Understanding a sentence means getting hold of its content; and the content of the sentence is in the sentence."
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18. We may now return to the ideas of 'recognizing' and 'familiarity', and in fact to that example of
recognition and familiarity which started our reflections on the use of these terms and of a multitude of terms
connected with them. I mean the example of reading, say, a written sentence in a well-known language.--I read such
a sentence to see what the experience of reading is like, what 'really happens' when one reads, and I get a particular
experience which I take to be the experience of reading. And, it seems, this doesn't simply consist in seeing and
pronouncing the words, but, besides, in an experience of an intimate character, as I should like to say. (I am as it
were on an intimate footing with the word 'I read'.)
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In reading the spoken words come in a particular way, I am inclined to say; and the written words themselves
which I read don't just look to me like any kind of scribbles. At the same time I am unable to point to, or get a grasp
on, that 'particular way'.
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The phenomenon of seeing and speaking the words seems enshrouded by a particular atmosphere. But I
don't recognize this atmosphere as one which always characterized the situation of reading. Rather, I notice it when I
read a line, trying to see what reading is like.
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When noticing this atmosphere I am in the situation of a man who is working in his room, reading, writing,
speaking, etc., and
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who suddenly concentrates his attention on some soft uniform noise, such as one can almost always hear,
particularly in a town (the dim noise resulting from all the various noises of the street, the sounds of wind, rain,
workshops, etc.). We could imagine that this man might think that a particular noise was a common element of all
the experiences he had in this room. We should then draw his attention to the fact that most of the time he hadn't
noticed any noises going on outside, and secondly, that the noise he could hear wasn't always the same (there was
sometimes wind, sometimes not, etc.).
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Now we have used a misleading expression when we said that besides the experiences of seeing and
speaking in reading there was another experience, etc. This is saying that to certain experiences another experience is
added.--Now take the experience of seeing a sad face, say in a drawing,--we can say that to see the drawing as a sad
face is not 'just' to see it as some complex of strokes (think of a puzzle picture). But the word 'just' here seems to
intimate that in seeing the drawing as a face some experience is added to the experience of seeing it as mere strokes;
as though I had to say that seeing the drawing as a face consisted of two experiences, elements.
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You should now notice the difference between the various cases in which we say that an experience consists
of several elements or that it is a compound experience. We might say to the doctor, "I don't have one pain; I have
two: toothache and headache". And one might express this by saying, "My experience of pain is not simple, but
compound, I have toothache and headache". Compare with this case that in which I say, "I have got both pains in
my stomach and a general feeling of sickness". Here I don't separate the constituent experiences by pointing to two
localities of pain. Or consider this statement: "When I drink sweet tea, my taste experience is a compound of the
taste of sugar and the taste of tea". Or again: "If I hear the C Major chord my experience is composed of hearing C,
E, and G". And, on the other hand, "I hear a piano playing and some noise in the street". A most instructive example
is this: In a song words are sung to certain notes. In what sense is the experience of hearing the vowel a sung to the
note C a composite one? Ask yourself in each of these cases: What is it like to single out the constituent experiences
in the compound experience?
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Now although the expression that seeing a drawing as a face is not merely seeing strokes seems to point to
some kind of addition of experiences, we certainly should not say that when we see the
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drawing as a face we also have the experience of seeing it as mere strokes and some other experience besides. And
this becomes still clearer when we imagine that someone said that seeing the drawing
as a cube consisted in seeing it as a plane figure plus having an experience of depth.
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Now when I felt that though while reading a certain constant experience went on and on, I could not in a
sense lay hold of that experience, my difficulty arose through wrongly comparing this case with one in which one
part of my experience can be said to be an accompaniment of another. Thus we are sometimes tempted to ask: "If I
feel this constant hum going on while I read, where is it?" I wish to make a pointing gesture, and there is nothing to
point to. And the words "lay hold of" express the same misleading analogy.
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Instead of asking the question "Where is this constant experience which seems to go on all through my
reading?", we should ask "What is it in saying 'A particular atmosphere enshrouds the words which I am reading',
that I am contrasting this case with?"
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I will try to elucidate this by an analogous case: We are inclined to be puzzled by the three-dimensional
appearance of the drawing
in a way expressed by the question "What does seeing it three-dimensionally consist in?" And this question really
asks 'What is it that is added to simply seeing the drawing when we see it three-dimensionally?' And yet what
answer can we expect to this question? It is the form of this question which produces the puzzlement. As Hertz
says: "Aber offenbar irrt die Frage in Bezug auf die Antwort, welche sie erwartet" (p. 9, Einleitung, Die Prinzipien
der Mechanik). The question itself keeps the mind pressing against a blank wall, thereby preventing it from ever
finding the outlet. To show a man how to get out you have first of all to free him from the misleading influence of
the question.
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Page 170
Look at a written word, say "read",--"It isn't just a scribble, it's 'read'", I should like to say, "it has one definite
physiognomy". But what is it that I am really saying about it? What is this statement, straightened out? "The word
falls", one is tempted to explain, "into a mould of my mind long prepared for it". But as I don't perceive both the
word and a mould, the metaphor of the word's fitting a mould can't allude to an experience of comparing the hollow
and the solid shape before they are fitted together, but rather to an experience of seeing the solid shape accentuated
by a particular background.
i) would be the picture of the hollow and the solid shape before they are fitted together. We here see two circles and
can compare them. ii) is the picture of the solid in the hollow. There is only one circle, and what we call the mould
only accentuates, or as we sometimes said, emphasizes it.
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I am tempted to say, "This isn't just a scribble, but it's this particular face".--But I can't say, "I see this as this
face", but ought to say "I see this as a face". But I feel I want to say, "I don't see this as a face, I see it as this face".
But in the second half of this sentence the word "face" is redundant, and it should have run, "I don't see this as a
face, I see it like this".
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Suppose I said "I see this scribble like this", and while saying "this scribble" I look at it as a mere scribble,
and while saying "like this", I see the face,--this would come to something like saying "What at one time appears to
me like this, at another appears to me like that", and here the "this" and the "that" would be accompanied by the two
different ways of seeing.--But we must ask ourselves in what game is this sentence with the processes
accompanying it to be used. E.g., whom am I telling this? Suppose the answer is "I'm saying it to myself". But that is
not enough. We are here in the grave danger of believing that we know what to do with a sentence if it looks more or
less like one of the common sentences of our language. But here in order not to be deluded we have to ask
ourselves: What is the use, say, of the words "this" and "that"?--or rather, What are the different uses which we
make of them? What we call their meaning is not anything which they have got in them or which is fastened to them
irrespective of what use we make of them. Thus it is one use of the
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word "this" to go along with a gesture pointing to something: We say "I am seeing the square with the diagonals like
this", pointing to a swastika. And referring to the square with diagonals I might have said, "What at one time appears
to me like this
at another time appears to me like that
And this is certainly not the use we made of the sentence in the above case.--One might think the whole difference
between the two cases is this, that in the first the pictures are mental, in the second, real drawings. We should here
ask ourselves in what sense we can call mental images pictures, for in some ways they are comparable to drawn or
painted pictures, and in others not. It is, e.g., one of the essential points about the use of a 'material' picture that we
say that it remains the same not only on the ground that it seems to us to be the same, that we remember that it
looked before as it looks now. In fact we shall say under certain circumstances that the picture hasn't changed
although it seems to have changed; and we say it hasn't changed because it has been kept in a certain way, certain
influences have been kept out. Therefore the expression "The picture hasn't changed" is used in a different way
when we talk of a material picture on the one hand, and of a mental one on the other. Just as the statement "These
ticks follow at equal intervals" has got one grammar if the ticks are the tick of a pendulum and the criterion for their
regularity is the result of measurements which we have made on our apparatus, and another grammar if the ticks are
ticks which we imagine. I might for instance ask the question: When I said to myself "What at one time appears to
me like this, at another...", did I recognize the two aspects, this and that, as the same which I got on previous
occasions? Or were they new to me and I tried to remember them for future occasions? Or was all that I meant to
say "I can change the aspect of this figure"?
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19. The danger of delusion which we are in becomes most clear if we propose to ourselves to give the aspects
'this' and 'that' names,
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say A and B. For we are most strongly tempted to imagine that giving a name consists in correlating in a peculiar and
rather mysterious way a sound (or other sign) with something. How we make use of this peculiar correlation then
seems to be almost a secondary matter. (One could almost imagine that naming was done by a peculiar sacramental
act, and that this produced some magic relation between the name and the thing.)
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But let us look at an example; consider this language game: A sends B to various houses in their town to
fetch goods of various sorts from various people. A gives B various lists. On top of every list he puts a scribble, and
B is trained to go to that house on the door of which he finds the same scribble, this is the name of the house. In the
first column of every list he then finds one or more scribbles which he has been taught to read out. When he enters
the house he calls out these words, and every inhabitant of the house has been trained to run up to him when a
certain one of these sounds is called out, these sounds are the names of the people. He then addresses himself to
each one of them in turn and shows to each two consecutive scribbles which stand on the list against his name. The
first of these two, people of that town have been trained to associate with some particular kind of object, say, apples.
The second is one of a series of scribbles which each man carries about him on a slip of paper. The person thus
addressed fetches say, five apples. The first scribble was the generic name of the objects required, the second, the
name of their number.
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What now is the relation between a name and the object named, say, the house and its name? I suppose we
could give either of two answers. The one is that the relation consists in certain strokes having been painted on the
door of the house. The second answer I meant is that the relation we are concerned with is established, not just by
painting these strokes on the door, but by the particular role which they play in the practice of our language as we
have been sketching it.--Again, the relation of the name of a person to the person here consists in the person having
been trained to run up to someone who calls out the name; or again, we might say that it consists in this and the
whole of the usage of the name in the language game.
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Look into this language game and see if you can find the mysterious relation of the object and its name.--The
relation of name and object we may say, consists in a scribble being written on an object (or some other such very
trivial relation), and that's all there is to it. But we are not satisfied with that, for we feel that a scribble written on an
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object in itself is of no importance to us, and interests us in no way. And this is true; the whole importance lies in the
particular use we make of the scribble written on the object, and we, in a sense, simplify matters by saying that the
name has a peculiar relation to its object, a relation other than that, say, of being written on the object, or of being
spoken by a person pointing to an object with his finger. A primitive philosophy condenses the whole usage of the
name into the idea of a relation, which thereby becomes a mysterious relation. (Compare the ideas of mental
activities, wishing, believing, thinking, etc., which for the same reason have something mysterious and inexplicable
about them.)
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Now we might use the expression "The relation of name and object does not merely consist in this kind of
trivial, 'purely external', connection", meaning that what we call the relation of name and object is characterized by
the entire usage of the name; but then it is dear that there is no one relation of name to object, but as many as there
are uses of sounds or scribbles which we call names.
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We can therefore say that if naming something is to be more than just uttering a sound while pointing to
something, there must also be, in some form or other, the knowledge of how in the particular case the sound or
scratch is to be used.
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Now when we proposed to give the aspects of a drawing names, we made it appear that by seeing the
drawing in two different ways, and each time saying something, we had done more than performing just this
uninteresting action; whereas we now see that it is the usage of the 'name' and in fact the detail of this usage which
gives the naming its peculiar significance.
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It is therefore not an unimportant question, but a question about the essence of the matter; "Are 'A' and 'B' to
remind me of these aspects; can I carry out such an order as 'See this drawing in the aspect A'; are there, in some
way, pictures of these aspects correlated with the names 'A' and 'B' (like
and );
are 'A' and 'B' used in communicating with other people, and what exactly is the game played with them?"
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Page 174
When I say "I don't see mere dashes (a mere scribble) but a face (or word) with this particular
physiognomy", I don't wish to assert any general characteristic of what I see, but to assert that I see that particular
physiognomy which I do see. And it is obvious that here my expression is moving in a circle. But this is so because
really the particular physiognomy which I saw ought to have entered my proposition.-When I find that "In reading a
sentence, a peculiar experience goes on all the while", I have actually to read over a fairly long stretch to get the
peculiar impression which makes one say this.
Page 174
I might then have said "I find that the same experience goes on all the time", but I wished to say: "I don't just
notice that it's the same experience throughout, I notice a particular experience". Looking at a uniformly coloured
wall I might say, "I don't just see that it has the same colour all over, but I see a particular colour". But in saying this
I am mistaking the function of a sentence.--It seems that you wish to specify the colour you see, but not by saying
anything about it, nor by comparing it with a sample,--but by pointing to it; using it at the same time as the sample
and that which the sample is compared with.
Page 174
Consider this example: You tell me to write a few lines, and while I am doing so you ask "Do you feel
something in your hand while you are writing?" I say, "Yes, I have a peculiar feeling".--Can't I say to myself when I
write, "I have this feeling"? Of course I can say it, and while saying "this feeling", I concentrate on the feeling.--But
what do I do with this sentence? What use is it to me? It seems that I am pointing out to myself what I am
feeling,--as though my act of concentration was an 'inward' act of pointing, one which no one else but me is aware
of, this however is unimportant. But I don't point to the feeling by attending to it. Rather, attending to the feeling
means producing or modifying it. (On the other hand, observing a chair does not mean producing or modifying the
chair.)
Page 174
Our sentence "I have this feeling while I'm writing" is of the kind of the sentence "I see this". I don't mean
the sentence when it is used to inform someone that I am looking at the object which I am pointing to, nor when it is
used, as above, to convey to someone that I see a certain drawing in the way A and not in the way B. I mean the
sentence, "I see this", as it is sometimes contemplated by us when we are brooding over certain philosophical
problems. We are then, say, holding on to a particular visual impression by staring at some object, and we feel it is
most natural to say to ourselves "I see this", though we know of no further use we can make of this sentence.
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Page 175
20. 'Surely it makes sense to say what I see, and how better could I do this than by letting what I see speak
for itself!'
Page 175
But the words "I see" in our sentence are redundant. I don't wish to tell myself that it is I who see this, nor
that I see it. Or, as I might put it, it is impossible that I should not see this. This comes to the same as saying that I
can't point out to myself by a visual hand what I am seeing; as this hand does not point to what I see but is part of
what I see.
Page 175
It is as though the sentence was singling out the particular colour I saw; as if it presented it to me.
Page 175
It seems as though the colour which I see was its own description.
Page 175
For the pointing with my finger was ineffectual. (And the looking is no pointing, it does not, for me, indicate
a direction, which would mean contrasting a direction with other directions.)
Page 175
What I see, or feel, enters my sentence as a sample does; but no use is made of this sample; the words of my
sentence don't seem to matter, they only serve to present the sample to me.
Page 175
I don't really speak about what I see, but to it.
Page 175
I am in fact going through the acts of attending which could accompany the use of a sample. And this is
what makes it seem as though I was making use of a sample. This error is akin to that of believing that an ostensive
definition says something about the object to which it directs our attention.
Page 175
When I said "I am mistaking the function of a sentence" it was because by its help I seemed to be pointing
out to myself which colour it is I see, whereas I was just contemplating a sample of a colour. It seemed to me that
the sample was the description of its own colour.
Page 175
21. Suppose I said to someone: "Observe the particular lighting of this room".--Under certain circumstances
the sense of this order will be quite clear, e.g., if the walls of the room were red with the setting sun. But suppose at
any other time when there is nothing striking about the lighting I said "Observe the particular lighting of this
room":--Well, isn't there a particular lighting? So what is the difficulty about observing it? But the person who was
told to observe the lighting when there was nothing striking about it would probably look about the room and say
"Well, what about it?" Now I might go on and say "It is exactly the same lighting as yesterday at this hour",
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or "It is just this slightly dim light which you see in this picture of the room".
Page 176
In the first case, when the room was lit a striking red, you could have pointed out the peculiarity which you
were meant, though not explicitly told, to observe. You could, e.g., have used a sample of the particular colour in
order to do so. We shall in this case be inclined to say that a peculiarity was added to the normal appearance of the
room.
Page 176
In the second case, when the room was just ordinarily lighted and there was nothing striking about its
appearance, you didn't know exactly what to do when you were told to observe the lighting of the room. All you
could do was to look about you waiting for something further to be said which would give the first order its full
sense.
Page 176
But wasn't the room, in both cases, lit in a particular way? Well, this question, as it stands, is senseless, and
so is the answer "It was...". The order "Observe the particular lighting of this room", does not imply any statement
about the appearance of this room. It seemed to say: "This room has a particular lighting, which I need not name;
observe it!" The lighting referred to, it seems, is given by a sample, and you are to make use of the sample; as you
would be doing in copying the precise shade of a colour sample on a palette. Whereas the order is similar to this:
"Get hold of this sample!"
Page 176
Imagine yourself saying "There is a particular lighting which I'm to observe". You could imagine yourself in
this case staring about you in vain, that is, without seeing the lighting.
Page 176
You could have been given a sample, e.g., a piece of colour material, and been asked: "Observe the colour of
this patch".-And we can draw a distinction between observing, attending to, the shape of the sample and attending to
its colour. But, attending to the colour can't be described as looking at a thing which is connected with the sample,
rather, as looking at the sample in a peculiar way.
Page 176
When we obey the order, "Observe the colour...", what we do is to open our eyes to colour. "Observe the
colour..." doesn't mean "See the colour you see". The order, "Look at so and so", is of the kind, "Turn your head in
this direction"; what you will see when you do so does not enter this order. By attending, looking, you produce the
impression; you can't look at the impression.
Page 176
Suppose someone answered to our order: "All right, I am now observing the particular lighting this room
has",--this would sound as though he could point out to us which lighting it was. The order, that is to say, may seem
to have told you to do something with this
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particular lighting, as opposed to another one (like "Paint this lighting, not that"). Whereas you obey the order by
taking in lighting, as opposed to dimensions, shapes, etc.
Page 177
(Compare, "Get hold of the colour of this sample" with "Get hold of this pencil", i.e., there it is, take hold of
it.)
Page 177
I return to our sentence: "this face has a particular expression". In this case too I did not compare or contrast
my impression with anything, I did not make use of the sample before me. The sentence was an utterance†1 of a
state of attention.
Page 177
What has to be explained is this: Why do we talk to our impression?--You read, put yourself into a state of
attention and say: "Something peculiar happens undoubtedly". You are inclined to go on: "There is a certain
smoothness about it"; but you feel that this is only an inadequate description and that the experience can only stand
for itself. "Something peculiar happens undoubtedly" is like saying, "I have had an experience". But you don't wish
to make a general statement independent of the particular experience you have had, but rather a statement into
which this experience enters.
Page 177
You are under an impression. This makes you say "I am under a particular impression", and this sentence
seems to say, to yourself at least, under what impression you are. As though you were referring to a picture ready in
your mind, and said "This is what my impression is like". Whereas you have only pointed to your impression. In our
case (p. 174), saying "I notice the particular colour of this wall" is like drawing, say, a black rectangle enclosing a
small patch of the wall and thereby designating that patch as a sample for further use.
Page 177
When you read, as it were attending closely to what happened in reading, you seemed to be observing
reading as under a magnifying glass and to see the reading process. (But the case is more like that of observing
something through a coloured glass.) You think you have noticed the process of reading, the particular way in which
signs are translated into spoken words.
Page 177
22. I have read a line with a peculiar attention; I am impressed by the reading, and this makes me say that I
have observed something besides the mere seeing of the written signs and the speaking of words. I have also
expressed it by saying that I have noticed a particular atmosphere round the seeing and speaking. How such a
metaphor as that embodied in the last sentence can come to suggest
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itself to me may be seen more clearly by looking at this example: If you heard sentences spoken in a monotone, you
might be tempted to say that the words were all enshrouded in a particular atmosphere. But wouldn't it be using a
peculiar way of representation to say that speaking the sentence in a monotone was adding something to the mere
saying of it? Couldn't we even conceive speaking in a monotone as the result of taking away from the sentence its
inflexion? Different circumstances would make us adopt different ways of representation. If, e.g., certain words had
to be read out in a monotone, this being indicated by a staff and a sustained note beneath the written words, this
notation would very strongly suggest the idea that something had been added to the mere speaking of the sentence.
Page 178
I am impressed by the reading of a sentence, and I say the sentence has shown me something, that I have
noticed something in it. This made me think of the following example: A friend and I once looked at beds of pansies.
Each bed showed a different kind. We were impressed by each in turn. Speaking about them my friend said "What a
variety of colour patterns, and each says something". And this was just what I myself wished to say.
Page 178
Compare such a statement with this: "Every one of these men says something".--
Page 178
If one had asked what the colour pattern of the pansy said, the right answer would have seemed to be that it
said itself. Hence we could have used an intransitive form of expression, say "Each of these colour patterns
impresses one".
Page 178
It has sometimes been said that what music conveys to us are feelings of joyfulness, melancholy, triumph,
etc., etc. and what repels us in this account is that it seems to say that music is an instrument for producing in us
sequences of feelings. And from this one might gather that any other means of producing such feelings would do for
us instead of music.--To such an account we are tempted to reply "Music conveys to us itself!"
Page 178
It is similar with such expressions as "Each of these colour patterns impresses one". We feel we wish to
guard against the idea that a colour pattern is a means to producing in us a certain impression--the colour pattern
being like a drug and we interested merely in the effect this drug produces.--We wish to avoid any form of
expression which would seem to refer to an effect produced by an object on a subject. (Here we are bordering on
the problem of idealism and realism and on the problem whether statements of aesthetics are subjective or
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objective.) Saying, "I see this and am impressed" is apt to make it seem as though the impression was some feeling
accompanying the seeing, and that the sentence said something like "I see this and feel a pressure".
Page 179
I could have used the expression "Each of these colour patterns has meaning"; but I didn't say "has
meaning", for this would provoke the question, "What meaning?", which in the case we are considering is senseless.
We are distinguishing between meaningless patterns and patterns which have meaning; but there is no such
expression in our game as "This pattern has the meaning so and so". Nor even the expression "These two patterns
have different meanings", unless this is to say: "These are two different patterns and both have meaning".
Page 179
It is easy to understand though why we should be inclined to use the transitive form of expression. For let us
see what use we make of such an expression as "This face says something", that is, what the situations are in which
we use this expression, what sentences would precede or follow it (what kind of conversation it is a part of). We
should perhaps follow up such a remark by saying, "Look at the line of these eyebrows" or "The dark eyes and the
pale face!", these expressions would draw attention to certain features. We should in the same connection use
comparisons, as for instance, "The nose is like a beak",--but also such expressions as "The whole face expresses
bewilderment", and here we have used "expressing" transitively.
Page 179
23. We can now consider sentences which, as one might say, give an analysis of the impression we get, say,
from a face. Take such a statement as, "The particular impression of this face is due to its small eyes and low
forehead". Here the words "the particular impression" may stand for a certain specification, e.g., "the stupid
expression". Or, on the other hand, they may mean 'what makes this expression a striking one' (i.e., an extraordinary
one); or, 'what strikes one about this face' (i.e., 'what draws one's attention'). Or again, our sentence may mean "If
you change these features in the slightest the expression will change entirely (whereas you might change other
features without changing the expression nearly so much)". The form of this statement, however, mustn't mislead us
into thinking that there is in every case a supplementing statement of the form "First the expression was this, after
the change it's that". We can, of course, say "Smith frowned, and his expression changed from this to that",
pointing, say, at two drawings
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of his face.--(Compare with this the two statements: "He said these words", and "His words said something".)
Page 180
When, trying to see what reading consisted in, I read a written sentence, let the reading of it impress itself
upon me, and said that I had a particular impression, one could have asked me such a question as whether my
impression was not due to the particular handwriting. This would be asking me whether my impression would not
be a different one if the writing had been a different one, or say, if each word of the sentence were written in a
different handwriting. In this sense we could also ask whether that impression wasn't due after all to the sense of the
particular sentence which I read. One might suggest: Read a different sentence (or the same one in a different
handwriting) and see if you would still say that you had the same impression. And the answer might be: "Yes, the
impression I had was really due to the handwriting".--But this would not imply that when I first said the sentence
gave me a particular impression I had contrasted one impression with another, or that my statement had not been of
the kind "This sentence has its own character". This will get clearer by considering the following example: Suppose
we have three faces drawn side by side:
I contemplate the first one, saying to myself "This face has a peculiar expression". Then I am shown the second one
and asked whether it has the same expression. I answer "Yes". Then the third one is shown to me and I say "It has a
different expression". In my two answers I might be said to have distinguished the face and its expression: for b) is
different from a) and still I say they have the same expression, whereas the difference between c) and a) corresponds
to a difference of expression; and this may make us think that also in my first utterance I distinguished between the
face and its expression.
Page 180
24. Let us now go back to the idea of a feeling of familiarity, which arises when I see familiar objects.
Pondering about the question whether there is such a feeling or not, we are likely to gaze at some object and say,
"Don't I have a particular feeling when I look at my old coat and hat?" But to this we now answer: What feeling do
you compare this with, or oppose it to? Should you say that your old coat gives you the same feeling as your old
friend A with whose
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appearance too you are well acquainted, or that whenever you happened to look at your coat you get that feeling,
say of intimacy and warmth?
Page 181
'But is there no such thing as a feeling of familiarity?'--I should say that there are a great many different
experiences, some of them feelings, which we might call "experiences (feelings) of familiarity".
Page 181
Different experiences of familiarity: a) Someone enters my room, I haven't seen him for a long time, and
didn't expect him. I look at him, say or feel "Oh, it's you".--Why did I in giving this example say that I hadn't seen
the man for a long time? Wasn't I setting out to describe experiences of familiarity? And whatever the experience
was I alluded to, couldn't I have had it even if I had seen the man half an hour ago? I mean, I gave the circumstances
of recognizing the man as a means to the end of describing the precise situation of the recognition. One might object
to this way of describing the experience, saying that it brought in irrelevant things, and in fact wasn't a description
of the feeling at all. In saying this one takes as the prototype of a description, say, the description of a table, which
tells you the exact shape, dimensions, the material which it is made of, and its colour. Such a description one might
say pieces the table together. There is on the other hand a different kind of description of a table, such as you might
find in a novel, e.g., "It was a small rickety table decorated in Moorish style, the sort that is used for smoker's
requisites". Such a description might be called an indirect one; but if the purpose of it is to bring a vivid image of the
table before your mind in a flash, it might serve this purpose incomparably better than a detailed 'direct'
description.--Now if I am to give the description of a feeling of familiarity or recognition,--what do you expect me to
do? Can I piece the feeling together? In a sense of course I could, giving you many different stages and the way my
feelings changed. Such detailed descriptions you can find in some of the great novels. Now if you think of
descriptions of pieces of furniture as you might find them in a novel, you see that to this kind of description you can
oppose another making use of drawings, measures such as one should give to a cabinet maker. This latter kind one is
inclined to call the only direct and complete description (though this way of expressing ourselves shows that we
forget that there are certain purposes which the 'real' description does not fulfil). These considerations should warn
you not to think that there is one real and direct description of, say, the feeling of recognition as opposed to the
'indirect' one which I have given.
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Page 182
b) The same as a), but the face is not familiar to me immediately. After a little, recognition 'dawns upon me'. I
say, "Oh, it's you", but with totally different inflexion than in a). (Consider tone of voice, inflexion, gestures, as
essential parts of our experience, not as inessential accompaniments or mere means of communication. (Compare
pp. 144-6.)) c) There is an experience directed towards people or things which we see every day when suddenly we
feel them to be 'old acquaintances' or 'good old friends'; one might also describe the feeling as one of warmth or of
being at home with them. d) My room with all the objects in it is thoroughly familiar to me. When I enter it in the
morning do I greet the familiar chairs, tables, etc., with a feeling of "Oh, hello!"? or have such a feeling as described
in c)? But isn't the way I walk about in it, take something out of a drawer, sit down, etc., different from my behaviour
in a room I don't know? And why shouldn't I say therefore, that I had experiences of familiarity whenever I lived
amongst these familiar objects? e) Isn't it an experience of familiarity when on being asked "Who is this man?" I
answer straight away (or after some reflection) "It is so and so"? Compare with this experience, f), that of looking at
the written word "feeling" and saying "This is A's handwriting" and on the other hand g) the experience of reading
the word, which also is an experience of familiarity.
Page 182
To e) one might object, saying that the experience of saying the man's name was not the experience of
familiarity, that he had to be familiar to us in order that we might know his name, and that we had to know his name
in order that we might say it. Or, we might say "Saying his name is not enough, for surely we might say the name
without knowing that it was his name". And this remark is certainly true if only we realize that it does not imply that
knowing the name is a process accompanying or preceding saying the name.
Page 182
25. Consider this example: What is the difference between a memory image, an image that comes with
expectation, and say, an image of a daydream. You may be inclined to answer, "There is an intrinsic difference
between the images".--Did you notice that difference, or did you only say there was one because you think there
must be one?
Page 182
But surely I recognize a memory image as a memory image, an image of a daydream as an image of a
daydream, etc.!--Remember that you are sometimes doubtful whether you actually saw a certain
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event happening or whether you dreamt it, or just had heard of it and imagined it vividly. But apart from that, what
do you mean by "recognizing an image as a memory image"? I agree that (at least in most cases) while an image is
before your mind's eye you are not in a state of doubt as to whether it is a memory image, etc. Also, if asked
whether your image was a memory image, you would (in most cases) answer the question without hesitation. Now
what if I asked you "When do you know what sort of an image it is?" Do you call knowing what sort of image it is
not being in a state of doubt, not wondering about it? Does introspection make you see a state or activity of mind
which you would call knowing that the image was a memory image, and which takes place while the image is before
your mind?--Further, if you answer the question what sort of image it was you had, do you do so by, as it were,
looking at the image and discovering a certain characteristic in it (as though you had been asked by whom a picture
was painted, looked at it, recognized the style, and said it was a Rembrandt)?
Page 183
It is easy, on the other hand, to point out experiences characteristic of remembering, expecting, etc.,
accompanying the images, and further differences in the immediate or more remote surrounding of them. Thus we
certainly say different things in the different cases, e.g., "I remember his coming into my room", "I expect his
coming into my room", "I imagine his coming into my room".-"But surely this can't be all the difference there is!" It
isn't all: There are the three different games played with these three words surrounding these statements.
Page 183
When challenged: do we understand the word "remember", etc.? is there really a difference between the
cases besides the mere verbal one? our thoughts move in the immediate surroundings of the image we had or the
expression we used. I have an image of dining in Hall with T. If asked whether this is a memory image, I say "Of
course", and my thoughts begin to move on paths starting from this image. I remember who sat next to us, what the
conversation was about, what I thought about it, what happened to T later on, etc., etc.
Page 183
Imagine two different games both played with chess men on a chess board. The initial positions of both are
alike. One of the games is always played with red and green pieces, the other with black and white. Two people are
beginning to play, they have the chess board between them with the green and red pieces in position. Someone asks
them "Do you know what game you're intending to play?" A player answers "Of course; we are playing no. 2".
"What is the
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difference now between playing no. 2 and no. 1?"--"Well, there are red and green pieces on the board and not black
and white ones, also we say that we are playing no. 2".--"But this couldn't be the only difference; don't you
understand what 'no. 2' means and what game the red and green pieces stand for?" Here we are inclined to say
"Certainly I do", and to prove this to ourselves we actually begin to move the pieces according to the rules of game
no. 2. This is what I should call moving in the immediate surrounding of our initial position.
Page 184
But isn't there also a peculiar feeling of pastness characteristic of images as memory images? There certainly
are experiences which I should be inclined to call feelings of pastness, although not always when I remember
something is one of these feelings present.--To get clear about the nature of these feelings it is again very useful to
remember that there are gestures of pastness and inflexions of pastness which we can regard as representing the
experiences of pastness.
Page 184
I will examine one particular case, that of a feeling which I shall roughly describe by saying it is the feeling of
'long, long ago'. These words and the tone in which they are said are a gesture of pastness. But I will specify the
experience which I mean still further by saying that it is that corresponding to a certain tune (Davids Bündler
Tänze--"Wie aus weiter Ferne"). I'm imagining this tune played with the right expression and thus recorded, say, for
a gramophone. Then this is the most elaborate and exact expression of a feeling of pastness which I can imagine.
Page 184
Now should I say that hearing this tune played with this expression is in itself that particular experience of
pastness, or should I say that hearing the tune causes the feeling of pastness to arise and that this feeling
accompanies the tune? I.e., can I separate what I call this experience of pastness from the experience of hearing the
tune? Or, can I separate an experience of pastness expressed by a gesture from the experience of making this
gesture? Can I discover something, the essential feeling of pastness, which remains after abstracting all those
experiences which we might call the experiences of expressing the feeling?
Page 184
I am inclined to suggest to you to put the expression of our experience in place of the experience. 'But these
two aren't the same'. This is certainly true, at least in the sense in which it is true to say that a railway train and a
railway accident aren't the same thing. And yet there is a justification for talking as though the expression "the
gesture
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'long, long ago'", and the expression "the feeling 'long, long ago'" had the same meaning. Thus I could give the rules
of chess in the following way: I have a chess board before me with a set of chess men on it. I give rules for moving
these particular chess men (these particular pieces of wood) on this particular board. Can these rules be the rules of
the game of chess? They can be converted into them by the usage of a single operator, such as the word "any". Or,
the rules for my particular set may stand as they are and be made into rules of the game of chess by changing our
standpoint towards them.
Page 185
There is the idea that the feeling, say, of pastness, is an amorphous something in a place, the mind, and that
this something is the cause or effect of what we call the expression of feeling. The expression of feeling then is an
indirect way of transmitting the feeling. And people have often talked of a direct transmission of feeling which
would obviate the external medium of communication..
Page 185
Imagine that I tell you to mix a certain colour and I describe the colour by saying that it is that which you get
if you let sulphuric acid react on copper. This might be called an indirect way of communicating the colour I meant.
It is conceivable that the reaction of sulphuric acid on copper under certain circumstances does not produce the
colour I wished you to mix, and that on seeing the colour you had got I should have to say "No, it's not this", and to
give you a sample.
Page 185
Now can we say that the communication of feelings by gestures is in this sense indirect? Does it make sense
to talk of a direct communication as opposed to that indirect one? Does it make sense to say "I can't feel his
toothache, but if I could I'd know what he feels like"?
Page 185
If I speak of communicating a feeling to someone else, mustn't I in order to understand what I say know
what I shall call the criterion of having succeeded in communicating?
Page 185
We are inclined to say that when we communicate a feeling to someone, something which we can never
know happens at the other end. All that we can receive from him is again an expression. This is closely analogous to
saying that we can never know when in Fizeau's experiment the ray of light reaches the mirror.
FOOTNOTES
Page 8
†1 See pp. 16, 44ff.
Page 11
†1 This promise is not kept.--Edd.
Page 12
†1 This he never does.--Edd.
Page 16
†1 See p. 47 for a few further remarks on this topic.--Edd.
Page 20
†1 Theaetetus 146D-7C.
Page 21
†1 See Tractatus 5.02.
Page 22
†1 Cf. Russell, Analysis of Mind, III.
Page 31
†1 He does not do this.--Edd.
Page 82
†1 Here Wittgenstein uses "sentence" and "proposition" interchangeably for the German "Satz".--Edd.
Page 102
†1 German lassen, i.e. 'make'.--Edd.
Page 139
†1 German "nach", i.e. "according to" or "in the light of".--Edd.
Page 161
†1 Language game no. 30 in Part I of the Brown Book.
Page 165
†1 § 15, Brown Book, Part II.
Page 177
†1 I.e. Äußerung. See Philosophical Investigations, § 256.--Edd.
REMARKS ON THE FOUNDATIONS OF MATHEMATICS
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Titlepage
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REMARKS ON THE FOUNDATIONS OF MATHEMATICS
By
LUDWIG WITTGENSTEIN
Edited by
G. H. von WRIGHT, R. RHEES, G. E. M. ANSCOMBE
Translated by
G. E. M. ANSCOMBE
BASIL BLACKWELL
OXFORD
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Copyright page
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First printed 1956
Second edition 1967
Third edition, revised and reset, 1978
Reprinted 1989, 1994, 1998
ISBN 0-631-12505-1 (Pbk)
Printed in Great Britain by Athenæum Press Ltd, Gateshead, Tyne & Wear
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TABLE OF CONTENTS
PART I
Page 5
1-5. Following a rule (Cf. Philosophical Investigations §§ 189-90 ff.).--Steps determined by a formula (1-2).
Continuation of a series (3). Inexorability of mathematics; mathematics and truth (4-5). Remark on measuring (5).
35
Page 5
6-23. Logical inference.--The word "all"; inference from '(x).fx' to 'fa' (10-16). Inference and truth (17-23).
39
Page 5
24-74. Proof.--Proof as pattern or model (paradigm). Example: hand and pentacle (25 ff.). Proof as picture of an
experiment (36). Example: 100 marbles (36 ff.). Construction of figures out of their parts (42-72). Mathematical
surprise. Proof and conviction. Mathematics and essence (32, 73, 74). The depth of the essence: the deep need for
the convention (74). 46
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75-105. Calculation and experiment.--The 'unfolding' of mathematical properties. Example of 100 marbles (75, 86,
88). Unfolding the properties of a polygon (76), and of a chain (79, 80, 91, 92). Measuring (93, 94). Geometrical
examples (96-98). Internal properties and relations (102-105); examples from the logic of colour.
65
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106-112. Mathematical belief. 76
Page 5
113-142. Logical compulsion.--In what sense does logical argument compel? (113-117). The inexorability of logic
compared with that of the law (118). The 'logical machine' and the kinematics of rigid bodies' (119-125). 'The
hardness of the logical must' (121). The machine as symbolizing its way of working (122). The employment
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of a word grasped in a flash (123-130). Possibility as a shadow of reality (125). The employment of a word
misunderstood and interpreted as a queer process (127). (For 122-130 see also Philosophical Investigations §§
191-197.) The laws of logic as 'laws of thought' (131-133). Logic interprets 'proposition' and 'language' (134). Going
wrong in a calculation (136-137). Remark on measuring (140). Logical impossibility (141). "What we are supplying
is really remarks on the natural history of man" (142). 79
Page 6
143-156. Foundation of a calculating procedure and of a logical inference.--Calculating without propositions
(143-145). Example: sale of timber (143-152). "Are our laws of inference eternal and unalterable"? (155). Logic
precedes truth (156). 92
Page 6
157-170. Mathematics, logic and experience.--Proof and experiment (157-169). What about mathematics is logic: it
moves in the rules of our language (165). The mathematician is an inventor, not a discoverer (168). The grounds of
the logical must (169-171). 96
Appendix I
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1-7. Two sorts of negation. One cancels out when repeated, the other is still a negation. How we mean the one or the
other sort in double negation. Meaning tested by the expression of meaning (3). Negation compared with rotation
through 180° in geometry (1, 6, 7). 102
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8-10. We can imagine a 'more primitive' logic, in which only sentences containing no negation can be negated (8).
The question whether negation has the same meaning for these people as for us (9). Does the figure one mean the
same both times in "This rod is 1 metre long" and "Here is 1 soldier" (10)? 105
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12-13. Two systems for measuring length. 1 foot = 1W, but 2W = 4 foot, etc. Do "W" and "foot" mean the same?
106
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16. Meaning as the function of a word in a sentence. 107
Page 7
17-27. Use of the word "is" as copula and as sign of identity. The personal union by the same word a mere accident.
Essential and inessential features of a notation. Comparison with the role of a piece in a game. A game does not just
have rules, it has a point (20). 108
Appendix II
Page 7
1-13. The surprising may play completely different roles in mathematics in two different ways.--A state of affairs
may appear astonishing because of the kind of way it is represented. Or the emergence of a particular result may be
seen as in itself surprising. The second kind of surprise not legitimate. Criticism of the conception that mathematical
demonstration brings something hidden to light (2). "No mystery here" means: Look around (4). A calculation
compared with a way of turning up a card (5). When we don't command a view over what we have been doing, it
strikes us as mysterious. We adopt a particular form of expression and become dominated by it in acting and
thinking (8). 111
Appendix III
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1-4. Kinds of proposition.--Arithmetic done without propositions (4). 116
Page 7
5-7. Truth and provability in the system of Principia Mathematica. 117
Page 7
8-19. Discussion of a proposition 'P' which asserts its own unprovability
Page Break 8
in the system of Principia Mathematica.--The role of contradiction in the language-game (11-14, 17).
118
Page 8
20. The propositions of logic. 'Proposition' and 'proposition-like formation'.
123
PART II
Page 8
1-22. ("Additions").--The diagonal procedure. What use can be made of the diagonal number? (3) "The motto here
is: Take a wider look round" (6). We should regard with suspicion the result of a calculation expressed in words (7).
The concept 'nondenumerable' (10-13). Comparison of the concepts 'real number' and 'cardinal number' from the
point of view of serial ordering (16-22). 125
Page 8
23. The sickness of a time. 132
Page 8
24-27. Discussion of the proposition "There is no greatest cardinal number." "We say of a licence that it does not
terminate" (26). 133
Page 8
28-34. Irrational numbers. There is no system of irrational numbers (33). But Cantor defines a higher order
difference, namely a difference between a development and a system of developments (34).
133
Page 8
35-39. ℵ0. From our having a use for a kind of numeral that gives the number of terms of an infinite series it doesn't
follow that it also makes any sense to talk about the number of the concept 'infinite series'. There is no grammatical
technique suggesting the use of such an expression. Such a use is not still to be discovered, but takes inventing (38).
135
Page 8
40-48. Discussion of the proposition "Fractions cannot be arranged in a series in order of magnitude."
137
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49-50. How do we compare games? 139
Page 9
51-57. Discussion of the proposition that fractions (number-pairs) can be ordered in an infinite series.
139
Page 9
58-60. "Is the word 'infinite' to be avoided in mathematics?" 141
Page 9
61. Finitism and Behaviourism are similar trends. Both deny the existence of something with a view to getting out of
a muddle. 142
Page 9
62. "What I am doing is not showing up calculations as wrong, but subjecting the interest of calculations to test."
142
PART III
Page 9
1-2. Proof.--Mathematical proof must be perspicuous. Role of definitions (2). 143
Page 9
3-8. Russell's logic and the idea of the reduction of arithmetic to symbolic logic.--The application of the calculation
must take care of itself (4). Proof in the Russellian calculus, in the decimal calculus, and in the stroke calculus. 144
Page 9
9-11. Proof.--Proof as a memorable picture (9). The reproduction of the pattern of a proof (10-11).
149
Page 9
12-20. Russell's logic and the problem of the mutual relation of different calculating techniques.--What is the
invention of the decimal system? (12.) Proof in the Russellian calculus and in the decimal system (13). Signs for
numbers that can, and that cannot, be taken in (16). Relation of abbreviated and unabbreviated calculating
techniques to one another (17-20). 151
Page 9
21-44. Proof.--Identity and reproducibility of a proof (21). The proof
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as model. Proof and experiment (22-24). Proof and mathematical conviction (25-26). In the proof we have won
through to a decision (27). The proved proposition as a rule. It serves to shew us what it makes sense to say (28).
The propositions of mathematics as 'instruments of language' (29). The mathematical must correspond to a track
which I lay down in language (30). The proof introduces a new concept (31). What concept does 'p ⊃ p' produce?
(32). 'p ⊃ p' as pivot of the linguistic method of representation (33). The proof as part of an institution (36).
Importance of the distinction between determining and using a sense (37). Acceptance of a proof; the 'geometrical'
conception of proof (38-40). The proof as adoption of a particular employment of signs (41). "The proof must be a
procedure plain to view" (42). "Logic as foundation of mathematics does not work if only because the cogency of
the logical proof stands and falls with its geometrical cogency" (43). In mathematics we can get away from logical
proofs (44). 158
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45-64. Russell's logic.--Relation between the ordinary and the Russellian technique of proof (45). Criticism of the
conception of logic as the 'foundation' of mathematics. Mathematics is a motley of calculating techniques. The
abbreviated technique as a new aspect of the unabbreviated (46-48). Remark on trigonometry (50). The decimal
notation is independent of calculation with unit strokes (51). Why Russell's logic does not teach us to divide (52).
Why mathematics is not logic (53). Recursive proof (54). Proof and experiment (55). The correspondence of
different calculi; stroke notation and decimal notation (56-57). Several proofs of one and the same proposition; proof
and the sense of a mathematical proposition (58-62). The exact correspondence of a convincing transition in music
and in mathematics (63). 175
Page 10
65-76. Calculation and experiment.--Are the propositions of mathematics anthropological propositions? (65).
Mathematical propositions conceived as prophecies of concordant results of calculating (66). Agreement is part of
the phenomenon of
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calculating (67). If a calculation is an experiment, what in that case is a mistake in calculation? (68). The calculation
as an experiment and as a path (69). A proof subserves mutual understanding. An experiment presupposes it (71).
Mathematics and the science of conditioned calculating reflexes (72). The concept of calculation excludes confusion
(75-76). 192
Page 11
77-90. Contradiction.--A game in which the one who moves first must always win (77). Calculating with (a - a). The
chasms in a calculus are not there if I do not see them (78). Discussion of the heterological paradox (79).
Contradiction regarded from the point of view of the language-game. Contradiction as a 'hidden sickness' of the
calculus (80). Contradiction and the usability of a calculus (81). The consistency proof and the misuse of the idea of
mechanical insurance against contradiction (82-89). "My aim is to change the attitude towards contradiction and the
consistency proof." (82) The role of the proposition: "I must have miscalculated"--the key to understanding the
'foundations' of mathematics (90). 202
PART IV
Page 11
1-7. On axioms.--The self-evidence of axioms (1-3). Self-evidence and use (2-3). Axiom and empirical proposition
(4-5). The negation of an axiom (5). The mathematical proposition stands on four legs, not on three (7). 223
Page 11
8-9. Following a rule.--Description by means of a rule (8). 227
Page 11
10. The arithmetical assumption is not tied to experience. 229
Page 11
11-13. The conception of arithmetic as the natural history of numbers.--Judging experience by means of the picture
(12). 229
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Page 12
14. External relation of the logical (mathematical) proposition. 231
Page 12
15-19. The possibility of doing applied mathematics without pure mathematics.--Mathematics need not be done in
propositions; the centre of gravity may lie in action (15). The commutative law as an example (16-17).
232
Page 12
20. Calculation as a mechanical activity. 234
Page 12
21. The picture as a proof. 235
Page 12
22-27. Intuition. 235
Page 12
26. What is the difference between not calculating and calculating wrong? 236
Page 12
29-33. Proof and mathematical concept formation.--Proof alters concept formation. Concept formation as the limit
of the empirical (29). Proof does not compel, but guides (30). Proof conducts our experience into definite channels
(31, 33). Proof and prediction (33). 237
Page 12
34. The philosophical problem is: how can we tell the truth and at the same time pacify these strong prejudices?
242
Page 12
35-36. The mathematical proposition.--We acknowledge it by turning our back on it (35). The effect of the proof: one
plunges into the new rule (36). 243
Page 12
39, 42. Synthetic character of mathematical propositions.--The distribution of primes as an example (42).
245
Page 12
40. The result set up as equivalent to the operation. 245
Page 12
41. That proof must be perspicuous means that causality plays no part in proof. 246
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Page 13
43-44. Intuition in mathematics. 246
Page 13
47. The mathematical proposition as determination of a concept, following upon the discovery of a new form.
248
Page 13
48. The working of the mathematical machine is only the picture of the working of a machine.
249
Page 13
49. The picture as a proof. 249
Page 13
50-51. Reversal of a word. 250
Page 13
52-53. Mathematical and empirical propositions.--The assumption of a mathematical concept expresses the
confident expectation of certain experiences; but the establishment of this measure is not equivalent to the
expression of the expectations (53). 253
Page 13
55-60. Contradiction.--The liar (58). Contradiction conceived as something supra-propositional, as a monument with
a Janus head enthroned above the propositions of logic (59). 254
PART V
Page 13
1-4. Mathematics as a game and as a machine-like activity.--Does the calculating machine calculate? (2). How far is it
necessary to have a concept of 'proposition' in order to understand Russell's mathematical logic? (4).
257
Page 13
5-8. Is a misunderstanding about the possible application of mathematics any objection to a calculation as part of
mathematics?--Set theory (7). 259
Page 13
9-13. The law of excluded middle in mathematics.--Where there is nothing to base a decision on, we must invent
something in order to give the application of the law of excluded middle a sense. 266
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Page 14
14-16 and 21-23. 'Alchemy' of the concept of infinity and of other mathematical concepts whose application is not
understood.--Infinite predictions (23). 272
Page 14
17-20. The law of excluded middle. The mathematical proposition as a commandment. Mathematical existence.
275
Page 14
24-27. Existence proofs in mathematics.--"The harmful invasion of mathematics by logic" (24; see also 46 and 48).
The mathematically general does not stand to the mathematically particular in the same relation as does the general
to the particular elsewhere (25). Existence proofs which do not admit of the construction of what exists (26-27). 281
Page 14
28. Proof by reductio ad absurdum. 285
Page 14
29-40. Extensional and intensional in mathematics; the Dedekind cut--geometrical illustration of analysis (2q) [[sic,
9?]]. Dedekind's theorem without irrational numbers (30). How does this theorem come by its deep content? (31)
The picture of the number-line (32, 37). Discussion of the concept "cut" (33, 34). The totality of functions is an
unordered totality (38). Discussion of the mathematical concept of a function; extension and intension in analysis
(39-40). 285
Page 14
41. Concepts occurring in 'necessary' propositions must also have a meaning in non-necessary ones.
295
Page 14
42-46. On proof and understanding of a mathematical proposition.--The proof conceived as a movement from one
concept to another (42). Understanding a mathematical proposition (45-46). The proof introduces a new concept.
The proof serves to convince one of something (45). Existence proof and construction (46).
295
Page 14
47. A concept is not essentially a predicate. 299
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Page 15
48. 'Mathematical logic' has completely blinded the thinking of mathematicians and philosophers.
300
Page 15
49. The numerical sign goes along with the sign for a concept and only together with this is it a measure.
300
Page 15
50. On the concept of generality. 300
Page 15
51. The proof shews how the result is yielded. 301
Page 15
52-53. General remarks.--The philosopher is the man who has to cure himself of many sicknesses of the
understanding before he can reach the notions of the healthy human understanding. 301
PART VI
Page 15
1. Proofs give propositions an order. 303
Page 15
2. Only within a technique of transformation is a proof a formal test. The addition of certain numbers is called a
formal test of the numerals, but only because adding is a practised technique. The proof also hangs together with the
application. 303
Page 15
3. When the proposition in application does not seem to be right, the proof must shew one why and how it must be
right. 305
Page 15
4. The proof shews how, and thus why the rule--e.g., that 8 × 9 makes 72--can be used.
305
Page 15
5. The queer thing is that the picture, not the reality, is supposed to prove a proposition.
306
Page 15
6. The Euclidean proof teaches us a technique of finding a prime number between p and p! And we become
convinced that this technique must always lead to a prime number > p. 307
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Page 16
7. The spectator sees the impressive procedure and judges "I realise that it must be like that."--This "must" shews
what kind of instruction he has drawn from the scene. 308
Page 16
8. This must shew that he has adopted a concept, i.e., a method; as opposed to the application of the method.
309
Page 16
9. "Shew me how 3 and 2 make 5." Child and abacus.--One would ask someone "how?" if one wanted to get him to
shew that he understands what is in question here. 310
Page 16
10. "Shew me how..." in distinction from "Shew me that..." 312
Page 16
11. The proof of a proposition doesn't mention the whole system of calculation behind the proposition, which gives
it sense. 313
Page 16
12. The domain of the tasks of philosophy. 314
Page 16
13. Ought we to say that mathematicians don't understand Fermat's last theorem? 314
Page 16
14. What would it be to "shew how there are infinitely many prime numbers"? 315
Page 16
15. The process of copying. The process of construction according to a rule. Must one always have a clear idea
whether his prediction is intended mathematically or otherwise? 316
Page 16
16. The inexorable proposition is that by this rule this number follows after that one. The result of the operation here
turns into the criterion for this operation's having been carried out. So we are able to judge in a new sense whether
someone has followed the rule. 318
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Page 17
17. The learning of a rule.--"It's supposed to keep on as I have shewn him."--What I understand by 'uniform' I would
explain by means of examples. 320
Page 17
18. Here definitions are no help to me. 321
Page 17
19. Paraphrasing the rule only makes it more intelligible to someone who can already obey the paraphrase.
321
Page 17
20. Teaching someone to multiply: we reject different patterns of multiplication with the same initial segment.
322
Page 17
21. It would be nonsense to say: just once in the history of the world, someone followed a rule.--Disputes do not
break out about whether a proceeding has been according to a rule or not.--This belongs to the structure from which
our language goes out to give a description, for example. 322
Page 17
22. As if we had hardened an empirical proposition into a rule and now experience is compared with it and it is used
to judge experience. 324
Page 17
23. In calculating everything depends on whether one calculates right or wrong.--The arithmetical proposition is
withdrawn from experiential checking. 324
Page 17
24. "If I follow the rule, then this is the only number I can get to from that one." That is how I proceed; don't ask for
a reason! 326
Page 17
25. Suppose someone were to work out the multiplication tables, log tables, etc., because he didn't trust them?--Does
it make any difference whether we utter a sentence of a calculus as a proposition of arithmetic or an empirical
proposition? 327
Page 17
26. "According to the rule that I see in this sequence, this is how it continues." Not: according to experience! Rather,
that just is the sense of the rule. 327
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Page 18
27. When I see a rule in the series--can this consist in my seeing an algebraic expression before me? Must it not
belong to a language? 328
Page 18
28. The certainty with which I call the colour "red" is not to be called in question when I give the description. For this
characterises what we call describing.--Following according to a rule is the bottom of our language-game. Because
this (e.g., 252 = 625) is the proceeding upon which we build all judging. 329
Page 18
29. Law, and the empirical proposition that we give this law.--"If I obey the order, I do this!" does not mean: If I
obey the order, I obey the order. So I must have a different way of identifying the "this."
330
Page 18
30. If humans who have been educated in this fashion calculate like this anyway, then what do we need the rule for?
"252 = 625" does not mean that human beings calculate like this, for 252 ≠ 625 would not be the proposition that
humans get not this but a different result.
"Apply the rule to those numbers."--If I want to follow it, have I any choice left? 332
Page 18
31. How far is it possible to describe the function of a rule?
If you aren't master of any rule, all I can do is train you.
But how can I explain the nature of the rule to myself? 333
Page 18
32. What surrounding is requisite for someone to be able to invent chess (e.g.)? Is regularity possible without
repetition? Not: how often must he have calculated right to prove to others that he can calculate, but: how often, to
prove it to himself? 334
Page 18
33. Can we imagine someone's knowing that he can calculate although he has never done any calculating?
335
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Page 19
34. In order to describe the phenomenon of language, one has to describe a practice.
A country that exists for two minutes, and is a projection of part of England, with everything that is going on
in two minutes. Someone is doing the same as a mathematician in England, who is calculating. In this
two-minute-man calculating? 335
Page 19
35. How do I know that the name of this colour is "green"? If I asked other people and they did not agree with me, I
should become totally confused and take them or myself for crazy. Here I react with this word ("green"); and that is
also how I know how I have to follow the rule now.
How a polygon of forces gets drawn according to the given arrows. 336
Page 19
36."The word OBEN has four sounds."--Is someone who counts the letters making an experiment? It may be one.
Compare: (1) The word there has 7 sounds
(2) The sound-picture "Dædalus" has 7 sounds.
The second proposition is tenseless. The employment of the two propositions must be different.
"By counting off the sounds one may obtain an empirical proposition--or again, a rule".
338
Page 19
37. Definitions--new ways of belonging together. 340
Page 19
38. "How can one follow a rule?"--Here we misunderstand the facts that stare us in the face.
341
Page 19
39. It is important that the enormous majority of us agree in certain things.
Languages of different tribes who all had the same vocabulary, but the meanings of the words were
different.--In order to communicate the people would have to agree about meanings--they would have to agree not
merely about definitions, but also in their judgments. 342
Page 19
40. The temptation to say "I can't understand language-game (2)
Page Break 20
because the explanation consists only in examples of the application." 343
Page 20
41. A cave-man who produces regular sequences of signs for himself. We do not say that he is acting according to a
rule because we can form the general expression of such a rule. 344
Page 20
42. Under what kind of circumstances should we say: someone is giving a rule by drawing this figure? Under what
kind of circumstances: that someone is following this rule, when he draws a series of such figures?
345
Page 20
43. Only within a particular technique of acting, speaking, thinking can anyone resolve upon something. (This "can"
is the "can" of grammar.) 345
Page 20
44. Two procedures: (1) deriving number after number in sequence according to an algebraic expression;
(2) this procedure: as he looks at a certain sign, someone has a digit occur to him; when he looks again at the
sign and the digit, again a digit occurs to him, and so on. Acting according to a rule presupposes some kind of
uniformity. 347
Page 20
45. Instruction in acting according to a rule. The effect of "and so on" will be that we almost all count and calculate
the same. It only makes sense to say "and so on" when the other continues in the same way as I.
348
Page 20
46. If something must come out, then it is a ground of judgment that I do not assail. 350
Page 20
47. Is it not enough that this certainty exists? Why should I seek a source for it? 350
Page 20
48. As we use the word "order" and "obey", gestures in less than
Page Break 21
words are engulfed in a set of many relationships. In a strange tribe, is the man whom the rest obey unconditionally
the chief?--What is the difference between making a wrong inference and not inferring at all?
351
Page 21
49. "'That logic belongs to the natural history of mankind'--is not combinable with the logical 'must'." The agreement
of human beings which is a presupposition of the phenomenon of logic, is not an agreement in opinions, much less
in opinions about questions of logic. 352
PART VII
Page 21
1. The role of propositions that treat of measures and are not empirical propositions. Such a proposition (e.g. 12
inches = 1 foot) is embedded in a technique, and so in the conditions of this technique; but it is not a statement of
those conditions. 355
Page 21
2. The role of a rule. It can also be used to make predictions. This depends on properties of the measuring rods and
of the people who use them. 355
Page 21
3. A mathematical proposition--a transformation of the expression. The rule considered from the point of view of
usefulness and from that of dignity. How are two arithmetical expressions supposed to say the same thing? They are
made equivalent in arithmetic. 357
Page 21
4. Someone who learns arithmetic simply by following my examples. If I say, "If you do with these numbers what I
did for you with the others, you will get such-and-such a result"--this seems to be both a prediction and a
mathematical proposition. 359
Page 21
5. The difference between being surprised that the figures on paper
Page Break 22
seem to behave like this and being surprised that this is what comes out. 361
Page 22
6. Isn't the contrast between rules of representation and descriptive propositions one which falls away on every side?
363
Page 22
7. What is common to a mathematical proposition and a mathematical proof, that they should both be called
"mathematical"? 364
Page 22
8. If we regard proving as the transformation of a proposition by an experiment, and the man who does the
calculation as an apparatus, then the intermediate steps are an uninteresting by-product.
364
Page 22
9. Proof as a picture. It is not approval alone which makes this picture into a calculation, but the consensus of
approvals. 365
Page 22
10. Does the sense of the proposition change when a proof has been found? The new proof gives the proposition a
place in a new system. 366
Page 22
11. Let us say we have got some of our results because of a hidden contradiction. Does that make them illegitimate?
Might we not let a contradiction stand? 369
Page 22
12. "A method for avoiding a contradiction mechanically." It is not bad mathematics that is amended here, but a new
bit of mathematics is invented. 371
Page 22
13. Must logical axioms always be convincing? 372
Page 22
14. The people who sometimes cancel out by expressions of value 0. 373
Page 22
15. If the calculation has lost its point for me, as soon as I know that
Page Break 23
I can get any arbitrary result from it--did it have no point as long as I did not know this?
One thinks that contradiction has to be senseless. 374
Page 23
16. What does mathematics need a foundation for?
A good angel will always be necessary. 378
Page 23
17. The practical value of calculating. Calculation and experiment.
A calculation as part of the technique of an experiment.
The activity of calculating may also be an experiment. 379
Page 23
18. Is mathematics supposed to bring facts to light? Does it not take mathematics to determine the character of what
we call a 'fact'? Doesn't it teach us how to enquire after facts?
In mathematics there are no causal connexions, only the connexions of the pattern. 381
Page 23
19. Remarks. 383
Page 23
20. The network of joins in a wall. Why do we call this a mathematical problem?
Does mathematics make experiments with units? 384
Page 23
21. "The proposition that says of itself that it is unprovable"--how is this to be understood?
385
Page 23
22. The construction of a propositional sign from axioms according to rules; it appears that we have demonstrated
the actual sense of the proposition to be false, and at the same time proved it.
387
Page 23
23. The question "How many?"
Measurement and unit. 389
Page 23
24. What we call the mathematical conception of a proposition belongs together with the special place calculation
has among our other activities. 389
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Page 24
25. What is the criterion that here I have followed the paradigm? 391
Page 24
26. Anyone who describes learning how to 'proceed according to the rule' for me, will himself be applying the
expression of a rule in his description and will presuppose that I understand it. 392
Page 24
27. Not allowing contradiction to stand characterizes the technique of our application of truth-functions--a man's
understanding of "and so on" is shown by his saying this and acting thus in certain cases.
393
Page 24
28. Does the "heterological" contradiction shew a logical property of this concept? 395
Page 24
29. A game. And after a certain move any attempt to go on playing proves to be against the rules.
396
Page 24
30. Logical inference is part of a language game.
Logical inference and non-logical inference.
The rules of logical inference cannot be either wrong or right.
They determine the meaning of the signs. 397
Page 24
31. A reasonable procedure with numerals need not be what we call "calculating". 398
Page 24
32. Is not a mathematics with an application that is sheer fantasy, still mathematics? 399
Page 24
33. The formation of concepts may be essential to a great part of mathematics; and have no role in other parts.
399
Page 24
34. A people who do not notice a contradiction, and draw conclusions from it.
Can it be a mathematical task to make mathematics into mathematics? 400
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35. If a contradiction were actually found in arithmetic, this would shew that an arithmetic with such a contradiction
can serve us very well. 401
Page 25
36. "The class of lions is not a lion, but the class of classes is a class." 401
Page 25
37. "I always lie." What part might this sentence play in human life? 404
Page 25
38. Logical inference. Is not a rule something arbitrary?
"It is impossible for human beings to recognize an object as different from itself." 404
Page 25
39. "Correct--i.e. it conforms to the rule." 405
Page 25
40. "Bringing the same"--how can I explain this to someone? 406
Page 25
41. When should we speak of a proof of the existence of '777' in an expansion? 407
Page 25
42. "Concept formation" may mean various things.
The concept of a rule for forming an infinite decimal. 408
Page 25
43. Is it essential to the concept of calculating, that people generally reach this result? 409
Page 25
44. If I ask, e.g., whether a certain body moves according to the equation of a parabola--what does mathematics do
in this case? 410
Page 25
45. Questions about the way in which mathematics forms concepts. 411
Page 25
46. Can one not make mathematical experiments after all? 413
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Page 26
47. The pupil has got hold of the rule when he reacts to it in such-and-such a way. This reaction presupposes a
surrounding of particular circumstances, forms of life and of language. 413
Page 26
48. "The line intimates to me how I am to go." 414
Page 26
49. In some circumstances: the line seems to intimate to him how he is to go, but it isn't a rule. 415
Page 26
50. How are we to decide if he is always doing the same thing? 415
Page 26
51. Whether he is doing the same, or a different thing every time--does not yet determine whether he is obeying a
rule. 416
Page 26
52. If you enumerate examples and then say "and so on", this last expression is not explained in the same way as the
examples. 416
Page 26
53. Suppose an inner voice, a sort of inspiration, tells me how to follow the line. What is the difference between this
procedure and following a rule? 417
Page 26
54. Doing the same thing is linked with following a rule. 418
Page 26
55. Can I play the language-game if I don't have such and such spontaneous reactions? 419
Page 26
56. What we do when following a rule, we see from the point of view of "Always the same thing."
419
Page 26
57. Calculating prodigies who get the right result but can't say how--are we to say they don't calculate?
419
Page 26
58. "Thinking you are following a rule." 420
Page 26
59. How can I explain the word "same"--how do I get the feeling that something lies in my understanding beyond
what I can say? 420
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Page 27
60. What is impalpable about intimation: there is nothing between the rule and my action.
421
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61. Adding shapes. Possibilities in folding a piece of paper. Suppose we did not separate geometrical and physical
possibility?
Might not people in certain circumstances calculate with figures, without a particular result's having to come
out?
If the calculation shews you a causal connexion, you are not calculating.
Mathematics is normative. 422
Page 27
62. The introduction of a new rule of inference as a transition to a new language game. 425
Page 27
63. Observation that a surface is red and blue, but not that it is red. Inferences from this.
Can logic tell us what we must observe? 425
Page 27
64. A surface with stripes of changing colours.
Could implications be observed? 426
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65. Someone says he sees a red and yellow star, but not anything yellow. 428
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66. "I hold to a rule." 429
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67. The mathematical must--the expression of an attitude towards the technique of calculating. The expression of the
fact that mathematics forms concepts. 430
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68. The case of seeing the complex formed from A and B, but seeing neither A nor B.
Can I see A and B, but only observe A ∨ B?
And vice versa. 431
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69. Experiences and timeless propositions. 432
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70. In what sense can a proposition of arithmetic be said to give us a concept? 432
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71. Not every language-game contains something that we want to call a "concept". 433
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72. Proof and picture. 434
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73. A language-game in which there are axioms, proofs and proved propositions. 435
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74. Any proof in applied mathematics can be regarded as a proof in pure mathematics proving that this proposition
can be got from these propositions by such and such operations--Any empirical proposition can serve as a rule if it
is made immovable and becomes a part of the system of coordinates.
436
Page Break 29
EDITORS' PREFACE TO THE REVISED EDITION
Page 29
THE posthumously published writings of Wittgenstein which first appeared in 1956 with the title "Remarks on the
Foundations of Mathematics" almost all take their origin from the period September 1937-April 1944. Wittgenstein
did not return to this subject matter in the last years of his life. On the other hand, he wrote a great deal on the
philosophy of mathematics and logic from 1929 to about 1934. A considerable part of this--together with other
material from these years--has been published under the titles "Philosophische Bermerkungen" (1964) and
"Philosophische Grammatik" (1969).
Page 29
The present revised edition of the "Remarks on the Foundations of Mathematics" contains the whole text of
the first (1956) edition. In editing it we have thus left out nothing that was already in print. On the other hand, we
have included additional material. Only Parts II and III of the first edition are here reprinted, practically unaltered, as
Parts III and IV.
Page 29
We have taken the second Appendix to Part I of the first edition, enlarged by a few additions from the MSS,
and placed it as an independent Part II of this edition.
Page 29
Part VI of the present edition is entirely new. The MS includes perhaps the most satisfactory presentation of
Wittgenstein's thoughts on the problem of following a rule--one of his most frequently recurring themes. The MS
(164) was written in the period 1941-1944; we have not been able to date it more precisely.†1 With the exception of
a few remarks at the end, which do not quite fit in with the circle of problems that are otherwise the topic of the MS,
it is here printed in extenso.
Page 29
Part I is the earliest of this collection and to a certain extent it has a peculiar position. It is the only part that
existed in typescript and is the most worked over of them all. The typescript itself goes back to manuscripts which
were composed for the most part in the period from September 1937 to about the end of that year (117, 118, 119).
But the remarks on negation form an exception; they stem from a MS belonging to about the turn of the year
1933-1934 (115).
Page 29
In its original form the typescript that is the basis for Part I formed
Page Break 30
the second half of an earlier version of "Philosophical Investigations". Wittgenstein then split up this half of that
version into clippings, supplied them with extensive alterations and additions, and only then constructed that order
of the individual remarks that is reproduced here. In a notebook as late as 1944 he proposed a few alterations to this
typescript. (See p. 80, n.)
Page 30
The last section of the rearranged collection consisted of papers that had not been cut up, though there were
many manuscript additions, and it is not quite clear whether Wittgenstein regarded them as belonging with the
preceding text. This section deals with the concept of negation, and as we have already mentioned, it was written 3-4
years earlier than the remainder of Part I. Its content occurs in great part in the "Investigations" §§ 547-568. The
editors left it out of the first edition, but have included it here as Appendix I to Part I.
Page 30
The collection had two further appendices. They come from the same typescript as the second half of the
(earlier) "Investigations"; nevertheless they were separated from the rest of the collection of clippings. The first deals
with 'mathematical surprise'. The second discusses among other things Gödel's theories of the existence of
unprovable but true propositions in the system of "Principia Mathematica." In the first edition we included only the
second appendix, but here both are published (Appendices II and III.)
Page 30
With the exception of a few remarks which Wittgenstein himself had left out in the arrangement of the
clippings, what is here published as Part I comprises the whole content of the second half of the early version of
"Philosophical Investigations."
Page 30
It must have been Wittgenstein's intention also to attach appendices on Cantor's theory of infinity and
Russell's logic to the contributions on problems of the foundations of mathematics that he planned to include in the
"Philosophical Investigations." Under the heading 'Additions' he wrote a certain amount on the problems connected
with set theory: about the diagonal procedure and the different kinds of number-concept. In the time from April
1938 to January 1939 he wrote a MS book where, together with other remarks on the philosophy of psychological
concepts, he put in a good deal on probability and truth (Gödel) and also on infinity and kinds of number (Cantor).
These writings he immediately continued in a notebook (162a and the beginning of 162b). In the later war-years too
he occasionally comes back to these topics. The confrontation with Cantor was never brought to a terminus.
Page Break 31
Page 31
What is here published as Part II consists of the above-mentioned "Additions" in 117 and of a selection of
remarks from 121. The whole presents an inconsiderable expansion of Appendix II of Part I of the earlier (1956)
edition. The arrangement of sentences and paragraphs into numbered Remarks corresponds to the original text
(which was not wholly the case in the 1956 edition). The sections have been numbered by the editors.
Page 31
Wittgenstein's confrontation with Russell, that is to say with the thought of the derivability of mathematics
from the calculi of logic, is found in Part III of this collection (Part II of the edition of 1956). These writings stem
from the period from October 1939 to April 1940. The MS (122, continued in the second half of 117) was the most
extensive of all the MSS which form the basis of this collection. Neither in style nor in content has it been perfected.
The author keeps on renewing the attempt to elucidate his thoughts on the nature of mathematical proof: what it
means, for example, to say that a proof must be surveyable; that it presents us with a new picture; that it creates a
new concept; and the like. His effort is to declare "the motley of mathematics" and to make clear the connexion
between the different techniques of calculation. In so striving he simultaneously sets his face against the idea of a
"foundation" of mathematics, whether in the form of a Russellian calculus or in that of the Hilbertian conception of a
meta-mathematics. The idea of contradiction and of a consistency proof is extensively discussed.
Page 31
The editors were of the opinion that this manuscript contained a wealth of valuable thoughts as they are
nowhere otherwise to be encountered in Wittgenstein's writings. On the other hand it also seemed clear to them that
this MS could not be published unabridged. Thus a selection was requisite. The task was difficult, and the editors are
not entirely satisfied with the result.
Page 31
In the autumn of 1940 Wittgenstein began to occupy himself anew with the philosophy of mathematics and
wrote something about the question of following a rule. These writings (MS 123) are not published here. In May
1941 the work was taken up again and soon led to investigations from which a considerable selection is published
here as Part VII.
Page 31
The first part of Part VII (§§ 1-23) was mostly written in June 1941. It discusses the relation between
mathematical and empirical propositions, between calculation and experiment, treats the concept of contradiction
and consistency anew and ends in the neighbourhood of the
Page Break 32
Gödelian problem. The second half was written in the spring of 1944. It deals principally with the concept of
following a rule, of mathematical proof and logical inference, and with the connexion between proof and concept
formation in mathematics. There are here numerous points of contact, on the one side with the manuscripts of the
intervening period (Parts IV and V) and with thoughts in the "Philosophical Investigations" on the other. §§ 47-60
essentially form an earlier version of what can now be found in the "Investigations" §§ 209 to 237. The sequence of
the remarks is different here; and some have not been taken up into the later version.--Both halves of this Part VII
were in the same MS book, which is one of the indications that the author regarded them as belonging together.
Page 32
Part V is taken from two MSS (126 and 127) belonging to the years 1942 and 1943--while Part IV derives
mainly from one MS (125) from the year 1942, with some additions from the two MSS on which Part V is based.
Much on these two parts has the character of "preliminary studies" for the second half of Part VII; but they also
contain a wealth of material that the author did not use there.
Page 32
In Part V Wittgenstein discusses topics that connect up with Brouwer and Intuitionism: the law of excluded
middle and mathematical existence; the Dedekind cut and the extensional and intensional way of looking at things in
mathematics. In the second half of this part there are remarks on the concept of generality in mathematics and
especially on a theme that makes its appearance still more strongly in Part VII: the role of concept-formation and the
relation between concept and truth in mathematics.
Page 32
The chronological arrangement of the material has the consequence that one and the same theme is
sometimes treated in different places. If Wittgenstein had put his remarks together into a book, he would presumably
have avoided many of these repetitions.
Page 32
It must once more be emphasized: Part I and, practically speaking also Part VI, but only these, are complete
reproductions of texts of Wittgenstein's. Thus what is here published as Parts II, III, IV, V and VII is a selection
from extensive MSS. In their preface to the first edition the editors conjectured that it might be desirable later to print
what they had omitted. They are still of the same opinion--but also of the opinion that the time has not yet come to
print the whole of Wittgenstein's MSS on these and other topics.
Page 32
The editors alone are responsible for the numbering of the selected paragraphs. (Even in Part I.) But the
articulation of the writing into
Page Break 33
"remarks"--here separated from one another by larger gaps--is Wittgenstein's own. With a few exceptions we did not
want to interfere with the order of the sections. Nevertheless we have sometimes (especially at the end of Part IV
and V) brought together material belonging to the same subject matter from different places.
Page 33
The list of contents and the index are meant to help the reader to look over the whole and to make it easier to
look things up. We alone are responsible for the thematic articulation of the material indicated in the list of contents.
Page Break 34
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PART I
Circa 1937-1938
Page 35
1. We use the expression: "The steps are determined by the formula...". How is it used?--We may perhaps
refer to the fact that people are brought by their education (training) so to use the formula y = x2, that they all work
out the same value for y when they substitute the same number for x. Or we may say: "These people are so trained
that they all take the same step at the same point when they receive the order 'add 3'". We might express this by
saying: for these people the order "add 3" completely determines every step from one number to the next. (In
contrast with other people who do not know what they are to do on receiving this order, or who react to it with
perfect certainty, but each one in a different way.)
Page 35
On the other hand we can contrast different kinds of formula, and the different kinds of use (different kinds
of training) appropriate to them. Then we call formulae of a particular kind (with the appropriate methods of use)
"formulae which determine a number y for a given value of x", and formulae of another kind, ones which "do not
determine the number y for a given value of x". (y = x2 + 1 would be of the first kind, y > x2 + 1, y = x2 ± 1, y = x2 + z
of the second.) The proposition "The formula... determines a number y" will then be a statement about the form of
the formulae--and now we must distinguish such a proposition as "The formula which I have written down
determines y", or "Here is a formula which determines y", from one of the following kind: "The formula, y = x2
determines the number y for a given value of x". The question "Is the formula written down there one that
determines y?" will then mean the same as "Is what is there a formula of this kind or that?"--but it is not clear
off-hand what we are to make of the question "Is y = x2 a formula which determines y for a given value of x?" One
might address this question to a pupil in order to test whether he understands the use of the word "to
Page Break 36
determine"; or it might be a mathematical problem to work out whether there was only one variable on the
right-hand side of the formula, as e.g. in the case: y = (x2 + z)2 - z(2x2 + z).
Page 36
2. "The way the formula is meant determines which steps are to be taken." What is the criterion for the way
the formula is meant? Presumably the way we always use it, the way we were taught to use it.
We say, for instance, to someone who uses a sign unknown to us: "If by 'x!2' you mean x2, then you get this
value for y, if you mean , that one".--Now ask yourself: how does one mean the one thing or the other by
"x!2"?
That will be how meaning it can determine the steps in advance.
3. How do I know that in working out the series + 2 I must write
"20004, 20006"
and not
"20004, 20008"?
--(The question: "How do I know that this colour is 'red'?" is similar.)
"But you surely know for example that you must always write the same sequence of numbers in the units: 2,
4, 6, 8, 0, 2, 4, etc."--Quite true: the problem must already appear in this sequence, and even in this one: 2, 2, 2, 2,
etc.--For how do I know that I am to write "2" after the five hundredth "2"? i.e. that 'the same figure' in that place is
"2"? And if I know it in advance, what use is this knowledge to me later on? I mean: how do I know what to do with
this earlier knowledge when the step actually has to be taken?
(If intuition is needed to continue the series + 1, then it is also needed to continue the series + 0.)
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Page 37
"But do you mean to say that the expression '+ 2' leaves you in doubt what you are to do e.g. after
2004?"--No; I answer "2006" without hesitation. But just for that reason it is superfluous to suppose that this was
determined earlier on. My having no doubt in face of the question does not mean that it has been answered in
advance.
"But I surely also know that whatever number I am given I shall be able, straight off and with certainty, to
give the next one.--Certainly my dying first is excluded, and a lot of other things too. But my being so certain of
being able to go on is naturally very important.--
Page 37
4. "But then what does the peculiar inexorability of mathematics consist in?"--Would not the inexorability
with which two follows one and three two be a good example?--But presumably this means: follows in the series of
cardinal numbers; for in a different series something different follows. And isn't this series just defined by this
sequence?--"Is that supposed to mean that it is equally correct whichever way a person counts, and that anyone can
count as he pleases?"--We should presumably not call it "counting" if everyone said the numbers one after the other
anyhow; but of course it is not simply a question of a name. For what we call "counting" is an important part of our
life's activities. Counting and calculating are not--e.g.--simply a pastime. Counting (and that means: counting like
this) is a technique that is employed daily in the most various operations of our lives. And that is why we learn to
count as we do: with endless practice, with merciless exactitude; that is why it is inexorably insisted that we shall all
say "two" after "one", "three" after "two" and so on.--But is this counting only a use, then; isn't there also some truth
corresponding to this sequence?" The truth is that counting has proved to pay.--"Then do you want to say that
'being true' means: being usable (or
Page Break 38
useful)?"--No, not that; but that it can't be said of the series of natural numbers--any more than of our language--that
it is true, but: that it is usable, and, above all, it is used.
Page 38
5. "But doesn't it follow with logical necessity that you get two when you add one to one, and three when
you add one to two? and isn't this inexorability the same as that of logical inference?"--Yes! it is the same.--"But isn't
there a truth corresponding to logical inference? Isn't it true that this follows from that?"--The proposition: "It is true
that this follows from that" means simply: this follows from that. And how do we use this proposition?--What
would happen if we made a different inference--how should we get into conflict with truth?
Page 38
How should we get into conflict with truth, if our footrules were made of very soft rubber instead of wood
and steel?--"Well, we shouldn't get to know the correct measurement of the table."--You mean: we should not get, or
could not be sure of getting, that measurement which we get with our rigid rulers. So if you had measured the table
with the elastic rulers and said it measured five feet by our usual way of measuring, you would be wrong; but if you
say that it measured five feet by your way of measuring, that is correct.--"But surely that isn't measuring at all!"--It is
similar to our measuring and capable, in certain circumstances, of fulfilling 'practical purposes'. (A shopkeeper might
use it to treat different customers differently.)
Page 38
If a ruler expanded to an extraordinary extent when slightly heated, we should say--in normal
circumstances--that that made it unusable. But we could think of a situation in' which this was just what was wanted.
I am imagining that we perceive the expansion with the naked eye; and we ascribe the same numerical measure of
length to bodies in rooms of different temperatures, if they measure the same by the ruler which to the eye is now
longer, now shorter.
Page Break 39
Page 39
It can be said: What is here called "measuring" and "length" and "equal length", is something different from
what we call those things. The use of these words is different from ours; but it is akin to it; and we too use these
words in a variety of ways.
Page 39
6. We must get clear what inferring really consists in: We shall perhaps say it consists in the transition from
one assertion to another. But does this mean that inferring is something that takes place when we are making a
transition from one assertion to another, and so before the second one is uttered--or that inferring consists in making
the one assertion follow upon the other, that is, e.g., in uttering it after the other? Misled by the special use of the
verb "infer" we readily imagine that inferring is a peculiar activity, a process in the medium of the understanding, as
it were a brewing of the vapour out of which the deduction arises. But let's look at what happens here.--There is a
transition from one proposition to another via other propositions, that is, a chain of inferences; but we don't need to
talk about this; for it presupposes another kind of transition, namely that from one link of the chain to the next. Now
a process of forming the transition may occur between the links. There is nothing occult about this process; it is a
derivation of one sentence from another according to a rule; a comparison of both with some paradigm or other,
which represents the schema of the transition; or something of the kind. This may go on on paper, orally, or 'in the
head'.--The conclusion may however also be drawn in such a way that the one proposition is uttered after the other,
without any such process; or the process may consist merely in our saying "Therefore" or "It follows from this", or
something of the kind. We call it a "conclusion" when the inferred proposition can in fact be derived from the
premise.
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Page 40
7. Now what does it mean to say that one proposition can be derived from another by means of a rule? Can't
anything be derived from anything by means of some rule--or even according to any rule, with a suitable
interpretation? What does it mean for me to say e.g.: this number can be got by multiplying these two numbers?
This is a rule telling us that we must get this number if we multiply correctly; and we can obtain this rule by
multiplying the two numbers, or again in a different way (though any procedure that leads to this result might be
called 'multiplication'). Now I am said to have multiplied when I have carried out the multiplication 265 × 463, and
also when I say: "twice four is eight", although here no calculating procedure led to the product (which, however, I
could also have worked out). And so we also say a conclusion is drawn, where it is not calculated.
Page 40
8. But still, I must only infer what really follows!--Is this supposed to mean: only what follows, going by the
rules of inference; or is it supposed to mean: only what follows, going by such rules of inference as somehow agree
with some (sort of) reality? Here what is before our minds in a vague way is that this reality is something very
abstract, very general, and very rigid. Logic is a kind of ultra-physics, the description of the 'logical structure' of the
world, which we perceive through a kind of ultra-experience (with the understanding e.g.). Here perhaps inferences
like the following come to mind: "The stove is smoking, so the chimney is out of order again". (And that is how this
conclusion is drawn! Not like this: "The stove is smoking, and whenever the stove smokes the chimney is out of
order; and so...".)
Page 40
9. What we call 'logical inference' is a transformation of our expression. For example, the translation of one
measure into another. One edge of a ruler is marked in inches, the other in centimetres. I
Page Break 41
measure the table in inches and go over to centimetres on the ruler.--And of course there is such a thing as right and
wrong in passing from one measure to the other; but what is the reality that 'right' accords with here? Presumably a
convention, or a use, and perhaps our practical requirements.
Page 41
10. "But doesn't e.g. 'fa' have to follow from '(x).fx' if '(x).fx' is meant in the way we mean it?"--And how does
the way we mean it come out? Doesn't it come out in the constant practice of its use? and perhaps further in certain
gestures--and similar things.--But it is as if there were also something attached to the word "all", when we say it;
something with which a different use could not be combined; namely, the meaning. "'All' surely means: all!" we
should like to say, when we have to explain this meaning; and we make a particular gesture and face.
Page 41
Cut down all these trees!--But don't you understand what 'all' means? (He had left one standing.) How did he
learn what 'all' means? Presumably by practice.--And of course this practice did not only bring it about that he does
this on receiving the order--it surrounded the word with a whole lot of pictures (visual and others) of which one or
another comes up when we hear and speak the word. (And if we are supposed to give an account of what the
'meaning' of the word is, we first pull out one from this mass of pictures--and then reject it again as non-essential
when we see that now this, now that, picture presents itself, and sometimes none at all.)
Page 41
One learns the meaning of "all" by learning that 'fa' follows from '(x).fx'.--The exercises which drill us in the
use of this word, which teach its meaning, always make it natural to rule out any exception.
Page 41
11. For how do we learn to infer? Or don't we learn it?
Page Break 42
Page 42
Does a child know that an affirmative follows from a double negative?--And how does one shew him that it
does? Presumably by shewing him a process (a double inversion, two turns through 180° and similar things) which
he then takes as a picture of negation.
Page 42
And the meaning of '(x).fx' is made clear by our insisting on 'fa''s following from it.
Page 42
12. "From 'all', if it is meant like this, this must surely follow!"--If it is meant like what? Consider how you
mean it. Here perhaps a further picture comes to your mind--and that is all you have got.--No, it is not true that it
must--but it does follow: we perform this transition.
And we say: If this does not follow, then it simply wouldn't be all--and that only shews how we react with
words in such a situation.--
Page 42
13. It strikes us as if something else, something over and above the use of the word "all", must have changed
if 'fa' is no longer to follow from '(x).fx'; something attaching to the word itself.
Isn't this like saying: "If this man were to act differently, his character would have to be different". Now this
may mean something in some cases and not in others. We say "behaviour flows from character" and that is how use
flows from meaning.
Page 42
14. This shews you--it might be said--how closely certain gestures, pictures, reactions, are linked with a
constantly practised use.
Page 42
'The picture forces itself on us....' It is very interesting that pictures do force themselves on us. And if it were
not so, how could
Page Break 43
such a sentence as "What's done cannot be undone" mean anything to us?
Page 43
15. It is important that in our language--our natural language--'all' is a fundamental concept and 'all but one'
less fundamental; i.e. there is not a single word for it, nor yet a characteristic gesture.
Page 43
16. The point of the word "all" is that it admits no exception.--True, that is the point of its use in our
language; but the kinds of use we feel to be the 'point' are connected with the role that such-and-such a use has in
our whole life.
Page 43
17. When we ask what inferring consists in, we hear it said e.g.: "If I have recognized the truth of the
propositions..., then I am justified in further writing down...".--In what sense justified? Had I no right to write that
down before?--"Those propositions convince me of the truth of this proposition." But of course that is not what is in
question either.--"The mind carries out the special activity of logical inference according to these laws." That is
certainly interesting and important; but then, is it true? Does the mind always infer according to these laws? And
what does the special activity of inferring consist in?--This is why it is necessary to look and see how we carry out
inferences in the practice of language; what kind of procedure in the language-game inferring is.
For example: a regulation says "All who are taller than five foot six are to join the ... section". A clerk reads
out the men's names and heights. Another allots them to such-and-such sections.--"N.N. five foot nine." "So N.N. to
the ... section." That is inference.
Page Break 44
Page 44
18. Now, what do we call 'inferences' in Russell or Euclid? Am I to say: the transitions from one proposition
to the next one in the proof? But where is the passage to be found?--I say that in Russell one proposition follows
from another if the one can be derived from the other according to the position of both in a proof and the appended
signs--when we read the book. For reading this book is a game that has to be learnt.
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19. One is often in the dark about what following and inferring really consists in; what kind of fact, and what
kind of procedure, it is. The peculiar use of these verbs suggests to us that following is the existence of a connexion
between propositions, which connexion we follow up when we infer. This comes out very instructively in Russell's
account (Principia Mathematica). That a proposition q follows from a proposition p ⊃ q.p is here a
fundamental law of logic:
p⊃q.p.⊃. q†1
Now this, one says, justifies us in inferring q from p⊃q.p. But what does 'inferring', the procedure that is now
justified, consist in? Surely in this: that in some language-game we utter, write down (etc.), the one proposition as an
assertion after the other; and how can the fundamental law justify me in this?
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20. Now Russell wants to say: "This is how I am going to infer, and it is right". So he means to tell us how
he means to infer: this is done by a rule of inference. How does it run? That this proposition implies that
one?--Presumably that in the proofs in this book a proposition like this is to come after a proposition like this.--But it
is supposed to be a fundamental law of logic that it is
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correct to infer in this way!--Then the fundamental law would have to run: "It is correct to infer ... from ..."; and this
fundamental law should presumably be self-evident--in which case the rule itself will self-evidently be correct, or
justified. "But after all this rule deals with sentences in a book, and that isn't part of logic!"--Quite correct, the rule is
really only a piece of information that in this book only this transition from one proposition to another will be used
(as it were a piece of information in the index); for the correctness of the transition must be evident where it is made;
and the expression of the 'fundamental law of logic' is then the sequence of propositions itself.
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21. In his fundamental law Russell seems to be saying of a proposition: "It already follows--all I still have to
do is, to infer it". Thus Frege somewhere says that the straight line which connects any two points is really already
there before we draw it; and it is the same when we say that the transitions, say in the series + 2, have really already
been made before we make them orally or in writing--as it were tracing them.
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22. One might reply to someone who said this: Here you are using a picture. One can determine the
transitions which someone is to make in a series, by doing them for him first. E.g. by writing down in another
notation the series which he is to write, so that all that remains for him to do is to translate it; or by actually writing it
down very faint, and he has to trace it. In the first case we can also say that we don't write down the series that he
has to write, and so that we do not ourselves make the transitions of that series; but in the second case we shall
certainly say that the series which he is to write is already there. We should also say this if we dictate what he has to
write down, although then we are producing a series of sounds and he a series of
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written signs. It is at any rate a sure way of determining the transitions that someone has to make, if we in some
sense make them first.--If, therefore, we determine these transitions in a quite different sense, namely, by subjecting
our pupil to such a training as e.g. children get in the multiplication tables and in multiplying, so that all who are so
trained do random multiplications (not previously done in the course of being taught) in the same way and with
results that agree--if, that is, the transitions which someone is to make on the order 'add 2' are so determined by
training that we can predict with certainty how he will go, even when he has never up to now taken this step--then it
may be natural to us to use this as a picture of the situation: the steps are all already taken and he is just writing them
down.
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23. "But we surely infer this proposition from that because it actually follows! We ascertain that it
follows."--We ascertain that what is written here follows from what is written there. And this proposition is being
used temporally.
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24. Separate the feelings (gestures) of agreement, from what you do with the proof.
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25. But how about when I ascertain that this pattern of lines:
(a)
is like-numbered with this pattern of angles:
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(b)
(I have made the patterns memorable on purpose) by correlating them:
(c)
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Now what do I ascertain when I look at this figure? What I see is a star with threadlike appendages.--
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26. But I can make use of the figure like this: five people stand arranged in a pentagon; against the wall are
wands, like the strokes in (a); I look at the figure (c) and say: "I can give each of the people a wand".
I could regard figure (c) as a schematic picture of my giving the five men a wand each.
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27. For if I first draw some arbitrary polygon:
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and then some arbitrary series of strokes
||||||||||||||||||||||||||||||||
I can find out by correlating them whether I have as many angles in the top figure as strokes in the bottom one. (I do
not know how it would turn out.) And so I can also say that by drawing projection-lines I have ascertained that there
are as many strokes at the top of figure (c) as the star beneath has points. (Temporally!) In this way of taking it the
figure is not like a mathematical proof (any more than it is a mathematical proof when I divide a bag of apples
among a group of people and find that each can have just one apple).
I can however conceive figure (c) as a mathematical proof. Let us give names to the shapes of the patterns (a)
and (b): let (a) be called a "hand", H, and (b) a "pentacle", P. I have proved that H has as many strokes as P has
angles. And this proposition is once more non-temporal.
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28. A proof--I might say--is a single pattern, at one end of which are written certain sentences and at the
other end a sentence (which we call the 'proved proposition'.)
To describe such a pattern we may say: in it the proposition... follows from.... This is one way of describing a
design, which might also be for example an ornament (a wallpaper design). I can say, then, "In the proof on that
blackboard the proposition p follows from q and r", and that is simply a description of what can be seen there. But it
is not the mathematical proposition that p follows from q and r. That has a different application. It says--as one
might put it--that it makes sense to speak of a proof (pattern) in which p follows from q and r. Just as one can say
that the proposition "white is lighter than black" asserts that it makes sense to speak of two objects, the lighter one
white and the other black, but not of two objects, the lighter one black and the other white.
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29. Let us imagine that we had given a paradigm of 'lighter' and 'darker' in the shape of a white and a black
patch, and now, so to speak, we use it to deduce that red is darker than white.
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30. The proposition proved by (c) now serves as a new prescription for ascertaining numerical equality: if one
set of objects has been arranged in the form of a hand and another as the angles of a pentacle, we say the two sets
are equal in number.
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31. "But isn't that merely because we have already correlated H and P and seen that they are the same in
number?"--Yes, but if they were so in one case, how do I know that they will be so again now?--"Why, because it is
of the essence of H and P to be the same in number."--But how can you have brought that out by correlating them?
(I thought the counting or correlation merely yielded the result that these two groups before me were--or were
not--the same in number.)
--"But now, if he has an H of things and a P of things, and he actually correlates them, it surely isn't possible
for him to get any result but that they are the same in number.--And that it is not possible can surely be seen from
the proof."--But isn't it possible? If, e.g., he--as someone else might say--omits to draw one of the correlating lines.
But I admit that in an enormous majority of cases he will always get the same result, and, if he did not get it, would
think something had put him out. And if it were not like this the ground would be cut away from under the whole
proof. For we decide to use the proof-picture instead of correlating the groups; we do not correlate them, but
instead compare the groups with those of the proof (in which indeed two groups are correlated with one another).
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32. I might also say as a result of the proof: "From now on an H and a P are called 'the same in number'".
Or: The proof doesn't explore the essence of the two figures, but it does express what I am going to count as
belonging to the essence of the figures from now on.--I deposit what belongs to the essence among the paradigms of
language.
The mathematician creates essences.
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33. When I say "This proposition follows from that one", that is to accept a rule. The acceptance is based on
the proof. That is to say, I find this chain (this figure) acceptable as a proof.--"But could I do otherwise? Don't I have
to find it acceptable?"--Why do you say you have to? Because at the end of the proof you say e.g.: "Yes--I have to
accept this conclusion". But that is after all only the expression of your unconditional acceptance.
I.e. (I believe): the words "I have to admit this" are used in two kinds of case: when we have got a proof--and
also with reference to the individual steps of the proof.
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34. And how does it come out that the proof compels me? Well, in the fact that once I have got it I go ahead
in such-and-such a way, and refuse any other path. All I should further say as a final argument against someone who
did not want to go that way, would be: "Why, don't you see...!"--and that is no argument.
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35. "But, if you are right, how does it come about that all men (or at any rate all normal men) accept these
patterns as proofs of these propositions?"--It is true, there is great--and interesting--agreement here.
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36. Imagine you have a row of marbles, and you number them with Arabic numerals, which run from 1 to
100; then you make a big gap after every 10, and in each 10 a rather smaller gap in the middle with 5 on either side:
this makes the 10 stand out clearly as 10; now you take the sets of 10 and put them one below another, and in the
middle of the column you make a bigger gap, so that you have five rows above and five below; and now you
number the rows from 1 to 10.--We have, so to speak, done drill with the marbles. I can say that we have unfolded
properties of the hundred marbles.--But now imagine that this whole process, this experiment with the hundred
marbles, were filmed. What I now see on the screen is surely not an experiment, for the picture of an experiment is
not itself an experiment.--But I see the 'mathematically essential' thing about the process in the projection too! For
here there appear first a hundred spots, and then they are arranged in tens, and so on and so on.
Thus I might say: the proof does not serve as an experiment; but it does serve as the picture of an
experiment.
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37. Put two apples on a bare table, see that no one comes near them and nothing shakes the table; now put
another two apples on the table; now count the apples that are there. You have made an experiment; the result of the
counting is probably 4. (We should present the result like this: when, in such-and-such circumstances, one puts first
2 apples and then another 2 on a table, mostly none disappear and none get added.) And analogous experiments can
be carried out, with the same result, with all kinds of solid bodies.--This is how our children learn sums; for one
makes them put down three beans and then another three beans and then count what is there. If the result at one
time were 5, at another 7 (say because, as we should now say, one sometimes got added, and one sometimes
vanished of itself), then the first thing we said would be that beans were no good for teaching
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sums. But if the same thing happened with sticks, fingers, lines and most other things, that would be the end of all
sums.
"But shouldn't we then still have 2 + 2 = 4?"--This sentence would have become unusable.
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38. "You only need to look at the figure
to see that 2 + 2 are 4."--Then I only need to look at the figure
to see that 2 + 2 + 2 are 4.
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39. What do I convince anyone of, if he has followed the film projection of the experiment with the hundred
marbles?
One might say, I convince him that it happened like that.--But this would not be a mathematical
conviction.--But can't I say: I impress a procedure on him? This procedure is the regrouping of 100 things in 10
rows of 10. And this procedure can as a matter of fact always be carried out again. And he can rightly be convinced
of that.
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40. And that is how the proof (25) impresses a procedure on us by drawing projection-lines: the procedure of
one-one correlation of the H and the P.--"But doesn't it also convince me of the fact that this†1 correlation is
possible?"--If that is supposed to mean: you can always carry it out--, then that doesn't have to be true at all. But the
drawing of the projection-lines convinces us that there are as many lines above as angles below; and it supplies us
with a model to use in correlating such patterns.--"But surely what the model shews in this way is that it does work,
not that it did work this time? In the sense in which it wouldn't have worked if the top figure had been |||||| instead of
"--How is that? doesn't it work then? Like this e.g.:
This figure too could be used to prove something. It could be used to shew that groups of these forms cannot be
given a 1-1 correlation.†2 'A 1-1 correlation is impossible here' means, e.g., "these figures and 1-1 correlation don't fit
together."
"I didn't mean it like that'"--Then shew me how you mean it, and I'll do it.
But can't I say that the figure shews how such a correlation is possible--and mustn't it for that reason also
shew that it is possible?--
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41. Now what was the point of our proposal to attach names to the five parallel strokes and the five-pointed
star? What is done by their having got names? It will be a means of indicating something about the kind of use these
figures have. Namely--that we recognize them as such-and-such at a glance. To do so, we don't count their strokes or
angles; for us they are typical shapes, like knife and fork, like letters and numerals.
Thus, when given the order "Draw an H" (for example)--I can produce this shape immediately.--Now the
proof teaches me a way of correlating the two shapes. (I should like to say that it is not merely these individual
figures that are correlated in the proof, but the shapes themselves. But this surely only means that these shapes are
well impressed on my mind; are impressed as paradigms.) Now isn't it possible for me to get into difficulties when I
want to correlate the shapes H and P--say by there being an angle too many at the bottom or a stroke too many at
the top?--"But surely not, if you have really drawn H and P again!--And that can be proved; look at this figure."
--This figure teaches me a new way of checking whether I have really drawn the same figures; but can't I still get into
difficulties when I now want to use this model as a guide? But I say that I am certain I shall
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not normally get into any difficulties.
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42. There is a puzzle which consists in making a particular figure, e.g. a rectangle, out of given pieces. The
division of the figure is such that we find it difficult to discover the right arrangement of the parts. Let it for example
be this:
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What do you discover when you succeed in arranging it?--You discover a position--of which you did not
think before.--Very well; but can't we also say: you find out that these triangles can be arranged like this?--But 'these
triangles': are they the actual ones in the rectangle above, or are they triangles which have yet to be arranged like
that?
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43. If you say: "I should never have thought that these shapes could be arranged like that", we can't point to
the solution of the puzzle and say: "Oh, you didn't think the pieces could be arranged like that?"--You would reply:
"I mean, I didn't think of this way of arranging them at all".
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44. Let us imagine the physical properties of the parts of the puzzle to be such that they can't come into the
desired position. Not, however, that one feels a resistance if one tries to put them in this
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position; but one simply tries everything else, only not this, and the pieces don't get into this position by accident
either. This position is as it were excluded from space. As if there were e.g. a 'blind spot' in our brain here.--And isn't
it like this when I believe I have tried all possible arrangements and have always passed this one by, as if bewitched?
Can't we say: the figure which shews you the solution removes a blindness, or again changes your geometry?
It as it were shews you a new dimension of space. (As if a fly were shewn the way out of the fly-bottle.)
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45. A demon has cast a spell round this position and excluded it from our space.
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46. The new position has as it were come to be out of nothingness. Where there was nothing, now there
suddenly is something.
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47. In what sense has the solution shewn you that such-and-such can be done? Before, you could not do
it--and now perhaps you can.--
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48. I said, "I accept such-and-such as proof of a proposition"--but is it possible for me not to accept the
figure shewing the arrangement of the pieces as proof that these pieces can be arranged to have this periphery?
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49. But now imagine that one of the pieces is lying so as to be the mirror-image of the corresponding part of
the pattern. Now you want to arrange the figure according to the pattern; you see it must work, but you never hit on
the idea of turning the piece over, and you find that you do not succeed in fitting the puzzle together.
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50. A rectangle can be made of two parallelograms and two triangles. Proof:
A child would find it difficult to hit on the composition of a rectangle with these parts, and would be surprised by
the fact that two sides of the parallelograms make a straight line, when the parallelograms are, after all, askew. It
might strike him as if the rectangle came out of these figures by something like magic. True, he has to admit that
they do form a rectangle, but it is by a trick, by a distorted arrangement, in an unnatural way.
I can imagine the child, after having put the two parallelograms together in this way, not believing his eyes
when he sees that they fit like that. 'They don't look as if they fitted together like that.' And I could imagine its being
said: It's only through some hocus-pocus that it looks to us as if they yielded the rectangle--in reality they have
changed their nature, they aren't the parallelograms any more.
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51. "You admit this--then you must admit this too."--He must admit it--and all the time it is possible that he
does not admit it! You want to say: "if he thinks, he must admit it".
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"I'll shew you why you have to admit it."--I shall bring a case before your eyes which will determine you to
judge this way if you think about it.
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52. Now, how can the manipulations of the proof make him admit anything?
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53. "Now you will admit that 5 consists of 3 and 2."
I will only admit it, if that is not to admit anything.--Except that I want to use this picture.
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54. One might for example take this figure
as a proof of the fact that 100 parallelograms arranged like this must yield a straight strip. Then, when one actually
does put 100 together, one gets e.g. a slightly curved strip.--But the proof has determined us to use this picture and
form of expression: if they don't yield a straight strip, they were not accurately constructed.
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55. Just think, how can the picture (or procedure) that you shew me now oblige me always to judge in
such-and-such a way?
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If what we have here is an experiment, then surely one is too little to bind me to any judgment.
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56. The one who is offering the proof says: "Look at this figure. What shall we say about it? Surely that a
rectangle consists of...?--"
Or again: "Now, surely you call this a 'parallelogram' and this a 'triangle', and this is what it is like for one
figure to consist of others".
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57. "Yes, you have convinced me that a rectangle always consists of..."--Should I also say: "Yes, you have
convinced me that this rectangle (the one in the proof) consists of..."? For wouldn't this be the more modest
proposition, which you ought to grant even if perhaps you don't yet grant the general proposition? But oddly
enough if that is what you grant, you seem to be granting, not the more modest geometrical proposition, but what is
not a proposition of geometry at all. Of course not--for as regards the rectangle in the proof he didn't convince me of
anything. (I shouldn't have been in any doubt about this figure, if I had seen it previously.) As far as concerns this
figure I acknowledged everything of my own accord. And he merely used it to make me realize something.--But on
the other hand, if he didn't convince me of anything as regards this rectangle, then how has he convinced me of a
property of other rectangles?
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58. "True, this shape doesn't look as if it could consist of two skew parts."
What are you surprised at? Surely not at seeing this figure. It is something in the figure that surprises
me.--But there isn't anything going on in the figure!
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What surprises me is the way straight and skew go together. It makes me as it were dizzy.
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59. But I do actually say: "I have convinced myself that this figure can be constructed with these pieces", e.g.
I have seen a picture of the solution of the puzzle.
Now if I say this to somebody it is surely supposed to mean: "Just try: these bits, properly arranged, really do
yield the figure". I want to encourage him to do something and I forecast that he will succeed. And the forecast is
founded on the ease with which we can construct the figure from the pieces as soon as we know how.
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60. You say you are astonished at what the proof shews you. But are you astonished at its having been
possible to draw these lines? No. You are only astonished when you tell yourself that two bits like this yield this
shape. When, that is, you think yourself into the situation of seeing the result after having expected something
different.
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61. "This follows inexorably from that."--True, in this demonstration this issues from that.
This is a demonstration for whoever acknowledges it as a demonstration. If anyone doesn't acknowledge it,
doesn't go by it as a demonstration, then he has parted company with us even before anything is said.
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62.
Here we have something thats looks inexorable--. And yet it can be 'inexorable' only in its consequences! For
otherwise it is nothing but a picture.
What does the action at a distance--as it might be called--of this pattern consist in?
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63. I have read a proof--and now I am convinced.--What if I straightway forgot this conviction?
For it is a peculiar procedure: I go through the proof and then accept its result.--I mean: this is simply what
we do. This is use and custom among us, or a fact of our natural history.
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64. 'If I have five, then I have three and two.'--But how do I know that I have five?--Well, if it looks like this:
|||||.--And is it also certain that when it looks like this, I can always split it up into groups like those?
It is a fact that we can play the following game: I teach someone what a group of two, three, four, or five, is
like, and I teach him how to put strokes into one-to-one correspondence; then I always make him carry out the order
"Draw a group of five" twice--and then I teach him to carry out the order: "Correlate these two groups"; and here it
proves that he practically always correlates the strokes without remainder.
Or again: it is a fact that I practically never get into difficulties in correlating what I have drawn as groups of
five.
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65. I have to assemble the puzzle, I try it this way and that, am doubtful whether I shall do it. Next someone
shows me a picture of the solution, and I say without any sort of doubt--"Now I can do it!"--Then am I certain to do
it now?--The fact, however, is: I don't have any doubt.
Suppose someone now asked: "What does the action at a distance of the picture consist in?"--In the fact that
I apply it.
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66. In a demonstration we get agreement with someone. If we do not, then we've parted ways before ever
starting to communicate in this language.
It is not essential that one should talk the other over by means of the demonstration. Both might see it (read
it), and accept it.
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67. "But you can see--there can't be any doubt, that a group like A consists essentially of one like B and one
like C."--I too say--i.e.
this is how I too express myself--that the group drawn there consists of the two smaller ones; but I don't know
whether every group which I should call the same in kind (or form) as the first will necessarily be composed of two
groups of the same kind as the smaller ones.--But I believe that it will probably always be so (perhaps experience has
taught me this), and that is why I am willing to accept the rule: I will say that a group is of the form A if and only if it
can be split up into two groups like B and C.
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68. And this too is how the drawing (50) works as a proof. "True enough! Two parallelograms together do
make this shape!" (That is very much as if I were to say: "Actually, a curve can consist of straight bits.")--I shouldn't
have thought it. Thought what? That the parts of this figure yield this figure? No, not that. For that doesn't mean
anything.--My surprise is only when I think of myself unwittingly fitting the top parallelogram on to the bottom one,
and then seeing the result.
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69. And it could be said: What the proof made me realize--that's what can surprise me.
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70. For why do I say that the figure (50) makes me realize something any more than this one:
After all it too shews that two bits like that yield a rectangle. "But that isn't interesting", we want to say. And why is
it uninteresting?
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71. When one says: "This shape consists of these shapes"--one is thinking of the shape as a fine drawing, a
fine frame of this shape, on which, as it were, things which have this shape are stretched. (Compare Plato's
conception of properties as ingredients of a thing.)
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72. "This shape consists of these shapes. You have shewn the essential property of this shape."--You have
shewn me a new picture.
It is as if God had constructed them like that.--So we are employing a simile. The shape becomes an ethereal
entity which has this shape; it is as if it had been constructed like this once for all (by whoever put the essential
properties into things). For if the shape is to be a thing consisting of parts, then the pattern-maker who made the
shape is he who also made light and dark, colour and hardness, etc. (Imagine someone asking: "The shape... is made
up of these parts; who made it? You?")
The word "being" has been used for a sublimed ethereal kind of existence. Now consider the proposition
"Red is" (e.g.). Of course no one ever uses it; but if I had to invent a use for it all the same, it would be this: as an
introductory formula to statements which went on to make use of the word "red". When I pronounce the formula I
look at a sample of the colour red.
One is tempted to pronounce a sentence like "red is" when one is looking attentively at the colour; that is, in
the same situation as that in which one observes the existence of a thing (of a leaflike insect, for example).
And I want to say: when one uses the expression, "the proof has taught me--shewn me--that this is the case",
one is still using this simile.
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73. I could also have said: it is not the property of an object that is ever 'essential', but rather the mark of a
concept.
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74. "If the form of the group was the same, then it must have had the same aspects, the same possibilities of
division. If it has different ones then it isn't the same form; perhaps it somehow made the same
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impression on you; but it is the same form only if you can divide it up in the same way."
It is as if this expressed the essence of form.--I say, however: if you talk about essence--, you are merely
noting a convention. But here one would like to retort: there is no greater difference than that between a proposition
about the depth of the essence and one about--a mere convention. But what if I reply: to the depth that we see in the
essence there corresponds the deep need for the convention.
Thus if I say: "It's as if this proposition expressed the essence of form"--I mean: it is as if this proposition
expressed a property of the entity form!--and one can say: the entity of which it asserts a property, and which I here
call the entity 'form', is the picture which I cannot help having when I hear the word "form".
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75. But what sort of properties of the hundred marbles did you unfold, or display?†1--Well, that these things
can be done with them.--But what things? Do you mean that you were able to move them about like that, that they
weren't glued on to the table top?--Not so much that, as that they have gone into these formations without any loss
or addition.--So you have shewn the physical properties of the row. But why did you use the expression
"unfolded"? You would not have spoken of unfolding the properties of a bar of iron by shewing that it melts at such
and such a temperature. And mightn't you as well say that you unfolded the properties of our memory for numbers,
as the properties of the row (e.g.)? For what you really do unfold, or lay out, is the row of marbles.--And you shew
e.g. that if a row looks thus and thus, or is numbered with roman numerals in this way, it can be brought into that
other memorable arrangement in a simple way, and without addition or loss of any marble. But this could after all
equally well have been a psychological experiment shewing that you now find memorable certain patterns into
which 100 spots are made merely by shifting them about.
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"I have shewn what can be done with 100 marbles."--You have shewn that these 100 marbles (or those
marbles over there), can be laid out in this way. The experiment was one of laying out (as opposed say to one of
burning).
And the psychological experiment might for example have shewn how easy it is for you to be deceived: i.e.
that you don't notice if marbles are smuggled into or out of the row. One could also say this: I have shewn what can
be made of a row of 100 spots by means of apparent shifts,--what figures can be got out of it by apparent
shifts.--But what did I unfold in this case?
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76. Imagine it were said: we unfold the properties of a polygon by using diagonals to take the sides together
three at a time. It then proves to be a figure with 24 angles. Do I want to say I have unfolded a property of the
24-angled polygon? No. I want to say I have unfolded a property of this polygon (the one drawn here). I now know
that there is drawn here a figure with 24 angles.
Is this an experiment? It shews me e.g. what kind of polygon is drawn here now. What I did can be called an
experiment in counting.
But what if I perform such an experiment on a pentagon, which I can already take in at a glance?--Well, let us
assume for a moment that I could not take it in at a glance,--which (e.g.) may be the case if it is very big. Then
drawing the diagonals would be a way of finding out that this is a pentagon. I could once more say I had unfolded
the properties of the polygon drawn here.--Now if I can take it in at a glance then surely nothing about it can be
changed. It was, perhaps, superfluous to unfold this property, as it is superfluous to count two apples which are
before my eyes.
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Ought I to say now: "It was an experiment again, but I was certain of the result"? But am I certain of the
result in the way I am certain of the result of the electrolysis of a mass of water? No, but in another way. If the
electrolysis of the liquid did not yield..., I should consider myself crazy, or say that I no longer have any idea what to
say.
Imagine I were to say: "Yes, here is a square,--but still let's look and see whether a diagonal divides it into
two triangles". Then I draw the diagonal and say: "Yes, here we have two triangles". Here I should be asked:
Couldn't you see that it could be divided into two triangles? Have you only just convinced yourself that there is a
square here; and why trust your eyes now rather than before?
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77. Exercises: Number of notes--the internal property of a tune; number of leaves--the external property of a
tree. How is this connected with the identity of the concept? (Ramsey.)
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78. If someone splits up four marbles into two and two, puts them together again, splits them up again and so
on, what is he shewing us? He is impressing on us a physiognomy and a typical alteration of this physiognomy.
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79. Think of the possible postures of a puppet. Or suppose you had a chain of, say, ten links, and you were
shewing what kind of characteristic (i.e. memorable) figures it can be made into. Let the links be numbered; in this
way they become an easily memorable structure even when they lie in a straight line.
So I impress characteristic positions and movements of this chain on you.
If I now say: "Look, this can be made of it too" and display it, am I shewing you an experiment?--It may be;
I am shewing for
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example that it can be got into this shape: but that you didn't doubt. And what interests you is not something to do
with this individual chain.--But all the same isn't what I am displaying a property of this chain? Certainly: but I only
display such movements, such transformations, as are of a memorable kind; and it interests you to learn these
transformations. But the reason why it interests you is that it is so easy to reproduce them again and again in
different objects.
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80. The words "look what I can make with it--" are indeed the same as I should use if I were shewing what I
can make with a lump of clay, for example. E.g. that I am clever enough to make such things with this lump. In
another case: that this material can be dealt with like this. Here I should hardly be said to be 'drawing your attention
to' the fact that I can do this, or that the material can stand this;--while in the case of the chain one would say: I draw
your attention to the fact that this can be done with it.--For you could also have imagined it. But of course you can't
get to know any property of the material by imagining.
The experimental character disappears when one looks at the process simply as a memorable picture.
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81. What I unfold may be said to be the role which '100' plays in our calculating system.
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82. (I once wrote: "In mathematics process and result are equivalent.")†1
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83. And yet I feel that it is a property of '100' that it is, or can be, produced in this way. But then how can it
be a property of the structure '100' to be produced in this way, if e.g. it didn't get produced in this way at all? If no
one multiplied in this way? Surely only if one could say, it is a property of this sign to be the subject of this rule. For
example it is the property of '5' to be the subject of the rule '3 + 2 = 5'. For only as the subject of the rule is this
number the result of the addition of the other numbers.
But suppose I now say: it is a property of the number... to be the result of the addition of... according to the
rule...?--So it is a property of the number that it arises when we apply this rule to these numbers. The question is:
should we call it 'application of the rule', if this number were not the result? And that is the same question as: "What
do you understand by 'application of this rule': what you e.g. do with it (and you may apply it at one time in this
way, at another in that), or is 'its application' otherwise explained?"
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84. "It is a property of this number that this process leads to it.--"But, mathematically speaking, a process
does not lead to it; it is the end of a process (is itself part of the process).
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85. But why do I feel that a property of the row is unfolded, is shewn?--Because I alternately look at what is
shewn as essential and as nonessential to the row. Or again: because I think of these properties alternately as external
and as internal. Because I alternately take something as a matter of course and find it noteworthy.
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86. "You surely unfold the properties of the hundred marbles
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when you shew what can be made of them."--Can be made of them how? For, that it can be made of them no one
has doubted, so the point must be the kind of way it is produced from them. But look at that, and see whether it does
not perhaps itself presuppose the result.--
For suppose that in that way you got one time this and another time a different result; would you accept
this? Would you not say: "I must have made a mistake; the same kind of way would always have to produce the
same result". This shows that you are incorporating the result of the transformation into the kind of way the
transforming is done.
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87. Exercise: am I to call it a fact of experience that this face turns into that through this alteration? (How
must 'this face', 'this alteration' be explained so as to...?)
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88. One says: this division makes it clear what kind of row of marbles we have here. Does it make it clear
what kind of row it was before the division, or does it make it clear what kind of row it is now?
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89. "I can see at a glance how many there are." Well, how many are there? Is the answer "so
many"?--(pointing to the group of objects). How does the answer go, though? There are '50', or '100', etc.
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90. "The division makes it clear to me what kind of row it is." Well, what kind of row is it? Is the answer "this
kind"? How does a significant answer run?
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91. Now I am surely also unfolding the geometrical properties of this chain, if I display the transformations
of another, similarly constructed, chain. What I do, however, does not shew what I can in fact do with the first one,
if it in fact should prove inflexible or in some other way physically unsuitable.
So after all I cannot say: I unfold the properties of this chain.
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92. Can one unfold properties of the chain which it doesn't possess at all?
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93. I measure a table; it is one yard long.--Now I put one yardstick up against another yardstick. Am I
measuring it by doing that? Am I finding out that the second yardstick is a yard long? Am I making the same
experiment of measuring, only with the difference that I am certain of the outcome?
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94. And when I put the ruler up against the table, am I always measuring the table; am I not sometimes
checking the ruler? And in what does the distinction between the one procedure and the other consist?
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95. The experiment of laying a row out may shew us, among other things, how many marbles the row
consists of, or on the other hand that we can move these (say) 100 marbles in such-and-such ways.
But when we calculate how the row can be laid out, the calculation shews us what we call a 'transformation
merely by laying out'.
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96. Examine this proposition: it is not an empiral [[sic]] fact that the tangent of a visual curve partly
coincides with the curve; and if a figure shews this, then it does not do so as the result of an experiment.
It could also be said: here you can see that segments of a continuous visual curve are straight.--But ought I not to
have said:--"Now you call this a 'curve'.--And do you call this little bit of it 'curved' or 'straight'?--Surely you call it a
'straight line'; and the curve contains this bit."
But why should one not use a new name for visual stretches of a curve which themselves exhibit no
curvature?
"But the experiment of drawing these lines has shewn that they do not touch at a point."--That they do not
touch at a point? How are 'they' defined? Or again: can you point to a picture of what it is like for them to 'touch at a
point'? Why shouldn't I simply say the experiment has yielded the result that they, i.e. a curved and a straight line,
touch one another? For isn't this what I call a "touching" of such lines?
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97. Let us draw a circle composed of black and white segments getting smaller and smaller.
"Which is the first of these segments--going from left to right--that strikes you as straight?" Here I am making an
experiment.
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98. What if someone were to say "Experience tells me that this line
is curved"?--Here it would have to be said that the words "this line" mean the line drawn on the paper. The
experiment can actually be made, one can show this line to different people and ask: "What do you see, a straight
line or a curved one?"--
But suppose someone were to say: "I am now imagining a curved line", whereupon we tell him: "So you see
that the line is a curved one"--what kind of sense would that make?
One can however also say: "I am imagining a circle made of black and white segments; one is big and
curved, the ones that come after it keep on getting smaller, the sixth is quite straight". Where is the experiment here?
I can calculate in the medium of imagination, but not experiment.
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99. What is the characteristic use of the derivation procedure as a calculation--as opposed to its use as an
experiment?
We regard the calculation as demonstrating an internal property (a property of the essence) of the structures.
But what does that mean?
The following might serve as a model of an 'internal property':
10 = 3 × 3 + 1
Now when I say: 10 strokes necessarily consist of 3 times 3 strokes and 1 stroke--that does not mean: if there are 10
strokes there, then
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they have always got these figures and loops round them.--But if I put them in, I say that I was only demonstrating
the nature of the group of strokes.--But are you certain that the group did not change while you were writing those
symbols in?--"I don't know; but a definite number of strokes was there; and if it was not 10 then it was another
number, and in that case it simply had different properties.--"
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100. One says: calculation 'unfolds' the property of a hundred.--What does it really mean to say that 100
consists of 50 and 50? One says: the contents of the box consist of 50 apples and 50 pears. But if someone were to
say: "The contents of the box consist of 50 apples and 50 apples"--, to begin with we shouldn't know what he
meant.--If one says: "The contents of the box consist of twice 50 apples", this means either that there are two
compartments each containing 50 apples; or what is in question is, say, a distribution in which each person is
supposed to get 50 apples, and now I hear that two people can be given their share out of this box.
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101. "The 100 apples in this box consist of 50 and 50"--here the non-temporal character of 'consist' is
important. For it doesn't mean that now, or just for a time, they consist of 50 and 50.
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102. For what is the characteristic mark of 'internal properties'? That they persist always, unalterably, in the
whole that they constitute; as it were independently of any outside happenings. As the construction of a machine on
paper does not break when the machine itself succumbs to external forces.--Or again, I should like to say that they
are not subject to wind and weather like physical things; rather are they unassailable, like shadows.
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103. When we say: "This proposition follows from that one" here again "to follow" is being used
non-temporally. (And this shews that the proposition does not express the result of an experiment.)
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104. Compare "White is lighter than black". This expression too is non-temporal and it too expresses the
existence of an internal relation.
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105. "But this relation holds"--one would like to say. The question is: has this proposition a use--and what
use? For at the moment all I know is that a picture comes before my mind as I say it (but that does not guarantee the
use for me) and that the words form an English sentence. But it sticks out that the words are being used otherwise
here than in the everyday case of a useful statement. (As, say, a wheelwright may notice that the statements that he
ordinarily makes about what is circular and straight are of a different kind from what are to be found in Euclid.) For
we say: this object is lighter than that one, or the colour of this thing is lighter than the colour of that one, and in this
case something is lighter now and may be darker later on.
Whence comes the feeling that "white is lighter than black" expresses something about the essence of the
two colours?--
But is this the right question to ask? For what do we mean by the 'essence' of white or black? We think
perhaps of 'the inside', 'the constitution', but this surely makes no sense here. We also say e.g.: "It is part of white to
be lighter than...".
Is it not like this: the picture of a black and a white patch
serves us simultaneously as a paradigm of what we understand by "lighter"
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and "darker" and as a paradigm for "white" and for "black". Now darkness 'is part of' black inasmuch as they are
both represented by this patch. It is dark by being black.--But to put it better: it is called "black" and hence in our
language "dark" too. That connexion, a connexion of the paradigms and the names, is set up in our language. And
our proposition is non-temporal because it only expresses the connexion of the words "white", "black" and "lighter"
with a paradigm.
Misunderstandings can be avoided by declaring it nonsense to say: "the colour of this body is brighter than
the colour of that one"; what would have to be said is "this body is brighter than that one". I.e. the former way of
putting it is excluded from our language.
Whom do we tell "White is lighter than black"? What information does it give?
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106. But can't I believe the geometrical proposition even without a proof, for example on someone else's
assurance?--And what does the proposition lose in losing its proof?--Here I presumably ought to ask: "What can I
do with it?", for that is the point. Accepting the proposition on someone else's assurance--how does my doing this
come out? I may for example use it in further calculating operations, or I use it in judging some physical fact. If
someone assures me, for example, that 13 × 13 are 196 and I believe him, then I shall be surprised that I can't arrange
196 nuts in 13 rows of 13 each, and I shall perhaps assume that the nuts have increased of themselves.
But I feel a temptation to say: one can't believe that 13 × 13 = 196, one can only accept this number
mechanically from somebody else. But why should I not say I believe it? For is believing it a mysterious act with as
it were an underground connexion with the correct calculation? At any rate I can say: "I believe it", and act
accordingly.
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One would like to ask: "What are you doing in believing that 13 × 13 = 196?" And the answer may be: Well,
that will depend on whether, for instance, you did the sum and made a slip of the pen in doing so,--or whether
somebody else did it, but you yourself know how such a calculation is done,--or whether you cannot multiply but
know that the product is the number of people to be found in 13 rows of 13 each,--in short it depends on what you
can do with the equation 13 × 13 = 196. For testing it is doing something with it.
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107. The thing is, if one thinks of an arithmetical equation as the expression of an internal relation, then one
would like to say: "You can't believe at all that 13 × 13 yields this, because that isn't a multiplication of 13 by 13, or
is not a case of something yielded, if 196 comes at the end." But that means that one is not willing to use the word
"believe" for the case of a calculation and its result,--or is willing only in the case in which one has a correct
calculation before one.
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108. "What are you believing if you believe 13 × 13 = 196?--"How deep do you penetrate, one might say,
with your belief, into the relation of these numbers? For--one wants to say--you cannot be penetrating all the way, or
you could not believe it.
But when have you penetrated into the relations of the numbers? Just while you say that you believe...? You
will not take your stand on that--for it is easy to see that this appearance is merely produced by the superficial form
of our grammar (as it might be called).
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109. For I want to say: "One can only see that 13 × 13 = 169, and even that one can't believe. And one
can--more or less blindly--accept a rule". And what am I doing if I say this? I am drawing a line between
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the calculation with its result (that is to say a particular picture, a particular model), and an experiment with its
outcome.
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110. I should like to say: "When I believe that a × b = c--and I do sometimes have such beliefs--do say that I
have them--I am not believing the mathematical proposition, for that comes at the end of a proof, is the end of a
proof; I am believing that this is the formula that comes in such-and-such a place, which I shall obtain in
such-and-such a way, and so on".--And this does sound as if I were penetrating the process of believing such a
proposition. Whereas I am merely--in an unskilful fashion--pointing to the fundamental difference, together with an
apparent similarity, between the roles of an arithmetical proposition and an empirical proposition.
For in certain circumstances I do say: "I believe that a × b = c". What do I mean by this?--What I say!--But
what is interesting is the question in what circumstances I say this and what is characteristic of them in contrast to
those of a statement like: "I believe it is going to rain". For what preoccupies us is this contrast. What we require is a
picture of the employment of mathematical propositions and of sentences beginning "I believe that...", where a
mathematical proposition is the object of belief.
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111. "But you surely don't believe a mathematical proposition."--That means: 'Mathematical proposition'
signifies a role for the proposition, a function, in which believing does not occur.
Compare: "If you say: 'I believe that castling takes place in such and such a way', then you are not believing
the rule of chess, but believing e.g. that a rule of chess runs like that".
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112. "One can't believe that the multiplication 13 × 13 yields 169, because the result is part of the
calculation."--What am I calling "the multiplication 13 × 13"? Only the correct pattern of multiplication, at the end of
which comes 169? Or a 'wrong multiplication' too?
How is it established which pattern is the multiplication 13 × 13?--Isn't it defined by the rules of
multiplication?--But what if, using these rules, you get different results today from what all the arithmetic books say?
Isn't that possible?--"Not if you apply the rules as they do!" Of course not! But that is a mere pleonasm. And where
does it say how they are to be applied--and if it does say somewhere, where does it say how that is to be applied?
And that does not mean only: in what book does it say, but also: in what head?--What then is the multiplication 13 ×
13--or what am I to take as a guide in multiplying--the rules, or the multiplication that comes in the arithmetic
books--if, that is, these two do not agree?--Well, it never in fact happens that somebody who has learnt to calculate
goes on obstinately getting different results, when he does a given multiplication, from what comes in the arithmetic
books. But if it should happen, then we should declare him abnormal, and take no further account of his calculation.
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113. "But am I not compelled, then, to go the way I do in a chain of inferences?"--Compelled? After all I can
presumably go as I choose!--"But if you want to remain in accord with the rules you must go this way."--Not at all, I
call this 'accord'.--"Then you have changed the meaning of the word 'accord', or the meaning of the rule."--No;--who
says what 'change' and 'remaining the same' mean here?
However many rules you give me--I give a rule which justifies my employment of your rules.
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114. We might also say: when we follow the laws of inference (inference-rules) then following always
involves interpretation too.
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115. "But you surely can't suddenly make a different application of the law now!"--If my reply is: "Oh yes of
course, that is how I was applying it!" or: "Oh! That's how I ought to have applied it--!"; then I am playing your
game. But if I simply reply: "Different?--But this surely isn't different!"--what will you do? That is: somebody may
reply like a rational person and yet not be playing our game.†1
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116. "Then according to you everybody could continue the series as he likes; and so infer anyhow!" In that
case we shan't call it "continuing the series" and also presumably not "inference". And thinking and inferring (like
counting) is of course bounded for us, not by an arbitrary definition, but by natural limits corresponding to the body
of what can be called the role of thinking and inferring in our life.
For we are at one over this, that the laws of inference do not compel him to say or to write such and such like
rails compelling a locomotive. And if you say that, while he may indeed say it, still he can't think it, then I am only
saying that that means, not: try as he may he can't think it, but: it is for us an essential part of 'thinking' that--in
talking, writing, etc.--he makes this sort of transition. And I say further that the line between what we include in
'thinking' and what we no longer include in 'thinking' is no more a hard and fast one than the line between what is
still and what is no longer called "regularity".
Nevertheless the laws of inference can be said to compel us; in the
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same sense, that is to say, as other laws in human society. The clerk who infers as in (17) must do it like that; he
would be punished if he inferred differently. If you draw different conclusions you do indeed get into conflict, e.g.
with society; and also with other practical consequences.
And there is even something in saying: he can't think it. One is trying e.g. to say: he can't fill it with personal
content; he can't really go along with it--personally, with his intelligence. It is like when one says: this sequence of
notes makes no sense, I can't sing it with expression. I cannot respond to it. Or, what comes to the same thing here: I
don't respond to it.
"If he says it"--one might say--"he can only say it without thinking". And here it merely needs to be noticed
that 'thoughtless' talk and other talk do indeed sometimes differ as regards what goes on in the talker, his images,
sensations and so on while he is talking, but that this accompaniment does not constitute the thinking, and the lack
of it is not enough to constitute 'thoughtlessness'.
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117. In what sense is logical argument a compulsion?--"After all you grant this and this; so you must also
grant this!" That is the way of compelling someone. That is to say, one can in fact compel people to admit
something in this way.--Just as one can e.g. compel someone to go over there by pointing over there with a bidding
gesture of the hand.
Suppose in such a case I point with two fingers at once in different directions, thus leaving it open to the man
to go in which of the two directions he likes,--and another time I point in only one direction; then this can also be
expressed by saying: my first order did not compel him to go just in one direction, while the second one did. But
this is a statement to tell us what kind of orders I gave; not the way they operate, not whether they do in fact compel
such-and-such a person, i.e. whether he obeys them.
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118. It looked at first as if these considerations were meant to shew that 'what seems to be a logical
compulsion is in reality only a psychological one'--only here the question arose: am I acquainted with both kinds of
compulsion, then?!
Imagine that people used the expression: "The law §... punishes a murderer with death". Now this could only
mean: this law runs so and so. That form of expression, however, might force itself on us, because the law is an
instrument when the guilty man is brought to punishment.--Now we talk of 'inexorability' in connexion with people
who punish. And here it might occur to us to say: "The law is inexorable--men can let the guilty go, the law executes
him". (And even: "the law always executes him".)--What is the use of such a form of expression?--In the first
instance, this proposition only says that such-and-such is to be found in the law, and human beings sometimes do
not go by the law. Then, however, it does give us a picture of a single inexorable judge, and many lax judges. That is
why it serves to express respect for the law. Finally, the expression can also be so used that a law is called inexorable
when it makes no provision for a possible act of grace, and in the opposite case it is perhaps called 'discriminating'.
Now we talk of the 'inexorability' of logic; and think of the laws of logic as inexorable, still more inexorable
than the laws of nature. We now draw attention to the fact that the word "inexorable" is used in a variety of ways.
There correspond to our laws of logic very general facts of daily experience. They are the ones that make it possible
for us to keep on demonstrating those laws in a very simple way (with ink on paper for example). They are to be
compared with the facts that make measurement with a yardstick easy and useful. This suggests the use of precisely
these laws of inference, and now it is we that are inexorable in applying these laws. Because we 'measure'; and it is
part of measuring for everybody to have the same measures. Besides this, however, inexorable, i.e. unambiguous
rules of inference can be distinguished from ones that are not unambiguous, I mean from such as leave an alternative
open to us.
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119. "But I can infer only what actually does follow."--That is to say: what the logical machine really does
produce. The logical machine--that would be an all-pervading ethereal mechanism.--We must give warning against
this picture.
Imagine a material harder and more rigid than any other. But if a rod made of this stuff is brought out of the
horizontal into the vertical, it shrinks; or it bends when set upright and at the same time it is so hard that there is no
other way of bending it.--(A mechanism made of this stuff, say a crank, connecting-rod and crosshead. The different
way the crosshead would move.)
Or again: a rod bends if one brings a certain mass near it; but it is completely rigid in face of all forces that we
subject it to. Imagine that the guide-rails of the crosshead bend and then straighten again as the crank approaches
and retreats. My assumption would be, however, that no particular external force is necessary to cause this. This
behaviour of the rails would give an impression as of something alive.
When we say: "If the parts of the mechanism were quite rigid, they would move so and so", what is the
criterion for their being quite rigid? Is it that they resist certain forces? Or that they do move so and so?
Suppose I say: "This is the law of motion of the crosshead (the correlation of its position and the position of
the crank perhaps) when the lengths of the crank and connecting-rod remain constant". This presumably means: If
the crank and crosshead keep these relative positions, I say that the length of the connecting-rod remains constant.
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120. "If the parts were perfectly rigid this is how they would move"; is that a hypothesis? It seems not. For
when we say: "Kinematics describes the movements of the mechanism on the assumption that its parts are
completely rigid", on the one hand we are admitting that this assumption never squares with reality, and on the other
hand it
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is not supposed to be in any way doubtful that completely rigid parts would move in this way. But whence this
certainty? The question here is not really one of certainty but of something stipulated by us. We do not know that
bodies would move in these ways if (by such and such criteria) they were quite rigid; but (in certain circumstances)
we should certainly call 'rigid' such parts as did move in those ways.--Always remember in such a case that
geometry (or kinematics) does not specify any method of measuring when it talks about the same, or constant,
length.
When therefore we call kinematics the theory, say, of the movement of perfectly rigid parts of a mechanism,
on the one hand this contains an indication as to (mathematical) method--we stipulate certain distances as the
lengths of machine parts that do not alter--and on the other hand an indication about the application of the calculus.
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121. The hardness of the logical must. What if one were to say: the must of kinematics is much harder than
the causal must compelling a machine part to move like this when another moves like this?--
Suppose we represented the movement of the 'perfectly rigid' mechanism by a cinematographic picture, a
cartoon film. Suppose this picture were said to be perfectly hard, and this meant that we had taken this picture as
our method of description--whatever the facts may be, however the parts of the real mechanism may bend or
expand.
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122. The machine (its structure) as symbolizing its action: the action of a machine--I might say at first--seems
to be there in it from the start. What does that mean?--
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If we know the machine, everything else, that is its movement, seems
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to be already completely determined.
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"We talk as if these parts could only move in this way, as if they could not do anything else."
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How is this--do we forget the possibility of their bending, breaking off, melting, and so on? Yes; in many
cases we don't think of that at all. We use a machine, or the picture of a machine, to symbolize a particular action of
the machine. For instance, we give someone such a picture and assume that he will derive the movement of the parts
from it. (Just as we can give someone a number by telling him that it is the twenty-fifth in the series 1, 4, 9, 16, ....)
Page 85
"The machine's action seems to be in it from the start" means: you are inclined to compare the future
movements of the machine in definiteness to objects which are already lying in a drawer and which we then take
out.
Page 85
But we do not say this kind of thing when we are concerned with predicting the actual behaviour of a
machine. Here we do not in general forget the possibility of a distortion of the parts and so on.
Page 85
We do talk like that, however, when we are wondering at the way we can use a machine to symbolize a given
way of moving--since it can also move in quite different ways.
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Now, we might say that a machine, or the picture of it, is the first of a series of pictures which we have learnt
to derive from this one.
Page 85
But when we remember that the machine could also have moved differently, it readily seems to us as if the
way it moves must be contained in the machine-as-symbol far more determinately than in the actual machine. As if
it were not enough here for the movements in question to be empirically determined in advance, but they had to be
really--in a mysterious sense--already present. And it is quite true: the movement of the machine-as-symbol is
predetermined in a different sense from that in which the movement of any given actual machine is predetermined.
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Page 86
123. "It is as if we could grasp the whole use of the word in a flash." Like what e.g.?--Can't the use--in a
certain sense--be grasped in a flash? And in what sense can it not? The point is, that it is as if we could 'grasp it in a
flash' in yet another and much more direct sense than that.--But have you a model for this? No. It is just that this
expression suggests itself to us. As the result of crossing similes.
Page 86
124. You have no model of this superlative fact, but you are seduced into using a super-expression.
Page 86
125. When does one have the thought: the possible movements of a machine are already there in it in some
mysterious way?--Well, when one is doing philosophy. And what leads us into thinking that? The way we talk about
machines. We say, for example, that a machine has (possesses) such-and-such possibilities of movement; we speak
of the ideally rigid machine which can only move in such-and-such a way.--What is this possibility of movement? It
is not the movement, but it does not seem to be the mere physical conditions for moving either, e.g. that there is a
certain space between socket and pin, the pin not fitting too tight in the socket. For while this is the empirical
condition for movement, one could also imagine it to be otherwise. The possibility of a movement is, rather,
supposed to be a shadow of the movement itself. But do you know of such a shadow? And by a shadow I do not
mean some picture of the movement; for such a picture would not necessarily be a picture of just this movement.
But the possibility of this movement must be the possibility of just this movement. (See how high the seas of
language run here!)
Page 86
The waves subside as soon as we ask ourselves: how do we use the
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phrase "possibility of movement" when we are talking about a given machine?--But then where did our queer ideas
come from? Well, I shew you the possibility of a movement, say by means of a picture of the movement: 'so
possibility is something which is like reality'. We say: "It isn't moving yet, but it already has the possibility of
moving"--'so possibility is something very near reality'. Though we may doubt whether such-and-such physical
conditions make this movement possible, we never discuss whether this is the possibility of this or of that
movement: 'so the possibility of the movement stands in a unique relation to the movement itself; closer than that of
a picture to its subject'; for it can be doubted whether a picture is the picture of this thing or that. We say
"Experience will shew whether this gives the pin this possibility of movement", but we do not say "Experience will
shew whether this is the possibility of this movement": 'so it is not an empirical fact that this possibility is the
possibility of precisely this movement'.
Page 87
We pay attention to the expressions we use concerning these things; we do not understand them, however,
but misinterpret them. When we do philosophy we are like savages, primitive people, who hear the expressions of
civilized men, put a false interpretation on them, and then draw queer conclusions from it.
Page 87
Imagine someone not understanding our past tense: "he has had it".--He says: "'he has'--that's present, so the
proposition says that in some sense the past is present."
Page 87
126. "But I don't mean that what I do now (in grasping a sense) determines the future use causally and as a
matter of experience, but that in a queer way, the use itself is in some sense present." But of course it is, 'in some
sense'! (And don't we also say: "the events of the years that are past are present to me"?) Really the only thing
wrong with what you say is the expression "in a queer way". The rest is
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correct; and the sentence only seems queer when one imagines a different language-game for it from the one in
which we actually use it. (Someone once told me that as a child he had wondered how a tailor 'sewed a dress'--he
thought this meant that a dress was produced just by sewing, by sewing one thread on to another.)
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127. In our failure to understand the use of a word we take it as the expression of a queer process. (As we
think of time as a queer medium, of the mind as a queer kind of being.)
The difficulty arises in all these cases through mixing up "is" and "is called".
Page 88
128. The connexion which is not supposed to be a causal, experiential one, but much stricter and harder, so
rigid even, that the one thing somehow already is the other, is always a connexion in grammar.
Page 88
129. How do I know that this picture is my image of the sun?--I call it an image of the sun. I use it as a
picture of the sun.
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130. "It's as if we could grasp the whole use of the word in a flash."--And that is just what we say we do.
That is to say: we sometimes describe what we do in these words. But there is nothing astonishing, nothing queer,
about what happens. It becomes queer when we are led to think that the future development must in some way
already be
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present in the act of grasping the use and yet isn't present.--For we say that there isn't any doubt that we understand
the word..., and on the other hand its meaning lies in its use. There is no doubt that I now want to play chess, but
chess is the game it is in virtue of all its rules (and so on). Don't I know, then, which game I wanted to play until I
have played it? Or are all the rules contained in my act of intending? Is it experience that tells me that this sort of
play usually follows this act of intention? So is it impossible for me to be certain what I am intending to do? And if
that is nonsense, what kind of super-strong connexion exists between the act of intending and the thing
intended?--Where is the connexion effected between the sense of the expression "Let's play a game of chess" and all
the rules of the game?--Well, in the list of rules of the game, in the teaching of it, in the day-to-day practice of
playing.
Page 89
131. The laws of logic are indeed the expression of 'thinking habits' but also of the habit of thinking. That is
to say they can be said to shew: how human beings think, and also what human beings call "thinking".
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132. Frege calls it 'a law about what men take for true' that 'It is impossible for human beings... to recognize
an object as different from itself".†1--When I think of this as impossible for me, then I think of trying to do it. So I
look at my lamp and say: "This lamp is different from itself". (But nothing stirs.) It is not that I see it is false, I can't
do anything with it at all. (Except when the lamp shimmers in sunlight; then I can quite well use the sentence to
Page Break 90
express that.) One can even get oneself into a thinking-cramp, in which one does someone trying to think the
impossible and not succeeding. Just as one can also do someone trying (vainly) to draw an object to himself from a
distance by mere willing (in doing this one makes e.g. certain faces, as if one were trying, by one's expression, to
give the thing to understand that it should come here.)
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133. The propositions of logic are 'laws of thought', 'because they bring out the essence of human
thinking'--to put it more correctly: because they bring out, or shew, the essence, the technique, of thinking. They
shew what thinking is and also shew kinds of thinking.
Page 90
134. Logic, it may be said, shews us what we understand by 'proposition' and by 'language'.
Page 90
135. Imagine the following queer possibility: we have always gone wrong up to now in multiplying 12 × 12.
True, it is unintelligible how this can have happened, but it has happened. So everything worked out in this way is
wrong!--But what does it matter? It does not matter at all!--And in that case there must be something wrong in our
idea of the truth and falsity of arithmetical propositions.
Page 90
136. But then, is it impossible for me to have gone wrong in my calculation? And what if a devil deceives me,
so that I keep on overlooking something however often I go over the sum step by step? So that if I were to awake
from the enchantment I should say: "Why, was I blind?"--But what difference does it make for me to 'assume'
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this? I might say: "Yes to be sure, the calculation is wrong--but that is how I calculate. And this is what I now call
adding, and this 'the sum of these two numbers'."
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137. Imagine someone bewitched so that he calculated:
i.e. 4 × 3 + 2 = 10.
Now he is to apply this calculation. He takes 3 nuts four times over, and then 2 more, and he divides them among 10
people and each one gets one nut; for he shares them out in a way corresponding to the loops of the calculation, and
as often as he gives someone a second nut it disappears.
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138. One might also say: in a proof you advance from one proposition to another; but do you also accept a
check on whether you have gone right?--Or do you merely say "It must be right" and measure everything else by the
proposition you arrive at?
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139. For if that is how it is, then you are only advancing from one picture to another.
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140. It might be practical to measure with a ruler which had the property of shrinking to, say, half its length
when it was taken from this room to that. A property which would make it useless as a ruler in other circumstances.
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Page 92
It might be practical, in certain circumstances, to leave numbers out when you were counting a set: to count
them: 1, 2, 4, 5, 7, 8, 10.
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141. What goes on when someone tries to make a shape coincide with its mirror-image by moving it about in
the plane, and does not succeed? He puts them one on top of the other in various ways; looks at the parts that don't
coincide; is dissatisfied; says perhaps: "But it must work", and puts the figures together again in another way.
What happens when someone tries to lift a weight and does not succeed because the weight is too heavy? He
assumes such and such a posture, takes hold of the weight, tenses such and such muscles, and then lets go and
perhaps shews dissatisfaction.
How does the geometrical, logical impossibility of the first task come out?
"Well, he could surely have shewn, in a picture or in some other way, what the thing he is attempting in the
second case looks like." But he asserts that he can do that in the first case too by putting two similar congruent
figures together so that they coincide.--What are we to say now? That the two examples are different? But so are the
picture and the reality in the second case.
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142. What we are supplying are really remarks on the natural history of man: not curiosities however, but
rather observations on facts which no one has doubted and which have only gone unremarked because they are
always before our eyes.
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143. We teach someone a method of sharing out nuts among
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people; a part of this method is multiplying two numbers in the decimal system.
We teach someone to build a house; and at the same time how he is to obtain a sufficient quantity of
material, boards, say; and for this purpose a technique of calculation. The technique of calculation is part of the
technique of house-building.
People pile up logs and sell them, the piles are measured with a ruler, the measurements of length, breadth
and height multiplied together, and what comes out is the number of pence which have to be asked and given. They
do not know 'why' it happens like this; they simply do it like this: that is how it is done.--Do these people not
calculate?
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144. If somebody calculates like this must he utter any 'arithmetical proposition'? Of course, we teach
children the multiplication tables in the form of little sentences, but is that essential? Why shouldn't they simply:
learn to calculate? And when they can do so haven't they learnt arithmetic?
Page 93
145. But in that case how is the foundation of a calculating procedure related to the calculation itself?
Page 93
146. "Yes, I understand that this proposition follows from that."--Do I understand why it follows or do I only
understand that it follows?
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147. Suppose I had said: those people pay for wood on the ground of calculation; they accept a calculation
as proof that they have to pay so much.--Well, that is simply a description of their procedure (of their behaviour).
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Page 94
148. Those people--we should say--sell timber by cubic measure--but are they right in doing so? Wouldn't it
be more correct to sell it by weight--or by the time that it took to fell the timber--or by the labour of felling measured
by the age and strength of the woodsman? And why should they not hand it over for a price which is independent of
all this: each buyer pays the same however much he takes (they have found it possible to live like that). And is there
anything to be said against simply giving the wood away?
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149. Very well; but what if they piled the timber in heaps of arbitrary, varying height and then sold it at a
price proportionate to the area covered by the piles?
And what if they even justified this with the words: "Of course, if you buy more timber, you must pay
more"?
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150. How could I shew them that--as I should say--you don't really buy more wood if you buy a pile
covering a bigger area?--I should, for instance, take a pile which was small by their ideas and, by laying the logs
around, change it into a 'big' one. This might convince them--but perhaps they would say: "Yes, now it's a lot of
wood and costs more"--and that would be the end of the matter.--We should presumably say in this case: they
simply do not mean the same by "a lot of wood" and "a little wood" as we do; and they have a quite different system
of payment from us.
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151. (A society acting in this way would perhaps remind us of the Wise Men of Gotham.)
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Page 95
152. Frege says in the preface to the Grundgesetze der Arithmetik†1: "... here we have a hitherto unknown
kind of insanity"--but he never said what this 'insanity' would really be like.
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153. What does people's agreement about accepting a structure as a proof consist in? In the fact that they use
words as language? As what we call "language".
Imagine people who used money in transactions; that is to say coins, looking like our coins, which are made
of gold and silver and stamped and are also handed over for goods--but each person gives just what he pleases for
the goods, and the merchant does not give the customer more or less according to what he pays. In short this
money, or what looks like money, has among them a quite different role from among us. We should feel much less
akin to these people than to people who are not yet acquainted with money at all and practise a primitive kind of
barter.--"But these people's coins will surely also have some purpose!"--Then has everything that one does a
purpose? Say religious actions--.
It is perfectly possible that we should be inclined to call people who behaved like this insane. And yet we
don't call everyone insane who acts similarly within the forms of our culture, who uses words 'without purpose'.
(Think of the coronation of a King.)
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154. Perspicuity is part of proof. If the process by means of which I get a result were not surveyable, I might
indeed make a note that this number is what comes out--but what fact is this supposed to confirm for me? I don't
know 'what is supposed to come out'.
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Page 96
155. Would it be possible that people should go through one of our calculations to-day and be satisfied with
the conclusions, but to-morrow want to draw quite different conclusions, and other ones again on another day?
Why, isn't it imaginable that it should regularly happen like that: that when we make this transition one time,
the next time, 'just for that reason', we make a different one, and therefore (say) the next time the first one again?
(As if in some language the colour which is called "red" one time is for that reason called another name the next
time, and "red" again the next time after that and so on; people might find this natural. It might be called a need for
variety.)
[Note in margin: Are our laws of inference eternal and immutable?]
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156. Isn't it like this: so long as one thinks it can't be otherwise, one draws logical conclusions. This
presumably means: so long as such-and-such is not brought in question at all.
The steps which are not brought in question are logical inferences. But the reason why they are not brought
in question is not that they 'certainly correspond to the truth'--or something of the sort,--no, it is just this that is
called 'thinking', 'speaking', 'inferring', 'arguing'. There is not any question at all here of some correspondence
between what is said and reality; rather is logic antecedent to any such correspondence; in the same sense, that is, as
that in which the establishment of a method of measurement is antecedent to the correctness or incorrectness of a
statement of length.
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157. Is it experimentally settled whether one proposition can be derived from another?--It looks as if it were.
For I write down certain sequences of signs, am guided in doing so by certain paradigms--in doing which it is indeed
essential that no sign should get overlooked
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or otherwise lost--and of what I get in this procedure I say: it follows.--One argument against this is: If 2 and 2 apples
add up to only 3 apples, i.e. if there are 3 apples there after I have put down two and again two, I don't say: "So after
all 2 + 2 are not always 4"; but "Somehow one must have gone".
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158. But how am I making an experiment when I merely follow a proof which has already been written out?
It might be said: "When you look at this chain of transformations,--don't they strike you as being in agreement with
the paradigms?".
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159. So if this is to be called an experiment it is presumably a psychological one. For the appearance of
agreement may of course be founded on sense-deception. And so it sometimes is when we make a slip in
calculating.
One also says: "This is my result". And what shews that this is my result is presumably an experiment.
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160. One might say: the result of the experiment is that at the end, having reached the result of the proof, I
say with conviction: "Yes, that's right".
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161. Is a calculation an experiment?--Is it an experiment for me to get out of bed in the morning? But might it
not be an experiment,--to shew whether I have the strength to raise myself up after so and so many hours' sleep?
And how does the action fall short of being this experiment?--Merely by not being carried out with this
purpose, i.e. in connexion
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with an investigation of this kind. It is the use that is made of something that turns it into an experiment.
Is an experiment in which we observe the acceleration of a freely falling body a physical experiment, or is it a
psychological one shewing what people see in such circumstances?--Can't it be either? Doesn't it depend on its
surroundings: on what we do with it, say about it?
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162. If a proof is conceived as an experiment, at any rate the result of the experiment is not what is called the
result of the proof. The result of the calculation is the proposition with which it concludes; the result of the
experiment is that from these propositions, by means of these rules, I was led to this proposition.
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163. But our interest does not attach to the fact that such-and-such (or all) human beings have been led this
way by these rules (or have gone this way); we take it as a matter of course that people--'if they can think
correctly'--go this way. We have now been given a road, as it were by means of the footsteps of those who have
gone this way. And the traffic now proceeds on this road--to various purposes.
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164. Certainly experience tells me how the calculation comes out; but that is not all there is to my accepting
it.
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165. I learned empirically that this came out this time, that it usually does come out; but does the proposition
of mathematics say
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that? I learned empirically that this is the road I travelled. But is that the mathematical statement?--What does it say,
though? What relation has it to these empirical propositions? The mathematical proposition has the dignity of a rule.
So much is true when it's said that mathematics is logic: its moves are from rules of our language to other
rules of our language. And this gives it its peculiar solidity, its unassailable position, set apart.
(Mathematics deposited among the standard measures.)
Page 99
166. What, then--does it just twist and turn about within these rules?--It forms ever new rules: is always
building new roads for traffic; by extending the network of the old ones.
Page 99
167. But then doesn't it need a sanction for this? Can it extend the network arbitrarily? Well, I could say: a
mathematician is always inventing new forms of description. Some, stimulated by practical needs, others, from
aesthetic needs,--and yet others in a variety of ways. And here imagine a landscape gardener designing paths for the
layout of a garden; it may well be that he draws them on a drawing-board merely as ornamental strips without the
slightest thought of someone's sometime walking on them.
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168. The mathematician is an inventor, not a discoverer.
Page 99
169. We know by experience that when we count anything off on the fingers of one hand, or on some group
of things that looks like this | | | | |, and say: I, you, I, you, etc., the first word is also the last.
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"But doesn't it have to be like that, then?"--Well, is it unimaginable for someone to see the group | | | | | (e.g.) as the
group | | || | | with the two middle strokes fused, and should accordingly count the middle stroke twice? (True, it is
not the usual case.--)
Page 100
170. But how about when I draw someone's attention for the first time to the fact that the result of counting
off is determined in advance by the beginning, and he understands and says: "Yes, of course,--that's how it has to
be". What sort of knowledge is this?--He e.g. drew himself the schema:
I Y I Y I
| | | | |
and his reasoning is e.g.: "That's what it's like when I count off.--So it has to...."
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(171. Connected with this: We should sometimes like to say "There must surely be a reason why--in a
movement of a sonata, for example--just this theme follows that one." What we should acknowledge as a reason
would be a certain relation between the themes, a kinship, a contrast or the like.--But we may even construct such a
relation: an operation, so to speak, that produces the one theme from the other; but this serves only when this
relation is one that we are familiar with. So it is as if the sequence of these themes had to correspond to a paradigm
that is already present in us.
Similarly one might say of a picture that shews two human figures: "There must be a reason why precisely
these two faces make such an impression upon us." That means: we should like to find this impression from the pair
of faces again somewhere else--in another region.--But could we?
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Page 101
One might ask: what arrangement of themes together has a point, and what has no point? Or again: Why has
this arrangement a point and this one none? That may not be easy to say! Often we may say: "This one corresponds
to a gesture, this one doesn't.")†1
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APPENDIX I
(1933-1934)
Page 102
1. Might I not say that two words--let's write them "non" and "ne"--had the same meaning, that they were
both negation signs--but
non non p = p
and
ne ne p = ne p
(In spoken language a double negation very often means a negation.) But then why do I call them both "negations"?
What have they in common with one another? Well, it is clear that a great part of their employment is common. But
that does not solve our problem. For we should after all like to say: "It must also hold for both of them that the
double negation is an affirmation, at least if the doubling is thought of appropriately". But how?--Well, as for
example we expressed it using brackets:
(ne ne)p = ne p, ne(ne p) = p
We think at once of an analogous case in geometry: "Two half turns added together cancel one another out,"
"Two half turns added together make one half turn."
It just depends how we add them. (Whether we put them side by side or one after the other.)
Page 102
2. (Here we stumble on a remarkable and characteristic phenomenon in philosophical investigation: The
difficulty--I might say--isn't one
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of finding the solution; it is one of recognizing something as the solution. We have already said everything. Not
something that follows from this; no, just this is the solution!
This, I believe, hangs together with our wrongly expecting an explanation; whereas a description is the
solution of the difficulty, if we give it the right place in our consideration. If we dwell upon it and do not try to get
beyond it.)
Page 103
3. "That's already all there is to say about it." Taking "non non p" as the negation of the negated proposition
in the particular case is, say, giving an explanation of the kind "non non p = non (non p)".
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4. "If 'ne' is a negation, then 'ne ne p' must be the same as p, if only it is taken appropriately."
"If one takes 'ne ne p' as negating p, one must be taking the doubling in a different way."
One would like to say: "'Doubling' means something different in this case and that's why it yields a negation
here;" and so, its yielding a negation here is the consequence of this difference of nature. "Now I mean it as a
strengthening", one would say. We use the expression of meaning to assess meaning.†1
Page 103
5. When I was uttering the double negation, what may it have
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consisted in that I meant it as a strengthening? In the circumstances in which I use the expression, perhaps in the
image that comes before my mind as I use it or which I employ, in my tone of voice (as I can even reproduce the
brackets in "ne (ne p) in my tone of voice). In that case, meaning the doubling as a strengthening corresponds to
pronouncing it as a strengthening. The activity of meaning the doubling as a cancellation was e.g., putting
brackets.--"Yes, but these brackets themselves may have a variety of roles; for who says that they are to be taken as
brackets in the ordinary sense in 'non (non p)' and not for example the first as a hyphen between the two 'non's and
the second as the full stop for the sentence? No one says it. And haven't you yourself replaced your conception by
words? What the brackets mean will come out in their use; and in another sense perhaps lies in the rhythm of the
optical impression of 'non (non p)'.
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6. Am I now to say: the meanings of "non" and "ne" are somewhat different? That they are different species
of negation?--That no one would say. For it would be objected, in that case won't "do not go into this room" perhaps
fail to mean exactly the same as usual if we set up the rule that "not not" is to be used as a negation?--But this might
be countered: "If the two propositions 'ne p' and 'non p' say exactly the same, then how can 'ne ne' not mean exactly
the same as 'non non'?" But here we are presupposing a symbolism--i.e., we are taking it as a model--in which from
"ne p = non p" it follows that "ne" and "non" are used in the same way in all cases.
Turning through 180° and negation are in fact the same in the particular case, and the application of 'non non
p = p' is of the same kind as the application of a particular geometry.
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7. What does one mean by saying: 'ne ne p', even if by convention it means 'ne p', could also be used as a
cancelled negation?--One would like to say: "with the meaning that we have given it, 'ne' could cancel itself, if only
we apply it right." What does one mean by that? (The two half turns in the same direction could cancel one another,
if they are put together appropriately.) "The movement of the negation 'not' is capable of cancelling itself." But where
is this movement? One would like of course to speak of a mental movement of negation, for the execution of which
the sign 'ne' merely gives the signal.
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8. We can imagine human beings with a 'more primitive' logic, in which only for certain sentences is there
anything corresponding to our negation: say for such as contain no negation. In the language of these people, then, a
sentence like "He is going into this house" could be negated; but they would understand a doubling of the negation
as mere repetition, never as cancelling the negation.
Page 105
9. The question whether negation had the same meaning for these people as for us would be analogous to the
question whether the digit '2' means the same, for people whose number series ends with 5, as it does for us.
Page 105
10. Suppose I were to ask: When we pronounce the proposition "this rod is 1 metre long" and 'here is 1
soldier', is it quite apparent to us that "1" has different meanings here?--It is not at all apparent. Especially when we
say a sentence like: "On every 1 metre there stands 1 soldier, every 2 metres 2 soldiers, and so on." Asked, "Do you
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mean the same by the two ones?" we should reply: "of course I mean the same:--one" (perhaps holding up one
finger).
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11. Whoever calls "~~p = p" (or again "~~p ≡ p") a "necessary proposition of logic" (not a stipulation about
the method of presentation that we adopt) also has a tendency to say that this proposition proceeds from the
meaning of negation. When double negation is used as negation in some dialect, as in "he found nothing nowhere",
we are inclined to say: really that would mean that he found something everywhere. Let us consider what this
"really" means.
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12. Suppose we had two systems for measuring length; in both a length is expressed by a numeral, followed
by a word that gives the system of measurement. One system designates a length as "n foot" and foot is a unit of
length in the ordinary sense; in the other system a length is designated by "n W" and
1 foot = 1 W
But: 2 W = 4 foot, 3 W = 9 foot and so on.
So the sentence "this post is 1 W long" says the same as "this post is 1 foot long".
Question: Have "W" and "foot" the same meaning in these two sentences?
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13. The question is framed wrong. One sees this when we express identity of meaning by means of an
equation. Then the question has to run: "Does W = foot or not?"--The sentences in which these signs occur
disappear in this way of looking at it. Of course in this terminology one can just as little ask whether "is" means the
same as "is";
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but one can ask whether "ε" means the same as "=". What we said was: 1 foot = 1 W, but: foot ≠ W.
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14. Has "ne" the same meaning as "non"?--Can I replace "non" by "ne"?--"Well, I can in certain places, but
not in others."--But I wasn't asking about that. My question was: can one, without any further qualification, use "ne"
in place of "non"?--No.
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15. "'ne' and 'non' mean exactly the same in this case."--And what do they mean?--"Well, one is not to do
such and such."--But by saying this you have only said that in this case ne p = non p and that we don't deny.
When you explain: ne ne p = ne p, non non p = p, you are indeed using the two words in different ways; and
if one holds on to the conception that what they yield in certain combinations 'depends' on their meaning, or the
meaning that they carry around with them, then one has to say that they must have different meanings if,
compounded in the same way, they may yet yield different results.
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16. One would like to speak of the function of the word, of what it does, in this sentence. As of the function
of a lever in a machine. But what does this function consist in? How does it come to light? For there isn't anything
hidden, is there? We see the whole sentence all right. The function must reveal itself in the course of the calculus.
But one wants to say: "'non' does the same with the proposition 'p' as 'ne' does: it reverses it". But that is just
"non p = ne p" in other words.†1 Over and over the thought, the picture, that what we
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see of the sign is only the exterior of some inner thing in which the real operations of meaning run on.
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17. But if the use of a sign is its meaning is it not remarkable that I say the word "is" gets used with two
different meanings (as 'ε' and '=') and should not like to say that its meaning is its use as copula and as sign of
identity?
One would like to say that these two kinds of use do not yield a single meaning; the personal union through
the same word is inessential, is mere accident.
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18. But how can I decide what is an essential and what is an inessential, accidental feature of the notation? Is
there a reality behind the notation, then, which its grammar is aiming at?
Think of a similar case in a game: In draughts a king is distinguished by putting two pieces one on top of the
other. Won't one say that it is inessential to draughts that this is the way a king is distinguished?
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19. Let us say: the meaning of a piece (a figure) is its role in the game.--Now before the start of any
chess-game let it be decided by lot which of the players gets white. For this purpose one player holds a king in each
closed hand and the other chooses one of the hands at random. Will it be reckoned as part of the role of the king in
chess that it is used for drawing by lot?
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20. Thus even in a game I am inclined to distinguish between essential and inessential. The game, I should
like to say, does not just have rules; it has a point.
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21. What is the word the same for? For in the calculus we make no use of this identity! What do both players
have the same pieces for? But what does "making use of the identity" mean here? For isn't it a use, if we do use the
same word?
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22. Here it looks now as if the use of the same word, the same piece, had a purpose--if the identity was not
accidental, not inessential. And as if the purpose were that one should recognize the piece and be able to tell how to
play. Is it a physical or a logical possibility that is in question here? If the latter, then the identity of the pieces does
indeed belong to the game.
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23. The game is supposed to be defined by the rules! So if a rule of the game prescribes that the kings are to
be taken for choosing by lot before the game starts then that belongs essentially to the game. What objection might
be made to this?--That one does not see the point of this rule. As, say, one wouldn't see the point of a prescription
either, that required one to turn any piece round three times before making a move with it. If we found this rule in a
board-game, we should be surprised, and form conjectures about the origin, the purpose, of such a rule. ("Is this
prescription supposed to prevent one from moving without consideration?")
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24. "If I understand the character of the game right," I might say, "this is not essential to it."
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25. But let us think of the two offices joined in one person as an old convention.
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26. One says: the use of the same word is inessential here, because the identity of the shape of the word does
not here serve to mediate a transition. But in saying that one is merely describing the character of the game that one
wants to play.
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27.†1 "What does the word 'a' mean in the sentence 'F(a)'?" "What does the word 'a' mean in the sentence 'Fa'
which you have just spoken?" "What does the word .... mean in this sentence?"
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APPENDIX II
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1. The surprising may play two completely different parts in mathematics.
One may see the value of a mathematical train of thought in its bringing to light something that surprises
us:--because it is of great interest, of great importance, to see how such and such a kind of representation of it makes
a situation surprising, or astonishing, even paradoxical.
But different from this is a conception, dominant at the present day, which values the surprising, the
astonishing, because it shews the depths to which mathematical investigation penetrates;--as we might measure the
value of a telescope by its shewing us things that we'd have had no inkling of without this instrument. The
mathematician says as it were: "Do you see, this is surely important, this you would never have known without me."
As if, by means of these considerations, as by means of a kind of higher experiment, astonishing, nay the most
astonishing facts were brought to light.
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2. But the mathematician is not a discoverer: he is an inventor.
"The demonstration has a surprising result!"--If you are surprised, then you have not understood it yet. For
surprise is not legitimate here, as it is with the issue of an experiment. There--I should like to say--it is permissible to
yield to its charm; but not when the surprise comes to you at the end of a chain of inference. For here it is only a
sign that unclarity or some misunderstanding still reigns.
"But why should I not be surprised that I have been led hither?"--Imagine you had a long algebraic
expression before you; at first it looks as if it could not be essentially shortened; but then you see a possibility of
shortening it and now it goes on until the expression
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is shrunk into a compact form. May we not be surprised at this result? (Something similar happens in playing
Patience.) Certainly, and it is a pleasant surprise; and it is of psychological interest, for it shows a phenomenon of
failure to command a clear view and of the change of aspect of a seen complex. It is interesting that one does not
always see in this complex that it can be shortened in this way; but if we are able to survey the way of shortening it,
the surprise disappears.
When one says that one just is surprised at having been led to this, that does not represent the situation quite
correctly. For one surely has this surprise only when one does not yet know the way. Not when one has the whole
of it before one's eyes. The fact that this way, that I have completely in view, begins where it begins and ends where
it ends, that's no surprise. The surprise and the interest, then come, so to speak, from outside. I mean: one can say
"This mathematical investigation is of great psychological interest" or "of great physical interest."
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3. I keep on being astonished at this turn of the theme; though I have heard it countless times and know it by
heart. It is perhaps its sense to arouse astonishment.
What is it supposed to mean, then, when I say "You oughtn't be astonished?"
Think of mathematical puzzles. They are framed because they surprise: that is their whole sense.
I want to say: You ought not to believe that there is something hidden here, into which one can get no
insight--as if we had walked through an underground passage and now come up somewhere into the light, without
being able to tell how we got here, or how the entry of the tunnel lay in relation to its exit.
But then how was it possible for us to fancy this at all? What is there about a calculation that is like a
movement underground? What can have suggested this picture to us? I believe it is this: no daylight
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falls on these steps; that we understand the starting point and the end of the calculation in a sense in which we do
not understand the remaining course of the calculation.
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4. "There's no mystery here!"--but then how can we have so much as believed that there was one?--Well, I
have retraced the path over and over again and over and over again been surprised; and I never had the idea that here
one can understand something.--So "There's no mystery here!" means "Just look about you!"
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5. Isn't it as if one saw a sort of turning up of a card in a calculation? One mixed up the cards; one doesn't
know what was going on among them; but in the end this card came out on top, and this means that rain is coming.
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6. The difference between casting lots and counting out before a game. But might not naive people even
when it is a serious matter use counting out instead of choosing a man by lot?
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7. What is someone doing when he makes us realize that in counting out the result is already fixed?
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8. I want to say: "We don't command a clear view of what we have done, and that is why it strikes us as
mysterious. For now there is a result in front of us and we no longer know how we got there,
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it is not perspicuous to us, but we say (we have learnt to say): "this is how it has to be"; and we accept it--and marvel
at it. Might we not imagine the following case: Someone has a series of orders of the form "You must now do such
and such", each one written on a card. He mixes these cards up together, reads the one that comes out on top and
says: so I must do that?--For the reading of a written order now makes a particular impression, has a particular effect
on him. And so equally has the reaching of the conclusion of an inference.--However, one might perhaps break the
spell of such an order, by bringing it clearly before the man's eyes again how he arrived at these words, and
comparing what happened here with other cases--by saying, e.g.: "After all, no one has given you the order!"
And isn't it like that, when I say "There's no mystery here"?--Indeed, in a certain sense he had not believed
that there was a mystery in the case. But he was under the impression of mystery (as the other was under the
impression of an order). In one sense he was indeed acquainted with the situation, but he related to it (in feeling and
in action) 'as if there were something else involved'--as we would say.
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9. "A definition surely only takes you one step further back, to something that is not defined." What does
that tell us? Did anyone not know that?--No--but may he not have lost sight of it?
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10. Or: "If you write
'1, 4, 9, 16....',
you have merely written down four numbers and four dots"--what are you bringing to our notice here? Could
anyone think anything else? It would also be natural to say: "You have written nothing there except four numerals
and a fifth sign--the dots." Well didn't he know
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this? Still, he might say, mightn't he, "I never really looked on the dots as one further sign in this series of signs--but
rather as a way of suggesting further numerals."
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11. Or suppose someone gets us to notice that a line, in Euclid's sense, is a boundary of two coloured
surfaces, and not a mark? and a point the intersection of such colour boundaries and not a dot? (How often has it
been said that you cannot imagine a point.)
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12. It is possible for one to live, to think, in the fancy that things are thus and so, without believing it; that is
to say, when one is asked, then one knows, but if one does not have to answer the question one does not know, but
acts and thinks according to another opinion.
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13. For a form of expression makes us act thus and so. When it dominates our thinking, then in spite of all
objections we should like to say: "But surely it is so in some sense." Although the 'some sense' is just what matters.
(Roughly like the way it signifies a man's dishonesty when we say "He's not a thief".)
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APPENDIX III
Page 116
1. It is easy to think of a language in which there is not a form for questions, or commands, but question and
command are expressed in the form of statements, e.g. in forms corresponding to our: "I should like to know if..."
and "My wish is that...".
No one would say of a question (e.g. whether it is raining outside) that it was true or false. Of course it is
English to say so of such a sentence as "I want to know whether...". But suppose this form were always used instead
of the question?--
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2. The great majority of sentences that we speak, write and read, are statement sentences.
And--you say--these sentences are true or false. Or, as I might also say, the game of truth-functions is played
with them. For assertion is not something that gets added to the proposition, but an essential feature of the game we
play with it. Comparable, say, to that characteristic of chess by which there is winning and losing in it, the winner
being the one who takes the other's king. Of course, there could be a game in a certain sense very near akin to chess,
consisting in making the chess moves, but without there being any winning and losing in it; or with different
conditions for winning.
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3. Imagine it were said: A command consists of a proposal ('assumption') and the commanding of the thing
proposed.
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4. Might we not do arithmetic without having the idea of uttering arithmetical propositions, and without ever
having been struck by the similarity between a multiplication and a proposition?
Should we not shake our heads, though, when someone shewed us a multiplication done wrong, as we do
when someone tells us it is raining, if it is not raining?--Yes; and here is a point of connexion. But we also make
gestures to stop our dog, e.g. when he behaves as we do not wish.
We are used to saying "2 times 2 is 4", and the verb "is" makes this into a proposition, and apparently
establishes a close kinship with everything that we call a 'proposition'. Whereas it is a matter only of a very
superficial relationship.
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5. Are there true propositions in Russell's system, which cannot be proved in his system?--What is called a
true proposition in Russell's system, then?
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6. For what does a proposition's 'being true' mean? 'p' is true = p. (That is the answer.)
So we want to ask something like: under what circumstances do we assert a proposition? Or: how is the
assertion of the proposition used in the language-game? And the 'assertion of the proposition' is here contrasted with
the utterance of the sentence e.g. as practice in elocution,--or as part of another proposition, and so on.
If, then, we ask in this sense: "Under what circumstances is a proposition asserted in Russell's game?" the
answer is: at the end of one of his proofs, or as a 'fundamental law' (Pp.). There is no other way in this system of
employing asserted propositions in Russell's symbolism.
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7. "But may there not be true propositions which are written in this symbolism, but are not provable in
Russell's system?"--'True propositions', hence propositions which are true in another system, i.e. can rightly be
asserted in another game. Certainly; why should there not be such propositions; or rather: why should not
propositions--of physics, e.g.--be written in Russell's symbolism? The question is quite analogous to: Can there be
true propositions in the language of Euclid, which are not provable in his system, but are true?--Why, there are even
propositions which are provable in Euclid's system, but are false in another system. May not triangles be--in another
system--similar (very similar) which do not have equal angles?--"But that's just a joke! For in that case they are not
'similar' to one another in the same sense!"--Of course not; and a proposition which cannot be proved in Russell's
system is "true" or "false" in a different sense from a proposition of Principia Mathematica.
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8. I imagine someone asking my advice; he says: "I have constructed a proposition (I will use 'P' to designate
it) in Russell's symbolism, and by means of certain definitions and transformations it can be so interpreted that it
says: 'P is not provable in Russell's system'. Must I not say that this proposition on the one hand is true, and on the
other hand is unprovable? For suppose it were false; then it is true that it is provable. And that surely cannot be! And
if it is proved, then it is proved that it is not provable. Thus it can only be true, but unprovable."
Just as we ask: "'provable' in what system?", so we must also ask: "'true' in what system?" 'True in Russell's
system' means, as was said: proved in Russell's system; and 'false in Russell's system' means: the opposite has been
proved in Russell's system.--Now what does your "suppose it is false" mean? In the Russell sense it means 'suppose
the opposite is proved in Russell's system'; if that is your assumption, you will now presumably give up the
interpretation that it is
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unprovable. And by 'this interpretation' I understand the translation into this English sentence.--If you assume that
the proposition is provable in Russell's system, that means it is true in the Russell sense, and the interpretation "P is
not provable" again has to be given up. If you assume that the proposition is true in the Russell sense, the same
thing follows. Further: if the proposition is supposed to be false in some other than the Russell sense, then it does
not contradict this for it to be proved in Russell's system. (What is called "losing" in chess may constitute winning in
another game.)
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9. For what does it mean to say that P and "P is unprovable" are the same proposition? It means that these
two English sentences have a single expression in such-and-such a notation.
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10. "But surely P cannot be provable, for, supposing it were proved, then the proposition that it is not
provable would be proved." But if this were now proved, or if I believed--perhaps through an error--that I had
proved it, why should I not let the proof stand and say I must withdraw my interpretation "unprovable"?
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11. Let us suppose I prove the unprovability (in Russell's system) of P; then by this proof I have proved P.
Now if this proof were one in Russell's system--I should in that case have proved at once that it belonged and did
not belong to Russell's system.--That is what comes of making up such sentences.--But there is a contradiction
here!--Well, then there is a contradiction here. Does it do any harm here?
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12. Is there harm in the contradiction that arises when someone says: "I am lying.--So I am not lying.--So I
am lying.--etc."? I mean: does it make our language less usable if in this case, according to the ordinary rules, a
proposition yields its contradictory, and vice versa?--the proposition itself is unusable, and these inferences equally;
but why should they not be made?--It is a profitless performance!--It is a language-game with some similarity to the
game of thumb-catching.
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13. Such a contradiction is of interest only because it has tormented people, and because this shews both
how tormenting problems can grow out of language, and what kind of things can torment us.
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14. A proof of unprovability is as it were a geometrical proof; a proof concerning the geometry of proofs.
Quite analogous e.g. to a proof that such-and-such a construction is impossible with ruler and compass. Now such a
proof contains an element of prediction, a physical element. For in consequence of such a proof we say to a man:
"Don't exert yourself to find a construction (of the trisection of an angle, say)--it can be proved that it can't be done".
That is to say: it is essential that the proof of unprovability should be capable of being applied in this way. It
must--we might say--be a forcible reason for giving up the search for a proof (i.e. for a construction of
such-and-such a kind).
A contradiction is unusable as such a prediction.
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15. Whether something is rightly called the proposition "X is unprovable" depends on how we prove this
proposition. The proof
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alone shews what counts as the criterion of unprovability. The proof is part of the system of operations, of the game,
in which the proposition is used, and shews us its 'sense'.
Thus the question is whether the 'proof of the unprovability of P' is here a forcible reason for the assumption
that a proof of P will not be found.
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16. The proposition "P is unprovable" has a different sense afterwards--from before it was proved.
If it is proved, then it is the terminal pattern in the proof of unprovability.--If it is unproved, then what is to
count as a criterion of its truth is not yet clear, and--we can say--its sense is still veiled.
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17. Now how am I to take P as having been proved? By a proof of unprovability? Or in some other way?
Suppose it is by a proof of unprovability. Now, in order to see what has been proved, look at the proof. Perhaps it
has here been proved that such-and-such forms of proof do not lead to P.--Or, suppose P has been proved in a direct
way--as I should like to put it--and so in that case there follows the proposition "P is unprovable", and it must now
come out how this interpretation of the symbols of P collides with the fact of the proof, and why it has to be given
up here.
Suppose however that not-P is proved.--Proved how? Say by P's being proved directly--for from that follows
that it is provable, and hence not-P. What am I to say now, "P" or "not-P"? Why not both? If someone asks me
"Which is the case, P, or not-P?" then I reply: P stands at the end of a Russellian proof, so you write P in the
Russellian system; on the other hand, however, it is then provable and this is expressed by not-P, but this
proposition does not stand at the end of a Russellian proof, and so does not belong to the Russellian system.
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--When the interpretation "P is unprovable" was given to P, this proof of P was not known, and so one cannot say
that P says: this proof did not exist.--Once the proof has been constructed, this has created a new situation: and
now we have to decide whether we will call this a proof (a further proof), or whether we will still call this the
statement of unprovability.
Suppose not-P is directly proved; it is therefore proved that P can be directly proved! So this is once more a
question of interpretation--unless we now also have a direct proof of P. If it were like that, well, that is how it would
be.
(The superstitious dread and veneration by mathematicians in face of contradiction.)
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18. "But suppose, now, that the proposition were false--and hence provable?"--Why do you call it 'false'?
Because you can see a proof?--Or for other reasons? For in that case it doesn't matter. For one can quite well call the
Law of Contradiction false, on the grounds that we very often make good sense by answering a question "Yes and
no". And the same for the proposition '~~p = p' because we employ double negation as a strengthening of the
negation and not merely as its cancellation.
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19. You say: "..., so P is true and unprovable". That presumably means: "Therefore P". That is all right with
me--but for what purpose do you write down this 'assertion'? (It is as if someone had extracted from certain
principles about natural forms and architectural style the idea that on Mount Everest, where no one can live, there
belonged a châlet in the Baroque style. And how could you make the truth of the assertion plausible to me, since
you can make no use of it except to do these bits of legerdemain?
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20. Here one needs to remember that the propositions of logic are so constructed as to have no application as
information in practice. So it could very well be said that they were not propositions at all; and one's writing them
down at all stands in need of justification. Now if we append to these 'propositions' a further sentence-like structure
of another kind, then we are all the more in the dark about what kind of application this system of sign-combinations
is supposed to have; for the mere ring of a sentence is not enough to give these connexions of signs any meaning.
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PART II
1938
Page 125
1. How far does the diagonal method prove that there is a number which--let's say--is not a square root? It is
of course extremely easy to shew that there are numbers that aren't square roots--but how does this method shew it?
Have we a general concept of what it means to shew that there is a number that is not included in this infinite set?
Let us suppose that someone had been given the task of naming a number different from every ; but
that he knew nothing of the diagonal procedure and had named the number ; and had shewn that it was not a
value of . Or that he had said: assume that = 1.4142... and subtract 1 from the first decimal, but have the
rest of the places agree with . 1.3142 cannot be a value of .
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2. "Name a number that agrees with at every second decimal place." What does this task demand? The
question is: is it performed by the answer: It is the number got by the rule: develop and add 1 or -1 to every
second decimal place?
It is the same as the way the task: Divide an angle into three can be
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regarded as carried out by laying 3 equal angles together.
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3. If someone says: "Shew me a number different from all these", and is given the rule of the diagonal for
answer, why should he not say: "But I didn't mean it like that!"? What you have given me is a rule for the
step-by-step construction of numbers that are successively different from each of these.
"But why aren't you willing to call this too a method of calculating a number?"--But what is the method of
calculating, and what the result, here? You will say that they are one, for it makes sense now to say: the number D is
bigger than... and smaller than...; it can be squared etc. etc.
Is the question not really: What can this number be used for? True, that sounds queer.--But what it means is:
what are its mathematical surroundings?
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4. So I am comparing methods of calculating--only here there are certainly very different ways of making
comparisons. However, I am supposed in some sense to be comparing the results of the methods with one another.
But this is enough to make everything unclear, for in one sense they don't each have a single result, or it is not clear
in advance what is to be regarded here as the result in each case. I want to say that here we are afforded every
opportunity of twisting and turning the meanings.
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5. Let us say--not: "This method gives a result", but rather: "it gives an infinite series of results". How do I
compare infinite series of results? Well, there are very different things that I may call doing that.
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6. The motto here is always: Take a wider look round.
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7. The result of a calculation expressed verbally is to be regarded with suspicion. The calculation illumines
the meaning of the expression in words. It is the finer instrument for determining the meaning. If you want to know
what the verbal expression means, look at the calculation; not the other way about. The verbal expression casts only
a dim general glow over the calculation: but the calculation a brilliant light on the verbal expression. (As if you
wanted to compare the heights of two mountains, not by the technique of measurement of heights, but by their
apparent relation when looked at from below.)
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8. "I want to shew you a method by which you can serially avoid all these developments." The diagonal
procedure is such a method.--"So it produces a series that is different from all of these." Is that right?--Yes; if, that is,
you want to apply these words to the described case.
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9. How would it be with the following method of construction? The diagonal number is produced by
addition or subtraction of 1, but whether to add or subtract is only found out by continuing the original series to
several places. Suppose it were now said: the development of the diagonal series never catches up with the
development of the other series:--certainly the diagonal series avoids each of those series when it encounters it, but
that is no help to it, as the development of the other series is again ahead of it. Here I can surely say: There is always
one of the series for which it is not determined whether or not it is different from the diagonal series. It may be said:
they run after one another to infinity, but the original series is always ahead.
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"But your rule already reaches to infinity, so you already know quite precisely that the diagonal series will be
different from any other!"--
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10. It means nothing to say: "Therefore the X numbers are not denumerable". One might say something like
this: I call number-concept X non-denumerable if it has been stipulated that, whatever numbers falling under this
concept you arrange in a series, the diagonal number of this series is also to fall under that concept.
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11. Since my drawing is after all only the indication of infinity, why must it be like this
and not like this
Here what we have is different pictures; and to them correspond different ways of talking. But does anything useful
emerge if we have a dispute about the justification of them? What is important must reside
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somewhere else; even though these pictures fire our imagination most strongly.
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12. What can the concept 'non-denumerable' be used for?
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13. Surely--if anyone tried day-in day-out 'to put all irrational numbers into a series' we could say: "Leave it
alone; it means nothing; don't you see, if you established a series, I should come along with the diagonal series!"
This might get him to abandon his undertaking. Well, that would be useful. And it strikes me as if this were the
whole and proper purpose of this method. It makes use of the vague notion of this man who goes on, as it were
idiotically, with his work, and it brings him to a stop by means of a picture. (But one could get him to resume his
undertaking by means of another picture.)
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14. The procedure exhibits something--which can in a very vague way be called the demonstration that these
methods of calculation cannot be ordered in a series. And here the meaning of "these" is just kept vague.
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15. A clever man got caught in this net of language! So it must be an interesting net.
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16. The mistake begins when one says that the cardinal numbers can be ordered in a series. For what concept
has one of this ordering? One has of course a concept of an infinite series, but here that gives
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us at most a vague idea, a guiding light for the formation of a concept. For the concept itself is abstracted from this
and from other series; or: the expression stands for a certain analogy between cases, and it can e.g. be used to define
provisionally a domain that one wants to talk about.
That, however, is not to say that the question: "Can the set R be ordered in a series?" has a clear sense. For
this question means e.g.: Can one do something with these formations, corresponding to the ordering of the cardinal
numbers in a series? Asked: "Can the real numbers be ordered in a series?" the conscientious answer might be: "For
the time being I can't form any precise idea of that".--"But you can order the roots and the algebraic numbers for
example in a series; so you surely understand the expression!"--To put it better, I have got certain analogous
formations, which I call by the common name 'series'. But so far I haven't any certain bridge from these cases to that
of 'all real numbers'. Nor have I any general method of trying whether such-and-such a set 'can be ordered in a
series'.
Now I am shewn the diagonal procedure and told: "Now here you have the proof that this ordering can't be
done here". But I can reply: "I don't know--to repeat--what it is that can't be done here". Though I can see that you
want to shew a difference between the use of "root", "algebraic number", etc. on the one hand, and "real number" on
the other. Such a difference as e.g. this: roots are called "real numbers", and so too is the diagonal number formed
from the roots. And similarly for all series of real numbers. For this reason it makes no sense to talk about a "series
of all real numbers", just because the diagonal number for each series is also called a "real number".--Would this not
be as if any row of books were itself ordinarily called a book, and now we said: "It makes no sense to speak of 'the
row of all books', since this row would itself be a book."
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17. Here it is very useful to imagine the diagonal procedure for the production of a real number as having
been well-known before the invention of set theory, and familiar even to school-children, as indeed might very well
have been the case. For this changes the aspect of Cantor's discovery. The discovery might very well have consisted
merely in the interpretation of this long familiar elementary calculation.
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18. For this kind of calculation is itself useful. The question set would be perhaps: write down a decimal
number which is different from the numbers:
0.1246798...
0.3469876...
0.0127649...
0.3426794...
............ (Imagine a long series.)
The child thinks to itself: how am I to do this, when I should have to look at all the numbers at once, to prevent what
I write down from being one of them? Now the method says: Not at all: change the first place of the first number,
the second of the second one etc. etc., and you are sure of having written down a number that does not coincide
with any of the given ones. The number got in this way might always be called the diagonal number.
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19. The dangerous, deceptive thing about the idea: "The real numbers cannot be arranged in a series", or
again "The set... is not denumerable" is that it makes the determination of a concept--concept formation--look like a
fact of nature.
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20. The following sentence sounds sober: "If something is called a series of real numbers, then the expansion
given by the diagonal
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procedure is also called a 'real number', and is moreover said to be different from all members of the series".
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21. Our suspicion ought always to be aroused when a proof proves more than its means allow it. Something
of this sort might be called 'a puffed-up proof'.
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22. The usual expression creates the fiction of a procedure, a method of ordering which, though applicable
here, nevertheless fails to reach its goal because of the number of objects involved, which is greater even than the
number of all cardinal numbers.
If it were said: "Consideration of the diagonal procedure shews you that the concept 'real number' has much
less analogy with the concept 'cardinal number' than we, being misled by certain analogies, are inclined to believe",
that would have a good and honest sense. But just the opposite happens: one pretends to compare the 'set' of real
numbers in magnitude with that of cardinal numbers. The difference in kind between the two conceptions is
represented, by a skew form of expression, as difference of extension. I believe, and hope, that a future generation
will laugh at this hocus pocus.
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23. The sickness of a time is cured by an alteration in the mode of life of human beings, and it was possible
for the sickness of philosophical problems to get cured only through a changed mode of thought and of life, not
through a medicine invented by an individual.
Think of the use of the motor-car producing or encouraging certain sicknesses, and mankind being plagued
by such sickness until, from some cause or other, as the result of some development or other, it abandons the habit
of driving.
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24. For how do we make use of the proposition: "There is no greatest cardinal number"? When and on what
occasion woud [[sic]] it be said? This use is at any rate quite different from that of the mathematical proposition '25
× 25 = 625'.
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25. First and foremost, notice that we ask this question at all; this points to the fact that the answer is not
ready to hand.
Moreover, if one tries to answer the question in a hurry, it is easy to trip up. The case is like that of the
question: what experience shews us that our space is three-dimensional?
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26. We say of a permission that it has no end.
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27. And it can be said that the permission to play language-games with cardinal numbers has no end. This
would be said e.g. to someone to whom we were teaching our language and language-games. So it would again be a
grammatical proposition, but of an entirely different kind from '25 × 25 = 625'. It would however be of great
importance if the pupil were, say, inclined to expect a definitive end to this series of language-games (perhaps
because he had been brought up in a different culture).
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28. Why should we say: The irrational numbers cannot be ordered?--We have a method by which to upset
any order.
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29. Cantor's diagonal procedure does not shew us an irrational
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number different from all in the system, but it gives sense to the mathematical proposition that the number
so-and-so is different from all those of the system. Cantor could say: You can prove that a number is different from
all the numbers in the system by proving that it differs in its first place from its first number and in its second place
from its second number and so on.
Cantor is saying something about the multiplicity of the concept "Real number different from all the ones of
a system".
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30. Cantor shews that if we have a system of expansions it makes sense to speak of an expansion that is
different from them all.--But that is not enough to determine the grammar of the word "expansion".
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31. Cantor gives a sense to the expression "expansion which is different from all the expansions in a system",
by proposing that an expansion should be so called when it can be proved that it is diagonally different from the
expansions in a system.
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32. Thus it can be set as a question: Find a number whose expansion is diagonally different from those in this
system.
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33. It might be said: Besides the rational points there are diverse systems of irrational points to be found in
the number line.
There is no system of irrational numbers--but also no super-system, no 'set of irrational numbers' of
higher-order infinity.
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34. Cantor defines a difference of higher order, that is to say a difference of an expansion from a system of
expansions. This definition can be used so as to shew that a number is in this sense different from a system of
numbers: let us say π from the system of algebraic numbers. But we cannot very well say that the rule of altering the
places in the diagonal in such-and-such a way is as such proved different from the rules of the system, because this
rule is itself of 'higher order'; for it treats of the alteration of a system of rules, and for that reason it is not clear in
advance in which cases we shall be willing to declare the expansion of such a rule different from all the expansions
of the system.
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35. 'These considerations may lead us to say that
That is to say: we can make the considerations lead us to that.
Or: we can say this and give this as our reason.
But if we do say it--what are we to do next? In what practice is this proposition anchored? It is for the time
being a piece of mathematical architecture which hangs in the air, and looks as if it were, let us say, an architrave, but
not supported by anything and supporting nothing.
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36. Certain considerations may lead us to say that 1010 souls fit into a cubic centimetre. But why do we
nevertheless not say it? Because it is of no use. Because, while it does conjure up a picture, the picture is one with
which we cannot go on to do anything.
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37. The proposition is worth as much as its grounds are.
It supports as much as the grounds that support it do.
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38. An interesting question is: what is the connexion of ℵ0 with the cardinal numbers whose number it is
supposed to be? ℵ0 would obviously be the predicate "infinite series" in its application to the series of cardinal
numbers and to similar mathematical formations. Here it is important to grasp the relationship between a series in
the nonmathematical sense and one in the mathematical sense. It is of course clear that in mathematics we do not
use the word "series of numbers" in the sense "series of numerical signs", even though, of course, there is also a
connexion between the use of the one expression and of the other. A railway is not a railway train; nor is it
something similar to a railway train. A 'series' in the mathematical sense is a method of construction for series of
linguistic expressions.
Thus we have a grammatical class "infinite sequence", and equivalent with this expression a word whose
grammar has (a certain) similarity with that of a numeral: "infinity" or "ℵ0". This is connected with the fact that
among the calculi of mathematics we have a technique which there is a certain justice in calling "1-1 correlation of
the members of two infinite series", since it has a similarity to such a mutual correlation of the members of what are
called 'finite' classes.
From the fact, however, that we have an employment for a kind of numeral which, as it were, gives the
number of the members of an infinite series, it does not follow that it also makes some kind of sense to speak of the
number of the concept 'infinite series'; that we have here some kind of employment for something like a numeral.
For there is no grammatical technique suggesting employment of such an expression. For I can of course form the
expression: "class of all classes which are equinumerous with the class 'infinite series'" (as also: "class of all angels
that can get on to a needlepoint") but this expression is empty so long as there is no employment for it. Such an
employment is not: yet to be discovered, but: still to be invented.
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39. Imagine that I put a playing-board divided into squares in front of you, and put pieces like chess pieces
on it--and stated: "This piece is the 'King', these are the 'Knights', these the 'Commoners'.--So far that's all we know
about the game; but that's always something.--And perhaps more will be discovered."
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40. "Fractions cannot be arranged in an order of magnitude."--First and foremost, this sounds extremely
interesting and remarkable.
It sounds interesting in a quite different way from, say, a proposition of the differential calculus. The
difference, I think, resides in the fact that such a proposition is easily associated with an application to physics,
whereas this proposition belongs simply and solely to mathematics, seems to concern as it were the natural history
of mathematical objects themselves.
One would like to say of it e.g.: it introduces us to the mysteries of the mathematical world. This is the aspect
against which I want to give a warning.
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41. When it looks as if..., we should look out.
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42. When, on hearing the proposition that the fractions cannot be arranged in a series in order of magnitude, I
form the picture of an unending row of things, and between each thing and its neighbour new things appear, and
more new ones again between each of these things and its neighbour, and so on without end, then certainly there is
something here to make one dizzy.
But once we see that this picture, though very exciting, is all the same not appropriate; that we ought not to
let ourselves be trapped by the words "series", "order", "exist", and others, we shall fall back on the technique of
calculating fractions, about which there is no longer anything queer.
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43. The fact that in a technique of calculating fractions the expression "the next greatest fraction" has no
sense, that we have not given it any sense, is nothing to marvel at.
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44. If we apply a technique of continuous interpolation of fractions, we shall not be willing to call any
fraction the "next biggest".
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45. To say that a technique is unlimited does not mean that it goes on without ever stopping--that it increases
immeasurably; but that it lacks the institution of the end, that it is not finished off. As one may say of a sentence that
it is not finished off if it has no period. Or of a playing-field that is unlimited, when the rules of the game do not
prescribe any boundaries--say by means of a line.
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46. For the point of a new technique of calculation is to supply us with a new picture, a new form of
expression; and there is nothing so absurd as to try and describe this new schema, this new kind of scaffolding, by
means of the old expressions.
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47. What is the function of such a proposition as: "A fraction has not a next biggest fraction but a cardinal
number has a next biggest cardinal number"? Well, it is as it were a proposition that compares two games. (Like: in
draughts pieces jump over one another, but not in chess.)
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48. We call something "constructing the next biggest cardinal number" but nothing "constructing the next
biggest fraction".
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49. How do we compare games? By describing them--by describing one as a variation of another--by
describing them and emphasizing their differences and analogies.
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50. "In draughts there isn't a King"--what does this mean? (It sounds childish.) Does it mean that none of the
pieces in draughts is called "King"; and if we did call one of the pieces that, would there be a King in draughts? But
what about this proposition: "In draughts all the pieces have the same rights, but not in chess"? Whom am I telling
this? One who already knows both games, or else someone who does not yet know them. Here it looks as if the first
one stands in no need of our information and the second can do nothing with it. But suppose I were to say: "See! In
draughts all the pieces have the same rights,..." or better still: "See! In these games all the pieces have the same
rights, in those not." But what does such a proposition do? It introduces a new concept, a new ground of
classification. I teach you to answer the question: "Name games of the first sort" etc. But in a similar way it would be
possible to set questions like: "Invent a game with a King".
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51. 'We cannot arrange fractions in a series in order of magnitude but we can order them in an infinite series.'
If someone did not know this, what has he now learnt? He has learnt a new kind of calculation, e.g.:
"Determine the number of the fraction
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52. He learns this technique--but doesn't he also learn that there is such a technique?
I have indeed, in an important sense, learned that there is such a technique; that is, I have got to know a
technique which can now be applied to all sorts of other things.
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53. 'What would you call this?'
Surely "a method of numbering the pairs of numbers"? And might I not also say: "of ordering pairs of numbers in a
series"?
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54. Now does mathematics teach me that I can order the pairs of numbers in a series? Can I say: it teaches
me that I can do this? For does it make sense to say that I teach a child that it is possible to multiply--by teaching
him to multiply? It would rather be natural to say I teach him that it is possible to multiply fractions, after he has
learned to multiply cardinal numbers together. For now, it might be said, he knows what "multiplying" means. But
wouldn't this be misleading too?
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55. If someone says I have proved the proposition that we can order pairs of numbers in a series, it should be
answered that this is not a mathematical proposition, since one doesn't calculate with the words "we", "can", "the",
"pairs of numbers", etc. The proposition
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"one can..." is rather a mere approximate description of the technique one is teaching, say a not unsuitable title, a
heading to this chapter. But a title with which it is not possible to calculate.
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56. But, you say, this is just what the logical calculus of Frege and Russell does: in it every word that is
spoken in mathematics has exact significance, is an element of the calculus. Thus in this calculus we can really prove
that "multiplying is possible". Very well, now it is a mathematical proposition; but who says that anything can be
done with this proposition? Who says what use it can be? For its sounding interesting is not enough.
Just because when we are teaching we may use the proposition "So you see, we can order the fractions in a
series", don't say that we have any other use for this proposition than of attaching a memorable picture to this sort of
calculation.
If the interest here attaches to the proposition that has been proved, then it attaches to a picture which has an
extremely weak justification, but which fascinates us by its queerness, like e.g. the picture of the "direction" of time.
It produces a slight reeling of one's thoughts.
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57. Here I can only say: depart as quickly as possible from this picture, and see the interest of this calculation
in its application. (It is as if we were at a masked ball at which every calculation appears in a queer guise.)
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58. "Ought the word 'infinite' to be avoided in mathematics?" Yes; where it appears to confer a meaning upon
the calculus; instead of getting one from it.
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59. This way of talking: "But when one examines the calculus there is nothing infinite there" is of course
clumsy--but it means: is it really necessary here to conjure up the picture of the infinite (of the enormously big)?
And how is this picture connected with the calculus? For its connexion is not that of the picture | | | | with 4.
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60. To act as if one were disappointed to have found nothing infinite in the calculus is of course funny; but
not to ask: what is the everyday employment of the word "infinite", which gives it its meaning for us; and what is its
connexion with these mathematical calculi?
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61. Finitism and behaviourism are quite similar trends. Both say, but surely, all we have here is.... Both deny
the existence of something, both with a view to escaping from a confusion.
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62. What I am doing is, not to shew that calculations are wrong, but to subject the interest of calculations to
a test. I test e.g. the justification for still using the word... here. Or really, I keep on urging such an investigation. I
shew that there is such an investigation and what there is to investigate there. Thus I must say, not: "We must not
express ourselves like this", or "That is absurd", or "That is uninteresting", but: "Test the justification of this
expression in this way". You cannot survey the justification of an expression unless you survey its employment;
which you cannot do by looking at some facet of its employment, say a picture attaching to it.
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PART III
1939-40
Page 143
1. 'A mathematical proof must be perspicuous.' Only a structure whose reproduction is an easy task is called
a "proof". It must be possible to decide with certainty whether we really have the same proof twice over, or not. The
proof must be a configuration whose exact reproduction can be certain. Or again: we must be sure we can exactly
reproduce what is essential to the proof. It may for example be written down in two different handwritings or
colours. What goes to make the reproduction of a proof is not anything like an exact reproduction of a shade of
colour or a hand-writing.
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It must be easy to write down exactly this proof again. This is where a written proof has an advantage over a
drawing. The essentials of the latter have often been misunderstood. The drawing of a Euclidian proof may be
inexact, in the sense that the straight lines are not straight, the segments of circles not exactly circular, etc. etc. and at
the same time the drawing is still an exact proof; and from this it can be seen that this drawing does
not--e.g.--demonstrate that such a construction results in a polygon with five equal sides; that what it proves is a
proposition of geometry, not one about the properties of paper, compass, ruler and pencil.
[Connects with: proof a picture of an experiment.]
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2. I want to say: if you have a proof-pattern that cannot be taken in, and by a change in notation you turn it
into one that can, then you are producing a proof, where there was none before.
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Page 144
Now let us imagine a proof for a Russellian proposition stating an addition like 'a + b = c', consisting of a few
thousand signs. You will say: Seeing whether this proof is correct or not is a purely external difficulty, of no
mathematical interest. ("One man takes in easily what someone else takes in with difficulty or not at all" etc. etc..)
The assumption is that the definitions serve merely to abbreviate the expression for the convenience of the
calculator; whereas they are part of the calculation. By their aid expressions are produced which could not have been
produced without it.
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3. But how about the following: "While it is true that we cannot--in the ordinary sense--multiply 234 by 537
in the Russellian calculus, still there is a Russellian calculation corresponding to this multiplication."--What kind of
correspondence is this? It might be like this: we can carry out this multiplication in the Russellian calculus too, only
in a different symbolism,--just as, as we should certainly say, we can carry it out in a different number system. In
that case, then, we could e.g. solve the practical problems for which we use that multiplication by means of the
calculation in the Russellian calculus too, only in a more roundabout way.
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Now let us imagine the cardinal numbers explained as 1, 1 + 1, (1 + 1) + 1, ((1 + 1) + 1) + 1, and so on. You
say that the definitions introducing the figures of the decimal system are a mere matter of convenience; the
calculation 703000 × 40000101 could be done in that wearisome notation too. But is that true?--"Of course it's true! I
can surely write down, construct, a calculation in that notation corresponding to the calculation in the decimal
notation."--But how do I know that it corresponds to it? Well, because I have derived it from the other by a given
method.--But now if I look at it again
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half an hour later, may it not have altered? For one cannot command a clear view of it.
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Now I ask: could we also find out the truth of the proposition 7034174 + 6594321 = 13628495 by means of a
proof carried out in the first notation?--Is there such a proof of this proposition?--The answer is: no.
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4. But still doesn't Russell teach us one way of adding?
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Suppose we proved by Russell's method that (∃ a ... g) (∃ a ... l) ⊃ (∃ a ... s) is a tautology; could we reduce
our result to g + l's being s? Now this presupposes that I can take the three bits of the alphabet as representatives of
the proof. But does Russell's proof shew this? After all I could obviously also have carried out Russell's proof with
groups of signs in the brackets whose sequence made no characteristic impression on me, so that it would not have
been possible to represent the group of signs between brackets by its last term.
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Even assuming that the Russellian proof were carried out with a notation such as x1x2... x10x11... x100... as in
the decimal notation, and there were 100 members in the first pair of brackets, 300 in the second and 400 in the third,
does the proof itself shew that 100 + 300 = 400?--What if this proof led at one time to this result, and at another to a
different one, for example 100 + 300 = 420? What is
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needed in order to see that the result of the proof, if it is correctly carried out, always depends solely on the last
figures of the first two pairs of brackets?
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But still for small numbers Russell does teach us to add; for then we take the groups of signs in the brackets
in at a glance and we can take them as numerals; for example 'xy', 'xyz', 'xyzuv'.
Thus Russell teaches us a new calculus for reaching 5 from 2 and 3; and that is true even if we say that a
logical calculus is only--frills tacked on to the arithmetical calculus.
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The application of the calculation must take care of itself. And that is what is correct about 'formalism'.
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The reduction of arithmetic to symbolic logic is supposed to shew the point of application of arithmetic, as it
were the attachment by means of which it is plugged in to its application. As if someone were shewn, first a trumpet
without the mouthpiece--and then the mouthpiece, which shews how a trumpet is used, brought into contact with
the human body. But the attachment which Russell gives us is on the one hand too narrow, on the other hand too
wide; too general and too special. The calculation takes care of its own application.
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We extend our ideas from calculations with small numbers to ones with large numbers in the same kind of
way as we imagine that, if the distance from here to the sun could be measured with a footrule, then we should get
the very result that, as it is, we get in a quite different way. That is to say, we are inclined to take the measurement of
length with a footrule as a model even for the measurement of the distance between two stars.
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And one says, e.g. at school: "If we imagine rulers stretching from here to the sun..." and seems in this way
to explain what we understand by the distance between the sun and the earth. And the use of such a picture is all
right, so long as it is clear to us that we can measure the distance from us to the sun, and that we cannot measure it
with footrules.
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5. Suppose someone were to say: "The only real proof of 1000 + 1000 = 2000 is after all the Russellian one,
which shews that the expression... is a tautology"? For can I not prove that a tautology results if I have 1000
members in each of the two first pairs of brackets and 2000 in the third? And if I can prove that, then I can look at it
as a proof of the arithmetical proposition.
Page 147
In philosophy it is always good to put a question instead of an answer to a question.
For an answer to the philosophical question may easily be unfair; disposing of it by means of another
question is not.
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Then should I put a question here, for example, instead of the answer that that arithmetical proposition
cannot be proved by Russell's method?
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6. The proof that is a tautology consists in always crossing out a term of the
third pair of brackets for a term of (1) or (2). And there are many methods for such collating. Or one might even
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say: there are many ways of establishing the success of a 1-1 correlation. One way, for example, would be to
construct a star-shaped pattern for the left-hand side of the implication and another one for the right-hand side and
then to compare these in their turn by making an ornament out of the two of them.
Thus the rule could be given: "If you want to know whether the numbers A and B together actually yield C,
write down an expression of the form... and correlate the variables in the brackets by writing down (or trying to) the
proof that the expression is a tautology".
My objection to this is not that it is arbitrary to prescribe just this way of collating, but that it cannot be
established in this way that 1000 + 1000 = 2000.
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7. Imagine that you had written down a 'formula' a mile long, and you shewed by transformation that it was
tautologous ('if it has not altered meanwhile', one would have to say). Now we count the terms in the brackets or we
divide them up and make the expression into one that can be taken in, and it comes out that there are 7566 terms in
the first pair of brackets, 2434 in the second, 10000 in the third. Now have I proved that 2434 + 7566 = 10000?--That
depends--one might say--on whether you are certain that the counting has really yielded the number of terms which
stood between the brackets in the course of the proof.
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Could one say: "Russell teaches us to write as many variables in the third pair of brackets as were in the first
two together"? But really: he teaches us to write a variable in (3) for every variable in (1) and (2).
But do we learn from this what number is the sum of two given
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numbers? Perhaps it is said: "Of course, for in the third pair of brackets we have the paradigm, the prototype of the
new number". But in what sense is | | | | | | | | | | | | | | | | the paradigm of a number? Consider how it can be used as
such.
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8. Above all, the Russellian tautology corresponding to the proposition a + b = c does not shew us in what
notation the number c is to be written, and there is no reason why it should not be written in the form a + b.--For
Russell does not teach us the technique of, say, adding in the decimal system.--But could we perhaps derive it from
his technique?
Let us just ask the following question: Can one derive the technique of the decimal system from that of the
system 1, 1 + 1, (1 + 1) + 1, etc.?
Could this question not also be formulated as follows: if one has one technique of calculation in the one
system and one in the other,--how is it shewn that the two are equivalent?
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9. "A proof ought to shew not merely that this is how it is, but this is how it has to be."
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In what circumstances does counting shew this?
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One would like to say: "When the figures and the thing being counted yield a memorable configuration.
When this configuration is now used in place of any fresh counting."--But here we seem to be talking only of spatial
configurations: but if we know a series of words by heart and then co-ordinate two such series, one to one, saying
for example: "First--Monday; second--Tuesday; third--Wednesday;
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etc."--can we not prove in this way that from Monday to Thursday is four days?
For the question is: What do we call a "memorable configuration"? What is the criterion for its being
impressed on our minds? Or is the answer to that: "That we use it as a paradigm of identity!"?
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10. We do not make experiments on a sentence or a proof in order to establish its properties.
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How do we reproduce, how do we copy, a proof?--Not e.g. by taking measurements of it.
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Suppose a proof were so hugely long that it could not possibly be taken in? Or let us look at a different case:
Let there be a long row of strokes engraved in hard rock which is our paradigm for the number that we call 1000. We
call this row the proto-thousand and if we want to know whether there are a thousand men in a square, we draw
lines or stretch threads. (1-1 correlation.)
Now here the sign of the number 1000 has the identity, not of a shape, but of a physical object. We could
imagine a 'proto-hundred' similarly, and a proof, which we could not take in at a glance, that 10 × 100 = 1000.
The figure for 1000 in the system of 1 + 1 + 1 + 1... cannot be recognized by its shape.
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11. | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | | |
Is this pattern a proof of 27 + 16 = 43, because one reaches '27' if one counts the strokes on the left-hand
side, '16' on the right-hand
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side, and '43' when one counts the whole row?
Where is the queerness of calling the pattern the proof of this proposition? It lies in the kind of way this
proof is to be reproduced or known again; in its not having any characteristic visual shape.
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Now even if that proof has not any such visual shape, still I can copy (reproduce) it exactly--so isn't the
figure a proof after all? I might e.g. have it engraved on a bit of steel and passed from hand to hand. So I should tell
someone: "Here you have the proof that 27 + 16 = 43".--Well, can't one say after all that he proves the proposition
with the aid of the pattern? Yes; but the pattern is not the proof.
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This, however, would surely be called a proof of 250 + 3220 = 3470: one counts on from 250 and at the same
time begins counting from 1 and co-ordinates the two counts:
251.... 1
252....2
253.... 3
etc.
3470.... 3220
That could be called a proof in 3220 steps. It is surely a proof--and can it be called perspicuous?
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12. What is the invention of the decimal system really? The invention of a system of abbreviations--but what
is the system of the abbreviations? Is it simply the system of the new signs or is it also a
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system of applying them for the purpose of abbreviation? And if it is the latter, then it is a new way of looking at the
old system of signs.
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Can we start from the system of 1 + 1 + 1 ... and learn to calculate in the decimal system through mere
abbreviations of the notation?
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13. Suppose that following Russell I have proved a proposition of the form (∃ xyz...) (∃ uvw...) ⊃ (∃
abc...)--and now 'I make it perspicuous' by writing signs x1, x2, x3... over the variables--am I to say that following
Russell I have proved an arithmetical proposition in the decimal system?
But for every proof in the decimal system there is surely a corresponding one in Russell's system!--How do
we know there is? Let us leave intuition on one side.--But it can be proved.--
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If a number in the decimal system is defined in terms of 1, 2, 3,... 9, 0, and the signs 0, 1... 9 in terms of 1, 1 +
1, (1 + 1) + 1,... can one then use the recursive explanation of the decimal system to reach a sign of the form 1 + 1 +
1... from any number?
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Suppose someone were to say: Russellian arithmetic agrees with ordinary arithmetic up to numbers less than
1010; but then it diverges from it. And now he produces a Russellian proof that 1010 + 1 = 1010. Now why should I
not trust such a proof? How will anybody convince me that I must have miscalculated in the Russellian proof?
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But then do I need a proof from another system in order to ascertain
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whether I have miscalculated in the first proof? Is it not enough for me to write down that proof in a way that makes
it possible to take it in?
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14. Is not my whole difficulty one of seeing how it is possible, without abandoning Russell's logical calculus,
to reach the concept of the set of variables in the expression '(∃ xyz...)', where this expression cannot be taken in?--
Well, but it can be made surveyable by writing: (∃ x1, x2, x3, etc.). And still there is something that I do not
understand: the criterion for the identity of such an expression has now surely been changed: I now see in a different
way that the set of signs in two such expressions is the same.
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What I am tempted to say is: Russell's proof can indeed be continued step by step, but at the end one does
not rightly know what one has proved--at least not by the old criteria. By making it possible to command a clear
view of the Russellian proof, I prove something about this proof.
I want to say: one need not acknowledge the Russellian technique of calculation at all--and can prove by
means of a different technique of calculation that there must be a Russellian proof of the proposition. But in that
case, of course, the proposition is no longer based upon the Russellian proof.
Or again: its being possible to imagine a Russellian proof for every proved proposition of the form m + n = l
does not shew that the proposition is based on this proof. For it is conceivable that the Russellian proof of one
proposition should not be distinguishable from the Russellian proof of another and should be called different only
because they are the translations of two recognizably different proofs.
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Page 154
Or again: something stops being a proof when it stops being a paradigm, for example Russell's logical
calculus; and on the other hand any other calculus which serves as a paradigm is acceptable.
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15. It is a fact that different methods of counting practically always agree.
Page 154
When I count the squares on a chess-board I practically always reach '64'.
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If I know two series of words by heart, for example numerals and the alphabet, and I put them into one-one
correspondence:
a 1
b 2
c 3
etc.
at 'z' I practically always reach '26'.
Page 154
There is such a thing as: knowing a series of words by heart. When am I said to know the poem... by heart?
The criteria are rather complicated. Agreement with a printed text is one. What would have to happen to make me
doubt that I really know the ABC by heart? It is difficult to imagine.
But I use reciting or writing down a series of words from memory as a criterion for equality of numbers,
equality of sets.
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Page 155
Ought I now to say: all that doesn't matter--logic still remains the fundamental calculus, only whether I have
the same formula twice is of course differently established in different cases?
Page 155
16. It is not logic--I should like to say--that compels me to accept a proposition of the form (∃) (∃ ) ⊃ (∃ ),
when there are a million variables in the first two pairs of brackets and two million in the third. I want to say: logic
would not compel me to accept any proposition at all in this case. Something else compels me to accept such a
proposition as in accord with logic.
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Logic compels me only so far as the logical calculus compels me.
Page 155
But surely it is essential to the calculus with 1000000 that this number must be capable of resolution into a
sum 1 + 1 + 1..., and in order to be certain that we have the right number of units before us, we can number the
units: . This notation would be like: '100,000.000,000' which also
makes the numeral surveyable. And I can surely imagine someone's having a great sum of money in pennies entered
in a book in which perhaps they appear as numbers of 100 places, with which I have to calculate. I should now begin
to translate them into a surveyable notation, but still I should call them 'numerals', should treat them as a record of
numbers. For I should even regard it as the record of a number if someone were to tell me that N has as many
shillings as this vessel will hold peas. Another case again: "He has as many shillings as the Song of Songs has
letters".
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Page 156
17. The notation 'x1, x2, x3,...' gives a shape to the expression '(∃...)', and so to the R-proved tautology.
Page 156
Let me ask the following question: Is it not conceivable that the 1-1 correlation could not be trustworthily
carried out in the Russellian proof, that when, for example, we try to use it for adding, we regularly get a result
contradicting the usual one, and that we blame this on fatigue, which makes us leave out certain steps unawares?
And might we not then say:--if only we didn't get tired we should get the same result--? Because logic demands it?
Does it demand it, then? Aren't we here rectifying logic by means of another calculus?
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Suppose we took 100 steps of the logical calculus at a time and now got trustworthy results, while we don't
get them if we try to take all the steps singly--one would like to say: the calculation is still based on unit steps, since
100 steps at a time is defined by means of unit steps.--But the definition says: to take 100 steps at a time is the same
thing as..., and yet we take the 100 steps at a time and not 100 unit steps.
Still, in the shortened calculation I am obeying a rule--and how was this rule justified?--What if the
shortened and the unshortened proof yielded different results?
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18. What I am saying surely comes to this: I can e.g. define '10' as 1 + 1 + 1 + 1...' and '100 × 2' as '2 + 2 + 2...'
but I cannot therefore necessarily define '100 × 10' as '10 + 10 + 10 ...', nor yet as '1 + 1 + 1 + 1 ....'
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Page 157
I can find out that 100 × 100 equals 10000 by means of a 'shortened' procedure. Then why should I not
regard that as the original proof procedure?
Page 157
A shortened procedure tells me what ought to come out with the unshortened one. (Instead of the other way
round.)
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19. "But the calculation is surely based on the unit steps...." Yes; but in a different way. For the procedure of
proof is a different one.
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I could say for example: 10 = 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 + 1 and in like manner 100 = 10 + 10 + 10 +
10 + 10 + 10 + 10 + 10 + 10 + 10. Have I not based the definition of 100 on the successive addition of 1? But in the
same way as if I had added 100 units? Is there any need at all in my notation for a sign of the form--'1 + 1 + 1...' with
100 components of the sum?
Page 157
The danger here seems to be one of looking at the shortened procedure as a pale shadow of the unshortened
one. The rule of counting is not counting.
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20. What does taking 100 steps of the calculus 'at once' consist in? Surely in one's regarding, not the unit
step, but a different step, as decisive.
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Page 158
In ordinary addition of whole numbers in the decimal system we make steps in units, steps in tens, etc. Can
one say that the procedure is founded on one of only making unit steps? One might justify it like this: the result of
the addition does indeed look so--'7583'; but the explanation of this sign, its meaning, which must ultimately receive
expression in its application too, is surely of this sort: 1 + 1 + 1 + 1 + 1 and so on. But is it so? Must the numerical
sign be explained in this way, or this explanation receive expression implicitly in its application? I believe that if we
reflect it turns out that that is not the case.
Page 158
Calculating with graphs or with a slide-rule.
Of course when we check the one kind of calculation by the other, we normally get the same result. But if
there are several kinds--who says, if they do not agree, which is the proper method of calculation, with its roots at
the source of mathematics?
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21. Where a doubt can make its appearance whether this is really the pattern of this proof, where we are
prepared to doubt the identity of the proof, the derivation has lost its proving power. For the proof serves as a
measure.
Page 158
Could one say: it is part of proof to have an accepted criterion for the correct reproduction of a proof?
Page 158
That is to say, e.g.: we must be able to be certain, it must hold as certain for us, that we have not overlooked a
sign in the course of the
Page Break 159
proof. That no demon can have deceived us by making a sign disappear without our noticing, or by adding one, etc.
Page 159
One might say: When it can be said: "Even if a demon had deceived us, still everything would be all right",
then the prank he wanted to play on us has simply failed of its purpose.
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22. Proof, one might say, does not merely shew that it is like this, but: how it is like this. It shows how 13 +
14 yield 27.
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"A proof must be capable of being taken in" means: we must be prepared to use it as our guide-line in
judging.
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When I say "a proof is a picture"--it can be thought of as a cinematographic picture.
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We construct the proof once for all. A proof must of course have the character of a model.
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The proof (the pattern of the proof) shews us the result of a procedure (the construction); and we are
convinced that a procedure regulated in this way always leads to this configuration.
(The proof exhibits a fact of synthesis to us.)
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23. When we say that a proof is a model,--we must, of course, not be saying anything new.
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Proof must be a procedure of which I say: Yes, this is how it has to be; this must come out if I proceed
according to this rule.
Page 160
Proof, one might say, must originally be a kind of experiment--but is then taken simply as a picture.
Page 160
If I pour two lots of 200 apples together and count them, and the result is 400, that is not a proof that 200 +
200 = 400. That is to say, we should not want to take this fact as a paradigm for judging all similar situations.
Page 160
To say: "these 200 apples and these 200 apples come to 400"--means: when one puts them together, none are
lost or added, they behave normally.
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24. "This is the model for the addition of 200 and 200"--not: "this is the model of the fact that 200 and 200
added together yield 400". The process of adding did indeed yield 400, but now we take this result as the criterion
for the correct addition--or simply: for the addition--of these numbers.
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Page 161
The proof must be our model, our picture, of how these operations have a result.
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The 'proved proposition' expresses what is to be read off from the proof-picture.
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The proof is now our model of correctly counting 200 apples and 200 apples together: that is to say, it defines
a new concept: 'the counting of 200 and 200 objects together'. Or, as we could also say: "a new criterion for nothing's
having been lost or added".
Page 161
The proof defines 'correctly counting together'.
Page 161
The proof is our model for a particular result's being yielded, which serves as an object of comparison
(yardstick) for real changes.
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25. The proof convinces us of something--though what interests us is, not the mental state of conviction, but
the applications attaching to this conviction.
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For this reason the assertion that the proof convinces us of the truth of this proposition leaves us cold,--since
this expression is capable of the most various constructions.
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Page 162
When I say: "the proof convinces me of something", still the proposition expressing this conviction need not
be constructed in the proof. As e.g. we multiply, but do not necessarily write down the result in the form of the
proposition '... × ... = ....' So we shall presumably say: the multiplication gives us this conviction without our ever
uttering the sentence expressing it.
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A psychological disadvantage of proofs that construct propositions is that they easily make us forget that the
sense of the result is not to be read off from this by itself, but from the proof. In this respect the intrusion of the
Russellian symbolism into the proofs has done a great deal of harm.
Page 162
The Russellian signs veil the important forms of proof as it were to the point of unrecognizability, as when a
human form is wrapped up in a lot of cloth.
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26. Let us remember that in mathematics we are convinced of grammatical propositions; so the expression,
the result, of our being convinced is that we accept a rule.
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Nothing is more likely than that the verbal expression of the result of a mathematical proof is calculated to
delude us with a myth.
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27. I am trying to say something like this: even if the proved mathematical proposition seems to point to a
reality outside itself, still it is only the expression of acceptance of a new measure (of reality).
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Page 163
Thus we take the constructability (provability) of this symbol (that is, of the mathematical proposition) as a
sign that we are to transform symbols in such and such a way.
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We have won through to a piece of knowledge in the proof? And the final proposition expresses this
knowledge? Is this knowledge now independent of the proof (is the navel string cut)?--Well, the proposition is now
used by itself and without having the proof attached to it.
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Why should I not say: in the proof I have won through to a decision?
Page 163
The proof places this decision in a system of decisions.
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(I might of course also say: "the proof convinces me that this rule serves my purpose". But to say this might
easily be misleading.)
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28. The proposition proved by means of the proof serves as a rule--and so as a paradigm. For we go by the
rule.
But does the proof only bring us to the point of going by this rule
Page Break 164
(accepting it), or does it also shew us how we are to go by it?
Page 164
For the mathematical proposition is to shew us what it makes SENSE to say.
Page 164
The proof constructs a proposition; but the point is how it constructs it. Sometimes, for example, it first
constructs a number and then comes the proposition that there is such a number. When we say that the construction
must convince us of the proposition, that means that it must lead us to apply this proposition in such-and-such a
way. That it must determine us to accept this as sense, that not.
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29. What is in common between the purpose of a Euclidean construction, say the bisection of a line, and the
purpose of deriving a rule from rules by means of logical inferences?
Page 164
The common thing seems to be that by the construction of a sign I compel the acceptance of a sign.
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Could we say: "mathematics creates new expressions, not new propositions"?
Inasmuch, that is, as mathematical propositions are instruments taken up into the language once for all--and
their proof shews the place where they stand.
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But in what sense are e.g. Russell's tautologies 'instruments of language'?
Russell at any rate would not have held them to be so. His mistake, if there was one, can however only have
consisted in his not paying attention to their application.
Page 165
The proof makes one structure generate another.
It exhibits the generation of one from others.
That is all very well--but still it does quite different things in different cases! What is the interest of this
transition?
Page 165
Even if I think of a proof as something deposited in the archives of language--who says how this instrument
is to be employed, what it is for?
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30. A proof leads me to say: this must be like this.--Now, I understand this in the case of a Euclidean proof or
the proof of '25 times 25 = 625', but is it also like this in the case of a Russellian proof, e.g. of ' p⊃q.p:⊃:q'? What
does 'it must be like this' mean here in contrast with 'it is like this'? Should I say: "Well, I accept this expression as a
paradigm for all non-informative propositions of this form"?
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I go through the proof and say: "Yes, this is how it has to be; I must fix the use of my language in this way".
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I want to say that the must corresponds to a track which I lay down in language.
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31. When I said that a proof introduces a new concept, I meant something like: the proof puts a new
paradigm among the paradigms of the language; like when someone mixes a special reddish blue, somehow settles
the special mixture of the colours and gives it a name.
But even if we are inclined to call a proof such a new paradigm--what is the exact similarity of the proof to
such a concept-model?
One would like to say: the proof changes the grammar of our language, changes our concepts. It makes new
connexions, and it creates the concept of these connexions. (It does not establish that they are there; they do not
exist until it makes them.)
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32. What concept is created by 'p⊃p'? And yet I feel as if it would be possible to say that 'p⊃p' serves as the
sign of a concept.
'p⊃p' is a formula. Does a formula establish a concept? One can say: "by the formula... such-and-such
follows from this". Or again: "such-and-such follows from this in the following way:..." But is that the sort of
proposition I want? What, however, about: "Draw the consequences of this in the following way:..."?
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33. If I call a proof a model (a picture), then I must also be able to say this of a Russellian primitive
proposition (as the egg-cell of a proof).
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It can be asked: how did we come to utter the sentence 'p⊃p' as
Page Break 167
a true assertion? Well, it was not used in practical linguistic intercourse,--but still there was an inclination to utter it in
particular circumstances (when for example one was doing logic) with conviction.
Page 167
But what about 'p⊃p'? I see in it a degenerate proposition, which is on the side of truth.
I fix it as an important point of intersection of significant sentences. A pivotal point of our method of
description.
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34. The construction of a proof begins with some signs or other, and among these some, the 'constants', must
already have meaning in the language. In this way it is essential that '∨' and '~' already possess a familiar application,
and the construction of a proof in Principia Mathematica gets its importance, its sense, from this. But the signs of
the proof do not enable us to see this meaning.
Page 167
The 'employment' of the proof has of course to do with that employment of its signs.
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35. To repeat, in a certain sense even Russell's primitive propositions convince me.
Thus the conviction produced by a proof cannot simply arise from the proof-construction.
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36. If I were to see the standard metre in Paris, but were not
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acquainted with the institution of measuring and its connexion with the standard metre--could I say, that I was
acquainted with the concept of the standard metre?
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Is a proof not also part of an institution in this way?
Page 168
A proof is an instrument--but why do I say "an instrument of language"?
Is a calculation necessarily an instrument of language, then?
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37. What I always do seems to be--to emphasize a distinction between the determination of a sense and the
employment of a sense.
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38. Accepting a proof: one may accept it as the paradigm of the pattern that arises when these rules are
correctly applied to certain patterns. One may accept it as the correct derivation of a rule of inference. Or as a correct
derivation from a correct empirical proposition; or as the correct derivation from a false empirical proposition; or
simply as the correct derivation from an empirical proposition, of which we do not know whether it is true or false.
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But now can I say that the conception of a proof as 'proof of constructability' of the proved proposition is in
some sense a simpler, more primary, one than any other conception?
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Page 169
Can I therefore say: "Any proof proves first and foremost that this formation of signs must result when I
apply these rules to these formations of signs"? Or: "The proof proves first and foremost that this formation can arise
when one operates with these signs according to these transformation-rules".--
This would point to a geometrical application. For the proposition whose truth, as I say, is proved here, is a
geometrical proposition--a proposition of grammar concerning the transformations of signs. It might for example be
said: it is proved that it makes sense to say that someone has got the sign ... according to these rules from ... and ...;
but no sense etc. etc.
Page 169
Or again: when mathematics is divested of all content, it would remain that certain signs can be constructed
from others according to certain rules.--
Page 169
The least that we have to accept would be: that these signs etc. etc.--and accepting this is a basis for accepting
anything else.
Page 169
I should now like to say: the sequence of signs in the proof does not necessarily carry with it any kind of
acceptance. If however it's to be a matter of accepting, this does not have to be 'geometrical' acceptance.
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A proof could surely consist of only two steps: say one proposition '(x).fx' and one 'fa'--does the correct
transition according to a rule play an important part here?
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39. What is unshakably certain about what is proved?
Page 170
To accept a proposition as unshakably certain--I want to say--means to use it as a grammatical rule: this
removes uncertainty from it.
Page 170
"Proof must be capable of being taken in" really means nothing but: a proof is not an experiment. We do not
accept the result of a proof because it results once, or because it often results. But we see in the proof the reason for
saying that this must be the result.
Page 170
What proves is not that this correlation leads to this result--but that we are persuaded to take these
appearances (pictures) as models for what it is like if.....
Page 170
The proof is our new model for what it is like if nothing gets added and nothing taken away when we count
correctly etc.. But these words shew that I do not quite know what the proof is a model of.
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I want to say: with the logic of Principia Mathematica it would be possible to justify an arithmetic in which
1000 + 1 = 1000; and all that would be necessary for this purpose would be to doubt the sensible correctness of
calculations. But if we do not doubt it, then it is not our conviction of the truth of logic that is responsible.
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Page 171
When we say in a proof: "This must come out"--then this is not for reasons that we do not see.
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It is not our getting this result, but its being the end of this route, that makes us accept it.
Page 171
What convinces us--that is the proof: a configuration that does not convince us is not the proof, even when it
can be shewn to exemplify the proved proposition.
Page 171
That means: it must not be necessary to make a physical investigation of the proof-configuration in order to
shew us what has been proved.
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40. If we have a picture of two men, we do not say first that the one appears smaller than the other, and then
that he seems to be further away. It is, one can say, perfectly possible that the one figure's being shorter should not
strike us at all, but only its being behind. (This seems to me to be connected with the question of the 'geometrical'
conception of proof.)
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41. "It (the proof) is the model for what is called such-and-such."
Page 171
But what is the transition from '(x).fx' to 'fa' supposed to be a model for? At most for how inferences can be
drawn from signs like '(x).fx'.
I thought of the model as a justification, but here it is not a justification.
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The pattern (x).fx ∴ fa does not justify the conclusion. If we want to talk about a justification of the conclusion, it
lies outside this schema of signs.
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And yet there is something in saying that a mathematical proof creates a new concept.--Every proof is as it
were an avowal of a particular employment of signs.
Page 172
But what is it an avowal of? Only of this employment of the rules of transition from formula to formula? Or
is it also an avowal in some sense, of the 'axioms'?
Page 172
Could I say: I avow p⊃p as a tautology?
Page 172
I accept 'p⊃p' as a maxim, e.g. of inference.
Page 172
The idea that proof creates a new concept might also be roughly put as follows: a proof is not its foundations
plus the rules of inference, but a new building--although it is an example of such and such a style. A proof is a new
paradigm.
Page 172
The concept which the proof creates may for example be a new concept of inference, a new concept of
correct inferring. But as for why I accept this as correct inferring, the reasons for that lie outside the proof.
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Page 173
The proof creates a new concept by creating or being a new sign. Or--by giving the proposition which is its
result a new place. (For the proof is not a movement but a route.)
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42. It must not be imaginable for this substitution in this expression to yield anything else. Or: I must
declare it unimaginable. (The result of an experiment, however, can turn out this way or that.)
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Still, the case could be imagined in which a proof altered in appearance--engraved in rock, it is stated to be
the same whatever the appearance says.
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Are you really saying anything but: a proof is taken as proof?
Page 173
Proof must be a procedure plain to view. Or again: the proof is the procedure that is plain to view.
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It is not something behind the proof, but the proof, that proves.
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43. When I say: "it must first and foremost be evident that this substitution really yields this expression"--I
might also say: "I must accept it as indubitable"--but then there must be good reasons for this: for example, that the
same substitution practically always yields
Page Break 174
the same result etc.. And isn't this exactly what surveyability consists in?
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I should like to say that where surveyability is not present, i.e. where there is room for a doubt whether what
we have really is the result of this substitution, the proof is destroyed. And not in some silly and unimportant way
that has nothing to do with the nature of proof.
Page 174
Or: logic as the foundation of all mathematics does not work, and to shew this it is enough that the cogency
of logical proof stands and falls with its geometrical cogency.†1
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We incline to the belief that logical proof has a peculiar, absolute cogency, deriving from the unconditional
certainty in logic of the fundamental laws and the laws of inference. Whereas propositions proved in this way can
after all not be more certain than is the correctness of the way those laws of inference are applied.
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That is to say: logical proof, e.g. of the Russellian kind, is cogent only so long as it also possesses geometrical
cogency.†1 And an abbreviation of such a logical proof may have this cogency and so be a proof, when the
Russellian construction, completely carried out, is not.
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The logical certainty of proofs--I want to say--does not extend beyond their geometrical certainty.
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44. Now if a proof is a model, then the point must be what is to count as a correct reproduction of the proof.
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If, for example, the sign '| | | | | | | | | |' were to occur in a proof, it is not clear whether merely 'the same number'
of strokes (or perhaps little crosses) should count as the reproduction of it, or whether some other, not too small,
number does equally well. Etc.
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But the question is what is to count as the criterion for the reproduction of a proof--for the identity of proofs.
How are they to be compared to establish the identity? Are they the same if they look the same?
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I should like, so to speak, to shew that we can get away from logical proofs in mathematics.
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45. "By means of suitable definitions, we can prove '25 × 25 = 625' in Russell's logic."--And can I define the
ordinary technique of proof by means of Russell's? But how can one technique of proof be defined by means of
another? How can one explain the essence of another? For if the one is an 'abbreviation' of the other, it must surely
be a systematic
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abbreviation. Proof is surely required that I can systematically shorten the long proofs and thus once more get a
system of proofs.
Long proofs at first always go along with the short ones and as it were tutor them. But in the end they can no
longer follow the short ones and these shew their independence.
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The consideration of long unsurveyable logical proofs is only a means of shewing how this technique--which
is based on the geometry of proving--may collapse, and new techniques become necessary.
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46. I should like to say: mathematics is a MOTLEY of techniques of proof.--And upon this is based its
manifold applicability and its importance.
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But that comes to the same thing as saying: if you had a system like that of Russell and produced systems
like the differential calculus out of it by means of suitable definitions, you would be producing a new bit of
mathematics.
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Now surely one could simply say: if a man had invented calculating in the decimal system--that would have
been a mathematical invention!--Even if he had already got Russell's Principia Mathematica.--
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What is it to co-ordinate one system of proofs with another? It involves a translation rule by means of which
proved propositions of
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the one can be translated into proved propositions of the other.
Now it is possible to imagine some--or all--of the proof systems of present-day mathematics as having been
co-ordinated in such a way with one system, say that of Russell. So that all proofs could be carried out in this
system, even though in a roundabout way. So would there then be only the single system--no longer the
many?--But then it must surely be possible to shew of the one system that it can be resolved into the many.--One
part of the system will possess the properties of trigonometry, another those of algebra, and so on. Thus one can say
that different techniques are used in these parts.
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I said: whoever invented calculation in the decimal notation surely made a mathematical discovery. But could
he not have made this discovery all in Russellian symbols? He would, so to speak, have discovered a new aspect.
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"But in that case the truth of true mathematical propositions can still be proved from those general
foundations."--It seems to me there is a snag here. When do we say that a mathematical proposition is true?--
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It seems to me as if we were introducing new concepts into the Russellian logic without knowing it.--For
example, when we settle what signs of the form '(∃ x, y, z...)' are to count as equivalent to one another, and what are
not to count as equivalent.
Is it a matter of course that '(∃ x, y, z)' is not the same sign as '(∃ x, y, z, n)'?
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But suppose I first introduce 'p ∨ q' and '~ p' and use them to construct some tautologies--and then produce
(say) the series ~p, ~~p, ~~~p, etc. and introduce a notation like ~1p, ~2p,... ~10p.... I should like to say: we should
perhaps originally never have thought of the possibility of such a sequence and we have now introduced a new
concept into our calculation. Here is a 'new aspect'.
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It is clear that I could have introduced the concept of number in this way, even though in a very primitive and
inadequate fashion--but this example gives me all I need.
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In what sense can it be correct to say that one would have introduced a new concept into logic with the series
~p, ~~p, ~~~p, etc.?--Well, first of all one could be said to have done it with the 'etc.'. For this 'etc.' stands for a law
of sign formation which is new to me. A characteristic mark of this is the fact that recursive definition is required for
the explanation of the decimal notation.
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A new technique is introduced.
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It can also be put like this: having the concept of the Russellian formation of proofs and propositions does
not mean you have the concept of every series of Russellian signs.
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I should like to say: Russell's foundation of mathematics postpones the introduction of new techniques--until
finally you believe that this is no longer necessary at all.
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(It would perhaps be as if I were to philosophize about the concept of measurement of length for so long that
people forgot that the actual fixing of a unit of length is necessary before you can measure length.)
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47. Can what I want to say be put like this: "If we had learnt from the beginning to do all mathematics in
Russell's system, the differential calculus, for example, would not have been invented just by our having Russell's
calculus. So if someone discovered this kind of calculation in Russell's calculus----."
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Suppose I had Russellian proofs of the propositions
'p ≡ ~~ p'
'~ p ≡ ~~~ p'
'p ≡ ~~~~ p'
and I were now to find a shortened way of proving the proposition
'p = ≡ ~ 10p'.
It is as if I had discovered a new kind of calculation within the old calculus. What does its having been discovered
consist in?
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Tell me: have I discovered a new kind of calculation if, having once learnt to multiply, I am struck by
multiplications with all the factors the same, as a special branch of these calculations, and so I introduce the notation
'an = ...'?
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Obviously the mere 'shortened', or different, notation--'162' instead of '16 × 16'--does not amount to that.
What is important is that we now merely count the factors.
Is '1615' merely another notation for '16 × 16 × 16 × 16 × 16 × 16 × 16 × 16 × 16 × 16 × 16 × 16 × 16 × 16 ×
16'?
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The proof that 1615 = ... does not simply consist in my multiplying 16 by itself fifteen times and getting this
result--the proof must shew that I take the number as a factor 15 times.
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When I ask "What is new about the 'new kind of calculation'--exponentiation"--that is difficult to say. The
expression 'new aspect' is vague. It means that we now look at the matter differently--but the question is: what is the
essential, the important manifestation of this 'looking at it differently'?
First of all I want to say: "It need never have struck anyone that in certain products all the factors are
equal"--or: "'Product of all equal factors' is a new concept"--or: "What is new consists in our classifying calculations
differently". In exponentiation the essential thing is evidently that we look at the number of the factors. But who
says we ever attended to the number of factors? It need not have struck us that there are products with 2, 3, 4 factors
etc. although we have often worked out such products. A new aspect--but once more: what is important about it?
For what purpose do I use what has struck me?--Well, first of all perhaps I put it down in a notation. Thus I write
e.g. 'a2' instead of 'a × a'. By this means I refer to the series of numbers (allude to it), which did not happen before.
So I am surely setting up a new connexion!--A connexion--between what objects? Between the technique of
counting factors and the technique of multiplying.
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Page 181
But in that way every proof, each individual calculation makes new connexions!
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But the same proof as shews that a × a × a × a... = b, surely also shews that an = b; it is only that we have to
make the transition according to the definition of 'an'.--
But this transition is exactly what is new. But if it is only a transition to the old proof, how can it be
important?
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'It is only a different notation.' Where does it stop being--just a different notation?
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Isn't it where only the one notation and not the other can be used in such-and-such a way?
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It might be called "finding a new aspect", if someone writes 'a (f)' instead of 'f (a)'; one might say: "He looks
at the function as an argument of its argument". Or if someone wrote '× (a)' instead of 'a × a' one could say: "he
looks at what was previously regarded as a special case of a function with two argument places as a function with
one argument place".
If anyone does this he has certainly altered the aspect in a sense, he has for example classified this
expression with others, compared it with others, with which it was not compared before.--But now, is that an
important change of aspect? No, not so long as it does not have certain consequences.
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Page 182
It is true enough that I changed the aspect of the logical calculation by introducing the concept of the number
of negations: "I never looked at it like that"--one might say. But this alteration only becomes important when it
connects with the application of the sign.
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Conceiving one foot as 12 inches would indeed mean changing the aspect of 'a foot', but this change would
only become important if one now also measured lengths in inches.
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If you introduce the counting of negation signs, you bring in a new way of reproducing signs.
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For arithmetic, which does talk about the equality of numbers, it is indeed a matter of complete indifference
how equality of number of two classes is established--but for its inferences it is not indifferent how its signs are
compared with one another, and so e.g. what is the method of establishing whether the number of figures in two
numerical signs is the same.
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It is not the introduction of numerical signs as abbreviations that is important, but the method of counting.
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48. I want to give an account of the motley of mathematics.
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49. "I can carry out the proof that 127 : 18 = 7·05 in Russell's system too." Why not.--But must the same
result be reached in the
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Russellian proof as in ordinary division? The two are of course connected by means of a type of calculation (by
rules of translation, say--); but still, is it not risky to work out the division by the new technique,--since the truth of
the result is now dependent on the geometry of the rendering?
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But now suppose someone says: "Nonsense--such considerations play no part in mathematics".--
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--But the question is not one of uncertainty, for we are certain of our conclusions, but of whether we are still
doing (Russellian) logic when we e.g. divide.
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50. Trigonometry has its original importance in connexion with measurements of lengths and angles: it is a
bit of mathematics adapted to employment on measurements of lengths and angles.
Applicability to this field might also be called an 'aspect' of trigonometry.
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When I divide a circle into equal sectors and determine the cosine of one of these sectors by measurement--is
that a calculation or an experiment?
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If a calculation--is it SURVEYABLE?
Is calculation with a slide-rule surveyable?
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If the cosine of an angle has to be determined by measurement, is a proposition of the form 'cos α = n' a
mathematical proposition? What is the criterion for this decision? Does the proposition say something external
about our rulers etc.; or something internal about our concepts?--How is this to be decided?
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Do the figures (drawings) in trigonometry belong to pure mathematics, or are they only examples of a
possible application?
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51. If there is something true about what I am trying to say, then--e.g.--calculating in the decimal notation
must have its own life.--One can of course represent any decimal number in the form:
and hence carry out the four species of calculation in this notation. But the life of the decimal notation would have to
be independent of calculating with unit strokes.
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52. In this connexion the following point constantly occurs to me: while indeed a proposition 'a: b = c' can be
proved in Russell's logic, still that logic does not teach us to construct a correct sentence of this form, i.e. does not
teach us to divide. The procedure of dividing would correspond e.g. to that of a systematic testing of Russellian
proofs with a view, say, to getting the proof of a proposition of the form '37 × 15 = x'. "But the technique of such a
systematic testing is in its turn founded on logic. It can surely be logically proved in turn that
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this technique must lead to the goal." So it is like proving in Euclid that such-and-such can be constructed in
such-and-such a way.
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53. If someone tries to shew that mathematics is not logic, what is he trying to shew? He is surely trying to
say something like:--If tables, chairs, cupboards, etc. are swathed in enough paper, certainly they will look spherical
in the end.
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He is not trying to shew that it is impossible that, for every mathematical proof, a Russellian proof can be
constructed which (somehow) 'corresponds' to it, but rather that the acceptance of such a correspondence does not
lean on logic.
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"But surely we can always go back to the primitive logical method!" Well, assuming that we can--how is it
that we don't have to? Or are we hasty, reckless, if we do not?
But how do we get back to the primitive expression? Do we e.g. take the route through the secondary proof
and back from the end of it into the primary system, and then look to see where we have got; or do we go forward in
both systems and then connect the end points? And how do we know that we reach the same result in the primary
system in the two cases?
Does not proceeding in the secondary system carry the power of conviction with it?
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"But at every step in the secondary system, we can imagine that it
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could be taken in the primary one too!"--That is just it: we can imagine that it could be done--without doing it.
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And why do we accept the one in place of the other? On grounds of logic?
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"But can't one prove logically that both transformations must lead to the same result?"--But what is in
question here is surely the result of transformations of signs! How can logic decide this?
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54. How can the proof in the stroke system prove that the proof in the decimal system is a proof?
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Well--isn't it the same for the proof in the decimal system, as it is for a construction in Euclid of which it is
proved that it really is the construction of such-and-such a figure?
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Can I put it like this: "The translation of the stroke system into the decimal system presupposes a recursive
definition. This definition, however, does not introduce the abbreviation of one expression to another. Yet of course
inductive proof in the decimal system does not contain the whole set of those signs which would have to be
translated by means of the recursive definition into stroke signs. Therefore this general proof cannot be translated by
recursive definition into a proof in the stroke system."?
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Page 187
Recursive definition introduces a new sign-technique.--It must therefore make the transition to a new
'geometry'. We are taught a new method of recognizing signs. A new criterion for the identity of signs is introduced.
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55. A proof shews us what OUGHT to come out.--And since every reproduction of the proof must
demonstrate the same thing, while on the one hand it must reproduce the result automatically, on the other hand it
must also reproduce the compulsion to get it.
That is: we reproduce not merely the conditions which once yielded this result (as in an experiment), but the
result itself. And yet the proof is not a stacked game, inasmuch as it must always be capable of guiding us.
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On the one hand we must be able to reproduce the proof in toto automatically, and on the other hand this
reproduction must once more be proof of the result.
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"Proof must be surveyable": this aims at drawing our attention to the difference between the concepts of
'repeating a proof', and 'repeating an experiment'. To repeat a proof means, not to reproduce the conditions under
which a particular result was once obtained, but to repeat every step and the result. And although this shews that
proof is something that must be capable of being reproduced in toto automatically, still every such reproduction
must contain the force of proof, which compels acceptance of the result.
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56. When do we say that one calculus 'corresponds' to another, is only an abbreviated form of the
first?--"Well, when the results of the latter can be translated by means of suitable definitions into the results of the
former." But has it been said how one is to calculate with these definitions? What makes us accept this translation?
Is it a stacked game in the end? It is, if we are decided on only accepting the translation that leads to the accustomed
result.
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Why do we call a part of the Russellian calculus the part corresponding to the differential calculus?--Because
the propositions of the differential calculus are proved in it.--But, ultimately, after the event.--But does that matter?
Sufficient that proofs of these propositions can be found in the Russellian system! But aren't they proofs of these
propositions only when their results can be translated only into these propositions? But is that true even in the case
of multiplying in the stroke system with numbered strokes?
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57. Now it must be clearly stated that calculations in the stroke notation will normally always agree with
those in the decimal notation. Perhaps, in order to make sure of agreement, we shall at some point have to take to
getting the stroke-calculation worked over by several people. And we shall do the same for calculations with still
higher numbers in the decimal system.
But that of course is enough to shew that it is not the proofs in the stroke notation that make the proofs in the
decimal system cogent.
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"Still, if we did not have the latter, we could use the former to prove the same thing."--The same thing? What
is the same thing?--Well, the stroke proof will convince me of the same thing, though not in the same way.--Suppose
I were to say: "The place to which a proof leads us cannot be determined independently of this proof."--Did a proof
in the stroke system demonstrate to me that the proved proposition possesses the applicability given it by the proof
in the decimal system--was it e.g. proved in the stroke system that the proposition is also provable in the decimal
system?
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58. Of course it would be nonsense to say that one proposition cannot have two proofs--for we do say just
that. But can we not say: this proof shews that... results when we do this; the other proof shews that this expression
results when we do something else?
For is e.g. the mathematical fact that 129 is divisible by 3 independent of the fact that this is the result in this
calculation? I mean: is the fact of this divisibility independent of the calculus in which it is a result; or is it a fact of
this calculus?
Page 189
Suppose it were said: "By calculating we get acquainted with the properties of numbers".
But do the properties of numbers exist outside the calculating?
Page 189
"Two proofs prove the same when what they convince me of is the same."--And when is what they convince
me of the same?--How do I know that what they convince me of is the same? Not of course by introspection.
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Page 190
I can be brought to accept this rule by a variety of paths.
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59. "Each proof proves not merely the truth of the proposition proved, but also that it can be proved in this
way."--But this latter can also be proved in another way.--"Yes, but the proof proves this in a particular way and in
doing so proves that it can be demonstrated in this way."--But even that could be shewn by means of a different
proof.--"Yes, but then not in this way."--
But this means e.g.: this proof is a mathematical entity that cannot be replaced by any other; one can say that
it can convince us of something that nothing else can, and this can be given expression by our assigning to it a
proposition that we do not assign to any other proof.
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60. But am I not making a crude mistake? For just this is essential to the propositions of arithmetic and to the
propositions of the Russellian logic: various proofs lead to them. Even: infinitely many proofs lead to any one of
them.
Page 190
Is it correct to say that every proof demonstrates something to us which it alone can demonstrate? Would
not--so to speak--the proved proposition then be superfluous, and the proof itself also be the thing proved?
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Is it only the proved proposition that the proof convinces me of?
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What is meant by: "A proof is a mathematical entity which cannot be replaced by any other"? It surely
means that every single proof has a usefulness which no other one has. It might be said: "--that every proof, even of
a proposition which has already been proved, is a contribution to mathematics". But why is it a contribution if its
only point was to prove the proposition? Well, one can say: "the new proof shews (or makes) a new connexion".
(But in that case is there not a mathematical proposition saying that this connexion exists?)
Page 191
What do we learn when we see the new proof--apart from the proposition, which we already know anyhow?
Do we learn something that cannot be expressed in a mathematical proposition?
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61. How far does the application of a mathematical proposition depend on what is allowed to count as a
proof of it and what is not?
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I can surely say: if the proposition '137 × 373 = 46792' is true in the ordinary sense, then there must be a
multiplication-sum, at the ends of which stand the two sides of this equation. And a multiplication-sum is a pattern
satisfying certain rules.
I want to say: if I did not accept the multiplication-sum as one proof of the proposition, then that would
mean that the application of the proposition to multiplication-sums would be gone.
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62. Let us remember that it is not enough that two proofs meet in the same propositional sign. For how do
we know that this sign
Page Break 192
says the same thing both times? That must proceed from other connexions.
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63. The exact correspondence of a correct (convincing) transition in music and in mathematics.
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64. Suppose I were to set someone the problem: "Find a proof of the proposition..."--The answer would
surely be to shew me certain signs. Very well: what condition must these signs satisfy? They must be a proof of that
proposition--but is that, say, a geometrical condition? Or a psychological one? Sometimes it could be called a
geometrical condition; where the means of proof are already prescribed and all that is being looked for is a particular
arrangement.
Page 192
65. Are the propositions of mathematics anthropological propositions saying how we men infer and
calculate?--Is a statute book a work of anthropology telling how the people of this nation deal with a thief
etc.?--Could it be said: "The judge looks up a book about anthropology and thereupon sentences the thief to a term
of imprisonment"? Well, the judge does not USE the statute book as a manual of anthropology.
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66. The prophecy does not run, that a man will get this result when he follows this rule in making a
transformation--but that he will get this result, when we say that he is following the rule.
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Page 193
What if we said that mathematical propositions were prophecies in this sense: they predict what result
members of a society who have learnt this technique will get in agreement with other members of the society? '25 ×
25 = 625' would thus mean that men, if we judge them to obey the rules of multiplication, will reach the result 625
when they multiply 25 × 25.--That this is a correct prediction is beyond doubt; and also that calculating is in essence
founded on such predictions. That is to say, we should not call something 'calculating' if we could not make such a
prophecy with certainty. This really means: calculating is a technique. And what we have said pertains to the essence
of a technique.
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67. This consensus belongs to the essence of calculation, so much is certain. I.e.: this consensus is part of
the phenomenon of our calculating.
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In a technique of calculating prophecies must be possible.
Page 193
And that makes the technique of calculating similar to the technique of a game, like chess.
Page 193
But what about this consensus--doesn't it mean that one human being by himself could not calculate? Well,
one human being could at any rate not calculate just once in his life.
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It might be said: all possible positions in chess can be conceived as propositions saying that they
(themselves) are possible positions, or again as prophecies that people will be able to reach these positions
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by moves which they agree in saying are in accordance with the rules. A position reached in this way is then a
proved proposition of this kind.
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"A calculation is an experiment."--A calculation can be an experiment. The teacher makes the pupil do a
calculation in order to see whether he can calculate; that is an experiment.
Page 194
When the stove is lit in the morning, is that an experiment? But it could be one.
And in the same way moves in chess are not proofs either, and chess positions are not propositions. And
mathematical propositions are not positions in a game. And in this way they are not prophecies either.
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68. If a calculation is an experiment, then what is a mistake in calculation? A mistake in the experiment?
Surely not; it would have been a mistake in the experiment, if I had not observed the conditions of the
experiment--if, e.g., I had made someone calculate when a terrible noise was going on.
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But why should I not say: while a mistake in calculating is not a mistake in the experiment, still, it is a
miscarriage of the experiment--sometimes explicable, sometimes inexplicable?
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69. "A calculation, for example a multiplication, is an experiment: we do not know what will result and we
learn it once the multiplication is done."
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--Certainly; nor do we know when we go for a walk where exactly we shall be in five minutes' time--but does that
make going for a walk into an experiment?--Very well; but in the calculation I surely wanted from the beginning to
know what the result was going to be; that was what I was interested in. I am, after all, curious about the result. Not,
however, as what I am going to say, but as what I ought to say.
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But isn't this just what interests you about this multiplication--how the generality of men will calculate?
No--at least not usually--even if I am running to a common meeting point with everybody else.
But surely this is just what the calculation shews me experimentally--where this meeting point is. I as it were
let myself unwind and see where I get. And the correct multiplication is the pattern of the way we all work, when we
are wound up like this.
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Experience teaches that we all find this calculation correct.
Page 195
We let ourselves unwind and get the result of the calculation. But now--I want to say--we aren't interested in
having--under such and such conditions say--actually produced this result, but in the pattern of our working; it
interests us as a convincing, harmonious, pattern--not, however, as the result of an experiment, but as a path.
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We say, not: "So that's how we go!", but: "So that's how it goes!"
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70. In what we accept we all work the same way, but we do not make use of this identity merely to predict
what people will accept. Just as we do not use the proposition "this notebook is red" only to predict that most people
will call it 'red'.
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"And that's what we call 'the same'." If there did not exist an agreement in what we call 'red', etc. etc.,
language would stop. But what about the agreement in what we call 'agreement'?
We can describe the phenomenon of a confusion of language; but what are our tokens of confusion of
language? Not necessarily tumult and muddle in action. But rather that, e.g. I am lost when people talk, I cannot
react in agreement with them.
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"For me this is not a language-game." But in this case I might also say: though they accompany their actions
with spoken sounds and I cannot call their actions 'confused', still they haven't a language.--But perhaps their
actions would become confused if they were prevented from emitting those sounds.
Page 196
71. It could be said: a proof helps communication. An experiment presupposes it.
Or even: a mathematical proof moulds our language.
Page 196
But it surely remains the case that we can use a mathematical proof to make scientific predictions about the
proving done by other people.--
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If someone asks me: "What colour is this book?" and I reply: "It's green"--might I as well have given the
answer: "The generality of English-speaking people call that 'green'"?
Might he not ask: "And what do you call it?" For he wanted to get my reaction.
Page 197
'The limits of empiricism.'†1
Page 197
72. But there is such a thing as a science of conditioned calculating reflexes;--is that mathematics? That
science will rely on experiments: and these experiments will be calculations. But suppose this science became quite
exact and in the end even a 'mathematical' science?
Now is the result of these experiments that human beings agree in their calculations, or that they agree in
what they call "agreeing"? And it goes on like that.
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It could be said: that science would not function if we did not agree regarding the idea of agreement.
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It is clear that we can make use of a mathematical work for a study in anthropology. But then one thing is not
clear:--whether we
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ought to say: "This writing shews us how operating with signs was done among these people", or: "This writing
shews us what parts of mathematics these people had mastered".
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73. Can I say, on reaching the end of a multiplication: "So this is what I agree with!--"?--But can I say it at a
single step of the multiplication? E.g. at the step '2 × 3 = 6'? Any more than I can say: "So this is what I call 'white'!",
looking at this paper?
Page 198
It seems to me it would be a similar case if someone were to say: "When I call to mind what I have done
to-day, I am making an experiment (starting myself off), and the memory that then comes serves to shew me what
other people, who saw me, will reply to the question what I did".
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What would happen if we rather often had this: we do a calculation and find it correct; then we do it again
and find it isn't right; we believe we overlooked something before--then when we go over it again our second
calculation doesn't seem right, and so on.
Now should I call this calculating or not?--At any rate he cannot use this calculation to predict that he will
land there again next time.--But could I say that he calculated wrong this time, because the next time he did not
calculate again the same way? I might say: where this uncertainty existed there would be no calculating.
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But on the other hand I say again: 'Calculating is right--as it is done'. There can be no mistake of calculation
in '12 × 12 = 144'. Why? This proposition has assumed a place among the rules.
But is '12 × 12 = 144' the assertion that it is natural to all men to work out 12 × 12 in such a way that the
answer is 144?
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74. If I go over a calculation several times so as to be sure of having done it right, and if I then accept it as
correct,--haven't I repeated an experiment so as to be sure that I shall tick the same way the next time?--But why
should going over the calculation three times convince me that I shall tick the same way the fourth time?--I'd say: I
went over the calculation 'so as to be sure of not having overlooked anything'.
The danger here, I believe, is one of giving a justification of our procedure where there is no such thing as a
justification and we ought simply to have said: that's how we do it.
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When somebody makes an experiment repeatedly 'always with the same result', has he at the same time
made an experiment which tells him what he will call 'the same result', i.e. how he uses the word "the same"? If you
measure a table with a yardstick, are you also measuring the yardstick? If you are measuring the yardstick, then you
cannot be measuring the table at the same time.
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Suppose I were to say: "When someone measures the table with a yardstick he is making an experiment
which tells him the results of measuring this table with all other yardsticks"? It is after all beyond doubt that a
measurement with one yardstick can be used to predict the results of measurement with others. And, further, that if
it could
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not--our whole system of measuring would collapse.
No yardstick, it might be said, would be correct, if in general they did not agree.--But when I say that, I do
not mean that then they would all be false.
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75. Calculating would lose its point, if confusion supervened. Just as the use of the words "green" and "blue"
would lose its point. And yet it seems to be nonsense to say--that a proposition of arithmetic asserts that there will
not be confusion.--Is the solution simply that the arithmetical proposition would not be false but useless, if
confusion supervened?
Just as the proposition that this room is 16 foot long would not become false, if rulers and measuring fell into
confusion. Its sense, not its truth, is founded on the regular working of measurements. (But don't be dogmatic here.
There are transitional cases which complicate the discussion.)
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Suppose I were to say: an arithmetical proposition expresses confidence that confusion will not supervene.
Then the use of all words expresses confidence that confusion will not supervene.
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We cannot say, however, that use of the word "green" signifies that confusion will not supervene--because
then the use of the word "confusion" would have in its turn to assert just the same thing about this word.
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If '25 × 25 = 625' expresses the confidence that we shall always find it easy to agree on taking the road that
ends with this proposition--
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then why doesn't this last clause express confidence in something different, viz. that we should always be able to
agree about its use?
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We do not play the same language-game with the two propositions.
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Or can one equally well be confident that one will see the same colour over there as here--and also: that one
will be inclined to call the colour the same, if it is the same?
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What I want to say is: mathematics as such is always measure, not thing measured.
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76. The concept of calculating excludes confusion.--Suppose someone were to get different results at
different times when he did a multiplication, and saw this, but found it all right?--But then surely he could not use
the multiplication for the same purposes as we!--Why not? Nor is there anything to say that he would necessarily
fare ill if he did.
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The conception of calculation as an experiment tends to strike us as the only realistic one.
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Everything else, we think, is moonshine. In an experiment we have
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something tangible. It is almost as if one were to say: "When a poet composes he is making a psychological
experiment; that is the only way of explaining how a poem can have value". We mistake the nature of
'experiment',--believing that whenever we are keen on knowing the end of a process, it is what we call an
"experiment".
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It looks like obscurantism to say that a calculation is not an experiment. And in the same way so does the
statement that mathematics does not treat of signs, or that pain is not a form of behaviour. But only because people
believe that one is asserting the existence of an intangible, i.e. a shadowy, object side by side with what we all can
grasp. Whereas we are only pointing to different modes of employment of words.
It is almost as if one were to say: 'Blue' has to stand for a blue object--otherwise we could not see what the
word was for.
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77. I have invented a game--realize that whoever begins must always win: so it isn't a game. I alter it; now it is
all right.
Did I make an experiment, whose result was that whoever begins must always win? Or that we are inclined
to play in such a way that this happens? No.--But the result was not what you would have expected! Of course not;
but that does not make the game into an experiment.
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But what does it mean not to know why it always has to work out like that? Well, it is because of the rules.--I
want to know how I must alter the rules in order to get a proper game.--But you can e.g. alter them entirely--and so
give a quite different game in place of this
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one.--But that is not what I want. I want to keep the general outline of the rules and only eliminate a mistake.--But
that is vague. It is now simply not clear what is to be considered as the mistake.
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It is almost like when one says: What is the mistake in this piece of music? It doesn't sound well on the
instruments.--Now the mistake is not necessarily to be looked for in the instrumentation; it could be looked for in the
themes.
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Let us suppose, however, that the game is such that whoever begins can always win by a particular simple
trick. But this has not been realized;--so it is a game. Now someone draws our attention to it;--and it stops being a
game.
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What turn can I give this, to make it clear to myself?--For I want to say: "and it stops being a game"--not:
"and we now see that it wasn't a game".
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That means, I want to say, it can also be taken like this: the other man did not draw our attention to
anything; he taught us a different game in place of our own.--But how can the new game have made the old one
obsolete?--We now see something different, and can no longer naïvely go on playing.
On the one hand the game consisted in our actions (our play) on the board; and these actions I could
perform as well now as before. But on the other hand it was essential to the game that I blindly tried to
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win; and now I can no longer do that.
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78. Let us suppose that people originally practised the four kinds of calculation in the usual way. Then they
began to calculate with bracketed expressions, including ones of the form (a - a). Then they noticed that
multiplications, for example, were becoming ambiguous. Would this have to throw them into confusion? Would
they have to say: "Now the solid ground of arithmetic seems to wobble"?
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And if they now demand a proof of consistency, because otherwise they would be in danger of falling into
the bog at every step--what are they demanding? Well, they are demanding a kind of order. But was there no order
before?--Well, they are asking for an order which appeases them now.--But are they like small children, that merely
have to be lulled asleep?
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Well, multiplication would surely become unusable in practice because of its ambiguity--that is for the
former normal purposes. Predictions based on multiplications would no longer hit the mark.--(If I tried to predict the
length of the line of soldiers that can be formed from a square 50 × 50, I should keep on arriving at wrong results.)
Is this kind of calculation wrong, then?--Well, it is unusable for these purposes. (Perhaps usable for other
ones.) Isn't it as if I were once to divide instead of multiplying? (As can actually happen.)
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What is meant by: "You have to multiply here; not divide!"--
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Now is ordinary multiplication a proper game; is it impossible to trip up? And is the calculation with (a - a)
not a proper game--is it impossible not to trip up?
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(What we want is to describe, not to explain.)
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Now, what is it for us not to know our way about in our calculus?
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We went sleepwalking along the road between abysses.--But even if we now say: "Now we are awake",--can
we be certain that we shall not wake up one day? (And then say:--so we were asleep again.)
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Can we be certain that there are not abysses now that we do not see?
But suppose I were to say: The abysses in a calculus are not there if I don't see them!
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Is no demon deceiving us at present? Well, if he is, it doesn't matter. What the eye doesn't see the heart
doesn't grieve over.
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Suppose I were to divide by 3 sometimes like this:
sometimes like this
without noticing it.--Then someone draws my attention to it.
To a mistake? Is it necessarily a mistake? And in what circumstances do we call it one?
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79.
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The propositions 'φ(φ)' and '~φ(φ)' sometimes seem to say the same thing and sometimes opposite things.
According as we look at it the proposition 'φ(φ)' sometimes seems to say ~ φ(φ), sometimes the opposite. And we
sometimes see it as the product of the substitution:
At other times as:
We should like to say: "'Heterological' is not heterological; so by definition it can be called 'heterological'." And it
sounds all right, goes quite smoothly, and the contradiction need not strike us at all. If we become aware of the
contradiction, we should at first like to say that we do not mean the same thing by the assertion, ξ is heterological, in
the two cases. The one time it is the unabbreviated assertion,
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the other time the assertion abbreviated according to the definition.
We should then like to get out of the thing by saying: '~φ(φ) = φ1(φ)'. But why should we lie to ourselves like
this? Here two contrary routes really do lead--to the same thing.
Or again:--it is equally natural in this case to say '~φ(φ)' and 'φ(φ)'.
According to the rule it is an equally natural expression to say that C lies to the right of the point A and that it
lies to the left.
According to this rule--which says that a place lies in the direction of the arrow if the street that begins in that
direction leads to it.
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Let us look at it from the point of view of the language-games.--
Originally we played the game only with straight streets.----
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80. Could it perhaps be imagined that where I see blue, this means that the object that I see is not blue--that
the colour that appears to me always counts as the one that is excluded? I might for example believe that God
always shews me a colour in order to say: not this.
Or does this work: the colour that I see merely tells me that this colour plays a part in the description of the
object. It corresponds, not to a proposition, but merely to the word "blue". And the description
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of the object can then equally well run: "it is blue", and "it is not blue". Then one says: the eye only shows me blue,
but not the role of this blue.--We compare seeing the colour with hearing the word "blue" when we have not heard
the rest of the sentence.
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I should like to shew that we could be led to want to describe something's being blue, both by saying it was
blue, and by saying it was not blue.
And so that it is in our hands to make such a shift in the method of projection that 'p' and '~p' get the same
sense. By which, however, they lose it, if I do not introduce something new as negation.
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Now a language-game can lose its sense through a contradiction, can lose the character of a language-game.
And here it is important to say that this character is not described by saying that the sounds must have a
certain effect. For our language-game (2)†1 would lose the character of a language-game if the builders kept on
uttering different sounds instead of the 4 orders; even if it could be shewn, say physiologically, that it was always
these noises that moved the assistant to bring the stones that he did bring.
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Even here it could be said that of course the examination of language-games gets its importance from the fact
that language-games continue to function. And so that it gets its importance from the fact that human beings can be
trained to such a reaction to sounds.
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It seems to me that there is a connexion between this and the question whether a calculation is an experiment
made with a view to predicting the course of calculations. For suppose that one did a calculation
and--correctly--predicted that one would calculate differently the next time, since the circumstances have changed
then precisely by one's already having done the calculation so-and-so many times.
Calculating is a phenomenon which we know from calculating. As language is a phenomenon which we
know from our language.
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Can we say: 'Contradiction is harmless if it can be sealed off'? But what prevents us from sealing it off? That
we do not know our way about in the calculus. Then that is the harm. And this is what one means when one says:
the contradiction indicates that there is something wrong about our calculus. It is merely the (local) symptom of a
sickness of the whole body. But the body is only sick if we do not know our way about.
The calculus has a secret sickness, means: what we have got is, as it is, not a calculus, and we do not know
our way about--i.e. cannot give a calculus which corresponds 'in essentials' to this simulacrum of a calculus, and
only excludes what is wrong in it.
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But how is it possible not to know one's way about in a calculus: isn't it there, open to view?
Let us imagine having been taught Frege's calculus, contradiction and all. But the contradiction is not
presented as a disease. It is, rather, an accepted part of the calculus, and we calculate with it. (The calculations
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do not serve the usual purpose of logical calculations.)--Now we are set the task of changing this calculus, of which
the contradiction is an entirely respectable part, into another one, in which this contradiction is not to exist, as the
new calculus is wanted for purposes which make a contradiction undesirable.--What sort of problem is this? And
what sort of inability is it, if we say: "We have not yet found a calculus satisfying this condition"?
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When I say: "I don't know my way about in the calculus"--I do not mean a mental state, but an inability to do
something.
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It is often useful, in order to help clarify a philosophical problem, to imagine the historical development, e.g.
in mathematics, as quite different from what it actually was. If it had been different no one would have had the idea
of saying what is actually said.
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I should like to ask something like: "Is it usefulness you are out for in your calculus?--In that case you do not
get any contradiction. And if you aren't out for usefulness--then it doesn't matter if you do get one."
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81. Our task is, not to discover calculi, but to describe the present situation.
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The idea of the predicate which is true of itself etc. does of course lean on examples--but these examples
were stupidities, for they were not thought out at all. But that is not to say that such predicates could not be applied,
and that the contradiction would not then have its application!
I mean: if one really fixes one's eye on the application, it does not occur to one at all to write 'f(f)'. On the
other hand, if one is using the signs in the calculus, without presuppositions so to speak, one may also write 'f(f)',
and must then draw the consequences and not forget that one has not yet an inkling of a possible practical
application of this calculus.
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Is the question this: "Where did we forsake the region of usability?"?--
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For might we not possibly have wanted to produce a contradiction? Have said--with pride in a mathematical
discovery: "Look, this is how we produce a contradiction"? Might not e.g. a lot of people possibly have tried to
produce a contradiction in the domain of logic, and then at last one person succeeded?
But why should people have tried to do this? Perhaps I cannot at present suggest the most plausible purpose.
But why not e.g. in order to show that everything in this world is uncertain?
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These people would then never actually employ expressions of the form f(f), but still would be glad to lead
their lives in the neighbourhood of a contradiction.
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"Can I see an order which prevents me from unwittingly arriving at a contradiction?" That is like saying:
shew me an order in my calculus to convince me that I can never in this way arrive at a number which.... Then I
shew him e.g. a recursive proof.
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But is it wrong to say: "Well, I shall go on. If I see a contradiction, then will be the time to do something
about it."?--Is that: not really doing mathematics? Why should that not be calculating? I travel this road untroubled;
if I should come to a precipice I shall try to turn round. Is that not 'travelling'?
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Let us imagine the following case: the people of a certain tribe only know oral calculation. They have no
acquaintance with writing. They teach their children to count in the decimal system. Among them mistakes in
counting are very frequent, digits get repeated or left out without their noticing. Now a traveller makes a gramophone
record of their counting. He teaches them writing and written calculation, and then shews them how often they make
mistakes when they calculate just by word of mouth.--Would these people now have to admit that they had not
really calculated before? That they had merely been groping about, whereas now they walk? Might they not perhaps
even say: our affairs went better before, our intuition was not burdened with the dead stuff of writing? You cannot
lay hold of the spirit with a machine. They say perhaps: "If we repeated a digit then, as your machine asserts--well,
that will have been right".
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We may trust 'mechanical' means of calculating or counting more than our memories. Why?--Need it be like
this? I may have
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miscounted, but the machine, once constructed by us in such-and-such a way, cannot have miscounted. Must I
adopt this point of view?--"Well, experience has taught us that calculating by machine is more trustworthy than by
memory. It has taught us that our life goes smoother when we calculate with machines." But must smoothness
necessarily be our ideal (must it be our ideal to have everything wrapped in cellophane)?
Might I not even trust memory and not trust the machine? And might I not mistrust the experience which
'gives me the illusion' that the machine is more trustworthy?
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82. Earlier I was not certain that, among the kinds of multiplication corresponding to this description, there
was none yielding a result different from the accepted one. But say my uncertainty is such as only to arise at a
certain distance from calculation of the normal kind; and suppose that we said: there it does no harm; for if I
calculate in a very abnormal way, then I must just reconsider everything. Wouldn't this be all right?
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I want to ask: must a proof of consistency (or of non-ambiguity) necessarily give me greater certainty than I
have without it? And, if I am really out for adventures, may I not go out for ones where this proof no longer offers
me any certainty?
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My aim is to alter the attitude to contradiction and to consistency proofs. (Not to shew that this proof shews
something unimportant. How could that be so?)
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If for example I were anxious to produce contradictions, say for aesthetic purposes, then I should now
unhesitatingly accept the inductive proof of consistency and say: it is hopeless to try and produce a contradiction in
this calculus; the proof shews that it won't work. (Proof in theory of harmony.)----
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83. It is a good way of putting things to say: "this order (this method) is unknown to this calculus, but not to
that one".
What if one said: "A calculus to which this order is unknown is really not a calculus"?
(An office system to which this order is unknown is not really an office system.)
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It is--I should like to say--for practical, not for theoretical purposes, that the disorder is avoided.
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A kind of order is introduced because one has fared ill without it--or again, it is introduced, like streamlining
in perambulators and lamps, because it has perhaps proved its value somewhere else and in this way has become the
style or fashion.
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The misuse of the idea of mechanical insurance against contradiction. But what if the parts of the mechanism
fuse together, break or bend?
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84. "Only the proof of consistency shews me that I can rely on the calculus."
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What sort of proposition is it, that only then can you rely on the calculus? But what if you do rely on it
without that proof! What sort of mistake have you made?
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I introduce order; I say: "There are only these possibilities: ...". It is like determining the set of possible
permutations of A, B, C: before the order was there, I had perhaps only a foggy idea of this set.--Am I now quite
certain that I have overlooked nothing? The order is a method for not overlooking anything. But--for not overlooking
any possibility in the calculus, or: for not overlooking any possibility in reality?--Is it now certain that people will
never want to calculate differently? That people will never look at our calculus as we look at the counting of
aborigines whose numbers only go up to 5?--that we shall never want to look at reality differently? But that is not at
all the certainty that this order is supposed to give us. It is not the eternal correctness of the calculus that is supposed
to be assured, but only, so to speak, the temporal.
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'But these are the possibilities that you mean!--Or do you mean other ones?'
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The order convinces me that I have overlooked nothing when I have these 6 possibilities. But does it also
convince me that nothing is going to be able to upset my present conception of such possibilities?
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85. Could I imagine our fearing a possibility of constructing the heptagon, like the construction of a
contradiction; and that the proof that the construction of the heptagon is impossible should have a settling effect, like
a consistency proof?
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How does it come about that we are at all tempted (or at any rate come near it) to divide through by (3 - 3) in
(3 - 3) × 2 = (3 - 3) × 5? How does it come about that by the rules this step looks plausible, and that even so it is still
unusable?
When one tries to describe this situation it is enormously easy to make a mistake in the description. (So it is
very difficult to describe.) The descriptions which immediately suggest themselves are all misleading--that is how
our language in this field is arranged.
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And there will be constant lapses from description into explanation here.
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It was, or appears to be, roughly like this: we have a calculus, let us say, with the beads of an abacus; we then
replace it by a calculus with written signs; this calculus suggests to us an extension of the method of calculating
which the first calculus did not suggest--or perhaps better: the second calculus obliterates a distinction which was
not to be overlooked in the first one. Now if it was the point of the first calculus that this distinction was made, and it
is not made in the second one then the latter thereby lost its usability as an equivalent of the former. And now--it
seems--the problem might arise: where did we depart from the original calculus, what frontiers in the new one
correspond to the natural frontiers of the old?
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I formed a system of rules of calculation which were modelled on those of another calculus. I took the latter
as a model. But exceeded its limits. This was even an advantage; but now the new calculus became unusable in
certain parts (at least for the former purposes). I therefore seek to alter it: that is, to replace it by one that is to some
extent different. And by one that has the advantages without the disadvantages of the new one. But is that a clearly
defined task?
Is there such a thing--it might also be asked--as the right logical calculus, only without the contradictions?
Could it be said, e.g., that while Russell's Theory of Types avoids the contradiction, still Russell's calculus is
not THE universal logical calculus but perhaps an artificially restricted, mutilated one? Could it be said that the pure,
universal logical calculus has yet to be found?
I was playing a game and in doing so I followed certain rules: but as for how I followed them, that depended
on circumstances and the way it so depended was not laid down in black and white. (This is to some extent a
misleading account.) Now I wanted to play this game in such a way as to follow rules 'mechanically' and I
'formalized' the game. But in doing this I reached positions where the game lost all point; I therefore wanted to avoid
these positions 'mechanically'.--The formalization of logic did not work out satisfactorily. But what was the attempt
made for at all? (What was it useful for?) Did not this need, and the idea that it must be capable of satisfaction, arise
from a lack of clarity in another place?
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The question "what was it useful for?" was a quite essential question. For the calculus was not invented for
some practical purpose, but in order 'to give arithmetic a foundation'. But who says that arithmetic is logic, or what
has to be done with logic to make it in some sense into a substructure for arithmetic? If we had e.g. been led to
attempt this
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by aesthetic considerations, who says that it can succeed? (Who says that this English poem can be translated into
German to our satisfaction?!)
(Even if it is clear that there is in some sense a translation of any English sentence into German.)
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Philosophical dissatisfaction disappears by our seeing more.
By my allowing the cancelling of (3 - 3) this type of calculation loses its point. But suppose that, for example,
I were to introduce a new sign of equality which was supposed to express: 'equal after this operation'? Would it,
however, make sense to say: "Won in this sense", if in this sense I should win every game?
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At certain places the calculus led me to its own abrogation. Now I want a calculus that does not do this and
that excludes these places.--Does this mean, however, that any calculus in which such an exclusion does not occur is
an uncertain one? "Well, the discovery of these places was a warning to us."--But did you not misunderstand this
'warning'?
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86. Can one prove that one has not overlooked anything?--Certainly. And must one not perhaps admit later:
"Yes, I did overlook something; but not in the field for which my proof held"?
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The proof of consistency must give us reasons for a prediction; and
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that is its practical purpose. That does not mean that this proof is a proof from the physics of our technique of
calculation--and so a proof from applied mathematics--but it does mean that that prediction is the application that
first suggests itself to us, and the one for whose sake we have this proof at heart. The prediction is not: "No disorder
will arise in this way" (for that would not be a prediction: it is the mathematical proposition) but: "no disorder will
arise".
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I wanted to say: the consistency-proof can only set our minds at rest, if it is a cogent reason for this
prediction.
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87. Where it is enough for me to get a proof that a contradiction or a trisection of the angle cannot be
constructed in this way, the recursive proof achieves what is required of it. But if I had to fear that something
somehow might at some time be interpreted as the construction of a contradiction, then no proof can take this
indefinite fear from me.
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The fence that I put round contradiction is not a super-fence.
How can a proof have put the calculus right in principle?
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How can it have failed to be a proper calculus until this proof was found?
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"This calculus is purely mechanical; a machine could carry it out." What sort of machine? One constructed of
the usual materials--or a super-machine? Are you not confusing the hardness of a rule with the hardness of a
material?
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We shall see contradiction in a quite different light if we look at its occurrence and its consequences as it
were anthropologically--and when we look at it with a mathematician's exasperation. That is to say, we shall look at
it differently, if we try merely to describe how the contradiction influences language-games, and if we look at it from
the point of view of the mathematical law-giver.
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88. But wait--isn't it clear that no one wants to reach a contradiction? And so that if you shew someone the
possibility of a contradiction, he will do everything to make such a thing impossible? (And so that if someone does
not do this, he is a sleepyhead.)
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But suppose he replied: "I can't imagine a contradiction in my
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calculus.--You have indeed shewn me a contradiction in another, but not in this one. In this there is none, nor can I
see the possibility of one."
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"If my conception of the calculus should sometime alter; if its aspect should alter because of some context
that I cannot see now--then we'll talk some more about it."
"I do not see the possibility of a contradiction. Any more than you--as it seems--see the possibility of there
being one in your consistency-proof."
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Do I know whether, if I ever should see a contradiction where at present I can see no possibility of such a
thing, it will then look dangerous to me?
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89. "What does a proof teach me, apart from its result?"--What does a new tune teach me? Am I not under a
temptation to say it teaches me something?--
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90. I have not yet made the role of miscalculating clear. The role of the proposition: "I must have
miscalculated". It is really the key to an understanding of the 'foundations' of mathematics.
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PART IV
1942-1944
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1. "The axioms of a mathematical axiom-system ought to be self-evident." How are they self-evident, then?
What if I were to say: this is how I find it easiest to imagine.
And here imagining is not a particular mental process during which one usually shuts one's eyes or covers
them with one's hands.
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2. What do we say when we are presented with such an axiom, e.g. the parallel axiom? Has experience
shewn us that this is how it is? Well perhaps; but what experience? I mean: experience plays a part; but not the one
that one would immediately expect. For we haven't made experiments and found that in reality only one straight line
through a given point fails to cut another. And yet the proposition is evident.--Suppose I now say: it is quite
indifferent why it is evident. It is enough that we accept it. All that is important is how we use it.
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The proposition describes a picture. Namely:
We find this picture acceptable. As we find it acceptable to indicate our rough knowledge of a number by rounding it
off at a multiple of 10.
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'We accept this proposition.' But as what do we accept it?
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3. I want to say: when the words of e.g. the parallel-axiom are given (and we understand the language) the
kind of use this proposition has and hence its sense are as yet quite undetermined. And when we say that it is
evident, this means that we have already chosen a definite kind of employment for the proposition without realizing
it. The proposition is not a mathematical axiom if we do not employ it precisely for this purpose.
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The fact, that is, that here we do not make experiments, but accept the self-evidence, is enough to fix the
employment. For we are not so naïf as to make the self-evidence count in place of experiment.
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It is not our finding the proposition self-evidently true, but our making the self-evidence count, that makes it
into a mathematical proposition.
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4. Does experience tell us that a straight line is possible between any two points? Or that two different
colours cannot be at the same place?
It might be said: imagination tells us it. And the germ of truth is here; only one must understand it right.
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Before the proposition the concept is still pliable.
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But might not experience determine us to reject the axiom?! Yes. And nevertheless it does not play the part
of an empirical proposition.
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Why are the Newtonian laws not axioms of mathematics? Because we could quite well imagine things being
otherwise. But--I want to say--this only assigns a certain role to those propositions in contrast to another one. I.e.: to
say of a proposition: 'This could be imagined otherwise' or 'We can imagine the opposite too', ascribes the role of an
empirical proposition to it.
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A proposition which it is supposed to be impossible to imagine as other than true has a different function
from one for which this does not hold.
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5. The functioning of the axioms of mathematics is such that, if experience moved us to give up an axiom,
that would not make its opposite into an axiom.
'2 × 2 ≠ 5' does not mean:
'2 × 2 = 5' has not worked.
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One might, so to speak, preface axioms with a special assertion sign.
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Something is an axiom, not because we accept it as extremely probable, nay certain, but because we assign it
a particular function, and
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one that conflicts with that of an empirical proposition.
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We give an axiom a different kind of acknowledgment from an empirical proposition. And by this I do not
mean that the 'mental act of acknowledgment' is a different one.
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An axiom, I should like to say, is a different part of speech.
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6. When one hears the mathematical axiom that such and such is possible, one assumes offhand that one
knows what 'being possible' means here; because this form of sentence is naturally familiar to us.
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We are not made aware how various the employment of the assertion "... is possible" is! And that is why it
does not occur to us to ask about the special employment in this case.
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Lacking the slightest survey of the whole use, we are here quite unable to doubt that we understand the
proposition.
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Does the proposition that there is no such thing as action at a distance belong to the family of mathematical
propositions? Here again one would like to say: the proposition is not designed to express any experience, but rather
to express the impossibility of imagining anything different.
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To say that between two points a straight line is--geometrically--always possible means: the proposition "The
points... lie on a straight line" is an assertion about the position of the points only if more than 2 points are involved.
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Just as one does not ask oneself, either, what is the meaning of a proposition of the form "There is no..." (e.g.
"there is no proof of this proposition") in a particular case. Asked what it means, one replies both to someone else
and to oneself with an example of nonexistence.
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7. A mathematical proposition stands on four feet, not on three; it is over-determined.
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8. When we describe what a man does, e.g., by means of a rule, we want the person to whom we give the
description to know, by applying the rule, what happens in the particular case. Now do I give him an indirect
description by means of the rule?
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There is of course such a thing as a proposition saying: if anyone tries to multiply the numbers... according to
such and such rules, he gets....
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One application of a mathematical proposition must always be the calculating itself. That determines the
relation of the activity of calculating to the sense of mathematical propositions.
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We judge identity and agreement by the results of our calculating; that is why we cannot use agreement to
explain calculating.
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We describe by means of the rule. What for? Why? That is another question.
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'The rule, applied to these numbers, yields those' might mean: the expression of the rule, applied to a human
being, makes him produce those numbers from these.
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One feels, quite rightly, that that would not be a mathematical proposition.
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The mathematical proposition determines a path, lays down a path for us.
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It is no contradiction of this that it is a rule, and not simply stipulated but produced according to rules.
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If you use a rule to give a description, you yourself do not know more than you say. I.e. you yourself do not
foresee the application that you will make of the rule in a particular case. If you say "and so on", you yourself do not
know more than "and so on".
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9. How could one explain to anybody what you have to do if you are to follow a rule?
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One is tempted to explain: first and foremost do the simplest thing (if the rule e.g. is always to repeat the
same thing). And there is of course something in this. It is significant that we can say that it is simpler to write down
a sequence of numbers in which each number is the same as its predecessor than a sequence in which each number
is greater by 1 than its predecessor. And again that this is a simpler law than that of alternately adding 1 and 2.
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10. Isn't it over-hasty to apply a proposition that one has tested on sticks and stones, to wavelengths of light?
I mean: that 2 × 5000 = 10,000.
Does one actually count on it that what has proved true in so many cases must hold for these too? Or is it not
rather that with the arithmetical assumption we have not committed ourselves at all?
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11. Arithmetic as the natural history (mineralogy) of numbers. But who talks like this about it? Our whole
thinking is penetrated with this idea.
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The numbers (I don't mean the numerals) are shapes, and arithmetic tells us the properties of these shapes.
But the difficulty here is that these properties of the shapes are possibilities, not the properties in respect to shape of
the things of this shape. And these possibilities in turn emerge as physical, or psychological possibilities (of
separation, arrangement, etc.). But the role of the shapes is merely that of pictures
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which are used in such-and-such a way. What we give is not properties of shapes, but transformations of shapes, set
up as paradigms of some kind or other.
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12. We do not judge the pictures, we judge by means of the pictures.
We do not investigate them, we use them to investigate something else.
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You get him to decide on accepting this picture. And you do so by means of the proof, i.e. by exhibiting a
series of pictures, or simply by shewing him the picture. What moves him to decide does not matter here. The main
thing is that it is a question of accepting a picture.
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The picture of combining is not a combining; the picture of separating is not a separating; the picture of
something's fitting not a case of fitting. And yet these pictures are of the greatest significance. That is what it is like,
if a combination is made; if a separation; and so on.
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13. What would it be for animals or crystals to have as beautiful properties as numbers? There would then be
e.g. a series of forms, each bigger than another by a unit.
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I should like to be able to describe how it comes about that mathematics appears to us now as the natural
history of the domain of numbers, now again as a collection of rules.
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But could one not study transformations of (e.g.) the forms of animals? But how 'study'? I mean: might it not
be useful to pass transformations of animal shapes in review? And yet this would not be a branch of zoology.
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It would then be a mathematical proposition (e.g.), that this shape is derived from this one by way of this
transformation. (The shapes and transformations being recognizable.)
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14. We must remember, however, that by its transformations a mathematical proof proves not only
propositions of sign-geometry, but propositions of the most various content.
In this way the transformation in a Russellian proof proves that this logical proposition can be formed from
the fundamental laws by the use of these rules. But the proof is looked at as a proof of the truth of the conclusion, or
as a proof that the conclusion says nothing.
Now this is possible only through a relation of the proposition to something outside itself; I mean, e.g.,
through its relation to other propositions and to their application.
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"A tautology (e.g. 'p∨~p') says nothing" is a proposition referring to the language-game in which the
proposition p has application. (E.g. "It is raining or it is not raining" tells us nothing about the weather.)
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Russellian logic says nothing about kinds of propositions--I don't mean logical propositions--and their
employment: and yet logic gets its whole sense simply from its presumed application to propositions.
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15. People can be imagined to have an applied mathematics without any pure mathematics. They can e.g.--let
us suppose--calculate the path described by certain moving bodies and predict their place at a given time. For this
purpose they make use of a system of co-ordinates, of the equations of curves (a form of description of actual
movement) and of the technique of calculating in the decimal system. The idea of a proposition of pure mathematics
may be quite foreign to them.
Thus these people have rules in accordance with which they transform the appropriate signs (in particular,
e.g., numerals) with a view to predicting the occurrence of certain events.
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But when they now multiply, for example, will they not arrive at a proposition saying that the result of the
multiplication is the same, however the factors are shifted round? That will not be a primary rule of notation, nor yet
a proposition of their physics.
They do not need to obtain any such proposition--even if they allow the shift of factors.
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I am imagining the matter as if this mathematics were done entirely in the form of orders. "You must do
such-and-such"--so as to get the answer, that is, to the question 'where will this body be at such-and-such a time?' (It
does not matter at all how these people have arrived at this method of prediction.)
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The centre of gravity of their mathematics lies for these people entirely in doing.
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16. But is this possible? Is it possible that they should not pronounce the commutative law (e.g.) to be a
proposition?
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But I want to say: these people are not supposed to arrive at the conception of making mathematical
discoveries--but only of making physical discoveries.
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Question: Must they make mathematical discoveries as discoveries? What do they miss if they make none?
Could they (for example) use the proof of the commutative law, but without the conception of its culminating in a
proposition, and so having a result which is in some way comparable with their physical propositions?
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17. The mere picture
regarded now as four rows of five dots, now as five columns of four dots, might convince someone of the
commutative law. And he might thereupon carry out multiplications, now in the one direction, now in the other.
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One look at the pattern and pieces convinces him that he will be able to make them into that shape, i.e. he
thereupon undertakes to do so.
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"Yes, but only if the pieces don't change."--If they don't change, and we don't make some unintelligible
mistake, or pieces disappear or get added without our noticing it.
"But it is surely essential that the pieces can as a matter of fact always be made into that shape! What would
happen if they could not?"--Perhaps we should think that something had put us out. But--what then?--Perhaps we
should even accept the thing as it was. And then Frege might say: "Here we have a new kind of insanity!"†1
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18. It is clear that mathematics as a technique for transforming signs for the purpose of prediction has
nothing to do with grammar.
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19. The people whose mathematics was only such a technique, are now also supposed to accept proofs
convincing them of the replaceability of one sign-technique by another. That is to say, they find transformations,
series of pictures, on the strength of which they can venture to use one technique in place of another.
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20. If calculating looks to us like the action of a machine, it is the human being doing the calculation that is
the machine.
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In that case the calculation would be as it were a diagram drawn by a part of the machine.
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21. And that brings me to the fact that a picture may very well convince us that a particular part of a
mechanism will move in such-and-such a way when the mechanism is set in motion.
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The effect of such a picture (or series of pictures) is like that of a proof. In this way I might e.g. make a
construction for how the point X of the mechanism
will move.
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Is it not queer that it is not instantly clear how the picture of the period in division convinces us of the
recurrence of that row of digits?
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(I find it so difficult to separate the inner from the outer--and the picture from the prediction.)
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The twofold character of the mathematical proposition--as law and as rule.
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22. Suppose that one were to say "guessing right" instead of "intuition"? This would shew the value of an
intuition in a quite
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different light. For the phenomenon of guessing is a psychological one, but not that of guessing right.
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23. Our having learned a technique brings it about that we now alter it in such and such a way after seeing
this picture.
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'We decide on a new language-game.'
'We decide spontaneously' (I should like to say) 'on a new language-game.'
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24. True;--it looks as though, if our memory functioned differently, we could not calculate as we do. But in
that case could we give definitions as we do; talk and write as we do?
But how can we describe the foundation of our language by means of empirical propositions?
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25. Suppose that when we worked out a division it did not lead to the same result as the copying of its
period. That might arise e.g. from our altering our tables, without being aware of it. (Though it might also arise from
our copying in a different way.)
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26. What is the difference between not calculating and calculating wrong?--Or: is there a sharp dividing line
between not measuring time and measuring it wrong? Not knowing any measurement of time and knowing a wrong
one?
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27. Pay attention to the patter by means of which we convince someone of the truth of a mathematical
proposition. It tells us something about the function of this conviction. I mean the patter by which intuition is
awakened.
By which, that is, the machine of a calculating technique is set in motion.
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28. Can it be said that if you learn a technique, that convinces you of the uniformity of its results?
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29. The limit of the empirical†1--is concept-formation.
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What is the transition that I make from "It will be like this" to "it must be like this"? I form a different
concept. One involving something that was not there before. When I say: "If these derivations are the same, then it
must be that...", I am making something into a criterion of identity. So I am recasting my concept of identity.
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But what if someone now says: "I am not aware of these two processes, I am only aware of the empirical, not
of a formation and transformation of concepts which is independent of it; everything seems to me to be in the
service of the empirical"?
In other words: we do not seem to become now more, now less, rational, or to alter the form of our thinking,
so as to alter what we call "thinking". We only seem always to be fitting our thinking to experience.
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So much is clear: when someone says: "If you follow the rule, it must be like this", he has not any clear
concept of what experience would correspond to the opposite.
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Or again: he has not any clear concept of what it would be like for it to be otherwise. And this is very
important.
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30. What compels us so to form the concept of identity as to say, e.g., "If you really do the same thing both
times, then the result must be the same too"?--What compels us to proceed according to a rule, to conceive
something as a rule? What compels us to talk to ourselves in the forms of the languages we have learnt?
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For the word "must" surely expresses our inability to depart from this concept. (Or ought I to say "refusal"?)
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And even if I have made the transition from one concept-formation to another, the old concept is still there in
the background.
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Can I say: "A proof induces us to make a certain decision, namely that of accepting a particular
concept-formation"?
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Do not look at the proof as a procedure that compels you, but as one
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that guides you.--And what it guides is your conception of a (particular) situation.
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But how does it come about that it guides each one of us in such a way that we agree in the influence it has
on us? Well, how does it come about that we agree in counting? "That is just how we are trained" one may say,
"and the agreement produced in this way is carried further by the proofs."
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In the course of this proof we formed our way of looking at the trisection of the angle, which excludes a
construction with ruler and compass.
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By accepting a proposition as self evident, we also release it from all responsibility in face of experience.
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In the course of the proof our way of seeing is changed--and it does not detract from this that it is connected
with experience.
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Our way of seeing is remodelled.
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31. It must be like this, does not mean: it will be like this. On the contrary: 'it will be like this' chooses
between one possibility and another. 'It must be like this' sees only one possibility.
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The proof as it were guides our experience into definite channels. Someone who has tried again and again to
do such-and-such gives the attempt up after the proof.
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Someone tries to arrange pieces to make a particular pattern. Now he sees a model in which one part of that
pattern is seen to be composed of all his pieces, and he gives up his attempt. The model was the proof that his
proposal is impossible.
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That model too, like the one that shews that he will be able to make a pattern of these pieces, changes his
concept. For, one might say, he never looked at the task of making the pattern of these pieces in this way before.
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Is it obvious that if anyone sees that part of the pattern can be made with these pieces, he realizes that there is
no way of making the whole pattern with them? May it not be that he goes on trying and trying whether after all
some arrangement of the pieces does not achieve this end? And may he not achieve it? (Use of one piece twice over,
e.g.)
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Must we not distinguish here between thinking and the practical success of the thinking?
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32. "... who do not have immediate knowledge of certain truths, as we do, but perhaps are reduced to the
roundabout path of
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induction", says Frege.†1 But what interests me is this immediate insight, whether it is of a truth or of a falsehood. I
am asking: what is the characteristic demeanour of human beings who 'have insight into' something 'immediately',
whatever the practical result of this insight is?
Page 241
What interests me is not having immediate insight into a truth, but the phenomenon of immediate insight.
Not indeed as a special mental phenomenon, but as one of human action.
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33. Yes: it is as if the formation of a concept guided our experience into particular channels, so that one
experience is now seen together with another one in a new way. (As an optical instrument makes light come from
various sources in a particular way to form a pattern.)
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Imagine that a proof was a work of fiction, a stage play. Cannot watching a play lead me to something?
Page 241
I did not know how it would go,--but I saw a picture and became convinced that it would go as it does in the
picture.
The picture helped me to make a prediction. Not as an experiment--it was only midwife to the prediction.
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For, whatever my experience is or has been, I surely still have to make the prediction. (Experience does not
make it for me.)
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No great wonder, then, that proof helps us to predict. Without this picture I should not have been able to say
how it will be, but when I see it I seize on it with a view to prediction.
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I cannot predict the colour of a chemical compound by means of a picture exhibiting the substances in the
test-tube and the reaction. If the picture shewed frothing, and finally red crystals, I should not be able to say: "Yes,
that is how it has to be" or "No, it cannot be like that". It is otherwise, however, when I see the picture of a
mechanism in motion; that can tell me how a part actually will move. Though if the picture represented a mechanism
whose parts were composed of a very soft material (dough, say), and hence bent about in various ways in the
picture, then this picture might not help me to make a prediction either.
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Can we say that a concept is so formed as to be adapted to a certain prediction, i.e. it enables it to be made in
the simplest terms--?
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34. The philosophical problem is: how can we tell the truth and pacify these strong prejudices in doing so?
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It makes a difference whether I think of something as a deception of my senses or an external event, whether
I take this object as a measure of that or the other way round, whether I resolve to make two criteria decide or only
one.
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35. If the calculation has been done right, then this must be the result. Must this always be the result, in that
case? Of course.
Page 243
By being educated in a technique, we are also educated to have a way of looking at the matter which is just
as firmly rooted as that technique.
Page 243
Mathematical propositions seem to treat neither of signs nor of human beings, and therefore they do not.
Page 243
They shew those connexions that we regard as rigid. But to a certain extent we look away from these
connexions and at something else. We turn our back upon them, so to speak. Or: we rest, or lean, on them.
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Once more: we do not look at the mathematical proposition as a proposition dealing with signs, and hence it
is not that.
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We acknowledge it by turning our back on it.
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What about e.g. the fundamental laws of mechanics? If you understand them you must know how
experience supports them. It is otherwise with the propositions of pure mathematics.
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36. A proposition may describe a picture and this picture be variously anchored in our way of looking at
things, and so in our way of living and acting.
Page 244
Is not the proof too flimsy a reason for entirely giving up the search for a construction of the trisection? You
have only gone through the sequence of signs once or twice; will you decide on the strength of that? Just because
you have seen this one transformation, will you give up the search?
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The effect of proof is, I believe, that we plunge into the new rule.
Page 244
Hitherto we have calculated according to such and such a rule; now someone shews us the proof that it can
also be done in another way, and we switch to the other technique--not because we tell ourselves that it will work
this way too, but because we feel the new technique as identical with the old one, because we have to give it the
same sense, because we recognize it as the same just as we recognize this colour as green.
That is to say: insight into mathematical relations has a role similar to that of seeing an identity. It might
almost be said to be a more complicated kind of identity.
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It might be said: the reasons why we now shift to a different technique are of the same kind as those which
make us carry out a new multiplication as we do; we accept the technique as the same as we have applied in doing
other multiplications.
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37. A human being is imprisoned in a room, if the door is unlocked but opens inwards; he, however, never
gets the idea of pulling instead of pushing against it.
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38. When white turns black some people say "Essentially it is still the same"; and others, when the colour
turns a shade darker: "It is completely different".
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39. The proposition 'a = a', 'p ⊃ p', "The word 'Bismarck' has 8 letters", "There is no such thing as
reddish-green", are all obvious and are propositions about essence: what have they in common? They are evidently
each of a different kind and differently used. The last but one is the most like an empirical proposition. And it can
understandably be called a synthetic a priori proposition.
It can be said: unless you put the series of numbers and the series of letters side by side, you cannot know
how many letters the word has.
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40. One pattern derived from another according to a rule. (Say the reversal of a theme.)
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Then the result put as equivalent to the operation.
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41. When I wrote "proof must be perspicuous" that meant: causality plays no part in the proof.
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Or again: a proof must be capable of being reproduced by mere copying.
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42. That, if you go on dividing 1 : 3, you must keep on getting 3 in the result is not known by intuition, any
more than that the multiplication 25 × 25 yields the same product every time it is repeated.
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43. It might perhaps be said that the synthetic character of the propositions of mathematics appears most
obviously in the unpredictable occurrence of the prime numbers.
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But their being synthetic (in this sense) does not make them any the less a priori. They could be said, I want
to say, not to be got out of their concepts by means of some kind of analysis, but really to determine a concept by
synthesis, e.g. as crossing prisms can be made to determine a body.
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The distribution of primes would be an ideal example of what could be called synthetic a priori, for one can
say that it is at any rate not discoverable by an analysis of the concept of a prime number.
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44. Might one not really talk of intuition in mathematics? Though it would not be a mathematical truth that
was grasped intuitively, but a physical or psychological one. In this way I know with great certainty that if I multiply
25 by 25 ten times I shall get 625 every time. That is to say I know the psychological fact that this calculation will
keep on seeming correct to me; as I know that if I write down the series of numbers from 1 to 20 ten times my lists
will prove identical on collation.--Now is that an empirical fact? Of course--and yet it would be difficult to mention
experiments that would convince me of it. Such a thing might be called an intuitively known empirical fact.
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45. You want to say that every new proof alters the concept of proof in one way or another.
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But then by what principle is something recognized as a new proof? Or rather there is certainly no 'principle'
here.
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46. Now ought I to say: "we are convinced that the same result will always come out"? No, that is not
enough. We are convinced that the same calculation will always come out, be calculated. Now is that a
mathematical conviction? No--for if it were not always the same that was calculated, we could not conclude that the
calculation yields at one time one result and at another time another.
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We are of course also convinced that when we repeat a calculation we shall repeat the pattern of the
calculation.--
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47. Might I not say: if you do a multiplication, in any case you do not find the mathematical fact, but you do
find the mathematical proposition? For what you find is the non-mathematical fact, and in this way the
mathematical proposition. For a mathematical proposition is the determination of a concept following upon a
discovery.
Page 248
You find a new physiognomy. Now you can e.g. memorize or copy it.
Page 248
A new form has been found, constructed. But it is used to give a new concept together with the old one.
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The concept is altered so that this had to be the result.
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I find, not the result, but that I reach it.
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And it is not this route's beginning here and ending here that is an empirical fact, but my having gone this
road, or some road to this end.
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48. But might it not be said that the rules lead this way, even if no one went it?
Page 249
For that is what one would like to say--and here we see the mathematical machine, which, driven by the rules
themselves, obeys only mathematical laws and not physical ones.
Page 249
I want to say: the working of the mathematical machine is only the picture of the working of a machine.
Page 249
The rule does not do work, for whatever happens according to the rule is an interpretation of the rule.
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49. Let us suppose that I have the stages of the movement of
in a picture in front of me; then this enables me to form a proposition, which I as it were read off from this picture.
The proposition contains the word "roughly" and is a proposition of geometry.
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It is queer that I should be able to read off a proposition from a picture.
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The proposition, however, does not treat of the picture that I see. It does not say that such-and-such can be
seen in this picture. But nor does it say what the actual mechanism will do, although it suggests it.
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But could I draw the movement of the mechanism in other ways too, if its parts do not alter? That is to say,
am I not compelled, under these conditions, to accept just this as the picture of the movement?
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Let us imagine the construction of the phases of the mechanism carried out with lines of changing colour.
Let the lines be partly black on a white ground, partly white on a black ground. Imagine the constructions in Euclid
carried out in this way; they will lose all obviousness.
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50. A word in reverse has a new face.
Page 250
What if it were said: If you reverse the sequence 1 2 3, you learn about it that it yields 3 2 1 when reversed?
And what you learn is not a property of these ink-marks, but of the sequence of forms. You learn a formal property
of forms. The proposition asserting this formal property is proved by experience, which shews you the one form
arising in this way out of the other.
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Now, if you learn this, do you have two impressions? One of the fact that the sequence is reversed, the other
of the fact that 3 2 1 arises? And could you not have the experience, the impression, of 1 2 3's being reversed and yet
not of 3 2 1's arising? Perhaps it will be said: "Only by a queer illusion".--
Page 251
The reason why one really cannot say that one learns that formal proposition from experience is--that one
only calls it this experience when this process leads to this result. The experience meant consists as such of this
process with this result.
Page 251
That is why it is more than the experience: seeing a pattern.
Page 251
Can one row of letters have two reverses?
Say one acoustic, and another, optical, reverse. Suppose I explain to someone what the reverse of a word on
paper is, what we call that. And now it turns out that he has an acoustic reverse of the word, i.e., something that he
would like to call that, but it does not quite agree with the written reverse. So that one can say: he hears this as the
reverse of the word. As if, as it were, the word got distorted for him in being turned round. And this might perhaps
occur if he pronounced the word and its reverse fluently, as opposed to the case of spelling it out. Or the reverse
might seem different when he spoke the word forwards and backwards in a single utterance.
Page 251
It might be that the exact mirror-image of a profile, seen immediately
Page Break 252
after it, was never pronounced to be the same thing, merely turned in the other direction; but that in order to give the
impression of exact reversal, the profile had to be altered a little in its measurements.
Page 252
But I want to say that we have no right to say: though we may indeed be in doubt about the correct reverse
of, for example, a long word, still we know that the word has only one reverse.
Page 252
"Yes, but if it is supposed to be a reverse in this sense there can be only one." Does 'in this sense' here mean:
by these rules, or: with this physiognomy? In the first case the proposition would be tautological, in the second it
need not be true.
Page 252
51. Think of a machine which 'is so constructed' that it reverses a row of letters. And now of the proposition
that in the case of
O V E R
the result is
R E V O.--
Page 252
The rule, as it is actually meant, seems to be a driving power which reverses an ideal sequence like
this,--whatever a human being may do with an actual sequence.
This is the mechanism which is the yardstick, the ideal, for the actual mechanism.
Page Break 253
Page 253
And that is intelligible. For if the result of the reversal becomes the criterion for the row's really having been
reversed, and if we express this as our imitating an ideal machine, then this machine must produce this result
infallibly.
Page 253
52. Now can it be said that the concepts which mathematics produces are a convenience, that essentially we
could do without them?
Page 253
First and foremost the adoption of these concepts expresses the sure expectation of certain experiences.
Page 253
We do not accept e.g. a multiplication's not yielding the same result every time.
Page 253
And what we expect with certainty is essential to our whole life.
Page 253
53. Why, then, should I not say that mathematical propositions just express those special expectations, i.e.,
therefore, that they express matters of experience? Only because they just do not. Perhaps I should not take the
measure of adopting a certain concept if I did not quite definitely expect the occurrence of certain facts; but for that
reason laying down this measure and expressing the expectations are not equivalent.
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Page 254
54. It is difficult to put the body of fact right side up: to regard the given as given. It is difficult to place the
body differently from the way one is accustomed to see it. A table in a lumber room may always lie upside down, in
order to save space perhaps. Thus I have always seen the body of fact placed like this, for reasons of various kinds;
and now I am supposed to see something else as its beginning and something else as its end. That is difficult. It as it
were will not stand like that, unless one supports it in this position by means of other contrivances.
Page 254
55. It is one thing to use a mathematical technique consisting in the avoidance of contradiction, and another
to philosophize against contradiction in mathematics.
Page 254
56. Contradiction. Why just this one bogy? That is surely very suspicious.
Page 254
Why should not a calculation made for a practical purpose, with a contradictory result, tell me: "Do as you
please, I, the calculation, do not decide the matter"?
Page 254
The contradiction might be conceived as a hint from the gods that I am to act and not consider.
Page 254
57. "Why should contradiction be disallowed in mathematics?" Well, why is it not allowed in our simple
language-games? (There is
Page Break 255
certainly a connexion here.) Is this then a fundamental law governing all thinkable language-games?
Page 255
Let us suppose that a contradiction in an order, e.g. produces astonishment and indecision--and now we say:
that is just the purpose of contradiction in this language-game.
Page 255
58. Someone comes to people and says: "I always lie". They answer: "Well, in that case we can trust
you!"--But could he mean what he said? Is there not a feeling of being incapable of saying something really true; let
it be what it may?--
Page 255
"I always lie!"--Well, and what about that?--"It was a lie too!"--But in that case you don't always lie!--"No,
it's all lies!"
Perhaps we should say of this man that he doesn't mean the same thing as we do by "true" and by "lying".
He means perhaps something like: What he says flickers; or nothing really comes from his heart.
Page 255
It might also be said: his "I always lie" was not really an assertion. It was rather an exclamation.
Page 255
And so it can be said: "If he was saying that sentence, not thoughtlessly--then he must have meant the words
in such-and-such a way, he cannot have meant them in the usual way"?
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Page 256
59. Why should Russell's contradiction not be conceived as something supra-propositional, something that
towers above the propositions and looks in both directions like a Janus head? N.B. the proposition F(F)--in which
F(ξ) = ~ξ(ξ)--contains no variables and so might hold as something supra-logical, as something unassailable, whose
negation itself in turn only asserts it. Might one not even begin logic with this contradiction? And as it were descend
from it to propositions.
Page 256
The proposition that contradicts itself would stand like a monument (with a Janus head) over the
propositions of logic.
Page 256
60. The pernicious thing is not: to produce a contradiction in the region in which neither the consistent nor
the contradictory proposition has any kind of work to accomplish; no, what is pernicious is: not to know how one
reached the place where contradiction no longer does any harm.
Page Break 257
PART V
1942-1944
Page 257
1. It is of course clear that the mathematician, in so far as he really is 'playing a game' does not infer. For here
'playing' must mean: acting in accordance with certain rules. And it would already be something outside the mere
game for him to infer that he could act in this way according to the general rule.
Page 257
2. Does a calculating machine calculate?
Page 257
Imagine that a calculating machine had come into existence by accident; now someone accidentally presses
its knobs (or an animal walks over it) and it calculates the product 25 × 20.
Page 257
I want to say: it is essential to mathematics that its signs are also employed in mufti.
It is the use outside mathematics, and so the meaning of the signs, that makes the sign-game into
mathematics.
Page 257
Just as it is not logical inference either, for me to make a change from one formation to another (say from
one arrangement of chairs to another) if these arrangements have not a linguistic function apart from this
transformation.
Page 257
3. But is it not true that someone with no idea of the meaning
Page Break 258
of Russell's symbols could work over Russell's proofs? And so could in an important sense test whether they were
right or wrong?
Page 258
A human calculating machine might be trained so that when the rules of inference were shewn it and perhaps
exemplified, it read through the proofs of a mathematical system (say that of Russell), and nodded its head after
every correctly drawn conclusion, but shook its head at a mistake and stopped calculating. One could imagine this
creature as otherwise perfectly imbecile.
Page 258
We call a proof something that can be worked over, but can also be copied.
Page 258
4. If mathematics is a game, then playing some game is doing mathematics, and in that case why isn't
dancing mathematics too?
Page 258
Imagine that calculating machines occurred in nature, but that people could not pierce their cases. And now
suppose that these people use these appliances, say as we use calculation, though of that they know nothing. Thus
e.g. they make predictions with the aid of calculating machines, but for them manipulating these queer objects is
experimenting.
Page 258
These people lack concepts which we have; but what takes their place?
Page Break 259
Page 259
Think of the mechanism whose movement we saw as a geometrical (kinematic) proof clearly it would not
normally be said of someone turning the wheel that he was proving something. Isn't it the same with someone who
makes and changes arrangements of signs as a game; even when what he produces could be seen as a proof?
Page 259
To say mathematics is a game is supposed to mean: in proving, we need never appeal to the meaning of the
signs, that is to their extra-mathematical application. But then what does appealing to this mean at all? How can such
an appeal be of any avail?
Page 259
Does it mean passing out of mathematics and returning to it again, or does it mean passing from one method
of mathematical inference to another?
Page 259
What does it mean to obtain a new concept of the surface of a sphere? How is it then a concept of the
surface of a sphere? Only in so far as it can be applied to real spheres.
Page 259
How far does one need to have a concept of 'proposition', in order to understand Russellian mathematical
logic?
Page 259
5. If the intended application of mathematics is essential, how about parts of mathematics whose
application--or at least what
Page Break 260
mathematicians take for their application--is quite fantastic? So that, as in set theory, one is doing a branch of
mathematics of whose application one forms an entirely false idea. Now, isn't one doing mathematics none the less?
Page 260
If the operations of arithmetic only served to construct a cipher, its application would of course be
fundamentally different from that of our arithmetic. But would these operations then be mathematical operations at
all?
Page 260
Can someone who is applying a decoding rule be said to be performing mathematical operations? And yet
his transformations can be so conceived. For he could surely say that he was calculating what had to come out in
decoding the symbols... with such-and-such a key. And the proposition: the signs..., decoded according to this rule,
yield... is a mathematical one. As is the proposition that you can get to this position from that one in chess.
Page 260
Imagine the geometry of four-dimensional space done with a view to learning about the living conditions of
spirits. Does that mean that it is not mathematics? And can I now say that it determines concepts?
Page 260
Would it not sound queer to say that a child could already do thousands and thousands of
multiplications--by which is supposed to be meant that it can already calculate in the unlimited number domain. And
indeed this might be reckoned an extremely modest way of putting it, as it says only 'thousands and thousands'
instead of 'infinitely many'.
Page Break 261
Page 261
Could people be imagined, who in their ordinary lives only calculated up to 1000 and kept calculations with
higher numbers for mathematical investigations about the world of spirits?
Page 261
"Whether or not this holds of the surface of a real sphere--it does hold for the mathematical one"--this makes
it look as if the special difference between the mathematical and an empirical proposition was that, while the truth of
the empirical proposition is rough and oscillating, the mathematical proposition describes its object precisely and
absolutely. As if, in fact, the 'mathematical sphere' were a sphere. And it might e.g. be asked whether there was only
one such sphere, or several (a Fregean question).
Page 261
Does a misunderstanding about the possible application constitute an objection to the calculation as a part of
mathematics?
Page 261
And apart from misunderstanding,--what about mere lack of clarity?
Page 261
Imagine someone who believes that mathematicians have discovered a queer thing, , which
when squared does yield - 1, can't he nevertheless calculate quite well with complex numbers, and apply such
calculations in physics? And does this make them any the less calculations?
In one respect of course his understanding has a weak foundation; but he will draw his conclusions with
certainty, and his calculus will have a solid foundation.
Page Break 262
Page 262
Now would it not be ridiculous to say this man wasn't doing mathematics?
Page 262
Someone makes an addition to mathematics, gives new definitions and discovers new theorems--and in a
certain respect he can be said not to know what he is doing.--He has a vague imagination of having discovered
something like a space (at which point he thinks of a room), of having opened up a kingdom, and when asked about
it he would talk a great deal of nonsense.
Page 262
Let us imagine the primitive case of someone carrying out enormous multiplications in order, as he says, to
conquer gigantic new provinces of the domain of numbers.
Page 262
Imagine calculating with invented by a madman, who, attracted merely by the paradox of the
idea, does the calculation as a kind of service, or temple ritual, of the absurd. He imagines that he is writing down the
impossible and operating with it.
Page 262
In other words: if someone believes in mathematical objects and their queer properties--can't he nevertheless
do mathematics? Or--isn't he also doing mathematics?
Page 262
'Ideal object.' "The symbol 'a' stands for an ideal object" is evidently supposed to assert something about the
meaning, and so about the use, of 'a'. And it means of course that this use is in a certain respect
Page Break 263
similar to that of a sign that has an object, and that it does not stand for any object. But it is interesting what the
expression 'ideal object' makes of this fact.
Page 263
6. In certain circumstances we might speak of an endless row of marbles.--Let us imagine such an endless
straight row of marbles at equal distances from one another; we calculate the force exerted by all these marbles on a
certain body according to a certain law of attraction. We regard the number yielded by this calculation as the ideal of
exactness for certain measurements.
Page 263
The feeling of something queer here comes from a misunderstanding. The kind of misunderstanding that is
produced by a thumb-catching of the intellect--to which I want to call a halt.
Page 263
The objection that 'the finite cannot grasp the infinite' is really directed against the idea of a psychological act
of grasping or understanding.
Page 263
Or imagine that we simply say: "This force corresponds to the attraction of an endless row of marbles which
we have arranged in such-and-such a way and which attract the body according to such-and-such a law of
attraction". Or again: "Calculate the force which an endless row of marbles of such-and-such a kind exerts on the
body".--It certainly makes sense to give such an order. It describes a particular calculation.
Page Break 264
Page 264
What about the following question: "Calculate the weight of a pillar composed of as many slabs lying on top
of one another as there are cardinal numbers; the undermost slab weighs 1 kg., and every higher one weighs half of
the one just below it".
Page 264
The difficulty is not that we can't form an image. It is easy enough to form some kind of image of an endless
row, for example. The question is what use the image is to us.
Page 264
Imagine infinite numbers used in a fairy tale. The dwarves have piled up as many gold pieces as there are
cardinal numbers--etc. What can occur in this fairy tale must surely make sense.--
Page 264
7. Imagine set theory's having been invented by a satirist as a kind of parody on mathematics.--Later a
reasonable meaning was seen in it and it was incorporated into mathematics. (For if one person†1 can see it as a
paradise of mathematicians, why should not another see it as a joke?)
Page 264
The question is: even as a joke isn't it evidently mathematics?--
Page Break 265
Page 265
And why is it evidently mathematics?--Because it is a game with signs according to rules?
Page 265
But isn't it evident that there are concepts formed here--even if we are not clear about their application?
But how is it possible to have a concept and not be clear about its application?
Page 265
8. Take the construction of the polygon of forces: isn't that a bit of applied mathematics? And where is the
proposition of pure mathematics which is invoked in connexion with this graphical calculation? Is this case not like
that of the tribe which has a technique of calculating in order to make certain predictions, but no propositions of
pure mathematics?
Page 265
Calculation that belongs to the performance of a ceremony. For example, let the number of words in a form
of blessing that is to be applied to a home be derived by a particular technique from the ages of the father and
mother and the number of their children. We could imagine procedures of calculating described in such a law as the
Mosaic law. And couldn't we imagine that the nation with these ceremonial prescriptions for calculating never
calculated in practical life?
Page 265
This would indeed be a case of applied calculation, but it would not serve the purpose of a prediction.
Page Break 266
Page 266
Would it be any wonder if the technique of calculating had a family of applications?
Page 266
9. We only see how queer the question is whether the pattern φ (a particular arrangement of digits e.g. '770')
will occur in the infinite expansion of π, when we try to formulate the question in a quite common or garden way:
men have been trained to put down signs according to certain rules. Now they proceed according to this training and
we say that it is a problem whether they will ever write down the pattern φ in following the given rule.
Page 266
But what are you saying if you say that one thing is clear: either one will come on φ in the infinite expansion,
or one will not?
Page 266
It seems to me that in saying this you are yourself setting up a rule or postulate.
Page 266
What if someone were to reply to a question: 'So far there is no such thing as an answer to this question'?
Page 266
So, e.g., the poet might reply when asked whether the hero of his poem has a sister or not--when, that is, he
has not yet decided anything about it.
Page 266
The question--I want to say--changes its status, when it becomes
Page Break 267
decidable. For a connexion is made then, which formerly was not there.
Page 267
Of someone who is trained we can ask 'How will he interpret the rule for this case?', or again 'How ought he
to interpret the rule for this case?'--but what if no decision about this question has been made?--Well, then the
answer is, not: 'he ought to interpret it in such a way that φ occurs in the expansion' or: 'he ought to interpret it in
such a way that it does not occur', but: 'nothing has so far been decided about this'.
Page 267
However queer it sounds, the further expansion of an irrational number is a further expansion of
mathematics.
Page 267
We do mathematics with concepts.--And with certain concepts more than with other ones.
Page 267
I want to say: it looks as if a ground for the decision were already there; and it has yet to be invented.
Page 267
Would this come to the same thing as saying: in thinking about the technique of expansion, which we have
learnt, we use the false picture of a completed expansion (of what is ordinarily called a "row") and this forces us to
ask unanswerable questions?
Page Break 268
Page 268
For after all in the end every question about the expansion of must be capable of formulation as a
practical question concerning the technique of expansion.
Page 268
And what is in question here is of course not merely the case of the expansion of a real number, or in general
the production of mathematical signs, but every analogous process, whether it is a game, a dance, etc., etc..
Page 268
10. When someone hammers away at us with the law of excluded middle as something which cannot be
gainsaid, it is clear that there is something wrong with his question.
Page 268
When someone sets up the law of excluded middle, he is as it were putting two pictures before us to choose
from, and saying that one must correspond to the fact. But what if it is questionable whether the pictures can be
applied here?
Page 268
And if you say that the infinite expansion must contain the pattern φ or not contain it, you are so to speak
shewing us the picture of an unsurveyable series reaching into the distance.
Page 268
But what if the picture began to flicker in the far distance?
Page 268
11. To say of an unending series that it does not contain a particular
Page Break 269
pattern makes sense only under quite special conditions.
Page 269
That is to say: this proposition has been given a sense for certain cases.
Page 269
Roughly, for those where it is in the rule for this series, not to contain the pattern....
Further: when I calculate the expansion further, I am deriving new rules which the series obeys.
Page 269
"Good,--then we can say: 'It must either reside in the rule for this series that the pattern occurs, or the
opposite'." But is it like that?--"Well, doesn't the rule of expansion determine the series completely? And if it does
so, if it allows of no ambiguity, then it must implicitly determine all questions about the structure of the
series."--Here you are thinking of finite series.
Page 269
"But surely all members of the series from the 1st up to 1,000th, up to the 1010-th and so on, are determined;
so surely all the members are determined." That is correct if it is supposed to mean that it is not the case that e.g. the
so-and-so-many'th is not determined. But you can see that that gives you no information about whether a particular
pattern is going to appear in the series (if it has not appeared so far). And so we can see that we are using a
misleading picture.
Page Break 270
Page 270
If you want to know more about the series, you have, so to speak, to get into another dimension (as it were
from the line into a surrounding plane).--But then isn't the plane there, just like the line, and merely something to be
explored, if one wants to know what the facts are?
No, the mathematics of this further dimension has to be invented just as much as any mathematics.
Page 270
In an arithmetic in which one does not count further than 5 the question what 4 + 3 makes doesn't yet make
sense. On the other hand the problem may very well exist of giving this question a sense. That is to say: the question
makes no more sense than does the law of excluded middle in application to it.
Page 270
12. In the law of excluded middle we think that we have already got something solid, something that at any
rate cannot be called in doubt. Whereas in truth this tautology has just as shaky a sense (if I may put it like that), as
the question whether p or ~ p is the case.
Page 270
Suppose I were to ask: what is meant by saying "the pattern... occurs in this expansion"? The reply would be:
"you surely know what it means. It occurs as the pattern... in fact occurs in the expansion."--So that is the way it
occurs?--But what way is that?
Imagine it were said: "Either it occurs in that way, or it does not occur in that way"!
Page 270
"But don't you really understand what is meant?"--But may I not believe I understand it, and be wrong?--
Page Break 271
Page 271
For how do I know what it means to say: the pattern... occurs in the expansion? Surely by way of
examples--which shew me what it is like for.... But these examples do not shew me what it is like for this pattern to
occur in the expansion!
Page 271
Might one not say: if I really had a right to say that these examples tell me what it is like for the pattern to
occur in the expansion, then they would also have to shew me what the opposite means.
Page 271
13. The general proposition that that pattern does not occur in the expansion can only be a commandment.
Page 271
Suppose we look at mathematical propositions as commandments, and even utter them as such? "Let 252 be
625."
Well--a commandment has an internal and an external negation.
Page 271
The symbols "(x).φx" and "(∃x).φx" are certainly useful in mathematics so long as one is acquainted with the
technique of the proofs of the existence or non-existence to which the Russellian signs here refer. If however this is
left open, then these concepts of the old logic are extremely misleading.
Page 271
If someone says: "But you surely know what 'this pattern occurs in the expansion' means, namely this"--and
points to a case of occurring,--then I can only reply that what he shews me is capable of illustrating a variety of
facts. For that reason I can't be said to know what the proposition means just from knowing that he will certainly use
it in this case.
Page Break 272
Page 272
The opposite of "there exists a law that p" is not: "there exists a law that ~p". But if one expresses the first by
means of P, and the second by means of ~ P, one will get into difficulties.
Page 272
14. Suppose children are taught that the earth is an infinite flat surface; or that God created an infinite
number of stars; or that a star keeps on moving uniformly in a straight line, without ever stopping.
Queer: when one takes something of this sort as a matter of course, as it were in one's stride, it loses its whole
paradoxical aspect. It is as if I were to be told: Don't worry, this series, or movement, goes on without ever stopping.
We are as it were excused the labour of thinking of an end.
Page 272
'We won't bother about an end.'
Page 272
It might also be said: 'for us the series is infinite'.
Page 272
'We won't worry about an end to this series; for us it is always beyond our ken.'
Page 272
15. The rational numbers cannot be enumerated, because they cannot be counted--but one can count with
them, as with the cardinal numbers. That squint-eyed way of putting things goes with the whole system of pretence,
namely that by using the new apparatus we deal with
Page Break 273
infinite sets with the same certainty as hitherto we had in dealing with finite ones.
Page 273
It should not have been called 'denumerable', but on the other hand it would have made sense to say
'numberable'. And this expression also informs us of an application of the concept. For one cannot set out to
enumerate the rational numbers, but one can perfectly well set out to assign numbers to them.
Page 273
But where is the problem here? Why should I not say that what we call mathematics is a family of activities
with a family of purposes?
Page 273
People might for example use calculating as a kind of competitive activity. As children do sometimes have
races in doing sums; only this use of sums plays a quite subordinate role among us.
Page 273
Or multiplication might strike us as much more difficult than it does--if e.g. we only calculated orally, and in
order to take note of, and so to grasp, a multiplication it were necessary to bring it into the form of rhyming verse.
When someone succeeded in doing this he would have the feeling of having discovered a wonderful great truth.
Every new multiplication would require a new individual piece of work.
If these people believed that numbers were spirits and that they were
Page Break 274
exploring the domain of spirits by means of their calculations, or compelling the spirits to manifest
themselves--would this now be arithmetic? Again--would it be arithmetic even in the case where these people used
the calculations for nothing else?
Page 274
16. The comparison with alchemy seems natural. We might speak of a kind of alchemy in mathematics.
Page 274
Is it already mathematical alchemy, that mathematical propositions are regarded as statements about
mathematical objects,--and mathematics as the exploration of these objects?
Page 274
In a certain sense it is not possible to appeal to the meaning of the signs in mathematics, just because it is
only mathematics that gives them their meaning.
Page 274
What is typical of the phenomenon I am talking about is that a mysteriousness about some mathematical
concept is not straight away interpreted as an erroneous conception, as a mistake of ideas; but rather as something
that is at any rate not to be despised, is perhaps even rather to be respected.
Page 274
All that I can do, is to shew an easy escape from this obscurity and this glitter of the concepts.
Page Break 275
Page 275
Strangely, it can be said that there is so to speak a solid core to all these glistening concept-formations. And I
should like to say that that is what makes them into mathematical productions.
Page 275
It might be said: what you see does of course look more like a gleaming Fata Morgana; but look at it from
another quarter and you can see the solid body, which only looks like a gleam without a corporeal substrate when
seen from that other direction.
Page 275
17. 'The pattern is in the series or it is not in the series' means: either the thing looks like this or it does not
look like this.
Page 275
How does one know what is meant by the opposite of the proposition "φ occurs in the series", or even of the
proposition "φ does not occur in the series"? This question sounds like nonsense, but does make sense all the same.
Namely: how do I know that I understand the proposition "φ occurs in this series"?
True, I can give examples illustrating the use of such statements, and also of the opposite ones. And they are
examples of there being a rule prescribing the occurrence in a definite region or series of regions, or determining that
such an occurrence is excluded.
Page 275
If "you do it" means: you must do it, and "you do not do it" means: you must not do it--then "Either you do
it, or you do not" is not the law of excluded middle.
Page Break 276
Page 276
Everyone feels uncomfortable at the thought that a proposition can state that such-and-such does not occur
in an infinite series--while on the other hand there is nothing startling about a command's saying that this must not
occur in this series however far it is continued.
Page 276
But what is the source of this distinction between: "however far you go you will never find this"--and
"however far you go you must never do this"?
Page 276
On hearing the proposition one can ask: "how can we know anything like that?" but nothing analogous holds
for the command.
Page 276
The statement seems to overreach itself, the command not at all.
Page 276
Can we imagine all mathematical propositions expressed in the imperative? For example: "Let 10 × 10 be
100".
Page 276
And if you now say: "Let it be like this, or let it not be like this", you are not pronouncing the law of
excluded middle--but you are pronouncing a rule. (As I have already said above.)
Page 276
18. But is this really a way out of the difficulty? For how about all
Page Break 277
the other mathematical propositions, say '252 = 625'; isn't the law of excluded middle valid for these inside
mathematics?
Page 277
How is the law of excluded middle applied?
Page 277
"Either there is a rule that prescribes it, or one that forbids it."
Page 277
Assuming that there is no rule forbidding the occurrence,--why is there then supposed to be one that
prescribes it?
Page 277
Does it make sense to say: "While there isn't any rule forbidding the occurrence, as a matter of fact the
pattern does not occur"?--And if this does not make sense, how can the opposite make sense, namely, that the
pattern does occur?
Page 277
Well, when I say it occurs, a picture of the series from its beginning up to the pattern floats before my
mind--but if I say that the pattern does not occur, then no such picture is of any use to me, and my supply of
pictures gives out.
Page 277
What if the rule should bend in use without my noticing it? What I mean is, that I might speak of different
spaces in which I use it.
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Page 278
The opposite of "it must not occur" is "it can occur". For a finite segment of the series, however, the opposite
of "it must not occur in it" seems to be: "it must occur in it".
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The queer thing about the alternative "φ occurs in the infinite series or it does not", is that we have to imagine
the two possibilities individually, that we look for a distinct idea of each, and that one is not adequate for the
negative and for the positive case, as it is elsewhere.
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19. How do I know that the general proposition "There is..." makes sense here? Well, if it can be used to tell
something about the technique of expansion in a language game.
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In one case what we are told is: "it must not occur"--i.e.: if it occurs you calculated wrong.
In one case what we are told is: "it can occur", i.e., no such interdict exists. In another: "it must occur in
such-and-such a region (always in this place in these regions)". But the opposite of this seems to be: "it must not
occur in such-and-such places"--instead of "it need not occur there".
But what if the rule were given that, e.g., everywhere where the formation rule for π yields 4, any arbitrary
digit other than 4 can be put in its place?
Consider also the rule which forbids one digit in certain places, but otherwise leaves the choice open.
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Isn't it like this? The concepts of infinite decimals in mathematical
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propositions are not concepts of series, but of the unlimited technique of expansion of series.
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We learn an endless technique: that is to say, something is done for us first, and then we do it; we are told
rules and we do exercises in following them; perhaps some expression like "and so on ad inf." is also used, but what
is in question here is not some gigantic extension.
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These are the facts. And now what does it mean to say: "φ either occurs in the expansion, or does not
occur"?
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20. But does this mean that there is no such problem as: "Does the pattern φ occur in this expansion?"?--To
ask this is to ask for a rule regarding the occurrence of φ. And the alternative of the existence or non-existence of
such a rule is at any rate not a mathematical one.
Page 279
Only within a mathematical structure which has yet to be erected does the question allow of a mathematical
decision, and at the same time become a demand for such a decision.
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21. Then is infinity not actual--can I not say: "these two edges of the slab meet at infinity"?
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Page 280
Say, not: "the circle has this property because it passes through the two points at infinity..."; but: "the
properties of the circle can be regarded in this (extraordinary) perspective".
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It is essentially a perspective, and a far-fetched one. (Which does not express any reproach.) But it must
always be quite clear how far-fetched this way of looking at it is. For otherwise its real significance is dark.
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22. What does it mean to say: "the mathematician does not know what he is doing", or: "he knows what he is
doing"?
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23. Can one make infinite predictions?--Well, why should one not for example call the law of inertia one? Or
the proposition that a comet describes a parabola?
In a certain sense of course the infinity of the prediction is not taken very seriously.
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Now what about a prediction that if you expand π, however far you go, you will never come across the
pattern φ?--Well, we could say that this is either a non-mathematical prediction, or alternatively a mathematical rule.
Page 280
Someone who has learned to expand goes to a fortune-teller, and she tells him that however far he may
expand he will never arrive at the pattern....--Is her soothsaying a mathematical proposition?
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No.--Unless she says: "If you always expand correctly you will never reach it". But is that still a prediction?
Page 281
Now it looks as if such a prediction of the correct expansion were imaginable and were distinct from a
mathematical law that it must be thus and thus. So that in the mathematical expansion there would be a distinction
between what as a matter of fact comes out like this--as it were accidentally--and what must come out.
Page 281
How is it to be decided whether an infinite prediction makes sense?
Page 281
At any rate not by one's saying: "I am certain I mean something when I say...".
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Besides, the question is not so much whether the prediction makes some kind of sense, as: what kind of
sense it makes. (That is, in what language games it occurs.)
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24. "The disastrous invasion" of mathematics by logic.
Page 281
In a field that has been prepared in this way this is a proof of existence.
Page 281
The harmful thing about logical technique is that it makes us forget the special mathematical technique.
Whereas logical technique is only an auxiliary technique in mathematics. For example it sets up certain
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connexions between different techniques.
Page 282
It is almost as if one tried to say that cabinet-making consisted in glueing.
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25. A proof convinces you that there is a root of an equation (without giving you any idea where)--how do
you know that you understand the proposition that there is a root? How do you know that you are really convinced
of anything? You may be convinced that the application of the proved proposition will turn up. But you do not
understand the proposition so long as you have not found the application.
Page 282
When a proof proves in a general way that there is a root, then everything depends on the form in which it
proves this. On what it is that here leads to this verbal expression, which is a mere shadow, and keeps mum about
essentials. Whereas to logicians it seems to keep mum only about incidentals.
Page 282
Generality in mathematics does not stand to particularity in mathematics in the same way as the general to
the particular elsewhere.
Page 282
Everything that I say really amounts to this, that one can know a proof thoroughly and follow it step by step,
and yet at the same time not understand what it was that was proved.
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Page 283
And this in turn is connected with the fact that one can form a mathematical proposition in a grammatically
correct way without understanding its meaning.
Now when does one understand it?--I believe: when one can apply it.
It might perhaps be said: when one has a clear picture of its application. For this, however, it is not enough to
connect a clear picture with it. It would rather have been better to say: when one commands a clear view of its
application. And even that is bad, for the matter is simply one of not imagining that the application is where it is not;
of not being deceived by the verbal form of the proposition.
Page 283
But how does it come about that one can fail to understand, or can misunderstand, a proposition or proof in
this way? And what is then necessary in order to produce understanding?
Page 283
There are here, I believe, cases in which someone can indeed apply the proposition (or proof), but is unable
to give a clear account of the kind of application. And the case in which he is even unable to apply the proposition.
(Multiplicative axiom.)†1
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How is it as regards 0 × 0 = 0?
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Page 284
One would like to say that the understanding of a mathematical proposition is not guaranteed by its verbal
form, as is the case with most non-mathematical propositions. This means--so it appears--that the words don't
determine the language-game in which the proposition functions.
Page 284
The logical notation swallows the structure.
Page 284
26. In order to see how something can be called an 'existence-proof', though it does not permit a construction
of what exists, think of the different meanings of the word "where". (For example the topological and the metrical.)
Page 284
For it is not merely that the existence-proof can leave the place of the 'existent' undetermined: there need not
be any question of such a place.
That is to say: when the proved proposition runs: "there is a number for which..." then it need not make
sense to ask "and which number is it?", or to say "and this number is...".
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27. A proof that 777 occurs in the expansion of π, without shewing where, would have to look at this
expansion from a totally new point of view, so that it shewed e.g. properties of regions of the expansion about which
we only knew that they lay very far out. Only the picture floats before one's mind of having to assume as it were a
dark zone of indeterminate length very far on in π, where we can no longer rely on our devices for calculating; and
then still further out a zone where in a different way we can once more see something.
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Page 285
28. We can always imagine proof by reductio ad absurdum used in argument with someone who puts
forward a non-mathematical assertion (e.g. that he has seen a checkmate with such-and-such pieces) which can be
mathematically refuted.
Page 285
The difficulty which is felt in connexion with reductio ad absurdum in mathematics is this: what goes on in
this proof? Something mathematically absurd, and hence unmathematical? How--one would like to ask--can one so
much as assume the mathematically absurd at all? That I can assume what is physically false and reduce it ad
absurdum gives me no difficulty. But how to think the--so to speak--unthinkable?
Page 285
What an indirect proof says, however, is: "If you want this then you cannot assume that: for only the
opposite of what you do not want to abandon would be combinable with that".
Page 285
29. The geometrical illustration of Analysis is indeed inessential; not, however, the geometrical application.
Originally the geometrical illustrations were applications of Analysis. Where they cease to be this they can be
wholly misleading.
What we have then is the imaginary application. The fanciful application.
The idea of a 'cut' is one such dangerous illustration.
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Page 286
Only in so far as the illustrations are also applications do they avoid producing that special feeling of
dizziness which the illustration produces in the moment at which it ceases to be a possible application; when, that is,
it becomes stupid.
Page 286
30. Dedekind's theorem could be derived, if what we call irrational numbers were quite unknown, but if there
were a technique of deciding the places of decimals by throwing dice. And this theorem would then have its
application even if the mathematics of irrational numbers did not exist. It is not as if Dedekind's expansions already
foresaw all the special real numbers. It merely looks like that as soon as Dedekind's calculus is joined to the calculi of
the special real numbers.
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31. It might be asked: what is there about the proof of Dedekind's theorem that a child 10 years old could not
understand?--For isn't this proof far simpler than all the calculations which the child has to master?--And if now
someone were to say: it can't understand the deeper content of the proposition--then I ask: how does this
proposition come to have a deep content?
Page 286
32. The picture of the number line is an absolutely natural one up to a certain point; that is to say so long as it
is not used for a general theory of real numbers.
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33. If you want to divide the real numbers into an upper and lower class, then do it first crudely by means of
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two rational points P and Q. Then halve PQ and decide in which half (if not at the point of division) the cut is
supposed to lie; if for example in the lower one, halve this and make a more exact decision and so on.
If you have a principle for unlimited repetition of this procedure then you can say that this principle executes
a cut, as it decides for each number whether it lies to the right or to the left.--Now the question is whether I can go all
the way by means of such a principle of division, or whether some other way of deciding is still needed; and again,
whether this would be after finishing the use of the principle, or before. Now in any case, not before the completion;
for so long as the question still is, in which finite bit of the straight line the point is supposed to lie, further division
may decide the matter.--But after the decision by a principle is there still room for a further decision?
Page 287
It is the same with Dedekind's theorem as with the law of excluded middle: it seems to exclude a third
possibility, whereas a third possibility is not in question here.
Page 287
The proof of Dedekind's theorem works with a picture which cannot justify it; which ought rather to be
justified by the theorem.
Page 287
You readily see a principle of division as an unendingly repeated division, for at any rate it does not
correspond to any finite division and seems to lead you on and on.
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Page 288
34. Couldn't we make a more extensional preparation for the theory of limits, functions, real numbers, than
we do? Even if this preparatory calculus should seem very trivial and in itself useless?
Page 288
The difficulty of looking at the matter now in an intensional, now again in an extensional way, is already
there with the concept of a 'cut'. That every rational number can be called a principle of division of the rational
numbers is perfectly clear. Now we discover something else that we can call a principle of division, e.g. what
corresponds to . Then other similar ones--and now we are already quite familiar with the possibility of such
divisions, and see them under the aspect of a cut made somewhere along the straight line, hence extensionally. For if
I cut, I can of course choose where I want to cut.
But if a principle of division is a cut, it surely is so only because it is possible to say of any arbitrary rational
number that it is on one side or the other of the cut. Can the idea of a cut now be said to have led us from the
rational to the irrational numbers? Are we for example led to by way of the concept of a cut?
Now what is a cut of the real numbers? Well, a principle of division into an upper and a lower class. Thus
such a principle yields every rational and irrational number. For even if we have no system of irrational numbers, still
those that we have divide into upper and lower by reference to the cut (so far, that is, as they are comparable with
it).
But now Dedekind's idea is that the division into an upper and lower class (under the known conditions) is
the real number.
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The cut is an extensional image.
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Page 289
It is of course true that, if I have a mathematical criterion for establishing, for any arbitrary rational number,
whether it belongs to the upper or the lower class, then it is easy for me systematically to approximate as close as I
like to the place where the two classes meet.
Page 289
In Dedekind we do not make a cut by cutting, i.e. pointing to the place, but--as in finding --by
approaching the adjacent ends of the upper and the lower class.
Page 289
The thing now is to prove that no other numbers except the real numbers can perform such a cut.
Page 289
Let us not forget that the division of the rational numbers into two classes did not originally have any
meaning, until we drew attention to a particular thing that could be so described. The concept is taken over from the
everyday use of language and that is why it immediately looks as if it had to have a meaning for numbers too.
Page 289
When the idea of a cut of the real numbers is now introduced by saying that we simply have to extend the
concept of a cut of the rational numbers to the real numbers--all that we need is a property dividing the real numbers
into two classes (etc.)--then first of all it is not clear what is meant by such a property, which thus divides all real
numbers. Now our attention can be drawn to the fact that any real number can serve this purpose. But that gets us
only so far and no further.
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Page 290
35. The extensional definitions of functions, of real numbers etc. pass over--although they
presuppose--everything intensional, and refer to the ever-recurring outward form.
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36. Our difficulty really already begins with the infinite straight line; although we learn even as children that a
straight line has no end, and I do not know that this idea has ever given anyone any difficulty. Suppose a finitist
were to try to replace this concept by that of a straight segment of definite length?!
But the straight line is a law for producing further.
Page 290
The concept of the limit and of continuity, as they are introduced nowadays, depend, without its being said,
on the concept of proof. For we say
= 1 when it can be proved that...
This means we use concepts which are infinitely harder to grasp than those that we make explicit.
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37. The misleading thing about Dedekind's conception is the idea that the real numbers are there spread out
in the number line. They may be known or not; that does not matter. And in this way all that one needs to do is to
cut or divide into classes, and one has dealt with them all.
Page 290
It is by combining calculation and construction that one gets the idea that there must be a point left out on
the straight line, namely P,
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if one does not admit as a measure of distance from O. 'For, if I were to construct really accurately, then the
circle would have to cut the straight line between its points.'
Page 291
This is a frightfully confusing picture.
Page 291
The irrational numbers are--so to speak--special cases.
Page 291
What is the application of the concept of a straight line in which a point is missing?! The application must be
'common or garden'. The expression "straight line with a point missing" is a fearfully misleading picture. The
yawning gulf between illustration and application.
Page 291
38. The generality of functions is so to speak an unordered generality. And our mathematics is built up on
such an unordered generality.
Page 291
39. If one imagines the general calculus of functions without the existence of examples, then the vague
explanations by means of value-tables
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and diagrams, such as are found in the textbooks, are in place as indications of how e.g. a sense might sometime be
given to this calculus.
Page 292
Imagine someone's saying: "I want to hear a composition which goes like this":
Page 292
Would that necessarily be senseless? Couldn't there be a composition whose correspondence to this line, in
some important sense, could be shewn?
Page 292
Or suppose one looked at continuity as a property of the sign 'x2 + y2 = z2'--of course only if this equation
and others were ordinarily subjected to a known method of testing. 'This is the relation of this rule (equation) to this
particular test.' A test, which goes with a sidelong glance at a kind of extension.
Page 292
In this test of the equation something is undertaken which is connected with certain extensions. Though not
as if what were in question here were an extension which would be somehow equivalent to the equation itself. It is
just that certain extensions are, so to speak, alluded to.--The real thing here is not the extension, which is only faute
de mieux described intensionally; rather is the intension described--or
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presented--by means of certain extensions, which are yielded by it then and there.
Page 293
The range of certain extensions casts a sidelight on the algebraic property of the function. In this sense, then,
the drawing of a hyperbola could be said to cast a sidelight on the equation of the hyperbola.
Page 293
It is no contradiction of this for those extensions to be the most important application of the rule; for it is one
thing to draw an ellipse, and another to construct it by means of its equation.--
Page 293
Suppose I were to say: extensional considerations (for example the Heine-Borel theorem) shew: This is how
to deal with intensions.
Page 293
The theorem gives us the main features of a method of proceeding with intensions. It says e.g.: 'this is what it
will have to be like'.
Page 293
And it will then be possible to attach a diagram as a particular illustration, e.g. to a procedure with particular
intensions. The illustration is a sign, a description, which is particularly easy to take in, particularly memorable.
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To give the illustration here will in fact be to give a procedure.
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Page 294
A theory of the placing of figures in a picture (a painting),--say on general aesthetic grounds--apart from
whether these figures are engaged in fighting, or love-making etc..
Page 294
The theory of functions as a schema, into which on the one hand a host of examples fits, and which on the
other hand is there as a standard for the classification of cases.
Page 294
The misleading thing about the usual account consists in its looking as if the general account could be quite
understood even without examples, without a thought of intensions (in the plural), since really everything could be
managed extensionally, if that were not impossible for external reasons.
Page 294
Compare the two forms of definition: "We say
= L when it can be shewn that..."
and
= L means: for every ε there is a δ..."
Page 294
40. Dedekind gives a general pattern of expression; so to speak a logical form of reasoning.
A general formulation of a procedure. The effect is similar to that of introducing the word "correlation" with a
view to the general definition of functions. A general way of talking is introduced, which is very useful for the
characterization of a mathematical procedure (as in Aristotelian logic). But the danger is that one will think one is in
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possession of the complete explanation of the individual cases when one has this general way of talking (the same
danger as in logic).
Page 295
We determine the concept of the rule for the construction of a non-terminating decimal further and further.
But the content of the concept?!--Well, can we not complete the construction of the concept as a receptacle
for whatever application may turn up? May I not complete the construction of the form (the form for which some
content has supplied me with the stimulus) and as it were prepare a form of language for possible employment? For,
so long as it remains empty, the form will contribute to determining the form of mathematics.
Page 295
For isn't the subject-predicate form open in this way; and waiting for the most various new applications?
Page 295
That is to say: is it true that the whole difficulty about the generality of the concept of a mathematical
function is already to be found in Aristotelian logic, since we can no more survey the generality of propositions and
of predicates than that of mathematical functions?
Page 295
41. Concepts which occur in 'necessary' propositions must also occur and have a meaning in non-necessary
ones.
Page 295
42. Would one say that someone understood the proposition '563 + 437 = 1000' if he did not know how it
can be proved? Can
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one deny that it is a sign of understanding a proposition, if a man knows how it could be proved?
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The problem of finding a mathematical decision of a theorem might with some justice be called the problem
of giving mathematical sense to a formula.
Page 296
An equation links two concepts; so that I can now pass from one to the other.
Page 296
An equation constructs a conceptual path. But is a conceptual path a concept? And if not, is there a sharp
distinction between them?
Page 296
Imagine that you have taught someone a technique of multiplying. He uses it in a language-game. In order
not to have to keep on multiplying afresh, he writes the multiplication in an abbreviated form as an equation, and he
uses this where he multiplied before.
Now he says that the technique of multiplying establishes connexions between the concepts. He will also say
the same thing about a multiplication as a picture of this transition. And finally he will say the same thing about the
equation itself: for it is essential that the transition should be capable of being represented simply by the pattern of
the equation. That is, that the transition should not always have to be made anew.
Now will he also be inclined to say that the process of multiplying is a concept?
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Page 297
It is surely a movement. It seems to be a movement between two stationary points; these are the concepts.
Page 297
If I conceive a proof as my movement from one concept to another, then I shall not want also to say that it is
a new concept. But can I not conceive the written multiplication as one picture, comparable to a number-sign, and
may not its functioning include functioning as a concept-sign?
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43. I should like to say: when we employ now the one, now the other side of the equation, we are employing
two sides of the same concept.
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44. Is the conceptual apparatus a concept?
Page 297
45. How does anyone shew that he understands a mathematical proposition? E.g. by applying it. So not also
by proving it?
Page 297
I should like to say: the proof shews me a new connexion, and hence it also gives me a new concept.
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Is not the new concept the proof itself?
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Page 298
But a proof certainly does enable you to form a new judgment. For you can after all say of a particular
pattern that it is or is not this proof.
Page 298
Yes, but is the proof, regarded, interpreted, as a proof, a pattern? As a proof, I might say, it has to convince
me of something. In consequence of it I will do or not do something. And in consequence of a new concept I don't
do or not do anything. So I want to say: the proof is the pattern of proof employed in a particular way.
And what it convinces me of can be of very various kinds. (Think of the proofs of Russellian tautologies, or
proofs in geometry and in algebra.)
Page 298
A mechanism can convince me of something (can prove something). But under what circumstances--in the
context of what activities and problems--shall I say that it convinces me of something?
Page 298
"But a concept surely does not convince me of anything, for it does not shew me a fact."--But why should a
concept not first and foremost convince me that I want to use it?
Page 298
Why should not the new concept, once formed, immediately license my transition to a judgment?
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46. 'Understanding a mathematical proposition'--that is a very vague concept.
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Page 299
But if you say "The point isn't understanding at all. Mathematical propositions are only positions in a game"
that too is nonsense! 'Mathematics' is not a sharply delimited concept.
Page 299
Hence the issue whether an existence-proof which is not a construction is a real proof of existence. That is,
the question arises: Do I understand the proposition "There is..." when I have no possibility of finding where it
exists? And here there are two points of view: as an English sentence for example I understand it, so far, that is, as I
can explain it (and note how far my explanation goes). But what can I do with it? Well, not what I can do with a
constructive proof. And in so far as what I can do with the proposition is the criterion of understanding it, thus far it
is not clear in advance whether and to what extent I understand it.
The curse of the invasion of mathematics by mathematical logic is that now any proposition can be
represented in a mathematical symbolism, and this makes us feel obliged to understand it. Although of course this
method of writing is nothing but the translation of vague ordinary prose.
Page 299
47. A concept is not essentially a predicate.†1 We do indeed sometimes say: "This thing is not a bottle" but it
is certainly not essential to the language-game with the concept 'bottle' that such judgments occur in it. The thing is
to pay attention to how a concept word ("slab", e.g.) is used in a language-game.
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Page 300
There need not e.g. be such a sentence as "This is a slab" at all; but e.g. merely: "Here is a slab."
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48. 'Mathematical logic' has completely deformed the thinking of mathematicians and of philosophers, by
setting up a superficial interpretation of the forms of our everyday language as an analysis of the structures of facts.
Of course in this it has only continued to build on the Aristotelian logic.
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49. It is quite true: the numerical sign belongs with a concept-sign, and only together with this is it, so to
speak, a measure.
Page 300
50. If you look into this mouse's jaw you will see two long incisor teeth.--How do you know?--I know that all
mice have them, so this one will too. (And one does not say: "And this thing is a mouse, so it too...") Why is this
such an important move? Well, we investigate e.g. animals, plants etc. etc.; we form general judgments and apply
them in particular cases.--But it surely is a truth that this mouse has the property, if all mice have it! That is a
determination about the application of the word "all". The factual generality is to be found somewhere else. Namely,
for example, in the general occurrence of that method of investigation and its application.
Page 300
Or: "This man is a student of mathematics." How do you know?--"All the people in this room are
mathematicians; only such people have been admitted."
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Page 301
The interesting case of generality is this: we often have a means of ascertaining the general proposition before
considering particular cases: and we then use the general method to judge the particular case.
We gave the porter the order only to admit people with invitations and now we count upon it that this man,
who has been admitted, has an invitation.
Page 301
The interesting generality in the case of the logical proposition is not the fact that it appears to express, but
the ever-recurring situation in which this transition is made.
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51. If it is said that the proof shews how (e.g.) 25 × 25 yield 625, that is of course a queer way of talking,
since for this to be the arithmetical result is not a temporal process. But the proof does not shew any temporal
process either.
Page 301
Imagine a sequence of pictures. They shew how two people fence with rapiers according to such-and-such
rules. A sequence of pictures can surely shew that. Here the picture refers to a reality. It cannot be said to shew that
fencing is done like this, but how fencing is done. In another sense we can say that the pictures shew how one can
get from this position into that in three movements. And now they also shew that one can get into that position in
this way.
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52. The philosopher must twist and turn about so as to pass by the mathematical problems, and not run up
against one,--which would have to be solved before he could go further.
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Page 302
His labour in philosophy is as it were an idleness in mathematics.
Page 302
It is not that a new building has to be erected, or that a new bridge has to be built, but that the geography, as
it now is, has to be described.
Page 302
We certainly see bits of the concepts, but we don't clearly see the declivities by which one passes into others.
Page 302
This is why it is of no use in the philosophy of mathematics to recast proofs in new forms. Although there is
a strong temptation here.
Page 302
Even 500 years ago a philosophy of mathematics was possible, a philosophy of what mathematics was then.
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53. The philosopher is the man who has to cure himself of many sicknesses of the understanding before he
can arrive at the notions of the sound human understanding.
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PART VI
ca. 1943/1944
Page 303
1. Proofs give propositions an order. They organize them.
Page 303
2. The concept of a formal test presupposes the concept of a transformation-rule, and hence of a technique.
Page 303
For only through a technique can we grasp a regularity.
Page 303
The technique is external to the pattern of the proof. One might have a perfectly accurate view of the proof,
yet not understand it as a transformation according to such-and-such rules.
Page 303
One will certainly call adding up the numbers ... to see whether they come to 1000, a formal test of the
numerals. But all the same that is only when adding is a practised technique. For otherwise how could the procedure
be called any kind of test?
Page 303
It is only within a technique of transformation that the proof is a formal test.
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Page 304
When you ask what right you have to pronounce this rule, the proof is the answer to your question.
Page 304
What right have you to say that? What right have you to say it?
Page 304
How do you test a theme for a contrapuntal property? You transform it according to this rule, you put it
together with another one in this way; and the like. In this way you get a definite result. You get it, as you would
also get it by means of an experiment. So far what you are doing may even have been an experiment. The word
"get" is here used temporally; you got the result at three o'clock.--In the mathematical proposition which I then
frame the verb ("get", "yields" etc.) is used non-temporally.
The activity of testing produced such and such a result.
So up to now the testing was, so to speak, experimental. Now it is taken as a proof. And the proof is the
picture of a test.
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The proof, like the application, lies in the background of the proposition. And it hangs together with the
application.
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The proof is the route taken by the test.
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The test is a formal one only in so far as we conceive the result as the result of a formal proposition.
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3. And if this picture justifies the prediction--that is to say, if you only have to see it and you are convinced
that a procedure will take such-and-such a course--then naturally this picture also justifies the rule. In this case the
proof stands behind the rule as a picture that justifies the rule.
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For why does the picture of the movement of the mechanism justify the belief that this kind of mechanism
will always move in this way?--It gives our belief a particular direction.
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When the proposition seems not to be right in application, the proof must surely shew me why and how it
must be right; that is, how I must reconcile it with experience.
Page 305
Thus the proof is a blue-print for the employment of the rule.
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4. How does the proof justify the rule?--It shews how, and therefore why, the rule can be used.
Page 305
The King's Bishop†1 shews us how 8 × 9 makes 72--but here the rule of counting is not acknowledged as a
rule.
The King's Bishop shews us that 8 × 9 makes 72: Now we are acknowledging the rule.
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Page 306
Or ought I to have said: the King's Bishop shews me how 8 × 9 can make 72; that is to say, it shews me a
way?
Page 306
The procedure shews me a How of 'making'.
Page 306
In so far as 8 × 9 = 72 is a rule, of course it means nothing to say that that shews me how 8 × 9 = 72; unless
this were to mean: someone shews me a process through the contemplation of which one is led to this rule.
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Now isn't going through any proof such a process?
Page 306
Would it mean anything to say: "I want to shew you how 8 × 9 originally made 72"?
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5. What is really queer is that the picture, not the reality, should be able to prove a proposition! As if here the
picture itself took over the role of reality.--But that's not how it is: for what I derive from the picture is only a rule.
And this rule does not stand to the picture as an empirical proposition stands to reality.--The picture of course does
not shew that such-and-such happens. It only shews that what does happen can be taken in this way.
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The proof shews how one proceeds according to the rule without a hitch.
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And so one may even say: the procedure, the proof, shews one how far 8 × 9 = 72.
Page 307
The picture shews one, not, of course, anything that happens, but that what ever does happen will allow of
being looked at like this.
Page 307
We are brought to the point of using this technique in this case. I am brought to this--and to that extent I am
convinced of something.
Page 307
See, in this way 3 and 2 make 5. Note this procedure. "In doing so you at once notice the rule."
Page 307
6. The Euclidean proof of the infinity of prime numbers might be so conducted that the investigation of the
numbers between p and p! + 1 was carried out on one or more examples, and in this way we learned a technique of
investigation. The force of the proof would of course in that case not reside in the fact that a prime number > p was
found in this example. And at first sight this is queer.
It will now be said that the algebraic proof is stricter than the one by way of examples, because it is, so to
speak, the extract of the effective principle of these examples. But after all, even the algebraic proof is not quite
naked. Understanding--I might say--is needed for both!
Page 307
The proof teaches us a technique of finding a prime number between p and p! + 1. And we become
convinced that this technique must
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always lead to a prime number > p. Or that we have miscalculated if it doesn't.
Page 308
Would one be inclined to say here that the proof shews us how there is an infinite series of prime numbers?
Well, one might say so. And at any rate: "What there being an infinity of primes amounts to." For it could also be
imagined that we had a proof that did indeed determine us to say that there were infinitely many primes, but did not
teach us to find a prime number > p.
Now perhaps it would be said: "Nevertheless, these two proofs prove the same proposition, the same
mathematical fact." There might be reason at hand for saying this, or again there might not.
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7. The spectator sees the whole impressive procedure. And he becomes convinced of something; that is the
special impression that he gets. He goes away from the performance convinced of something. Convinced that (for
example) he will end up the same way with other numbers. He will be ready to express what he is convinced of in
such-and-such a way. Convinced of what? Of a psychological fact?--
Page 308
He will say that he has drawn a conclusion from what he has seen.--Not, however as one does from an
experiment. (Think of periodic division.)
Page 308
Could he say: "What I have seen was very impressive. I have drawn a conclusion from it. In future I shall..."?
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Page 309
(E.g.: In future I shall always calculate like this.)
Page 309
He tells us: "I saw that it must be like that."
Page 309
"I realised that it must be like that"--that is his report.
Page 309
He will now perhaps run through the proof procedure in his mind.
Page 309
But he does not say: I realised that this happens. Rather: that it must be like that. This "must" shews what
kind of lesson he has drawn from the scene.
The "must" shews that he has gone in a circle.
Page 309
I decide to see things like this. And so, to act in such-and-such a way.
Page 309
I imagine that whoever sees the process also draws a moral from it.
Page 309
'It must be so' means that this outcome has been defined to be essential to this process.
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8. This must shews that he has adopted a concept.
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Page 310
This must signifies that he has gone in a circle.
Page 310
He has read off from the process, not a proposition of natural science but, instead of that, the determination
of a concept.
Let concept here mean method. In contrast to the application of the method.
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9. See, 50 and 50 make 100 like this. One has, say, added 10 to 50 five times in succession. And one goes on
with the increase of the number until it grows to 100. Here of course the observed process would be a process of
calculating in some fashion (on the abacus, perhaps); a proof.
Page 310
The meaning of that "like this" is of course not that the proposition "50 + 50 = 100" says: this takes place
somewhere. So it is not as when I say: "See, a horse canters like this"--and shew him a picture.
Page 310
One could however say: "See, this is why I say 50 + 50 = 100".
Page 310
Or: See, this is how one gets 50 + 50 = 100.
But if I now say: See, this is how 3 + 2 make 5, laying 3 apples on the table and then 2 more, here I mean to
say: 3 apples and 2 apples
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make 5 apples, if none are added or taken away.--Or one might even tell someone: If you put 3 apples and then 2
more on the table (as I am doing), then what you see now almost always happens--and there are now 5 apples lying
there.
I want perhaps to shew him that 3 apples and 2 apples don't make 5 apples in such a way as they might make
6 (because e.g. one makes a sudden appearance). This is really an explanation, a definition of the operation of
adding. This is indeed how one might actually explain adding with the abacus.
Page 311
"If we put 3 things by 2 things, that may yield various counts of things. But we see as a norm the procedure
that 3 things and 2 things make 5 things. See, this is how it looks when they make 5."
Page 311
Couldn't one say to a child: "Shew me how 3 and 2 make 5." And the child would then have to calculate 3 +
2 on the abacus.
Page 311
When, in teaching the child to calculate, one asks, "How do 3 + 2 make 5?"--what is he supposed to shew?
Well, obviously he is supposed to move three beads up to 2 beads and then to count the beads (or something like
that).
Page 311
Might one not say "Shew me how this theme makes a canon." And someone asked this would have to prove
that it does make a canon.--One would ask someone "how" if one wanted to to get him to shew that he does grasp
what is in question here.
Page Break 312
Page 312
And if the child now shews how 3 and 2 make 5, then he shews a procedure that can be regarded as a ground
for the rule "2 + 3 = 5."
Page 312
10. But suppose one asks the pupil: "Shew me how there are infinitely many prime numbers."--Here the
grammar is doubtful! But it would be appropriate to say: "Shew me in how far one may say that there are infinitely
many prime numbers."
Page 312
When one says "Shew me that it is...," then the question whether it is is already put and it remains only to
answer "yes" or "no". But if one says "Shew me how it is that...", then here the language-game itself needs to be
explained. At any rate, one has so far no clear concept of what one is supposed to be at with this assertion. (One is
asking, so to speak; "How can such an assertion be justified at all?")
Page 312
Now am I meant to give different answers to the question: "Shew me how..." and "Shew me that..."?
Page 312
From the proof you derive a theory. If you derive a theory from the proof, then the sense of the theory must
be independent of the proof; for otherwise the theory could never have been separated from the proof.
In the same way as I can remove auxiliary construction lines in a drawing and leave the rest.
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Thus it is as if the proof did not determine the sense of the proposition proved; and yet as if it did determine
it.
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Page 313
But isn't it like that with any verification of any proposition?
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11. I believe this: Only in a large context can it be said at all that there are infinitely many prime numbers.
That is to say: For this to be possible there must already exist an extended technique of calculating with cardinal
numbers. That proposition only makes sense within this technique. A proof of the proposition locates it in the whole
system of calculations. And its position therein can now be described in more than one way, as of course the whole
complicated system in its background is presupposed.
Page 313
If for example 3 co-ordinate systems are given a definite mutual arrangement, I can determine the position of
a point for all of them by giving it for any one.
Page 313
The proof of a proposition certainly does not mention, certainly does not describe, the whole system of
calculation that stands behind the proposition and gives it its sense.
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Assume that an adult with intelligence and experience has learnt only the first elements of calculation, say
the four fundamental operations with numbers up to 20. In doing so he has also learnt the word "prime number".
And suppose someone said to him "I am going to prove to you that there are infinitely many prime numbers." Now,
how can he prove it to him? He has got to teach him to calculate. That is here part of the proof. It takes that, so to
speak, to give the question "Are there infinitely many prime numbers?" any sense.
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12. Philosophy has to work things out in face of the temptations to misunderstand on this level of
knowledge. (On another level there are again new temptations.) But that doesn't make philosophising any easier!
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13. Now isn't it absurd to say that one doesn't understand the sense of Fermat's last theorem?--Well, one
might reply: the mathematicians are not completely blank and helpless when they are confronted by this proposition.
After all, they try certain methods of proving it; and, so far as they try methods, so far do they understand the
proposition.--But is that correct? Don't they understand it just as completely as one can possibly understand it?
Page 314
Now let us assume that, quite contrary to mathematicians' expectations, its contrary were proved. So now it
is shewn that it cannot be so at all.
Page 314
But, if I am to know what a proposition like Fermat's last theorem says, must I not know what the criterion is,
for the proposition to be true? And I am of course acquainted with criteria for the truth of
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similar propositions, but not with any criterion for the truth of this proposition.
Page 315
'Understanding' is a vague concept.
Page 315
In the first place, there is such a thing as belief that one understands a proposition.
And if understanding is a psychical process--why should it interest us so much? Unless experience connects
it with the capacity to make use of the proposition.
Page 315
"Shew me how..." means: shew me the connexions in which you are using this proposition (this
machine-part).
Page 315
14. "I am going to shew you how there are infinitely many prime numbers" presupposes a condition in which
the proposition that there are infinitely many prime numbers had no, or only the vaguest, meaning. It might have
been merely a joke to him, or a paradox.
Page 315
If this procedure convinces you of that, then it must be very impressive.--But is it?--Not particularly. Why is
it not more so? I believe it would only be impressive if one were to explain it quite radically. If for example one did
not merely write p! + 1, but first explained it and illustrated it with examples. If one did not presuppose the
techniques as something obvious, but gave an account of them.
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15.
We keep on copying the last figure "2" going round to the right. If we copy correctly, the last figure is in turn a copy
of the first one.
Page 316
A language-game:
Page 316
One person (A) predicts the result to another (B). The other follows the arrows with excitement, as it were
curious how they will conduct him, and is pleased at the way they end by leading him to the predicted result. He
reacts to it perhaps as one reacts to a joke.
A may have constructed the result before, or merely have guessed it. B knows nothing about it and it does
not interest him.
Page 316
Even if he was acquainted with the rule, still he had never followed it thus. He is now doing something new.
But there is also such a thing as curiosity and surprise when one has already travelled this road. In this way one can
read a story again and again, even know it by heart, and yet keep on being surprised at a particular turn that it takes.
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Page 317
Before I have followed the two arrows like this , I don't know how the route or the
result will look. I do not know what face I shall see. Is it strange that I did not know it? How should I have known it?
I had never seen it! I knew the rule and had mastered it and I saw the sheaf of arrows.
Page 317
But why wasn't this a genuine prediction: "If you follow the rule, you will produce this"? Whereas the
following is certainly a genuine prediction: "If you follow the rule as best you can, you will..." The answer is: the first
is not a prediction because I might also have said: "If you follow the rule, you must produce this." It is not a
prediction if the concept of following the rule is so determined, that the result is the criterion for whether the rule was
followed.
Page 317
A says: "If you follow the rule you will get this" or he says simply: "You will get this." At the same time he
draws the resulting arrow there.
Page 317
Now was what A said in this game a prediction? Well damn it, Yes--in a certain sense. Does that not become
particularly clear if we make the suggestion that the prediction was wrong? It was only not a prediction in the case
where the condition turned the proposition into a pleonasm.
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Page 318
A might have said: "If you are in agreement with each of your steps, then you will arrive at this."
Page 318
Suppose that while B is deriving the polygon, the arrows of the sheaf were to alter their direction a little. B
always draws an arrow parallel, as it is just at this moment. He is now just as surprised and excited as in the
foregoing game, although here the result is not that of a calculation. So he had taken the first game in the same way
as the second.
Page 318
The reason why "If you follow the rule, this is where you'll get to" is not a prediction is that this proposition
simply says: "The result of this calculation is..." and that is a true or false mathematical proposition: The allusion to
the future and to yourself is mere clothing.
Page 318
Now must A have a clear idea at all, of whether his prediction is meant mathematically or otherwise? He
simply says "If you follow the rule... will result" and enjoys the game. If for example the predicted result does not
come out, he does not investigate any further.
Page 318
16. ... And this series is defined by a rule. Or again by the training in proceeding according to the rule. And
the inexorable proposition is that according to this rule this number is the successor of this one.†1
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Page 319
And this proposition is not an empirical one. But why not an empirical one? A rule is surely something that
we go by, and we produce one numeral out of another. Is it not matter of experience, that this rule takes someone
from here to there?
Page 319
And if the rule + 1 carries him one time from 4 to 5, perhaps another time it carries him from 4 to 7. Why is
that impossible?
Page 319
The question arises, what we take as criterion of going according to the rule. Is it for example a feeling of
satisfaction that accompanies the act of going according to the rule? Or an intuition (intimation) that tells me I have
gone right? Or is it certain practical consequences of proceeding that determine whether I have really followed the
rule?--In that case it would be possible that 4 + 1 sometimes made 5 and sometimes something else. It would be
thinkable, that is to say, that an experimental investigation would shew whether 4 + 1 always makes 5.
Page 319
If it is not supposed to be an empirical proposition that the rule leads from 4 to 5, then this, the result, must
be taken as the criterion for one's having gone by the rule.
Page 319
Thus the truth of the proposition that 4 + 1 makes 5 is, so to speak, overdetermined. Overdetermined by this,
that the result of the operation is defined to be the criterion that this operation has been carried out.
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Page 320
The proposition rests on one too many feet to be an empirical proposition. It will be used as a determination
of the concept 'applying the operation + 1 to 4'. For we now have a new way of judging whether someone has
followed the rule.
Page 320
Hence 4 + 1 = 5 is now itself a rule, by which we judge proceedings.
This rule is the result of a proceeding that we assume as decisive for the judgment of other proceedings. The
rule-grounding proceeding is the proof of the rule.
Page 320
17. How does one describe the process of learning a rule?--If A claps his hands, B is always supposed to do
it too.
Page 320
Remember that the description of a language-game is already a description.
Page 320
I can train someone in a uniform activity. E.g. in drawing a line like this with a pencil on paper:
Now I ask myself, what is it that I want him to do, then? The answer is: He is always to go on as I have shewn him.
And what do I really mean by: he is always to go on in that way? The best answer to this that I can give myself, is an
example like the one I have just given.
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Page 321
I would use this example in order to shew him, and also to shew myself, what I mean by uniform.
Page 321
We talk and act. That is already presupposed in everything that I am saying.
Page 321
I say to him "That's right," and this expression is the bearer of a tone of voice, a gesture. I leave him to it. Or I
say "No!" and hold him back.
Page 321
18. Does this mean that 'following a rule' is indefinable? No. I can surely define it in countless ways. Only
definitions are no use to me in these considerations.
Page 321
19. I might also teach him to understand an order of the form:
or
(Let the reader guess what I mean.)
Page 321
Now what do I mean him to do? The best answer that I can give myself to this is to carry these orders on a
bit further. Or do you believe that an algebraic expression of this rule presupposes less?
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And now I train him to follow the rule
etc.
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And again I don't myself know any more about what I want from him, than what the example itself shews. I can of
course paraphrase the rule in all sorts of different forms, but that makes it more intelligible only for someone who
can already follow these paraphrases.
Page 322
20. This, then, is how I have taught someone to count and to multiply in the decimal system, for example.
"365 × 428" is an order and he complies with it by carrying out the multiplication.
Page 322
Here we insist on this, that the same sum that is set always has the same multiplication-pattern in its train,
and so the same result. Different patterns of multiplication for the same set sum we reject.
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The situation will now arise, of a calculator making mistakes in calculation; and also of his correcting
mistakes.
Page 322
A further language-game is this: He gets asked "How much is '365 × 428'?" And he may act on this question
in two different ways. Either he does the multiplication, or if he has already done it before, he reads off the previous
result.
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21. The application of the concept 'following a rule' presupposes a custom. Hence it would be nonsense to
say: just once in the history of
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the world someone followed a rule (or a signpost; played a game, uttered a sentence, or understood one; and so on).
Page 323
Here there is nothing more difficult than to avoid pleonasms and only to say what really describes
something.
Page 323
For here there is an overwhelming temptation to say something more, when everything has already been
described.
Page 323
It is of the greatest importance that a dispute hardly ever arises between people about whether the colour of
this object is the same as the colour of that, the length of this rod the same as the length of that, etc. This peaceful
agreement is the characteristic surrounding of the use of the word "same".
Page 323
And one must say something analogous about proceeding according to a rule.
Page 323
No dispute breaks out over the question whether a proceeding was according to the rule or not. It doesn't
come to blows, for example.
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This belongs to the framework, out of which our language works (for example, gives a description).
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Page 324
22. Now someone says that in the series of cardinal numbers that obeys the rule + 1, the technique of which
was taught to us in such-and-such a way, 450 succeeds 449. That is not the empirical proposition that we come from
449 to 450 when it strikes us that we have applied the operation + 1 to 449. Rather is it a stipulation that only when
the result is 450 have we applied this operation.
Page 324
It is as if we had hardened the empirical proposition into a rule. And now we have, not an hypothesis that
gets tested by experience, but a paradigm with which experience is compared and judged. And so a new kind of
judgment.
Page 324
For one judgment is: "He worked out 25 × 25, was attentive and conscientious in doing so and made it 615";
and another: "He worked out 25 × 25 and got 615 out instead of 625."
Page 324
But don't the two judgments come to the same thing in the end?
Page 324
The arithmetical proposition is not the empirical proposition: "When I do this, I get this"--where the criterion
for my doing this is not supposed to be what results from it.
Page 324
23. Might we not imagine that the main point in multiplying was the concentration of the mind in a definite
way, and that indeed one didn't always work out the same sums the same way, but for the particular practical
problems that we want to solve, just these differences of result were advantageous?
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Page 325
Is the main thing not this: that in calculating the main weight would be placed on whether one has calculated
right or wrong, quite prescinding from the psychical condition etc. of the person who is doing the calculation?
Page 325
The justification of the proposition 25 × 25 = 625 is, naturally, that if anyone has been trained in
such-and-such a way, then under normal circumstances he gets 625 as the result of multiplying 25 by 25. But the
arithmetical proposition does not assert that. It is so to speak an empirical proposition hardened into a rule. It
stipulates that the rule has been followed only when that is the result of the multiplication. It is thus withdrawn from
being checked by experience, but now serves as a paradigm for judging experience.
Page 325
If we want to make practical use of a calculation, we convince ourselves that it has been "worked out right",
that the correct result has been obtained. And there can be only one correct result of (e.g.) the multiplication; it
doesn't depend on what you get when you apply the calculation. Thus we judge the facts by the aid of the
calculation and quite differently from the way in which we should do so, if we did not regard the result of the
calculation as something determined once for all.
Page 325
Not empiricism and yet realism in philosophy, that is the hardest thing. (Against Ramsey.)
Page 325
You do not yourself understand any more of the rule than you can explain.
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Page 326
24. "I have a particular concept of the rule. If in this sense one follows it, then from that number one can only
arrive at this one". That is a spontaneous decision.
Page 326
But why do I say "I must", if it is my decision? Well, may it not be that I must decide?
Page 326
Doesn't its being a spontaneous decision merely mean: that's how I act; ask for no reason!
Page 326
You say you must; but cannot say what compels you.
Page 326
I have a definite concept of the rule. I know what I have to do in any particular case. I know, that is I am in
no doubt: it is obvious to me. I say "Of course". I can give no reason.
Page 326
When I say "I decide spontaneously", naturally that does not mean: I consider which number would really be
the best one here and then plump for...
Page 326
We say: "First the calculations must be done right, and then it will be possible to pass some judgment on the
facts of nature."
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25. Someone has learned the rule of counting in the decimal system. Now he takes pleasure in writing down
number after number in the "natural" number series.
Or he follows the rule in the language-game "Write down the successor of the number .... in the series
...."--How can I explain this language-game to anyone? Well, I can describe an example (or examples).--In order to
see whether he has understood the language-game, I may make him work out examples.
Page 327
Suppose someone were to verify the multiplication tables, the logarithm tables etc., because he did not trust
them. If he reaches a different result, he trusts it, and says that his mind had been so concentrated on the rule that the
result it gets must count as the right one. If someone points out a mistake to him he says that he would rather doubt
the trustworthiness of his own understanding and his own meaning now than then when he first made the
calculation.
Page 327
We can take agreement for granted in all questions of calculation. But now, does it make any difference
whether we utter the proposition used in calculating as an empirical proposition or as a rule?
Page 327
26. Should we acknowledge the rule 252 = 625, if we did not all arrive at this result? Well, why then should
we not be able to make use of the empirical proposition instead of the rule?--Is the answer to that: Because the
contrary of the empirical proposition does not correspond to the contrary of the rule?
Page 327
When I write down a bit of a series for you, that you then see this
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regularity in it may be called an empirical fact, a psychological fact. But, if you have seen this law in it, that you then
continue the series in this way--that is no longer an empirical fact.
But how is it not an empirical fact?--for "seeing this in it" was presumably not the same as: continuing it like
this.
One can only say that it is not an empirical proposition, by defining the step on this level as the one that
corresponds to the expression of the rule.
Page 328
Thus you say: "By the rule that I see in this sequence, it goes on in this way." Not: according to experience!
Rather: that just is the meaning of this rule.
Page 328
I understand: You say "that is not according to experience"--but still isn't it according to experience?
Page 328
"By this rule it goes like this": i.e., you give this rule an extension.
Page 328
But why can't I give it this extension today, that one tomorrow?
Page 328
Well, so I can. I might for example alternately give one of two interpretations.
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27. If I have once grasped a rule I am bound in what I do further. But of course that only means that I am
bound in my judgment about
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what is in accord with the rule and what not.
Page 329
If I now see a rule in the sequence that is given me--can that simply consist in, for example, my seeing an
algebraic expression before me? Must it not belong to a language?
Page 329
Someone writes up a sequence of numbers. At length I say: "Now I understand it; I must always..." And this
is the expression of a rule. But, only within a language!
Page 329
For when do I say that I see the rule--or a rule--in this sequence? When, for example, I can talk to myself
about this sequence in a particular way. But surely also when I simply can continue it? No, I give myself or someone
else a general explanation of how it is to be continued. But might I not give this explanation purely in the mind, and
so without any real language?
Page 329
28. Someone asks me: What is the colour of this flower? I answer: "red".--Are you absolutely sure? Yes,
absolutely sure! But may I not have been deceived and called the wrong colour "red"? No. The certainty with which
I call the colour "red" is the rigidity of my measuring-rod, it is the rigidity from which I start. When I give
descriptions, that is not to be brought into doubt. This simply characterizes what we call describing.
(I may of course even here assume a slip of the tongue, but nothing else.)
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Page 330
Following according to the rule is FUNDAMENTAL to our language-game. It characterizes what we call
description.
Page 330
This is the similarity of my treatment with relativity-theory, that it is so to speak a consideration about the
clocks with which we compare events.
Page 330
Is 252 = 625 a fact of experience? You'd like to say: "No".--Why isn't it?--"Because, by the rules, it can't be
otherwise."--And why so?--Because that is the meaning of the rules. Because that is the procedure on which we
build all judging.
Page 330
29. When we carry out a multiplication, we give a law. But what is the difference between the law and the
empirical proposition that we give this law?
Page 330
When I have been taught the rule of repeating the ornament
and now I have been told "Go on like that": how do I know what I have to do the next time?--Well, I do it with
certainty, I shall also know how to defend what I do--that is, up to a certain point. If that does not count as a defence
then there is none.
Page 330
"As I understand the rule, this comes next."
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Page 331
Following a rule is a human activity.
Page 331
I give the rule an extension.
Page 331
Might I say: See here, if I follow the order I draw this line? Well in certain cases I shall say that. When for
example I have constructed a curve according to an equation.
Page 331
"See here! if I follow the order I do this!" That is naturally not supposed to mean: if I follow the order I
follow the order. So I must have a different identification for the "this".
Page 331
"So that's what following this order looks like!"
Page 331
Can I say: "Experience teaches me: if I take the rule like this then this is how I must go on?"
Page 331
Not if I make 'taking it so' one and the same with 'continuing so'.
Page 331
Following a rule of transformation is not more problematic than following the rule: "keep on writing the
same". For the transformation is a kind of identity.
Page Break 332
Page 332
30. It might however be asked: if all humans that are educated like this also calculate like this, or at least
agree to this calculation as the right one; then what does one need the law for?
Page 332
"252 = 625" cannot be the empirical proposition that people calculate like that, because 252 ≠ 626 would in
that case not be the proposition that people get not this but another result; and also it could be true if people did not
calculate at all.
Page 332
The agreement of people in calculation is not an agreement in opinions or convictions.
Page 332
Could it be said: "In calculating, the rules strike you as inexorable; you feel that you can only do that and
nothing else if you want to follow the rule"?
Page 332
"As I see the rule, this is what it requires." It does not depend on whether I am disposed this way or that.
Page 332
I feel that I have given the rule an interpretation before I have followed it; and that this interpretation is
enough to determine what I have to do in order to follow it in the particular case.
If I take the rule as I have taken it, then only doing this will correspond to it.
Page Break 333
Page 333
"Have you understood the rule?"--Yes, I have understood--"Then apply it now to the numbers....." If I want
to follow the rule, have I now any choice left?
Page 333
Assuming that he orders me to follow the rule and that I am frightened not to obey him: am I now not
compelled?
But that is surely so too if he orders me: "Bring me this stone." Am I compelled less by these words?
Page 333
31. To what extent can the function of language be described? If someone is not master of a language, I may
bring him to a mastery of it by training. Someone who is master of it, I may remind of the kind of training, or I may
describe it; for a particular purpose; thus already using a technique of the language.
To what extent can the function of a rule be described? Someone who is master of none, I can only train. But
how can I explain the nature of a rule to myself?
The difficult thing here is not, to dig down to the ground; no, it is to recognize the ground that lies before us
as the ground.
Page 333
For the ground keeps on giving us the illusory image of a greater depth, and when we seek to reach this, we
keep on finding ourselves on the old level.
Page 333
Our disease is one of wanting to explain.
Page 333
"Once you have got hold of the rule, you have the route traced for you."
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Page 334
32. What sort of public must there be if a game is to exist, if a game can be invented?
Page 334
What surrounding is needed for someone to be able to invent, say, chess?
Of course I might invent a board-game today, which would never actually be played. I should simply
describe it. But that is possible only because there already exist similar games, that is because such games are
played.
Page 334
One might also ask: is regularity possible without repetition?
Page 334
I may give a new rule today, which has never been applied, and yet is understood. But would that be
possible, if no rule had ever actually been applied?
Page 334
And if it is now said: "Isn't it enough for there to be an imaginary application?" the answer is: No. (Possibility
of a private language.)
Page 334
A game, a language, a rule is an institution.
Page 334
"But how often must a rule have actually been applied, in order for one to have the right to speak of a rule?"
How often must a human being have added, multiplied, divided, before we can say that he has
Page Break 335
mastered the technique of these kinds of calculation? And by that I don't mean: how often must he have calculated
right in order to convince others that he can calculate? No, I mean: in order to prove it to himself.
Page 335
33. But couldn't we imagine that someone without any training should see a sum that was set to do, and
straightway find himself in the mental state that in the normal course of things is only produced by training and
practice? So that he knew he could calculate although he had never calculated. (One might, then, it seems, say; The
training would merely be history, and merely as a matter of empirical fact would it be necessary for the production
of knowledge.)--But suppose now he is in that state of certainty and he calculates wrong? What is he supposed to
say himself? And suppose he then multiplied sometimes right, sometimes again quite wrong.--The training may of
course be overlooked as mere history, if he now always calculates right. But that he can calculate he shews, to
himself as well as to others only by this, that he calculates correctly.
Page 335
What, in a complicated surrounding, we call "following a rule" we should certainly not call that if it stood in
isolation.
Page 335
34. Language, I should like to say, relates to a way of living.
Page 335
In order to describe the phenomenon of language, one must describe a practice, not something that happens
once, no matter of what kind.
Page Break 336
Page 336
It is very hard to realize this.
Page 336
Let us imagine a god creating a country instantaneously in the middle of the wilderness, which exists for two
minutes and is an exact reproduction of a part of England, with everything that is going on there in two minutes. Just
like those in England, the people are pursuing a variety of occupations. Children are in school. Some people are
doing mathematics. Now let us contemplate the activity of some human being during these two minutes. One of
these people is doing exactly what a mathematician in England is doing, who is just doing a calculation.--Ought we
to say that this two-minute-man is calculating? Could we for example not imagine a past and a continuation of these
two minutes, which would make us call the processes something quite different?
Page 336
Suppose that these beings did not speak English but apparently communicated with one another in a
language that we are not acquainted with. What reason should we have to say that they were speaking a language?
And yet could one not conceive what they were doing as that?
Page 336
And suppose that they were doing something that we were inclined to call "calculating"; perhaps because its
outward appearance was similar.--But is it calculating; and do (say) the people who are doing it know, though we do
not?
Page 336
35. How do I know that the colour that I am now seeing is called "green"? Well, to confirm it I might ask
other people; but if they did not agree with me, I should become totally confused and should perhaps
Page Break 337
take them or myself for crazy. That is to say: I should either no longer trust myself to judge, or no longer react to
what they say as to a judgement.
If I am drowning and I shout "Help!", how do I know what the word Help means? Well, that's how I react in
this situation.--Now that is how I know what "green" means as well and also know how I have to follow the rule in
the particular case.
Page 337
Is it imaginable that the polygon of forces of
looks, not like this:
but otherwise? Well, is it imaginable that the parallel to a should not look to have the direction of a' but a different
direction? That is to say: is it imaginable that I should regard not a' but a differently directed arrow as parallel to a?
Well, I might for example imagine that I was somehow seeing the parallel lines in perspective and so I call
parallel arrows, and that it never occurs to me that I have been using a different way of looking at them. Thus, then,
it is imaginable that I should draw a different polygon of forces corresponding to the arrows.
Page Break 338
Page 338
36. What sort of proposition is this: "There are four sounds in the word OBEN"?
Is it an empirical proposition?
Page 338
Before we have counted the letters, we don't know it.
Page 338
Someone who counts the letters in the word 'OBEN' in order to find out how many sounds there are in a
sequence that sounds like that, does just the same thing as someone who counts in order to find out how many
letters there are in the word that is written in such-and-such a place. So the former is doing something that might
also be an experiment. And that might be reason to call the proposition that 'OBEN' has four letters synthetic a
priori.
Page 338
The word "Plato" has as many sounds in it as the pentacle has corners. Is that a proposition of logic?--Is it an
empirical proposition?
Page 338
Is counting an experiment? It may be one.
Page 338
Imagine a language-game in which someone has to count the sounds in a word. Now it might be that a word
apparently always had the same sound, but that when we count its sounds we come to different numbers on
different occasions. It might be, for example, that a word did seem to us to sound the same in different contexts (as
it were by an acoustical illusion) but the difference emerged when we counted the sounds. In such a case we shall
perhaps keep on counting the
Page Break 339
sounds of a word on different occasions, and this will perhaps be a kind of experiment.
On the other hand it may be that we count the sounds in words once for all, make a calculation, and make
use of the result of this counting.
The resulting proposition will in the first case be a temporal one, in the second it will be non-temporal.
Page 339
When I count the sounds in the word "Daedalus" I can regard the result in two different ways: (1) The word
that is written there (or looks like this or was just now pronounced or etc.) has 7 sounds. (2) The sound-pattern
"Dædalus" has 7 sounds.
The second proposition is timeless.
The employment of the two propositions must be different.
The counting is the same in the two cases. Only, what we reach by means of it is different.
Page 339
The timelessness of the second proposition is not e.g. a result of the counting, but of the decision to employ
the result of counting in a particular way.
Page 339
In English the word Dædalus has 7 sounds. That is surely an empirical proposition.
Page 339
Imagine that someone counted the sounds in words in order to find or test a linguistic law, say a law of
development of language. He says: "'Dædalus' has 7 sounds". That's an empirical proposition. Consider
Page Break 340
here the identity of the word. The same word may here have now this, now that number of sounds.
Page 340
Now I tell someone: "Count the sounds in these words and write down the number by each word."
Page 340
I should like to say: "Through counting the sounds one may get an empirical proposition--but also one may
get a rule."
Page 340
To say: "The word.... has.... sounds--in the timeless sense" is a determination about the identity of the
concept 'The word....'. Hence the timelessness.
Page 340
Instead of "The word.... has.... sounds--in the timeless sense," one might also say: "The word.... has
essentially.... sounds."
Page 340
37.
Page 340
Definitions would not at all need to be abbreviations; they might make new connexions in another way. Say
by means of brackets or the use of different colours for the signs.
Page Break 341
Page 341
I may for example prove a proposition by using colours to indicate that it has the form of one of my axioms,
lengthened by a certain substitution.
Page 341
38. "I know how I have to go" means: I am in no doubt how I have to go.
Page 341
"How can one follow a rule?" That is what I should like to ask.
Page 341
But how does it come about that I want to ask that, when after all I find no kind of difficulty in following a
rule?
Page 341
Here we obviously misunderstand the facts that lie before our eyes.
Page 341
How can the word "Slab" indicate what I have to do, when after all I can bring any action into accord with
any interpretation?
Page 341
How can I follow a rule, when after all whatever I do can be interpreted as following it?
Page 341
What must I know, in order to be able to obey the order? Is there some knowledge, which makes the rule
followable only in this way?
Page Break 342
Sometimes I must know something, sometimes I must interpret the rule before I apply it.
Now, how was it possible for the rule to have been given an interpretation during instruction, an
interpretation which reaches as far as to any arbitrary step?
And if this step was not named in the explanation, how then can we agree about what has to happen at this
step, since after all whatever happens can be brought into accord with the rule and the examples?
Thus, you say, nothing definite has been said about these steps.
Page 342
Interpretation comes to an end.
Page 342
39. It is true that anything can be somehow justified. But the phenomenon of language is based on
regularity, on agreement in action.
Page 342
Here it is of the greatest importance that all or the enormous majority of us agree in certain things. I can, e.g.,
be quite sure that the colour of this object will be called 'green' by far the most of the human beings who see it.
Page 342
It would be imaginable that humans of different stocks possessed languages that all had the same
vocabulary, but the meanings of the words were different. The word that meant green among one tribe, meant same
among another, table for a third and so on. We could even
Page Break 343
imagine that the same sentences were used by the tribes, only with entirely different senses.
Page 343
Now in this case I should not say that they spoke the same language.
Page 343
We say that, in order to communicate, people must agree with one another about the meanings of words. But
the criterion for this agreement is not just agreement with reference to definitions, e.g., ostensive definitions--but
also an agreement in judgments. It is essential for communication that we agree in a large number of judgments.
Page 343
40. Language-game (2),†1 how can I explain it to someone, or to myself? Whenever A shouts "Slab" B
brings this kind of object.--I might also ask: how can I understand it? Well, only as far as I can explain it.
Page 343
But there is here a queer temptation which expresses itself in my inclination to say: I cannot understand it,
because the interpretation of the explanation is still vague.
Page Break 344
Page 344
That is to say, both to you and to myself I can only give examples of the application.
Page 344
41. The word "agreement" and the word "rule" are related, they are cousins. The phenomena of agreement
and of acting according to a rule hang together.
Page 344
There might be a cave-man who produced regular sequences of marks for himself. He amused himself, e.g.,
by drawing on the wall of the cave:
or
But he is not following the general expression of a rule. And when we say that he acts in a regular way that is not
because we can form such and expression.
Page 344
But suppose he now developed π! (I mean without a general expression of the rule.)
Page 344
Only in the practice of a language can a word have meaning.
Page 344
Certainly I can give myself a rule and then follow it.--But is it not a rule only for this reason, that it is
analogous to what is called 'rule' in human dealings?
Page Break 345
Page 345
When a thrush always repeats the same phrase several times in its song, do we say that perhaps it gives itself
a rule each time, and then follows the rule?
Page 345
42. Let us consider very simple rules. Let the expression be a figure, say this one:
| - - |
and one follows the rule by drawing a straight sequence of such figures (perhaps as an ornament).
| - - || - - || - - || - - || - - |
Under what circumstances should we say: someone gives a rule by writing down such a figure? Under what
circumstances: someone is following this rule when he draws that sequence? It is difficult to describe this.
Page 345
If one of a pair of chimpanzees once scratched the figure | - - | in the earth and thereupon the other the series |
- - | | - - | etc., the first would not have given a rule nor would the other be following it, whatever else went on at the
same time in the mind of the two of them.
If however there were observed, e.g., the phenomenon of a kind of instruction, of shewing how and of
imitation, of lucky and misfiring attempts, of reward and punishment and the like; if at length the one who had been
so trained put figures which he had never seen before one after another in sequence as in the first example, then we
should probably say that the one chimpanzee was writing rules down, and the other was following them.
Page 345
43. But suppose that already the first time the one chimpanzee had purposed to repeat this procedure? Only
in a particular technique of
Page Break 346
acting, speaking, thinking, can someone purpose something. (This 'can' is the grammatical 'can'.)
Page 346
It is possible for me to invent a card-game today, which however never gets played. But it means nothing to
say: in the history of mankind just once was a game invented, and that game was never played by anyone. That
means nothing. Not because it contradicts psychological laws. Only in a quite definite surrounding do the words
"invent a game" "play a game" make sense.
Page 346
In the same way it cannot be said either that just once in the history of mankind did someone follow a
sign-post. Whereas it can be said that just once in the history of mankind did some walk parallel with a board. And
that first impossibility is again not a psychological one.
Page 346
The words "language", "proposition", "order", "rule", "calculation", "experiment", "following a rule" relate to
a technique, a custom.
Page 346
A preliminary step towards acting according to a rule would be, say, pleasure in simple regularities such as
the tapping out of simple rhythms or drawing or looking at simple ornaments. So one might train someone to obey
the order: "draw something regular", "tap regularly". And here again one must imagine a particular technique.
Page Break 347
Page 347
You must ask yourself: under what special circumstances do we say that someone has "made a mere slip of
the pen" or "he could perfectly well have gone on, but on purpose did not do so" or "he had meant to repeat the
figure that he drew, but he happened not to do it".
Page 347
The concept "regular tapping", "regular figure", is taught us in the same way as 'light-coloured' or 'dirty' or
'gaudy'.
Page 347
44. But aren't we guided by the rule? And how can it guide us, when its expression can after all be interpreted
by us both thus and otherwise? I.e. when after all various regularities correspond to it. Well, we are inclined to say
that an expression of the rule guides us, i.e., we are inclined to use this metaphor.
Page 347
Now what is the difference between the proceeding according to a rule (say an algebraic expression) in which
one derives number after number according to the series, and the following proceeding: When we shew someone a
certain sign, e.g. , a numeral occurs to him; if he looks at the numeral and the sign, another numeral occurs
to him and so on. And each time we engage in this experiment the same series of numerals occurs to him. Is the
difference between this proceeding and that of going on according to the rule the psychological one that in the
second case we have something occurring to him? Might I not say: When he was following the rule "|- -|", then "| - |"
kept on occurring to him?
Page 347
Well in our own case we surely have intuition, and people say that intuition underlies acting according to a
rule.
Page Break 348
Page 348
So let us assume that that, so to speak, magical sign produces the series 123123123 etc.: is the sign then not
the expression of a rule? No.
Acting according to a rule presupposes the recognition of a uniformity and the sign "123123123 etc." was the
natural expression of a uniformity.
Page 348
Now perhaps it will be said that | 22 || 22 || 22 | is indeed a uniform sequence of marks but surely
| 2 || 22 || 222 || 2222 |
is not.
Page 348
Well, I might call this another kind of uniformity.
Page 348
45. Suppose however there were a tribe whose people apparently had an understanding of a kind of
regularity which I do not grasp. That is they would also have learning and instruction, quite analogous to that in § 42.
If one watches them one would say that they follow rules, learn to follow rules. The instruction effects, e.g.,
agreement in actions on the part of pupil and teacher. But if we look at one of their series of figures we can see no
regularity of any kind.
What should we say now? We might say: "They appear to be following a rule which escapes us," but also
"Here we have a phenomenon of behaviour on the part of human beings, which we don't understand".
Page 348
Instruction in acting according to the rule can be described without employing "and so on".
What can be described in this description is a gesture, a tone of voice, a sign which the teacher uses in a
particular way in giving instruction, and which the pupils imitate. The effect of these expressions can also
Page Break 349
be described, again without calling 'and so on' to our aid, i.e. finitely. The effect of "and so on" will be to produce
agreement going beyond what is done in the lessons, with the result that we all or nearly all count the same and
calculate the same.
Page 349
It would be possible, though, to imagine the very instruction without any "and so on" in it. But on leaving
school the people would still all calculate the same beyond the examples in the instruction they had had.
Page 349
Suppose one day instruction no longer produced agreement?
Page 349
Could there be arithmetic without agreement on the part of calculators?
Page 349
Could there be only one human being that calculated? Could there be only one that followed a rule?
Page 349
Are these questions like, say, this one: "Can one man alone engage in commerce?"
Page 349
It only makes sense to say "and so on" when "and so on" is understood. I.e., when the other is as capable of
going on as I am, i.e., does go on just as I do.
Page Break 350
Page 350
Could two people engage in trade with one another?
Page 350
46. When I say: "If you follow the rule, this must come out," that doesn't mean: it must, because it always
has. Rather, that it comes out is one of my foundations.
Page 350
What must come out is a foundation of judgment, which I do not touch.
On what occasion will it be said: "If you follow the rule this must come out"?
This may be a mathematical definition given in the train of a proof that a particular route branches. It may
also be that one says it to someone in order to impress the nature of a rule upon him, in order to tell him something
like: "You are not making an experiment here".
Page 350
47. "But at every step I know absolutely what I have to do; what the rule demands of me." The rule, as I
conceive it. I don't reason. The picture of the rule makes it clear how the picture of the series is to be continued.
"But I know at every step what I have to do. I see it quite clear before me. It may be boring, but there is no
doubt what I have to do."
Whence this certainty? But why do I ask that question? Is it not enough that this certainty exists? What for
should I look for a source of it? (And I can indeed give causes of it.)
Page Break 351
Page 351
When someone, whom we fear to disobey, orders us to follow the rule... which we understand, we shall
write down number after number without any hesitation. And that is a typical kind of reaction to a rule.
Page 351
"You already know how it is"; "You already know how it goes on."
Page 351
I can now determine to follow the rule .
Page 351
Like this:
But it is remarkable that I don't lose the meaning of the rule as I do it. For how do I hold it fast?
But--how do I know that I do hold it fast, that I do not lose it?! It makes no sense at all to say I have held it
fast unless there is such a thing as an outward mark of this. (If I were falling through space I might hold something,
but not hold it still.)
Page 351
Language just is a phenomenon of human life.
Page 351
48. One person makes a bidding gesture, as if he meant to say "Go!" The other slinks off with a frightened
expression. Might I not call this procedure "order and obedience", even if it happened only once?
Page 351
What is this supposed to mean: "Might I not call the proceeding----"? Against any such naming the objection
could naturally be made, that among human beings other than ourselves a quite different
Page Break 352
gesture corresponds to "Go away!" and that perhaps our gesture for this order has among them the significance of
our extending the hand in token of friendship. And whatever interpretation one has to give to a gesture depends on
other actions, which precede and follow the gesture.
Page 352
As we employ the word "order" and "obey", gestures no less than words are intertwined in a net of
multifarious relationships. If I am now construing a simplified case, it is not clear whether I ought still to call the
phenomenon "ordering" and "obeying".
Page 352
We come to an alien tribe whose language we do not understand. Under what circumstances shall we say
that they have a chief? What will occasion us to say that this man is the chief even if he is more poorly clad than
others? The one whom the others obey--is he without question the chief?
What is the difference between inferring wrong and not inferring? between adding wrong and not adding?
Consider this.
Page 352
49. What you say seems to amount to this, that logic belongs to the natural history of man. And that is not
combinable with the hardness of the logical "must".
Page Break 353
Page 353
But the logical "must" is a component part of the propositions of logic, and these are not propositions of
human natural history. If what a proposition of logic said was: Human beings agree with one another in such and
such ways (and that would be the form of the natural-historical proposition), then its contradictory would say that
there is here a lack of agreement. Not, that there is an agreement of another kind.
Page 353
The agreement of humans that is a presupposition of logic is not an agreement in opinions, much less in
opinions on questions of logic.
Page Break 354
Page Break 355
PART VII
1941 and 1944
Page 355
1. The role of propositions which deal with measures and are not 'empirical propositions'.--Someone tells me:
"this stretch is two hundred and forty inches long". I say: "that's twenty foot, so it's roughly seven paces" and now I
have got an idea of the length.--The transformation is founded on arithmetical propositions and on the proposition
that 12 inches = 1 foot.
Page 355
No one will ordinarily see this last proposition as an empirical proposition. It is said to express a convention.
But measuring would entirely lose its ordinary character if, for example, putting 12 bits each one inch long end to
end didn't ordinarily yield a length which can in its turn be preserved in a special way.
Page 355
Does this mean that I have to say that the proposition '12 inches = 1 foot' asserts all those things which give
measuring its present point?
No. The proposition is grounded in a technique. And, if you like, also in the physical and psychological facts
that make the technique possible. But it doesn't follow that its sense is to express these conditions. The opposite of
that proposition, 'twelve inches = one foot' does not say that rulers are not rigid enough or that we don't all count and
calculate in the same way.
Page 355
2. The proposition has the typical (but that doesn't mean simple) role of a rule.
Page Break 356
Page 356
I can use the proposition '12 inches = 1 foot' to make a prediction; namely that twelve inch-long pieces of
wood laid end to end will turn out to be of the same length as one piece measured in a different way. Thus the point
of that rule is, e.g., that it can be used to make certain predictions. Does it lose the character of a rule on that
account?
Page 356
Why can one make those predictions? Well,--all rulers are made alike; they don't alter much in length; nor do
pieces of wood cut up into inch lengths; our memory is good enough for us not to take numbers twice in counting
up to '12', and not to leave any out; and so on.
Page 356
But then can the rule not be replaced by an empirical proposition saying that rulers are made in such and
such ways, that people do this with them? One might give an ethnological account of this human institution.
Page 356
Now it is evident that this account could take over the function of a rule.
Page 356
If you know a mathematical proposition, that's not to say you yet know anything. If there is confusion in our
operations, if everyone calculates differently, and each one differently at different times, then there isn't any
calculating yet; if we agree, then we have only set our watches, but not yet measured any time.
If you know a mathematical proposition, that's not to say you yet know anything.
I.e., the mathematical proposition is only supposed to supply a framework for a description.
Page Break 357
Page 357
3. How can the mere transformation of an expression be of practical consequence?
Page 357
The fact that I have 25 × 25 nuts can be verified by my counting 625 nuts, but it can also be discovered in
another way which is closer to the form of expression '25 × 25'. And of course it is in the linking of these two ways
of determining a number that one point of multiplying lies.
Page 357
A rule qua rule is detached, it stands as it were alone in its glory; although what gives it importance is the
facts of daily experience.
Page 357
What I have to do is something like describing the office of a king;--in doing which I must never fall into the
error of explaining the kingly dignity by the king's usefulness, but I must leave neither his usefulness nor his dignity
out of account.
Page 357
I am guided in practical work by the result of transforming an expression.
Page 357
But in that case how can I still say that it means the same thing whether I say "here are 625 nuts", or "here are
25 × 25 nuts"?
Page Break 358
Page 358
If you verify the proposition "here are 625 ..." then in doing that you are also verifying "here are 25 × 25 ...";
etc. But the one form is closer to one kind of verification, the other closer to another.
Page 358
How can you say that "... 625 ..." and "... 25 × 25 ..." say the same thing?--Only through our arithmetic do
they become one.
Page 358
I can at one time arrive at the one, and at another time at the other kind of description, e.g. by counting. That
is to say, I can arrive at either of these forms in either way; but by different routes.
Page 358
It might now be asked: if the proposition "... 625..." was verified at one time in this way and at another time
in a different way, then did it mean the same thing both times?
Or: what happens if one method of verification gives '625', but the other not '25 × 25'?--Is "... 625..." true and
"... 25 times 25..." false? No.--To doubt the one means to doubt the other: that is the grammar given to these signs by
our arithmetic.
Page 358
If both ways of counting are supposed to justify giving a number then giving one number, even though in
different forms, is all that is provided for. On the other hand there is no contradiction in saying: "By one method of
counting I get 25 × 25 (and so 625), by the other not 625 (and so not 25 × 25)". Arithmetic has no objection to this.
Page Break 359
Page 359
For arithmetic to equate the two expressions is, one might say, a grammatical trick.
In this way arithmetic bars a particular kind of description and conducts description into other channels.
(And it goes without saying that this is connected with the facts of experience.)
Page 359
4. Suppose I have taught somebody to multiply; not, however, by using an explicit general rule, but only by
his seeing how I work out examples for him. I can then set him a new question and say: "Do the same with these
two numbers as I did with the previous ones". But I can also say: "If you do with these two what I did with the
others, then you will arrive at the number...". What kind of proposition is that?
"You will write such-and-such" is a prediction. 'If you write such-and-such, then you will have done it as I
shewed you' determines what he calls "following his example".
Page 359
'The solution to this problem is...'.--If I read this before I have worked out the sum,--what sort of proposition
is it?
Page 359
"If you do with these numbers what I did with the others, you will get..."--that surely means: "The result of
this calculation is..."--and that is not a prediction but a mathematical proposition. But it is none the less a prediction
too--A prediction of a special kind. Just as someone who at the end finds that he really does get such-and-such when
he adds up the column may be really surprised; for example may exclaim: Good Lord, it does come out!
Just think of this procedure of prediction and confirmation as a
Page Break 360
special language-game--I mean: isolated from the rest of arithmetic and its application.
Page 360
What is so singular about this game of prediction? What strikes me as singular would disappear if the
prediction ran: "If you believe that you have gone by my example, then you will have produced this" or: "If
everything seems correct to you, this will be the result". This game could be imagined in connexion with the
administration of a particular poison and the prediction would be that the injection affects our faculties, our memory
for example, in such-and-such a way.--But if we can imagine the game with the administration of a poison, then why
not with the administration of a medicine? But even then the weight of the prediction may still always rest on the
fact that the healthy man sees this as the result. Or perhaps: that this satisfies the healthy man.
Page 360
"Do as I do, and this is what you will get" doesn't of course mean: "If you do as I do then you will do as I
do"--nor: "Calculate like this, and you will calculate like this".--But what does "Do as I do" mean? In the language
game--it can simply be an order: "Now do as I do!"
Page 360
What is the difference between these predictions: "If you calculate correctly you will get this result"--and: "If
you believe you are calculating correctly you will get this result"?
Now who says that the prediction in my language-game above does not mean the latter? It seems not to--but
what shews this? Ask yourself in what circumstances the prediction would seem to predict the one thing and in
what circumstances the other. For it is clear that it all depends on the rest of the circumstances.
Page Break 361
Page 361
If you predict that I shall get this, are you not simply predicting that I shall take this result as
correct?--"But"--perhaps you say--"only because it really is correct!"--But what does it mean to say: "I take the
calculation as correct because it is correct"?
Page 361
And yet we can say: the person who is calculating in my language-game does not think of it as a peculiarity
of his nature that he gets this; the fact does not appear to him as a psychological one.
I am imagining him as under the impression that he has only followed a thread that is already there, and
accepting the How of the following as something that is a matter of course; and only knowing one explanation of his
action, namely: how the thread runs.
Page 361
He does just let himself go on when he follows the rule or the examples; however, he does not regard what
he does as a peculiarity of his course; he says, not: "so that's how I went", but: "so that's how it goes".
Page 361
But now, suppose someone did say at the end of the calculation in our language-game: "so that's how I
went"--or: "so this course satisfies me"--can I say he has misunderstood the whole language-game? Certainly not!
So long as he does not make some further unwelcome application of it.
Page 361
5. Isn't it the application that elicits that conception: that it is not we, but the calculation, that takes a certain
course?
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Page 362
The different 'conceptions' must correspond to different applications.
Page 362
For there is indeed a distinction between these two things: being surprised that the figures on the paper seem
to behave like this; and being surprised that this is what comes out as the result. In each case, however, I see the
calculation in a different context.
Page 362
I think of the feeling of its 'coming out' when for instance we add up a rather long column of numbers of
various patterns, and a round number of 1,000,000 comes out, as we had been told it would before, "Yes, by Jove,
another nought--" we say.
Page 362
"One wouldn't guess it from looking at the numbers", I might say.
Page 362
How would it be if we said--instead of '6 × 6 gives 36':--'The number 36's being given by 6 × 6'?--Replacing
the proposition by a substantival expression. (The proof shews the being given.)
Page 362
Why do you always want to look at mathematics under the aspect of finding and not of doing?
Page 362
It must have a great influence, that we use the words "right" and "true" and "wrong" and the form of
statement, in calculating. (Head-shaking and nodding.)
Page Break 363
Page 363
Why should I say that the knowledge that this is the way in which all human beings who have learned to
calculate do calculate isn't mathematical knowledge? Because it seems to point in the direction of a different
context.
Page 363
Then is working out what someone will get out by a calculation already applied mathematics?--and hence
also: working out what I myself get out?
Page 363
6. There is no doubt at all that in certain language-games mathematical propositions play the part of rules of
description, as opposed to descriptive propositions.
Page 363
But that is not to say that this contrast does not shade off in all directions. And that in turn is not to say that
the contrast is not of the greatest importance.
Page 363
We feel that mathematics stands on a pedestal--this pedestal it has because of a particular role that its
propositions play in our language games.
Page 363
What is proved by a mathematical proof is set up as an internal relation and withdrawn from doubt.
Page Break 364
Page 364
7. What is common to a mathematical proposition and a mathematical proof, that they are both called
"mathematical"?
Not, that the mathematical proposition has to be proved mathematically; not, that the mathematical proof has
to prove a mathematical proposition.
What is mathematical about an unproved proposition (an axiom)? what has it in common with a
mathematical proof?
Page 364
Should I answer: "The inference rules of mathematical proof are always mathematical propositions"? Or:
"Mathematical propositions and proofs are used in inference"? That would be getting closer to the truth.
Page 364
8. Proof must shew an internal relation, not an external one. For we might also imagine a process of
transforming a sentence by experiment, and a transformation which would be used to predict what would be
asserted by the transformed sentence. One might imagine, e.g., signs getting shifted through adding other signs to
them, in such fashion that they form a true prediction on the basis of the conditions expressed in their initial
position. And if you like, you may regard the calculating human being as an apparatus for such an experiment.
For, that a human being works out the result, in the sense that he doesn't write down the result at once, but
only after he has written down various other things--doesn't make him any the less a physical-chemical means of
producing one sequence of signs from another.
Page Break 365
Page 365
Thus I should have to say: The proved proposition is not: that sequence of signs which the man who has
received such-and-such schooling produces under such-and-such conditions.
Page 365
When we think of proving in that way, what we see in it changes entirely. The intermediate steps become an
uninteresting by-product. (Like a rattle in the insides of the automatic machine before it discharges its wares for us.)
Page 365
9. We say that a proof is a picture. But this picture stands in need of ratification, and that we give it when we
work over it.--
Page 365
True enough; but if it got ratification from one person, but not from another, and they could not come to any
understanding--would what we had here be calculation?
So it is not the ratification by itself that makes it calculation but the agreement of ratifications.
Page 365
For another game could quite well be imagined, in which people were prompted by expressions (similar
perhaps to general rules) to let sequences of signs come to them for particular practical purposes, i.e. ad hoc; and
that this even proved to pay. And here the 'calculations' if we choose to call them that, do not have to agree with one
another. (Here we might speak of 'intuition'.)
Page 365
The agreement of ratifications is the pre-condition of our language-game, it is not affirmed in it.
Page Break 366
Page 366
If a calculation is an experiment and the conditions are fulfilled, then we must accept whatever comes, as
the result; and if a calculation is an experiment then the proposition that it yields such and such a result is after all the
proposition that under such conditions this kind of sign makes its appearance. And if under these conditions one
result appears at one time and another at another, we have no right to say "there's something wrong here" or "both
calculations cannot be all right", but we should have to say: this calculation does not always yield the same result
(why need not be known). But although the procedure is now just as interesting, perhaps even more interesting, what
we have here now is no longer calculation. And this is of course a grammatical remark about the use of the word
"calculation". And this grammar has of course a point.
Page 366
What does it mean to reach an understanding about a difference in the result of a calculation? It surely
means to arrive at a calculation that is free of discrepancy. And if we can't reach an understanding, then the one
cannot say that the other is calculating too, only with different results.
Page 366
10. Now how about this--ought I to say that the same sense can only have one proof? Or that when a proof is
found the sense alters?
Of course some people would oppose this and say: "Then the proof of a proposition cannot ever be found,
for, if it has been found, it is no longer the proof of this proposition". But to say this is so far to say nothing at all.--
Page 366
It all depends what settles the sense of a proposition, what we choose
Page Break 367
to say settles its sense. The use of the signs must settle it; but what do we count as the use?--
Page 367
That these proofs prove the same proposition means, e.g.: both demonstrate it as a suitable instrument for the
same purpose.
Page 367
And the purpose is an allusion to something outside mathematics.
Page 367
I once said: 'If you want to know what a mathematical proposition says, look at what its proof proves'.†1
Now is there not both truth and falsehood in this? For is the sense, the point, of a mathematical proposition really
clear as soon as we can follow the proof?
Page 367
What Russell's '~f(f)' lacks above all is application, and hence meaning.
If we do apply this form, however, that is not to say that 'f(f)' need be a proposition in any ordinary sense or
'f(ξ)' a propositional function. For the concept of a proposition, apart from that of a proposition of logic, is only
explained in Russell in its general conventional features.
Here one is looking at language without looking at the language-game.
Page 367
When we say of different sequences of configuration that they shew e.g. that 25 × 25 = 625, it is easy enough
to recognize what fixes the
Page Break 368
place of this proposition, which is reached by the two routes.
Page 368
A new proof gives the proposition a place in a new system; here there is often a translation of one kind of
operation into a quite different kind. As when we translate equations into curves. And then we realize something
about curves and, by means of that, about equations. But what right have we to be convinced by lines of thought
which are apparently quite remote from the object of our thought?
Well, our operations are not more remote from that object than is, say, dividing in the decimal system from
sharing out nuts. Especially if one imagines (what is quite easy to imagine) that operation as originally invented for a
different purpose from that of making divisions and the like.
Page 368
If you ask: "What right have we?" the answer is: perhaps none.--What right have you to say that the
development of this system will always run parallel with that one? (It is as if you were to fix both inch and foot as
units, and assert that 12n inches will always be the same length as n feet.)
Page 368
When two proofs prove the same proposition it is possible to imagine circumstances in which the whole
surrounding connecting these proofs fell away, so that they stood naked and alone, and there were no cause to say
that they had a common point, proved the same proposition.
One has only to imagine the proofs without the organism of applications which envelopes and connects the
two of them: as it were stark naked. (Like two bones separated from the surrounding manifold
Page Break 369
context of the organism; in which alone we are accustomed to think of them.)
Page 369
11. Suppose that people calculated with numbers, and sometimes did divisions by expressions of the form (n
- n), and in this way occasionally got results different from the normal results of multiplying etc. But that nobody
minded this.--Compare with this: lists, rolls, of people are prepared, but not alphabetically as we do it; and in this
way it happens that in some lists the same name appears more than once.--But now it can be supposed that this does
not strike anyone; or that people see it, but accept it without worrying. As we could imagine people of a tribe who,
when they dropped coins on the ground, did not think it worth while to pick them up. (They have, say, an idiom for
these occasions: "It belongs to the others" or the like.)
Page 369
But now times have changed and people (at first only a few) begin to demand exactness. Rightly,
wrongly?--Were the earlier lists not really lists?--
Page 369
Say we quite often arrived at the results of our calculations through a hidden contradiction. Does that make
them illegitimate?--But suppose that we now absolutely refuse to accept such results, but still are afraid that some
might slip through.--Well then, in that case we have an idea which might serve as a model for a new calculus. As one
can have the idea of a new game.
Page Break 370
Page 370
The Russellian contradiction is disquieting, not because it is a contradiction, but because the whole growth
culminating in it is a cancerous growth, seeming to have grown out of the normal body aimlessly and senselessly.
Page 370
Now can we say: "We want a calculus which more certainly tells us the truth"?
Page 370
But you can't allow a contradiction to stand!--Why not? We do sometimes use this form in our talk, of
course not often--but one could imagine a technique of language in which it was a regular instrument.
It might for example be said of an object in motion that it existed and did not exist in this place; change
might be expressed by means of contradiction.
Page 370
Take a theme like that of Haydn's (St. Antony Chorale), take the part of one of Brahms's variations
corresponding to the first part of the theme, and set the task of constructing the second part of the variation in the
style of its first part. That is a problem of the same kind as mathematical problems are. If the solution is found, say
as Brahms gives it, then one has no doubt;--that is the solution.
Page 370
We are agreed on this route. And yet, it is obvious here that there may easily be different routes, on each of
which we can be in agreement, each of which we might call consistent.
Page Break 371
Page 371
'We take a number of steps, all legitimate--i.e. allowed by the rules--and suddenly a contradiction results. So
the list of rules, as it is, is of no use, for the contradiction wrecks the whole game!' Why do you have it wreck the
game?
But what I want is that one should be able to go on inferring mechanically according to the rule without
reaching any contradictory results. Now, what kind of provision do you want? One that your present calculus does
not allow? Well, that does not make that calculus a bad piece of mathematics,--or not mathematics in the fullest
sense. The meaning of the word "mechanical" misleads you.
Page 371
12. When, for some practical purpose, you want to avoid a contradiction mechanically, as your calculus so
far cannot do, this is e.g. like looking for a construction of the ...-gon, which you have up to now only been able to
draw by trial and error; or for a solution of a third degree equation, to which you have so far only approximated.
Page 371
What is done here is not to improve bad mathematics, but to create a new bit of mathematics.
Suppose I wanted to determine that the pattern '777' did not occur in the expansion of an irrational number. I
might take π and settle that if that pattern occurs, we replace it by '000'. Now I am told: that is not enough, for
whoever is calculating the places is prevented from looking back to the earlier ones. Now I need another calculus;
one in which I can be assured in advance that it cannot yield '777'. A mathematical problem.
Page Break 372
Page 372
'So long as freedom from contradiction has not been proved I can never be quite certain that someone who
calculates without thinking, but according to the rules, won't work out something wrong.' Thus so long as this
provision has not been obtained the calculus is untrustworthy.--But suppose that I were to ask: "How
untrustworthy?"--If we spoke of degrees of untrustworthiness mightn't this help us to take the metaphysical sting
out of it?
Were the first rules of the calculus not good? Well, we gave them only because they were good.--If a
contradiction results later,--have they failed in their office? No, they were not given for this application.
Page 372
I may want to supply my calculus with a particular kind of provision. This does not make it into a proper
piece of mathematics, but e.g. into one that is more useful for a certain purpose.
Page 372
The idea of the mechanization of mathematics. The fashion of the axiomatic system.
Page 372
13. But suppose the 'axioms' and 'methods of inference' are not just some kind of construction, but are
absolutely convincing. Well, this means that there are cases in which a construction out of these elements is not
convincing.
And the logical axioms are in fact not at all convincing if for the propositional variables we substitute
structures which no one originally foresaw as possible values, when, that is, we began by acknowledging the truth of
the axioms absolutely.
Page Break 373
Page 373
But what about saying: the axioms and methods of inference surely ought to be so chosen that they cannot
prove any false proposition?
Page 373
'We want, not just a fairly trustworthy, but an absolutely trustworthy calculus. Mathematics must be
absolute.'
Page 373
Suppose I had erected rules for a game of 'hare and hounds'--fancying it to be a nice amusing game.--Later,
however, I find that the hounds can always win once one knows how.
Now, let's say, I am dissatisfied with my game. The rules which I gave brought forth a result which I did not
foresee and which spoils the game for me.
Page 373
14. "N. came upon the fact that in their calculations people had often reduced by expressions of the form '(n n)'.
He pointed out the consequent discrepancy of results and shewed how this way of calculating had led to the loss
of human life."
Page 373
But let us suppose that other people too had noticed these contradictions, only they had not been able to give
any account of their source. They calculated as it were with a bad conscience. They had chosen one among
contradictory results but with uncertainty, whereas N's discovery would have made them quite certain.--But did they
tell themselves: "There's something wrong with our calculus"? Was their uncertainty of the same kind as ours when
we do a physical calculation
Page Break 374
but are not certain whether these formulae really give the correct result here? Or was it a doubt whether their
calculating was really calculating? In this case: what did they do to get over the difficulty?
Page 374
So far these people have only rarely made use of reduction by expressions of values. But some time
somebody discovers that they can actually arrive at any arbitrary result in this way.--What do they do now? Well,
we could imagine very different things. They may now, e.g., state that this kind of calculation has lost its point, and
that in future people are not to calculate in this way any more.
Page 374
'He believes that he is calculating'--one would like to say--'but as a matter of fact he is not calculating.'
Page 374
15. If the calculation lost its point for me as soon as I knew I could work out any arbitrary result--did it have
none so long as I did not know that?
Page 374
I may of course now declare all these calculations to be null--for I have given up doing them now--but does
that mean that they weren't calculations?
Page 374
I at one time inferred via a contradiction without realizing it. Is my result then wrong, or at any rate wrongly
got?
Page Break 375
Page 375
If the contradiction is so well hidden that no one notices it, why shouldn't we call what we do now proper
calculation?
Page 375
We say that the contradiction would destroy the calculus. But suppose it only occurred in tiny doses in
lightning flashes as it were, not as a constant instrument of calculation, would it nullify the calculus?
Page 375
Imagine people had fancied that (a + b)2 must be equal to a2 + b2. (Is this a fancy of the same kind as that
there must be a trisection of the angle by ruler and compass?) Is it possible, then, to fancy that two ways of
calculating had to yield the same result, if it is not the same?
Page 375
I add up a column, doing it in a variety of ways (e.g. I take the numbers in a different order), and I keep on
getting random different results.--I shall perhaps say: "I am in a complete muddle, either I am making random
mistakes in calculating, or I am making certain mistakes in particular connexions: e.g. always saying '7 + 7 = 15' after
'6 + 3 = 9'."
Or I might imagine that suddenly, once in the sum, I subtract instead of adding, but don't think I am doing
anything different.
Page 375
Now it might be that I didn't find the mistake and thought I had lost my wits. But this would not have to be
my reaction.
Page Break 376
Page 376
'Contradiction destroys the calculus'--what gives it this special position? With a little imagination, I believe, it
can certainly be shaken.
Page 376
To resolve these philosophical problems one has to compare things which it has never seriously occurred to
anyone to compare.
Page 376
In this field one can ask all sorts of things which, while they belong to the topic, still do not lead through its
centre.
A particular series of questions leads through the centre and out into the open. The rest get answered
incidentally.
It is enormously difficult to find the path through the centre.
Page 376
It goes via new examples and comparisons. The hackneyed ones don't shew us it.
Page 376
Let us suppose that the Russellian contradiction had never been found. Now--is it quite clear that in that case
we should have possessed a false calculus? For aren't there various possibilities here?
Page 376
And suppose the contradiction had been discovered but we were not excited about it, and had settled e.g.
that no conclusions were to be drawn from it. (As no one does draw conclusions from the 'Liar'.) Would this have
been an obvious mistake?
Page Break 377
Page 377
"But in that case it isn't a proper calculus! It loses all strictness!" Well, not all. And it is only lacking in full
strictness, if one has a particular ideal of rigour, wants a particular style in mathematics.
'But a contradiction in mathematics is incompatible with its application.
Page 377
'If it is consistently applied, i.e. applied to produce arbitrary results, it makes the application of mathematics
into a farce, or some kind of superfluous ceremony. Its effect is e.g. that of non-rigid rulers which permit various
results of measuring by being expanded and contracted.' But was measuring by pacing not measuring at all? And if
people worked with rulers made of dough, would that of itself have to be called wrong?
Page 377
Couldn't reasons be easily imagined, on account of which a certain elasticity in rulers might be desirable?
Page 377
"But isn't it right to manufacture rulers out of ever harder, more unalterable material?" Certainly it is right; if
that is what one wants!
Page 377
'Then are you in favour of contradiction?' Not at all; any more than of soft rulers.
Page 377
There is one mistake to avoid: one thinks that a contradiction must be senseless: that is to say, if e.g. we use
the signs 'p', '~', '.' consistently,
Page Break 378
then 'p.~p' cannot say anything.--But think: what does it mean to continue such and such a use 'consistently'? ('A
consistent continuation of this bit of a curve.')
Page 378
16. What does mathematics need a foundation for? It no more needs one, I believe, than propositions about
physical objects--or about sense impressions, need an analysis. What mathematical propositions do stand in need of
is a clarification of their grammar, just as do those other propositions.
Page 378
The mathematical problems of what is called foundations are no more the foundation of mathematics for us
than the painted rock is the support of a painted tower.
Page 378
'But didn't the contradiction make Frege's logic useless for giving a foundation to arithmetic?' Yes, it did. But
then, who said that it had to be useful for this purpose?
Page 378
One could even imagine a savage's having been given Frege's logic as an instrument with which to derive
arithmetical propositions. He derived the contradiction unawares, and now he derives arbitrary true and false
propositions from it.
Page 378
'Up to now a good angel has preserved us from going this way.' Well, what more do you want? One might
say, I believe: a good angel will always be necessary, whatever you do.
Page Break 379
Page 379
17. One says that calculation is an experiment, in order to shew how it is that it can be so practical. For we do
know that an experiment really does have practical value. Only one forgets that it possesses this value in virtue of a
technique which is a fact of natural history, but whose rules do not play the part of propositions of natural history.
Page 379
"The limits of empiricism."†1--(Do we live because it is practical to live? Do we think because thinking is
practical?)
Page 379
He knows that an experiment is practical; and so calculation is an experiment.
Page 379
Our experimental activities have indeed a characteristic physiognomy. If I see somebody in a laboratory
pouring a liquid into a test tube and heating it over a Bunsen burner, I am inclined to say he is making an
experiment.
Page 379
Let us suppose that people, who know how to count, want--just as we do--to know numbers for practical
purposes of various kinds. And to this end they ask certain people who, having had the practical problem explained
to them, shut their eyes, and let the appropriate number occur to them--here there wouldn't be any calculation,
however trustworthy the numbers given might be. This way of determining numbers might be even more
trustworthy in practice than any calculation.
Page Break 380
Page 380
A calculation--it might be said--is perhaps a part of the technique of an experiment, but is by itself not an
experiment.
Page 380
Do we forget that a particular application is part of a procedure's being an experiment? And the calculation
is an instrument of the application.
Page 380
For would anyone think of calling the translation of a cipher by means of a key an experiment?
Page 380
When I doubt whether n and m multiplied yield l, my doubt isn't about whether our calculating is going to
fall into confusion, and e.g. half of mankind say one thing is right and the other half another.
Page 380
An action is an 'experiment' only as seen from a certain point of view. And it is obvious that the action of
calculating can also be an experiment.
I may for example want to test what this man calculates, in such-and-such circumstances, when set this
question.--But isn't that exactly what you are asking when you want to know what 52 × 63 is? I may very well ask
that--my question may even be expressed in these words. (Compare: is the sentence "Listen, she's groaning!" a
proposition about her behaviour or about her suffering?)
But suppose I work over his calculation?--'Well, then I am making a further experiment so as to find out with
complete certainty that all normal human beings react like that.'--And if they do not react uniformly--which one is
the mathematical result?
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Page 381
18. "If calculation is to be practical, then it must uncover facts. And only experiment can do that."
But what things are 'facts'? Do you believe that you can shew what fact is meant by, e.g., pointing to it with
your finger? Does that of itself clarify the part played by 'establishing' a fact?--Suppose it takes mathematics to
define the character of what you are calling a 'fact'!
'It is interesting to know how many vibrations this note has! But it took arithmetic to teach you this question.
It taught you to see this kind of fact.
Page 381
Mathematics--I want to say--teaches you, not just the answer to a question, but a whole language-game with
questions and answers.
Page 381
Are we to say that mathematics teaches us to count?
Page 381
Can mathematics be said to teach us experimental methods of investigation? Or to help us to discover such
methods of investigation?
'To be practical, mathematics must tell us facts.'--But do these facts have to be the mathematical facts?--But
why should not mathematics, instead of 'teaching us facts', create the forms of what we call facts?
Page 381
"Yes but surely it remains an empirical fact that men calculate like this!"--Yes, but that does not make the
propositions used in calculating into empirical propositions.
Page Break 382
Page 382
"Yes, but surely our calculating must be founded on empirical facts!" Certainly. But what empirical facts are
you now thinking of? The psychological and physiological ones that make it possible, or those that make it a useful
activity? The connexion with the latter consists in the fact that the calculation is the picture of an experiment as it
practically always turns out. From the former it gets its point, its physiognomy; but that is certainly not to say that
the propositions of mathematics have the functions of empirical propositions. (That would almost be as if someone
were to believe that because only the actors appear in the play, no other people could usefully be employed upon the
stage of the theatre.)
Page 382
There are no causal connexions in a calculation, only the connexions of the pattern. And it makes no
difference to this that we work over the proof in order to accept it. That we are therefore tempted to say that it arose
as the result of a psychological experiment. For the psychical course of events is not psychologically investigated
when we calculate.
Page 382
'There are 60 seconds to a minute.' This proposition is very like a mathematical one. Does its truth depend on
experience?--Well, could we talk about minutes and hours, if we had no sense of time; if there were no clocks, or
could be none for physical reasons; if there did not exist all the connexions that give our measures of time meaning
and importance? In that case--we should say--the measure of time would have lost its meaning (like the action of
delivering check-mate if the game of chess were to disappear)--or it would have some quite different meaning. But
suppose our experience were like that--then would experience make the proposition false; and the contrary
experience make it true? No; that would not describe its function. It functions quite differently.
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Page 383
'Calculating, if it is to be practical, must be grounded in empirical facts.'--Why should it not rather determine
what empirical facts are?
Page 383
Consider: 'Our mathematics turns experiments into definitions'.
Page 383
19. But can't we imagine a human society in which calculating quite in our sense does not exist, any more
than measuring quite in our sense?--Yes.--But then why do I want to take the trouble to work out what mathematics
is?
Because we have a mathematics, and a special conception of it, as it were an ideal of its position and
function,--and this needs to be clearly worked out.
Page 383
Don't demand too much, and don't be afraid that your just demand will dwindle into nothing.
Page 383
It is my task, not to attack Russell's logic from within, but from without.
Page 383
That is to say: not to attack it mathematically--otherwise I should be doing mathematics--but its position, its
office.
Page 383
My task is, not to talk about (e.g.) Gödel's proof, but to by-pass it.
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Page 384
20. The problem: find the number of ways in which we can trace the joins in this wall:
continuously and without repetition, will be recognized by everyone as a mathematical problem.--If the drawing
were much bigger and more complicated, and could not be taken in at a glance, it could be supposed to change
without our noticing; and then the problem of finding that number (which perhaps changes according to some law)
would no longer be a mathematical one. But even if it does not change, the problem is, in this case, still not
mathematical.--But even when the wall can be taken in at a glance, that cannot be said to make the question
mathematical, as when we say: this question is now a question in embryology. Rather: here we need a mathematical
solution. (Like: here what we need is a model.)
Page 384
Did we 'recognize' the problem as a mathematical one because mathematics treats of making tracings from
drawings?
Page 384
Why, then, are we inclined to call this problem straight away a 'mathematical' one? Because we see at once
that here the answer to a mathematical question is practically all we need. Although the problem could easily be
seen as, for example, a psychological one.
Similarly with the task of folding a piece of paper in such-and-such a way.
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Page 385
It may look as if mathematics were here a science that makes experiments with units; experiments, that is, in
which it does not matter what kind of units they are, whether for instance they are peas, glass marbles, strokes and
so on.--Mathematics discovers only what holds for all these things. And so it does not discover anything about e.g.
their melting point, but that 2 and 2 of them are 4. And the first problem of the wall is a mathematical one, i.e. can be
solved by means of this kind of experiment.--And what does the mathematical experiment consist in? Well, in
setting things out and moving them about, in drawing lines, writing down expressions, propositions, etc. And we
must not be disturbed by the fact that the outward appearance of these experiments is not that of physical or
chemical experiments, etc.; they just are of a different kind. Only there is a difficulty here: the procedure is easy
enough to see, to describe,--but how is it to be looked at as an experiment? What is the head and what the tail of the
experiment here? What are the conditions of the experiment, what its result? Is the result what is yielded by the
calculation; or the pattern of calculation; or the assent (whatever that consists in) of the person doing the calculation?
Page 385
But does it make the principles of dynamics, say, into propositions of pure mathematics if we leave their
interpretation open, and then use them to produce a system of measurement?
Page 385
'A mathematical proof must be perspicuous'--this is connected with the perspicuousness of that figure.
Page 385
21. Do not forget that the proposition asserting of itself that it is unprovable is to be conceived as a
mathematical assertion--for that is not a matter of course.
It is not a matter of course that the proposition that such-and-such
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a structure cannot be constructed is to be conceived as a mathematical proposition.
Page 386
That is to say: when we said: "it asserts of itself"--this has to be understood in a special way. For here it is
easy for confusion to occur through the variegated use of the expression "this proposition asserts something of...".
Page 386
In this sense the proposition '625 = 25 × 25' also asserts something about itself: namely that the left-hand
number is got by the multiplication of the numbers on the right.
Page 386
Gödel's proposition, which asserts something about itself, does not mention itself.
Page 386
'The proposition says that this number cannot be got from these numbers in this way.'--But are you also
certain that you have translated it correctly into English? Certainly it looks as if you had.--But isn't it possible to go
wrong here?
Page 386
Could it be said: Gödel says that one must also be able to trust a mathematical proof when one wants to
conceive it practically, as the proof that the propositional pattern can be constructed according to the rules of proof?
Or: a mathematical proposition must be capable of being conceived as a proposition of a geometry which is
actually applicable to itself. And if one does this it comes out that in certain cases it is not possible to rely on a proof.
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Page 387
The limits of empiricism†1 are not assumptions unguaranteed, or intuitively known to be correct: they are
ways in which we make comparisons and in which we act.
Page 387
22. 'Let us assume that we have an arithmetical proposition saying that a particular number... cannot be
obtained from the numbers ..., ..., ..., by means of such and such operations. And let us assume that a rule of
translation can be given according to which this arithmetical proposition is translatable into the figures of the first
number--the axioms from which we are trying to prove it, into the figures of the other numbers--and our rules of
inference into the operations mentioned in the proposition.--If we had then derived the arithmetical proposition
from the axioms according to our rules of inference, then by this means we should have demonstrated its
derivability, but we should also have proved a proposition which, by that translation rule, can be expressed: this
arithmetical proposition (namely ours) is not derivable.'
What would have to be done here? I am supposing that we trust our construction of the propositional sign;
i.e. we trust the geometrical proof. So we say that this 'propositional pattern' can be obtained from those in such and
such ways. And, merely translated into another notation, this means: this number can be got from those by means of
these operations. So far the proposition and its proof have nothing to do with any special logic. Here the constructed
proposition was simply another way of writing the constructed number; it had the form of a proposition but we don't
compare it with other propositions as a sign saying this or that, making sense.
Page 387
But it must of course be said that that sign need not be regarded either as a propositional sign or as a number
sign.--Ask yourself: what makes it into the one, and what into the other?
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Page 388
If we now read the constructed proposition (or the figures) as a proposition of mathematical language (in
English, say) then it says the opposite of what we regard as proved. Thus we have demonstrated the falsity of the
real sense of the proposition and at the same time proved it--if, that is, we look on its construction from the admitted
axioms by means of the admitted rules of inference as a proof.
Page 388
If someone objects to us that we couldn't make such assumptions, for they would be logical or
mathematical assumptions, then we reply that we need only assume that someone has made a mistake in calculating
and so has reached the result we 'assume', and that for the time being he cannot find the mistake.
Page 388
Here once more we come back to the expression "the proof convinces us". And what interests us about
conviction here is neither its expression by voice or gesture, nor yet the feeling of satisfaction or anything of that
kind; but its ratification in the use of what is proved.
Page 388
It might justly be asked what importance Gödel's proof has for our work. For a piece of mathematics cannot
solve problems of the sort that trouble us.--The answer is that the situation, into which such a proof brings us, is of
interest to us. 'What are we to say now?'--That is our theme.
Page 388
However queer it sounds, my task as far as concerns Gödel's proof seems merely to consist in making clear
what such a proposition as:
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Page 389
"Suppose this could be proved" means in mathematics.
Page 389
23. We take it much too much for granted that we ask "How many?" and thereupon count and calculate.
Page 389
Do we count because it is practical to count? We count!--And in the same way we calculate.
Page 389
An experiment--or whatever one likes to call it--can be what we go on, sometimes in determining the
measurement of the thing measured, and sometimes even in determining the appropriate measure.
Page 389
Then is the unit of measurement in this way the result of measurements? Yes and no. Not the result reached
in measuring but perhaps the consequence of measurements.
Page 389
"Has experience taught us to calculate in this way?" would be one question and: "Is calculation an
experiment?" another.
Page 389
24. But isn't it possible to derive anything from anything according to some rule or other--nay, according to
any rule with a suitable interpretation? What does it mean for me to say, for example: this number can be got from
that pair of numbers by multiplying? Ask yourself: When does one use this proposition? Well, it isn't, e.g., a
psychological proposition saying what humans will do under certain conditions;
Page Break 390
what will satisfy them; nor is it a physical proposition concerning the behaviour of marks on paper. That is, it will be
applied in a surrounding other than a psychological or physical one.
Page 390
Assume that human beings learn to calculate, roughly as they in fact do; but now imagine different
'surroundings' which turn the calculating, now into a psychological experiment, now into a physical experiment with
the marks used in calculating, now into something else!
We assume that children learn counting and the simple kinds of sum by means of imitation, encouragement
and correction. But now, from a certain point of view, the nonagreement of the one who is doing the sums (i.e., the
mistakes) get treated, not as something bad, but as something psychologically interesting. "So you took that for
correct then, did you? The rest of us did it like this."
Page 390
I want to say that what we call mathematics, the mathematical conception of the proposition 13 × 14 = 182,
hangs together with the special position that we assign to the activity of calculating. Or, the special position that the
calculation... has in our life, in the rest of our activities. The language-game in which it is found.
Page 390
One may learn a piece of music by heart in order to be able to play it correctly; but also as part of a
psychological experiment, in order to investigate the working of musical memory. But one might also impress it on
one's memory in order thereby to judge some alterations in the score.
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Page 391
25. A language-game: I am doing multiplication and I say to the other: if you calculate right you will get
such-and-such a result; whereupon he carries out the calculation and is pleased at the correctness, and sometimes
the incorrectness, of my prediction. What does this language-game presuppose? That 'mistakes in calculating' are
easy to discover, and that agreement about the rightness or wrongness of the calculation is always quickly achieved.
Page 391
"If you agree with each step, you will arrive at this result."
Page 391
What is the criterion for a step in the calculation's being right; isn't it that the step seems right to me, and
other things of the same sort?
What is the criterion for my working out the same figure twice? Isn't it things like the figures' appearing to
me to be the same?
Page 391
What is the criterion for my having followed the paradigm here?
Page 391
"If you say that each step is correct, this is what will come out."
Page 391
The prediction really is: where you hold what you do to be right, this is what you will do.
Where you hold each step to be right, you will go this way.--And so you will reach this end.
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Page 392
A logical conclusion is being drawn, when no experience can contradict the conclusion without
contradicting the premises. I.e., when the inference is only a movement within the means of representation.
Page 392
26. In some language-game sentences are used; reports, orders and so on. And now the people also employ
calculating propositions. They say them to themselves perhaps, in between the orders and the reports.
Page 392
A language-game, in which someone calculates according to a rule and places the blocks of a building
according to the results of the calculation. He has learnt to operate with written signs according to rules.--Once you
have described the procedure of this teaching and learning, you have said everything that can be said about acting
correctly according to a rule. We can go no further. It is no use, for example, to go back to the concept of agreement,
because it is no more certain that one proceeding is in agreement with another, than that it has happened in
accordance with a rule. Admittedly going according to a rule is also founded on an agreement.
Page 392
To repeat, what the correct following of a rule consists in cannot be described more closely than by
describing the learning of 'proceeding according to the rule.' And this description is an everyday one, like that of
cooking and sewing, for example. It presupposes as much as these. It distinguishes one thing from another, and so it
informs a human being who is ignorant of something particular. (Cf. the remark: Philosophy doesn't use a
preparatory language, etc.)
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Page 393
For if you give me a description of how people are trained in following a rule and how they react correctly to
the training, you will yourself employ the expression of a rule in the description and will presuppose that I
understand it.
Page 393
We have, then, taught someone the technique of multiplying. So we employ expressions of acquiescence
and rejection. We shall also sometimes write down the goal of the multiplication: "You must get this, if it is to be
right," we may say to him.
Page 393
But now, can the pupil contradict and say: "How do you know that? And what do you want?--Do you want
me to follow the rule, or to get this result? For there's no need for the two to coincide." Well, we do not assume that
the pupil can say that; we assume that he accepts the rule as valid when approached from either side. That he
conceives both the individual step and the multiplication-pattern--and therefore the result of the multiplication--as
criteria of correctness, and if these are not in accord with one another, he believes there is some confusion of his
senses.
Page 393
27. Now is it imaginable for someone to follow the rule right and nevertheless to work out different results at
different times in multiplying 15 × 13? It all depends on what criteria one allows to count for correct following of the
rule. In mathematics the result itself is also a criterion for correct calculation. Here then it is unthinkable that one
should follow the rule right and should produce different patterns of multiplication.
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Page 394
Not letting a contradiction stand is something that characterises the technique of our employment of our
truth-functions. If we do let the contradiction stand in our language-games, we alter that technique--as, if we
departed from regarding a double negative as an affirmative. And this alteration would be significant, because the
technique of our logic is connected in its character with the conception of the truth-functions.
Page 394
"The rules compel me to..."--this can be said if only for the reason that it is not all a matter of my own will
what seems to me to agree with the rule. And that is why it can even happen that I memorize the rules of a
board-game and subsequently find out that in this game whoever starts must win. And it is something like this, when
I discover that the rules lead to a contradiction.
Page 394
I am now compelled to acknowledge that this is not a proper game.
Page 394
'The rules of multiplication, once adopted, compel me to acknowledge that ... × ... = ....' Suppose it were
disagreeable for me to acknowledge this proposition. Am I to say: "Well, this arises from that type of training.
Human beings who are so trained, so conditioned, then get into this kind of difficulty"?
Page 394
'How does one count in the decimal system?'--"We write 2 after 1, 3 after 2, ... 14 after 13 ... 124 after 123
and so on."--That is an explanation for someone who, while there is indeed something he doesn't know, does
understand 'and so on'. And understanding it
Page Break 395
means not understanding it as an abbreviation: it does not mean that he now sees a much longer series in his mind
than that of my examples. That he understands it comes out in his now making certain applications, in his saying
this and acting so in particular cases.
Page 395
"How do we count in the decimal system?"-- ......... --Now is that not an answer?--But it isn't one for
someone who did not understand the 'and so on'.--But may our explanation not have made it intelligible to him?
May he not, through it, have got hold of the idea of the rule?--Ask yourself what are the criteria for his having got
hold of the idea now.
Page 395
What is it that compels me?--the expression of the rule?--Yes, once I have been educated in this way. But
can I say it compels me to follow it? Yes: if here one thinks of the rule, not as a line that I trace, but rather as a spell
that holds us in thrall.
(("plain nonsense, and bumps..."))
Page 395
28. Why shouldn't it be said that such a contradiction as: 'heterological' ∈ heterological ≡ ~ ('heterological' ∈
heterological), shews a logical property of the concept 'heterological'?
Page 395
"'Two-syllabled' is heterological", or "'Four-syllabled' is not heterological" are empirical propositions. It might
be important in some contexts to find out whether adjectives possess the properties they stand for or not. The word
"heterological" would in that case
Page Break 396
be used in a language-game. But now, is the proposition "'h' ∈ h" supposed to be an empirical proposition? It
obviously is not one, nor should we admit it as a proposition in our language-game even if we had not discovered
the contradiction.
Page 396
'h' ∈ h ≡ ~ ('h' ∈ h) might be called 'a true contradiction'.--But this contradiction is not a significant
proposition! Agreed, but the tautologies of logic aren't either.
Page 396
"The contradiction is true" means: it is proved; derived from the rules for the word "h". Its employment is, to
shew that "'h'" is one of those words which do not yield a proposition when inserted into 'ξ ∈ h'.
Page 396
"The contradiction is true" means: this really is a contradiction, and so you cannot use the word "'h'" as an
argument in 'ξ ∈ h'.
Page 396
29. I am defining a game and I say: "If you move like this, then I move like this, and if you do that, then I do
this.--Now play." And now he makes a move, or something that I have to accept as a move and when I want to
reply according to my rules, whatever I do proves to conflict with the rules. How can this have come about? When I
set the rules up, I said something: I was following a certain use. I did not foresee what we should go on to do, or I
saw only a particular possibility. It was just as if I had said to somebody: "Give up the game; you can't mate with
these pieces" and had overlooked an existing possibility of mating.
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Page 397
The various half joking guises of logical paradox are only of interest in so far as they remind anyone of the
fact that a serious form of the paradox is indispensable if we are to understand its function properly. The question is:
what part can such a logical mistake play in a language-game?
Page 397
You may instruct someone what to do in such-and-such a case; and these instructions later prove
nonsensical.
Page 397
30. Logical inference is part of a language-game. And someone who carries out logical inferences in the
language-game follows certain instructions which were given him in the actual learning of the language-game. If,
say, a builder's mate is building a house in accordance with certain orders, he has to interrupt his cartage of materials
etc. from time to time and carry out certain operations with signs on paper; and then he takes up his work again in
conformity with the result.
Page 397
Imagine a procedure in which someone who is pushing a wheelbarrow comes to realize that he must clean
the axle of the wheel when the wheelbarrow gets too difficult to push. I don't mean that he says to himself.
"Whenever the wheelbarrow can't be pushed...", but he simply acts in this way. And he happens to shout to
someone else: "The wheelbarrow won't push; clean the axle", or again: "The wheelbarrow won't push. So the axle
needs cleaning." Now this is an inference. Not a logical one, of course.
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Page 398
Can I now say: "Non-logical inference can prove wrong; but logical inference not"?
Page 398
Is logical inference correct when it has been made according to rules; or when it is made according to correct
rules? Would it be wrong, for example, if it were said that p should always be inferred from ~p? But why should one
not rather say: such a rule would not give the signs '~p' and 'p' their usual meaning?
Page 398
We can conceive the rules of inference--I want to say--as giving the signs their meaning, because they are
rules for the use of these signs. So that the rules of inference are involved in the determination of the meaning of the
signs. In this sense rules of inference cannot be right or wrong.
Page 398
In the course of building A has measured the length and breadth of an area and gives B the order: "bring 15 ×
18 slabs". B is trained to multiply and to count out a number of slabs in conformity with the result.†1
Page 398
The sentence '15 × 18 = 270' need of course never be uttered.
Page 398
It might be said: experiment--calculation are poles between which human activities move.
Page 398
31. We condition a man in such-and-such ways; then bring a question to bear on him; and get a
number-sign. We go on to use
Page Break 399
this for our purposes and it proves practical. That is calculating.--No, it isn't enough! It might be an eminently
sensible procedure--but need not be what we call 'calculating'. As one could imagine sounds being emitted for
purposes now served by language, which sounds yet did not form a language.
It is essential to calculating that everyone who calculates right produces the same pattern of calculation. And
'calculating right' does not mean calculating with a clear understanding or smoothly; it means calculating like this.
Page 399
Every mathematical proof gives the mathematical edifice a new leg to stand on. (I was thinking of the legs of
a table.)
Page 399
32. I have asked myself: if mathematics has a purely fanciful application, isn't it still mathematics?--But the
question arises: don't we call it 'mathematics' only because e.g. there are transitions, bridges from the fanciful to
non-fanciful applications? That is to say: should we say that people possessed a mathematics if they used
calculating, operating with signs, merely for occult purposes?
Page 399
33. But in that case isn't it incorrect to say: the essential thing about mathematics is that it forms
concepts?--For mathematics is after all an anthropological phenomenon. Thus we can recognize it as the essential
thing about a great part of mathematics (of what is called 'mathematics') and yet say that it plays no part in other
regions. This insight by itself will of course have some influence on people once they learn to see mathematics in
this way. Mathematics is, then, a family; but that is not to say that we shall not mind what is incorporated into it.
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Page 400
We might say: if you did not understand any mathematical proposition better than you understand the
Multiplicative Axiom,†1 then you would not understand mathematics.
Page 400
34. --There is a contradiction here. But we don't see it and we draw conclusions from it. E.g. we infer
mathematical propositions; and wrong ones. But we accept these inferences.--And now if a bridge collapses, which
we built on the basis of these calculations, we find some other cause for it, or we call it an Act of God. Now was our
calculation wrong; or was it not a calculation?
Certainly, if we are explorers observing the people who do this we shall perhaps say: these people don't
calculate at all. Or: there is an element of arbitrariness in their calculations, which distinguishes the nature of their
mathematics from ours. And yet we should not be able to deny that these people have a mathematics.
Page 400
What kind of rules must the king†2 give so as to escape henceforward from the awkward position, which his
prisoner has put him in?--What sort of problem is this?--It is surely like the following one: how must I change the
rules of this game, so that such-and-such a situation cannot occur? And that is a mathematical problem.
Page 400
But can it be a mathematical problem to make mathematics into mathematics?
Page 400
Can one say: "After this mathematical problem was solved, human beings began really to calculate"?
Page Break 401
Page 401
35. What sort of certainty is it that is based on the fact that in general there won't actually be a run on the
banks by all their customers; though they would break if it did happen?! Well, it is a different kind of certainty from
the more primitive one, but it is a kind of certainty all the same.
I mean: if a contradiction were now actually found in arithmetic--that would only prove that an arithmetic
with such a contradiction in it could render very good service; and it will be better for us to modify our concept of
the certainty required, than to say that it would really not yet have been a proper arithmetic.
Page 401
"But surely this isn't ideal certainty!"--Ideal for what purpose?
Page 401
The rules of logical inference are rules of the language-game.
Page 401
36. What sort of proposition is: "The class of lions is not a lion, but the class of classes is a class"? How is it
verified? How could it be used?--So far as I can see, only as a grammatical proposition. To draw someone's attention
to the fact that the word "lion" is used in a fundamentally different way from the name of a lion; whereas the class
word "class" is used like the designation of one of the classes, say the class lion.
Page 401
One may say that the word "class" is used reflexively, even if for instance one accepts Russell's theory of
types. For it is used reflexively there too.
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Page 402
Of course to say in this sense that the class of lions is not a lion etc. is like saying one has taken an "e" for an
"a" when one has taken a ball for a bell.
Page 402
The sudden change of aspect in the picture of a cube and the impossibility of seeing 'lion' and 'class' as
comparable concepts.
Page 402
The contradiction says: "Look out....".
Page 402
But suppose that one gives a particular lion (the king of lions) the name "Lion"? Now you will say: But it is
clear that in the sentence "Lion is a lion" the word "lion" is being used in two different ways. (Tractatus
Logico-philosophicus.†1) But can't I count them as one kind of use?
Page 402
But if the sentence "Lion is a lion" is used in this way: shouldn't I be drawing your attention to anything, if I
drew your attention to the difference of employment of the two "lion"s?
Page 402
One can examine an animal to see if it is a cat. But at any rate the concept cat cannot be examined in this
way.
Page 402
Even though "the class of lions is not a lion" seems like nonsense,
Page Break 403
to which one can only ascribe a sense out of politeness; still I do not want to take it like that, but as a proper
sentence, if only it is taken right. (And so not as in the Tractatus.) Thus my conception is a different one here. Now
this means that I am saying: there is a language-game with this sentence too.
Page 403
"The class of cats is not a cat."--How do you know?
Page 403
The fable says: "The lion went for a walk with the fox", not a lion with a fox; nor yet the lion so-and-so with
the fox so-and-so. And here it actually is as if the species lion came to be seen as a lion. (It isn't as Lessing†1 says, as
if a particular lion were put in the place of some lion or other. "Reynard the Fox" does not mean: a fox of the name
"Reynard".)
Page 403
Imagine a language in which the class of lions is called "the lion of all lions", the class of trees "the tree of all
trees", etc.--Because people imagine all lions as forming one big lion. (We say: "God created man".)
Then it would be possible to set up the paradox that there isn't a definite number of all lions. And so on.
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Page 404
But would it be impossible to count and calculate in such a language?
Page 404
37. We might ask: What role can a sentence like "I always lie" have in human life? And here we can imagine
a variety of things.
Page 404
38. Is turning inches into centimetres logical inference? "The cylinder is 2 inches long.--So it is about 5 cm.
long." Is that a logical inference?†1
Page 404
But isn't a rule something arbitrary? Something that I lay down? And could I lay it down that the
multiplication 18 × 15 shall not yield 270?--Why not?--But then it just hasn't taken place according to the rule which
I first laid down, and whose use I have practised.
Is something that follows from a rule itself in turn a rule? And if not,--what kind of proposition am I to call
it?
Page 404
"It is... impossible for human beings to recognize an object as different from itself." Well, if only I had an
inkling how it is done,--I should try at once!--But, if it is impossible for us to recognize an object as different from
itself, is it quite possible to recognize two objects as different from one another? I have e.g. two chairs before me and
I recognize that they are two. But here I may sometimes believe that they are only one; and in that sense I can also
take one for two.--
Page Break 405
But that doesn't mean that I recognize the chair as different from itself! Very well; but then neither have I recognized
the two as different from one another. If you think you can do this and you are playing a kind of psychological
game, then translate it into a game with gestures. When you have two objects before you, point with each hand at
one of them; as if, as it were, you wanted to indicate that they were independent. If you only have one object before
you then you point to it with both hands in order to indicate that no difference between it and itself can be
made.--But now, why should one not play the game the opposite way?
Page 405
39. The words "right" and "wrong" are used when giving instruction in proceeding according to a rule. The
word "right" makes the pupil go on, the word "wrong" holds him back. Now could one explain these words to a
pupil by saying instead: "this agrees with the rule--that not"? Well yes, if he has a concept of agreement. But what if
this has yet to be formed? (The point is how he reacts to the word "agree".)
Page 405
One does not learn to obey a rule by first learning the use of the word "agreement".
Page 405
Rather, one learns the meaning of "agreement" by learning to follow a rule.
Page 405
If you want to understand what it means "to follow a rule", you have already to be able to follow a rule.
Page Break 406
Page 406
"If you accept this rule you must do this."--This may mean: the rule doesn't leave two paths open to you
here. (A mathematical proposition.) But I mean: the rule conducts you like a gangway with rigid walls. But against
this one can surely object that the rule could be interpreted in all sorts of ways.--Here is the rule, like an order! And
like an order too in its effect.
Page 406
40. A language-game: to bring something else; to bring the same. Now, we can imagine how it is
played.--But how can I explain it to anyone? I can give him this training.--But then how does he know what he is to
bring the next time as 'the same'--with what justice can I say that he has brought the right thing or the wrong?--Of
course I know very well that in certain cases people would turn on me with signs of opposition.
And does this mean e.g. that the definition of "same" would be this: same is what all or most human beings
with one voice take for the same?--Of course not.
Page 406
For of course I don't make use of the agreement of human beings to affirm identity. What criterion do you
use, then? None at all.
Page 406
To use the word without a justification does not mean to use it wrongfully.
Page 406
The problem of the preceding language-game exists also here: Bring me something red. For what shews me
that something is red? The
Page Break 407
agreement of the colour with a sample?--What right have I to say: "Yes, that's red"? Well, I say it; and it cannot be
justified. And it is characteristic of this language-game as of the other that all men consent in it without question.
Page 407
An undecided proposition of mathematics is something that is accepted neither as a rule nor as the opposite
of a rule, and which has the form of a mathematical statement.--But is this form a sharply circumscribed concept?
Page 407
Imagine as a property of a piece of music (say). But of course not as if the piece
went on endlessly, but as a property that can be recognized by the ear (as it were an algebraic property) of the piece.
Page 407
Imagine equations used as ornaments (wallpaper patterns), and now a test of these ornaments with a view to
discovering what kind of curves they correspond to. The test would be analogous to that of the contrapuntal
properties of a piece of music.
Page 407
41. A proof that shews that the pattern '777' occurs in the expansion of π, but does not shew where.†1 Well,
proved in this way this 'existential proposition' would, for certain purposes, not be a rule. But might it not serve e.g.
as a means of classifying expansion rules? It would perhaps be proved in an analogous way that '777' does not occur
in π2 but it does occur in π × e etc. The question would simply be: is it reasonable to say of the proof concerned: it
proves the existence
Page Break 408
of '777' in this expansion? This can be simply misleading. It is in fact the curse of prose, and particularly of Russell's
prose, in mathematics.
Page 408
What harm is done e.g. by saying that God knows all irrational numbers? Or: that they are already all there,
even though we only know certain of them? Why are these pictures not harmless?
Page 408
For one thing, they hide certain problems.--
Page 408
Suppose that people go on and on calculating the expansion of π. So God, who knows everything, knows
whether they will have reached '777' by the end of the world. But can his omniscience decide whether they would
have reached it after the end of the world? It cannot. I want to say: Even God can determine something
mathematical only by mathematics. Even for him the mere rule of expansion cannot decide anything that it does not
decide for us.
Page 408
We might put it like this: if the rule for the expansion has been given us, a calculation can tell us that there is
a '2' at the fifth place. Could God have known this, without the calculation, purely from the rule of expansion? I want
to say: No.
Page 408
42. When I said that the propositions of mathematics determine concepts, that is vague; for '2 + 2 = 4' forms
a concept in a different sense from 'p ⊃ p', '(x).fx ⊃ fa', or Dedekind's Theorem. The point is, there is a family of
cases.
Page Break 409
Page 409
The concept of the rule for the formation of an infinite decimal is--of course--not a specifically mathematical
one. It is a concept connected with a rigidly determined activity in human life. The concept of this rule is not more
mathematical than that of: following the rule. Or again: this latter is not less sharply defined than the concept of such
a rule itself.--For the expression of the rule and its sense is only a part of the language-game: following the rule.
Page 409
One has the same right to speak of such rules in general, as of the activities of following them.
Page 409
Of course, we say: "all this is involved in the concept itself", of the rule for example--but what that means is
that we incline to these determinations of the concept. For what have we in our heads, which of itself contains all
these determinations?
Page 409
A number is, as Frege says, a property of a concept--but in mathematics it is a mark of a mathematical
concept. ℵ0 is a mark of the concept of a cardinal number; and the property of a technique. 2ℵ0 is a mark of the
concept of an infinite decimal, but what is this number a property of? That is to say: of what kind of concept can one
assert it empirically?
Page 409
43. The proof of a proposition shews me what I am prepared to stake on its truth. And different proofs can
perfectly well cause me to stake the same thing.
Page Break 410
Page 410
Something surprising, a paradox, is a paradox only in a particular, as it were defective, surrounding. One
needs to complete this surrounding in such a way that what looked like a paradox no longer seems one.
Page 410
If I have proved that 18 × 15 = 270, I have thereby also proved the geometrical proposition that we get the
sign '270' by applying certain transformation rules to the sign '18 × 15'.--Now suppose that people, having their
vision or memory impaired (as we now put it) by some harmful drug, did not get '270' when they did this
calculation.--If we cannot use it to make a correct prediction of the result anyone is going to get under normal
circumstances, isn't the calculation useless? Well, even if it is, that does not shew that the proposition '18 × 15 = 270'
is the empirical proposition: people in general calculate like this.
Page 410
On the other hand it is not clear that the general agreement of people doing calculations is a characteristic
mark of all that is called "calculating". I could imagine that people who had learned to calculate might in particular
circumstances, say under the influence of opium, begin to calculate differently from one another, and might make
use of these calculations; and that they were not said not to be calculating at all and to be deranged--but that their
calculations were accepted as a reasonable procedure.
But must they not at least be trained to do the same calculations? Doesn't this belong essentially to the
concept of calculating? I believe that we could imagine deviations here too.
Page 410
44. Can we say that mathematics teaches an experimental method of investigation, teaches us to formulate
empirical questions (cf. p. 381).†1
Page Break 411
Can't it be said to teach me e.g. to ask whether a particular body moves according to the equation of a
parabola?--What does mathematics do in this case? Without it, or without the mathematicians, we should of course
not have arrived at the definition of this curve. But was defining this curve itself a piece of mathematics? Would it
for instance imply mathematics for people to investigate the movement of a body so as to see whether its path can
be represented by the construction of an ellipse with two pegs and a string? Was whoever invented this inquiry
doing mathematics?
He did create a new concept. But was it in the same way as mathematics does? Was it like the way the
multiplication 18 × 15 = 270 gives us a new concept?
Page 411
45. Then can't one say that mathematics teaches us to count? But if it teaches us to count, then why doesn't
it also teach us to compare colours?
Page 411
It is clear that if someone teaches us the equation of an ellipse he is teaching us a new concept. But if
someone proves to us that this ellipse and this straight line intersect at these points--he too is giving us a new
concept.
Page 411
Teaching us the equation of an ellipse is like teaching us to count. But it is also like teaching us to ask the
question: "Are there a hundred times as many marbles here as there?".
Page 411
Now if I had taught someone this question in a language-game, and a method of answering it, should I have
taught him mathematics? Or
Page Break 412
would it have been that only if he operated with signs?
Page 412
(Would that be like asking: "Would it be geometry, even if it only consisted of the Euclidian axioms?")
Page 412
If arithmetic teaches us the question "how many?", then why doesn't it also teach the question "how dark?"?
Page 412
Is a new conceptual connexion a new concept? And does mathematics create conceptual connexions?
Page 412
But the question "are there a hundred times as many marbles here as there?" is surely not a mathematical
question. And the answer to it is not a mathematical proposition. A mathematical question would be: "are 170
marbles a hundred times as many as 3 marbles?" (And this is a question of pure, not of applied mathematics.)
Page 412
Now ought I to say that whoever teaches us to count etc. gives us new concepts; and also whoever uses such
concepts to teach us pure mathematics?
Page 412
The word "concept" is too vague by far.
Page Break 413
Page 413
Mathematics teaches us to operate with concepts in a new way. And hence it can be said to change the way
we work with concepts.
Page 413
But only a mathematical proposition that has been proved or that is assumed as a postulate does this, not a
problematic proposition.
Page 413
46. But can we not experiment mathematically? for instance, try whether a square bit of paper can be folded
into a cat's head, where the physical properties of the paper, such as stiffness or elasticity, don't come into the
question? Now certainly we speak of trying here. And why not of experimenting too? This case is like one in which
we substitute pairs of numbers in the equation x2 + y2 = 25 in order to find by trial and error one that satisfies the
equation. And if one finally arrives at 32 + 42 = 25, is this proposition now the result of an experiment? For why did
we call our procedure "trying"? Should we also have called it that if someone always solved such problems first time
off with complete certainty (giving the signs of certainty) but without calculating? What did the experiment consist
in here? Suppose that before he gives the solution, he has a vision of it.--
Page 413
47. If a rule does not compel you, then you aren't following a rule.
Page 413
But how am I supposed to be following it; if I can after all follow it as I like?
Page Break 414
Page 414
How am I supposed to follow a sign-post, if whatever I do is a way of following it?
Page 414
But, that everything can (also) be interpreted as following, doesn't mean that everything is following.
Page 414
But how then does the teacher interpret the rule for the pupil? (For he is certainly supposed to give it a
particular interpretation.)--Well, how but by means of words and training?
And if the pupil reacts to it thus and thus; he possesses the rule inwardly.
But this is important, namely that this reaction, which is our guarantee of understanding, presupposes as a
surrounding particular circumstances, particular forms of life and speech. (As there is no such thing as a facial
expression without a face.)
(This is an important movement of thought.)
Page 414
48. Does a line compel me to trace it?--No; but if I have decided to use it as a model in this way, then it
compels me.--No; then I compel myself to use it in this way. I as it were cleave to it.--But here it is surely important
that I can form the decision with the (general) interpretation so to speak once for all, and can hold by it, and do not
interpret afresh at every step.
Page 414
The line, it might be said, intimates to me how I am to go. But that is of course only a picture. And if I judge
that it intimates this or that to me as it were irresponsibly, then I would not say that I was following it as a rule.
Page Break 415
Page 415
"The line intimates to me how I am to go": that is merely a paraphrase for:--it is my last court of appeal for
how I am to go.
Page 415
49. Imagine someone was following a line as a rule in this way: he holds a pair of compasses, one point of
which he carries along the rule, while the other point draws the line that follows the rule. And as he goes along the
rule-line in this fashion, he opens and doses the compasses, to all appearances with great exactness; as he does this,
he keeps on looking at the rule, as if it determined what he was doing. Now we, who are watching him, can see no
regularity of any kind in this opening and shutting. Hence we cannot learn his way of following the rule from him
either. But we believe him when he tells us that the line intimated to him to do what he did.
Page 415
Here we should perhaps really say: "The model seems to intimate to him how he has to go. But it isn't a
rule."
Page 415
50. Suppose someone follows the series "1, 3, 5, 7, ... in writing the series 2x + 1; and he asked himself "But
am I always doing the same thing, or something different every time?"
If from one day to the next someone promises: "Tomorrow I will give up smoking," does he say the same
thing every day, or every day something different?
Page 415
How is it to be decided whether he always does the same, when the line intimates to him how he is to go?
Page Break 416
Page 416
51. Didn't I want to say: Only the total picture of the use of the word "same" as it is interwoven with the uses
of the other words, can determine whether he does use the word as we do?
Page 416
Doesn't he always do the same, namely, let the line intimate to him how he is to go? But suppose he says
that the line intimates now this to him and now that? Couldn't he now say: in one sense he is always doing the same
thing, but still he isn't following a rule? And cannot the one who is following a rule nevertheless also say that in a
certain sense he does something different every time? Thus whether he does the same thing or keeps on doing
something different does not determine whether he is following a rule.
Page 416
The procedure of following a rule can be described only like this: by describing in a different way what we do
in the course of it.
Would it make sense to say: "If he did something different every time, we should not say he was following a
rule"? That does not make sense.
Page 416
52. Following a rule is a particular language-game. How can it be described? When do we say he has
understood the description?--We do this and that; if he now reacts in such-and-such a way, he understood the game.
And this "this and that," "in such-and-such a way" doesn't contain an "and so on."--Or: if I used an "and so on" in
the description, and were to be asked what that meant, I should have to explain that in turn by the narration of
examples; or perhaps by means of a gesture. And I should then regard it as a sign of understanding, if
Page Break 417
he, say, repeated the gesture with an intelligent expression; and in special cases acted in such-and-such a way.
Page 417
"But then doesn't the understanding reach beyond all the examples?" A very remarkable expression, and one
that is entirely natural.
Page 417
When one recounts examples and then says "and so on", this latter expression does not get explained in the
same way as the examples.
Page 417
For the "and so on" might on the one hand be replaced by an arrow, which indicates that the end of the series
of examples is not supposed to signify an end of their application. On the other hand "and so on" also means: that's
enough, you've understood; we don't need any more examples.
Page 417
If we replace the expression by a gesture, it might easily be that people only took our series of examples as
they were supposed to (only followed it correctly) when we made this gesture at the end. Thus it would be quite
analogous to pointing to an object or a place.
Page 417
53. Let us imagine a line intimating to me how I am to follow it; that is, as my eye travels along the line a
voice within me says: "This way!"--What is the difference between this process of obeying a kind of inspiration and
that of obeying a rule? For they are surely not
Page Break 418
the same. In the case of inspiration I await direction. I shall not be able to teach anyone else my 'technique' of
following the line. Unless, indeed, I teach him some way of hearkening, some kind of receptivity. But then, of
course, I cannot require him to follow the line in the same way as I do.
Page 418
It would also be possible to imagine such instruction in a sort of calculating. The children can calculate, each
in his own way--as long as they listen to their inner voice and obey it. Calculating in this way would be something
like composing.
Page 418
For doesn't the technique (the possibility) of training someone else in following it belong to the following of
a rule? To be sure, by means of examples. And the criterion of his understanding must be the agreement of their
individual actions. Hence it is not as it is with instruction in receptivity.
Page 418
54. How do you follow the rule?--"I do it like this;..." and now there follow general explanations and
examples.--How do you follow the voice of the line?--"I look at it, exclude all thoughts, etc., etc."
Page 418
"I wouldn't say that it kept on intimating something else to me--if I were following it as a rule". Can one say
that? "Doing the same" is tied up with "following the rule."
Page Break 419
Page 419
55. Can you imagine having absolute pitch, if you don't have it? Can you imagine it, if you do?--Can a blind
man imagine the seeing of red? Can I imagine it? Can I imagine spontaneously reacting in such-and-such a way if I
don't do so? Can I imagine it better, if I do do so?
Page 419
But can I play the language-game, if I don't react in this way?
Page 419
56. One does not feel that one must always be awaiting the tip-off of the rule. On the contrary. We are not
excited about what it will tell us to do next, rather it always tells us the same thing, and we do what it says.
Page 419
It might be said: we look at what we do in following according to the rule from the point of view: always the
same.
Page 419
You might say to someone you were beginning to train: "See, I always do the same:..."
Page 419
57. When do we say: "the line intimates this to me as a rule--always the same." And on the other hand: "It
keeps on intimating to me what I have to do--it is not a rule."
In the first case the story is: I have no further court of appeal for what I have to do. The rule does it all by
itself; I need only follow it (and following just is one thing). I don't feel for example that it's queer that the line
always tells me something.--
Page Break 420
Page 420
The other proposition says: I don't know what I shall do; the line will tell me.
Page 420
Calculating prodigies, who reach the right result, but cannot tell how. Are we to say: they don't calculate? (A
family of cases.)
Page 420
These things are finer spun than crude hands have any inkling of.
Page 420
58. May I not believe I am following a rule? Doesn't this case exist?
Page 420
And if so, then may I not also believe I am not following a rule and yet be following a rule? Isn't there
something that we should call that, too?
Page 420
59. How can I explain the word "same"?--Well, by means of examples.--But is that all? Isn't there a still
deeper explanation; or must not the understanding of the explanation be deeper?--Well, have I myself a deeper
understanding? Have I more than I give in the explanation?
But whence arises the feeling, as if I had more than I can say?
Page 420
Is it that I interpret the not-limited as length which reaches further than any given length? (The permission
that is not limited, as a permission for something limitless.)
Page 420
The image that goes with the limitless, is of something so big that we can't see its end.
Page Break 421
Page 421
The employment of the word "rule" is interwoven with the employment of the word "same".
Page 421
Consider: Under what circumstances will the explorer say: The word "..." of this tribe means the same as our
"and so on"? Imagine the details of their life and their language, which would justify him in this.
Page 421
"But I know what 'same' means!"--I have no doubt of that; I know it too.
Page 421
60. "The line intimates to me..." Here the emphasis is on the impalpability of the intimating. On this: that
nothing stands between the rule and my action.
Page 421
One could however imagine that someone multiplied, multiplied correctly, with such feelings; kept on
saying: "I don't know--now suddenly the rules intimates this to me!" and that we reply: "Of course; for you are
going ahead perfectly in accordance with the rule."
Page 421
Following a rule: this can be contrasted with various things. Among other things the explorer will also
describe the circumstances under which someone of these people doesn't want to say he is following a rule. Even
when in this or that respect it looks as if he were.
Page Break 422
Page 422
But might we not also calculate as we do calculate (all agreeing), etc., and yet at every step have the feeling of
being guided by the rule as if by a spell; astonished maybe at agreeing with one another? (Thanking the deity
perhaps for this agreement.)
Page 422
From this you can just see how much there is to the physiognomy of what we call "following a rule" in
everyday life!
Page 422
One follows the rule mechanically. Hence one compares it with a mechanism.
Page 422
"Mechanical"--that means: without thinking. But entirely without thinking? Without reflecting.
Page 422
The explorer might say: "they follow rules, but it looks different from the way it looks among us."
Page 422
"It--for no reason--intimates this or that to me" means: I can't teach you how I follow the line. I make no
presumption that you will follow it as I do, even if you do follow it.
Page 422
61. An addition of shapes together, so that some of the edges fuse, plays a very small part in our life.--As
when
Page Break 423
and
yield the figure
But if this were an important operation, our ordinary concept of arithmetical addition would perhaps be different.
Page 423
It is natural for us to regard it as a geometrical fact, not as a fact of physics, that a square piece of paper can
be folded into a boat or hat. But is not geometry, so understood, part of physics? No; we split geometry off from
physics. The geometrical possibility from the physical one. But what if we left them together? If we simply said: "If
you do this and this and this with the piece of paper then this will be the result"? What has to be done might be told
in a rhyme. For might it not be that someone did not distinguish at all between the two possibilities? As e.g. a child
who learns this technique does not. It does not know and does not consider whether these results of folding are
possible only because the paper stretches, is pulled out of shape, when it is folded in such-and-such a way, or
because it is not pulled out of shape.
Page 423
And now isn't it like this in arithmetic too? Why shouldn't it be possible for people to learn to calculate
without having the concepts of a mathematical and a physical fact? They merely know that this
Page Break 424
is always the result when they take care and do what they have learnt.
Let us imagine that while we were calculating the figures on paper altered erratically. A 1 would suddenly
become a 6 and then a 5 and then again a 1 and so on. And I want to assume that this does not make any difference
to the calculation because, as soon as I read a figure in order to calculate with it or to apply it, it once more becomes
the one that we have in our calculating. At the same time, one would see how the figures change during the
calculation; but we are trained not to worry about this.
Of course, even if we do not make the above assumption, this calculation could lead to useful results.
Here we calculate strictly according to rules, yet this result does not have to come out.--I am assuming that
we see no sort of regularity in the alteration of the figures.
Page 424
I want to say: this calculating could really be conceived as an experiment, and we might for example say:
"Let's try what will come out now if I apply this rule".
Page 424
Or again: "Let us make the following experiment: we'll write the figures with ink of such-and-such a
composition... and calculate according to the rule...."
Page 424
Now you might of course say: "In this case the manipulation of figures according to rules is not calculation."
Page 424
"We are calculating only when there is a must behind the result."--But suppose we don't know this must,--is
it contained in the calculation
Page Break 425
all the same? Or are we not calculating, if we do it quite naïvely?
Page 425
How about the following: You aren't calculating if, when you get now this, now that result, and cannot find a
mistake, you accept this and say: this simply shews that certain circumstances which are still unknown have an
influence on the result.
Page 425
This might be expressed: if calculation reveals a causal connexion to you, then you are not calculating.
Page 425
Our children are not only given practice in calculation but are also trained to adopt a particular attitude
towards a mistake in calculating.†1
Page 425
What I am saying comes to this, that mathematics is normative. But "norm" does not mean the same thing
as "ideal".
Page 425
62. The introduction of a new rule of inference can be conceived as a transition to a new language-game. I
can imagine one in which for example one person pronounces: 'p ⊃ q', another 'p' and a third draws the conclusion.
Page 425
63. Is it possible to observe that a surface is coloured red and blue;
Page Break 426
and not to observe that it is red? Imagine that a kind of colour adjective were used for things that are half red and
half blue: they are said to be 'bu'. Now might not someone be trained to observe whether something is bu; and not to
observe whether it is also red? Such a man would then only know how to report: "bu", or "not bu". And from the
first report we could draw the conclusion that the thing was partly red.
Page 426
I am imagining that the observation happens by means of a psychological sieve, which for example only lets
through the fact that the surface is blue-white-red (the French tricolour) or that it is not.
Page 426
Now if it is a special observation that the surface is partly red, how can this follow logically from the
preceding? Surely logic cannot tell us what to observe.
Page 426
Someone is counting apples in a box; he counts up to 100. Someone else says: "so there are at any rate 50
apples in the box" (that is all that interests him). This is surely a logical conclusion; but isn't it also a special piece of
experience?
Page 426
64. A surface which is divided into a number of strips is observed by several people. The colours of the strips
change every minute, all at the same time.
Page Break 427
Now the colours are: red, green, blue, white, black, blue.
It is observed:
red. blue ⊃ black. ⊃. white.
Page 427
It is also observed:
~green ⊃ ~white
and someone draws the conclusion:
~ green ⊃: red. blue. ~black.
And these implications are 'material implications' in Russell's sense.
Page 427
But then is it possible to observe that
red. blue ⊃ black. ⊃. white?
Isn't one observing arrangements of colours, and so for example that red.blue.black.white; and then deducing that
proposition?
But may not someone who is observing a surface be quite preoccupied with the question whether it is going
to turn green or not green; and if he now sees: ~green, need he be attentive to the particular colour that the surface
is?
And might not someone be preoccupied with the aspect red.blue ⊃ black.⊃. white? If, for example, he has
been taught to forget everything else, and only to look at the surface from this point of view. (In particular
circumstances it might be all one to people whether objects were red or green but important whether they had one of
these colours or some third one. And in this case there might be a colour word for "red or green".)
Page 427
But if one can observe that
red. blue:⊃ black. ⊃. white
and
~green ⊃ ~white
Page Break 428
then one can also observe, and not merely infer, that
~green ⊃: red. blue. ~black.
Page 428
If these are three observations then it must also be possible for the third observation not to agree with the
logical conclusion from the first two.
Page 428
Then is it imaginable that someone observing a surface should see the combination red-black (say as a flag),
but if he now sets himself to see one of the two halves, he sees blue instead of red? Well, you have just described
it.--It would perhaps be as if someone were to look at a group of apples and always see it as two groups of two
apples each, but as soon as he tried to take the whole lot in at a glance, they seemed to him to be five. This would be
a very remarkable phenomenon. And it is not one of whose possibility we take any notice.
Page 428
Remember that a rhombus, seen as a diamond, does not look like a parallelogram. Not that the opposite sides
seem not to be parallel, only the parallelism does not strike us.
Page 428
65. I could imagine someone saying that he saw a red and yellow star but did not see anything
yellow--because he sees the star as, so to speak, a conjunction of coloured parts, which he cannot separate.
Page 428
For example he had figures like these before him:
Page Break 429
Asked whether he sees a red pentagon he would say "yes"; asked whether he sees a yellow one, "no". In the same
way he says that he sees a blue triangle but not a red one.--When his attention was drawn to it perhaps he said: "Yes,
now I see it; I had not taken the star like that."
Page 429
And it might seem to him that you can't separate the colours in the star, because you can't separate the
shapes.
Page 429
You cannot learn to view the geography of a landscape as a whole, if you move on in it so slowly that you
have already forgotten one bit when you come to another.
Page 429
66. Why do I always speak of being compelled by a rule; why not of the fact that I can choose to follow it?
For that is equally important.
But I don't want to say, either, that the rule compels me to act like this; but that it makes it possible for me to
hold by it and let it compel me.
Page 429
And if e.g. you play a game, you keep to its rules. And it is an interesting fact that people set up rules for the
fun of it, and then keep to them.
Page Break 430
Page 430
My question really was: "How can one keep to a rule†1?" And the picture that might occur to someone here
is that of a short bit of handrail, by means of which I am to let myself be guided further than the rail reaches. [But
there is nothing there; but there isn't nothing there!] For when I ask "How can one...", that means that something
here looks paradoxical to me; and so a picture is confusing me.
Page 430
"I never thought of its being red too; I only saw it as part of a multi-coloured ornament."
Page 430
Logical inference is a transition that is justified if it follows a particular paradigm and its rightness is not
dependent on anything else.
Page 430
67. We say: "If you really follow the rule in multiplying, you must all get the same result." Now if this is only
the somewhat hysterical way of putting things that you get in university talk, it need not interest us overmuch.
It is however the expression of an attitude towards the technique of calculation, which comes out everywhere
in our life. The emphasis of the must corresponds only to the inexorableness of this attitude both to the technique of
calculating and to a host of related techniques.
Page 430
The mathematical Must is only another expression of the fact that mathematics forms concepts.
Page Break 431
Page 431
And concepts help us to comprehend things. They correspond to a particular way of dealing with situations.
Page 431
Mathematics forms a network of norms.
Page 431
68. It is possible to see the complex formed of A and B, without seeing A or B. It is even possible to call the
complex a "complex of A and B" and to think that this name points to some kind of kinship of this whole with A
and with B. Thus it is possible to say that one is seeing a complex formed from A and B but neither A nor B. As for
example one might say that there is a reddish yellow here but neither red nor yellow.
Page 431
Now can I have A and B before me and also see them both, but only observe A ∨ B? Well, in a certain sense
this is surely possible. I was thinking of it like this: the observer is preoccupied with a particular aspect; for example,
he has a special kind of paradigm before him; he is engaged in a particular routine of application.--And just as he can
be adjusted to A ∨ B, so he can also be adjusted to A.B. Thus only A.B strikes him, and not, for example, A. To be
adjusted to A ∨ B might be said to mean: to react to such-and-such a situation with the concept 'A ∨ B'. And one can
of course do exactly the same thing with A.B too.
Page 431
Say someone is interested only in A.B, and so whatever happens he judges merely either "A.B", or "~(A.B)";
then I can imagine his judging "A.B" and saying "No, I see A.B" when he is asked "Do you see B?" As for example
some people who see A.B will not concede that they see A ∨ B.
Page Break 432
Page 432
69. But 'seeing' the surface 'blue all over' and 'seeing' it 'red all over' are surely 'genuine' experiences, and yet
we say that a man could not have them at the same time.
Page 432
Now suppose he assured us that he saw this surface really red all over and blue all over at the same time? We
should have to say: "You aren't making yourself intelligible to us."
Page 432
With us the proposition "1 foot = ... cm." is timeless. But we could imagine the case in which the foot and the
metre gradually altered somewhat, and kept on having to be compared anew in order for us to calculate their
translations into one another.
Page 432
But have we not determined the relative length of foot and metre experimentally? Yes; but the result was
given the character of a rule.
Page 432
70. In what sense can a proposition of arithmetic be said to give us a concept? Well let us interpret it, not as a
proposition, as something that decides a question, but as a--somehow accepted--connexion of concepts.
Page 432
The equating of 252 and 625 could be said to give me a new concept. And the proof shews what the position
is regarding this equality.--"To give a new concept" can only mean to introduce a new employment of a concept, a
new practice.
Page Break 433
Page 433
"How can the proposition be separated from its proof?" This question betrays a false conception.
Page 433
The proof is part of the surroundings of the proposition.
Page 433
'Concept' is a vague concept.
Page 433
71. It is not in every language-game that there occurs something that one would call a concept.
Page 433
Concept is something like a picture with which one compares objects.
Page 433
Are there concepts in language-game (2)†1? Still, it would be easy to add to it in such a way that "slab",
"block" etc. became concepts. For example, by means of a technique of describing or portraying those objects.
There is of course no sharp dividing line between language-games which work with concepts and others. What is
important is that the word "concept" refers to one kind of expedient in the mechanism of language-games.
Page Break 434
Page 434
72. Consider a mechanism. For example this one:
While the point A describes a circle, B describes a figure eight. Now we write this down as a proposition of
kinematics.
When I work the mechanism its movement proves the proposition to me; as would a construction on paper.
The proposition corresponds e.g. to a picture of the mechanism with the paths of the points A and B drawn in. Thus
it is in a certain respect a picture of that movement. It holds fast what the proof shews me. Or--what it persuades me
of.
Page 434
If the proof registers the procedure according to the rule, then by doing this it produces a new concept.
Page 434
In producing a new concept it convinces me of something. For it is essential to this conviction that the
procedure according to these rules must always produce the same configuration. ('Same', that is, by our ordinary
rules of comparison and copying.)
Page 434
With this is connected the fact that we can say that proof must shew the existence of an internal relation. For
the internal relation is the operation producing one structure from another, seen as equivalent to the picture of the
transition itself--so that now the transition
Page Break 435
according to this series of configurations is eo ipso a transition according to those rules for operating.
Page 435
In producing a concept, the proof convinces me of something: what it convinces me of is expressed in the
proposition that it has proved.
Page 435
Problem: Does the adjective "mathematical" always mean the same: when we speak of "mathematical"
concepts, of "mathematical" propositions and of mathematical proofs?
Page 435
Now what has the proved proposition got to do with the concept created by the proof? Again: what has the
proved proposition got to do with the internal relations demonstrated by the proof?
Page 435
The picture (proof-picture) is an instrument producing conviction.
Page 435
It is clear that one can also apply an unproved mathematical proposition; even a false one.
Page 435
The mathematical proposition says to me: Proceed like this!
Page 435
73. "If the proof convinces us, then we must also be convinced of the axioms." Not as by empirical
propositions, that is not their role. In the language-game of verification by experience they are excluded. Are, not
empirical propositions, but principles of judgement.
Page Break 436
Page 436
A language-game: How have I to imagine one in which axioms, proofs and proved propositions occur?
Someone who hears a bit of logic for the first time at school is straightway convinced when he is told that a
proposition implies itself, or when he hears the law of contradiction, or of excluded middle.--Why is he immediately
convinced? Well, these laws fit entirely into the use of language that he is so familiar with. Then he learns perhaps to
prove more complicated propositions of logic. The proofs are exhibited to him, and he is again convinced; or he
invents proofs himself.
In this way he learns new techniques of inference. And also, what account to lay it to, if now errors appear.
Page 436
The proof convinces him that he must hold fast to the proposition, to the technique that it prescribed; but it
also shews him how he can hold fast to the proposition without running any risk of getting into conflict with
experience.
Page 436
74. Any proof in applied mathematics may be conceived as a proof in pure mathematics which proves that
this proposition follows from these propositions, or can be got from them by means of such and such operations;
etc.
Page 436
The proof is a particular path. When we describe it, we do not mention causes.
Page 436
I act on the proof.--But how?--I act according to the proposition that got proved.
Page Break 437
Page 437
The proof taught me e.g., a technique of approximation. But still it proved something, convinced me of
something. That is expressed by the proposition: It says what I shall now do on the strength of the proof.
Page 437
The proof belongs to the background of the proposition. To the system in which the proposition has an
effect.
Page 437
See, this is how 3 and 2 yield 5. Note this proceeding.
Page 437
Every empirical proposition may serve as a rule if it is fixed, like a machine part, made immovable, so that
now the whole representation turns around it and it becomes part of the coordinate system, independent of facts.
Page 437
"This is how it is, if this proposition is derived from these ones. That you have to admit."--What I admit is,
this is what I call such a procedure.
FOOTNOTES
Page 29
†1 The numbering of Wittgenstein's manuscripts and typescripts follows that of the list given in the article
"The Wittgenstein Papers" by G. H. von Wright in: The Philosophical Review, Vol. LXXVIII, 1969.
Page 44
†1 Principia Mathematica: What is implied by a true premiss is true. Pp. (Eds.)
Page 53
†1 Is 'this correlation' here the correlation of the patterns in the proof itself? A thing cannot be at the same
time the measure and the thing measured. (Note in margin.)
Page 53
†2 On the strength of the figure I shall e.g. try to effect one correlation, but not the other, and shall say that
that one is not possible. (Note in margin.)
Page 65
†1 See above, § 36. (Eds.)
Page 68
†1 Cf. Tractatus 6.1261: In logic process and result are equivalent. (Eds.)
Page 80
†1 The last sentence added in March, 1944. (Eds.)
Page 89
†1 Grundgesezte der Arithmetik I, xviii. (Eds.)
Page 95
†1 ibid., I, XVI.
Page 101
†1 This remark came at the end of the cut-up typescript which is the source of this Part I and the following
Appendix I. (Cf. Preface, p. 33). But its place in the collection of cuttings is not quite clear--and for that reason the
editors did not include the remark in the first edition. It is uncertain whether the words "Connected with this" relate
to the preceding remarks 169 and 170. The remark was put between brackets in the typescript too. (Eds.)
Page 103
†1 Several alternatives to the last sentence were indicated in the MS. "We fix our eye on the expression of
meaning." "We investigate the expression of meaning." "We focus on the expression of meaning." "Focus on the
expression of meaning." (Eds.)
Page 107
†1 [Marginal note:] What is meant by "ne non p" and "non ne p"?
Page 110
†1 This remark was in handwriting on the back of the page. (Eds.)
Page 174
†1 But compare § 38. (Eds.)
Page 197
†1 Probably refers to Bertrand Russell's article The Limits of Empiricism. Proceedings of the Aristotelian
Society, 1935-1936. (Eds.)
Page 208
†1 Cf. Grundgesetze der Arithmetik, Vol. I, Preface p. XVI. Compare also above, p. 95. (Eds.)
Page 234
†1 Philosophical Investigations, § 2 quoted here infra, VI § 40n. (Eds.)
Page 237
†1 See note, p. 197. (Eds.)
Page 241
†1 Grundgesetze der Arithmetik, Vorwort, p. XVI.
Page 264
†1 D. Hilbert. Über das Unendliche. Mathematische Annalen 95 (1926) (Translated in: Putnam and
Bennaceraf, Philosophy of Mathematics; also in Heijenoort, From Frege to Gödel.) Eds.
Page 283
†1 I.e. Axiom of choice. (Eds.)
Page 299
†1 Cf. Frege, Die Grundlagen der Arithmetik, § 65n; "Begriff ist für mich ein mögliches Prädikat..."; also:
Grundgesetze der Arithmetik II, p. 69: "Eine Definition eines Begriffes (möglichen Pradikates) muss vollständig
sein...". (Eds.)
Page 305
†1 It is possible to devise a 'rule of counting' to fit the text--e.g. one of counting up the positions commanded
from eight of the ten squares commanding nine (the colour of the bishop's diagonals being determinate). But we
have not been able to find out what routine Wittgenstein did have in mind. (Eds.)
Page 318
†1 An amendment of and addition to the fourth sentence of § 4, Part I, which runs: "And isn't this series just
defined by this sequence?" (p. 37). In a revision belonging to about the same period as the present passage there
then comes "Not by the sequence; but by a rule; or by the training in the use of a rule." (Eds.)
Page 343
†1 §2 of Philosophical Investigations. An imaginary language 'is supposed to serve for communication
between a builder A and an assistant B. A is constructing a building out of building stones; there are cubes, pillars,
slabs and beams available. B has to pass him the blocks, and in the order that A needs them in. To this end they
make use of a language consisting of the words "cube", "pillar", "slab" and "beam". A calls them out;--B brings the
block that he has learnt to bring at this call. Conceive this as a complete primitive language.' (Eds.)
Page 367
†1 cf. Philosophical Grammar, p. 369 f; cf. also Philosophical Remarks, pp. 183, 184. (Eds.)
Page 379
†1 cf. p. 197 n.
Page 387
†1 cf. note p. 197. (Eds.)
Page 398
†1 cf. Philosophical Investigations, § 2, § 8. (Eds.)
Page 400
†1 I.e. the Axiom of Choice. (Eds.)
Page 400
†2 Presumably the king who made the law that all who came to his city must state their business and be
hanged if they lied. A sophist said he came to be hanged under that law.--(Eds.)
Page 402
†1 Cf. Tractatus 3.323. (Eds.)
Page 403
†1 Abbandlungen [[sic]] über die Fabeln, in: G. E. Lessing, Fabeln, 1759. (Eds.)
Page 404
†1 Cf. Part I, § 9. (Eds.)
Page 407
†1 Cf. Part V, § 27. (Eds.)
Page 410
†1 cf. p. 381.
Page 425
†1 [Variant]:... towards a departure from the norm. (Ed.)
Page 430
†1 Cf. Part VI, § 47. (Eds.)
Page 433
†1 Philosophical Investigations, § 2; here, above, p. 343n. (Eds.)
<tabs before most paragraphs need to be removed in Culture and Value>
CULTURE AND VALUE
Page i
Page Break ii
Page ii
Ludwig Wittgenstein from Blackwell Publishers
The Wittgenstein Reader
Edited by Anthony Kenny
Philosophical Investigations
Translated by G. E. M. Anscombe
Zettel
Second Edition
Edited by G. E. M. Anscombe and G. H. von Wright
Remarks on Colour
Edited by G. E. M. Anscombe
Remarks on the Foundations of Mathematics
Third edition
Edited by G. H. von Wright, Rush Rhees and G. E. M. Anscombe
Philosophical Remarks
Edited by Rush Rhees
Philosophical Grammar
Edited by Rush Rhees
On Certainty
Edited by G. E. M. Anscombe and G. H. von Wright
Lectures and Conversations
Edited by Cyril Barrett
Culture and Value
Edited by G. H. von Wright
The Blue and Brown Books
Preliminary Studies for the Philosophical Investigations
Remarks on the Philosophy of Psychology, Vol. I
Edited by G. E. M. Anscombe and G. H. von Wright
Remarks on the Philosophy of Psychology, Vol. II
Edited by G. E. M. Anscombe and Heikki Nyman
Last Writings on the Philosophy of Psychology, Vol. I
Edited by G. H. von Wright and Heikki Nyman
Last Writings on the Philosophy of Psychology, Vol. II
Edited by G. H. von Wright and Heikki Nyman
Page Break iii
Page Break iv
Page Break v
Titlepage
Page v
LUDWIG WITTGENSTEIN
Culture and Value
A Selection from the Posthumous Remains
Edited by
Georg Henrik von Wright
in Collaboration with Heikki Nyman
Revised Edition of the Text by
Alois Pichler
Translated by
Peter Winch
Page Break vi
Copyright page
Page vi
English translation copyright © Blackwell Publishers Ltd, 1998
German edition copyright © Suhrkamp Verlag, 1994
First edition of Vermischte Bemerkungen 1977
Second edition 1978
Amended second edition with English translation 1980
Reprinted 1984, 1988, 1989, 1992
Revised second edition of Vermischte Bemerkungen 1994
Revised second edition with English translation 1998
Reprinted 1998
Blackwell Publishers Ltd
108 Cowley Road
Oxford OX4 1JF
UK
Blackwell Publishers Inc.
350 Main Street
Malden, Massachusetts 02148
USA
All rights reserved. Except for the quotation of short passages for the purposes of criticism and review, no part of
this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means,
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Except in the United States of America, this book is sold subject to the condition that it shall not, by way of trade or
otherwise, be lent, resold, hired out, or otherwise circulated without the publisher's prior consent in any form of
binding or cover other than that in which it is published and without a similar condition including this condition
being imposed on the subsequent purchaser.
British Library Cataloguing in Publication Data
A CIP catalogue record for this book is available from the British Library.
Library of Congress Cataloging-in-Publication Data
Wittgenstein. Ludwig. 1889-1951.
[Vermischte Bemerkungen. English & German]
Culture and value: a selection from the posthumous remains: Ludwig Wittgenstein: edited by Georg Henrik von Wright
in collaboration with Heikki Nyman: revised edition of the text by Alois Pichler: translated by Peter Winch.--Rev. 2nd
ed. with English translation.
p. cm.
Includes bibliographical references and indexes.
ISBN 0-631-20570-5 (alk. paper)--ISBN 0-631-20571-3 (pbk: alk. paper)
1. Philosophy-Miscellanea. I. Wright, G. H. von (Georg Henrik), 1916--. II. Nyman, Heikki. III. Pichler, Alois, 1966--.
IV. Title.
B3376.W562E5 1997
192-dc21 96-37738
Page vi
CIP
Typeset in 10 on 12pt Sabon
by Ace Filmsetting Ltd, Frome, Somerset
Printed and bound by Athenaeum Press Ltd, Gateshead, Tyne & Wear
This book is printed on acid-free paper
Page Break vii
Page Break viii
Contents
Page viii
Foreword to the Edition of 1977 ixe
Foreword to the 1994 Edition xiie
Editorial Note xive
Note by Translator xviiie
Culture and Value 2e
A Poem 100e
Notes 101e
Appendix 105
List of Sources 107
List of Sources, Arranged Alphanumerically 113
Index of Beginning Remarks 120e
Page Break ix
Foreword to the Edition of 1977
Page ix
In the manuscript material left by Wittgenstein there are numerous notes which do not belong directly with his
philosophical works although they are scattered among the philosophical texts. Some of these notes are
autobiographical, some are about the nature of philosophical activity, and some concern subjects of a general sort,
such as questions about art or about religion. It is not always possible to separate them sharply from the
philosophical text; in many cases, however, Wittgenstein himself hinted at such a separation--by the use of brackets
or in other ways.
Page ix
Some of these notes are ephemeral; others on the other hand--the majority--are of great interest. Sometimes
they are strikingly beautiful or profound. It was evident to the literary executors that a number of these notes would
have to be published. G.H. von Wright was commissioned to make a selection and arrange it.
Page ix
It was a decidedly difficult task; at various times I had different ideas about how best to accomplish it. To
begin with, for example, I imagined that the remarks could be arranged according to the topics of which they
treated--such as "music", "architecture", "Shakespeare", "aphorisms of practical wisdom", "philosophy", and the
like. Sometimes the remarks can be arranged into such groupings without strain, but by and large, splitting up the
material in this way would probably give an impression of artificiality. At one time moreover I had thought of
including already published material. For many of Wittgenstein's most impressive "aphorisms" are to be found in his
philosophical works--in the Notebooks from the First World War, in the Tractatus, and in the Investigations too. I
should like to say that it is when they are embedded in such contexts that Wittgenstein's remarks really have their
most powerful effect. But for that very reason it did not seem to me right to tear them from their surroundings.
Page Break x
Page x
At one time too I played with the idea of not making a very extensive selection, but including only the "best"
remarks. The impression made by the good remarks would, I thought, only be weakened by a great mass of material.
That, presumably, is true--but it was not my job to be an arbiter of taste. Furthermore, I did not trust myself to
choose between repeated formulations of the same, or nearly the same, thought. Often the repetitions themselves
seemed to me to have a substantial point.
Page x
In the end I decided on the only principle of selection that seemed to me unconditionally right. I excluded
from the collection notes of a purely "personal" sort--i.e. notes in which Wittgenstein is commenting on the external
circumstances of his life, his state of mind and relations with other people--some of whom are still living. Generally
speaking these notes were easy to separate from the rest and they are on a different level of interest from those
which are printed here. Only in a few cases where these two conditions seemed not to be met did I include notes of
an autobiographical nature as well.
Page x
The remarks are published here in chronological order with an indication of their year of origin. It is
conspicuous that nearly half the remarks stem from the period after the completion (in 1945) of Part One of
Philosophical Investigations.
Page x
In the absence of further explanation some of the remarks will be obscure or enigmatic to a reader who is not
familiar with the circumstances of Wittgenstein's life or with what he was reading. In many cases it would have been
possible to provide explanatory comments in footnotes. I have nevertheless, with very few exceptions, refrained
from adding comments. I ought to add that all the footnotes are the editor's.†1
Page x
It is unavoidable that a book of this sort will reach the hands of readers to whom otherwise Wittgenstein's
philosophical work is, and will remain, unknown. This need not necessarily be harmful or useless. I am all the same
convinced that these notes can be properly understood and appreciated only against the background of
Wittgenstein's philosophy and, furthermore, that they make a contribution to our understanding of that philosophy.
Page x
I began making my selection from the manuscripts in the years 1965-6. I then laid the work aside until 1974. Mr
Heikki Nyman helped me with the final selection and arrangement of the collection. He also checked that the next
agreed exactly with the manuscripts and removed many errors and
Page Break xi
gaps from my typescript. I am very grateful to him for his work, which he carried out with great care and good taste.
Without his help I should probably not have been able to bring myself to complete the collection for the press. I am
also deeply indebted to Mr Rush Rhees for making corrections in the text which I produced and for giving me
valuable advice on matters of selection.
Helsinki, January 1977 Georg Henrik von Wright
Page Break xii
Foreword to New Edition 1994
Page xii
The present new version of the text of Vermischte Bemerkungen is the work of Alois Pichler. Mr Pichler, who works
at the Wittgenstein Archive of the University of Bergen†*, has newly transcribed from the manuscripts all the
remarks. In the process a few mistakes in the earlier editions were corrected, mainly places which had been difficult
to read correctly. Some of these corrections had already been noticed by the original editor in the course of the
years.
Page xii
The new edition contains all the remarks of the earlier editions and only those. However they appear here edited
more completely and more faithfully to the original. Wittgenstein usually wrote his remarks in short sections,
separated from each other by one or more blank lines. Some of the remarks printed in the earlier editions consisted
simply of "extracts" from these sections, i.e. often parts were left out which did not seem relevant to the editor. This
is a judgement that might appear controversial to some; for this reason in the present edition all such passages have
been completed so as to comprise the totality of the section. Another new feature is that variants are retained in
footnotes--formerly the editor had made a choice. The musical notation and the drawings are this time reproduced in
facsimile; thanks are due to Michael Biggs of the University of Hertfordshire for his advice and help. Many will be
pleased that the sources of the remarks are cited.†** (See Alois Pichler's Editorial Note for more details about
editing.)
Page Break xiii
At the end of the remarks there is a poem which was in the possession of Hofrat Ludwig Hänsel, to whom
Wittgenstein had given it. We assume that it was written by Wittgenstein. Here the poem is reproduced as a
facsimile of the surviving typescript. There is supposed to have been a handwritten version too, which has probably
been lost. It is not known when the poem was composed. The Wittgenstein Trustees thank Prof. Dr Hermann
Hänsel, Vienna, for making this unique document available.
Page xiii
Alois Pichler and I thank the Wittgenstein Archive of the University of Bergen for professional and technical
support.
Helsinki, November 1993 Georg Henrik von Wright
Page Break xiv
Editorial Note
Page xiv
History of the Edition: The Vermischte Bemerkungen were first published in 1977. In 1978 a new edition appeared
with supplementary remarks. The edition of 1978 was corrected and expanded in Volume 8 in the complete works,
1984; between "Es ist als hätte ich mich verirrt (...)" and "Sind alle Leute große Menschen? (...)" (1978: p. 93) was
inserted the remark "Je weniger sich Einer selbst kennt (...)" (1984: p. 516). For this reason the latter remark is not
included in the English edition of 1980. The present new edition contains all the remarks of the edition of 1984
and--apart from the completion of the context to include the full section and the noting of the variants--only those
remarks.
Page xiv
Sources: The manuscript source is given after each remark: "MS #" refers to the manuscript number in Georg Henrik
von Wright's catalogue of the Nachla߆*. Following the manuscript number the page is cited on which the remark
begins (Folio pagination is distinguished according to recto and verso by "r" and "v"; identical pagination is
distinguished according to left and right by "a" and "b".) The date of writing is also given, where this can be
ascertained.
Page xiv
Arrangement: The remarks are arranged chronologically; this has led to fairly extensive rearrangements in relation to
the earlier editions. For remarks which follow immediately one after another in the original, the source has been
given after the last of those remarks, so that the original grouping may be clear in this edition.
Page xiv
Context: All remarks, i.e. all passages separated from each other by blank
Page Break xv
lines and not indented, correspond to whole sections in the original; whereas the former editions sometimes
contained only parts of sections, the complete section is reproduced here. Completions of this sort are marked in the
citation of sources with an asterisk * (see e.g. the first remark). Further sections were added to the remark
"Architecture immortalizes (...)" (1978: p. 133; 1984: p. 548; here p. 74), as they constitute different versions of it and
in the manuscript are on the same or the previous side. In the earlier editions this passage was marked with "Several
variants in the manuscript". With remarks which belong together and which are here edited under the same
indication of source, but which in the original are separated from each other by one or more sections not published
here, the omission of the relevant section(s) is indicated with (...) (e.g. as on p. 75).
Page xv
Code: Several of the published remarks are written in the original partly or wholly in Wittgenstein's code; this is
registered in the indication of source with a "c" for "code", following the page number. The code, roughly, consists
in the reversal of the alphabet, in detail in the following correlations (read "a is z", "b is y", etc.):
a z h s n n u f
b y hh ss/ß o m v e
c x i r p l w d
d w k q q k x d
e v l p r i/j y b
f u m o s h z a
&funk; ü &munk; ö t g &zunk; ä
g t
"Rsx", e.g., is when decoded "Ich".
Page xv
Completeness: In the originals many of the remarks are marked with working signs and lines in the margin: these
marks were not included, as their significance belongs to the context of a work process that is not present here. The
same holds for (curved and pointed) brackets, bracketing a section as a whole: where the bracketed sections are
edited here without preceding or following section(s) that are not bracketed, the brackets have been omitted. This is
because such brackets have the function of delimiting the context and so are meaningful only where the context is
also included. Also omitted are text deleted by Wittgenstein words and punctuation marks, which do not fit
syntactically into the remark (this applies to crossings-out
Page Break xvi
in cases where Wittgenstein has neglected to cross out the whole text which belongs syntactically together with the
deleted material, and to duplications of words and punctuation marks). Insertions and rearrangements were arranged
or followed without this being indicated. When there was a date by a section this was included in the indication of
source in standardized form. Text underlined once in the original is printed in italics; text underlined twice in the
original is in SMALL CAPITALS. Passages underlined in the original with a wavy line (expressing doubts about the
expression, cf. collected works, Vol. 3, p. 166) are here underlined; e.g. p. 5 "Das", p. 35 ";".
Page xvi
Indentation: Wittgenstein has used indentations of various lengths for the separate paragraphs of his sections. The
extent of the indentation is not rendered here; it would have been meaningful to do this in a context reaching beyond
section and page. The first line of a section is printed here without indentation; all following paragraphs of a section
are indented.
Page xvi
Orthography, Grammar and Punctuation: Wittgenstein's orthographic habits especially in the use of upper and
lower case (e.g. "pointen"†i), in separations and runnings-together of words (e.g. "jeder so & sovielte"†ii) and
historically or regionally restricted orthographies (like "c" for "z" and "k" in words of Latin origin, "stätig" for "stetig"
and "alchemistisch" for "alchimistisch") have been respected. The ampersand "&" for "und" or "and" is retained. The
use of "ss"/"ß" and of the apostrophe in genitives ("Goethe's") has been consistently corrected in accordance with
modern usage. Extra punctuation marks have been supplied only where their absence would make reading difficult;
and brackets were completed where in the original brackets are opened, but not closed. Quotation marks are printed
in standardized form, ",", ','. Otherwise the punctuation has been left as in the original. All expansions at the level of
words (i.e. expansions which constitute a new word) and at the level of punctuation are shown with pointed
brackets, e.g. p. 7 "<)>"; (orthographic) expansions and omissions below the level of words are not indicated.
Indication is also made where an abbreviation has been expanded into the full word, as in the case of "B" into
"B<unyan>".
Page xvi
Variants: The remarks are printed inclusive of (undeleted) variants. Except
Page Break xvii
in cases like "(...) während die eigentlich(e) philosophische Überlegung (...)" (alternatives of "eigentliche" and
"eigentlich": both versions in the main text, p. 53), the version written first is given in the main text, the other(s)
however in footnotes. The various ways of marking variants ("[(...)]", "//(...)//" etc.) are not printed. Variants within
variants are separated by "|". When variants required repetition of what was written only once in the original, this is
indicated by "<(...)>". Alternative punctuation is also counted as a variant.
Page xvii
Graphics: The musical notation on p. 19 and the figures on pages 44 and 60 are facsimiles (reduced). In the original
they are on lined paper; here the lines are suppressed for the sake of greater clarity. The facsimiles were produced by
Michael Biggs.
Page xvii
Notes: Comments, explanations and textual notes are in end notes. "Unklar" means that the passage's content is
unclear; "Nicht klar leserlich" means that the passage is not clearly legible.†i
Page xvii
Appendix: The Appendix contains three lists: the first lists the sources of the remarks in the order in which they are
published here; the second lists the sources alphanumerically; the third lists the beginnings of the remarks (with page
references). The remarks which, in contrast to earlier editions, have been edited as whole sections are marked in the
lists with an asterisk *.
Alois Pichler
Page Break xviii
Note by Translator
Page xviii
The present translation is a quite extensive revision of my original translation published in 1980. This is of course
partly to take account of the new material included in Alois Pichler's revised edition, but there are other changes too.
Some of these changes relate to my dissatisfaction with my earlier renderings; but there are others which are
consequent on the somewhat different character of Mr Pichler's edition as compared with earlier editions.
Page xviii
Professor von Wright's earlier editions were intended for a readership broader than that to be expected for
Wittgenstein's more technically philosophical works. They did not, partly for that reason, attempt to include the kind
of textual detail that Mr Pichler has aimed at. One important feature of the newly included material is the detailed
noting of the many variant readings that Wittgenstein included in his manuscripts and typescripts. In order even to
begin any attempt to translate these variants, it was necessary to stick much more closely to the original grammatical
structure of Wittgenstein's texts than I had thought appropriate in my earlier version. I have done this while still
trying as far as possible to produce a text that reads like English and not a word for word representation into a weird
"translatese".
Page xviii
Sometimes Wittgenstein's variant readings can be captured more or less satisfactorily; but by no means
always. This is because the relative values of words which are roughly synonymous in German are not mirrored in
the English counterparts of these words. In these cases there is no reason to suppose that Wittgenstein would have
wished to present anything like the same variant readings had he been writing in English. It is important for the
reader to bear this in mind.
Page xviii
I have added some footnotes of my own. These are numbered in small Roman numerals thus: i, ii, etc.
Page Break xix
Page xix
I wish to reiterate my gratitude to the people who gave me generous help with the earlier translation: Marina
Barabas, Steven Burns, S. Ellis, Stephan Körner, Norman Malcolm, Heiki Nyman, Rush Rhees, Helen Widdess,
Erika Winch and G. H. von Wright. I now wish to add my thanks to two people who have helped me with my
revised version: Helen Geyer and Lars Hertzberg. Lars Hertzberg in particular took enormous pains to go through
the first draft of my revision. He made valuable suggestions, some of which I followed; though in other cases I
obstinately stuck to my original versions.
University of Illinois at Urbana/Champaign Peter Winch
November 1995
Page Break 1
Page Break 2
Culture and Value
Page 2
Page Break 3
Page 3
The Lieutenant†1 & I have already talked about all kinds of thing; a very nice man. He is able to get along with the
greatest scoundrels & be friendly without compromising himself. If we hear a Chinese we tend to take his speech for
inarticulate gurgling. Someone who understands Chinese will recognize language in what he hears. Similarly I often
cannot recognize the human being in someone etc. Worked a bit, but without success. MS 101 7 c: 21.8.1914*
Page 3
There is no religious denomination in which so much sin has been committed through the misuse of metaphorical
expressions as in mathematics. MS 106 58: 1929
Page 3
The human gaze has the power of making things precious; though it's true that they become more costly too. MS
106 247: 1929
Page 3
I myself still find my way of philosophizing new, & it keeps striking me so afresh, & that is why I have to repeat
myself so often. It will have become part of the flesh & blood of a new generation & it will find the repetitions
boring. For me they are necessary.--This method consists essentially in leaving the question of truth and asking
about sense instead. MS 105 46 c: 1929*
Page 3
It's a good thing I don't let myself be influenced! MS 105 67 c: 1929
Page 3
A good simile refreshes the intellect. MS 105 73 c: 129
Page 3
It is hard to tell someone who is shortsighted how to get to a place. Because you can't say "Look at that church
tower ten miles away over there and go in that direction.<"> MS 105 85 c: 1929
Page 3
Just let nature speak & acknowledge only one thing higher than nature, but not what others might think. MS 107 70
c: 1929
Page 3
You get tragedy where the tree, instead of bending, breaks. Tragedy is something unjewish. Mendelssohn is perhaps
the most untragic of composers. Tragically holding on, defiantly holding on to a tragic situation in love always seems
to me quite alien to my ideal. Does that mean my ideal
Page Break 4
is feeble? I cannot & should not judge. If it is feeble then it is bad. I believe that fundamentally I have a gentle &
calm ideal. But may God protect my ideal from feebleness & mawkishness! MS 107 72 c: 1929*
Page 4
A new word is like a fresh seed thrown on the ground of the discussion. MS 107 82 [[sic, c?]]: 1929
Page 4
Each morning you have to break through the dead rubble afresh so as to reach the living, warm seed. MS 107 82
c: 1929
Page 4
With my full philosophical rucksack I can climb only slowly up the mountain of mathematics. MS 107 97 c: 1929
Page 4
Mendelssohn is not a peak, but a plateau. His Englishness. MS 107 98 c: 1929
Page 4
No one can think a thought for me in the way no one can don my hat for me. MS 107 100 c: 1929
Page 4
Anyone who listens to a child's crying with understanding will know that psychic forces, terrible forces, sleep within
it, different from anything commonly assumed. Profound rage & pain & lust for destruction. MS 107 116 c:
1929
Page 4
Mendelssohn is like a man who is cheerful only when everything is cheerful anyway, or good only when everyone
around him is good, & not self-sufficient like a tree that stands firmly in its place, whatever may be going on around
it. I too am like that & tend to be so. MS 107 120 c: 1929
Page 4
My ideal is a certain coolness. A temple providing a setting for the passions without meddling with them.MS 107
130 c: 1929
Page 4
I often wonder whether my cultural ideal is a new one, i.e. contemporary, or whether it comes from the time of
Schumann. At least it strikes me as a continuation of that ideal, though not the continuation that actually followed it
then. That is to say, the second half of the 19th Century has been left out. This, I ought to say, has happened quite
instinctively & and was not the result of reflection. MS 107 156 c: 10.10.1929
Page Break 5
If we think of the world's future, we always mean the place it will get to if it keeps going as we see it going now and
it doesn't occur to us that it is not going in a straight line but in a curve & that its direction is constantly changing.
MS 107 176 c: 24.10.1929
Page 5
I think good Austrian work (Grillparzer, Lenau, Bruckner, Labor) is particularly hard to understand. There is a sense
in which it is subtler than anything else and its truth never leans towards plausibility. MS 107 184 c: 7.11.1929
Page 5
What is Good is Divine too. That, strangely enough, sums up my ethics.
Page 5
Only something supernatural can express the Supernatural. MS 107 192 c: 10.11.1929
Page 5
You cannot lead people to the good; you can only lead them to some place or other; the good lies outside the space
of facts. MS 107 196: 15.11.1929
Page 5
I recently said to Arvid,†2 after I had been watching a very old film with him in the cinema: A modern film is to an
old one as a present-day motor car is to one built 25 years ago. The impression it makes is just as ridiculous and
clumsy & the way film-making has improved is comparable to the sort of technical improvement we see in cars. It is
not to be compared with the improvement--if it's right to call it that--of an artistic style. It must be much the same
with modern dance music too. A jazz dance, like a film, must be something that can be improved. What
distinguishes all these developments from the formation of a style is that spirit plays no part in them.
Page 5
Today the difference between a good & a poor architect consists in the fact that the poor architect succumbs to
every temptation while the good one resists it.
Page 5
I once said, & perhaps rightly: The earlier culture will become a heap of rubble & finally a heap of ashes; but spirits
will hover over the ashes. MS 107 229: 10.-11.1.1930
Page 5
One uses straw to try to stuff the cracks which show in the work of art's
Page Break 6
organic unity, but to quiet one's conscience one uses the best straw. MS 107 242: 16.1.1930
Page 6
If anyone should think he has solved the problem of life & feels like telling himself everything is quite easy now, he
need only tell himself, in order to see that he is wrong, that there was a time when this "solution" had not been
discovered; but it must have been possible to live then too & the solution which has now been discovered appears in
relation to how things were then like†a an accident. And it is the same for us in logic too. If there were a "solution to
the problems of logic (philosophy)" we should only have to caution ourselves that there was a time when they had
not been solved (and then too it must have been possible to live and think)-- MS 108 207: 29.6.1930
Page 6
Engelmann told me that when he rummages round at home in a drawer full of his own manuscripts, they strike him
as so glorious that he thinks they would be worth presenting to other people. (He said it's the same when he is
reading through letters from his dead relations.) But when he imagines a selection of them published he said the
whole business loses its charm & value & becomes impossible I said this case was like the following one: Nothing
could be more remarkable than seeing someone who thinks himself unobserved engaged in some quite simple
everyday activity. Let's imagine a theatre, the curtain goes up & and we see someone alone in his room walking up
and down, lighting a cigarette, seating himself etc. so that suddenly we are observing a human being from outside in
a way that ordinarily we can never observe ourselves; as if we were watching a chapter from a biography with our
own eyes,--surely this would be at once uncanny and wonderful. More wonderful than anything that a playwright
could cause to be acted or spoken on the stage. We should be seeing life itself.--But then we do see this every day &
it makes not the slightest impression on us! True enough, but we do not see it from that point of view.--Similarly
when E. looks at his writings and finds them splendid†b (even though he would not care to publish any of the pieces
individually) he is seeing his life as God's work of art, & and as such it is certainly worth contemplating, as is every
life & everything whatever. But only the artist can represent the individual thing so that it appears to us as a work of
art; those manuscripts rightly lose their value if we contemplate them singly & in any case without prejudice, i.e.
without being enthusiastic about them in
Page Break 7
advance. The work of art compels us--as one might say--to see it in the right perspective, but without art the object is
a piece of nature like any other & the fact that we may exalt it through our enthusiasm does not give anyone the right
to display it to us. (I am always reminded of one of those insipid photographs of a piece of scenery which is
interesting to the person who took it because he was there himself, experienced something, but which a third party
looks at with justifiable coldness; insofar as it is ever justifiable to look at something with coldness.<)>
But now it seems to me too that besides the work†a†b of the artist there is another through which the world
may be captured sub specie æterni. It is--as I believe--the way of thought which as it were flies above the world and
leaves it the way it is, contemplating it from above in its†c flight.†d†e
MS 109 28: 22.8.1930
Page 7
In Renan's Peuple d'Israël I read: "Birth, sickness, death, madness, catalepsy, sleep, dreams, all made an infinite
impression and, even nowadays, it is given to only a small number to see clearly that these phenomena have causes
within our constitution<.">†3 On the contrary there is absolutely no reason to marvel at such things; because they
are such everyday occurrences. If primitive human beings must marvel at them, how much more so dogs &
monkeys. Or is it being assumed that human beings suddenly awoke as it were & noticed these things which had
always been there & were understandably amazed? Well, one might even assume something like this; not however
that they became aware of these things for the first time, but rather that they suddenly began to marvel at them. But
that too has nothing to do with their being primitive. Unless we call it primitive not to marvel at things, in which case
it is precisely the people of today & Renan himself who are primitive, if he believes that scientific explanation could
enhance wonderment.
As though today lightning were more commonplace or less astounding than 2000 years ago.
In order to marvel human beings--and perhaps peoples--have to wake up. Science is a way of sending them
off to sleep again.
Page 7
I.e. it is simply false to say: of course, these primitive peoples had to marvel
Page Break 8
at everything. But perhaps right that these people did marvel at everything around them.--To think they had to
marvel at them is a primitive superstition. (Like that of thinking that they had to fear all the forces of nature & that
we of course do not have†a to fear. On the other hand experience may show that certain primitive tribes are very
strongly inclined to fear natural phenomena.--But we cannot exclude the possibility that highly civilized peoples will
become liable to this very same fear again & their civilization and the knowledge of science will†b not protect them
from this. All the same it is true that the spirit in which science is carried on nowadays is not compatible with fear of
this kind)
Page 8
What Renan calls the bon sens précoce of the semitic races (an idea that I †c already entertained a long time ago) is
their unpoetic mentality, which heads straight for what is concrete. Which is characteristic of my philosophy.
Things are right before our eyes†d, not covered by any veil.--This is where religion & art part company.
MS 109 200: 5.11.1930
Page 8
Sketch for a Foreword†4
This book is written for those†e who are in sympathy with the spirit in which it is written.†f This spirit is, I believe,
different from that of the†g prevailing European and American civilization. The spirit of this civilization the
expression of which is the industry, architecture, music, of present day†h fascism & socialism, is a spirit that is alien
& uncongenial†i to the author. This is not a value judgement. It is not as though I did not know that†j what today
represents itself as architecture is not architecture & not†k as though he did not approach what is called modern
music with the greatest mistrust (without understanding its language), but the disappearance of the arts does not
justify a disparaging judgement on a whole segment of humanity. For in these times genuine & strong characters
simply turn away from the field of the arts & towards other things & somehow the value of the individual finds
expression. Not, to be sure, in the way it would at a time of Great Culture. Culture is like a great organization which
assigns to each of its members his place, at which he
Page Break 9
can work in the spirit of the whole, and his strength can with a certain justice be measured by his success as
understood within that whole. In a time without culture, however, forces are fragmented and the strength of the
individual is wasted through the overcoming of opposing forces & frictional resistances; it is not manifest in the
distance travelled but rather perhaps in the heat generated through the overcoming of frictional resistances. But
energy is still energy & even if the spectacle afforded by this age is not the coming into being of a great work of
culture in which the best contribute to the same great end, so much as the unimposing spectacle of a crowd whose
best members pursue purely private ends, still we must not forget that the spectacle is not what matters.
Even if it is clear to me then that the disappearance of a culture does not signify the disappearance of human
value but simply of certain means of expressing this value, still the fact remains that I contemplate the current of
European civilization without sympathy, without understanding its aims if any. So I am really writing for friends
who are scattered throughout the corners of the globe.
Page 9
It is all one to me whether the typical western scientist understands or appreciates my work since in any case he does
not understand the spirit in which I write.
Page 9
Our civilization is characterized by the word progress. Progress is its form, it is not one of its properties that it makes
progress. Typically it constructs. Its activity is to construct a more and more complicated structure. And even clarity
is only a means to this end & not an end in itself.
For me on the contrary clarity, transparency, is an end in itself
I am not interested in erecting a building but in having the foundations of possible buildings transparently
before me.
So I am aiming at something different than are the scientists & my thoughts move differently than do theirs.
Page 9
Each sentence that I write is trying to say the whole thing, that is, the same thing over and over again & it is as
though they were†a views of one object seen from different angles.
Page Break 10
I might say: if the place I want to reach could only be climbed up to by a ladder, I would give up trying to get there.
For the place to which I really have to go is one that I must actually be at already.
Anything that can be reached with a ladder does not interest me.
Page 10
One movement orders one thought to the others in a series, the other keeps aiming at the same place.
Page 10
One movement constructs & takes (in hand) one stone after another,†a the other keeps reaching for the same one.
Page 10
The danger in a long foreword is that the spirit of a book has to be evident in the book itself & cannot be described.
For if a book has been written for only a few readers that will be clear just from the fact that only a few understand
it. The book must automatically separate those who understand it & those who do not. The foreword too is written
just for such as†b understand the book.
Telling someone something he does not understand is pointless, even if you add that he will not be able to
understand it. (That so often happens with someone you love.)
If you do not want certain people to get into a room, put a lock on it for which they do not have the key. But
it is senseless to talk with them about it, unless you want them all the same to admire the room from outside!
The decent thing to do is: put a lock on the doors that attracts only those†c who are able to open it & is not
noticed by the rest.
But it's alright to say that the book in my opinion has nothing to do with the progressive civilization of
Europe & America.
That this civilization is perhaps an environment necessary for its spirit but that they have different aims.
Everything ritualistic (everything that, as it were, smacks of the high priest) is strictly to be avoided because it
straightaway turns rotten†d.
Of course a kiss is a ritual too & it isn't rotten; but no more ritual is permissible than is as genuine as a kiss.
Page Break 11
Page 11
It is a great temptation to want to make the spirit explicit.
MS 109 204: 6-7.11.1930
Page 11
When you bump against the limits of your own decency it is as though a whirlpool of thoughts is generated, (&) an
endless regress: you may say what you like, it gets you no further. MS 109 212: 8.11.1930
Page 11
I am reading Lessing (on the Bible): "Add to this the verbal clothing and the style.... absolutely full of tautologies,
but of a kind to exercise one's wits by seeming sometimes to say something different while really saying the same
thing, and at other times seeming to say the same thing while at bottom meaning, or being capable of meaning,
something different..."†5 MS 110 5:12.12.1930
Page 11
If I do not quite know how to begin a book that is because something is still unclear. For I should like to begin with
the original data of philosophy, written & spoken sentences, with books as it were
And here we encounter the difficulty of "Everything is in flux". And perhaps that is the very point at which to begin.
MS 110 10: 13.12.1930
Page 11
If someone is merely ahead of his time, it will catch him up one day.
MS 110 11: 25.12.1930
Page 11
Music, with its few notes & rhythms, seems to some people a primitive art. But only its surface†a is simple, while
the body which makes possible the interpretation of this manifest content has all the infinite complexity that is
suggested in the external forms of other arts & which music conceals. In a certain sense it is the most sophisticated
art of all.
Page 11
There are problems I never tackle, which do not lie in my path or belong to my world. Problems of the intellectual
world of the West which Beethoven (& perhaps Goethe to a certain extent) tackled & wrestled with but which no
philosopher has ever confronted (perhaps Nietzsche passed close to them)
And perhaps they are lost to western philosophy, that is there will be no one there who experiences and so can
describe the development of this culture as an epic. Or more precisely it just is no longer an epic, or is one
Page Break 12
only for someone who observes it from outside & perhaps Beethoven did this with prevision (as Spengler hints in
one place) It might be said that civilization can only have its epic poet in advance. Just as one can only foresee one's
own death and describe it as something lying in the future, not report it as it happens. So it might be said: If you
want to see the epic of a whole culture written†a you will have to seek it in the works of its greatest figures and
hence seek it at a time when the end of this culture can only be foreseen, for later there is no one there any more to
describe it. So it is not to be wondered at that it should be written in the dark language of prevision†b & intelligible
only to the very few.
Page 12
But I do not get to these problems at all. When I "have done with the world" I have created an amorphous
(transparent) mass & and the world in all its variety is left on one side like an uninteresting lumber room.
Or perhaps more precisely: the whole outcome of the entire work is for the world to be set on one side. (A
throwing-into-the-lumber-room of the whole world)
Page 12
In this world (mine) there is no tragedy & with that all the endlessness that gives rise to tragedy (as its result†c) is
lacking
It is as though everything were soluble in the ether;†d there are no harnesses.
This means that hardness & conflict do not become something splendid†e but a defect.
Page 12
Conflict is dissipated in much the same way as is the tension of a spring in a mechanism that you melt (or dissolve in
nitric acid). In this†f solution tensions no longer exist. MS 110 12: 12.-16.1.1931
Page 12
If I say that my book is meant for only a small circle of people (if that can be called a circle) I do not mean to say
that this circle is in my view the élite of mankind but it is the circle to which†g I turn (not because they are better or
worse than the others but) because they form my cultural circle, as it
Page Break 13
were my fellow countrymen in contrast to the others who are foreign to me.
MS 110 18: 18.1.1931
Page 13
The limit of language manifests itself in the impossibility of describing the fact that corresponds to (is the translation
of) a sentence without simply repeating the sentence.
Page 13
(We are involved here with the Kantian solution of the problem of philosophy.)
MS 110 61: 10.2.1931
Page 13
Can I say that drama has its own time which is not a segment of historical time. I.e. I can speak of earlier and later
within it but there is no sense to the question whether the events in it took place, say, before or after Caesar's death.
MS 110 67: 12.2.1931
Page 13
The charming difference in temperature between the parts of a human body. MS 153a 4v: 10.5.1931
Page 13
It is humiliating having to present oneself as an empty tube only inflated by the mind.
MS 153a 12v: 1931
Page 13
No one likes having offended another person; that is why it does everyone good when the other person doesn't
show that he has been offended. Nobody likes being confronted by a wounded spaniel. Remember that. It is much
easier patiently--& tolerantly†i--to avoid the person that offended you than to approach him as a friend. You need
courage too for that. MS 153a 18v 1931
Page 13
To treat well somebody who does not like you requires not just great good nature but great tact too.
MS 153a 29v: 1931
Page 13
We are struggling with language.
Page 13
We are engaged in a struggle with language. MS 153a 35r: 1931
Page 13
Compare the solution of philosophical problems with the fairy tale gift that seems magical in the enchanted castle
and if it is looked at in daylight
Page Break 14
is nothing but an ordinary bit of iron (or something of the sort). MS 153a 35v: 1931
Page 14
A thinker is very similar to a draughtsman. Who wants†a to represent all the interconnections. MS 153a 90v: 1931
Page 14
Pieces of music composed at the keyboard, those by thinking with the pen & those composed just with imagined
sounds must be of quite a different kind†b and make quite different kinds of impression.
I am sure that Bruckner composed just in his head, imagining the orchestra playing, Brahms with his pen. Of course
this is an oversimplification. But it does highlight one feature. MS 153a 127v: 1931
Page 14
A tragedy might really always start with the words: "Nothing at all would have happened, had it not been that..."
Page 14
(Had he not been caught in the machine by a corner of his clothing?)
Page 14
But isn't it a one-sided view of tragedy to think of it merely as showing, that an encounter can decide one's whole
life.
Page 14
I think today there could be a form of theatre played in masks. The characters would be just stylized human
beings.†c†6 In Kraus's writings this can be clearly seen. His pieces could be, or should be, performed in masks. Of
course this goes with a certain abstractness in these works. And masked theatre, as I believe, is in any case the
expression of an intellectual character. Perhaps (too) for this reason only Jews will be attracted to this theatre. MS
153a 128v: 1931
Frida Schanz:
Foggy day. Grey autumn haunts us.
Laughter seems tainted;
Page Break 15
the world is mute today,
as though last night it died.
In the red-gold hedge
are brewing fog dragons;
and sleeping lies the day.
The day will not awaken.
(...)
I took the poem on the previous page from a "Rösselsprung"†i in which of course the punctuation was not shown.
So I do not know if the words "Foggy day" form the title, or belong to the first line, as I have written it. And it is
remarkable how trivial the poem sounds if it does not begin with "Foggy day" but with "Grey". This changes the
rhythm of the whole poem.†a MS 153a 136r: 1931
Page 15
What you have achieved cannot mean†b more to others than to you.
Page 15
As much (as) it has cost you, that is what they will pay. MS 153a 141r: 1931
Page 15
The Jew is a desert region under whose thin layer of rock lies the molten lava of spirit. MS 153a 160v: 1931
Page 15
Grillparzer: "How easy it is to move about in broad distant regions, how hard to grasp what is individual & near at
hand..." MS 153b 3r: 1931
Page 15
How should we feel if we had never heard of Christ?
Should we feel left alone in the dark?
Do we not feel like that only in the way a child doesn't when he knows there is someone in the room with
him?
Religious madness is madness springing from irreligiousness. MS 153b 29r: 1931
Page 15
I look at the photographs of Corsican brigands and reflect: these faces are too hard & mine too soft for Christianity
to be able to write on them. The
Page Break 16
faces of the brigands are terrible to behold & yet they are certainly no more distant from a good life & are simply
situated on a different side of it than am I. MS 153b 39v: 1931
Page 16
A confession has to be part of one's new life. MS 154 1r: 1931
Page 16
I never more than half succeed in expressing what I want to express. Indeed not even so much,†7 but perhaps only
one tenth. That must mean something. My writing is often nothing but "stammering".
MS 154 1v: 1931
Page 16
The saint is the only Jewish "genius". Even the greatest Jewish thinker is no more than talented. (Myself for
instance.)
Page 16
I think there is some truth in my idea that I am really only reproductive in my thinking. I think I have never invented
a line of thinking but that it was always provided for me by someone else & I have done no more than passionately
take it up for my work of clarification. That is how Boltzmann Hertz Schopenhauer Frege, Russell, Kraus, Loos
Weininger Spengler, Sraffa†8 have influenced me. Can one take Breuer & Freud as an example of Jewish
reproductive thinking?--What I invent are new comparisons.
Page 16
At the time I modelled the head for Drobil too the stimulus was essentially a work of Drobil's & my work was again
really one of clarification. I believe that what is essential is for the activity of clarification to be carried out with
COURAGE; without this it becomes a mere clever game.
Page 16
The Jew must in a real sense "make nothing his business".†i But for him especially this is particularly hard because
he, as it were, has nothing. It is much harder to be poor voluntarily if you can't help being poor, than when you
might also be rich.
Page 16
It might be said (rightly or wrongly) that the Jewish mind is not in a
Page Break 17
position to produce even so much as a tiny blade of grass or flower but that its way is to make a drawing of the blade
of grass or the flower that has grown in the mind of another & then use it to sketch a comprehensive picture. This is
not to allege a vice & everything is all right as long as what is being done is quite clear. Danger arises only when
someone confuses the nature of a Jewish work with that of a non-Jewish work & especially when the author of the
former does so himself, as he so easily may. ("Doesn't he look as proud as though he were being milked
himself."†9<)>
It is typical of the Jewish mind to understand someone else's work better than he understands it himself.
Page 17
When I have had a picture suitably framed or have hung it in the right surroundings I have often caught myself being
as proud as though I had painted the picture. Actually that's not right: not "as proud as though I had painted it" but
as proud as though I had helped to paint it, as though I had so to speak painted a little bit of it. It is as if an
exceptional arranger of grasses were at last to think that he too had produced at least a quite tiny blade of grass
himself. Whereas it ought to be clear to him that his work lies in a different region altogether.
The process through which even the tiniest & meanest blade of grass comes into being is quite foreign & unknown
to him.
Page 17
A picture of a complete apple tree, however accurate, in a certain sense resembles it infinitely less than does the
smallest daisy. And in this sense a symphony by Bruckner is infinitely more closely related to a symphony from the
heroic period than is one by Mahler. If the latter is a work of art it is one of a totally different sort. (But this
observation itself is actually Spenglerian.)
Page 17
Anyway when I was in Norway during the year 1913-14 I had some thoughts of my own, or so at least it seems to
me now. I mean that I have the impression of having given birth to new lines of thinking at that time (But perhaps I
am mistaken). Whereas now I seem just to apply old ones. MS 154 15v: 1931
Page 17
There is something Jewish in Rousseau's character. MS 154 20v: 1931
Page 17
If it is said on occasion that (someone's) philosophy is a matter of temperament, there is some truth in this. A
preference for certain comparisons is something we call a matter of temperament & far more disagreements
Page Break 18
rest on this than appears at first sight.†a
Page 18
"Look on this wart†b as a regular limb of your body!" Can one do that, to order?
Do I have the power to decide at will to have, or not to have, a certain ideal conception of my body?
Within the history of the peoples of Europe the history of the Jews is†10 not treated so circumstantially as
their intervention in European affairs would actually merit, because within this history they are experienced as a sort
of disease, anomaly, & nobody wants to put a disease on the same level as normal life†c [[sic . ?]]
We may say: this bump can be regarded as a limb of one's body only if our whole feeling for the body
changes (if the whole national feeling for the body changes). Otherwise the best we can do is put up with it.
You may expect an individual to display this sort of tolerance or even to disregard such things; but you
cannot expect this of a nation since it is only a nation by virtue of not disregarding such things. I.e. there is a
contradiction in expecting someone to retain the original aesthetic feeling for his body & also to make the swelling
welcome.
Page 18
Power & possession are not the same thing. Even though possession also gives us power. If Jews are said not to
have any sense for possession that is presumably compatible with their liking to be rich; for money is for them a
particular sort of power not possession. (I should for instance not like my people to be poor, since I wish them to
have a certain power. Naturally I wish them to use this power properly too.)
Page 18
There is definitely a certain kinship between Brahms & Mendelssohn; but I do not mean that shown by the
individual passages in Brahms's works that are reminiscent of passages in Mendelssohn but the kinship of which I
am speaking could be expressed by saying that Brahms does with complete rigour what Mendelssohn did
half-rigorously. Or: Brahms is often Mendelssohn without the flaws.
Page Break 19
Page 19
†11 That must be the end of a theme which I cannot place†i. It occurred to me today as I was thinking about my
work in philosophy & said to myself: "I destroy, I destroy, I destroy--" MS 154 21v: 1931
Page 19
It has sometimes been said that the Jews' secretive & cunning nature is a result of their long persecution. That is
certainly untrue; on the other hand it is certain that, despite this persecution, they continue to exist only because
they have the inclination towards this secretiveness. As we may say that such & such an animal has escaped
extinction only because it has the possibility or capability of concealing itself. Of course I do not mean that one
should commend this ability for such a reason, not by any means.
Page 19
In Bruckner's music nothing is left of the long & slender (nordic?) face of Nestroy, Grillparzer, Haydn, etc. but it has
in full measure a round full (alpine?) face even purer in type than was Schubert's.
Page 19
The power of language to make everything look the same which appears in its crassest form in the dictionary &
which makes it possible to personify time, something which is no less remarkable than would have been making
divinities of the logical constants. MS 154 25v: 1931
Page 19
A beautiful garment that changes (coagulates as it were) into worms &
Page Break 20
serpents if its wearer smugly smartens himself up†12 in it in the mirror. MS 155 29r: 1931
Page 20
The pleasure I take in my thoughts is pleasure in my own strange life. Is this joi de vivre? MS 155 46r: 1931
Page 20
By the way in†a the old conception--roughly that of the (great) western philosophers--there were two†b sorts of
problem in the scientific sense: essential, great, universal, & inessential, as it were accidental, problems. Our
conception on the contrary is that there is no great essential problem in the scientific sense.†i MS 110 200:
22.6.1931
Page 20
Structure & feeling in music. Feelings accompany our grasp of a piece of music as they accompany events in our
life. MS 110 226: 25.6.1931
Page 20
Labor's seriousness is a very late seriousness. MS 110 231 c: 29.6.1931
Page 20
Talent is a spring from which fresh water is constantly flowing. But this spring loses its value if it is not used†c in the
right way. MS 110 238: 30.6.1931
Page 20
"What a sensible man knows is hard to know." Does Goethe's contempt for laboratory experiment and his
exhortation to go out into uncontrolled nature & learn from that, does this have some connection with the idea that a
hypothesis (wrongly conceived) is already a falsification of the truth? And with the beginning I am now thinking of
for my book which might consist of a description of nature?†d MS 110 257: 2.7.1931
Page 20
If people find a flower or an animal ugly they always have an impression as though they were artifacts. "It looks like
a ..." they say. This sheds light on the meaning of the words "ugly" & "beautiful". MS 110 260 c: 2.7.1931
Page Break 21
Page 21
Labor, when he writes good music, is absolutely unromantic. That is a very remarkable & significant indication. MS
111 2c: 7.7.1931
Page 21
Reading the Socratic dialogues, one has the feeling: what a frightful waste of time! What's the point of these
arguments that prove nothing & clarify nothing. MS 111 55: 30.7.1931
Page 21
The story of Peter Schlemihl†13 should, it seems to me, go†a like this: He makes over his soul to the Devil for
money. Then he repents it & now the Devil demands his shadow as ransom. But Peter Schlemihl still has a choice
between giving the Devil his soul or sacrificing along with his shadow life in community with human beings. MS
111 77:11.8.1931
Page 21
In Christianity it is as though God said to human beings: Don't act a tragedy, that is to say, don't enact heaven & hell
on earth, heaven & hell are my affair. MS 111 115: 19.8.1931
Page 21
Spengler could be better understood if he said: I am comparing different periods of culture with the lives of families;
within the family there is a family resemblance, while you will also find a resemblance between members of different
families; family resemblance differs from the other sort of resemblance in such & such ways etc.. What I mean is:
We have to be told the object of comparison, the object from which this approach is derived, so that prejudices do
not constantly slip into the discussion. Because then we shall willy nilly ascribe what is true†b of the prototype of
the approach†c to the object to which we are applying the approach as well; & we claim "it must always be..."
This comes about because we want to give the prototype's characteristics a foothold in the approach. But since we
confuse prototype & object we find ourselves dogmatically conferring on the object properties which only the
prototype necessarily possesses. On the other hand we think the approach will lack the†d generality we want to give
it if it really holds only of the one case. But the prototype must just be presented for what it is; as characterizing the
whole approach and determining its form. In this way it stands at the head & is generally valid by virtue of
determining the form of approach, not by virtue of a claim that everything which is true only of it
Page Break 22
holds for all the objects to which the approach is applied.
One should thus always ask when exaggerated dogmatic claims are made: What is actually true in this. Or
again: In what case is that actually true MS 111 119: 19.8.1931
Page 22
From Simplicissimus: riddles of technology.
(Picture: two professors in front of a bridge under construction) Voice from above: "Fotch it dahn--coom on--fotch
it dahn A tell tha--we'll turn it t'other rooad sooin!"†i --"It really is quite incomprehensible, my dear colleague, how
such complicated & precise work can be carried out in this language." MS 111 132: 23.8.1931
Page 22
We keep hearing the remark that philosophy really does not progress, that we are still occupied with the same
philosophical problems as were the Greeks. Those who say this however don't understand why it is so.†a It is
because our language has remained the same & keeps seducing us into asking the same questions. As long as there
is still a verb 'to be' that looks as though it functions in the same way as 'to eat' and 'to drink', as long as we still have
the adjectives 'identical', 'true', 'false', 'possible', as long as we continue to talk of a river of time & an expanse of
space, etc., etc., people will keep stumbling over the same cryptic difficulties & staring at something that no
explanation seems capable of clearing up.
And this satisfies besides a longing for the supernatural†b for in so far as people think they can see the "limit
of human understanding", they believe of course that they can see beyond it.
Page 22
I read: "philosophers are no nearer to the meaning of 'Reality' than Plato got;..." What a singular†c situation. How
singular then that Plato has been able to get†d even as far as he did! Or that we could get no further afterwards! Was
it because Plato was so clever? MS 111 133: 24.8.1931
Page Break 23
Kleist wrote somewhere†14 that what the poet would most of all like to be able to do, would be to convey thoughts
in themselves†a without words. (What a strange avowal.) MS 111 173: 13.9.1931
Page 23
It is often said that a new religion brands the gods of the old one as devils. But in reality they have presumably by
that time already become devils. MS 111 180:13.9.1931
Page 23
The works of the great masters are stars†b which rise and set around us. So the time will come again for every great
work that is now in the descendent. MS 111 194: 13.9.1931
Page 23
(Mendelssohn's music, when it is flawless, consists of musical arabesques. That is why we feel embarrassed at every
lack of rigour in his work.)
Page 23
In Western Civilization the Jew is always being measured according to calibrations which do not fit him. That the
Greek thinkers were neither philosophers in the western sense, nor scientists in the western sense, that those who
took part in the Olympic Games were not sportsmen & fit into <no> western occupation, is clear to many people.
But it is the same with the Jews too†c
And insofar as the words of our <language> seem to us the only possible standards of measurement we are always
doing him†d injustice. And he is†e first overestimated then underestimated. In this context Spengler is quite right
not to classify Weininger with the western philosophers.†f†15
Page 23
Nothing we do can be defended definitively. But only by reference to something else that is established.
I.e. no reason can be given why you should act (or should have acted) like this, except that by doing so you bring
about such and such a situation, which again you have to accept as an aim. MS 111 195: 13.9.1931
Page 23
The inexpressible (what I find enigmatic & cannot express) perhaps provides the background, against which
whatever I was able to express acquires meaning. MS 112 1: 5.10.1931
Page Break 24
Work on philosophy--like work in architecture in many respects--is really more work†a on oneself. On one's own
conception. On how one sees things. (And what one expects of them.) MS 112 46: 14.10.1931
Page 24
The philosopher easily gets into the position of an incompetent manager who, instead of doing his own work &
simply seeing to it†b that his employees do theirs properly takes over their work & so finds himself one day
overloaded with other people's work, while the employees look on and criticize him. MS 112 60: 15.10.1931
Page 24
The idea is worn out by now & no longer usable. (I once heard Labor make a similar remark about musical ideas.) In
the way silver paper, once crumpled, can never quite be smoothed out again. Nearly all my ideas are a bit crumpled.
MS 112 76: 24.10.1931
Page 24
I really do think with my pen, for my head often knows nothing of what my hand is writing.
Page 24
(Philosophers are often like little children who first scribble some marks on a piece of paper at random and now†c
ask the grown-up "what's that?"--It happened like this: The grown-up had often drawn something for the child &
said: "this is a man", "this is a house" etc. And now the child makes some marks too and asks: and what's this then?"
MS 112 114: 27.10.1931
Page 24
Ramsey was a bourgeois thinker. I.e. he thought with the aim of clearing up the affairs of some particular
community. He did not reflect on the essence of the state--or at least he did not like doing so--but on how this state
might reasonable be organized. The idea that this state might not be the only possible one partly disquieted him and
partly bored him. He wanted to get down as quickly as possible to reflecting on the foundations--of this state. This
was what he was good at & what really interested him; whereas real philosophical reflection disquieted him until he
put its result (if it had one) on one side as trivial. MS 112 139: 1.11.1931
Page Break 25
A curious analogy could be based on the fact that the eye-piece of even the hugest telescope cannot be bigger†a than
our eye. MS 112 153: 11.11.1931
Page 25
Tolstoy: the meaning (importance) of something lies in its being something everyone can understand. That is both
true & false. What makes the object hard to understand--if it's significant, important--is not that you have to be
instructed in abstruse matters in order to understand it, but the antithesis between understanding the object & what
most people want to see. Because of this precisely what is most obvious may be what is most difficult to
understand. It is not a difficulty for the intellect but one for the will that has to be overcome. MS 112 221:
22.11.1931
Page 25
Someone who teaches philosophy nowadays gives his pupil foods, not†b because they are to his taste, but in order
to change his taste. MS 112 223: 22.11.1931
Page 25
I must be nothing more than the mirror in which my reader sees his own thinking with all its deformities & with this
assistance can set it in order. MS 112 225: 22.11.1931
Page 25
Language sets everyone the same traps; it is an immense network of well kept†c wrong turnings. And hence we see
one person after another walking down the same paths & we know in advance the point at which they will branch
off, at which they will walk straight on without noticing the turning, etc., etc. So what I should do is erect signposts
at all the junctions where there are wrong turnings, to help people past the danger points.
Page 25
What Eddington says about the 'direction of time' & the principle of entropy amounts to saying that time would
reverse its direction if people began one day to walk backwards. If you like you can by all means call it that; but then
you must be clear in your mind that you have said no more than that people have changed the direction in which
they walk. MS 112 231: 22.11.1931
Page Break 26
Someone divides human beings into buyers & sellers, & forgets that buyers are sellers as well. If I remind him of
this,†a is his grammar changed? MS 112 232: 22.11.1931
Page 26
The real achievement of a Copernicus or a Darwin was not the discovery of a true theory but of a fertile new point of
view. MS 112 233: 22.11.1931
Page 26
I believe that what Goethe was really seeking was not a physiological but a psychological theory of colours. MS
112 255: 26.11.1931
Page 26
Philosophers who say: "after death a timeless state will supervene", or "at death a timeless state supervenes" & do
not notice that they have used in a temporal sense the words "after" & "at" & "supervenes" & that temporality is
embedded in their grammar. MS 113 80: 29.2.1932
Page 26
Remember the impression made by good architecture, that it expresses a thought. One would like to respond to it
too with a gesture. MS 156a 25r: ca. 1932-1934
Page 26
Don't play with what lies deep in another person! MS 156a 30v: ca. 1932-1934
Page 26
The face is the soul of the body. MS 156a 49r: ca. 1932-1934
Page 26
One cannot view†b one's own character from outside any more than one's own handwriting.
I have a one-sided relation to my handwriting that prevents me from seeing & comparing it with the writing of others
on the same footing. MS 156a 49v: ca. 1932-1934
Page 26
In art it is hard to say anything, that is as good as: saying nothing. MS 156a 57r: ca. 1932-1934
Page Break 27
My thinking, like everyone's, has sticking to it the shrivelled husks†a of my earlier (withered) thoughts.
MS 156a 58v: ca. 1932-1934
Page 27
The strength of the musical thinking in Brahms. MS 156b 14v: ca. 1932-1934
Page 27
The various plants & their human character: rose, ivy, grass, oak, apple tree, corn palm. Compared with the diverse
character of words MS 156b 23v: ca. 1932-1934
Page 27
If one wanted to characterize the essence of Mendelssohn's music one could do it by saying that there is perhaps no
music by Mendelssohn that is hard to understand. MS 156b 24v: ca. 1932-1934
Page 27
Every artist has been influenced by others & shows (the) traces of that influence in his works; but what we get from
him is all the same only his own personality†b. What is inherited from others can be nothing but egg shells. We
should treat the fact of their presence with indulgence but they will not give us Spiritual nourishment.
Page 27
It seems to me (sometimes) as though I were already†c philosophizing with toothless gums & as though I took
speaking without teeth for the right way, the more worthwhile way. I detect something similar in Kraus. Instead of
my recognizing it as a deterioration. MS 156b 32r: ca. 1932-1934
Page 27
If someone says, let's suppose, "A's eyes have a more beautiful expression than B's", then I want to say that he
certainly does not mean by the word beautiful what is common to everything that we call "beautiful". Rather he is
playing a game with this word that has quite narrow bounds. But what shows this? Did I have in mind some
particular restricted explanation of the word "beautiful"? Certainly not.--But perhaps I shall not even want to
compare the beauty of expression in a pair of eyes with the beauty in the shape of a nose.
Page Break 28
Page 28
Indeed we might perhaps say: If a language had two words so that there was no indication of anything common to
these cases I should have no trouble taking one of these two specialized words for my case & nothing would be lost
from the sense of what I wanted to say.
One might say: how would I explain the word 'rule' or 'plant' in the particular case then? that will show 'what
I mean by it'.
Suppose I had said: "the gardener raises very beautiful plants in this greenhouse". I want to communicate
something to my hearer with this & the question arises: for this does he have to know what is common to everything
that we call "plant"? No. I could quite well have given him the explanation for the case in hand by means of a few
examples or a few pictures.
In the same way if I say: "I will just explain the rules of this game to you", do I presuppose that the other
knows everything that is common to what we call "rule"? MS 145 14r: Autumn 1933*
Page 28
If I say A. has beautiful eyes I may be asked: What do you find beautiful about his eyes & perhaps I will answer: the
almond shape, the long lashes, the delicate lids.
What do these eyes have in common with a Gothic church that I also find beautiful? Am I to say they make a similar
impression on me? What if I said: what they have in common is that in both cases my hand is tempted to draw
them? That at any rate would be a narrow definition of the†a beautiful.
It will often be possible to say: ask what your reasons are for calling something good or beautiful & the
particular grammar of the word "good" in this case will be apparent. MS 145 17v: 1933
Page 28
I believe I summed up where I stand in relation to philosophy when I said: really one should write philosophy only
as one writes a poem. That, it seems to me, must reveal how far my thinking belongs to the present, the future, or
the past. For I was acknowledging myself, with these words, to be someone who cannot quite do what he would like
to be able to do. MS 146 25v: 1933-1934
Page 28
If you use a trick in logic, whom can you be tricking but yourself? MS 146 35v: 1933-1934
Page Break 29
Composers' names. Sometimes it is the method of projection that we treat as given. When we, say, ask What name
would hit off this person's character But sometimes we project the character into the name & treat that as given.
Thus we get the impression that the great masters we know so well have just the names that suit their work. MS
146 44v: 1933-1934
Page 29
If someone prophesies that the generation to come will take up these problems & solve them that is usually a sort of
wishful thinking, a way of excusing oneself for what one should have accomplished & hasn't. A father would like his
son to achieve what he has not achieved so that the task he left unresolved should find a resolution nevertheless. But
his son is faced with a new task. I mean: the wish that the task should not remain unfinished disguises itself as a
prediction that it will be taken further by the next generation. MS 147 16r: 1934
Page 29
The overwhelming skill in Brahms. MS 147 22r: 1934
Page 29
If I say: "that is the only thing that is really seen" I point in front of me. If I were to point sideways or behind me
however--at things that I did not see--this pointing would lose all its sense for me. That means, though that my
pointing in front of me is not contrasted with anything.
(Someone who is in a hurry will when sitting in a car push involuntarily, even though he may tell himself that he is
not pushing the car at all.) MS 157a 2r: 1934*
Page 29
By the way, in my artistic activities I have merely good manners. MS 157a 22v: 1934
Page 29
In the days of silent films all the classics were played with the films, except Brahms & Wagner.
Not Brahms because he is too abstract. I can imagine an exciting scene in a film accompanied with music by
Beethoven or Schubert & might gain some sort of understanding of the music from the film. But not an
understanding of music by Brahms. Bruckner on the other hand does go with a film. MS 157a 44v: 1934 or
1937
Page 29
The queer resemblance between a philosophical investigation (perhaps especially in mathematics <)> and one in
aesthetics. (E.g. what is bad about this garment, how it should be, etc..) MS 116 56: 1937
Page Break 30
The edifice of your pride has to be dismantled. And that means frightful work. MS 157a 57r: 1937
Page 30
In one day you can experience the horrors of hell; that is plenty of time. MS 157a 57r: 1937
Page 30
There is a big difference between the effect of a script that you can read fluently & one that you can write but not
decipher†a easily. The thoughts are enclosed,†16 as in a casket. MS 157a 58r: 1937
Page 30
The greater "purity" of objects that do not affect the senses, numbers for instance. MS 157a 62v: 1937
Page 30
If you offer a sacrifice & then are conceited about it, you will be cursed along with your†b sacrifice. MS 157a
66v c: 1937
Page 30
The light shed by work is a beautiful light, but it only shines with real beauty if it is illuminated by yet another light.
MS 157a 67v c: 1937
Page 30
"Yes, that's how it is," you say, "because that's how it must be!"
Page 30
(Schopenhauer: the real life span of the human being is 100 years.)
"Of course, it must be like that!" It is as though you have understood a creator's purpose. You have understood the
system.
You do not ask yourself 'How long do human beings actually live then?', that seems now a superficial matter;
whereas you have understood something more profound. MS 157b 9v: 1937
Page 30
The†17 only way namely for us†c to avoid prejudice†18 --or vacuity in our claims, is to posit†d the ideal as what it
is, namely as an object of comparison--a measuring rod as it were--within our way of looking at things, & not†e as a
preconception to which everything must†f conform. This namely is†g the dogmatism into which philosophy†h can
so easily degenerate.
Page Break 31
But then†19 what is the relation between an approach like Spengler's & mine?
Injustice in Spengler: The ideal loses none of its dignity if it is posited as the principle determining the form
of one's approach. A good unit of measurement.-- --†20 MS 157b 15v: 1937
Page 31
Slept a bit better. Vivid dreams. A bit depressed; weather & state of health.
The solution of the problem you see in life is a way of living which makes what is problematic disappear.
The fact that life is problematic means that your life does not fit life's shape. So you must change your life, &
once it fits the shape, what is problematic will disappear.
But don't we have the feeling that someone who doesn't see a problem there is blind to something important,
indeed to what is most important of all?
Wouldn't I like to say he is living aimlessly--just blindly like a mole as it were; & if he could only see†a, he
would see the problem?
Or shouldn't I say: someone who lives rightly does not experience the problem as sorrow, hence not after all
as a problem, but rather as joy, that is so to speak as a bright halo round his life, not a murky background. MS
118 17r c: 27.8.1937*
Page 31
Almost in the same way as earlier physicists are said to have found suddenly that they had too little mathematical
understanding to be able to master physics; we may say that young people today are suddenly in the position that
ordinary common sense no longer suffices to meet the strange demands life makes. Everything has become so
intricate that for its mastery†b an exceptional degree of understanding is required. For it is not enough any longer to
be able to play the game well; but the question is again and again: what sort of game is to be played now anyway?†c
MS 118 20r: 27.8.1937
Page 31
There is much that is excellent in Macaulay's essays; only his value judgements on people are tiresome, &
superfluous. One would like to say to him: stop gesticulating! & just say what you have to say. MS 118 21v:
27.8.1937
Page Break 32
Ideas too sometimes fall from the tree before they are ripe. MS 118 35r c: 29.8.1937
Page 32
In philosophizing it is important for me to keep changing my position, not to stand too long on one leg, so as not to
get stiff.
Like someone on a long up-hill climb who walks backwards for a while to revive himself, stretch some
different muscles. MS 118 45r c: 1.9.1937
Page 32
Caught a bit of a chill & unable to think. Ghastly weather.--
Christianity is not a doctrine, not, I mean, a theory about what has happened & will happen to the human soul, but a
description of something that actually takes place in human life. For 'recognition of sin' is an actual occurrence & so
is despair & so is redemption through faith. Those who speak of it (like Bunyan), are simply describing what has
happened to them; whatever gloss someone may want to put on it! MS 118 56r c: 4.9.1937*
Page 32
When I imagine a piece of music, something I do every day & often, I--always I think--rhythmically grind my upper
& lower front teeth together. I have noticed it before but usually it takes place quite unconsciously. Moreover it's as
though the notes in my imagination were produced by this movement.
I think this way of hearing music in the imagination may be very common. I can of course also imagine
music without moving my teeth, but then the notes are much more blurred, much less clear, less pronounced. MS
118 71v c: 9.9.1937
Page 32
If certain graphic propositions for instance are laid down for human beings as dogmas governing thinking, namely in
such a way that opinions are not thereby determined, but the expression of opinions†a is completely controlled, this
will have a very strange effect. People will live under an absolute, palpable tyranny, yet without being able to say
they are not free.†i I think the Catholic Church does something like this. For dogma is expressed in the form of an
assertion & is unshakable, & at the same time any practical
Page Break 33
opinion can be made to accord with it; admittedly this is easier in some cases, more difficult in others. It is not a wall
setting limits to belief, but like a brake which in practice however serves the same purpose;†a†b almost as though
someone attached a weight to your foot to limit your freedom of movement.†c This is how dogma becomes
irrefutable & beyond the reach of attack. MS 118 86v: 11.9.1937
Page 33
With thinking too there is a time for ploughing & a time for harvesting.
It gives me satisfaction to write a lot every day. This is childish but that's how it is.MS 118 87r c: 11.9.1937*
Page 33
If I am thinking just for myself without wanting to write a book, I jump about all round the topic; that is the only
way of thinking that is natural to me. Forcing my thoughts into an ordered sequence is a torment for me. Should I
even attempt it now??
I squander untold effort making an arrangement of my thoughts that may have no value whatever. MS 118
94v c: 15.9.1937
Page 33
People have sometimes said to me†d they cannot make any judgement about this or that because they have never
learnt philosophy. This is irritating nonsense,†e it is†f being assumed that philosophy is some sort of science. And
people speak of it as they might speak of medicine.--What one can say, however, is that people who have never
carried out an investigation of a philosophical sort, like most mathematicians for instance, are not equipped with the
right optical instruments for that sort of investigation or scrutiny. Almost,†g as someone who is not used to
searching in the forest for berries†h will not find any because his eye has not been sharpened for such things & he
does not know where you have to be particularly on the lookout for them. Similarly someone unpractised in
philosophy passes by all the spots where difficulties lie hidden under the grass, while someone with practice pauses
& senses that there is a difficulty here, even though†i he does not yet
Page Break 34
see it.--And no wonder, if one knows how long even the practised investigator, who realizes there is a difficulty, has
to search in order to find it.
If something is well hidden it is hard to find. MS 118 113r: 24.9.1937
Page 34
Religious similes can be said to move on the edge of the abyss. B<unyan>'s allegory for instance. For what if we
simply add: "and all these traps, swamps, wrong turnings, were planted by the Lord of the Road, the monsters,
thieves, robbers were created by him?"
Without doubt, that is not the sense of the simile! but this sequel is too obvious! For many & for me it robs the
simile of its power.
But more especially if this is--as it were--suppressed. It would be different if it were said openly at every turn:
'I am using this as a simile, but look: it doesn't fit here'. Then you wouldn't feel you were being cheated, that
someone were trying to convince you by trickery. You can say to someone for instance: "Thank God for the good
you receive but don't complain about the evil,†a as you would of course do if a human being were to do you good
and evil by turns." Rules of life are dressed up in pictures. And these pictures can only serve to describe what we are
supposed to do, but not to justify it. Because to be a justification they would have to hold good in other respects too.
I can say: "Thank these bees for their honey as though they†i were good people who have prepared it for you"; that
is intelligible & describes how I wish you to behave. But not: <">. Thank them, for look how good they are!"--since
the next moment they may sting you.
Religion says: Do this!--Think like that! but it cannot justify this and it only need try to do so to become
repugnant; since for every reason it gives, there is a cogent counter-reason.
It is more convincing to say: "Think like this!--however strange it may seem.--" Or: "Won't you do
this?--repugnant as it is.--"
Page 34
Election by grace: It is only permissible to write like this out of the most frightful suffering--& then it means
something quite different. But for this reason it is not permissible for anyone to cite it as truth, unless he himself says
it in torment.--It simply isn't a theory.--Or as one might also say: if this is truth, it is not the truth it appears at first
glance to express. It's less
Page Break 35
a theory than a sigh, or a cry. MS 118 117v: 24.9.1937
Page 35
In the course of our conversations Russell would often exclaim: "Logic's hell!"--And this fully expresses what we†a
experienced while†b thinking about the problems of logic; namely their immense difficulty. Their hardness--their
hard & slippery texture.
The primary ground of this experience, I think, was this fact: that each new†c phenomenon of language that we
might retrospectively think of†d could show our earlier explanation to be unworkable.†e†f But that is the difficulty
Socrates gets caught up in when he tries to give the definition of a concept. Again and again an application of the
word emerges that seems not to be compatible with the concept to which other applications have led us. We say: but
that isn't how it is!--it is like that though!--& all we can do is keep repeating these antitheses. MS 119 59:
1.10.1937
Page 35
The spring that flows quietly & clearly†g in the Gospels seems to foam in Paul's Epistles. Or that is how it seems to
me. Perhaps it is just my own impurity that reads muddiness into it; for why shouldn't this impurity be able to
pollute what is clear? But for me it's as though I saw human passion, something like pride or anger, which does not
square with the humility of the Gospels. It is as though he really is insisting here on his own person, & doing so
moreover as a religious act, something which is foreign to the Gospel. I want to ask--& may this be no
blasphemy--: "What would Christ perhaps have said to Paul?"
But a fair rejoinder to that would be: What business is that of yours? Look after making yourself more decent! In
your present state, you are quite incapable of understanding what may be the truth here.
Page 35
In the Gospels--as it seems to me--everything is less pretentious, humbler, simpler. There you find huts;--with Paul
a church. There all human beings are equal & God himself is a human being; with Paul there is already something
like a hierarchy; honours, and official positions.--That is, as it were, what my NOSE tells me. MS 119 71:
4.10.1937
Page Break 36
Let us be human.--
Page 36
I just took some apples out of a paper bag where they had been lying for a long time; I had to cut off & throw away
half of many of them. Afterwards as I was copying out a sentence of mine the second half of which was bad, I at
once saw it as a half-rotten apple. And that's how it always is with me. Everything that comes my way becomes for
me†a a picture of what I am thinking about. (Is there something feminine about this outlook?) MS 119 83:
7.10.1937
Page 36
Doing this work I am†b in the same state as that of many people when they struggle in vain†c to recall a name; we
say in such a case: "think of something else, then it will come to you"--& similarly I had constantly to think of
something else†i so that what I had long been searching for could occur†d to me. MS 119 108: 14.10.1937
Page 36
The origin & the primitive form of the language game is a reaction; only from this can the more complicated forms
grow.
Language--I want to say--is a refinement, 'in the beginning was the deed'†ii. MS 119 146: 21.10.1937
Page 36
Kierkegaard writes: If Christianity were so easy and cosy, why would God have moved Heaven & Earth in his
Scripture, threatened eternal punishments--. --Question: But why is this Scripture so unclear then? If we want to
warn someone of a terrible danger, do we do it by giving him a riddle to solve, whose solution is perhaps the
warning?--But who is to say that the Scripture really is unclear: isn't it possible that it was essential in this case to tell
a riddle? That a more direct warning, on the other hand, would necessarily have had the wrong effect? God has four
people recount the life of the incarnate God, each one differently, & contradicting each other--but can't we say: It is
important that this narrative should not have more than quite middling historical plausibility, just so that this†21
should not be taken as the essential, decisive thing. So that the letter should not be believed more strongly than is
proper & the spirit should receive its due. I.e.: What you are supposed to see cannot be communicated even by the
Page Break 37
best, most accurate, historian; therefore a mediocre account suffices, is even to be preferred. For that too can tell
you what you are supposed to be told. (Roughly in the way a mediocre stage set can be better than a sophisticated
one, painted trees better than real ones,--which distract attention from what matters.)
The Spirit puts what is essential, essential for your life, into these words. The point is precisely that you are
SUPPOSED to see clearly only what even this representation clearly shows. (I am not sure how far all this is exactly
in the spirit of Kierkegaard.) MS 119 151: 22.10.1937
Page 37
In religion it must be the case that corresponding to every level of devoutness there is a form of expression that has
no sense at a lower level. For those still at the lower level this doctrine, which means something at the higher level, is
null & void; it can only be understood wrongly, & so these words are not valid for such a person.
Paul's doctrine of election by grace for instance is at my level irreligiousness, ugly non-sense. So it is not meant for
me since I can only apply wrongly the picture offered me. If it is a holy & good picture, then it is so for a quite
different level, where it must be applied in life quite differently than I could apply it. MS 120 8: 20.11.1937
Page 37
Christianity is not based on a historical truth, but presents us with a (historical) narrative & says: now believe! But
not believe this report with the belief that†22 is appropriate to a historical report,--but rather: believe, through thick &
thin & you can do this only as the outcome of a life. Here you have a message!--don't treat it as you would another
historical message! Make a quite different place for it in your life.--There is no paradox about that!
Page 37
If I realized how mean & petty I am, I should become more modest.
Nobody can say with truth of himself that he is filth. For if I do say it, though it can be true in a sense, still I cannot
myself be penetrated by this truth: otherwise I should have to go mad, or change myself.
Had coffee with A.R.†23; it was not as it used to be, but it was not bad either.
Queer as it sounds: the historical accounts of the Gospels might, in the historical sense, be demonstrably
false, & yet belief would lose nothing through this: but not because it has to do with 'universal truths of reason'!
Page Break 38
rather, because historical proof (the historical proof-game) is irrelevant to belief. This message (the Gospels) is seized
on by a human being believingly (i.e. lovingly): That is the certainty of this "taking-for-true", nothing else.
The believer's relation to these messages is neither a relation to historical truth (probability) nor yet that to a
doctrine consisting of 'truths of reason'. There is such a thing.--(We have quite different attitudes even to different
species of what we call fiction!) MS 120 83 c: 8-9.12.1937*
Page 38
You cannot write more truly about yourself than you are. That is the difference between writing about yourself and
writing about external things. You write about yourself from your own height. Here you don't stand on stilts or on a
ladder but on your bare feet. MS 120 103 c: 12.12.1937
Page 38
A great blessing for me to be able to work today. But I so easily forget all my blessings!
I am reading: "& no man can say that Jesus is the Lord, but by the Holy Ghost."†i And it is true: I cannot call him
Lord; because that says absolutely nothing to me. I could call him "the paragon", "God" even or rather: I can
understand it when he is so called; but I cannot utter the word "Lord" meaningfully. Because I do not believe that he
will come to judge me; because that says nothing to me. And it could only say something to me if I were to live
quite differently.
What inclines even me to believe in Christ's resurrection? I play as it were with the thought.--If he did not rise
from the dead, then he decomposed in the grave like every human being. He is dead & decomposed. In that case he
is a teacher, like any other & can no longer help; & we are once more orphaned & alone. And have to make do with
wisdom & speculation. It is as though we are in a hell, where we can only†a dream & are shut out from heaven,
roofed in as it were. But if I am to be REALLY redeemed,--I need certainty--not wisdom, dreams, speculation--and
this certainty is faith. And faith is faith in what my heart, my soul, needs, not my speculative intellect. For my soul,
with its passions, as it were with its flesh & blood, must be redeemed, not my abstract mind. Perhaps one may
Page Break 39
say: Only love can believe the Resurrection. Or: it is love that believes the Resurrection. One might say: redeeming
love believes even in the Resurrection; holds fast even to the Resurrection. What fights doubt is as it were
redemption. Holding fast to it must be holding fast to this belief. So this means: first be redeemed & hold on tightly
to your redemption (keep hold of your redemption)--then you will see that what you are holding on to is this belief.
So this can only come about if you no longer support yourself on this†a earth but suspend yourself from heaven.
Then everything is different and it is 'no wonder' if you can then do what now you cannot do. (It is true that
someone who is suspended looks like someone who is standing but the interplay of forces within him is nevertheless
a quite different one & hence he is able to do quite different things than can one who stands.) MS 120 108 c:
12.12.1937*
Page 39
Freud's idea: in madness the lock is not destroyed, only altered; the old key can no longer open it, but a differently
configured key could do so. MS 120 113: 2.1.1938
Page 39
A Bruckner symphony can be said to have two beginnings: the beginning of the first idea & the beginning of the
second idea. These two ideas stand to each other not as blood relations, but†b as man & wife.
Page 39
Bruckner's Ninth is a sort of protest against Beethoven's, & because of this†c becomes bearable, which as a sort of
imitation it would not be. It stands to Beethoven's Ninth very much as Lenau's Faust to Goethe's, which means as
the Catholic to the Enlightenment Faust. etc. etc. MS 120 142: 19.2.1938
Page 39
Nothing is so difficult as not deceiving yourself. MS 120 283: 7.4.1938
Longfellow:
In the elder days of art,
Builders wrought with greatest care
Each minute & unseen part,
For the gods are†i everywhere.
(This might serve as my motto.)
Page 39
MS 120 289: 20.4.1938
Page Break 40
Phenomena akin to language in music or architecture. Significant irregularity--in Gothic e.g. (I have in mind too the
towers of St. Basil's Cathedral.) Bach's music is more like language than Mozart's & Haydn's. The double bass
recitative in the 4th movement of Beethoven's 9th Symphony. (Compare too Schopenhauer's remark about
universal music composed to a particular text.)†24 MS 121 26v: 25.5.1938
Page 40
In philosophy the winner of the race is the one who can run most slowly. Or: the one who gets to the winning post
last.†a†b MS 121 35v: 11.6.1938
Page 40
Being psychoanalyzed is in a way like eating from the tree of knowledge.†c The knowledge we acquire sets us (new)
ethical problems; but contributes nothing to their solution. MS 122 129: 30.12.1939
Page 40
What is lacking in Mendelssohn's music? A 'courageous' melody? MS 162a 18: 1939-1940
Page 40
The Old Testament seen as the body without its head; the New T.: the head; the Epistles of the Apostles: the crown
on the head.
If I think of the Jewish Bible, the Old Testament on its own, I should like to say: the head is (still) missing from this
body The solution to these problems is missing The fulfilment of these hopes is missing. But I do not necessarily
think of a head as having a crown. MS 162b 16v: 1939-1940
Page 40
Envy is something superficial--i.e.: the typical colour of envy does not go down deep--farther down passion has a
different colouring. (That does not, of course, make envy any less real.) MS 162 21v: 1939-1940
Page 40
The measure of genius is character,--even if character on its own does not amount to genius
Genius is not 'talent and character', but character manifesting itself in the form of a special talent. Where one man
will show courage by jumping into the water, another will show courage by writing a symphony. (This is a weak
example.) MS 162b 22r c: 1939-1940
Page Break 41
There is no more light in a genius than in any other honest human being--but the genius concentrates this light into a
burning point by means of a particular kind of lens.
Page 41
Why is the soul moved by idle thoughts,--since they are after all idle? Well, it is moved by them.
(How can the wind move a tree, since it is after all just wind†a ? Well, it does move it; & don't forget it.) MS 162b
24r: 1939-1940
Page 41
One cannot speak the truth;--if one has not yet conquered oneself. One cannot speak it--but not, because one is still
not clever enough.
Page 41
The truth can be spoken only by someone who is already at home in it; not by someone who still lives in
untruthfulness, & does no more than reach out towards it from within untruthfulness. MS 162b 37r c: 1939-1940
Page 41
Resting on your laurels is as dangerous as resting when hiking through snow. You doze off & die in your sleep. MS
162b 42v c: 1939-1940
Page 41
The monstrous vanity of wishes is revealed for instance in†b my wish to fill a nice notebook with writing as soon as
possible. I get nothing from this; it's not that I wish it because, say, it will be evidence of my productivity; it is
simply a longing to rid myself of something familiar as soon as I can; although of course, as soon as I am rid of it, I
must start a fresh one & the whole business will have to be repeated. MS 162b 53r: 1939-1940
Page 41
One could call Schopenhauer a quite crude mind. I.e.,†c He does have refinement, but at a certain level this
suddenly comes to an end & he is as crude as the crudest. Where real depth starts, his finishes.
Page 41
One might say of Schopenhauer: he never takes stock of himself.
Page Break 42
I sit astride life like a bad rider on his mount. I owe it solely to the horse's good nature that I am not thrown off right
now. MS 162b 55v: 1939-1940
Page 42
'The impression (made by this melody) is completely indescribable.'- That means: a description is no use (for my
purpose); you have to hear the melody.
If art serves 'to arouse feelings', is, perhaps, perceiving it with the senses included amongst these feelings? MS
162b 59r: 1939-1940*
Page 42
My originality (if that is the right word) is, I believe, an originality that belongs to the soil, not the seed. (Perhaps I
have no seed of my own.) Sow a seed in my soil, & it will grow differently than it would in any other soil.
Freud's originality too was like this, I think. I have always believed--without knowing why--that the original seed of
psychoanalysis was due to Breuer, not Freud. Of course Breuer's seed-grain can only have been quite tiny.
(Courage is always original.)
Page 42
People nowadays think, scientists are there to instruct them, poets, musicians etc. to entertain them. That the latter
have something to teach them; that never occurs to them.
Page 42
Piano playing, a dance of human fingers. MS 162b 59v: 1939-1940
Page 42
Shakespeare, one might say, displays the dance of human passions. For this reason he has to be objective, otherwise
he would not so much display the dance of human passions--as perhaps talk about it. But he shows us them in a
dance, not naturalistically. (I got this idea from Paul Engelmann.) MS 162b 61r: 1939-1940
Page 42
The comparisons of the N.T.†i leave room for as much depth of interpretation†a as you like. They are bottomless.†b
Page Break 43
They have less style than the first speech of a child. Even a work of supreme art has something that can be called
'style', yes even something that can be called 'fashion'. MS 162b 63r: 1939-1940
Page 43
Within all great art there is a WILD animal: tamed.
Not, e.g., in Mendelssohn. All great art has primitive human drives as its ground bass. They are not the melody (as
they are, perhaps, in Wagner), but they are what gives the melody depth†a & power.
In this sense one may call Mendelssohn a 'reproductive' artist.--
In the same sense: my house for Gretl†25 is the product of a decidedly sensitive ear, good manners, the
expression of great understanding (for a culture, etc.). But primordial life, wild life striving to erupt into the open--is
lacking. And so you might say,†b health is lacking (Kierkegaard). (Hothouse plant.) MS 122 175 c: 10.1.1940
Page 43
A teacher who can show good, or indeed†c astounding results while he is teaching, is still not on that account a good
teacher, for it may be that, while his pupils are under his immediate influence, he raises them to a level which is not
natural to them, without developing their own capacities for work at this level, so that they immediately decline again
once the teacher leaves the schoolroom. Perhaps this holds for me; I have thought about this. (When Mahler was
himself conducting, his private performances†26 were excellent; the orchestra seemed to collapse at once if†d he
was not conducting it himself.) MS 122 190 c: 13.1.1940
Page 43
'The aim of music: to communicate feelings.'
Connected with this: We may rightly say "he has†e now the same face as before"--although measurement gave
different results in the two cases.
How are the words "the same facial expression" used?--How do we know that someone is using these words
correctly? But how do I know that I am using them correctly? MS 122 235: 1.2.1940
Page 43
Not funk but funk conquered is what is worthy of admiration & makes life
Page Break 44
worth having been lived. Courage, not cleverness; not even inspiration, is the grain of mustard that grows up to be a
great tree. To the extent there is courage, there is connection with life & death. (I was thinking of Labor's &
Mendelssohn's organ music.) But it is not by recognizing the want of courage in someone else, that you acquire
courage yourself. MS 117 151 c: 4.2.1940
Page 44
One might say: "Genius is courage in one's talent". MS 117 152 c: 4.2.1940
Page 44
Try to be loved & not-admired.†27 MS 117 153 c: 4.2.1940
Page 44
Sometimes you have to take an expression out of the language,†a to send it for cleaning,--& then you can put it back
into circulation. MS 117 156: 5.2.1940
Page 44
How hard it is for me to see what is right in front of my eyes! MS 117 160 c: 10.2.1940
Page 44
You can't be reluctant to give up your lie & still tell the truth. MS 117 168 c: 17.2.1940
Page 44
Writing the right style means, setting the carriage precisely†b on the rails. MS 117 225: 2.3.1940
Page 44
If this stone won't budge at present, if it is wedged in, first move other stones around it.--
Page 44
We are only going to set you straight on the track, if your carriage stands on the rails crookedly; driving†c is
something we shall leave you to do by yourself.†d MS 117 237: 6.3.1940
Page 44
Scraping away mortar is much easier than moving a stone. Well, you have to do the one, before you can do the
other. MS 117 253: 11.3.1940
Page Break 45
What is insidious about the causal approach is that it leads one to say: "Of course,†a that's how it has to†b happen".
Whereas one ought to say: It may have happened like that, & in many other ways.†c
Page 45
If we use the ethnological approach does that mean we are saying philosophy is ethnology? No it only means we are
taking up our position far outside, in order to see the things more objectively. MS 162b 67r: 2.7.1940
Page 45
One of my most important methods is to imagine a historical development of our ideas different from what has†d
actually occurred. If we do that the problem shows us a quite new side. MS 162b 68v: 14.8.1940
Page 45
What I am resisting is the concept of an ideal exactness thought as it were to be given us a priori. At different times
our ideals of exactness are different; & none of them is preeminent. MS 162b 69v: 19.8.1940
Page 45
It is often only very slightly more disagreeable to tell the truth than a lie; only about as much as is drinking bitter
rather than sweet coffee; & yet even then I have a strong inclination to tell the lie. MS 162b 70r: 21.8.1940
Page 45
(My style is like bad musical composition.)
Page 45
Don't apologize for anything, don't obscure anything, look & tell how it really is--but you must see something that
sheds a new light on the facts. MS 123 112: 1.6.1941
Page 45
Our greatest stupidities may be very wise. MS 124 3 c: 6.6.1941
Page 45
It is incredible how helpful a new drawer can be, suitably placed in our filing cabinet. MS 124 25: 11.6.1941
Page 45
You must say something new & yet nothing but what is old. (N.)
Page Break 46
You must indeed say only what is old--but all the same something new!
Page 46
Different 'interpretations' must correspond to different applications.
Page 46
The poet too must always be asking himself: 'is what I am writing really true then?' which does not necessarily mean:
'is this how it happens in reality?'.
Page 46
(...)
Page 46
It's true you must assemble old material. But for a building.--(W.)†28 MS 124 28: 11.6.1941
Page 46
As we get old the problems slip through our fingers again, as in our youth. It is not just that we cannot crack them
open,†a we can't even keep hold of them. MS 124 31: 12.6.1941
Page 46
What a curious attitude scientists have--: "We still don't know that;†b but it is knowable & it is only a question of
time till we know it"! As if that went without saying.-- MS 124 49: 16.6.1941
Page 46
I could imagine someone thinking the names "Fortnum" & "Mason"†29 fitted together. MS 124 56: 18.6.1941
Page 46
Don't demand too much, & don't be afraid that your just demand will melt into nothing. MS 124 82: 27.6.1941
Page 46
People who are constantly asking 'why' are like tourists, who stand in front of a building, reading Baedeker, &
through reading about the history of the building's construction etc etc are prevented from seeing it. MS 124 93:
3.7.1941
Page 46
Counterpoint might represent an extraordinarily difficult problem for a composer; the problem namely: given my
propensities what should be my relation towards counterpoint. He may have found a conventional relation
Page Break 47
to it yet feel perhaps that it is not his. That it is not clear what the importance of counterpoint to him ought to be.
(I was thinking of Schubert in this connection; of his still wanting to take lessons in counterpoint at the end
of his life. I think his aim may have been not simply learning more counterpoint, but rather determining where he
stood in relation to it.) MS 163 25r: 4.7.1941
Page 47
Wagner's motifs might be called musical prose sentences. And just as there is such a thing as "rhyming prose", so
too these motifs can certainly be put together into melodic form, but without their constituting one melody.
Page 47
Wagnerian drama too is not drama, but a stringing together of situations as if on a thread, which for its part is only
cleverly spun but not, like the motifs & situations, inspired. MS 163 34r: 7.7.1941
Page 47
Don't let yourself be guided by the example of others, but by nature! MS 163 39r c: 8.7.1941
Page 47
The language used by philosophers is already deformed, as though by shoes that are too tight. MS 163 47v:
11.7.1941
Page 47
The characters in a drama arouse our sympathy, they are like people we know, often like people we love or hate: The
characters in the second part of Faust don't arouse our sympathy at all! We don't feel as though we knew them. They
file past us like thoughts not like human beings. MS 163 64v c: 6.9.1941
Page 47
The mathematician (Pascal) who admires the beauty of a theorem in number theory†a; it is as though he were
admiring some natural beauty. It's wonderful, he says, what splendid properties numbers have. It's as though he
were admiring the conformity to laws of a crystal†b.
Page 47
One might say: what splendid laws the Creator has built into numbers!
Page Break 48
Page 48
You can't construct clouds. And that is why the future you dream of never comes true.
Page 48
Before there was an aeroplane people dreamed about aeroplanes & what a world with them would look like. But, as
the reality was nothing like this dream, so we have no reason to believe that reality will develop in the way we
dream. For our dreams are full of tinsel, like paper hats & costumes. MS 125 2v: 4.1.1942 or later
Page 48
The popular scientific writings of our scientists are not the expression of hard work but of resting on their
laurels.†a†b†30
Page 48
If you already have someone's love, no sacrifice is too high a price to pay for it but any sacrifice is too great†c to buy
it. MS 125 21r: 1942
Page 48
Virtually as there is such a thing as a deep & a shallow sleep, there are thoughts which occur deep within one &
thoughts which romp about on the surface. MS 125 42r: 1942
Page 48
You cannot draw the seed up out of the earth. You can only give it warmth†d, moisture & light & then it must grow.
(You mustn't even touch†e it except with care.) MS 125 44r: 1942
Page 48
What is pretty cannot be beautiful.------ MS 125 58r: 1942
Page 48
Someone is imprisoned in a room if the door is unlocked, opens inwards; but it doesn't occur to him to pull, rather
than push against it.
Page 48
Put someone in the wrong atmosphere & nothing will function as it should. He will seem unhealthy in every part.
Bring him back into his right element, & everything will blossom and look healthy. But if he is not in his
Page Break 49
right element, what then? Well he just has to make the best of looking like a cripple.
Page 49
If white turns to black some say: "Essentially it is still the same". And others, if the colour becomes†a one degree
darker, say "It has changed completely." MS 125 58v: 18.5.1942
Page 49
Architecture is a gesture. Not every purposive movement of the human body is a gesture. Just as little as every
functional building is architecture. MS 126 15r: 28.10.1942
Page 49
At present we are combatting a trend. But this trend will die out, superseded by others. And then people will no
longer understand our arguments against it; will not see why all that needed saying. MS 126 64r: 15.12.1942
Page 49
Looking for the fallacy in a fishy argument & hunt-the-thimble. MS 126 65v: 17.12.1942
Page 49
Suppose that 2000 years ago someone had invented the shape
& said that one day it would be the shape of an instrument of locomotion.
Or perhaps: that someone had constructed the complete mechanism of the steam engine without having the
least†b idea how it could be used as a motor.†c MS 127 14r: 20.1.1943
Page 49
What you are taking for a gift is a problem you have to solve.
Page 49
Genius is what makes us forget the master's talent.
Page 49
Genius is what makes us forget talent.†d
Page 49
Where genius wears thin skill may show through.†e (Overture to the Mastersingers.
Page Break 50
Page 50
Genius is what makes us unable to see the master's talent.
Page 50
Only where genius wears thin can you see the talent. MS 127 35v: 4.4.1943
Page 50
Why shouldn't I apply words in opposition to their original usage? Doesn't e.g. Freud†a do that when he calls even
an anxiety dream a wish-fulfilment dream? Where is the difference? In the scientific approach the new use is
justified through a theory. And if this theory is false then the new extended use has to be given up too. But in
philosophy the extended use is not supported by true or false opinions about natural processes. No fact†b justifies it
(&)†c non can overturn it.
Page 50
We say:†d "You understand this expression, don't you? Well, the way you always understand it†e is the way I too
am using it."†f [Not: "... in that meaning..."]
As though meaning were a halo which the word carries over†g†h into every sort of application MS 127 36v:
27.2.1944
Page 50
Thoughts at peace. That is the goal someone who philosophizes longs for. MS 127 41v: 4.3.1944
Page 50
The philosopher is someone who has to cure many diseases of the understanding in himself, before he can arrive at
the notions of common sense. MS 127 76r: 1944
Page 50
If in life we are surrounded by death, so too in the health of our understanding by madness.†i†31 MS 127 77v: 1944
Page 50
Wanting to think is one thing, having a talent for thinking another. MS 127 78v: 1944
Page Break 51
If there is anything in the Freudian theory of dream interpretation; then it shows how complicated is the way the
human mind makes†a pictures of the facts.
Page 51
So complicated, so irregular is the mode of representation that it can barely be called representation any more. MS
127 84r: 1944
Page 51
It will be hard to follow my portrayal: for it says something new, but still has eggshells of the old material sticking to
it. MS 129 181: 1944 or later
Page 51
Is it some frustrated longing that makes someone mad? (I was thinking of Schumann, but of myself too.)MS 165
200 c: ca. 1941-1944
Page 51
The revolutionary will be the one who can revolutionize himself. MS 165 204: ca. 1944
Page 51
People are religious to the extent that they believe themselves to be not so much imperfect as sick.
Page 51
Anyone who is half-way decent will think himself utterly imperfect, but the religious person thinks himself wretched
Page 51
What's ragged should be left ragged.
Page 51
A miracle is, as it were, a gesture which God makes. As a man sits quietly & then makes an impressive gesture, God
lets the world run on smoothly & then accompanies the words of a Saint by a symbolic occurrence, a gesture of
nature. It would be an instance if, when a saint has spoken, the trees around him bowed, as if in reverence.--Now, do
I believe that this happens? I don't.
Page 51
The only way for me to believe in a miracle in this sense would be to be impressed by an occurrence in this
particular way. So that I should say e.g.: "It was impossible to see these trees & not to feel that they were responding
to the words." Just as I might say "It is impossible to see the face of this dog & not to see that he is alert & full of
attention to what his master is
Page Break 52
doing <">. And I can imagine that the mere report of the words & life of a saint can make someone believe the
reports that the trees bowed. But I am not so impressed.
Page 52
When I came home I expected a surprise & there was no surprise for me, so, of course, I was surprised. MS 128 46:
ca. 1944
Page 52
Go on, believe! It does no harm.
Page 52
'Believing' means, submitting to an authority. Having once submitted to it, you cannot then, without rebelling against
it, first call it in question & then once again find it convincing.
Page 52
A cry of distress cannot be greater than that of one human being.
Page 52
Or again no distress can be greater than what a single person can suffer.
Hence one human being can be in infinite distress & so need infinite help.
The Christian religion is only for the one who needs infinite help, that is only for the one who suffers infinite
distress.
Page 52
The whole Earth cannot be in greater distress than one soul.
Page 52
Christian faith--so I believe--is refuge in this ultimate distress.
Someone to whom it is given in such distress to open his heart instead of contracting it, absorbs the remedy into his
heart.
Someone who in this way opens his heart to God in remorseful confession opens it for others too. He
thereby loses his dignity as someone special†a & so becomes like a child. That means without office, dignity &
aloofness from others. You can open yourself to others only out of a particular kind of love. Which acknowledges as
it were that we are all wicked children.
Page 52
It might also be said: hate between human beings comes from our cutting ourselves off from each other. Because we
don't want anyone else to see inside us, since it's not a pretty sight in there.
Page Break 53
Page 53
Of course you must continue to feel ashamed of what's within you, but not ashamed of yourself before your fellow
human beings.
Page 53
There is no greater distress to be felt than that of One human being. For if someone feels himself lost, that is the
ultimate distress. MS 128 49: ca. 1944
Page 53
Words are deeds.†32 MS 179 20: ca. 1945
Page 53
Only someone very unhappy has the right to pity someone else. MS 179 26: ca. 1945
Page 53
It isn't reasonable to be furious even at Hitler; let alone at God. MS 179 27: ca. 1945
Page 53
When people have died we see their life in a conciliatory light. His life looks well-rounded through a haze. For him it
was not well-rounded however, but jagged & incomplete. For him there was no conciliation; his life is naked &
wretched. MS 180a 30: ca. 1945
Page 53
It is as though I had lost my way & asked someone the way home. He says he will show me and walks with me
along a nice smooth path. This suddenly comes to an end. And now my friend says: "All you have to do now is to
find the rest of the way home from here.<">†33 MS 180a 67: ca. 1945
Page 53
The less somebody knows & understands himself the less great he is, however great may be his talent. For this
reason our scientists are not great. For this reason Freud, Spengler, Kraus, Einstein are not great. MS 130 239:
1.8.1946*
Page 53
Schubert is irreligious & melancholy. MS 130 283: 5.8.1946
Page 53
Are all people great human beings? No.--Well then, what hope can you have of being a great human being! Why
should something be given you that is not given your fellows? To what purpose?!--If it isn't your wish to be rich that
makes you think you are rich, then it must be some observation some experience that shows you it! And what
experience do you have (except that of vanity)? Simply that you have a talent. And my conceit of
Page Break 54
being an extraordinary human being is of course much older than my experience, of my particular talent. MS 130
291 c: 9.8.1946
Page 54
Schubert's melodies can be said to be full of climaxes, & this cannot be said of Mozart's; Schubert is baroque. You
can point to particular places in a Schubert melody & say: look, that is the point of this melody, this is where the
idea comes to a head.
Page 54
The melodies of different composers can be approached by applying the principle: every species of tree is a 'tree' in a
different sense of the word. I.e. Don't let yourself be misled by our saying they are all melodies. They are steps along
a path that leads from something you would not call a melody to something else that you again would not call one.
If you simply look at the sequences of notes & the changes of key all these structures no doubt appear on the same
level. But if you look at the field of force in which they stand (and hence at their significance), you will be inclined to
say: Here melody is something quite different than there (here it has a different origin, plays a different role, inter
alia. MS 131 2: 10.8.1946
Page 54
The idea working its way towards the light. MS 131 19: 11.8.1946
Page 54
The remark by Jucundus in 'The Lost Laugh'†34, that his religion consisted in: his knowing, if things are going well
for him now,†a that his fate could take a turn for the worse--this actually is an expression of the same religion as the
saying "The Lord hath given, the Lord hath taken away". MS 131 27: 12.8.1946
Page 54
It is hard to understand yourself properly since something that you might be doing out of generosity & goodness is
the same as you may be doing out of cowardice or indifference. To be sure, one may act in such & such a way from
true love, but also from deceitfulness & from a cold heart too. Similarly not all moderation is goodness. And only if I
could be submerged in religion might these doubts be silenced. For only religion could destroy vanity & penetrate
every nook & cranny. MS 131 38: 14.8.1946
Page 54
What I want to say then is: Someone who--e.g.--cannot EXPERIENCE
Page Break 55
the word "pas" in "je ne sais pas" as "step", cannot be taught an expression of voice†a by being told "speak it with
this meaning".
If you are reading aloud and want to read well, you accompany the words with more vivid images. At least it
is often like that. Sometimes though ["To Corinth from Athens..."]†35 it is the punctuation, i.e., the precise
intonation & the length of the pauses that is all that matters to us. MS 131 43: 14.8.1946*
Page 55
It is remarkable how hard we find it to believe something the truth of which we do not see for ourselves. If e.g. I hear
expressions of admiration for Shakespeare made by the distinguished men of several centuries, I can never rid
myself of a suspicion that praising him has been a matter of convention, even though I have to tell myself that this is
not the case. I need the authority of a Milton to be really convinced. In his case I take it for granted that he was
incorruptible.--But of course I don't mean to deny by this that an enormous amount of praise has been & still is
lavished on Shakespeare without understanding & for specious reasons by a thousand professors of literature. MS
131 46: 15.8.1946
Page 55
Grasping the difficulty in its depth is what is hard.
For if you interpret it in a shallow way the difficulty just remains. It has to be pulled out by the root; & that means,
you have to start thinking about these things in a new way. The change is as decisive e.g. as that from the alchemical
to the chemical way of thinking.--The new way of thinking is what is so hard to establish.
Page 55
Once it†b is established the old problems disappear; indeed it becomes hard to recapture them. For they are
embedded in the way we express ourselves; & if we clothe ourselves in a new form of expression, the old problems
are discarded along with the old garment.†c MS 131 48: 15.8.1946
Page 55
The hysterical fear of the atom bomb the public now has, or at least expresses, is almost a sign that here for once a
really salutary discovery has been made. At least the fear gives the impression of being fear in the face of a really
effective bitter medicine. I cannot rid myself of the thought: if there
Page Break 56
were not something good here, the philistines would not be making an outcry. But perhaps this too is a childish
idea. For all I can mean really is that the bomb creates the prospect of the end, the destruction of a ghastly evil, of
disgusting soapy water science and certainly that is not an unpleasant thought; but who is to say what would come
after such a destruction? The people now making speeches against the production of the bomb are undoubtedly the
dregs of the intelligentsia, but even that does not prove beyond question that what they abominate is to be
welcomed. MS 131 66c: 19.8.1946
Page 56
In former times people entered monasteries. Were they perhaps simple-minded, or obtuse people?--Well, if people
like that took such measures so as to be able to go on living, the problem cannot be an easy one! MS 131 79 c:
20.8.1946
Page 56
The human being is the best picture of the human soul.†36 MS 131 80:20.8.1946
Page 56
Shakespeare's similes are, in the ordinary sense, bad. So if they are nevertheless good--& I don't know whether they
are or not--they must be a law to themselves. Perhaps e.g. their ring makes them convincing & gives them truth.
It might be the case that with S. the essential thing is his effortlessness, his arbitrariness, so that if you are to be able
really to admire him, you just have to accept him as he is in the way you accept nature, a piece of scenery e.g.
If I am right about this, that would mean that the style of his whole work, I mean, of his complete works†a is
in this case what is essential, & provides the justification.
Page 56
That I do not understand him could then be explained by the fact that I cannot read him with ease. Not, that is, as
one views a splendid piece of scenery. MS 131 163:31.8.1946
Page 56
A man sees well enough what he has, but not what he is. What he is can be compared with his height above sea
level, which you cannot for the most part judge straight off. And the greatness, or triviality, of a work depends on
where its creator stands.
Page Break 57
Page 57
But you can equally say: someone who misjudges himself is never great: someone who throws dust in his own eyes.
MS 131 176: 1.9.1946
Page 57
How small a thought it takes to fill a whole life!
Page 57
Just as someone may travel around the same little country throughout his whole life, & think there is nothing outside
it!
You see everything in a queer perspective (or projection): the country that you ceaselessly keep covering, strikes you
as enormously big; the surrounding countries seem to you like narrow border regions.†a MS 131 180: 2.9.1946
Page 57
To go down into the depths you don't need to travel far; you can do it in your own backgarden.†b MS 131
182: 2.9.1946
Page 57
It is very remarkable, that we should be inclined to think of civilization--houses, streets, cars, etc--as separating man
from his origin, from the lofty, eternal, etc. Our civilized environment, even its trees & plants, seems to us then
cheap, wrapped†c in cellophane, & isolated from everything great & from God as it were. It is a remarkable picture
that forces itself on us here. MS 131 186: 3.9.1946
Page 57
My 'achievement' is very much like a mathematician's,†d who invents a new calculus.†e MS 131 218: 8.9.1946
Page 57
If people did not sometimes commit stupidities, nothing intelligent at all would ever happen. MS 131 219:
8.9.1946
Page 57
The purely corporeal can be uncanny. Compare the way†f angels & devils are portrayed. A so-called "miracle" must
be connected with this. It must be as it were a sacred gesture. MS 131 221: 8.9.1946
Page Break 58
The way you use the word "God" does not show whom you mean,†a but what you mean. MS 132 8: 11.9.1946
Page 58
In a bullfight the bull is the hero of a tragedy. First driven mad by suffering, he dies a slow & terrible death. MS
132 12: 12.9.1946
Page 58
A hero looks death in the face, real death, not just a picture of death. Behaving decently in a crisis does not mean
being able to act the part of a hero well, as in the theatre, it means rather being able to look death itself in the eye.
For an actor may play a multitude of roles, but in the end it is after all he himself, the human being, who has to die.
MS 132 46 c: 22.9.1946
Page 58
What does it consist in: following a musical phrase with understanding? Observing a face with a feeling†b for its
expression? Drinking in the expression on the face?
Page 58
Think of the demeanour of someone who draws the face with understanding for its expression. Think of the
sketcher's face his movements;--what shows that every stroke he makes is†i dictated by the face, that nothing in his
sketch is arbitrary, that he is a delicate instrument?
Is that really an experience? I mean: can we say that this expresses an experience?
Page 58
Once again: what does it consist in, following a musical phrase with understanding, or, playing it with
understanding? Don't look inside yourself. Ask yourself rather, what makes you say that's what someone else is
doing. And what prompts you to say he has a particular experience? Indeed, do we ever actually say that? Wouldn't
I be more likely to say of someone else that he's having a whole host of experiences†ii?
I would perhaps say: "He is experiencing the theme intensely"; but ask yourself, what the expression of this†c
is?†d†e
Page Break 59
Then again you might think intensive experiencing of the theme 'consists' in the sensations of the movements etc.
with which we accompany it. And that seems (again) like a soothing explanation. But have you any reason to think
it true? I mean, e.g., a recollection of this experience? Is not this theory again merely a picture? No, this is not how
things are: the theory is no more than an attempt to link up the expressive movements with an 'experience'.
Page 59
If you ask: how I experienced the theme, I shall perhaps say "As a question" or something of the sort, or I shall
whistle it with expression etc. MS 132 51: 22.9.1946
Page 59
"He is experiencing the theme intensely. Something is happening in him when†a he hears it." Well, what?
Page 59
Does the theme point to nothing beyond itself? Oh yes! But that means:--The impression it makes on me is
connected with things in its surroundings--e.g. with the existence of the German language & of its intonation, but
that means with the whole field†i of our language games.†37
If I say e.g.: it's as if here a conclusion were being drawn, or, as if here something were being confirmed, or, as if this
were a reply to what came earlier,--then the way I understand it clearly presupposes familiarity with conclusions,
confirmations, replies, etc.
Page 59
A theme, no less than a face, wears an expression.
Page 59
"The repeat is necessary" In what respect is it necessary? Well, sing it, then you will see that it is only the repeat that
gives it its tremendous power.--Don't we feel then as though a model for this theme must in this case exist in reality,
& as though the theme only approached it, corresponded to it, once this part were repeated Or am I to utter the
inanity: 'It just sounds more beautiful with the repeat'? (You see there by the way what an inane role the word
"beautiful" plays in aesthetics) And yet there just is no paradigm there other than the theme. And yet again there is a
paradigm other than the theme: namely the rhythm of our language, of our thinking & feeling. And furthermore the
theme is a new part of our language, it becomes incorporated in it; we learn a new gesture.
Page Break 60
Page 60
The theme interacts with language.
Page 60
It is one thing to sow in thought, another to reap in thought.
Page 60
The last two bars of the "Death & the Maiden" theme, the ; you may think first that this figure is
conventional, ordinary, until you understand its deeper expression. I.e. until you understand that here the ordinary is
filled with significance. MS 132 59: 25.9.1946
Page 60
"Fare well!"
"A whole world of pain lies in these words" How can it live†a in them?--It is bound up with them. The words are like
the acorn from which an oak tree can grow.
But where is the law laid down, according to which the tree grows out of the acorn? Well, the picture is
incorporated into our thinking as a result of experience.†b MS 132 62: 25.9.1946*
Page 60
Esperanto. Our feeling of disgust, when we utter an invented word with invented derivative syllables. The word is
cold, has no associations & yet plays at 'language'. A system of purely written signs would not disgust us like this.
MS 132 69: 26.9.1946
Page 60
You could attach prices to ideas. Some cost a lot some little. [Broad's ideas all cost very little.] And how do you pay
for ideas? I believe: with courage. MS 132 75: 28.9.1946*
Page 60
If life becomes hard to bear we think of improvements†c. But the most important & effective improvement,†d in our
own attitude, hardly occurs to us, & we can decide on this only with the utmost difficulty.†e MS 132 136:
7.10.1946
Page 60
It's possible to write in a style that is unoriginal in form--like mine--but with well chosen words; or on the other hand
in one that is original in form, freshly grown from within oneself. (And also of course in one which is botched
together just anyhow out of old furnishings†f.) MS 32 145: 8.10.1946
Page Break 61
Amongst other things Christianity says, I believe, that sound doctrines are all useless. That you have to change your
life. (Or the direction of your life.)
That all wisdom is cold; & that you can no more use it for setting your life to rights, than you can forge iron
when it is cold.
For a sound doctrine need not seize you; you can follow it, like a doctor's prescription.--But here you have to
be seized & turned around by something.--(I.e. this is how I understand it.) Once turned round, you must stay
turned round.
Wisdom is passionless. By contrast Kierkegaard calls faith a passion. MS 132 167: 11.10.1946
Page 61
Religion is as it were the calm sea bottom at its deepest, remaining calm, however high the waves rise on the
surface.-- MS 132 190: 16.10.1946
Page 61
"I never before believed in God"--that I understand. But not: "I never before really believed in Him." MS 132
191: 18.10.1946
Page 61
I often fear madness. Have I any reason to assume that this fear does not spring from, so to speak, an optical
illusion: of seeing something as an abyss that is close by, when it isn't? The only experience I know of that speaks
for its not being an illusion, is the case of Lenau. For in his "Faust" there are thoughts of a kind I too am familiar
with. Lenau puts them into Faust's mouth, but they are no doubt his own about himself. What is important is what
Faust says about his loneliness or isolation.
Page 61
His talent too strikes me as similar to mine: A lot of froth--but a few fine thoughts. The stories in his Faust are all
bad, but the observations often true & great. MS 132 197: 19.10.1946
Page 61
Lenau's Faust is remarkable in that here man has dealings only with the Devil. God does not stir himself. MS 132
202: 20.10.1946
Page 61
In my view Bacon was not a precise thinker. He had large, as it were broad, visions But someone who has nothing
but these is bound to be generous with promises, inadequate in keeping them. You may envision a flying machine
without being precise about its details. Outwardly you may†a imagine it as very similar to a proper aeroplane, &
describe its functioning
Page Break 62
graphically. Nor is it obvious, that such an invention†a has to be worthless. Perhaps it spurs others to a different sort
of work.--So while these others make preparations, a long time in advance as it were, for building an aeroplane that
really flies, the former occupies himself with dreaming what this aeroplane has to look like & what it will be capable
of. This so far says nothing about the value of these activities. The dreamer's may be worthless--& so may the
others. MS 132 205: 22.10.1946
Page 62
Madness doesn't have to be regarded as an illness. Why not as a sudden--more or less sudden--change of character?
Everybody is (or most are) mistrustful, & perhaps more so towards their relations, than towards others. Is their any
reason for mistrust? Yes & no. Reasons can be given for it, but they are not compelling. Why shouldn't someone
suddenly become much more mistrustful of people? Why not much more withdrawn? or devoid of love? Don't
people get like this even in the ordinary course of events?--Where is the line to be drawn here between will & ability?
Is it that I will not open my heart to anyone any longer, or that I cannot? If so much can lose its attraction, why not
everything? If someone is wary even in ordinary life, why shouldn't he--& perhaps suddenly--become much more
wary? And much more inaccessible. MS 133 2: 23.10.1946
Page 62
The lesson in a poem is overstated, if the intellectual points are nakedly exposed, not clothed by the heart. MS
133 6: 24.10.1946
Page 62
Oh a key can†38 lie for ever where the locksmith placed it, & never be used to open†b the lock for which the master
forged it. MS 133 12: 24.10.1946
Page 62
"It is high time for us to compare this phenomenon†c with something different"--one may say.--I am thinking, e.g.
of mental illnesses. MS 133 18: 29.10.1946
Page 62
Freud's fanciful pseudo-explanations (just because they are so brilliant) performed a disservice.
Page Break 63
Page 63
(Now every ass has them†a within reach for 'explaining' symptoms of illness with their help.) MS 133 21:
31.10.1946
Page 63
Irony in music. E.g. Wagner in the Mastersingers. Incomparably deeper in the first movement of the IXth in the
fugato. Here is something, that corresponds to the expression of bitter irony in speech.
Page 63
I could equally well have said: the distorted in music. In the sense in which we speak of features distorted by grief.
When Grillparzer says Mozart countenanced only the "beautiful" in music, that means, I think, that he did not
countenance the distorted, frightful, that there is nothing in his music corresponding to this. I am not saying that is
quite true; but assuming it to be so, it is a prejudice on Grillparzer's part, to think that by rights it ought not to be
otherwise. The fact that music since Mozart (of course especially through Beethoven) has extended the range of its
language is to be neither commended nor deplored, rather: that's how it is.†b Grillparzer's attitude involves a certain
ingratitude. Did he want another Mozart? Could he imagine something†39 that such a being might compose?
Would he have been able to imagine Mozart if he had not known him?
The concept of "the beautiful" has done a lot of mischief here too.
Page 63
Concepts may alleviate mischief or make it worse; foster it or check it. MS 133 30: 1.-2.11.1946
Page 63
The fundamental insecurity of life. Misery, everywhere you look.
The grinning faces of idiots may, it is true, make us think they do not really suffer; but they do, only not in the same
place as the more intelligent. They do not have, as one might say, headache, but as much other wretchedness as
anyone else. Not all wretchedness need after all evoke the same facial expression. A nobler person who suffers will
look different from me. MS 133 68 c: 12.11.1946*
Page 63
I cannot kneel to pray, because it's as though my knees were stiff. I am afraid of dissolution (of my own dissolution)
should I become soft. MS 133 82: 24.11.1946
Page Break 64
I am showing my pupils sections of an immense landscape, which they cannot possibly find their way around. MS
133 82: 24.11.1946
Page 64
The truly apocalyptic view of the world is that things do not repeat themselves. It is not e.g. absurd to believe that
the scientific & technological age is the beginning of the end for humanity, that the idea of Great Progress is a
bedazzlement, along with the idea that the truth will ultimately be known; that there is nothing good or desirable
about scientific knowledge & that humanity, in seeking it, is falling into a trap. It is by no means clear that this is not
how things are. MS 133 90: 7.1.1947
Page 64
A man's dreams are virtually never realized. MS 133 118: 19.1.1947
Page 64
Socrates, who always reduces the Sophist to silence--does he reduce him to silence rightfully?--It's true, the Sophist
does not know what he thinks he knows; but that is no triumph for Socrates. It can neither be a case of "You see!
You don't know it!"--nor, triumphantly, "So none of us knows anything!"
Because I don't want to think just to convict myself, or even someone else, of unclarity I am not trying to
understand something,†a simply in order to see that I still do not understand it. MS 133 188: 27.2.1947*
Page 64
Wisdom is something cold, & to that extent foolish. (Faith, on the other hand, a passion.) We might also say:
wisdom merely conceals life from you. (Wisdom is like cold, grey ash covering the glowing embers.) MS 134 9:
3.3.1947
Page 64
Don't for heaven's sake, be afraid of talking nonsense! Only don't fail to pay attention to your nonsense. MS 134 20:
5.3.1947
Page 64
The miracles of nature,
We might say: art discloses the miracles of nature to us. It is based on the concept of the miracles of nature. (The
blossom, just opening out. What is marvellous about it?) We say: "Look, how it's opening out!"
Page 64
It could only be by accident that someone's dreams about the future of
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philosophy, art, science would come true. What he sees is a continuation of his own world in his dream, that is to
say PERHAPS his wish (and perhaps not) but not reality.
It might still happen that a person's photograph, e.g., changed with time, almost as if he were aging on it. But
its changes then take place according to their own laws & why should they lead in a parallel direction to the
development of the real person?
Page 65
The mathematician too can of course marvel at the miracles (the crystal) of nature; but can he do it, once a problem
has arisen about what he sees?†a Is it really possible as long as the object he finds awe-inspiring or gazes at with awe
is shrouded in a philosophical fog?
I could imagine someone admiring trees, & also the shadows, or reflections of trees, which he mistakes for trees. But
if he should once tell himself that these†b are not after all trees & if it becomes a problem for him what they are, or
what relation they have to trees, then his admiration will have suffered a rupture, that will now need healing. MS
134 27: 10.-15.3.1947*
Page 65
Sometimes a sentence can be understood only if it is read at the right tempo. My sentences are all to be read slowly.
MS 134 76: 28.3.1947
Page 65
The 'necessity' with which the second idea succeeds the first. (Overture to Figaro.) Nothing could be more idiotic
than to say it's 'pleasing' to hear the second after the first!--But the paradigm according to which everything there is
right is certainly obscure. 'It is the natural development.' You gesture with your hand, would like to say: "of
course!"--You could too compare the transition to a transition (the entry of a new character) in a story, e.g., or a
poem. That is how this piece fits into the world of our thoughts & feelings.MS 134 78: 30.3.1947
Page 65
The folds of my heart all the time tend to stick together & to open it I should need to keep tearing them apart. MS
134 80 30.3.1947
Page 65
A foolish & naïve American film can in all its foolishness & by means of it be instructive. A fatuous, non-naïve†c
English film can teach nothing. I have
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often drawn a lesson from a foolish American film. MS 134 89: 2.4.1947
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Is what I am doing in any way worth the effort? Well only, if it receives a light from above. And if that
happens,--why should I take care, not to be robbed of the fruits of my labour? If what I write really has value, how
were anyone to steal the value from me? If the light from above is lacking, then I can in any case be no more than
clever. MS 134 95: 3.4.1947
Page 66
I completely understand, how someone may hate it, if the priority of his invention or discovery is called in question,
how†a he may be willing to defend†b this priority with tooth & claw. And yet it is only a chimera. To be sure it
seems to me too cheap, all too easy for Claudius to scoff at the priority disputes between Newton & Leibniz; but I
think it is true all the same that this quarrel springs only from vile weaknesses & is nourished by VILE people. What
would Newton have lost if he had acknowledged Leibniz's originality? Absolutely nothing! He would have gained a
lot. And yet, how hard is such an acknowledgement, seeming,†c to someone who attempts it, like a confession of
his own incapacity. Only people who esteem one, & at the same time love one, can make such behaviour†d easy for
one.†e
It's a question of envy of course. And anyone who feels it, ought to keep saying to himself: 'It's a mistake! It's a
mistake!--" MS 134 100: 4.4.1947
Page 66
In the train of every idea that costs a lot come a host of cheap ones: amongst them even a few that are useful.
Page 66
Sometimes one sees ideas, as an astronomer sees stars in the far distance. (Or at least it seems so.) MS 134
105: 5.4.1947
Page 66
If I had written a good sentence, & they were by accident two rhyming lines,†f†g†h this would be a blemish.
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There is much that could be learned from Tolstoy's false†a theorizing that the work of art conveys 'a feeling'.--And
you really might call it, if not the expression of a feeling, an expression of feeling, or a felt expression. And you
might say too that people who understand it to that extent 'resonate' with it, respond to it. You might say: The work
of art does not seek to convey something else, just itself. As, if I pay someone a visit, I don't wish simply to produce
such & such feelings in him, but above all to pay him a visit, & naturally I also want to be well received.
And it does start to be really absurd, to say, the artist wishes that, what he feels when writing, the other
should feel when reading. Presumably I can think I understand a poem (e.g.), understand it in the way its author
would wish,--but what he may have felt in writing it, that doesn't concern me at all. MS 134 106: 5.4.1947
Page 67
Just as I cannot write verse, so too I can write prose only up to a certain point, & no further. There is a quite
definite limit to my prose, & I can no more overstep it, than I would be able to write a poem. This is how my
equipment is constituted; it is the only equipment available to me. It is like someone's saying: In this game I can
attain only this level of perfection, & not that. MS 134 108: 5.4.1947
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It is possible†b, that everyone who executes an important work, sees before his mind's eye, dreams of, a
continuation of, a sequel to, his work; but it would be remarkable all the same if it really turned out as he dreamed.
Nowadays not believing in your own dreams is of course easy. MS 134 120: 7.4.1947
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Nietzsche writes somewhere,†40 that even the best poets & thinkers have written mediocre & bad stuff, but have
just separated off the good. But it is not quite like that. It's true that along with the roses a gardener has manure &
sweepings & straw in his garden, but they are distinguished not only†c by value, but above all too by†d function in
the garden.
What looks like a bad sentence can be the germ of a good one. MS 134 124: 8.4.1947
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The faculty of 'taste' cannot create a new organism, only rectify one that is already there. Taste loosens screws &
tightens screws, it doesn't create a new original work.†41
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Taste rectifies, it doesn't give birth.†a
Page 68
Taste makes ACCEPTABLE.
Page 68
(Hence, I think, a great creator needs no†b taste: the child is born into the world well formed.)
Polishing is sometimes the job of taste, sometimes not.
I have taste.
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The†c most refined taste has nothing to do with creative power.
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Taste is refinement of sensibility; but sensibility does not act, it merely assimilates.
Page 68
I cannot†d judge whether I have only taste, or originality as well. The former I can see distinctly, but not the latter, or
only quite indistinctly. And perhaps it has to be like that, & you see only what you have, not what you are. Someone
who does not lie is original enough. For, after all, the originality that would be worth wishing for, cannot be a sort of
trick, or an idiosyncracy, however marked.
In fact it is already a seed of good originality not to want to be what you are not. And all that has been said before
much better by others.
Page 68
Taste can delight, but not seize. MS 134 129: 9.4.1947
Page 68
You can as it were restore an old style in a new†e language; perform it afresh so to speak in a manner†f that†g suits
our times. In doing so you really only reproduce. I have done this in my building work.
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Page 69
What I mean is not however giving an old style a new trim. You don't take the old forms & fix them up to suit
today's taste. No, you are really†a speaking, maybe unconsciously, the old language, but speaking it in a manner that
belongs to the newer world, though not on that account necessarily one that is to its taste. MS 134 133: 10.4.1947
Page 69
Someone reacts like this: he says "Not that!"--& resists it. Out of this situations perhaps develop which are equally
intolerable; & perhaps by then strength for any further revolt is exhausted. We say "If he hadn't done that, the evil
would not have come about." But with what justification? Who knows the laws according to which society unfolds?
I am sure even the cleverest has no idea. If you fight, you fight. If you hope, you hope.
Someone can fight, hope & even believe, without believing scientifically.
Page 69
Science: enrichment & impoverishment. The one method elbows all others aside. Compared with this they all seem
paltry, preliminary stages at best. You must climb down to the sources to see them all side by side, the disregarded
& the preferred. MS 134 141: 13.4.1947
Page 69
Is it just I who cannot found a school, or can a philosopher never do so? I cannot found a school, because I actually
want not to be imitated.†i In any case not by those who publish articles in philosophical journals.
Page 69
The use of the word "fate". Our attitude to the future & the past. To what extent do we hold ourselves responsible
for the future? How much do we speculate about the future? How do we think about past & future? If something
unwelcome happens:--do we ask "Who's to blame?", do we say "Someone must be to blame for it"?,--or do we say
"It was God's will", "It was fate"?
In the way in which asking a question, insisting on an answer, or not asking it, expresses a different attitude, a
different way of living, so too, in this sense, an utterance like "It is God's will" or "We are not masters of our fate".
What this sentence does, or at least something similar, a commandment
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too could do. Including one that you give to yourself. And conversely a commandment, e.g. "Do not grumble!" can
be uttered like the affirmation of a truth.
Now why am I so anxious to keep apart these ways of using "declarative sentences"? Is it really necessary?
Did people in former times really not properly understand what they wanted to do with a sentence? Is it
pedantry?--It is simply an attempt to see that every usage gets its due. Perhaps then a reaction against the
overestimation of science. The use of the word "science"†i for "everything that can be said without nonsense"
already betrays this over-estimation. For this amounts in reality to dividing utterances into two classes: good & bad;
& the danger is already there. It is similar to dividing all animals, plants & rocks into the useful & the harmful.
But of course the words "see that they get their due" & "overestimation" express my point of view. I could
have said instead: "I want to help this & this to regain respect."; only I don't see it like that.
Page 70
Fate is the antithesis of natural law. A natural law is something you try to fathom, & make use of, fate is not.
Page 70
It is not by any means clear to me, that I wish for a continuation of my work by others, more than a change in the
way we live, making all these questions superfluous. (For this reason I could never found a school.)
Page 70
The philosopher says "Look at things like this!"--but first, that is not to say that people will look at things like this,
second, he may be altogether too late with his admonition, & it's possible too that such an admonition can achieve
absolutely nothing & that the impulse towards such a change in the way things are perceived must come from
another direction. For instance it is quite unclear whether Bacon started anything moving, except the surface of his
readers' minds.
Page 70
Nothing seems to me more unlikely than that a scientist or mathematician, who reads me, should be seriously
influenced thereby in the way he works. (In that respect my warnings†a are like the posters on the ticket offices at
English railway stations†42 "Is your journey really necessary?" As if anyone
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Page 71
reading that would say to himself "On second thoughts, no"†i.) Quite different artillery is needed here from anything
I am in a position to muster. Most likely I could still achieve an effect in that, above all, a whole lot of garbage is
written in response to†a my stimulus & that perhaps this provides†b the stimulus for something good. I ought
always to hope only for the most indirect of influences.
Page 71
E.g. nothing more stupid than the chatter about cause & effect in history books; nothing more wrong-headed, more
half-baked.--But who could put a stop to it by saying that? (It is†c as though I wanted to change men's and women's
fashions†d by talking.) MS 134 143: 13.-14.4.1947
Page 71
Think about how it was said of Labor's playing "He is speaking". How curious! What was it about this playing that
was so reminiscent of speaking? And how remarkable that this†e similarity with speaking is not something we find
incidental, but an important & big matter!--We should like to call music, & certainly some music, a language; but no
doubt not some music.†f (Not that this need involve a judgement of value!) MS 134 156: 11.5.1947
Page 71
The book is full of life--not like a human being, but like an ant-heap. MS 134 157: 11.5.1947
Page 71
One keeps forgetting to go down to the foundations. One doesn't put the question marks deep enough down.
Page 71
The labour pains at the birth of new concepts. MS 134 180: 27.6.1947
Page 71
"Wisdom is grey." Life on the other hand & religion are full of colour. MS 134 181: 27.6.1947
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It may be that science & industry, & their progress, are the most enduring thing in the world today. That any guess
at a coming collapse of science & industry were†a for now, & for a long time to come, simply a dream, & that
science & industry after†43 & with infinite misery will unite the world, I mean integrate it into a single empire, in
which†i to be sure peace is the last thing that will then find a home.
For science & industry do decide wars, or so it seems. MS 135 14: 14.7.1947
Page 72
Do not interest yourself in what, presumably, only you are doing! MS 135 23: 16.7.1947
Page 72
My thoughts probably move in a far narrower circle than I suspect! MS 135 85: 24.7.1947
Page 72
Thoughts rise to the surface slowly, like bubbles.
Page 72
Sometimes it's as though you could see a thought, an idea, as an indistinct point far away on the horizon; & then it
often comes closer†b with surprising speed. MS 135 101: 26.7.1947
Page 72
Where there is bad management in the state, I believe, bad management is fostered in families too. A worker who is
ready for a strike†44 at any time will not bring up his children to respect order either. MS 135 102: 27.7.1947
Page 72
God grant the philosopher insight into what lies in front of everyone's eyes. MS 135 103 c: 27.7.1947
Page 72
Life is like a path along a mountain ridge; right & left smooth†c slopes down which you slide in this or that†d
direction without being able to stop yourself. I keep seeing people slip like this & I say: "How could anyone help
himself in that situation!" And that is what "denying free will" comes to. That is the attitude that expresses itself in
this 'belief'. But it is not a scientific belief, has nothing to do with scientific convictions.
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Page 73
Denying responsibility means, not holding anyone responsible. MS 135 110: 28.7.1947
Page 73
Some people have a taste that is related to an educated taste as is the visual impression of a purblind eye to that of a
normal eye. Where a normal eye see clear articulation, the weak one sees blurred patches of colour. MS 135
133: 2.8.1947
Page 73
Someone who knows too much finds it hard not to lie. MS 135 191: 17.12.1947
Page 73
I am so afraid of someone's playing the piano in the house that, when it happens & the strumming has stopped, I still
have a sort of hallucination, that it's continuing. I can hear it then quite clearly, although I know that it is all in my
imagination. MS 135 192: 17.12.1947
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It appears to me as though a religious belief could only be (something like) passionately committing oneself to a
system of coordinates†a. Hence although it's belief, it is really a way of living, or a way of judging life. Passionately
taking up this interpretation. And so instructing in a religious belief would have to be portraying, describing that
system of reference & at the same time appealing to the conscience. And these together would have to result finally
in the one under instruction himself, of his own accord, passionately taking up that system of reference. It would be
as though someone were on the one hand to let me see my hopeless situation, on the other depict the
rescue-anchor†b, until of my own accord, or at any rate not†c†d led by the hand by the instructor, I were to rush up
& seize it.†45 MS 136 16b: 21.12.1947
Page 73
Perhaps one day a culture will arise out of this civilization.
Then there will be a real history of the discoveries of the 18th, 19th & 20th centuries, which will be of profound
interest. MS 136 18b: 21.12.1947
Page 73
In the course of a scientific investigation we say all kinds of things; we make many utterances the role of which in
our investigation we do not
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understand. For of course not everything we say has a conscious purpose, but our tongues just keep going. Our
thoughts run in established routines, we make, automatically, transitions†a according to the techniques†b we have
learned. And now comes the time for us to survey what we have said. We have made a whole lot of movements that
do not further our purpose, even impede it, and now we have to clarify our thought processes philosophically. MS
136 31a: 24.12.1947
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It seems to me I am still a long way from understanding these things; from the point, that is, at which I know what I
have to talk about, & what I don't need to talk about. I still keep getting entangled in details without knowing
whether I ought to be talking about such things at all; & I have the impression that I may be inspecting a large area,
simply to exclude it eventually from consideration. But even in this case these reflections would not be worthless; as
long, that is, as they are not just going round in a circle. MS 136 37a: 25.12.1947
Page 74
Architecture glorifies something (because it endures).†c It glorifies its purpose.
Page 74
(...)
Page 74
Architecture immortalizes & glorifies something. Hence there can be no architecture where there is nothing (to
immortalize &) glorify.
Page 74
(...)
Page 74
Architecture immortalizes & glorifies something. Hence there can be no architecture where there is nothing to
glorify.
Page 74
Architecture glorifies something (because it endures). Hence there can be no architecture where there is nothing to
glorify. MS 167 10v: ca. 1947-1948
Page 74
When philosophizing you have to descend into the old chaos†i & feel at home there. MS 136 51a: 3.1.1948
Page Break 75
Genius is talent in which character makes itself heard. For that reason, I would like to say, Kraus has talent, an
extraordinary talent, but not genius.
To be sure, there are flashes of genius where, despite the great application of talent, you do not notice the
talent. Example: "For the ox & the ass can do things too..."†46 It is curious that this e.g. is so much greater than
anything Kraus ever wrote. Here you see not merely an intellectual skeleton, but a whole human being.
That is the reason too why the greatness of what someone writes depends on everything else he writes &
does. MS 136 59a: 4.1.1948
Page 75
In a dream, & even long after we wake up, dream words can seem to us to have the greatest significance. Isn't the
same illusion possible too in waking life? It seems to me as though I am sometimes subject to it these days. It often
appears to be like this with the insane. MS 136 60b: 4.1.1948
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What I am writing here may be feeble stuff; well, in that case I am just not capable of getting out the big, important
thing. But there are great prospects hidden in these feeble remarks. MS 136 62a: 4.1.1948
Page 75
Schiller writes in a letter (to Goethe, I think)†47 of a 'poetic mood'. I think I know what he means, I think I am
familiar with it myself. It is the mood of receptivity to nature & one in which one's thoughts seem as vivid as nature
itself. But it is strange that Schiller did not produce anything better (or so it seems to me) & so I am furthermore not
entirely convinced that what I produce in such a mood is worth anything. It is quite possible that what gives my
thoughts their lustre on such occasions is a light that they receive from behind them. That they do not themselves
glow.
Page 75
Where others go on ahead, I remain standing. MS 136 80a: 8.1.1948
Page 75
[For the Foreword.]†48 It is not without†a reluctance that I offer the book to the public. The hands into which it will
fall are for the most part not those in which I like to imagine it. May it soon--this is what I wish for it--be completely
forgotten by the philosophical journalists & thus perhaps be kept for a more upright†b kind of reader. MS 136
81a: 8.1.1948
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Only every so often does one of the sentences I am writing here make a step forward; the rest are like the snipping of
the barber's scissors, which he has to keep in motion so as to be able to make a cut with them at the right moment.
MS 136 81b: 8.1.1948
Page 76
As I again & again come across questions in more remote regions that I cannot answer, it becomes clear†a why I still
cannot find my way round regions that are less remote. For how do I know that what stands in the way of an answer
here is not precisely what prevents me from clearing away the fog over there? MS 136 89a: 10.1.1948
Page 76
Raisins may be the best part of a cake; but a bag of raisins is not better than a cake; & someone who is in a position
to give us a bag full of raisins still cannot bake a cake with them, let alone do something better.
I am thinking of Kraus & his aphorisms, but of myself too & my philosophical remarks.
A cake is not as it were: thinned out raisins. MS 136 91b: 11.1.1948
Page 76
Colours are a stimulus to philosophizing. Perhaps that explains Goethe's passion for the theory of colours.
Colours seem to present us with a riddle, a riddle that stimulates us,--not one that exasperates†i us. MS 136
92b: 11.1.1948
Page 76
Human beings can regard all the evil within them as blindness. MS 136 107a: 14.1.1948
Page 76
If it is true, as I believe, that Mahler's music is worthless, the question is what I think he should have done with his
talent. For quite obviously it took a string of very rare talents to produce this bad music. Should he, say, have
written his symphonies & burnt them? Or should he have done himself violence & not have written them? Should
he have written them & realized that they were worthless? But how could he have realized that? I see it because I can
compare his music with that of the great composers. But he could not do that; for someone to whom that has
occurred may perhaps have misgivings about the value of his production, because he no doubt sees that he does
not, so to speak, have the nature of the other great
Page Break 77
composers,--but that does not mean that he will grasp the worthlessness; because he can always tell himself that he
is, it is true, different from the rest (whom however he admires) but excellent in another way. We could perhaps say:
If nobody whom you admire is like you, then presumably you believe in your own value only because you are
you.--Even someone who struggles against vanity, but not entirely successfully, will always deceive himself about
the value of what he produces.
But what seems most dangerous is to put your work into the position of being compared, first by yourself &
then by others, with the great works of former times. You should not entertain such a comparison at all. For if
today's circumstances are really so different, from what they once were, that you cannot compare your work with
earlier works in respect of its genre, then you equally cannot compare its value with that of the other work. I myself
am constantly making the mistake under discussion.
Incorruptibility is everything!
Page 77
Conglomeration: national sentiment, e.g. MS 136 110b: 14.1.1948*
Page 77
Animals come when their names are called. Just like human beings. MS 136 113a: 15.1.1948
Page 77
I ask countless irrelevant questions. If only I can beat my way through this forest! MS 136 117a: 15.1.1948
Page 77
Really I want to slow down the speed of reading with continual†a punctuation marks. For I should like to be read
slowly. (As I myself read.) MS 136 128b: 18.1.1948
Page 77
I think Bacon got bogged down in his philosophy & this danger threatens me too. He had a vivid image of a huge
building, it disappeared however when he really wanted to get down to details. It was as though his contemporaries
had begun to build a great building from the foundations; & as though he had seen something similar in his
imagination, the vision of such a building; had seen it as even more imposing than, perhaps, those who were
working on the construction. For this an inkling of the method was necessary, but by no means a talent for building
work. But the bad
Page Break 78
thing was that he launched polemical attacks against the real builders & either did not recognize, or did not want to
recognize, his limitations.
On the other hand it is tremendously hard to see these limitations, & that means, to delineate them clearly.
That is, as it were, to find†a a way of painting to depict this fuzziness. For I want to keep saying to myself: "Paint
nothing more than what you see!" MS 136 129b: 19.1.1948
Page 78
In Freudian analysis the dream is as it were dismantled. It loses its original sense completely. You might think of it as
performed on the stage, with a plot that is sometimes fairly incomprehensible but also in part quite comprehensible,
or at least apparently so, & as though this plot were then torn into little pieces & each part given a completely
different meaning. You could also think of it like this: a picture is drawn on a big sheet of paper & the sheet is then
folded in such a way that pieces which do not belong together at all in the original picture collide in appearance & a
new picture, which may make sense or may not, is formed (this would be the manifest dream, the first picture the
'latent dream thought').
Now I could imagine that someone, who sees the unfolded picture, might exclaim "Yes, that is the solution, that is
what I dreamed, but without gaps & distortions." It would then be this acknowledgement that made this solution the
solution. Just as, if you are searching for a word while writing & then say: "That's it, that says what I wanted to
say!"--Your acknowledgement stamps the word as having been found, i.e. the one you were looking for. (In this
case it might really be said: only when you have found it, do you know what you were looking for--much as Russell
said about wishing.)
Page 78
What is intriguing about a dream, is not its causal connection with events in my life, etc., but rather this, that it
affects us like part of a story,†b & indeed a very vivid part, the rest lying in darkness. (We would like to say†c:
"Where on earth did this image come from, & what has become of it?") Yes, and if someone now shows me that this
story was not the right story; that in reality quite a different one underlay it, so that I want to say†d, disenchanted,
"Oh, that's how it was?", I have seemingly really been robbed of something. Certainly, the first story now
disintegrates, as the paper is unfolded; the man I saw was taken from here, his words from
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there, the surroundings in the dream from somewhere else again; but the dream story all the same has its own
charm, like a painting that attracts & inspires us.
It can certainly be said that we contemplate the dream picture with inspiration, that we just are inspired. For
if we tell the dream to someone else, the picture usually does not inspire him. The dream is like an idea pregnant
with possible implications†a MS 136 137a: 22.1.1948
Page 79
Strike a coin from every mistake. MS 137 17a: 10.2.1948
Page 79
Understanding & explaining a musical phrase.--The simplest explanation is sometimes a gesture; another might be a
dance step, or words describing a dance.--But isn't our understanding of the phrase an experience we have while
hearing it? & what function, in that case, has the explanation? Are we supposed to think of it while we hear the
music? Are we supposed to imagine the dance, or whatever it may be, as we listen? And supposing we do,--why
should that be called hearing the music with understanding?? If seeing the dance is what matters, it would be better
that, rather than the music, were performed. But that is all a misunderstanding.
I give someone an explanation, say to him: "It's as though..."; then he says "Yes now I understand it" or "Yes now I
know how it is to be played". Above all he did not have to accept the explanation; it is not after all as though I had
given him compelling reasons for comparing this passage with this & that. I did not e.g. explain to him that remarks
made by the composer show that this passage is supposed to represent this & that.
Page 79
If I now ask "What do I actually experience then, if I hear the theme & hear it with understanding?"--nothing but
inanities†b occur to me by way of reply. Such as images, kinaesthetic sensations, thoughts†c and the like.
Sure enough I say "I go along with it"--but what does that mean? It might mean roughly that I accompany the music
with gestures. And if we point out that after all this happens for the most part only in very rudimentary measure, we
shall perhaps receive the answer that the rudimentary movements are supplemented with images. But let us
nevertheless assume that someone does accompany the music with movements in full
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measure,--in what sense does that amount to understanding it? And do I want to say, the movements are the
understanding; or his kinaesthetic sensations? (What do I know about them?)--What is true is, that, in certain
circumstances, I shall regard his movements as signs of his understanding.
Page 80
But am I to say (if I reject images, kinaesthetic sensations, etc. as an explanation) that understanding is just a specific
experience that cannot be analysed further? Well, that would be passable, as long as it is not supposed to mean,†a it
is a specific experiential content. For these words make one think of distinctions like those between seeing, hearing
& smelling.†b
Page 80
How then do we explain to someone what it means "to understand music"? By naming the images, kinaesthetic
sensations, etc. experienced by someone who understands? More likely, by pointing out the expressive movements
of one who understands.--Anyway, there is also the question, what function does explanation have here? & what
does it mean: to understand what it means to understand music? Some indeed would say: to understand that means:
to understand music oneself. And so the question would be "Then can we teach someone to understand music", for
only that kind of teaching could be called an explanation of music.
Appreciation†c of music is expressed in a certain way, both in the course of hearing & playing and at other times†d
too. This expression sometimes includes movements, but sometimes only the way the one who understands plays,
or hums, occasionally too parallels he draws & images which, as it were, illustrate the music. Someone who
understands music will listen differently (with a different facial expression, e.g.), play differently, hum differently,
talk differently about the piece than someone who does not understand. His appreciation of a theme will not
however be shown only in phenomena that accompany the hearing or playing of the theme, but also in an
appreciation for music in general.
Page 80
Appreciating music is a manifestation of human life. How could it be described to someone? Well, above all I
suppose we should have to describe music. Then we could describe the relation human beings have to it. But is that
all that is necessary, or is it also part of the process to teach
Page Break 81
him to appreciate it for himself? Well, developing his appreciation will teach him what appreciation is in a different
sense, than a teaching†a that does not do this. And again, teaching him to appreciate poetry or painting can be part
of an explanation of what music is. MS 137 20b: 15.2.1948
Page 81
Our children learn in school already that water consists of the gases hydrogen & oxygen, or sugar of carbon,
hydrogen & oxygen. Anyone who does not understand is stupid. The most important questions are concealed. MS
137 30b: 8.3.1948
Page 81
The beauty of a star-shaped figure--of a hexagonal star perhaps--is spoiled if we see it as symmetrical relative to a
given axis. MS 137 34b: 10.3.1948
Page 81
Bach said that everything he achieved was the result of industry. But industry like that presupposes humility & an
enormous capacity for suffering, strength then. And anyone who in addition can express himself perfectly, simply
addresses us in the language of a great human being. MS 137 40b: 28.5.1948
Page 81
I think that present day education of human beings†49 aims at decreasing the capacity for suffering. Nowadays a
school counts as good, if the children have a Good time. And formerly that was not the yardstick. And parents
would like children to become the way they themselves are (only more so) & yet they give them an education which
is quite different from their own.--Capacity for suffering is not rated highly, since there are not supposed to be any
sufferings, really they are out of date.
Page 81
"The cussedness of things"†i--An unnecessary anthropomorphism. We might speak of a malice of the world; easily
imagine the devil created the
Page Break 82
world, or part of it. And we need not imagine the demon intervening in particular situations; everything may happen
'in accordance with the laws of nature': it is just that the whole plan is directed at evil from the start. But a human
being exists in this world in which things break, slide about, cause every possible mischief. And of course he†a is
one of the things.--The 'malice' of the object is a stupid anthropomorphism. For the truth is much graver than this
fiction. MS 137 42a: 30.5.1948
Page 82
A stylistic device may be useful & yet I may be barred from using it. Schopenhauer's "as which" e.g. It would
sometimes make for much more comfortable, clearer expression, but cannot be used by someone who perceives it
as archaic; & he must not disregard this perception†b. MS 137 43a: 30.5.1948
Page 82
Religious faith & superstition are quite different. The one springs from fear & is a sort of false science. The other is a
trusting. MS 137 48b: 4.6.1948
Page 82
It would almost be strange if there did not exist animals with the mental life of plants. I.e. lacking mental life. MS
137 49a: 4.6.1948
Page 82
I think it might be regarded as a fundamental law of natural history that, whenever something in nature 'has a
function', 'serves a purpose', the same thing also occurs in circumstances where it serves none, is even
'dysfunctional'.
If dreams sometimes protect sleep, you can count on their sometimes disturbing it; if dream hallucination sometimes
serves a plausible end (imagined†c wish fulfilment), count on its doing the opposite as well. There is no 'dynamic
theory of dreams'. MS 137 49b: 4.6.1948
Page 82
What is important about depicting anomalies precisely? If you cannot do it, that†d shows you do not know your
way around the concepts. MS 137 51a: 15.6.1948
Page Break 83
I am too soft, too weak, & so too lazy, to achieve anything important. The industry of the great is, amongst other
things, a sign of their strength, quite apart from their inner wealth. MS 137 54b: 25.6.1948
Page 83
If God really does choose those who are to be saved, there is no reason why he should not choose them according
to their nationalities, races, or temperaments. Why the choice should not be expressed in the laws of nature. (He was
of course also able so to choose, that the choice follows a law.)
I have been reading extracts from the writings of St. John of the Cross†50, in which it is written that people have
gone to their ruin, because they did not have the good fortune to find a wise spiritual director at the right moment.
And how can you†a say then that God does not try people beyond their strengths?
I am inclined to say here, it is true, that crooked concepts have done a lot of mischief, but the truth is, that I
do not know at all, what does good & what does mischief. MS 137 57a: 26.6.1948
Page 83
We must not forget: even our more refined, more philosophical, scruples have a foundation in instinct. E.g. the 'We
can never know...' Remaining receptive to further arguments. People who couldn't be taught this would strike us as
mentally inferior. Still incapable of forming a certain concept. MS 137 57b: 30.6.1948
Page 83
If the dreams of sleep have a similar function to daydreams, then they partly serve <to> prepare people for any
eventuality (including the worst). MS 137 65b: 3.7.1948
Page 83
If someone can believe in God with complete certainty, why not in Other Minds? MS 137 67a: 3.7.1948
Page 83
This musical phrase is a gesture for me. It creeps into my life. I make it my own.
Page 83
Life's infinite variations are an essential part of our life. And so precisely of the habitual character of life. Expression
consists for us <in> incalculability. If I knew exactly how he would grimace, move, there would be no facial
expression, no gesture.--But is that true?--I can after all listen again
Page Break 84
& again to a piece of music that I know (completely) by heart; & it could even be played on†a a musical box. Its
gestures would still remain gestures for me although I know all the time, what comes next. Indeed I may even be
surprised afresh again & again. (In a certain sense.) MS 137 67a: 4.7.1948
Page 84
The honest religious thinker is like a tightrope walker. It almost looks as though he were walking on nothing but air.
His support†b†i is the slenderest imaginable. And yet it really is possible to walk on it. MS 137 67b: 5.7.1948
Page 84
Unshakable faith. (E.g. in a promise.) Is it less certain than being convinced of a mathematical truth?--(But does that
make the language games any more alike!) MS 137 70b: 7.7.1948
Page 84
It is important for our approach, that someone may feel concerning certain people, that he will never know what
goes on inside them. He will never understand them. (Englishwomen for Europeans.) MS 137 71a: 9.7.1948
Page 84
I think it is an important & remarkable fact, that a musical theme, if it is played <at> (very) different tempi, changes
its character. Transition from quantity to quality. MS 137 72b: 14.7.1948
Page 84
The problems of life are insoluble on the surface, & can only be solved in depth. In surface dimensions they are
insoluble. MS 137: 73b: 25.7.1948
Page 84
In a conversation: One person throws a ball; the other does not know: is he to throw it back, throw it to a third
person, or leave it lying, or pick it up & put it in his pocket, etc.. MS 137 75b: 23.8.1948
Page 84
A great architect in a bad period (Van der Nüll) has a quite different task from that of a great architect in a good
period. You must again not let yourself be deceived†c by the generic term. Don't take comparability, but rather
incomparability, as a matter of course. MS 137 76a: 19.10.1948
Page Break 85
Nothing is more important though than the construction of fictional concepts,†a which will teach us at last to
understand our own. MS 137 78b: 24.10.1948
Page 85
"Thinking is difficult." (Ward) What does that really mean? Why is it difficult?--It is almost like saying "Looking is
difficult". For looking intently is difficult. And you may look intently & yet see nothing, or keep thinking you see
something & yet not be able to see clearly†b. You can tire from looking, even if you see nothing. MS 137 81b:
27.10.1948
Page 85
If you cannot unravel a tangle, the most sensible thing you can do†c is to recognize this; & the most decent, to admit
it. [Antisemitism.]
What you should do to cure the evil is not clear. What is not permissible is clear from one case to another. MS
137 88a: 4.11.1948
Page 85
It is remarkable that Busch's drawings can often be called 'metaphysical'. There is then a way of drawing that is
metaphysical.--"Seen, with the eternal as background"†51 one might say. However these strokes mean this only in a
whole language. And it is a language without grammar, you couldn't say what its rules are. MS 137 88b:
4.11.1948
Page 85
When he was old Charlemagne tried unsuccessfully to learn to write: & someone may be similarly unsuccessful in
trying to learn a new line of thinking. He never becomes fluent in it. MS 137 89b: 5.11.1948
Page 85
A language, which is spoken in strict tempo, so that you can also speak according to the metronome. It does not go
without saying that music can be performed, like ours, at least occasionally, to the metronome. (Playing the theme
from the 8th Symph.†52 exactly according to the metronome.) MS 137 97b: 14.11.1948
Page 85
It would already be enough, that all members of a community had the same facial features, for us not to be able to
fathom them. MS 137 97b: 16.11.1948
Page Break 86
If a false thought is so much as expressed boldly & clearly, a great deal has already been gained. MS 137 100a:
19.11.1948
Page 86
Only by thinking much more crazily even than the philosophers, can you solve their problems. MS 137 102a:
20.11.1948
Page 86
Imagine someone watching a pendulum & thinking: God makes it move like that. Well, doesn't God have the right†a
even to act in accordance with a calculation?
Page 86
A writer far more talented than I would still have little talent. MS 137 104a: 21.11.1948
Page 86
Human beings have a physical need to tell themselves when at work: "Let's have done with it now", & it's having
constantly to go on thinking in the face of this need when philosophizing, that makes this work so strenuous. MS
137 104b: 22.11.1948
Page 86
You must accept the faults in your own style. Almost like the blemishes in your own face. MS 137 106b:
23.11.1948
Page 86
Always come down from the barren heights of cleverness into the green valleys of folly. MS 137 111b: 28.11.1948
Page 86
I have one of those talents that has constantly to make a virtue out of necessity.
Page 86
Tradition is not something that anyone†b can pick up†c, it's not a thread, that someone can pick up, if & when he
pleases; any more than you can choose your own ancestors.†d
Someone who has no tradition & would like to have it, is like an unhappy lover.
Page 86
The happy lover & the unhappy lover both have their particular pathos.
But it is harder to bear yourself well as an unhappy lover than as a happy one. MS 137 112b: 29.11.1948
Page Break 87
Moore poked into a philosophical wasp nest with his paradox; & if the wasps did not duly fly out, that's only
because they were too listless. MS 137 120a: 10.12.1948
Page 87
In the realm of the mind a project can usually not be begun again, nor should it be. These thoughts fertilize the soil
for fresh thoughts.†a MS 137 122a: 11.12.1948
Page 87
Are you a bad philosopher then, if what you write is hard to understand. If you were better, then you would make it
easy to understand what is difficult.--But who says that is possible?! [Tolstoy] MS 137 127a: 16.12.1948
Page 87
The greatest happiness for a human being is love. Suppose you say of the schizophrenic: he does not love, he cannot
love, he refuses to love--where is the difference?
Page 87
"He refuses to..." means: it is in his power. And who wants to say that?!
Page 87
Well, of what do we say "it is in his power"?--We say it in cases where we want to draw a distinction. I can lift this
weight, but I will not lift it; that weight I cannot lift.
Page 87
"God has commanded it, therefore we must be able to do it." That means nothing. There is no "therefore" about it.
The two expressions might at most mean the same.
Page 87
"He has commanded it" means here roughly: He will punish anyone who does not do it. And nothing follows from
that about being able. And that is the sense of 'election by grace'.
Page 87
But that does not mean that it is right to say: "He punishes, although we cannot act otherwise."--Perhaps, though,
one might say: here there is punishment, where punishment by human beings would be impermissible. And the
whole concept of 'punishment' changes here. For the old illustrations can no longer be applied, or now have to be
applied quite differently. Just look at an allegory like "The Pilgrim's Progress"†53 & see how nothing
Page Break 88
--in human terms--is right.--But isn't it right all the same? i.e. can it not be applied? Indeed, it has been applied. (At
railway stations there are dials with two hands, they indicate when the next train leaves. They look like clocks &
aren't; but they have a†a use.) (There should be a better comparison here.)
Page 88
To someone who is upset by this†b allegory it might be said: Apply it differently or don't bother with it! (But some
will be far more confused than helped by it.) MS 137 130a: 22.12.1948
Page 88
Anything the reader can do for himself, leave it to the reader. MS 137 134b: 25.12.1948
Page 88
Almost the whole time I am writing conversations with myself. Things I say to myself tête-à-tête. MS 137 134b:
26.12.1948
Page 88
Ambition is the death of thought. MS 137 135a: 27.12.1948
Page 88
Humour is not a mood, but a way of looking at the world. So, if it's right to say that humour was eradicated in Nazi
Germany, that does not mean that people were not in good spirits or anything of that sort, but something much
deeper & more important. MS 137 135a: 28.12.1948
Page 88
Two people who are laughing together, at a joke perhaps. One of them has said†c certain somewhat unusual words
& now they both break out into a sort of bleating. That might appear very bizarre to someone arriving among us
from a quite different background. Whereas we find it quite reasonable.
(I witnessed this scene recently on a bus & was able to think myself into the skin of someone not accustomed to it. It
struck me then as quite irrational & like the reactions of an outlandish animal.) MS 137 136b: 31.12.1948
Page 88
Recounting a dream, a medley of recollections. Often forming a significant & enigmatic whole. As it were a
fragment, that makes a powerful impression on us (sometimes that is), so that we look for an explanation, for
connections.
Page Break 89
Page 89
But why did these recollections come now? Who will say?--It may be connected with our present life, and so too
with our wishes, fears, etc. "But do you mean to say that this phenomenon must exist in the particular causal
interconnection?"--I mean to say that it does not necessarily make sense to speak of discovering its cause.
Page 89
Shakespeare & the dream. A dream is all wrong, absurd, composite, & yet completely right: in this strange
concoction it makes an impression. Why? I don't know. And if Shakespeare is great, as he is said to be, then we
must be able to say of him: Everything is wrong, things aren't like that--& is all the same completely right according
to a law of its own.
It could be put like this too: If Shakespeare is great, then he can be so only in the whole corpus of his plays, which
create their own language & world. So he is completely unrealistic. (Like the dream.) MS 168 1r: January 1949
Page 89
If Christianity is the truth, then all the philosophy about it is false. MS 169 58v: 1949
Page 89
Culture is an observance. Or at least presupposes an observance. MS 169 62v: 1949
Page 89
The concept of a 'festivity'. Connected for us with merrymaking; perhaps in another age only with fear & dread.
What we call "wit" & what we call "humour" doubtless did not exist in other ages. And both these are perpetually†a
changing.†b MS 137 137a: 1.1.1949
Page 89
"Le style c'est l'homme." "Le style c'est l'homme même." The first expression has a cheap epigrammatic brevity. The
second, correct, one opens up a quite different perspective. It says that style is the picture of the man.
Page 89
There are remarks that sow, & remarks that reap. MS 137 140a: 4.1.1949
Page Break 90
To piece together†a the landscape of these conceptual relationships out of their individual fragments†b is too
difficult for me. I can make only a very imperfect job of it. MS 137 141a: 6.1.1949
Page 90
If I prepare myself for some eventuality, you can be pretty sure that it won't happen. Perhaps.†i
Page 90
It is difficult to know something, & to act as though you didn't know it. MS 137 143a: 7.-8.1.1949
Page 90
There really are cases in which one†c has the sense of what one†d wants to say much more clearly in mind than
he†e†54 can express in words. (This happens to me very often.) It is as though one remembered a dream very
clearly, but could not give a good account of it.†f Indeed the image often stays there behind the words for the writer
(me), so that they seem to describe it to me.
Page 90
A mediocre writer must beware of too quickly replacing a crude, incorrect expression with a correct one. By doing
so he kills the original idea, which was still at least a living seedling. And now it is shrivelled & no longer worth
anything. He may now just as well throw it on the rubbish heap. Whereas the pitiful seedling still had a certain
usefulness. MS 138 2a: 17.1.1949
Page 90
That writers, who after all were something, go out of date is connected with the fact that their writings, when
complemented by the setting of their own age, speak strongly to people, but that they die without this
complementation, as if bereft of the lighting that gave them colour.
And I believe that the beauty of mathematical demonstrations, as experienced by Pascal too, is connected with this.
Within this way of
Page Break 91
looking at the world these demonstrations did have beauty--not what superficial people call beauty. A crystal too is
not beautiful in every 'setting'--though perhaps everywhere attractive.--
Page 91
The way whole periods are incapable of freeing themselves from the grip of certain concepts--e.g. the concept
'beautiful' & 'beauty'. MS 138 3a: 18.1.1949
Page 91
My own thinking about art & values is far more disillusioned, than would have been possible for people 100 years
ago. However that does not mean that it is more correct on that account. It only means that there are examples of
decline in the forefront of my mind, which were not in the forefront for those people then. MS 138 4a:
18.1.1949
Page 91
Troubles are like illnesses; you have to put up with them: the worst thing you can do is, rebel against them.
They come in attacks too, triggered by inner, or outer causes. And then you must tell yourself: "Another attack". MS
138 4b: 19.1.1949
Page 91
Scientific questions may interest me, but they never really grip†a me. Only conceptual & aesthetic questions have
that effect on me. At bottom it leaves me cold whether scientific problems are solved; but not those other questions.
MS 138 5b: 21.1.1949
Page 91
Even if we are not thinking in circles, still, we sometimes walk straight through the thicket of questions out into the
open country, sometimes†b along tortuous, or zigzagging paths, which don't take us into the open country. MS
138 8a: 22.1.1949
Page 91
The Sabbath is not simply a time to rest, to recuperate. We are supposed to look at our work from the outside, not
just from within. MS 138 8b: 23.1.1949
Page 91
This is how philosophers should salute each other: "Take your time!" MS 138 9a: 24.1.1949
Page Break 92
Page 92
For a human being the eternal, the consequential is often hidden behind an impenetrable veil. He knows: there is
something under there, but he cannot see it; the veil reflects the daylight. MS 138 9a: 24.1.1949
Page 92
Why shouldn't someone become desperately unhappy? It is one human possibility. As in 'Corinthian Bagatelle', this
is one of the possible paths for the balls. And perhaps not even one of the rarest. MS 138 9b: 25.1.1949
Page 92
The valleys of foolishness have more grass growing in them for the philosopher than do the barren heights of
cleverness. MS 138 11a: 28.1.1949
Page 92
Isochronism according to the clock & isochronism in music. They are by no means equivalent concepts. Playing in
strict time, does not mean playing exactly according to the metronome. But it would be possible that a certain kind
of music should be played according to the metronome. (Is the opening theme <of the second movement> of the
8th symphony of this kind?) MS 138 12a: 30.1.1949
Page 92
Could the concept of the punishments of hell be explained in some other way than by way of the concept of
punishment? Or the concept of God's goodness in some other way than by way of the concept of goodness?
If you want to achieve the right effect with your words, doubtless not.
Page 92
Suppose someone were taught: There is a being who, if you do this & that, live in such & such a way, will take you
after your death to a place of eternal torment; most people end up there, a few get to a place of eternal joy.--This
being has picked out in advance those who are to get to the good place; &, since only those who have lived a certain
sort of life get to the place of torment, he has also picked out in advance those who are to lead that sort of life.
What might be the effect of such a doctrine?
Page 92
Well, there is no mention of punishment here, but rather a kind of natural law. And anyone to whom it is
represented in such a light, could derive only despair or incredulity from it.†a
Page Break 93
Page 93
Teaching this could not be an ethical training. And if you wanted to train anyone ethically & yet teach him like this,
you would have to teach the doctrine after the ethical training, and represent it as a sort of incomprehensible
mystery. MS 138 13b: 2.2.1949
Page 93
"He has chosen them, in his goodness, & you he will punish" really makes no sense. The two halves belong to
different kinds of perspective. The second half is ethical & the first not. And taken together with the first the second
is absurd. MS 138 14a: 2.2.1949
Page 93
It is an accident that 'last' rhymes with 'fast'.†i But a lucky accident, & you can discover†a this lucky accident. MS
138 25a: 25a: 22.2.1949
Page 93
In Beethoven's music what one might call the expression of irony is to be found for the first time. E.G. in the first
movement of the Ninth. With him, moreover, it is a terrible irony, that of fate perhaps.--In Wagner irony reappears,
but turned into something bourgeois.
You could no doubt say that Wagner & Brahms, each in his own way, imitated Beethoven; but what with him was
cosmic, is†b earthly with them.
The same expressions are to be found in him, but they follow different laws.
Page 93
In Mozart's or Haydn's music again fate plays no sort of role. That is not the concern of this music.
That ass Tovey says somewhere that this, or something similar, is connected with the fact that Mozart has no access
to literature of a certain sort. As though it were established, that only books had made the music of the masters what
it was. Naturally, books & music are connected. But if Mozart found no great tragedy in his reading, does that mean
that he did not find it in his life? And do composers always see solely through the spectacles of poets? MS 138
28a: 27.2.1949
Page 93
Only in a quite particular musical context is there such a thing as three-part counterpoint. MS 138 28b: 27.2.1949
Page Break 94
Soulful expression in music. It is not to be described in terms of degrees of loudness & of tempo. Any more than is
a soulful facial expression describable in terms of the distribution of matter in space. Indeed it is not even to be
explained by means of a paradigm, since the same piece can be played with genuine expression in innumerable
ways. MS 138 29a: 1.3.1949
Page 94
God's essence is said to guarantee his existence--what this really means is that here what is at issue is not the
existence of something.
Page 94
For could one not equally say that the essence of colour guarantees its existence? As opposed, say, to the white
elephant. For it really only means: I cannot explain what 'colour' is, what the word "colour" means, without the help
of a colour sample. So in this case there is no such thing as explaining 'what it would be like if colours were to exist'.
Page 94
And now we might say: There can be a description of what it would be like if there were gods on Olympus--but not:
'what it would be like if there were God'. And this determines the concept 'God' more precisely,
Page 94
How are we taught the word "God" (its use, that is)? I cannot give an exhaustive systematic description. But I can as
it were make contributions towards the description; I can say something about it & perhaps in time assemble a sort
of collection of examples.
Page 94
Reflect in this connection that in a dictionary one would perhaps like to give such descriptions of use, but in reality
one gives only a few examples & explanations. But also that no more than this is necessary. What use could we
make of an enormously long description?--Well, it would be no use to us if it dealt with the use of words in
languages already familiar to us. But what if we came across such a description of the use of an Assyrian word? And
in what language? Let's say in another language already known to us.--In this description the word "sometimes" will
frequently occur, or "often", or "usually", or "nearly always" or "almost never".
Page 94
It is difficult to form a good picture of a description of this sort.
Page Break 95
And what I basically am after all is a painter, & often a very bad painter. MS 138 30b: 17.3.1949
Page 95
What is it like when people do not have the same sense of humour? They do not react properly to each other. It is as
though there were a custom among certain people to throw someone a ball, which he is supposed to catch & throw
back; but certain people might not throw it back, but put it in their pocket instead.
Page 95
Or what is it like for someone to have no idea how to fathom another's taste? MS 138 32b: 20.5.1949
Page 95
A picture that is firmly rooted in us may indeed be compared to superstition, but it may be said too that we always
have to reach some sort of firm ground, be it a picture, or not, so that a picture at the root of all our thinking is to be
respected & not treated as a superstition. MS 138 32b: 20.5.1949
Page 95
It is not unheard of†a that†b someone's character may be influenced by the external world (Weininger). For that
only means that, as we know from experience, people change with circumstances. If someone asks: How could the
environment coerce someone, the ethical in someone?--the answer is that he may indeed say "No human being has
to give way to coercion",†i but all the same under such circumstances†c someone will do such & such.
'You don't HAVE to, I can show you a (different) way out,--but you won't take it.' MS 173 17r: 30.3.1950
Page 95
I do not think that Shakespeare can be set alongside any other poet.
Was he perhaps a creator of language rather than a poet?
Page 95
I could only stare in wonder at Shakespeare; never do anything with him.
Page 95
I am deeply suspicious of most of Shakespeare's admirers. I think the trouble is that, in western culture at least, he
stands alone, & so, one can only place him by placing him wrongly.
Page Break 96
It is not as though S. portrayed human types well & were in that respect true to life. He is not true to life. But he has
such a supple hand & such individual brush strokes. [[sic , ?]] that each one of his characters looks significant,
worth looking at.
Page 96
"Beethoven's great heart"--no one could say "Shakespeare's great heart". 'The supple hand that created new natural
forms of language' would seem to me nearer the mark.
Page 96
The poet cannot really say of himself "I sing as the bird sings"--but perhaps S. could have said it of himself. MS
173 35r: 12.4.1950 or later
Page 96
One & the same theme has a different character in the minor than in the major, but it is quite wrong to speak
generally of a character belonging to the minor. (In Schubert the major often sounds sadder than the minor.†a)
And similarly, I think, it is idle & futile for the understanding of painting to speak of the characters of the individual
colours. In doing so one really thinks only of special applications. The fact that green has one effect as the colour of
a table cloth, red another, licenses no conclusion about their effect in a picture. MS 173 69r: 1950
Page 96
I do not think Shakespeare could have reflected on the 'lot of the poet'.
Page 96
Neither could he regard himself as a prophet or teacher of humanity.
People regard him with amazement almost as a spectacle of nature. They do not have the feeling that this brings
them into contact with a great human being. Rather with a phenomenon.
Page 96
I think that, in order to enjoy a poet, you have to like the culture to which he belongs as well. If you are indifferent to
this or repelled by it, your admiration cools off.†bMS 173 75v: 1950
Page 96
If the believer in God looks around & asks "Where does everything I see come from?" "Where does all that come
from?", what he hankers after is
Page Break 97
not a (causal) explanation; and the point of his question is that it is the expression of this hankering.†i He is
expressing, then, a stance†a towards all explanations.--But how is this manifested in his life?
It is the attitude of taking a certain matter seriously, but then at a certain point not taking it seriously after all,
& declaring that something else is still more serious.
Someone may for instance say that it is a very grave matter that such & such a person has died before he
could complete a certain piece of work; & in another sense that is not what matters. At this point one uses the words
"in a deeper sense".
Really what I should like to say is that here too what is important is not the words you use or what you think
while saying them, so much as the difference that they make at different points in your life. How do I know that two
people mean the same thing when each says he believes in God? And just the same thing goes for the Trinity.
Theology that insists on certain†b words & phrases & prohibits others makes nothing clearer. (Karl Barth)
It gesticulates with words, as it were, because it wants to say something & does not know how to express
it.†c Practice gives the words their sense. MS 173 92r: 1950
Page 97
A proof of God ought really to be something by means of which you can convince yourself of God's existence. But
I think that believers who offered such proofs wanted to analyse & make a case for their 'belief' with their intellect,
although they themselves would never have arrived at belief by way of such proofs. "Convincing someone of God's
existence" is something you might do by means of a certain upbringing, shaping his life in such & such a way.
Life can educate you to "believing in God". And experiences too are what do this but not visions, or other sense
experiences, which show us the "existence of this being", but e.g. sufferings of various sorts. And they do not show
us God as a sense experience does an object, nor do they give rise to conjectures about him. Experiences,
thoughts,--life can force this concept on us.
Page 97
So perhaps it is similar to the concept 'object'. MS 174 1v: 1950
Page Break 98
The reason I cannot understand Shakespeare is that I want to find symmetry in all this asymmetry.
Page 98
It seems to me as though his pieces are, as it were, enormous sketches, not paintings; as though they were dashed
off by someone who could permit himself anything, so to speak. And I understand how someone may admire this
& call it supreme art, but I don't like it.--So I can understand someone who stands before those pieces speechless;
but someone who admires him as one admires Beethoven, say, seems to me to misunderstand Shakespeare. MS
174 5r: 24.4.1950 or later
Page 98
One age misunderstands another; and a petty age misunderstands all the others in its own ugly way. MS 174 5v:
1950
Page 98
How God judges people is something we cannot imagine at all. If he really takes the strength of temptation & the
frailty of nature into account, whom can he condemn? But if not, then these two forces simply yield as a result the
end for which this person was predestined. In that case he was created so as either to conquer or succumb as a result
of the interplay of forces. And that is not a religious idea at all, so much as a scientific hypothesis.
So if you want to stay within the religious sphere, you must struggle. MS 174 7v: 1950
Page 98
Look at human beings: One is poison for the other. A mother for her son, and vice versa, etc. etc. But the mother is
blind & the son too. Perhaps they have a guilty conscience, but what good does that do them? The child is wicked,
but nobody teaches it to be different, & the parents only spoil it with their foolish affection; & how are they
supposed to understand this, & how is the child to understand it? They are, so to speak, all wicked & all innocent.
MS 174 8r: 1950
Page 98
Philosophy hasn't made any progress?--If someone scratches where it itches, do we have to see progress? isn't it
genuine scratching otherwise, or genuine itching? And can't this reaction to the irritation†a go on†b like this for a
long time, before a cure for the itching is found? MS 174 10r: 1950
Page Break 99
Page 99
God may say to me: "I am judging you out of your own mouth. You have shuddered with disgust at your own
actions when you have seen them in other people". MS 175 56r: 15.3.1951
Page 99
Is the sense of belief in the Devil this, that not everything that comes to us as an inspiration is good†a? MS 175
63v: 17.3.1951
Page 99
You cannot judge yourself, if you are not versed in the categories. (Frege's style of writing is sometimes great;
Freud writes excellently, & it is a pleasure to read him, but his writing is never great)†55 MS 176 55v: 6.4.1951
Page Break 100
A Poem†a
Page 100
If you throw the fragrant veil of
true love on my head,
at the moving of the hands
the soft stirring of the limbs
bereft of sense becomes the soul.
Page 100
Can you grasp it as it's drifting
as it stirs with scarce a sound
and deep within the heart its imprint fixes.
Page 100
At the sounding of the morning's bell
The gardener through the garden's space is passing
Touching with light feet his ground //the ground//
the flowers rouse themselves and gaze
inquiring on his radiant,
peaceful face:
Who was it then who wove the veil around your foot
touching us gently like a breath of wind
Is even Zephyr too your servant?
Was it the spider, or was it the silkworm?
Page Break 101
Notes
Page 101
†1 The Lieutenant in question is most probably Vojeslav Molé.
Page 101
†2 Arvid Sjögren, a friend and relation of Ludwig Wittgenstein, married to his niece Clara Salzer.
Page 101
†3 Ernest Renan: History of the people of Israel, Vol 1, Chapter III.
Page 101
†4 An earlier draft of the printed foreword to Philosophical Remarks, edited by Rush Rhees and translated
by Raymond Hargreaves and Roger White (Oxford, Blackwell 1975).
Page 101
†5 G.E. Lessing, The Education of the Human Race, §§48-49.
Page 101
†6 Not clearly legible, unclear whether: "types of human being." or "human beings. types." and "types." as a
variant on "Human beings.".
Page 101
†7 The editor has corrected an obvious slip in the punctuation of the original, which results in nonsense.
Page 101
†8 Wittgenstein first wrote "Frege, Russell, Spengler, Sraffa" and added the other names later without adding
the necessary commas.
Page 101
†9 The sentence is from Wilhelm Busch's prose poem "Edward's dream", The editor is indebted to MR.
Robert Löffler for this information.
Page 101
†10 In the original "are": Wittgenstein first wrote "the Jews", and then replaced it with "the history of the
Jews", without correcting the "are" to "is".
Page 101
†11 The time signature is not in the MS. The editor is very grateful to Mr Fabian Dahlström for professional
help in interpreting the notes, which are very hard to read. Mr Dahlström has suggested the following interpretation:
Page Break 102
Page 102
†12 Not clearly legible. Unclear whether it reads: "if its wearer looks smugly at himself in it in the mirror" or
"if its wearer smugly smartens himself up in it in the mirror".
Page 102
†13 Adelbert von Chamisso, The Strange Tale of Peter Schlemihl.
Page 102
†14 Heinrich von Kleist: "Letter from One poet to Another", 5th January 1811.
Page 102
†15 The words "no" in "<no> western occupation" and "language" in "the words of our <language>" were
supplied from the corresponding text of the notebook in MS 153a: S. 122r.
Page 102
†16 Not clearly legible: comma after "enclose".
Page 102
†17 Cf. Philosophical Investigations, Part I, § 131.
Page 102
†18 It is unclear whether the text reads: "Nur so nämlich können wir unsere <n> Behauptungen der
Ungerechtigkeit (...) entgehen" or whether "unsere Behauptungen" ("our claims") is a variant for "wir" ("we").
"unsere Behauptungen" was inserted between "wir" und "der Ungerechtigkeit" ("prejudice").
Page 102
†19 Not clearly legible: either "for" or "then".
Page 102
†20 Not clearly legible. "Maßeinheit--" Wittgenstein seems first to have written "Ein guter Maßstab" ("a good
measuring rod"), then, changing the gender of the indefinite article appropriately, to have changed "Maßstab" to
"Maßeinheit" ("unit of measurement"). The gender of "guter" (good") was not changed.
Page 102
†21 It is unclear whether the text reads "diese" ("this" in its feminine form which would make it refer to
"plausibility") or "dieser" ("this" in its masculine form which would make it refer to "narrative").
Page 102
†22 In the text there is an intrusive comma which makes no grammatical sense and is clearly an error.
Page 102
†23 Anna Rebni, schoolteacher from Skjolden, Norway, where Wittgenstein had a hut.
Page 102
†24 Schopenhauer, "The Metaphysics of Music", The World as Will and as Idea, Chapter 39.
Page 102
†25 Wittgenstein's sister, Margarete Stonborough, for whom he built the house at 19 Kundmanngasse,
Vienna.
Page 102
†26 It is unclear in the text whether this should read "private performances" ("Leeraufführungen"--literally
empty performances) or "training performances" ("Lehraufführungen").
Page 102
†27 Wittgenstein expressly notes above "not-admired": "hyphen".
Page 102
†28 The sense of "(N.)" and "(W.)" is unclear.
Page 102
†29 A department store in London.
Page 102
†30 "scientists are not (...) on their laurels" is crossed out, which shows that Wittgenstein preferred the two
subsequent alternatives.
Page 102
†31 Cf. Editor's note on p. 302 of Remarks on the Foundations of Mathematics, Second Edition.
Page 102
†32 Cf. Philosophical Investigations, Part I, § 546.
Page 102
†33 The German text is unclear as between "an" and "aus". [These involve a slight difference in emphasis:
perhaps roughly, between "to find the rest of the way home from here" and "to find the rest of the way home from
here onwards". But the difference, such as it is, can hardly be reproduced in English. (PW)]
Page 102
†34 Gottfried Keller, The Lost Laugh.
Page Break 103
Page 103
†35 Goethe, The Bride of Corinth.
Page 103
†36 Philosophical Investigations, Part II, Section IV.
Page 103
†37 Cf. Zettel § 175.
Page 103
†38 Text unclear: either Section mark "S", "Oh es kann ein Schlüssel (...)" ["Oh a key can (...)"] or "Soh es
kann ein Schlüssel (...)"--with the "h" in "Soh" crossed out ["So a key can (...)"].
Page 103
†39 Not clearly legible: either "etwas" ("something") or "etwa" ("perhaps").
Page 103
†40 Friedrich Nietzsche, Human, All-Too Human, I, § 155.
Page 103
†41 "Urwerk". It is unclear whether this should read "Uhrwerk" [= "original work" or perhaps even
"prototype"] or "Uhrwerk" ["clockwork" or simply "piece of machinery"].
Page 103
†42 During the Second World War and immediately after.
Page 103
†43 Not clearly legible: either "nach" (= "after") or noch (= "yet or "still").
Page 103
†44 In the original: "Strike": unclear whether English "strike" or German "Streike" (= "strikes").
Page 103
†45 In the original "anchor" in "rescue-anchor" is crossed out; "the rescue-anchor" was replaced by the
variant "the <rescue>-instrument". However in what follows ("rush up to it & seize it") the masculine pronoun "ihn"
(corresponding to "anchor" is not replaced by the neuter pronoun "es" (corresponding to "instrument").
Page 103
†46 Georg Christoph Lichtenberg, Timorus, Perface[[sic]]. The complete sentence reads: "For the ox & the
ass can do things too, but up to now only a human being can give you an assurance."
Page 103
†47 Letter to Goethe, 17th December 1795
Page 103
†48 For Philosophical Investigations.
Page 103
†49 Not clearly legible: whether singular or plural.
Page 103
†50 St. John of the Cross, Juan de Yepes, 1542-91.
Page 103
†51 Cf. Notebooks, 7.10.1916.
Page 103
†52 Beethoven's 8th Symphony.
Page 103
†53 John Bunyan, The Pilgrim's Progress from this World to that which is to come.
Page 103
†54 It may be that the upper case "Einer" ("someone") is connected with the personal pronoun "er" ("he")
and the lower case "einer" ("one") with the impersonal "man" ("one").
Page 103
†55 Cf. Zettel, § 712.
Page Break 104
Page Break 105
Appendix
Page Break 106
Page Break 107
List of Sources
(The remarks that in this edition have been completed to comprise a whole section are marked with an asterisk [*].)
MS 101 7 c : 21.8.1914*
MS 106 58 : 1929
MS 106 247 : 1929
MS 105 46 c : 1929*
MS 105 67 c : 1929
MS 105 73 c : 1929
MS 105 85 c : 1929
MS 107 70 c : 1929
MS 107 72 c : 1929*
MS 107 82 : 1929
MS 107 82 c : 1929
MS 107 97 c : 1929
MS 107 98 c : 1929
MS 107 100 c : 1929
MS 107 116 c : 1929
MS 107 120 c : 1929
MS 107 130 c : 1929
MS 107 156 c : 10.10.1929
MS 107 176 c : 24.10.1929
MS 107 184 c : 7.11.1929
MS 107 192 c : 10.11.1929
MS 107 196 : 15.11.1929
MS 107 229 : 10.-11.1.1930
MS 107 242 : 16.1.1930
MS 108 207 : 29.6.1930
MS 109 28 : 22.8.1930
MS 109 200 : 5.11.1930
MS 109 204 : 6.-7.11.1930
MS 109 212 : 8.11.1930
MS 110 5 : 12.12.1930
MS 110 10 : 13.12.1930
MS 110 11 : 25.12.1930
MS 110 12 : 12.-16.1.1931
MS 110 18 : 18.1.1931
MS 110 61 : 10.2.1931
MS 110 67 : 12.2.1931
MS 153a 4v : 10.5.1931
MS 153a 12v : 1931
MS 153a 18v : 1931
MS 153a 29v : 1931
MS 153a 35r : 1931
MS 153a 35v : 1931
MS 153a 90v : 1931
MS 153a 127v : 1931
MS 153a 128v : 1931
MS 153a 136r : 1931
MS 153a 141r : 1931
MS 153a 160v : 1931
MS 153b 3r : 1931
MS 153b 29r : 1931
MS 153b 39v : 1931
MS 154 1r : 1931
MS 154 1v : 1931
MS 154 15v : 1931
MS 154 20v : 1931
MS 154 21v : 1931
MS 154 25v : 1931
MS 155 29r : 1931
MS 155 46r : 1931
MS 110 200 : 22.6.1931
MS 110 226 : 25.6.1931
MS 110 231 c : 29.6.1931
MS 110 238 : 30.6.1931
MS 110 257 : 2.7.1931
Page Break 108
MS 110 260 c : 2.7.1931
MS 111 2c : 7.7.1931
MS 111 55 : 30.7.1931
MS 111 77 : 11.8.1931
MS 111 115 : 19.8.1931
MS 111 119 : 19.8.1931
MS 111 132 : 23.8.1931
MS 111 133 : 24.8.1931
MS 111 173 : 13.9.1931
MS 111 180 : 13.9.1931
MS 111 194 : 13.9.1931
MS 111 195 : 13.9.1931
MS 112 1 : 5.10.1931
MS 112 46 : 14.10.1931
MS 112 60 : 15.10.1931
MS 112 76 : 24.10.1931
MS 112 114 : 27.10.1931
MS 112 139 : 1.11.1931
MS 112 153 : 11.11.1931
MS 112 221 : 22.11.1931
MS 112 223 : 22.11.1931
MS 112 225 : 22.11.1931
MS 112 231 : 22.11.1931
MS 112 232 : 22.11.1931
MS 112 233 : 22.11.1931
MS 112 255 : 26.11.1931
MS 113 80 : 29.2.1932
MS 156a 25r : ca. 1932-1934
MS 156a 30v : ca. 1932-1934
MS 156a 49r : ca. 1932-1934
MS 156a 49v : ca. 1932-1934
MS 156a 57r : ca. 1932-1934
MS 156a 58v : ca. 1932-1934
MS 156b 14v : ca. 1932-1934
MS 156b 23v : ca. 1932-1934
MS 156b 24v : ca. 1932-1934
MS 156b 32r : ca. 1932-1934
MS 145 14r : Herbst 1933*
MS 145 17v : 1933
MS 146 25v : 1933-1934
MS 146 35v : 1933-1934
MS 146 44v : 1933-1934
MS 147 16r : 1934
MS 147 22r : 1934
MS 157a 2r : 1934*
MS 157a 22v : 1934
MS 157a 44v : 1934 oder 1937
MS 116 56 : 1937
MS 157a 57r : 1937
MS 157a 57r : 1937
MS 157a 58r : 1937
MS 157a 62v : 1937
MS 157a 66v c:1937
MS 157a 67v c:1937
MS 157b 9v : 1937
MS 157b 15v : 1937
MS 118 17r c : 27.8.1937*
MS 118 20r : 27.8.1937
MS 118 21v : 27.8.1937
MS 118 35r c : 29.8.1937
MS 118 45r c : 1.9.1937
MS 118 56r c : 4.9.1937*
MS 118 71v c : 9.9.1937
MS 118 86v : 11.9.1937
MS 118 87r c : 11.9.1937*
MS 118 94v : 15.9.1937
MS 118 113r : 24.9.1937
MS 118 117v : 24.9.1937
MS 119 59 : 1.10.1937
MS 119 71 : 4.10.1937
MS 119 83 : 7.10.1937
MS 119 108 : 14.10.1937
MS 119 146 : 21.10.1937
MS 119 151 : 22.10.1937
MS 120 8 : 20.11.1937
MS 120 83 c : 8.-9.12.1937*
MS 120 103 c : 12.12.1937
MS 120 108 c : 12.12.1937*
Page Break 109
MS 120 113 : 2.1.1938
MS 120 142 : 19.2.1938
MS 120 283 : 7.4.1938
MS 120 289 : 20.4.1938
MS 121 26v : 25.5.1938
MS 121 35v : 11.6.1938
MS 122 129 : 30.12.1939
MS 162a 18 : 1939-1940
MS 162b 16v : 1939-1940
MS 162b 21v : 1939-1940
MS 162b 22r c : 1939-1940
MS 162b 24r : 1939-1940
MS 162b 37r c : 1939-1940
MS 162b 42v c :1939-1940
MS 162b 53r : 1939-1940
MS 162b 55v : 1939-1940
MS 162b 59r : 1939-1940*
MS 162b 59v : 1939-1940
MS 162b 61r : 1939-1940
MS 162b 63r : 1939-1940
MS 122 175 c : 10.1.1940
MS 122 190 c : 13.1.1940
MS 122 235 : 1.2.1940
MS 117 151 c : 4.2.1940
MS 117 152 c : 4.2.1940
MS 117 153 c : 4.2.1940
MS 117 156 : 5.2.1940
MS 117 160 c : 10.2.1940
MS 117 168 c : 17.2.1940
MS 117 225 : 2.3.1940
MS 117 237 : 6.3.1940
MS 117 253 : 11.3.1940
MS 162b 67r : 2.7.1940
MS 162b 68v : 14.8.1940
MS 162b 69v : 19.8.1940
MS 162b 70r : 21.8.1940
MS 123 112 : 1.6.1941
MS 124 3 c : 6.6.1941
MS 124 25 :11.6.1941
MS 124 30 :11.6.1941
MS 124 31 : 12.6.1941
MS 124 49 : 16.6.1941
MS 124 56 : 18.6.1941
MS 124 82 : 27.6.1941
MS 124 93 : 3.7.1941
MS 163 25r : 4.7.1941
MS 163 34r : 7.7.1941
MS 163 39r c : 8.7.1941
MS 163 47v : 11.7.1941
MS 163 64v c : 6.9.1941
MS 125 2v : 4.1.1942 oder später
MS 125 21r : 1942
MS 125 42r : 1942
MS 125 44r : 1942
MS 125 58r : 1942
MS 125 58v : 18.5.1942
MS 126 15r : 28.10.1942
MS 126 64r : 15.12.1942
MS 126 65v : 17.12.1942
MS 127 14r : 20.1.1943
MS 127 35v : 4.4.1943
MS 127 36v : 27.2.1944
MS 127 41v : 4.3.1944
MS 127 76r : 1944
MS 127 77v : 1944
MS 127 78v : 1944
MS 127 84r : 1944
MS 129 181 : 1944 oder später
MS 165 200 c : ca. 1941-1944
MS 165 204 : ca. 1944
MS 128 46 : ca. 1944
MS 128 49 : ca. 1944
MS 179 20 : ca. 1945
MS 179 26 : ca. 1945
MS 179 27 : ca. 1945
MS 180a 30 : ca. 1945
Page Break 110
MS 180a 67 : ca. 1945
MS 130 239 : 1.8.1946
MS 130 283 : 5.8.1946
MS 130 291c : 9.8.1946
MS 131 2 : 10.8.1946
MS 131 19 : 11.8.1946
MS 131 27 : 12.8.1946
MS 131 38 : 14.8.1946
MS 131 43 : 14.8.1946*
MS 131 46 : 15.8.1946
MS 131 48 : 15.8.1946
MS 131 66 c : 19.8.1946
MS 131 79 c : 20.8.1946
MS 131 80 : 20.8.1946
MS 131 163 : 31.8.1946
MS 131 176 : 1.9.1946
MS 131 180 : 2.9.1946
MS 131 182 : 2.9.1946
MS 131 186 : 3.9.1946
MS 131 218 : 8.9.1946
MS 131 219 : 8.9.1946
MS 131 221 : 8.9.1946
MS 132 8 : 11.9.1946
MS 132 12 : 12.9.1946
MS 132 46 c : 22.9.1946
MS 132 51 : 22.9.1946
MS 132 59 : 25.9.1946
MS 132 62 : 25.9.1946*
MS 132 69 : 26.9.1946
MS 132 75 : 28.9.1946*
MS 132 136 : 7.10.1946
MS 132 145 : 8.10.1946
MS 132 167 : 11.10.1946
MS 132 190 : 16.10.1946
MS 132 191 : 18.10.1946
MS 132 197 : 19.10.1946
MS 132 202 : 20.10.1946
MS 132 205 : 22.10.1946
MS 133 2 : 23.10.1946
MS 133 6 : 24.10.1946
MS 133 12 :24.10.1946
MS 133 18 :29.10.1946
MS 133 21 :31.10.1946
MS 133 30 :1.-2.11.1946
MS 133 68 c : 12.11.1946*
MS 133 82 : 24.11.1946
MS 133 82 : 24.11.1946
MS 133 90 : 7.1.1947
MS 133 118 : 19.1.1947
MS 133 188 : 27.2.1947*
MS 134 9 : 3.3.1947
MS 134 20 : 5.3.1947
MS 134 27 : 10.-15.3.1947*
MS 134 76 : 28.3.1947
MS 134 78 : 30.3.1947
MS 134 80 : 30.3.1947
MS 134 89 : 2.4.1947
MS 134 95 : 3.4.1947
MS 134 100 : 4.4.1947
MS 134 105 : 5.4.1947
MS 134 106 : 5.4.1947
MS 134 108 : 5.4.1947
MS 134 120 : 7.4.1947
MS 134 124 : 8.4.1947
MS 134 129 : 9.4.1947
MS 134 133 : 10.4.1947
MS 134 141 : 13.4.1947
MS 134 143 : 13.-14.4.1947*
MS 134 156 : 11.5.1947
MS 134 157 : 11.5.1947
MS 134 180 : 27.6.1947
MS 134 181 : 27.6.1947
MS 135 14 : 14.7.1947
MS 135 23 : 16.7.1947
MS 135 85 : 24.7.1947
MS 135 101 : 26.7.1947
MS 135 102 : 27.7.1947
MS 135 103 c : 27.7.1947
Page Break 111
MS 135 110 : 28.7.1947
MS 135 133 : 2.8.1947
MS 135 191 : 17.12.1947
MS 135 192 : 17.12.1947
MS 136 16b : 21.12.1947
MS 136 18b : 21.12.1947
MS 136 31a : 24.12.1947
MS 136 37a : 25.12.1947
MS 167 10v : ca. 1947-1948
MS 136 51a : 3.1.1948
MS 136 59a : 4.1.1948
MS 136 60b : 4.1.1948
MS 136 62a : 4.1.1948
MS 136 80a : 8.1.1948
MS 136 81a : 8.1.1948
MS 136 81b : 8.1.1948
MS 136 89a : 10.1.1948
MS 136 91b : 11.1.1948
MS 136 92b : 11.1.1948
MS 136 107a : 14.1.1948
MS 136 110b : 14.1.1948*
MS 136 113a : 15.1.1948
MS 136 117a : 15.1.1948
MS 136 128b : 18.1.1948
MS 136 129b : 19.1.1948
MS 136 137a : 22.1.1948
MS 137 17a : 10.2.1948
MS 137 20b : 15.2.1948
MS 137 30b : 8.3.1948
MS 137 34b : 10.3.1948
MS 137 40b : 28.5.1948
MS 137 42a : 30.5.1948
MS 137 43a : 30.5.1948
MS 137 48b : 4.6.1948
MS 137 49a : 4.6.1948
MS 137 49b : 4.6.1948
MS 137 51a : 15.6.1948
MS 137 54b : 25.6.1948
MS 137 57a : 26.6.1948
MS 137 57b : 30.6.1948
MS 137 65b : 3.7.1948
MS 137 67a : 3.7.1948
MS 137 67a : 4.7.1948
MS 137 67b : 5.7.1948
MS 137 70b : 7.7.1948
MS 137 71a : 9.7.1948
MS 137 72b : 14.7.1948
MS 137 73b : 25.7.1948
MS 137 75b : 23.8.1948
MS 137 76a : 19.10.1948
MS 137 78b : 24.10.1948
MS 137 81b : 27.10.1948
MS 137 88a : 4.11.1948
MS 137 88b : 4.11.1948
MS 137 89b : 5.11.1948
MS 137 97b : 14.11.1948
MS 137 97b : 16.11.1948
MS 137 100a : 19.11.1948
MS 137 102a : 20.11.1948
MS 137 104a : 21.11.1948
MS 137 104b : 22.11.1948
MS 137 106b : 23.11.1948
MS 137 111b : 28.11.1948
MS 137 112b : 29.11.1948
MS 137 120a : 10.12.1948
MS 137 122a : 11.12.1948
MS 137 127a : 16.12.1948
MS 137 130a : 22.12.1948
MS 137 134b : 25.12.1948
MS 137 134b : 26.12.1948
MS 137 135a c:27.12.1948
MS 137 135a : 28.12.1948
MS 137 136b : 31.12.1948
MS 168 lr : Jänner 1949
MS 169 58v : 1949
MS 169 62v : 1949
MS 137 137a : 1.1.1949
MS 137 140a : 4.1.1949
Page Break 112
MS 137 141a : 6.1.1949
MS 137 143a : 7.-8.1.1949
MS 138 2a : 17.1.1949
MS 138 3a : 18.1.1949
MS 138 4a : 18.1.1949
MS 138 4b : 19.1.1949
MS 138 5b : 21.1.1949
MS 138 8a : 22.1.1949
MS 138 8b : 23.1.1949
MS 138 9a : 24.1.1949
MS 138 9a : 24.1.1949
MS 138 9b : 25.1.1949
MS 138 11a : 28.1.1949
MS 138 12a : 30.1.1949
MS 138 13b : 2.2.1949
MS 138 14a : 2.2.1949
MS 138 25a : 22.2.1949
MS 138 28a : 27.2.1949
MS 138 28b : 27.2.1949
MS 138 29a : 1.3.1949
MS 138 30b : 17.3.1949
MS 138 32b : 20.5.1949
MS 138 32b : 20.5.1949
MS 173 17r : 30.3.1950
MS 173 35r : 12.4.1950 oder später
MS 173 69r : 1950
MS 173 75v : 1950
MS 173 92r : 1950
MS 174 1v : 1950
MS 174 5r : 24.4.1950 oder später
MS 174 5v : 1950
MS 174 7v : 1950
MS 174 8r : 1950
MS 174 10r : 1950
MS 175 56r : 15.3.1951
MS 175 63v : 17.3.1951
MS 176 55v : 16.4.1951
Page Break 113
List of Sources, Arranged Alphanumerically
(The remarks that in this edition have been completed to comprise a whole section are marked with an asterisk [*].)
MS 101 7 c : 21.8.1914*
MS 105 46 c : 1929*
MS 105 67 c : 1929
MS 105 73 c : 1929
MS 105 85 c : 1929
MS 106 247 : 1929
MS 106 58 : 1929
MS 107 100 c : 1929
MS 107 116 c : 1929
MS 107 120 c : 1929
MS 107 130 c : 1929
MS 107 156 c : 10.10.1929
MS 107 176 c : 24.10.1929
MS 107 184 c : 7.11.1929
MS 107 192 c : 10.11.1929
MS 107 196 : 15.11.1929
MS 107 229 : 10.-11.1.1930
MS 107 242 : 16.1.1930
MS 107 70 c : 1929
MS 107 72 c : 1929*
MS 107 82 : 1929
MS 107 82 c : 1929
MS 107 97 c : 1929
MS 107 98 c : 1929
MS 108 207 : 29.6.1930
MS 109 200 : 5.11.1930
MS 109 204 : 6.-7.11.1930
MS 109 212 : 8.11.1930
MS 109 28 : 22.8.1930
MS 110 10 : 13.12.1930
MS 110 11 : 25.12.1930
MS 110 12 : 12.-16.1.1931
MS 110 18 : 18.1.1931
MS 110 200 : 22.6.1931
MS 110 226 : 25.6.1931
MS 110 231 c : 29.6.1931
MS 110 238 : 30.6.1931
MS 110 257 : 2.7.1931
MS 110 260 c : 2.7.1931
MS 110 5 : 12.12.1930
MS 110 61 : 10.2.1931
MS 110 67 : 12.2.1931
MS 111 115 : 19.8.1931
MS 111 119 : 19.8.1931
MS 111 132 : 23.8.1931
MS 111 133 : 24.8.1931
MS 111 173 : 13.9.1931
MS 111 180 : 13.9.1931
MS 111 194 : 13.9.1931
MS 111 195 : 13.9.1931
MS 111 2 c : 7.7.1931
MS 111 55 : 30.7.1931
MS 111 77 : 11.8.1931
MS 112 1 : 5.10.1931
MS 112 114 : 27.10.1931
MS 112 139 : 1.11.1931
MS 112 153 : 11.11.1931
MS 112 221 : 22.11.1931
MS 112 223 : 22.11.1931
MS 112 225 : 22.11.1931
MS 112 231 : 22.11.1931
MS 112 232 : 22.11.1931
MS 112 233 : 22.11.1931
MS 112 255 : 26.11.1931
Page Break 114
MS 112 46 : 14.10.1931
MS 112 60 : 15.10.1931
MS 112 76 : 24.10.1931
MS 113 80 : 29.2.1932
MS 116 56 : 1937
MS 117 151 c : 4.2.1940
MS 117 152 c : 4.2.1940
MS 117 153 c : 4.2.1940
MS 117 156 : 5.2.1940
MS 117 160 c : 10.2.1940
MS 117 168 c : 17.2.1940
MS 117 225 : 2.3.1940
MS 117 237 : 6.3.1940
MS 117 253 : 11.3.1940
MS 118 113r : 24.9.1937
MS 118 117v : 24.9.1937
MS 118 17r c : 27.8.1937*
MS 118 20r : 27.8.1937
MS 118 21v : 27.8.1937
MS 118 35r c : 29.8.1937
MS 118 45r c : 1.9.1937
MS 118 56r c : 4.9.1937*
MS 118 71v c : 9.9.1937
MS 118 86v : 11.9.1937
MS 118 87r c : 11.9.1937*
MS 118 94v : 15.9.1937
MS 119 108 : 14.10.1937
MS 119 146 : 21.10.1937
MS 119 151 : 22.10.1937
MS 119 59 : 1.10.1937
MS 119 71 : 4.10.1937
MS 119 83 : 7.10.1937
MS 120 103 c : 12.12.1937
MS 120 108 c : 12.12.1937*
MS 120 113 : 2.1.1938
MS 120 142 : 19.2.1938
MS 120 283 : 7.4.1938
MS 120 289 : 20.4.1938
MS 120 8 : 20.11.1937
MS 120 83 c : 8.-9.12.1937*
MS 121 26v : 25.5.1938
MS 121 35v : 11.6.1938
MS 122 129 : 30.12.1939
MS 122 175c : 10.1.1940
MS 122 190 c : 13.1.1940
MS 122 235 : 1.2.1940
MS 123 112 : 1.6.1941
MS 124 25 : 11.6.1941
MS 124 3 c : 6.6.1941
MS 124 30 : 11.6.1941
MS 124 31 : 12.6.1941
MS 124 49 : 16.6.1941
MS 124 56 : 18.6.1941
MS 124 82 : 27.6.1941
MS 124 93 : 3.7.1941
MS 125 21r : 1942
MS 125 2v : 4.1.1942 oder später
MS 125 42r : 1942
MS 125 44r : 1942
MS 125 58r : 1942
MS 125 58v : 18.5.1942
MS 126 15r : 28.10.1942
MS 126 64r : 15.12.1942
MS 126 65v : 17.12.1942
MS 127 14r : 20.1.1943
MS 127 35v : 4.4.1943
MS 127 36v : 27.2.1944
MS 127 41v : 4.3.1944
MS 127 76r : 1944
MS 127 77v : 1944
MS 127 78v : 1944
MS 127 84r : 1944
MS 128 46 : ca. 1944
MS 128 49 : ca. 1944
MS 129 181 : 1944 oder später
MS 130 239 : 1.8.1946
Page Break 115
MS 130 283 : 5.8.1946
MS 130 291 c : 9.8.1946
MS 131 163 : 31.8.1946
MS 131 176 : 1.9.1946
MS 131 180 : 2.9.1946
MS 131 182 : 2.9.1946
MS 131 186 : 3.9.1946
MS 131 19 : 11.8.1946
MS 131 2 : 10.8.1946
MS 131 218 : 8.9.1946
MS 131 219 : 8.9.1946
MS 131 221 : 8.9.1946
MS 131 27 : 12.8.1946
MS 131 38 : 14.8.1946
MS 131 43 : 14.8.1946*
MS 131 46 : 15.8.1946
MS 131 48 : 15.8.1946
MS 131 66 c : 19.8.1946
MS 131 79 c : 20.8.1946
MS 131 80 : 19.8.1946
MS 132 12 : 12.9.1946
MS 132 136 : 7.10.1946
MS 132 145 : 8.10.1946
MS 132 167 : 11.10.1946
MS 132 190 : 16.10.1946
MS 132 191 : 18.10.1946
MS 132 197 : 19.10.1946
MS 132 202 : 20.10.1946
MS 132 205 : 22.10.1946
MS 132 46 c : 22.9.1946
MS 132 51 : 22.9.1946
MS 132 59 : 25.9.1946
MS 132 62 : 25.9.1946*
MS 132 69 : 26.9.1946
MS 132 75 : 28.9.1946*
MS 132 8 : 11.9.1946
MS 133 118 : 19.1.1947
MS 133 12 : 24.10.1946
MS 133 18 : 29.10.1946
MS 133 188 : 27.2.1947*
MS 133 2 : 23.10.1946
MS 133 21 : 31.10.1946
MS 133 30 : 1.-2.11.1946
MS 133 6 : 24.10.1946
MS 133 68 c : 12.11.1946*
MS 133 82 : 24.11.1946
MS 133 82 : 24.11.1946
MS 133 90 : 7.1.1947
MS 134 100 : 4.4.1947
MS 134 105 : 5.4.1947
MS 134 106 : 5.4.1947
MS 134 108 : 5.4.1947
MS 134 120 : 7.4.1947
MS 134 124 : 8.4.1947
MS 134 129 : 9.4.1947
MS 134 133 : 10.4.1947
MS 134 141 : 13.4.1947
MS 134 143 : 13.-14.4.1947*
MS 134 156 : 11.5.1947
MS 134 157 : 11.5.1947
MS 134 180 : 27.6.1947
MS 134 181 : 27.6.1947
MS 134 20 : 5.3.1947
MS 134 27 : 10.-15.3.1947*
MS 134 76 : 28.3.1947
MS 134 78 : 30.3.1947
MS 134 80 : 30.3.1947
MS 134 89 : 2.4.1947
MS 134 9 : 3.3.1947
MS 134 95 : 3.4.1947
MS 135 101 : 26.7.1947
MS 135 102 : 27.7.1947
MS 135 103 c : 27.7.1947
MS 135 110 : 28.7.1947
MS 135 133 : 2.8.1947
MS 135 14 : 14.7.1947
MS 135 191 : 17.12.1947
MS 135 192 : 17.12.1947
Page Break 116
MS 135 23 : 16.7.1947
MS 135 85 : 24.7.1947
MS 136 107a : 14.1.1948
MS 136 110b : 14.1.1948*
MS 136 113a : 15.1.1948
MS 136 117a : 15.1.1948
MS 136 128b : 18.1.1948
MS 136 129b : 19.1.1948
MS 136 137a : 22.1.1948
MS 136 16b : 21.12.1947
MS 136 18b : 21.12.1947
MS 136 31a : 24.12.1947
MS 136 37a : 25.12.1947
MS 136 51a : 3.1.1948
MS 136 59a : 4.1.1948
MS 136 60b : 4.1.1948
MS 136 62a : 4.1.1948
MS 136 80a : 8.1.1948
MS 136 81a : 8.1.1948
MS 136 81b : 8.1.1948
MS 136 89a : 10.1.1948
MS 136 91b : 11.1.1948
MS 136 92b : 11.1.1948
MS 137 100a : 19.11.1948
MS 137 102a : 20.11.1948
MS 137 104a : 21.11.1948
MS 137 104b : 22.11.1948
MS 137 106b : 23.11.1948
MS 137 111b : 28.11.1948
MS 137 112b : 29.11.1948
MS 137 120a : 10.12.1948
MS 137 122a : 11.12.1948
MS 137 127a : 16.12.1948
MS 137 130a : 22.12.1948
MS 137 134b : 25.12.1948
MS 137 134b : 26.12.1948
MS 137 135a c: 28.12.1948
MS 137 135a c: 27.12.1948
MS 137 136b : 31.12.1948
MS 137 137a : 1.1.1949
MS 137 140a : 4.1.1949
MS 137 141a : 6.1.1949
MS 137 143a : 7.-8.1.1949
MS 137 17a : 10.2.1948
MS 137 20b : 15.2.1948
MS 137 30b : 8.3.1948
MS 137 34b : 10.3.1948
MS 137 40b : 28.5.1948
MS 137 42a : 30.5.1948
MS 137 43a : 30.5.1948
MS 137 48b : 4.6.1948
MS 137 49a : 4.6.1948
MS 137 49b : 4.6.1948
MS 137 51a : 15.6.1948
MS 137 54b : 25.6.1948
MS 137 57a : 26.6.1948
MS 137 57b : 30.6.1948
MS 137 65b : 3.7.1948
MS 137 67a : 3.7.1948
MS 137 67a : 4.7.1948
MS 137 67b : 5.7.1948
MS 137 70b : 7.7.1948
MS 137 71a : 9.7.1948
MS 137 72b : 14.7.1948
MS 137 73b : 25.7.1948
MS 137 75b : 23.8.1948
MS 137 76a : 19.10.1948
MS 137 78b : 24.10.1948
MS 137 81b : 27.10.1948
MS 137 88a : 4.11.1948
MS 137 88b : 4.11.1948
MS 137 89b : 5.11.1948
MS 137 97b : 14.11.1948
MS 137 97b : 16.11.1948
MS 138 11a : 28.1.1949
MS 138 12a : 30.1.1949
MS 138 13b : 2.2.1949
MS 138 14a : 2.2.1949
Page Break 117
MS 138 25a : 22.2.1949
MS 138 28a : 27.2.1949
MS 138 28b : 27.2.1949
MS 138 29a : 1.3.1949
MS 138 2a : 17.1.1949
MS 138 30b : 17.3.1949
MS 138 32b : 20.5.1949
MS 138 32b : 20.5.1949
MS 138 3a : 18.1.1949
MS 138 4a : 18.1.1949
MS 138 4b : 19.1.1949
MS 138 5b : 21.1.1949
MS 138 8a : 22.1.1949
MS 138 8b : 23.1.1949
MS 138 9a : 24.1.1949
MS 138 9a : 24.1.1949
MS 138 9b : 25.1.1949
MS 145 14r : Herbst 1933*
MS 145 17v : 1933
MS 146 25v : 1933-1934
MS 146 35v : 1933-1934
MS 146 44v : 1933-1934
MS 147 16r : 1934
MS 147 22r : 1934
MS 153a 127v : 1931
MS 153a 128v : 1931
MS 153 12v : 1931
MS 153a 136r : 1931
MS 153a 141r : 1931
MS 153a 160v : 1931
MS 153a 18v : 1931
MS 153a 29v : 1931
MS 153a 35r : 1931
MS 153a 35v : 1931
MS 153a 4v : 10.5.1931
MS 153b 90v : 1931
MS 153 29r : 1931
MS 153b 39v : 1931
MS 153b 3r : 1931
MS 154 15v : 1931
MS 154 1r : 1931
MS 154 1v : 1931
MS 154 20v : 1931
MS 154 21v : 1931
MS 154 25v : 1931
MS 155 29r : 1931
MS 155 46r : 1931
MS 156a 25r : ca. 1932-1934
MS 156a 30v : ca. 1932-1934
MS 156a 49r : ca. 1932-1934
MS 156a 49v : ca. 1932-1934
MS 156a 57r : ca. 1932-1934
MS 156a 58b : ca. 1932-1934
MS 156b 14v : ca. 1932-1934
MS 156b 23v : ca. 1932-1934
MS 156b 24v : ca. 1932-1934
MS 156b 32r : ca. 1932-1934
MS 157a 22v : 1934
MS 157a 2r : 1934*
MS 157a 44v : 1934 oder 1937
MS 157a 57r : 1937
MS 157a 57r : 1937
MS 157a 58r : 1937
MS 157a 62v : 1937
MS 157a 66v c: 1937
MS 157a 67v c: 1937
MS 157b 15v : 1937
MS 157b 9v : 1937
MS 162a 18 : 1939-1940
MS 162b 16v : 1939-1940
MS 162b 21v : 1939-1940
MS 162b 22r c : 1939-1940
MS 162b 24r : 1939-1940
MS 162b 37r c : 1939-1940
MS 162b 42v c : 1939-1940
MS 162b 53r : 1939-1940
MS 162b 55v : 1939-1940
Page Break 118
MS 162b 59r : 1939-1940*
MS 162b 59v : 1939-1940
MS 162b 61r : 1939-1940
MS 162b 63r : 1939-1940
MS 162b 67r : 2.7.1940
MS 162b 68v : 14.8.1940
MS 162b 69v : 19.8.1940
MS 162b 70r : 21.8.1940
MS 163 25r : 4.7.1941
MS 163 34r : 7.7.1941
MS 163 39r c : 8.7.1941
MS 163 47v : 11.7.1941
MS 163 64v c : 6.9.1941
MS 165 200 c : ca. 1941-1944
MS 165 204 : ca. 1944
MS 167 10v : ca. 1947-1948
MS 168 1r : Jänner 1949
MS 169 58v : 1949
MS 169 62v : 1949
MS 173 17r : 30.3.1950
MS 173 35r : 12.4.1950 oder später
MS 173 69r : 1950
MS 173 75v : 1950
MS 173 92r : 1950
MS 174 10r : 1950
MS 174 1v : 1950
MS 174 5r : 24.4.1950 oder später
MS 174 5v : 1950
MS 174 7v : 1950
MS 174 8r : 1950
MS 175 56r : 15.3.1951
MS 175 63v : 17.3.1951
MS 176 55v : 16.4.1951
MS 179 20 : ca. 1945
MS 179 26 : ca. 1945
MS 179 27 : ca. 1945
MS 180a 30 : ca. 1945
MS 180a 67 : ca. 1945
Page Break 119
Page Break 120
Index of Beginnings of Remarks
(The remarks that in this edition have been completed to comprise a whole section are marked with an asterisk [*].)
Page 120
A beautiful garment that changes 19e
Page 120
A Bruckner symphony can be said 39e
Page 120
A confession has to be part 16e
Page 120
A cry of distress cannot be greater 52e
Page 120
A curious analogy could be based 25e
Page 120
A foolish & naïve American film 65e
Page 120
A teacher who can show good, or 43e
Page 120
A good simile refreshes 3e
Page 120
A great architect in a bad period 84e
Page 120
A great blessing for me to be able 38e
Page 120
A hero looks death in the face, 58e
Page 120
At the time I modelled the head 16e
Page 120
A theme, no less than a face, wears 59e
Page 120
A thinker is very similar 14e
Page 120
A language, which is spoken in strict 85e
Page 120
A man sees well enough what he has, 56e
Page 120
A man's dreams are virtually never 64e
Page 120
A mediocre writer must beware of too 90e
Page 120
A miracle is, as it were, a 51e
Page 120
A new word is like a fresh seed 4e
Page 120
A picture of a complete apple tree, 17e
Page 120
A picture that is firmly rooted95e
Page 120
At present we are combatting a trend 49e
Page 120
A proof of God ought really to be 97e
Page 120
A stylistic device may be useful 82e
Page 120
A tragedy might really always start 14e
Page 120
A writer far more talented than I 86e
Page 120
Almost in the same way as earlier 31e
Page 120
Almost the whole time I am writing 88e
Page 120
Always come down from the barren 86e
Page 120
Ambition is the death of thought 88e
Page 120
Amongst other things Christianity 61e
Page 120
And now we might say: There can be94e
Page 120
And what I basically am after all 95e
Page 120
Animals come when their names 77e
Page 120
Anyone who is half-way decent will 51e
Page 120
Anyone who listens to a child's 4e
Page 120
Anything the reader can do 88e
Page Break 121
Page 121
Anyway when I was in Norway 17e
Page 121
Appreciating music is a manifestation 80e
Page 121
Architecture glorifies something 74e
Page 121
Architecture immortalizes & glorifies 74e
Page 121
Architecture is a gesture 49e
Page 121
Are all people great human beings? 53e
Page 121
Are you a bad philosopher then, 87e
Page 121
As I again & again come across questions 76e
Page 121
As much (as) it has cost you, that 15e
Page 121
As we get old the problems slip 46e
Page 121
Bach said that everything he achieved 81e
Page 121
"Beethoven's great heart"-- 96e
Page 121
Before there was an aeroplane 48e
Page 121
Being psychoanalyzed is in a way 40e
Page 121
'Believing' means, submitting to 52e
Page 121
Bruckner's Ninth is a sort of protest 39e
Page 121
But am I to say (if I reject images, 80e
Page 121
But I do not get to these problems 12e
Page 121
But isn't it a one-sided view of tragedy 14e
Page 121
But that does not mean that it is right 87e
Page 121
But then what is the relation between 31e
Page 121
By the way, in my artistic activities 29e
Page 121
By the way in the old conception 20e
Page 121
Can I say that drama has its own time 13e
Page 121
Caught a bit of a chill & unable to 32e
Page 121
Christian faith--so I believe--is 52e
Page 121
Christianity is not based on a historical 37e
Page 121
Colours are a stimulus to philosophizing 76e
Page 121
Compare the solution of philosophical 13e
Page 121
Composers' names. Sometimes it is the method 29e
Page 121
Concepts may alleviate mischief or 63e
Page 121
Conflict is dissipated in much12e
Page 121
Conglomeration: national sentiment, 77e
Page 121
Could the concept of the punishments of hell 92e
Page 121
Counterpoint might represent an 46e
Page 121
Culture is an observance 89e
Page 121
Different 'interpretations' must correspond 46e
Page 121
Does the theme point to nothing beyond 59e
Page 121
Doing this work I am in the same state 36e
Page 121
Do not interest yourself in what 72e
Page 121
Don't apologize for anything, don't obscure 45e
Page 121
Don't demand too much, & don't be 46e
Page Break 122
Page 122
Don't for heaven's sake, be afraid 64e
Page 122
Don't let yourself be guided by the example 47e
Page 122
Don't play with what lies deep 26e
Page 122
Each morning you have to break 4e
Page 122
Each sentence that I write 9e
Page 122
E.g. nothing more stupid than the chatter 71e
Page 122
Election by grace: it is only permissible 34e
Page 122
Engelmann told me that when he rummages 6e
Page 122
Envy is something superficial-- 40e
Page 122
Esperanto. Our feeling of disgust, 60e
Page 122
Even if we are not thinking in circles, 91e
Page 122
Every artist has been influenced 27e
Page 122
"Fare well!" 60e
Page 122
Fate is the antithesis of natural law 70e
Page 122
For a human being the eternal, 92e
Page 122
For could one not equally say94e
Page 122
[For the Foreword.] It is not 75e
Page 122
Freud's fanciful pseudo-explanations 62e
Page 122
Freud's idea: in madness the lock 39e
Page 122
Frida Schanz: 14e
Page 122
From Simplicissimus: riddles of 22e
Page 122
Genius is talent in which character 75e
Page 122
Genius is what makes us forget talent49e
Page 122
Genius is what makes us forget the 49e
Page 122
Genius is what makes us unable to see 50e
Page 122
God grant the philosopher insight into 72e
Page 122
"God has commanded it, therefore we must 87e
Page 122
God may say to me: "I am judging you 99e
Page 122
God's essence is said to guarantee his94e
Page 122
Go on, believe! It does no harm 52e
Page 122
Grasping the difficulty in its depth 55e
Page 122
Grillparzer: "How easy it is to move 15e
Page 122
(Had he not been caught in the machine 14e
Page 122
He has chosen them, in his goodness, 93e
Page 122
"He has commanded it" means here roughly:87e
Page 122
"He is experiencing the theme intensely 59e
Page 122
(Hence, I think, a great creator needs 68e
Page 122
"He refuses to..." means: it is in his 87e
Page 122
His talent too strikes me as 61e
Page 122
How are we taught the word "God" 94e
Page 122
How God judges people is something 98e
Page Break 123
Page 123
How hard it is for me to see what 44e
Page 123
How should we feel if we had never 15e
Page 123
How small a thought it takes 57e
Page 123
How then do we explain to someone 80e
Page 123
Human beings can regard all the evil 76e
Page 123
Human beings have a physical need 86e
Page 123
Humour is not a mood, but a way of 88e
Page 123
I am deeply suspicious of most of Shakespeare's 95e
Page 123
I am too soft, too weak, & so too lazy, 83e
Page 123
I am reading Lessing (on the Bible) 11e
Page 123
I am showing my pupils sections of 64e
Page 123
I am so afraid of someone's playing the piano 73e
Page 123
I ask countless irrelevant questions 77e
Page 123
I believe that what Goethe was really seeking26e
Page 123
I believe I summed up where I stand 28e
Page 123
I cannot judge whether I have only taste, 68e
Page 123
I cannot kneel to pray, because 63e
Page 123
I completely understand, how someone may hate 66e
Page 123
I could equally well have said: the distorted 63e
Page 123
I could imagine someone thinking the names46e
Page 123
I could only stare in wonder at Shakespeare; 95e
Page 123
Ideas too sometimes fall from the tree 32e
Page 123
I do not think that Shakespeare 95e
Page 123
I do not think Shakespeare could have 96e
Page 123
I.e. it is simply false to say 7e
Page 123
If a false thought is so much as 86e
Page 123
If anyone should think he has solved 6e
Page 123
If certain graphic propositions 32e
Page 123
If Christianity is the truth, 89e
Page 123
If God really does choose those 83e
Page 123
If the believer in God looks around 96e
Page 123
If the dreams of sleep have a similar 83e
Page 123
If there is anything in the Freudian 50e
Page 123
If this stone won't budge at present, 44e
Page 123
If I am thinking just for myself 33e
Page 123
If I do not quite know how to begin 11e
Page 123
If I had written a good sentence, 66e
Page 123
If in life we are surrounded by death, 50e
Page 123
If I now ask "What do I actually experience 79e
Page 123
If I prepare myself for some eventuality, 90e
Page 123
If I realized how mean & petty I am 37e
Page 123
If I say A. has beautiful eyes 28e
Page 123
If I say: "that is the only thing that 29e
Page 123
If I say that my book is meant 12e
Page Break 124
Page 124
If it is said on occasion that 17e
Page 124
If it is true, as I believe, that Mahler's 76e
Page 124
If life becomes hard to bear we think 60e
Page 124
If one wanted to characterize the essence 27e
Page 124
If people did not sometimes commit stupidities, 57e
Page 124
If people find a flower or an animal 20e
Page 124
If someone can believe in God with complete 83e
Page 124
If someone is merely ahead of his time 11e
Page 124
If someone prophesies that the generation 29e
Page 124
If someone says, let's suppose, "A's eyes 27e
Page 124
If we think of the world's future 5e
Page 124
If we use the ethnological approach 45e
Page 124
If white turns to black some say: 49e
Page 124
If you already have someone's love, 48e
Page 124
If you ask: how I experienced the theme, 59e
Page 124
If you cannot unravel a tangle, 85e
Page 124
If you offer a sacrifice & then 30e
Page 124
If you use a trick in logic, 28e
Page 124
I have one of those talents that has 86e
Page 124
I just took some apples out of a paper bag 36e
Page 124
I look at the photographs of Corsican15e
Page 124
Imagine someone watching a pendulum 86e
Page 124
I might say: if the place I want to reach 10e
Page 124
I must be nothing more than the mirror 25e
Page 124
I myself still find my way of philosophizing 3e
Page 124
In a bullfight the bull is the hero 58e
Page 124
In a conversation: One person throws 84e
Page 124
In a dream, & even long after we wake up, 75e
Page 124
In art it is hard to say anything, 26e
Page 124
In Beethoven's music what one might call 93e
Page 124
In Bruckner's music nothing is left 19e
Page 124
In Christianity it is as though God said" 21e
Page 124
I never before believed in God" 61e
Page 124
I never more than half succeed in expressing 16e
Page 124
In former times people entered monasteries 56e
Page 124
In Freudian analysis the dream is 78e
Page 124
In Mozart's or Haydn's music again fate 93e
Page 124
In my view Bacon was not a precise 61e
Page 124
In one day you can experience the horrors 30e
Page 124
In philosophizing it is important 31e
Page 124
In philosophy the winner of the race 40e
Page 124
In religion it must be the case 37e
Page 124
In Renan's Peuple d'Israël 7e
Page 124
In the course of a scientific investigation 73e
Page 124
In the course of our conversation Russell 35e
Page Break 125
Page 125
In the days of silent films 29e
Page 125
In the Gospels--as it seems to me 35e
Page 125
In the train of every idea that costs 66e
Page 125
In the realm of the mind a project 87e
Page 125
In this world (mine) there is not tragedy 12e
Page 125
In Western Civilization the Jew 23e
Page 125
I often fear madness. Have I any reason 61e
Page 125
I often wonder whether my cultural 4e
Page 125
I once said, & perhaps rightly 5e
Page 125
I read: "philosophers are no nearer 22e
Page 125
I really do think with my pen,24e
Page 125
I recently said to Arvid 5e
Page 125
Irony in music. E.g. Wagner 63e
Page 125
I sit astride life like a bad rider42e
Page 125
Is it just I who cannot found a school, 69e
Page 125
Is it some frustrated longing that 51e
Page 125
Isochronism according to the clock 92e
Page 125
Is the sense of belief in the Devil 99e
Page 125
Is what I am doing in any way worth 66e
Page 125
It appears to me as though a religious73e
Page 125
It could only be by accident that someone's 64e
Page 125
It has sometimes been said that the Jews' 19e
Page 125
I think Bacon got bogged down in 77e
Page 125
I think good Austrian work 5e
Page 125
I think it is an important & remarkable 84e
Page 125
I think it might be regarded as a fundamental82e
Page 125
I think that, in order to enjoy a poet, 96e
Page 125
I think that present day education 81e
Page 125
I think there is some truth in my idea 16e
Page 125
I think today there could be a form of theatre14e
Page 125
It is a great temptation to want 11e
Page 125
It is all one to me whether the typical 9e
Page 125
It is an accident that 'last' rhymes with 93e
Page 125
It is as though I had lost my way 53e
Page 125
It is difficult to form a good picture 94e
Page 125
It is difficult to know something, & to act 90e
Page 125
It is hard to tell someone who is shortsighted3e
Page 125
It is hard to understand yourself properly 54e
Page 125
"It is high time for us to compare 62e
Page 125
It is humiliating having to present oneself 13e
Page 125
It is important for our approach, that someone 84e
Page 125
It is incredible how helpful a new 45e
Page 125
It is not as though S. portrayed human 96e
Page 125
It is not by any means clear to me, that I wish 70e
Page 125
It is not unheard of that someone's character 95e
Page Break 126
Page 126
It isn't reasonable to be furious even 53e
Page 126
It is often only very slightly more 45e
Page 126
It is often said that a new religion 23e
Page 126
It is one thing to sow in thought, 60e
Page 126
It is possible, that everyone who executes 67e
Page 126
It is remarkable how hard we find it 55e
Page 126
It is remarkable that Busch's drawings 85e
Page 126
It is very remarkable, that we should 57e
Page 126
It may be that science & industry, 72e
Page 126
It might also be said: hate between human 52e
Page 126
It might be said (rightly or wrongly) 16e
Page 126
I took the poem on the previous page15e
Page 126
It's a good thing I don't let myself 3e
Page 126
It seems to me as though his pieces are 98e
Page 126
It seems to me I am still a long way 74e
Page 126
It seems to me (sometimes) as though 27e
Page 126
It's possible to write in a style 60e
Page 126
It's true you must assemble old material 46e
Page 126
It will be hard to follow my portrayal: 51e
Page 126
It would almost be strange if there 82e
Page 126
It would already be enough, that all members 85e
Page 126
Just as I cannot write verse 67e
Page 126
Just as someone may travel around 57e
Page 126
Just let nature speak 3e
Page 126
Kierkegaard writes: If Christianity 36e
Page 126
Kleist wrote somewhere that 23e
Page 126
Labor's seriousness is a very late 20e
Page 126
Labor, when he writes good music, 21e
Page 126
Language sets everyone the same traps; 25e
Page 126
"Le style c'est l'homme." 89e
Page 126
Lenau's Faust is remarkable in that 61e
Page 126
Let us be human.-- 36e
Page 126
Life is like a path along a mountain 72e
Page 126
Life's infinite variations are an83e
Page 126
Longfellow: In the elder days 39e
Page 126
Look at human beings: One is poison98e
Page 126
Looking for the fallacy in a fishy 49e
Page 126
"Look on this wart as a regular limb 18e
Page 126
Madness doesn't have to be regarded 62e
Page 126
Mendelssohn is like a man who is cheerful 4e
Page 126
Mendelssohn is not a peak 4e
Page Break 127
Page 127
(Mendelssohn's music, when it is flawless 23e
Page 127
Moore poked into a philosophical wasp nest 87e
Page 127
Music, with its few notes & rhythms 11e
Page 127
My 'achievement' is very much like 57e
Page 127
My ideal is a certain coolness 4e
Page 127
My originality (if that is the right word) 42e
Page 127
My own thinking about art & values 91e
Page 127
(My style is like bad musical composition.) 45e
Page 127
My thinking, like everyone's, has sticking 27e
Page 127
My thoughts probably move in a far narrower 72e
Page 127
Neither could he regard himself as a prophet 96e
Page 127
Nietzsche writes somewhere, that even the 67e
Page 127
No one can think a thought for me 4e
Page 127
No one likes having offended 13e
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Not funk but funk conquered is what 43e
Page 127
Nothing is more important though than the 84e
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Nothing is so difficult as not deceiving 39e
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Nothing seems to me more unlikely 70e
Page 127
Nothing we do can be defended definitively 23e
Page 127
Oh a key can lie for ever where 62e
Page 127
Once again: what does it consist in, 58e
Page 127
Once it is established the old problems 55e
Page 127
One age misunderstands another; 98e
Page 127
One & the same theme has a different 96e
Page 127
One cannot speak the truth;--if 41e
Page 127
One cannot view one's own character26e
Page 127
One could call Schopenhauer a quite crude 41e
Page 127
One keeps forgetting to go down to the 71e
Page 127
One might say: "Genius is courage 44e
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One might say of Schopenhauer: he never 41e
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One might say: what splendid laws 47e
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One movement constructs 10e
Page 127
One movement orders one thought to the 10e
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One of my most important methods 45e
Page 127
One uses straw to try to stuff 5e
Page 127
Only by thinking much more crazily 86e
Page 127
Only every so often does one of the sentences 76e
Page 127
Only in a quite particular musical context 93e
Page 127
Only someone very unhappy 53e
Page 127
Only something supernatural 5e
Page 127
Only where genius wears thin can 50e
Page 127
Or again no distress can be greater 52e
Page 127
Or what is it like for someone to have no 95e
Page Break 128
Page 128
Our children learn in school 81e
Page 128
Our civilization is characterized 9e
Page 128
Our greatest stupidities may be 45e
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People are religious to the extent 51e
Page 128
People have sometimes said to me 33e
Page 128
People nowadays think, scientists 42e
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People who are constantly asking 'why' 46e
Page 128
Perhaps one day a culture will arise 73e
Page 128
Phenomena akin to language in music 40e
Page 128
(Philosophers are often like little children 24e
Page 128
Philosophers who say: "after death 26e
Page 128
Philosophy hasn't made any progress? 98e
Page 128
Piano playing, a dance of human fingers 42e
Page 128
Pieces of music composed at the keyboard 14e
Page 128
Power and possession are not the same 18e
Page 128
Put someone in the wrong atmosphere 48e
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Raisins may be the best part of a cake 76e
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Ramsey was a bourgeois thinker 24e
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Reading the Socratic dialogues, 21e
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Really I want to slow down the speed 77e
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Recounting a dream, a medley of 88e
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Reflect in this connection that in 94e
Page 128
Religion is as it were the calm sea 61e
Page 128
Religious faith & superstition 82e
Page 128
Religious similes can be said to move34e
Page 128
Remember the impression made by good 26e
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Resting on your laurels is as dangerous 41e
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Schiller writes in a letter (to Goethe, 75e
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(Schopenhauer: the real life span 30e
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Schubert is irreligious & melancholy 53e
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Schubert's melodies can be said 54e
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Science, enrichment & impoverishment 69e
Page 128
Scientific questions may interest me, 91e
Page 128
Scraping away mortar is much 44e
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Shakespeare & the dream. A dream is all 89e
Page 128
Shakespeare, one might say, 42e
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Shakespeare's similes are, 56e
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Sketch for a Forward 8e
Page 128
Slept a bit better. Vivid dreams 31e
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So complicated, so irregular is the mode 51e
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Socrates, who always reduces the Sophist 64e
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Some people have a taste 73e
Page Break 129
Page 129
Sometimes a sentence can be understood 65e
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Sometimes it's as though you could see a 72e
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Sometimes one sees ideas, as an astronomer 66e
Page 129
Sometimes you have to take an expression 44e
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Someone divides human beings into buyers 26e
Page 129
Someone is imprisoned in a room 48e
Page 129
Someone reacts like this: he says "Not that!" 69e
Page 129
Someone who teaches philosophy nowadays 25e
Page 129
Someone who knows too much finds it hard 73e
Page 129
So perhaps it is similar to the concept97e
Page 129
Soulful expression in music 94e
Page 129
Spengler could be better understood 21e
Page 129
Strike a coin from every mistake 79e
Page 129
Structure & feeling in music 20e
Page 129
Suppose someone were taught: There is a being 92e
Page 129
Suppose that 2000 years ago someone 49e
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Talent is a spring: 20e
Page 129
Taste can delight, but not seize 68e
Page 129
Taste is refinement of sensibility 68e
Page 129
Taste makes ACCEPTABLE 68e
Page 129
Taste rectifies, it doesn't give birth 68e
Page 129
Teaching this could not be an ethical 93e
Page 129
That I do not understand him could then 56e
Page 129
That must be the end of a theme 19e
Page 129
That writers, who after all were something, 90e
Page 129
'The aim of music: to communicate feelings.' 43e
Page 129
The beauty of a star-shaped figure 81e
Page 129
The book is full of life--not 71e
Page 129
The characters in a drama arouse our 47e
Page 129
The charming difference in temperature 13e
Page 129
The comparisons of the N.T. 42e
Page 129
The concept of a 'festivity' 89e
Page 129
"The cussedness of things" 81e
Page 129
The danger in a long foreword 10e
Page 129
The edifice of your pride has to be 30e
Page 129
The face is the soul 26e
Page 129
The faculty of 'taste' cannot create 68e
Page 129
The folds of my heart all the time 65e
Page 129
The fundamental insecurity of life 63e
Page 129
The greater "purity" of objects 30e
Page 129
The greatest happiness for a human being 87e
Page 129
The happy lover & the unhappy lover 86e
Page 129
The honest religious thinker is like 84e
Page 129
The human being is the best picture 56e
Page Break 130
Page 130
The human gaze has the power 3e
Page 130
The hysterical fear of the atom bomb 55e
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The idea is worn out by now 24e
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The idea working its way towards the light 54e
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The impression (made by this melody) 42e
Page 130
The inexpressible (what I find enigmatic 23e
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The Jew is a desert region 15e
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The Jew must in a real sense "make nothing 16e
Page 130
The labour pains at the birth of new 71e
Page 130
The language used by philosophers 47e
Page 130
The last two bars of the "Death & the60e
Page 130
The lesson in a poem is overstated, 62e
Page 130
The less somebody knows & understands 53e
Page 130
The Lieutenant & I have already 3e
Page 130
The light shed by work is a beautiful 30e
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The limit of language manifests itself 13e
Page 130
The mathematician (Pascal) who admires 47e
Page 130
The mathematician too can of course marvel 65e
Page 130
The measure of genius is character 40e
Page 130
The melodies of different composers 54e
Page 130
The miracles of nature, 64e
Page 130
The monstrous vanity of wishes is 41e
Page 130
The most refined taste68e
Page 130
Then again you might think intensive 59e
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The 'necessity' with which the second idea 65e
Page 130
The Old Testament seen as the body 40e
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The only way for me to believe in a miracle 51e
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The only way namely for us to avoid prejudice 30e
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The origin & the primitive form 36e
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The overwhelming skill in Brahms 29e
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The philosopher easily gets into the position 24e
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The philosopher is someone who has to cure50e
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The philosopher says "Look at things70e
Page 130
The pleasure I take in my thoughts 20e
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The poet cannot really say of himself 96e
Page 130
The poet too must always be asking himself: 46e
Page 130
The popular scientific writings 48e
Page 130
The power of language to make everything 19e
Page 130
The problems of life are insoluble 84e
Page 130
The purely corporeal can be uncanny57e
Page 130
The queer resemblance between a philosophical 29e
Page 130
The real achievement of a Copernicus 26e
Page 130
There are problems I never tackle 11e
Page 130
There are remarks that sow, 89e
Page 130
The reason I cannot understand Shakespeare98e
Page Break 131
Page 131
There is a big difference between the effect 30e
Page 131
There is definitely a certain kinship 18e
Page 131
There is much that could be learned from 67e
Page 131
There is much that is excellent in Macaulay's 31e
Page 131
There is no greater distress to be felt 53e
Page 131
There is no more light in a genius 41e
Page 131
There is no religious denomination 3e
Page 131
There is something Jewish in Rousseau's 17e
Page 131
The remark by Jucundus in 'The Lost Laugh'54e
Page 131
"The repeat is necessary" In what respect 59e
Page 131
There really are cases in which one has the sense 90e
Page 131
The revolutionary will be the one who 51e
Page 131
The Sabbath is not simply a time to rest, 91e
Page 131
The saint is the only Jewish "genius" 16e
Page 131
The story of Peter Schlemihl 21e
Page 131
The spring that flows quietly & clearly 35e
Page 131
The strength of the musical thinking 27e
Page 131
The theme interacts with language 60e
Page 131
The truly apocalyptic view of the world 64e
Page 131
The truth can be spoken only by someone 41e
Page 131
The use of the word "fate". Our 69e
Page 131
The valleys of foolishness have more grass 92e
Page 131
The various plants & their human 27e
Page 131
The way whole periods are incapable of freeing 91e
Page 131
The way you use the word "God" 58e
Page 131
The whole Earth cannot be in greater distress52e
Page 131
The works of the great masters 23e
Page 131
They have less style than the first speech 43e
Page 131
Think about how it was said of Labor's playing 71e
Page 131
"Thinking is difficult." (Ward)85e
Page 131
Think of the demeanour of someone who draws 58e
Page 131
This is how philosophers should salute 91e
Page 131
This musical phrase is a gesture 83e
Page 131
Thoughts at peace 50e
Page 131
Thoughts rise to the surface slowly, 72e
Page 131
Today the difference between a good & a poor 5e
Page 131
To go down into the depths you don't need 57e
Page 131
Tolstoy: the meaning (importance) of25e
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To piece together the landscape 90e
Page 131
To someone who is upset by this allegory 88e
Page 131
To treat well somebody who does not like you 13e
Page 131
Tradition is not something that 86e
Page 131
Troubles are like illnesses; 91e
Page 131
Try to be loved & not-admired 44e
Page 131
Two people who are laughing together, 88e
Page Break 132
Page 132
Understanding & explaining a musical 79e
Page 132
Unshakable faith. (E.g. in a promise.) 84e
Page 132
Virtually as there is such a thing as a deep 48e
Page 132
Wagner's motifs might be called 47e
Page 132
Wanting to think is one thing,50e
Page 132
We are engaged in a struggle with 13e
Page 132
(We are involved here with the Kantian 13e
Page 132
We are only going to set you straight 44e
Page 132
We are struggling with language 13e
Page 132
We keep hearing the remark that philosophy 22e
Page 132
Well, of what do we say "it is in his power"? 87e
Page 132
Well, there is no mention of punishment here, 92e
Page 132
We must not forget: even our more refined, 83e
Page 132
We say: "You understand this expression 50e
Page 132
What a curious attitude scientists 46e
Page 132
"What a sensible man knows is hard 20e
Page 132
What does it consist in: following a musical 58e
Page 132
What Eddington says about the 'direction 25e
Page 132
What I am resisting is the concept 45e
Page 132
What I am writing here may be feeble 75e
Page 132
What is Good is Divine too 5e
Page 132
What is it like when people do not have 95e
Page 132
What is important about depicting anomalies82e
Page 132
What is intriguing about a dream, 78e
Page 132
What is insidious about the causal 45e
Page 132
What is lacking in Mendelssohn's music? 40e
Page 132
What is pretty cannot be beautiful 48e
Page 132
What I want to say then is: 54e
Page 132
What Renan calls the bon sens précoce 8e
Page 132
What's ragged should be left 51e
Page 132
What you are taking for a gift is a problem 49e
Page 132
What you have achieved cannot mean 15e
Page 132
When he was old Charlemagne tried 85e
Page 132
When I came home I expected a surprise 52e
Page 132
When I have had a picture suitably framed 17e
Page 132
When I imagine a piece of music, 32e
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When people have died we see their life 53e
Page 132
When philosophizing you have to descend 74e
Page 132
When you bump against the limits 11e
Page 132
Where genius wears thin skill may 49e
Page 132
Where others go on ahead, I remain 75e
Page 132
Where there is bad management in the state, 72e
Page 132
Why is the soul moved by idle thoughts 41e
Page Break 133
Page 133
Why shouldn't I apply words in opposition 50e
Page 133
Why shouldn't someone become desperately 92e
Page 133
"Wisdom is grey." Life on the other hand 71e
Page 133
Wisdom is something cold, & to that extent 64e
Page 133
Within all great art there is a WILD 43e
Page 133
With my full philosophical rucksack 4e
Page 133
With thinking too there is a time for 33e
Page 133
Words are deeds. Only someone very unhappy 53e
Page 133
Work on philosophy--like work in 24e
Page 133
Writing the right style means, 44e
Page 133
"Yes, that's how it is," you say, 30e
Page 133
You can as it were restore an old style 68e
Page 133
You cannot draw the seed up out 48e
Page 133
You cannot judge yourself, if you are not 99e
Page 133
You cannot lead people to the good 5e
Page 133
You cannot write more truly about yourself 38e
Page 133
You can't be reluctant to give up your lie 44e
Page 133
You can't construct clouds 48e
Page 133
You could attach prices to ideas 60e
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You get tragedy where the tree 3e
Page 133
You must accept the faults in your own 86e
Page 133
You must indeed say only what is 46e
Page 133
You must say something new & yet 45e
FOOTNOTES
Page x
†1 But see Note by Translator.
Page xii
†* The Wittgenstein Archive of the University of Bergen is producing a machine-readable version of the
complete philosophical remains of Wittgenstein.
Page xii
†** Mr Pichler published a list of the sources of Vermischte Bemerkungen in Spring 1991: Alois Pichler,
"Ludwig Wittgenstein, Vermischte Bemerkungen: Liste der Manuskriptquellen. Ludwig Wittgenstein, Culture and
Value: A List of Source Manuscripts", Skriftserie fra Wittgensteinarkivet ved Universitetet i Bergen 1 (1991)
Page xiv
†* Georg Henrik von Wright
Page 6
†a as
Page 6
†b wonderful
Page 7
†a activity
Page 7
†b function
Page 7
†c from its
Page 7
†d contemplating it from above from its flight
Page 7
†e contemplating it from its flight
Page 8
†a have no need
Page 8
†b can
Page 8
†c have
Page 8
†d us
Page 8
†e the ones
Page 8
†f its spirit
Page 8
†g the current of the
Page 8
†h our day's
Page 8
†i alien and uncongenial
Page 8
†j he believed that--
Page 8
†k were architecture & not
Page 9
†a they are as it were
Page 10
†a picks up one stone after another
Page 10
†b those who
Page 10
†c is noticed only by those
Page 10
†d because it immediately | at once putrefies
Page 11
†a its foreground
Page 12
†a described
Page 12
†b presentiment
Page 12
†c outcome
Page 12
†d world ether
Page 12
†e become nothing splendid
Page 12
†f a
Page 12
†g these are the people to whom
Page 14
†a would like
Page 14
†b wear a quite different character
Page 14
†c <stylized> types.
Page 15
†a The whole rhythm of the poem...
Page 15
†b be
Page 18
†a ... could be called a matter of temperament & a much larger proportion of disagreements rest on this than
may appear.
Page 18
†b swelling
Page 18
†c & nobody wants to speak of a disease as though it had the same rights as healthy bodily processes (even
painful ones).
Page 20
†a according to
Page 20
†b two different
Page 20
†c made use of
Page 20
†d And with the beginning I am now thinking of for my book, the description of nature with which it is to
start?
Page 21
†a read
Page 21
†b holds
Page 21
†c of the comparison
Page 21
†d will not have the
Page 22
†a has to be so
Page 22
†b transcendent
Page 22
†c strange
Page 22
†d could <get>
Page 23
†a themselves
Page 23
†b suns
Page 23
†c But with the Jews it is just the same.
Page 23
†d them
Page 23
†e they are
Page 23
†f thinkers
Page 24
†a a <work>
Page 24
†b while seeing <to it>
Page 24
†c then
Page 25
†a is no bigger
Page 25
†b --teacher of philosophy is like a person, someone, who gives his pupil foods, not |
Page 25
†c accessible
Page 26
†a draw his attention to this
Page 26
†b know
Page 27
†a remains
Page 27
†b but what he means to us is all the same only his personality
Page 27
†c already now
Page 28
†a one of the narrow def<initions [[sic > ?]] o<f> t<the>
Page 30
†a read
Page 30
†b the
Page 30
†c our claims
Page 30
†d regard
Page 30
†e rather than
Page 30
†f would have to
Page 30
†g In this <namely> lies
Page 30
†h our philosophy
Page 31
†a look up
Page 31
†b , to master it,
Page 31
†c but the question constantly arises: should this game be played at all now & what is the right game?
Page 32
†a of all <opinions>
Page 33
†a a hindrance which nullifies the movement by friction
Page 33
†b has the same effect
Page 33
†c but a weight attached to one's foot, which will not allow us to walk far.
Page 33
†d say from time to time
Page 33
†e ;|:
Page 33
†f as it is
Page 33
†g Similarly,
Page 33
†h flowers, berries or herbs
Page 33
†i although
Page 34
†a :
Page 35
†a both he and I
Page 35
†b in
Page 35
†c additional
Page 35
†d that might retrospectively be thought of,
Page 35
†e Our experience was that language could continually make new, & impossible, demands; & in this way
every explanation was frustrated.
Page 35
†f could <show our earlier explanation to be unworkable>--frustrating every attempt at explanation.--
Page 35
†g transparently
Page 36
†a in me
Page 36
†b was
Page 36
†c as that of someone <when> he <struggle>s <in vain>
Page 36
†d occurred
Page 38
†a are permitted <only> to
Page 39
†a the
Page 39
†b are not blood relations but stand to each other
Page 39
†c written against Beethoven's & because of this
Page 40
†a he who gets there last.
Page 40
†b arrives last.
Page 40
†c may be similar to eating from the tree of knowledge.
Page 41
†a air
Page 41
†b by
Page 41
†c :
Page 42
†a of understanding
Page 42
†b have no bottom.
Page 43
†a its depth
Page 43
†b :
Page 43
†c even
Page 43
†d on the occasions when
Page 43
†e is making
Page 44
†a withdraw <an expression from the language>
Page 44
†b straight
Page 44
†c Driving
Page 44
†d , that is if your carriage stands | stood on the rails crookedly. You can drive then by yourself.
Page 45
†a Page
45
†b had to
Page 45
†c manners
Page 45
†d had
Page 46
†a break them open
Page 46
†b ,
Page 47
†a about numbers
Page 47
†b the regularities of a sort of crystal
Page 48
†a ... scientists do not express (hard) work, but resting on laurels.
Page 48
†b ... do not express hard work, but are the expression of resting on laurels.
Page 48
†c much
Page 48
†d You give it warmth...
Page 48
†e take hold of
Page 49
†a became
Page 49
†b any
Page 49
†c without any idea that, & how, it could be used as a motor
Page 49
†d skill
Page 49
†e look <through>
Page 50
†a the scientist
Page 50
†b experience
Page 50
†c ,
Page 50
†c [[sic, d?]] Someone says to us:
Page 50
†d [[sic, e?]] with the meaning known to you
Page 50
†f Well then, I too am using it with the meaning that you know."
Page 50
†g , which the word takes with it & carries over into whatever | every kind of application.
Page 50
†h takes with it
Page 50
†i so too in our everyday understanding by madness.
Page 51
†a paints
Page 52
†a outstanding
Page 54
†a --<if things are going well for him now>--
Page 55
†a a nuance of stress
Page 55
†b the new way of thinking
Page 55
†c , the old problems are put on one side along with the old garment
Page 56
†a of his total output
Page 57
†a seem to you like narrow borders.
Page 57
†b indeed for this you need not even leave your most immediate & familiar surroundings I need not for this
<leave> your most immediate...
Page 57
†c tucked away
Page 57
†d .
Page 57
†e a mathematician's.
Page 57
†f the forms in which
Page 58
†a ,--
Page 58
†b <with feeling>
Page 58
†c of that
Page 58
†d the outward manifestation is?.
Page 58
†e ask yourself, what the expression of that is.
Page 59
†a while
Page 60
†a lie
Page 60
†b Experience has incorporated the picture into our thinking.
Page 60
†c a change of situation
Page 60
†d change,
Page 60
†e <we can> hardly | <only> with difficulty decide <on this>.
Page 60
†f bits & pieces
Page 61
†a Someone might <fantasize a flying machine, without being precise about its details. Outwardly> he
<may>
Page 62
†a <a> vision
Page 62
†b to unlock
Page 62
†c <these phenom>ena
Page 63
†a these pictures
Page 63
†b rather: that's how it has changed.
Page 64
†a :
Page 65
†a then he is looking at?
Page 65
†b they
Page 65
†c affected
Page 66
†a that
Page 66
†b would like to defend
Page 66
†c appearing
Page 66
†d attitude
Page 66
†e <esteem> you <& at the same time love> you <can make this behaviour easy> for you you
Page 66
†f , & if | they had by accident become a pair of lines that rhymed (with each other),...
Page 66
†g , & it by accident they had become two rhyming lines,...
Page 66
†h , & it turned out by accident to read as two rhyming lines,...
Page 67
†d [[sic a?]] bad
Page 67
†a [[sic b?]] not impossible
Page 67
†b [[sic c?]] simply
Page 67
†c [[sic d?]] <by> their
Page 68
†a Taste rectifies. Giving birth is not its affair.
Page 68
†b <does> not <need>
Page 68
†c Even the
Page 68
†d am not able to
Page 68
†e newer
Page 68
†f <an> interpretation
Page 68
†g in a tempo that
Page 69
†a in reality
Page 70
†a observations
Page 71
†a through
Page 71
†b becomes
Page 71
†c would be
Page 71
†d change women's & men's dress
Page 71
†e the
Page 71
†f We should like to call music a 'language'; & no doubt this does apply to some music--& to some no doubt
not.
Page 72
†a is
Page 72
†b very close
Page 72
†c slippery
Page 72
†d one or the other
Page 73
†a a system of reference
Page 73
†b the <rescue>-instrument
Page 73
†c <but> certainly not
Page 73
†d certainly not however
Page 74
†a transitions between thoughts
Page 74
†b forms
Page 74
†c because it is a gesture that endures
Page 75
†a with
Page 75
†b better
Page 76
†a understandable
Page 77
†a my copious
Page 78
†a discover
Page 78
†b tale
Page 78
†c ask
Page 78
†d exclaim
Page 79
†a idea pregnant with further developments
Page 79
†b banalities
Page 79
†c recollections
Page 80
†a :
Page 80
†b a distinction like that between seeing, hearing & smelling.
Page 80
†c understanding
Page 80
†d at another time
Page 81
†a <an> explanation
Page 82
†a his body
Page 82
†b does not have the right to <disregard this perception>
Page 82
†c delusive
Page 82
†d it
Page 83
†a <do you> have a right to
Page 84
†a by
Page 84
†b footing
Page 84
†c seduced
Page 85
†a than fictional concepts,
Page 85
†b distinctly
Page 85
†c you can <not unravel a tangle, then the most sensible thing that> you can <do>
Page 86
†a the freedom
Page 86
†b someone
Page 86
†c learn
Page 86
†d can choose the ancestors from whom you would like to be descended.
Page 87
†a for a new sowing.
Page 88
†a their
Page 88
†b that
Page 88
†c used
Page 89
†a constantly
Page 89
†b are in a process of perpetual change.
Page 90
†a Piecing together
Page 90
†b Piecing together <the landscape of these conceptual relationships> out of their individual fragments, as
language reveals them to us,
Page 90
†c S<omeone>
Page 90
†d he
Page 90
†e one
Page 90
†f as though one saw a dream image quite clearly before one's mind's eye, but could not describe it in such a
way that someone else sees it too.
Page 91
†a intrigue
Page 91
†b through the questions into the open, sometimes...
Page 92
†a derive <only despair or incredulity> from this doctrine.
Page 93
†a find
Page 93
†b becomes
Page 95
†a <There is> nothing unheard of in the idea
Page 95
†b nothing unheard of
Page 95
†c <circumstances> of this nature
Page 96
†a Major & minor in Schubert
Page 96
†b only a cold admiration is possible.
Page 97
†a <an> attitude
Page 97
†b particular
Page 97
†c & does not know how it can be expressed.
Page 98
†a response to the itch
Page 98
†b continue
Page 99
†a comes from the good
Page 100
†a The rhythms of the German are only roughly suggested; and no attempt has been made to reproduce the
rhymes in the original.
TRANSLATOR'S NOTES
Page xvi
†i A noun, which in conventional German orthography would begin with an upper case "P".
Page xvi
†ii More conventionally "jeder so und so vielte".
Page xvii
†i Such messages have in the main been rendered somewhat differently in the translation. Footnotes (like this
one) numbered with small Roman numerals have been added by the translator.
Page 13
†i In German the irony is intensified by a play on the words geduldig and duldend.
Page 15
†i This is something like a crossword puzzle. Each space is occupied by a separate syllable. These are joined
together to form a meaningful passage by making transpositions according to the knight's move (= Rösselsprung) in
chess.
Page 16
†i The phrase in quotation marks is adapted from the first line of Goethe's poem, "Vanitas! Vanitatis vanitas"
which in its turn is the title of the first chapter of Max Stirner's Der Einzige und sein Eigentum. Wittgenstein is here
probably alluding more directly to Stirner than to Goethe, the sense of whose poem hardly fits the present context.
The translator is indebted to the late Rush Rhees for drawing his attention to these allusions.
Page 19
†i Literally: "which I do not know".
Page 20
†i "Wissenschaft" and "wissenschaftlich" in this sentence have been translated as "science" and "scientific".
However, Wittgenstein probably did not mean this in the sense of natural science (which is the most common
English usage) but, according to natural German usage, was thinking of intellectual questions in a much more
general way.
Page 22
†i The translator is grateful for this rendering to Mr S. Ellis of the Institute of Dialect and Folk Life Studies at
the University of Leeds.
Page 32
†i The variant Wittgenstein wrote at this point consists of putting the clause expressing indirect speech into
the subjunctive--which is grammatically correct in German but not, at least in this context, in English. One might
translate the variant by rewriting the sentence: "People would live under an absolute, palpable tyranny, yet without
being able to say they were not free."
Page 34
†i The alternative versions "es" and "sie" have, in this context, no grammatical equivalent in English.
Page 36
†i Wittgenstein's variant here consists in supplying a capital A for the noun form "Anderes". There is no
English counterpart for this grammatical move.
Page 36
†ii Goethe, Faust, Part I (In the Study).
Page 38
†i 1 Corinthians, 3.
Page 39
†i Lars Hertzberg has pointed out to me that Wittgenstein misquotes here. The last line of the stanza reads:
"For the Gods see everywhere."
Page 42
†i New Testament
Page 58
†i Wittgenstein's alternative reading "wird" for "ist" does not correspond to any meaningful distinction in
English.
Page 58
†ii The grammatical variant in Wittgenstein's German text has no meaningful counterpart in English.
Page 59
†i I think "field" here should be understood in the sense of a "field of force", as in physics.
Page 69
†i This could also be translated by the weaker "I actually do not want to be imitated".
Page 70
†i See translator's note on p. 46.
Page 71
†i The variant Wittgenstein gives here is simply a German version of the English phrase he originally used.
Page 72
†i Wittgenstein's alternative versions, "in dem" or "in welchem", do not correspond to any distinction in
English.
Page 74
†i This could also be rendered as "the former chaos".
Page 76
†i The German text plays on the two congnate verbs "anregt" (= "stimulates") "aufregt" (= exasperates).
Page 81
†i This is the idiomatic phrase corresponding to the German. However, the reader's attention is drawn to two
points. [1] "Cussedness" here translates the German "Tücke", which in other contexts sounds rather stronger than
would "cussedness" in English. In this passage, therefore, outside the context of the particular idiom it has been
rendered as "malice". [2] The plural "things" corresponds to the German singular "des Objekts". This may be
important insofar as Wittgenstein interprets the idiomatic phrase under discussion as implying a demonic
intervention in particular cases; whereas, in English at least, the phrase is quite compatible with the conception
Wittgenstein here develops in opposition to such an implication.
Page 84
†i The justification for translating the variants thus is the slenderest imaginable...
Page 90
†i I am reading "U.u." as an idiosyncratic version of "U.U." (= "Unter Umständen"). It has been suggested to
me that it could stand for "Und umgekehrt". However, (a) "And vice versa" seems to make little sense applied to the
preceding sentence; and (b) "U.u." is not a recognized abbreviation for "Und umgekehrt"--in Germany at least; I am
not sure about Austria.
Page 93
†i In the German 'Rast' (= 'rest') and 'Hast' (= 'haste').
Page 95
†i Literally: "No human being must must"
Page 97
†i "Craving" is too strong for "Verlangen" in this context; "desire" is too weak. The vulgarism "hankering"
strikes me as just right.
ZETTEL
Titlepage
LUDWIG WITTGENSTEIN
ZETTEL
Second Edition
Edited by
G. E. M. ANSCOMBE
and G. H. von WRIGHT
Translated by
G. E. M. ANSCOMBE
BASIL BLACKWELL
OXFORD
Page Break ii
Copyright page
Page ii
Copyright © Blackwell Publishers Ltd 1967, 1981
First edition 1967
First published in paperback 1975
Second edition 1981
Reprinted 1988, 1990, 1998
Blackwell Publishers Ltd
108 Cowley Road
Oxford OX4 1JF, UK
All rights reserved. Except for the quotation of short passages for the purposes of criticism and review, no part of
this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher.
Except in the United States of America, this book is sold subject to the condition that it shall not, by way of trade or
otherwise, be lent, resold, hired out, or otherwise circulated without the publisher's prior consent in any form of
binding or cover other than that in which it is published and without a similar condition including this condition
being imposed on the subsequent purchaser.
ISBN 0 631 12813 1
ISBN 0 631 12823 9 (pbk)
Printed in Great Britain by Athenaeum Press Ltd,
Gateshead, Tyne & Wear
Page Break iii
EDITORS' PREFACE
Page iii
WE publish here a collection of fragments made by Wittgenstein himself and left by him in a box-file. They were for
the most part cut from extensive typescripts of his, other copies of which still exist. Some few were cut from
typescripts which we have not been able to trace and which it is likely that he destroyed but for the bits that he put in
the box. Others again were in manuscript, apparently written to add to the remarks on a particular matter preserved
in the box.
Page iii
The earliest time of composition of any of these fragments was, so far as we can judge, 1929. The date at
which the latest datable fragment was written was August 1948. By far the greatest number came from typescripts
which were dictated from 1945-1948.
Page iii
Often fragments on the same topic were clipped together; but there were also a large number lying loose in
the box. Some years ago Peter Geach made an arrangement of this material, keeping together what were in single
bundles, and otherwise fitting the pieces as well as he could according to subject matter. This arrangement we have
retained with a very few alterations. We hereby express our indebtedness to him for the work that he did, which was
laborious and exacting. Though the arrangement is not the kind of arrangement that Wittgenstein made of his
'remarks', we found that it made a very instructive and readable compilation.
Page iii
We were naturally at first rather puzzled to account for this box. Were its contents an accidental collection of
left-overs? Was it a receptacle for random deposits of casual scraps of writing? Should the large works which were
some of its sources be published and it be left on one side? One of these works was one of two total rearrangements
of Investigations and other material; another was an extremely long and repetitive early typescript presenting great
editorial problems. Another--though there are only a few cuttings from this source--has already been published
under the title Philosophische Bemerkungen.
Page Break iv
Page iv
After most of the typed fragments had been traced to their sources, comparison of them with their original
forms, together with certain physical features, shewed clearly that Wittgenstein did not merely keep these fragments,
but worked on them, altered and polished them in their cut-up condition. This suggested that the addition of separate
MS pieces to the box was calculated; the whole collection had a quite different character from the various bundles of
more or less 'stray' bits of writing which were also among his Nachlass.
Page iv
We therefore came to the conclusion that this box contained remarks which Wittgenstein regarded as
particularly useful and intended to weave into finished work if places for them should appear. Now we know that his
method of composition was in part to make an arrangement of such short, almost independent pieces as, in the
enormous quantity that he wrote, he was fairly satisfied with.
Page iv
Not every one of these remarks is quite of this kind; very occasionally the cutting was grammatically
incomplete, so that it looked as if it were preserved only for the sake of the idea or expression to be seen on it. Here
we have supplied the missing words from the original typescript where we could; once we had to supply the last few
words ourselves. Very rarely, there is a pronoun or the like demanding a previous reference to explain it; once we
supplied the preceding remark from the original typescript; once or twice we have put in appropriate words. Square
brackets are the editors'; e.g. when Wittgenstein wrote a mere note on his text in the margin we have printed it,
prefaced by the words 'Marginal note', between square brackets. Elsewhere words between square brackets have
been supplied by us.
G. E. M. ANSCOMBE
G. H. VON WRIGHT
Page Break v
TRANSLATOR'S NOTE
Page v
I am indebted for the avoidance of many errors in the translation and for various helpful suggestions to Dr. L.
Labowsky, Professor G. H. von Wright, Mr. R. Rhees and Professor P. T. Geach. Only I am to blame for any errors that
remain.
G. E. M. A.
PREFACE TO THE SECOND EDITION
Page v
A CLOSE comparison of the printed text with the original cuttings has shewn that the first edition was marked by
many inaccuracies, and even some misunderstandings of the original text. (See, e.g., § 671.) They have been
corrected. The editors are grateful to Mr. Heikki Nyman for his laborious and conscientious work towards the
production of a printed text that is faithful to the sources.
G. E. M. ANSCOMBE
G. H. VON WRIGHT
Page Break vi
Page Break 1
[Zettel: Main Body]
Page 1
1. William James: The thought is already complete at the beginning of the sentence. How can one know that?--But
the intention of uttering the thought may already exist before the first word has been said. For if you ask someone:
"Do you know what you mean to say?" he will often say yes.
Page 1
2. I tell someone: "I'm going to whistle you the theme... ", it is my intention to whistle it, and I already know what I
am going to whistle.
It is my intention to whistle this theme: have I then already, in some sense, whistled it in thought?
Page 1
3. "I'm not just saying this, I mean something by it."--Should one thereupon ask "What?" then one gets another
sentence in reply. Or perhaps one cannot ask that, as the sentence meant, e.g. "I'm not just saying this, it moves me
too".
Page 1
4. (The question "What do I mean by that?" is one of the most misleading of expressions. In most cases one might
answer: "Nothing at all--I say...".)
Page 1
5. Can I then not use words to mean what I like? Look at the door of your room, utter a sequence of random sounds
and mean by them a description of that door.
Page 1
6. "Say 'a b c d' and mean: the weather is fine."--Should I say, then, that the utterance of a sentence in a familiar
language is a quite different experience from the utterance of sounds which are not familiar to us as a sentence? So if
I learnt the language in which "abcd" meant that--should I come bit by bit to have the familiar experience when I
pronounced the letters? Yes and no.--A major difference between the two cases is that in the first one I can't move.
It is as if one of my joints were in splints, and I not yet familiar with the possible movements, so that I as it were
keep on stumbling.
Page 1
7. If I have two friends with the same name and am writing one of them a letter, what does the fact that I am not
writing it to the other consist in? In the content? But that might fit either. (I haven't yet written the address.) Well, the
connexion might be in the antecedents. But in that case it may also be in what follows the writing. If someone asks
me "Which of the two are you
Page Break 2
writing to?" and I answer him, do I infer the answer from the antecedents? Don't I give it almost as I say "I have
toothache"?--Could I be in doubt which of the two I was writing to? And what is a case of such a doubt
like?--Indeed, couldn't there also be an illusion of this kind: I believe I am writing to one of them when in fact I am
writing to the other? And what would such a case of illusion look like?
Page 2
8. (One sometimes says: "What was I going to look for in this drawer?--Oh yes, the photograph!" Once this has
occurred to us, we recall the connexion between our actions and what was happening before. But the following is
also possible: I open the drawer and routle around in it; at last I come to and ask myself "Why am I rummaging in
this drawer?" And then the answer comes, "I want to look at the photograph of...". "I want to", not "I wanted to".
Opening the drawer, etc. happened so to speak automatically and got interpreted subsequently.)
Page 2
9. "That remark of mine was aimed at him." If I hear this I can imagine a situation and a history that fit it. I could
present it on the stage, project myself into the state of mind in which I 'aim at him'.--But how is this state of mind to
be described, i.e. to be identified?--I think myself into the situation, assume a certain expression and tone, etc.. What
connects my words with him? The situation and my thoughts. And my thoughts in just the same way as things I say
out loud.
Page 2
10. Suppose I wanted to replace all the words of my language at once by other ones; how could I tell the place
where one of the new words belongs? Is it images that keep the places of the words?
Page 2
11. I am inclined to say: I 'point' in different senses to this body, to its shape, to its colour, etc..--What does that
mean?
What does it mean to say I 'hear' in a different sense the piano, its sound, the piece, the player, his fluency? I
'marry', in one sense a woman, in another her money.
Page 2
12. Here meaning gets imagined as a kind of mental pointing, indicating.
Page Break 3
Page 3
13. In some spiritualistic procedures it is essential to think of a particular person. And here we have the impression
that 'thinking of him' is as it were nailing him with my thought. Or it is as if I kept on thrusting at him in thought. For
the thoughts keep on swerving slightly away from him.
Page 3
14. "Suddenly I had to think of him." Say a picture of him suddenly floated before me. Did I know that it was a
picture of him, N.? I did not tell myself it was. What did its being of him consist in, then? Perhaps what I later said or
did.
Page 3
15. When Max says "The prince has a fatherly concern for the troops", he means Wallenstein.--Suppose someone
said: We don't know that he means Wallenstein: he might mean some other prince in this sentence.†1
Page 3
16. "Your meaning the piano-playing consisted in your thinking of the piano-playing."
"That you meant that man by the word 'you' in that letter consisted in this, that you were writing to him."
The mistake is to say that there is anything that meaning something consists in.
Page 3
17. "When I said that, I only wanted to drop a hint."--How can I know that I said it only to drop a hint? Well, the
words "When I said it, etc." describe a particular intelligible situation. What is that situation like? In order to describe
it, I must describe a context.
Page 3
18. How does he enter into these proceedings:
I thrust at him,
I spoke to him,
I called him,
I spoke of him
I imagined him,
I esteem him?
Page 3
19. It is wrong to say: I meant him by looking at him. "Meaning" does not stand for an activity which wholly or
partly consists in the 'utterances' [outward expressions] of meaning.
Page Break 4
Page 4
20. Hence it would be stupid to call meaning a 'mental activity', because that would encourage a false picture of the
function of the word.
Page 4
21. I say "Come here" and point towards A. B, who is standing by him, takes a step towards me. I say "No; A is to
come". Will that be taken as a communication about my mental state? Certainly not.--Yet couldn't inferences be
made from it about processes going on in me when I pronounced the summons "Come here"?
But what kind of processes? Mightn't it be conjectured that I looked at A as I gave the order? That I directed
my thoughts towards him? But perhaps I don't know B at all; I am in touch only with A. In that case the man who
guessed at my mental processes might have been quite wrong, but all the same would have understood that I meant
A and not B.
Page 4
22. I point with my hand and say "Come here". A asks "Did you mean me?" I say: "No, B".--What went on when I
meant B (since my pointing left it in doubt which I meant)?--I said those words, made that gesture. Must still more
have taken place, in order for the language-game to take place? But didn't I already know, while I was pointing,
whom I meant? Know? Of course--going by the usual criteria for knowledge.
Page 4
23. "What I wanted to get at in my account was....". This was the objective I had before me. In my mind I could see
the passage in the book, that I was aiming at.
Describing an intention means describing what went on from a particular point of view, with a particular
purpose. I paint a particular portrait of what went on.
Page 4
24. Instead of "I meant him" one may also say "I was speaking of him". And how does one do that, how does one
speak of him in speaking those words? Why does it sound wrong to say "I spoke of him by pointing to him as I
spoke those words"?
"To mean him" means, say, "to talk of him". Not: to point to him. And if I talk of him, of course there is a
connexion between my talk and him, but this connexion resides in the application of the talk, not in an act of
pointing. Pointing is itself only a sign,
Page Break 5
and in the language-game it may direct the application of the sentence, and so shew what is meant.
Page 5
25. If I say "I saw a chair in this room", I can mostly recall the particular visual impression only very roughly, nor
does it have any importance in most cases. The use that is made of the sentence bypasses this particular feature.
Now is it also like that when I say "I meant N"? Does this sentence bypass the particular features of the proceeding
in the same way?
Page 5
26. When I make a remark with an allusion to N., I may let this appear--given particular circumstances--in my
glance, my expression, etc.
Page 5
You can shew that you understand the expression "to allude to N." by describing examples of alluding. What
will you describe? First of all, circumstances. Then what someone says. Perhaps his glance etc. as well. Then what
someone making an allusion is trying to do.
Page 5
And if I go on to tell someone the feelings, images etc. which I had while I was making the remark, these
may fill out the typical picture of an allusion (or one such picture). But it doesn't follow that the expression "alluding
to N" means: behaving like this, feeling this, imagining this, etc. And here some will say: "Of course not! We knew
that all along. A red thread must run through all these phenomena. It is, so to speak, entangled with them and so it is
difficult to pick out."--And that is not true either.
Page 5
But it would also be wrong to say that "alluding" stands for a family of mental and other processes.--For one
can well ask "Which was your allusion to N.?" "How did you give others to understand that you meant N.?"; but
not: "How did you mean this utterance as an allusion to N.?".
Page 5
"I alluded to him in my talk."--"When you said what?"--"I was alluding to him when I spoke of a man
who...."
Page 5
"I was alluding to him" means roughly: "I wanted someone to think of him at these words. But "I wanted" is
not the description of a state of mind. Neither is "understanding that N was meant" such a description. [Marginal
note: But one does ask: "In which remark did you allude to him?", "Which was the remark where you meant him?"]
Page Break 6
Page 6
27. When the situation is ambiguous, is it doubtful whether I mean him? When I say that I did or did not mean him,
I don't go by the situation. And if not by the situation, what do I go by? Apparently not by anything at all. For I do
remember the situation, but I interpret it. I may, e.g., now imitate my sidelong glance at him; but meaning him
appears as a quite impalpable fine atmosphere of the speaking and acting. (A fishy picture!)
Page 6
28. In the course of a conversation I want to point at something; I have already begun a pointing movement, but I
don't complete it. Later on I say "I was going to point to it then. I still remember quite clearly that I was already
raising my finger." In the current of those events, thoughts, and experiences, that was the beginning of a pointing
gesture.
And if I completed the gesture and said: "He is lying over there", this would not be a case of pointing unless
these words belonged to a language.
Page 6
29. "You moved your hand--did you mean anything by the movement?--I thought you meant me to come to you."
So he may have meant something or nothing. And if the former, then was it the movement that he meant--or
something else? When he used an expression, did he mean something other than the expression or did he simply
mean the expression?
Page 6
30. Could one also reply: "I meant something by this movement, which I can only express by this movement"?
(Music, musical thought.)
Page 6
31. "Of course I was thinking of him: I saw him in my mind's eye!"--But I did not recognize him by his appearance.
Page 6
32. Imagine someone you know.--Now say who it was.--Sometimes the picture comes first and the name afterwards.
But do I guess the name by the picture's likeness to the man?--And if the name only comes after the picture--was the
idea of that man there as soon as the picture was, or was it only complete when I had the name? I did not infer the
name from the picture; and just for that reason I can say that the idea of him was already there once the picture was
there.
Page Break 7
Page 7
33. It is as if one experienced a tendency, a readiness (James). And why shouldn't I call it that? (And some would
explain what happens here by innervations of muscles, dispositions to move, or even images of movement.) Only
there is no need to see the experience of a tendency as a not quite complete experience.
It often strikes us as if in grasping meaning the mind made small rudimentary movements, like someone
irresolute who does not know which way to go--i.e. it tentatively reviews the field of possible applications.
Page 7
34. Imagine humans who from childhood up scribble very fast as they talk: as it were illustrating what they say.
Must I assume that if someone draws or describes or imitates something from memory, he reads off his
representation from something or other?!--What supports this?
Page 7
35. Guessing thoughts. There are playing-cards on the table. I want the other man to touch one. I shut my eyes and
think of one of the cards; the other is supposed to guess which I mean.--He makes himself think of a card, and at the
same time wills to hit on the one I mean. He touches the card and I say "Yes, that was it"; or else it wasn't. A variant
of this game would be for me to look at a particular card, but so that the other can't see the direction of my gaze; he
has to guess the card I am looking at. It is important that this is a variant of the other game. Here it may be important
how I think of the card, because it might turn out that the reliability of the guessing depended on that. But if I say in
ordinary life: "I thought of so-and-so" I am not asked "How did you think of him?"
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36. One would like to ask: "Would someone who could look into your mind have been able to see that you meant to
say that?"
Suppose I had written my intention down on a slip of paper, then someone else could have read it there. And
can I imagine that he might in some way have found it out more surely than that? Certainly not.
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37. (At the beginning of a piece of music it says = 88, written there by the composer. But in order to play it right
nowadays it must be played = 94: which is the tempo intended by the composer?)
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38. Interrupt a man in quite unpremeditated and fluent talk. Then ask him what he was going to say; and in many
cases he will be able to continue the sentence he had begun.--"For that, what he was going to say must already have
swum into view before his mind."--Is not that phenomenon perhaps the ground of our saying that the continuation
had swum into his mental view?
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39. But is it not peculiar that there is such a thing as this reaction, this confession of intention? Is it not an extremely
remarkable instrument of language? What is really remarkable about it? Well--it is difficult to imagine how a human
being learns this use of words. It is so very subtle.
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40. But is it really subtler than that of the phrase "I imagined him", for example? Yes, every such use of language is
remarkable, peculiar, if one is adjusted only to consider the description of physical objects.
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41. If I say "I was then going to do such-and-such", and if this statement is based on the thoughts, images etc. which
I remember, then someone else to whom I tell only these thoughts, images etc. ought to be able to infer with as great
certainty as mine that I was then going to do such-and-such.--But often he could not do so. Indeed, were I myself to
infer my intention from the evidence, other people would be right to say that this conclusion was very uncertain.
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42. And how does [a child] learn to use the expression "I was just on the point of throwing then"? And how do we
tell that he was then really in that state of mind then which I call "being on the point of"?
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43. Suppose a human being never learnt the expression "I was on the point of" or "I was just going to..." and could
not learn their use? A man can after all think a good deal without thinking that. He can master a great field of
language-games without mastering this one.
But isn't it odd that among all the diversity of mankind we do not encounter defective humans of this sort?
Or are such people just to be found among the feeble-minded, only it is not closely enough observed which uses of
language such people are capable of and which not?
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44. "I had the intention of ..." does not express the memory of an experience. (Any more than "I was on the point of
...".)
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45. Intention is neither an emotion, a mood, nor yet a sensation or image. It is not a state of consciousness. It does
not have genuine duration.
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46. "I have the intention of going away tomorrow."--When have you that intention? The whole time; or
intermittently?
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47. Look in the drawer where you think you'll find it. The drawer is empty.--I believe you were looking for it among
the sensations.
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Consider what it would really mean "to have an intention intermittently". It would mean: to have the
intention, to abandon it, to resume it, and so on.
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48. In what circumstances does one say "This appliance is a brake, but it doesn't work"? That surely means: it does
not fulfil its purpose. What is it for it to have this purpose? It might also be said: "It was the intention that this
should work as a brake." Whose intention? Here intention as a state of mind entirely disappears from view.
Might it not even be imagined that several people had carried out an intention without any one of them
having it? In this way a government may have an intention that no man has.
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49. There might be a verb which meant: to formulate an intention in words or other signs, out loud or in one's
thoughts. This word would not mean the same as our "intend".
There might be a verb which meant: to act according to an intention; and neither would this word mean the
same as our "intend".
Yet another might mean: to brood over an intention; or to turn it over and over in one's head.
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50. One may disturb someone in thinking-but in intending?--Certainly in planning. Also in keeping to an intention,
that is in thinking or acting.
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51. Application of the imperative. Compare these orders:
Raise your arm.
Imagine...
Work... out in your head
Consider...
Concentrate your attention on...
See this figure as a cube
with these:
Intend...
Mean... by these words
Suspect that this is the case
Believe that it is so
Be of the firm conviction...
Remember that this happened
Doubt whether it has happened
Hope for his return.
Is this the difference, that the first are voluntary, the second involuntary mental movements? I may rather say that
the verbs of the second group do not stand for actions. (Compare with this the order: "Laugh heartily at this joke.")
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52. Can one order someone to understand a sentence? Why can't one tell someone: "Understand that!" Couldn't I
obey the order "Understand this Greek sentence" by learning Greek?--Similarly: one can say "Produce pain in
yourself", but not "Have pain". One says "Bring yourself into this condition" but not "Be in this condition".
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53. I expect an explosion any moment. I can't give my full attention to anything else; I look in a book, but without
reading. Asked why I seem distracted or tense, I say I am expecting the explosion any moment.--Now how was it:
did this sentence describe that behaviour? But then how is the process of expecting the explosion distinguished from
the process of expecting some quite different event, e.g. a particular signal? And how is the expectation of one signal
distinguished from the expectation of a slightly different one? Or was my behaviour only a side-effect of the real
expectation, and was that a special mental process? And was this process homogeneous or articulated like a
sentence (with an internal beginning and end)?--But how does
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the person in whom it goes on know which event the process is the expectation of? For he does not seem to be in
uncertainty about it. It is not as if he observed a mental or other condition and formed a conjecture about its cause.
He may well say: "I don't know whether it is only this expectation that makes me so uneasy today"; but he will not
say: "I don't know whether this state of mind, in which I now am, is the expectation of an explosion or of something
else."
The statement "I am expecting a bang at any moment" is an expression of expectation. This verbal reaction is
the movement of the pointer, which shows the object of expectation.
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54. It seems as if the expectation and the fact satisfying the expectation fitted together somehow. Now one would
like to describe an expectation and a fact which fit together, so as to see what this agreement consists in. Here one
thinks at once of the fitting of a solid into a corresponding hollow. But when one wants to describe these two one
sees that, to the extent that they fit, a single description holds for both. (On the other hand compare the meaning of:
"These trousers don't go with this jacket".)
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55. Like everything metaphysical the harmony between thought and reality is to be found in the grammar of the
language.
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56. Here my thought is: If someone could see the expectation itself--he would have to see what is being expected.
(But in such a way that it doesn't further require a method of projection, a method of comparison, in order to pass
from what he sees to the fact that is expected.)
But that's how it is: if you see the expression of expectation you see 'what is expected'.
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57. The idea that it takes finding to show what we were looking for, and fulfilment of a wish to show what we
wanted, means one is judging the process like the symptoms of expectation or search in someone else. I see him
uneasily pacing up and down his room; then someone comes in at the door and he relaxes and gives signs of
satisfaction. And I say "Obviously he was expecting this person".
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58. We say that the expression of expectation describes the expected fact and think of this as of an object or
complex which makes its appearance as fulfilment of the expectation.--But it is not the expected thing that is the
fulfilment, but rather: its coming about.
The mistake is deeply rooted in our language: we say "I expect him" and "I expect his arrival", and "I expect
he is coming".
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59. It is difficult for us to shake off this comparison: a man makes his appearance--an event makes its appearance. As
if an event even now stood in readiness before the door of reality and were then to make its appearance in
reality--like coming into a room.
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60. Reality is not a property still missing in what is expected and which accedes to it when one's expectation comes
about.--Nor is reality like the daylight that things need to acquire colour, when they are already there, as it were
colourless, in the dark.
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61. One may say of the bearer of a name that he does not exist; and of course that is not an activity, although one
may compare it with one and say: he must be there all the same, if he does not exist. (And this has certainly already
been written some time by a philosopher.)
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62. The shadowy anticipation of a fact consists in this: something is only going to happen, but we can think that it is
going to happen. Or, as it misleadingly goes: we can now think what (or of what) is only going to happen.
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63. Some will perhaps want to say "An expectation is a thought". And we need to remember that the process of
thinking may be very various.
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64. I whistle and someone asks me why I am so cheerful. I reply "I'm hoping N. will come today".--But while I
whistled I wasn't thinking of him. All the same, it would be wrong to say: I stopped hoping when I began to whistle.
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65. If I say "I am expecting...",--am I remarking that the situation, my actions, thoughts etc. are those of expectancy
of this event; or are the words: "I am expecting..." part of the process of expecting?
In certain circumstances these words will mean (will be replaceable by) "I believe such-and-such will occur".
Sometimes also: "Be prepared for this to happen...."
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66. Psychological--trivial--discussions about expectation, association etc. always pass over what is really noteworthy
and it is noticeable that they talk around, without touching, the punctum saliens.†1
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67. An expectation is embedded in a situation from which it takes its rise. The expectation of an explosion for
example, may arise from a situation in which an explosion is to be expected. The man who expects it had heard two
people whispering: "Tomorrow at ten o'clock the fuse will be lit". Then he thinks: perhaps someone means to blow
up a house here. Towards ten o'clock he becomes uneasy, jumps at every sound, and at last answers the question
why he is so tense: "I'm expecting...". This answer will e.g. make his behaviour intelligible. It will enable us to fill out
the picture of his thoughts and feelings.†2
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68. Fulfilment of expectation doesn't consist in this: a third thing happens which can be described otherwise than as
"the fulfilment of this expectation", i.e. as a feeling of satisfaction or joy or whatever it may be. The expectation that
something will be the case is the same as the expectation of the fulfilment of that expectation. [Marginal note:
Expectation of what is not.]†3
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69. Socrates to Theaetetus: "If you have an idea, must it not be an idea of something?"--Theaetetus:
"Necessarily".--Socrates: "And if you have an idea of something mustn't it be of something real?"--Theaetetus: "It
seems so".†4
If we put the word "kill", say, in place of "have an idea of" in this argument, then there is a rule for the use of
this word: it makes no sense to say "I am killing something that does not exist". I can imagine a stag that is not there,
in this meadow,
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but not kill one that is not there. And "to imagine a stag in this meadow" means to imagine that a stag is there. But to
kill a stag does not mean to kill that.... But if someone says "In order for me to be able to imagine a stag it must after
all exist in some sense",--the answer is: no, it does not have to exist in any sense. And if it should be replied: "But
the colour brown at any rate must exist, for me to be able to have an idea of it"--then we can say: "The colour brown
exists" means nothing at all; except that not it exists here or there as the colouring of an object, and that is necessary
in order for me to be able to imagine a brown stag.
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70. Being able to do something seems like a shadow of the actual doing, just as the sense of a sentence seems like
the shadow of a fact, or the understanding of an order the shadow of its execution. In the order the fact as it were
"casts its shadow before". But this shadow, whatever it may be, is not the event.
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71. Compare the applications of:
"I have been in pain since yesterday"
"I have been expecting him since yesterday"
"I have known since yesterday"
"Since yesterday I've known how to integrate".
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72. The general differentiation of all states of consciousness from dispositions seems to me to be that one cannot
ascertain by spotcheck whether they are still going on.
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73. Some sentences have to be read several times to be understood as sentences.
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74. A sentence is given me in code together with the key. Then of course in one way everything required for
understanding the sentence has been given me. And yet I should answer the question "Do you understand this
sentence?": No, not yet; I must first decode it. And only when e.g. I had translated it into English would I say "Now
I understand it".
If now we raise the question "At what moment of translating do I understand the sentence?", we shall get a
glimpse into the nature of what is called "understanding".
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75. I can attend to the course of my pains, but not in the same way to that of my belief, or my translation, or my
knowledge.
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76. One can observe the duration of a phenomenon by uninterrupted observation or by trials. The observation of
duration may be continuous or intermittent.
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77. How do I observe my knowledge, my opinions? And on the other hand an after-image, a pain? Is there such a
thing as uninterrupted observation of my capacity to carry out the multiplication...?
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78. Is "I hope..." a description of a state of mind? A state of mind has duration. So "I have been hoping for the whole
day" is such a description; but suppose I say to someone: "I hope you come"--what if he asks me "For how long
have you been hoping that?" Is the answer "For as long as I've been saying so"? Supposing I had some answer or
other to that question, would it not be quite irrelevant to the purpose of the words "I hope you'll come"?
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79. We say "I hope you'll come", but not "I believe I hope you'll come", but we may well say: "I believe I still hope
he'll come."
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80. What is the past tense of "You are coming, aren't you?"?
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81. Where there is genuine duration one can tell someone: "Pay attention and give me a signal when the thing you
are experiencing (the picture, the rattling etc.) alters."
Here there is such a thing as paying attention. Whereas one cannot follow with attention the forgetting of
what one knew or the like. [Not right, for one also cannot follow one's own mental images with attention.]
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82. Think of this language-game: Determine how long an impression lasts by means of a stop-watch. The duration of
knowledge, ability, understanding, could not be determined in this way.
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83. "But the difference between knowing and hearing surely doesn't reside simply in such a characteristic as the kind
of duration they have. They are surely wholly and utterly distinct!" Of course. But one can't say: "Know and hear,
and you will notice the difference".
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84. "Pain is a state of consciousness, understanding is not."--"Well, the thing is, I don't feel my understanding."--But
this explanation achieves nothing. Nor would it be any explanation to say: What one in some sense feels is a state of
consciousness. For that would only mean: State of consciousness = feeling. (One word would merely have been
replaced by another.)
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85. Really one hardly ever says that one has believed, understood, intended something "uninterruptedly" since
yesterday. An interruption of belief would be a period of unbelief, not e.g. the withdrawal of attention from what one
believes--e.g. sleep.
(Difference between 'knowing' and 'being aware of'.)
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86. The most important thing here is this: there is a difference; one notices the difference which is 'a
category-difference'--without being able to say what it consists in. That is the case where it is usually said that we
know the difference by introspection.
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87. This is likely to be the point at which it is said that only form, not content, can be communicated to others.--So
one talks to oneself about the content!--(But how do my words 'relate' to the content I know? And to what
purpose?)
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88. It is very noteworthy that what goes on in thinking practically never interests us. It is noteworthy, but not queer.
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89. ((Thoughts, as it were only hints.))†1
Isn't it the same here as with a calculating prodigy?--He has calculated right if he has got the right answer.
Perhaps he himself cannot say what went on in him. And if we were to hear it, it would perhaps seem like a queer
caricature of calculation.
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90. What do I know of what goes on within someone who is reading a sentence attentively? And can he describe it
to me afterwards, and, if he does describe something, will it be the characteristic process of attention?
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91. Ask: What result am I aiming at when I tell someone: "Read attentively"? That, e.g., this and that should strike
him, and he be able to give an account of it.--Again, it could, I think, be said that if you read a sentence with
attention you will often be able to
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give an account of what has gone on in your mind, e.g. the occurrence of images. But that does not mean that these
things are what we call "attention".
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92. "Did you think as you read the sentence?"--"Yes, I did think as I read it; every word was important to me."
That is not the usual experience.†1 One is not usually half-astonished to hear oneself say something; one
doesn't follow one's own talk with attention; for one ordinarily talks voluntarily, not involuntarily.
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93. If a normal human is holding a normal conversation under normal circumstances, and I were to be asked what
distinguishes thinking from not-thinking in such a case,--I should not know what answer to give. And I could
certainly not say that the difference lay in something that goes on or fails to go on while he is speaking.
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94. The boundary-line that is drawn here between 'thinking' and 'not thinking' would run between two conditions
which are not distinguished by anything in the least resembling a play of images. (For the play of images is
admittedly the model according to which one would like to think of thinking.)
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95. Only under quite special circumstances does the question arise whether one spoke thinkingly or not.
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96. Sure, if we are to speak of an experience of thinking, the experience of speaking is as good as any. But the
concept 'thinking' is not a concept of an experience. For we don't compare thoughts in the same way as we compare
experiences.
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97. What one mimics is, say, a man's tone in speaking, his expression and similar things; and that suffices us. This
proves that the important accompanying phenomena of talking are found here.
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98. Do we say that anyone who is speaking significantly is thinking? For example the builder in language-game no.
2?†2 Couldn't we imagine him building and calling out the words in surroundings in which we should not connect
this even remotely with thinking?
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99. (On language-game no. 2†1) "You are just tacitly assuming that these people think; that they are like people as
we know them in that respect; that they do not carry on that language-game merely mechanically. For if you
imagined them doing that, you yourself would not call it the use of a rudimentary language."
What am I to reply to this? Of course it is true that the life of those men must be like ours in many respects,
and I said nothing about this similarity. But the important thing is that their language, and their thinking too, may be
rudimentary, that there is such a thing as 'primitive thinking' which is to be described via primitive behaviour. The
surroundings are not the 'thinking accompaniment' of speech.
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100. Let us imagine someone doing work that involves comparison, trial, choice. Say he is constructing an appliance
out of various bits of stuff with a given set of tools. Every now and then there is the problem "Should I use this
bit?"--The bit is rejected, another is tried. Bits are tentatively put together, then dismantled; he looks for one that fits
etc., etc.. I now imagine that this whole procedure is filmed. The worker perhaps also produces sound-effects like
"hm" or "ha!" As it were sounds of hesitation, sudden finding, decision, satisfaction, dissatisfaction. But he does not
utter a single word. Those sound-effects may be included in the film. I have the film shewn me, and now I invent a
soliloquy for the worker, things that fit his manner of work, its rhythm, his play of expression, his gestures and
spontaneous noises; they correspond to all this. So I sometimes make him say "No, that bit is too long, perhaps
another'll fit better." Or "What am I to do now?" "Got it!"-- Or "That's not bad" etc.
If the worker can talk--would it be a falsification of what actually goes on if he were to describe that precisely
and were to say e.g. "Then I thought: no, that won't do, I must try it another way" and so on--although he had
neither spoken during the work nor imagined these words?
I want to say: May he not later give his wordless thoughts in words? And in such a fashion that we, who
might see the work in progress, could accept this account?--And all the more, if we had often watched the man
working, not just once?
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101. Of course we cannot separate his 'thinking' from his activity. For the thinking is not an accompaniment of the
work, any more than of thoughtful speech.
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102. Were we to see creatures at work whose rhythm of work, play of expression etc. was like our own, but for their
not speaking, perhaps in that case we should say that they thought, considered, made decisions. For there would be
a great deal there corresponding to the action of ordinary humans. And there is no deciding how close the
correspondence must be to give us the right to use the concept 'thinking' in their case too.
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103. And anyhow what should we come to this decision for?
We shall be making an important distinction between creatures that can learn to do work, even complicated
work, in a 'mechanical' way, and those that make trials and comparisons as they work.--But what should be called
"making trials" and "comparisons" can in turn be explained only by giving examples, and these examples will be
taken from our life or from a life that is like ours.
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104. If he has made some combination in play or by accident and he now uses it as a method of doing this and that,
we shall say he thinks.--In considering he would mentally review ways and means. But to do this he must already
have some in stock. Thinking gives him the possibility of perfecting his methods. Or rather: He 'thinks' when, in a
definite kind of way, he perfects a method he has. [Marginal note: What does the search look like?]
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105. It could also be said that a man thinks when he learns in a particular way.
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106. And this too could be said: Someone who thinks as he works will intersperse his work with auxiliary activities.
The word "thinking" does not now mean these auxiliary activities, just as thinking is not talking either. Although the
concept 'thinking' is formed on the model of a kind of imaginary auxiliary activity. (Just as we might say that the
concept of the differential quotient is formed on the model of a kind of ideal quotient.)
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107. These auxiliary activities are not the thinking; but one imagines thinking as the stream which must be flowing
under the surface of these expedients, if they are not after all to be mere mechanical procedures.
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108. Suppose it were a question of buying and selling creatures (anthropoid brutes) which we use as slaves. They
cannot learn to talk, but the cleverer among them can be taught to do quite complicated work; and some of these
creatures work 'thinkingly', others quite mechanically. For a thinking one we pay more than for one that is merely
mechanically clever.
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109. If there were only quite few people who could get the answer to a sum without speaking or writing, they could
not be adduced as testimony to the fact that calculating can be done without signs. The reason is that it would not be
clear that these people were 'calculating' at all. Equally Ballard's testimony (in James) cannot convince one that it is
possible to think without a language.
Indeed, where no language is used, why should one speak of 'thinking'? If this is done, it shows something
about the concept of thinking.
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110. 'Thinking', a widely ramified concept. A concept that comprises many manifestations of life. The phenomena of
thinking are widely scattered.
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111. We are not at all prepared for the task of describing the use of e.g. the word "to think" (And why should we
be? What is such a description useful for?)
And the naïve idea that one forms of it does not correspond to reality at all. We expect a smooth contour and
what we get to see is ragged. Here it might really be said that we have constructed a false picture.
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112. It is not to be expected of this word that it should have a unified employment; we should rather expect the
opposite.
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113. Where do we get the concept 'thinking' from which we want to consider here? From everyday language. What
first fixes the direction of our attention is the word "thinking". But
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the use of this word is confused. Nor can we expect anything else. And that can of course be said of all
psychological verbs. Their employment is not so clear or so easy to get a synoptic view of, as that of terms in
mechanics, for example.
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114. One learns the word "think", i.e. its use, under certain circumstances, which, however, one does not learn to
describe.
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115. But I can teach a person the use of the word! For a description of those circumstances is not needed for that.
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116. I just teach him the word under particular circumstances.
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117. We learn to say it perhaps only of human beings; we learn to assert or deny it of them. The question "Do fishes
think?" need not exist among their applications of language, it is not raised. (What can be more natural than such a
set-up, such a use of language?)
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118. "No one thought of that case"--we may say. Indeed, I cannot enumerate the conditions under which the word
"to think" is to be used-but if a circumstance makes the use doubtful, I can say so, and also say how the situation is
deviant from the usual ones.
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119. If I have learned to carry out a particular activity in a particular room (putting the room in order, say) and am
master of this technique, it does not follow that I must be ready to describe the arrangement of the room; even if I
should at once notice, and could also describe, any alteration in it.
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120. "This law was not given with such cases in view." Does that mean it is senseless?
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121. It could very well be imagined that someone knows his way around a city perfectly, i.e. would confidently find
the shortest way from any place in it to any other,--and yet would be quite incompetent to draw a map of the city.
That, as soon as he tries, he produces nothing that is not completely wrong. (Our concept of 'instinct'.)
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122. Remember that our language might possess a variety of different words: one for 'thinking out loud'; one for
thinking as one talks to oneself in the imagination; one for a pause during which something or other floats before the
mind, after which, however, we are able to give a confident answer.
One word for a thought expressed in a sentence; one for the lightning thought which I may later 'clothe in
words'; one for wordless thinking as one works.
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123. "Thinking is a mental activity"--Thinking is not a bodily activity. Is thinking an activity? Well, one may tell
someone: "Think it over". But if someone in obeying this order talks to himself or even to someone else, does he
then carry out two activities?
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124. Concern with what we say has its own specific signs. It also has its own specific consequences and
preconditions. Concern is something experienced; we attribute it to ourselves, not on grounds of observation. It is
not an accompaniment of what we say. What would make an accompaniment of a sentence into concern about the
content of that sentence? (Logical condition.)
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125. Compare the phenomenon of thinking with the phenomenon of burning. May not burning, flame, seem
mysterious to us? And why flame more than furniture?--And how do you clear up the mystery?
And how is the riddle of thinking to be solved?--Like that of flame?
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126. Isn't flame mysterious because it is impalpable? All right--but why does that make it mysterious? Why should
something impalpable be more mysterious than something palpable? Unless it's because we want to catch hold of
it.--
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127. The soul is said to leave the body. Then, in order to exclude any similarity to the body, any sort of idea that
some gaseous thing is meant, the soul is said to be incorporeal, non-spatial; but with the word "leave" one has
already said it all. Shew me how you use the word "spiritual" and I shall see whether the soul is non-corporeal and
what you understand by "spirit".
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128. I am inclined to speak of a lifeless thing as lacking something. I see life definitely as a plus, as something added
to a lifeless thing. (Psychological atmosphere.)
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129. We don't say of a table and a chair: "Now they are thinking," nor "Now they are not thinking," nor yet "They
never think"; nor do we say it of plants either, nor of fishes; hardly of dogs; only of human beings. And not even of
all human beings.
"A table doesn't think" is not assimilable to an expression like "a table doesn't grow". (I shouldn't know 'what
it would be like if' a table were to think.) And here there is obviously a gradual transition to the case of human
beings.
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130. We only speak of 'thinking' in quite particular circumstances.
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131. How then can the sense and the truth (or the truth and the sense) of sentences collapse together? (Stand or fall
together?)
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132. And isn't it as if you wanted to say: "If such-and-such is not the case, then it makes no sense to say it is the
case"?
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133. Like this, e.g.: "If all moves were always false, it would make no sense to speak of a 'false move'." But that is
only a paradoxical way of putting it. The nonparadoxical way would be: "The general description ... makes no
sense".
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134. Do not say "one cannot", but say instead: "it doesn't exist in this game". Not: "one can't castle in draughts"
but--"there is no castling in draughts"; and instead of "I can't exhibit my sensation"--"in the use of the word
'sensation', there is no such thing as exhibiting what one has got"; instead of "one cannot enumerate all the cardinal
numbers"--"there is no such thing here as enumerating all the members".
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135. Conversation flows on, the application and interpretation of words, and only in its course do words have their
meaning.
"He has gone away" "Why?"--What did you mean, when you uttered the word "why"? What did you think
of?
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136. Think of putting your hand up in school. Need you have rehearsed the answer silently to yourself, in order to
have the right to put your hand up? And what must have gone on inside you?--Nothing need have. But it is
important that you usually know an answer when you put your hand up; and that is the criterion for one's
understanding of putting one's hand up.
Nothing need have gone on in you; and yet you would be remarkable if on such occasions you never had
anything to report about what went on in you.
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137. Sometimes when I say "Just then I had the thought...." I may report that I had said those very words to myself,
out loud or silently; or if not those, then others of which the present ones reproduce the gist. Surely that often
happens. But it does also happen that my present words are 'not a reproduction', for they are only a reproduction if
they are so by rules of projection.
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138. It looks as if a sentence with e.g. the word "ball" in it already contained the shadow of other uses of this word.
That is to say, the possibility of forming those other sentences.--To whom does it look like that? And under what
circumstances?
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139. We don't get free of the idea that the sense of a sentence accompanies the sentence: is there alongside of it.
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140. One wants to say e.g.: "The one negation does the same thing with the proposition as the other, it excludes what
it describes." But that is only another way of expressing one's assimilation of the two negative propositions. (Which
is valid only when the negated proposition is not itself a negative proposition.) Ever and again comes the thought
that what we see of a sign is only the outside of something within, in which the real operations of sense and meaning
go on.†1
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141. Our problem could be (very clearly) formulated like this: suppose we had two systems for measuring length; in
both a length is expressed by a numeral which is followed by a word giving the unit. One system designates a length
as "n ft", and a foot is a unit of length in the ordinary sense; in the other system a length is designated by "n W" and
1 ft = 1 W. But 2 W = 4 ft,
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3 W = 9 ft and so on.--So the sentence "This stick is 1 W long" means the same as "This stick is 1 ft long". Question:
have "W" and "ft" the same meaning in these two sentences?
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142. The question is wrongly framed. We can see this, if we express the identity of meaning by an equation. The
question can only run: "Does W = ft or not?"--The sentences in which these signs occur do not come in here.--No
more, of course, can we ask in this terminology whether "is" in one place means the same as "is" in another; what
we can ask is whether the copula means the same as the equals sign. Well, what we said was: 1 ft = 1 W; but ft ≠ W.
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143. We might say: in all cases what one means by "thought" is what is alive in the sentence. That without which it
is dead, a mere sequence of sounds or written shapes.
If however I were to speak in the same way of a something that gives meaning to a configuration of chess
pieces, that is to say distinguishes them from any old arrangement of little bits of wood--couldn't I mean all sorts of
things? The rules that make the chess arrangement into a situation in a game; the special experiences that we
associate with such positions in a game; the usefulness of the game.
Or suppose we were to speak of a something that distinguishes paper money from mere printed slips of
paper and gives it its meaning, its life!
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144. How words are understood is not told by words alone. (Theology.)
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145. There could also be a language in whose use the impression made on us by the signs played no part; in which
there was no such thing as understanding, in the sense of such an impression. The signs are e.g. written and
transmitted to us, and we are able to take notice of them. (That is to say, the only impression that comes in here is
the pattern of the sign.) If the sign is an order, we translate it into action by means of rules, tables. It does not get as
far as an impression, like that of a picture; nor are stories written in this language.
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146. In this case one might say: "Only in the system has the sign any life."
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147. We could of course also imagine that we had to use rules, and translate a verbal sentence into a drawing in order
to get an impression from it. (That only the picture had a soul.)
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148. We could imagine a language in which the meanings of expressions changed according to definite rules, e.g.: in
the morning the expression A means this, in the afternoon it means that.
Or a language in which the individual words altered every day; each day each letter of the previous day
would be replaced by the next one in the alphabet (and z by a).
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149. Imagine the following language: its vocabulary and grammar are those of English, but the words occur in the
sentences in reverse order. So a sentence of this language sounds like an English sentence read from the full stop
back to the beginning. Thus the possibilities of expression have the same multiplicity as in English. But the familiar
ring of our sentences is done away with.
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150. Someone who doesn't know English hears me say on certain occasions: "What marvellous light!" He guesses
the sense and now uses the exclamation himself, as I use it, but without understanding the three individual words.
Does he understand the exclamation?
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151. I intentionally chose an example in which a man gives expression to his sensation. For in this case sounds
belonging to no language are said to be full of meaning.
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152. Would it be equally easy to imagine the analogous case for this sentence: "If the train does not arrive punctually
at five o'clock, he'll miss the connexion"? What would guessing the sense mean in this case?
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153. It somehow worries us that the thought in a sentence is not wholly present at any one moment. We regard it as
an object which we are making and have never got all there, for no sooner does one part appear than another
vanishes.
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154. (On no. 150) A language may easily be imagined in which people use a single word for the exclamation. But
what about one word for the sentence "If the train...."? In what kind of case should we say that the word actually
stood for that sentence?
Say in this one: people begin by using a sentence like ours, but then circumstances arise in which the
sentence has to be uttered so often that they contract it to a single word. So these people could still explain the word
by means of the sentence.
But is the further case possible in which people possess only a single word with that sense, that is for that
use? Why not? One needs to imagine how the use of this word is learnt, and in what circumstances we should say
that the word really stands in place of that sentence.
But remember this: in our language someone says "He is arriving at five o'clock"; someone else replies "No,
at ten past five". Is there also this sort of exchange in the other language?
That is why sense and reference are vague concepts.
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155. A poet's words can pierce us. And that is of course causally connected with the use that they have in our life.
And it is also connected with the way in which, conformably to this use, we let our thoughts roam up and down in
the familiar surroundings of the words.
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156. Is there a difference of meaning that can be explained and another that does not come out in an explanation?
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157. Soulful expression in music--this cannot be recognized by rules. Why can't we imagine that it might be, by
other beings?
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158. If a theme, a phrase, suddenly means something to you, you don't have to be able to explain it. Just this gesture
has been made accessible to you.
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159. But you do speak of understanding music. You understand it, surely, while you hear it! Ought we to say this is
an experience which accompanies the hearing?
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160. The way music speaks. Do not forget that a poem, even though it is composed in the language of information,
is not used in the language-game of giving information.
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161. Mightn't we imagine a man who, never having had any acquaintance with music, comes to us and hears
someone playing a reflective piece of Chopin and is convinced that this is a language and people merely want to
keep the meaning secret from him?
There is a strongly musical element in verbal language. (A sigh, the intonation of voice in a question, in an
announcement, in longing; all the innumerable gestures made with the voice.)
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162. But if I hear a tune with understanding, doesn't something special go on in me--which does not go on if I hear it
without understanding? And what?--No answer comes; or anything that occurs to me is insipid. I may indeed say:
"Now I've understood it," and perhaps talk about it, play it, compare it with others etc. Signs of understanding may
accompany hearing.
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163. It is wrong to call understanding a process that accompanies hearing. (Of course its manifestation, expressive
playing, cannot be called an accompaniment of hearing either.)
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164. For how can it be explained what 'expressive playing' is? Certainly not by anything that accompanies the
playing.--What is needed for the explanation? One might say: a culture.--If someone is brought up in a particular
culture--and then reacts to music in such-and-such a way, you can teach him the use of the phrase "expressive
playing".
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165. The understanding of music is neither sensation nor a sum of sensations. Nevertheless it is correct to call it an
experience inasmuch as this concept of understanding has some kinship with other concepts of experience. You say
"I experienced that passage quite differently". But still this expression tells you 'what happened' only if you are at
home in the special conceptual world that belongs to these situations. (Analogy: "I won the match".)
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166. This floats before my mind as I read. So does something go on in reading...?--This question doesn't get us
anywhere.
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167. But how can it float before me?--Not in the dimensions you are thinking of.
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168. How do I know that someone is enchanted? How does one learn the linguistic expression of enchantment?
What does it connect up with? With the expression of bodily sensations? Do we ask someone what he feels in his
breast and facial muscles in order to find out whether he is feeling enjoyment?
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169. But does that mean that there are not sensations which often return when one is enjoying music? Certainly not.
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170. A poem makes an impression on us as we read it. "Do you feel the same while you read it as when you read
something indifferent?"--How have I learnt to answer this question? Perhaps I shall say "Of course not!"--which is
as much as to say: this takes hold of me, and the other not.
"I experience something different"--And what kind of thing?--I can give no satisfactory answer. For the
answer I give is not in itself of any importance.--"But didn't you enjoy it during the reading?" Of course--for the
opposite answer would mean: I enjoyed it earlier or later, and I don't want to say that.
But now, surely you remember sensations and images as you read, and they are such as to connect up with
the enjoyment, with the impression.--But they got their significance only from the surroundings: through the reading
of this poem, from my familiarity with its language, with its metre and with innumerable associations.
You must ask how we learnt the expression "Isn't that glorious!" at all.--No one explained it to us by
referring to sensations, images or thoughts that accompany hearing! Nor should we doubt whether he had enjoyed it
if he had no account to give of such experiences; though we should, if he shewed that he did not understand certain
tie-ups.
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171. But isn't understanding shewn e.g. in the expression with which someone reads the poem, sings the tune?
Certainly. But what is the experience during the reading? About that you would just have to say: you enjoy and
understand it if you hear it well read, or feel it well read in your speech-organs.
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172. Understanding a musical phrase may also be called understanding a language.
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173. I think of a quite short phrase, consisting of only two bars. You say "What a lot that's got in it!" But it is only,
so to speak, an optical illusion if you think that what is there goes on as we hear it. ("It all depends who says it.")
(Only in the stream of thought and life do words have meaning.)
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174. What contains the illusion is not this: "Now I've understood"--followed perhaps by a long explanation of what I
have understood.
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175. Doesn't the theme point to anything outside itself? Yes, it does! But that means:--it makes an impression on me
which is connected with things in its surroundings--e.g. with our language and its intonations; and hence with the
whole field of our language-games.
If I say for example: Here it's as if a conclusion were being drawn, here as if something were being
confirmed, this is like an answer to what was said before,-then my understanding presupposes a familiarity with
inferences, with confirmation, with answers.
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176. The words "Gottlob! Noch etwas Weniges hat man geflüchtet--vor den Fingern der Kroaten,"†1 and the tone
and glance that go with them seem indeed to carry within themselves every last nuance of the meaning they have.
But only because we know them as part of a particular scene. But it would be possible to construct an entirely
different scene around these words so as to shew that the special spirit they have resides in the story in which they
come.
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177. If I hear someone say: "Away!" with a gesture of repulsion, do I have an 'experience' of meaning here as I do in
the game where I pronounce that to myself meaning it now in one sense, now in another?--For he could also have
said "Get away from me!" and then perhaps I'd have experienced the whole phrase in such-and-such a way--but the
single word? Perhaps it was the supplementary words that made the impression on me.
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178. The peculiar experience of meaning is characterized by the fact that we come out with an explanation and use
the past tense: just as if we were explaining the meaning of a word for practical purposes.
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179. Forget, forget that you have these experiences yourself!
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180. How could he hear the word with that meaning? How was it possible?! It just wasn't--not in these dimensions.--
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181. But isn't it true, then, that the word means that to me now? Why not? For this sense doesn't come into conflict
with the rest of the use of the word.
Someone says: "Give him the news that..., and mean by it...."--What sense would this order make?
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182. "When I uttered the word just now, it meant... to me." Why should that not be mere lunacy? Because I
experienced that? That is not a reason.
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183. The man I call meaning-blind will understand the instruction "Tell him he is to go to the bank--I mean the river
bank," but not "Say the word bank and mean the bank of a river". What concerns this investigation is not the forms
of mental defect that are found among men; but the possibility of such forms. We are interested, not in whether
there are men incapable of a thought of the type: "I was then going to..."--but in how the concept of such a defect
should be worked out.
If you assume that someone cannot do this, how about that? Are you supposing he can't do that
either?--Where does this concept take us? For what we have here are of course paradigms.
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184. Different people are very different in their sensitiveness about changes in the orthography of a word. And the
feeling is not just piety towards an old use.--If for you spelling is just a practical question, the feeling you are lacking
in is not unlike the one that a 'meaning-blind' man would lack. (Goethe on people's names. Prisoners' numbers.)
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185. It's just like the way some people do not understand the question "What colour has the vowel a for you?"--If
someone did not understand this, if he were to declare it was nonsense--
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could we say he did not understand English, or the meaning of the individual words "colour", "vowel" etc.?
On the contrary: Once he has learned to understand these words, then it is possible for him to react to such
questions 'with understanding' or 'without understanding'.
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186. Misunderstanding-non-understanding. Understanding is effected by explanation; but also by training.
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187. Why can't a cat be taught to retrieve? Doesn't it understand what one wants? And what constitutes
understanding or failure to understand here?
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188. "I read each word with the feeling appropriate to it. The word 'but' e.g. with the but-feeling--and so on."--And
even if that is true--what does it really signify? What is the logic of the concept 'but-feeling'?--It certainly isn't a
feeling just because I call it "a feeling".
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189. Is lying a particular experience? Well, can I tell someone "I am going to tell you a lie" and straightway do it?
Page 32
190. To what extent am I aware of lying while I'm telling a lie? Just in so far as I don't 'only realise it later on', and all
the same I do know later that I was lying. The awareness that one is lying is a knowing-how. It is no contradiction of
this that there are characteristic feelings of lying. [Marginal note: Intention.]
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191. Knowledge is not translated into words when it is expressed. The words are not a translation of something else
that was there before they were.
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192. "To purpose to do something is a special inner process."--But what sort of process--even if you could dream
one upcould satisfy our requirements about purpose?
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193. Isn't it just like this with the verb "to understand"? Someone tells me the route I have to take to some place and
from there on. He asks "Did you understand?" I reply "Yes I did". --Do I mean to tell him what was going on within
me during his explanation?--And after all that could be told him too.
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194. Imagine the following game: A list of words from various languages and of senseless sound-sequences is read
out to me. I am to say after each whether I understand it or not; and also what went on within me as I understood or
failed to understand.--At the word "tree" I shall answer "yes" without reflection (perhaps an image floats before my
mind); at a collocation of sounds that I have never heard before, I answer "No" equally without reflection. At words
which stand for particular shades of colour, the answer will often be preceded by an image, at a few words
("continuum," say) there will be consideration; at words like the article "the" perhaps a shrug of the shoulders; words
of a foreign language I shall sometimes translate into English; when images rise in my mind they are sometimes
images of the objects that are designated by the words (in turn a host of cases), sometimes different pictures.
This game might be supplemented by one in which someone calls out the names of activities and at each
one asks: "Can you do that?"--The subject is to give his reasons for answering the question "yes" or "no".
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195. Let us imagine a kind of puzzle picture: there is not one particular object to find; at first glance it appears to us
as a jumble of meaningless lines, and only after some effort do we see it as, say, a picture of a landscape.--What
makes the difference between the look of the picture before and after the solution? It is clear that we see it differently
the two times. But what does it amount to to say that after the solution the picture means something to us, whereas it
meant nothing before?
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196. We can also put this question like this: What is the general mark of the solution's having been found?
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197. I will assume that as soon as it is solved I make the solution obvious by strongly tracing certain lines in the
picture and perhaps putting in some shadows. Why do you call the picture you have sketched in a solution?
(a) Because it is the clear representation of a group of spatial objects.
(b) Because it is the representation of a regular solid.
(c) Because it is a symmetrical figure.
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(d) Because it is a shape that makes an ornamental impression on me.
(e) Because it is the representation of a body I am familiar with.
(f) Because there is a list of solutions and this shape (this body) is on the list.
(g) Because it represents a kind of object that I am very familiar with; for it gives me an instantaneous
impression of familiarity, I instantly have all sorts of associations in connexion with it; I know what it is called; I
know I have often seen it; I know what it is used for etc.
(h) Because I seem to be familiar with the object, a word occurs to me at once as its name (although the word
does not belong to any existent language); I tell myself "Of course that's a..." and give myself a nonsensical
explanation, which at that moment seems to me to make sense. (Like in a dream.)
(i) Because it represents a face which strikes me as familiar.
(j) Because it represents a face which I recognize; it is the face of my friend N; it is a face which I have often
seen pictures of, etc.
(k) Because it represents an object which I remember having seen at some time.
(l) Because it is an ornament that I know well (though I don't remember where I have seen it).
(m) Because it is an ornament that I know well; I know its name, I know where I have seen it.
(n) Because it represents part of the furniture of my room.
(o) Because I instinctively traced out those lines and now feel easy.
(p) Because I remember that this object has been described to me. And so on.
(Anyone who does not understand why we talk about these things must feel what we say to be mere trifling.)
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198. Can I think away the impression of familiarity where it exists; and think it into a situation where it does not?
And what does that mean? I see e.g. the face of a friend and ask myself: What does this face look like if I see it as a
strange face (as if I were seeing it now for the first time)? What remains, as it were,
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of the look of this face, if I think away, subtract, the impression of familiarity from it? Here I am inclined to say: "It is
very difficult to separate the familiarity from the impression of the face". But I also feel that this is a bad way of
putting things. For I have no notion how I should so much as try to separate these two things. The expression "to
separate them" does not have any clear sense for me.
I know what this means: "Imagine this table black instead of brown". To this there corresponds: "Paint this
table, but black instead of brown".
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199. Suppose someone were to say: "Imagine this butterfly exactly as it is, but ugly instead of beautiful"?!
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200. In this case we have not determined what thinking the familiarity away is to mean.
It might mean, say, to recall the impression which I had when I saw the face for the first time.
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201. For someone who has no knowledge of such things a diagram representing the inside of a radio receiver will be
a jumble of meaningless lines. But if he is acquainted with the apparatus and its function, that drawing will be a
significant picture for him.
Given some solid figure (say in a picture) that means nothing to me at present--can I at will imagine it as
meaningful? That's as if I were asked: Can I imagine an object of any old shape as an appliance? But to be applied to
what?
One class of corporeal shapes might readily be imagined as dwellings for beasts or men. Another class as
weapons. Another as models of landscapes. Etc. etc. So here I know how I can ascribe meaning to a meaningless
shape.
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202. Consider well how we use the word "recognize". I recognize the furniture in my room, my friend whom I see
every day. But no 'act of recognition takes place'.
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203. Someone might say: I should have no impression of the room as a whole, if I could not let my glance roam
rapidly to and fro in it and myself move about in it freely. (Stream of thought.) But now the question is: How is it
manifested that I 'have an
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impression of the room as a whole'? For one thing, in its being a matter of course that I find my way around in it; in
the absence of hunting about, hesitation, and surprise. In there being no end of activities that are encompassed by its
walls and in all this being covered by the expression "my room" when I'm talking. In my finding it useful and
necessary to keep on using the idea 'my room', as opposed to its walls, its corners etc.
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204. What is the description of an 'attitude' like?
One says e.g. "Disregard these spots and this little irregularity, and look at it as a picture of a...."
"Think that away. Would you dislike the thing even without this...." Of course it will be said that I alter my
visual image--as by blinking or blocking out a detail. This "Disregarding..." does indeed play a part quite like, say, the
production of a new picture.
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205. Very well--that is good reason to say that we altered our visual impression through our attitude. That is to say,
there are good reasons to delimit the concept 'visual impression' in this way.
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206. "But in seeing I can obviously take elements together (lines for example)." But why does one call it "taking
together"? Why does one here--essentially--need a word that already has another meaning? (This is of course like
the case of the phrase "calculating in one's head".)
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207. If I tell someone "Take these lines (or something else) together" what will he do? Well, various things, according
to the circumstances. Perhaps he is supposed to count them two by two, or to put them in a drawer, or to look at
them etc.
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208. Let us consider what is said about a phenomenon like this:
Seeing the figure &unk; now as an F, now as the mirror image of an F.
I want to ask: what constitutes seeing the figure now like this, now another way?--Do I really see something
different every time? Or do I merely interpret what I see in a different way?--I am inclined to say the first. But why?
Well, interpreting is a procedure. It may for example consist in somebody's saying
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"That is supposed to be an F"; or not saying it, but replacing the sign with an F in copying; or again considering:
"What can that be? It'll be an F that the writer did not hit off."--Seeing is not an action but a state. (A grammatical
remark.) And if I have never read the figure as anything but an F, or considered what it might be, we shall say that I
see it as F; if, that is, we know that it can also be seen differently. I should call it "interpretation" if I were to say
"That is certainly supposed to be an F; the writer does all his F's like that."
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For how do we arrive at the concept 'seeing this as that' at all? On what occasions does it get formed, when is
there need of it? (Very frequently in art.) Where, for example, there is a question of phrasing by eye or ear. We say
"You have to hear these bars as an introduction," "You must hear it as in this key." "Once you have seen this figure
as... it is difficult to see it otherwise", "I hear the French 'ne ... pas' as a negation in two parts, not as 'not a step'" etc.,
etc. Now, is it a real case of seeing or hearing? Well, we call it that; we react with these words in particular situations.
And we react to these words in turn by particular actions.
209. This shape that I see--I want to say--is not simply a shape; it is one of the shapes I know; it is a shape marked
out in advance. It is one of those shapes of which I already had a pattern in me; and only because it corresponds to
such a pattern is it this familiar shape. (I as it were carry a catalogue of such shapes around with me, and the objects
portrayed in it are the familiar ones.)
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210. But my already carrying the pattern around with me would be only a causal explanation of the present
impression. It is like saying: this movement is made as easily as if it had been practised.
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211. "When I am asked 'Do you see a ball over there' and another time 'Do you see half a ball over there?' what I see
may be the same both times, and if I answer 'Yes', still I distinguish between the two hypotheses. As I distinguish
between pawn and king in chess, even if the present move is one that either might make, and even if an actual
king-piece were being
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used as a pawn."--In philosophy one is in constant danger of producing a myth of symbolism, or a myth of mental
processes. Instead of simply saying what anyone knows and must admit.
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212. Does introspection tell me whether what I have here is a genuine case of seeing, or one of interpretation after
all? First of all I must make dear to myself what I should call an interpretation; how to tell whether something is to
be called a case of interpreting or of seeing. [Marginal note: Seeing according to an interpretation.]
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213. Don't I see the figure now like this, now another way, even when I do not react verbally or otherwise?
But "now like this" "now another way" are words, and what right have I to use them here? Can I shew my
right to you or to myself? (Unless by a further reaction.)
But I surely know that they are two impressions, even if I don't say so. [Marginal note: But how do I know
that what I say is what I knew? What consequences follow from my interpreting this as that, or from my seeing this
as that?]
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214. Experience of the real size. We see a picture showing a chair-shape; we are told it represents a construction the
size of a house. Now we see it differently.
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215. Imagine someone watching the sun and suddenly having the feeling that it is not the sun that moves--but we
that move past it. Now he wants to say he has seen a new state of motion that we are in; imagine him showing by
gestures which movement he means, and that it is not the sun's movement.--We should here be dealing with two
different applications of the word "movement".
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216. We see, not change of aspect, but change of interpretation.
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217. You see it conformably, not to an interpretation, but to an act of interpreting.
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218. I interpret words; yes--but do I also interpret looks? Do I interpret a facial expression as threatening or
kind?--That may happen.
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Suppose I said: "It is not enough to perceive the threatening face, I have to interpret it."--Someone whips out
a knife at me and I say "I conceive that as a threat."
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219. We don't understand Chinese gestures any more than Chinese sentences.
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220. Consciousness in another's face. Look into someone else's face, and see the consciousness in it, and a particular
shade of consciousness. You see on it, in it, joy, indifference, interest, excitement, torpor and so on. The light in
other people's faces.
Do you look into yourself in order to recognize the fury in his face? It is there as clearly as in your own
breast.
(And what do we want to say now? That someone else's face stimulates me to imitate it, and that I therefore
feel little movements and muscle-contractions in my own face and mean the sum of these? Nonsense.
Nonsense,--because you are making assumptions instead of simply describing. If your head is haunted by
explanations here, you are neglecting to remind yourself of the most important facts.)
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221. "Consciousness is as clear in his face and behaviour, as in myself."
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222. We do not see the human eye as a receiver, it appears not to let anything in, but to send something out. The ear
receives; the eye looks. (It casts glances, it flashes, radiates, gleams.) One can terrify with one's eyes, not with one's
ear or nose. When you see the eye you see something going out from it. You see the look in the eye.
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223. "If you only shake free from your physiological prejudices, you will find nothing queer about the fact that the
glance of the eye can be seen too." For I also say that I see the look that you cast at someone else. And if someone
wanted to correct me and say that I don't really see it, I should take that for pure stupidity.
On the other hand I have not made any admissions by using that manner of speaking, and I should
contradict anyone who told me I saw the glance 'just the way' I see the shape and colour of the eye.
For 'naïve language', that is to say our naïve, normal way of expressing ourselves, does not contain any
theory of seeing--does not show you a theory but only a concept of seeing.
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224. Get a human being to give angry, proud, ironical looks; and now veil the face so that only the eyes remain
uncovered--in which the whole expression seemed concentrated: their expression is now surprisingly ambiguous.
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225. "We see emotion."--As opposed to what?--We do not see facial contortions and make inferences from them
(like a doctor framing a diagnosis) to joy, grief, boredom. We describe a face immediately as sad, radiant, bored,
even when we are unable to give any other description of the features.--Grief, one would like to say, is personified in
the face.
This belongs to the concept of emotion.
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226. (The ugliness of a human being can repel in a picture, in a painting, as in reality, but so it can too in a
description, in words.)
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227. How curious: we should like to explain our understanding of a gesture by means of a translation into words,
and the understanding of words by translating them into a gesture. (Thus we are tossed to and fro when we try to
find out where understanding properly resides.)
And we really shall be explaining words by a gesture, and a gesture by words.
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228. Explain to someone that the position of the clock-hands that you have just noted down is supposed to mean:
the hands of this clock are now in this position.--The awkwardness of the sign in getting its meaning across, like a
dumb person who uses all sorts of suggestive gestures--this disappears when we know that it all depends on the
system to which the sign belongs.
We wanted to say: only the thought can say it, not the sign.
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229. But an interpretation is something that is given in signs. It is this interpretation as opposed to a different one
(running differently).--So when we wanted to say "Any sentence still stands in need of an interpretation", that
meant: no sentence can be understood without a rider.
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230. It would almost be like settling how much a toss is to be worth by another toss.
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231. By "intention" I mean here what uses a sign in a thought. The intention seems to interpret, to give the final
interpretation; which is not a further sign or picture, but something else--the thing that cannot be further interpreted.
But what we have reached is a psychological, not a logical terminus.
Think of a sign language, an 'abstract' one, I mean one that is strange to us, in which we do not feel at home,
in which, as we should say, we do not think; and let us imagine this language interpreted by a translation into--as we
should like to say--an unambiguous picture-language, a language consisting of pictures painted in perspective. It is
quite clear that it is much easier to imagine different interpretations of the written language than of a picture painted
in the usual way. Here we shall also be inclined to think that there is no further possibility of interpretation.
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232. Here we might also say we didn't enter into the sign-language, but did enter into the painted picture.
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233. "Only the intended picture reaches up to reality like a yardstick. Looked at from outside, there it is, lifeless and
isolated."--It is as if at first we looked at a picture so as to enter into it and the objects in it surrounded us like real
ones; and then we stepped back, and were now outside it; we saw the frame, and the picture was a painted surface.
In this way, when we intend, we are surrounded by our intention's pictures, and we are inside them. But when we
step outside intention, they are mere patches on a canvas, without life and of no interest to us. When we intend, we
exist in the space of intention, among the pictures (shadows) of intention, as well as with real things. Let us imagine
we are sitting in a darkened cinema and entering into the film. Now the lights are turned on, though the film
continues on the screen. But suddenly we are outside it and see it as movements of light and dark patches on a
screen.
(In dreams it sometimes happens that we first read a story and then are ourselves participants in it. And after
waking up after a dream it is sometimes as if we had stepped back out of the dream and now see it before us as an
alien picture.) And it also means something to speak of "living in the pages of a book."
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234. What happens is not that this symbol cannot be further interpreted, but: I do no interpreting. I do not interpret,
because I feel at home in the present picture. When I interpret, I step from one level of thought to another.
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235. If I see the thought symbol 'from outside', I become conscious that it could be interpreted thus or thus; if it is a
step in the course of my thoughts, then it is a stopping-place that is natural to me, and its further interpretability does
not occupy (or trouble) me. As I have a time-table and use it without being concerned with the fact that a table is
susceptible of various interpretations.
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236. If I try to describe the process of intention, I feel first and foremost that it can do what it is supposed to only by
containing an extremely faithful picture of what it intends. But further, that that too does not go far enough, because
a picture, whatever it may be, can be variously interpreted; hence this picture too in its turn stands isolated. When
one has the picture in view by itself it is suddenly dead, and it is as if something had been taken away from it, which
had given it life before. It is not a thought, not an intention; whatever accompaniments we imagine for it, articulate or
inarticulate processes, or any feeling whatsoever, it remains isolated, it does not point outside itself to a reality
beyond.
Now one says: "Of course it is not the picture that intends, but we who use it to intend something." But if
this intending, this meaning, is in turn something that is done with the picture, then I cannot see why that has to
involve a human being. The process of digestion can also be studied as a chemical process, independently of
whether it takes place in a living being. We want to say "Meaning is surely essentially a mental process, a process of
conscious life, not of dead matter." But what will give such a thing the specific character of what goes on?--so long
as we think of it as a process. And now it seems to us as if intending could not be any process at all, of any kind
whatever.--For what we are dissatisfied with here is the grammar of process, not the specific kind of process.--It
could be said: we should call any process "dead" in this sense.
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237. It might almost be said: "Meaning moves, whereas a process stands still."
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238. One says: How can these gestures, this way of holding the hand, this picture, be the wish that such and such
were the case? It is nothing more than a hand over a table and there it is, alone and without a sense. Like a single bit
of scenery from the production of a play, which has been left by itself in a room. It had life only in the play.
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239. "At that moment the thought was before my mind."--And how?--"I had this picture."--So was the picture the
thought? No; for if I had just told someone the picture, he would not have got the thought.
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240. The picture was the key. Or it seemed like a key.
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241. Let us imagine a picture story in schematic pictures, and thus more like the narrative in a language than a series
of realistic pictures. Using such a picture-language we might in particular e.g. keep our hold on the course of battles.
(Language-game.) And a sentence of our word-language approximates to a picture in this picture language much
more closely than we think.
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242. Let us remember too that we don't have to translate such pictures into realistic ones in order to 'understand'
them, any more than we ever translate photographs or film pictures into coloured pictures, although black-and-white
men or plants in reality would strike us as unspeakably strange and frightful.
Suppose we were to say at this point: "Something is a picture only in a picture-language"?
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243. Certainly I read a story and don't give a hang about any system of language. I simply read, have impressions,
see pictures in my mind's eye, etc.. I make the story pass before me like pictures, like a cartoon story. (Of course I do
not mean by this that every sentence summons up one or more visual images, and that that is, say, the purpose of a
sentence.)
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244. "Sentences serve to describe how things are", we think. The sentence as a picture.
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245. I understand the picture exactly, I could model it in clay.--I understand this description exactly, I could make a
drawing from it.
In many cases we might set it up as a criterion of understanding, that one had to be able to represent the
sense of a sentence in a drawing (I am thinking of an officially instituted test of understanding). How is one
examined in map-reading, for example?
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246. And the significant picture is what can not merely be drawn, but also represented plastically. And saying this
would make sense. But the thinking of a sentence is not an activity which one does from the words (like, say, singing
from the notes). The following example shews this. Does it make sense to say "I have as many friends as the number
yielded by a solution of the equation... "? One can't immediately see whether this makes sense, from the equation.
And so while one is reading the sentence one doesn't know whether or not it can be thought. Whether or not it can
be understood.
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247. For what does it mean "to discover that a sentence does not make sense"?
And what does this mean: "if I mean something by it, surely it must make sense"?
The first presumably means: not to be misled by the appearance of a sentence and to investigate its
application in the language-game.
And "if I mean something by it"--does that mean something like: "if I can imagine something in connexion
with it"?--An image often leads on to a further application.
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248. (Something that at first sight looks like a sentence and is not one.) The following design for the construction of
a steamroller. The motor is in the inside of the hollow roller. The crank-shaft runs through the middle of the roller
and is connected at both ends by spokes with the wall of the roller. The cylinder of the motor is fixed onto the inside
of the roller. At first glance this construction looks like a machine. But it is a rigid system and the piston cannot
move to and fro in the cylinder. Unwittingly we have deprived it of all possbility [[sic]] of movement.
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249. "Nothing easier than to imagine a four-dimensional cube! It looks like this:†1
--But I don't mean that, I mean something like
but with four dimensions!--"But isn't what I showed you like
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only with four dimensions?"--No; I don't mean that!--But what do I mean? What is my picture? Well, it is not the
four-dimensional cube as you drew it. I have now for a picture only the words and my rejection of anything you can
show me.
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250. Are roses red in the dark?--One can think of the rose in the dark as red.--
(That one can 'imagine' something does not mean that it makes sense to say it.)
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251. "The supposition that this person--who behaves quite normally--is nevertheless blind, surely makes
sense!"--That means: 'after all it is a supposition.' 'I surely can actually suppose something like that.' And that means:
I picture the thing I am supposing. Very well: but does it go any further than that? If in other circumstances, I
suppose that someone is blind, I never assure myself that this assumption really makes sense. And my actually
imagining something, picturing something, as I make the assumption, plays no part at all in that case. This picture
only becomes important here, where it is so to speak the only thing that gives a handle for thinking that I really have
supposed something. That is all that is left of there being an assumption here.
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252. "I can quite well imagine someone acting like that and nevertheless seeing nothing shameful in the action."
There follows a description shewing how this is to be imagined.
"I can imagine a human society in which it counts as dishonest to calculate, except as a pastime." That means
roughly the same as: I could easily fill this picture out with more detail.
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253. "I haven't ever in fact seen a black patch gradually getting lighter until it was white and then the white turning
more and more reddish, until it was red. But I know that it is possible, because I can imagine it."
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254. (When you are talking with someone about some division of time, you often take out your watch, not to see
what time it is, but to help form a picture of the division being considered.)
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255. How can one learn the truth by thinking? As one learns to see a face better if one draws it.
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256. Philosophers who think that one can as it were use thought to make an extension of experience, should think
about the fact that one can transmit talk, but not measles, by telephone.
Nor can I experience time as limited, when I want to, or my visual field as homogeneous etc.†1
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257. Would it be possible to discover a new colour? For a colour-blind man is in the same situation as we are, his
colours form just as complete a system as ours do; he doesn't see any gaps where the remaining colours belong.
(Comparison with mathematics.)†2
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258. Generality in logic cannot be extended any further than our logical foresight reaches. Or better: than our logical
vision reaches.
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259. "But how can human understanding outstrip reality and itself think the unverifiable?"--Why should we not say
the unverifiable? For we ourselves made it unverifiable.
A false appearance is produced? And how can it so much as look like that? For don't you want to say that
this like that is not a description at all? Well, then it isn't a false appearance either, but rather one that robs us of our
orientation. So that we clutch our brows and ask: How can that be?
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260. It is only apparently possible "to transcend any possible experience", even these words only seem to make
sense, because they are arranged on the analogy of significant expressions.
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261. The "philosophy of as if" itself rests wholly on this shifting between simile and reality.
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262. "But I can't anticipate reality in my thoughts, using words to sneak in something I am not acquainted with."
(Nihil est in intellectu....)
As if I could as it were get round and approach from behind in thought, and so snatch a glimpse of what it is
impossible to see from the front."
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263. Hence there is something right about saying that unimaginability is a criterion for nonsensicality.
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264. Suppose someone were to say "I can't imagine what it is like for someone to see a chair, except precisely when I
see it"? Would he be justified in saying this?
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265. Am I justified in saying: "I cannot see |||||||||| as a shape"?
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What justifies me? (What justifies the blind man in saying he cannot see?)
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266. Can you imagine absolute pitch, if you have not got it?--Can you imagine it if you have it?--Can a blind man
imagine seeing? Can I imagine it?--Can I imagine spontaneously reacting thus and so, if I don't do it? Can I imagine
it any better, if I do do it? ((Belongs to the question: can I imagine someone seeing |||||||||| as an articulated shape.))
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267. Is it supposed to be an empirical fact that someone who has had an experience can imagine it, and that
someone else can not? (How do I know that a blind man can imagine colours?) But: he cannot play a certain
language game (cannot learn it). But is this empirical, or is it the case eo ipso? The latter.
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268. What should we say to someone who asserted that he could imagine exactly what it is like to have absolute
pitch without having it?
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269. If we think we can imagine four-dimensional space, why not also four-dimensional colours, that is colours
which, besides degree of saturation, hue, and brightness, allowed of a fourth determination?†1
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270. "How can it make sense to speak of a kind of sense-perception which is quite new to me, which I shall perhaps
have some time? If that is, you do not want to speak of a sense organ."
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271. What purpose is served by a sentence like: "We can't in the least imagine the sensations of a conjurer like
Rastelli"?
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272. "It makes sense to speak of an endless row of trees; I can surely imagine a row of trees going on without end."
That means something like: If it makes sense to say the row of trees comes to an end here, then it makes sense to say
[it doesn't come to an end here, and so also that it never comes to an end.]†1 Ramsey used to reply to such
questions: "But it just is possible to think of such a thing." As, perhaps, one says: "Technology achieves things
nowadays which you can't imagine at all."--Well here one has to find out what you are thinking. (Your asseveration
that this phrase can be thought--what can I do with that? For that's not the point. Its purpose is not that of causing a
fog to rise in your mind.) What you mean--how is that to be discovered? We must patiently examine how this
sentence is supposed to be applied. What things look like round about it. Then its tense will come to light.
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273. Hardy: "That 'the finite cannot understand the infinite' should surely be a theological and not a mathematical
war-cry." True, the expression is inept. But what people are using it to try and say is: "We mustn't have any juggling!
How comes this leap from the finite to the infinite?" Nor is the expression all that nonsensical--only the 'finite' that
can't conceive the infinite is not 'man' or 'our understanding', but the calculus. And HOW this conceives the infinite
is well worth an investigation. This may be compared to the way a chartered accountant precisely investigates and
clarifies the conduct of a business undertaking. The aim is a synoptic comparative account of all the applications,
illustrations, conceptions of the calculus. The complete survey of everything that may produce unclarity. And this
survey must extend over a wide domain, for the roots of our ideas reach a long way.--"The finite cannot understand
the infinite" means here: It cannot work in the way you, with characteristic superficiality, are presenting it.
Thought can as it were fly, it doesn't have to walk. You do not understand your own transactions, that is to
say you do not have a synoptic view of them, and you as it were project your lack of understanding into the idea of
a medium in which the most astounding things are possible.
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274. The 'actual infinite' is a 'mere word'. It would be better to say: for the moment this expression merely produces a
picture -which still hangs in the air: you owe us an account of its application.
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275. An infinitely long row of marbles, an infinitely long rod. Imagine these coming in in some kind of fairy tale.
What application--even though a fictitious one---might be made of this concept? Let us ask now, not "Can there be
such a thing?" but "What do we imagine?" So give free rein to your imagination. You can have things now just as
you choose. You only need to say how you want them. So (just) make a verbal picture, illustrate it as you
choose--by drawing, comparisons, etc.! Thus you can--as it were--prepare a blueprint.--And now there remains the
question how to work from it. [Marginal note: Belongs with 'It depends on the service'.]
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276. I believe I see a design drawn very fine in a bit of a series; which only stands in need of "and so on" to reach to
infinity.
"I see a distinctive character in it."--Well, presumably something that corresponds to the algebraic
expression.--"Yes, only nothing written, but positively something ethereal."--What a queer picture.--"Something that
is not the algebraic expression, something for which this is only the expression!"†1
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277. I see something in it--like a shape in a puzzle picture. And if I see that, I say "That is all I need."--If you find the
signpost, you don't now look for further instruction--you walk. (And if instead of "you walk" I were to say "you go
by it" the difference between the two expressions might be only that the second one alludes to certain psychological
accompaniments.)
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278. What does it mean to say: a straight line can be arbitrarily produced? Is there not an "and so on ad inf." here
which is quite different from that of mathematical induction? According to the foregoing there would exist the
expression for the possibility of producing the line, in the sense of the description of the produced part, or of its
production. Here at first sight there does not seem to be any question of numbers. I can imagine the pencil that
draws the line continuing its movement and keeping
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on going the same way. But is it also conceivable that there should be no possibility of accompanying this process
with some countable process? I believe not.†1
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279. When do we say: "The line intimates this to me like a rule--always the same"? And on the other hand: "It keeps
on intimating to me what I have to do--it is not a rule"?
In the first case the thought is: I have no further court of appeal for what I have to do. The rule does it all by
itself: I only have to obey it--(and obeying is one thing!). I do not feel, for example: it is queer that the line always
tells me something.--The other proposition says: I do not know what I am going to do: the line will tell me.
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280. One might imagine someone multiplying, correctly multiplying, with such feelings as these; he keeps on saying
"I don't know--now the rule suddenly intimates this to me!"--and we answer "Of course; you are going on quite in
accord with the rule."
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281. To say that the points yielded in this experiment lie roughly on this line, e.g. on a straight line, means something
like: "Seen from this distance they seem to lie on a straight line."
I may say that a stretch gives the general impression of a straight line; but I cannot say this of the line
; although it would be possible to see it as a bit of a longer line in which the deviations from the
straight were lost. I cannot say: "This bit of line looks straight, for it may be a bit of a line that as a whole gives me
the impression of being straight." (Mountains on the earth and the moon. The earth a ball.)†2
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282. "It intimates this or that to me, irresponsibly" means: I cannot teach you how I follow the line. I do not
presuppose that you will follow it as I do, even when you do follow it.
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283. What does it mean to understand that something is an order, although one does not yet understand the order
itself? ("He means I am to do something--but I don't know what he wants.")
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284. Of course the proposition "I must understand the order before I can act on it" makes good sense: but not a
metalogical sense.
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285. The idea that one has of understanding in this connexion is roughly that through it one gets a stage nearer from
the words to the execution.--In what sense is this right?
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286. "But I must understand an order to be able to act according to it." Here the "must" is fishy.
Think too of the question "How long before obeying it must you understand the order?"
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287. "I cannot carry out the order because I don't understand what you mean.--Yes, I understand you now."--What
went on when I suddenly understood him? Here there were various possibilities. The order may for example have
been given with a wrong emphasis; and the right emphasis suddenly occurred to me. In that case I should say to a
third party "Now I understand him, he means..." and should repeat the order with the right emphasis. And now, with
the right emphasis, I should understand him; that is, I did not now have further to grasp a sense (something outside
the sentence, hence something ethereal) but the familiar sound of English words perfectly suffices me.--Or the order
was given me in comprehensible English, but seemed preposterous. Then an explanation occurs to me; and now I
can carry it out.--Or several interpretations may have passed through my mind, and I eventually decide on one of
them.
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288. If the order is not executed--where in that case is that shadow of its execution which you think you see;
because the form: "He ordered such-and-such" swam before your mind.
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289. If the meaning-connexion can be set up before the order, then it can also be set up afterwards.
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290. "He did what I told him."--Why should one not say here: There is an identity between action and WORD?!
Why should I interpose a shadow between the two? Indeed we have a method of projection.--Only there is a
different identity in: "I did what he did" and in: "I did what he ordered."
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291. The lines of projection might be called the "connexion between the picture and what it depicts"; but so too
might the technique of projection.
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292. The ambiguity of our ways of expressing ourselves: If an order were given us in code with the key for
translating it into English, we might call the procedure of constructing the English form of the order "derivation of
what we have to do from the code" or "derivation of what executing the order is". If on the other hand we act
according to the order, obey it, here too in certain cases one may speak of a derivation of the execution.
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293. I give the rules of a game. The other party makes a move, perfectly in accord with the rules, whose possibility I
had not foreseen, and which spoils the game, that is, as I had wanted it to be. I now have to say: "I gave bad rules; I
must change or perhaps add to my rules."
So in this way have I a picture of the game in advance? In a sense: Yes.
It was surely possible, for example, for me not to have foreseen that a quadratic equation need have no real
root.
Thus the rule leads me to something of which I say: "I did not expect this pattern: I imagined a solution
always like this...."
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294. In one case we make a move in an existent game, in the other we establish a rule of the game. Moving a piece
could be conceived in these two ways: as a paradigm for future moves, or as a move in an actual game.
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295. You must remember that there may be such a language-game as 'continuing a series of digits' in which no rule,
no expression of a rule is ever given, but learning happens only through examples. So that the idea that every step
should be justified by a something--a sort of pattern--in our mind, would be alien to these people.
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296. How queer: It looks as if a physical (mechanical) form of guidance could misfire and let in something
unforeseen, but not a rule! As if a rule were, so to speak, the only reliable form of
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guidance. But what does guidance not allowing a movement, and a rule's not allowing it, consist in?--How does one
know the one and how the other?
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297. "How do I manage always to use a word correctly--i.e. significantly; do I keep on consulting a grammar? No;
the fact that I mean something--the thing I mean, prevents me from talking nonsense."--"I mean something by the
words" here means: I know that I can apply them.
I may however believe I can apply them, when it turns out that I was wrong.
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298. From this it does not follow that understanding is the activity by which we shew that we understand. It is
misleading to ask whether it is this activity. The question ought not to be conceived as: "Is understanding this
activity then, isn't it a different one instead?"--But rather as: "Is 'understanding' used to designate this activity--isn't
its use different?"
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299. We say: "If you really follow the rule in multiplying, it MUST come out the same." Now, when this is merely
the slightly hysterical style of university talk, we have no need to be particularly interested. It is however the
expression of an attitude towards the technique of multiplying, which comes out everywhere in our lives. The
emphasis of the 'must' corresponds only to the inexorability of this attitude, not merely towards the technique of
calculating, but also towards innumerable related practices.†1
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300. With the words "This number is the right continuation of this series" I may bring it about that for the future
someone calls such-and-such the "right continuation". What 'such-and-such' is I can only show in examples. That is,
I teach him to continue a series (basic series), without using any expression of the 'law of the series'; rather, I am
forming a substratum for the meaning of algebraic rules or what is like them.
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301. He must go on like this without a reason. Not, however, because he cannot yet grasp the reason but
because--in this system--there is no reason. ("The chain of reasons comes to an end.")
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And the like this (in "go on like this") is signified by a number, a value. For at this level the expression of the
rule is explained by the value, not the value by the rule.
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302. For just where one says "But don't you see...?" the rule is no use, it is what is explained, not what does the
explaining.
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303. "He grasps the rule intuitively."--But why the rule? Why not how he is to continue?
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304. "Once he has seen the right thing, seen the one of infinitely many references which I am trying to push him
towards--once he has got hold of it, he will continue the series right without further ado. I grant that he can only
guess (intuitively guess) the reference that I mean--but once he has managed that the game is won." But this 'right
thing' that I mean does not exist. The comparison is wrong. There is no such thing here as, so to say, a wheel that he
is to catch hold of, the right machine which, once chosen, will carry him on automatically. It could be that something
of the sort happens in our brain but that is not our concern.
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305. "Do the same." But in saying this I must point to the rule. So its application must already have been learnt. For
otherwise what meaning will its expression have for him?
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306. To guess the meaning of a rule, to grasp it intuitively, could surely mean nothing but: to guess its application.
And that can't now mean: to guess the kind of application; the rule for it. Nor does guessing come in here.
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307. I might e.g. guess what continuation will give the other pleasure (by his expression, perhaps). The application of
a rule can be guessed only when there is already a choice between different applications.
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308. We might in that case also imagine that, instead of 'guessing the application of the rule,' he invents it. Well,
what would that look like? Ought he perhaps to say "Following the rule + 1 may mean writing 1, 1 + 1, 1 + 1 + 1, and
so on"? But what does he mean by that? For the "and so on" presupposes that one has already mastered a
technique.
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Instead of "and so on" he might also have said: "Now you know what I mean." And his explanation would
simply be a definition of the expression "following the rule + 1". This would have been his discovery.
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309. We copy the numerals from 1 to 100, say, and this is the way we infer, think.
I might put it this way: If I copy the numerals from 1 to 100--how do I know that I shall get a series of
numerals that is right when I count them? And here what is a check on what? Or how am I to describe the important
empirical fact here? Am I to say experience teaches that I always count the same way? Or that none of the numerals
gets lost in copying? Or that the numerals remain on the paper as they are, even when I don't watch them? Or all
these facts? Or am I to say that we simply don't get into difficulties? Or that almost always everything seems all right
to us?
This is how we think. This is how we act. This is how we talk about it.
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310. Imagine you had to describe how humans learn to count (in the decimal system, for example). You describe
what the teacher says and does and how the pupil reacts to it. What the teacher says and does will include e.g. words
and gestures which are supposed to encourage the pupil to continue a sequence; and also expressions such as "Now
he can count". Now should the description which I give of the process of teaching and learning include, in addition
to the teacher's words, my own judgment: the pupil can count now, or: now the pupil has understood the numeral
system? If I do not include such a judgment in the description--is it incomplete? And if I do include it, am I going
beyond pure description?--Can I refrain from that judgment, giving as my ground: "That is all that happens"?
Page 56
311. Must I not rather ask: "What does the description do anyway? What purpose does it serve?"--In another
context, indeed, we know what is a complete and what an incomplete description. Ask yourself: How do we use the
expressions "complete" and "incomplete description"?
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Giving a complete (or incomplete) report of a speech. Is it part of this to report the tone of voice, the play of
expression, the genuineness or falsity of feeling, the intentions of the speaker, the strain of speaking? Whether this or
that belongs to a complete description will depend on the purpose of the description, on what the recipient does with
the description.
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312. The expression "that is all that happens" sets limits to what we call "happening".
Page 57
313. Here the temptation is overwhelming to say something further, when everything has already been
described.--Whence this pressure? What analogy, what wrong interpretation produces it?
Page 57
314. Here we come up against a remarkable and characteristic phenomenon in philosophical investigation: the
difficulty--I might say--is not that of finding the solution but rather that of recognizing as the solution something that
looks as if it were only a preliminary to it. "We have already said everything.--Not anything that follows from this,
no, this itself is the solution!"
This is connected, I believe, with our wrongly expecting an explanation, whereas the solution of the difficulty
is a description, if we give it the right place in our considerations. If we dwell upon it, and do not try to get beyond it.
The difficulty here is: to stop.
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315. "Why do you demand explanations? If they are given you, you will once more be facing a terminus. They
cannot get you any further than you are at present."
Page 57
316. A red object can be used as a sample for painting a reddish white or a reddish yellow (etc.)--but can it also be
used as a sample for painting a shade of bluish green?--Suppose I saw someone, with all the outward signs of
making an exact copy, 'reproducing' a red patch bluish green?--I should say: "I don't know how he's doing it" or
even "I don't know what he's doing."--But supposing that he now 'copied' this shade of red bluish green on various
occasions, and perhaps other shades of red systematically other shades of bluish green--ought I now to say he's
copying, or he isn't copying?
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But what does it mean to say that I don't know 'what he's doing'? For can't I see what he's doing?--But I can't
see into him.--Avoid that comparison! Suppose I see him copying red as red--what do I know here? Do I know how
I do it? Of course one says: "I'm just painting the same colour."--But suppose he says: "And I'm painting the fifth of
this colour"? Can I see a special mediating process when I paint the 'same' colour?
Assume I know him for an honest man; he reproduces a red, as I described, by a bluish green--but now not
the same shade by always the same shade, but sometimes one, sometimes another. Am I to say "I don't know what
he's doing?"--He does what I can see--but I should never do it; I don't know, why he does it; his proceeding 'is
unintelligible to me'.
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317. We might imagine a negative portrait, that is one that is supposed to represent how Herr N does not look (and
so is a bad portrait if it looks like N).
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318. I cannot describe how (in general) to employ rules, except by teaching you, training you to employ rules.
Page 58
319. I may now e.g. make a talkie of such instruction. The teacher will sometimes say "That's right." If the pupil
should ask him "Why?"--he will answer nothing, or at any rate nothing relevant, not even: "Well, because we all do it
like that"; that will not be the reason.
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320. Why don't I call cookery rules arbitrary, and why am I tempted to call the rules of grammar arbitrary? Because
'cookery' is defined by its end, whereas 'speaking' is not. That is why the use of language is in a certain sense
autonomous, as cooking and washing are not. You cook badly if you are guided in your cooking by rules other than
the right ones; but if you follow other rules than those of chess you are playing another game; and if you follow
grammatical rules other than such-and-such ones, that does not mean you say something wrong, no, you are
speaking of something else.
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321. When a rule concerning a word in it is appended to a sentence, the sense does not change.
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322. Language is not defined for us as an arrangement fulfilling a definite purpose. Rather "language" is for us a
name for a collection, and I understand it as including German, English and so on, and further various systems of
signs which have more or less affinity with these languages.
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323. Being acquainted with many languages prevents us from taking quite seriously a philosophy which is laid down
in the forms of any one. But here we are blind to the fact that we ourselves have strong prejudices for, and against,
certain forms of expression; that this very piling up of a lot of languages results in our having a particular picture.
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324. Does a child learn only to talk, or also to think? Does it learn the sense of multiplication before--or after it learns
multiplication?
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325. How did I arrive at the concept 'sentence' or 'language'? Surely only through the languages that I have
learnt.--But they seem to me in a certain sense to have led beyond themselves, for I am now able to construct new
language, e.g. to invent words.--So such construction also belongs to the concept of language. But only because that
is how I want to fix the concept.
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326. The concept of a living being has the same indeterminacy as that of a language.
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327. Compare: inventing a game--inventing language--inventing a machine.
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328. In philosophy it is significant that such-and-such a sentence makes no sense; but also that it sounds funny.
Page 59
329. I make a plan not merely so as to make myself understood but also in order to get clear about the matter myself.
(I.e. language is not merely a means of communication.)
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330. What does it mean to say: "But that's no longer the same game!" How do I use this sentence? As information?
Well, perhaps to introduce some information in which differences are enumerated and their consequences explained.
But also to express that just for that reason I don't join in here, or at any rate take up a different attitude to the game.
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331. One is tempted to justify rules of grammar by sentences like "But there really are four primary colours". And
the saying that the rules of grammar are arbitrary is directed against the possibility of this justification, which is
constructed on the model of justifying a sentence by pointing to what verifies it.
Yet can't it after all be said that in some sense or other the grammar of colour-words characterizes the world
as it actually is? One would like to say: May I not really look in vain for a fifth primary colour? Doesn't one put the
primary colours together because there is a similarity among them, or at least put colours together, contrasting them
with e.g. shapes or notes, because there is a similarity among them? Or, when I set this up as the right way of
dividing up the world, have I a preconceived idea in my head as a paradigm? Of which in that case I can only say:
"Yes, that is the kind of way we look at things" or "We just do want to form this sort of picture." For if I say "there is
a particular similarity among the primary colours"--whence do I derive the idea of this similarity? Just as the idea
'primary colour' is nothing else but 'blue or red or green or yellow'--is not the idea of that similarity too given simply
by the four colours? Indeed, aren't they the same?--"Then might one also take red, green and circular
together?"--Why not?!
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332. Do not believe that you have the concept of colour within you because you look at a coloured object--however
you look.
(Any more than you possess the concept of a negative number by having debts.)
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333. "Red is something specific"--that would have to mean the same as: "That is something specific"--said while
pointing to something red. But for that to be intelligible, one would have already to mean our concept 'red', to mean
the use of that sample.
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334. I can indeed obviously express an expectation at one time by the words "I'm expecting a red circle," and at
another by putting a coloured picture of a red circle in the place of the last few words. But in this expression there
are not two things corresponding to the two separate words "red" and "circle". So the expression in the second
language is of a completely different kind.
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335. There might be another language, besides this last one, in which 'red circle' was expressed by the juxtaposition
of a circle and a red patch.
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336. Now if I have two signs at hand, the expression "red circle" and the coloured picture, or image, of the red circle,
then surely the question would be: How is the one word correlated with the shape, the other with the colour?
For it seems possible to say: one word turns the attention to the colour, the other to the shape. But what does
that mean? How can these separate words be translated into this pattern?
Or again: If the word "red" summons up a colour in my memory, it must surely be in connexion with a
shape; in that case how can I abstract from the shape?
The important question here is never: how does he know what to abstract from? but: how is this possible at
all? or: what does it mean?
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337. Perhaps it becomes clearer if we compare these two languages: in one the phrase "red circle" is replaced by a
red slip and a slip with a circle on it (say black on a white ground); and in the other by a red circle.
For how does translating proceed here? Say one looks at the red slip and chooses a red pencil, then at the
circle, and now one makes a circle with this pencil.
It would first of all have been learnt that the first slip always determines the choice of the pencil, and the
second what we should draw with it. Thus the two slips belong to two different parts of speech (say noun and
activity-word). But in the other language there would be nothing that could be called two different words.
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338. If someone were to say: "Red is complex"--we could not guess what he was alluding to, what he was trying to
do with this sentence. But if he says "This chair is complex," we may indeed not know straight off which kind of
complexity he is talking about, but we can straight away think of more than one sense for his assertion.
Now what kind of fact am I drawing attention to here?
At any rate it is an important fact.--We are not familiar with any technique, to which that sentence might be
alluding.
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339. We are here describing a language-game that we cannot learn.
Page 62
340. "In that case something quite different must be going on in him, something that we are not acquainted
with."--This shews us what we go by in determining whether something that takes place 'in another' is different from,
or the same as in ourselves. This shews us what we go by in judging inner processes.
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341. Can you imagine what a red-green colour blind man sees? Can you paint a picture of the room as he sees it?
Can he paint it as he sees it? Then can I paint it as I see it? In what sense can I?
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342. "If someone saw only grey, black and white, he would have to be given something if he were to know what red,
green etc. are." And what would he have to be given? Well, the colours. And so, for example, this and this and this.
(Imagine, e.g. that coloured patterns had to be introduced into his brain in addition to the merely grey and black
ones.) But would this have to happen for the purpose of future action? Or does this action itself involve these
patterns? Am I trying to say: "Something would have to be given him, for it is clear that otherwise he could
not..."--or: His seeing behaviour contains new constituents?
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343. Again: what should we call an "explanation of seeing"? Is one to say: Well, you surely know what
"explanation" means elsewhere; so employ this concept here too.
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344. Can I say: "Look at it! Then you'll see that it can't be explained!"--Or: "Drink in the colour red, then you'll see
that it can never be presented by anything else."--And if the other man now agrees with me, does that shew that he
has drunk in the same as I?--And what is the significance of our inclination to say this? Red seems to stand there,
isolated. Why? What is the value of this appearance, this inclination of ours?
But one might ask: What peculiarity of the concept does this inclination point to?
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345. Think of the sentence: "Red is not a mixed colour" and of its function.
For the language-game with colours is characterized by what we can do and what we cannot do.
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346. "There is no such thing as a reddish green" is akin to the sentences that we use as axioms in mathematics.
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347. The fact that we calculate with certain concepts and not with others only shews how various in kind conceptual
tools are (how little reason we have here ever to assume uniformity). [Marginal Note: On propositions about colours
that are like mathematical ones e.g. Blue is darker than white. On this Goethe's Theory of Colour.]
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348. "The possibility of agreement involves some sort of agreement already."--Suppose someone were to say: "Being
able to play chess is a sort of playing chess"!
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349. It is very difficult to describe paths of thought where there are already many lines of thought laid down,--your
own or other people's--and not to get into one of the grooves. It is difficult to deviate from an old line of thought just
a little.
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350. "It is as if our concepts involved a scaffolding of facts."
That would presumably mean: If you imagine certain facts otherwise, describe them otherwise, than the way
they are, then you can no longer imagine the application of certain concepts, because the rules for their application
have no analogue in the new circumstances.--So what I am saying comes to this: A law is given for human beings,
and a jurisprudent may well be capable of drawing consequences for any case that ordinarily comes his way; thus
the law evidently has its use, makes sense. Nevertheless its validity presupposes all sorts of things, and if the being
that he is to judge is quite deviant from ordinary human beings, then e.g. the decision whether he has done a deed
with evil intent will become not difficult but (simply) impossible.
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351. "If humans were not in general agreed about the colours of things, if undetermined cases were not exceptional,
then our concept of colour could not exist." No:--our concept would not exist.
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352. Do I want to say, then, that certain facts are favourable to the formation of certain concepts; or again
unfavourable? And does experience teach us this? It is a fact of experience that human beings alter their concepts,
exchange them for others
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when they learn new facts; when in this way what was formerly important to them becomes unimportant, and vice
versa. (It is discovered e.g. that what formerly counted as a difference in kind, is really only a difference in degree.)
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353. But may it not be said: "If there were only one substance, there would be no use for the word 'substance'"? That
however presumably means: The concept 'substance' presupposes the concept 'difference of substance'. (As that of
the king in chess presupposes that of a move in chess, or that of colour that of colours.)
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354. I want to say that there is a geometrical gap, not a physical one, between green and red.†1
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355. But doesn't anything physical correspond to it? I do not deny that. (And suppose it were merely our habituation
to these concepts, to these language-games? But I am not saying that it is so.) If we teach a human being
such-and-such a technique by means of examples,--that he then proceeds like this and not like that in a particular
new case, or that in this case he gets stuck, and thus that this and not that is the 'natural' continuation for him: this of
itself is an extremely important fact of nature.
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356. "But if by 'bluish yellow' I mean green, I am taking this expression in a different way from the original one. The
original conception signifies a different road, a no thoroughfare."
But what is the right simile here? That of a road that is physically impassable, or of the non-existence of a
road? i.e. is it one of physical or of mathematical impossibility?
Page 64
357. We have a colour system as we have a number system.
Do the systems reside in our nature or in the nature of things? How are we to put it?--Not in the nature of
numbers or colours.
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358. Then is there something arbitrary about this system? Yes and no. It is akin both to what is arbitrary and to what
is non-arbitrary.
Page 65
359. It is obvious at a glance that we aren't willing to acknowledge anything as a colour intermediate between red and
green. (Nor does it matter whether this is always obvious, or whether it takes experience and education to make it
so.)
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360. 'a is between b and c, and nearer to b than to c': this is a characteristic relation between sensations of the same
kind. That is, there is e.g. a language-game with the order "Produce a sensation between this and this, and nearer the
first than the second." And also "Name two sensations which this is between."
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361. And here it is important that e.g. with grey one will get "black and white" for answer, with purple "blue and
red", with pink "red and white", but with olive green one will not get "red and green."
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362. These people are acquainted with reddish green--"But there is no such thing!"--What an extraordinary
sentence.--(How do you know?)
Page 65
363. Let's just put it like this: Must these people notice the discrepancy? Perhaps they are too stupid. And again:
perhaps not that either.--
Page 65
364. Yes, but has nature nothing to say here? Indeed she has--but she makes herself audible in another way.
"You'll surely run up against existence and non-existence somewhere!" But that means against facts, not
concepts.
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365. It is an extremely important fact that a colour which we are inclined to call (e.g.) "reddish yellow" can really be
produced (in various ways) by a mixture of red and yellow. And that we are not able to recognize straight off a
colour that has come about by mixing red and green as one that can be produced in that way. (But what does
"straight off" signify here?)
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366. Confusion of tastes: I say "This is sweet", someone else "This is sour" and so on. So someone comes along and
says: "You have none of you any idea what you are talking about. You no longer know at all what you once called a
taste." What would be the sign of our still knowing? ((Connects with a question about confusion in calculating.))
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367. But might we not play a language-game even in this 'confusion'?--But is it still the earlier one?--†1
Page 66
368. Let us imagine men who express a colour intermediate between red and yellow, say by means of a fraction in a
kind of binary notation like this: R, LLRL and the like, where we have (say) yellow on the right, and red on the
left.--These people learn how to describe shades of colour in this way in the kindergarten, how to use such
descriptions in picking colours out, in mixing them, etc. They would be related to us roughly as people with absolute
pitch are to those who lack it. They can do what we cannot.
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369. And here one would like to say: "But then, is it imaginable? Of course, the behaviour is! But is the inner
process, the experience of colour?" And it is difficult to see what to say in answer to such a question. Could people
without absolute pitch have guessed at the existence of people with absolute pitch?
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370. High-lights or reflections: when a child paints it will never paint these. Indeed it is quite hard to believe that they
can be represented by ordinary oil or water colours.
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371. What would a society all of deaf men be like? Or a society of the 'feebleminded'? An important question! What
then of a society that never played many of our customary language-games?
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372. One imagines the feeble-minded under the aspect of the degenerate, the essentially incomplete, as it were in
tatters. And so under that of disorder instead of a more primitive order (which would be a far more fruitful way of
looking at them).
We just don't see a society of such people.
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373. Concepts other than though akin to ours might seem very queer to us; deviations from the usual in an unusual
direction.
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374. Concepts with fixed limits would demand a uniformity of behaviour. But where I am certain, someone else is
uncertain. And that is a fact of nature.
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375. These are the fixed rails along which all our thinking runs, and so our judgment and action goes according to
them too.
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376. Where e.g. a certain type is only seldom to be found, no concept of that type will be formed. People do not feel
this as a unity, as a particular physiognomy.
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377. They make no picture of it and always recognize it just in the particular cases.
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378. Must people be acquainted with the concept of modesty or of swaggering, wherever there are modest and
swaggering men? Perhaps nothing hangs on this difference for them.
For us, too, many differences are unimportant, which we might find important.
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379. And others have concepts that cut across ours.
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380. A tribe has two concepts, akin to our 'pain'. One is applied where there is visible damage and is linked with
tending, pity etc. The other is used for stomachache for example, and is tied up with mockery of anyone who
complains. "But then do they really not notice the similarity?"--Do we have a single concept everywhere where there
is a similarity? The question is: Is the similarity important to them? And need it be so? And why should their
concept 'pain' not split ours up?
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381. But in that case isn't this man overlooking something that is there?--He takes no notice of it, and why should
he?--But in that case his concept just is fundamentally different from ours.--Fundamentally different?
Different.--But in that case it surely is as if his word could not designate the same as ours. Or only part
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of that.--But of course it must look like that, if his concept is different. For the indefiniteness of our concept may be
projected for us into the object that the word designates. So that if the indefiniteness were missing we should also
not have 'the same thing meant'. The picture that we employ symbolizes the indefiniteness.
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382. In philosophizing we may not terminate a disease of thought. It must run its natural course, and slow cure is all
important. (That is why mathematicians are such bad philosophers.)
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383. Imagine that the people of a tribe were brought up from early youth to give no expression of feeling of any
kind. They find it childish, something to be got rid of. Let the training be severe. 'Pain' is not spoken of; especially
not in the form of a conjecture "Perhaps he has got...." If anyone complains, he is ridiculed or punished. There is no
such thing as the suspicion of shamming. Complaining is so to speak already shamming.
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384. "Shamming," these people might say, "What a ridiculous concept!" (As if one were to distinguish between a
murder with one shot and one with three.)
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385. Complaining is already so bad that there is no room at all for shamming as something worse.
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386. One disgrace is invisible to them because of the other.
Page 68
387. I want to say: an education quite different from ours might also be the foundation for quite different concepts.
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388. For here life would run on differently.--What interests us would not interest them. Here different concepts
would no longer be unimaginable. In fact, this is the only way in which essentially different concepts are imaginable.
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389. [Someone] might surely be taught e.g. to mime pain (not with the intention of deceiving). But could this be
taught to just anyone? I mean: someone might well learn to give certain crude tokens of pain, but without ever
spontaneously giving a
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finer imitation out of his own insight. (Talent for languages.) (A clever dog might perhaps be taught to give a kind of
whine of pain but it would never get as far as conscious imitation.)
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390. 'These men would have nothing human about them.' Why?--We could not possibly make ourselves understood
to them. Not even as we can to a dog. We could not find our feet with them.
And yet there surely could be such beings, who in other respects were human.
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391. I really want to say that scruples in thinking begin with (have their roots in) instinct. Or again: a language-game
does not have its origin in consideration. Consideration is part of a language-game.
And that is why a concept is in its element within the language-game.
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392. 'Heap of sand' is a concept without sharp boundaries--but why isn't one with sharp boundaries used instead of
it?--Is the reason to be found in the nature of the heaps? What is the phenomenon whose nature is definitive for our
concept?
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393. It is easy to imagine and work out in full detail events which, if they actually came about, would throw us out in
all our judgments.
If I were sometime to see quite new surroundings from my window instead of the long familiar ones, if
things, humans and animals were to behave as they never did before, then I should say something like "I have gone
mad"; but that would merely be an expression of giving up the attempt to know my way about. And the same thing
might befall me in mathematics. It might e.g. seem as if I kept on making mistakes in calculating, so that no answer
seemed reliable to me.
But the important thing about this for me is that there isn't any sharp line between such a condition and the
normal one. [Marginal note: Hangs together with the concept of 'knowledge'.]
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394. What would it mean for me to be wrong about his having a mind, having consciousness? And what would it
mean for me to be wrong about myself and not have any? What would it mean
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to say "I am not conscious"?--But don't I know that there is a consciousness in me?--Do I know it then, and yet the
statement that it is so has no purpose?
And how remarkable that one can learn to make oneself understood to others in these matters!
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395. A man can pretend to be unconscious; but conscious?
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396. What would it be like for someone to tell me with complete seriousness that he (really) did not know whether
he was dreaming or awake?--
Is the following situation possible: Someone says "I believe I am now dreaming"; he actually wakes up soon
afterwards, remembers that utterance in his dream and says "So I was right!"--This narrative can surely only signify:
Someone dreamt that he had said he was dreaming.
Imagine an unconscious man (anaesthetised, say) were to say "I am conscious"--should we say "He ought to
know"?
And if someone talked in his sleep and said "I am asleep"--should we say "He's quite right"?
Is someone speaking untruth if he says to me "I am not conscious"? (And truth, if he says it while
unconscious? And suppose a parrot says "I don't understand a word", or a gramophone: "I am only a machine"?)
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397. Suppose it were part of my day-dream to say: "I am merely engaged in phantasy", would this be true? Suppose
I write such a phantasy or narrative, an imaginary dialogue, and in it I say "I am engaged in phantasy"--but, when I
write it down,--how does it come out that these words belong to the phantasy and that I have not emerged from the
phantasy?
Might it not actually happen that a dreamer, as it were emerging from the dream, said in his sleep "I am
dreaming"? It is quite imaginable there should be such a language-game.
This hangs together with the problem of 'meaning'. For I can write "I am healthy" in the dialogue of a play,
and so not mean it, although it is true. The words belong to this and not that language-game.
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398. 'True' and 'false' in a dream. I dream that it is raining, and that I say "It is raining"--on the other hand: I dream
that I say "I am dreaming".
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399. Has the verb "to dream" a present tense? How does a person learn to use this?
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400. Suppose I were to have an experience like waking up, were then to find myself in quite different surroundings,
with people who assure me that I have been asleep. Suppose further I insisted that I had not been dreaming, but
living in some way outside my sleeping body. What function has this assertion?
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401. "'I have consciousness'--that is a statement about which no doubt is possible." Why should that not say the
same as: "'I have consciousness' is not a proposition"?
It might also be said: What's the harm if someone says that "I have consciousness" is a statement admitting
of no doubt? How do I come into conflict with him? Suppose someone were to say this to me--why shouldn't I get
used to making no answer to him instead of starting an argument? Why shouldn't I treat his words like his whistling
or humming?
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402. "Nothing is so certain as that I possess consciousness." In that case, why shouldn't I let the matter rest? This
certainty is like a mighty force whose point of application does not move, and so no work is accomplished by it.
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403. Remember: most people say one feels nothing under anaesthetic. But some say: It could be that one feels, and
simply forgets it completely.
If then there are here some who doubt and some whom no doubt assails, still the lack of doubt might after all
be far more general.
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404. Or doubt might after all have a different and much less indefinite form than in our world of thought.
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405. No one but a philosopher would say "I know that I have two hands"; but one may well say: "I am unable to
doubt that I have two hands."
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406. "Know," however, is not ordinarily used in this sense. "I know what 97 × 78 is." "I know that 97 × 78 is 432." In
the first case I tell someone that I can do something, that I possess something; in the second I simply asseverate that
97 × 78 is 432.
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For doesn't "97 × 78 is quite definitely 432" say: I know it is so? The first sentence is not an arithmetical one, nor can
it be replaced by an arithmetical one; an arithmetical sentence could be used in place of the second one.
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407. Can someone believe that 25 × 25 = 625? What does it mean to believe that? How does it come out that he
believes it?
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408. But isn't there a phenomenon of knowing, as it were quite apart from the sense of the phrase "I know"? Is it not
remarkable that a man can know something, can as it were have the fact within him?--But that is a wrong
picture.--For, it is said, it's only knowledge if things really are as he says. But that is not enough. It mustn't be just an
accident that they are. For he has got to know that he knows: for knowing is a state of his own mind; he cannot be in
doubt or error about it--apart from some special sort of blindness. If then knowledge that things are so is only
knowledge if they really are so; and if knowledge is in him so that he cannot go wrong about whether it is
knowledge; in that case, then, he is also infallible about things being so, just as he knows his knowledge; and so the
fact which he knows must be within him just like the knowledge.
And this does indeed point to one kind of use for "I know". "I know that it is so" then means: It is so, or else
I'm crazy.
So: when I say, without lying: "I know that it is so", then only through a special sort of blindness can I be
wrong.
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409. How does it come about that doubt is not subject to arbitrary choice?--And that being so--might not a child
doubt everything because it was so remarkably talented?
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410. A person can doubt only if he has learnt certain things; as he can miscalculate only if he has learnt to calculate.
In that case it is indeed involuntary.
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411. Imagine that a child was quite specially clever, so clever that he could at once be taught the doubtfulness of the
existence of all things. So he learns from the beginning: "That is probably a chair."
And now how does he learn the question: "Is it also really a chair?"--
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412. Am I doing child psychology?--I am making a connexion between the concept of teaching and the concept of
meaning.
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413. One man is a convinced realist, another a convinced idealist and teaches his children accordingly. In such an
important matter as the existence or non-existence of the external world they don't want to teach their children
anything wrong.
What will the children be taught? To include in what they say: "There are physical objects" or the opposite?
If someone does not believe in fairies, he does not need to teach his children "There are no fairies": he can
omit to teach them the word "fairy". On what occasion are they to say: "There are..." or "There are no..."? Only when
they meet people of the contrary belief.
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414. But the idealist will teach his children the word "chair" after all, for of course he wants to teach them to do this
and that, e.g. to fetch a chair. Then where will be the difference between what the idealist-educated children say and
the realist ones? Won't the difference only be one of battle cry?
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415. For doesn't the game "That is probably a..." begin with disillusion? And can the first attitude of all be directed
towards a possible disillusion?
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416. "So does he have to begin by being taught a false certainty?"
There isn't any question of certainty or uncertainty yet in their language-game. Remember: they are learning
to do something.
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417. The language-game "What is that?"--"A chair."--is not the same as: "What do you take that for?"--"It might be a
chair."
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418. To begin by teaching someone "That looks red" makes no sense. For he must say that spontaneously once he
has learnt what "red" means, i.e. has learnt the technique of using the word.
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419. Any explanation has its foundation in training. (Educators ought to remember this.)
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420. "It looks red to me."--"And what is red like?"--"Like this." Here the right paradigm must be pointed to.
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421. When he first learns the names of colours--what is taught him? Well, he learns e.g. to call out "red" on seeing
something red.--But is that the right description; or ought it to have gone: "He learns to call 'red' what we too call
'red'"?--Both descriptions are right.
What differentiates this from the language-game "How does it strike you?"?
But someone might be taught colour-vocabulary by being made to look at white objects through coloured
spectacles. What I teach him however must be a capacity. So he can now bring something red at an order; or
arrange objects according to colour. But then what is something red?
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422. Why doesn't one teach a child the language-game "It looks red to me" from the first? Because it is not yet able
to understand the rather fine distinction between seeming and being?
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423. The red visual impression is a new concept.
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424. The language-game that we teach him then is: "It looks to me..., it looks to you..." In the first language-game a
person does not occur as perceiving subject.
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425. You give the language game a new joint. Which does not mean, however, that now it is always used.
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426. The inward glance at the sensation--what connexion is this supposed to set up between words and sensation;
and what purpose is served by this connexion? Was I taught that when I learned to use this sentence, to think this
thought? (Thinking it really was something I had to learn.)
This is indeed something further that we learn, namely to turn our attention on to things and on to
sensations. We learn to observe and to describe observations. But how am I taught this; how is my 'inner activity'
checked in this case? How will it be judged whether I really have paid attention?
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427. "The chair is the same whether I am looking at it or not"--that need not have been true. People are often
embarrassed when one looks at them. "The chair goes on existing, whether I look at it or not." This might be treated
as an empirical proposition or it might be that we took it as a grammatical one. But it is also possible in this
connexion simply to think of the conceptual difference between sense-impression and object [Objekt].
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428. But isn't human agreement essential to the game? Must not anybody who learns it first know the meaning of
"same", and do not the presuppositions of this include agreement? And so on.
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429. You say "That is red," but how is it decided if you are right? Doesn't human agreement decide?--But do I
appeal to this agreement in my judgments of colour? Then is what goes on like this: I get a number of people to look
at an object; to each of them there occurs one of a certain group of words (what are called the "names of colours"); if
the word "red" occurred to the majority of the spectators (I myself need not belong to this majority) the predicate
"red" belongs to the object by rights. Such a technique might have its importance.
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430. Colour-words are explained like this: "That's red" e.g.--Our language game only works, of course, when a
certain agreement prevails, but the concept of agreement does not enter into the language-game. If agreement were
universal, we should be quite unacquainted with the concept of it.
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431. Does human agreement decide what is red? Is it decided by appeal to the majority? Were we taught to
determine colour in that way?
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432. For I describe the language-game "Bring something red" to someone who can himself already play it. Others I
might at most teach it. (Relativity.)
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433. "What I perceive is THIS--" and now follows a form of DESCRIPTION. The word "this" might also be
explained as follows: Let us imagine a direct transfer of experience.--But now what is our criterion for the
experience's really having been transferred? "Well, he just does have what I have."--But how does he 'have' it?
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434. What does it mean "to use a word as a designation, a name, of a sensation"? Isn't there something to investigate
here?
Imagine you were starting from a language-game with physical objects--and then it was said, from now on
sensations are going to be named too. Wouldn't that be as if first there were talk of
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transferring possessions and then suddenly of transferring joy in possession or pride in possession? Don't we have
to learn something new here? Something new, which we call "transferring" too.
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435. The description of what is subjectively seen is more or less akin to the description of an object, but just for that
reason does not function as a description of an object. How are visual sensations compared? How do I compare my
visual sensations with someone else's?
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436. "Verifying by inspection" is a wholly misleading expression. For it says that first of all a procedure, the
inspection, takes place (it might be compared with looking through a microscope, or with the procedure of turning
one's head round in order to see something). And that then the seeing has to succeed. One might speak of "seeing
by turning round," or "seeing by looking". But in that case the turning round (or looking) is a process external to the
seeing, a process that is thus of only practical concern. What one would like to say is "seeing by seeing".
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437. The causes of our belief in a proposition are indeed irrelevant to the question what we believe. Not so the
grounds, which are grammatically related to the proposition, and tell us what proposition it is.
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438. Nothing is commoner than for the meaning of an expression to oscillate, for a phenomenon to be regarded
sometimes as a symptom, sometimes as a criterion, of a state of affairs. And mostly in such a case the shift of
meaning is not noted. In science it is usual to make phenomena that allow of exact measurement into defining
criteria for an expression; and then one is inclined to think that now the proper meaning has been found.
Innumerable confusions have arisen in this way.
There are for example degrees of pleasure, but it is stupid to speak of a measurement of pleasure. It is true
that in certain cases a measurable phenomenon occupies the place previously occupied by a non-measurable one.
Then the word designating this place changes its meaning, and its old meaning has become more or less obsolete.
We are soothed by the fact that the one
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concept is the more exact, the other the more inexact one, and do not notice that here in each particular case a
different relation between the 'exact' and the 'inexact' concept is in question: it is the old mistake of not testing
particular cases.
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439. Sufficient evidence passes over into insufficient without a definite borderline. Shall I say that a natural
foundation for the way this concept is formed is the complex nature and the variety of human contingencies?
Then given much less variety, a sharply bounded conceptual structure would have to seem natural. And why
does it seem so difficult to imagine the simplified case?
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440. How should we have to imagine a complete list of rules for the employment of a word?--What do we mean by
a complete list of rules for the employment of a piece in chess? Couldn't we always construct doubtful cases, in
which the normal list of rules does not decide? Think e.g. of such a question as: how to determine who moved last, if
a doubt is raised about the reliability of the players' memories?
The regulation of traffic in the streets permits and forbids certain actions on the part of drivers and
pedestrians; but it does not attempt to guide the totality of their movements by prescription. And it would be
senseless to talk of an 'ideal' ordering of traffic which should do that; in the first place we should have no idea what
to imagine as this ideal. If someone wants to make traffic regulations stricter on some point or other, that does not
mean that he wants to approximate to such an ideal.
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441. Consider also the following proposition: "The rules of a game may well allow a certain freedom, but all the
same they must be quite definite rules." That is as if one were to say: "You may indeed leave a person enclosed by
four walls a certain liberty of movement, but the walls must be perfectly rigid"--and that is not true.--"Well, the walls
may be elastic all right, but in that case they have a perfectly determinate degree of elasticity"--But what does this
say? It seems to say that it must be possible to state the elasticity, but that again is not true. "The wall always has
some determinate degree of elasticity--whether I know it or
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not": that is really the avowal of adherence to a form of expression. The one that makes use of the form of an ideal of
accuracy. As it were like the form of a parameter of representation.
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442. The avowal of adherence to a form of expression, if it is formulated in the guise of a proposition dealing with
objects (instead of signs) must be 'a priori'. For its opposite will really be unthinkable, inasmuch as there
corresponds to it a form of thought, a form of expression, that we have excluded.
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443. Suppose people used always to point to objects in the following way: they describe a circle as it were round the
object with their finger in the air; in that case a philosopher could be imagined who said: "All things are circular, for
the table looks like this, the stove like this, the lamp like this", etc., drawing a circle round the thing each time.
Page 78
444. We now have a theory, a 'dynamic theory'†* of the proposition; of language, but it does not present itself to us
as a theory. For it is the characteristic thing about such a theory that it looks at a special clearly intuitive case and
says: "That shews how things are in every case; this case is the exemplar of all cases."--"Of course! It has to be like
that" we say, and are satisfied. We have arrived at a form of expression that strikes us as obvious. But it is as if we
had now seen something lying beneath the surface.
The tendency to generalize the case seems to have a strict justification in logic: here one seems completely
justified in inferring: "If one proposition is a picture, then any proposition must be a picture, for they must all be of
the same nature." For we are under the illusion that what is sublime, what is essential, about our investigation
consists in its grasping one comprehensive essence.
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445. How can I understand a proposition now, if it is for analysis to shew what I really understand?--Here there
sneaks in the idea of understanding as a special mental process.
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446. But don't think of understanding as a 'mental process' at all.--For that is the way of speaking that is confusing
you. Rather ask yourself: in what kind of case, under what circumstances do we say "Now I can go on," if the
formula has occurred to us?†1
That way of speaking is what prevents us from seeing the facts without prejudice. Consider the
pronunciation of a word as its spelling presents it. How easy it is to persuade oneself that two words--e.g. "fore" and
"four" sound different in everyday use--because one pronounces them differently when one has the difference in
spelling directly in view. Comparable with this is the opinion that a violin player with a fine sense of pitch always
strikes F somewhat higher than E sharp. Reflect on such cases.--That is how it can come about that the means of
representation produces something imaginary. So let us not think we must find a specific mental process, because
the verb "to understand" is there and because one says: Understanding is an activity of mind.
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447. Disquiet in philosophy might be said to arise from looking at philosophy wrongly, seeing it wrong, namely as if
it were divided into (infinite) longitudinal strips instead of into (finite) cross strips. This inversion in our conception
produces the greatest difficulty. So we try as it were to grasp the unlimited strips and complain that it cannot be
done piecemeal. To be sure it cannot, if by a piece one means an infinite longitudinal strip. But it may well be done,
if one means a cross-strip.--But in that case we never get to the end of our work!--Of course not, for it has no end.
(We want to replace wild conjectures and explanations by quiet weighing of linguistic facts.)
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448. And does one say that the sentence "It's raining" says: such-and-such is the case? What is the everyday use of
this expression in ordinary language? For you learned it from this use. If you now use it contrary to its original use,
and think you are still playing the old game with it, that is as if you were to play draughts with chess-pieces and
imagine that your game had kept something of the spirit of chess.
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449. Extension of a concept in a theory (e.g. 'wish-fulfilment dream').
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450. One who philosophizes often makes the wrong, inappropriate gesture for a verbal expression.
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451. (One says the ordinary thing--with the wrong gesture.)
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452. How does it come about that philosophy is so complicated a structure? It surely ought to be completely simple,
if it is the ultimate thing, independent of all experience, that you make it out to be.--Philosophy unties knots in our
thinking; hence its result must be simple, but philosophizing has to be as complicated as the knots it unties.†1
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453. (As one can sometimes reproduce music only in one's inward ear, and cannot whistle it, because the whistling
drowns out the inner voice, so sometimes the voice of a philosophical thought is so soft that the noise of spoken
words is enough to drown it and prevent it from being heard, if one is questioned and has to speak.)
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454. Plato: "--What? he said, it be of no use? If wisdom is the knowledge of knowledge and is prior to other
knowledges, then it must also be prior to that knowledge which relates to the good and in that way must be of use to
us.--Does it make us healthy, I said, and not medicine? And similarly with the rest of the arts; does it direct their
business, and not rather each of them its own? Again, have we not long since allowed that it would only be the
knowledge of knowledges and ignorances and not of any other matter?--We have indeed.--So it will not produce
health in us?--Presumably not.--Because health belongs to a different art?--Yes--Then, friend, neither will it produce
utility for us. For this is a business we have too assigned to another art.--Of course--So how can wisdom be useful, if
it does not bring us any utility?"†2
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455. (The philosopher is not a citizen of any community of ideas. That is what makes him into a philosopher.)
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456. Some philosophers (or whatever you like to call them) suffer from what may be called "loss of problems". Then
everything seems quite simple to them, no deep problems seem to exist any more, the world becomes broad and flat
and loses all depth, and what they write becomes immeasurably shallow and trivial. Russell and H. G. Wells suffer
from this.
Page 81
457. ...quia plus loquitur inquisitio quam inventio... (Augustine.)†1
Page 81
458. Philosophical investigations: conceptual investigations. The essential thing about metaphysics: it obliterates the
distinction between factual and conceptual investigations.
Page 81
459. The fundamental thing expressed grammatically: What about the sentence: "One cannot step into the same river
twice"?
Page 81
460. In a certain sense one cannot take too much care in handling philosophical mistakes, they contain so much
truth.†2
"Such irritations are wont to stimulate the powers of thought." How does thought remedy an irritation?
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461. I should like you to say: "Yes, it's true, that can be imagined, that may even have happened!" But was I trying to
draw your attention to the fact that you are able to imagine this? I wanted to put this picture before your eyes, and
your acceptance of this picture consists in your being inclined to regard a given case differently; that is, to compare
it with this series of pictures. I have changed your way of seeing. (I once read somewhere that a geometrical figure,
with the words "Look at this", serves as a proof for certain Indian mathematicians. This looking too effects an
alteration in one's way of seeing.)†3
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462. (The classifications of philosophers and psychologists: they classify clouds by their shape.)
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463. On mathematics: "Your concept is wrong.--However, I cannot illumine the matter by fighting against your
words, but only by trying to turn your attention away from certain expressions, illustrations, images, and towards
the employment of the words."
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464. The pedigree of psychological concepts: I strive not after exactness, but after a synoptic view.
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465. The treatment of all these phenomena of mental life is not of importance to me because I am keen on
completeness. Rather because each one casts light on the correct treatment of all.
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466. And here what is in question is not symptoms, but logical criteria. That these are not always sharply
differentiated does not prevent them from being differentiated.
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467. Our investigation does not try to find the real, exact meaning of words; though we do often give words exact
meanings in the course of our investigation.
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468. "Man thinks, is afraid etc. etc.": that is the reply one might give to someone who asked what chapters a book on
psychology should contain.
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469. Imagine someone saying: "Man hopes". How should this general phenomenon of natural history be
described?--One might observe a child and wait until one day he manifests hope; and then one could say "Today he
hoped for the first time". But surely that sounds queer! Although it would be quite natural to say "Today he said 'I
hope' for the first time". And why queer?--One does not say that a suckling hopes that.... nor yet that he has no hope
that.... and one does say it of a grown man--Well, bit by bit daily life becomes such that there is a place for hope in
it.
But now it is said: We can't be certain when a child really begins to hope, for hope is an inner process. What
nonsense! For then how do we know what we are talking about at all?
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470. Or might he offer this example: "I, e.g., see, am not blind"? Even that sounds queer.
It would be correct to say: "You can observe the phenomena of thinking, hoping, seeing, etc., in my case as
well".
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471. The psychological verbs to see, to believe, to think, to wish, do not signify phenomena [appearances]. But
psychology observes the phenomena of seeing, believing, thinking, wishing.
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472. Plan for the treatment of psychological concepts.
Psychological verbs characterized by the fact that the third person of the present is to be verified by
observation, the first person not.
Sentences in the third person present: information. In the first person present: expression. ((Not quite right.))
The first person of the present akin to an expression.
Sensations: their inner connexions and analogies.
All have genuine duration. Possibility of giving the beginning and the end. Possibility of their being
synchronized, of simultaneous occurrence.
All have degrees and qualitative mixtures. Degree: scarcely perceptible--unendurable.
In this sense there is not a sensation of position or movement. Place of feeling in the body: differentiates
seeing and hearing from sense of pressure, temperature, taste and pain.
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473. We need to reflect that a state of language is possible (and presumably has existed) in which it does not possess
the general concept of sensation, but does have words corresponding to our "see", "hear", "taste".
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474. We call seeing, hearing,... sense-perception. There are analogies and connexions between these concepts; these
are our justification for so taking them together.
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475. It can, then, be asked: what kind of connexions and analogies exist between seeing and hearing? Between
seeing and touching? Between seeing and smelling? Etc.
Page 83
476. And if we ask this, then the senses at once shift further apart from one another than they seemed to be at first
sight.
Page 83
477. What is common to sense-experiences?--The answer that they give us knowledge of the external world is partly
wrong and partly right. It is right in so far as it is supposed to point to a logical criterion.
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478. The duration of sensation. Compare the duration of a sense-experience of sound with the duration of the
sensation of touch which informs you that you have a ball in your hand; and with the "feeling" that informs you that
your knees are bent.
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Page 84
479. We feel our movements. Yes, we really feel them; the sensation is similar, not to a sensation of taste or heat, but
to one of touch: to the sensation when skin and muscles are squeezed, pulled, displaced.
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480. I feel my arm and in a queer way I should now like to say: I feel it in a definite position in space; as if the feeling
of my body in a space were disposed in the shape of an arm, so that in order to represent it I should have to model
my arm, in plaster say, in the right position.
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481. It is queer. My lower arm is now resting horizontally, and I should like to say that I feel it; not, however, as if I
had a feeling that always accompanies this position (as one would feel ischaemia or engorgement)--but as if the
'body-feeling' of the arm were arranged or disposed horizontally as e.g. a vapour or dust-particles on the surface of
my arm are so disposed in space. So it is not really as if I felt the position of my arm, but as if I felt my arm, and the
feeling had such-andsuch a position. But that means only: I simply know how it is--without knowing it because....
Just as I also know where I feel pain--but do not know it because....
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482. It positively seems to us as if pain had a body, as if it were a thing, a body with shape and colour. Why? Has it
the shape of the part of the body that hurts? One would like to say for example "I could describe the pain if I only
had the necessary words and elementary meanings". One feels: all that is lacking is the requisite nomenclature.
(James). As if one could even paint the sensation, if only other people would understand this language.--And one
really can describe pain spatially and temporally.
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483. (If sensations are characteristic of the position and movements of the limbs, at any rate their place is not the
joint.)
One knows the position of one's limbs and their movements. One can give them if asked, for example. Just as
one also knows the place of a sensation (pain) in the body.
Reaction of touching the painful place.
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No local sign about the sensation. Any more than a temporal sign about a memory-image. (Temporal signs in
a photograph.)
Pain differentiated from other sensations by a characteristic expression. This makes it akin to joy (which is
not a sense-experience).
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484. Is it hair-splitting to say:--joy, enjoyment, delight, are not sensations?--Let us at least ask ourselves: How much
analogy is there between delight and what we call e.g. "sensation"?
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485. The connecting link between them would be pain. For this concept resembles that of e.g. tactile sensation
(through the characteristics of localisation, genuine duration, intensity, quality) and at the same time that of the
emotions through its expression (facial expressions, gestures, noises).
Page 85
486. "I feel great joy"--Where?--that sounds like nonsense. And yet one does say "I feel a joyful agitation in my
breast".--But why is joy not localized? Is it because it is distributed over the whole body? Even where the feeling
that arouses joy is localized, joy is not: if for example we rejoice in the smell of a flower.--Joy is manifested in facial
expression, in behaviour. (But we do not say that we are joyful in our faces).
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487. "But I do have a real feeling of joy!" Yes, when you are glad you really are glad. And of course joy is not joyful
behaviour, nor yet a feeling round the corners of the mouth and the eyes.
"But 'joy' surely designates an inward thing." No. "Joy" designates nothing at all. Neither any inward nor any
outward thing.
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488. Continuation of the classification of psychological concepts.
Page 85
Emotions. Common to them: genuine duration, a course. (Rage flares up, abates, vanishes, and likewise joy,
depression, fear.)
Distinction from sensations: they are not localized (nor yet diffuse!).
Common: they have characteristic expression-behaviour. (Facial expression.) And this itself implies
characteristic sensations too. Thus sorrow often goes with weeping, and characteristic
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sensations with the latter. (The voice heavy with tears). But these sensations are not the emotions. (In the sense in
which the numeral 2 is not the number 2).
Among emotions the directed might be distinguished from the undirected. Fear at something, joy over
something.
This something is the object, not the cause of the emotion.
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489. The language game "I am afraid" already contains the object.
"Anxiety" is what undirected fear might be called, in so far as its manifestations resemble or are the same as
those of fear.
The content of an emotion--here one imagines something like a picture, or something of which a picture can
be made. (The darkness of depression which descends on a man, the flames of anger.)
Page 86
490. The human face too might be called such a picture and its alterations might represent the course of a passion.
Page 86
491. What goes to make them different from sensations: they do not give us any information about the external
world. (A grammatical remark.)
Love and hate might be called emotional dispositions, and so might fear in one sense.
Page 86
492. It is one thing to feel acute fear, and another to have a 'chronic' fear of someone. But fear is not a sensation.
'Horrible fear': is it the sensations that are so horrible?
Typical causes of pain on the one hand, and of depression, sorrow, joy on the other. Cause of these also their
object.
Pain-behaviour and the behaviour of sorrow.--These can only be described along with their external
occasions. (If a child's mother leaves it alone it may cry because it is sad; if it falls down, from pain.) Behaviour and
kind of occasion belong together.
Page 86
493. Thoughts may be fearful, hopeful, joyful, angry, etc.
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Page 87
494. Emotions are expressed in thoughts. A man talks angrily, timidly, sadly, joyfully etc., not lumbagoishly.
A thought rouses emotions in me (fear, sorrow etc.) not bodily pain.
Page 87
495. I should almost like to say: One no more feels sorrow in one's body than one feels seeing in one's eyes.
Page 87
496. ((The horribleness of fear is not in the sensations of fear.)) This matter also calls to mind hearing a sound from a
particular direction. It is almost as if one felt the heaviness around the stomach, the constraint of breath, from the
direction of the fear. That means really that "I am sick with fear" does not assign a cause of fear.
Page 87
497. "Where do you feel grief?"--In the mind.--What kind of consequences do we draw from this assignment of
place? One is that we do not speak of a bodily place of grief. Yet we do point to our body, as if the grief were in it. Is
that because we feel a bodily discomfort? I do not know the cause. But why should I assume it is a bodily
discomfort?
Page 87
498. Consider the following question: Can a pain be thought of, say with the quality of rheumatic pain, but
unlocalized? Can one imagine this?
If you begin to think this over, you see how much you would like to change the knowledge of the place of
pain into a characteristic of what is felt, into a characteristic of a sense-datum, of the private object that I have before
my mind.
Page 87
499. If fear is frightful and if while it goes on I am conscious of my breathing and of a tension in my facial
muscles--is that to say that I find these feelings frightful? Might they not even be a mitigation? ((Dostoievsky)).
Page 87
500. Why does one use the word "suffering" for pain as well as for fear? Well, there are plenty of tie-ups.--
Page 87
501. To the utterance: "I can't think of it without fear" one replies: "There's no reason for fear, for...." That is at any
rate one way of dismissing fear. Contrast with pain.
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Page 88
502. That there is a fear-syndrome of sensations, thoughts etc. (for example) does not mean that fear is a syndrome.
Page 88
503. If someone acts grief in the study, he will indeed readily become aware of the tensions in his face. But be really
sad, or follow a sorrowful action in a film, and ask yourself if you were aware of your face.
Page 88
504. Love is not a feeling. Love is put to the test, pain not. One does not say: "That was not true pain, or it would not
have gone off so quickly".
Page 88
505. One connexion between moods and sense-impressions is that we use mood-concepts to describe
sense-impressions and images. We say of a theme, a landscape, that it is sad, glad etc. But much more important, of
course, is our using all the mood-concepts to describe human faces, actions, behaviours.
Page 88
506. A friendly mouth, friendly eyes. How would one think of a friendly hand?--Probably open and not as a
fist.--And could one think of the colour of a man's hair as an expression of friendliness or the opposite?--But put like
that the question seems to be whether we can manage to. The question ought to run: Do we want to call anything a
friendly or unfriendly hair-colour? If we wanted to give such words a sense, we should perhaps imagine a man
whose hair darkened when he got angry. The reading of an angry expression into dark hair, however, would work
via a previously existent conception.
It may be said: The friendly eyes, the friendly mouth, the wagging of a dog's tail, are among the primary and
mutually independent symbols of friendliness; I mean: they are parts of the phenomena that are called friendliness.
If one wants to imagine further appearances as expressions of friendliness, one reads these symbols into them. We
say: "He has a black look", perhaps because the eyes are more strongly shadowed by the eyebrows; and now we
transfer the idea of darkness to the colour of the hair.
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Page 89
507. If anyone asks whether pleasure is a sensation, he probably does not distinguish between reason and cause, for
otherwise it would occur to him that one takes pleasure in something, which does not mean that this something
produces a sensation in us.
Page 89
508. But pleasure does at any rate go with a facial expression; and while we do not see that in ourselves, all the same
we notice it.
Page 89
And just try to think over something very sad with an expression of radiant joy!
Page 89
509. It is quite possible that the glands of a sad person secrete differently from those of someone who is glad; and
also that their secretion is the or a cause of sadness. But does it follow that the sadness is a sensation produced by
this secretion?
Page 89
510. But here the thought is: "After all, you feel sadness--so you must feel it somewhere; otherwise it would be a
chimera". But if you want to think that, remember the difference between seeing and pain. I feel pain in the
wound--but colour in the eye? If we try to use a schema here, instead of merely noting what is really common, we
see everything falsely simplified.
Page 89
511. But if one wanted to find an analogy to the place of pain, it would of course not be the mind (as, of course, the
place of bodily pain is not the body) but the object of regret.
Page 89
512. Suppose it were said: Gladness is a feeling, and sadness consists in not being glad.--Is the absence of a feeling a
feeling?
Page 89
513. One speaks of a feeling of conviction because there is a tone of conviction. For the characteristic mark of all
'feelings' is that there is expression of them, i.e. facial expression, gestures, of feeling.
Page 89
514. Now one might say this: A man's face has by no means a constant appearance. It alters from one moment to the
next; sometimes only a little, sometimes up to the point of unrecognizability. Nevertheless it is possible to draw a
picture of his physiognomy. Of course a picture in which the face smiles does
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not shew how it looks when weeping. But it does permit inferences.--And in this way it would also be possible to
describe a kind of average physiognomy of belief (e.g.).
Page 90
515. I give signs of delight and comprehension.
Page 90
516. Can 'knowing one's way about' be called an experience? Surely not. But there are experiences characteristic of
the condition of knowing one's way about and not knowing one's way about. (Not knowing one's way about and
lying.)
Page 90
517. But it is surely important that all these paraphrases exist. That care can be described in such words as: "the
descent of a permanent cloud". I have perhaps never stressed the importance of this paraphrasing enough.
Page 90
518. Why can a dog feel fear but not remorse? Would it be right to say "Because he can't talk"?
Page 90
519. Only someone who can reflect on the past can repent. But that does not mean that as a matter of empirical fact
only such a one is capable of the feeling of remorse.
Page 90
520. There is nothing astonishing about certain concepts' only being applicable to a being that e.g. possesses a
language.
Page 90
521. "The dog means something by wagging his tail."--What grounds would one give for saying this?--Does one also
say: "By drooping its leaves, the plant means that it needs water"?--
Page 90
522. We should hardly ask if the crocodile means something when it comes at a man with open jaws. And we
should declare that since the crocodile cannot think there is really no question of meaning here.
Page 90
523. Let us just forget entirely that we are interested in the state of mind of a frightened man. It is certain that under
given circumstances we may also be interested in his behaviour as an indication of how he will behave in the future.
So why should we not have a word for this?
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Page 91
It might now be asked whether this word would really relate simply to behaviour, simply to bodily changes.
And this may be denied. There is no future in simplifying the use of this word in this way. It relates to the behaviour
under certain external circumstances. If we observe these circumstances and that behaviour we say that a man is... or
has....
Page 91
524. There might be a concept of fear that had application only to beasts, and hence only through observation.--But
you don't want to say that such a concept would have no use. The verb that would roughly correspond to the word
"to fear" would then have no first person and none of its forms would be an expression of fear.
Page 91
525. I now want to say that humans who employ such a concept would not have to be able to describe its use. And
were they to try, it is possible that they would give a quite inadequate description. (Like most people, if they tried to
describe the use of money correctly). (They are not prepared for such a task).
Page 91
526. If someone behaves in such-and-such a way under such-and-such circumstances, we say that he is sad. (We
say it of a dog too). To this extent it cannot be said that the behaviour is the cause of the sadness: it is its symptom.
Nor would it be beyond cavil to call it the effect of sadness.--If he says it of himself (that he is sad) he will not in
general give his sad face as a reason. But what about this: "Experience has taught me that I get sad as soon as I start
sitting about sadly, etc." This might have two different meanings. Firstly: "As soon as, following a slight inclination, I
set out to carry and conduct myself in such-and-such a way, I get into a state in which I have to persist in this
behaviour". For it might be that toothache got worse by groaning.--Secondly, however, that proposition might
contain a speculation about the cause of human sadness; the content being that if you could somehow or other
produce certain bodily states, you would make the man sad. But here arises the difficulty that we should not call a
man sad, if he looked and acted sad in all circumstances. If we taught such a one the expression "I am sad" and he
constantly kept on saying this with an expression of sadness, these words, like the other signs, would have lost their
normal sense.
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Page 92
527. Isn't it as if one were trying to imagine a facial expression not susceptible of alterations which were gradual and
difficult to catch hold of, but which had, say, just five positions; when it changed it would snap straight from one to
another. Now would this fixed smile, for example, really be a smile? And why not?--"Smiling" is our name for an
expression in a normal play of expressions.--I might not be able to react as I do to a smile. It would e.g. not make me
smile myself. One wants to say: "No wonder we have this concept in these circumstances".
Page 92
528. An auxiliary construction. A tribe that we want to enslave. The government and the scientists give it out that the
people of this tribe have no souls; so they can be used for any arbitrary purpose. Naturally we are interested in their
language nevertheless; for we certainly want to give them orders and to get reports from them. We also want to
know what they say to one another, as this ties up with the rest of their behaviour. But we must also be interested in
what corresponds in them to our 'psychological utterances', since we want to keep them fit for work; for that reason
their manifestations of pain, of being unwell, of depression, of joy in life, are important to us. We have even found
that it has good results to use these people as experimental subjects in physiological and psychological laboratories,
since their reactions--including their linguistic reactions--are quite those of mind-endowed human beings. Let it also
have been found out that these automata can have our language imparted to them instead of their own by a method
which is very like our 'instruction'.
Page 92
529. These creatures now learn e.g. to calculate, they learn paper or oral calculation. But by some method we make
them able to give the result of a multiplication after behaving in a 'reflective' manner for a time, though without
writing or speaking. If one considers the kind of way they learn this 'calculating in the head', together with the
surrounding phenomena, this suggests the picture of the process of calculating as, so to speak, submerged and going
on under the surface.
Of course for various purposes we need an order like "Work this out in your head"; a question like "Have
you worked it out?"; and even "How far have you got?"; a statement "I have worked
Page Break 93
... out" on the part of the automaton; etc. In short: everything that we say among ourselves about calculating in the
head is of interest to us when they say it. And what goes for calculating in the head goes for all other forms of
thinking as well.--If one of us gives vent to the opinion that these beings must after all have some kind of mind, we
jeer at him.
Page 93
530. The slaves also say: "When I heard the word 'bank' it meant... to me." Question: what technique of using
language is the background for their saying this? For everything depends on that. What have we taught them, what
use for the word "mean"? And what, if anything at all, do we gather from their utterance? For if we can do nothing
with it, still it might interest us as a curiosity.--Let us imagine a tribe of men, unacquainted with dreams, who hear
our narrations of dreams. One of us had come to these non-dreaming people and learnt bit by bit to make himself
understood to them.--Perhaps one thinks they would never understand the word "to dream". But they would soon
find a use for it. And their doctors might very well be interested in the phenomenon and might make important
inferences from the dreams of the stranger.--Nor can it be said that for these people the verb "to dream" could mean
nothing but: to tell a dream. For the stranger would of course use both expressions, both "to dream" and "to tell a
dream", and the people of that tribe would not be allowed to confuse "I dreamt..." with "I told the dream...."
Page 93
531. "I assume that a picture swims before him".--Could I also assume that a picture swims before this stove?--And
why does this seem impossible? Is the human shape necessary for this?--
Page 93
532. The concept of pain is characterized by its particular function in our life.
Page 93
533. Pain has this position in our life; has these connexions; (That is to say: we only call "pain" what has this
position, these connexions).
Page 93
534. Only surrounded by certain normal manifestations of life, is there such a thing as an expression of pain. Only
surrounded
Page Break 94
by an even more far-reaching particular manifestation of life, such a thing as the expression of sorrow or affection.
And so on.
Page 94
535. If I, and if anyone else, can imagine a pain, or at least we say we can--how is it to be found out whether we are
imagining it right, and how accurately we are imagining it?
Page 94
536. I may know that he is in pain, but I never know the exact degree of his pain. So here is something that he knows
and that his expression of pain does not tell me. Something purely private.
He knows exactly how severe his pain is? (Isn't that much as if one were to say he always knows exactly
where he is? Namely here.) Is the concept of the degree given with the pain?
Page 94
537. You say you attend to a man who groans because experience has taught you that you yourself groan when you
feel such-and-such. But as you don't in fact make any such inference, we can abandon the justification by analogy.
Page 94
538. Nor does it make sense to say: "I don't bother about my own groaning because I know that I am in pain"--or
"because I feel my pain".
Yet this is perfectly true:--"I don't bother about my groaning".
Page 94
539. I infer that he needs to go to the doctor from observation of his behaviour; but I do not make this inference in
my own case from observation of my behaviour. Or rather: I do this too sometimes, but not in parallel cases.
Page 94
540. It is a help here to remember that it is a primitive reaction to tend, to treat, the part that hurts when someone else
is in pain; and not merely when oneself is--and so to pay attention to other people's pain-behaviour, as one does not
pay attention to one's own pain behaviour.
Page 94
541. But what is the word "primitive" meant to say here? Presumably that this sort of behaviour is pre-linguistic:
that a language-game is based on it, that it is the prototype of a way of thinking and not the result of thought.
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Page 95
542. "Putting the cart before the horse" may be said of an explanation like the following: we tend someone else
because by analogy with our own case we believe that he is experiencing pain too.--Instead of saying: Get to know a
new aspect from this special chapter of human behaviour--from this use of language.
Page 95
543. My relation to the appearances here is part of my concept.
Page 95
544. When we tell a doctor that we have been having pains--in what cases is it useful for him to imagine pain?--And
doesn't this happen in a variety of ways? (As great a variety as: recalling pain). (Knowing what a man looks like).
Page 95
545. Suppose someone explains how a child learns the use of the word "pain" in the following way: When the child
behaves in such-and-such a way on particular occasions, I think he's feeling what I feel in such cases; and if it is so
then the child associates the word with his feeling and uses the word when the feeling reappears.--What does this
explanation explain? Ask yourself: What sort of ignorance does it remove?--Being sure that someone is in pain,
doubting whether he is, and so on, are so many natural, instinctive, kinds of behaviour towards other human beings,
and our language is merely an auxiliary to, and further extension of, this relation. Our language-game is an extension
of primitive behaviour. (For our language-game is behaviour.) (Instinct).
Page 95
546. "I am not certain whether he is in pain."--Suppose now someone always pricked himself with a pin when he
said this, in order to have the meaning of the word "pain" vividly before his mind (so as not to have to rest content
with imagination) and to know what he is in doubt of about the other man.--Would the sense of his statement now
be assured?
Page 95
547. So he is having genuine pain, and it is the possession of this by someone else that he feels doubt of.--But how
does he do this?--It is as if I were told: "Here is a chair. Can you see it clearly?--Good--now translate it into French!"
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Page 96
548. So he has real pain; and now he knows what he is to doubt in someone else's case. He has the object before him
and it is not some piece of behaviour or the like. (But now!) In order to doubt whether someone else is in pain he
needs, not pain, but the concept 'pain'.
Page 96
549. To call the expression of a sensation a statement is misleading because 'testing', 'justification', 'confirmation',
'reinforcement' of the statement are connected with the word "statement" in the language-game.
Page 96
550. What purpose is served by the statement: "I do have something, if I have a pain"?
Page 96
551. "The smell is marvellous!" Is there a doubt whether it is the smell that is marvellous?
Is it a property of the smell?--Why not? It is a property of ten to be divisible by two and also to be the
number of my fingers.
There might however be a language in which the people merely shut their eyes and say "Oh, this smell!" and
there is no subject-predicate sentence equivalent to it. That is simply a 'specific' reaction.
Page 96
552. It's not merely the picture of the behaviour that belongs to the language-game with the words "he is in
pain"--one would like to say--but also the picture of the pain.--But here one must take care: think of my example of
the private tables that don't belong to the game.--The impression of a 'private table' in the game arises through the
absence of a table and through the similarity of the game to one that is played with a table.†1
Page 96
553. Remember: we often use the phrase "I don't know" in a queer way; when for example we say that we don't
know whether this man really feels more than that other, or merely gives stronger expression to his feeling. In that
case it is not clear what sort of investigation might settle the question. Of course the expression is not quite idle: we
want to say that we certainly can compare the feelings of A and B with one another, but that the circumstances of a
comparison of A with C throw us out.
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Page 97
554. That the evidence makes someone else's feeling ((i.e. what is within)) merely probable is not what matters to us;
what we are looking at is the fact that this is taken as evidence for something (important); that we base a judgment
on this involved sort of evidence, and hence that such evidence has a special importance in our lives and is made
prominent by a concept. ((The 'inner' and the 'outer', a picture)).
Page 97
555. The 'uncertainty' relates not to the particular case, but to the method, to the rules of evidence.
Page 97
556. The uncertainty is not founded on the fact that he does not wear his pain on his sleeve. And there is not an
uncertainty in each particular case. If the frontier between two counties were in dispute, would it follow that the
county to which any individual resident belonged was dubious?
Page 97
557. Imagine that people could observe the functioning of the nervous system in others. In that case they would
have a sure way of distinguishing genuine and simulated feeling.--Or might they after all doubt in turn whether
someone feels anything when these signs are present?,--What they see there could at any rate readily be imagined to
determine their reaction without their having any qualms about it.
And now this can be transferred to outward behaviour.
This observation fully determines their attitude to others and doubt does not occur.
Page 97
558. There is indeed the case where someone later reveals his inmost heart to me by a confession: but that this is so
cannot offer me any explanation of outer and inner, for I have to give credence to the confession.
For confession is of course something exterior.
Page 97
559. Look at people who doubt even in these circumstances, and at ones who do not doubt.
Page 97
560. Only God sees the most secret thoughts. But why should these be all that important? Some are important, not
all. And need all human beings count them as important?
Page Break 98
Page 98
561. One kind of uncertainty is that with which we might face an unfamiliar mechanism. In another we should
possibly be recalling an occasion in our life. It might be e.g. that someone who has just escaped the fear of death
would shrink from swatting a fly, though he would otherwise do it without thinking twice about it. Or on the other
hand that, having this experience in his mind's eye, he does with hesitancy what otherwise he does unhesitatingly.
Page 98
562. Even when I 'do not rest secure in my sympathy' I need not think of uncertainty about his later behaviour.
Page 98
563. The one uncertainty stems from you, so to speak, the other from him.
The one could surely be said to connect up with an analogy, then, but not the other. Not, however, as if I
were drawing a conclusion from the analogy!
Page 98
564. If however I doubt whether a spider feels pain, it is not because I don't know what to expect.
Page 98
565. But we cannot get away from forming the picture of a mental process. And not because we are acquainted with
it in our own case!
Page 98
566. Might not the attitude, the behaviour, of trusting, be quite universal among a group of humans? So that a doubt
about manifestations of feeling is quite foreign to them?†1
Page 98
567. How could human behaviour be described? Surely only by sketching the actions of a variety of humans, as
they are all mixed up together. What determines our judgment, our concepts and reactions, is not what one man is
doing now, an individual action, but the whole hurly-burly of human actions, the background against which we see
any action.
Page 98
568. Seeing life as a weave, this pattern (pretence, say) is not always complete and is varied in a multiplicity of ways.
But we, in our conceptual world, keep on seeing the same, recurring with variations. That is how our concepts take
it. For concepts are not for use on a single occasion.†2
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Page 99
569. And one pattern in the weave is interwoven with many others.
Page 99
570. "One can't pretend like that".--This may be a matter of experience--namely that no one who behaves like that
will later behave in such-and-such a way; but it also may be a conceptual stipulation ("That wouldn't still be
pretence"); and the two may be connected.
That can no longer be called "pretence".
(For it wouldn't have been said that the planets had to move in circles, if it had not appeared that they move
in circles).
((Compare: "One cannot talk like that without thinking", "One cannot act like that involuntarily")).
Page 99
571. Couldn't you imagine a further surrounding in which this too could be interpreted as pretence? Must not any
behaviour allow of such an interpretation?
But what does it mean to say that all behaviour might always be pretence? Has experience taught us this?
How else can we be instructed about pretence? No, it is a remark about the concept 'pretence'. But then this concept
would be unusable, for pretending would have no criteria in behaviour.
Page 99
572. Isn't there something here like the relation between Euclidean geometry and the experience of the senses? (I
mean that there is a profound resemblance here). For Euclidean geometry too corresponds to experience only in
some way that is not at all easy to understand, not merely as something more exact to its less exact counterpart.
Page 99
573. There is such a thing as trust and mistrust in behaviour!
If anyone complains, e.g., I may be trustful and react with perfect confidence, or I may be uncertain, like
someone who has his suspicions. Neither words nor thoughts are needed for this.
Page 99
574. Is what he says he has, and I say I have, without our inferring this from any sort of observation--is it the same
as what we gather from observing the behaviour of someone else and from his expressions of conviction?
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Page 100
575. Can it be said that I infer that he will act as he intends to act? ((Case of the wrong gesture)).
Page 100
576. Why do I never infer my probable actions from my words? For the same reason as I do not infer my probable
behaviour from my facial expression.--For the interesting thing is not that I do not infer my emotion from my
expression of emotion, but that I also do not infer my later behaviour from that expression, as other people do, who
observe me.
Page 100
577. What is voluntary is certain movements with their normal surrounding of intention, learning, trying, acting.
Movements of which it makes sense to say that they are sometimes voluntary and sometimes involuntary are
movements in a special surrounding.
Page 100
578. If someone were now to tell us that he eats involuntarily--what evidence would make one believe this?
Page 100
579. One produces a sneeze or a fit of coughing in oneself, but not a voluntary movement. And the will does not
produce sneezing, nor yet walking.
Page 100
580. My expression†1 came from my thinking of willing as a sort of producing--not however as a case of causation,
but--I should like to say--as a direct, non-causal producing. And the basis of this idea is our imagining that the causal
nexus is the connexion of two machine parts by means of a mechanism, say a train of cogwheels.
Page 100
581. Is "I am doing my utmost" the expression of an experience?--One difference: one says "Do your utmost".
Page 100
582. If someone meets me in the street and asks "Where are you going?" and I reply "I don't know", he assumes I
have no definite intention; not that I do not know whether I shall be able to carry out my intention. (Hebel.)†2
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Page 101
583. What is the difference between these two things: Following a line involuntarily--Following a line intentionally?
What is the difference between these two things: Tracing a line with care and great attention--Attentively
observing how my hand follows a line?
Page 101
584. Certain differences are easy to give. One lies in foreseeing what the hand will do.
Page 101
585. The experience of getting to know a new experience. E.g. in writing. When does one say one has become
acquainted with a new experience? How does one use such a proposition?
Page 101
586. Writing is certainly a voluntary movement, and yet an automatic one. And of course there is no question of a
feeling of each movement in writing. One feels something, but could not possibly analyse the feeling. One's hand
writes; it does not write because one wills, but one wills what it writes.
One does not watch it in astonishment or with interest while writing; does not think "What will it write now?"
But not because one had a wish it should write that. For that it writes what I want might very well throw me into
astonishment.
Page 101
587. A child learns to walk, to crawl, to play. It does not learn to play voluntarily and involuntarily. But what makes
its movements in play into voluntary movements?--What would it be like if they were involuntary?--Equally, I might
ask: what makes this movement into a game?--Its character and its surroundings.
Page 101
588. Active and passive. Can one give an order to do it or not? This perhaps seems, but is not, a far-fetched
distinction. It is like: "Can one (logical possibility) decide to do it or not?"--And that means: How is it surrounded
by thoughts, feelings, etc.?
Page 101
589. "When I make an effort, I surely do something, I don't merely have a sensation". And that is so; for one tells
someone "Make an effort" and he may express his intention: "Now I will make an effort". And if he says "I can't
keep it up"--that does not mean "I can't put up with the feeling in my limbs"--e.g. of
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pain.--On the other hand, however, one suffers from making an effort as from pain. "I am completely exhausted"--if
someone said that, but moved as briskly as ever, we should not understand him.
Page 102
590. The connexion of our main problem with the epistemological problem of willing has occurred to me before.
When such an obstinate problem makes its appearance in psychology, it is never a question about facts of
experience (such a problem is always much more tractable), but a logical, and hence properly a grammatical
question.
Page 102
591. My own behaviour is sometimes--but rarely--the object of my own observation. And this is connected with the
fact that I intend my behaviour. Even if an actor observes his own expressions in a glass, or a musician pays close
attention to every note he plays and judges it, this is done so as to direct his action accordingly.
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592. What does it mean to say e.g. that self-observation makes my action, my movements, uncertain?
I cannot observe myself unobserved. And I do not observe myself for the same purpose as I do someone
else.
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593. If a child stamps and roars with fury--who would say it was doing so involuntarily? And why? Why does one
assume it not to be doing this involuntarily? What are the tokens of involuntary action? Are there such
tokens?--Then what are the tokens of involuntary movement? It does not obey orders, as voluntary movement does.
We have "Come here", "Go over there", "Make this arm-movement"; but not "Make your heart beat".
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594. There is a particular interplay of movements, words, expressions of face, as of manifestations of reluctance or
readiness, which are characteristic of the voluntary movements of a normal human being. If one calls a child, he
does not come automatically: there is e.g. the gesture "I don't want to!" Or coming cheerfully, the decision to come,
running away with signs of fear, the effects of being addressed, all the reactions of the game, the signs and the
effects of consideration.
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595. How could I prove to myself that I can move my arm voluntarily? Say by telling myself: "Now I will move it"
after which it moves? Or should I say "Simply by moving it"? But how do I know that I did it, and that it did not
move just by accident? Do I in the end feel it after all? And suppose my memory of previous feelings were to
deceive me and these were not the right feelings to go by! (And which are the right ones?) And how does someone
else know whether I have moved my arm voluntarily? Perhaps I shall say to him "Order me to make whatever
movement you like and, to convince you, I will do it".--And what do you feel in your arm? "Well, the usual feeling".
There is nothing unusual about the feelings--the arm isn't without feeling, for example (as if it had 'gone to sleep').
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596. If I do not know that a movement of my body is taking place or has taken place, that movement will be called
involuntary.--But how about when I merely try to lift a weight, and so there is a movement that does not take place?
What would it be like if someone involuntarily strained to lift a weight? In what circumstances would one call this
behaviour 'involuntary'?
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597. May not rest be just as voluntary as motion? May not the cessation of movement be voluntary? What better
argument against a feeling of innervation?
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598. What a queer concept 'to attempt', 'to try', is: what can one not 'try to do'! (One tries to remember, to lift a
weight, to attend, to think of nothing). But then it might also be said: What a remarkable concept 'doing' is! What are
the relations of affinity between 'talking' and 'thinking', between 'speaking' and 'speaking inwardly'? (Compare the
relation of affinity between the kinds of numbers).
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599. One draws quite different conclusions from an involuntary movement and from a voluntary: this characterizes
voluntary movement.
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600. But how do I know that this movement was voluntary?--I don't know this, I manifest it.
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601. "I am pulling as hard as I can". How do I know that? Does the feeling in my muscles tell me so? The words are
a signal; and they have a function.
But then, don't I experience anything? Do I experience something, then? Something specific? A specific
feeling of straining and not-being-able-to-do-anymore, of reaching the limit? Of course, but these expressions say no
more than "I am pulling as hard as I can".
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602. Compare this case: Someone is to say what he feels when a weight is resting on his flat hand. Now I can
imagine a split here: On the one hand he tells himself that what he feels is a pressure against the surface of his hand
and a tension in the muscles of his arm; on the other hand he wants to say: "But that isn't all! I surely feel a pull, a
drive downwards on the part of the weight."--Does he then have a sensation of such a 'drive'? Yes: when he thinks of
the 'drive'. With the word 'drive' there goes here a particular picture, a gesture, a tone of voice; and in this you can see
the experience of the drive.
(Think also of this: Some people say Such-and-such a person 'gives forth a fluid'--This is the source of the
word "influence".)
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603. The unpredictability of human behaviour. But for this--would one still say that one can never know what is
going on in anyone else?
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604. But what would it be like if human behaviour were not unpredictable? How are we to imagine this? (That is to
say: how should we depict it in detail, what are the connexions we should assume?)
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605. One of the most dangerous of ideas for a philosopher is, oddly enough, that we think with our heads or in our
heads.
Page 104
606. The idea of thinking as a process in the head, in a completely enclosed space, gives him something occult.
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607. Is thinking a specific organic process of the mind, so to speak--as it were chewing and digesting in the mind?
Can we replace it by an inorganic process that fulfils the same end,
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as it were use a prosthetic apparatus for thinking? How should we have to imagine a prosthetic organ of thought?
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608. No supposition seems to me more natural than that there is no process in the brain correlated with associating
or with thinking; so that it would be impossible to read off thought-processes from brain-processes. I mean this: if I
talk or write there is, I assume, a system of impulses going out from my brain and correlated with my spoken or
written thoughts. But why should the system continue further in the direction of the centre? Why should this order
not proceed, so to speak, out of chaos? The case would be like the following--certain kinds of plants multiply by
seed, so that a seed always produces a plant of the same kind as that from which it was produced--but nothing in the
seed corresponds to the plant which comes from it; so that it is impossible to infer the properties or structure of the
plant from those of the seed that it comes out of--this can only be done from the history of the seed. So an organism
might come into being even out of something quite amorphous, as it were causelessly; and there is no reason why
this should not really hold for our thoughts, and hence for our talking and writing.
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609. It is thus perfectly possible that certain psychological phenomena cannot be investigated physiologically,
because physiologically nothing corresponds to them.
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610. I saw this man years ago: now I have seen him again, I recognize him, I remember his name. And why does
there have to be a cause of this remembering in my nervous system? Why must something or other, whatever it may
be, be stored up there in any form? Why must a trace have been left behind? Why should there not be a
psychological regularity to which no physiological regularity corresponds? If this upsets our concepts of causality
then it is high time they were upset.
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611. The prejudice in favour of psychophysical parallelism is a fruit of primitive interpretations of our concepts. For
if one allows a causality between psychological phenomena which is not mediated physiologically, one thinks one is
making profession that there exists a soul side by side with the body, a ghostly soul-nature.
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612. Imagine the following phenomenon. If I want someone to take note of a text that I recite to him, so that he can
repeat it to me later, I have to give him paper and pencil; while I am speaking he makes lines, marks, on the paper; if
he has to reproduce the text later he follows those marks with his eyes and recites the text. But I assume that what he
has jotted down is not writing, it is not connected by rules with the words of the text; yet without these jottings he is
unable to reproduce the text; and if anything in it is altered, if part of it is destroyed, he sticks in his 'reading' or
recites the text uncertainly or carelessly, or cannot find the words at all.--This can be imagined!--What I called
jottings would not be a rendering of the text, not so to speak a translation with another symbolism. The text would
not be stored up in the jottings. And why should it be stored up in our nervous system?
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613. Why should there not be a natural law connecting a starting and a finishing state of a system, but not covering
the intermediary state? (Only one must not think of causal efficacy.)
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614. "How does it come about that I see the tree standing up straight even if I incline my head to one side, and so the
retinal image is that of an obliquely standing tree?" Well, how does it come about that I speak of the tree as standing
up straight even in these circumstances?--"Well, I am conscious of the inclination of my head, and so I supply the
requisite correction in the way I take my visual impression."--But doesn't that mean confusing what is primary and
what is secondary? Imagine that we know nothing at all of the inner structure of the eye--would this problem
altogether disappear? We do not supply any correction here--that explanation is gratuitous.
Well--but now that the structure of the eye is known--how does it come about that we act, react, in this way?
But must there be a physiological explanation here? Why don't we just leave explaining alone?--But you would
never talk like that, if you were examining the behaviour of a machine!--Well, who says that a living creature, an
animal body, is a machine in this sense?--
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615. (I have never yet read a comment on the fact that when one shuts one eye and "only sees with one eye" one
does not
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simultaneously see darkness (blackness) with the one that is shut.)
Page 107
616. The limitlessness of the visual field is clearest when we are seeing nothing in complete darkness.†1
Page 107
617. How is it with a blind man; can one part of language not be explained to him? Or rather not be described?
Page 107
618. A blind man can say that he is blind and the people around him sighted. "Yes, but doesn't he after all mean
something different from a sighted man when he uses the words 'blind' and 'sighted'?" What is the ground of one's
inclination to say so? Well, if someone did not know what a leopard looked like, still he could say and understand
"That place is very dangerous, there are leopards there". He would perhaps all the same be said not to know what a
leopard is, and so not to know, or not completely, what the word "leopard" means, until he is shewn such an animal.
Now it strikes us as being the same with the blind. They don't know, so to speak, what seeing is like.--Now is 'not
knowing fear' analogous to 'never having seen a leopard'? That, of course, I want to deny.
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619. Might I not e.g. make the assumption that he sees something red when I hit him on the head? This might
correspond to an experience in the case of sighted people.
Assuming this, still he is blind for all practical purposes. That is to say he does not react like a normal human
being. If, however, someone were blind of eye, but on the other hand so conducted himself that we were forced to
say that he saw with the palms of his hands (this behaviour is easy to work out), we should treat him as sighted and
should also find possible the use of a sample slip to explain the word 'red' to him.
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620. You give someone a signal when you imagine something: you use different signals for different images.--How
have you agreed together what each signal is to mean?
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621. Auditory images, visual images--how are they distinguished from sensations? Not by "vivacity".
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Images tell us nothing, either right or wrong, about the external world. (Images are not hallucinations, nor yet
fancies).
While I am looking at an object I cannot imagine it.
Difference between the language-games: "Look at this figure!" and: "Imagine this figure!"
Images are subject to the will.
Images are not pictures. I do not tell what object I am imagining by the resemblance between it and the
image.
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Asked "What image have you?" one can answer with a picture.
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622. One would like to say: The imaged is in a different space from the heard sound. (Question: Why?) The seen in a
different space from the imaged.
Hearing is connected with listening; forming an image of a sound is not.
That is why the heard sound is in a different space from the imagined sound.
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623. I read a story and have all sorts of images while I read, i.e. while I am looking attentively, and hence seeing
clearly.
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624. People might exist who never use the expression "seeing something with the inner eye" or anything like it, and
these people might be able to draw and model 'out of imagination' or from memory, to mimic others etc. Such a
person might also shut his eyes or stare into vacancy as if blind before drawing something from memory. And yet
he might deny that he then sees before him what he goes on to draw. But what value need I set on this utterance?
Should I judge by it whether he has a visual image? (Not it alone. Think of the expression: "Now I see it before
me--now no longer". Here is genuine duration.)
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625. I might also have said earlier: The tie-up between imaging and seeing is close; but there is no similarity.
The language-games employing these concepts are radically different--but hang together.
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626. Difference between: 'trying to see something' and 'trying to form an image of something'. In the first case one
says: "Look, just over there!", in the second "Shut your eyes!"
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627. It is just because forming images is a voluntary activity that it does not instruct us about the external world.
Page 109
628. What is imaged is not in the same space as what is seen. Seeing is connected with looking.
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629. "Seeing and imaging are different phenomena".--The words "seeing" and "imaging" have different meanings.
Their meanings relate to a host of important kinds of human behaviour, to phenomena of human life.
Page 109
630. If someone insists that what he calls a "visual image" is like a visual impression, say to yourself once more that
perhaps he is making a mistake. Or: Suppose he is making a mistake. That is to say: What do you know about the
resemblance of his visual impression and his visual image?! (I speak of others because what goes for them goes for
me too).
So what do you know about this resemblance? It is manifested only in the expressions which he is inclined
to use; not in something he uses those expressions to say.
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631. "There's no doubt at all: visual images and visual impressions are of the same kind!" That must be something
you know from your own experience; and in that case it is something that may be true for you and not for other
people. (And this of course holds for me too, if I say it).
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632. When we form an image of something we are not observing. The coming and going of the pictures is not
something that happens to us. We are not surprised by these pictures, saying "Look!" (Contrast with e.g.
after-images).
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633. We do not 'banish' visual impressions, as we do images. And we don't say of the former, either, that we might
not banish them.
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634. If someone really were to say "I don't know whether I am now seeing a tree or having an image of it", I should
at first think he meant: "or just fancying that there is a tree over there".
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If he does not mean this, I couldn't understand him at all--but if someone tried to explain this case to me and said
"His images are of such extraordinary vivacity that he can take them for impressions of sense"--should I understand
it then?
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635. But must one not distinguish here: (a) forming the image of a human face, for example, but not in the space that
surrounds me--(b) forming an image of a picture on that wall over there?
At the request "Imagine a round spot over there" one might fancy that one really was seeing one there.
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636. The 'imagination-picture' does not enter the language-game in the place where one would like to surmise its
presence.
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637. I learn the concept 'seeing' along with the description of what I see. I learn to observe and to describe what I
observe. I learn the concept 'to have an image' in a different context. The descriptions of what is seen and what is
imaged are indeed of the same kind, and a description might be of the one just as much as of the other; but
otherwise the concepts are thoroughly different. The concept of imaging is rather like one of doing than of receiving.
Imagining might be called a creative act. (And is of course so called).
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638. "Yes, but the image itself, like the visual impression, is surely the inner picture, and you are talking only of
differences in the production, the coming to be, and in the treatment of the picture." The image is not a picture, nor is
the visual impression one. Neither 'image' nor 'impression' is the concept of a picture, although in both cases there is
a tie-up with a picture, and a different one in either case.
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639. What do you call the "experiential content" of seeing, what the "experiential content" of imaging?
Page 110
640. "But couldn't I imagine an experiential content of the same kind as visual images, but not subject to the will,
and so in this respect like visual impressions?"
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641. (Clearly the voluntary act of forming images cannot be compared with moving the body; for someone else is
also competent to judge whether the movement has taken place; whereas with the movement of my images the
whole point would always
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be what I said I saw--whatever anyone else sees. So really moving objects would drop out of consideration, since no
such thing would be in question).
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642. If then one said: "Images are inner pictures, resembling or exactly like my visual impressions, only subject to
my will"--the first thing is that this doesn't yet make sense.
For if someone has learnt to report what he sees over there, or what seems to him to be over there, it surely
isn't clear to him what it would mean if he were ordered now to see this over there, or now to have this seem to him
to be over there.
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643. "To move by pure will"--What does that mean? That the image-pictures always exactly obey my will, whereas
my hand in drawing, my pencil, does not? All the same in that case it would be possible to say: "Usually I form
images of exactly what I want to; today it has turned out differently". Is there such a thing as 'images not coming
off'?
Page 111
644. A language-game comprises the use of several words.
Page 111
645. Nothing could be more mistaken than to say: seeing and forming an image are different activities. That is as if
one were to say that moving and losing in chess were different activities.
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646. When we learn as children to use the words "see", "look", "image", voluntary actions and orders play a part in
this training. But a different one for each of the three words. The language-game "Look" and "Form an image
of..."--how am I ever to compare them?--If we want to train someone to react to the order "Look..." and to
understand the order "Form an image of..." we must obviously teach him quite differently. Reactions which belong
to the latter language-game do not belong to the former. There is of course a close tie-up of these language-games;
but a resemblance?--Bits of one resemble bits of the other, but the resembling bits are not homologous.
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647. I could imagine something similar for actual games.
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648. One language-game analogous to a fragment of another. One space projected into a limited extent of another. A
'gappy' space. (For "inner and outer".)
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649. Let us imagine a variant of tennis: it is included in the rules of this game that the player has to form
such-and-such images as he performs certain moves in the game. (Let the purpose of this rule be to make the game
more difficult.) The first objection is: it is too easy to cheat in this game. But this is met by the assumption that the
game is played only by honest and reliable people. So here we have a game with inner moves of the game.--
What sort of move is the inner move of the game, what does it consist in? In this, that--according to the
rule--he forms an image of....--But might it not also be said: We do not know what kind of inner move of the game
he does perform according to the rule; we only know its manifestations. Let the inner move of the game be an X,
whose nature we do not know. Or again: Even here there are only external moves of the game; the communication
of the rule and what is called the 'manifestation of the inner process'.--Now may one not describe the game in all
three different ways? Even the one with the 'unknown' X is a quite possible kind of description. One man says that
the so-called 'inner' move in the game is not comparable with a move in the game in the ordinary sense--the next
says it is so comparable--and the third: it is comparable only with an action happens in secret and which no one
knows but the agent.
It is important for us to see the dangers of the expression "inner move of the game". It is dangerous because
it produces confusion.
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650. Memory: "I see us still, sitting at that table".--But have I really the same visual image--or one of those that I had
then? Do I also certainly see the table and my friend from the same point of view as then, and so not see
myself?--My memory-image is not evidence for that past situation, like a photograph which was taken then and
convinces me now that this was how things were then. The memory-image and the memory-words stand on the
same level.
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651. Shrugging of shoulders, head-shakes, nods and so on we call signs first and foremost because they are
embedded in the use of our verbal language.
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652. If one takes it as obvious that a man takes pleasure in his own fantasies, let it be remembered that fantasy does
not correspond to a painted picture, to a sculpture or a film, but to a complex formation out of heterogeneous
components--signs and pictures.
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653. Some men recall a musical theme by having an image of the score rise before them, and reading it off.
It could be imagined that what we call "memory" in some man consisted in his seeing himself looking things
up in a book in spirit, and that what he read in that book was what he remembered. (How do I react to a memory?)
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654. Can a memory-experience be described?--Certainly.--But can what is memory-like about this experience be
described? What does that mean?--((The indescribable aroma.))
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655. "A picture (imagination-picture, memory-image) of longing." One thinks one has already done everything by
speaking of a 'picture'; for longing is a content of consciousness, and a picture of it is something that is (very) like it,
even though less clear than the original.
And it might well be said of someone who plays longing on the stage, that he experiences or has a picture of
longing: not as an explanation of his action, but as a description of it.
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656. To be ashamed of a thought. Is one ashamed at the fact that one has spoken such-and-such a sentence in one's
imagination?
Language is variously rooted; it has roots, not a single root. [Marginal note: ((Remembering a thought, an
intention.)) A seed.]
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657. "It tastes exactly like sugar." How is it I can be so sure of this? Even if it turns out wrong.--And what astonishes
me about it? That I bring the concept sugar into so firm a connexion with the taste sensation. That I seem to
recognize the substance sugar directly in the taste.
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But instead of the expression "It tastes exactly..." I might more primitively exclaim "Sugar!" And can it be
said that 'the substance sugar comes before my mind' at the word? How does it do that?
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658. Can I say that this taste brought the name "sugar" along with it in a peremptory fashion? Or the picture of a
lump of sugar? Neither seems right. The demand for the concept 'sugar' is indeed peremptory, just as much so,
indeed, as the demand for the concept 'red' when we use it to describe what we see.
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659. I remember that sugar tasted like this. The experience returns to consciousness. But, of course: how do I know
that this was the earlier experience? Memory is no more use to me here. No, in those words--that the experience
returns to consciousness....,--I am only transcribing my memory, not describing it.
But when I say "It tastes exactly like sugar", in an important sense no remembering takes place. So I do not
have grounds for my judgment or my exclamation. If someone asks me "What do you mean by 'sugar'?"--I shall
indeed try to shew him a lump of sugar. And if someone asks "How do you know that sugar tastes like that?" I shall
indeed answer him "I've eaten sugar thousands of times"--but that is not a justification that I give myself.
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660. "It tastes like sugar". One remembers exactly and with certainty what sugar tastes like. I do not say "I believe
sugar tastes like this". What a remarkable phenomenon! It just is the phenomenon of memory.--But is it right to call
it a remarkable phenomenon?
It is anything but remarkable. That certainty is not (by a hair's breadth) more remarkable than uncertainty
would be. For what is remarkable? My saying with certainty "This tastes like sugar", or its then really being sugar?
Or that other people find the same thing?
If the certain recognition of sugar is remarkable, then the failure to recognize it would be less so.
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661. "What a queer and frightful sound. I shall never forget it". And why should one not be able to say that of
remembering
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("What a queer... experience...") when one has seen into the past for the first time?--
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662. Remembering: a seeing into the past. Dreaming might be called that, when it presents the past to us. But not
remembering; for, even if it shewed scenes with hallucinatory clarity, still it takes remembering to tell us that this is
past.
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663. But if memory shews us the past, how does it shew us that it is the past?
It does not shew us the past. Any more than our senses shew us the present.
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664. Nor can it be said to communicate the past to us. For even supposing that memory were an audible voice that
spoke to us--how could we understand it? If it tells us e.g. "Yesterday the weather was fine", how can I learn what
"yesterday" means?
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665. I give myself an exhibition of something only in the same way as I give one to other people.
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666. I can display my good memory to someone else and also to myself. I can subject myself to an examination.
(Vocabulary, dates).
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667. But how do I give myself an exhibition of remembering? Well, I ask myself "How did I spend this morning?"
and give myself an answer.--But what have I really exhibited to myself? Remembering? That is, what it's like to
remember something?--Should I have exhibited remembering to someone else by doing that?
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668. Forgetting the meaning of a word--and then remembering it again. What sort of processes go on here? What
does one remember, what occurs to one, when one recalls what the French word "peut-être" means?
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669. If I am asked "Do you know the ABC?" and I answer "Yes" I am not saying that I am now going through the
ABC in my mind, or that I am in a special mental state that is somehow equivalent to the recitation of the ABC.†1
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670. One can own a mirror; does one then own the reflection that can be seen in it?
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671. Saying something is an activity, being inclined to say something a state. "But what does it consist in?"--Give
yourself an account of how the expression is used.
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672. "So long as the temperature of the rod does not fall below... it can be forged." So it makes sense to say: "I can
forge it from five till six o'clock." Or: "I can play chess from five till six," i.e. I have time from five till six.--"So long
as my pulse does not fall below... I can do the calculation." This calculation takes one and a half minutes; but how
long does being able to do it take? And if you can do it for an hour, do you keep on starting afresh?
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673. Attention is dynamic, not static--one would like to say. I begin by comparing attention to gazing but that is not
what I call attention; and now I want to say that I find it is impossible that one should attend statically.
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674. If in a particular case I say: attention consists in preparedness to follow each smallest movement that may
appear--that is enough to shew you that attention is not a fixed gaze: no, this is a concept of a different kind.
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675. States: 'Being able to climb a mountain' may be called a state of my body. I say: "I can climb it--I mean I am
strong enough." Compare with this the following condition of being able. "Yes, I can go there"--I mean I have
enough time.
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676. What is the role of false propositions in a language-game? I believe there are various cases.
(1) Someone has to observe the signal lights at a street-crossing and to tell someone else what colours are
shewing. He makes a slip of the tongue and says the wrong colour.
(2) Meteorological observations are put in train and the weather for the next day is predicted from them by
certain rules. The prediction comes off or does not.
In the first case one can say he plays wrong; in the second one cannot--as at one time I thought.
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Here one is tormented by a question running something like: Is the verification too part of the
language-game?
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677. I assert: "If this happens, that will happen. If I am right, you pay me a shilling, if I am wrong, I pay you one, if
it remains undecided, neither pays". This might also be expressed like this: The case in which the antecedent does
not come true does not interest us, we aren't talking about it. Or again: we do not find it natural to use the words
"yes" and "no" in the same way as in the case (and there are such cases) in which we are interested in the material
implication. By "No" we mean here "p and not q", by "Yes", only "p and q". There is no law of excluded middle
running: Either you win the bet or you lose it--there is no third possibility.
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678. Someone playing at dice throws first five, then four and says "If only I'd thrown a four instead of the five I'd
have won!" The condition is not physical but only mathematical, for one might reply: "If you had thrown a four
first,--who knows what you would have thrown after?"
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679. If you say now "The use of the subjunctive is founded on belief in natural law--the rejoinder may be: "It is not
founded on that belief; it and that belief are on the same level". (In a film I heard a father tell his daughter that he
ought to have married a different woman: "She ought to have been your mother"! Why is this wrong?)
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680. Fate stands in contrast to natural law. One seeks to find grounds for natural law and to use it, but not fate.
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681. "If p occurs, then q occurs" might be called a conditional prediction. That is, I make no prediction for the case
not-p. But for that reason what I say also remains unverified by "not-p and not-q".
Or even: there are conditional predictions and "p implies q" is not one. ((Relates to Vol. Q, p. 14.))†1
Page 117
682. I will call the sentence "If p occurs then q occurs" "S".--"S or not S" is a tautology; but is it (also) the law of
excluded middle?--Or again: If I want to say that the prediction "S"
Page Break 118
may be right, wrong, or undecided, is that expressed by the sentence "not (S or not-S)"?
Page 118
683. Is the negation of a proposition identical with the disjunction of the cases it does not exclude? In some cases it
is. (E.g. in this one: "The permutation of the elements ABC that he wrote down was not ACB").
Page 118
684. The important sense of Frege's assertion-sign is perhaps best grasped if we say: it signalises clearly the
beginning of the sentence.--This is important: for our philosophical difficulties about the nature of 'negation' and of
'thinking' hang together with the fact that a proposition "&unk; not-p" or "&unk; I believe p" does contain the
proposition "p" but not "&unk; p". (For if I hear someone say: "it's raining" I don't know what he said if I don't know
whether I heard the beginning of the sentence).†1
Page 118
685. A contradiction prevents me from getting to act in the language-game.
Page 118
686. But suppose the language-game consisted in my continuously being driven from one decision to the contrary
one!
Page 118
687. Contradiction is to be regarded, not as a catastrophe, but as a wall indicating that we can't go on here.
Page 118
688. I should like to ask, not so much "What must we do to avoid a contradiction?" as "What ought we to do if we
have arrived at a contradiction?"
Page 118
689. Why is a contradiction more to be feared than a tautology?
Page 118
690. Our motto might be: "Let us not be bewitched".
Page 118
691†2. "The Cretan Liar". He might have written "This proposition is false" instead of "I am lying". The
answer would be: "Very well, but which proposition do you mean?"--"Well, this proposition".--"I understand, but
which is the proposition mentioned in it?"--"This one"--"Good, and which proposition
Page Break 119
does it refer to?" and so on. Thus he would be unable to explain what he means until he passes to a complete
proposition.--We may also say: The fundamental error lies in one's thinking that a phrase e.g. "This proposition" can
as it were allude to its object (point to it from far off) without having to go proxy for it.
Page 119
692. Let us raise the question: What practical purpose can be served by Russell's Theory of Types?--Russell makes
us realize that we must sometimes put restrictions on the expression of generality in order to avoid having
undesirable consequences drawn from it.
Page 119
693. The reasoning that leads to an infinite regress is to be given up not 'because in this way we can never reach the
goal', but because here there is no goal; so it makes no sense to say "we can never reach it".
We readily think that we must run through a few steps of the regress and then so to speak give it up in
despair. Whereas its aimlessness (the lack of a goal in the calculus) can be derived from the starting position.
Page 119
694. A variant of Cantor's diagonal proof:
Let N = F (k, n) be the form of a law for the development of decimal fractions. N is the nth decimal position
of the kth development. The law of the diagonal is then: N = F (n, n) = Def. F'(n).
To prove: that F' (n) cannot be one of the rules F' (k, n). Assume it is the 100th. Then the rule for the
construction
of F' (1) runs F (1, 1)
of F' (2) F (2, 2) etc.
but the rule for the construction of the 100th position of F' (n) becomes F (100, 100) i.e. it shews only that the 100th
place is supposed to be the same as itself, and so for n = 100 it is not a rule.
The rule of the game runs "Do the same as..."--and in the special case it becomes "Do the same as what you
do".
Page 119
695. The understanding of a mathematical question. How do we know if we understand a mathematical question?
A question--it may be said--is a commission. And understanding a commission means: knowing what one
has got
Page Break 120
to do. Naturally, a commission can be quite vague--e.g., if I say "Bring him something that'll do him good". But that
may mean: think about him, about his state etc. in a friendly way and then bring him something corresponding to
your sentiment towards him.
Page 120
696. A mathematical question is a challenge. And we might say: it makes sense, if it spurs us on to some
mathematical activity.
Page 120
697. We might then also say that a question in mathematics makes sense if it stimulates the mathematical
imagination.
Page 120
698. Translating from one language into another is a mathematical task, and the translation of a lyrical poem, for
example, into a foreign language is quite analogous to a mathematical problem. For one may well frame the problem
"How is this joke (e.g.) to be translated (i.e. replaced) by a joke in the other language?" and this problem can be
solved; but there was no systematic method of solving it.
Page 120
699. Imagine human beings who calculate with 'extremely complicated' numerals. These present themselves as
figures which arise if our numerals are written on top of one another. They write e.g. π up to the fifth place like this:
. If you watch them you find it difficult to guess what they are doing. And they themselves perhaps
cannot explain it. For this numeral, written in a somewhat different notation, may alter its appearance to the point of
unrecognizability by us. And what the people were doing would seem to us purely intuitive.
Page 120
700. Why do we count? Has it proved practical? Do we have the concepts we have, e.g. our psychological concepts,
because it has proved advantageous?--And yet we do have certain concepts on that account, we have introduced
them on that account.
Page 120
701. At any rate the difference between what are called propositions of mathematics and empirical propositions
comes to light if one reflects whether it makes sense to say: "I wish twice two were five!"
Page Break 121
Page 121
702. If one considers that 2 + 2 = 4 is a proof of the proposition "there are even numbers", one sees how loosely the
word "proof" is used here. The proposition "there are even numbers" is supposed to proceed from the equation 2 + 2
= 4!--And what is the proof of the existence of prime numbers?--The method of reduction to prime factors. But in
this method nothing is said, not even about "prime numbers".
Page 121
703. "To understand sums in the elementary school the children would have to be important philosophers; failing
that, they need practice".†1
Page 121
704. Russell and Frege take concepts as, as it were, properties of things. But it is very unnatural to take the words
man, tree, treatise, circle, as properties of a substrate.†2
Page 121
705. Dirichlet's conception of a function is only possible where it does not seek to express an infinite rule by a list,
for there is no such thing as an infinite list.
Page 121
706. Numbers are not fundamental to mathematics.
Page 121
707. The concept of the 'order' of the rational numbers, e.g., and of the impossibility of so ordering the irrational
numbers. Compare this with what is called an 'ordering' of digits. Likewise the difference between the 'co-ordination'
of one digit (or nut) with another and the 'co-ordination' of all whole numbers with the even numbers; etc.
Everywhere distortion of concepts.
Page 121
708. There is obviously a method of making a straight-edge. This method involves an ideal, I mean an
approximation-procedure of unlimited possibility, for this very procedure is the ideal.
Or rather: only if there is an approximation-procedure of unlimited possibility can (not must) the geometry of
this procedure be Euclidean.†3
Page 121
709. To regard a calculation as an ornament is also formalism, but of a good sort.
Page 121
710. A calculation can be regarded as an ornament. A figure in a plane may fit another one or not, may be taken with
other ones
Page Break 122
in various ways. If further the figure is coloured, there is a further fit according to colour. (Colour is only another
dimension).
Page 122
711. There is a way of looking at electrical machines and installations (dynamos, radio stations, etc., etc.) which sees
these objects as arrangements of copper, iron, rubber etc. in space, without any preliminary understanding. And this
way of looking at them might lead to some interesting results. It is quite analogous to looking at a mathematical
proposition as an ornament.--It is of course an absolutely strict and correct conception; and the characteristic and
difficult thing about it is that it looks at the object without any preconceived idea (as it were from a Martian point of
view), or perhaps more correctly: it upsets the normal preconceived idea, explanation (runs athwart it).
Page 122
712. (The style of my sentences is extraordinarily strongly influenced by Frege. And if I wanted to, I could establish
this influence where at first sight no one would see it).
Page 122
713. "Put it here"--indicating the place with one's finger--that is giving an absolute spatial position. And if someone
says that space is absolute he might produce this as an argument for it: "There is a place: here". [Marginal note:
((Perhaps belongs with the first language-games.))]
Page 122
714. A mental illness could be imagined in which one can use and understand a name only in the presence of the
bearer.
Page 122
715. There might be a use of signs made, such that they become useless (they are destroyed) as soon as the bearer
has ceased to exist.
In this language-game the name has the object on a string, so to speak; and if the object ceases to exist, the
name, which has done its work in conjunction with the object, can be thrown away. (The word "handle" for a proper
name).
Page 122
716. What about these two sentences: "This sheet is red" and "this sheet is the colour called 'red' in English"? Do
they both say the same?
Page Break 123
Page 123
Doesn't this depend on what the criterion is for a colour's being called 'red' in English?
Page 123
717. "You can't hear God speak to someone else, you can hear him only if you are being addressed".--That is a
grammatical remark.
FOOTNOTES
Page 3
†1 Schiller, Wallenstein, Die Piccolomini, Act I, Scene 2. The words are in fact said by Illo, and not by Max.
Page 13
†1 See Philosophical [[sic]] Bemerkungen § 31. Eds.
Page 13
†2 See Philosophical Investigations § 581. Eds.
Page 13
†3 See Philosophische Bemerkungen § 25. Eds.
Page 13
†4 Theaetetus 189a. Eds.
Page 16
†1 In the original this sentence is in square brackets. Here as elsewhere we have shown this as double
brackets. Eds.
Page 17
†1 "Experience" is crossed out in the typescript. Eds.
Page 17
†2 See Philosophical Investigations § 2. Eds.
Page 18
†1 See Philosophical Investigations § 2. Eds.
Page 24
†1 See Philosophical Investigations § 556. Eds.
Page 30
†1 "Heaven be praised! A little slipped--out of the Croat clutches." Schiller, Wallenstein, Die Piccolomini,
Act 1, Scene 2.
Page 45
†1 No drawing appears in the MS. at this place; the reader may imagine something appropriate. Of various
possibilities, we have adopted a drawing by Dr. R. B. O. Richards. Eds.
Page 47
†1 See Philosophische Bemerkungen § 66. Eds.
Page 47
†2 See Philosophische Bemerkungen § 95. Eds.
Page 48
†1 See Philosophische Bemerkungen § 66. Ed&.
Page 49
†1 Supplied from source-typescript. Eds.
Page 50
†1 See Philosophical Investigations § 229. Eds.
Page 51
†1 See Philosophical Remarks § 130. Eds.
Page 51
†2 See Philosophische Bemerkungen §§ 235, 236. Eds.
Page 54
†1 See Remarks on the Foundations of Mathematics V--§ 46; new edition VII--§ 67. Eds.
Page 64
†1 This remark was not among those preserved in the box of cuttings; we have supplied it from the
typescript from another copy of which the succeeding remark was cut. Eds.
Page 66
†1 The cuttings include the following, omitted from the 1st edition: (Relates to what Frege, and occasionally
Ramsey, said about recognition as a condition of symbolizing. What is the criterion for my having recognized colour
right? Something like the experience of joy in recognizing?) Eds.
Page 78
†* Freud speaks of his 'dynamic' theory of dreams.
Page 79
†1 See Philosophical Investigations § 154. Eds.
Page 80
†1 Philosophische Bemerkungen § 2. Eds.
Page 80
†2 Charmides, 174d-175a. Eds.
Page 81
†1 .... because the search says more than the discovery... Eds.
Page 81
†2 After this remark there occur in the typescript the words: "Such irritations are wont to stimulate the
powers of thought." How does thought remedy an irritation?--The passage is missing from the first edition of Zettel.
Eds.
Page 81
†3 See Philosophical Investigations § 144. Eds.
Page 96
†1 See Philosophical Investigations § 300. Eds.
Page 98
†1 In the typescript of the Zettel there is here the further remark: How could you explain what it means "to
simulate pain", "to put on a show of pain"? Eds.
Page 98
†2 See Philosophical lnvestigations p. 174. Eds.
Page 100
†1 See Philosophical Investigations § 613. Eds.
Page 100
†2 Schatzkästlein, Zwei Erzählungen. Eds.
Page 107
†1 Philosophische Bemerkungen § 224. Eds.
Page 115
†1 After this remark there occur in the typescript the words: How does one teach anyone to read to himself?
How does one know if he can do so? How does he himself know that he is doing what is required of him?--This is
paragraph 375 in PI. Eds.
Page 117
†1 The reference is to MS. 136. Eds.
Page 118
†1 See Philosophical Investigations § 22. Eds.
Page 118
†2 See Philosophical Investigations § 16. Eds.
Page 121
†1 Here in the typescript there occurs: When we read an explanation say of logical consequence, we rely on
what he writes; on the signs ((on (for me) the calculus)). Eds.
Page 121
†2 See Philosophical Remarks § 96. Eds.
Page 121
†3 See Philosophische Bemerkungen § 178. Eds.
PHILOSOPHICAL INVESTIGATIONS
Page i
Page Break ii
Page ii
Ludwig Wittgenstein from Blackwell Publishers
The Wittgenstein Reader
Edited by Anthony Kenny
Zettel
Second Edition
Edited by G.E.M. Anscombe and G.H. von Wright
Remarks on Colour
Edited by G.E.M. Anscombe
Remarks on the Foundations of Mathematics
Third edition
Edited by G.H. von Wright, Rush Rhees and G.E.M. Anscombe
Philosophical Remarks
Edited by Rush Rhees
Philosophical Grammar
Edited by Rush Rhees
On Certainty
Edited by G.E.M. Anscombe and G.H. von Wright
Lectures and Conversations
Edited by Cyril Barrett
Culture and Value
Edited by G.H. von Wright
The Blue and Brown Books
Preliminary Studies for the Philosophical Investigations
Remarks on the Philosophy of Psychology, Vol. I
Edited by G.E.M. Anscombe and G.H. von Wright
Remarks on the Philosophy of Psychology, Vol. II
Edited by G.E.M. Anscombe and Heikki Nyman
Last Writings on the Philosophy of Psychology, Vol. I
Edited by G.H. von Wright and Heikki Nyman
Last Writings on the Philosophy of Psychology, Vol. II
Edited by G.H. von Wright and Heikki Nyman
Page Break iii
Titlepage
Page iii
PHILOSOPHICAL INVESTIGATIONS
Second Edition
By
LUDWIG WITTGENSTEIN
Translated by
G. E. M. ANSCOMBE
Page Break iv
Copyright page
Page iv
Copyright © Blackwell Publishers Ltd, 1953, 1958, 1997
First published 1953
Second edition 1958
Reprinted 1963, 1967, 1997, 1998, 1999
Blackwell Publishers Ltd
108 Cowley Road
Oxford OX4 1JF
UK
Blackwell Publishers Inc.
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Malden, Massachusetts 02148
USA
All rights reserved. Except for the quotation of short passages for the purposes of criticism and review, no part of
this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher.
Except in the United States of America, this book is sold subject to the condition that it shall not, by way of trade or
otherwise, be lent, re-sold, hired out, or otherwise circulated without the publisher's prior consent in any form of
binding or cover other than that in which it is published and without a similar condition including this condition
being imposed on the subsequent purchaser.
British Library Cataloguing in Publication Data
A CIP catalogue record for this book is available from the British Library.
Library of Congress Cataloging-in-Publication Data
Wittgenstein, Ludwig, 1889-1951.
[Philosophische Untersuchungen. English & German]
Philosophical investigations / by Ludwig Wittgenstein: translated by G.E.M. Anscombe.--2nd ed. rept. p. cm.
Includes index.
Added title page title: Philosophische Untersuchungen.
ISBN 0-631-20569-1 (pbk: alk. paper)
1. Philosophy. 2. Language and languages--Philosophy.
2. Semantics (Philosophy) I. Anscombe, G. E. M. (Gertrude Elizabeth Margaret) II. Title. III. Title: Philosophische
Untersuchungen
B3376.W563P53 1997
192-dc2196-37331
CIP
Printed in Great Britain by T. J. International, Padstow, Cornwall
Page Break v
TRANSLATOR'S NOTE
Page v
MY acknowledgements are due to the following, who either checked the translation or allowed me to consult them
about German and Austrian usage or read the translation through and helped me to improve the English: Mr. R.
Rhees, Professor G. H. von Wright, Mr. P. Geach, Mr. G. Kreisel, Miss L. Labowsky, Mr. D. Paul, Miss I. Murdoch.
NOTE TO SECOND EDITION
Page v
THE text has been revised for the new edition. Alterations on the German side have been few, and have been
confined to mistakes in spelling or punctuation. A large number of small changes have been made in the English
text. The following passages have been significantly altered:
Page v
In Part I: §§ 108, 109, 116, 189, 193, 251, 284, 352, 360, 393, 418, 426, 442, 456, 493, 520, 556, 582, 591, 644,
690, 692.
Page v
In Part II: pp. 193e, 211e, 216e, 217e, 220e, 232e.
NOTE TO THE RE-ISSUED SECOND EDITION
Page v
THE graphical representations have been reassessed by Michael A. R. Biggs of the University of Hertfordshire, UK.
Reference has been made to both the manuscript sources for this volume and to the broader context of
Wittgenstein's writings and graphical practice as a whole. Such a synoptic view was not available when this volume
was first published and the editors have taken the opportunity to incorporate recommendations on the grounds of
improved legibility, felicity or perspicuity. Revisions have been made to the graphics on the following pages: 23, 138,
193, 203 and 210. The parallel German and English text is identical to the 1967 issue, with the exception of the
substitution of the word 'sigma' by the sign 'σ' on page 67.
Page Break vi
EDITORS' NOTE
Page vi
WHAT appears as Part I of this volume was complete by 1945. Part II was written between 1947 and 1949. If
Wittgenstein had published his work himself, he would have suppressed a good deal of what is in the last thirty
pages or so of Part I and worked what is in Part II, with further material, into its place.
Page vi
We have had to decide between variant readings for words and phrases throughout the manuscript. The
choice never affected the sense.
Page vi
The passages printed beneath a line at the foot of some pages are written on slips which Wittgenstein had cut
from other writings and inserted at these pages, without any further indication of where they were to come in.
Page vi
Words standing between double brackets are Wittgenstein's references to remarks either in this work or in
other writings of his which we hope will appear later.
Page vi
We are responsible for placing the final fragment of Part II in its present position.
G. E. M. ANSCOMBE
R. RHEES
Page Break vii
Page vii
"Überhaupt hat der Fortschritt das an sich, daß er viel großer ausschaut, als er wirklich ist".
NESTROY
Page Break viii
PREFACE
Page viii
THE thoughts which I publish in what follows are the precipitate of philosophical investigations which have
occupied me for the last sixteen years. They concern many subjects: the concepts of meaning, of understanding, of a
proposition, of logic, the foundations of mathematics, states of consciousness, and other things. I have written down
all these thoughts as remarks, short paragraphs, of which there is sometimes a fairly long chain about the same
subject, while I sometimes make a sudden change, jumping from one topic to another.--It was my intention at first
to bring all this together in a book whose form I pictured differently at different times. But the essential thing was
that the thoughts should proceed from one subject to another in a natural order and without breaks.
Page viii
After several unsuccessful attempts to weld my results together into such a whole, I realized that I should
never succeed. The best that I could write would never be more than philosophical remarks; my thoughts were soon
crippled if I tried to force them on in any single direction against their natural inclination.--And this was, of course,
connected with the very nature of the investigation. For this compels us to travel over a wide field of thought
criss-cross in every direction.--The philosophical remarks in this book are, as it were, a number of sketches of
landscapes which were made in the course of these long and involved journeyings.
Page viii
The same or almost the same points were always being approached afresh from different directions, and new
sketches made. Very many of these were badly drawn or uncharacteristic, marked by all the defects of a weak
draughtsman. And when they were rejected a number of tolerable ones were left, which now had to be arranged and
sometimes cut down, so that if you looked at them you could get a picture of the landscape. Thus this book is really
only an album.
Page viii
Up to a short time ago I had really given up the idea of publishing my work in my lifetime. It used, indeed, to
be revived from time to time: mainly because I was obliged to learn that my results (which I had communicated in
lectures, typescripts and discussions), variously
Page Break ix
misunderstood, more or less mangled or watered down, were in circulation. This stung my vanity and I had
difficulty in quieting it.
Page ix
Four years ago I had occasion to re-read my first book (the Tractatus Logico-Philosophicus) and to explain
its ideas to someone. It suddenly seemed to me that I should publish those old thoughts and the new ones together:
that the latter could be seen in the right light only by contrast with and against the background of my old way of
thinking.†1
Page ix
For since beginning to occupy myself with philosophy again, sixteen years ago, I have been forced to
recognize grave mistakes in what I wrote in that first book. I was helped to realize these mistakes--to a degree which
I myself am hardly able to estimate--by the criticism which my ideas encountered from Frank Ramsey, with whom I
discussed them in innumerable conversations during the last two years of his life. Even more than to this--always
certain and forcible--criticism I am indebted to that which a teacher of this university, Mr. P. Sraffa, for many years
unceasingly practised on my thoughts. I am indebted to this stimulus for the most consequential ideas of this book.
Page ix
For more than one reason what I publish here will have points of contact with what other people are writing
to-day.--If my remarks do not bear a stamp which marks them as mine,--I do not wish to lay any further claim to
them as my property.
Page ix
I make them public with doubtful feelings. It is not impossible that it should fall to the lot of this work, in its
poverty and in the darkness of this time, to bring light into one brain or another--but, of course, it is not likely.
Page ix
I should not like my writing to spare other people the trouble of thinking. But, if possible, to stimulate
someone to thoughts of his own.
Page ix
I should have liked to produce a good book. This has not come about, but the time is past in which I could
improve it.
CAMBRIDGE,
January 1945.
Page Break x
Page Break 1
PART I
Page 1
Page Break 2
Page 2
1. "Cum ipsi (majores homines) appellabant rem aliquam, et cum secundum eam vocem corpus ad aliquid
movebant, videbam, et tenebam hoc ab eis vocari rem illam, quod sonabant, cum eam vellent ostendere. Hoc autem
eos velle ex motu corporis aperiebatur: tamquam verbis naturalibus omnium gentium, quae fiunt vultu et nutu
oculorum, ceterorumque membrorum actu, et sonitu vocis indicante affectionem animi in petendis, habendis,
rejiciendis, fugiendisve rebus. Ita verba in variis sententiis locis suis posita, et crebro audita, quarum rerum signa
essent, paulatim colligebam, measque jam voluntates, edomito in eis signis ore, per haec enuntiabam." (Augustine,
Confessions, I. 8.)†1
Page 2
These words, it seems to me, give us a particular picture of the essence of human language. It is this: the
individual words in language name objects--sentences are combinations of such names.--In this picture of language
we find the roots of the following idea: Every word has a meaning. This meaning is correlated with the word. It is the
object for which the word stands.
Page 2
Augustine does not speak of there being any difference between kinds of word. If you describe the learning
of language in this way you are, I believe, thinking primarily of nouns like "table", "chair", "bread", and of people's
names, and only secondarily of the names of certain actions and properties; and of the remaining kinds of word as
something that will take care of itself.
Page 2
Now think of the following use of language: I send someone shopping. I give him a slip marked "five red
apples". He takes the slip to
Page Break 3
the shopkeeper, who opens the drawer marked "apples"; then he looks up the word "red" in a table and finds a
colour sample opposite it; then he says the series of cardinal numbers--I assume that he knows them by heart--up to
the word "five" and for each number he takes an apple of the same colour as the sample out of the drawer.--It is in
this and similar ways that one operates with words.--"But how does he know where and how he is to look up the
word 'red' and what he is to do with the word 'five'?"--Well, I assume that he acts as I have described. Explanations
come to an end somewhere.--But what is the meaning of the word "five"?--No such thing was in question here, only
how the word "five" is used.
Page 3
2. That philosophical concept of meaning has its place in a primitive idea of the way language functions. But
one can also say that it is the idea of a language more primitive than ours.
Page 3
Let us imagine a language for which the description given by Augustine is right. The language is meant to
serve for communication between a builder A and an assistant B. A is building with building-stones: there are
blocks, pillars, slabs and beams. B has to pass the stones, and that in the order in which A needs them. For this
purpose they use a language consisting of the words "block", "pillar", "slab", "beam". A calls them out;--B brings the
stone which he has learnt to bring at such-and-such a call.--Conceive this as a complete primitive language.
Page 3
3. Augustine, we might say, does describe a system of communication; only not everything that we call
language is this system. And one has to say this in many cases where the question arises "Is this an appropriate
description or not?" The answer is: "Yes, it is appropriate, but only for this narrowly circumscribed region, not for
the whole of what you were claiming to describe."
Page 3
It is as if someone were to say: "A game consists in moving objects about on a surface according to certain
rules..."--and we replied: You seem to be thinking of board games, but there are others. You can make your
definition correct by expressly restricting it to those games.
Page 3
4. Imagine a script in which the letters were used to stand for sounds, and also as signs of emphasis and
punctuation. (A script can be conceived as a language for describing sound-patterns.) Now imagine someone
interpreting that script as if there were simply a
Page Break 4
correspondence of letters to sounds and as if the letters had not also completely different functions. Augustine's
conception of language is like such an over-simple conception of the script.
Page 4
5. If we look at the example in §1, we may perhaps get an inkling how much this general notion of the
meaning of a word surrounds the working of language with a haze which makes clear vision impossible. It disperses
the fog to study the phenomena of language in primitive kinds of application in which one can command a clear
view of the aim and functioning of the words.
Page 4
A child uses such primitive forms of language when it learns to talk. Here the teaching of language is not
explanation, but training.
Page 4
6. We could imagine that the language of §2 was the whole language of A and B; even the whole language of
a tribe. The children are brought up to perform these actions, to use these words as they do so, and to react in this
way to the words of others.
Page 4
An important part of the training will consist in the teacher's pointing to the objects, directing the child's
attention to them, and at the same time uttering a word; for instance, the word "slab" as he points to that shape. (I do
not want to call this "ostensive definition", because the child cannot as yet ask what the name is. I will call it
"ostensive teaching of words".--I say that it will form an important part of the training, because it is so with human
beings; not because it could not be imagined otherwise.) This ostensive teaching of words can be said to establish an
association between the word and the thing. But what does this mean? Well, it can mean various things; but one
very likely thinks first of all that a picture of the object comes before the child's mind when it hears the word. But
now, if this does happen--is it the purpose of the word?--Yes, it can be the purpose.--I can imagine such a use of
words (of series of sounds). (Uttering a word is like striking a note on the keyboard of the imagination.) But in the
language of §2 it is not the purpose of the words to evoke images. (It may, of course, be discovered that that helps to
attain the actual purpose.)
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But if the ostensive teaching has this effect,--am I to say that it effects an understanding of the word? Don't
you understand the call "Slab!" if you act upon it in such-and-such a way?--Doubtless the ostensive teaching helped
to bring this about; but only together with a particular
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training. With different training the same ostensive teaching of these words would have effected a quite different
understanding.
Page 5
"I set the brake up by connecting up rod and lever."--Yes, given the whole of the rest of the mechanism.
Only in conjunction with that is it a brake-lever, and separated from its support it is not even a lever; it may be
anything, or nothing.
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7. In the practice of the use of language (2) one party calls out the words, the other acts on them. In
instruction in the language the following process will occur: the learner names the objects; that is, he utters the word
when the teacher points to the stone.--And there will be this still simpler exercise: the pupil repeats the words after
the teacher--both of these being processes resembling language.
Page 5
We can also think of the whole process of using words in (2) as one of those games by means of which
children learn their native language. I will call these games "language-games" and will sometimes speak of a primitive
language as a language-game.
Page 5
And the processes of naming the stones and of repeating words after someone might also be called
language-games. Think of much of the use of words in games like ring-a-ring-a-roses.
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I shall also call the whole, consisting of language and the actions into which it is woven, the
"language-game".
Page 5
8. Let us now look at an expansion of language (2). Besides the four words "block", "pillar", etc., let it
contain a series of words used as the shopkeeper in (1) used the numerals (it can be the series of letters of the
alphabet); further, let there be two words, which may as well be "there" and "this" (because this roughly indicates
their purpose), that are used in connexion with a pointing gesture; and finally a number of colour samples. A gives
an order like: "d--slab--there". At the same time he shews the assistant a colour sample, and when he says "there" he
points to a place on the building site. From the stock of slabs B takes one for each letter of the alphabet up to "d", of
the same colour as the sample, and brings them to the place indicated by A.--On other occasions A gives the order
"this--there". At "this" he points to a building stone. And so on.
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9. When a child learns this language, it has to learn the series of 'numerals' a, b, c,... by heart. And it has to
learn their use.--Will this training include ostensive teaching of the words?--Well, people
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will, for example, point to slabs and count: "a, b, c slabs".--Something more like the ostensive teaching of the words
"block", "pillar", etc. would be the ostensive teaching of numerals that serve not to count but to refer to groups of
objects that can be taken in at a glance. Children do learn the use of the first five or six cardinal numerals in this way.
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Are "there" and "this" also taught ostensively?--Imagine how one might perhaps teach their use. One will
point to places and things--but in this case the pointing occurs in the use of the words too and not merely in learning
the use.--
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10. Now what do the words of this language signify?--What is supposed to shew what they signify, if not the
kind of use they have? And we have already described that. So we are asking for the expression "This word signifies
this" to be made a part of the description. In other words the description ought to take the form: "The word....
signifies....".
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Of course, one can reduce the description of the use of the word "slab" to the statement that this word
signifies this object. This will be done when, for example, it is merely a matter of removing the mistaken idea that the
word "slab" refers to the shape of building-stone that we in fact call a "block"--but the kind of 'referring' this is, that
is to say the use of these words for the rest, is already known.
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Equally one can say that the signs "a", "b", etc. signify numbers; when for example this removes the
mistaken idea that "a", "b", "c", play the part actually played in language by "block", "slab", "pillar". And one can
also say that "c" means this number and not that one; when for example this serves to explain that the letters are to
be used in the order a, b, c, d, etc. and not in the order a, b, d, c.
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But assimilating the descriptions of the uses of words in this way cannot make the uses themselves any more
like one another. For, as we see, they are absolutely unlike.
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11. Think of the tools in a tool-box: there is a hammer, pliers, a saw, a screw-driver, a rule, a glue-pot, glue,
nails and screws.--The functions of words are as diverse as the functions of these objects. (And in both cases there
are similarities.)
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Of course, what confuses us is the uniform appearance of words when we hear them spoken or meet them in
script and print. For their application is not presented to us so clearly. Especially when we are doing philosophy!
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12. It is like looking into the cabin of a locomotive. We see handles all looking more or less alike. (Naturally,
since they are all supposed to be handled.) But one is the handle of a crank which can be moved continuously (it
regulates the opening of a valve); another is the handle of a switch, which has only two effective positions, it is either
off or on; a third is the handle of a brake-lever, the harder one pulls on it, the harder it brakes; a fourth, the handle of
a pump: it has an effect only so long as it is moved to and fro.
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13. When we say: "Every word in language signifies something" we have so far said nothing whatever;
unless we have explained exactly what distinction we wish to make. (It might be, of course, that we wanted to
distinguish the words of language (8) from words 'without meaning' such as occur in Lewis Carroll's poems, or
words like "Lilliburlero" in songs.)
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14. Imagine someone's saying: "All tools serve to modify something. Thus the hammer modifies the position
of the nail, the saw the shape of the board, and so on."--And what is modified by the rule, the glue-pot, the
nails?--"Our knowledge of a thing's length, the temperature of the glue, and the solidity of the box."--Would
anything be gained by this assimilation of expressions?--
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15. The word "to signify" is perhaps used in the most straightforward way when the object signified is
marked with the sign. Suppose that the tools A uses in building bear certain marks. When A shews his assistant such
a mark, he brings the tool that has that mark on it.
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It is in this and more or less similar ways that a name means and is given to a thing.--It will often prove useful
in philosophy to say to ourselves: naming something is like attaching a label to a thing.
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16. What about the colour samples that A shews to B: are they part of the language? Well, it is as you please.
They do not belong among the words; yet when I say to someone: "Pronounce the word 'the'", you will count the
second "the" as part of the sentence. Yet it has a role just like that of a colour-sample in language-game (8); that is, it
is a sample of what the other is meant to say.
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It is most natural, and causes least confusion, to reckon the samples among the instruments of the language.
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((Remark on the reflexive pronoun "this sentence".))
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Page 8
17. It will be possible to say: In language (8) we have different kinds of word. For the functions of the word
"slab" and the word "block" are more alike than those of "slab" and "d". But how we group words into kinds will
depend on the aim of the classification,--and on our own inclination.
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Think of the different points of view from which one can classify tools or chess-men.
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18. Do not be troubled by the fact that languages (2) and (8) consist only of orders. If you want to say that
this shews them to be incomplete, ask yourself whether our language is complete;--whether it was so before the
symbolism of chemistry and the notation of the infinitesimal calculus were incorporated in it; for these are, so to
speak, suburbs of our language. (And how many houses or streets does it take before a town begins to be a town?)
Our language can be seen as an ancient city: a maze of little streets and squares, of old and new houses, and of
houses with additions from various periods; and this surrounded by a multitude of new boroughs with straight
regular streets and uniform houses.
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19. It is easy to imagine a language consisting only of orders and reports in battle.--Or a language consisting
only of questions and expressions for answering yes and no. And innumerable others.--And to imagine a language
means to imagine a form of life.
Page 8
But what about this: is the call "Slab!" in example (2) a sentence or a word?--If a word, surely it has not the
same meaning as the like-sounding word of our ordinary language, for in §2 it is a call. But if a sentence, it is surely
not the elliptical sentence: "Slab!" of our language.--As far as the first question goes you can call "Slab!" a word and
also a sentence; perhaps it could be appropriately called a 'degenerate sentence' (as one speaks of a degenerate
hyperbola); in fact it is our 'elliptical' sentence.--But that is surely only a shortened form of the sentence "Bring me a
slab", and there is no such sentence in example (2).--But why should I not on the contrary have called the sentence
"Bring me a slab" a lengthening of the sentence "Slab!"?--Because if you shout "Slab!" you really mean: "Bring me
a slab".--But how do you do this: how do you mean that while you say "Slab!"? Do you say the unshortened
sentence to yourself? And why should I translate the call "Slab!" into a different expression in order to say
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what someone means by it? And if they mean the same thing--why should I not say: "When he says 'Slab!' he
means 'Slab!'"? Again, if you can mean "Bring me the slab", why should you not be able to mean "Slab!"?--But
when I call "Slab!", then what I want is, that he should bring me a slab!--Certainly, but does 'wanting this' consist in
thinking in some form or other a different sentence from the one you utter?--
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20. But now it looks as if when someone says "Bring me a slab" he could mean this expression as one long
word corresponding to the single word "Slab!"--Then can one mean it sometimes as one word and sometimes as
four? And how does one usually mean it?--I think we shall be inclined to say: we mean the sentence as four words
when we use it in contrast with other sentences such as "Hand me a slab", "Bring him a slab", "Bring two slabs",
etc.; that is, in contrast with sentences containing the separate words of our command in other combinations.--But
what does using one sentence in contrast with others consist in? Do the others, perhaps, hover before one's mind?
All of them? And while one is saying the one sentence, or before, or afterwards?--No. Even if such an explanation
rather tempts us, we need only think for a moment of what actually happens in order to see that we are going astray
here. We say that we use the command in contrast with other sentences because our language contains the
possibility of those other sentences. Someone who did not understand our language, a foreigner, who had fairly
often heard someone giving the order: "Bring me a slab!", might believe that this whole series of sounds was one
word corresponding perhaps to the word for "building-stone" in his language. If he himself had then given this order
perhaps he would have pronounced it differently, and we should say: he pronounces it so oddly because he takes it
for a single word.--But then, is there not also something different going on in him when he pronounces
it,--something corresponding to the fact that he conceives the sentence as a single word?--Either the same thing may
go on in him, or something different. For what goes on in you when you give such an order? Are you conscious of
its consisting of four words while you are uttering it? Of course you have a mastery of this language--which contains
those other sentences as well--but is this having a mastery something that happens while you are uttering the
sentence?--And I have admitted that the foreigner will probably pronounce a sentence differently if he conceives it
differently; but what we call his wrong conception need not lie in anything that accompanies the utterance of the
command.
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The sentence is 'elliptical', not because it leaves out something that we think when we utter it, but because it
is shortened--in comparison with a particular paradigm of our grammar.--Of course one might object here: "You
grant that the shortened and the unshortened sentence have the same sense.--What is this sense, then? Isn't there a
verbal expression for this sense?"--But doesn't the fact that sentences have the same sense consist in their having the
same use?--(In Russian one says "stone red" instead of "the stone is red"; do they feel the copula to be missing in
the sense, or attach it in thought?)
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21. Imagine a language-game in which A asks and B reports the number of slabs or blocks in a pile, or the
colours and shapes of the building-stones that are stacked in such-and-such a place.--Such a report might run: "Five
slabs". Now what is the difference between the report or statement "Five slabs" and the order "Five slabs!"?--Well, it
is the part which uttering these words plays in the language-game. No doubt the tone of voice and the look with
which they are uttered, and much else besides, will also be different. But we could also imagine the tone's being the
same--for an order and a report can be spoken in a variety of tones of voice and with various expressions of
face--the difference being only in the application. (Of course, we might use the words "statement" and "command"
to stand for grammatical forms of sentence and intonations; we do in fact call "Isn't the weather glorious to-day?" a
question, although it is used as a statement.) We could imagine a language in which all statements had the form and
tone of rhetorical questions; or every command the form of the question "Would you like to...?". Perhaps it will then
be said: "What he says has the form of a question but is really a command",--that is, has the function of a command
in the technique of using the language. (Similarly one says "You will do this" not as a prophecy but as a command.
What makes it the one or the other?)
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22. Frege's idea that every assertion contains an assumption, which is the thing that is asserted, really rests on
the possibility found in our language of writing every statement in the form: "It is asserted that such-and-such is the
case."--But "that such-and-such is the case" is not a sentence in our language--so far it is not a move in the
language-game. And if I write, not "It is asserted that....", but "It is asserted: such-and-such is the case", the words "It
is asserted" simply become superfluous.
Page 10
We might very well also write every statement in the form of a
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question followed by a "Yes"; for instance: "Is it raining? Yes!" Would this shew that every statement contained a
question?
Page 11
Of course we have the right to use an assertion sign in contrast with a question-mark, for example, or if we
want to distinguish an assertion from a fiction or a supposition. It is only a mistake if one thinks that the assertion
consists of two actions, entertaining and asserting (assigning the truth-value, or something of the kind), and that in
performing these actions we follow the propositional sign roughly as we sing from the musical score. Reading the
written sentence loud or soft is indeed comparable with singing from a musical score, but 'meaning' (thinking) the
sentence that is read is not.
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Frege's assertion sign marks the beginning of the sentence. Thus its function is like that of the full-stop. It
distinguishes the whole period from a clause within the period. If I hear someone say "it's raining" but do not know
whether I have heard the beginning and end of the period, so far this sentence does not serve to tell me anything.
Page 11
------------------
Imagine a picture representing a boxer in a particular stance. Now, this picture can be used to tell someone
how he should stand, should hold himself; or how he should not hold himself; or how a particular man did stand in
such-and-such a place; and so on. One might (using the language of chemistry) call this picture a proposition-radical.
This will be how Frege thought of the "assumption".
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23. But how many kinds of sentence are there? Say assertion, question, and command?--There are countless
kinds: countless different kinds of use of what we call "symbols", "words", "sentences". And this multiplicity is not
something fixed, given once for all; but new types of language, new language-games, as we may say, come into
existence, and others become obsolete and get forgotten. (We can get a rough picture of this from the changes in
mathematics.)
Page 11
Here the term "language-game" is meant to bring into prominence the fact that the speaking of language is
part of an activity, or of a form of life.
Page 11
Review the multiplicity of language-games in the following examples, and in others:
Giving orders, and obeying them--
Describing the appearance of an object, or giving its measurements--
Constructing an object from a description (a drawing)--
Reporting an event--
Speculating about an event--
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Page 12
Forming and testing a hypothesis--
Presenting the results of an experiment in tables and diagrams--
Making up a story; and reading it--
Play-acting--
Singing catches--
Guessing riddles--
Making a joke; telling it--
Solving a problem in practical arithmetic--
Translating from one language into another--
Asking, thanking, cursing, greeting, praying.
--It is interesting to compare the multiplicity of the tools in language and of the ways they are used, the multiplicity
of kinds of word and sentence, with what logicians have said about the structure of language. (Including the author
of the Tractatus Logico-Philosophicus.)
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24. If you do not keep the multiplicity of language-games in view you will perhaps be inclined to ask
questions like: "What is a question?"--Is it the statement that I do not know such-and-such, or the statement that I
wish the other person would tell me....? Or is it the description of my mental state of uncertainty?--And is the cry
"Help!" such a description?
Page 12
Think how many different kinds of thing are called "description": description of a body's position by means
of its co-ordinates; description of a facial expression; description of a sensation of touch; of a mood.
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Of course it is possible to substitute the form of statement or description for the usual form of question: "I
want to know whether...." or "I am in doubt whether...."--but this does not bring the different language-games any
closer together.
Page 12
The significance of such possibilities of transformation, for example of turning all statements into sentences
beginning "I think" or "I believe" (and thus, as it were, into descriptions of my inner life) will become clearer in
another place. (Solipsism.)
Page 12
25. It is sometimes said that animals do not talk because they lack the mental capacity. And this means: "they
do not think, and that is why they do not talk." But--they simply do not talk. Or to put it better: they do not use
language--if we except the most primitive forms of language.--Commanding, questioning, recounting, chatting, are
as much a part of our natural history as walking, eating, drinking, playing.
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26. One thinks that learning language consists in giving names to objects. Viz, to human beings, to shapes, to
colours, to pains, to
Page Break 13
moods, to numbers, etc.. To repeat--naming is something like attaching a label to a thing. One can say that this is
preparatory to the use of a word. But what is it a preparation for?
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27. "We name things and then we can talk about them: can refer to them in talk."--As if what we did next
were given with the mere act of naming. As if there were only one thing called "talking about a thing". Whereas in
fact we do the most various things with our sentences. Think of exclamations alone, with their completely different
functions.
Water!
Away!
Ow!
Help!
Fine!
No!
Are you inclined still to call these words "names of objects"?
Page 13
In languages (2) and (8) there was no such thing as asking something's name. This, with its correlate,
ostensive definition, is, we might say, a language-game on its own. That is really to say: we are brought up, trained,
to ask: "What is that called?"--upon which the name is given. And there is also a language-game of inventing a name
for something, and hence of saying, "This is...." and then using the new name. (Thus, for example, children give
names to their dolls and then talk about them and to them. Think in this connexion how singular is the use of a
person's name to call him!)
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28. Now one can ostensively define a proper name, the name of a colour, the name of a material, a numeral,
the name of a point of the compass and so on. The definition of the number two, "That is called 'two'"--pointing to
two nuts--is perfectly exact.--But how can two be defined like that? The person one gives the definition to doesn't
know what one wants to call "two"; he will suppose that "two" is the name given to this group of nuts!--He may
suppose this; but perhaps he does not. He might make the opposite mistake; when I want to assign a name to this
group of nuts, he might understand it as a numeral. And he might equally well take the name of a person, of which I
give an ostensive definition, as that of a colour, of a race, or even of a point
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of the compass. That is to say: an ostensive definition can be variously interpreted in every case.
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29. Perhaps you say: two can only be ostensively defined in this way: "This number is called 'two'". For the
word "number" here shews what place in language, in grammar, we assign to the word. But this means that the word
"number" must be explained before the ostensive definition can be understood.--The word "number" in the
definition does indeed shew this place; does shew the post at which we station the word. And we can prevent
misunderstandings by saying: "This colour is called so-and-so", "This length is called so-and-so", and so on. That is
to say: misunderstandings are sometimes averted in this way. But is there only one way of taking the word "colour"
or "length"?--Well, they just need defining.--Defining, then, by means of other words! And what about the last
definition in this chain? (Do not say: "There isn't a 'last' definition". That is just as if you chose to say: "There isn't a
last house in this road; one can always build an additional one".)
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Whether the word "number" is necessary in the ostensive definition depends on whether without it the other
person takes the definition otherwise than I wish. And that will depend on the circumstances under which it is given,
and on the person I give it to.
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And how he 'takes' the definition is seen in the use that he makes of the word defined.
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Page 14
Could one define the word "red" by pointing to something that was not red? That would be as if one were
supposed to explain the word "modest" to someone whose English was weak, and one pointed to an arrogant man
and said "That man is not modest". That it is ambiguous is no argument against such a method of definition. Any
definition can be misunderstood.
Page 14
But it might well be asked: are we still to call this "definition"?--For, of course, even if it has the same
practical consequences, the same effect on the learner, it plays a different part in the calculus from what we
ordinarily call "ostensive definition" of the word "red".
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30. So one might say: the ostensive definition explains the use--the meaning--of the word when the overall
role of the word in language is clear. Thus if I know that someone means to explain a colour-word to me the
ostensive definition "That is called 'sepia'" will help me to understand the word.--And you can say this, so long as
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you do not forget that all sorts of problems attach to the words "to know" or "to be clear".
Page 15
One has already to know (or be able to do) something in order to be capable of asking a thing's name. But
what does one have to know?
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31. When one shews someone the king in chess and says: "This is the king", this does not tell him the use of
this piece--unless he already knows the rules of the game up to this last point: the shape of the king. You could
imagine his having learnt the rules of the game without ever having been shewn an actual piece. The shape of the
chessman corresponds here to the sound or shape of a word.
Page 15
One can also imagine someone's having learnt the game without ever learning or formulating rules. He might
have learnt quite simple board-games first, by watching, and have progressed to more and more complicated ones.
He too might be given the explanation "This is the king",--if, for instance, he were being shewn chessmen of a shape
he was not used to. This explanation again only tells him the use of the piece because, as we might say, the place for
it was already prepared. Or even: we shall only say that it tells him the use, if the place is already prepared. And in
this case it is so, not because the person to whom we give the explanation already knows rules, but because in
another sense he is already master of a game.
Page 15
Consider this further case: I am explaining chess to someone; and I begin by pointing to a chessman and
saying: "This is the king; it can move like this,.... and so on."--In this case we shall say: the words "This is the king"
(or "This is called the 'king'") are a definition only if the learner already 'knows what a piece in a game is'. That is, if
he has already played other games, or has watched other people playing 'and understood'--and similar things.
Further, only under these conditions will he be able to ask relevantly in the course of learning the game: "What do
you call this?"--that is, this piece in a game.
Page 15
We may say: only someone who already knows how to do something with it can significantly ask a name.
Page 15
And we can imagine the person who is asked replying: "Settle the name yourself"--and now the one who
asked would have to manage everything for himself.
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32. Someone coming into a strange country will sometimes learn the language of the inhabitants from
ostensive definitions that they give him; and he will often have to guess the meaning of these definitions; and will
guess sometimes right, sometimes wrong.
Page 15
And now, I think, we can say: Augustine describes the learning
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of human language as if the child came into a strange country and did not understand the language of the country;
that is, as if it already had a language, only not this one. Or again: as if the child could already think, only not yet
speak. And "think" would here mean something like "talk to itself".
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33. Suppose, however, someone were to object: "It is not true that you must already be master of a language
in order to understand an ostensive definition: all you need--of course!--is to know or guess what the person giving
the explanation is pointing to. That is, whether for example to the shape of the object, or to its colour, or to its
number, and so on."--And what does 'pointing to the shape', 'pointing to the colour' consist in? Point to a piece of
paper.--And now point to its shape--now to its colour--now to its number (that sounds queer).--How did you do
it?--You will say that you 'meant' a different thing each time you pointed. And if I ask how that is done, you will say
you concentrated your attention on the colour, the shape, etc. But I ask again: how is that done?
Page 16
Suppose someone points to a vase and says "Look at that marvellous blue--the shape isn't the point."--Or:
"Look at the marvellous shape--the colour doesn't matter." Without doubt you will do something different when
you act upon these two invitations. But do you always do the same thing when you direct your attention to the
colour? Imagine various different cases. To indicate a few:
Page 16
"Is this blue the same as the blue over there? Do you see any difference?"--
Page 16
You are mixing paint and you say "It's hard to get the blue of this sky."
Page 16
"It's turning fine, you can already see blue sky again."
Page 16
"Look what different effects these two blues have."
Page 16
"Do you see the blue book over there? Bring it here."
Page 16
"This blue signal-light means...."
Page 16
"What's this blue called?--Is it 'indigo'?"
You sometimes attend to the colour by putting your hand up to keep the outline from view; or by not looking at the
outline of the thing; sometimes by staring at the object and trying to remember where you saw that colour before.
Page 16
You attend to the shape, sometimes by tracing it, sometimes by screwing up your eyes so as not to see the
colour clearly, and in many other ways. I want to say: This is the sort of thing that happens while one 'directs one's
attention to this or that'. But it isn't these things by
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themselves that make us say someone is attending to the shape, the colour, and so on. Just as a move in chess
doesn't consist simply in moving a piece in such-and-such a way on the board--nor yet in one's thoughts and
feelings as one makes the move: but in the circumstances that we call "playing a game of chess", "solving a chess
problem", and so on.
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34. But suppose someone said: "I always do the same thing when I attend to the shape: my eye follows the
outline and I feel....". And suppose this person to give someone else the ostensive definition "That is called a 'circle'",
pointing to a circular object and having all these experiences--cannot his hearer still interpret the definition
differently, even though he sees the other's eyes following the outline, and even though he feels what the other feels?
That is to say: this 'interpretation' may also consist in how he now makes use of the word; in what he points to, for
example, when told: "Point to a circle".--For neither the expression "to intend the definition in such-and-such a way"
nor the expression "to interpret the definition in such-and-such a way" stands for a process which accompanies the
giving and hearing of the definition.
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35. There are, of course, what can be called "characteristic experiences" of pointing to (e.g.) the shape. For
example, following the outline with one's finger or with one's eyes as one points.--But this does not happen in all
cases in which I 'mean the shape', and no more does any other one characteristic process occur in all these
cases.--Besides, even if something of the sort did recur in all cases, it would still depend on the circumstances--that
is, on what happened before and after the pointing--whether we should say "He pointed to the shape and not to the
colour".
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For the words "to point to the shape", "to mean the shape", and so on, are not used in the same way as these:
"to point to this book (not to that one), "to point to the chair, not to the table", and so on.--Only think how
differently we learn the use of the words "to point to this thing", "to point to that thing", and on the other hand "to
point to the colour, not the shape", "to mean the colour", and so on.
Page 17
To repeat: in certain cases, especially when one points 'to the shape' or 'to the number' there are characteristic
experiences and ways of pointing--'characteristic' because they recur often (not always) when shape or number are
'meant'. But do you also know of an experience characteristic of pointing to a piece in a game as a piece in a game?
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All the same one can say: "I mean that this piece is called the 'king', not this particular bit of wood I am pointing to".
(Recognizing, wishing, remembering, etc..)
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36. And we do here what we do in a host of similar cases: because we cannot specify any one bodily action
which we call pointing to the shape (as opposed, for example, to the colour), we say that a spiritual [mental,
intellectual] activity corresponds to these words.
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Where our language suggests a body and there is none: there, we should like to say, is a spirit.
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37. What is the relation between name and thing named?--Well, what is it? Look at language-game (2) or at
another one: there you can see the sort of thing this relation consists in. This relation may also consist, among many
other things, in the fact that hearing the name calls before our mind the picture of what is named; and it also
consists, among other things, in the name's being written on the thing named or being pronounced when that thing is
pointed at.
------------------
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What is it to mean the words "That is blue" at one time as a statement about the object one is pointing to--at
another as an explanation of the word "blue"? Well, in the second case one really means "That is called
'blue'".--Then can one at one time mean the word "is" as "is called" and the word "blue" as "'blue'", and another time
mean "is" really as "is"?
Page 18
It is also possible for someone to get an explanation of the words out of what was intended as a piece of
information. [Marginal note: Here lurks a crucial superstition.]
Page 18
Can I say "bububu" and mean "If it doesn't rain I shall go for a walk"?--It is only in a language that I can
mean something by something. This shews clearly that the grammar of "to mean" is not like that of the expression
"to imagine" and the like.
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38. But what, for example, is the word "this" the name of in language-game (8) or the word "that" in the
ostensive definition "that is called...."?--If you do not want to produce confusion you will do best not to call these
words names at all.--Yet, strange to say, the word "this" has been called the only genuine name; so that anything
else we call a name was one only in an inexact, approximate sense.
Page 18
This queer conception springs from a tendency to sublime the logic of our language--as one might put it. The
proper answer to it is: we call very different things "names"; the word "name" is used to
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characterize many different kinds of use of a word, related to one another in many different ways;--but the kind of
use that "this" has is not among them.
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It is quite true that, in giving an ostensive definition for instance, we often point to the object named and say
the name. And similarly, in giving an ostensive definition for instance, we say the word "this" while pointing to a
thing. And also the word "this" and a name often occupy the same position in a sentence. But it is precisely
characteristic of a name that it is defined by means of the demonstrative expression "That is N" (or "That is called
'N'"). But do we also give the definitions: "That is called 'this'", or "This is called 'this'"?
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This is connected with the conception of naming as, so to speak, an occult process. Naming appears as a
queer connexion of a word with an object.--And you really get such a queer connexion when the philosopher tries
to bring out the relation between name and thing by staring at an object in front of him and repeating a name or even
the word "this" innumerable times. For philosophical problems arise when language goes on holiday. And here we
may indeed fancy naming to be some remarkable act of mind, as it were a baptism of an object. And we can also say
the word "this" to the object, as it were address the object as "this"--a queer use of this word, which doubtless only
occurs in doing philosophy.
Page 19
39. But why does it occur to one to want to make precisely this word into a name, when it evidently is not a
name?--That is just the reason. For one is tempted to make an objection against what is ordinarily called a name. It
can be put like this: a name ought really to signify a simple. And for this one might perhaps give the following
reasons: The word "Excalibur", say, is a proper name in the ordinary sense. The sword Excalibur consists of parts
combined in a particular way. If they are combined differently Excalibur does not exist. But it is clear that the
sentence "Excalibur has a sharp blade" makes sense whether Excalibur is still whole or is broken up. But if
"Excalibur" is the name of an object, this object no longer exists when Excalibur is broken in pieces; and as no
object would then correspond to the name it would have no meaning. But then the sentence "Excalibur has a sharp
blade" would contain a word that had no meaning, and hence the sentence would be nonsense. But it does make
sense; so there must always be something corresponding to the words of which it consists. So the word "Excalibur"
must disappear when the sense is
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analysed and its place be taken by words which name simples. It will be reasonable to call these words the real
names.
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40. Let us first discuss this point of the argument: that a word has no meaning if nothing corresponds to
it.--It is important to note that the word "meaning" is being used illicitly if it is used to signify the thing that
'corresponds' to the word. That is to confound the meaning of a name with the bearer of the name. When Mr. N. N.
dies one says that the bearer of the name dies, not that the meaning dies. And it would be nonsensical to say that, for
if the name ceased to have meaning it would make no sense to say "Mr. N. N. is dead."
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41. In §15 we introduced proper names into language (8). Now suppose that the tool with the name "N" is
broken. Not knowing this, A gives B the sign "N". Has this sign meaning now or not?--What is B to do when he is
given it?--We have not settled anything about this. One might ask: what will he do? Well, perhaps he will stand there
at a loss, or shew A the pieces. Here one might say: "N" has become meaningless; and this expression would mean
that the sign "N" no longer had a use in our language-game (unless we gave it a new one). "N" might also become
meaningless because, for whatever reason, the tool was given another name and the sign "N" no longer used in the
language-game.--But we could also imagine a convention whereby B has to shake his head in reply if A gives him
the sign belonging to a tool that is broken.--In this way the command "N" might be said to be given a place in the
language-game even when the tool no longer exists, and the sign "N" to have meaning even when its bearer ceases to
exist.
Page 20
42. But has for instance a name which has never been used for a tool also got a meaning in that game?--Let
us assume that "X" is such a sign and that A gives this sign to B--well, even such signs could be given a place in the
language-game, and B might have, say, to answer them too with a shake of the head. (One could imagine this as a
sort of joke between them.)
Page 20
43. For a large class of cases--though not for all--in which we employ the word "meaning" it can be defined
thus: the meaning of a word is its use in the language.
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Page 21
And the meaning of a name is sometimes explained by pointing to its bearer.
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44. We said that the sentence "Excalibur has a sharp blade" made sense even when Excalibur was broken in
pieces. Now this is so because in this language-game a name is also used in the absence of its bearer. But we can
imagine a language-game with names (that is, with signs which we should certainly include among names) in which
they are used only in the presence of the bearer; and so could always be replaced by a demonstrative pronoun and
the gesture of pointing.
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45. The demonstrative "this" can never be without a bearer. It might be said: "so long as there is a this, the
word 'this' has a meaning too, whether this is simple or complex."--But that does not make the word into a name.
On the contrary: for a name is not used with, but only explained by means of, the gesture of pointing.
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46. What lies behind the idea that names really signify simples?--
Page 21
Socrates says in the Theaetetus: "If I make no mistake, I have heard some people say this: there is no
definition of the primary elements--so to speak--out of which we and everything else are composed; for everything
that exists†1 in its own right can only be named, no other determination is possible, neither that it is nor that it is
not..... But what exists†1 in its own right has to be..... named without any other determination. In consequence it is
impossible to give an account of any primary element; for it, nothing is possible but the bare name; its name is all it
has. But just as what consists of these primary elements is itself complex, so the names of the elements become
descriptive language by being compounded together. For the essence of speech is the composition of names."
Page 21
Both Russell's 'individuals' and my 'objects' (Tractatus Logico-Philosophicus) were such primary elements.
Page 21
47. But what are the simple constituent parts of which reality is composed?--What are the simple constituent
parts of a chair?--The bits of wood of which it is made? Or the molecules, or the atoms?--"Simple" means: not
composite. And here the point is: in what sense 'composite'? It makes no sense at all to speak absolutely of the
'simple parts of a chair'.
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Page 22
Again: Does my visual image of this tree, of this chair, consist of parts? And what are its simple component
parts? Multi-colouredness is one kind of complexity; another is, for example, that of a broken outline composed of
straight bits. And a curve can be said to be composed of an ascending and a descending segment.
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If I tell someone without any further explanation: "What I see before me now is composite", he will have the
right to ask: "What do you mean by 'composite'? For there are all sorts of things that that can mean!"--The question
"Is what you see composite?" makes good sense if it is already established what kind of complexity--that is, which
particular use of the word--is in question. If it had been laid down that the visual image of a tree was to be called
"composite" if one saw not just a single trunk, but also branches, then the question "Is the visual image of this tree
simple or composite?", and the question "What are its simple component parts?", would have a clear sense--a clear
use. And of course the answer to the second question is not "The branches" (that would be an answer to the
grammatical question: "What are here called 'simple component parts'?") but rather a description of the individual
branches.
Page 22
But isn't a chessboard, for instance, obviously, and absolutely, composite?--You are probably thinking of the
composition out of thirty-two white and thirty-two black squares. But could we not also say, for instance, that it was
composed of the colours black and white and the schema of squares? And if there are quite different ways of
looking at it, do you still want to say that the chessboard is absolutely 'composite'?--Asking "Is this object
composite?" outside a particular language-game is like what a boy once did, who had to say whether the verbs in
certain sentences were in the active or passive voice, and who racked his brains over the question whether the verb
"to sleep" meant something active or passive.
Page 22
We use the word "composite" (and therefore the word "simple") in an enormous number of different and
differently related ways. (Is the colour of a square on a chessboard simple, or does it consist of pure white and pure
yellow? And is white simple, or does it consist of the colours of the rainbow?--Is this length of 2 cm. simple, or does
it consist of two parts, each 1 cm. long? But why not of one bit 3 cm. long, and one bit 1 cm. long measured in the
opposite direction?)
Page 22
To the philosophical question: "Is the visual image of this tree
Page Break 23
composite, and what are its component parts?" the correct answer is: "That depends on what you understand by
'composite'." (And that is of course not an answer but a rejection of the question.)
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48. Let us apply the method of §2 to the account in the Theaetetus. Let us consider a language-game for
which this account is really valid. The language serves to describe combinations of coloured squares on a surface.
The squares form a complex like a chessboard. There are red, green, white and black squares. The words of the
language are (correspondingly) "R", "G", "W", "B", and a sentence is a series of these words. They describe an
arrangement of squares in the order:
And so for instance the sentence "RRBGGGRWW" describes an arrangement of this sort:
Here the sentence is a complex of names, to which corresponds a complex of elements. The primary elements are
the coloured squares. "But are these simple?"--I do not know what else you would have me call "the simples", what
would be more natural in this language-game. But under other circumstances I should call a monochrome square
"composite", consisting perhaps of two rectangles, or of the elements colour and shape. But the concept of
complexity might also be so extended that a smaller area was said to be 'composed' of a greater area and another one
subtracted from it. Compare the 'composition of
Page Break 24
forces', the 'division' of a line by a point outside it; these expressions shew that we are sometimes even inclined to
conceive the smaller as the result of a composition of greater parts, and the greater as the result of a division of the
smaller.
Page 24
But I do not know whether to say that the figure described by our sentence consists of four or of nine
elements! Well, does the sentence consist of four letters or of nine?--And which are its elements, the types of letter,
or the letters? Does it matter which we say, so long as we avoid misunderstandings in any particular case?
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49. But what does it mean to say that we cannot define (that is, describe) these elements, but only name
them? This might mean, for instance, that when in a limiting case a complex consists of only one square, its
description is simply the name of the coloured square.
Page 24
Here we might say--though this easily leads to all kinds of philosophical superstition--that a sign "R" or "B",
etc. may be sometimes a word and sometimes a proposition. But whether it 'is a word or a proposition' depends on
the situation in which it is uttered or written. For instance, if A has to describe complexes of coloured squares to B
and he uses the word "R" alone, we shall be able to say that the word is a description--a proposition. But if he is
memorizing the words and their meanings, or if he is teaching someone else the use of the words and uttering them
in the course of ostensive teaching, we shall not say that they are propositions. In this situation the word "R", for
instance, is not a description; it names an element--but it would be queer to make that a reason for saying that an
element can only be named! For naming and describing do not stand on the same level: naming is a preparation for
description. Naming is so far not a move in the language-game--any more than putting a piece in its place on the
board is a move in chess. We may say: nothing has so far been done, when a thing has been named. It has not even
got a name except in the language-game. This was what Frege meant too, when he said that a word had meaning
only as part of a sentence.
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50. What does it mean to say that we can attribute neither being nor non-being to elements?--One might say:
if everything that we call "being" and "non-being" consists in the existence and non-existence of connexions
between elements, it makes no sense to speak of an element's being (non-being); just as when everything that we call
"destruction" lies in the separation of elements, it makes no sense to speak of the destruction of an element.
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One would, however, like to say: existence cannot be attributed to an element, for if it did not exist, one
could not even name it and so one could say nothing at all of it.--But let us consider an analogous case. There is one
thing of which one can say neither that it is one metre long, nor that it is not one metre long, and that is the standard
metre in Paris.--But this is, of course, not to ascribe any extraordinary property to it, but only to mark its peculiar
role in the language-game of measuring with a metre-rule.--Let us imagine samples of colour being preserved in
Paris like the standard metre. We define: "sepia" means the colour of the standard sepia which is there kept
hermetically sealed. Then it will make no sense to say of this sample either that it is of this colour or that it is not.
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We can put it like this: This sample is an instrument of the language used in ascriptions of colour. In this
language-game it is not something that is represented, but is a means of representation.--And just this goes for an
element in language-game (48) when we name it by uttering the word "R": this gives this object a role in our
language-game; it is now a means of representation. And to say "If it did not exist, it could have no name" is to say
as much and as little as: if this thing did not exist, we could not use it in our language-game.--What looks as if it had
to exist, is part of the language. It is a paradigm in our language-game; something with which comparison is made.
And this may be an important observation; but it is none the less an observation concerning our language-game--our
method of representation.
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51. In describing language-game (48) I said that the words "R", "B", etc. corresponded to the colours of the
squares. But what does this correspondence consist in; in what sense can one say that certain colours of squares
correspond to these signs? For the account in (48) merely set up a connexion between those signs and certain words
of our language (the names of colours).--Well, it was presupposed that the use of the signs in the language-game
would be taught in a different way, in particular by pointing to paradigms. Very well; but what does it mean to say
that in the technique of using the language certain elements correspond to the signs?--Is it that the person who is
describing the complexes of coloured squares always says "R" where there is a red square; "B" when there is a black
one, and so on? But what if he goes wrong in the description and mistakenly says "R" where he sees a black
square--what is the criterion by which this is a mistake?--Or does "R"s standing for a red square consist in this, that
when the
Page Break 26
people whose language it is use the sign "R" a red square always comes before their minds?
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In order to see more clearly, here as in countless similar cases, we must focus on the details of what goes on;
must look at them from close to.
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52. If I am inclined to suppose that a mouse has come into being by spontaneous generation out of grey rags
and dust, I shall do well to examine those rags very closely to see how a mouse may have hidden in them, how it
may have got there and so on. But if I am convinced that a mouse cannot come into being from these things, then
this investigation will perhaps be superfluous.
Page 26
But first we must learn to understand what it is that opposes such an examination of details in philosophy.
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53. Our language-game (48) has various possibilities; there is a variety of cases in which we should say that a
sign in the game was the name of a square of such-and-such a colour. We should say so if, for instance, we knew
that the people who used the language were taught the use of the signs in such-and-such a way. Or if it were set
down in writing, say in the form of a table, that this element corresponded to this sign, and if the table were used in
teaching the language and were appealed to in certain disputed cases.
Page 26
We can also imagine such a table's being a tool in the use of the language. Describing a complex is then done
like this: the person who describes the complex has a table with him and looks up each element of the complex in it
and passes from this to the sign (and the one who is given the description may also use a table to translate it into a
picture of coloured squares). This table might be said to take over here the role of memory and association in other
cases. (We do not usually carry out the order "Bring me a red flower" by looking up the colour red in a table of
colours and then bringing a flower of the colour that we find in the table; but when it is a question of choosing or
mixing a particular shade of red, we do sometimes make use of a sample or table.)
Page 26
If we call such a table the expression of a rule of the language-game, it can be said that what we call a rule of
a language-game may have very different roles in the game.
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54. Let us recall the kinds of case where we say that a game is played according to a definite rule.
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Page 27
The rule may be an aid in teaching the game. The learner is told it and given practice in applying it.--Or it is
an instrument of the game itself.--Or a rule is employed neither in the teaching nor in the game itself; nor is it set
down in a list of rules. One learns the game by watching how others play. But we say that it is played according to
such-and-such rules because an observer can read these rules off from the practice of the game--like a natural law
governing the play.--But how does the observer distinguish in this case between players' mistakes and correct
play?--There are characteristic signs of it in the players' behaviour. Think of the behaviour characteristic of correcting
a slip of the tongue. It would be possible to recognize that someone was doing so even without knowing his
language.
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55. "What the names in language signify must be indestructible; for it must be possible to describe the state
of affairs in which everything destructible is destroyed. And this description will contain words; and what
corresponds to these cannot then be destroyed, for otherwise the words would have no meaning." I must not saw off
the branch on which I am sitting.
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One might, of course, object at once that this description would have to except itself from the
destruction.--But what corresponds to the separate words of the description and so cannot be destroyed if it is true,
is what gives the words their meaning--is that without which they would have no meaning.--In a sense, however, this
man is surely what corresponds to his name. But he is destructible, and his name does not lose its meaning when the
bearer is destroyed.--An example of something corresponding to the name, and without which it would have no
meaning, is a paradigm that is used in connexion with the name in the language-game.
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56. But what if no such sample is part of the language, and we bear in mind the colour (for instance) that a
word stands for?--"And if we bear it in mind then it comes before our mind's eye when we utter the word. So, if it is
always supposed to be possible for us to remember it, it must be in itself indestructible."--But what do we regard as
the criterion for remembering it right?--When we work with a sample instead of our memory there are circumstances
in which we say that the sample has changed colour and we judge of this by memory. But can we not sometimes
speak of a darkening (for example) of our memory-image? Aren't we as much at the mercy of memory as of a
sample? (For someone might feel like saying: "If we
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had no memory we should be at the mercy of a sample".)--Or perhaps of some chemical reaction. Imagine that you
were supposed to paint a particular colour "C", which was the colour that appeared when the chemical substances X
and Y combined.--Suppose that the colour struck you as brighter on one day than on another; would you not
sometimes say: "I must be wrong, the colour is certainly the same as yesterday"? This shews that we do not always
resort to what memory tells us as the verdict of the highest court of appeal.
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57. "Something red can be destroyed, but red cannot be destroyed, and that is why the meaning of the word
'red' is independent of the existence of a red thing."--Certainly it makes no sense to say that the colour red is torn up
or pounded to bits. But don't we say "The red is vanishing"? And don't clutch at the idea of our always being able to
bring red before our mind's eye even when there is nothing red any more. That is just as if you chose to say that
there would still always be a chemical reaction producing a red flame.--For suppose you cannot remember the
colour any more?--When we forget which colour this is the name of, it loses its meaning for us; that is, we are no
longer able to play a particular language-game with it. And the situation then is comparable with that in which we
have lost a paradigm which was an instrument of our language.
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58. "I want to restrict the term 'name' to what cannot occur in the combination 'X exists'.--Thus one cannot
say 'Red exists', because if there were no red it could not be spoken of at all."--Better: If "X exists" is meant simply
to say: "X" has a meaning,--then it is not a proposition which treats of X, but a proposition about our use of
language, that is, about the use of the word "X".
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It looks to us as if we were saying something about the nature of red in saying that the words "Red exists" do
not yield a sense. Namely that red does exist 'in its own right'. The same idea--that this is a metaphysical statement
about red--finds expression again when we say such a thing as that red is timeless, and perhaps still more strongly in
the word "indestructible".
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But what we really want is simply to take "Red exists" as the statement: the word "red" has a meaning. Or
perhaps better: "Red does not exist" as "'Red' has no meaning". Only we do not want to say that that expression
says this, but that this is what it would have to be saying if it meant anything. But that it contradicts itself in the
attempt
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to say it--just because red exists 'in its own right'. Whereas the only contradiction lies in something like this: the
proposition looks as if it were about the colour, while it is supposed to be saying something about the use of the
word "red".--In reality, however, we quite readily say that a particular colour exists; and that is as much as to say
that something exists that has that colour. And the first expression is no less accurate than the second; particularly
where 'what has the colour' is not a physical object.
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59. "A name signifies only what is an element of reality. What cannot be destroyed; what remains the same
in all changes."--But what is that?--Why, it swam before our minds as we said the sentence! This was the very
expression of a quite particular image: of a particular picture which we want to use. For certainly experience does
not shew us these elements. We see component parts of something composite (of a chair, for instance). We say that
the back is part of the chair, but is in turn itself composed of several bits of wood; while a leg is a simple component
part. We also see a whole which changes (is destroyed) while its component parts remain unchanged. These are the
materials from which we construct that picture of reality.
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60. When I say: "My broom is in the corner",--is this really a statement about the broomstick and the brush?
Well, it could at any rate be replaced by a statement giving the position of the stick and the position of the brush.
And this statement is surely a further analysed form of the first one.--But why do I call it "further analysed"?--Well,
if the broom is there, that surely means that the stick and brush must be there, and in a particular relation to one
another; and this was as it were hidden in the sense of the first sentence, and is expressed in the analysed sentence.
Then does someone who says that the broom is in the corner really mean: the broomstick is there, and so is the
brush, and the broomstick is fixed in the brush?--If we were to ask anyone if he meant this he would probably say
that he had not thought specially of the broomstick or specially of the brush at all. And that would be the right
answer, for he meant to speak neither of the stick nor of the brush in particular. Suppose that, instead of saying
"Bring me the broom", you said "Bring me the broomstick and the brush which is fitted on to it."!--Isn't the answer:
"Do you want the broom? Why do you put it so oddly?"--Is he going to understand the further analysed sentence
better?--This sentence, one might say, achieves the same as the ordinary one, but in a more roundabout way.--
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Imagine a language-game in which someone is ordered to bring certain objects which are composed of several parts,
to move them about, or something else of the kind. And two ways of playing it: in one (a) the composite objects
(brooms, chairs, tables, etc.) have names, as in (15); in the other (b) only the parts are given names and the wholes
are described by means of them.--In what sense is an order in the second game an analysed form of an order in the
first? Does the former lie concealed in the latter, and is it now brought out by analysis?--True, the broom is taken to
pieces when one separates broomstick and brush; but does it follow that the order to bring the broom also consists
of corresponding parts?
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61. "But all the same you will not deny that a particular order in (a) means the same as one in (b); and what
would you call the second one, if not an analysed form of the first?"--Certainly I too should say that an order in (a)
had the same meaning as one in (b); or, as I expressed it earlier: they achieve the same. And this means that if I were
shewn an order in (a) and asked: "Which order in (b) means the same as this?" or again "Which order in (b) does this
contradict?" I should give such-and-such an answer. But that is not to say that we have come to a general agreement
about the use of the expression "to have the same meaning" or "to achieve the same". For it can be asked in what
cases we say: "These are merely two forms of the same game."
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62. Suppose for instance that the person who is given the orders in (a) and (b) has to look up a table
co-ordinating names and pictures before bringing what is required. Does he do the same when he carries out an
order in (a) and the corresponding one in (b)?--Yes and no. You may say: "The point of the two orders is the same".
I should say so too.--But it is not everywhere clear what should be called the 'point' of an order. (Similarly one may
say of certain objects that they have this or that purpose. The essential thing is that this is a lamp, that it serves to
give light;--that it is an ornament to the room, fills an empty space, etc., is not essential. But there is not always a
sharp distinction between essential and inessential.)
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63. To say, however, that a sentence in (b) is an 'analysed' form of one in (a) readily seduces us into thinking
that the former is the more fundamental form; that it alone shews what is meant by the other, and so on. For
example, we think: If you have only the unanalysed form you miss the analysis; but if you know the analysed form
that
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gives you everything.--But can I not say that an aspect of the matter is lost on you in the latter case as well as the
former?
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64. Let us imagine language game (48) altered so that names signify not monochrome squares but rectangles
each consisting of two such squares. Let such a rectangle, which is half red half green, be called "U"; a half green
half white one, "V"; and so on. Could we not imagine people who had names for such combinations of colour, but
not for the individual colours? Think of the cases where we say: "This arrangement of colours (say the French
tricolor) has a quite special character."
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In what sense do the symbols of this language-game stand in need of analysis? How far is it even possible to
replace this language-game by (48)?--It is just another language-game; even though it is related to (48).
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65. Here we come up against the great question that lies behind all these considerations.--For someone might
object against me: "You take the easy way out! You talk about all sorts of language-games, but have nowhere said
what the essence of a language-game, and hence of language, is: what is common to all these activities, and what
makes them into language or parts of language. So you let yourself off the very part of the investigation that once
gave you yourself most headache, the part about the general form of propositions and of language."
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And this is true.--Instead of producing something common to all that we call language, I am saying that these
phenomena have no one thing in common which makes us use the same word for all,--but that they are related to
one another in many different ways. And it is because of this relationship, or these relationships, that we call them all
"language". I will try to explain this.
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66. Consider for example the proceedings that we call "games". I mean board-games, card-games,
ball-games, Olympic games, and so on. What is common to them all?--Don't say: "There must be something
common, or they would not be called 'games'"--but look and see whether there is anything common to all.--For if
you look at them you will not see something that is common to all, but similarities, relationships, and a whole series
of them at that. To repeat: don't think, but look!--Look for example at board-games, with their multifarious
relationships. Now pass to card-games; here you find many correspondences with the first group, but many
common
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features drop out, and others appear. When we pass next to ball-games, much that is common is retained, but much
is lost.--Are they all 'amusing'? Compare chess with noughts and crosses. Or is there always winning and losing, or
competition between players? Think of patience. In ball games there is winning and losing; but when a child throws
his ball at the wall and catches it again, this feature has disappeared. Look at the parts played by skill and luck; and at
the difference between skill in chess and skill in tennis. Think now of games like ring-a-ring-a-roses; here is the
element of amusement, but how many other characteristic features have disappeared! And we can go through the
many, many other groups of games in the same way; can see how similarities crop up and disappear.
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And the result of this examination is: we see a complicated network of similarities overlapping and
criss-crossing: sometimes overall similarities, sometimes similarities of detail.
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67. I can think of no better expression to characterize these similarities than "family resemblances"; for the
various resemblances between members of a family: build, features, colour of eyes, gait, temperament, etc. etc.
overlap and criss-cross in the same way.--And I shall say: 'games' form a family.
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And for instance the kinds of number form a family in the same way. Why do we call something a
"number"? Well, perhaps because it has a--direct--relationship with several things that have hitherto been called
number; and this can be said to give it an indirect relationship to other things we call the same name. And we extend
our concept of number as in spinning a thread we twist fibre on fibre. And the strength of the thread does not reside
in the fact that some one fibre runs through its whole length, but in the overlapping of many fibres.
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But if someone wished to say: "There is something common to all these constructions--namely the
disjunction of all their common properties"--I should reply: Now you are only playing with words. One might as
well say: "Something runs through the whole thread--namely the continuous overlapping of those fibres".
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68. "All right: the concept of number is defined for you as the logical sum of these individual interrelated
concepts: cardinal numbers, rational numbers, real numbers, etc.; and in the same way the concept of a game as the
logical sum of a corresponding set of sub-concepts."--It need not be so. For I can give the concept 'number' rigid
limits
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in this way, that is, use the word "number" for a rigidly limited concept, but I can also use it so that the extension of
the concept is not closed by a frontier. And this is how we do use the word "game". For how is the concept of a
game bounded? What still counts as a game and what no longer does? Can you give the boundary? No. You can
draw one; for none has so far been drawn. (But that never troubled you before when you used the word "game".)
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"But then the use of the word is unregulated, the 'game' we play with it is unregulated."--It is not everywhere
circumscribed by rules; but no more are there any rules for how high one throws the ball in tennis, or how hard; yet
tennis is a game for all that and has rules too.
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69. How should we explain to someone what a game is? I imagine that we should describe games to him,
and we might add: "This and similar things are called 'games'". And do we know any more about it ourselves? Is it
only other people whom we cannot tell exactly what a game is?--But this is not ignorance. We do not know the
boundaries because none have been drawn. To repeat, we can draw a boundary--for a special purpose. Does it take
that to make the concept usable? Not at all! (Except for that special purpose.) No more than it took the definition: 1
pace = 75 cm. to make the measure of length 'one pace' usable. And if you want to say "But still, before that it wasn't
an exact measure", then I reply: very well, it was an inexact one.--Though you still owe me a definition of exactness.
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Someone says to me: "Shew the children a game." I teach them gaming with dice, and the other says "I didn't
mean that sort of game." Must the exclusion of the game with dice have come before his mind when he gave me the
order?
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70. "But if the concept 'game' is uncircumscribed like that, you don't really know what you mean by a
'game'."--When I give the description: "The ground was quite covered with plants"--do you want to say I don't know
what I am talking about until I can give a definition of a plant?
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My meaning would be explained by, say, a drawing and the words "The ground looked roughly like this".
Perhaps I even say "it looked exactly like this."--Then were just this grass and these leaves there, arranged just like
this? No, that is not what it means. And I should not accept any picture as exact in this sense.
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71. One might say that the concept 'game' is a concept with blurred edges.--"But is a blurred concept a
concept at all?"--Is an indistinct photograph a picture of a person at all? Is it even always an advantage to replace an
indistinct picture by a sharp one? Isn't the indistinct one often exactly what we need?
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Frege compares a concept to an area and says that an area with vague boundaries cannot be called an area at
all. This presumably means that we cannot do anything with it.--But is it senseless to say: "Stand roughly there"?
Suppose that I were standing with someone in a city square and said that. As I say it I do not draw any kind of
boundary, but perhaps point with my hand--as if I were indicating a particular spot. And this is just how one might
explain to someone what a game is. One gives examples and intends them to be taken in a particular way.--I do not,
however, mean by this that he is supposed to see in those examples that common thing which I--for some
reason--was unable to express; but that he is now to employ those examples in a particular way. Here giving
examples is not an indirect means of explaining--in default of a better. For any general definition can be
misunderstood too. The point is that this is how we play the game. (I mean the language-game with the word
"game".)
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72. Seeing what is common. Suppose I shew someone various multi-coloured pictures, and say: "The colour
you see in all these is called 'yellow ochre'".--This is a definition, and the other will get to understand it by looking
for and seeing what is common to the pictures. Then he can look at, can point to, the common thing.
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Compare with this a case in which I shew him figures of different shapes all painted the same colour, and
say: "What these have in common is called 'yellow ochre'".
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And compare this case: I shew him samples of different shades of blue and say: "The colour that is common
to all these is what I call 'blue'".
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73. When someone defines the names of colours for me by pointing to samples and saying "This colour is
called 'blue', this 'green'....." this case can be compared in many respects to putting a table in my hands, with the
words written under the colour-samples.--Though this comparison may mislead in many ways.--One is now inclined
to extend the comparison: to have understood the definition means to have in one's mind an idea of the thing
defined, and that is a sample or picture. So if I am shewn various different leaves and told
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"This is called a 'leaf'", I get an idea of the shape of a leaf, a picture of it in my mind.--But what does the picture of a
leaf look like when it does not shew us any particular shape, but 'what is common to all shapes of leaf'? Which shade
is the 'sample in my mind' of the colour green--the sample of what is common to all shades of green?
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"But might there not be such 'general' samples? Say a schematic leaf, or a sample of pure green?"--Certainly
there might. But for such a schema to be understood as a schema, and not as the shape of a particular leaf, and for a
slip of pure green to be understood as a sample of all that is greenish and not as a sample of pure green--this in turn
resides in the way the samples are used.
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Ask yourself: what shape must the sample of the colour green be? Should it be rectangular? Or would it then
be the sample of a green rectangle?--So should it be 'irregular' in shape? And what is to prevent us then from
regarding it--that is, from using it--only as a sample of irregularity of shape?
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74. Here also belongs the idea that if you see this leaf as a sample of 'leaf shape in general' you see it
differently from someone who regards it as, say, a sample of this particular shape. Now this might well be
so--though it is not so--for it would only be to say that, as a matter of experience, if you see the leaf in a particular
way, you use it in such-and-such a way or according to such-and-such rules. Of course, there is such a thing as
seeing in this way or that; and there are also cases where whoever sees a sample like this will in general use it in this
way, and whoever sees it otherwise in another way. For example, if you see the schematic drawing of a cube as a
plane figure consisting of a square and two rhombi you will, perhaps, carry out the order "Bring me something like
this" differently from someone who sees the picture three-dimensionally.
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75. What does it mean to know what a game is? What does it mean, to know it and not be able to say it? Is
this knowledge somehow equivalent to an unformulated definition? So that if it were formulated I should be able to
recognize it as the expression of my knowledge? Isn't my knowledge, my concept of a game, completely expressed
in the explanations that I could give? That is, in my describing examples of various kinds of game; shewing how all
sorts of other games can be constructed on the analogy of these; saying that I should scarcely include this or this
among games; and so on.
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76. If someone were to draw a sharp boundary I could not acknowledge it as the one that I too always
wanted to draw, or had drawn in my mind. For I did not want to draw one at all. His concept can then be said to be
not the same as mine, but akin to it. The kinship is that of two pictures, one of which consists of colour patches with
vague contours, and the other of patches similarly shaped and distributed, but with clear contours. The kinship is
just as undeniable as the difference.
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77. And if we carry this comparison still further it is clear that the degree to which the sharp picture can
resemble the blurred one depends on the latter's degree of vagueness. For imagine having to sketch a sharply defined
picture 'corresponding' to a blurred one. In the latter there is a blurred red rectangle: for it you put down a sharply
defined one. Of course--several such sharply defined rectangles can be drawn to correspond to the indefinite
one.--But if the colours in the original merge without a hint of any outline won't it become a hopeless task to draw a
sharp picture corresponding to the blurred one? Won't you then have to say: "Here I might just as well draw a circle
or heart as a rectangle, for all the colours merge. Anything--and nothing--is right."--And this is the position you are
in if you look for definitions corresponding to our concepts in aesthetics or ethics.
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In such a difficulty always ask yourself: How did we learn the meaning of this word ("good" for instance)?
From what sort of examples? in what language-games? Then it will be easier for you to see that the word must have
a family of meanings.
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78. Compare knowing and saying:
how many feet high Mont Blanc is--
how the word "game" is used--
how a clarinet sounds.
If you are surprised that one can know something and not be able to say it, you are perhaps thinking of a case like
the first. Certainly not of one like the third.
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79. Consider this example. If one says "Moses did not exist", this may mean various things. It may mean: the
Israelites did not have a single leader when they withdrew from Egypt--or: their leader was not called Moses--or:
there cannot have been anyone who accomplished all that the Bible relates of Moses--or: etc. etc.--We may say,
following Russell: the name "Moses" can be defined by
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means of various descriptions. For example, as "the man who led the Israelites through the wilderness", "the man
who lived at that time and place and was then called 'Moses'", "the man who as a child was taken out of the Nile by
Pharaoh's daughter" and so on. And according as we assume one definition or another the proposition "Moses did
not exist" acquires a different sense, and so does every other proposition about Moses.--And if we are told "N did
not exist", we do ask: "What do you mean? Do you want to say...... or...... etc.?"
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But when I make a statement about Moses,--am I always ready to substitute some one of these descriptions
for "Moses"? I shall perhaps say: By "Moses" I understand the man who did what the Bible relates of Moses, or at
any rate a good deal of it. But how much? Have I decided how much must be proved false for me to give up my
proposition as false? Has the name "Moses" got a fixed and unequivocal use for me in all possible cases?--Is it not
the case that I have, so to speak, a whole series of props in readiness, and am ready to lean on one if another should
be taken from under me and vice versa?--Consider another case. When I say "N is dead", then something like the
following may hold for the meaning of the name "N": I believe that a human being has lived, whom I (1) have seen
in such-and-such places, who (2) looked like this (pictures), (3) has done such-and-such things, and (4) bore the
name "N" in social life.--Asked what I understand by "N", I should enumerate all or some of these points, and
different ones on different occasions. So my definition of "N" would perhaps be "the man of whom all this is
true".--But if some point now proves false?--Shall I be prepared to declare the proposition "N is dead" false--even if
it is only something which strikes me as incidental that has turned out false? But where are the bounds of the
incidental?--If I had given a definition of the name in such a case, I should now be ready to alter it.
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And this can be expressed like this: I use the name "N" without a fixed meaning. (But that detracts as little
from its usefulness, as it detracts from that of a table that it stands on four legs instead of three and so sometimes
wobbles.)
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Should it be said that I am using a word whose meaning I don't know, and so am talking nonsense?--Say
what you choose, so long as it does not prevent you from seeing the facts. (And when you see them there is a good
deal that you will not say.)
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(The fluctuation of scientific definitions: what to-day counts as an
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observed concomitant of a phenomenon will to-morrow be used to define it.)
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80. I say "There is a chair". What if I go up to it, meaning to fetch it, and it suddenly disappears from
sight?--"So it wasn't a chair, but some kind of illusion".--But in a few moments we see it again and are able to touch
it and so on.--"So the chair was there after all and its disappearance was some kind of illusion".--But suppose that
after a time it disappears again--or seems to disappear. What are we to say now? Have you rules ready for such
cases--rules saying whether one may use the word "chair" to include this kind of thing? But do we miss them when
we use the word "chair"; and are we to say that we do not really attach any meaning to this word, because we are not
equipped with rules for every possible application of it?
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81. F. P. Ramsey once emphasized in conversation with me that logic was a 'normative science'. I do not
know exactly what he had in mind, but it was doubtless closely related to what only dawned on me later: namely,
that in philosophy we often compare the use of words with games and calculi which have fixed rules, but cannot say
that someone who is using language must be playing such a game.--But if you say that our languages only
approximate to such calculi you are standing on the very brink of a misunderstanding. For then it may look as if
what we were talking about were an ideal language. As if our logic were, so to speak, a logic for a
vacuum.--Whereas logic does not treat of language--or of thought--in the sense in which a natural science treats of a
natural phenomenon, and the most that can be said is that we construct ideal languages. But here the word "ideal" is
liable to mislead, for it sounds as if these languages were better, more perfect, than our everyday language; and as if
it took the logician to shew people at last what a proper sentence looked like.
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All this, however, can only appear in the right light when one has attained greater clarity about the concepts
of understanding, meaning, and thinking. For it will then also become clear what can lead us (and did lead me) to
think that if anyone utters a sentence and means or understands it he is operating a calculus according to definite
rules.
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82. What do I call 'the rule by which he proceeds'?--The hypothesis that satisfactorily describes his use of
words, which we observe; or the rule which he looks up when he uses signs; or the one which he
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gives us in reply if we ask him what his rule is?--But what if observation does not enable us to see any clear rule, and
the question brings none to light?--For he did indeed give me a definition when I asked him what he understood by
"N", but he was prepared to withdraw and alter it.--So how am I to determine the rule according to which he is
playing? He does not know it himself.--Or, to ask a better question: What meaning is the expression "the rule by
which he proceeds" supposed to have left to it here?
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83. Doesn't the analogy between language and games throw light here? We can easily imagine people
amusing themselves in a field by playing with a ball so as to start various existing games, but playing many without
finishing them and in between throwing the ball aimlessly into the air, chasing one another with the ball and
bombarding one another for a joke and so on. And now someone says: The whole time they are playing a ball-game
and following definite rules at every throw.
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And is there not also the case where we play and--make up the rules as we go along? And there is even one
where we alter them--as we go along.
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84. I said that the application of a word is not everywhere bounded by rules. But what does a game look like
that is everywhere bounded by rules? whose rules never let a doubt creep in, but stop up all the cracks where it
might?--Can't we imagine a rule determining the application of a rule, and a doubt which it removes--and so on?
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But that is not to say that we are in doubt because it is possible for us to imagine a doubt. I can easily
imagine someone always doubting before he opened his front door whether an abyss did not yawn behind it, and
making sure about it before he went through the door (and he might on some occasion prove to be right)--but that
does not make me doubt in the same case.
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85. A rule stands there like a sign-post.--Does the sign-post leave no doubt open about the way I have to go?
Does it shew which direction I am to take when I have passed it; whether along the road or the footpath or
cross-country? But where is it said which way I am to follow it; whether in the direction of its finger or (e.g.) in the
opposite one?--And if there were, not a single sign-post, but a chain of adjacent ones or of chalk marks on the
ground--is there only one way of interpreting them?--So I can say, the sign-post does after all
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leave no room for doubt. Or rather: it sometimes leaves room for doubt and sometimes not. And now this is no
longer a philosophical proposition, but an empirical one.
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86. Imagine a language-game like (2) played with the help of a table. The signs given to B by A are now
written ones. B has a table; in the first column are the signs used in the game, in the second pictures of building
stones. A shews B such a written sign; B looks it up in the table, looks at the picture opposite, and so on. So the
table is a rule which he follows in executing orders.--One learns to look the picture up in the table by receiving a
training, and part of this training consists perhaps in the pupil's learning to pass with his finger horizontally from left
to right; and so, as it were, to draw a series of horizontal lines on the table.
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Suppose different ways of reading a table were now introduced; one time, as above, according to the
schema:
another time like this:
or in some other way.--Such a schema is supplied with the table as the rule for its use.
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Can we not now imagine further rules to explain this one? And, on the other hand, was that first table
incomplete without the schema of arrows? And are other tables incomplete without their schemata?
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87. Suppose I give this explanation: "I take 'Moses' to mean the man, if there was such a man, who led the
Israelites out of Egypt, whatever he was called then and whatever he may or may not have done besides."--But
similar doubts to those about "Moses" are possible about the words of this explanation (what are you calling
"Egypt", whom the "Israelites" etc.?). Nor would these questions come to an end when we got down to words like
"red", "dark", "sweet".--"But then how does an explanation help me to understand,
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if after all it is not the final one? In that case the explanation is never completed; so I still don't understand what he
means, and never shall!"--As though an explanation as it were hung in the air unless supported by another one.
Whereas an explanation may indeed rest on another one that has been given, but none stands in need of
another--unless we require it to prevent a misunderstanding. One might say: an explanation serves to remove or to
avert a misunderstanding--one, that is, that would occur but for the explanation; not every one that I can imagine.
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It may easily look as if every doubt merely revealed an existing gap in the foundations; so that secure
understanding is only possible if we first doubt everything that can be doubted, and then remove all these doubts.
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The sign-post is in order--if, under normal circumstances, it fulfils its purpose.
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88. If I tell someone "Stand roughly here"--may not this explanation work perfectly? And cannot every other
one fail too?
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But isn't it an inexact explanation?--Yes; why shouldn't we call it "inexact"? Only let us understand what
"inexact" means. For it does not mean "unusable". And let us consider what we call an "exact" explanation in
contrast with this one. Perhaps something like drawing a chalk line round an area? Here it strikes us at once that the
line has breadth. So a colour-edge would be more exact. But has this exactness still got a function here: isn't the
engine idling? And remember too that we have not yet defined what is to count as overstepping this exact boundary;
how, with what instruments, it is to be established. And so on.
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We understand what it means to set a pocket watch to the exact time or to regulate it to be exact. But what if
it were asked: is this exactness ideal exactness, or how nearly does it approach the ideal?--Of course, we can speak
of measurements of time in which there is a different, and as we should say a greater, exactness than in the
measurement of time by a pocket-watch; in which the words "to set the clock to the exact time" have a different,
though related meaning, and 'to tell the time' is a different process and so on.--Now, if I tell someone: "You should
come to dinner more punctually; you know it begins at one o'clock exactly"--is there really no question of exactness
here? because it is possible to say: "Think of the determination of time in the laboratory or the observatory; there
you see what 'exactness' means"?
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"Inexact" is really a reproach, and "exact" is praise. And that is to say that what is inexact attains its goal less
perfectly than what is more exact. Thus the point here is what we call "the goal". Am I inexact when I do not give
our distance from the sun to the nearest foot, or tell a joiner the width of a table to the nearest thousandth of an
inch?
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No single ideal of exactness has been laid down; we do not know what we should be supposed to imagine
under this head--unless you yourself lay down what is to be so called. But you will find it difficult to hit upon such a
convention; at least any that satisfies you.
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89. These considerations bring us up to the problem: In what sense is logic something sublime?
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For there seemed to pertain to logic a peculiar depth--a universal significance. Logic lay, it seemed, at the
bottom of all the sciences.--For logical investigation explores the nature of all things. It seeks to see to the bottom of
things and is not meant to concern itself whether what actually happens is this or that.--It takes its rise, not from an
interest in the facts of nature, nor from a need to grasp causal connexions: but from an urge to understand the basis,
or essence, of everything empirical. Not, however, as if to this end we had to hunt out new facts; it is, rather, of the
essence of our investigation that we do not seek to learn anything new by it. We want to understand something that
is already in plain view. For this is what we seem in some sense not to understand.
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Augustine says in the Confessions "quid est ergo tempus? si nemo ex me quaerat scio; si quaerenti explicare
velim, nescio".--This could not be said about a question of natural science ("What is the specific gravity of
hydrogen?" for instance). Something that we know when no one asks us, but no longer know when we are supposed
to give an account of it, is something that we need to remind ourselves of. (And it is obviously something of which
for some reason it is difficult to remind oneself.)
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90. We feel as if we had to penetrate phenomena: our investigation, however, is directed not towards
phenomena, but, as one might say, towards the 'possibilities' of phenomena. We remind ourselves, that is to say, of
the kind of statement that we make about phenomena. Thus Augustine recalls to mind the different statements that
are made about the duration, past present or future, of events. (These are, of course, not philosophical statements
about time, the past, the present and the future.)
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Our investigation is therefore a grammatical one. Such an investigation sheds light on our problem by
clearing misunderstandings away. Misunderstandings concerning the use of words, caused, among other things, by
certain analogies between the forms of expression in different regions of language.--Some of them can be removed
by substituting one form of expression for another; this may be called an "analysis" of our forms of expression, for
the process is sometimes like one of taking a thing apart.
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91. But now it may come to look as if there were something like a final analysis of our forms of language,
and so a single completely resolved form of every expression. That is, as if our usual forms of expression were,
essentially, unanalysed; as if there were something hidden in them that had to be brought to light. When this is done
the expression is completely clarified and our problem solved.
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It can also be put like this: we eliminate misunderstandings by making our expressions more exact; but now
it may look as if we were moving towards a particular state, a state of complete exactness; and as if this were the real
goal of our investigation.
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92. This finds expression in questions as to the essence of language, of propositions, of thought.--For if we
too in these investigations are trying to understand the essence of language--its function, its structure,--yet this is not
what those questions have in view. For they see in the essence, not something that already lies open to view and that
becomes surveyable by a rearrangement, but something that lies beneath the surface. Something that lies within,
which we see when we look into the thing, and which an analysis digs out.
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'The essence is hidden from us': this is the form our problem now assumes. We ask: "What is language?",
"What is a proposition?" And the answer to these questions is to be given once for all; and independently of any
future experience.
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93. One person might say "A proposition is the most ordinary thing in the world" and another: "A
proposition--that's something very queer!"--And the latter is unable simply to look and see how propositions really
work. The forms that we use in expressing ourselves about propositions and thought stand in his way.
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Why do we say a proposition is something remarkable? On the one hand, because of the enormous
importance attaching to it. (And that is correct). On the other hand this, together with a misunderstanding
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of the logic of language, seduces us into thinking that something extraordinary, something unique, must be achieved
by propositions.--A misunderstanding makes it look to us as if a proposition did something queer.
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94. 'A proposition is a queer thing!' Here we have in germ the subliming of our whole account of logic. The
tendency to assume a pure intermediary between the propositional signs and the facts. Or even to try to purify, to
sublime, the signs themselves.--For our forms of expression prevent us in all sorts of ways from seeing that nothing
out of the ordinary is involved, by sending us in pursuit of chimeras.
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95. "Thought must be something unique". When we say, and mean, that such-and-such is the case, we--and
our meaning--do not stop anywhere short of the fact; but we mean: this--is--so. But this paradox (which has the
form of a truism) can also be expressed in this way: Thought can be of what is not the case.
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96. Other illusions come from various quarters to attach themselves to the special one spoken of here.
Thought, language, now appear to us as the unique correlate, picture, of the world. These concepts: proposition,
language, thought, world, stand in line one behind the other, each equivalent to each. (But what are these words to
be used for now? The language-game in which they are to be applied is missing.)
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97. Thought is surrounded by a halo.--Its essence, logic, presents an order, in fact the a priori order of the
world: that is, the order of possibilities, which must be common to both world and thought. But this order, it seems,
must be utterly simple. It is prior to all experience, must run through all experience; no empirical cloudiness or
uncertainty can be allowed to affect it--It must rather be of the purest crystal. But this crystal does not appear as an
abstraction; but as something concrete, indeed, as the most concrete, as it were the hardest thing there is (Tractatus
Logico-Philosophicus No. 5.5563).
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We are under the illusion that what is peculiar, profound, essential, in our investigation, resides in its trying to
grasp the incomparable essence of language. That is, the order existing between the concepts of proposition, word,
proof, truth, experience, and so on. This order is a super-order between--so to speak--super-concepts. Whereas, of
course, if the words "language", "experience", "world", have a use, it must be as humble a one as that of the words
"table", "lamp", "door".
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98. On the one hand it is clear that every sentence in our language 'is in order as it is'. That is to say, we are
not striving after an ideal, as if our ordinary vague sentences had not yet got a quite unexceptionable sense, and a
perfect language awaited construction by us.--On the other hand it seems clear that where there is sense there must
be perfect order.--So there must be perfect order even in the vaguest sentence.
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99. The sense of a sentence--one would like to say--may, of course, leave this or that open, but the sentence
must nevertheless have a definite sense. An indefinite sense--that would really not be a sense at all.--This is like: An
indefinite boundary is not really a boundary at all. Here one thinks perhaps: if I say "I have locked the man up fast in
the room--there is only one door left open"--then I simply haven't locked him in at all; his being locked in is a sham.
One would be inclined to say here: "You haven't done anything at all". An enclosure with a hole in it is as good as
none.--But is that true?
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100. "But still, it isn't a game, if there is some vagueness in the rules".--But does this prevent its being a
game?--"Perhaps you'll call it a game, but at any rate it certainly isn't a perfect game." This means: it has impurities,
and what I am interested in at present is the pure article.--But I want to say: we misunderstand the role of the ideal in
our language. That is to say: we too should call it a game, only we are dazzled by the ideal and therefore fail to see
the actual use of the word "game" clearly.
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101. We want to say that there can't be any vagueness in logic. The idea now absorbs us, that the ideal 'must'
be found in reality. Meanwhile we do not as yet see how it occurs there, nor do we understand the nature of this
"must". We think it must be in reality; for we think we already see it there.
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102. The strict and clear rules of the logical structure of propositions appear to us as something in the
background--hidden in the medium of the understanding. I already see them (even though through a medium): for I
understand the propositional sign, I use it to say something.
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103. The ideal, as we think of it, is unshakable. You can never get outside it; you must always turn back.
There is no outside; outside you cannot breathe.--Where does this idea come from? It is like a pair of glasses on our
nose through which we see whatever we look at. It never occurs to us to take them off.
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104. We predicate of the thing what lies in the method of representing it. Impressed by the possibility of a
comparison, we think we are perceiving a state of affairs of the highest generality.
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105. When we believe that we must find that order, must find the ideal, in our actual language, we become
dissatisfied with what are ordinarily called "propositions", "words", "signs".
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The proposition and the word that logic deals with are supposed to be something pure and clear-cut. And we
rack our brains over the nature of the real sign.--It is perhaps the idea of the sign? or the idea at the present
moment?
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106. Here it is difficult as it were to keep our heads up,--to see that we must stick to the subjects of our
every-day thinking, and not go astray and imagine that we have to describe extreme subtleties, which in turn we are
after all quite unable to describe with the means at our disposal. We feel as if we had to repair a torn spider's web
with our fingers.
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107. The more narrowly we examine actual language, the sharper becomes the conflict between it and our
requirement. (For the crystalline purity of logic was, of course, not a result of investigation: it was a requirement.)
The conflict becomes intolerable; the requirement is now in danger of becoming empty.--We have got on to slippery
ice where there is no friction and so in a certain sense the conditions are ideal, but also, just because of that, we are
unable to walk. We want to walk: so we need friction. Back to the rough ground!
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Faraday in The Chemical History of a Candle: "Water is one individual thing--it never changes."
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108. We see that what we call "sentence" and "language" has not the formal unity that I imagined, but is the
family of structures more or less related to one another.--But what becomes of logic now? Its rigour seems to be
giving way here.--But in that case doesn't logic altogether disappear?--For how can it lose its rigour? Of course not
by our bargaining any of its rigour out of it.--The preconceived idea of crystalline purity can only be removed by
turning our whole examination round. (One might say: the axis of reference of our examination must be rotated, but
about the fixed point of our real need.)
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The philosophy of logic speaks of sentences and words in exactly the sense in which we speak of them in
ordinary life when we say e.g.
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"Here is a Chinese sentence", or "No, that only looks like writing; it is actually just an ornament" and so on.
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We are talking about the spatial and temporal phenomenon of language, not about some non-spatial,
non-temporal phantasm. [Note in margin: Only it is possible to be interested in a phenomenon in a variety of ways].
But we talk about it as we do about the pieces in chess when we are stating the rules of the game, not describing
their physical properties.
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The question "What is a word really?" is analogous to "What is a piece in chess?"
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109. It was true to say that our considerations could not be scientific ones. It was not of any possible interest
to us to find out empirically 'that, contrary to our preconceived ideas, it is possible to think such-and-such'--whatever
that may mean. (The conception of thought as a gaseous medium.) And we may not advance any kind of theory.
There must not be anything hypothetical in our considerations. We must do away with all explanation, and
description alone must take its place. And this description gets its light, that is to say its purpose, from the
philosophical problems. These are, of course, not empirical problems; they are solved, rather, by looking into the
workings of our language, and that in such a way as to make us recognize those workings: in despite of an urge to
misunderstand them. The problems are solved, not by giving new information, but by arranging what we have
always known. Philosophy is a battle against the bewitchment of our intelligence by means of language.
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110. "Language (or thought) is something unique"--this proves to be a superstition (not a mistake!), itself
produced by grammatical illusions.
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And now the impressiveness retreats to these illusions, to the problems.
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111. The problems arising through a misinterpretation of our forms of language have the character of depth.
They are deep disquietudes; their roots are as deep in us as the forms of our language and their significance is as
great as the importance of our language.--Let us ask ourselves: why do we feel a grammatical joke to be deep? (And
that is what the depth of philosophy is.)
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112. A simile that has been absorbed into the forms of our language produces a false appearance, and this
disquiets us. "But this isn't how it is!"--we say. "Yet this is how it has to be!"
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113. "But this is how it is--" I say to myself over and over again. I feel as though, if only I could fix my gaze
absolutely sharply on this fact, get it in focus, I must grasp the essence of the matter.
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114. (Tractatus Logico-Philosophicus, 4.5): "The general form of propositions is: This is how things
are."--That is the kind of proposition that one repeats to oneself countless times. One thinks that one is tracing the
outline of the thing's nature over and over again, and one is merely tracing round the frame through which we look at
it.
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115. A picture held us captive. And we could not get outside it, for it lay in our language and language
seemed to repeat it to us inexorably.
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116. When philosophers use a word--"knowledge", "being", "object", "I", "proposition", "name"--and try to
grasp the essence of the thing, one must always ask oneself: is the word ever actually used in this way in the
language-game which is its original home?--
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What we do is to bring words back from their metaphysical to their everyday use.
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117. You say to me: "You understand this expression, don't you? Well then--I am using it in the sense you
are familiar with."--As if the sense were an atmosphere accompanying the word, which it carried with it into every
kind of application.
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If, for example, someone says that the sentence "This is here" (saying which he points to an object in front of
him) makes sense to him, then he should ask himself in what special circumstances this sentence is actually used.
There it does make sense.
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118. Where does our investigation get its importance from, since it seems only to destroy everything
interesting, that is, all that is great and important? (As it were all the buildings, leaving behind only bits of stone and
rubble.) What we are destroying is nothing but houses of cards and we are clearing up the ground of language on
which they stand.
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119. The results of philosophy are the uncovering of one or another piece of plain nonsense and of bumps
that the understanding has got by running its head up against the limits of language. These bumps make us see the
value of the discovery.
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120. When I talk about language (words, sentences, etc.) I must speak the language of every day. Is this
language somehow too coarse and material for what we want to say? Then how is another one to be
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constructed?--And how strange that we should be able to do anything at all with the one we have!
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In giving explanations I already have to use language full-blown (not some sort of preparatory, provisional
one); this by itself shews that I can adduce only exterior facts about language.
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Yes, but then how can these explanations satisfy us?--Well, your very questions were framed in this
language; they had to be expressed in this language, if there was anything to ask!
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And your scruples are misunderstandings.
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Your questions refer to words; so I have to talk about words.
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You say: the point isn't the word, but its meaning, and you think of the meaning as a thing of the same kind
as the word, though also different from the word. Here the word, there the meaning. The money, and the cow that
you can buy with it. (But contrast: money, and its use.)
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121. One might think: if philosophy speaks of the use of the word "philosophy" there must be a second-order
philosophy. But it is not so: it is, rather, like the case of orthography, which deals with the word "orthography"
among others without then being second-order.
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122. A main source of our failure to understand is that we do not command a clear view of the use of our
words.--Our grammar is lacking in this sort of perspicuity. A perspicuous representation produces just that
understanding which consists in 'seeing connexions'. Hence the importance of finding and inventing intermediate
cases.
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The concept of a perspicuous representation is of fundamental significance for us. It earmarks the form of
account we give, the way we look at things. (Is this a 'Weltanschauung'?)
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123. A philosophical problem has the form: "I don't know my way about".
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124. Philosophy may in no way interfere with the actual use of language; it can in the end only describe it.
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For it cannot give it any foundation either.
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It leaves everything as it is.
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It also leaves mathematics as it is, and no mathematical discovery can advance it. A "leading problem of
mathematical logic" is for us a problem of mathematics like any other.
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125. It is the business of philosophy, not to resolve a contradiction by means of a mathematical or
logico-mathematical discovery, but to make it possible for us to get a clear view of the state of mathematics that
troubles us: the state of affairs before the contradiction is resolved. (And this does not mean that one is sidestepping
a difficulty.)
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The fundamental fact here is that we lay down rules, a technique, for a game, and that then when we follow
the rules, things do not turn out as we had assumed. That we are therefore as it were entangled in our own rules.
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This entanglement in our rules is what we want to understand (i.e. get a clear view of).
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It throws light on our concept of meaning something. For in those cases things turn out otherwise than we
had meant, foreseen. That is just what we say when, for example, a contradiction appears: "I didn't mean it like that."
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The civil status of a contradiction, or its status in civil life: there is the philosophical problem.
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126. Philosophy simply puts everything before us, and neither explains nor deduces anything.--Since
everything lies open to view there is nothing to explain. For what is hidden, for example, is of no interest to us.
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One might also give the name "philosophy" to what is possible before all new discoveries and inventions.
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127. The work of the philosopher consists in assembling reminders for a particular purpose.
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128. If one tried to advance theses in philosophy, it would never be possible to debate them, because
everyone would agree to them.
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129. The aspects of things that are most important for us are hidden because of their simplicity and
familiarity. (One is unable to notice something--because it is always before one's eyes.) The real foundations of his
enquiry do not strike a man at all. Unless that fact has at some time struck him.--And this means: we fail to be struck
by what, once seen, is most striking and most powerful.
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130. Our clear and simple language-games are not preparatory studies for a future regularization of
language--as it were first approximations, ignoring friction and air-resistance. The language-games are rather set up
as objects of comparison which are meant to throw light on the facts of our language by way not only of similarities,
but also of dissimilarities.
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131. For we can avoid ineptness or emptiness in our assertions only by presenting the model as what it is, as
an object of comparison--as, so to speak, a measuring-rod; not as a preconceived idea to which reality must
correspond. (The dogmatism into which we fall so easily in doing philosophy.)
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132. We want to establish an order in our knowledge of the use of language: an order with a particular end in
view; one out of many possible orders; not the order. To this end we shall constantly be giving prominence to
distinctions which our ordinary forms of language easily make us overlook. This may make it look as if we saw it as
our task to reform language.
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Such a reform for particular practical purposes, an improvement in our terminology designed to prevent
misunderstandings in practice, is perfectly possible. But these are not the cases we have to do with. The confusions
which occupy us arise when language is like an engine idling, not when it is doing work.
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133. It is not our aim to refine or complete the system of rules for the use of our words in unheard-of ways.
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For the clarity that we are aiming at is indeed complete clarity. But this simply means that the philosophical
problems should completely disappear.
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The real discovery is the one that makes me capable of stopping doing philosophy when I want to.--The one
that gives philosophy peace, so that it is no longer tormented by questions which bring itself in question.--Instead,
we now demonstrate a method, by examples; and the series of examples can be broken off.--Problems are solved
(difficulties eliminated), not a single problem.
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There is not a philosophical method, though there are indeed methods, like different therapies.
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134. Let us examine the proposition: "This is how things are."--How can I say that this is the general form of
propositions?--It is first and foremost itself a proposition, an English sentence, for it has a subject and a predicate.
But how is this sentence applied--that is, in our everyday language? For I got it from there and nowhere else.
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We may say, e.g.: "He explained his position to me, said that this was how things were, and that therefore he
needed an advance". So far, then, one can say that that sentence stands for any statement. It is employed as a
propositional schema, but only because it has the
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construction of an English sentence. It would be possible to say instead "such and such is the case", "this is the
situation", and so on. It would also be possible here simply to use a letter, a variable, as in symbolic logic. But no
one is going to call the letter "p" the general form of propositions. To repeat: "This is how things are" had that
position only because it is itself what one calls an English sentence. But though it is a proposition, still it gets
employed as a propositional variable. To say that this proposition agrees (or does not agree) with reality would be
obvious nonsense. Thus it illustrates the fact that one feature of our concept of a proposition is, sounding like a
proposition.
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135. But haven't we got a concept of what a proposition is, of what we take "proposition" to mean?--Yes; just
as we also have a concept of what we mean by "game". Asked what a proposition is--whether it is another person or
ourselves that we have to answer--we shall give examples and these will include what one may call inductively
defined series of propositions. This is the kind of way in which we have such a concept as 'proposition'. (Compare
the concept of a proposition with the concept of number.)
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136. At bottom, giving "This is how things are" as the general form of propositions is the same as giving the
definition: a proposition is whatever can be true or false. For instead of "This is how things are" I could have said
"This is true". (Or again "This is false".) But we have
'p' is true = p
'p' is false = not-p.
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And to say that a proposition is whatever can be true or false amounts to saying: we call something a
proposition when in our language we apply the calculus of truth functions to it.
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Now it looks as if the definition--a proposition is whatever can be true or false--determined what a
proposition was, by saying: what fits the concept 'true', or what the concept 'true' fits, is a proposition. So it is as if
we had a concept of true and false, which we could use to determine what is and what is not a proposition. What
engages with the concept of truth (as with a cogwheel), is a proposition.
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But this is a bad picture. It is as if one were to say "The king in chess is the piece that one can check." But
this can mean no more than that in our game of chess we only check the king. Just as the proposition that only a
proposition can be true or false can say no more than
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that we only predicate "true" and "false" of what we call a proposition. And what a proposition is is in one sense
determined by the rules of sentence formation (in English for example), and in another sense by the use of the sign
in the language-game. And the use of the words "true" and "false" may be among the constituent parts of this game;
and if so it belongs to our concept 'proposition' but does not 'fit' it. As we might also say, check belongs to our
concept of the king in chess (as so to speak a constituent part of it). To say that check did not fit our concept of the
pawns, would mean that a game in which pawns were checked, in which, say, the players who lost their pawns lost,
would be uninteresting or stupid or too complicated or something of the kind.
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137. What about learning to determine the subject of a sentence by means of the question "Who or
what....?"--Here, surely, there is such a thing as the subject's 'fitting' this question; for otherwise how should we find
out what the subject was by means of the question? We find it out much as we find out which letter of the alphabet
comes after 'K' by saying the alphabet up to 'K' to ourselves. Now in what sense does 'L' fit on to this series of
letters?--In that sense "true" and "false" could be said to fit propositions; and a child might be taught to distinguish
between propositions and other expressions by being told "Ask yourself if you can say 'is true' after it. If these
words fit, it's a proposition." (And in the same way one might have said: Ask yourself if you can put the words "This
is how things are:" in front of it.)
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138. But can't the meaning of a word that I understand fit the sense of a sentence that I understand? Or the
meaning of one word fit the meaning of another?--Of course, if the meaning is the use we make of the word, it
makes no sense to speak of such 'fitting.' But we understand the meaning of a word when we hear or say it; we
grasp it in a flash, and what we grasp in this way is surely something different from the 'use' which is extended in
time!
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Must I know whether I understand a word? Don't I also sometimes imagine myself to understand a word (as
I may imagine I understand a kind of calculation) and then realize that I did not understand it? ("I thought I knew
what 'relative' and 'absolute' motion meant, but I see that I don't know.")
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139. When someone says the word "cube" to me, for example, I know what it means. But can the whole use
of the word come before my mind, when I understand it in this way?
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Well, but on the other hand isn't the meaning of the word also determined by this use? And can't these ways
of determining meaning conflict? Can what we grasp in a flash accord with a use, fit or fail to fit it? And how can
what is present to us in an instant, what comes before our mind in an instant, fit a use?
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What really comes before our mind when we understand a word?--Isn't it something like a picture? Can't it
be a picture?
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Well, suppose that a picture does come before your mind when you hear the word "cube", say the drawing
of a cube. In what sense can this picture fit or fail to fit a use of the word "cube"?--Perhaps you say: "It's quite
simple;--if that picture occurs to me and I point to a triangular prism for instance, and say it is a cube, then this use
of the word doesn't fit the picture."--But doesn't it fit? I have purposely so chosen the example that it is quite easy to
imagine a method of projection according to which the picture does fit after all.
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The picture of the cube did indeed suggest a certain use to us, but it was possible for me to use it differently.
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(a) "I believe the right word in this case is....". Doesn't this shew that the meaning of a word is a something
that comes before our mind, and which is, as it were, the exact picture we want to use here? Suppose I were
choosing between the words "imposing", "dignified", "proud", "venerable"; isn't it as though I were choosing
between drawings in a portfolio?--No: the fact that one speaks of the appropriate word does not shew the existence
of a something that etc.. One is inclined, rather, to speak of this picture-like something just because one can find a
word appropriate; because one often chooses between words as between similar but not identical pictures; because
pictures are often used instead of words, or to illustrate words; and so on.
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(b) I see a picture; it represents an old man walking up a steep path leaning on a stick.--How? Might it not
have looked just the same if he had been sliding downhill in that position? Perhaps a Martian would describe the
picture so. I do not need to explain why we do not describe it so.
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140. Then what sort of mistake did I make; was it what we should like to express by saying: I should have
thought the picture forced a particular use on me? How could I think that? What did I think? Is there such a thing as
a picture, or something like a picture, that forces a particular application on us; so that my mistake lay in confusing
one picture with another?--For we might also be inclined to express ourselves like this: we are at most under a
psychological, not a logical, compulsion. And now it looks quite as if we knew of two kinds of case.
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What was the effect of my argument? It called our attention to (reminded us of) the fact that there are other
processes, besides the one we originally thought of, which we should sometimes be prepared to call "applying the
picture of a cube". So our 'belief that the picture forced a particular application upon us' consisted in the fact that
only the one case and no other occurred to us. "There is another solution as well" means: there is something else that
I am also prepared to call a "solution"; to which I am prepared to apply such-and-such a picture, such-and-such an
analogy, and so on.
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What is essential is to see that the same thing can come before our minds when we hear the word and the
application still be different. Has it the same meaning both times? I think we shall say not.
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141. Suppose, however, that not merely the picture of the cube, but also the method of projection comes
before our mind?--How am I to imagine this?--Perhaps I see before me a schema shewing the method of projection:
say a picture of two cubes connected by lines of projection.--But does this really get me any further? Can't I now
imagine different applications of this schema too?--Well, yes, but then can't an application come before my
mind?--It can: only we need to get clearer about our application of this expression. Suppose I explain various
methods of projection to someone so that he may go on to apply them; let us ask ourselves when we should say that
the method that I intend comes before his mind.
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Now clearly we accept two different kinds of criteria for this: on the one hand the picture (of whatever kind)
that at some time or other comes before his mind; on the other, the application which--in the course of time--he
makes of what he imagines. (And can't it be clearly seen here that it is absolutely inessential for the picture to exist in
his imagination rather than as a drawing or model in front of him; or again as something that he himself constructs
as a model?)
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Can there be a collision between picture and application? There can, inasmuch as the picture makes us
expect a different use, because people in general apply this picture like this.
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I want to say: we have here a normal case, and abnormal cases.
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142. It is only in normal cases that the use of a word is clearly prescribed; we know, are in no doubt, what to
say in this or that case. The more abnormal the case, the more doubtful it becomes what we are to say. And if things
were quite different from what they actually are--if there were for instance no characteristic expression of pain, of
fear, of joy; if rule became exception and exception rule; or if both became phenomena of roughly equal
frequency--this would make our normal language-games lose their point.--The procedure of putting a lump of
cheese on a balance and fixing the price by the turn of the scale would lose its point if it frequently happened for
such lumps to suddenly grow or shrink for no obvious reason. This remark will become clearer when we discuss
such things as the relation of expression to feeling, and similar topics.
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What we have to mention in order to explain the significance, I mean the importance, of a concept, are often
extremely general facts of nature: such facts as are hardly ever mentioned because of their great generality.
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143. Let us now examine the following kind of language-game: when A gives an order B has to write down
series of signs according to a certain formation rule.
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The first of these series is meant to be that of the natural numbers in decimal notation.--How does he get to
understand this notation?--First of all series of numbers will be written down for him and he will be required to copy
them. (Do not balk at the expression "series of numbers"; it is not being used wrongly here.) And here already there
is a normal and an abnormal learner's reaction.--At first perhaps we guide his hand in writing out the series 0 to 9;
but then the possibility of getting him to understand will depend on his going on to write it down
independently.--And here we can imagine, e.g., that he does copy the figures independently, but not in the right
order: he writes sometimes one sometimes another at random. And then communication stops at that point.--Or
again, he makes 'mistakes'
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in the order.--The difference between this and the first case will of course be one of frequency.--Or he makes a
systematic mistake; for example, he copies every other number, or he copies the series 0, 1, 2, 3, 4, 5,.... like this: 1,
0, 3, 2, 5, 4,..... Here we shall almost be tempted to say that he has understood wrong.
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Notice, however, that there is no sharp distinction between a random mistake and a systematic one. That is,
between what you are inclined to call "random" and what "systematic".
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Perhaps it is possible to wean him from the systematic mistake (as from a bad habit). Or perhaps one accepts
his way of copying and tries to teach him ours as an offshoot, a variant of his.--And here too our pupil's capacity to
learn may come to an end.
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144. What do I mean when I say "the pupil's capacity to learn may come to an end here"? Do I say this from
my own experience? Of course not. (Even if I have had such experience.) Then what am I doing with that
proposition? Well, I should like you to say: "Yes, it's true, you can imagine that too, that might happen too!"--But
was I trying to draw someone's attention to the fact that he is capable of imagining that?--I wanted to put that picture
before him, and his acceptance of the picture consists in his now being inclined to regard a given case differently:
that is, to compare it with this rather than that set of pictures. I have changed his way of looking at things. (Indian
mathematicians: "Look at this.")
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145. Suppose the pupil now writes the series 0 to 9 to our satisfaction.--And this will only be the case when
he is often successful, not if he does it right once in a hundred attempts. Now I continue the series and draw his
attention to the recurrence of the first series in the units; and then to its recurrence in the tens. (Which only means
that I use particular emphases, underline figures, write them one under another in such-and-such ways, and similar
things.)--And now at some point he continues the series independently--or he does not.--But why do you say that?
so much is obvious!--Of course; I only wished to say: the effect of any further explanation depends on his reaction.
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Now, however, let us suppose that after some efforts on the teacher's part he continues the series correctly,
that is, as we do it. So now we can say he has mastered the system.--But how far need he continue
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the series for us to have the right to say that? Clearly you cannot state a limit here.
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146. Suppose I now ask: "Has he understood the system when he continues the series to the hundredth
place?" Or--if I should not speak of 'understanding' in connection with our primitive language-game: Has he got the
system, if he continues the series correctly so far?--Perhaps you will say here: to have got the system (or, again, to
understand it) can't consist in continuing the series up to this or that number: that is only applying one's
understanding. The understanding itself is a state which is the source of the correct use.
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What is one really thinking of here? Isn't one thinking of the derivation of a series from its algebraic formula?
Or at least of something analogous?--But this is where we were before. The point is, we can think of more than one
application of an algebraic formula; and every type of application can in turn be formulated algebraically; but
naturally this does not get us any further.--The application is still a criterion of understanding.
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147. "But how can it be? When I say I understand the rule of a series, I am surely not saying so because I
have found out that up to now I have applied the algebraic formula in such-and-such a way! In my own case at all
events I surely know that I mean such-and-such a series; it doesn't matter how far I have actually developed it."--
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Your idea, then, is that you know the application of the rule of the series quite apart from remembering actual
applications to particular numbers. And you will perhaps say: "Of course! For the series is infinite and the bit of it
that I can have developed finite."
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148. But what does this knowledge consist in? Let me ask: When do you know that application? Always? day
and night? or only when you are actually thinking of the rule? do you know it, that is, in the same way as you know
the alphabet and the multiplication table? Or is what you call "knowledge" a state of consciousness or a process--say
a thought of something, or the like?
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149. If one says that knowing the ABC is a state of the mind, one is thinking of a state of a mental apparatus
(perhaps of the brain) by means of which we explain the manifestations of that knowledge. Such a state is called a
disposition. But there are objections to speaking
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of a state of the mind here, inasmuch as there ought to be two different criteria for such a state: a knowledge of the
construction of the apparatus, quite apart from what it does. (Nothing would be more confusing here than to use the
words "conscious" and "unconscious" for the contrast between states of consciousness and dispositions. For this
pair of terms covers up a grammatical difference.)
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150. The grammar of the word "knows" is evidently closely related to that of "can", "is able to". But also
closely related to that of "understands". ('Mastery' of a technique,)
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(a) "Understanding a word": a state. But a mental state?--Depression, excitement, pain, are called mental
states. Carry out a grammatical investigation as follows: we say
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"He was depressed the whole day".
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"He was in great excitement the whole day".
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"He has been in continuous pain since yesterday".--
We also say "Since yesterday I have understood this word". "Continuously", though?--To be sure, one can speak of
an interruption of understanding. But in what cases? Compare: "When did your pains get less?" and "When did you
stop understanding that word?"
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(b) Suppose it were asked: "When do you know how to play chess? All the time? or just while you are
making a move? And the whole of chess during each move?--How queer that knowing how to play chess should
take such a short time, and a game so much longer!
------------------
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151. But there is also this use of the word "to know": we say "Now I know!"--and similarly "Now I can do
it!" and "Now I understand!"
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Let us imagine the following example: A writes series of numbers down; B watches him and tries to find a
law for the sequence of numbers. If he succeeds he exclaims: "Now I can go on!"--So this capacity, this
understanding, is something that makes its appearance in a moment. So let us try and see what it is that makes its
appearance here.--A has written down the numbers 1, 5, 11, 19, 29; at this point B says he knows how to go on.
What happened here? Various things may have happened; for example, while A was slowly putting one number
after another, B was occupied with trying various algebraic formulae on the numbers which had been written down.
After A had written the number 19 B tried the formula an = n² + n - 1; and the next number confirmed his
hypothesis.
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Or again, B does not think of formulae. He watches A writing his numbers down with a certain feeling of
tension, and all sorts of vague thoughts go through his head. Finally he asks himself: "What is the series of
differences?" He finds the series 4, 6, 8, 10 and says: Now I can go on.
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Or he watches and says "Yes, I know that series"--and continues it, just as he would have done if A had
written down the series 1, 3, 5, 7, 9.--Or he says nothing at all and simply continues the series. Perhaps he had what
may be called the sensation "that's easy!". (Such a sensation is, for example, that of a light quick intake of breath, as
when one is slightly startled.)
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152. But are the processes which I have described here understanding?
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"B understands the principle of the series" surely doesn't mean simply: the formula "an = ...." occurs to B. For
it is perfectly imaginable that the formula should occur to him and that he should nevertheless not understand. "He
understands" must have more in it than: the formula occurs to him. And equally, more than any of those more or
less characteristic accompaniments or manifestations of understanding.
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153. We are trying to get hold of the mental process of understanding which seems to be hidden behind
those coarser and therefore more readily visible accompaniments. But we do not succeed; or, rather, it does not get
as far as a real attempt. For even supposing I had found something that happened in all those cases of
understanding,--why should it be the understanding? And how can the process of understanding have been hidden,
when I said "Now I understand" because I understood?! And if I say it is hidden--then how do I know what I have
to look for? I am in a muddle.
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154. But wait--if "Now I understand the principle" does not mean the same as "The formula.... occurs to me"
(or "I say the formula", "I write it down", etc.)--does it follow from this that I employ the sentence "Now I
understand....." or "Now I can go on" as a description of a process occurring behind or side by side with that of
saying the formula?
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If there has to be anything 'behind the utterance of the formula' it is particular circumstances, which justify
me in saying I can go on--when the formula occurs to me.
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Try not to think of understanding as a 'mental process' at all.--For that is the expression which confuses you.
But ask yourself: in what sort of case, in what kind of circumstances, do we say, "Now I know how to go on," when,
that is, the formula has occurred to me?--
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In the sense in which there are processes (including mental processes) which are characteristic of
understanding, understanding is not a mental process.
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(A pain's growing more and less; the hearing of a tune or a sentence: these are mental processes.)
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155. Thus what I wanted to say was: when he suddenly knew how to go on, when he understood the
principle, then possibly he had a special experience--and if he is asked: "What was it? What took place when you
suddenly grasped the principle?" perhaps he will describe it much as we described it above--but for us it is the
circumstances under which he had such an experience that justify him in saying in such a case that he understands,
that he knows how to go on.
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156. This will become clearer if we interpolate the consideration of another word, namely "reading". First I
need to remark that I am not counting the understanding of what is read as part of 'reading' for purposes of this
investigation: reading is here the activity of rendering out loud what is written or printed; and also of writing from
dictation, writing out something printed, playing from a score, and so on.
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The use of this word in the ordinary circumstances of our life is of course extremely familiar to us. But the
part the word plays in our life, and therewith the language-game in which we employ it, would be difficult to
describe even in rough outline. A person, let us say an Englishman, has received at school or at home one of the
kinds of education usual among us, and in the course of it has learned to read his native language. Later he reads
books, letters, newspapers, and other things.
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Now what takes place when, say, he reads a newspaper?--His eye passes--as we say--along the printed
words, he says them out loud--or only to himself; in particular he reads certain words by taking in their printed
shapes as wholes; others when his eye has taken in the first syllables; others again he reads syllable by syllable, and
an occasional one perhaps letter by letter.--We should also say that he had read a sentence if he spoke neither aloud
nor to himself during the reading but was afterwards able to repeat the sentence word for word or nearly so.--He
may attend to what he reads, or again--as we
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might put it--function as a mere reading-machine: I mean, read aloud and correctly without attending to what he is
reading; perhaps with his attention on something quite different (so that he is unable to say what he has been reading
if he is asked about it immediately afterwards).
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Now compare a beginner with this reader. The beginner reads the words by laboriously spelling them
out.--Some however he guesses from the context, or perhaps he already partly knows the passage by heart. Then his
teacher says that he is not really reading the words (and in certain cases that he is only pretending to read them).
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If we think of this sort of reading, the reading of a beginner, and ask ourselves what reading consists in, we
shall be inclined to say: it is a special conscious activity of mind.
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We also say of the pupil: "Of course he alone knows if he is really reading or merely saying the words off by
heart". (We have yet to discuss these propositions: "He alone knows....".)
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But I want to say: we have to admit that--as far as concerns uttering any one of the printed words--the same
thing may take place in the consciousness of the pupil who is 'pretending' to read, as in that of the practised reader
who is 'reading' it. The word "to read" is applied differently when we are speaking of the beginner and of the
practised reader.--Now we should of course like to say: What goes on in that practised reader and in the beginner
when they utter the word can't be the same. And if there is no difference in what they happen to be conscious of
there must be one in the unconscious workings of their minds, or, again, in the brain.--So we should like to say:
There are at all events two different mechanisms at work here. And what goes on in them must distinguish reading
from not reading.--But these mechanisms are only hypotheses, models designed to explain, to sum up, what you
observe.
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157. Consider the following case. Human beings or creatures of some other kind are used by us as
reading-machines. They are trained for this purpose. The trainer says of some that they can already read, of others
that they cannot yet do so. Take the case of a pupil who has so far not taken part in the training: if he is shewn a
written word he will sometimes produce some sort of sound, and here and there it happens 'accidentally' to be
roughly right. A third person hears this pupil on such an occasion and says: "He is reading". But the teacher says:
"No, he isn't reading; that was just an accident".--But let us suppose that this pupil continues to react correctly to
further words
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that are put before him. After a while the teacher says: "Now he can read!"--But what of that first word? Is the
teacher to say: "I was wrong, and he did read it"--or: "He only began really to read later on"?--When did he begin to
read? Which was the first word that he read? This question makes no sense here. Unless, indeed, we give a
definition: "The first word that a person 'reads' is the first word of the first series of 50 words that he reads correctly"
(or something of the sort).
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If on the other hand we use "reading" to stand for a certain experience of transition from marks to spoken
sounds, then it certainly makes sense to speak of the first word that he really read. He can then say, e.g. "At this
word for the first time I had the feeling: 'now I am reading'."
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Or again, in the different case of a reading machine which translated marks into sounds, perhaps as a pianola
does, it would be possible to say: "The machine read only after such-and-such had happened to it--after
such-and-such parts had been connected by wires; the first word that it read was....".
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But in the case of the living reading-machine "reading" meant reacting to written signs in such-and-such
ways. This concept was therefore quite independent of that of a mental or other mechanism.--Nor can the teacher
here say of the pupil: "Perhaps he was already reading when he said that word". For there is no doubt about what he
did.--The change when the pupil began to read was a change in his behaviour; and it makes no sense here to speak
of 'a first word in his new state'.
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158. But isn't that only because of our too slight acquaintance with what goes on in the brain and the nervous
system? If we had a more accurate knowledge of these things we should see what connexions were established by
the training, and then we should be able to say when we looked into his brain: "Now he has read this word, now the
reading connexion has been set up".--And it presumably must be like that--for otherwise how could we be so sure
that there was such a connexion? That it is so is presumably a priori--or is it only probable? And how probable is it?
Now, ask yourself: what do you know about these things?--But if it is a priori, that means that it is a form of account
which is very convincing to us.
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159. But when we think the matter over we are tempted to say: the one real criterion for anybody's reading is
the conscious act of reading, the act of reading the sounds off from the letters. "A man
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surely knows whether he is reading or only pretending to read!"--Suppose A wants to make B believe he can read
Cyrillic script. He learns a Russian sentence by heart and says it while looking at the printed words as if he were
reading them. Here we shall certainly say that A knows he is not reading, and has a sense of just this while
pretending to read. For there are of course many more or less characteristic sensations in reading a printed sentence;
it is not difficult to call such sensations to mind: think of sensations of hesitating, of looking closer, of misreading, of
words following on one another more or less smoothly, and so on. And equally there are characteristic sensations in
reciting something one has learnt by heart. In our example A will have none of the sensations that are characteristic
of reading, and will perhaps have a set of sensations characteristic of cheating.
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160. But imagine the following case: We give someone who can read fluently a text that he never saw before.
He reads it to us--but with the sensation of saying something he has learnt by heart (this might be the effect of some
drug). Should we say in such a case that he was not really reading the passage? Should we here allow his sensations
to count as the criterion for his reading or not reading?
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Or again: Suppose that a man who is under the influence of a certain drug is presented with a series of
characters (which need not belong to any existing alphabet). He utters words corresponding to the number of the
characters, as if they were letters, and does so with all the outward signs, and with the sensations, of reading. (We
have experiences like this in dreams; after waking up in such a case one says perhaps: "It seemed to me as if I were
reading a script, though it was not writing at all.") In such a case some people would be inclined to say the man was
reading those marks. Others, that he was not.--Suppose he has in this way read (or interpreted) a set of five marks as
A B O V E--and now we shew him the same marks in the reverse order and he reads E V O B A; and in further tests
he always retains the same interpretation of the marks: here we should certainly be inclined to say he was making up
an alphabet for himself ad hoc and then reading accordingly.
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161. And remember too that there is a continuous series of transitional cases between that in which a person
repeats from memory what he is supposed to be reading, and that in which he spells out every word without being
helped at all by guessing from the context or knowing by heart.
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Try this experiment: say the numbers from 1 to 12. Now look at the dial of your watch and read
them.--What was it that you called "reading" in the latter case? That is to say: what did you do, to make it into
reading?
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162. Let us try the following definition: You are reading when you derive the reproduction from the original.
And by "the original" I mean the text which you read or copy; the dictation from which you write; the score from
which you play; etc. etc.--Now suppose we have, for example, taught someone the Cyrillic alphabet, and told him
how to pronounce each letter. Next we put a passage before him and he reads it, pronouncing every letter as we have
taught him. In this case we shall very likely say that he derives the sound of a word from the written pattern by the
rule that we have given him. And this is also a clear case of reading. (We might say that we had taught him the 'rule
of the alphabet'.)
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But why do we say that he has derived the spoken from the printed words? Do we know anything more than
that we taught him how each letter should be pronounced, and that he then read the words out loud? Perhaps our
reply will be: the pupil shews that he is using the rule we have given him to pass from the printed to the spoken
words.--How this can be shewn becomes clearer if we change our example to one in which the pupil has to write out
the text instead of reading it to us, has to make the transition from print to handwriting. For in this case we can give
him the rule in the form of a table with printed letters in one column and cursive letters in the other. And he shews
that he is deriving his script from the printed words by consulting the table.
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163. But suppose that when he did this he always wrote b for A, c for B, d for C, and so on, and a for
Z?--Surely we should call this too a derivation by means of the table.--He is using it now, we might say, according to
the second schema in §86 instead of the first.
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It would still be a perfectly good case of derivation according to the table, even if it were represented by a
schema of arrows without any simple regularity.
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Suppose, however, that he does not stick to a single method of transcribing, but alters his method according
to a simple rule: if he has once written n for A, then he writes o for the next A, p for the next, and so on.--But where
is the dividing line between this procedure and a random one?
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But does this mean that the word "to derive" really has no meaning, since the meaning seems to disintegrate
when we follow it up?
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164. In case (162) the meaning of the word "to derive" stood out clearly. But we told ourselves that this was
only a quite special case of deriving; deriving in a quite special garb, which had to be stripped from it if we wanted to
see the essence of deriving. So we stripped those particular coverings off; but then deriving itself disappeared.--In
order to find the real artichoke, we divested it of its leaves. For certainly (162) was a special case of deriving; what is
essential to deriving, however, was not hidden beneath the surface of this case, but his 'surface' was one case out of
the family of cases of deriving.
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And in the same way we also use the word "to read" for a family of cases. And in different circumstances we
apply different criteria for a person's reading.
------------------
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The grammar of the expression "a quite particular" (atmosphere). One says "This face has a quite particular
expression," and maybe looks for words to characterize it.
------------------
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165. But surely--we should like to say--reading is a quite particular process! Read a page of print and you can
see that something special is going on, something highly characteristic.--Well, what does go on when I read the
page? I see printed words and I say words out loud. But, of course, that is not all, for I might see printed words and
say words out loud and still not be reading. Even if the words which I say are those which, going by an existing
alphabet, are supposed to be read off from the printed ones.--And if you say that reading is a particular experience,
then it becomes quite unimportant whether or not you read according to some generally recognized alphabetical
rule.--And what does the characteristic thing about the experience of reading consist in?--Here I should like to say:
"The words that I utter come in a special way." That is, they do not come as they would if I were for example making
them up.--They come of themselves.--But even that is not enough; for the sounds of words may occur to me while I
am looking at printed words, but that does not mean that I have read them.--In addition I might say here, neither do
the spoken words occur to me as if, say, something reminded me of them. I should for example not wish to say: the
printed word "nothing" always reminds me of the sound "nothing"--but the spoken words as it were slip in as one
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reads. And if I so much as look at a German printed word, there occurs a peculiar process, that of hearing the sound
inwardly.
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166. I said that when one reads the spoken words come 'in a special way': but in what way? Isn't this a
fiction? Let us look at individual letters and attend to the way the sound of the letter comes. Read the letter A.--Now,
how did the sound come?--We have no idea what to say about it.--Now write a small Roman a.--How did the
movement of the hand come as you wrote? Differently from the way the sound came in the previous
experiment?--All I know is, I looked at the printed letter and wrote the cursive letter.--Now look at the mark
and let a sound occur to you as you do so; utter it. The sound 'U' occurred to me; but I could not say that
there was any essential difference in the kind of way that sound came. The difference lay in the difference of
situation. I had told myself beforehand that I was to let a sound occur to me; there was a certain tension present
before the sound came. And I did not say 'U' automatically as I do when I look at the letter U. Further, that mark was
not familiar to me in the way the letters of the alphabet are. I looked at it rather intently and with a certain interest in
its shape; as I looked I thought of a reversed σ--Imagine having to use this mark regularly as a letter; so that you got
used to uttering a particular sound at the sight of it, say the sound "sh". Can we say anything but that after a while
this sound comes automatically when we look at the mark? That is to say: I no longer ask myself on seeing it "What
sort of letter is that?"--nor, of course, do I tell myself "This mark makes me want to utter the sound 'sh'", nor yet
"This mark somehow reminds me of the sound 'sh'".
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(Compare with this the idea that memory images are distinguished from other mental images by some
special characteristic.)
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167. Now what is there in the proposition that reading is 'a quite particular process'? It presumably means
that when we read one particular process takes place, which we recognize.--But suppose that I at one time read a
sentence in print and at another write it in Morse code--is the mental process really the same?--On the other hand,
however, there is certainly some uniformity in the experience of reading a page of print. For the process is a uniform
one. And it is quite easy to understand that there is a difference between this process and one of, say, letting words
occur to one at the sight of arbitrary marks.--For the mere look of a printed line is itself extremely
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characteristic--it presents, that is, a quite special appearance, the letters all roughly the same size, akin in shape too,
and always recurring; most of the words constantly repeated and enormously familiar to us, like well-known
faces.--Think of the uneasiness we feel when the spelling of a word is changed. (And of the still stronger feelings that
questions about the spelling of words have aroused.) Of course, not all signs have impressed themselves on us so
strongly. A sign in the algebra of logic for instance can be replaced by any other one without exciting a strong
reaction in us.--
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Remember that the look of a word is familiar to us in the same kind of way as its sound.
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168. Again, our eye passes over printed lines differently from the way it passes over arbitrary pothooks and
flourishes. (I am not speaking here of what can be established by observing the movement of the eyes of a reader.)
The eye passes, one would like to say, with particular ease, without being held up; and yet it doesn't skid. And at the
same time involuntary speech goes on in the imagination. That is how it is when I read German and other languages,
printed or written, and in various styles.--But what in all this is essential to reading as such? Not any one feature that
occurs in all cases of reading. (Compare reading ordinary print with reading words which are printed entirely in
capital letters, as solutions of puzzles sometimes are. How different it is!--Or reading our script from right to left.)
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169. But when we read don't we feel the word-shapes somehow causing our utterance?--Read a
sentence.--And now look along the following line
&8§≠ §≠?ß +% 8!'§*
and say a sentence as you do so. Can't one feel that in the first case the utterance was connected with seeing the
signs and in the second went on side by side with the seeing without any connexion?
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But why do you say that we felt a causal connexion? Causation is surely something established by
experiments, by observing a regular concomitance of events for example. So how could I say that I felt something
which is established by experiment? (It is indeed true that observation of regular concomitances is not the only way
we establish causation.) One might rather say, I feel that the letters are the reason why I read such-and-such. For if
someone asks me "Why
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do you read such-and-such?"--I justify my reading by the letters which are there.
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This justification, however, was something that I said, or thought: what does it mean to say that I feel it? I
should like to say: when I read I feel a kind of influence of the letters working on me--but I feel no influence from
that series of arbitrary flourishes on what I say.--Let us once more compare an individual letter with such a flourish.
Should I also say I feel the influence of "i" when I read it? It does of course make a difference whether I say "i" when
I see "i" or when I see "§". The difference is, for instance, that when I see the letter it is automatic for me to hear the
sound "i" inwardly, it happens even against my will; and I pronounce the letter more effortlessly when I read it than
when I am looking at "§". That is to say: this is how it is when I make the experiment; but of course it is not so if I
happen to be looking at the mark "§" and at the same time pronounce a word in which the sound "i" occurs.
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170. It would never have occurred to us to think that we felt the influence of the letters on us when reading, if
we had not compared the case of letters with that of arbitrary marks. And here we are indeed noticing a difference.
And we interpret it as the difference between being influenced and not being influenced.
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In particular, this interpretation appeals to us especially when we make a point of reading slowly--perhaps in
order to see what does happen when we read. When we, so to speak, quite intentionally let ourselves be guided by
the letters. But this 'letting myself be guided' in turn only consists in my looking carefully at the letters--and perhaps
excluding certain other thoughts.
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We imagine that a feeling enables us to perceive as it were a connecting mechanism between the look of the
word and the sound that we utter. For when I speak of the experiences of being influenced, of causal connexion, of
being guided, that is really meant to imply that I as it were feel the movement of the lever which connects seeing the
letters with speaking.
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171. I might have used other words to hit off the experience I have when I read a word. Thus I might say that
the written word intimates the sound to me.--Or again, that when one reads, letter and sound form a unity--as it
were an alloy. (In the same way e.g. the faces of famous men and the sound of their names are fused together. This
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name strikes me as the only right one for this face.) When I feel this unity, I might say, I see or hear the sound in the
written word.--
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But now just read a few sentences in print as you usually do when you are not thinking about the concept of
reading; and ask yourself whether you had such experiences of unity, of being influenced and the rest, as you
read.--Don't say you had them unconsciously! Nor should we be misled by the picture which suggests that these
phenomena came in sight 'on closer inspection'. If I am supposed to describe how an object looks from far off, I
don't make the description more accurate by saying what can be noticed about the object on closer inspection.
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172. Let us consider the experience of being guided, and ask ourselves: what does this experience consist in
when for instance our course is guided?--Imagine the following cases:
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You are in a playing field with your eyes bandaged, and someone leads you by the hand, sometimes left,
sometimes right; you have constantly to be ready for the tug of his hand, and must also take care not to stumble
when he gives an unexpected tug.
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Or again: someone leads you by the hand where you are unwilling to go, by force.
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Or: you are guided by a partner in a dance; you make yourself as receptive as possible, in order to guess his
intention and obey the slightest pressure.
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Or: someone takes you for a walk; you are having a conversation; you go wherever he does.
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Or: you walk along a field-track, simply following it.
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All these situations are similar to one another; but what is common to all the experiences?
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173. "But being guided is surely a particular experience!"--The answer to this is: you are now thinking of a
particular experience of being guided.
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If I want to realize the experience of the person in one of the earlier examples, whose writing is guided by the
printed text and the table, I imagine 'conscientious' looking-up, and so on. As I do this I assume a particular
expression of face (say that of a conscientious bookkeeper). Carefulness is a most essential part of this picture; in
another the exclusion of every volition of one's own would be essential. (But take something normal people do quite
unconcernedly and imagine someone accompanying it with the expression--and why not the
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feelings?--of great carefulness.--Does that mean he is careful? Imagine a servant dropping the tea-tray and
everything on it with all the outward signs of carefulness.) If I imagine such a particular experience, it seems to me to
be the experience of being guided (or of reading). But now I ask myself: what are you doing?--You are looking at
every letter, you are making this face, you are writing the letters with deliberation (and so on).--So that is the
experience of being guided?--Here I should like to say: "No, it isn't that; it is something more inward, more
essential."--It is as if at first all these more or less inessential processes were shrouded in a particular atmosphere,
which dissipates when I look closely at them.
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174. Ask yourself how you draw a line parallel to a given one 'with deliberation'--and another time, with
deliberation, one at an angle to it. What is the experience of deliberation? Here a particular look, a gesture, at once
occur to you--and then you would like to say: "And it just is a particular inner experience". (And that is, of course,
to add nothing).
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(This is connected with the problem of the nature of intention, of willing.)
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175. Make some arbitrary doodle on a bit of paper.--And now make a copy next to it, let yourself be guided
by it.--I should like to say: "Sure enough, I was guided here. But as for what was characteristic in what happened--if
I say what happened, I no longer find it characteristic."
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But now notice this: while I am being guided everything is quite simple, I notice nothing special; but
afterwards, when I ask myself what it was that happened, it seems to have been something indescribable. Afterwards
no description satisfies me. It's as if I couldn't believe that I merely looked, made such-and-such a face, and drew a
line.--But don't I remember anything else? No; and yet I feel as if there must have been something else; in particular
when I say "guidance", "influence", and other such words to myself. "For surely," I tell myself, "I was being
guided."--Only then does the idea of that ethereal, intangible influence arise.
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176. When I look back on the experience I have the feeling that what is essential about it is an 'experience of
being influenced', of a connexion--as opposed to any mere simultaneity of phenomena: but at the same time I
should not be willing to call any experienced phenomenon the "experience of being influenced". (This contains the
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germ of the idea that the will is not a phenomenon.) I should like to say that I had experienced the 'because', and yet
I do not want to call any phenomenon the "experience of the because".
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177. I should like to say: "I experience the because". Not because I remember such an experience, but
because when I reflect on what I experience in such a case I look at it through the medium of the concept 'because'
(or 'influence' or 'cause' or 'connexion').--For of course it is correct to say I drew the line under the influence of the
original: this, however, does not consist simply in my feelings as I drew the line--under certain circumstances, it may
consist in my drawing it parallel to the other--even though this in turn is not in general essential to being guided.--
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178. We also say: "You can see that I am guided by it"--and what do you see, if you see this?
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When I say to myself: "But I am guided"--I make perhaps a movement with my hand, which expresses
guiding.--Make such a movement of the hand as if you were guiding someone along, and then ask yourself what the
guiding character of this movement consisted in. For you were not guiding anyone. But you still want to call the
movement one of 'guiding'. This movement and feeling did not contain the essence of guiding, but still this word
forces itself upon you. It is just a single form of guiding which forces the expression on us.
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179. Let us return to our case (151). It is clear that we should not say B had the right to say the words "Now I
know how to go on", just because he thought of the formula--unless experience shewed that there was a connexion
between thinking of the formula--saying it, writing it down--and actually continuing the series. And obviously such a
connexion does exist.--And now one might think that the sentence "I can go on" meant "I have an experience which
I know empirically to lead to the continuation of the series." But does B mean that when he says he can go on? Does
that sentence come to his mind, or is he ready to produce it in explanation of what he meant?
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No. The words "Now I know how to go on" were correctly used when he thought of the formula: that is,
given such circumstances as that he had learnt algebra, had used such formulae before.--But that does not mean that
his statement is only short for a description of all the circumstances which constitute the scene for our
language-game.--Think how we learn to use the expressions "Now I know how to go
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on", "Now I can go on" and others; in what family of language-games we learn their use.
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We can also imagine the case where nothing at all occurred in B's mind except that he suddenly said "Now I
know how to go on"--perhaps with a feeling of relief; and that he did in fact go on working out the series without
using the formula. And in this case too we should say--in certain circumstances--that he did know how to go on.
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180. This is how these words are used. It would be quite misleading, in this last case, for instance, to call the
words a "description of a mental state".--One might rather call them a "signal"; and we judge whether it was rightly
employed by what he goes on to do.
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181. In order to understand this, we need also to consider the following: suppose B says he knows how to go
on--but when he wants to go on he hesitates and can't do it: are we to say that he was wrong when he said he could
go on, or rather that he was able to go on then, only now is not?--Clearly we shall say different things in different
cases. (Consider both kinds of case.)
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182. The grammar of "to fit", "to be able", and "to understand". (Exercises: (1) When is a cylinder C said to
fit into a hollow cylinder H? Only while C is stuck into H? (2) Sometimes we say that C ceased to fit into H at
such-and-such a time. What criteria are used in such a case for its having happened at that time? (3) What does one
regard as criteria for a body's having changed its weight at a particular time if it was not actually on the balance at
that time? (4) Yesterday I knew the poem by heart; today I no longer know it. In what kind of case does it make
sense to ask: "When did I stop knowing it?" (5) Someone asks me "Can you lift this weight?" I answer "Yes". Now
he says "Do it!"--and I can't. In what kind of circumstances would it count as a justification to say "When I
answered 'yes' I could do it, only now I can't"?
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The criteria which we accept for 'fitting', 'being able to', 'understanding', are much more complicated than
might appear at first sight. That is, the game with these words, their employment in the linguistic intercourse that is
carried on by their means, is more involved--the role of these words in our language other--than we are tempted to
think.
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(This role is what we need to understand in order to resolve philosophical paradoxes. And hence definitions
usually fail to
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resolve them; and so, a fortiori does the assertion that a word is 'indefinable'.)
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183. But did "Now I can go on" in case (151) mean the same as "Now the formula has occurred to me" or
something different? We may say that, in those circumstances, the two sentences have the same sense, achieve the
same thing. But also that in general these two sentences do not have the same sense. We do say: "Now I can go on,
I mean I know the formula", as we say "I can walk, I mean I have time"; but also "I can walk, I mean I am already
strong enough"; or: "I can walk, as far as the state of my legs is concerned", that is, when we are contrasting this
condition for walking with others. But here we must be on our guard against thinking that there is some totality of
conditions corresponding to the nature of each case (e.g. for a person's walking) so that, as it were, he could not but
walk if they were all fulfilled.
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184. I want to remember a tune and it escapes me; suddenly I say "Now I know it" and I sing it. What was it
like to suddenly know it? Surely it can't have occurred to me in its entirety in that moment!--Perhaps you will say:
"It's a particular feeling, as if it were there"--but is it there? Suppose I now begin to sing it and get stuck?--But may I
not have been certain at that moment that I knew it? So in some sense or other it was there after all!--But in what
sense? You would say that the tune was there, if, say, someone sang it through, or heard it mentally from beginning
to end. I am not, of course, denying that the statement that the tune is there can also be given a quite different
meaning--for example, that I have a bit of paper on which it is written.--And what does his being 'certain', his
knowing it, consist in?--Of course we can say: if someone says with conviction that now he knows the tune, then it
is (somehow) present to his mind in its entirety at that moment--and this is a definition of the expression "the tune is
present to his mind in its entirety".
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185. Let us return to our example (143). Now--judged by the usual criteria--the pupil has mastered the series
of natural numbers. Next we teach him to write down other series of cardinal numbers and get him to the point of
writing down series of the form
0, n, 2n, 3n, etc.
at an order of the form "+ n"; so at the order "+ 1" he writes
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down the series of natural numbers.--Let us suppose we have done exercises and given him tests up to 1000.
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Now we get the pupil to continue a series (say + 2) beyond 1000--and he writes 1000, 1004, 1008, 1012.
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We say to him: "Look what you've done!"--He doesn't understand. We say: "You were meant to add two:
look how you began the series!"--He answers: "Yes, isn't it right? I thought that was how I was meant to do it."--Or
suppose he pointed to the series and said: "But I went on in the same way."--It would now be no use to say: "But
can't you see....?"--and repeat the old examples and explanations.--In such a case we might say, perhaps: It comes
natural to this person to understand our order with our explanations as we should understand the order: "Add 2 up
to 1000, 4 up to 2000, 6 up to 3000 and so on."
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Such a case would present similarities with one in which a person naturally reacted to the gesture of pointing
with the hand by looking in the direction of the line from finger-tip to wrist, not from wrist to finger-tip.
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186. "What you are saying, then, comes to this: a new insight--intuition--is needed at every step to carry out
the order '+ n' correctly."--To carry it out correctly! How is it decided what is the right step to take at any particular
stage?--"The right step is the one that accords with the order--as it was meant."--So when you gave the order + 2 you
meant that he was to write 1002 after 1000--and did you also mean that he should write 1868 after 1866, and 100036
after 100034, and so on--an infinite number of such propositions?--"No: what I meant was, that he should write the
next but one number after every number that he wrote; and from this all those propositions follow in turn."--But that
is just what is in question: what, at any stage, does follow from that sentence. Or, again, what, at any stage we are to
call "being in accord" with that sentence (and with the mean-ing you then put into the sentence--whatever that may
have consisted in). It would almost be more correct to say, not that an intuition was needed at every stage, but that a
new decision was needed at every stage.
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187. "But I already knew, at the time when I gave the order, that he ought to write 1002 after
1000."--Certainly; and you can also say you meant it then; only you should not let yourself be misled by the
grammar of the words "know" and "mean". For you don't want
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to say that you thought of the step from 1000 to 1002 at that time--and even if you did think of this step, still you did
not think of other ones. When you said "I already knew at the time....." that meant something like: "If I had then
been asked what number should be written after 1000, I should have replied '1002'." And that I don't doubt. This
assumption is rather of the same kind as: "If he had fallen into the water then, I should have jumped in after
him".--Now, what was wrong with your idea?
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188. Here I should first of all like to say: your idea was that that act of meaning the order had in its own way
already traversed all those steps: that when you meant it your mind as it were flew ahead and took all the steps
before you physically arrived at this or that one.
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Thus you were inclined to use such expressions as: "The steps are really already taken, even before I take
them in writing or orally or in thought." And it seemed as if they were in some unique way predetermined,
anticipated--as only the act of meaning can anticipate reality.
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189. "But are the steps then not determined by the algebraic formula?"--The question contains a mistake.
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We use the expression: "The steps are determined by the formula.....". How is it used?--We may perhaps
refer to the fact that people are brought by their education (training) so to use the formula y = x², that they all work
out the same value for y when they substitute the same number for x. Or we may say: "These people are so trained
that they all take the same step at the same point when they receive the order 'add 3'". We might express this by
saying: for these people the order "add 3" completely determines every step from one number to the next. (In
contrast with other people who do not know what they are to do on receiving this order, or who react to it with
perfect certainty, but each one in a different way.)
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On the other hand we can contrast different kinds of formula, and the different kinds of use (different kinds
of training) appropriate to them. Then we call formulae of a particular kind (with the appropriate methods of use)
"formulae which determine a number y for a given value of x", and formulae of another kind, ones which "do not
determine the number y for a given value of x". (y = x² would be of the first kind, y ≠ x²> of the second.) The
proposition "The formula.... determines a number y" will then be a statement about
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the form of the formula--and now we must distinguish such a proposition as "The formula which I have written
down determines y", or "Here is a formula which determines y", from one of the following kind: "The formula y = x2
determines the number y for a given value of x". The question "Is the formula written down there one that
determines y?" will then mean the same as "Is what is there a formula of this kind or that?"--but it is not clear
off-hand what we are to make of the question "Is y = x2 a formula which determines y for a given value of x?" One
might address this question to a pupil in order to test whether he understands the use of the word "to determine"; or
it might be a mathematical problem to prove in a particular system that x has only one square.
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190. It may now be said: "The way the formula is meant determines which steps are to be taken". What is the
criterion for the way the formula is meant? It is, for example, the kind of way we always use it, the way we are
taught to use it.
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We say, for instance, to someone who uses a sign unknown to us: "If by 'x!2' you mean x2, then you get this
value for y, if you mean 2x, that one."--Now ask yourself: how does one mean the one thing or the other by "x!2"?
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That will be how meaning it can determine the steps in advance.
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191. "It is as if we could grasp the whole use of the word in a flash." Like what e.g.?--Can't the use--in a
certain sense--be grasped in a flash? And in what sense can it not?--The point is, that it is as if we could 'grasp it in a
flash' in yet another and much more direct sense than that.--But have you a model for this? No. It is just that this
expression suggests itself to us. As the result of the crossing of different pictures.
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192. You have no model of this superlative fact, but you are seduced into using a super-expression. (It might
be called a philosophical superlative.)
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193. The machine as symbolizing its action: the action of a machine--I might say at first--seems to be there in
it from the start. What does that mean?--If we know the machine, everything else, that is its movement, seems to be
already completely determined.
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We talk as if these parts could only move in this way, as if they could not do anything else. How is this--do
we forget the possibility of their bending, breaking off, melting, and so on? Yes; in many cases
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we don't think of that at all. We use a machine, or the drawing of a machine, to symbolize a particular action of the
machine. For instance, we give someone such a drawing and assume that he will derive the movement of the parts
from it. (Just as we can give someone a number by telling him that it is the twenty-fifth in the series 1, 4, 9, 16, ....)
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"The machine's action seems to be in it from the start" means: we are inclined to compare the future
movements of the machine in their definiteness to objects which are already lying in a drawer and which we then
take out.--But we do not say this kind of thing when we are concerned with predicting the actual behaviour of a
machine. Then we do not in general forget the possibility of a distortion of the parts and so on.--We do talk like that,
however, when we are wondering at the way we can use a machine to symbolize a given way of moving--since it can
also move in quite different ways.
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We might say that a machine, or the picture of it, is the first of a series of pictures which we have learnt to
derive from this one.
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But when we reflect that the machine could also have moved differently it may look as if the way it moves
must be contained in the machine-as-symbol far more determinately than in the actual machine. As if it were not
enough for the movements in question to be empirically determined in advance, but they had to be really--in a
mysterious sense--already present. And it is quite true: the movement of the machine-as-symbol is predetermined in
a different sense from that in which the movement of any given actual machine is predetermined.
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194. When does one have the thought: the possible movements of a machine are already there in it in some
mysterious way?--Well, when one is doing philosophy. And what leads us into thinking that? The kind of way in
which we talk about machines. We say, for example, that a machine has (possesses) such-and-such possibilities of
movement; we speak of the ideally rigid machine which can only move in such-and-such a way.--What is this
possibility of movement? It is not the movement, but it does not seem to be the mere physical conditions for moving
either--as, that there is play between socket and pin, the pin not fitting too tight in the socket. For while this is the
empirical condition for movement, one could also imagine it to be otherwise. The possibility of a movement is,
rather, supposed to be like a shadow of the movement itself. But do you know of such a shadow? And by a shadow
I do not mean some picture of the movement--for such a
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picture would not have to be a picture of just this movement. But the possibility of this movement must be the
possibility of just this movement. (See how high the seas of language run here!)
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The waves subside as soon as we ask ourselves: how do we use the phrase "possibility of movement" when
we are talking about a given machine?--But then where did our queer ideas come from? Well, I shew you the
possibility of a movement, say by means of a picture of the movement: 'so possibility is something which is like
reality'. We say: "It isn't moving yet, but it already has the possibility of moving"--'so possibility is something very
near reality'. Though we may doubt whether such-and-such physical conditions make this movement possible, we
never discuss whether this is the possibility of this or of that movement: 'so the possibility of the movement stands
in a unique relation to the movement itself; closer than that of a picture to its subject'; for it can be doubted whether
a picture is the picture of this thing or that. We say "Experience will shew whether this gives the pin this possibility
of movement", but we do not say "Experience will shew whether this is the possibility of this movement": 'so it is
not an empirical fact that this possibility is the possibility of precisely this movement'.
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We mind about the kind of expressions we use concerning these things; we do not understand them,
however, but misinterpret them. When we do philosophy we are like savages, primitive people, who hear the
expressions of civilized men, put a false interpretation on them, and then draw the queerest conclusions from it.
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195. "But I don't mean that what I do now (in grasping a sense) determines the future use causally and as a
matter of experience, but that in a queer way, the use itself is in some sense present."--But of course it is, 'in some
sense'! Really the only thing wrong with what you say is the expression "in a queer way". The rest is all right; and
the sentence only seems queer when one imagines a different language-game for it from the one in which we
actually use it. (Someone once told me that as a child he had been surprised that a tailor could 'sew a dress'--he
thought this meant that a dress was produced by sewing alone, by sewing one thread on to another.)
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196. In our failure to understand the use of a word we take it as the expression of a queer process. (As we
think of time as a queer medium, of the mind as a queer kind of being.)
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197. "It's as if we could grasp the whole use of a word in a flash."--And that is just what we say we do. That
is to say: we sometimes describe what we do in these words. But there is nothing astonishing, nothing queer, about
what happens. It becomes queer when we are led to think that the future development must in some way already be
present in the act of grasping the use and yet isn't present.--For we say that there isn't any doubt that we understand
the word, and on the other hand its meaning lies in its use. There is no doubt that I now want to play chess, but
chess is the game it is in virtue of all its rules (and so on). Don't I know, then, which game I want to play until I have
played it? or are all the rules contained in my act of intending? Is it experience that tells me that this sort of game is
the usual consequence of such an act of intending? so is it impossible for me to be certain what I am intending to
do? And if that is nonsense--what kind of super-strong connexion exists between the act of intending and the thing
intended?--Where is the connexion effected between the sense of the expression "Let's play a game of chess" and all
the rules of the game?--Well, in the list of rules of the game, in the teaching of it, in the day-to-day practice of
playing.
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198. "But how can a rule shew me what I have to do at this point? Whatever I do is, on some interpretation,
in accord with the rule."--That is not what we ought to say, but rather: any interpretation still hangs in the air along
with what it interprets, and cannot give it any support. Interpretations by themselves do not determine meaning.
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"Then can whatever I do be brought into accord with the rule?"--Let me ask this: what has the expression of
a rule--say a sign-post--got to do with my actions? What sort of connexion is there here?--Well, perhaps this one: I
have been trained to react to this sign in a particular way, and now I do so react to it.
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But that is only to give a causal connexion; to tell how it has come about that we now go by the sign-post;
not what this going-by-the-sign really consists in. On the contrary; I have further indicated that a person goes by a
sign-post only in so far as there exists a regular use of sign-posts, a custom.
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199. Is what we call "obeying a rule" something that it would be possible for only one man to do, and to do
only once in his life?--This is of course a note on the grammar of the expression "to obey a rule".
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It is not possible that there should have been only one occasion on which someone obeyed a rule. It is not
possible that there should have been only one occasion on which a report was made, an order given or understood;
and so on.--To obey a rule, to make a report, to give an order, to play a game of chess, are customs (uses,
institutions).
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To understand a sentence means to understand a language. To understand a language means to be master of
a technique.
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200. It is, of course, imaginable that two people belonging to a tribe unacquainted with games should sit at a
chess-board and go through the moves of a game of chess; and even with all the appropriate mental
accompaniments. And if we were to see it we should say they were playing chess. But now imagine a game of chess
translated according to certain rules into a series of actions which we do not ordinarily associate with a game--say
into yells and stamping of feet. And now suppose those two people to yell and stamp instead of playing the form of
chess that we are used to; and this in such a way that their procedure is translatable by suitable rules into a game of
chess. Should we still be inclined to say they were playing a game? What right would one have to say so?
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201. This was our paradox: no course of action could be determined by a rule, because every course of action
can be made out to accord with the rule. The answer was: if everything can be made out to accord with the rule, then
it can also be made out to conflict with it. And so there would be neither accord nor conflict here.
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It can be seen that there is a misunderstanding here from the mere fact that in the course of our argument we
give one interpretation after another; as if each one contented us at least for a moment, until we thought of yet
another standing behind it. What this shews is that there is a way of grasping a rule which is not an interpretation,
but which is exhibited in what we call "obeying the rule" and "going against it" in actual cases.
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Hence there is an inclination to say: every action according to the rule is an interpretation. But we ought to
restrict the term "interpretation" to the substitution of one expression of the rule for another.
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202. And hence also 'obeying a rule' is a practice. And to think one is obeying a rule is not to obey a rule.
Hence it is not possible to obey a rule 'privately': otherwise thinking one was obeying a rule would be the same thing
as obeying it.
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203. Language is a labyrinth of paths. You approach from one side and know your way about; you approach
the same place from another side and no longer know your way about.
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204. As things are I can, for example, invent a game that is never played by anyone.--But would the
following be possible too: mankind has never played any games; once, however, someone invented a game--which
no one ever played?
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205. "But it is just the queer thing about intention, about the mental process, that the existence of a custom,
of a technique, is not necessary to it. That, for example, it is imaginable that two people should play chess in a world
in which otherwise no games existed; and even that they should begin a game of chess--and then be interrupted."
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But isn't chess defined by its rules? And how are these rules present in the mind of the person who is
intending to play chess?
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206. Following a rule is analogous to obeying an order. We are trained to do so; we react to an order in a
particular way. But what if one person reacts in one way and another in another to the order and the training? Which
one is right?
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Suppose you came as an explorer into an unknown country with a language quite strange to you. In what
circumstances would you say that the people there gave orders, understood them, obeyed them, rebelled against
them, and so on?
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The common behaviour of mankind is the system of reference by means of which we interpret an unknown
language.
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207. Let us imagine that the people in that country carried on the usual human activities and in the course of
them employed, apparently, an articulate language. If we watch their behaviour we find it intelligible, it seems
'logical'. But when we try to learn their language we find it impossible to do so. For there is no regular connexion
between what they say, the sounds they make, and their actions; but still these sounds are not superfluous, for if we
gag one of the people, it has the same consequences as with us; without the sounds their actions fall into
confusion--as I feel like putting it.
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Are we to say that these people have a language: orders, reports, and the rest?
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There is not enough regularity for us to call it "language".
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208. Then am I defining "order" and "rule" by means of "regularity"?--How do I explain the meaning of
"regular", "uniform",
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"same" to anyone?--I shall explain these words to someone who, say, only speaks French by means of the
corresponding French words. But if a person has not yet got the concepts, I shall teach him to use the words by
means of examples and by practice.--And when I do this I do not communicate less to him than I know myself.
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In the course of this teaching I shall shew him the same colours, the same lengths, the same shapes, I shall
make him find them and produce them, and so on. I shall, for instance, get him to continue an ornamental pattern
uniformly when told to do so.--And also to continue progressions. And so, for example, when given: . .. ... to go on:
.... ..... ...... .
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I do it, he does it after me; and I influence him by expressions of agreement, rejection, expectation,
encouragement. I let him go his way, or hold him back; and so on.
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Imagine witnessing such teaching. None of the words would be explained by means of itself; there would be
no logical circle.
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The expressions "and so on", "and so on ad infinitum" are also explained in this teaching. A gesture, among
other things, might serve this purpose. The gesture that means "go on like this", or "and so on" has a function
comparable to that of pointing to an object or a place.
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We should distinguish between the "and so on" which is, and the "and so on" which is not, an abbreviated
notation. "And so on ad inf." is not such an abbreviation. The fact that we cannot write down all the digits of π is not
a human shortcoming, as mathematicians sometimes think.
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Teaching which is not meant to apply to anything but the examples given is different from that which 'points
beyond' them.
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209. "But then doesn't our understanding reach beyond all the examples?"--A very queer expression, and a
quite natural one!--
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But is that all? Isn't there a deeper explanation; or mustn't at least the understanding of the explanation be
deeper?--Well, have I myself a deeper understanding? Have I got more than I give in the explanation?--But then,
whence the feeling that I have got more?
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Is it like the case where I interpret what is not limited as a length that reaches beyond every length?
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210. "But do you really explain to the other person what you yourself understand? Don't you get him to
guess the essential thing? You give him examples,--but he has to guess their drift, to guess your
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intention."--Every explanation which I can give myself I give to him too.--"He guesses what I intend" would mean:
various interpretations of my explanation come to his mind, and he lights on one of them. So in this case he could
ask; and I could and should answer him.
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211. How can he know how he is to continue a pattern by himself--whatever instruction you give him?--Well,
how do I know?--If that means "Have I reasons?" the answer is: my reasons will soon give out. And then I shall act,
without reasons.
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212. When someone whom I am afraid of orders me to continue the series, I act quickly, with perfect
certainty, and the lack of reasons does not trouble me.
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213. "But this initial segment of a series obviously admitted of various interpretations (e.g. by means of
algebraic expressions) and so you must first have chosen one such interpretation."--Not at all. A doubt was possible
in certain circumstances. But that is not to say that I did doubt, or even could doubt. (There is something to be said,
which is connected with this, about the psychological 'atmosphere' of a process.)
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So it must have been intuition that removed this doubt?--If intuition is an inner voice--how do I know how I
am to obey it? And how do I know that it doesn't mislead me? For if it can guide me right, it can also guide me
wrong.
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((Intuition an unnecessary shuffle.))
Page 84
214. If you have to have an intuition in order to develop the series 1 2 3 4... you must also have one in order
to develop the series 2 2 2 2....
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215. But isn't the same at least the same?
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We seem to have an infallible paradigm of identity in the identity of a thing with itself. I feel like saying:
"Here at any rate there can't be a variety of interpretations. If you are seeing a thing you are seeing identity too."
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Then are two things the same when they are what one thing is? And how am I to apply what the one thing
shews me to the case of two things?
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216. "A thing is identical with itself."--There is no finer example of a useless proposition, which yet is
connected with a certain play of the imagination. It is as if in imagination we put a thing into its own shape and saw
that it fitted.
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We might also say: "Every thing fits into itself." Or again: "Every thing fits into its own shape." At the same
time we look at a thing and imagine that there was a blank left for it, and that now it fits into it exactly.
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Does this spot 'fit' into its white surrounding?--But that is just how it would look if there had at first
been a hole in its place and it then fitted into the hole. But when we say "it fits" we are not simply describing this
appearance; not simply this situation.
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"Every coloured patch fits exactly into its surrounding" is a rather specialized form of the law of identity.
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217. "How am I able to obey a rule?"--if this is not a question about causes, then it is about the justification
for my following the rule in the way I do.
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If I have exhausted the justifications I have reached bedrock, and my spade is turned. Then I am inclined to
say: "This is simply what I do."
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(Remember that we sometimes demand definitions for the sake not of their content, but of their form. Our
requirement is an architectural one; the definition a kind of ornamental coping that supports nothing.)
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218. Whence comes the idea that the beginning of a series is a visible section of rails invisibly laid to infinity?
Well, we might imagine rails instead of a rule. And infinitely long rails correspond to the unlimited application of a
rule.
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219. "All the steps are really already taken" means: I no longer have any choice. The rule, once stamped with
a particular meaning, traces the lines along which it is to be followed through the whole of space.--But if something
of this sort really were the case, how would it help?
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No; my description only made sense if it was to be understood symbolically.--I should have said: This is
how it strikes me.
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When I obey a rule, I do not choose.
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I obey the rule blindly.
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220. But what is the purpose of that symbolical proposition? It was supposed to bring into prominence a
difference between being causally determined and being logically determined.
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221. My symbolical expression was really a mythological description of the use of a rule.
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222. "The line intimates to me the way I am to go."--But that is of course only a picture. And if I judged that
it intimated this or that as it were irresponsibly, I should not say that I was obeying it like a rule.
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223. One does not feel that one has always got to wait upon the nod (the whisper) of the rule. On the
contrary, we are not on tenterhooks about what it will tell us next, but it always tells us the same, and we do what it
tells us.
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One might say to the person one was training: "Look, I always do the same thing: I....."
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224. The word "agreement" and the word "rule" are related to one another, they are cousins. If I teach
anyone the use of the one word, he learns the use of the other with it.
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225. The use of the word "rule" and the use of the word "same" are interwoven. (As are the use of
"proposition" and the use of "true".)
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226. Suppose someone gets the series of numbers 1, 3, 5, 7,.... by working out the series 2x + 1†1. And now
he asks himself: "But am I always doing the same thing, or something different every time?"
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If from one day to the next you promise: "To-morrow I will come and see you"--are you saying the same
thing every day, or every day something different?
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227. Would it make sense to say "If he did something different every day we should not say he was obeying
a rule"? That makes no sense.
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228. "We see a series in just one way!"--All right, but what is that way? Clearly we see it algebraically, and as
a segment of an expansion. Or is there more in it than that?--"But the way we see it surely gives us everything!"--But
that is not an observation about the segment of the series; or about anything that we notice in it; it gives expression
to the fact that we look to the rule for instruction and do something, without appealing to anything else for guidance.
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229. I believe that I perceive something drawn very fine in a segment of a series, a characteristic design,
which only needs the addition of "and so on", in order to reach to infinity.
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230. "The line intimates to me which way I am to go" is only a paraphrase of: it is my last arbiter for the way
I am to go.
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231. "But surely you can see....?" That is just the characteristic expression of someone who is under the
compulsion of a rule.
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232. Let us imagine a rule intimating to me which way I am to obey it; that is, as my eye travels along the
line, a voice within me says: "This way!"--What is the difference between this process of obeying a kind of
inspiration and that of obeying a rule? For they are surely not the same. In the case of inspiration I await direction. I
shall not be able to teach anyone else my 'technique' of following the line. Unless, indeed, I teach him some way of
hearkening, some kind of receptivity. But then, of course, I cannot require him to follow the line in the same way as I
do.
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These are not my experiences of acting from inspiration and according to a rule; they are grammatical notes.
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233. It would also be possible to imagine such a training in a sort of arithmetic. Children could calculate, each
in his own way--as long as they listened to their inner voice and obeyed it. Calculating in this way would be like a
sort of composing.
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234. Would it not be possible for us, however, to calculate as we actually do (all agreeing, and so on), and
still at every step to have a feeling of being guided by the rules as by a spell, feeling astonishment at the fact that we
agreed? (We might give thanks to the Deity for our agreement.)
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235. This merely shews what goes to make up what we call "obeying a rule" in everyday life.
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236. Calculating prodigies who get the right answer but cannot say how. Are we to say that they do not
calculate? (A family of cases.)
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237. Imagine someone using a line as a rule in the following way: he holds a pair of compasses, and carries
one of its points along the line that is the 'rule', while the other one draws the line that follows the rule. And while he
moves along the ruling line he alters the opening of the compasses, apparently with great precision, looking at the
rule the whole time as if it determined what he did. And watching him we see no kind of regularity in this opening
and shutting of the compasses. We cannot learn his way of following the line from it. Here perhaps one really would
say: "The original seems to intimate to him which way he is to go. But it is not a rule."
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238. The rule can only seem to me to produce all its consequences in advance if I draw them as a matter of
course. As much as it is a matter
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of course for me to call this colour "blue". (Criteria for the fact that something is 'a matter of course' for me.)
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239. How is he to know what colour he is to pick out when he hears "red"?--Quite simple: he is to take the
colour whose image occurs to him when he hears the word.--But how is he to know which colour it is 'whose image
occurs to him'? Is a further criterion needed for that? (There is indeed such a procedure as choosing the colour which
occurs to one when one hears the word "....")
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"'Red' means the colour that occurs to me when I hear the word 'red'"--would be a definition. Not an
explanation of what it is to use a word as a name.
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240. Disputes do not break out (among mathematicians, say) over the question whether a rule has been
obeyed or not. People don't come to blows over it, for example. That is part of the framework on which the working
of our language is based (for example, in giving descriptions).
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241. "So you are saying that human agreement decides what is true and what is false?"--It is what human
beings say that is true and false; and they agree in the language they use. That is not agreement in opinions but in
form of life.
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242. If language is to be a means of communication there must be agreement not only in definitions but also
(queer as this may sound) in judgments. This seems to abolish logic, but does not do so.--It is one thing to describe
methods of measurement, and another to obtain and state results of measurement. But what we call "measuring" is
partly determined by a certain constancy in results of measurement.
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243. A human being can encourage himself, give himself orders, obey, blame and punish himself; he can ask
himself a question and answer it. We could even imagine human beings who spoke only in monologue; who
accompanied their activities by talking to themselves.--An explorer who watched them and listened to their talk
might succeed in translating their language into ours. (This would enable him to predict these people's actions
correctly, for he also hears them making resolutions and decisions.)
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But could we also imagine a language in which a person could write down or give vocal expression to his
inner experiences--his feelings, moods, and the rest--for his private use?--Well, can't we do so in our ordinary
language?--But that is not what I mean. The
Page Break 89
individual words of this language are to refer to what can only be known to the person speaking; to his immediate
private sensations. So another person cannot understand the language.
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244. How do words refer to sensations?--There doesn't seem to be any problem here; don't we talk about
sensations every day, and give them names? But how is the connexion between the name and the thing named set
up? This question is the same as: how does a human being learn the meaning of the names of sensations?--of the
word "pain" for example. Here is one possibility: words are connected with the primitive, the natural, expressions of
the sensation and used in their place. A child has hurt himself and he cries; and then adults talk to him and teach him
exclamations and, later, sentences. They teach the child new pain-behaviour.
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"So you are saying that the word 'pain' really means crying?"--On the contrary: the verbal expression of pain
replaces crying and does not describe it.
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245. For how can I go so far as to try to use language to get between pain and its expression?
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246. In what sense are my sensations private?--Well, only I can know whether I am really in pain; another
person can only surmise it.--In one way this is wrong, and in another nonsense. If we are using the word "to know"
as it is normally used (and how else are we to use it?), then other people very often know when I am in pain.--Yes,
but all the same not with the certainty with which I know it myself!--It can't be said of me at all (except perhaps as a
joke) that I know I am in pain. What is it supposed to mean--except perhaps that I am in pain?
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Other people cannot be said to learn of my sensations only from my behaviour,--for I cannot be said to learn
of them. I have them.
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The truth is: it makes sense to say about other people that they doubt whether I am in pain; but not to say it
about myself.
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247. "Only you can know if you had that intention." One might tell someone this when one was explaining
the meaning of the word "intention" to him. For then it means: that is how we use it.
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(And here "know" means that the expression of uncertainty is senseless.)
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248. The proposition "Sensations are private" is comparable to: "One plays patience by oneself".
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249. Are we perhaps over-hasty in our assumption that the smile of an unweaned infant is not a
pretence?--And on what experience is our assumption based?
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(Lying is a language-game that needs to be learned like any other one.)
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250. Why can't a dog simulate pain? Is he too honest? Could one teach a dog to simulate pain? Perhaps it is
possible to teach him to howl on particular occasions as if he were in pain, even when he is not. But the
surroundings which are necessary for this behaviour to be real simulation are missing.
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251. What does it mean when we say: "I can't imagine the opposite of this" or "What would it be like, if it
were otherwise?"--For example, when someone has said that my images are private, or that only I myself can know
whether I am feeling pain, and similar things.
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Of course, here "I can't imagine the opposite" doesn't mean: my powers of imagination are unequal to the
task. These words are a defence against something whose form makes it look like an empirical proposition, but
which is really a grammatical one.
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But why do we say: "I can't imagine the opposite"? Why not: "I can't imagine the thing itself"?
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Example: "Every rod has a length." That means something like: we call something (or this) "the length of a
rod"--but nothing "the length of a sphere." Now can I imagine 'every rod having a length'? Well, I simply imagine a
rod. Only this picture, in connexion with this proposition, has a quite different role from one used in connexion with
the proposition "This table has the same length as the one over there". For here I understand what it means to have a
picture of the opposite (nor need it be a mental picture).
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But the picture attaching to the grammatical proposition could only shew, say, what is called "the length of a
rod". And what should the opposite picture be?
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((Remark about the negation of an a priori proposition.))
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252. "This body has extension." To this we might reply: "Nonsense!"--but are inclined to reply "Of
course!"--Why is this?
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253. "Another person can't have my pains."--Which are my pains? What counts as a criterion of identity
here? Consider what makes it possible in the case of physical objects to speak of "two exactly the same", for
example, to say "This chair is not the one you saw here yesterday, but is exactly the same as it".
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In so far as it makes sense to say that my pain is the same as his, it is also possible for us both to have the
same pain. (And it would also be imaginable for two people to feel pain in the same--not just the
corresponding--place. That might be the case with Siamese twins, for instance.)
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I have seen a person in a discussion on this subject strike himself on the breast and say: "But surely another
person can't have THIS pain!"--The answer to this is that one does not define a criterion of identity by emphatic
stressing of the word "this". Rather, what the emphasis does is to suggest the case in which we are conversant with
such a criterion of identity, but have to be reminded of it.
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254. The substitution of "identical" for "the same" (for instance) is another typical expedient in philosophy.
As if we were talking about shades of meaning and all that were in question were to find words to hit on the correct
nuance. That is in question in philosophy only where we have to give a psychologically exact account of the
temptation to use a particular kind of expression. What we 'are tempted to say' in such a case is, of course, not
philosophy; but it is its raw material. Thus, for example, what a mathematician is inclined to say about the
objectivity and reality of mathematical facts, is not a philosophy of mathematics, but something for philosophical
treatment.
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255. The philosopher's treatment of a question is like the treatment of an illness.
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256. Now, what about the language which describes my inner experiences and which only I myself can
understand? How do I use words to stand for my sensations?--As we ordinarily do? Then are my words for
sensations tied up with my natural expressions of sensation? In that case my language is not a 'private' one.
Someone else might understand it as well as I.--But suppose I didn't have any natural expression for the sensation,
but only had the sensation? And now I simply associate names with sensations and use these names in
descriptions.--
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257. "What would it be like if human beings shewed no outward signs of pain (did not groan, grimace, etc.)?
Then it would be impossible to teach a child the use of the word 'tooth-ache'."--Well, let's assume the child is a
genius and itself invents a name for the sensation!--But then, of course, he couldn't make himself understood when
he used the word.--So does he understand the name, without being able to explain its meaning to anyone?--But
what does it mean to say that he has 'named his pain'?--How has he done this naming of pain?! And whatever he
did, what was its purpose?--When one says "He gave a name to his sensation" one forgets that a great deal of
stage-setting in the language is presupposed if the mere act of naming is to make sense. And when we speak of
someone's having given a name to pain, what is presupposed is the existence of the grammar of the word "pain"; it
shews the post where the new word is stationed.
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258. Let us imagine the following case. I want to keep a diary about the recurrence of a certain sensation. To
this end I associate it with the sign "S" and write this sign in a calendar for every day on which I have the
sensation.--I will remark first of all that a definition of the sign cannot be formulated.--But still I can give myself a
kind of ostensive definition.--How? Can I point to the sensation? Not in the ordinary sense. But I speak, or write the
sign down, and at the same time I concentrate my attention on the sensation--and so, as it were, point to it
inwardly.--But what is this ceremony for? for that is all it seems to be! A definition surely serves to establish the
meaning of a sign.--Well, that is done precisely by the concentrating of my attention; for in this way I impress on
myself the connexion between the sign and the sensation.--But "I impress it on myself" can only mean: this process
brings it about that I remember the connexion right in the future. But in the present case I have no criterion of
correctness. One would like to say: whatever is going to seem right to me is right. And that only means that here we
can't talk about 'right'.
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259. Are the rules of the private language impressions of rules?--The balance on which impressions are
weighed is not the impression of a balance.
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260. "Well, I believe that this is the sensation S again."--Perhaps you believe that you believe it!
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Then did the man who made the entry in the calendar make a note
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of nothing whatever?--Don't consider it a matter of course that a person is making a note of something when he
makes a mark--say in a calendar. For a note has a function, and this "S" so far has none.
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(One can talk to oneself.--If a person speaks when no one else is present, does that mean he is speaking to
himself?)
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261. What reason have we for calling "S" the sign for a sensation? For "sensation" is a word of our common
language, not of one intelligible to me alone. So the use of this word stands in need of a justification which
everybody understands.--And it would not help either to say that it need not be a sensation; that when he writes
"S", he has something--and that is all that can be said. "Has" and "something" also belong to our common
language.--So in the end when one is doing philosophy one gets to the point where one would like just to emit an
inarticulate sound.--But such a sound is an expression only as it occurs in a particular language-game, which should
now be described.
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262. It might be said: if you have given yourself a private definition of a word, then you must inwardly
undertake to use the word in such-and-such a way. And how do you undertake that? Is it to be assumed that you
invent the technique of using the word; or that you found it ready-made?
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263. "But I can (inwardly) undertake to call THIS 'pain' in the future."--"But is it certain that you have
undertaken it? Are you sure that it was enough for this purpose to concentrate your attention on your feeling?"--A
queer question.--
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264. "Once you know what the word stands for, you understand it, you know its whole use."
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265. Let us imagine a table (something like a dictionary) that exists only in our imagination. A dictionary can
be used to justify the translation of a word X by a word Y. But are we also to call it a justification if such a table is to
be looked up only in the imagination?--"Well, yes; then it is a subjective justification."--But justification consists in
appealing to something independent.--"But surely I can appeal from one memory to another. For example, I don't
know if I have remembered the time of departure of a train right and to check it I call to mind how a page of the
time-table looked. Isn't it the same here?"--No; for this process has got to produce a memory which is
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actually correct. If the mental image of the time-table could not itself be tested for correctness, how could it confirm
the correctness of the first memory? (As if someone were to buy several copies of the morning paper to assure
himself that what it said was true.)
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Looking up a table in the imagination is no more looking up a table than the image of the result of an
imagined experiment is the result of an experiment.
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266. I can look at the clock to see what time it is: but I can also look at the dial of a clock in order to guess
what time it is; or for the same purpose move the hand of a clock till its position strikes me as right. So the look of a
clock may serve to determine the time in more than one way. (Looking at the clock in imagination.)
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267. Suppose I wanted to justify the choice of dimensions for a bridge which I imagine to be building, by
making loading tests on the material of the bridge in my imagination. This would, of course, be to imagine what is
called justifying the choice of dimensions for a bridge. But should we also call it justifying an imagined choice of
dimensions?
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268. Why can't my right hand give my left hand money?--My right hand can put it into my left hand. My
right hand can write a deed of gift and my left hand a receipt.--But the further practical consequences would not be
those of a gift. When the left hand has taken the money from the right, etc., we shall ask: "Well, and what of it?"
And the same could be asked if a person had given himself a private definition of a word; I mean, if he has said the
word to himself and at the same time has directed his attention to a sensation.
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269. Let us remember that there are certain criteria in a man's behaviour for the fact that he does not
understand a word: that it means nothing to him, that he can do nothing with it. And criteria for his 'thinking he
understands', attaching some meaning to the word, but not the right one. And, lastly, criteria for his understanding
the word right. In the second case one might speak of a subjective understanding. And sounds which no one else
understands but which I 'appear to understand' might be called a "private language".
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270. Let us now imagine a use for the entry of the sign "S" in my diary. I discover that whenever I have a
particular sensation a manometer
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shews that my blood-pressure rises. So I shall be able to say that my blood-pressure is rising without using any
apparatus. This is a useful result. And now it seems quite indifferent whether I have recognized the sensation right
or not. Let us suppose I regularly identify it wrong, it does not matter in the least. And that alone shews that the
hypothesis that I make a mistake is mere show. (We as it were turned a knob which looked as if it could be used to
turn on some part of the machine; but it was a mere ornament, not connected with the mechanism at all.)
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And what is our reason for calling "S" the name of a sensation here? Perhaps the kind of way this sign is
employed in this language-game.--And why a "particular sensation," that is, the same one every time? Well, aren't
we supposing that we write "S" every time?
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271. "Imagine a person whose memory could not retain what the word 'pain' meant--so that he constantly
called different things by that name--but nevertheless used the word in a way fitting in with the usual symptoms and
presuppositions of pain"--in short he uses it as we all do. Here I should like to say: a wheel that can be turned though
nothing else moves with it, is not part of the mechanism.
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272. The essential thing about private experience is really not that each person possesses his own exemplar,
but that nobody knows whether other people also have this or something else. The assumption would thus be
possible--though unverifiable--that one section of mankind had one sensation of red and another section another.
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273. What am I to say about the word "red"?--that it means something 'confronting us all' and that everyone
should really have another word, besides this one, to mean his own sensation of red? Or is it like this: the word "red"
means something known to everyone; and in addition, for each person, it means something known only to him? (Or
perhaps rather: it refers to something known only to him.)
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274. Of course, saying that the word "red" "refers to" instead of "means" something private does not help us
in the least to grasp its function; but it is the more psychologically apt expression for a particular experience in doing
philosophy. It is as if when I uttered the word I cast a sidelong glance at the private sensation, as it were in order to
say to myself: I know all right what I mean by it.
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275. Look at the blue of the sky and say to yourself "How blue the sky is!"--When you do it
spontaneously--without philosophical intentions--the idea never crosses your mind that this impression of colour
belongs only to you. And you have no hesitation in exclaiming that to someone else. And if you point at anything as
you say the words you point at the sky. I am saying: you have not the feeling of pointing-into-yourself, which often
accompanies 'naming the sensation' when one is thinking about 'private language'. Nor do you think that really you
ought not to point to the colour with your hand, but with your attention. (Consider what it means "to point to
something with the attention".)
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276. But don't we at least mean something quite definite when we look at a colour and name our
colour-impression? It is as if we detached the colour-impression from the object, like a membrane. (This ought to
arouse our suspicions.)
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277. But how is even possible for us to be tempted to think that we use a word to mean at one time the
colour known to everyone--and at another the 'visual impression' which I am getting now? How can there be so
much as a temptation here?--I don't turn the same kind of attention on the colour in the two cases. When I mean the
colour impression that (as I should like to say) belongs to me alone I immerse myself in the colour--rather like when
I 'cannot get my fill of a colour'. Hence it is easier to produce this experience when one is looking at a bright colour,
or at an impressive colour-scheme.
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278. "I know how the colour green looks to me"--surely that makes sense!--Certainly: what use of the
proposition are you thinking of?
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279. Imagine someone saying: "But I know how tall I am!" and laying his hand on top of his head to prove it.
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280. Someone paints a picture in order to shew how he imagines a theatre scene. And now I say: "This
picture has a double function: it informs others, as pictures or words inform--but for the one who gives the
information it is a representation (or piece of information?) of another kind: for him it is the picture of his image, as it
can't be for anyone else. To him his private impression of the picture means what he has imagined, in a sense in
which the picture cannot mean this to others."--And what right have I to speak in this second
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case of a representation or piece of information--if these words were rightly used in the first case?
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281. "But doesn't what you say come to this: that there is no pain, for example, without pain-behaviour?"--It
comes to this: only of a living human being and what resembles (behaves like) a living human being can one say: it
has sensations; it sees; is blind; hears; is deaf; is conscious or unconscious.
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282. "But in a fairy tale the pot too can see and hear!" (Certainly; but it can also talk.)
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"But the fairy tale only invents what is not the case: it does not talk nonsense."--It is not as simple as that. Is
it false or nonsensical to say that a pot talks? Have we a clear picture of the circumstances in which we should say of
a pot that it talked? (Even a nonsense-poem is not nonsense in the same way as the babbling of a child.)
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We do indeed say of an inanimate thing that it is in pain: when playing with dolls for example. But this use of
the concept of pain is a secondary one. Imagine a case in which people ascribed pain only to inanimate things; pitied
only dolls! (When children play at trains their game is connected with their knowledge of trains. It would
nevertheless be possible for the children of a tribe unacquainted with trains to learn this game from others, and to
play it without knowing that it was copied from anything. One might say that the game did not make the same sense
to them as to us.)
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283. What gives us so much as the idea that living beings, things, can feel?
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Is it that my education has led me to it by drawing my attention to feelings in myself, and now I transfer the
idea to objects outside myself? That I recognize that there is something there (in me) which I can call "pain" without
getting into conflict with the way other people use this word?--I do not transfer my idea to stones, plants, etc.
Page 97
Couldn't I imagine having frightful pains and turning to stone while they lasted? Well, how do I know, if I
shut my eyes, whether I have not turned into a stone? And if that has happened, in what sense will the stone have
the pains? In what sense will they be ascribable to the stone? And why need the pain have a bearer at all here?!
Page 97
And can one say of the stone that it has a soul and that is what has the pain? What has a soul, or pain, to do
with a stone?
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Only of what behaves like a human being can one say that it has pains.
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For one has to say it of a body, or, if you like of a soul which some body has. And how can a body have a
soul?
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284. Look at a stone and imagine it having sensations.--One says to oneself: How could one so much as get
the idea of ascribing a sensation to a thing? One might as well ascribe it to a number!--And now look at a wriggling
fly and at once these difficulties vanish and pain seems able to get a foothold here, where before everything was, so
to speak, too smooth for it.
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And so, too, a corpse seems to us quite inaccessible to pain.--Our attitude to what is alive and to what is
dead, is not the same. All our reactions are different.--If anyone says: "That cannot simply come from the fact that a
living thing moves about in such-and-such a way and a dead one not", then I want to intimate to him that this is a
case of the transition 'from quantity to quality'.
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285. Think of the recognition of facial expressions. Or of the description of facial expressions--which does
not consist in giving the measurements of the face! Think, too, how one can imitate a man's face without seeing
one's own in a mirror.
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286. But isn't it absurd to say of a body that it has pain?--And why does one feel an absurdity in that? In what
sense is it true that my hand does not feel pain, but I in my hand?
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What sort of issue is: Is it the body that feels pain?--How is it to be decided? What makes it plausible to say
that it is not the body?--Well, something like this: if someone has a pain in his hand, then the hand does not say so
(unless it writes it) and one does not comfort the hand, but the sufferer: one looks into his face.
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287. How am I filled with pity for this man? How does it come out what the object of my pity is? (Pity, one
may say, is a form of conviction that someone else is in pain.)
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288. I turn to stone and my pain goes on.--Suppose I were in error and it was no longer pain?--But I can't be
in error here; it means nothing to doubt whether I am in pain!--That means: if anyone said "I do not know if what I
have got is a pain or something else", we should think something like, he does not know what the
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English word "pain" means; and we should explain it to him.--How? Perhaps by means of gestures, or by pricking
him with a pin and saying: "See, that's what pain is!" This explanation, like any other, he might understand right,
wrong, or not at all. And he will shew which he does by his use of the word, in this as in other cases.
Page 99
If he now said, for example: "Oh, I know what 'pain' means; what I don't know is whether this, that I have
now, is pain"--we should merely shake our heads and be forced to regard his words as a queer reaction which we
have no idea what to do with. (It would be rather as if we heard someone say seriously: "I distinctly remember that
some time before I was born I believed.....".)
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That expression of doubt has no place in the language-game; but if we cut out human behaviour, which is the
expression of sensation, it looks as if I might legitimately begin to doubt afresh. My temptation to say that one
might take a sensation for something other than what it is arises from this: if I assume the abrogation of the normal
language-game with the expression of a sensation, I need a criterion of identity for the sensation; and then the
possibility of error also exists.
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289. "When I say 'I am in pain' I am at any rate justified before myself."--What does that mean? Does it
mean: "If someone else could know what I am calling 'pain', he would admit that I was using the word correctly"?
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To use a word without a justification does not mean to use it without right.
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290. What I do is not, of course, to identify my sensation by criteria: but to repeat an expression. But this is
not the end of the language-game: it is the beginning.
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But isn't the beginning the sensation--which I describe?--Perhaps this word "describe" tricks us here. I say "I
describe my state of mind" and "I describe my room". You need to call to mind the differences between the
language-games.
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291. What we call "descriptions" are instruments for particular uses. Think of a machine-drawing, a
cross-section, an elevation with measurements, which an engineer has before him. Thinking of a description as a
word-picture of the facts has something misleading about it: one tends to think only of such pictures as hang on our
walls: which seem simply to portray how a thing looks, what it is like. (These pictures are as it were idle.)
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292. Don't always think that you read off what you say from the facts; that you portray these in words
according to rules. For even so you would have to apply the rule in the particular case without guidance.
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293. If I say of myself that it is only from my own case that I know what the word "pain" means--must I not
say the same of other people too? And how can I generalize the one case so irresponsibly?
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Now someone tells me that he knows what pain is only from his own case!--Suppose everyone had a box
with something in it: we call it a "beetle". No one can look into anyone else's box, and everyone says he knows what
a beetle is only by looking at his beetle.--Here it would be quite possible for everyone to have something different in
his box. One might even imagine such a thing constantly changing.--But suppose the word "beetle" had a use in
these people's language?--If so it would not be used as the name of a thing. The thing in the box has no place in the
language-game at all; not even as a something: for the box might even be empty.--No, one can 'divide through' by
the thing in the box; it cancels out, whatever it is.
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That is to say: if we construe the grammar of the expression of sensation on the model of 'object and
designation' the object drops out of consideration as irrelevant.
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294. If you say he sees a private picture before him, which he is describing, you have still made an
assumption about what he has before him. And that means that you can describe it or do describe it more closely. If
you admit that you haven't any notion what kind of thing it might be that he has before him--then what leads you
into saying, in spite of that, that he has something before him? Isn't it as if I were to say of someone: "He has
something. But I don't know whether it is money, or debts, or an empty till."
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295. "I know.... only from my own case"--what kind of proposition is this meant to be at all? An experiential
one? No.--A grammatical one?
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Suppose everyone does say about himself that he knows what pain is only from his own pain.--Not that
people really say that, or are even prepared to say it. But if everybody said it--it might be a kind of exclamation. And
even if it gives no information, still it is a picture, and why should we not want to call up such a picture? Imagine an
allegricoal [[sic]] painting taking the place of those words.
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When we look into ourselves as we do philosophy, we often get to
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see just such a picture. A full-blown pictorial representation of our grammar. Not facts; but as it were illustrated
turns of speech.
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296. "Yes, but there is something there all the same accompanying my cry of pain. And it is on account of
that that I utter it. And this something is what is important--and frightful."--Only whom are we informing of this?
And on what occasion?
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297. Of course, if water boils in a pot, steam comes out of the pot and also pictured steam comes out of the
pictured pot. But what if one insisted on saying that there must also be something boiling in the picture of the pot?
Page 101
298. The very fact that we should so much like to say: "This is the important thing"--while we point privately
to the sensation--is enough to shew how much we are inclined to say something which gives no information.
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299. Being unable--when we surrender ourselves to philosophical thought--to help saying such-and-such;
being irresistibly inclined to say it--does not mean being forced into an assumption, or having an immediate
perception or knowledge of a state of affairs.
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300. It is--we should like to say--not merely the picture of the behaviour that plays a part in the
language-game with the words "he is in pain", but also the picture of the pain. Or, not merely the paradigm of the
behaviour, but also that of the pain.--It is a misunderstanding to say "The picture of pain enters into the
language-game with the word 'pain'." The image of pain is not a picture and this image is not replaceable in the
language-game by anything that we should call a picture.--The image of pain certainly enters into the language game
in a sense; only not as a picture.
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301. An image is not a picture, but a picture can correspond to it.
Page 101
302. If one has to imagine someone else's pain on the model of one's own, this is none too easy a thing to do:
for I have to imagine pain which I do not feel on the model of the pain which I do feel. That is, what I have to do is
not simply to make a transition in imagination from one place of pain to another. As, from pain in the hand to pain
in the arm. For I am not to imagine that I feel pain in some region of his body. (Which would also be possible.)
Page 101
Pain-behaviour can point to a painful place--but the subject of pain is the person who gives it expression.
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303. "I can only believe that someone else is in pain, but I know it if I am."--Yes: one can make the decision
to say "I believe he is in pain" instead of "He is in pain". But that is all.--What looks like an explanation here, or like a
statement about a mental process, is in truth an exchange of one expression for another which, while we are doing
philosophy, seems the more appropriate one.
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Just try--in a real case--to doubt someone else's fear or pain.
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304. "But you will surely admit that there is a difference between pain-behaviour accompanied by pain and
pain-behaviour without any pain?"--Admit it? What greater difference could there be?--"And yet you again and
again reach the conclusion that the sensation itself is a nothing."--Not at all. It is not a something, but not a nothing
either! The conclusion was only that a nothing would serve just as well as a something about which nothing could
be said. We have only rejected the grammar which tries to force itself on us here.
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The paradox disappears only if we make a radical break with the idea that language always functions in one
way, always serves the same purpose: to convey thoughts--which may be about houses, pains, good and evil, or
anything else you please.
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305. "But you surely cannot deny that, for example, in remembering, an inner process takes place."--What
gives the impression that we want to deny anything? When one says "Still, an inner process does take place
here"--one wants to go on: "After all, you see it." And it is this inner process that one means by the word
"remembering".--The impression that we wanted to deny something arises from our setting our faces against the
picture of the 'inner process'. What we deny is that the picture of the inner process gives us the correct idea of the
use of the word "to remember". We say that this picture with its ramifications stands in the way of our seeing the use
of the word as it is.
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306. Why should I deny that there is a mental process? But "There has just taken place in me the mental
process of remembering...." means nothing more than: "I have just remembered....". To deny the mental process
would mean to deny the remembering; to deny that anyone ever remembers anything.
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307. "Are you not really a behaviourist in disguise? Aren't you at bottom really saying that everything except
human behaviour is
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a fiction?"--If I do speak of a fiction, then it is of a grammatical fiction.
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308. How does the philosophical problem about mental processes and states and about behaviourism
arise?--The first step is the one that altogether escapes notice. We talk of processes and states and leave their nature
undecided. Sometime perhaps we shall know more about them--we think. But that is just what commits us to a
particular way of looking at the matter. For we have a definite concept of what it means to learn to know a process
better. (The decisive movement in the conjuring trick has been made, and it was the very one that we thought quite
innocent.)--And now the analogy which was to make us understand our thoughts falls to pieces. So we have to deny
the yet uncomprehended process in the yet unexplored medium. And now it looks as if we had denied mental
processes. And naturally we don't want to deny them.
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309. What is your aim in philosophy?--To shew the fly the way out of the fly-bottle.
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310. I tell someone I am in pain. His attitude to me will then be that of belief; disbelief; suspicion; and so on.
Page 103
Let us assume he says: "It's not so bad."--Doesn't that prove that he believes in something behind the
outward expression of pain?--His attitude is a proof of his attitude. Imagine not merely the words "I am in pain" but
also the answer "It's not so bad" replaced by instinctive noises and gestures.
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311. "What difference could be greater?"--In the case of pain I believe that I can give myself a private
exhibition of the difference. But I can give anyone an exhibition of the difference between a broken and an unbroken
tooth.--But for the private exhibition you don't have to give yourself actual pain; it is enough to imagine it--for
instance, you screw up your face a bit. And do you know that what you are giving yourself this exhibition of is pain
and not, for example, a facial expression? And how do you know what you are to give yourself an exhibition of
before you do it? This private exhibition is an illusion.
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312. But again, aren't the cases of the tooth and the pain similar? For the visual sensation in the one
corresponds to the sensation of pain in the other. I can exhibit the visual sensation to myself as little or as well as the
sensation of pain.
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Let us imagine the following: The surfaces of the things around us (stones, plants, etc.) have patches and
regions which produce pain in our skin when we touch them. (Perhaps through the chemical composition of these
surfaces. But we need not know that.) In this case we should speak of pain-patches on the leaf of a particular plant
just as at present we speak of red patches. I am supposing that it is useful to us to notice these patches and their
shapes; that we can infer important properties of the objects from them.
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313. I can exhibit pain, as I exhibit red, and as I exhibit straight and crooked and trees and stones.--That is
what we call "exhibiting".
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314. It shews a fundamental misunderstanding, if I am inclined to study the headache I have now in order to
get clear about the philosophical problem of sensation.
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315. Could someone understand the word "pain", who had never felt pain?--Is experience to teach me
whether this is so or not?--And if we say "A man could not imagine pain without having sometime felt it"--how do
we know? How can it be decided whether it is true?
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316. In order to get clear about the meaning of the word "think" we watch ourselves while we think; what we
observe will be what the word means!--But this concept is not used like that. (It would be as if without knowing how
to play chess, I were to try and make out what the word "mate" meant by close observation of the last move of some
game of chess.)
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317. Misleading parallel: the expression of pain is a cry--the expression of thought, a proposition.
Page 104
As if the purpose of the proposition were to convey to one person how it is with another: only, so to speak,
in his thinking part and not in his stomach.
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318. Suppose we think while we talk or write--I mean, as we normally do--we shall not in general say that we
think quicker than we talk; the thought seems not to be separate from the expression. On the other hand, however,
one does speak of the speed of thought; of how a thought goes through one's head like lightning; how problems
become clear to us in a flash, and so on. So it is natural to ask if the same thing happens in lightning-like
thought--only extremely accelerated--as when we talk and 'think while we talk.' So that in the
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first case the clockwork runs down all at once, but in the second bit by bit, braked by the words.
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319. I can see or understand a whole thought in a flash in exactly the sense in which I can make a note of it in
a few words or a few pencilled dashes.
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What makes this note into an epitome of this thought?
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320. The lightning-like thought may be connected with the spoken thought as the algebraic formula is with
the sequence of numbers which I work out from it.
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When, for example, I am given an algebraic function, I am CERTAIN that I shall be able to work out its
values for the arguments 1, 2, 3,... up to 10. This certainty will be called 'well-founded', for I have learned to compute
such functions, and so on. In other cases no reasons will be given for it--but it will be justified by success.
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321. "What happens when a man suddenly understands?"--The question is badly framed. If it is a question
about the meaning of the expression "sudden understanding", the answer is not to point to a process that we give
this name to.--The question might mean: what are the tokens of sudden understanding; what are its characteristic
psychical accompaniments?
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(There is no ground for assuming that a man feels the facial movements that go with his expression, for
example, or the alterations in his breathing that are characteristic of some emotion. Even if he feels them as soon as
his attention is directed towards them.) ((Posture.))
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322. The question what the expression means is not answered by such a description; and this misleads us
into concluding that understanding is a specific indefinable experience. But we forget that what should interest us is
the question: how do we compare these experiences; what criterion of identity do we fix for their occurrence?
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323. "Now I know how to go on!" is an exclamation; it corresponds to an instinctive sound, a glad start. Of
course it does not follow from my feeling that I shall not find I am stuck when I do try to go on.--Here there are
cases in which I should say: "When I said I knew how to go on, I did know." One will say that if, for example, an
unforeseen interruption occurs. But what is unforeseen must not simply be that I get stuck.
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We could also imagine a case in which light was always seeming to dawn on someone--he exclaims "Now I
have it!" and then can never justify himself in practice.--It might seem to him as if in the twinkling of an eye he
forgot again the meaning of the picture that occurred to him.
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324. Would it be correct to say that it is a matter of induction, and that I am as certain that I shall be able to
continue the series, as I am that this book will drop on the ground when I let it go; and that I should be no less
astonished if I suddenly and for no obvious reason got stuck in working out the series, than I should be if the book
remained hanging in the air instead of falling?--To that I will reply that we don't need any grounds for this certainty
either. What could justify the certainty better than success?
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325. "The certainty that I shall be able to go on after I have had this experience--seen the formula, for
instance,--is simply based on induction." What does this mean?--"The certainty that the fire will burn me is based on
induction." Does that mean that I argue to myself: "Fire has always burned me, so it will happen now too?" Or is the
previous experience the cause of my certainty, not its ground? Whether the earlier experience is the cause of the
certainty depends on the system of hypotheses, of natural laws, in which we are considering the phenomenon of
certainty.
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Is our confidence justified?--What people accept as a justification--is shewn by how they think and live.
Page 106
326. We expect this, and are surprised at that. But the chain of reasons has an end.
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327. "Can one think without speaking?"--And what is thinking?--Well, don't you ever think? Can't you
observe yourself and see what is going on? It should be quite simple. You do not have to wait for it as for an
astronomical event and then perhaps make your observation in a hurry.
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328. Well, what does one include in 'thinking'? What has one learnt to use this word for?--If I say I have
thought--need I always be right?--What kind of mistake is there room for here? Are there circumstances in which
one would ask: "Was what I was doing then really thinking; am I not making a mistake?" Suppose someone takes a
measurement in the middle of a train of thought: has he interrupted the thought if he says nothing to himself during
the measuring?
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329. When I think in language, there aren't 'meanings' going through my mind in addition to the verbal
expressions: the language is itself the vehicle of thought.
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330. Is thinking a kind of speaking? One would like to say it is what distinguishes speech with thought from
talking without thinking.--And so it seems to be an accompaniment of speech. A process, which may accompany
something else, or can go on by itself.
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Say: "Yes, this pen is blunt. Oh well, it'll do." First, thinking it; then without thought; then just think the
thought without the words.--Well, while doing some writing I might test the point of my pen, make a face--and then
go on with a gesture of resignation.--I might also act in such a way while taking various measurements that an
onlooker would say I had--without words--thought: If two magnitudes are equal to a third, they are equal to one
another.--But what constitutes thought here is not some process which has to accompany the words if they are not
to be spoken without thought.
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331. Imagine people who could only think aloud. (As there are people who can only read aloud.)
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332. While we sometimes call it "thinking" to accompany a sentence by a mental process, that
accompaniment is not what we mean by a "thought".--Say a sentence and think it; say it with understanding.--And
now do not say it, and just do what you accompanied it with when you said it with understanding!--(Sing this tune
with expression. And now don't sing it, but repeat its expression!--And here one actually might repeat something.
For example, motions of the body, slower and faster breathing, and so on.)
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333. "Only someone who is convinced can say that."--How does the conviction help him when he says
it?--Is it somewhere at hand by the side of the spoken expression? (Or is it masked by it, as a soft sound by a loud
one, so that it can, as it were, no longer be heard when one expresses it out loud?) What if someone were to say "In
order to be able to sing a tune from memory one has to hear it in one's mind and sing from that"?
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334. "So you really wanted to say...."--We use this phrase in order to lead someone from one form of
expression to another. One is tempted to use the following picture: what he really 'wanted to say', what he 'meant'
was already present somewhere in his mind even
Page Break 108
before we gave it expression. Various kinds of thing may persuade us to give up one expression and to adopt
another in its place. To understand this, it is useful to consider the relation in which the solutions of mathematical
problems stand to the context and ground of their formulation. The concept 'trisection of the angle with ruler and
compass', when people are trying to do it, and, on the other hand, when it has been proved that there is no such
thing.
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335. What happens when we make an effort--say in writing a letter--to find the right expression for our
thoughts?--This phrase compares the process to one of translating or describing: the thoughts are already there
(perhaps were there in advance) and we merely look for their expression. This picture is more or less appropriate in
different cases.--But can't all sorts of things happen here?--I surrender to a mood and the expression comes. Or a
picture occurs to me and I try to describe it. Or an English expression occurs to me and I try to hit on the
corresponding German one. Or I make a gesture, and ask myself: What words correspond to this gesture? And so
on.
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Now if it were asked: "Do you have the thought before finding the expression?" what would one have to
reply? And what, to the question: "What did the thought consist in, as it existed before its expression?"
Page 108
336. This case is similar to the one in which someone imagines that one could not think a sentence with the
remarkable word order of German or Latin just as it stands. One first has to think it, and then one arranges the words
in that queer order. (A French politician once wrote that it was a peculiarity of the French language that in it words
occur in the order in which one thinks them.)
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337. But didn't I already intend the whole construction of the sentence (for example) at its beginning? So
surely it already existed in my mind before I said it out loud!--If it was in my mind, still it would not normally be
there in some different word order. But here we are constructing a misleading picture of 'intending', that is, of the use
of this word. An intention is embedded in its situation, in human customs and institutions. If the technique of the
game of chess did not exist, I could not intend to play a game of chess. In so far as I do intend the construction of a
sentence in advance, that is made possible by the fact that I can speak the language in question.
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338. After all, one can only say something if one has learned to talk. Therefore in order to want to say
something one must also have mastered a language; and yet it is clear that one can want to speak without speaking.
Just as one can want to dance without dancing.
Page 109
And when we think about this, we grasp at the image of dancing, speaking, etc..
Page 109
339. Thinking is not an incorporeal process which lends life and sense to speaking, and which it would be
possible to detach from speaking, rather as the Devil took the shadow of Schlemiehl from the ground.--But how "not
an incorporeal process"? Am I acquainted with incorporeal processes, then, only thinking is not one of them? No; I
called the expression "an incorporeal process" to my aid in my embarrassment when I was trying to explain the
meaning of the word "thinking" in a primitive way.
Page 109
One might say "Thinking is an incorporeal process", however, if one were using this to distinguish the
grammar of the word "think" from that of, say, the word "eat". Only that makes the difference between the meanings
look too slight. (It is like saying: numerals are actual, and numbers non-actual, objects.) An unsuitable type of
expression is a sure means of remaining in a state of confusion. It as it were bars the way out.
Page 109
340. One cannot guess how a word functions. One has to look at its use and learn from that.
Page 109
But the difficulty is to remove the prejudice which stands in the way of doing this. It is not a stupid
prejudice.
Page 109
341. Speech with and without thought is to be compared with the playing of a piece of music with and
without thought.
Page 109
342. William James, in order to shew that thought is possible without speech, quotes the recollection of a
deaf-mute, Mr. Ballard, who wrote that in his early youth, even before he could speak, he had had thoughts about
God and the world.--What can he have meant?--Ballard writes: "It was during those delightful rides, some two or
three years before my initiation into the rudiments of written language, that I began to ask myself the question: how
came the world into being?"--Are you sure--one would like to ask--that this is the correct translation of your
wordless thought into words? And why does this question--which otherwise seems not to exist--raise its head here?
Do I want to say that the writer's memory deceives
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him?--I don't even know if I should say that. These recollections are a queer memory phenomenon,--and I do not
know what conclusions one can draw from them about the past of the man who recounts them.
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343. The words with which I express my memory are my memory-reaction.
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344. Would it be imaginable that people should never speak an audible language, but should still say things
to themselves in the imagination?
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"If people always said things only to themselves, then they would merely be doing always what as it is they
do sometimes."--So it is quite easy to imagine this: one need only make the easy transition from some to all. (Like:
"An infinitely long row of trees is simply one that does not come to an end.") Our criterion for someone's saying
something to himself is what he tells us and the rest of his behaviour; and we only say that someone speaks to
himself if, in the ordinary sense of the words, he can speak. And we do not say it of a parrot; nor of a gramophone.
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345. "What sometimes happens might always happen."--What kind of proposition is that? It is like the
following: If "F (a)" makes sense "(x).F(x)" makes sense.
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"If it is possible for someone to make a false move in some game, then it might be possible for everybody to
make nothing but false moves in every game."--Thus we are under a temptation to misunderstand the logic of our
expressions here, to give an incorrect account of the use of our words.
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Orders are sometimes not obeyed. But what would it be like if no orders were ever obeyed? The concept
'order' would have lost its purpose.
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346. But couldn't we imagine God's suddenly giving a parrot understanding, and its now saying things to
itself?--But here it is an important fact that I imagined a deity in order to imagine this.
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347. "But at least I know from my own case what it means 'to say things to oneself'. And if I were deprived
of the organs of speech, I could still talk to myself."
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If I know it only from my own case, then I know only what I call that, not what anyone else does.
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348. "These deaf-mutes have learned only a gesture-language, but each of them talks to himself inwardly in a
vocal language."--Now,
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don't you understand that?--But how do I know whether I understand it?!--What can I do with this information (if it
is such)? The whole idea of understanding smells fishy here. I do not know whether I am to say I understand it or
don't understand it. I might answer "It's an English sentence; apparently quite in order--that is, until one wants to do
something with it; it has a connexion with other sentences which makes it difficult for us to say that nobody really
knows what it tells us; but everyone who has not become calloused by doing philosophy notices that there is
something wrong here."
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349. "But this supposition surely makes good sense!"--Yes; in ordinary circumstances these words and this
picture have an application with which we are familiar.--But if we suppose a case in which this application falls away
we become as it were conscious for the first time of the nakedness of the words and the picture.
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350. "But if I suppose that someone has a pain, then I am simply supposing that he has just the same as I
have so often had."--That gets us no further. It is as if I were to say: "You surely know what 'It is 5 o'clock here'
means; so you also know what 'It's 5 o'clock on the sun' means. It means simply that it is just the same time there as
it is here when it is 5 o'clock."--The explanation by means of identity does not work here. For I know well enough
that one can call 5 o'clock here and 5 o'clock there "the same time", but what I do not know is in what cases one is to
speak of its being the same time here and there.
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In exactly the same way it is no explanation to say: the supposition that he has a pain is simply the
supposition that he has the same as I. For that part of the grammar is quite clear to me: that is, that one will say that
the stove has the same experience as I, if one says: it is in pain and I am in pain.
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351. Yet we go on wanting to say: "Pain is pain--whether he has it, or I have it; and however I come to know
whether he has a pain or not."--I might agree.--And when you ask me "Don't you know, then, what I mean when I
say that the stove is in pain?"--I can reply: These words may lead me to have all sorts of images; but their usefulness
goes no further. And I can also imagine something in connexion with the words: "It was just 5 o'clock in the
afternoon on the sun"--such as a grandfather clock which points to 5.--But a still better example would be that of the
application of "above" and "below" to the earth. Here we all have a quite clear idea of what
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"above" and "below" mean. I see well enough that I am on top; the earth is surely beneath me! (And don't smile at
this example. We are indeed all taught at school that it is stupid to talk like that. But it is much easier to bury a
problem than to solve it.) And it is only reflection that shews us that in this case "above" and "below" cannot be
used in the ordinary way. (That we might, for instance, say that the people at the antipodes are 'below' our part of the
earth, but it must also be recognized as right for them to use the same expression about us.)
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352. Here it happens that our thinking plays us a queer trick. We want, that is, to quote the law of excluded
middle and to say: "Either such an image is in his mind, or it is not; there is no third possibility!"--We encounter this
queer argument also in other regions of philosophy. "In the decimal expansion of π either the group "7777" occurs,
or it does not--there is no third possibility." That is to say: "God sees--but we don't know." But what does that
mean?--We use a picture; the picture of a visible series which one person sees the whole of and another not. The law
of excluded middle says here: It must either look like this, or like that. So it really--and this is a truism--says nothing
at all, but gives us a picture. And the problem ought now to be: does reality accord with the picture or not? And this
picture seems to determine what we have to do, what to look for, and how--but it does not do so, just because we do
not know how it is to be applied. Here saying "There is no third possibility" or "But there can't be a third
possibility!"--expresses our inability to turn our eyes away from this picture: a picture which looks as if it must
already contain both the problem and its solution, while all the time we feel that it is not so.
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Similarly when it is said "Either he has this experience, or not"--what primarily occurs to us is a picture
which by itself seems to make the sense of the expressions unmistakable: "Now you know what is in question"--we
should like to say. And that is precisely what it does not tell him.
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353. Asking whether and how a proposition can be verified is only a particular way of asking "How d'you
mean?" The answer is a contribution to the grammar of the proposition.
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354. The fluctuation in grammar between criteria and symptoms makes it look as if there were nothing at all
but symptoms. We say,
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for example: "Experience teaches that there is rain when the barometer falls, but it also teaches that there is rain
when we have certain sensations of wet and cold, or such-and-such visual impressions." In defence of this one says
that these sense-impressions can deceive us. But here one fails to reflect that the fact that the false appearance is
precisely one of rain is founded on a definition.
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355. The point here is not that our sense-impressions can lie, but that we understand their language. (And this
language like any other is founded on convention.)
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356. One is inclined to say: "Either it is raining, or it isn't--how I know, how the information has reached me,
is another matter." But then let us put the question like this: What do I call "information that it is raining"? (Or have I
only information of this information too?) And what gives this 'information' the character of information about
something? Doesn't the form of our expression mislead us here? For isn't it a misleading metaphor to say: "My eyes
give me the information that there is a chair over there"?
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357. We do not say that possibly a dog talks to itself. Is that because we are so minutely acquainted with its
soul? Well, one might say this: If one sees the behaviour of a living thing, one sees its soul.--But do I also say in my
own case that I am saying something to myself, because I am behaving in such-and-such a way?--I do not say it
from observation of my behaviour. But it only makes sense because I do behave in this way.--Then it is not because
I mean it that it makes sense?
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358. But isn't it our meaning it that gives sense to the sentence? (And here, of course, belongs the fact that
one cannot mean a senseless series of words.) And 'meaning it' is something in the sphere of the mind. But it is also
something private! It is the intangible something; only comparable to consciousness itself.
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How could this seem ludicrous? It is, as it were, a dream of our language.
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359. Could a machine think?--Could it be in pain?--Well, is the human body to be called such a machine? It
surely comes as close as possible to being such a machine.
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360. But a machine surely cannot think!--Is that an empirical statement? No. We only say of a human being
and what is like one that it thinks. We also say it of dolls and no doubt of spirits too. Look at the word "to think" as a
tool.
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361. The chair is thinking to itself:.....
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WHERE? In one of its parts? Or outside its body; in the air around it? Or not anywhere at all? But then what
is the difference between this chair's saying something to itself and another one's doing so, next to it?--But then how
is it with man: where does he say things to himself? How does it come about that this question seems senseless; and
that no specification of a place is necessary except just that this man is saying something to himself? Whereas the
question where the chair talks to itself seems to demand an answer.--The reason is: we want to know how the chair is
supposed to be like a human being; whether, for instance, the head is at the top of the back and so on.
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What is it like to say something to oneself; what happens here?--How am I to explain it? Well, only as you
might teach someone the meaning of the expression "to say something to oneself". And certainly we learn the
meaning of that as children.--Only no one is going to say that the person who teaches it to us tells us 'what takes
place'.
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362. Rather it seems to us as though in this case the instructor imparted the meaning to the pupil--without
telling him it directly; but in the end the pupil is brought to the point of giving himself the correct ostensive
definition. And this is where our illusion is.
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363. "But when I imagine something, something certainly happens!" Well, something happens--and then I
make a noise. What for? Presumably in order to tell what happens.--But how is telling done? When are we said to
tell anything?--What is the language-game of telling?
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I should like to say: you regard it much too much as a matter of course that one can tell anything to anyone.
That is to say: we are so much accustomed to communication through language, in conversation, that it looks to us
as if the whole point of communication lay in this: someone else grasps the sense of my words--which is something
mental: he as it were takes it into his own mind. If he then does something further with it as well, that is no part of
the immediate purpose of language.
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One would like to say "Telling brings it about that he knows that I am in pain; it produces this mental
phenomenon; everything else is inessential to the telling." As for what this queer phenomenon of knowledge
is--there is time enough for that. Mental processes just are queer. (It is as if one said: "The clock tells us the time.
What time is, is not yet settled. And as for what one tells the time for--that doesn't come in here.")
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364. Someone does a sum in his head. He uses the result, let's say, for building a bridge or a machine.--Are
you trying to say that he has not really arrived at this number by calculation? That it has, say, just 'come' to him in
the manner of a kind of dream? There surely must have been calculation going on, and there was. For he knows that,
and how, he calculated; and the correct result he got would be inexplicable without calculation.--But what if I said:
"It strikes him as if he had calculated. And why should the correct result be explicable? Is it not incomprehensible
enough, that without saying a word, without making a note, he was able to CALCULATE?"--
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Is calculating in the imagination in some sense less real than calculating on paper? It is
real--calculation-in-the-head.--Is it like calculation on paper?--I don't know whether to call it like. Is a bit of white
paper with black lines on it like a human body?
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365. Do Adelheid and the Bishop play a real game of chess?--Of course. They are not merely
pretending--which would also be possible as part of a play.--But, for example, the game has no beginning!--Of
course it has; otherwise it would not be a game of chess.--
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366. Is a sum in the head less real than a sum on paper?--Perhaps one is inclined to say some such thing; but
one can get oneself to think the opposite as well by telling oneself: paper, ink, etc. are only logical constructions out
of our sense-data.
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"I have done the multiplication..... in my head"--do I perhaps not believe such a statement?--But was it really
a multiplication? It was not merely 'a' multiplication, but this one--in the head. This is the point at which I go wrong.
For I now want to say: it was some mental process corresponding to the multiplication on paper. So it would make
sense to say: "This process in the mind corresponds to this process on paper." And it would then make sense to talk
of a method of projection according to which the image of the sign was a representation of the sign itself.
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367. The mental picture is the picture which is described when someone describes what he imagines.
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368. I describe a room to someone, and then get him to paint an impressionistic picture from this description
to shew that he has understood it.--Now he paints the chairs which I described as green, dark red; where I said
"yellow", he paints blue.--That is the impression
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which he got of that room. And now I say: "Quite right! That's what it's like."
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369. One would like to ask: "What is it like--what happens--when one does a sum in one's head?"--And in a
particular case the answer may be "First I add 17 and 18, then I subtract 39.....". But that is not the answer to our
question. What is called doing sums in one's head is not explained by such an answer.
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370. One ought to ask, not what images are or what happens when one imagines anything, but how the word
"imagination" is used. But that does not mean that I want to talk only about words. For the question as to the nature
of the imagination is as much about the word "imagination" as my question is. And I am only saying that this
question is not to be decided--neither for the person who does the imagining, nor for anyone else--by pointing; nor
yet by a description of any process. The first question also asks for a word to be explained; but it makes us expect a
wrong kind of answer.
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371. Essence is expressed by grammar.
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372. Consider: "The only correlate in language to an intrinsic necessity is an arbitrary rule. It is the only thing
which one can milk out of this intrinsic necessity into a proposition."
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373. Grammar tells what kind of object anything is. (Theology as grammar.)
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374. The great difficulty here is not to represent the matter as if there were something one couldn't do. As if
there really were an object, from which I derive its description, but I were unable to shew it to anyone.--And the best
that I can propose is that we should yield to the temptation to use this picture, but then investigate how the
application of the picture goes.
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375. How does one teach anyone to read to himself? How does one know if he can do so? How does he
himself know that he is doing what is required of him?
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376. When I say the ABC to myself, what is the criterion of my doing the same as someone else who silently
repeats it to himself? It might be found that the same thing took place in my larynx and in his. (And similarly when
we both think of the same thing, wish the same, and so on.) But then did we learn the use of the words: "to
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say such-and-such to oneself" by someone's pointing to a process in the larynx or the brain? Is it not also perfectly
possible that my image of the sound a and his correspond to different physiological processes? The question is: How
do we compare images?
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377. Perhaps a logician will think: The same is the same--how identity is established is a psychological
question. (High is high--it is a matter of psychology that one sometimes sees, and sometimes hears it.)
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What is the criterion for the sameness of two images?--What is the criterion for the redness of an image? For
me, when it is someone else's image: what he says and does. For myself, when it is my image: nothing. And what
goes for "red" also goes for "same".
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378. "Before I judge that two images which I have are the same, I must recognize them as the same." And
when that has happened, how am I to know that the word "same" describes what I recognize? Only if I can express
my recognition in some other way, and if it is possible for someone else to teach me that "same" is the correct word
here.
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For if I need a justification for using a word, it must also be one for someone else.
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379. First I am aware of it as this; and then I remember what it is called.--Consider: in what cases is it right to
say this?
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380. How do I recognize that this is red?--"I see that it is this; and then I know that that is what this is called."
This?--What?! What kind of answer to this question makes sense?
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(You keep on steering towards the idea of the private ostensive definition.)
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I could not apply any rules to a private transition from what is seen to words. Here the rules really would
hang in the air; for the institution of their use is lacking.
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381. How do I know that this colour is red?--It would be an answer to say: "I have learnt English".
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382. At these words I form this image. How can I justify this?
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Has anyone shewn me the image of the colour blue and told me that this is the image of blue?
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What is the meaning of the words: "This image"? How does one point to an image? How does one point
twice to the same image?
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383. We are not analysing a phenomenon (e.g. thought) but a concept (e.g. that of thinking), and therefore
the use of a word. So it may look as if what we were doing were Nominalism. Nominalists make the mistake of
interpreting all words as names, and so of not really describing their use, but only, so to speak, giving a paper draft
on such a description.
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384. You learned the concept 'pain' when you learned language.
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385. Ask yourself: Would it be imaginable for someone to learn to do sums in his head without ever doing
written or oral ones?--"Learning it" will mean: being made able to do it. Only the question arises, what will count as
a criterion for being able to do it?--But is it also possible for some tribe to know only of calculation in the head, and
of no other kind? Here one has to ask oneself: "What will that be like?"--And so one will have to depict it as a
limiting case. And the question will then arise whether we are still willing to use the concept of 'calculating in the
head' here--or whether in such circumstances it has lost its purpose, because the phenomena gravitate towards
another paradigm.
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386. "But why have you so little confidence in yourself? Ordinarily you always know well enough what it is
to 'calculate.' So if you say you have calculated in imagination, then you will have done so. If you had not
calculated, you would not have said you had. Equally, if you say that you see something red in imagination, then it
will be red. You know what 'red' is elsewhere.--And further: you do not always rely on the agreement of other
people; for you often report that you have seen something no one else has."--But I do have confidence in myself--I
say without hesitation that I have done this sum in my head, have imagined this colour. The difficulty is not that I
doubt whether I really imagined anything red. But it is this: that we should be able, just like that, to point out or
describe the colour we have imagined, that the translation of the image into reality presents no difficulty at all. Are
they then so alike that one might mix them up?--But I can also recognize a man from a drawing straight off.--Well,
but can I ask: "What does a correct image of this colour look like?" or "What sort of thing is it?"; can I learn this?
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(I cannot accept his testimony because it is not testimony. It only tells me what he is inclined to say.)
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387. The deep aspect of this matter readily eludes us.
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388. "I don't see anything violet here, but I can shew it you if you give me a paint box." How can one know
that one can shew it if...., in other words, that one can recognize it if one sees it?
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How do I know from my image, what the colour really looks like?
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How do I know that I shall be able to do something? that is, that the state I am in now is that of being able to
do that thing?
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389. "The image must be more like its object than any picture. For, however like I make the picture to what it
is supposed to represent, it can always be the picture of something else as well. But it is essential to the image that it
is the image of this and of nothing else." Thus one might come to regard the image as a super-likeness.
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390. Could one imagine a stone's having consciousness? And if anyone can do so--why should that not
merely prove that such image-mongery is of no interest to us?
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391. I can perhaps even imagine (though it is not easy) that each of the people whom I see in the street is in
frightful pain, but is artfully concealing it. And it is important that I have to imagine an artful concealment here. That
I do not simply say to myself: "Well, his soul is in pain: but what has that to do with his body?" or "After all it need
not shew in his body!"--And if I imagine this--what do I do; what do I say to myself; how do I look at the people?
Perhaps I look at one and think: "It must be difficult to laugh when one is in such pain", and much else of the same
kind. I as it were play a part, act as if the others were in pain. When I do this I am said for example to be
imagining....
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392. "When I imagine he is in pain, all that really goes on in me is...." Then someone else says: "I believe I
can imagine it without thinking '....'" ("I believe I can think without words.") This leads to nothing. The analysis
oscillates between natural science and grammar.
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393. "When I imagine that someone who is laughing is really in pain I don't imagine any pain-behaviour, for I
see just the opposite. So what do I imagine?"--I have already said what. And I do not necessarily imagine my being
in pain.--"But then what is the process of imagining it?"--Where (outside philosophy) do we use the
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words "I can imagine his being in pain" or "I imagine that...." or "Imagine that...."?
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We say, for example, to someone who has to play a theatrical part: "Here you must imagine that this man is
in pain and is concealing it"--and now we give him no directions, do not tell him what he is actually to do. For this
reason the suggested analysis is not to the point either.--We now watch the actor who is imagining this situation.
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394. In what sort of circumstances should we ask anyone: "What actually went on in you as you imagined
this?"--And what sort of answer do we expect?
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395. There is a lack of clarity about the role of imaginability in our investigation. Namely about the extent to
which it ensures that a proposition makes sense.
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396. It is no more essential to the understanding of a proposition that one should imagine anything in
connexion with it, than that one should make a sketch from it.
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397. Instead of "imaginability" one can also say here: representability by a particular method of
representation. And such a representation may indeed safely point a way to further use of a sentence. On the other
hand a picture may obtrude itself upon us and be of no use at all.
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398. "But when I imagine something, or even actually see objects, I have got something which my neighbour
has not."--I understand you. You want to look about you and say: "At any rate only I have got THIS."--What are
these words for? They serve no purpose.--Can one not add: "There is here no question of a 'seeing'--and therefore
none of a 'having'--nor of a subject, nor therefore of 'I' either"? Might I not ask: In what sense have you got what you
are talking about and saying that only you have got it? Do you possess it? You do not even see it. Must you not
really say that no one has got it? And this too is clear: if as a matter of logic you exclude other people's having
something, it loses its sense to say that you have it.
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But what is the thing you are speaking of? It is true I said that I knew within myself what you meant. But that
meant that I knew how one thinks to conceive this object, to see it, to make one's looking and pointing mean it. I
know how one stares ahead and looks about
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one in this case--and the rest. I think we can say: you are talking (if, for example, you are sitting in a room) of the
'visual room'. The 'visual room' is the one that has no owner. I can as little own it as I can walk about it, or look at it,
or point to it. Inasmuch as it cannot be any one else's it is not mine either. In other words, it does not belong to me
because I want to use the same form of expression about it as about the material room in which I sit. The description
of the latter need not mention an owner, in fact it need not have any owner. But then the visual room cannot have
any owner. "For"--one might say--"it has no master, outside or in."
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Think of a picture of a landscape, an imaginary landscape with a house in it.--Someone asks "Whose house is
that?"--The answer, by the way, might be "It belongs to the farmer who is sitting on the bench in front of it". But
then he cannot for example enter his house.
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399. One might also say: Surely the owner of the visual room would have to be the same kind of thing as it
is; but he is not to be found in it, and there is no outside.
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400. The 'visual room' seemed like a discovery, but what its discoverer really found was a new way of
speaking, a new comparison; it might even be called a new sensation.
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401. You have a new conception and interpret it as seeing a new object. You interpret a grammatical
movement made by yourself as a quasi-physical phenomenon which you are observing. (Think for example of the
question: "Are sense-data the material of which the universe is made?")
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But there is an objection to my saying that you have made a 'grammatical' movement. What you have
primarily discovered is a new way of looking at things. As if you had invented a new way of painting; or, again, a
new metre, or a new kind of song.--
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402. "It's true I say 'Now I am having such-and-such an image', but the words 'I am having' are merely a sign
to someone else; the description of the image is a complete account of the imagined world."--You mean: the words
"I am having" are like "I say!...." You are inclined to say it should really have been expressed differently. Perhaps
simply by making a sign with one's hand and then giving a description.--When as in this case, we disapprove of the
expressions of ordinary language (which are after all performing their office), we have got a picture in our heads
which conflicts with the picture of our ordinary
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way of speaking. Whereas we are tempted to say that our way of speaking does not describe the facts as they really
are. As if, for example the proposition "he has pains" could be false in some other way than by that man's not
having pains. As if the form of expression were saying something false even when the proposition faute de mieux
asserted something true.
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For this is what disputes between Idealists, Solipsists and Realists look like. The one party attack the normal
form of expression as if they were attacking a statement; the others defend it, as if they were stating facts recognized
by every reasonable human being.
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403. If I were to reserve the word "pain" solely for what I had hitherto called "my pain", and others "L.W.'s
pain", I should do other people no injustice, so long as a notation were provided in which the loss of the word "pain"
in other connexions were somehow supplied. Other people would still be pitied, treated by doctors and so on. It
would, of course, be no objection to this mode of expression to say: "But look here, other people have just the same
as you!"
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But what should I gain from this new kind of account? Nothing. But after all neither does the solipsist want
any practical advantage when he advances his view!
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404. "When I say 'I am in pain', I do not point to a person who is in pain, since in a certain sense I have no
idea who is." And this can be given a justification. For the main point is: I did not say that such-and-such a person
was in pain, but "I am....." Now in saying this I don't name any person. Just as I don't name anyone when I groan
with pain. Though someone else sees who is in pain from the groaning.
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What does it mean to know who is in pain? It means, for example, to know which man in this room is in
pain: for instance, that it is the one who is sitting over there, or the one who is standing in that corner, the tall one
over there with the fair hair, and so on.--What am I getting at? At the fact that there is a great variety of criteria for
personal 'identity'.
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Now which of them determines my saying that 'I' am in pain? None.
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405. "But at any rate when you say 'I am in pain', you want to draw the attention of others to a particular
person."--The answer might be: No, I want to draw their attention to myself.--
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406. "But surely what you want to do with the words 'I am....' is to distinguish between yourself and other
people."--Can this be said in every case? Even when I merely groan? And even if I do 'want to distinguish' between
myself and other people--do I want to distinguish between the person L.W. and the person N.N.?
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407. It would be possible to imagine someone groaning out: "Someone is in pain--I don't know who!"--and
our then hurrying to help him, the one who groaned.
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408. "But you aren't in doubt whether it is you or someone else who has the pain!"--The proposition "I don't
know whether I or someone else is in pain" would be a logical product, and one of its factors would be: "I don't
know whether I am in pain or not"--and that is not a significant proposition.
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409. Imagine several people standing in a ring, and me among them. One of us, sometimes this one,
sometimes that, is connected to the poles of an electrical machine without our being able to see this. I observe the
faces of the others and try to see which of us has just been electrified.--Then I say: "Now I know who it is; for it's
myself." In this sense I could also say: "Now I know who is getting the shocks; it is myself." This would be a rather
queer way of speaking.--But if I make the supposition that I can feel the shock even when someone else is
electrified, then the expression "Now I know who...." becomes quite unsuitable. It does not belong to this game.
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410. "I" is not the name of a person, nor "here" of a place, and "this" is not a name. But they are connected
with names. Names are explained by means of them. It is also true that it is characteristic of physics not to use these
words.
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411. Consider how the following questions can be applied, and how settled:
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(1) "Are these books my books?"
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(2) "Is this foot my foot?"
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(3) "Is this body my body?"
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(4) "Is this sensation my sensation?"
Each of these questions has practical (non-philosophical) applications.
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(2) Think of cases in which my foot is anaesthetized or paralysed. Under certain circumstances the question
could be settled by determining whether I can feel pain in this foot.
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(3) Here one might be pointing to a mirror-image. Under certain circumstances, however, one might touch a
body and ask the question. In others it means the same as: "Does my body look like that?"
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(4) Which sensation does one mean by 'this' one? That is: how is one using the demonstrative pronoun here?
Certainly otherwise than in, say, the first example! Here confusion occurs because one imagines that by directing
one's attention to a sensation one is pointing to it.
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412. The feeling of an unbridgeable gulf between consciousness and brain-process: how does it come about
that this does not come into the considerations of our ordinary life? This idea of a difference in kind is accompanied
by slight giddiness,--which occurs when we are performing a piece of logical sleight-of-hand. (The same giddiness
attacks us when we think of certain theorems in set theory.) When does this feeling occur in the present case? It is
when I, for example, turn my attention in a particular way on to my own consciousness, and, astonished, say to
myself: THIS is supposed to be produced by a process in the brain!--as it were clutching my forehead.--But what
can it mean to speak of "turning my attention on to my own consciousness"? This is surely the queerest thing there
could be! It was a particular act of gazing that I called doing this. I stared fixedly in front of me--but not at any
particular point or object. My eyes were wide open, the brows not contracted (as they mostly are when I am
interested in a particular object). No such interest preceded this gazing. My glance was vacant; or again like that of
someone admiring the illumination of the sky and drinking in the light.
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Now bear in mind that the proposition which I uttered as a paradox (THIS is produced by a brain-process!)
has nothing paradoxical about it. I could have said it in the course of an experiment whose purpose was to shew that
an effect of light which I see is produced by stimulation of a particular part of the brain.--But I did not utter the
sentence in the surroundings in which it would have had an everyday and unparadoxical sense. And my attention
was not such as would have accorded with making an experiment. (If it had been, my look would have been intent,
not vacant.)
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413. Here we have a case of introspection, not unlike that from which William James got the idea that the
'self' consisted mainly of 'peculiar motions in the head and between the head and throat'.
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And James' introspection shewed, not the meaning of the word "self" (so far as it means something like
"person", "human being", "he himself", "I myself"), nor any analysis of such a thing, but the state of a philosopher's
attention when he says the word "self" to himself and tries to analyse its meaning. (And a good deal could be learned
from this.)
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414. You think that after all you must be weaving a piece of cloth: because you are sitting at a loom--even if it
is empty--and going through the motions of weaving.
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415. What we are supplying are really remarks on the natural history of human beings; we are not
contributing curiosities however, but observations which no one has doubted, but which have escaped remark only
because they are always before our eyes.
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416. "Human beings agree in saying that they see, hear, feel, and so on (even though some are blind and
some are deaf). So they are their own witnesses that they have consciousness."--But how strange this is! Whom do I
really inform, if I say "I have consciousness"? What is the purpose of saying this to myself, and how can another
person understand me?--Now, expressions like "I see", "I hear", "I am conscious" really have their uses. I tell a
doctor "Now I am hearing with this ear again", or I tell someone who believes I am in a faint "I am conscious again",
and so on.
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417. Do I observe myself, then, and perceive that I am seeing or conscious? And why talk about observation
at all? Why not simply say "I perceive I am conscious"?--But what are the words "I perceive" for here?--why not say
"I am conscious"?--But don't the words "I perceive" here shew that I am attending to my consciousness?--which is
ordinarily not the case.--If so, then the sentence "I perceive I am conscious" does not say that I am conscious, but
that my attention is disposed in such-and-such a way.
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But isn't it a particular experience that occasions my saying "I am conscious again"?--What experience? In
what situations do we say it?
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418. Is my having consciousness a fact of experience?--
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But doesn't one say that a man has consciousness, and that a tree or a stone does not?--What would it be like
if it were otherwise?--Would human beings all be unconscious?--No; not in the ordinary sense of the word. But I,
for instance, should not have consciousness--as I now in fact have it.
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419. In what circumstances shall I say that a tribe has a chief? And the chief must surely have consciousness.
Surely we can't have a chief without consciousness!
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420. But can't I imagine that the people around me are automata, lack consciousness, even though they
behave in the same way as usual?--If I imagine it now--alone in my room--I see people with fixed looks (as in a
trance) going about their business--the idea is perhaps a little uncanny. But just try to keep hold of this idea in the
midst of your ordinary intercourse with others, in the street, say! Say to yourself, for example: "The children over
there are mere automata; all their liveliness is mere automatism." And you will either find these words becoming
quite meaningless; or you will produce in yourself some kind of uncanny feeling, or something of the sort.
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Seeing a living human being as an automaton is analogous to seeing one figure as a limiting case or variant of
another; the cross-pieces of a window as a swastika, for example.
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421. It seems paradoxical to us that we should make such a medley, mixing physical states and states of
consciousness up together in a single report: "He suffered great torments and tossed about restlessly". It is quite
usual; so why do we find it paradoxical? Because we want to say that the sentence deals with both tangibles and
intangibles at once.--But does it worry you if I say: "These three struts give the building stability"? Are three and
stability tangible?--Look at the sentence as an instrument, and at its sense as its employment.
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422. What am I believing in when I believe that men have souls? What am I believing in, when I believe that
this substance contains two carbon rings? In both cases there is a picture in the foreground, but the sense lies far in
the background; that is, the application of the picture is not easy to survey.
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423. Certainly all these things happen in you.--And now all I ask is to understand the expression we
use.--The picture is there. And I am not disputing its validity in any particular case.--Only I also want to understand
the application of the picture.
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424. The picture is there; and I do not dispute its correctness. But what is its application? Think of the
picture of blindness as a darkness in the soul or in the head of the blind man.
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425. In numberless cases we exert ourselves to find a picture and once it is found the application as it were
comes about of itself. In
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this case we already have a picture which forces itself on us at every turn,--but does not help us out of the difficulty,
which only begins here.
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If I ask, for example: "How am I to imagine this mechanism going into this box?"--perhaps a drawing
reduced in scale may serve to answer me. Then I can be told: "You see, it goes in like this"; or perhaps even: "Why
are you surprised? See how it goes here; it is the same there". Of course the latter does not explain anything more: it
simply invites me to apply the picture I am given.
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426. A picture is conjured up which seems to fix the sense unambiguously. The actual use, compared with
that suggested by the picture, seems like something muddied. Here again we get the same thing as in set theory: the
form of expression we use seems to have been designed for a god, who knows what we cannot know; he sees the
whole of each of those infinite series and he sees into human consciousness. For us, of course, these forms of
expression are like pontificals which we may put on, but cannot do much with, since we lack the effective power
that would give these vestments meaning and purpose.
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In the actual use of expressions we make detours, we go by side-roads. We see the straight highway before
us, but of course we cannot use it, because it is permanently closed.
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427. "While I was speaking to him I did not know what was going on in his head." In saying this, one is not
thinking of brain-processes, but of thought-processes. The picture should be taken seriously. We should really like to
see into his head. And yet we only mean what elsewhere we should mean by saying: we should like to know what
he is thinking. I want to say: we have this vivid picture--and that use, apparently contradicting the picture, which
expresses the psychical.
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428. "This queer thing, thought"--but it does not strike us as queer when we are thinking. Thought does not
strike us as mysterious while we are thinking, but only when we say, as it were retrospectively: "How was that
possible?" How was it possible for thought to deal with the very object itself? We feel as if by means of it we had
caught reality in our net.
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429. The agreement, the harmony, of thought and reality consists in this: if I say falsely that something is red,
then, for all that, it isn't red.
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And when I want to explain the word "red" to someone, in the sentence "That is not red", I do it by pointing to
something red.
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430. "Put a ruler against this body; it does not say that the body is of such-and-such a length. Rather is it in
itself--I should like to say--dead, and achieves nothing of what thought achieves."--It is as if we had imagined that
the essential thing about a living man was the outward form. Then we made a lump of wood in that form, and were
abashed to see the stupid block, which hadn't even any similarity to a living being.
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431. "There is a gulf between an order and its execution. It has to be filled by the act of understanding."
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"Only in the act of understanding is it meant that we are to do THIS. The order--why, that is nothing but
sounds, ink-marks.--"
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432. Every sign by itself seems dead. What gives it life?--In use it is alive. Is life breathed into it there?--Or is
the use its life?
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433. When we give an order, it can look as if the ultimate thing sought by the order had to remain
unexpressed, as there is always a gulf between an order and its execution. Say I want someone to make a particular
movement, say to raise his arm. To make it quite clear, I do the movement. This picture seems unambiguous till we
ask: how does he know that he is to make that movement?--How does he know at all what use he is to make of the
signs I give him, whatever they are?--Perhaps I shall now try to supplement the order by means of further signs, by
pointing from myself to him, making encouraging gestures, etc.. Here it looks as if the order were beginning to
stammer.
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As if the signs were precariously trying to produce understanding in us.--But if we now understand them, by
what token do we understand?
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434. The gesture--we should like to say--tries to portray, but cannot do it.
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435. If it is asked: "How do sentences manage to represent?"--the answer might be: "Don't you know? You
certainly see it, when you use them." For nothing is concealed.
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How do sentences do it?--Don't you know? For nothing is hidden.
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But given this answer: "But you know how sentences do it, for nothing is concealed" one would like to retort
"Yes, but it all goes by so quick, and I should like to see it as it were laid open to view."
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436. Here it is easy to get into that dead-end in philosophy, where one believes that the difficulty of the task
consists in our having to describe phenomena that are hard to get hold of, the present experience that slips quickly
by, or something of the kind. Where we find ordinary language too crude, and it looks as if we were having to do,
not with the phenomena of every-day, but with ones that "easily elude us, and, in their coming to be and passing
away, produce those others as an average effect". (Augustine: Manifestissima et usitatissima sunt, et eadem rusus
nimis latent, et nova est inventio eorum.)
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437. A wish seems already to know what will or would satisfy it; a proposition, a thought, what makes it
true--even when that thing is not there at all! Whence this determining of what is not yet there? This despotic
demand? ("The hardness of the logical must.")
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438. "A plan as such is something unsatisfied." (Like a wish, an expectation, a suspicion, and so on.)
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By this I mean: expectation is unsatisfied, because it is the expectation of something; belief, opinion, is
unsatisfied, because it is the opinion that something is the case, something real, something outside the process of
believing.
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439. In what sense can one call wishes, expectations, beliefs, etc. "unsatisfied"? What is our prototype of
nonsatisfaction? Is it a hollow space? And would one call that unsatisfied? Wouldn't this be a metaphor too?--Isn't
what we call nonsatisfaction a feeling--say hunger?
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In a particular system of expressions we can describe an object by means of the words "satisfied" and
"unsatisfied". For example, if we lay it down that we call a hollow cylinder an "unsatisfied cylinder" and the solid
cylinder that fills it "its satisfaction".
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440. Saying "I should like an apple" does not mean: I believe an apple will quell my feeling of
nonsatisfaction. This proposition is not an expression of a wish but of nonsatisfaction.
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441. By nature and by a particular training, a particular education, we are disposed to give spontaneous
expression to wishes in certain circumstances. (A wish is, of course, not such a 'circumstance'.) In this game the
question whether I know what I wish before my wish is
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fulfilled cannot arise at all. And the fact that some event stops my wishing does not mean that it fulfills it. Perhaps I
should not have been satisfied if my wish had been satisfied.
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On the other hand the word "wish" is also used in this way: "I don't know myself what I wish for." ("For
wishes themselves are a veil between us and the thing wished for.")
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Suppose it were asked "Do I know what I long for before I get it?" If I have learned to talk, then I do know.
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442. I see someone pointing a gun and say "I expect a report". The shot is fired.--Well, that was what you
expected; so did that report somehow already exist in your expectation? Or is it just that there is some other kind of
agreement between your expectation and what occurred; that that noise was not contained in your expectation, and
merely accidentally supervened when the expectation was being fulfilled?--But no, if the noise had not occurred, my
expectation would not have been fulfilled; the noise fulfilled it; it was not an accompaniment of the fulfilment like a
second guest accompanying the one I expected.--Was the thing about the event that was not in the expectation too
an accident, an extra provided by fate?--But then what was not an extra? Did something of the shot already occur in
my expectation?--Then what was extra? for wasn't I expecting the whole shot?
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"The report was not so loud as I had expected."--"Then was there a louder bang in your expectation?"
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443. "The red which you imagine is surely not the same (not the same thing) as the red which you see in
front of you; so how can you say that it is what you imagined?"--But haven't we an analogous case with the
propositions "Here is a red patch" and "Here there isn't a red patch"? The word "red" occurs in both; so this word
cannot indicate the presence of something red.
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444. One may have the feeling that in the sentence "I expect he is coming" one is using the words "he is
coming" in a different sense from the one they have in the assertion "He is coming". But if it were so how could I
say that my expectation had been fulfilled? If I wanted to explain the words "he" and "is coming", say by means of
ostensive definitions, the same definitions of these words would go for both sentences.
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But it might now be asked: what's it like for him to come?--The door opens, someone walks in, and so
on.--What's it like for me to
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expect him to come?--I walk up and down the room, look at the clock now and then, and so on.--But the one set of
events has not the smallest similarity to the other! So how can one use the same words in describing them?--But
perhaps I say as I walk up and down: "I expect he'll come in"--Now there is a similarity somewhere. But of what
kind?!
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445. It is in language that an expectation and its fulfilment make contact.
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446. It would be odd to say: "A process looks different when it happens from when it doesn't happen." Or "A
red patch looks different when it is there from when it isn't there--but language abstracts from this difference, for it
speaks of a red patch whether it is there or not."
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447. The feeling is as if the negation of a proposition had to make it true in a certain sense, in order to negate
it.
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(The assertion of the negative proposition contains the proposition which is negated, but not the assertion of
it.)
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448. "If I say I did not dream last night, still I must know where to look for a dream; that is, the proposition 'I
dreamt', applied to this actual situation, may be false, but mustn't be senseless."--Does that mean, then, that you did
after all feel something, as it were the hint of a dream, which made you aware of the place which a dream would
have occupied?
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Again: if I say "I have no pain in my arm", does that mean that I have a shadow of the sensation of pain,
which as it were indicates the place where the pain might be?
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In what sense does my present painless state contain the possibility of pain?
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If anyone says: "For the word 'pain' to have a meaning it is necessary that pain should be recognized as such
when it occurs"--one can reply: "It is not more necessary than that the absence of pain should be recognized."
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449. "But mustn't I know what it would be like if I were in pain?"--We fail to get away from the idea that
using a sentence involves imagining something for every word.
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We do not realize that we calculate, operate, with words, and in the course of time translate them sometimes
into one picture, sometimes into another.--It is as if one were to believe that a written order for a
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cow which someone is to hand over to me always had to be accompanied by an image of a cow, if the order was not
to lose its meaning.
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450. Knowing what someone looks like: being able to call up an image--but also: being able to mimic his
expression. Need one imagine it in order to mimic it? And isn't mimicking it just as good as imagining it?
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451. Suppose I give someone the order "Imagine a red circle here"--and now I say: understanding the order
means knowing what it is like for it to have been carried out--or even: being able to imagine what it is like.....?
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452. I want to say: "If someone could see the mental process of expectation, he would necessarily be seeing
what was being expected."--But that is the case: if you see the expression of an expectation, you see what is being
expected. And in what other way, in what other sense would it be possible to see it?
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453. Anyone who perceived my expectation would necessarily have a direct perception of what was being
expected. That is to say, he would not have to infer it from the process he perceived!--But to say that someone
perceives an expectation makes no sense. Unless indeed it means, for example, that he perceives the expression of
an expectation. To say of an expectant person that he perceives his expectation instead of saying that he expects,
would be an idiotic distortion of the expression.
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454. "Everything is already there in...." How does it come about that this arrow
points? Doesn't it seem to carry in it something besides itself?--"No, not the dead line on paper; only the psychical
thing, the meaning, can do that."--That is both true and false. The arrow points only in the application that a living
being makes of it.
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This pointing is not a hocus-pocus which can be performed only by the soul.
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455. We want to say: "When we mean something, it's like going up to someone, it's not having a dead picture
(of any kind)." We go up to the thing we mean.
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456. "When one means something, it is oneself meaning"; so one is oneself in motion. One is rushing ahead
and so cannot also observe oneself rushing ahead. Indeed not.
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457. Yes: meaning something is like going up to someone.
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458. "An order orders its own execution." So it knows its execution, then, even before it is there?--But that
was a grammatical proposition and it means: If an order runs "Do such-and-such" then executing the order is called
"doing such-and-such."
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459. We say "The order orders this--" and do it; but also "The order orders this: I am to...." We translate it at
one time into a proposition, at another into a demonstration, and at another into action.
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460. Could the justification of an action as fulfilment of an order run like this: "You said 'Bring me a yellow
flower', upon which this one gave me a feeling of satisfaction; that is why I have brought it"? Wouldn't one have to
reply: "But I didn't set you to bring me the flower which should give you that sort of feeling after what I said!"?
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461. In what sense does an order anticipate its execution? By ordering just that which later on is carried
out?--But one would have to say "which later on is carried out, or again is not carried out." And that is to say
nothing.
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"But even if my wish does not determine what is going to be the case, still it does so to speak determine the
theme of a fact, whether the fact fulfils the wish or not." We are--as it were--surprised, not at anyone's knowing the
future, but at his being able to prophesy at all (right or wrong).
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As if the mere prophecy, no matter whether true or false, foreshadowed the future; whereas it knows nothing
of the future and cannot know less than nothing.
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462. I can look for him when he is not there, but not hang him when he is not there.
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One might want to say: "But he must be somewhere there if I am looking for him."--Then he must be
somewhere there too if I don't find him and even if he doesn't exist at all.
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463. "You were looking for him? You can't even have known if he was there!"--But this problem really does
arise when one looks for something in mathematics. One can ask, for example, how was it possible so much as to
look for the trisection of the angle?
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464. My aim is: to teach you to pass from a piece of disguised nonsense to something that is patent
nonsense.
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465. "An expectation is so made that whatever happens has to accord with it, or not."
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Suppose you now ask: then are facts defined one way or the other by an expectation--that is, is it defined for
whatever event may occur whether it fulfils the expectation or not? The answer has to be: "Yes, unless the
expression of the expectation is indefinite; for example, contains a disjunction of different possibilities."
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466. What does man think for? What use is it?--Why does he make boilers according to calculations and not
leave the thickness of their walls to chance? After all it is only a fact of experience that boilers do not explode so
often if made according to these calculations. But just as having once been burnt he would do anything rather than
put his hand into a fire, so he would do anything rather than not calculate for a boiler.--But as we are not interested
in causes,--we shall say: human beings do in fact think: this, for instance, is how they proceed when they make a
boiler.--Now, can't a boiler produced in this way explode? Oh, yes.
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467. Does man think, then, because he has found that thinking pays?--Because he thinks it advantageous to
think?
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(Does he bring his children up because he has found it pays?)
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468. What would shew why he thinks?
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469. And yet one can say that thinking has been found to pay. That there are fewer boiler explosions than
formerly, now that we no longer go by feeling in deciding the thickness of the walls, but make such-and-such
calculations instead. Or since each calculation done by one engineer got checked by a second one.
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470. So we do sometimes think because it has been found to pay.
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471. It often happens that we only become aware of the important facts, if we suppress the question "why?";
and then in the course of our investigations these facts lead us to an answer.
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472. The character of the belief in the uniformity of nature can perhaps be seen most clearly in the case in
which we fear what we expect. Nothing could induce me to put my hand into a flame--although after all it is only in
the past that I have burnt myself.
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473. The belief that fire will burn me is of the same kind as the fear that it will burn me.
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474. I shall get burnt if I put my hand in the fire: that is certainty.
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That is to say: here we see the meaning of certainty. (What it amounts to, not just the meaning of the word
"certainty.")
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475. On being asked for the grounds of a supposition, one bethinks oneself of them. Does the same thing
happen here as when one considers what may have been the causes of an event?
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476. We should distinguish between the object of fear and the cause of fear.
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Thus a face which inspires fear or delight (the object of fear or delight), is not on that account its cause,
but--one might say--its target.
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477. "Why do you believe that you will burn yourself on the hot-plate?"--Have you reasons for this belief;
and do you need reasons?
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478. What kind of reason have I to assume that my finger will feel a resistance when it touches the table?
What kind of reason to believe that it will hurt if this pencil pierces my hand?--When I ask this, a hundred reasons
present themselves, each drowning the voice of the others. "But I have experienced it myself innumerable times, and
as often heard of similar experiences; if it were not so, it would .......; etc."
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479. The question: "On what grounds do you believe this?" might mean: "From what you are now deducing
it (have you just deduced it)?" But it might also mean: "What grounds can you produce for this assumption on
thinking it over?"
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480. Thus one could in fact take "grounds" for an opinion to mean only what a man had said to himself
before he arrived at the opinion. The calculation that he has actually carried out. If it is now asked: But how can
previous experience be a ground for assuming that such-and-such will occur later on?--the answer is: What general
concept have we of grounds for this kind of assumption? This sort of statement about the past is simply what we call
a ground for assuming that this will happen in the future.--And if you are surprised at our playing such a game I refer
you to the effect of a past experience (to the fact that a burnt child fears the fire).
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481. If anyone said that information about the past could not convince him that something would happen in
the future, I should not understand him. One might ask him: What do you expect to be told, then? What sort of
information do you call a ground for such a belief? What do you call "conviction"? In what kind of way do you
expect to be convinced?--If these are not grounds, then what are grounds?--If you say these are not grounds, then
you must surely be able to state what must be the case for us to have the right to say that there are grounds for our
assumption.
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For note: here grounds are not propositions which logically imply what is believed.
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Not that one can say: less is needed for belief than for knowledge.--For the question here is not one of an
approximation to logical inference.
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482. We are misled by this way of putting it: "This is a good ground, for it makes the occurrence of the event
probable." That is as if we had asserted something further about the ground, which justified it as a ground; whereas
to say that this ground makes the occurrence probable is to say nothing except that this ground comes up to a
particular standard of good grounds--but the standard has no grounds!
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483. A good ground is one that looks like this.
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484. One would like to say: "It is a good ground only because it makes the occurrence really probable".
Because it, so to speak, really has an influence on the event; as it were an experiential one.
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485. Justification by experience comes to an end. If it did not it would not be justification.
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486. Does it follow from the sense-impressions which I get that there is a chair over there?--How can a
proposition follow from sense-impressions? Well, does it follow from the propositions which describe the
sense-impressions? No.--But don't I infer that a chair is there from impressions, from sense-data?--I make no
inference!--and yet I sometimes do. I see a photograph for example, and say "There must have been a chair over
there" or again "From what I can see here I infer that there is a chair over there." That is an inference; but not one
belonging to logic. An inference is a transition to an assertion; and so also to the behaviour that corresponds to the
assertion. 'I draw the consequences' not only in words, but also in action.
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Page 137
Was I justified in drawing these consequences? What is called a justification here?--How is the word
"justification" used? Describe language-games. From these you will also be able to see the importance of being
justified.
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487. "I am leaving the room because you tell me to."
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"I am leaving the room, but not because you tell me to."
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Does this proposition describe a connexion between my action and his order; or does it make the
connexion?
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Can one ask: "How do you know that you do it because of this, or not because of this?" And is the answer
perhaps: "I feel it"?
Page 137
488. How do I judge whether it is so? By circumstantial evidence?
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489. Ask yourself: On what occasion, for what purpose, do we say this?
Page 137
What kind of actions accompany these words? (Think of a greeting.) In what scenes will they be used; and
what for?
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490. How do I know that this line of thought has led me to this action?--Well, it is a particular picture: for
example, of a calculation leading to a further experiment in an experimental investigation. It looks like this--and now
I could describe an example.
Page 137
491. Not: "without language we could not communicate with one another"--but for sure: without language
we cannot influence other people in such-and-such ways; cannot build roads and machines, etc.. And also: without
the use of speech and writing people could not communicate.
Page 137
492. To invent a language could mean to invent an instrument for a particular purpose on the basis of the
laws of nature (or consistently with them); but it also has the other sense, analogous to that in which we speak of the
invention of a game.
Page 137
Here I am stating something about the grammar of the word "language", by connecting it with the grammar
of the word "invent".
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493. We say: "The cock calls the hens by crowing"--but doesn't a comparison with our language lie at the
bottom of this?--Isn't the aspect quite altered if we imagine the crowing to set the hens in motion by some kind of
physical causation?
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But if it were shewn how the words "Come to me" act on the person addressed, so that finally, given certain
conditions, the muscles of his
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legs are innervated, and so on--should we feel that that sentence lost the character of a sentence?
Page 138
494. I want to say: It is primarily the apparatus of our ordinary language, of our word-language, that we call
language; and then other things by analogy or comparability with this.
Page 138
495. Clearly, I can establish by experience that a human being (or animal) reacts to one sign as I want him to,
and to another not. That, e.g., a human being goes to the right at the sign "→" and goes to the left at the sign "←";
but that he does not react to the sign " ", as to "→".
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I do not even need to fabricate a case, I only have to consider what is in fact the case; namely, that I can
direct a man who has learned only German, only by using the German language. (For here I am looking at learning
German as adjusting a mechanism to respond to a certain kind of influence; and it may be all one to us whether
someone else has learned the language, or was perhaps from birth constituted to react to sentences in German like a
normal person who has learned German.)
Page 138
496. Grammar does not tell us how language must be constructed in order to fulfil its purpose, in order to
have such-and-such an effect on human beings. It only describes and in no way explains the use of signs.
Page 138
497. The rules of grammar may be called "arbitrary", if that is to mean that the aim of the grammar is nothing
but that of the language.
Page 138
If someone says "If our language had not this grammar, it could not express these facts"--it should be asked
what "could" means here.
Page 138
498. When I say that the orders "Bring me sugar" and "Bring me milk" make sense, but not the combination
"Milk me sugar", that does not mean that the utterance of this combination of words has no effect. And if its effect is
that the other person stares at me and gapes, I don't on that account call it the order to stare and gape, even if that
was precisely the effect that I wanted to produce.
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499. To say "This combination of words makes no sense" excludes it from the sphere of language and
thereby bounds the domain of language. But when one draws a boundary it may be for various kinds of reason. If I
surround an area with a fence or a line or otherwise, the purpose may be to prevent someone from getting in or out;
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but it may also be part of a game and the players be supposed, say, to jump over the boundary; or it may shew
where the property of one man ends and that of another begins; and so on. So if I draw a boundary line that is not
yet to say what I am drawing it for.
Page 139
500. When a sentence is called senseless, it is not as it were its sense that is senseless. But a combination of
words is being excluded from the language, withdrawn from circulation.
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501. "The purpose of language is to express thoughts."--So presumably the purpose of every sentence is to
express a thought. Then what thought is expressed, for example, by the sentence "It's raining"?--
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502. Asking what the sense is. Compare:
"This sentence makes sense."--"What sense?"
"This set of words is a sentence."--"What sentence?"
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503. If I give anyone an order I feel it to be quite enough to give him signs. And I should never say: this is
only words, and I have got to get behind the words. Equally, when I have asked someone something and he gives
me an answer (i.e. a sign) I am content--that was what I expected--and I don't raise the objection: but that's a mere
answer.
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504. But if you say: "How am I to know what he means, when I see nothing but the signs he gives?" then I
say: "How is he to know what he means, when he has nothing but the signs either?"
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505. Must I understand an order before I can act on it?--Certainly, otherwise you wouldn't know what you
had to do!--But isn't there in turn a jump from knowing to doing?--
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506. The absent-minded man who at the order "Right turn!" turns left, and then, clutching his forehead, says
"Oh! right turn" and does a right turn.--What has struck him? An interpretation?
Page 139
507. "I am not merely saying this, I mean something by it."--When we consider what is going on in us when
we mean (and don't merely say) words, it seems to us as if there were something coupled to these words, which
otherwise would run idle.--As if they, so to speak, connected with something in us.
Page 139
508. I say the sentence: "The weather is fine"; but the words are after all arbitrary signs--so let's put "a b c d"
in their place. But now when I read this, I can't connect it straight away with the above sense.--
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I am not used, I might say, to saying "a" instead of "the", "b" instead of "weather", etc.. But I don't mean by that that
I am not used to making an immediate association between the word "the" and "a", but that I am not used to using
"a" in the place of "the"--and therefore in the sense of "the". (I have not mastered this language.)
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(I am not used to measuring temperatures on the Fahrenheit scale. Hence such a measure of temperature
'says' nothing to me.)
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509. Suppose we asked someone "In what sense are these words a description of what you are seeing?"--and
he answers: "I mean this by these words." (Say he was looking at a landscape.) Why is this answer "I mean this...."
no answer at all?
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How does one use words to mean what one sees before one?
Page 140
Suppose I said "a b c d" and meant: the weather is fine. For as I uttered these signs I had the experience
normally had only by someone who had year-in year-out used "a" in the sense of "the", "b" in the sense of
"weather", and so on.--Does "a b c d" now mean: the weather is fine?
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What is supposed to be the criterion for my having had that experience?
Page 140
510. Make the following experiment: say "It's cold here" and mean "It's warm here". Can you do it?--And
what are you doing as you do it? And is there only one way of doing it?
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511. What does "discovering that an expression doesn't make sense" mean?--and what does it mean to say:
"If I mean something by it, surely it must make sense"?--If I mean something by it?--If I mean what by it?!--One
wants to say: a significant sentence is one which one can not merely say, but also think.
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512. It looks as if we could say: "Word-language allows of senseless combinations of words, but the
language of imagining does not allow us to imagine anything senseless."--Hence, too, the language of drawing
doesn't allow of senseless drawings? Suppose they were drawings from which bodies were supposed to be
modelled. In this case some drawings make sense, some not.--What if I imagine senseless combinations of words?
Page 140
513. Consider the following form of expression: "The number of pages in my book is equal to a root of the
equation x3 + 2x - 3 = 0." Or: "I have n friends and n2 + 2n + 2 = 0". Does this sentence make sense? This cannot be
seen immediately. This example shews
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how it is that something can look like a sentence which we understand, and yet yield no sense.
Page 141
(This throws light on the concepts 'understanding' and 'meaning'.)
Page 141
514. A philosopher says that he understands the sentence "I am here", that he means something by it, thinks
something--even when he doesn't think at all how, on what occasions, this sentence is used. And if I say "A rose is
red in the dark too" you positively see this red in the dark before you.
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515. Two pictures of a rose in the dark. One is quite black; for the rose is invisible. In the other, it is painted in
full detail and surrounded by black. Is one of them right, the other wrong? Don't we talk of a white rose in the dark
and of a red rose in the dark? And don't we say for all that that they can't be distinguished in the dark?
Page 141
516. It seems clear that we understand the meaning of the question: "Does the sequence 7777 occur in the
development of π?" It is an English sentence; it can be shewn what it means for 415 to occur in the development of
π; and similar things. Well, our understanding of that question reaches just so far, one may say, as such explanations
reach.
Page 141
517. The question arises: Can't we be mistaken in thinking that we understand a question?
Page 141
For many mathematical proofs do lead us to say that we cannot imagine something which we believed we
could imagine. (E.g., the construction of the heptagon.) They lead us to revise what counts as the domain of the
imaginable.
Page 141
518. Socrates to Theaetetus: "And if someone thinks mustn't he think something?"--Th.: "Yes, he
must."--Soc.: "And if he thinks something, mustn't it be something real?"--Th.: "Apparently."
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And mustn't someone who is painting be painting something--and someone who is painting something be
painting something real!--Well, tell me what the object of painting is: the picture of a man (e.g.), or the man that the
picture portrays?
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519. One wants to say that an order is a picture of the action which was carried out on the order; but also that
it is a picture of the action which is to be carried out on the order.
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520. "If a proposition too is conceived as a picture of a possible state of affairs and is said to shew the
possibility of the state of affairs,
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still the most that the proposition can do is what a painting or relief or film does: and so it can at any rate not set
forth what is not the case. So does it depend wholly on our grammar what will be called (logically) possible and
what not,--i.e. what that grammar permits?"--But surely that is arbitrary!--Is it arbitrary?--It is not every sentence-like
formation that we know how to do something with, not every technique has an application in our life; and when we
are tempted in philosophy to count some quite useless thing as a proposition, that is often because we have not
considered its application sufficiently.
Page 142
521. Compare 'logically possible' with 'chemically possible'. One might perhaps call a combination
chemically possible if a formula with the right valencies existed (e.g. H - O - O - O - H). Of course such a
combination need not exist; but even the formula HO2 cannot have less than no combination corresponding to it in
reality.
Page 142
522. If we compare a proposition to a picture, we must think whether we are comparing it to a portrait (a
historical representation) or to a genre-picture. And both comparisons have point.
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When I look at a genre-picture, it 'tells' me something, even though I don't believe (imagine) for a moment
that the people I see in it really exist, or that there have really been people in that situation. But suppose I ask: "What
does it tell me, then?"
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523. I should like to say "What the picture tells me is itself." That is, its telling me something consists in its
own structure, in its own lines and colours. (What would it mean to say "What this musical theme tells me is
itself"?)
Page 142
524. Don't take it as a matter of course, but as a remarkable fact, that pictures and fictitious narratives give us
pleasure, occupy our minds.
Page 142
("Don't take it as a matter of course" means: find it surprising, as you do some things which disturb you.
Then the puzzling aspect of the latter will disappear, by your accepting this fact as you do the other.)
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((The transition from patent nonsense to something which is disguised nonsense.))
Page 142
525. "After he had said this, he left her as he did the day before."--Do I understand this sentence? Do I
understand it just as I should if I heard it in the course of a narrative? If it were set down in isolation
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I should say, I don't know what it's about. But all the same I should know how this sentence might perhaps be used;
I could myself invent a context for it.
Page 143
(A multitude of familiar paths lead off from these words in every direction.)
Page 143
526. What does it mean to understand a picture, a drawing? Here too there is understanding and failure to
understand. And here too these expressions may mean various kinds of thing. A picture is perhaps a still-life; but I
don't understand one part of it: I cannot see solid objects there, but only patches of colour on the canvas.--Or I see
everything as solid but there are objects that I am not acquainted with (they look like implements, but I don't know
their use).--Perhaps, however, I am acquainted with the objects, but in another sense do not understand the way
they are arranged.
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527. Understanding a sentence is much more akin to understanding a theme in music than one may think.
What I mean is that understanding a sentence lies nearer than one thinks to what is ordinarily called understanding a
musical theme. Why is just this the pattern of variation in loudness and tempo? One would like to say "Because I
know what it's all about." But what is it all about? I should not be able to say. In order to 'explain' I could only
compare it with something else which has the same rhythm (I mean the same pattern). (One says "Don't you see,
this is as if a conclusion were being drawn" or "This is as it were a parenthesis", etc. How does one justify such
comparisons?--There are very different kinds of justification here.)
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528. It would be possible to imagine people who had something not quite unlike a language: a play of
sounds, without vocabulary or grammar. ('Speaking with tongues.')
Page 143
529. "But what would the meaning of the sounds be in such a case?"--What is it in music? Though I don't at
all wish to say that this language of a play of sounds would have to be compared to music.
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530. There might also be a language in whose use the 'soul' of the words played no part. In which, for
example, we had no objection to replacing one word by another arbitrary one of our own invention.
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531. We speak of understanding a sentence in the sense in which it can be replaced by another which says
the same; but also in the sense
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in which it cannot be replaced by any other. (Any more than one musical theme can be replaced by another.)
Page 144
In the one case the thought in the sentence is something common to different sentences; in the other,
something that is expressed only by these words in these positions. (Understanding a poem.)
Page 144
532. Then has "understanding" two different meanings here?--I would rather say that these kinds of use of
"understanding" make up its meaning, make up my concept of understanding.
Page 144
For I want to apply the word "understanding" to all this.
Page 144
533. But in the second case how can one explain the expression, transmit one's comprehension? Ask
yourself: How does one lead anyone to comprehension of a poem or of a theme? The answer to this tells us how
meaning is explained here.
Page 144
534. Hearing a word in a particular sense. How queer that there should be such a thing!
Page 144
Phrased like this, emphasized like this, heard in this way, this sentence is the first of a series in which a
transition is made to these sentences, pictures, actions.
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((A multitude of familiar paths lead off from these words in every direction.))
Page 144
535. What happens when we learn to feel the ending of a church mode as an ending?
Page 144
536. I say: "I can think of this face (which gives an impression of timidity) as courageous too." We do not
mean by this that I can imagine someone with this face perhaps saving someone's life (that, of course, is imaginable
in connexion with any face). I am speaking rather of an aspect of the face itself. Nor do I mean that I can imagine
that this man's face might change so that, in the ordinary sense, it looked courageous; though I may very well mean
that there is a quite definite way in which it can change into a courageous face. The reinterpretation of a facial
expression can be compared to the reinterpretation of a chord in music, when we hear it as a modulation first into
this, then into that key.
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537. It is possible to say "I read timidity in this face" but at all events the timidity does not seem to be merely
associated, outwardly connected, with the face; but fear is there, alive, in the features. If the features change slightly,
we can speak of a corresponding change in the
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fear. If we were asked "Can you think of this face as an expression of courage too?"--we should, as it were, not
know how to lodge courage in these features. Then perhaps I say "I don't know what it would mean for this to be a
courageous face." But what would an answer to such a question be like? Perhaps one says: "Yes, now I understand:
the face as it were shews indifference to the outer world." So we have somehow read courage into the face. Now
once more, one might say, courage fits this face. But what fits what here?
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538. There is a related case (though perhaps it will not seem so) when, for example, we (Germans) are
surprised that in French the predicative adjective agrees with the substantive in gender, and when we explain it to
ourselves by saying: they mean: "the man is a good one."
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539. I see a picture which represents a smiling face. What do I do if I take the smile now as a kind one, now
as malicious? Don't I often imagine it with a spatial and temporal context which is one either of kindness or malice?
Thus I might supply the picture with the fancy that the smiler was smiling down on a child at play, or again on the
suffering of an enemy.
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This is in no way altered by the fact that I can also take the at first sight gracious situation and interpret it
differently by putting it into a wider context.--If no special circumstances reverse my interpretation I shall conceive a
particular smile as kind, call it a "kind" one, react correspondingly.
Page 145
((Probability, frequency.))
Page 145
540. "Isn't it very odd that I should be unable--even without the institution of language and all its
surroundings--to think that it will soon stop raining?"--Do you want to say that it is queer that you should be unable
to say these words and mean them without those surroundings?
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Suppose someone were to point at the sky and come out with a number of unintelligible words. When we
ask him what he means he explains that the words mean "Thank heaven, it'll soon stop raining." He even explains to
us the meaning of the individual words.--I will suppose him suddenly to come to himself and say that the sentence
was completely senseless, but that when he spoke it it had seemed to him like a sentence in a language he knew.
(Positively
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like a familiar quotation.)--What am I to say now? Didn't he understand the sentence as he was saying it? Wasn't the
whole meaning there in the sentence?
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541. But what did his understanding, and the meaning, consist in? He uttered the sounds in a cheerful voice
perhaps, pointing to the sky, while it was still raining but was already beginning to clear up; later he made a
connexion between his words and the English words.
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542. "But the point is, the words felt to him like the words of a language he knew well."--Yes: a criterion for
that is that he later said just that. And now do not say: "The feel of the words in a language we know is of a quite
particular kind." (What is the expression of this feeling?)
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543. Can I not say: a cry, a laugh, are full of meaning?
Page 146
And that means, roughly: much can be gathered from them.
Page 146
544. When longing makes me cry "Oh, if only he would come!" the feeling gives the words 'meaning'. But
does it give the individual words their meanings?
Page 146
But here one could also say that the feeling gave the words truth. And from this you can see how the
concepts merge here. (This recalls the question: what is the meaning of a mathematical proposition?)
Page 146
545. But when one says "I hope he'll come"--doesn't the feeling give the word "hope" its meaning? (And
what about the sentence "I do not hope for his coming any longer"?) The feeling does perhaps give the word "hope"
its special ring; that is, it is expressed in that ring.--If the feeling gives the word its meaning, then here "meaning"
means point. But why is the feeling the point?
Page 146
Is hope a feeling? (Characteristic marks.)
Page 146
546. In this way I should like to say the words "Oh, let him come!" are charged with my desire. And words
can be wrung from us,--like a cry. Words can be hard to say: such, for example, as are used to effect a renunciation,
or to confess a weakness. (Words are also deeds.)
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547. Negation: a 'mental activity'. Negate something and observe what you are doing.--Do you perhaps
inwardly shake your head? And if you do--is this process more deserving of our interest than, say,
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that of writing a sign of negation in a sentence? Do you now know the essence of negation?
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548. What is the difference between the two processes: wishing that something should happen--and wishing
that the same thing should not happen?
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If we want to represent it pictorially, we shall treat the picture of the event in various ways: cross it out, put a
line round it, and so on. But this strikes us as a crude method of expression. In word-language indeed we use the
sign "not". But this is like a clumsy expedient. We think that in thought it is arranged differently.
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549. "How can the word 'not' negate?"--"The sign 'not' indicates that you are to take what follows negatively."
We should like to say: The sign of negation is our occasion for doing something--possibly something very
complicated. It is as if the negation-sign occasioned our doing something. But what? That is not said. It is as if it only
needed to be hinted at; as if we already knew. As if no explanation were needed, for we are in any case already
acquainted with the matter.
------------------
Page 147
(a) "The fact that three negatives yield a negative again must already be contained in the single negative that I
am using now." (The temptation to invent a myth of 'meaning'.)
Page 147
It looks as if it followed from the nature of negation that a double negative is an affirmative. (And there is
something right about this. What? Our nature is connected with both.)
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(b) There cannot be a question whether these or other rules are the correct ones for the use of "not". (I mean,
whether they accord with its meaning.) For without these rules the word has as yet no meaning; and if we change
the rules, it now has another meaning (or none), and in that case we may just as well change the word too.
------------------
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550. Negation, one might say, is a gesture of exclusion, of rejection. But such a gesture is used in a great
variety of cases!551. "Does the same negation occur in: 'Iron does not melt at a hundred degrees Centigrade' and
'Twice two is not five'?" Is this to be decided by introspection; by trying to see what we are thinking as we utter the
two sentences?
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552. Suppose I were to ask: is it clear to us, while we are uttering the sentences "This rod is one yard long"
and "Here is one soldier",
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that we mean different things by "one", that "one" has different meanings?--Not at all.--Say e.g. such a sentence as
"One yard is occupied by one soldier, and so two yards are occupied by two soldiers." Asked "Do you mean the
same thing by both 'ones'?" one would perhaps answer: "Of course I mean the same thing: one!" (Perhaps raising
one finger.)
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553. Now has "1" a different meaning when it stands for a measure and when it stands for a number? If the
question is framed in this way, one will answer in the affirmative.
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554. We can easily imagine human beings with a 'more primitive' logic, in which something corresponding to
our negation is applied only to certain sorts of sentence; perhaps to such as do not themselves contain any negation.
It would be possible to negate the proposition "He is going into the house", but a negation of the negative
proposition would be meaningless, or would count only as a repetition of the negation. Think of means of
expressing negation different from ours: by the pitch of one's voice, for instance. What would a double negation be
like there?
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555. The question whether negation had the same meaning to these people as to us would be analogous to
the question whether the figure "5" meant the same to people whose numbers ended at 5 as to us.
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556. Imagine a language with two different words for negation, "X" and "Y". Doubling "X" yields an
affirmative, doubling "Y" a strengthened negative. For the rest the two words are used alike.--Now have "X" and "Y"
the same meaning in sentences where they occur without being repeated?--We could give various answers to this.
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(a) The two words have different uses. So they have different meanings. But sentences in which they occur
without being repeated and which for the rest are the same make the same sense.
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(b) The two words have the same function in language-games, except for this one difference, which is just a
trivial convention. The use of the two words is taught in the same way, by means of the same actions, gestures,
pictures and so on; and in explanations of the words the difference in the ways they are used is appended as
something incidental, as one of the capricious features of the language. For this reason we shall say that "X" and "Y"
have the same meaning.
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(c) We connect different images with the two negatives. "X" as it
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were turns the sense through 180°. And that is why two such negatives restore the sense to its former position. "Y"
is like a shake of the head. And just as one does not annul a shake of the head by shaking it again, so also one
doesn't cancel one "Y" by a second one. And so even if, practically speaking, sentences with the two signs of
negation come to the same thing, still "X" and "Y" express different ideas.
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557. Now, when I uttered the double negation, what constituted my meaning it as a strengthened negative
and not as an affirmative? There is no answer running: "It consisted in the fact that....." In certain circumstances
instead of saying "This duplication is meant as a strengthening," I can pronounce it as a strengthening. Instead of
saying "The duplication of the negative is meant to cancel it" I can e.g. put brackets.--"Yes, but after all these
brackets may themselves have various roles; for who says that they are to be taken as brackets?" No one does. And
haven't you explained your own conception in turn by means of words? The meaning of the brackets lies in the
technique of applying them. The question is: under what circumstances does it make sense to say "I meant....", and
what circumstances justify me in saying "He meant...."?
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558. What does it mean to say that the "is" in "The rose is red" has a different meaning from the "is" in "twice
two is four"? If it is answered that it means that different rules are valid for these two words, we can say that we have
only one word here.--And if all I am attending to is grammatical rules, these do allow the use of the word "is" in both
connexions.--But the rule which shews that the word "is" has different meanings in these sentences is the one
allowing us to replace the word "is" in the second sentence by the sign of equality, and forbidding this substitution in
the first sentence.
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559. One would like to speak of the function of a word in this sentence. As if the sentence were a mechanism
in which the word had a particular function. But what does this function consist in? How does it come to light? For
there isn't anything hidden--don't we see the whole sentence? The function must come out in operating with the
word. ((Meaning-body.))
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560. "The meaning of a word is what is explained by the explanation of the meaning." I.e.: if you want to
understand the use of the word "meaning", look for what are called "explanations of meaning".
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561. Now isn't it queer that I say that the word "is" is used with two different meanings (as the copula and as
the sign of equality), and should not care to say that its meaning is its use; its use, that is, as the copula and the sign
of equality?
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One would like to say that these two kinds of use do not yield a single meaning; the union under one head is
an accident, a mere inessential.
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562. But how can I decide what is an essential, and what an inessential, accidental, feature of the notation? Is
there some reality lying behind the notation, which shapes its grammar?
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Let us think of a similar case in a game: in draughts a king is marked by putting one piece on top of another.
Now won't one say it is inessential to the game for a king to consist of two pieces?
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563. Let us say that the meaning of a piece is its role in the game.--Now let it be decided by lot which of the
players gets white before any game of chess begins. To this end one player holds a king in each closed fist while the
other chooses one of the two hands at random. Will it be counted as part of the role of the king in chess that it is
used to draw lots in this way?
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564. So I am inclined to distinguish between the essential and the inessential in a game too. The game, one
would like to say, has not only rules but also a point.
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565. Why the same word? In the calculus we make no use of this identity!--Why the same piece for both
purposes?--But what does it mean here to speak of "making use of the identity"? For isn't it a use, if we do in fact
use the same word?
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566. And now it looks as if the use of the same word or the same piece, had a purpose--if the identity is not
accidental, inessential. And as if the purpose were that one should be able to recognize the piece and know how to
play.--Are we talking about a physical or a logical possibility here? If the latter then the identity of the piece is
something to do with the game.
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567. But, after all, the game is supposed to be defined by the rules! So, if a rule of the game prescribes that
the kings are to be used for drawing lots before a game of chess, then that is an essential part of the game. What
objection might one make to this? That one does not see the point of this prescription. Perhaps as one wouldn't see
the point either of a rule by which each piece had to be turned round three times
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before one moved it. If we found this rule in a board-game we should be surprised and should speculate about the
purpose of the rule. ("Was this prescription meant to prevent one from moving without due consideration?")
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568. If I understand the character of the game aright--I might say--then this isn't an essential part of it.
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((Meaning is a physiognomy.))
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569. Language is an instrument. Its concepts are instruments. Now perhaps one thinks that it can make no
great difference which concepts we employ. As, after all, it is possible to do physics in feet and inches as well as in
metres and centimetres; the difference is merely one of convenience. But even this is not true if, for instance,
calculations in some system of measurement demand more time and trouble than it is possible for us to give them.
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570. Concepts lead us to make investigations; are the expression of our interest, and direct our interest.
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571. Misleading parallel: psychology treats of processes in the psychical sphere, as does physics in the
physical.
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Seeing, hearing, thinking, feeling, willing, are not the subject of psychology in the same sense as that in
which the movements of bodies, the phenomena of electricity etc., are the subject of physics. You can see this from
the fact that the physicist sees, hears, thinks about, and informs us of these phenomena, and the psychologist
observes the external reactions (the behaviour) of the subject.
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572. Expectation is, grammatically, a state; like: being of an opinion, hoping for something, knowing
something, being able to do something. But in order to understand the grammar of these states it is necessary to ask:
"What counts as a criterion for anyone's being in such a state?" (States of hardness, of weight, of fitting.)
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573. To have an opinion is a state.--A state of what? Of the soul? Of the mind? Well, of what object does one
say that it has an opinion? Of Mr. N.N. for example. And that is the correct answer.
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One should not expect to be enlightened by the answer to that question. Others go deeper: What, in
particular cases, do we regard as criteria for someone's being of such-and-such an opinion? When do we say: he
reached this opinion at that time? When: he has altered his opinion? And so on. The picture which the answers to
these questions give us shews what gets treated grammatically as a state here.
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574. A proposition, and hence in another sense a thought, can be the 'expression' of belief, hope, expectation,
etc. But believing is not thinking. (A grammatical remark.) The concepts of believing, expecting, hoping are less
distantly related to one another than they are to the concept of thinking.
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575. When I sat down on this chair, of course I believed it would bear me. I had no thought of its possibly
collapsing.
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But: "In spite of everything that he did, I held fast to the belief...." Here there is thought, and perhaps a
constant struggle to renew an attitude.
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576. I watch a slow match burning, in high excitement follow the progress of the burning and its approach to
the explosive. Perhaps I don't think anything at all or have a multitude of disconnected thoughts. This is certainly a
case of expecting.
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577. We say "I am expecting him", when we believe that he will come, though his coming does not occupy
our thoughts. (Here "I am expecting him" would mean "I should be surprised if he didn't come" and that will not be
called the description of a state of mind.) But we also say "I am expecting him" when it is supposed to mean: I am
eagerly awaiting him. We could imagine a language in which different verbs were consistently used in these cases.
And similarly more than one verb where we speak of 'believing', 'hoping' and so on. Perhaps the concepts of such a
language would be more suitable for understanding psychology than the concepts of our language.
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578. Ask yourself: What does it mean to believe Goldbach's theorem? What does this belief consist in? In a
feeling of certainty as we state, hear, or think the theorem? (That would not interest us.) And what are the
characteristics of this feeling? Why, I don't even know how far the feeling may be caused by the proposition itself.
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Am I to say that belief is a particular colouring of our thoughts? Where does this idea come from? Well,
there is a tone of belief, as of doubt.
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I should like to ask: how does the belief connect with this proposition? Let us look and see what are the
consequences of this belief, where it takes us. "It makes me search for a proof of the proposition."--Very well; and
now let us look and see what your searching really consists in. Then we shall know what belief in the proposition
amounts to.
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579. The feeling of confidence. How is this manifested in behaviour?
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580. An 'inner process' stands in need of outward criteria.
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581. An expectation is imbedded in a situation, from which it arises. The expectation of an explosion may,
for example, arise from a situation in which an explosion is to be expected.
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582. If someone whispers "It'll go off now", instead of saying "I expect the explosion any moment", still his
words do not describe a feeling; although they and their tone may be a manifestation of his feeling.
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583. "But you talk as if I weren't really expecting, hoping, now--as I thought I was. As if what were
happening now had no deep significance."--What does it mean to say "What is happening now has significance" or
"has deep significance"? What is a deep feeling? Could someone have a feeling of ardent love or hope for the space
of one second--no matter what preceded or followed this second?--What is happening now has significance--in
these surroundings. The surroundings give it its importance. And the word "hope" refers to a phenomenon of human
life. (A smiling mouth smiles only in a human face.)
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584. Now suppose I sit in my room and hope that N.N. will come and bring me some money, and suppose
one minute of this state could be isolated, cut out of its context; would what happened in it then not be
hope?--Think, for example, of the words which you perhaps utter in this space of time. They are no longer part of
this language. And in different surroundings the institution of money doesn't exist either.
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A coronation is the picture of pomp and dignity. Cut one minute of this proceeding out of its surroundings:
the crown is being placed on the head of the king in his coronation robes.--But in different surroundings gold is the
cheapest of metals, its gleam is thought vulgar. There the fabric of the robe is cheap to produce. A crown is a parody
of a respectable hat. And so on.
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585. When someone says "I hope he'll come"--is this a report about his state of mind, or a manifestation of
his hope?--I can, for example, say it to myself. And surely I am not giving myself a report. It may be a sigh; but it
need not. If I tell someone "I can't keep my mind on my work today; I keep on thinking of his coming"--this will be
called a description of my state of mind.
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586. "I have heard he is coming; I have been waiting for him all day." That is a report on how I have spent the
day.--In conversation I came to the conclusion that a particular event is to be expected, and I draw this conclusion in
the words: "So now I must expect him to come". This may be called the first thought, the first act, of this
expectation.--The exclamation "I'm longing to see him!" may be called an act of expecting. But I can utter the same
words as the result of self-observation, and then they might mean: "So, after all that has happened, I am still longing
to see him." The point is: what led up to these words?
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587. Does it make sense to ask "How do you know that you believe?"--and is the answer: "I know it by
introspection"?
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In some cases it will be possible to say some such thing, in most not.
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It makes sense to ask: "Do I really love her, or am I only pretending to myself?" and the process of
introspection is the calling up of memories; of imagined possible situations, and of the feelings that one would have
if....
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588. "I am revolving the decision to go away to-morrow." (This may be called a description of a state of
mind.)--"Your arguments don't convince me; now as before it is my intention to go away tomorrow." Here one is
tempted to call the intention a feeling. The feeling is one of a certain rigidity; of unalterable determination. (But there
are many different characteristic feelings and attitudes here.)--I am asked: "How long are you staying here?" I reply:
"To-morrow I am going away; it's the end of my holidays."--But over against this: I say at the end of a quarrel "All
right! Then I leave to-morrow!"; I make a decision.
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589. "In my heart I have determined on it." And one is even inclined to point to one's breast as one says it.
Psychologically this way of speaking should be taken seriously. Why should it be taken less seriously than the
assertion that belief is a state of mind? (Luther: "Faith is under the left nipple.")
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590. Someone might learn to understand the meaning of the expression "seriously meaning what one says"
by means of a gesture of pointing at the heart. But now we must ask: "How does it come out that he has learnt it?"
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591. Am I to say that any one who has an intention has an experience of tending towards something? That
there are particular experiences of 'tending'?--Remember this case: if one urgently wants to make some remark, some
objection, in a discussion, it often happens that one opens one's mouth, draws a breath and holds it; if one then
decides to let the objection go, one lets the breath out. The experience of this process is evidently the experience of
veering towards saying something. Anyone who observes me will know that I wanted to say something and then
thought better of it. In this situation, that is.--In a different one he would not so interpret my behaviour, however
characteristic of the intention to speak it may be in the present situation. And is there any reason for assuming that
this same experience could not occur in some quite different situation--in which it has nothing to do with any
'tending'?
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592. "But when you say 'I intend to go away', you surely mean it! Here again it just is the mental act of
meaning that gives the sentence life. If you merely repeat the sentence after someone else, say in order to mock his
way of speaking, then you say it without this act of meaning."--When we are doing philosophy it can sometimes
look like that. But let us really think out various different situations and conversations, and the ways in which that
sentence will be uttered in them.--"I always discover a mental undertone; perhaps not always the same one." And
was there no undertone there when you repeated the sentence after someone else? And how is the 'undertone' to be
separated from the rest of the experience of speaking?
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593. A main cause of philosophical disease--a one-sided diet: one nourishes one's thinking with only one
kind of example.
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594. "But the words, significantly uttered, have after all not only a surface, but also the dimension of depth!"
After all, it just is the case that something different takes place when they are uttered significantly from when they
are merely uttered.--How I express this is not the point. Whether I say that in the first case they have depth; or that
something goes on in me, inside my mind, as I utter them; or that they have an atmosphere--it always comes to the
same thing.
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"Well, if we all agree about it, won't it be true?"
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(I cannot accept someone else's testimony, because it is not testimony. It only tells me what he is inclined to
say.)
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595. It is natural for us to say a sentence in such-and-such surroundings, and unnatural to say it in isolation.
Are we to say that
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there is a particular feeling accompanying the utterance of every sentence when we say it naturally?
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596. The feeling of 'familiarity' and of 'naturalness'. It is easier to get at a feeling of unfamiliarity and of
unnaturalness. Or, at feelings. For not everything which is unfamiliar to us makes an impression of unfamiliarity
upon us. Here one has to consider what we call "unfamiliar". If a boulder lies on the road, we know it for a boulder,
but perhaps not for the one which has always lain there. We recognize a man, say, as a man, but not as an
acquaintance. There are feelings of old acquaintance: they are sometimes expressed by a particular way of looking or
by the words: "The same old room!" (which I occupied many years before and now returning find unchanged).
Equally there are feelings of strangeness. I stop short, look at the object or man questioningly or mistrustfully, say "I
find it all strange."--But the existence of this feeling of strangeness does not give us a reason for saying that every
object which we know well and which does not seem strange to us gives us a feeling of familiarity.--We think that,
as it were, the place once filled by the feeling of strangeness must surely be occupied somehow. The place for this
kind of atmosphere is there, and if one of them is not in possession of it, then another is.
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597. Just as Germanisms creep into the speech of a German who speaks English well although he does not
first construct the German expression and then translate it into English; just as this makes him speak English as if he
were translating 'unconsciously' from the German--so we often think as if our thinking were founded on a
thought-schema: as if we were translating from a more primitive mode of thought into ours.
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598. When we do philosophy, we should like to hypostatize feelings where there are none. They serve to
explain our thoughts to us.
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'Here explanation of our thinking demands a feeling!' It is as if our conviction were simply consequent upon
this requirement.
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599. In philosophy we do not draw conclusions. "But it must be like this!" is not a philosophical proposition.
Philosophy only states what everyone admits.
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600. Does everything that we do not find conspicuous make an impression of inconspicuousness? Does
what is ordinary always make the impression of ordinariness?
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601. When I talk about this table,--am I remembering that this object is called a "table"?
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602. Asked "Did you recognize your desk when you entered your room this morning?"--I should no doubt
say "Certainly!" And yet it would be misleading to say that an act of recognition had taken place. Of course the desk
was not strange to me; I was not surprised to see it, as I should have been if another one had been standing there, or
some unfamiliar kind of object.
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603. No one will say that every time I enter my room, my long-familiar surroundings, there is enacted a
recognition of all that I see and have seen hundreds of times before.
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604. It is easy to have a false picture of the processes called "recognizing"; as if recognizing always consisted
in comparing two impressions with one another. It is as if I carried a picture of an object with me and used it to
perform an identification of an object as the one represented by the picture. Our memory seems to us to be the agent
of such a comparison, by preserving a picture of what has been seen before, or by allowing us to look into the past
(as if down a spy-glass).
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605. And it is not so much as if I were comparing the object with a picture set beside it, but as if the object
coincided with the picture. So I see only one thing, not two.
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606. We say "The expression in his voice was genuine". If it was spurious we think as it were of another one
behind it.--This is the face he shews the world, inwardly he has another one.--But this does not mean that when his
expression is genuine he has two the same.
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(("A quite particular expression."))
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607. How does one judge what time it is? I do not mean by external evidences, however, such as the position
of the sun, the lightness of the room, and so on.--One asks oneself, say, "What time can it be?", pauses a moment,
perhaps imagines a clock-face, and then says a time.--Or one considers various possibilities, thinks first of one time,
then of another, and in the end stops at one. That is the kind of way it is done.--But isn't the idea accompanied by a
feeling of conviction; and doesn't that mean that it accords with an inner clock?--No, I don't read the time off from
any clock; there is a feeling of conviction inasmuch as I say a time to myself without feeling any doubt, with calm
assurance.--But doesn't something click as I say
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this time?--Not that I know of; unless that is what you call the coming-to-rest of deliberation, the stopping at one
number. Nor should I ever have spoken of a 'feeling of conviction' here, but should have said: I considered a while
and then plumped for its being quarter past five.--But what did I go by? I might perhaps have said: "simply by feel",
which only means that I left it to what should suggest itself.--But you surely must at least have disposed yourself in
a definite way in order to guess the time; and you don't take just any idea of a time of day as yielding the correct
time!--To repeat: I asked myself "I wonder what time it is?" That is, I did not, for example, read this question in
some narrative, or quote it as someone else's utterance; nor was I practising the pronunciation of these words; and so
on. These were not the circumstances of my saying the words.--But then, what were the circumstances?--I was
thinking about my breakfast and wondering whether it would be late today. These were the kind of
circumstances.--But do you really not see that you were all the same disposed in a way which, though impalpable, is
characteristic of guessing the time, like being surrounded by a characteristic atmosphere?--Yes; what was
characteristic was that I said to myself "I wonder what time it is?"--And if this sentence has a particular atmosphere,
how am I to separate it from the sentence itself? It would never have occurred to me to think the sentence had such
an aura if I had not thought of how one might say it differently--as a quotation, as a joke, as practice in elocution,
and so on. And then all at once I wanted to say, then all at once it seemed to me, that I must after all have meant the
words somehow specially; differently, that is, from in those other cases. The picture of the special atmosphere
forced itself upon me; I can see it quite clear before me--so long, that is, as I do not look at what my memory tells
me really happened.
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And as for the feeling of certainty: I sometimes say to myself "I am sure it's .... o'clock", and in a more or less
confident tone of voice, and so on. If you ask me the reason for this certainty I have none.
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If I say, I read it off from an inner clock,--that is a picture, and the only thing that corresponds to it is that I
said it was such-and-such a time. And the purpose of the picture is to assimilate this case to the other one. I am
refusing to acknowledge two different cases here.
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608. The idea of the intangibility of that mental state in estimating the time is of the greatest importance. Why
is it intangible? Isn't it
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because we refuse to count what is tangible about our state as part of the specific state which we are postulating?
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609. The description of an atmosphere is a special application of language, for special purposes.
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((Interpreting 'understanding' as atmosphere; as a mental act. One can construct an atmosphere to attach to
anything. 'An indescribable character.'))
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610. Describe the aroma of coffee.--Why can't it be done? Do we lack the words? And for what are words
lacking?--But how do we get the idea that such a description must after all be possible? Have you ever felt the lack of
such a description? Have you tried to describe the aroma and not succeeded?
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((I should like to say: "These notes say something glorious, but I do not know what." These notes are a
powerful gesture, but I cannot put anything side by side with it that will serve as an explanation. A grave nod. James:
"Our vocabulary is inadequate." Then why don't we introduce a new one? What would have to be the case for us to
be able to?))
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611. "Willing too is merely an experience," one would like to say (the 'will' too only 'idea'). It comes when it
comes, and I cannot bring it about.
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Not bring it about?--Like what? What can I bring about, then? What am I comparing willing with when I say
this?
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612. I should not say of the movement of my arm, for example: it comes when it comes, etc.. And this is the
region in which we say significantly that a thing doesn't simply happen to us, but that we do it. "I don't need to wait
for my arm to go up--I can raise it." And here I am making a contrast between the movement of my arm and, say,
the fact that the violent thudding of my heart will subside.
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613. In the sense in which I can ever bring anything about (such as stomach-ache through over-eating), I can
also bring about an act of willing. In this sense I bring about the act of willing to swim by jumping into the water.
Doubtless I was trying to say: I can't will willing; that is, it makes no sense to speak of willing willing. "Willing" is
not the name of an action; and so not the name of any voluntary action either. And my use of a wrong expression
came from our wanting to think of willing as an immediate non-causal bringing-about. A misleading analogy lies at
the root of this idea; the causal
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nexus seems to be established by a mechanism connecting two parts of a machine. The connexion may be broken if
the mechanism is disturbed. (We think only of the disturbances to which a mechanism is normally subject, not, say,
of cog-wheels suddenly going soft, or passing through one another, and so on.)
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614. When I raise my arm 'voluntarily' I do not use any instrument to bring the movement about. My wish is
not such an instrument either.
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615. "Willing, if it is not to be a sort of wishing, must be the action itself. It cannot be allowed to stop
anywhere short of the action." If it is the action, then it is so in the ordinary sense of the word; so it is speaking,
writing, walking, lifting a thing, imagining something. But it is also trying, attempting, making an effort,--to speak, to
write, to lift a thing, to imagine something etc..
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616. When I raise my arm, I have not wished it might go up. The voluntary action excludes this wish. It is
indeed possible to say: "I hope I shall draw the circle faultlessly". And that is to express a wish that one's hand
should move in such-and-such a way.
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617. If we cross our fingers in a certain special way we are sometimes unable to move a particular finger
when someone tells us to do so, if he only points to the finger--merely shews it to the eye. If on the other hand he
touches it, we can move it. One would like to describe this experience as follows: we are unable to will to move the
finger. The case is quite different from that in which we are not able to move the finger because someone is, say,
holding it. One now feels inclined to describe the former case by saying: one can't find any point of application for
the will till the finger is touched. Only when one feels the finger can the will know where it is to catch hold.--But this
kind of expression is misleading. One would like to say: "How am I to know where I am to catch hold with the will,
if feeling does not shew the place?" But then how is it known to what point I am to direct the will when the feeling is
there?
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That in this case the finger is as it were paralysed until we feel a touch on it is shewn by experience; it could
not have been seen a priori.
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618. One imagines the willing subject here as something without any mass (without any inertia); as a motor
which has no inertia in itself to overcome. And so it is only mover, not moved. That is:
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One can say "I will, but my body does not obey me"--but not: "My will does not obey me." (Augustine.)
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But in the sense in which I cannot fail to will, I cannot try to will either.
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619. And one might say: "I can always will only inasmuch as I can never try to will."
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620. Doing itself seems not to have any volume of experience. It seems like an extensionless point, the point
of a needle. This point seems to be the real agent. And the phenomenal happenings only to be consequences of this
acting. "I do..." seems to have a definite sense, separate from all experience.
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621. Let us not forget this: when 'I raise my arm', my arm goes up. And the problem arises: what is left over if
I subtract the fact that my arm goes up from the fact that I raise my arm?
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((Are the kinaesthetic sensations my willing?))
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622. When I raise my arm I do not usually try to raise it.
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623. "At all costs I will get to that house."--But if there is no difficulty about it--can I try at all costs to get to
the house?
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624. In the laboratory, when subjected to an electric current, for example, someone says with his eyes shut "I
am moving my arm up and down"--though his arm is not moving. "So," we say, "he has the special feeling of
making that movement."--Move your arm to and fro with your eyes shut. And now try, while you do so, to tell
yourself that your arm is staying still and that you are only having certain queer feelings in your muscles and joints!
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625. "How do you know that you have raised your arm?"--"I feel it." So what you recognize is the feeling?
And are you certain that you recognize it right?--You are certain that you have raised your arm; isn't this the
criterion, the measure, of the recognition?
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626. "When I touch this object with a stick I have the sensation of touching in the tip of the stick, not in the
hand that holds it." When someone says "The pain isn't here in my hand, but in my wrist", this has the consequence
that the doctor examines the wrist. But what difference does it make if I say that I feel the hardness of the
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object in the tip of the stick or in my hand? Does what I say mean "It is as if I had nerve-endings in the tip of the
stick?" In what sense is it like that?--Well, I am at any rate inclined to say "I feel the hardness etc. in the tip of the
stick." What goes with this is that when I touch the object I look not at my hand but at the tip of the stick; that I
describe what I feel by saying "I feel something hard and round there"--not "I feel a pressure against the tips of my
thumb, middle finger, and index finger...." If, for example, someone asks me "What are you now feeling in the
fingers that hold the probe?" I might reply: "I don't know--I feel something hard and rough over there."
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627. Examine the following description of a voluntary action: "I form the decision to pull the bell at 5 o'clock,
and when it strikes 5, my arm makes this movement."--Is that the correct description, and not this one: "..... and
when it strikes 5, I raise my arm"?--One would like to supplement the first description: "and see! my arm goes up
when it strikes 5." And this "and see!" is precisely what doesn't belong here. I do not say "See, my arm is going up!"
when I raise it.
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628. So one might say: voluntary movement is marked by the absence of surprise. And now I do not mean
you to ask "But why isn't one surprised here?"
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629. When people talk about the possibility of foreknowledge of the future they always forget the fact of the
prediction of one's own voluntary movements.
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630. Examine these two language-games:
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(a) Someone gives someone else the order to make particular movements with his arm, or to assume
particular bodily positions (gymnastics instructor and pupil). And here is a variation of this language-game: the pupil
gives himself orders and then carries them out.
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(b) Someone observes certain regular processes--for example, the reactions of different metals to acids--and
thereupon makes predictions about the reactions that will occur in certain particular cases.
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There is an evident kinship between these two language-games, and also a fundamental difference. In both
one might call the spoken words "predictions". But compare the training which leads to the first technique with the
training for the second one.
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631. "I am going to take two powders now, and in half-an-hour I shall be sick."--It explains nothing to say
that in the first case I am the agent, in the second merely the observer. Or that in the first case I see the causal
connexion from inside, in the second from outside. And much else to the same effect.
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Nor is it to the point to say that a prediction of the first kind is no more infallible than one of the second kind.
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It was not on the ground of observations of my behaviour that I said I was going to take two powders. The
antecedents of this proposition were different. I mean the thoughts, actions and so on which led up to it. And it can
only mislead you to say: "The only essential presupposition of your utterance was just your decision."
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632. I do not want to say that in the case of the expression of intention "I am going to take two powders" the
prediction is a cause--and its fulfilment the effect. (Perhaps a physiological investigation could determine this.) So
much, however, is true: we can often predict a man's actions from his expression of a decision. An important
language-game.
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633. "You were interrupted a while ago; do you still know what you were going to say?"--If I do know now,
and say it--does that mean that I had already thought it before, only not said it? No. Unless you take the certainty
with which I continue the interrupted sentence as a criterion of the thought's already having been completed at that
time.--But, of course, the situation and the thoughts which I had contained all sorts of things to help the continuation
of the sentence.
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634. When I continue the interrupted sentence and say that this was how I had been going to continue it, this
is like following out a line of thought from brief notes.
Page 163
Then don't I interpret the notes? Was only one continuation possible in these circumstances? Of course not.
But I did not choose between interpretations. I remembered that I was going to say this.
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635. "I was going to say....."--You remember various details. But not even all of them together shew your
intention. It is as if a snapshot of a scene had been taken, but only a few scattered details of it were to be seen: here a
hand, there a bit of a face, or a hat--the rest is dark. And now it is as if we knew quite certainly what the whole
picture represented. As if I could read the darkness.
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636. These 'details' are not irrelevant in the sense in which other circumstances which I can remember equally
well are irrelevant. But if I tell someone "For a moment I was going to say...." he does not learn those details from
this, nor need he guess them. He need not know, for instance, that I had already opened my mouth to speak. But he
can 'fill out the picture' in this way. (And this capacity is part of understanding what I tell him.)
Page 164
637. "I know exactly what I was going to say!" And yet I did not say it.--And yet I don't read it off from
some other process which took place then and which I remember.
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Nor am I interpreting that situation and its antecedents. For I don't consider them and don't judge them.
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638. How does it come about that in spite of this I am inclined to see an interpretation in saying "For a
moment I was going to deceive him"?
Page 164
"How can you be certain that for the space of a moment you were going to deceive him? Weren't your
actions and thoughts much too rudimentary?"
Page 164
For can't the evidence be too scanty? Yes, when one follows it up it seems extraordinarily scanty; but isn't
this because one is taking no account of the history of this evidence? Certain antecedents were necessary for me to
have had a momentary intention of pretending to someone else that I was unwell.
Page 164
If someone says "For a moment....." is he really only describing a momentary process?
Page 164
But not even the whole story was my evidence for saying "For a moment....."
Page 164
639. One would like to say that an opinion develops. But there is a mistake in this too.
Page 164
640. "This thought ties on to thoughts which I have had before."--How does it do so? Through a feeling of
such a tie? But how can a feeling really tie thoughts together?--The word "feeling" is very misleading here. But it is
sometimes possible to say with certainty: "This thought is connected with those earlier thoughts", and yet be unable
to shew the connexion. Perhaps that comes later.
Page 164
641. "My intention was no less certain as it was than it would have been if I had said 'Now I'll deceive
him'."--But if you had said the words, would you necessarily have meant them quite seriously? (Thus
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the most explicit expression of intention is by itself insufficient evidence of intention.)
Page 165
642. "At that moment I hated him."--What happened here? Didn't it consist in thoughts, feelings, and
actions? And if I were to rehearse that moment to myself I should assume a particular expression, think of certain
happenings, breathe in a particular way, arouse certain feelings in myself. I might think up a conversation, a whole
scene in which that hatred flared up. And I might play this scene through with feelings approximating to those of a
real occasion. That I have actually experienced something of the sort will naturally help me to do so.
Page 165
643. If I now become ashamed of this incident, I am ashamed of the whole thing: of the words, of the
poisonous tone, etc..
Page 165
644. "I am not ashamed of what I did then, but of the intention which I had."--And didn't the intention lie
also in what I did? What justifies the shame? The whole history of the incident.
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645. "For a moment I meant to...." That is, I had a particular feeling, an inner experience; and I remember
it.--And now remember quite precisely! Then the 'inner experience' of intending seems to vanish again. Instead one
remembers thoughts, feelings, movements, and also connexions with earlier situations.
Page 165
It is as if one had altered the adjustment of a microscope. One did not see before what is now in focus.
Page 165
646. "Well, that only shews that you have adjusted your microscope wrong. You were supposed to look at a
particular section of the culture, and you are seeing a different one."
Page 165
There is something right about that. But suppose that (with a particular adjustment of the lenses) I did
remember a single sensation; how have I the right to say that it is what I call the "intention"? It might be that (for
example) a particular tickle accompanied every one of my intentions.
Page 165
647. What is the natural expression of an intention?--Look at a cat when it stalks a bird; or a beast when it
wants to escape.
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((Connexion with propositions about sensations.))
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648. "I no longer remember the words I used, but I remember my intention precisely; I meant my words to
quiet him." What does my memory shew me; what does it bring before my mind? Suppose it did
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nothing but suggest those words to me!--and perhaps others which fill out the picture still more exactly.--("I don't
remember my words any more, but I certainly remember their spirit.")
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649. "So if a man has not learned a language, is he unable to have certain memories?" Of course--he cannot
have verbal memories, verbal wishes or fears, and so on. And memories etc., in language, are not mere threadbare
representations of the real experiences; for is what is linguistic not an experience?
Page 166
650. We say a dog is afraid his master will beat him; but not, he is afraid his master will beat him to-morrow.
Why not?
Page 166
651. "I remember that I should have been glad then to stay still longer."--What picture of this wish came
before my mind? None at all. What I see in my memory allows no conclusion as to my feelings. And yet I
remember quite clearly that they were there.
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652. "He measured him with a hostile glance and said...." The reader of the narrative understands this; he has
no doubt in his mind. Now you say: "Very well, he supplies the meaning, he guesses it."--Generally speaking: no.
Generally speaking he supplies nothing, guesses nothing.--But it is also possible that the hostile glance and the
words later prove to have been pretence, or that the reader is kept in doubt whether they are so or not, and so that he
really does guess at a possible interpretation.--But then the main thing he guesses at is a context. He says to himself
for example: The two men who are here so hostile to one another are in reality friends, etc. etc.
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(("If you want to understand a sentence, you have to imagine the psychical significance, the states of mind
involved."))
Page 166
653. Imagine this case: I tell someone that I walked a certain route, going by a map which I had prepared
beforehand. Thereupon I shew him the map, and it consists of lines on a piece of paper; but I cannot explain how
these lines are the map of my movements, I cannot tell him any rule for interpreting the map. Yet I did follow the
drawing with all the characteristic tokens of reading a map. I might call such a drawing a 'private' map; or the
phenomenon that I have described "following a private map". (But this expression would, of course, be very easy to
misunderstand.)
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Could I now say: "I read off my having then meant to do such-and-such,
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as if from a map, although there is no map"? But that means nothing but: I am now inclined to say "I read the
intention of acting thus in certain states of mind which I remember."
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654. Our mistake is to look for an explanation where we ought to look at what happens as a
'proto-phenomenon'. That is, where we ought to have said: this language-game is played.
Page 167
655. The question is not one of explaining a language-game by means of our experiences, but of noting a
language-game.
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656. What is the purpose of telling someone that a time ago I had such-and-such a wish?--Look on the
language-game as the primary thing. And look on the feelings, etc., as you look on a way of regarding the
language-game, as interpretation.
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It might be asked: how did human beings ever come to make the verbal utterances which we call reports of
past wishes or past intentions?
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657. Let us imagine these utterances always taking this form: "I said to myself: 'if only I could stay longer!'"
The purpose of such a statement might be to acquaint someone with my reactions. (Compare the grammar of
"mean" and "vouloir dire".)
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658. Suppose we expressed the fact that a man had an intention by saying "He as it were said to himself 'I
will....'"--That is the picture. And now I want to know: how does one employ the expression "as it were to say
something to oneself"? For it does not mean: to say something to oneself.
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659. Why do I want to tell him about an intention too, as well as telling him what I did?--Not because the
intention was also something which was going on at that time. But because I want to tell him something about
myself, which goes beyond what happened at that time.
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I reveal to him something of myself when I tell him what I was going to do.--Not, however, on grounds of
self-observation, but by way of a response (it might also be called an intuition).
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660. The grammar of the expression "I was then going to say...." is related to that of the expression "I could
then have gone on."
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In the one case I remember an intention, in the other I remember having understood.
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661. I remember having meant him. Am I remembering a process or state?--When did it begin, what was its
course; etc.?
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662. In an only slightly different situation, instead of silently beckoning, he would have said to someone "Tell
N. to come to me." One can now say that the words "I wanted N. to come to me" describe the state of my mind at
that time; and again one may not say so.
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663. If I say "I meant him" very likely a picture comes to my mind, perhaps of how I looked at him, etc.; but
the picture is only like an illustration to a story. From it alone it would mostly be impossible to conclude anything at
all; only when one knows the story does one know the significance of the picture.
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664. In the use of words one might distinguish 'surface grammar' from 'depth grammar'. What immediately
impresses itself upon us about the use of a word is the way it is used in the construction of the sentence, the part of
its use--one might say--that can be taken in by the ear.--And now compare the depth grammar, say of the word "to
mean", with what its surface grammar would lead us to suspect. No wonder we find it difficult to know our way
about.
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665. Imagine someone pointing to his cheek with an expression of pain and saying "abracadabra!"--We ask
"What do you mean?" And he answers "I meant toothache".--You at once think to yourself: How can one 'mean
toothache' by that word? Or what did it mean to mean pain by that word? And yet, in a different context, you would
have asserted that the mental activity of meaning such-and-such was just what was most important in using
language.
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But--can't I say "By 'abracadabra' I mean toothache"? Of course I can; but this is a definition; not a
description of what goes on in me when I utter the word.
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666. Imagine that you were in pain and were simultaneously hearing a nearby piano being tuned. You say
"It'll soon stop." It certainly makes quite a difference whether you mean the pain or the piano-tuning!--Of course;
but what does this difference consist in? I admit, in many cases some direction of the attention will correspond to
your meaning one thing or another, just as a look often does, or a gesture, or a way of shutting one's eyes which
might be called "looking into oneself".
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667. Imagine someone simulating pain, and then saying "It'll get better soon". Can't one say he means the
pain? and yet he is not concentrating his attention on any pain.--And what about when I finally say "It's stopped
now"?
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668. But can't one also lie in this way: one says "It'll stop soon", and means pain--but when asked "What did
you mean?" one answers "The noise in the next room"? In this sort of case one may say: "I was going to answer....
but thought better of it and did answer....."
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669. One can refer to an object when speaking by pointing to it. Here pointing is a part of the language-game.
And now it seems to us as if one spoke of a sensation by directing one's attention to it. But where is the analogy? It
evidently lies in the fact that one can point to a thing by looking or listening.
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But in certain circumstances, even pointing to the object one is talking about may be quite inessential to the
language-game, to one's thought.
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670. Imagine that you were telephoning someone and you said to him: "This table is too tall", and pointed to
the table. What is the role of pointing here? Can I say: I mean the table in question by pointing to it? What is this
pointing for, and what are these words and whatever else may accompany them for?
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671. And what do I point to by the inner activity of listening? To the sound that comes to my ears, and to the
silence when I hear nothing?
Page 169
Listening as it were looks for an auditory impression and hence can't point to it, but only to the place where
it is looking for it.
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672. If a receptive attitude is called a kind of 'pointing' to something--then that something is not the sensation
which we get by means of it.
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673. The mental attitude doesn't 'accompany' what is said in the sense in which a gesture accompanies it. (As
a man can travel alone, and yet be accompanied by my good wishes; or as a room can be empty, and yet full of
light.)
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674. Does one say, for example: "I didn't really mean my pain just now; my mind wasn't on it enough for
that?" Do I ask myself, say: "What did I mean by this word just now? My attention was divided between my pain
and the noise--"?
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675. "Tell me, what was going on in you when you uttered the words....?"--The answer to this is not: "I was
meaning....."!
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676. "I meant this by that word" is a statement which is differently used from one about an affection of the
mind.
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677. On the other hand: "When you were swearing just now, did you really mean it?" This is perhaps as
much as to say: "Were you really angry?"--And the answer may be given as a result of introspection and is often
some such thing as: "I didn't mean it very seriously", "I meant it half jokingly" and so on. There are differences of
degree here.
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And one does indeed also say "I was half thinking of him when I said that."
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678. What does this act of meaning (the pain, or the piano-tuning) consist in? No answer comes--for the
answers which at first sight suggest themselves are of no use.--"And yet at the time I meant the one thing and not
the other." Yes,--now you have only repeated with emphasis something which no one has contradicted anyway.
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679. "But can you doubt that you meant this?"--No; but neither can I be certain of it, know it.
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680. When you tell me that you cursed and meant N. as you did so it is all one to me whether you looked at a
picture of him, or imagined him, uttered his name, or what. The conclusions from this fact that interest me have
nothing to do with these things. On the other hand, however, someone might explain to me that cursing was
effective only when one had a clear image of the man or spoke his name out loud. But we should not say "The point
is how the man who is cursing means his victim."
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681. Nor, of course, does one ask: "Are you sure that you cursed him, that the connexion with him was
established?"
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Then this connexion must be very easy to establish, if one can be so sure of it?! Can know that it doesn't fail
of its object!--Well, can it happen to me, to intend to write to one person and in fact write to another? and how
might it happen?
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682. "You said, 'It'll stop soon'.--Were you thinking of the noise or of your pain?" If he answers "I was
thinking of the piano-tuning"--is he observing that the connexion existed, or is he making it by means
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of these words?--Can't I say both? If what he said was true, didn't the connexion exist--and is he not for all that
making one which did not exist?
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683. I draw a head. You ask "Whom is that supposed to represent?"--I: "It's supposed to be N."--You: "But it
doesn't look like him; if anything, it's rather like M."--When I said it represented N.--was I establishing a connexion
or reporting one? And what connexion did exist?
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684. What is there in favour of saying that my words describe an existing connexion? Well, they relate to
various things which didn't simply make their appearance with the words. They say, for example, that I should have
given a particular answer then, if I had been asked.
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And even if this is only conditional, still it does say something about the past.
Page 171
685. "Look for A" does not mean "Look for B"; but I may do just the same thing in obeying the two orders.
Page 171
To say that something different must happen in the two cases would be like saying that the propositions
"Today is my birthday" and "My birthday is on April 26th" must refer to different days, because they do not make
the same sense.
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686. "Of course I meant B; I didn't think of A at all!"
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"I wanted B to come to me, so as to..."--All this points to a wider context.
Page 171
687. Instead of "I meant him" one can, of course, sometimes say "I thought of him"; sometimes even "Yes,
we were speaking of him." Ask yourself what 'speaking of him' consists in.
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688. In certain circumstances one can say "As I was speaking, I felt I was saying it to you". But I should not
say this if I were in any case talking with you.
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689. "I am thinking of N." "I am speaking of N."
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How do I speak of him? I say, for instance, "I must go and see N today"--But surely that is not enough! After
all, when I say "N" I might mean various people of this name.--"Then there must surely be a further, different
connexion between my talk and N, for otherwise I should still not have meant HIM.
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Certainly such a connexion exists. Only not as you imagine it: namely by means of a mental mechanism.
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(One compares "meaning him" with "aiming at him".)
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690. What about the case where I at one time make an apparently innocent remark and accompany it with a
furtive sidelong glance at someone; and at another time, without any such glance, speak of somebody present
openly, mentioning his name--am I really thinking specially about him as I use his name?
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691. When I make myself a sketch of N's face from memory, I can surely be said to mean him by my
drawing. But which of the processes taking place while I draw (or before or afterwards) could I call meaning him?
Page 172
For one would naturally like to say: when he meant him, he aimed at him. But how is anyone doing that,
when he calls someone else's face to mind?
Page 172
I mean, how does he call HIM to mind?
Page 172
How does he call him?
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692. Is it correct for someone to say: "When I gave you this rule, I meant you to..... in this case"? Even if he
did not think of this case at all as he gave the rule? Of course it is correct. For "to mean it" did not mean: to think of
it. But now the problem is: how are we to judge whether someone meant such-and-such?--The fact that he has, for
example, mastered a particular technique in arithmetic and algebra, and that he taught someone else the expansion
of a series in the usual way, is such a criterion.
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693. "When I teach someone the formation of the series.... I surely mean him to write.... at the hundredth
place."--Quite right; you mean it. And evidently without necessarily even thinking of it. This shews you how
different the grammar of the verb "to mean" is from that of "to think". And nothing is more wrong-headed than
calling meaning a mental activity! Unless, that is, one is setting out to produce confusion. (It would also be possible
to speak of an activity of butter when it rises in price, and if no problems are produced by this it is harmless.)
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PART II
Page 173
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i
Page 174
One can imagine an animal angry, frightened, unhappy, happy, startled. But hopeful? And why not?
Page 174
A dog believes his master is at the door. But can he also believe his master will come the day after
to-morrow?--And what can he not do here?--How do I do it?--How am I supposed to answer this?
Page 174
Can only those hope who can talk? Only those who have mastered the use of a language. That is to say, the
phenomena of hope are modes of this complicated form of life. (If a concept refers to a character of human
handwriting, it has no application to beings that do not write.)
Page 174
"Grief" describes a pattern which recurs, with different variations, in the weave of our life. If a man's bodily
expression of sorrow and of joy alternated, say with the ticking of a clock, here we should not have the characteristic
formation of the pattern of sorrow or of the pattern of joy.
Page 174
"For a second he felt violent pain."--Why does it sound queer to say: "For a second he felt deep grief"? Only
because it so seldom happens?
Page 174
But don't you feel grief now? ("But aren't you playing chess now?" The answer may be affirmative, but that
does not make the concept of grief any more like the concept of a sensation.--The question was really, of course, a
temporal and personal one, not the logical question which we wanted to raise.
Page 174
"I must tell you: I am frightened."
Page 174
"I must tell you: it makes me shiver."--
And one can say this in a smiling tone of voice too.
Page 174
And do you mean to tell me he doesn't feel it? How else does he know it?--But even when he says it as a
piece of information he does not learn it from his sensations.
Page 174
For think of the sensations produced by physically shuddering: the words "it makes me shiver" are
themselves such a shuddering reaction; and if I hear and feel them as I utter them, this belongs among the rest of
those sensations. Now why should the wordless shudder be the ground of the verbal one?
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ii
Page 175
In saying "When I heard this word, it meant.... to me" one refers to a point of time and to a way of using the
word. (Of course, it is this combination that we fail to grasp.)
Page 175
And the expression "I was then going to say....." refers to a point of time and to an action.
Page 175
I speak of the essential references of the utterance in order to distinguish them from other peculiarities of the
expression we use. The references that are essential to an utterance are the ones which would make us translate
some otherwise alien form of expression into this, our customary form.
Page 175
If you were unable to say that the word "till" could be both a verb and a conjunction, or to construct
sentences in which it was now the one and now the other, you would not be able to manage simple schoolroom
exercises. But a schoolboy is not asked to conceive the word in one way or another out of any context, or to report
how he has conceived it.
Page 175
The words "the rose is red" are meaningless if the word "is" has the meaning "is identical with".--Does this
mean: if you say this sentence and mean the "is" as the sign of identity, the sense disintegrates?
Page 175
We take a sentence and tell someone the meaning of each of its words; this tells him how to apply them and
so how to apply the sentence too. If we had chosen a senseless sequence of words instead of the sentence, he would
not learn how to apply the sequence. And if we explain the word "is" as the sign of identity, then he does not learn
how to use the sentence "the rose is red".
Page 175
And yet there is something right about this 'disintegration of the sense'. You get it in the following example:
one might tell someone: if you want to pronounce the salutation "Hail!" expressively, you had better not think of
hailstones as you say it.
Page 175
Experiencing a meaning and experiencing a mental image. "In both cases", we should like to say, "we are
experiencing something, only something different. A different content is proffered--is present--to
consciousness."--What is the content of the experience of imagining? The answer is a picture, or a description. And
what is the content
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of the experience of meaning? I don't know what I am supposed to say to this.--If there is any sense in the above
remark, it is that the two concepts are related like those of 'red' and 'blue'; and that is wrong.
Page 176
Can one keep hold of an understanding of meaning as one can keep hold of a mental image? That is, if one
meaning of a word suddenly strikes me,--can it also stay there in my mind?
Page 176
"The whole scheme presented itself to my mind in a flash and stayed there like that for five minutes." Why
does this sound odd? One would like to think: what flashed on me and what stayed there in my mind can't have
been the same.
Page 176
I exclaimed "Now I have it!"--a sudden start, and then I was able to set the scheme forth in detail. What is
supposed to have stayed in this case? A picture, perhaps. But "Now I have it" did not mean, I have the picture.
Page 176
If a meaning of a word has occurred to you and you have not forgotten it again, you can now use the word
in such-and-such a way.
Page 176
If the meaning has occurred to you, now you know it, and the knowing began when it occurred to you. Then
how is it like an experience of imagining something?
Page 176
If I say "Mr. Scot is not a Scot", I mean the first "Scot" as a proper name, the second one as a common
name. Then do different things have to go on in my mind at the first and second "Scot"? (Assuming that I am not
uttering the sentence 'parrot-wise'.)--Try to mean the first "Scot" as a common name and the second one as a proper
name.--How is it done? When I do it, I blink with the effort as I try to parade the right meanings before my mind in
saying the words.--But do I parade the meanings of the words before my mind when I make the ordinary use of
them?
Page 176
When I say the sentence with this exchange of meanings I feel that its sense disintegrates.--Well, I feel it, but
the person I am saying it to does not. So what harm is done?--"But the point is, when one utters the sentence in the
usual way something else, quite definite, takes place."--What takes place is not this 'parade of the meanings before
one's mind'.
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iii
Page 177
What makes my image of him into an image of him?
Page 177
Not its looking like him.
Page 177
The same question applies to the expression "I see him now vividly before me" as to the image. What makes
this utterance into an utterance about him?--Nothing in it or simultaneous with it ('behind it'). If you want to know
whom he meant, ask him.
Page 177
(But it is also possible for a face to come before my mind, and even for me to be able to draw it, without my
knowing whose it is or where I have seen it.)
Page 177
Suppose, however, that someone were to draw while he had an image or instead of having it, though it were
only with his finger in the air. (This might be called "motor imagery.") He could be asked: "Whom does that
represent?" And his answer would be decisive.--It is quite as if he had given a verbal description: and such a
description can also simply take the place of the image.
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iv
Page 178
"I believe that he is suffering."--Do I also believe that he isn't an automaton?
Page 178
It would go against the grain to use the word in both connexions.
Page 178
(Or is it like this: I believe that he is suffering, but am certain that he is not an automaton? Nonsense!)
Page 178
Suppose I say of a friend: "He isn't an automaton".--What information is conveyed by this, and to whom
would it be information? To a human being who meets him in ordinary circumstances? What information could it
give him? (At the very most that this man always behaves like a human being, and not occasionally like a machine.)
Page 178
"I believe that he is not an automaton", just like that, so far makes no sense.
Page 178
My attitude towards him is an attitude towards a soul. I am not of the opinion that he has a soul.
Page 178
Religion teaches that the soul can exist when the body has disintegrated. Now do I understand this
teaching?--Of course I understand it--I can imagine plenty of things in connexion with it. And haven't pictures of
these things been painted? And why should such a picture be only an imperfect rendering of the spoken doctrine?
Why should it not do the same service as the words? And it is the service which is the point.
Page 178
If the picture of thought in the head can force itself upon us, then why not much more that of thought in the
soul?
Page 178
The human body is the best picture of the human soul.
Page 178
And how about such an expression as: "In my heart I understood when you said that", pointing to one's
heart? Does one, perhaps, not mean this gesture? Of course one means it. Or is one conscious of using a mere
figure? Indeed not.--It is not a figure that we choose, not a simile, yet it is a figurative expression.
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Page 179
Suppose we were observing the movement of a point (for example, a point of light on a screen). It might be
possible to draw important consequences of the most various kinds from the behaviour of this point. And what a
variety of observations can be made here!--The path of the point and certain of its characteristic measures (amplitude
and wave-length for instance), or the velocity and the law according to which it varies, or the number or position of
the places at which it changes discontinuously, or the curvature of the path at these places, and innumerable other
things.--Any of these features of its behaviour might be the only one to interest us. We might, for example, be
indifferent to everything about its movements except for the number of loops it made in a certain time.--And if we
were interested, not in just one such feature, but in several, each might yield us special information, different in kind
from all the rest. This is how it is with the behaviour of man; with the different characteristic features which we
observe in this behaviour.
Page 179
Then psychology treats of behaviour, not of the mind?
Page 179
What do psychologists record?--What do they observe? Isn't it the behaviour of human beings, in particular
their utterances? But these are not about behaviour.
Page 179
"I noticed that he was out of humour." Is this a report about his behaviour or his state of mind? ("The sky
looks threatening": is this about the present or the future?) Both; not side-by-side, however, but about the one via
the other.
Page 179
A doctor asks: "How is he feeling?" The nurse says: "He is groaning". A report on his behaviour. But need
there be any question for them whether the groaning is really genuine, is really the expression of anything? Might
they not, for example, draw the conclusion "If he groans, we must give him more analgesic"--without suppressing a
middle term? Isn't the point the service to which they put the description of behaviour?
Page 179
"But then they make a tacit presupposition." Then what we do in our language-game always rests on a tacit
presupposition.
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Page 180
I describe a psychological experiment: the apparatus, the questions of the experimenter, the actions and
replies of the subject--and then I say that it is a scene in a play.--Now everything is different. So it will be said: If this
experiment were described in the same way in a book on psychology, then the behaviour described would be
understood as the expression of something mental just because it is presupposed that the subject is not taking us in,
hasn't learnt the replies by heart, and other things of the kind.--So we are making a presupposition?
Page 180
Should we ever really express ourselves like this: "Naturally I am presupposing that....."?--Or do we not do
so only because the other person already knows that?
Page 180
Doesn't a presupposition imply a doubt? And doubt may be entirely lacking. Doubting has an end.
Page 180
It is like the relation: physical object--sense-impressions. Here we have two different language-games and a
complicated relation between them.--If you try to reduce their relations to a simple formula you go wrong.
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Page 181
Suppose someone said: every familiar word, in a book for example, actually carries an atmosphere with it in
our minds, a 'corona' of lightly indicated uses.--Just as if each figure in a painting were surrounded by delicate
shadowy drawings of scenes, as it were in another dimension, and in them we saw the figures in different
contexts.--Only let us take this assumption seriously!--Then we see that it is not adequate to explain intention.
Page 181
For if it is like this, if the possible uses of a word do float before us in half-shades as we say or hear it--this
simply goes for us. But we communicate with other people without knowing if they have this experience too.
Page 181
How should we counter someone who told us that with him understanding was an inner process?--How
should we counter him if he said that with him knowing how to play chess was an inner process?--We should say
that when we want to know if he can play chess we aren't interested in anything that goes on inside him.--And if he
replies that this is in fact just what we are interested in, that is, we are interested in whether he can play chess--then
we shall have to draw his attention to the criteria which would demonstrate his capacity, and on the other hand to
the criteria for the 'inner states'.
Page 181
Even if someone had a particular capacity only when, and only as long as, he had a particular feeling, the
feeling would not be the capacity.
Page 181
The meaning of a word is not the experience one has in hearing or saying it, and the sense of a sentence is
not a complex of such experiences.--(How do the meanings of the individual words make up the sense of the
sentence "I still haven't seen him yet"?) The sentence is composed of the words, and that is enough.
Page 181
Though--one would like to say--every word has a different character in different contexts, at the same time
there is one character it always has: a single physiognomy. It looks at us.--But a face in a painting looks at us too.
Page 181
Are you sure that there is a single if-feeling, and not perhaps several? Have you tried saying the word in a
great variety of contexts? For
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example, when it bears the principal stress of the sentence, and when the word next to it does.
Page 182
Suppose we found a man who, speaking of how words felt to him, told us that "if" and "but" felt the
same.--Should we have the right to disbelieve him? We might think it strange. "He doesn't play our game at all", one
would like to say. Or even: "This is a different type of man."
Page 182
If he used the words "if" and "but" as we do, shouldn't we think he understood them as we do?
Page 182
One misjudges the psychological interest of the if-feeling if one regards it as the obvious correlate of a
meaning; it needs rather to be seen in a different context, in that of the special circumstances in which it occurs.
Page 182
Does a person never have the if-feeling when he is not uttering the word "if"? Surely it is at least remarkable
if this cause alone produces this feeling. And this applies generally to the 'atmosphere' of a word;--why does one
regard it so much as a matter of course that only this word has this atmosphere?
Page 182
The if-feeling is not a feeling which accompanies the word "if".
Page 182
The if-feeling would have to be compared with the special 'feeling' which a musical phrase gives us. (One
sometimes describes such a feeling by saying "Here it is as if a conclusion were being drawn", or "I should like to
say 'hence.....'", or "Here I should always like to make a gesture--" and then one makes it.)
Page 182
But can this feeling be separated from the phrase? And yet it is not the phrase itself, for that can be heard
without the feeling.
Page 182
Is it in this respect like the 'expression' with which the phrase is played?
Page 182
We say this passage gives us a quite special feeling. We sing it to ourselves, and make a certain movement,
and also perhaps have some special sensation. But in a different context we should not recognize these
accompaniments--the movement, the sensation--at all. They are quite empty except just when we are singing this
passage.
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Page 183
"I sing it with a quite particular expression." This expression is not something that can be separated from the
passage. It is a different concept. (A different game.)
Page 183
The experience is this passage played like this (that is, as I am doing it, for instance; a description could only
hint at it).
Page 183
Thus the atmosphere that is inseparable from its object--is not an atmosphere.
Page 183
Closely associated things, things which we have associated, seem to fit one another. But what is this seeming
to fit? How is their seeming to fit manifested? Perhaps like this: we cannot imagine the man who had this name, this
face, this handwriting, not to have produced these works, but perhaps quite different ones instead (those of another
great man).
Page 183
We cannot imagine it? Do we try?--
Page 183
Here is a possibility: I hear that someone is painting a picture "Beethoven writing the ninth symphony". I
could easily imagine the kind of thing such a picture would shew us. But suppose someone wanted to represent
what Goethe would have looked like writing the ninth symphony? Here I could imagine nothing that would not be
embarrassing and ridiculous.
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Page 184
People who on waking tell us certain incidents (that they have been in such-and-such places, etc.). Then we
teach them the expression "I dreamt", which precedes the narrative. Afterwards I sometimes ask them "did you
dream anything last night?" and am answered yes or no, sometimes with an account of a dream, sometimes not.
That is the language-game. (I have assumed here that I do not dream myself. But then, nor do I ever have the feeling
of an invisible presence; other people do, and I can question them about their experiences.)
Page 184
Now must I make some assumption about whether people are deceived by their memories or not; whether
they really had these images while they slept, or whether it merely seems so to them on waking? And what meaning
has this question?--And what interest? Do we ever ask ourselves this when someone is telling us his dream? And if
not--is it because we are sure his memory won't have deceived him? (And suppose it were a man with a quite
specially bad memory?--)
Page 184
Does this mean that it is nonsense ever to raise the question whether dreams really take place during sleep, or
are a memory phenomenon of the awakened? It will turn on the use of the question.
Page 184
"The mind seems able to give a word meaning"--isn't this as if I were to say "The carbon atoms in benzene
seem to lie at the corners of a hexagon"? But this is not something that seems to be so; it is a picture.
Page 184
The evolution of the higher animals and of man, and the awakening of consciousness at a particular level.
The picture is something like this: Though the ether is filled with vibrations the world is dark. But one day man
opens his seeing eye, and there is light.
Page 184
What this language primarily describes is a picture. What is to be done with the picture, how it is to be used,
is still obscure. Quite clearly, however, it must be explored if we want to understand the sense of what we are saying.
But the picture seems to spare us this work: it already points to a particular use. This is how it takes us in.
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Page 185
"My kinaesthetic sensations advise me of the movement and position of my limbs."
Page 185
I let my index finger make an easy pendulum movement of small amplitude. I either hardly feel it, or don't
feel it at all. Perhaps a little in the tip of the finger, as a slight tension. (Not at all in the joint.) And this sensation
advises me of the movement?--for I can describe the movement exactly.
Page 185
"But after all, you must feel it, otherwise you wouldn't know (without looking) how your finger was
moving." But "knowing" it only means: being able to describe it.--I may be able to tell the direction from which a
sound comes only because it affects one ear more strongly than the other, but I don't feel this in my ears; yet it has
its effect: I know the direction from which the sound comes; for instance, I look in that direction.
Page 185
It is the same with the idea that it must be some feature of our pain that advises us of the whereabouts of the
pain in the body, and some feature of our memory image that tells us the time to which it belongs.
Page 185
A sensation can advise us of the movement or position of a limb. (For example, if you do not know, as a
normal person does, whether your arm is stretched out, you might find out by a piercing pain in the elbow.)--In the
same way the character of a pain can tell us where the injury is. (And the yellowness of a photograph how old it is.)
Page 185
What is the criterion for my learning the shape and colour of an object from a sense-impression?
Page 185
What sense-impression? Well, this one; I use words or a picture to describe it.
Page 185
And now: what do you feel when your fingers are in this position?--"How is one to define a feeling? It is
something special and indefinable." But it must be possible to teach the use of the words!
Page 185
What I am looking for is the grammatical difference.
Page 185
Let us leave the kinaesthetic feeling out for the moment.--I want to describe a feeling to someone, and I tell
him "Do this, and then you'll
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get it," and I hold my arm or my head in a particular position. Now is this a description of a feeling? and when shall I
say that he has understood what feeling I meant?--He will have to give a further description of the feeling
afterwards. And what kind of description must it be?
Page 186
I say "Do this, and you'll get it". Can't there be a doubt here? Mustn't there be one, if it is a feeling that is
meant?
Page 186
This looks so; this tastes so; this feels so. "This" and "so" must be differently explained.
Page 186
Our interest in a 'feeling' is of a quite particular kind. It includes, for instance, the 'degree of the feeling', its
'place', and the extent to which one feeling can be submerged by another. (When a movement is very painful, so that
the pain submerges every other slight sensation in the same place, does this make it uncertain whether you have
really made this movement? Could it lead you to find out by looking?)
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Page 187
If you observe your own grief, which senses do you use to observe it? A particular sense; one that feels
grief? Then do you feel it differently when you are observing it? And what is the grief that you are observing--is it
one which is there only while it is being observed?
Page 187
'Observing' does not produce what is observed. (That is a conceptual statement.)
Page 187
Again: I do not 'observe' what only comes into being through observation. The object of observation is
something else.
Page 187
A touch which was still painful yesterday is no longer so today.
Page 187
Today I feel the pain only when I think about it. (That is: in certain circumstances.)
Page 187
My grief is no longer the same; a memory which was still unbearable to me a year ago is now no longer so.
Page 187
That is a result of observation.
Page 187
When do we say that any one is observing? Roughly: when he puts himself in a favourable position to
receive certain impressions in order (for example) to describe what they tell him.
Page 187
If you trained someone to emit a particular sound at the sight of something red, another at the sight of
something yellow, and so on for other colours, still he would not yet be describing objects by their colours. Though
he might be a help to us in giving a description. A description is a representation of a distribution in a space (in that
of time, for instance).
Page 187
If I let my gaze wander round a room and suddenly it lights on an object of a striking red colour, and I say
"Red!"--that is not a description.
Page 187
Are the words "I am afraid" a description of a state of mind?
Page 187
I say "I am afraid"; someone else asks me: "What was that? A cry of fear; or do you want to tell me how you
feel; or is it a reflection on your present state?"--Could I always give him a clear answer? Could I never give him
one?
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Page 188
We can imagine all sorts of things here, for example:
Page 188
"No, no! I am afraid!"
Page 188
"I am afraid. I am sorry to have to confess it."
Page 188
"I am still a bit afraid, but no longer as much as before."
Page 188
"At bottom I am still afraid, though I won't confess it to myself."
Page 188
"I torment myself with all sorts of fears."
Page 188
"Now, just when I should be fearless, I am afraid!"
Page 188
To each of these sentences a special tone of voice is appropriate, and a different context.
Page 188
It would be possible to imagine people who as it were thought much more definitely than we, and used
different words where we use only one.
Page 188
We ask "What does 'I am frightened' really mean, what am I referring to when I say it?" And of course we
find no answer, or one that is inadequate.
Page 188
The question is: "In what sort of context does it occur?"
Page 188
I can find no answer if I try to settle the question "What am I referring to?" "What am I thinking when I say
it?" by repeating the expression of fear and at the same time attending to myself, as it were observing my soul out of
the corner of my eye. In a concrete case I can indeed ask "Why did I say that, what did I mean by it?"--and I might
answer the question too; but not on the ground of observing what accompanied the speaking. And my answer
would supplement, paraphrase, the earlier utterance.
Page 188
What is fear? What does "being afraid" mean? If I wanted to define it at a single shewing--I should play-act
fear.
Page 188
Could I also represent hope in this way? Hardly. And what about belief?
Page 188
Describing my state of mind (of fear, say) is something I do in a particular context. (Just as it takes a
particular context to make a certain action into an experiment.)
Page 188
Is it, then, so surprising that I use the same expression in different games? And sometimes as it were between
the games?
Page 188
And do I always talk with very definite purpose?--And is what I say meaningless because I don't?
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Page 189
When it is said in a funeral oration "We mourn our...." this is surely supposed to be an expression of
mourning; not to tell anything to those who are present. But in a prayer at the grave these words would in a way be
used to tell someone something.
Page 189
But here is the problem: a cry, which cannot be called a description, which is more primitive than any
description, for all that serves as a description of the inner life.
Page 189
A cry is not a description. But there are transitions. And the words "I am afraid" may approximate more, or
less, to being a cry. They may come quite close to this and also be far removed from it.
Page 189
We surely do not always say someone is complaining, because he says he is in pain. So the words "I am in
pain" may be a cry of complaint, and may be something else.
Page 189
But if "I am afraid" is not always something like a cry of complaint and yet sometimes is, then why should it
always be a description of a state of mind?
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Page 190
How did we ever come to use such an expression as "I believe..."? Did we at some time become aware of a
phenomenon (of belief)?
Page 190
Did we observe ourselves and other people and so discover belief?
Page 190
Moore's paradox can be put like this: the expression "I believe that this is the case" is used like the assertion
"This is the case"; and yet the hypothesis that I believe this is the case is not used like the hypothesis that this is the
case.
Page 190
So it looks as if the assertion "I believe" were not the assertion of what is supposed in the hypothesis "I
believe"!
Page 190
Similarly: the statement "I believe it's going to rain" has a meaning like, that is to say a use like, "It's going to
rain", but the meaning of "I believed then that it was going to rain", is not like that of "It did rain then".
Page 190
"But surely 'I believed' must tell of just the same thing in the past as 'I believe' in the present!"--Surely
must mean just the same in relation to - 1, as means in relation to 1! This means nothing at all.
Page 190
"At bottom, when I say 'I believe...' I am describing my own state of mind--but this description is indirectly
an assertion of the fact believed."--As in certain circumstances I describe a photograph in order to describe the thing
it is a photograph of.
Page 190
But then I must also be able to say that the photograph is a good one. So here too: "I believe it's raining and
my belief is reliable, so I have confidence in it."--In that case my belief would be a kind of sense-impression.
Page 190
One can mistrust one's own senses, but not one's own belief.
Page 190
If there were a verb meaning 'to believe falsely', it would not have any significant first person present
indicative.
Page 190
Don't look at it as a matter of course, but as a most remarkable thing, that the verbs "believe", "wish", "will"
display all the inflexions possessed by "cut", "chew", "run".
Page 190
The language-game of reporting can be given such a turn that a report is not meant to inform the hearer about
its subject matter but about the person making the report.
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Page 191
It is so when, for instance, a teacher examines a pupil. (You can measure to test the ruler.)
Page 191
Suppose I were to introduce some expression--"I believe", for instance--in this way: it is to be prefixed to
reports when they serve to give information about the reporter. (So the expression need not carry with it any
suggestion of uncertainty. Remember that the uncertainty of an assertion can be expressed impersonally: "He might
come today".)--"I believe...., and it isn't so" would be a contradiction.
Page 191
"I believe...." throws light on my state. Conclusions about my conduct can be drawn from this expression.
So there is a similarity here to expressions of emotion, of mood, etc..
Page 191
If, however, "I believe it is so" throws light on my state, then so does the assertion "It is so". For the sign "I
believe" can't do it, can at the most hint at it.
Page 191
Imagine a language in which "I believe it is so" is expressed only by means of the tone of the assertion "It is
so". In this language they say, not "He believes" but "He is inclined to say...." and there exists also the hypothetical
(subjunctive) "Suppose I were inclined etc.", but not the expression "I am inclined to say".
Page 191
Moore's paradox would not exist in this language; instead of it, however, there would be a verb lacking one
inflexion.
Page 191
But this ought not to surprise us. Think of the fact that one can predict one's own future action by an
expression of intention.
Page 191
I say of someone else "He seems to believe...." and other people say it of me. Now, why do I never say it of
myself, not even when others rightly say it of me?--Do I myself not see and hear myself, then?--That can be said.
Page 191
"One feels conviction within oneself, one doesn't infer it from one's own words or their tone."--What is true
here is: one does not infer one's own conviction from one's own words; nor yet the actions which arise from that
conviction.
Page 191
"Here it looks as if the assertion 'I believe' were not the assertion of what is supposed in the hypothesis."--So
I am tempted to look for a different development of the verb in the first person present indicative.
Page 191
This is how I think of it: Believing is a state of mind. It has duration; and that independently of the duration
of its expression in a sentence, for example. So it is a kind of disposition of the believing person. This is shewn me in
the case of someone else by his behaviour; and
Page Break 192
by his words. And under this head, by the expression "I believe...' as well as by the simple assertion.--What about
my own case: how do I myself recognize my own disposition?--Here it will have been necessary for me to take
notice of myself as others do, to listen to myself talking, to be able to draw conclusions from what I say!
Page 192
My own relation to my words is wholly different from other people's.
Page 192
That different development of the verb would have been possible, if only I could say "I seem to believe".
Page 192
If I listened to the words of my mouth, I might say that someone else was speaking out of my mouth.
Page 192
"Judging from what I say, this is what I believe." Now, it is possible to think out circumstances in which
these words would make sense.
Page 192
And then it would also be possible for someone to say "It is raining and I don't believe it", or "It seems to me
that my ego believes this, but it isn't true." One would have to fill out the picture with behaviour indicating that two
people were speaking through my mouth.
Page 192
Even in the hypothesis the pattern is not what you think.
Page 192
When you say "Suppose I believe...." you are presupposing the whole grammar of the word "to believe", the
ordinary use, of which you are master.--You are not supposing some state of affairs which, so to speak, a picture
presents unambiguously to you, so that you can tack on to this hypothetical use some assertive use other than the
ordinary one.--You would not know at all what you were supposing here (i.e. what, for example, would follow from
such a supposition), if you were not already familiar with the use of "believe".
Page 192
Think of the expression "I say....", for example in "I say it will rain today", which simply comes to the same
thing as the assertion "It will....". "He says it will...." means approximately "He believes it will....". "Suppose I say...."
does not mean: Suppose it rains today.
Page 192
Different concepts touch here and coincide over a stretch. But you need not think that all lines are circles.
Page 192
Consider the misbegotten sentence "It may be raining, but it isn't".
Page 192
And here one should be on one's guard against saying that "It may be raining" really means "I think it'll be
raining." For why not the other way round, why should not the latter mean the former?
Page 192
Don't regard a hesitant assertion as an assertion of hesitancy.
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xi
Page 193
Two uses of the word "see".
Page 193
The one: "What do you see there?"--"I see this" (and then a description, a drawing, a copy). The other: "I see
a likeness between these two faces"--let the man I tell this to be seeing the faces as clearly as I do myself.
Page 193
The importance of this is the difference of category between the two 'objects' of sight.
Page 193
The one man might make an accurate drawing of the two faces, and the other notice in the drawing the
likeness which the former did not see.
Page 193
I contemplate a face, and then suddenly notice its likeness to another. I see that it has not changed; and yet I
see it differently. I call this experience "noticing an aspect".
Page 193
Its causes are of interest to psychologists.
Page 193
We are interested in the concept and its place among the concepts of experience.
Page 193
You could imagine the illustration
appearing in several places in a book, a text-book for instance. In the relevant text something different is in question
every time: here a glass cube, there an inverted open box, there a wire frame of that shape, there three boards
forming a solid angle. Each time the text supplies the interpretation of the illustration.
Page 193
But we can also see the illustration now as one thing now as another.--So we interpret it, and see it as we
interpret it.
Page 193
Here perhaps we should like to reply: The description of what is got immediately, i.e. of the visual
experience, by means of an interpretation--is an indirect description. "I see the figure as a box" means: I have a
particular visual experience which I have found that I always have when I interpret the figure as a box or when I look
at
Page Break 194
a box. But if it meant this I ought to know it. I ought to be able to refer to the experience directly, and not only
indirectly. (As I can speak of red without calling it the colour of blood.)
Page 194
I shall call the following figure, derived from Jastrow†1, the duck-rabbit. It can be seen as a rabbit's head or
as a duck's.
And I must distinguish between the 'continuous seeing' of an aspect and the 'dawning' of an aspect.
Page 194
The picture might have been shewn me, and I never have seen anything but a rabbit in it.
Page 194
Here it is useful to introduce the idea of a picture-object. For instance
would be a 'picture-face'.
Page 194
In some respects I stand towards it as I do towards a human face. I can study its expression, can react to it as
to the expression of the human face. A child can talk to picture-men or picture-animals, can treat them as it treats
dolls.
Page 194
I may, then, have seen the duck-rabbit simply as a picture-rabbit from the first. That is to say, if asked
"What's that?" or "What do you see here?" I should have replied: "A picture-rabbit". If I had further been asked what
that was, I should have explained by pointing to all sorts of pictures of rabbits, should perhaps have pointed to real
rabbits, talked about their habits, or given an imitation of them.
Page 194
I should not have answered the question "What do you see here?" by saying: "Now I am seeing it as a
picture-rabbit". I should simply
Page Break 195
have described my perception: just as if I had said "I see a red circle over there."--
Page 195
Nevertheless someone else could have said of me: "He is seeing the figure as a picture-rabbit."
Page 195
It would have made as little sense for me to say "Now I am seeing it as..." as to say at the sight of a knife and
fork "Now I am seeing this as a knife and fork". This expression would not be understood.--Any more than: "Now
it's a fork" or "It can be a fork too".
Page 195
One doesn't 'take' what one knows as the cutlery at a meal for cutlery; any more than one ordinarily tries to
move one's mouth as one eats, or aims at moving it.
Page 195
If you say "Now it's a face for me", we can ask: "What change are you alluding to?"
Page 195
I see two pictures, with the duck-rabbit surrounded by rabbits in one, by ducks in the other. I do not notice
that they are the same. Does it follow from this that I see something different in the two cases?--It gives us a reason
for using this expression here.
Page 195
"I saw it quite differently, I should never have recognized it!" Now, that is an exclamation. And there is also a
justification for it.
Page 195
I should never have thought of superimposing the heads like that, of making this comparison between them.
For they suggest a different mode of comparison.
Page 195
Nor has the head seen like this the slightest similarity to the head seen like this--although they are congruent.
Page 195
I am shewn a picture-rabbit and asked what it is; I say "It's a rabbit". Not "Now it's a rabbit". I am reporting
my perception.--I am shewn the duck-rabbit and asked what it is; I may say "It's a duck-rabbit". But I may also react
to the question quite differently.--The answer that it is a duck-rabbit is again the report of a perception; the answer
"Now it's a rabbit" is not. Had I replied "It's a rabbit", the ambiguity would have escaped me, and I should have been
reporting my perception.
Page 195
The change of aspect. "But surely you would say that the picture is altogether different now!"
Page 195
But what is different: my impression? my point of view?--Can I say? I describe the alteration like a
perception; quite as if the object had altered before my eyes.
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Page 196
"Now I am seeing this", I might say (pointing to another picture, for example). This has the form of a report
of a new perception.
Page 196
The expression of a change of aspect is the expression of a new perception and at the same time of the
perception's being unchanged.
Page 196
I suddenly see the solution of a puzzle-picture. Before, there were branches there; now there is a human
shape. My visual impression has changed and now I recognize that it has not only shape and colour but also a quite
particular 'organization'.--My visual impression has changed;--what was it like before and what is it like now?--If I
represent it by means of an exact copy--and isn't that a good representation of it?--no change is shewn.
Page 196
And above all do not say "After all my visual impression isn't the drawing; it is this--which I can't shew to
anyone."--Of course it is not the drawing, but neither is it anything of the same category, which I carry within
myself.
Page 196
The concept of the 'inner picture' is misleading, for this concept uses the 'outer picture' as a model; and yet
the uses of the words for these concepts are no more like one another than the uses of 'numeral' and 'number'. (And
if one chose to call numbers 'ideal numerals', one might produce a similar confusion.)
Page 196
If you put the 'organization' of a visual impression on a level with colours and shapes, you are proceeding
from the idea of the visual impression as an inner object. Of course this makes this object into a chimera; a queerly
shifting construction. For the similarity to a picture is now impaired.
Page 196
If I know that the schematic cube has various aspects and I want to find out what someone else sees, I can
get him to make a model of what he sees, in addition to a copy, or to point to such a model; even though he has no
idea of my purpose in demanding two accounts.
Page 196
But when we have a changing aspect the case is altered. Now the only possible expression of our experience
is what before perhaps seemed, or even was, a useless specification when once we had the copy.
Page 196
And this by itself wrecks the comparison of 'organization' with colour and shape in visual impressions.
Page 196
If I saw the duck-rabbit as a rabbit, then I saw: these shapes and colours (I give them in detail)--and I saw
besides something like this:
Page Break 197
and here I point to a number of different pictures of rabbits.--This shews the difference between the concepts.
Page 197
'Seeing as....' is not part of perception. And for that reason it is like seeing and again not like.
Page 197
I look at an animal and am asked: "What do you see?" I answer: "A rabbit".--I see a landscape; suddenly a
rabbit runs past. I exclaim "A rabbit!"
Page 197
Both things, both the report and the exclamation, are expressions of perception and of visual experience. But
the exclamation is so in a different sense from the report: it is forced from us.--It is related to the experience as a cry
is to pain.
Page 197
But since it is the description of a perception, it can also be called the expression of thought.--If you are
looking at the object, you need not think of it; but if you are having the visual experience expressed by the
exclamation, you are also thinking of what you see.
Page 197
Hence the flashing of an aspect on us seems half visual experience, half thought.
Page 197
Someone suddenly sees an appearance which he does not recognize (it may be a familiar object, but in an
unusual position or lighting); the lack of recognition perhaps lasts only a few seconds. Is it correct to say he has a
different visual experience from someone who knew the object at once?
Page 197
For might not someone be able to describe an unfamiliar shape that appeared before him just as accurately
as I, to whom it is familiar? And isn't that the answer?--Of course it will not generally be so. And his description will
run quite differently. (I say, for example, "The animal had long ears"--he: "There were two long appendages", and
then he draws them.)
Page 197
I meet someone whom I have not seen for years; I see him clearly, but fail to know him. Suddenly I know
him, I see the old face in the altered one. I believe that I should do a different portrait of him now if I could paint.
Page 197
Now, when I know my acquaintance in a crowd, perhaps after looking in his direction for quite a while,--is
this a special sort of seeing? Is it a case of both seeing and thinking? or an amalgam of the two, as I should almost
like to say?
Page 197
The question is: why does one want to say this?
Page Break 198
Page 198
The very expression which is also a report of what is seen, is here a cry of recognition.
Page 198
What is the criterion of the visual experience?--The criterion? What do you suppose?
Page 198
The representation of 'what is seen'.
Page 198
The concept of a representation of what is seen, like that of a copy, is very elastic, and so together with it is
the concept of what is seen. The two are intimately connected. (Which is not to say that they are alike.)
Page 198
How does one tell that human beings see three-dimensionally?--I ask someone about the lie of the land (over
there) of which he has a view. "Is it like this?" (I shew him with my hand)--"Yes."--"How do you know?"--"It's not
misty, I see it quite clear."--He does not give reasons for the surmise. The only thing that is natural to us is to
represent what we see three-dimensionally; special practice and training are needed for two-dimensional
representation whether in drawing or in words. (The queerness of children's drawings.)
Page 198
If someone sees a smile and does not know it for a smile, does not understand it as such, does he see it
differently from someone who understands it?--He mimics it differently, for instance.
Page 198
Hold the drawing of a face upside down and you can't recognize the expression of the face. Perhaps you can
see that it is smiling, but not exactly what kind of smile it is. You cannot imitate the smile or describe it more
exactly.
Page 198
And yet the picture which you have turned round may be a most exact representation of a person's face.
Page 198
The figure (a) is the reverse of the figure (b)
As (c) is the reverse of (d) But--I should like to say--there
is a different difference between my impressions of (c) and (d) and between those of (a) and (b). (d), for example,
looks neater than (c). (Compare a remark of Lewis Carroll's.) (d) is easy, (c) hard to copy.
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Page 199
Imagine the duck-rabbit hidden in a tangle of lines. Now I suddenly notice it in the picture, and notice it
simply as the head of a rabbit. At some later time I look at the same picture and notice the same figure, but see it as
the duck, without necessarily realizing that it was the same figure both times.--If I later see the aspect change--can I
say that the duck and rabbit aspects are now seen quite differently from when I recognized them separately in the
tangle of lines? No.
Page 199
But the change produces a surprise not produced by the recognition.
Page 199
If you search in a figure (1) for another figure (2), and then find it, you see (1) in a new way. Not only can
you give a new kind of description of it, but noticing the second figure was a new visual experience.
Page 199
But you would not necessarily want to say "Figure (1) looks quite different now; it isn't even in the least like
the figure I saw before, though they are congruent!"
Page 199
There are here hugely many interrelated phenomena and possible concepts.
Page 199
Then is the copy of the figure an incomplete description of my visual experience? No.--But the
circumstances decide whether, and what, more detailed specifications are necessary.--It may be an incomplete
description; if there is still something to ask.
Page 199
Of course we can say: There are certain things which fall equally under the concept 'picture-rabbit' and under
the concept 'picture-duck'. And a picture, a drawing, is such a thing.--But the impression is not simultaneously of a
picture-duck and a picture-rabbit.
Page 199
"What I really see must surely be what is produced in me by the influence of the object"--Then what is
produced in me is a sort of copy, something that in its turn can be looked at, can be before one; almost something
like a materialization.
Page 199
And this materialization is something spatial and it must be possible to describe it in purely spatial terms. For
instance (if it is a face) it can smile; the concept of friendliness, however, has no place in an account of it, but is
foreign to such an account (even though it may subserve it).
Page 199
If you ask me what I saw, perhaps I shall be able to make a sketch which shews you; but I shall mostly have
no recollection of the way my glance shifted in looking at it.
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Page 200
The concept of 'seeing' makes a tangled impression. Well, it is tangled.--I look at the landscape, my gaze
ranges over it, I see all sorts of distinct and indistinct movement; this impresses itself sharply on me, that is quite
hazy. After all, how completely ragged what we see can appear! And now look at all that can be meant by
"description of what is seen".--But this just is what is called description of what is seen. There is not one genuine
proper case of such description--the rest being just vague, something which awaits clarification, or which must just
be swept aside as rubbish.
Page 200
Here we are in enormous danger of wanting to make fine distinctions.--It is the same when one tries to define
the concept of a material object in terms of 'what is really seen'.--What we have rather to do is to accept the
everyday language-game, and to note false accounts of the matter as false. The primitive language-game which
children are taught needs no justification; attempts at justification need to be rejected.
Page 200
Take as an example the aspects of a triangle. This triangle
can be seen as a triangular hole, as a solid, as a geometrical drawing; as standing on its base, as hanging from its
apex; as a mountain, as a wedge, as an arrow or pointer, as an overturned object which is meant to stand on the
shorter side of the right angle, as a half parallelogram, and as various other things.
Page 200
"You can think now of this now of this as you look at it, can regard it now as this now as this, and then you
will see it now this way, now this."--What way? There is no further qualification.
Page 200
But how is it possible to see an object according to an interpretation?--The question represents it as a queer
fact; as if something were being forced into a form it did not really fit. But no squeezing, no forcing took place here.
Page 200
When it looks as if there were no room for such a form between other ones you have to look for it in another
dimension. If there is no room here, there is room in another dimension.
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Page 201
(It is in this sense too that there is no room for imaginary numbers in the continuum of real numbers. But
what this means is: the application of the concept of imaginary numbers is less like that of real numbers than appears
from the look of the calculations. It is necessary to get down to the application, and then the concept finds a
different place, one which, so to speak, one never dreamed of.)
Page 201
How would the following account do: "What I can see something as, is what it can be a picture of"?
Page 201
What this means is: the aspects in a change of aspects are those ones which the figure might sometimes have
permanently in a picture.
Page 201
A triangle can really be standing up in one picture, be hanging in another, and can in a third be something
that has fallen over.--That is, I who am looking at it say, not "It may also be something that has fallen over", but
"That glass has fallen over and is lying there in fragments". This is how we react to the picture.
Page 201
Could I say what a picture must be like to produce this effect? No. There are, for example, styles of painting
which do not convey anything to me in this immediate way, but do to other people. I think custom and upbringing
have a hand in this.
Page 201
What does it mean to say that I 'see the sphere floating in the air' in a picture?
Page 201
Is it enough that this description is the first to hand, is the matter-of-course one? No, for it might be so for
various reasons. This might, for instance, simply be the conventional description.
Page 201
What is the expression of my not merely understanding the picture in this way, for instance, (knowing what
it is supposed to be), but seeing it in this way?--It is expressed by: "The sphere seems to float", "You see it floating",
or again, in a special tone of voice, "It floats!"
Page 201
This, then, is the expression of taking something for something. But not being used as such.
Page 201
Here we are not asking ourselves what are the causes and what produces this impression in a particular case.
Page 201
And is it a special impression?--"Surely I see something different when I see the sphere floating from when I
merely see it lying there."--This really means: This expression is justified!--(For taken literally it is no more than a
repetition.)
Page Break 202
Page 202
(And yet my impression is not that of a real floating sphere either. There are various forms of
'three-dimensional seeing'. The three-dimensional character of a photograph and the three-dimensional character of
what we see through a stereoscope.)
Page 202
"And is it really a different impression?"--In order to answer this I should like to ask myself whether there is
really something different there in me. But how can I find out?--I describe what I am seeing differently.
Page 202
Certain drawings are always seen as flat figures, and others sometimes, or always, three-dimensionally.
Page 202
Here one would now like to say: the visual impression of what is seen three-dimensionally is
three-dimensional; with the schematic cube, for instance, it is a cube. (For the description of the impression is the
description of a cube.)
Page 202
And then it seems queer that with some drawings our impression should be a flat thing, and with some a
three-dimensional thing. One asks oneself "Where is this going to end?"
Page 202
When I see the picture of a galloping horse--do I merely know that this is the kind of movement meant? Is it
superstition to think I see the horse galloping in the picture?--And does my visual impression gallop too?
Page 202
What does anyone tell me by saying "Now I see it as....."? What consequences has this information? What
can I do with it?
Page 202
People often associate colours with vowels. Someone might find that a vowel changed its colour when it was
repeated over and over again. He finds a 'now blue--now red', for instance.
Page 202
The expression "Now I am seeing it as..." might have no more significance for us than: "Now I find a red".
Page 202
(Linked with physiological observations, even this change might acquire importance for us.)
Page 202
Here it occurs to me that in conversation on aesthetic matters we use the words: "You have to see it like this,
this is how it is meant"; "When you see it like this, you see where it goes wrong"; "You have to hear this bar as an
introduction"; "You must hear it in this key"; "You must phrase it like this" (which can refer to hearing as well as to
playing).
Page Break 203
Page 203
This figure
is supposed to represent a convex step and to be used in some kind of topological demonstration. For this purpose
we draw the straight line a through the geometric centres of the two surfaces.--Now if anyone's three-dimensional
impression of the figure were never more than momentary, and even so were now concave, now convex, that might
make it difficult for him to follow our demonstration. And if he finds that the flat aspect alternates with a
three-dimensional one, that is just as if I were to shew him completely different objects in the course of the
demonstration.
Page 203
What does it mean for me to look at a drawing in descriptive geometry and say: "I know that this line appears
again here, but I can't see it like that"? Does it simply mean a lack of familiarity in operating with the drawing; that I
don't 'know my way about' too well?--This familiarity is certainly one of our criteria. What tells us that someone is
seeing the drawing three-dimensionally is a certain kind of 'knowing one's way about'. Certain gestures, for instance,
which indicate the three-dimensional relations: fine shades of behaviour.
Page 203
I see that an animal in a picture is transfixed by an arrow. It has struck it in the throat and sticks out at the
back of the neck. Let the picture be a silhouette.--Do you see the arrow--or do you merely know that these two bits
are supposed to represent part of an arrow?
Page 203
(Compare Köhler's figure of the interpenetrating hexagons.)
Page 203
"But this isn't seeing!"--"But this is seeing!"--It must be possible to give both remarks a conceptual
justification.
Page 203
But this is seeing! In what sense is it seeing?
Page 203
"The phenomenon is at first surprising, but a physiological explanation of it will certainly be found."--
Page 203
Our problem is not a causal but a conceptual one.
Page 203
If the picture of the transfixed beast or of the interpenetrating hexagons were shewn to me just for a moment
and then I had to describe it, that would be my description; if I had to draw it I should
Page Break 204
certainly produce a very faulty copy, but it would shew some sort of animal transfixed by an arrow, or two
hexagons interpenetrating. That is to say: there are certain mistakes that I should not make.
Page 204
The first thing to jump to my eye in this picture is: there are two hexagons.
Page 204
Now I look at them and ask myself: "Do I really see them as hexagons?"--and for the whole time they are
before my eyes? (Assuming that they have not changed their aspect in that time.)--And I should like to reply: "I am
not thinking of them as hexagons the whole time."
Page 204
Someone tells me: "I saw it at once as two hexagons. And that's the whole of what I saw." But how do I
understand this? I think he would have given this description at once in answer to the question "What are you
seeing?", nor would he have treated it as one among several possibilities. In this his description is like the answer "A
face" on being shewn the figure
Page 204
The best description I can give of what was shewn me for a moment is this:.....
Page 204
"The impression was that of a rearing animal." So a perfectly definite description came out.--Was it seeing,
or was it a thought?
Page 204
Do not try to analyse your own inner experience.
Page 204
Of course I might also have seen the picture first as something different, and then have said to myself "Oh,
it's two hexagons!" So the aspect would have altered. And does this prove that I in fact saw it as something definite?
Page 204
"Is it a genuine visual experience?" The question is: in what sense is it one?
Page 204
Here it is difficult to see that what is at issue is the fixing of concepts.
Page 204
A concept forces itself on one. (This is what you must not forget.)
Page 204
For when should I call it a mere case of knowing, not seeing?--Perhaps when someone treats the picture as a
working drawing, reads it like a blueprint. (Fine shades of behaviour.--Why are they important? They have
important consequences.)
Page Break 205
Page 205
"To me it is an animal pierced by an arrow." That is what I treat it as; this is my attitude to the figure. This is
one meaning in calling it a case of 'seeing'.
Page 205
But can I say in the same sense: "To me these are two hexagons"? Not in the same sense, but in a similar
one.
Page 205
You need to think of the role which pictures such as paintings (as opposed to working drawings) have in our
lives. This role is by no means a uniform one.
Page 205
A comparison: texts are sometimes hung on the wall. But not theorems of mechanics. (Our relation to these
two things.)
Page 205
If you see the drawing as such-and-such an animal, what I expect from you will be pretty different from what
I expect when you merely know what it is meant to be.
Page 205
Perhaps the following expression would have been better: we regard the photograph, the picture on our wall,
as the object itself (the man, landscape, and so on) depicted there.
Page 205
This need not have been so. We could easily imagine people who did not have this relation to such pictures.
Who, for example, would be repelled by photographs, because a face without colour and even perhaps a face
reduced in scale struck them as inhuman.
Page 205
I say: "We regard a portrait as a human being,"--but when do we do so, and for how long? Always, if we see
it at all (and do not, say, see it as something else)?
Page 205
I might say yes to this, and that would determine the concept of regarding-as.--The question is whether yet
another concept, related to this one, is also of importance to us: that, namely, of a seeing--as which only takes place
while I am actually concerning myself with the picture as the object depicted.
Page 205
I might say: a picture does not always live for me while I am seeing it.
Page 205
"Her picture smiles down on me from the wall." It need not always do so, whenever my glance lights on it.
Page 205
The duck-rabbit. One asks oneself: how can the eye--this dot--be looking in a direction?--"See, it is looking!"
(And one 'looks' oneself as one says this.) But one does not say and do this the whole time one is looking at the
picture. And now, what is this "See, it's looking!"--does it express a sensation?
Page Break 206
Page 206
(In giving all these examples I am not aiming at some kind of completeness, some classification of
psychological concepts. They are only meant to enable the reader to shift for himself when he encounters conceptual
difficulties.)
Page 206
"Now I see it as a...." goes with "I am trying to see it as a...." or "I can't see it as a.... yet". But I cannot try to
see a conventional picture of a lion as a lion, any more than an F as that letter. (Though I may well try to see it as a
gallows, for example.)
Page 206
Do not ask yourself "How does it work with me?"--Ask "What do I know about someone else?"
Page 206
How does one play the game: "It could be this too"? (What a figure could also be--which is what it can be
seen as--is not simply another figure. If someone said "I see as
", he might still be meaning very different things.)
Page 206
Here is a game played by children: they say that a chest, for example, is a house; and thereupon it is
interpreted as a house in every detail. A piece of fancy is worked into it.
Page 206
And does the child now see the chest as a house?
Page 206
"He quite forgets that it is a chest; for him it actually is a house." (There are definite tokens of this.) Then
would it not also be correct to say he sees it as a house?
Page 206
And if you knew how to play this game, and, given a particular situation, you exclaimed with special
expression "Now it's a house!"--you would be giving expression to the dawning of an aspect.
Page 206
If I heard someone talking about the duck-rabbit, and now he spoke in a certain way about the special
expression of the rabbit's face I should say, now he's seeing the picture as a rabbit.
Page 206
But the expression in one's voice and gestures is the same as if the object had altered and had ended by
becoming this or that.
Page 206
I have a theme played to me several times and each time in a slower tempo. In the end I say "Now it's right",
or "Now at last it's a march", "Now at last it's a dance".--The same tone of voice expresses the dawning of an aspect.
Page Break 207
Page 207
'Fine shades of behaviour.'--When my understanding of a theme is expressed by my whistling it with the
correct expression, this is an example of such fine shades.
Page 207
The aspects of the triangle: it is as if an image came into contact, and for a time remained in contact, with the
visual impression.
Page 207
In this, however, these aspects differ from the concave and convex aspects of the step (for example). And
also from the aspects of the figure
(which I shall call a "double cross") as a white cross on a black ground and as a black cross on a white ground.
Page 207
You must remember that the descriptions of the alternating aspects are of a different kind in each case.
Page 207
(The temptation to say "I see it like this", pointing to the same thing for "it" and "this".) Always get rid of the
idea of the private object in this way: assume that it constantly changes, but that you do not notice the change
because your memory constantly deceives you.
Page 207
Those two aspects of the double cross (I shall call them the aspects A) might be reported simply by pointing
alternately to an isolated white and an isolated black cross.
Page 207
One could quite well imagine this as a primitive reaction in a child even before it could talk.
Page 207
(Thus in reporting the aspects A we point to a part of the double cross.--The duck and rabbit aspects could
not be described in an analogous way.)
Page 207
You only 'see the duck and rabbit aspects' if you are already conversant with the shapes of those two
animals. There is no analogous condition for seeing the aspects A.
Page 207
It is possible to take the duck-rabbit simply for the picture of a rabbit, the double cross simply for the picture
of a black cross, but not to take the bare triangular figure for the picture of an object that has fallen over. To see this
aspect of the triangle demands imagination.
Page Break 208
Page 208
The aspects A are not essentially three-dimensional; a black cross on a white ground is not essentially a cross
with a white surface in the background. You could teach someone the idea of the black cross on a ground of
different colour without shewing him anything but crosses painted on sheets of paper. Here the 'background' is
simply the surrounding of the cross.
Page 208
The aspects A are not connected with the possibility of illusion in the same way as are the three-dimensional
aspects of the drawing of a cube or step.
Page 208
I can see the schematic cube as a box;--but can I also see it now as a paper, now as a tin, box?--What ought I
to say, if someone assured me he could?--I can set a limit to the concept here.
Page 208
Yet think of the expression "felt" in connexion with looking at a picture. ("One feels the softness of that
material.") (Knowing in dreams. "And I knew that... was in the room.")
Page 208
How does one teach a child (say in arithmetic) "Now take these things together!" or "Now these go
together"? Clearly "taking together" and "going together" must originally have had another meaning for him than
that of seeing in this way or that.--And this is a remark about concepts, not about teaching methods.
Page 208
One kind of aspect might be called 'aspects of organization'. When the aspect changes parts of the picture go
together which before did not.
Page 208
In the triangle I can see now this as apex, that as base--now this as apex, that as base.--Clearly the words
"Now I am seeing this as the apex" cannot so far mean anything to a learner who has only just met the concepts of
apex, base, and so on.--But I do not mean this as an empirical proposition.
Page 208
"Now he's seeing it like this", "now like that" would only be said of someone capable of making certain
applications of the figure quite freely.
Page 208
The substratum of this experience is the mastery of a technique.
Page 208
But how queer for this to be the logical condition of someone's having such-and-such an experience! After
all, you don't say that one only 'has toothache' if one is capable of doing such-and-such.--From this it follows that
we cannot be dealing with the same concept of experience here. It is a different though related concept.
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Page 209
It is only if someone can do, has learnt, is master of, such-and-such, that it makes sense to say he has had
this experience.
Page 209
And if this sounds crazy, you need to reflect that the concept of seeing is modified here. (A similar
consideration is often necessary to get rid of a feeling of dizziness in mathematics.)
Page 209
We talk, we utter words, and only later get a picture of their life.
Page 209
For how could I see that this posture was hesitant before I knew that it was a posture and not the anatomy of
the animal?
Page 209
But surely that only means that I cannot use this concept to describe the object of sight, just because it has
more than purely visual reference?--Might I not for all that have a purely visual concept of a hesitant posture, or of a
timid face?
Page 209
Such a concept would be comparable with 'major' and 'minor' which certainly have emotional value, but can
also be used purely to describe a perceived structure.
Page 209
The epithet "sad", as applied for example to the outline face, characterizes the grouping of lines in a circle.
Applied to a human being it has a different (though related) meaning. (But this does not mean that a sad expression
is like the feeling of sadness!)
Page 209
Think of this too: I can only see, not hear, red and green,--but sadness I can hear as much as I can see it.
Page 209
Think of the expression "I heard a plaintive melody". And now the question is: "Does he hear the plaint?"
Page 209
And if I reply: "No, he doesn't hear it, he merely has a sense of it"--where does that get us? One cannot
mention a sense-organ for this 'sense'.
Page 209
Some would like to reply here: "Of course I hear it!"--Others: "I don't really hear it."
Page 209
We can, however, establish differences of concept here.
Page 209
We react to the visual impression differently from someone who does not recognize it as timid (in the full
sense of the word).--But I do not want to say here that we feel this reaction in our muscles and joints and that this is
the 'sensing'.--No, what we have here is a modified concept of sensation.
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Page 210
One might say of someone that he was blind to the expression of a face. Would his eyesight on that account
be defective?
Page 210
This is, of course, not simply a question for physiology. Here the physiological is a symbol of the logical.
Page 210
If you feel the seriousness of a tune, what are you perceiving?--Nothing that could be conveyed by
reproducing what you heard.
Page 210
I can imagine some arbitrary cipher--this, for instance: --to be a strictly correct letter of some foreign
alphabet. Or again, to be a faultily written one, and faulty in this way or that: for example, it might be slap-dash, or
typical childish awkwardness, or like the flourishes in a legal document. It could deviate from the correctly written
letter in a variety of ways.--And I can see it in various aspects according to the fiction I surround it with. And here
there is a close kinship with 'experiencing the meaning of a word'.
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I should like to say that what dawns here lasts only as long as I am occupied with the object in a particular
way. ("See, it's looking!")--'I should like to say'--and is it so?--Ask yourself "For how long am I struck by a
thing?"--For how long do I find it new?
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The aspect presents a physiognomy which then passes away. It is almost as if there were a face there which
at first I imitate, and then accept without imitating it.--And isn't this really explanation enough?--But isn't it too
much?
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"I observed the likeness between him and his father for a few minutes, and then no longer."--One might say
this if his face were changing and only looked like his father's for a short time. But it can also mean that after a few
minutes I stopped being struck by the likeness.
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"After the likeness had struck you, how long were you aware of it?" What kind of answer might one give to
this question?--"I soon stopped thinking about it", or "It struck me again from time to time", or "I several times had
the thought, how like they are!", or "I marvelled at the likeness for at least a minute"--That is the sort of answer you
would get.
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I should like to put the question "Am I aware of the spatial character, the depth of an object (of this
cupboard for instance), the whole time
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I am seeing it?" Do I, so to speak, feel it the whole time?--But put the question in the third person.--When would you
say of someone that he was aware of it the whole time, and when the opposite?--Of course, one could ask him,--but
how did he learn how to answer such a question?--He knows what it means "to feel pain continuously". But that will
only confuse him here (as it confuses me).
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If he now says he is continuously aware of the depth--do I believe him? And if he says he is aware of it only
occasionally (when talking about it, perhaps)--do I believe that? These answers will strike me as resting on a false
foundation.--It will be different if he says that the object sometimes strikes him as flat, sometimes as
three-dimensional.
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Someone tells me: "I looked at the flower, but was thinking of something else and was not conscious of its
colour." Do I understand this?--I can imagine a significant context, say his going on: "Then I suddenly saw it, and
realized it was the one which......".
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Or again: "If I had turned away then, I could not have said what colour it was."
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"He looked at it without seeing it."--There is such a thing. But what is the criterion for it?--Well, there is a
variety of cases here.
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"Just now I looked at the shape rather than at the colour." Do not let such phrases confuse you. Above all,
don't wonder "What can be going on in the eyes or brain?"
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The likeness makes a striking impression on me; then the impression fades.
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It only struck me for a few minutes, and then no longer did.
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What happened here?--What can I recall? My own facial expression comes to mind; I could reproduce it. If
someone who knew me had seen my face he would have said "Something about his face struck you just
now".--There further occurs to me what I say on such an occasion, out loud or to myself. And that is all.--And is this
what being struck is? No. These are the phenomena of being struck; but they are 'what happens'.
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Is being struck looking plus thinking? No. Many of our concepts cross here.
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('Thinking' and 'inward speech'--I do not say 'to oneself'--are different concepts.)
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The colour of the visual impression corresponds to the colour of the object (this blotting paper looks pink to
me, and is pink)--the shape of the visual impression to the shape of the object (it looks rectangular to me, and is
rectangular)--but what I perceive in the dawning of an aspect is not a property of the object, but an internal relation
between it and other objects.
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It is almost as if 'seeing the sign in this context' were an echo of a thought.
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"The echo of a thought in sight"--one would like to say.
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Imagine a physiological explanation of the experience. Let it be this: When we look at the figure, our eyes
scan it repeatedly, always following a particular path. The path corresponds to a particular pattern of oscillation of
the eyeballs in the act of looking. It is possible to jump from one such pattern to another and for the two to alternate.
(Aspects A.) Certain patterns of movement are physiologically impossible; hence, for example, I cannot see the
schematic cube as two interpenetrating prisms. And so on. Let this be the explanation.--"Yes, that shews it is a kind
of seeing."--You have now introduced a new, a physiological, criterion for seeing. And this can screen the old
problem from view, but not solve it.--The purpose of this paragraph however, was to bring before our view what
happens when a physiological explanation is offered. The psychological concept hangs out of reach of this
explanation. And this makes the nature of the problem clearer.
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Do I really see something different each time, or do I only interpret what I see in a different way? I am
inclined to say the former. But why?--To interpret is to think, to do something; seeing is a state.
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Now it is easy to recognize cases in which we are interpreting. When we interpret we form hypotheses,
which may prove false.--"I am seeing this figure as a....." can be verified as little as (or in the same sense as) "I am
seeing bright red". So there is a similarity in the use of "seeing" in the two contexts. Only do not think you knew in
advance what the "state of seeing" means here! Let the use teach you the meaning.
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We find certain things about seeing puzzling, because we do not find the whole business of seeing puzzling
enough.
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If you look at a photograph of people, houses and trees, you do not feel the lack of the third dimension in it.
We should not find it easy to describe a photograph as a collection of colour-patches on a flat surface; but what we
see in a stereoscope looks three-dimensional in a different way again.
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(It is anything but a matter of course that we see 'three-dimensionally' with two eyes. If the two visual images
are amalgamated, we might expect a blurred one as a result.)
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The concept of an aspect is akin to the concept of an image. In other words: the concept 'I am now seeing it
as....' is akin to 'I am now having this image'.
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Doesn't it take imagination to hear something as a variation on a particular theme? And yet one is perceiving
something in so hearing it.
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"Imagine this changed like this, and you have this other thing." One can use imagining in the course of
proving something.
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Seeing an aspect and imagining are subject to the will. There is such an order as "Imagine this", and also:
"Now see the figure like this"; but not: "Now see this leaf green".
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The question now arises: Could there be human beings lacking in the capacity to see something as
something--and what would that be like? What sort of consequences would it have?--Would this defect be
comparable to colour-blindness or to not having absolute pitch?--We will call it "aspect-blindness"--and will next
consider what might be meant by this. (A conceptual investigation.) The aspect-blind man is supposed not to see the
aspects A change. But is he also supposed not to recognize that the double cross contains both a black and a white
cross? So if told "Shew me figures containing a black cross among these examples" will he be unable to manage it?
No, he should be able to do that; but he will not be supposed to say: "Now it's a black cross on a white ground!"
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Is he supposed to be blind to the similarity between two faces?--And so also to their identity or approximate
identity? I do not want to settle this. (He ought to be able to execute such orders as "Bring me something that looks
like this.")
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Ought he to be unable to see the schematic cube as a cube?--It would not follow from that that he could not
recognize it as a representation (a working drawing for instance) of a cube. But for him it
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would not jump from one aspect to the other.--Question: Ought he to be able to take it as a cube in certain
circumstances, as we do?--If not, this could not very well be called a sort of blindness.
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The 'aspect-blind' will have an altogether different relationship to pictures from ours.
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(Anomalies of this kind are easy for us to imagine.)
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Aspect-blindness will be akin to the lack of a 'musical ear'.
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The importance of this concept lies in the connexion between the concepts of 'seeing an aspect' and
'experiencing the meaning of a word'. For we want to ask "What would you be missing if you did not experience the
meaning of a word?"
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What would you be missing, for instance, if you did not understand the request to pronounce the word "till"
and to mean it as a verb,--or if you did not feel that a word lost its meaning and became a mere sound if it was
repeated ten times over?
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In a law-court, for instance, the question might be raised how someone meant a word. And this can be
inferred from certain facts.--It is a question of intention. But could how he experienced a word--the word "bank" for
instance--have been significant in the same way?
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Suppose I had agreed on a code with someone; "tower" means bank. I tell him "Now go to the tower"--he
understands me and acts accordingly, but he feels the word "tower" to be strange in this use, it has not yet 'taken on'
the meaning.
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"When I read a poem or narrative with feeling, surely something goes on in me which does not go on when I
merely skim the lines for information."--What processes am I alluding to?--The sentences have a different ring. I pay
careful attention to my intonation. Sometimes a word has the wrong intonation, I emphasize it too much or too little.
I notice this and shew it in my face. I might later talk about my reading in detail, for example about the mistakes in
my tone of voice. Sometimes a picture, as it were an illustration, comes to me. And this seems to help me to read
with the correct expression. And I could mention a good deal more of the same kind.--I can also give a word a tone
of voice which brings out the meaning of the rest, almost as if this word were a picture of the whole thing. (And this
may, of course, depend on sentence-formation.)
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When I pronounce this word while reading with expression it is completely filled with its meaning.--"How
can this be, if meaning is the use of the word?" Well, what I said was intended figuratively. Not that I chose the
figure: it forced itself on me.--But the figurative employment of the word can't get into conflict with the original one.
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Perhaps it could be explained why precisely this picture suggests itself to me. (Just think of the expression,
and the meaning of the expression: "the word that hits it off".)
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But if a sentence can strike me as like a painting in words, and the very individual word in the sentence as like
a picture, then it is no such marvel that a word uttered in isolation and without purpose can seem to carry a particular
meaning in itself.
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Think here of a special kind of illusion which throws light on these matters.--I go for a walk in the environs of
a city with a friend. As we talk it comes out that I am imagining the city to lie on our right. Not only have I no
conscious reason for this assumption, but some quite simple consideration was enough to make me realize that the
city lay rather to the left ahead of us. I can at first give no answer to the question why I imagine the city in this
direction. I had no reason to think it. But though I see no reason still I seem to see certain psychological causes for
it. In particular, certain associations and memories. For example, we walked along a canal, and once before in similar
circumstances I had followed a canal and that time the city lay on our right.--I might try as it were psychoanalytically
to discover the causes of my unfounded conviction.
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"But what is this queer experience?"--Of course it is not queerer than any other; it simply differs in kind from
those experiences which we regard as the most fundamental ones, our sense impressions for instance.
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"I feel as if I knew the city lay over there."--"I feel as if the name 'Schubert' fitted Schubert's works and
Schubert's face."
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You can say the word "March" to yourself and mean it at one time as an imperative at another as the name of
a month. And now say "March!"--and then "March no further!"--Does the same experience accompany the word
both times--are you sure?
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If a sensitive ear shews me, when I am playing this game, that I have now this now that experience of the
word--doesn't it also shew
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me that I often do not have any experience of it in the course of talking?--For the fact that I then also mean it, intend
it, now like this now like that, and maybe also say so later is, of course, not in question.
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But the question now remains why, in connexion with this game of experiencing a word, we also speak of
'the meaning' and of 'meaning it'.--This is a different kind of question.--It is the phenomenon which is characteristic
of this language-game that in this situation we use this expression: we say we pronounced the word with this
meaning and take this expression over from that other language-game.
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Call it a dream. It does not change anything.
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Given the two ideas 'fat' and 'lean', would you be rather inclined to say that Wednesday was fat and Tuesday
lean, or the other way round? (I incline to choose the former.) Now have "fat" and "lean" some different meaning
here from their usual one?--They have a different use.--So ought I really to have used different words? Certainly not
that.--I want to use these words (with their familiar meanings) here.--Now, I say nothing about the causes of this
phenomenon. They might be associations from my childhood. But that is a hypothesis. Whatever the
explanation,--the inclination is there.
Page 216
Asked "What do you really mean here by 'fat' and 'lean'?"--I could only explain the meanings in the usual
way. I could not point to the examples of Tuesday and Wednesday.
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Here one might speak of a 'primary' and 'secondary' sense of a word. It is only if the word has the primary
sense for you that you use it in the secondary one.
Page 216
Only if you have learnt to calculate--on paper or out loud--can you be made to grasp, by means of this
concept, what calculating in the head is.
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The secondary sense is not a 'metaphorical' sense. If I say "For me the vowel e is yellow" I do not mean:
'yellow' in a metaphorical sense,--for I could not express what I want to say in any other way than by means of the
idea 'yellow'.
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Someone tells me: "Wait for me by the bank". Question: Did you, as you were saying the word, mean this
bank?--This question is of the same kind as "Did you intend to say such-and-such to him on your way to meet
him?" It refers to a definite time (the time of walking, as the former question refers to the time of speaking)--but not
to an
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experience during that time. Meaning is as little an experience as intending.
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But what distinguishes them from experience?--They have no experience-content. For the contents (images
for instance) which accompany and illustrate them are not the meaning or intending.
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The intention with which one acts does not 'accompany' the action any more than the thought 'accompanies'
speech. Thought and intention are neither 'articulated' nor 'non-articulated'; to be compared neither with a single note
which sounds during the acting or speaking, nor with a tune.
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'Talking' (whether out loud or silently) and 'thinking' are not concepts of the same kind; even though they are
in closest connexion.
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The interest of the experiences one has while speaking and of the intention is not the same. (The experiences
might perhaps inform a psychologist about the 'unconscious' intention.)
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"At that word we both thought of him." Let us assume that each of us said the same words to himself--and
how can it mean MORE than that?--But wouldn't even those words be only a germ? They must surely belong to a
language and to a context, in order really to be the expression of the thought of that man.
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If God had looked into our minds he would not have been able to see there whom we were speaking of.
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"Why did you look at me at that word, were you thinking of....?"--So there is a reaction at a certain moment
and it is explained by saying "I thought of...." or "I suddenly remembered...."
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In saying this you refer to that moment in the time you were speaking. It makes a difference whether you
refer to this or to that moment.
Page 217
Mere explanation of a word does not refer to an occurrence at the moment of speaking.
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The language-game "I mean (or meant) this" (subsequent explanation of a word) is quite different from this
one: "I thought of.... as I said it." The latter is akin to "It reminded me of...."
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"I have already remembered three times today that I must write to him." Of what importance is it what went
on in me then?--On the
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other hand what is the importance, what the interest, of the statement itself?--It permits certain conclusions.
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"At these words he occurred to me."--What is the primitive reaction with which the language-game
begins--which can then be translated into these words? How do people get to use these words?
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The primitive reaction may have been a glance or a gesture, but it may also have been a word.
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"Why did you look at me and shake your head?"--"I wanted to give you to understand that you....." This is
supposed to express not a symbolic convention but the purpose of my action.
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Meaning it is not a process which accompanies a word. For no process could have the consequences of
meaning.
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(Similarly, I think, it could be said: a calculation is not an experiment, for no experiment could have the
peculiar consequences of a multiplication.)
Page 218
There are important accompanying phenomena of talking which are often missing when one talks without
thinking, and this is characteristic of talking without thinking. But they are not the thinking.
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"Now I know!" What went on here?--So did I not know, when I declared that now I knew?
Page 218
You are looking at it wrong.
Page 218
(What is the signal for?)
Page 218
And could the 'knowing' be called an accompaniment of the exclamation?
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The familiar physiognomy of a word, the feeling that it has taken up its meaning into itself, that it is an actual
likeness of its meaning--there could be human beings to whom all this was alien. (They would not have an
attachment to their words.)--And how are these feelings manifested among us?--By the way we choose and value
words.
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How do I find the 'right' word? How do I choose among words? Without doubt it is sometimes as if I were
comparing them by fine differences of smell: That is too......, that is too......,--this is the right one.--But I do not
always have to make judgments, give explanations; often I might only say: "It simply isn't right yet". I am
dissatisfied, I go on looking. At last a word comes: "That's it!" Sometimes I can say why. This is simply what
searching, this is what finding, is like here.
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But doesn't the word that occurs to you somehow 'come' in a special way? Just attend and you'll
see!--Careful attention is no use to me. All it could discover would be what is now going on in me.
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And how can I, precisely now, listen for it at all? I ought to have to wait until a word occurs to me anew.
This, however, is the queer thing: it seems as though I did not have to wait on the occasion, but could give myself an
exhibition of it, even when it is not actually taking place. How?--I act it.--But what can I learn in this way? What do I
reproduce?--Characteristic accompaniments. Primarily: gestures, faces, tones of voice.
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It is possible--and this is important--to say a great deal about a fine aesthetic difference.--The first thing you
say may, of course, be just: "This word fits, that doesn't"--or something of the kind. But then you can discuss all the
extensive ramifications of the tie-up effected by each of the words. That first judgment is not the end of the matter,
for it is the field of force of a word that is decisive.
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"The word is on the tip of my tongue." What is going on in my consciousness? That is not the point at all.
Whatever did go on was not what was meant by that expression. It is of more interest what went on in my
behaviour.--"The word is on the tip of my tongue" tells you: the word which belongs here has escaped me, but I
hope to find it soon. For the rest the verbal expression does no more than certain wordless behaviour.
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James, in writing of this subject, is really trying to say: "What a remarkable experience! The word is not there
yet, and yet in a certain sense is there,--or something is there, which cannot grow into anything but this word."--But
this is not experience at all. Interpreted as experience it does indeed look odd. As does intention, when it is
interpreted as the accompaniment of action; or again, like minus one interpreted as a cardinal number.
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The words "It's on the tip of my tongue" are no more the expression of an experience than "Now I know
how to go on!"--We use them in certain situations, and they are surrounded by behaviour of a special kind, and
also by some characteristic experiences. In particular they are frequently followed by finding the word. (Ask
yourself: "What would it be like if human beings never found the word that was on the tip of their tongue?")
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Silent 'internal' speech is not a half hidden phenomenon which is as it were seen through a veil. It is not
hidden at all, but the concept may easily confuse us, for it runs over a long stretch cheek by jowl with the concept of
an 'outward' process, and yet does not coincide with it.
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(The question whether the muscles of the larynx are innervated in connexion with internal speech, and
similar things, may be of great interest, but not in our investigation.)
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The close relationship between 'saying inwardly' and 'saying' is manifested in the possibility of telling out
loud what one said inwardly, and of an outward action's accompanying inward speech. (I can sing inwardly, or read
silently, or calculate in my head, and beat time with my hand as I do so.)
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"But saying things inwardly is surely a certain activity which I have to learn!" Very well; but what is 'doing'
and what is 'learning' here?
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Let the use of words teach you their meaning. (Similarly one can often say in mathematics: let the proof
teach you what was being proved.)
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"So I don't really calculate, when I calculate in my head?"--After all, you yourself distinguish between
calculation in the head and perceptible calculation! But you can only learn what 'calculating in the head' is by
learning what 'calculating' is; you can only learn to calculate in your head by learning to calculate.
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One can say things in one's head very 'distinctly', when one reproduces the tone of voice of one's sentences
by humming (with closed lips). Movements of the larynx help too. But the remarkable thing is precisely that one
then hears the talk in one's imagination and does not merely feel the skeleton of it, so to speak, in one's larynx. (For
human beings could also well be imagined calculating silently with movements of the larynx, as one can calculate on
one's fingers.)
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A hypothesis, such as that such-and-such went on in our bodies when we made internal calculations, is only
of interest to us in that it points to a possible use of the expression "I said.... to myself"; namely that of inferring the
physiological process from the expression.
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That what someone else says to himself is hidden from me is part of the concept 'saying inwardly'. Only
"hidden" is the wrong word
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here; for if it is hidden from me, it ought to be apparent to him, he would have to know it. But he does not 'know' it;
only, the doubt which exists for me does not exist for him.
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"What anyone says to himself within himself is hidden from me" might of course also mean that I can for the
most part not guess it, nor can I read it off from, for example, the movements of his throat (which would be a
possibility.)
Page 221
"I know what I want, wish, believe, feel,......." (and so on through all the psychological verbs) is either
philosophers' nonsense, or at any rate not a judgment a priori.
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"I know..." may mean "I do not doubt..." but does not mean that the words "I doubt..." are senseless, that
doubt is logically excluded.
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One says "I know" where one can also say "I believe" or "I suspect"; where one can find out. (If you bring up
against me the case of people's saying "But I must know if I am in pain!", "Only you can know what you feel", and
similar things, you should consider the occasion and purpose of these phrases. "War is war" is not an example of the
law of identity, either.)
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It is possible to imagine a case in which I could find out that I had two hands. Normally, however, I cannot
do so. "But all you need is to hold them up before your eyes!"--If I am now in doubt whether I have two hands, I
need not believe my eyes either. (I might just as well ask a friend.)
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With this is connected the fact that, for instance, the proposition "The Earth has existed for millions of years"
makes clearer sense than "The Earth has existed in the last five minutes". For I should ask anyone who asserted the
latter: "What observations does this proposition refer to; and what observations would count against it?"--whereas I
know what ideas and observations the former proposition goes with.
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"A new-born child has no teeth."--"A goose has no teeth."--"A rose has no teeth."--This last at any rate--one
would like to say--is obviously true! It is even surer than that a goose has none.--And yet it is none so clear. For
where should a rose's teeth have been? The goose has none in its jaw. And neither, of course, has it any in its
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wings; but no one means that when he says it has no teeth.--Why, suppose one were to say: the cow chews its food
and then dungs the rose with it, so the rose has teeth in the mouth of a beast. This would not be absurd, because one
has no notion in advance where to look for teeth in a rose. ((Connexion with 'pain in someone else's body'.))
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I can know what someone else is thinking, not what I am thinking.
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It is correct to say "I know what you are thinking", and wrong to say "I know what I am thinking."
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(A whole cloud of philosophy condensed into a drop of grammar.)
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"A man's thinking goes on within his consciousness in a seclusion in comparison with which any physical
seclusion is an exhibition to public view."
Page 222
If there were people who always read the silent internal discourse of others--say by observing the
larynx--would they too be inclined to use the picture of complete seclusion?
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If I were to talk to myself out loud in a language not understood by those present my thoughts would be
hidden from them.
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Let us assume there was a man who always guessed right what I was saying to myself in my thoughts. (It
does not matter how he manages it.) But what is the criterion for his guessing right? Well, I am a truthful person and
I confess that he has guessed right.--But might I not be mistaken, can my memory not deceive me? And might it not
always do so when--without lying--I express what I have thought within myself?--But now it does appear that 'what
went on within me' is not the point at all. (Here I am drawing a construction-line.)
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The criteria for the truth of the confession that I thought such-and-such are not the criteria for a true
description of a process. And the importance of the true confession does not reside in its being a correct and certain
report of a process. It resides rather in the special consequences which can be drawn from a confession whose truth
is guaranteed by the special criteria of truthfulness.
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(Assuming that dreams can yield important information about the dreamer, what yielded the information
would be truthful accounts of dreams. The question whether the dreamer's memory deceives him when he reports
the dream after waking cannot arise, unless indeed we introduce a completely new criterion for the report's 'agreeing'
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with the dream, a criterion which gives us a concept of 'truth' as distinct from 'truthfulness' here.)
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There is a game of 'guessing thoughts'. A variant of it would be this: I tell A something in a language that B
does not understand. B is supposed to guess the meaning of what I say.--Another variant: I write down a sentence
which the other person cannot see. He has to guess the words or their sense.--Yet another: I am putting a jig-saw
puzzle together; the other person cannot see me but from time to time guesses my thoughts and utters them. He
says, for instance, "Now where is this bit?"--"Now I know how it fits!"--"I have no idea what goes in here,"--"The
sky is always the hardest part" and so on--but I need not be talking to myself either out loud or silently at the time.
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All this would be guessing at thoughts; and the fact that it does not actually happen does not make thought
any more hidden than the unperceived physical proceedings.
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"What is internal is hidden from us."--The future is hidden from us. But does the astronomer think like this
when he calculates an eclipse of the sun?
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If I see someone writhing in pain with evident cause I do not think: all the same, his feelings are hidden from
me.
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We also say of some people that they are transparent to us. It is, however, important as regards this
observation that one human being can be a complete enigma to another. We learn this when we come into a strange
country with entirely strange traditions; and, what is more, even given a mastery of the country's language. We do
not understand the people. (And not because of not knowing what they are saying to themselves.) We cannot find
our feet with them.
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"I cannot know what is going on in him" is above all a picture. It is the convincing expression of a
conviction. It does not give the reasons for the conviction. They are not readily accessible.
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If a lion could talk, we could not understand him.
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It is possible to imagine a guessing of intentions like the guessing of thoughts, but also a guessing of what
someone is actually going to do.
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To say "He alone can know what he intends" is nonsense: to say "He alone can know what he will do",
wrong. For the prediction contained in my expression of intention (for example "When it strikes
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five I am going home") need not come true, and someone else may know what will really happen.
Page 224
Two points, however, are important: one, that in many cases someone else cannot predict my actions,
whereas I foresee them in my intentions; the other, that my prediction (in my expression of intention) has not the
same foundation as his prediction of what I shall do, and the conclusions to be drawn from these predictions are
quite different.
Page 224
I can be as certain of someone else's sensations as of any fact. But this does not make the propositions "He
is much depressed", "25 × 25 = 625" and "I am sixty years old" into similar instruments. The explanation suggests
itself that the certainty is of a different kind.--This seems to point to a psychological difference. But the difference is
logical.
Page 224
"But, if you are certain, isn't it that you are shutting your eyes in face of doubt?"--They are shut.
Page 224
Am I less certain that this man is in pain than that twice two is four?--Does this shew the former to be
mathematical certainty?--'Mathematical certainty' is not a psychological concept.
Page 224
The kind of certainty is the kind of language-game.
Page 224
"He alone knows his motives"--that is an expression of the fact that we ask him what his motives are.--If he
is sincere he will tell us them; but I need more than sincerity to guess his motives. This is where there is a kinship
with the case of knowing.
Page 224
Let yourself be struck by the existence of such a thing as our language-game of: confessing the motive of my
action.
Page 224
We remain unconscious of the prodigious diversity of all the everyday language-games because the clothing
of our language makes everything alike.
Page 224
Something new (spontaneous, 'specific') is always a language-game.
Page 224
What is the difference between cause and motive?--How is the motive discovered, and how the cause?
Page 224
There is such a question as: "Is this a reliable way of judging people's motives?" But in order to be able to ask
this we must know what "judging a motive" means; and we do not learn this by being told what 'motive' is and what
'judging' is.
Page Break 225
Page 225
One judges the length of a rod and can look for and find some method of judging it more exactly or more
reliably. So--you say--what is judged here is independent of the method of judging it. What length is cannot be
defined by the method of determining length.--To think like this is to make a mistake. What mistake?--To say "The
height of Mont Blanc depends on how one climbs it" would be queer. And one wants to compare 'ever more
accurate measurement of length' with the nearer and nearer approach to an object. But in certain cases it is, and in
certain cases it is not, clear what "approaching nearer to the length of an object" means. What "determining the
length" means is not learned by learning what length and determining are; the meaning of the word "length" is
learnt by learning, among other things, what it is to determine length.
Page 225
(For this reason the word "methodology" has a double meaning. Not only a physical investigation, but also a
conceptual one, can be called "methodological investigation".)
Page 225
We should sometimes like to call certainty and belief tones, colourings, of thought; and it is true that they
receive expression in the tone of voice. But do not think of them as 'feelings' which we have in speaking or thinking.
Page 225
Ask, not: "What goes on in us when we are certain that....?"--but: How is 'the certainty that this is the case'
manifested in human action?
Page 225
"While you can have complete certainty about someone else's state of mind, still it is always merely
subjective, not objective, certainty."--These two words betoken a difference between language-games.
Page 225
There can be a dispute over the correct result of a calculation (say of a rather long addition). But such
disputes are rare and of short duration. They can be decided, as we say, 'with certainty'.
Page 225
Mathematicians do not in general quarrel over the result of a calculation. (This is an important fact.)--If it
were otherwise, if for instance one mathematician was convinced that a figure had altered unperceived, or that his or
someone else's memory had been deceived, and so on--then our concept of 'mathematical certainty' would not exist.
Page 225
Even then it might always be said: "True we can never know what the result of a calculation is, but for all that
it always has a quite
Page Break 226
definite result. (God knows it.) Mathematics is indeed of the highest certainty--though we only have a crude
reflection of it."
Page 226
But am I trying to say some such thing as that the certainty of mathematics is based on the reliability of ink
and paper? No. (That would be a vicious circle.)--I have not said why mathematicians do not quarrel, but only that
they do not.
Page 226
It is no doubt true that you could not calculate with certain sorts of paper and ink, if, that is, they were
subject to certain queer changes--but still the fact that they changed could in turn only be got from memory and
comparison with other means of calculation. And how are these tested in their turn?
Page 226
What has to be accepted, the given, is--so one could say--forms of life.
Page 226
Does it make sense to say that people generally agree in their judgments of colour? What would it be like for
them not to?--One man would say a flower was red which another called blue, and so on.--But what right should we
have to call these people's words "red" and "blue" our 'colour-words'?"--
Page 226
How would they learn to use these words? And is the language-game which they learn still such as we call
the use of 'names of colour'? There are evidently differences of degree here.
Page 226
This consideration must, however, apply to mathematics too. If there were not complete agreement, then
neither would human beings be learning the technique which we learn. It would be more or less different from ours
up to the point of unrecognizability.
Page 226
"But mathematical truth is independent of whether human beings know it or not!"--Certainly, the
propositions "Human beings believe that twice two is four" and "Twice two is four" do not mean the same. The latter
is a mathematical proposition; the other, if it makes sense at all, may perhaps mean: human beings have arrived at
the mathematical proposition. The two propositions have entirely different uses.--But what would this mean: "Even
though everybody believed that twice two was five it would still be four"?--For what would it be like for everybody
to believe that?--Well, I could imagine, for instance, that people had a different calculus, or a technique which we
should
Page Break 227
not call "calculating". But would it be wrong? (Is a coronation wrong? To beings different from ourselves it might
look extremely odd.)
Page 227
Of course, in one sense mathematics is a branch of knowledge,--but still it is also an activity. And 'false
moves' can only exist as the exception. For if what we now call by that name became the rule, the game in which
they were false moves would have been abrogated.
Page 227
"We all learn the same multiplication table." This might, no doubt, be a remark about the teaching of
arithmetic in our schools,--but also an observation about the concept of the multiplication table. ("In a horse-race the
horses generally run as fast as they can.")
Page 227
There is such a thing as colour-blindness and there are ways of establishing it. There is in general complete
agreement in the judgments of colours made by those who have been diagnosed normal. This characterizes the
concept of a judgment of colour.
Page 227
There is in general no such agreement over the question whether an expression of feeling is genuine or not.
Page 227
I am sure, sure, that he is not pretending; but some third person is not. Can I always convince him? And if
not is there some mistake in his reasoning or observations?
Page 227
"You're all at sea!"--we say this when someone doubts what we recognize as clearly genuine--but we cannot
prove anything.
Page 227
Is there such a thing as 'expert judgment' about the genuineness of expressions of feeling?--Even here, there
are those whose judgment is 'better' and those whose judgment is 'worse'.
Page 227
Correcter prognoses will generally issue from the judgments of those with better knowledge of mankind.
Page 227
Can one learn this knowledge? Yes; some can. Not, however, by taking a course in it, but through
'experience'.--Can someone else be a man's teacher in this? Certainly. From time to time he gives him the right
tip.--This is what 'learning' and 'teaching' are like here.--What one acquires here is not a technique; one learns correct
judgments. There are also rules, but they do not form a system, and only experienced people can apply them right.
Unlike calculating-rules.
Page 227
What is most difficult here is to put this indefiniteness, correctly and unfalsified, into words.
Page Break 228
Page 228
"The genuineness of an expression cannot be proved; one has to feel it."--Very well,--but what does one go
on to do with this recognition of genuineness? If someone says "Voila ce que peut dire un cœur vraiment épris"--and
if he also brings someone else to the same mind,--what are the further consequences? Or are there none, and does
the game end with one person's relishing what another does not?
Page 228
There are certainly consequences, but of a diffuse kind. Experience, that is varied observation, can inform us
of them, and they too are incapable of general formulation; only in scattered cases can one arrive at a correct and
fruitful judgment, establish a fruitful connexion. And the most general remarks yield at best what looks like the
fragments of a system.
Page 228
It is certainly possible to be convinced by evidence that someone is in such-and-such a state of mind, that,
for instance, he is not pretending. But 'evidence' here includes 'imponderable' evidence.
Page 228
The question is: what does imponderable evidence accomplish?
Page 228
Suppose there were imponderable evidence for the chemical (internal) structure of a substance, still it would
have to prove itself to be evidence by certain consequences which can be weighed.
Page 228
(Imponderable evidence might convince someone that a picture was a genuine.... But it is possible for this to
be proved right by documentary evidence as well.)
Page 228
Imponderable evidence includes subtleties of glance, of gesture, of tone.
Page 228
I may recognize a genuine loving look, distinguish it from a pretended one (and here there can, of course, be
a 'ponderable' confirmation of my judgment). But I may be quite incapable of describing the difference. And this not
because the languages I know have no words for it. For why not introduce new words?--If I were a very talented
painter I might conceivably represent the genuine and the simulated glance in pictures.
Page 228
Ask yourself: How does a man learn to get a 'nose' for something? And how can this nose be used?
Page 228
Pretending is, of course, only a special case of someone's producing (say) expressions of pain when he is not
in pain. For if this is possible
Page Break 229
at all, why should it always be pretending that is taking place--this very special pattern in the weave of our lives?
Page 229
A child has much to learn before it can pretend. (A dog cannot be a hypocrite, but neither can he be sincere.)
Page 229
There might actually occur a case where we should say "This man believes he is pretending."
Page Break 230
xii
Page 230
If the formation of concepts can be explained by facts of nature, should we not be interested, not in
grammar, but rather in that in nature which is the basis of grammar?--Our interest certainly includes the
correspondence between concepts and very general facts of nature. (Such facts as mostly do not strike us because of
their generality.) But our interest does not fall back upon these possible causes of the formation of concepts; we are
not doing natural science; nor yet natural history--since we can also invent fictitious natural history for our purposes.
Page 230
I am not saying: if such-and-such facts of nature were different people would have different concepts (in the
sense of a hypothesis). But: if anyone believes that certain concepts are absolutely the correct ones, and that having
different ones would mean not realizing something that we realize--then let him imagine certain very general facts of
nature to be different from what we are used to, and the formation of concepts different from the usual ones will
become intelligible to him.
Page 230
Compare a concept with a style of painting. For is even our style of painting arbitrary? Can we choose one at
pleasure? (The Egyptian, for instance.) Is it a mere question of pleasing and ugly?
Page Break 231
xiii
Page 231
When I say: "He was here half an hour ago"--that is, remembering it--this is not the description of a present
experience.
Page 231
Memory-experiences are accompaniments of remembering.
Page 231
Remembering has no experiential content.--Surely this can be seen by introspection? Doesn't it shew
precisely that there is nothing there, when I look about for a content?--But it could only shew this in this case or that.
And even so it cannot shew me what the word "to remember" means, and hence where to look for a content!
Page 231
I get the idea of a memory-content only because I assimilate psychological concepts. It is like assimilating
two games. (Football has goals, tennis not.)
Page 231
Would this situation be conceivable: someone remembers for the first time in his life and says "Yes, now I
know what 'remembering' is, what it feels like to remember".--How does he know that this feeling is 'remembering'?
Compare: "Yes, now I know what 'tingling' is". (He has perhaps had an electric shock for the first time.)--Does he
know that it is memory because it is caused by something past? And how does he know what the past is? Man
learns the concept of the past by remembering.
Page 231
And how will he know again in the future what remembering feels like?
Page 231
(On the other hand one might, perhaps, speak of a feeling "Long, long ago", for there is a tone, a gesture,
which go with certain narratives of past times.)
Page Break 232
xiv
Page 232
The confusion and barrenness of psychology is not to be explained by calling it a "young science"; its state is
not comparable with that of physics, for instance, in its beginnings. (Rather with that of certain branches of
mathematics. Set theory.) For in psychology there are experimental methods and conceptual confusion. (As in the
other case conceptual confusion and methods of proof.)
Page 232
The existence of the experimental method makes us think we have the means of solving the problems which
trouble us; though problem and method pass one another by.
Page 232
An investigation is possible in connexion with mathematics which is entirely analogous to our investigation
of psychology. It is just as little a mathematical investigation as the other is a psychological one. It will not contain
calculations, so it is not for example logistic. It might deserve the name of an investigation of the 'foundations of
mathematics'.
FOOTNOTES
Page ix
†1 It was hoped to carry out this plan in a purely German edition of the present work.
Page 2
†1 "When they (my elders) named some object, and accordingly moved towards something, I saw this and I
grasped that the thing was called by the sound they uttered when they meant to point it out. Their intention was
shewn by their bodily movements, as it were the natural language of all peoples: the expression of the face, the play
of the eyes, the movement of other parts of the body, and the tone of voice which expresses our state of mind in
seeking, having, rejecting, or avoiding something. Thus, as I heard words repeatedly used in their proper places in
various sentences, I gradually learnt to understand what objects they signified; and after I had trained my mouth to
form these signs, I used them to express my own desires."
Page 21
†1 I have translated the German translation which Wittgenstein used rather than the original. Tr.
Page 86
†1 The MSS. have: .... der Reihe x = 1, 3, 5, 7, .... indem er die Reihe der x2 + 1 hinschreibt.--Ed.
Page 194
†1 Fact and Fable in Psychology.
REMARKS ON THE PHILOSOPHY OF PSYCHOLOGY
Volume I
Page i
Page Break ii
Titlepage
Page ii
Ludwig Wittgenstein:
REMARKS ON THE PHILOSOPHY OF PSYCHOLOGY:
VOLUME I
Edited by
G. E. M. ANSCOMBE
and
G. H. von WRIGHT
Translated by
G. E. M. ANSCOMBE
BASIL BLACKWELL
OXFORD
Page Break iii
Copyright page
Page iii
Copyright © Blackwell Publishers Ltd, 1980
First published in 1980
Reprinted 1988, 1990, 1998
Blackwell Publishers Ltd
108 Cowley Road
Oxford OX4 1JF, UK
All rights reserved. Except for the quotation of short passages for the purposes of criticism and review, no part of
this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher.
Page iii
Except in the United States of America, this book is sold subject to the condition that it shall not, by way of trade or
otherwise, be lent, resold, hired out, or otherwise circulated without the publisher's prior consent in any form of
binding or cover other than that in which it is published and without a similar condition including this condition
being imposed on the subsequent purchaser.
British Library Cataloguing in Publication Data
Wittgenstein, Ludwig
Remarks on the philosophy of psychology
Vol. I
1. Cognition 2. Knowledge, Theory of
I. Title II. Anscombe, Gertrude Elizabeth Margaret. III. Wright, Georg Henrik von
153.4 BF311
ISBN 0-631-13061-6
Printed in Great Britain by Athenaeum Press Ltd, Gateshead, Tyne & Wear
This book is printed on acid-free paper
Page Break iv
PREFACE
Page iv
After Wittgenstein had finished Part I of Philosophical Investigations in the form in which it was later printed, he
was writing remarks from May 1946 to May 1949 in new MS volumes (MSS 130-138), remarks almost exclusively
concerned with the nature of psychological concepts. During this period he twice dictated a selection of the MS
material to a typist, in late autumn 1947 (TS 229) and in early autumn 1948 (TS 232). For the last third of the MS
entries there is no corresponding extant typescript. Probably in the middle of 1949 Wittgenstein put together a MS
selection (MS 144), mainly from what he had written since October 1948, but partly also from earlier MS volumes
and typescripts. He then had a fair copy of this MS typed out, and it was printed as Part II of Philosophical
Investigations. This typescript has unfortunately disappeared.
Page iv
What Wittgenstein wrote in MS books 130-138 may with some justification be described as preparatory
studies for Part II of the Investigations. He cut up the two typescripts 229 and 232 into slips, and preserved in all
three hundred and sixty-nine of the fragments for further use. They are printed in the collection Zettel. (They
amount to more than half of the remarks in that work.) But by far the greater part of the remarks in TSS 229 and 232
and in the MS volumes 137 and 138 have thus far remained unpublished.
Page iv
The editors considered it right to publish the two typescripts 229 and 232 in their entirety in two volumes
under the title "Remarks on the Philosophy of Psychology".
Page iv
Typescript no. 229 is here published as the first volume. The underlying MSS cover the time from May 10th
1946 to October 11th 1947. There were two versions of the typescript, one probably copied from the other. Both
were marred with many spelling mistakes and other faults. An exact collation with the MS sources was carried out.
The MSS for the most part do not contain drawings, and so we have taken them from the corresponding MSS.
Page iv
Mr. André Maury and Mr. Heikki Nyman helped in working over the sources to make an accurate and
complete text. Heikki Nyman made the index for the book. The editors would like to take this opportunity of
thanking their collaborators for this laborious work.
G. E. M. Anscombe
G. H. von Wright
Page iv
Wittgenstein's spelling is sometimes old-fashioned, sometimes vacillating--and sometimes obviously
incorrect. The punctuation also often
Page Break v
departs very much from normal punctuation. Care was necessary in making corrections. In general we have tried to
follow the readings of the typescript, e.g. in the case of the initial letters of adjectives functioning substantivally. In a
few cases only we have chosen to disturb the punctuation of the typescript. We are greatly indebted to Joachim
Schulte for helpful advice in correction of the printed text.
Page Break 1
Page Break 2
I
Page 2
1. Let's consider what is said about such a phenomenon as this: seeing the figure sometimes as an F,
sometimes as the mirror-image of an F.
I want to ask: what does seeing the figure now this way now that consist in?--Do I actually see something
different each time; or do I only interpret what I see in a different way?--I am inclined to say the former. But why?
Well, interpreting is an action. It may consist, e.g., in someone's saying "That's supposed to be an F", or he doesn't
say it, but when he copies the sign he replaces it by an F; or he considers: "What may that be? It'll be an F that the
writer slipped with". Seeing isn't an action but a state. (A grammatical remark.) And if I have never read the figure as
anything but "F", never considered what it might be, then we shall say that I see it as an F; if, that is, we know that it
can also be seen differently.
For how have we arrived at the concept of 'seeing this as this'? On what occasions does it get formed, is it felt
as a need? (Very frequently, when we are talking about a work of art.) Where, for example, what is in question is a
phrasing by the eye or ear. We say "You must hear this bar as an introduction", "You must listen for this mode", but
also "I hear the French 'ne... pas' as a negation with two parts to it, not as 'not a step'" etc. Now is it an actual seeing
or hearing? Well: that's what we call it; we react with these words in particular situations. And in turn we react to
these words with particular actions. [Zettel, 208.]
Page 2
2. Is it introspection that tells me whether I have to do with a genuine seeing, or rather with an act of interpreting? To
start with I must get clear about what I would call a case of interpreting; what it is that tells me whether something is
interpreting or seeing.
Page 2
(Seeing in accordance with an interpretation.) [Z 212.]
Page 2
3. I should like to say: "I see the figure as the mirror-image of an F" is only an indirect description of my experience.
That there is a direct one; namely: I see the figure like this (here I point for myself at my visual impression). Whence
this temptation here?--There is an important fact here, namely that we are prepared to allow for a number of different
descriptions of our visual impression, e.g. "Now the figure is looking to the right, now to the left."
Page Break 3
Page 3
4. Suppose we were to ask someone: What similarity is there between this figure and an F? Now one person answers
"The figure is a reversed F", and another "It is an F with the horizontals made too long". Are we to say "These two
see the figure differently"?
Page 3
5. Don't I see the figure sometimes this way, sometimes otherwise, even when I don't react with words or any other
signs?
But "sometimes this way", "sometimes otherwise" are after all words, and what right have I to use them
here? Can I prove my right to you, or to myself? (Unless by a further reaction.)
But surely I know that there are two impressions, even if I don't say so! But how do I know that what I say
then, is the thing that I knew? [Z 213.]
Page 3
6. The familiar face of a word; the feeling that a word is as it were a picture of its meaning; that it has as it were taken
its meaning up into itself--it's possible for there to be a language to which all that is alien. And how are these feelings
expressed among us? By the way we choose and value words. [Cf. P.I. p. 218f.]
Page 3
7. It is easy to describe the cases in which we are right to say we interpret what we see, as such-and-such. [Cf. P.I.
p. 212e.]
Page 3
8. When we interpret, we make a conjecture, we express a hypothesis, which may subsequently turn out false. If we
say "I see this figure as an F", there isn't any verification or falsification for that, just as there isn't for "I see a
luminous red". This is the kind of similarity that we must look for, in order to justify the use of the word "see" in that
context. If someone says that he knows by introspection that it is a case of 'seeing', the answer is: "And how do I
know what you are calling introspection? You explain one mystery to me by another." [Cf. P.I. p. 212e.]
Page 3
9. In different places in a book, a text-book of physics say, we see the illustration: . In the
accompanying text what is in question is one time a glass cube, another a wire frame, another a lidless open box,
another time it's three boards making a solid angle. The text interprets the illustration every time.
But we can also say that we see the illustration now as one thing, now as another.--Now how remarkable it is,
that we are able to use
Page Break 4
the words of the interpretation also to describe what is immediately perceived!
Here at first we should like to reply: This description of the immediate experience by means of an
interpretation is only an indirect description. That the truth is this: We can give the figure one time interpretation A,
one time interpretation B, one time interpretation C; and there are also three direct experiences--three ways of seeing
the figure--A', B', C', such that A' favours interpretation A, B' interpretation B, C' interpretation C. That is why we
use interpretation A as a description of the way of seeing which is favourable to it. [Cf. P.I. p. 193f, g.]
Page 4
10. But what does it mean to say that the experience A' favours interpretation A? What is the experience A'? How is
it identified?
Page 4
11. Let us assume that someone makes the following discovery. He investigates the processes in the retina of human
beings who are seeing the figure now as a glass cube, now as a wire frame etc., and he finds out that these processes
are like the ones that he observes when the subject sees now a glass cube, now a wire frame etc.... One would be
inclined to regard such a discovery as a proof that we actually see the figure differently each time.
But with what right? How can the experiment make any pronouncement upon the nature of the immediate
experience?--It puts it in a particular class of phenomena.
Page 4
12. How is the experience A' identified? How does it come about that I know of this experience at all?
How does one teach anyone the expression of this experience: "Now I am seeing the figure as a wire frame"?
Many have learnt the word "see" and never made any such use of it.
Now if I shew our figure to such a one, and tell him "Now just try to see it as a wire frame"--must he
understand me? What if he says: "Do you mean anything but that I am to follow the text of the book, which is about
a wire frame, and to use the figure as an aid in doing so?" And if he doesn't understand me, what can I do? And if he
does understand me, how is that manifested? Isn't it just in this, that he too says he is now seeing the figure as a wire
frame?
Page 4
13. Thus the inclination to use that form of verbal expression is a characteristic utterance of the experience. (And an
utterance is not a symptom.)
Page Break 5
Page 5
14. Are there still other utterances of this experience? Wouldn't the following proceeding be conceivable: I put a wire
frame, a glass cube, a box, etc. in front of someone and ask him "Which of these things does the figure represent?"
He replies "The wire frame".
Page 5
15. Ought we now to say he saw the figure as a wire frame--though he did not have the experience of seeing it now
as this, now as something else?
Page 5
16. Suppose someone asked "Do we all see a printed F the same way?" Well, one might try the following: We shew
various people an F and put the question: "Which way does an F look, to the right or to the left?"
Or we ask: "If you were supposed to compare an F with a face in profile, where would be the front and
where the back?"
But maybe some would not understand these questions. They are analogous to questions like "What colour
is the sound a for you?" or "Does a strike you as yellow or white?" etc.
If someone didn't understand this question, if he called it nonsense,--could we say he didn't understand
English, or the meanings of the words "colour", "sound", etc.?
On the contrary: it's when he has learnt to understand these words that he can react to those questions 'with
comprehension' or 'uncomprehendingly'.
Page 5
17. "Do we all see an F the same way?"--That doesn't yet mean a thing, so long as it isn't settled how we learn 'what
way' someone sees it. But if now, e.g., I also say "For me an F looks towards the right and a J towards the
left",--does that allow me to say: Whenever I see an F, it looks in this direction, or in any direction? What reason
would I have to say anything of the kind?
Page 5
18. Let us assume that the question "Which direction does an F look in?" had never been put--but only this one: "If
you had to paint an eye and a nose onto an F and a J, would it look to the right or to the left?" This too would surely
be a psycological[[sic]] question. And it would not involve anything about a 'seeing' this way, or otherwise! What is
in question is an inclination to do one thing or the other.
Page 5
19. One employment of the concept 'looking in this direction' is, e.g., as follows: One says, perhaps to an architect:
"This distribution of the windows makes the façade look in that direction." Similarly one uses the expression "This
arm interrupts the movement of the sculpture" or "The movement should go like this" (here one makes a gesture).
Page Break 6
Page 6
20. The question whether what is involved is a seeing or an act of interpreting arises because an interpretation
becomes an expression of experience. And the interpretation is not an indirect description; no, it is the primary
expression of the experience.
Page 6
21. But why don't we see that at once, but think rather that there must be an immediate expression here, and that the
phenomenon just is too intangible, can't really be described, and in any case we have to grope for an indirect
representation to communicate with other people?
We tell ourselves: Unless we supply something extra to the figure in our fancy, we can't possibly have an
experience essentially tied up with things that are quite outside the sphere of immediate perception.
One might say, e.g.: "You assert that you see the figure as a wire frame. Do you perhaps also know if it is a
copper wire or an iron wire? And why then has it got to be a wire?--This shews that the word 'wire' doesn't actually
belong essentially to the description of the experience."
Page 6
22. But now let us imagine the following kind of explanation: If one holds one's nose while eating, foods lose all
their taste, except for sweet, bitter, salt and sour. So, we want to say, the special taste of bread, say, consists of this
'taste' in the narrower sense and the aroma, which is what gets lost when we don't breathe through our nose. Now
why shouldn't there be something like that going on in connection with seeing? Perhaps in this way: The eye doesn't
distinguish the figure as a wire frame from the figure as a box, etc. That is so to speak the aroma, which the brain
supplies to what is seen. On the other hand the eye does distinguish various aspects: it as it were phrases the visual
picture; and one phrasing is more in accord with one interpretation, the other with the other. (More in accord as a
matter of experience.)
Think, for example, of certain involuntary interpretations that we give to one or another passage in a piece of
music. We say: This interpretation forces itself on us. (That is surely an experience.) And the interpretation can be
explained by purely musical relationships:--Very well, but our purpose is, not to explain, but to describe.
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23.
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See the triangle in such a way that c is the base and C the apex; and now, so that b is the base and B the
apex.--What do you do?--First of all:--do you know what you do? No.
"Well, perhaps it is the glance, which first fixes on the 'base' and then goes to the 'apex'. But can you say that
your glance couldn't shift in just the same way, in another context, without your having seen the triangle that way?
Make this experiment too: See the triangle in such a way that (like an arrowhead) it points now in direction A,
now in direction B.
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24. Of whom do we say that he is seeing the triangle as an arrow that points to the right? Of one who has simply
learned to use it as such an arrow and has always used it like that? No. Naturally, that does not mean that such a one
is said to be seeing it differently, or that we wouldn't know how he is seeing it. Seeing this way or otherwise doesn't
come in here yet.--But what about a case in which I correct someone else and say "What is over there is not an
arrow pointing to the right, but one pointing upwards", and now I confront him with some practical consequence of
this interpretation. He says: "I always took the triangle as an arrow pointing to the right."--Is a seeing in question
here? No: for of course it may mean "When I have encountered this sign I have always followed it this way."
Someone who says that need not have the least understanding of the question: "But: were you seeing it as an arrow
pointing to the right?"
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25. We say that a man sees the triangle now this way, now that, if he says it of himself; if he pronounces, or hears,
these words with signs of understanding; but also, if he says, e.g., "Now the triangle is pointing in this direction;
before, it pointed in the other direction," and then, when asked whether the triangle has changed its form or position,
answers: "It's not like that". And so on.
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26. Let us consider the case of the picture of wheels rotating in opposite directions. Firstly, I may again see the
movement in the picture as one or as the other movement. Secondly, I may also take it for the one or the other.
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27. The somewhat queer phenomenon of seeing this way or that surely makes its first appearance when someone
recognizes that the optical picture in one sense remains the same, while something else, which one might call
"conception", may change. If I take the picture for this or that, let's say for two wheels turning opposite ways, there
is so far no question of a division of the impression into optical picture and conception--Should I say, then, that this
division is the phenomenon that interests me?
Or let us ask this: What reaction am I interested in? The one that shews that someone takes a bowl for a
bowl (and so also the one that shews that he takes a bowl for something else)? Or the one that shews that he
observes a change and yet shews at the same time that nothing has altered in his optical picture?
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28. Another possibility is that I say: I have always taken that for a bowl; now I see that it isn't one--without being
conscious of any change of 'aspect'. I mean simply: I now see something different, now have a different visual
impression.
Suppose someone shews something to me and asks me what it is. I say: "It's a cube." At which he says "So
that's how you see it". Would I have to understand these words in any other sense than: "So that's what you take it
to be"?
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29. When I contemplate the objects around me, I am not conscious of there being such a thing as a visual
conception.
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30. "I see this figure as a solid angle": why don't you simply accept that as true if, that is, he knows English and is
reliable?--I don't doubt that it is the truth. But what he said is a tensed sentence. Not one about the nature of this
phenomenon: no, but one saying: this happened.
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31. The expression of the experience is: "Now I'm seeing it as a pyramid; now as a square with its diagonals."
Now, what is the "it" which I see now this way, now that? Is it the drawing? And how do I know it is the
same drawing both times? Do I merely know this, or do I see it as well?--How would it be, if it were subsequently
proved that the drawing always altered slightly when it was seen as
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something else; or that the optical picture was then slightly different? One line, for example, looks a little heavier, or
thinner, then than before.
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32. Shall I say that the various aspects of the figure are associations? And how does that help me?
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33. Something about the optical picture of the figure seems to alter here; and then again, nothing alters after all. And
I cannot say "A new interpretation keeps on striking me". Indeed it does; but it also incorporates itself straight away
in what is seen. There keeps on striking me a new aspect of the drawing--which I see remains the same. It is as if a
new garment kept on being put on it, and as if all the same each garment was the same as the other.
One might also say "I do not merely interpret the figure, but I clothe it with the interpretation".
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34. I say to myself: "What is this? What does this phrase say? Just what does it express?"--I feel as if there must still
be a much clearer understanding of it than the one I have. And this understanding would be reached by saying a
great deal about the surrounding of the phrase. As if one were trying to understand an expressive gesture in a
ceremony. And in order to explain it I should need as it were to analyse the ceremony. E.g., to alter it and shew what
influence that would have on the role of that gesture.
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35. I might also say: I feel as if there must be parallels to this musical expression in other fields.
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36. The question is really: are these notes not the best expression for what is expressed here? Presumably. But that
does not mean that they aren't to be explained by working on their surounding[[sic]].
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37. Is it a contradiction if I say: "This is beautiful and this is not beautiful" (pointing at different objects)? And ought
one to say that it isn't a contradiction, because the two words "This" mean different things? No; the two "This's"
have the same meaning. "Today" has the same meaning today as it had yesterday, "here" the same meaning here and
there. It is not here as with the sentence "Mr. White turned white".
"This is beautiful and this is not beautiful" is a contradiction, but it has a use.
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38. The basic evil of Russell's logic, as also of mine in the Tractatus, is that what a proposition is is illustrated by a
few commonplace examples, and then pre-supposed as understood in full generality.
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39. But isn't it clear that the two 'this's' have different meanings, since they can be replaced by different proper
names?--Replaced? "This" just doesn't now mean "A", now "B".--Of course not by itself, but together with the
pointing gesture.--Very well; that is only to say that a sign consisting of the word "this" and a gesture has a different
meaning from a sign consisting of "this" and another gesture.
But that is of course mere juggling with words. What you are saying is that your sentence "This is beautiful
and this is not beautiful" is not a complete sentence, because these words have to have gestures going with
them.--But why is it not a complete sentence in that case? It is a sentence of a different kind from, say "The sun is
rising"; it has a very different kind of employment. But such are the differences that there are, this is the profusion
that there is in the realm of sentences.
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40. "Scot is not a Scot." If I say this, I mean the first S. as a proper name, the second as a class-name. Is there
something different going on in my mind, when I pronounce the two words "S."?--The word functions in the
proposition in a different way in the two cases. That would be to make a comparison of the word to a machine-part
and of the sentence to the machine. Quite ineptly. Rather one might say: the language is the machine, the sentence
the machine part. It would then go something like this: This crank has two holes of the same size. With one it is
attached to the shaft, while the crank pin sticks into the other. [Cf. P.I. p. 176f.]
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41. Try to mean the first "S." as a class-name, the second as a proper name! How do you make the attempt? [Cf. P.I.
p. 176f.]
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42. "The concept Scot is not a Scot." Is this nonsense? Well, I do not know what anyone who says that is trying to
say, that is, how he is intending to use this sentence. I can think out several uses for it, which are ready to hand. "But
you just can't use it, nor can you think it, in such a way that the same thing is meant by the words 'the concept Scot'
and the second 'Scot', as you ordinarily mean by these words." Here is the mistake. Here one is thinking as if this
comparison came into one's mind: words fit together in the sentence, i.e. senseless
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sequences of words may be written down; but the meaning of each word is an invisible body, and these
meaning-bodies do not fit together. (("Meaning it gives the sentence a further dimension."))
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43. Hence the idea that the sentence can't be thought, for in thought I should have to fit the meanings of the words
together into a sense, and it doesn't work. (Jigsaw puzzle.)†1
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44. But isn't contradiction forbidden by the law of contradiction?--At any rate "Non (p and non p)" doesn't forbid
anything. It is a tautology. But if we forbid a contradiction, then we are excluding forms of contradiction from our
language. We expunge these forms.
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45. One may have the thought: "How remarkable that the single meaning of the word "to feel" (and of the other
psychological verbs) is compounded of heterogeneous components, the meanings of the first and of the third
person."
But what can be more different than the profile and the front view of a face; and yet the concepts of our
language are so formed, that the one appears merely as a variation of the other. And of course it is easy to give a
ground in facts of nature for this structure of concepts. (Heterogeneous things; arrow-head and arrow-shaft.)
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46. If we can find a ground for the structures of concepts among the facts of nature (psychological and physical),
then isn't the description of the structure of our concepts really disguised natural science; ought we not in that case
to concern ourselves not with grammar, but with what lies at the bottom of grammar in nature?
Indeed the correspondence between our grammar and general (seldom mentioned) facts of nature does
concern us. But our interest does not fall back on these possible causes. We are not pursuing a natural science; our
aim is not to predict anything. Nor natural history either, for we invent facts of natural history for our own purposes.
[Cf. P.I. p. 230a.]
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47. It is interesting, for example, to observe that particular shapes are not tied to particular colours in our
environment; that, for example, we do not always see green in connection with round, red in connection with
square. If we imagined a world in which shapes and colours were always tied to one another in such ways, we'd find
intelligible a system of concepts, in which the fundamental division--shape and colour--did not hold.
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Some further examples:
It is important, for example, that we are accustomed to draw with pencil, pen or the like, and that therefore
the elements of our representation are strokes and points (in the sense of dots). Had human beings never drawn, but
always painted (so that the concept of the contour of shapes did not play a big part), if there were a word in
common use, let's call it "line", at which no one thought of a stroke, i.e. of something very thin, but always thought
only of the boundary of two colours, and if at the word "point" one never thought of something tiny, but only of the
intersection of two colour boundaries, then perhaps much of the development of geometry would not have
occurred.
If we only saw one of our primary colours, red say, extremely seldom and only in tiny expanses, if we could
not prepare colours for painting, if red occurred only in particular connections with other colours, say only at the
very tips of leaves of certain trees, these tips gradually changing from green to red in the autumn, then nothing
would be more natural than to call red a degenerate green.
Think of the circumstances under which white and black appear to us as colours and on the other hand as the
lack of any colour. Imagine its being possible to wash all colours away, and that then the base was always white, and
that there was no such thing as white paint.
It is easier for us to reproduce and recognize a pure red, green, etc. from memory, than say, a shade of
reddish brown.
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48. But I am not saying: if the facts of nature were different we should have different concepts. That is an
hypothesis. I have no use for it and it does not interest me.
I am only saying if you believe that our concepts are the right ones, the ones suited to intelligent human
beings; that anyone with different ones would not realize something that we realize, then imagine certain general
facts of nature different from the way they are, and conceptual structures different from our own will appear natural
to you. [Cf. P.I. p. 230b.]
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49. 'Natural', not 'necessary'. For is everything that we do a means to an end? Is everything inappropriate, that can't
be called a means to an end?
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50. (On 33) The explanation: "I associate this object with the figure", makes nothing clearer.
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51. How is "will" actually used? In philosophy one is unaware of having invented a quite new use of the word, by
assimilating its use to
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that of, e.g., the word "wish". It is interesting that one constructs certain uses of words specially for philosophy,
wanting to claim a more elaborated use than they have, for words that seem important to us.
"Want" is sometimes used with the meaning 'try': "I wanted to get up, but was too weak." On the other hand
one wants to say that whereever a voluntary movement is made, there is volition. Thus if I walk, speak, eat, etc., etc.,
then I am supposed to will to do so. And here it can't mean trying. For when I walk, that doesn't mean that I try to
walk and it succeeds. Rather, in the ordinary way I walk without trying to. Of course it can also be said "I walk
because I want to", if that distinguishes the ordinary case of walking from that in which I am shoved, or electric
currents move my leg muscles.
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52. Philosophy has tried to fix itself up with a use of the word which presents as it were a more consistent following
up of certain features of the ordinary use.
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53. "The word 'x' has two meanings" means: it has two kinds of use.
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Ought I to say "If you describe the use of this word in our language, you will see that it has two uses, not just
one"?
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54. Might we not imagine people declaring that the word "bench" always has the same meaning? That a bench is
always something like this: But that they did, nevertheless, also use the word for a legal institution;
but of that they say that since it is a bench, it is something of the kind we have drawn in our picture.
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55. Have the words "go" and "went" the same meaning?
Have the words "go" and "goest" the same meaning?
Has the word "go"†1 the same meaning in "I go"†1 and in "You go"†1?
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56. Should I say: "With two different meanings there go two different explanations of meaning"?
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57. Imagine a group of sentences in a language each consisting of three signs. The sentences describe the work
carried out by a particular man. The first sign (from left to right) is the man's name, the second signifies an activity
(such as sawing, boring, filing) the third what is worked on.
Such a sentence might run "a a a". If, that is, "a" is the name of a person, an action, and what is worked on.
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58. Now what does it mean to say "The sign 'a' has a different meaning in 'x a y' and in 'a x y'"? It might even be said
to have a different meaning according to its position. (Like a digit in the decimal system.)
Imagine chess played with pieces all the same shape. One would then always have to remember where a
particular piece was at the beginning of the game. And it might be said: "This piece and that have different
meanings;" I can't make the same moves with one as with the other. Just so I gather from the "a" in the first position
that the matter concerns this man (perhaps I point to him); from the one in the second position, that he is doing this
work, etc. The "a" might occur in three tables in which it is correlated with certain pictures that explain its meaning.
And in that case I should look the sign 'a' up in a different table according to its position in order to interpret the
sentence.
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59. What does it mean "to investigate whether 'f(f)' makes sense when 'f' has the same meaning in both places"?
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60. One is looking for something, hasn't found it yet, but knows what one is looking for. But it may also happen that
one looks around searchingly and cannot say what one is searching for; finally one lights upon something and says
"That's what I wanted". "Looking", it might be called, "without knowing what one is looking for".
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61. We might speak of "functional states". (E.g.: Today I am very irritable. If I am told such-and-such today, I keep
on reacting in such-and-such a way. In contrast with this: I have a headache the whole day.)
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62. How did such an expression as "I believe..." ever come to be used? Did a phenomenon, that of belief, suddenly
get noticed? [Cf. P.I. p. 190a.]
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63. Did we observe ourselves and discover this phenomenon in that way?
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64. Did we observe ourselves and other men and so discover the phenomenon of belief?
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65. In the language of a tribe there might be a pronoun, such as we do not possess and for which we have no
practical use, which 'refers' to the propositional sign in which it occurs. I will write it like this: I. The proposition "I
am ten centimetres long" will then be tested for truth by measuring the written sign. The proposition "I contain four
words" for example is true; and so is "I do not contain four words". "I am false" corresponds to the paradox of the
Cretan Liar.--The question is: What do people use this pronoun for? Well, the proposition "I am ten centimetres
long" might serve as a ruler, the proposition "I am beautifully written" as a paradigm of beautiful script.
What interests us is: How does the word "I" get used in a language-game"? For the proposition is a paradox
only when we abstract from its use. Thus I might imagine that the proposition "I am false" was used in the
kindergarten. When the children read it, they begin to infer "If that's false, it's true, so it is false, etc. etc." People have
perhaps discovered that this inferring is a useful exercise for children.†1
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What interests us is: how this pronoun gets used in a language-game. It is possible, though not quite easy, to
fill out a picture of a language-game with this word. A proposition like "I contain four words" might, for example, be
used as a paradigm for the number four, and in another sense so might the proposition "I do not contain four
words". A proposition is a paradox only if we abstract from its use.†1
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66. How would people differ from us who were not able, like us, to see a triangle now this way, now that?--If we
came to a tribe that did not have these experiences, how should we notice this?
How should we notice it, if the people could not see depth? So that they were as Berkeley thought we are.
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67. How many squares go in a square when the scale in which to take the small square has not
been determined? Suppose someone came and said: one can't say for sure how many will go in, but one can at any
rate make an estimate!
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68. "The expression like the feeling"--the bitter food like the bitter sorrow. "As like as like can be",--how would it be
if they were not merely like, but the same?
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69. "Sorrow and care are similar feelings." Is that an empirical fact?
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70. Ought I to say: "A rabbit may look like a duck"?
Would it be conceivable that someone who knows rabbits but not ducks should say: "I can see the drawing
as a rabbit and also in another way, although I have no word for the second aspect"? Later he gets to
know ducks and says: "That's what I saw the drawing as that time!" Why is that not possible?
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71. Or suppose someone said: This rabbit has a complacent expression.--If someone knew nothing about a
complacent expression--might something strike him here, and he later on, having learnt to recognize complacency,
say that that was the expression that struck him then?
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72. The appropriate word. How do we find it? Describe this! In contrast to this: I find the right term for a curve,
after I have made particular measurements of it.
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73. I see that the word is appropriate even before I know, and even when I never know, why it is appropriate.
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74. I should not understand someone who said that he had seen the picture as that of a rabbit, but had not been able
to say so, because at that time he had not been aware of the existence of such a creature.
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75. Should I say: "The picture-rabbit and the picture-duck look just the same"?! Something militates against
that--But can't I say: they look just the same, namely like this--and now I produce the ambiguous drawing. (The draft
of water, the draft of a treaty.) But if I now wanted to offer reasons against this way of putting things what would I
have to say? That one sees the picture differently each time, if it is now a duck and now a rabbit--or, that what is the
beak in the duck is the ears in the rabbit, etc?
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76. Imagine the ambiguous picture being used in a strip cartoon. Then it is not possible, for example, that some other
animal should meet the duck and take it for a rabbit; but it would be possible for someone in the twilight to take the
duck in profile for a rabbit.
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77. "I can no more see the rabbit and the duck at the same time than I can mean the words 'Weiche Wotan, weiche!'
in their two meanings."--But that would not be right; what is right is that it is not natural for us to pronounce these
words in order to tell Wotan he should depart, and in saying so to tell him that we prefer our eggs soft boiled.†1 And
yet it would be possible to imagine such a use of words.
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78. The facts of human natural history that throw light on our problem, are difficult for us to find out, for our talk
passes them by, it is occupied with other things. (In the same way we tell someone: "Go into the shop and
buy..."--not: "Put your left foot in front of your right foot etc. etc., then put coins down on the counter, etc. etc.")
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79. If I do not believe in an inner state of seeing and the other says: "I see...", then I believe that he does not know
English, or is lying.
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80. What has been said, if it is said that anyone who sees the drawing now as a rabbit now as a duck has quite
different visual experiences? The inclination to say this becomes very great, if, e.g., one adds a line to the drawing
that perhaps emphasizes the mouth of the rabbit, and then sees how this line plays a quite different part in the
picture of the duck.--Or think of the facial expression of the rabbit, which completely disappears in the other picture.
At first, for example, I see a haughty face, and then I don't see a haughty face.
And what is done by someone, if he admits that I see something quite different each time?
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81. "How do I know that I am smiling at this facial expression?"
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82. "I see a quite particular facial expression, which I call that of the rabbit, and a completely different one which I
call that of the duck." Let me merely call one A and the other one B: How could I now explain the meaning of A and
B to someone without making any reference to a rabbit and a duck?
It would be possible, e.g. like this: I say "A" to him and give an imitation of a rabbit's face with my own face
etc.
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83. "'Seeing this' doesn't mean: reacting in this way,--for I can see without reacting." Of course. For neither does "I
see" mean: I react, nor "he sees": he reacts, nor "I saw": I reacted, etc.
And even if I said "I see" whenever I saw, these words wouldn't say "I say 'I see'".
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84. I point to a particular spot in the picture and say "That is the eye of the rabbit or of the duck." Now how can
something in this drawing be an eye?
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85. "Can depth really be seen?"--"Why should one not be able to see depth, when one can see colours and shapes?!
The retina's being two-dimensional is no reason for saying the opposite."--Certainly not; but the answer does not
meet the problem. The problem arises from this, that the description of the seen, what we call the description of what
is seen, is of a different kind, if one time I represent colour and shape, perhaps using a transparency, another time the
dimension of depth by means of a gesture or a profile.
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86. It is unhelpful to remark that the arrangement in the dimension of depth is, like any other, a property of the 'seen'.
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87. What does it mean to say that the cavity in a tooth that the dentist is probing feels much bigger than it is to the
patient? I shew with my fingers, e.g., and say: "I would have thought it was as big as this." What do I go by in
measuring the distance apart of the fingers?--Do I measure it at all? Can one say: "First, I know how big the cavity
strikes me as being, then I shew it with my fingers"? Well, in some cases that could be said, when, for example, I
think to myself that the cavity is 5 mm and explain this to someone by shewing him the distance--Suppose I were
asked: "Did you know how big the diameter struck you as being before you shewed it?"--Here I might reply: "Yes.
For if you had asked me earlier, I should have given you this answer."--Knowing something just isn't: thinking a
thought.
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88. When I say what I know, how is what I say what I know?
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89. What is the description of what I see? (This doesn't mean only: In what words am I to describe what I see?--but
also "What does a description of what I see look like? What am I to call by that name?")
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90. The peculiar feeling that the recurrence of a refrain gives us. I should like to make a gesture. But the gesture isn't
really at all characteristic precisely of the recurrence of a refrain. Perhaps I might find that a phrase characterizes the
situation better; but it too would fail to explain why the refrain strikes one as a joke, why its recurrence elicits a laugh
or grin from me. If I could dance to the music, that would be my best way of expressing just how the refrain moves
me. Certainly there couldn't be any better expression than that.--
I might, for example, put the words "To repeat", before the refrain. And that would certainly be apt; but it
does not explain why the refrain makes a strongly comic impression on me. For I don't always laugh when a "To
repeat" is appropriate.
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91. The 'content' of experience, of experiencing: I know what toothaches are like, I am acquainted with them, *I
know what it's like to see red, green, blue, yellow, I know what it's like to feel sorrow, hope fear, joy, affection, to
wish to do something, to remember having done something, to intend doing something, to see a drawing alternately
as the head of a rabbit and of a duck, to take a word in one meaning and not in another etc.*†1 I know how it is to
see the vowel a grey and the vowel ü dark purple.--I know, too, what it means to parade these experiences before
one's mind. When I do that, I don't parade kinds of behaviour or situations before my mind.--So I know, do I, what
it means to parade these experiences before one's mind? And what does it mean? How can I explain it to anyone
else, or to myself?
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92. The concept 'word' in linguistics. How does one use "the same word"?
'"have" and "had" are the same word.'
'He's saying the same word, once out loud, once silently.'
'Are "bank" (money) and "bank" (river) the same word?
'Is it the same word "have" both times when one says "I have a house" and "I have built a house"?
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93. Speculation: A tribe that we have brought into subjection, which we want to make into a slave-race. The
behaviour, the bearing of these people is of interest to us just for that reason. We want to describe it, to describe
various aspects of this behaviour. We watch and observe, e.g. pain-behaviour, joy-behaviour etc. Their behaviour
also includes the use of a language. And generally it includes such behaviour as is learned, no less than such as is not
learned, like a child's crying. Nor do they merely have a language, they have one containing psychological forms of
expression.--Ask yourself: How do these get taught to the children of this tribe?--
Now I assume that these people possess expressions like the following: "I have black hair", "He has black
hair"; "I have money", "He has money"; "I have a wound", "He has a wound". And now they use this grammatical
construction in psychological ascriptions.
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94. "As I heard 'bank' the meaning money-bank came to mind." It is as if a germ of meaning were experienced, and
then got interpreted. Now is that an experience?
One might precisely say: "I had an experience which was the germ for this use." That might be a form of
expression that was natural to us.
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95. Having a favourite... is also a movement of thought that one can learn.
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96. A tribe that we want to enslave. The government and the scientists give it out that the people of this tribe have no
souls; so they can be used without scruple for any purpose whatever. Naturally we are interested in their language all
the same; for of course we need to give them orders and to get reports from them. We want to know too what they
say among themselves, as this hangs together with the rest of their behaviour. But also we must be interested in what
in them corresponds to our 'psychological utterances', for we want to keep them capable of work, and so their
expressions of pain, of feeling unwell, of depression, of pleasure in life etc. etc. are of importance to us. Indeed, we
have also found that these people can be used successfully as experimental objects in physiological and
psychological laboratories; since their reactions--including speech-reactions--are altogether those of men endowed
with souls. I assume that it has also been found that these automata can be taught our language instead of their own
by a method very like our 'instruction'. [Cf. Z 528.]
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97. These beings now learn, e.g. to calculate, to calculate on paper or orally. But somehow we bring them to the
point of being able to say the result of a multiplication after they have sat still for a while without writing or speaking.
When one considers the kind of way in which they learn this 'calculating in the head', and the phenomena that
surround it, the picture suggests itself, that the process of calculating is as it were submerged, and goes on under the
mirror surface of the water. (Think of the sense in which water 'consists' of H and O.)
Naturally there are various purposes for which we need to have an order of the form "Calculate this in your
head"; a question "Have you calculated it?"; and even "How far have you got?"; a statement of the automaton "I
have calculated..."; etc. etc. In short: all that we say among ourselves about calculating in the head, is also of interest
to us when they say it. And what goes for calculating in the head also goes for other forms of thinking.--If anyone
among us voices the idea that something must surely be going on in these beings, something mental, this is laughed
at like a stupid superstition. And if it does happen that the slaves spontaneously form the expression that this or that
has taken place in them, that strikes us as especially comical. [Cf. Z 529.]
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98. With these beings we also play the game "Think of a number Multiply it by 5--..."--Does that prove that after all
something has taken place in them?Page
21
99. And now we observe a phenomenon,--which we might interpret as the expression of the experience: seeing a
figure now this way now that. Now we shew them, e.g., a puzzle picture. They find the solution; and then they say
something, point to something, draw something etc., and we can teach them our expression: "Now I always see the
picture this way." Or they have learnt our language and the ordinary use of the word "to see" and now they invent
that form spontaneously.
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100. What interest, what importance has this phenomenon, this reaction? It may be quite unimportant, quite
uninteresting, or again important and interesting. Some people associate certain colours with our vowels; some can
answer the question which days of the week are fat and which are thin. These experiences play a very subordinate
part in our lives; but I can easily think out circumstances, in which what is unimportant to us would acquire great
importance.
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101. The slaves also say "When I heard the word 'bank' it meant... to me". Question: Against the background of what
technique of language do they say this? For everything turns on that. What had we taught them, what employment
of the word "mean"? And what, if anything at all, do we gather from their utterance? For if we can do nothing with
it, it might interest us as a curiosity. Let us just imagine human beings who are unacquainted with dreams and who
hear our narrations of dreams. Imagine one of us coming to this non-dreaming tribe and gradually learning to
communicate with the people.--Perhaps you think they would never understand the word "dream". But they would
soon find a use for it. And the doctors of the tribe might very well be interested in our dreams and draw important
conclusions from the dreams of these strangers.--Nor can it be said that for these people the verb "to dream" could
mean nothing but to tell a dream. For the stranger would use both expressions, "to dream" and "to tell a dream" and
the people of our tribe wouldn't be allowed to confuse "I dreamt..." with "I told the dream...". [Cf. Z 530.]
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102. We ask ourselves: "What interests us about the psychological utterances of human beings?"--Don't see it so
much as a matter of course that these verbal reactions do interest us.
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103. Why does the chemical formula of a substance interest us? "Well, or [[sic, of?]] course its composition interests
us."--Here we have a similar case. The answer might also have been "Because its inner nature interests us".
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104. "You are surely not going to deny that rust and water and sugar have an inner nature!" "If one didn't know it
already, science would surely have shewn it beyond cavil."
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105. Is the hearing or thinking of a word in this or that meaning a genuine experience?--How is that to be
judged?--What speaks against it? Well, that one cannot discover any content for this experience. It's as if one were
expressing an experience, but then could not think what the experience really was. As if one could indeed
sometimes think of an experience that was simultaneous with the one we are looking for, but what we get to see then
is merely a garment, and, where what it clothes should be, there is a vacuum. And then one is inclined to say: "You
must not look for another content". The content of the experience just is to be described by the specific expression
(of the
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experience). But that does not satisfy us either. For why do we feel nevertheless that there just is no content there?
And is it like that only with the experience of meaning? Isn't it so also with, e.g., that of remembering? If
someone asks me what I have been doing in the last two hours, I answer him straight off and I don't read the answer
off from an experience I am having. And yet one says that I remembered, and that this is a mental process.
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106. One might almost marvel that one can answer the question "What did you do this morning?"--without looking
up historical traces of activity or the like. Yes; I answer, and wouldn't even know that this was only possible through
a special mental process, remembering, if I were not told so.
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107. But there is also such a thing as "I believe I remember that", whether rightly or wrongly--and here there comes
into view what is subjective about the psychological.
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108. If I now say that the experience of remembering and the experience of pain are different in kind, that is
misleading: for "experiences of different kinds" makes one think perhaps of a difference like that between a pain, a
tickle, and a feeling of familiarity. Whereas the difference of which we are speaking is comparable, rather, to that
between the numbers 1 and .
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109. Where do we get the concept of the 'content' of an experience from? Well, the content of an experience is the
private object, the sense-datum, the 'object' that I grasp immediately with the mental eye, ear, etc. The inner
picture.--But where does one find one needs this concept?
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110. Why, when I communicate my subjective memory, am I not inclined to say I was describing the content of my
experience?
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111. Of course, when I say "Memories of that day rose up in me" it looks different. Here I am inclined to speak of a
content of the experience, and I imagine something like words and pictures which rise up before my mind.
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112. I can shew someone what a particular pain, an itch, a tingle etc. is, by producing the feeling in him and
observing his reaction, the
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description that he gives, etc. But can I do anything like that in the case of memory-experience?--In such a way, that
is, that he can now say "Yes, now I know what it is to remember something". Of course, I can teach him what we
call "remembering something"; I can teach him the use of these words. But can he then say "Yes, now I have
experienced what that is!" (("Yes, now I know what shuddering is!"†1)) If he were to say so, we should be
astonished, and think "What can he have experienced?" For we experience nothing special. [Cf. P.I. p. 231c.]
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113. When someone says "Now I know what a tingle is," we know that he knows through his 'expression of the
sensation'; he jerks, makes a particular noise, says what we too say in this case, finds the same description apt as we
do. [Cf P.I. p. 231c.]
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114. And in this way we might actually speak of a feeling "Long, long ago" and these words are an expression of the
feeling; but not these: "I remember that I often met him." [Cf P.I. p. 231c.]
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115. "If it passes, then it was not true love." Why was it not in that case? Is it our experience, that only this feeling
and not that endures? Or are we using a picture: we test love for its inner character, which the immediate feeling
does not discover. Still, this picture is important to us. Love, what is important, is not a feeling, but something
deeper, which merely manifests itself in the feeling.
We have the word "love" and now we give this title to the most important thing. (As we confer the title
"Philosophy" on a particular intellectual activity.)
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116. We confer individual words as we confer already existing titles.
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117. "A new born child has no teeth."--"A goose has no teeth." "A rose has no teeth."--This last at any rate--one
would like to say--is obviously true! It is even surer than that a goose has none.--And yet it is none so clear. For
where should a rose's teeth have been? The goose has none in its beak. Nor, of course, has it any in its wings; but
that's not what anyone means when he says it has no teeth--Why, suppose one were to say: the cow chews its food
and then dungs the
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rose with it, so the rose has teeth in the mouth of a beast. This is not absurd, because one wouldn't have any idea in
advance, where to look for teeth in a rose. ((This hangs together somehow with the problem that the proposition
"The earth has existed for more than 100,000 years" has a clearer sense than "The earth has existed for the last five
minutes". For if you were to say that, I should ask you: "What observations are you referring to? What observations
would go against your proposition? Whereas I probably know the thinking and observations to which the first
proposition belongs.)) [Cf. P.I. p. 221h, g.]
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118. "You see, this is what it's like when one remembers something." This? What?--Can one imagine someone
saying: "I shall never forget this experience (namely of remembering)!"?
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119. Is memory an experience? What do I experience? And is it an experience, when the word "bank" means one
thing or the other to me?
Again: What do I experience?--One is inclined to answer: I saw this or that before me, I imagined it.
Well, do I merely say it?--that is, that this word meant this to me--and did nothing happen? It was mere
words?--Not mere words; and it can also be said that something happened, which corresponded to them--but one
cannot explain that it wasn't mere words by saying that something corresponding to them happened. For the two
expressions mean the same thing.
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120. The feeling of having been in just the same situation before. I have never had this feeling.
When I see someone I know well, his is a well-known face; it is far more intimately known to me, than when
it merely 'strikes me as familiar'. But wherein consists this familiar knowledge? Have I the feeling of familiar
knowledge the whole time when I am seeing him? And why does one not want to say that? One would like to say: "I
have no special feeling of familiar knowledge, no feeling that corresponds to my familiarity with him." When I say
that I know him extremely well, that I have seen him and talked with him countless times, that isn't meant to
describe a feeling. And what shews that this does not describe a feeling?--If, say, someone were to assert that he had
such a feeling the whole time he was seeing some intimately known object--or if he says he believes he has such a
feeling,--should I say I don't believe him?--Or should I say I don't know what sort of feeling that is?
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I see someone I know well, and someone asks me whether his face strikes me as familiar. I shall say: no. I
shall say that the face is that of a human being I have seen thousands of times. "And do you not have the experience
of familiarity--when you do have it with a face you hardly know?"!
How does it come out that I am not expressing a feeling, when I say: Of course his face is familiar to me, it is
as familiar as can be?
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121. Why is it ridiculous to speak of a continuous feeling of familiar acquaintance?--"Well, because you don't feel
one." But is that the answer?
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122. A feeling of familiar acquaintance; that would be something like a feeling of well-being. Why does it seem
correct to speak of a feeling here, and not there?--Here there occurs to me the special expression of well-being. A
cat's purr, say.
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123. And can I not also imagine a case, in which I should say someone has a constant feeling of familiar
acquaintance with an object? Think of someone going round a room in which he had not been for a long time, and
enjoying his familiar acquaintance with all the old things? Could one not speak of a feeling of familiarity here? And
why?--Do I know this feeling in myself? Is that why I find that here it makes sense to speak of the feeling?
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124. I imagine that all his doings have a familiar feel to him.--But how shall I know this?--Well, by his saying it. So
he must use certain words, he must, e.g., say "everything feels so familiar," or give some other specific expression of
the feeling.
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125. The feeling of the unreality of one's surroundings. This feeling I have had once, and many have it before the
onset of mental illness. Everything seems somehow not real; but not as if one saw things unclear or blurred;
everything looks quite as usual. And how do I know that another has felt what I have? Because he uses the same
words as I find appropriate.
But why do I choose precisely the word "unreality" to express it? Surely not because of its sound. (A word
of very like sound but different meaning would not do.) I choose it because of its meaning.
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But I surely did not learn to use the word to mean: a feeling. No; but I learned to use it with a particular meaning and
now I use it spontaneously like this. One might say--though it may mislead--: When I have learnt the word in its
ordinary meaning, then I choose that meaning as a simile for my feeling. But of course what is in question here is
not a simile, not a comparison of the feeling with something else.
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126. The fact is simply that I use a word, the bearer of another technique, as the expression of a feeling. I use it in a
new way. And wherein consists this new kind of use? Well, one thing is that I say: I have a 'feeling of
unreality'--after I have, of course, learnt the use of the word "feeling" in the ordinary way. Also: the feeling is a state.
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127. Anger. "I hate..." is obviously the expression of hate, "I am angry" seldom the expression of anger. Is anger a
feeling? And why not?--First and foremost: what does someone do, if he is angry? How does he conduct himself? In
other words: when does one say that someone is angry? In such cases he learns to use the expression "I am angry".
It is the expression of a feeling?--And why should it be the expression of a feeling, or of feelings?
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128. Then is anger not an experience?--Is clenching my fist, say, an experience, or pronouncing or writing down a
sentence?
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129. Take the various psychological phenomena: thinking, pain, anger, joy, wish, fear, intention, memory etc..,--and
compare the behaviour corresponding to each.--But what does behaviour include here? Only the play of facial
expression and the gestures? Or also the surrounding, so to speak the occasion of this expression? And if one does
include the surrounding as well,--how is the behaviour to be compared in the case of anger and in that of memory,
for example?
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130. Isn't this as if someone were to say: "Compare different states of water"--and by that he means its temperature,
the speed with which it is flowing, its colour, etc.?
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131. The behaviour of humans includes of course not only what they do without ever having learned the behaviour,
but also what they do (and so, e.g. say) after having received a training. And this behaviour has its importance in
relation to the special training.--If, e.g., someone
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has learnt to use the words "I am glad" as someone else has learnt to use the words "I am frightened", we shall draw
unlike conclusions from like behaviour.
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132. "But may he not be frightened, even though he never reveals it?" What does this "may" mean? Is it supposed to
mean "Does it sometimes happen that someone is frightened without ever saying so?"--No. Rather: "Is there any
sense in e.g. that question?"--Or: does it make sense, if a novelist narrates that someone was frightened but never
revealed it? Well, it does make sense. But what sense? I mean:--Where and how will such a sentence be used? When
I ask "What sense does it make?"--I want someone to answer me not with a picture or a series of pictures, but with
the description of situations.
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133. "But depression is surely a feeling; you surely don't want to say that you are depressed and don't feel it? And
where do you feel it?" That depends on what you call "feeling it". If I direct my attention to my bodily feelings, I
notice a very slight headache, a slight discomfort in the region of the stomach, perhaps a certain tiredness. But do I
mean that, when I say I am severely depressed?--And yet I say again: "I feel a burden weighing on my soul." "Well,
I can't express it any differently!"--But how remarkable that I say it that way and cannot express it differently!
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134. My difficulty is altogether like that of a man who is inventing a new calculus (say the differential calculus) and
is looking for a symbolism.
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135. Depression is not a bodily feeling; for we do not learn the expression "I feel depressed" in the circumstances
that are characteristic of a particular bodily feeling.
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136. "But depression, anger, is surely a particular feeling!"--What sort of proposition is that? Where is it used?
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137. Uncertainty: whether a man really has this feeling, or is merely putting up an appearance of it. But of course it is
also uncertain whether he is not merely putting up an appearance of pretending. This pretence is merely rarer and
does not have grounds that are so easily understood.--But what does this uncertainty consist in? Am I really always
in some uncertainty whether someone is really angry, sad, glad etc. etc.? No. Any more than whether I have a
notebook in
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front of me and a pen in my hand, or whether this book will fall if I let go of it, or whether I have made a
miscalculation when I say 25 × 25 is 625. The following, however, is true: I can't give criteria which put the presence
of the sensation beyond doubt; that is to say: there are no such criteria.--But what sort of fact is that? A
psychological one, concerning sensations? One will want to say it resides in the nature of sensation, or of the
expression of sensation. I might say: it is a peculiarity of our language-game.--But even if that is true, it passes over a
main point: In certain cases I am in some uncertainty whether someone else is in pain or not, I am not secure in my
sympathy with him--and no expression on his part can remove this uncertainty.--In that case I say, e.g.: "He might
be pretending this too." But why should he necessarily be pretending? For pretence is only one quite special case of
someone's expressing pain and not feeling it. A particular drug might put him into a state in which he 'acts like an
automaton', is not pretending, but feels nothing, though he expresses feelings. I am imagining, e.g., that the drug has
the effect that some time after a real illness he repeats all the actions of his period of illness, while the objective
illness, the causes of pain, for example, have ceased to exist. In this case we have as little sympathy with him as with
someone under a narcotic. We say that he repeats all the expressions of pain, etc. purely automatically, but that of
course he isn't pretending.
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138. "I can never know what is going on in him; he always knows": When one thinks philosophically, one would like
to say that. But what situation does this statement correspond to? Every day we hear one man saying of another that
he is in pain, is sad, is merry, etc. without a trace of doubt, and we relatively seldom hear that he does not know
what is going on in the other. In this way, then, the uncertainty is not so bad. And it also happens that one says "I
know that you felt like this then, even if you won't admit it now".
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139. The picture "He knows--I don't know" is one that makes our lack of knowledge appear in an especially irritating
light. It is like when one looks for an object in various drawers, and tells oneself that God knows the whole time
where it actually is, and that we are searching this drawer quite futilely.
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140. "Any human knows he is in pain"--and does he also know exactly how severe his pain is?
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141. The uncertainty of the ascription "He's got a pain" might be called a constitutional certainty.
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142. The child that is learning to speak learns the use of the words "having pain", and also learns that one can
simulate pain. This belongs to the language-game that it learns.
Or again: It doesn't just learn the use of "He has pain" but also that of "I believe he has pain". (But naturally
not of "I believe I have pain".)
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143. "He can also simulate pain"--that is to say: he can behave as if he had pains without having them. Certainly; and
such a proposition underlines a particular picture; but is the employment of "He has pain" influenced by this?
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144. But how if someone were to say "Having pain and shamming pain are very different states of mind, which
might have the same expression in behaviour"?
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145. So do sham pain and true pain have the same expression? And in that case how does one distinguish them?
How do I know that the child I teach the use of the word "pain" does not misunderstand me and so always call
"pain" what I call "sham pain"?
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146. Suppose someone explains the teaching of the word "pain" in this way: when a child behaves in such-and-such
a way on particular occasions, I think it feels what I feel in such cases; and if I am not mistaken in this, then the child
associates the word with the feeling and uses the word when the feeling reappears.--
This explanation is correct enough; but what does it explain? Or: what sort of ignorance does it remove?--It
tells us, e.g., that the person does not associate the word with a behaviour or an 'occasion'. So if anyone did not
know whether the word "pain" names a feeling or a behaviour, the explanation would be instructive to him. It also
says that the word is not used now for this feeling now for that--as of course might also be the case. [Cf. Z 545.]
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147. The explanation says that I use the word wrong if I later use it for a different feeling. A whole cloud of
philosophy condensed into a droplet of symbolic practice. [Cf. P.I. p. 222b.]
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148. Why should the words "I believe he is in pain" not be mere lunacy? Somewhat as if someone were to say "I
believe my teeth are in his mouth".
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149. A tribe: the people often pretend, they lie in the road looking ill and in pain; if someone comes to their aid, they
attack him. For this behaviour the tribe has a particular word.
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150. Instead of "It is uncertain whether he is in pain" one might say "Be mistrustful in face of his manifestation of
pain".--And how does one do that?
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151. Believing that someone else is in pain, doubting whether he is, are so many natural kinds of behaviour towards
other human beings; and our language is but an auxiliary to and extension of this behaviour. I mean: our language is
an extension of the more primitive behaviour. (For our language-game is a piece of behaviour.) [Cf. Z 545.]
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152. I am not certain whether he is in pain."--Suppose someone were to stick a pin into himself whenever he said
this, in order to have the meaning of the word "pain" vividly before his mind and to know what he was in doubt
about! Would the sense of his statement be assured by his providing himself with pain while he makes it? Surely he
knows now what it is he doubts about the other?--But how will he doubt what he now feels, about the other? How
will he attach the doubt to his feelings? For what is the route from his pain to the other? For can he really better
doubt the pain of the other, if he himself feels pain at the time? Need I myself have a cow in order to be able to
doubt whether someone else has one? [Cf. Z 546.]
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153. So he has true pain; and what he doubts about another, is the possession of this.--But how does he do so?--It is
as if I were to say to someone: "Here you have a chair; do you see it? Now translate it into French!" [Cf. Z 547.]
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154. So he has true pain;--and now he knows what he is to doubt about the other. He has the object before him; and
it isn't any such thing as 'behaviour'. (But now!) In order to doubt whether the other is feeling pain now, I need the
concept of pain; not pain. And it is probably true that this concept might be imparted to me by providing me with
pain. [Cf. Z 548.]
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155. It would be just as wrong to use an experience of meaning to explain the concept of understanding meaning as
to explain reality and unreality by the experience of unreality, or the concept of the presence of a human being by
the feeling of a presence. One might just as well try to explain what check is in chess by a check-feeling.
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156. "But one can surely see the figure as an arrow and as a bird's foot, even when one never tells anyone."
And that in turn means: It makes sense to say: Someone saw the figure now this way, now that, without telling
anyone.--I don't want to say it makes no sense, but the sense is not clear straight off:--I know, for example, that
people talk of a feeling of unreality, they say everything seems unreal to them; and now one says: everything might
strike people as unreal even if they had never told anyone. How does one know straight off that it makes sense to
say "perhaps everything strikes this person as unreal, although he never speaks of it"?
Of course I have here purposely chosen a very rare experience. For because it is not one of the everyday
experiences, one looks more sharply at the use of the words.--I should like to say: In some pressing trouble it makes
sense to cry out: "It's all unreal!"--and so one knows that that other statement makes sense too!--Or again: Someone
says to me "Everything seems unreal to me". I hardly know what that means--and yet I know already, that it would
make sense to say, etc. etc. Now this of course depends on his using this sentence to describe an experience, i.e. on
its being a psychological statement.
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157. That is to say: when someone manifests a state of mind, he might also have had that state of mind without
manifesting it. That is a bit of talk.†1 But what is the purpose of a sentence saying: perhaps N had the experience E
but never gave any sign of it? Well, it is at any rate possible to think of an application for the sentence. Suppose, for
example, that a trace of the experience were to be found in the brain, and then we say it has turned out that before
his death he had thought or seen such and such etc. Such an application might be held to be artificial and
far-fetched; but it is important that it is possible.
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158. If there is such a thing as a temptation to regard the differential calculus as a calculus with infinitely small
magnitudes, it's conceivable that in another case there may be an analogous temptation, a still more powerful
one--when, that is, it gets nourishment on every side from the forms of language; and one can imagine it becoming
irresistible.
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159. "I had toothache"--when I say this I don't remember my behaviour, I remember my pain. And how does that
happen? A faint copy of the pain comes into my mind?--So is it as if one were ever so slightly in pain? "No; it is
another kind of copy; something specific." So is it as if one had never seen a painted picture but only busts, and one
said "No, a painting is quite different from a bust, it is a quite different sort of copy". It might be, for example, that
one would find it far harder to make it intelligible to a blind man what a painting is than what a bust is.
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160. But the word "specific" (or an analogous one), which one would very much like to use here, does not help. It is
as little of a resource as the word "indefinable" when one says that the word "good" is indefinable.
What we want to know, to get a bird's-eye view of, is the use of the word "good", and equally that of the
word "remember".
For one can't say: "Afer all, you are acquainted with the specific thing, the memory-image." I am not
acquainted with it.--To be sure I may say "I can't describe Mr. N., but I am acquainted with him"; but that means
that I recognize him, not that I believe I recognize him.
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161. Its making sense to say that someone had a feeling, without ever revealing it, hangs together with its making
sense to say "I felt this then; I remember it".
The explanation might be as follows: After all, one isn't going to say: "If I had never said that I had pains at
that time, then I wouldn't have had them."
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162. "I surely know what this means: 'He was in pain'!" Does that mean that I can imagine it? What would make this
imagining important?
It is indeed important that, in order to explain this proposition I can turn to the memory of my own pains at
any time, or to summoning pains up in myself, etc.
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163. How does anyone learn to call a lump of sugar "sugar"? How, to obey the request "Give me a lump of sugar?"
And how does he learn the words "A lump of sugar, please"--i.e. the expression of a wish?! How, to understand the
order "Throw!"; and how, the expression of intention "Now I am going to throw"? Well--the grown-ups may
perform before the child, may pronounce the word
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and straightway throw,--but now the child must imitate that. ("But that is the expression of intention only if the
child really has the intention in its mind"--But then when does one say that that is the case?)
And how does it learn to use the expression "I was just about to throw"? And how does one know that it was
then really in the state of mind that I call "being about to throw"? After such-and-such language games have been
taught it, then on such-and-such occasions it uses the words that the grown-ups spoke in such cases, or it uses a
more primitive form of expression, which contains the essential relations to what it has previously learnt, and the
grown-ups substitute the regular form of expression for the more primitive one.
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164. The new (spontaneous, 'specific' is a language-game). [Cf. P.I. p. 224h.]
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165. "But weren't there all these appearances--of pain, of wishing, of intention, of memory etc., before there was any
language?" What is the appearance of pain?--"What is a table?"--"Well, that, for example!" And that is of course an
explanation, but what it teaches is the technique of the use of the word "table". And now the question is: What
explanation corresponds to it in the case of an 'appearance' of mental life? Well, there is no such thing as an
explanation which one can recognize straight away as the homologous explanation.
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166. It may be asked: Does something always come into my head when I understand a word?! (The following
question is similar: "When I look at a familiar object, does an act of recognition always take place?")
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167. But there is the phenomenon that when a word is heard outside any context--for example--for a fleeting
moment it has one meaning, and the next moment another; that if one pronounces the word over and over it loses all
'meaning'; and so on. And here it is a matter of something's coming into one's head.
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168. What should we say about men who didn't understand the words "Now I'm seeing this figure as..., now as..."?
Would they be lacking in an important sense; is it as if they were blind; or colour-blind; or without absolute pitch?
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169. Well, it is easy to imagine men who could not 'phrase' drawings thus and so; but would they not all the same
take a drawing now for this, now for something else? Or am I to assume that in this case they would not say that the
optical picture has in an important sense stayed the same? Thus, when the schematic representation of a cube looked
now this way now that to them, would they believe that the lines had altered their position?
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170. Imagine someone who did not like to see a drawing or a photograph, because he says that a colourless human
being is ugly. Or there might be someone who found that men, houses, etc. all tiny as they are in pictures, were
uncanny or ridiculous. This would certainly be a very queer attitude. ('Thou shalt make thyself no image.')
Think of our reactions towards a good photograph, towards the facial expression in the photograph. There
might be people who at most saw a kind of diagram in a photograph, as we consider a map; from it we can gather
various things about the landscape; but we can't, e.g., admire the landscape in looking at the map, or exclaim "What
a glorious view!"
The 'form-blind' man must be abnormal in this kind of way. [Cf. P.I. p. 205f.]
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171. How can the non-occurrence of an experience in hearing the word hinder our calculating with words, or
influence it?
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172. Imagine people who only think out loud and only imagine by drawing on paper. Or perhaps it would be better
to say: who draw, where we imagine. Then the case where I imagine my friend N does not correspond to the case
where someone else draws him; rather he must draw him and say or write that it is his friend N.--But suppose he has
two friends who are like one another and have the same name? and I ask him "Which did you mean, the clever one
or the stupid one?"--He could not answer this. But he could answer the question "Which of them does that
present?"--In this case the answer is simply a further use of the picture, not a statement about an experience.
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173. Compare James' idea that the thought is already complete at the beginning of the sentence, with the idea of the
lightning speed of thought and the concept of the intention of saying such-and-such. That
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the thought is already complete at the beginning of the sentence (and why not at the beginning of the previous one?)
means the same as: If someone is interrupted after the first word and you ask him later on "What were you wanting
to say then?", he can answer the question at least, he often can. But here too James says something that sounds like
a psychological statement and is not one. For, whether the thought was already complete at the beginning of the
sentence would surely have to be proved by the experience of individuals. [Cf. Z 1.]
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174. However, we also often cannot answer the question what we meant to say then. But in this case we say we have
forgotten it. Would it be imaginable that in such cases people should reply: "I merely said these words; how am I to
know what would have come after them?"
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175. If you say "As I heard this word, it meant... for me" you refer to a point of time and to an employment of the
word.--The remarkable thing about it is of course the relation to the point of time.
The 'meaning-blind' would lose that relation. [Cf. P.I. p. 175a.]
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176. And if you say "I was wanting to go on..."--you refer to a point of time and to an action. [Cf. P.I. p. 175b.]
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177. If I speak of the essential references of the utterance, that is because this pushes the inessential special
expressions of our language into the background. The essential references are the ones that would lead us to
translate an otherwise unaccustomed expression into the customary one. [Cf. P.I. p. 175c.]
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178. Suppose someone never said "I was going to do this then" and could not be taught to use such an expression
either? It is surely clear that a person can think a lot without thinking that. He can master a great area of language,
without mastering this one. I mean: he remembers his expressions, including perhaps that he said such-and-such to
himself. So he will say, e.g., "I said to myself 'I want to go there'" and perhaps also "I imagined the house and went
on the path that led there". What is characteristic here is that he has his intentions in the form of thoughts or pictures
and hence that they would always be replaceable by the speaking of a sentence or the seeing of a picture. The
"lightning speed" of thought is missing in him.--But now, is that supposed to mean that he often moves like an
automaton; walks in the street, perhaps, and makes purchases; but when one meets
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him and asks "Where are you going?"--he stares at one as if he were sleep-walking?--He won't answer "I don't
know" either. Or will his proceedings strike him, or us, as planless? I don't see why!
When I go to the baker, say, perhaps I say to myself "I need bread" and I go the usual way. If someone asks
him "Where are you going?" I want to assume that he answers with the expression of intention just as we do.--But
will he also say: "As I left the house, I was meaning to go to the baker, but now..."? No; but ought we to say that on
this account he set out on his way as it were sleepwalking?
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179. But isn't it then remarkable that, in all the great variety of mankind we do not meet such people as this? Or are
there such people among the mental defectives; and it is merely not sufficiently observed which language-games
these are capable of and which not?
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180. Plato says that thinking is a conversation. If it really were a conversation, then one could only report the words
of the conversation and the external circumstances under which it was carried on, but not also the meaning
(Meinung) that these words then had for the speaker. If someone said to himself (or out loud) "I hope to see N
soon", it would make no sense to ask: "And which person of that name did you mean then?" For all that he did was
say these words.
But could I not imagine that he, nevertheless, wants to go on in a particular way; so that I could ask him:
"And do you now mean someone by this name, and whom?"
And suppose that usually he could go on now, could explain his words--where would be the difference
between him and us? He could give a verbal report of any thought-process. So if he said "I just thought of N" and
we asked him "How did you think of him?" he can always answer us, unless he says he has forgotten.
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181. If someone says to me "N has written to me", I can ask him "which N do you mean?--and must he refer to an
experience in speaking the name if he is to answer me?--And if he now simply pronounces the name N--perhaps as
an introduction to a statement about N,--can't I equally well ask him "Whom do you mean?" and he equally well
answer?
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182. One does actually often simply pronounce someone's name; perhaps in a sigh. And now someone else asks
"Whom did you mean?"
And how will our meaning-blind man act? Will he not sigh like that; or not be able to answer anything to the
question; or answer "I mean..." instead of "I meant..."?
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183. Imagine one of your acquaintances. Now say who it was.--Sometimes the picture comes first and the name
after. But does that mean that I guess the name according to whom the picture resembles?--And if the name only
comes after, am I to say that the idea of the acquaintance was already there with the picture, or that it was only
complete with the name? For I did not infer the name from the likeness of the picture; and that is the very reason
why I can say that the idea had already been there with the picture.
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184. "I must go to the bank and get some money."--How did you understand that sentence? Need this question
mean anything but "How would you explain this sentence, what action to expect when you hear it?" etc? If the
sentence is uttered under different circumstances, so that the word "bank" obviously sometimes means this,
sometimes something else--must something special go on in hearing the sentence if you are to understand it? Don't
all experiences of understanding get covered up by the use, by the practice of the language-game? And that merely
means: here such experiences aren't of the slightest interest to us.
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185. When I see the milkman coming, I fetch my jug and go to meet him. Do I experience an intending? Not that I
knew of. (Any more, perhaps, than I try to walk, in order to walk.) But if I were stopped and asked "Where are you
going with that jug?" I should express my intention.
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186. If now I say, e.g. "I got up to go to the milk van,"--is this to be called the description of an experience of
intending? And why is that misleading? Is it because there was here no 'expression' of an experience?
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187. But if I say "I got up to..., but then I recollected myself and..."--where is the experience here, and when did it
take place? Was the experience only the 'recollecting myself', 'changing my mind'?
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188. I take the milk-jug, go a few steps, then I see that it isn't clean, say "No!" and go the [[sic to?]] the water-tap.
Then I describe what happened and name my intentions. Now didn't I have these? Of course! But once again: isn't it
misleading to call them experiences? if, that is, one also calls by that name what I said to myself, imagined etc.! (It
would also be misleading to call intention a "feeling".)
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189. And now the question arises whether for the same reasons it wouldn't be totally misleading to speak of
'form-blindness' or 'meaning-blindness' (as though one were to talk of 'will-blindness', when someone behaves
passively). For a blind man just is someone who does not have a sense. (The mental defective--e.g.--can't be
compared to the blind man.)
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190. When I drew the first it was a half circle; the second was a half S; the third was a whole.
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191. "I have no doubt that that often happens."--If you say this in a conversation, can you really believe that in
speaking you distinguish between the meanings of the two words 'that'?
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192. One might want to make the following objection against the fiction about people who only think out loud:
Suppose such a one were to say "As I left the house, I said to myself 'I must go to the baker,'" couldn't he be asked
"Did you really mean those words? For you might have said them as practice in elocution, or as a quotation, or as a
joke, or in order to mislead someone." That is true. But was what he was doing a matter of the experience that
accompanied the words? What speaks for such an assertion? Presumably, that the one who is asked may reply "I
meant the sentence like this" without inferring this from external circumstances.
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193. Of course one wants to say that if someone remembers having meant these words, he is remembering the
experience of a certain depth, of a resonance. (If he had not meant it, he wouldn't have had this resonance.) But is
that not simply an illusion (like that in which someone believes that he feels thinking in his head)? One uses
inappropriate concepts to form the picture of the processes. (Cf. James.)
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194. Make the following experiment: Say some ambiguous word to yourself ("till"); if you now experience it as a
verb, try to hang on to
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this experience, so that it lasts.--If you say the word to yourself several times, it loses its meaning for you; and now
ask yourself whether, when you are using it as a verb in ordinary speech, the word does not perhaps feel as it does
when it has lost its meaning through being often repeated.--You certainly can't testify from your memory that the
contrary is true. But one merely finds that a priori it can't be otherwise.
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195. It is indifferent whether one says that the interpretation of the word 'till' is projected later into the experience
had while pronouncing it. For here there is no difference between projecting and describing.
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196. One may take a drawing for a real cube; can one also, in the same sense, take a triangle as lying down or
standing up--"When I came nearer, I saw that it was only a drawing." But not: "When I looked closer I saw that this
was the base line and this the apex."
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197. My words "When you began to speak, I thought you meant..." tie up with the beginning of his speech and with
an idea that I had then.--And it is of course possible for someone never to do anything like that. But I will assume
that at the end he can answer the question "Which N was I speaking of?" And it is of course possible that he would
have answered it differently if I had put the question after the very first words of my story. Isn't he then supposed to
understand the question: "Did you know right at the beginning whom I was talking about?"--And now if he does not
understand such a question--shall we not simply judge him to be mentally defective? I mean: shall we not simply
assume that his thinking is not really clear, or that he no longer remembers what he was thinking then? That is to
say, here we shall ordinarily use a different picture from the one which I was proposing.
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198. But it is true: with mental defectives we often feel as if they talked more automatically than we do, and if
someone were what we called 'meaning-blind', we should picture him as making a less lively impression than we do,
behaving more 'like an automaton'. (One also says: "God knows what goes on in his mind", and one thinks of
something ill-defined, disorderly.)
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199. It might be that when one pronounced a word to some people, they immediately formed some sentence or
other with this word, and that others did not, that the former was a sign of intelligence, the latter of dullness.
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200. What can be adduced against the expressions 'specific psychological phenomenon' or 'irreducible
phenomenon'? They are misleading: but what is their source? One wants to say: "If someone is unacquainted with
sweet, bitter, red, green, notes and colours, one cannot make the meaning of these words intelligible to him." On the
other hand, if someone hasn't yet eaten a sour apple, what is meant can be explained to him. For red is this, and
bitter this and pain this. And if one says that, one must now actually exhibit what these words mean; that is, one
must point to something red; taste, or make the other taste, something bitter; give oneself or the other pain etc. Not
think that one can privately point to pain within oneself. But how in that case will one exhibit what "imagining",
"remembering", "intending", "believing" mean? The expression "specific psychological phenomenon" corresponds
to that of the private ostensive definition.
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201. Is it (in the end) an illusion, if I believed that the other's words had this sense for me at that time? Of course
not! Any more than it is an illusion to believe that one has dreamed something before waking up.
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202. When I supposed the case of a 'meaning-blind' man, this was because the experience of meaning seems to have
no importance in the use of language. And so because it looks as if the meaning-blind could not lose much. But it
conflicts with this, that we sometimes say that some word in a communication meant one thing to us until we saw
that it meant something else. First, however, we don't feel in this case that the experience of the meaning took place
while we were hearing the word. Secondly, here one might speak of an experience rather of the sense of the
sentence, than of the meaning of a word.
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203. The picture that one perhaps connects with the utterance of the sentence "The bank is far away", is an
illustration of it and not of one of its words.
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204. If someone insists that when he hears and understands an order, a piece of information etc., he mostly does not
experience anything at all, at least not anything that determines the sense for him--might this
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man not say nevertheless in some form or other that he had taken the first words of the sentence like this and later
altered the way he took them?--But what would he say that for? It might explain a particular reaction on his part. He
heard, e.g., that N was dead and believed that this meant his friend N; then he realizes that it is not so. At first he
looks upset; then relieved.--And it is easy to see what kind of interest such an explanation may have.
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205. What am I to say now--that the meaning-blind man is not in a position to react like that? Or that he merely does
not assert that he then experienced the meaning--and so, that he merely does not use a particular picture?
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206. Is the meaning-blind man then one who does not say: "The whole course of thought was before my mind in a
flash"? But is that to say that he can't say "Now I've got it!"--
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207. "In that place there was no tree, no shrub"--how does this sentence function? Well, "tree" stands for a thing that
looks like this. Of course, that's what a tree looks like; but is the idea of a word's going proxy for a thing really so
easy to understand? If I am planning a garden, I can have a peg go proxy for a tree here. Where the peg stands now,
the tree will be set later.--But still, one might say that the word "tree" in a sentence goes proxy there for the picture of
a tree (and of course even a tree can be used as that). For in a picture-language one might put the picture in the place
of the word "tree", and the word "tree" will in any case be connected with the picture by means of the ostensive
definition. In that case, then, it is the ostensive definition that determines what the word 'goes proxy for'. And now
apply this, e.g. to the word "pain".--But does not the sign " " in a map go proxy for a house? Surely only in so
far as a house too might serve as a sign! But the sign surely does not go proxy for the house for which it
stands.--"Well, it corresponds to it."--So when I walk with the map in my hand and come to this house, I point to
the place on the map and say "That's the house". "The sign goes proxy for the house" would mean: "Because I can't
place the house itself on the map, I put this sign instead of it." But what would the house itself be doing on the map
anyway? Proxy is something provisional but if the sign corresponds to the house, there is nothing provisional here;
for it isn't going to be replaced by the house when we get to the house.
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And as the sign never will be replaced by its bearer, one might ask: How can a house be replaced by an ink line?
No: the peg is a substitute for the tree; the picture may be a substitute for the person, when one would rather
see him but must be content with the picture; but the sign on the map is not a substitute for the object that it means.
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208. While I write, do I feel anything in my hand or in my wrist? Not generally. But still, wouldn't it feel different, if
my hand were anaesthetized? Yes. And is that now a proof that I nevertheless do feel something when I move my
hand in the normal way? No, I believe not.
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209. "I give you my full confidence." If someone who is saying this pauses after the word "you", perhaps I am able
to continue; the situation yields what he wants to say. But if to my surprise he now goes on: "a gold watch" and I
say "I was prepared for something else"--does that mean: while he was saying the first words I experienced
something that may be called that way of taking the words?? I believe that this can't be said.
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210. Imagine this conversation: He: "I give you--." "I know. But in this case, all the same, you do not trust me."--I
interrupted him because I knew what he was going to say. But did I necessarily fill out the continuation in thought?
When I see a sketch, do I fill it out in my imagination?
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211. "I found myself going..."
saying..." etc.†1
This description does not always apply when I say something, take a path etc.
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212. Introspection can never lead to a definition. It can only lead to a psychological statement about the introspector.
If, e.g., someone says: "I believe that when I hear a word that I understand I always feel something that I don't feel
when I don't understand the word"--that is a statement about his peculiar experiences. Someone else perhaps feels
something quite different; and if both of them make correct use of the word "understand" the essence of
understanding lies in this use, and not in what they may say about what they experience.
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213. What must the man be called, who cannot understand the concept 'God', cannot see how a reasonable man may
use this word seriously? Are we to say he suffers from some blindness?
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214. One suddenly understands, suddenly repeats a word that the other has said. He tells me "It is seven o'clock"; at
first I don't react; suddenly I cry out: "Seven o'clock! Then I'm already too late..." What he said had only just reached
my consciousness. But now, what happened, when I repeated the words "Seven o'clock!"? I can give no answer to
this that would be of any interest. Only, to repeat, I had just grasped what he had said, and so on; and that gets us no
further. Of course the talk (the idea) of a 'specific process' is based on this "Only, to repeat". (The absent minded
man who at the order "Right about turn!" turns left about....)
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215. Does something happen when I understand this word, intend this or that?--Does nothing happen?--That is not
the point; but rather: why should what happens within you interest me? (His soul may boil or freeze, turn red or
blue: what do I care?)
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216. A mental defective will certainly not say: "When you began to speak, I thought you meant...."--Now it will be
asked: Is that because he always understands right at once? Or because he never corrects himself? Or does the same
go on in him as in me, and he merely can't express it?
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217. "When you began to speak, I thought you were going to.... That was why I made the movement... too." So one
explains what one did by means of the thoughts that one had at the time. Now do I really think this explanation out
only after the event? Didn't I really make this movement because I thought...?--What sort of question is that? Of
course the "because" does not relate to a cause.
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218. "I am going to explain to you why I stood up; it was because I thought you meant...."--Yes, now I understand
it!--But wherein lies the importance of this understanding? Well, for example: If the explanation had been another
one, I should have had to react differently with words or actions. To that extent his thought is like an action, or a
process in his body. The report about this thought is like one about such processes.--What interest have the words
"At first I thought you meant..."? Often none. It may be said to disclose his
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world of thoughts. But what purpose does that serve? Why isn't this disclosure empty talk, or mere fantasy?
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219. One might (of course) call the report of such a conception the report of a tendency. (James.) But here the
experience of a tendency must not be seen under the aspect of an experience which isn't quite complete! As if
experiences yielded a coloured picture, and certain colours were laid on in full strength, others merely indicated, i.e.
put on much more faintly.
In itself, however, a faint colour is not a hint at a stronger one.
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220. An event leaves a trace in the memory: one sometimes imagines this as if it consisted in the event's having left a
trace, an impression, a consequence, in the nervous system. As if one could say: even the nerves have a memory.
But then when someone remembered an event, he would have to infer it from this impression, this trace. Whatever
the event does leave behind in the organism, it isn't the memory.
The organism compared with a dictaphone spool; the impression, the trace, is the alteration in the spool that
the voice leaves behind. Can one say that the dictaphone (or the spool) is remembering what was spoken all over
again, when it reproduces what it took?
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221. The feeling of dependence. How can one feel dependent? How can one feel 'It doesn't depend on me'? But what
a queer expression of a feeling this is anyway!
But if, e.g., every morning one had difficulty in making certain movements at first, in raising one's arm and
the like, and had to wait till the paralysis passed off, and that that sometimes took a long time, and sometimes
happened quickly, and one could not foresee it or adopt any means of speeding it up--wouldn't that be the very thing
to give us a consciousness of dependence? Isn't it the failure of the regular, or the vivid imagination of its failing, that
lies at the bottom of this consciousness?
It is the consciousness: "It didn't have to go like that!" When I get up from a chair, I don't ordinarily tell
myself "So I can get up". I say it after an illness, perhaps. But someone who did habitually say that to himself, or
who said afterwards: "So it worked this time"--of him one might say he had a peculiar attitude to life.
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222. Why does one say "He knows what he means"? How does one know that he knows it?
If he knows it, but I don't know what he means--what would it be like, if I did know it? And suppose I knew
it and he didn't? How would someone have to behave for us to say: "He knows what the other is experiencing"?
But must there be a case that we should describe in that way if we were consistent? It isn't clear that any
appearance must be described by the words "A has pains in the body of B".
That is to say: one can indeed say: "Wouldn't that be a consequent application of this expression?" but I may
or may not be inclined to call it consequent.
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223. Remember especially the expression in a dream narrative: "And I knew that...." One might think: 'It's surely
remarkable that one can dream that one knew." One also says: "and in the dream I knew that...."
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224. Not all that I do, do I do with some intention. (I whistle as I go along etc. etc.) But if I were now to stand up and
go out of the house, and then come back inside, and to the question "Why did you do that?" I answered: "For no
particular reason" or "I just did" this would be found queer, and someone who often did this with nothing particular
in mind would deviate very much from the norm. Would he have to be what is called "feeble minded"?
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225. Imagine someone of whom one would say: he can never remember an intention except by remembering the
expression of an intention.
What we normally do 'with a definite intention' someone might do without any, but it might nevertheless
prove useful. And perhaps in such a case we should say that he acted with unconscious intention. E.g., he suddenly
climbs on a chair and then gets down again. To the question "Why"? he has no answer; but then he reports having
noticed this and that from the chair, and that it seems as if he climbed up in order to observe this.
Might a 'meaning-blind' person not behave likewise?
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226. "When I said 'He is an ass' I was speaking of...." What sort of connexion have these sounds with this
man?--Asked, "Whom do you mean?" I shall mention his name, describe him, shew his photograph etc. Is there a
further connexion here? One that held
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particularly at the time of the utterance? During the whole time of uttering the sentence, though, or only while I said
"he"? No answer!
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227. The experience during those words--I should like to say--grows naturally towards this explanation.
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228. But it is surely like this: I shall sometimes say "He is an ass", in conversation perhaps; and if I were asked
"Would you have experienced anything different during these words if we had been speaking of N instead of M?" I
shall have to grant that this need not be the case. On the other hand it sometimes seems to me as if I had an
experience, while pronouncing the words, that pertains unambiguously to him.
The experiences while speaking seems to be connected intimately with him.
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229. "Of course I was thinking of him: I saw him before me!"--but I didn't recognize him from my picture of him.
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230. I suddenly say "He is an ass". A: "Whom did you mean?" I: "N." A: "Did you think of him while you were
saying the sentence, or only when you gave the explanation?"--I might now reply that my words had been the
terminus of a rather long course of thought. I had already been thinking of N the whole time. And could I now say:
the words themselves were not tied up with him through any special experience, but the whole course of thought
was? Thus I might easily have meant someone else by those words, and who was their reference was a matter of
what preceded them.
In order, however, to be able to say that I was speaking of him, meant him, thought of him, must I really be
able to remember an experience that unconditionally ties up with him? So might it not perhaps always strike one as
if nothing had happened while my words were going on, that could only point to him? So I am imagining that I am
always conscious that my images are ambiguous. At the same time however--this is what I am assuming--I still say
"I meant...". But is this not a contradictory assumption? No: for that really is how things are. I say "I meant...": that
is how I go on.
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231. I was speaking to my neighbours about their doctor; as I did so a picture of this man came into my mind--but I
had never seen him, merely knew his name, and perhaps formed a picture of him from the
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name. How can this picture characterize my speaking of him? And yet this is how it struck me, till I recalled that I
don't know at all what the man looks like. So his picture represents him for me not a whit better than does his name.
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232. If I compare the coming of the meaning into one's mind to a dream, then our talk is ordinarily dreamless.
The 'meaning-blind' man would then be one who would always talk dreamlessly.
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233. And one really can ask: What do his dreams matter to me? Why need I be interested in what he dreams and
whether he dreams while he speaks to me or hears me?--Naturally that does not mean that these dreams can never
interest me. But why should they be the most important thing in linguistic traffic?
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234. The use of the word 'dream' here is useful, but only if one sees that it still conceals an error within itself.
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235. "I thought the whole time that you were talking about...."--Only how was it?--Surely not otherwise, than if he
really had been speaking of that man. My later realization that I understood him wrong does not alter anything about
what happened as I was understanding the words.--
If, then, the sentence "At that point I believed that you meant..." is the report of a 'dream', that means that I
always 'dream' when I understand a sentence.
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236. We also say "I assumed you were talking about..." and that sounds still less like the report of an experience.
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237. "I thought you were speaking of... and wondered at your saying... of him."--This wondering in turn is in like
case: Here too we again have the feeling as if it took the pronouncing of this thought to fill out the rudimentary
experience.
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238. Well, it is surely true! For sometimes, when I say "I thought..." I can report that I did say these words to myself
out loud or silently; or that I used, not these but other words, of which the present ones reproduce the gist. This does
surely sometimes happen! In contrast with this, however, is the case in which my present expression is not the
reproduction of anything. For it is a 'reproduction' only if there are rules of projection making it one.
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239. Someone who was unable to say: the word "till" may be a verb and a conjunction, or to frame sentences in
which it is the one or the other--such a one could not master simple school exercises. But what a school child is not
required to do is to take the word outside any context in this way and that, or to report how he has taken it. [Cf. P.I.
p. 175b.]
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240. I should like to say: conversation, the application and further interpretation of words flows on and only in this
current does a word have its meaning. "He has left."--"Why?" What did you mean as you pronounced the word
"Why"? What did you think of?
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241. "I thought you were meaning him"--Now, that does not mean the same as "I think you meant him". Don't let
the comparison with another use of the past tense confuse you.
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242. We play this game: There are pictures here and words are pronounced and we have to point to the picture
corresponding to the word. Among the words there are also ambiguous ones. At the word... only one meaning
occurs to me and I point to a picture, later only another one and I point to another picture. Will the meaning-blind
man be able to do this? Of course.--But how about this? A word is mentioned, one of its meanings occurs to me. I
do not say it, but look for the picture. Before I have found it, I am struck by a further meaning of the word; I say: "A
second meaning has just occurred to me." And then I explain: "First this meaning occurred to me, and afterwards
that one." Can the meaning-blind do that?--Can't he say he knows the meaning of the word but isn't saying it? Or
can't he say that it has just occurred to him but that he isn't saying it?--It strikes me that he can say both. But in that
case surely also: "As you said the word, this meaning occurred to me." And now why not "When I said that word I
meant it at first in this meaning."?
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243. It's as if the word that I understand had a definite slight aroma that corresponds to my understanding of it. As if
two familiar words were distinguished for me not merely by their sound or their appearance, but by an atmosphere
as well, even when I don't imagine anything in connexion with them.--But remember how the names of famous
poets and composers seem to have taken up a peculiar meaning into themselves. So that one can say: the names
"Beethoven" and "Mozart" don't merely sound different; no, they are also accompanied by a different character.
But if you had to describe this
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character more closely--would you point to their portraits, or to their music?
And now the meaning-blind man again: He would not feel that the names, when heard or seen, were
distinguished by an imponderable Something. And what would he have lost by this?--And yet, when he hears a
name, first one bearer of it, and then other, may occur to him.--
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244. I said, the words "Now I can do it!" don't express an experience. Any more than these: "Now I am going to
raise my arm."--But why don't they express any experience, any feeling--Well, how are they used? Both, e.g., are
preliminary to an action. The fact that a statement makes reference to a point of time, at which time, however,
nothing that it means, nothing of which it speaks, happens in the outer world, does not shew us that it spoke of an
experience.
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245. Think of children putting up their hands in class when they know the answer to a question. Must one of them
have said the answer silently to himself, for putting up his hand to make sense? What must have gone on in him for
this?--Nothing. But it is important that he ordinarily gives an answer when he has put up his hand; and that is the
criterion for his understanding putting up one's hand. [Cf. Z 136a.]
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246. "The words 'the rose is red' are senseless if the word 'is' has the meaning of 'is the same as'." We have the idea
that if someone tried to pronounce the words "the rose is red" with these meanings for the words, he could not but
get stuck in thinking it. (As also, that one cannot think a contradiction, because so to speak the thought collapses for
one.)
One would like to say: "You can't mean these words like this and still connect a sense with the whole." [Cf.
P.I. p. 175c.]
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247. Could one say that a meaning-blind man would reveal himself in this: One can have no success in saying to
such a man: "You must hear this word as..., then you will say the sentence properly." That is the direction one gives
someone in playing a piece of music. "Play this as if it were the answer"--and one perhaps adds a gesture.
But how does anyone translate this gesture into playing? If he understands me, he now plays it more as I
want him to.
But could you not give just such a direction but using the words "louder", "softer", "quicker", "slower"? No: I
could not. For even if he does now play this note louder, that one more softly, I don't even
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realize it. In the same way I can also tell him "Make a crafty face", and I would know if he had made one, without
being able to describe the geometrical alterations of the face beforehand, or afterwards.
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248. When one asks "Is experiencing meaning analogous to experiencing a mental image?", one means: isn't the
difference simply that of a different content? Now, what is the content of the experience of imagining? "It is
this"--but here I must point to a picture or a description.--"In both of these cases one has an experience" (one would
like to say)--"Only it's different. A different content is presented to consciousness--stands before it." And this is of
course a very misleading picture. For it is the illustration of a turn of speech and it explains nothing. One might as
well try to explain the chemical symbolism of a formula by drawing pictures in which the elements were represented
as people who stretch out their hands to one another. (Illustrations of the alchemists.) [Cf. P.I. p. 175e.]
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249. If someone says he has had an image of a shining gold ball, we shall understand him; but not if he says that the
ball was hollow. But in a dream a man might see a ball and know it was hollow.
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250. The direction: "Wie aus weiter Ferne"†1 in Schumann. Must everyone understand such a direction? Everyone,
for example, who would understand the direction "Not too quick"? Isn't the capacity that is supposed to be absent in
the meaning-blind one of this kind?
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251. Can one keep hold of the understanding of a meaning, as one can keep hold of a mental image? So if a meaning
of the word suddenly strikes me--can it also stand still before my mind? [Cf. P.I. p. 176b.]
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252. "The whole plan came before my mind in a flash and stayed still like that for one minute." Here one would like
to think that what stayed still can't be the same as what flashed upon one. (As one can't extend a diphthong.) [Cf.
P.I. p. 176c.]
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253. For if it happened that I said "Now I have it!" (i.e. a sudden start), of course one can't talk about this as staying
still.
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254. "Yes, I know the word. It's on the tip of my tongue.--"Here the idea forces itself on one, of the gap which James
speaks of, which only this word will fit into, and so on.--One is somehow as it were already experiencing the word,
although it is not yet there.--One experiences a growing word.--And I might of course also say that I experienced a
growing meaning, or growing explanation of meaning.--Only it is queer that we don't want to say that there was
something there, which then grew up into this explanation. For when you 'put your hand up' you say you already
know it.--Very well; but you might also say "Now I can say it", and whether this ability grows into a saying is
something you don't know. And what if it were now said "The saying is the fruit of this ability, if it grew out of this
ability"?
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255. When I was going to say it, was able to say it, I had not yet said it.
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256. Of course there is something wrong too about the explanation that the meaning or its explanation has grown
out of a certain germ. In fact we do not perceive such a growth; or at any rate only in very rare cases. And this
explanation springs from the tendency to explain instead of merely describing.
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257. Mere description is so difficult because one believes that one needs to fill out the facts in order to understand
them. It is as if one saw a screen with scattered colour-patches, and said: the way they are here, they are
unintelligible; they only make sense when one completes them into a shape.--Whereas I want to say: Here is the
whole. (If you complete it, you falsify it.)
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258. Of course the meaning occurred to me then! Not at the time when I reported it, nor in the interval.
This just is what one calls it: This just
is the way we use the words "The meaning occurred to me" ("in this so-called twentieth century"†1).
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259. "The meaning is surely not something that one can experience!"--Why not?--The meaning isn't a
sense-impression. But what are sense-impressions? Something like a smell, a taste, a pain, a noise etc. But what is
'something like' all these things? What is common to them? This question cannot of course be answered by
immersing oneself in these sense-impressions. But one might ask this: "In what
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circumstances shall we say that someone has a kind of sense-impression that we lack?"--We say for example of
beasts, that they have an organ with which they perceive such-and-such, and such a sense-organ need not be similar
to ours.
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260. Could a kind of sense perception be imagined, through which we grasped the form of a solid body, the whole
form, not just what can be seen from a certain point of view? Such a person would, e.g., be able to model a body in
clay without walking round it or touching it.
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261. Is it the multiplicity of the possible explanations of a meaning that lies at the bottom of our not experiencing a
meaning 'in the same sense' as a visual image?
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262. What makes my image of him into an image of him?--What makes this portrait into his portrait? The intention
of the painter? And does that mean: his state of mind?--And what makes a photograph into a picture of him? The
intention of the photographer? And suppose a painter had the intention of drawing N from memory, but, guided by
forces in his unconscious, draws an excellent picture of M.--Would we now call it a bad picture of N? And imagine
people trained in the drawing of likenesses who draw the person sitting in front of them 'mechanically'. (Human
reading-machines.)
And now--what makes my image of him into an image of him? Nothing of what holds for a portrait holds for
the image. The question makes a mistake. [Cf. P.I. p. 177.]
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263. If the meaning has struck you, and you have not forgotten it again, you can now use the word in this way.
If the meaning has occurred to you, you know it now, and its occurring to you was simply the beginning of
knowing. Here there is no analogy with the experiencing of a mental image. [Cf. P.I. p. 176e.]
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264. How is it, though, that when pointing to a particular figure I tell myself that I should like to call this
such-and-such ('x')? I may even say the ostensive definition "'x' means this" out loud to myself. But I must surely
also understand it myself! So I must know how, according to what technique, I think of using the sign "x".--If
someone asks me, say, "Do you know how you are going to use the word?" I shall answer: yes.
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265. How about religion's teaching that the soul can exist when the body has disintegrated? Do I understand what it
teaches? Of course I understand it--I can imagine a lot here. (Pictures of these things have been painted too. And
why should such a picture be only the incomplete reproduction of the spoken thought? Why should it not perform
the same service as what we say? And this service is the point.) [Cf. P.I. p. 178.]
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266. But you aren't a pragmatist? No. For I am not saying that a proposition is true if it is useful.
The usefulness, i.e. the use, gives the proposition its special sense, the language-game gives it.
And in so far as a rule is often given in such a way that it proves useful, and mathematical propositions are
essentially akin to rules, usefulness is reflected in mathematical truths.
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267. The expression of soul in a face. One really needs to remember that a face with a soulful expression can be
painted, in order to believe that it is merely shapes and colours that make this impression. It isn't to be believed, that
it is merely the eyes--eyeball, lids, eyelashes etc.--of a human being, that one can be lost in the gaze of, into which
one can look with astonishment and delight. And yet human eyes just do affect one like this. "From which you may
see...."
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268. Do I believe in a soul in someone else, when I look into his eyes with astonishment and delight?
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269. The proposition "if p, then q", as, e.g. "if he comes, he will bring something", is not the same as "p ⊃ q". For the
proposition "if p then q" can go into the subjunctive, but the proposition "p ⊃ q" cannot.--If someone replies to the
proposition "If he comes,..." with "that's not true", he doesn't mean to say "He will come and not bring anything" but
rather: "He may come and not bring anything."
From "p ⊃ q" there does not follow: "if p then q"; for I can very well assert the former (I know, e.g. that
~p.~q is the case) and deny the second proposition.
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270. Am I now to say that the proposition "If... then..." is either true, or false, or undecided? (So the law of excluded
middle is not valid?)
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271. Another answer to the statement "If he comes, he will bring something" is: "Not necessarily."--Also: "That
doesn't follow."--One may also say "There isn't that connexion."--Russell said that when one says "If... then..." one
ordinarily means not material but formal implication; but that is not correct either. "If... then..." can't be reproduced
in expressions belonging to Russellian logic.
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272. One may, however, very well say that the proposition "If... then..." is either true, or false, or undecided.--But on
what occasion will one say this? I think: as an introduction to a further exposition. One treats the matter under these
three headings. I divide the field of possibilities into three parts.
It will perhaps now be said: a proposition divides it into two parts. But why? Unless that is part of the
definition of a proposition. Why shouldn't I also call something a proposition that makes a three-fold division?
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273. Now take a twofold division. I say: "Either he comes, or not. If he comes, then..., if he doesn't come, then..."
May I not apply this treatment to the proposition "If... gets into contact with... there will be an explosion"? If
someone has asserted this--may I not reply "Either you are right about that or not: if it is as you say, then...; if not,
then..."?
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274. The law of excluded middle does not say, as its form suggests: There are only these two possibilities, Yes and
No, and no third one. But rather: "Yes" and "No" divide the field of possibilities into two parts.--And that of course
need not be so. ("Have you stopped beating your wife?")
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275. 'Wish is a stance of the mind, the soul, in relation to an object.' 'Wish is a state of mind that relates to an object.'
In order to make this more intelligible, one thinks perhaps of yearning, and of the object of our yearning's being
before our eyes and that we look at it longingly. If it is not there in front of us, perhaps its picture goes proxy for it,
and if there is no picture there, then an image. And so the wish is a stance of the soul towards an image. But one
really always thinks of the stance of the body towards an object. The stance of the soul to the image is just what one
might represent in a picture: the man's soul, as
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it leans with gestures of longing towards the picture (an actual picture) of an object.
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276. And in this way of course one might also represent someone in whose bearing there is no kind of expression of
the wish, but whose soul has this longing.
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277. "The sentence 'If only he would come' may be laden with our longing."--What was it laden with there? It is as if
a weight were loaded on to it from our spirit. I should indeed like to say all of that. And doesn't it matter, that I want
to say that?
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278. Doesn't it matter that I want to say that? Isn't it important? Is it not important that for me hope lives in the
breast? Isn't this a picture of one or another important bit of human behaviour? Why does a human being believe a
thought comes into his head? Or, more correctly, he does not believe it; he lives it. For he clutches at his head; he
shuts his eyes in order to be alone with himself in his head. He tilts his head back and makes a movement as a sign
that nothing should disturb the process in the head.--Now are these not important kinds of behaviour?
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279. And if the picture of the thought in the head can force itself upon us, why not much more that of thought in the
soul? [Cf. P.I. p. 178f.]
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280. What better picture of believing could there be, than the human being who, with the expression of belief, says
"I believe..."?
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281. The human being is the best picture of the human soul. [Cf. P.I. p. 178g.]
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282. It is, of course, important that a man wanting an apple can easily be represented in a picture of desire without
putting words of desire into his mouth--but that the conviction that something is thus and so cannot be so
represented.
Important, because it shews the difference, the essential difference, between psychological phenomena; and
the kind of way this difference is to be described.
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283. Why did I say "essential difference"? Is it a difference like that between carbon, gravitation, the velocity of light
and ultra-violet rays? All of which are 'objects' treated of by natural science.--
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284. Suppose we are talking of the phenomena we get in connection with human speech. We might be interested in:
the speed of talk, the change of intonation, the gestures, the length or shortness of sentences etc. etc.--Now when
one says of a human being that he has a mental life: he thinks, wishes, fears, believes, doubts, has images, is sad,
merry etc.,--is that analogous to: he eats, drinks, speaks, writes, runs,--or analogous to: he moves now fast, now
slow, now towards a goal, now without any goal, now continuously, now in jerks?
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285. Think of what may be called the character of a line, and of all that must be called a description of its character.
What a lot of things one may ask, if one is interested in the character of a line!
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286. Imagine we were observing the movement of a dot, say a black dot on a white paper surface. Important
conclusions of every conceivable kind might be drawn from the character of the movements. But what a host of
different things we might observe!--Whether the dot moves uniformly or non-uniformly; whether its velocity alters
periodically; whether it alters continuously or in jerks; whether the dot describes a closed line; how close this gets to
being a circle; whether the dot describes the line of a wave and what its amplitude and wave length are; and
innumerable other things. And any of these might be the one thing that interested us. We might, e.g., be indifferent
to everything about this movement except the number of angles of the path in a definite time. And that means that if
what interests us is not just a single characteristic, but rather many, then any one of them may yield us special
information quite different from all the rest. And that's how it is with the behaviour of human beings, with the
various characteristics of this behaviour, which we observe. [Cf. P.I. p. 179a.]
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287. So does psychology deal with behaviour (say), not with human states of mind? If someone does a
psychological experiment--what will he report?--What the subject says, what he does, what has happened, to him in
the past and how he has reacted to it.--And not: what the subject thinks, what he sees, feels, believes,
experiences?--If you describe a painting, do you describe the arrangement of paint strokes on the canvas--and not
what someone looking at it sees?
But now how about this: The observer in the experiment will sometimes say: "The subject said 'I feel...', and I
had the impression
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that this was true."--Or he says: "The subject seemed tired." Is that a statement about his behaviour? One would
perhaps like to say: Of course, what else should it be?--It may also be reported: "The subject said 'I am tired'"--but
the cash value of these words will depend on whether they are plausible, whether they were repeating what someone
else said, whether they were a translation from French, etc.
Now think of this: I recount: "He made a dejected impression." I am asked: "What was it that made this
impression on you?" I say: "I don't know."--Can it now be said that I described his behaviour? Well, can one not say
I have described his face if I say "His face changed to sadness"? Even though I cannot say what spatial alterations in
the face made this impression?
It will perhaps be replied: "If you had looked closer, you would have been able to describe the characteristic
changes of colour and position." But who says that I or anyone could do this? [Cf. P.I. p. 179b.]
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288. Once more: When I report "He was put out", am I reporting a behaviour or a state of mind? (When I say "The
sky looks threatening", am I talking about the present or the future?) Both. But not side by side; rather one in one
sense, the other in another. But what does that mean? (Is this not mythology? No.) [Cf. P.I. p. 179c.]
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289. It is here quite as it is with talk of physical objects and sense-impressions. We have two language-games, and
their mutual relations are complicated. If one tries to describe these relations in a simple fashion, one goes wrong.
[Cf. P.I. p. 180c.]
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290. Suppose I describe a psychological experiment: the apparatus, the questions of the experimenter, the answers
and actions of the subject. And then I say: all that is a scene in such-and-such a play. Now all is altered. So it will be
said: If this experiment were described in the same way in a book on psychology, in that case the description of the
behaviour of the subject would be understood as expression of the state of mind, because one presupposes that the
subject is speaking the truth, is not pulling our legs, has not learnt the answers by heart.--So we make an
assumption? [Cf. P.I. p. 180a.]
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291. The nurse says to the doctor "He's groaning"--one time she means to say "He is in severe pain"; another "He's
groaning--although there's nothing wrong"; another "He's groaning; I don't know whether he is in pain or is merely
making this noise."
We form a presupposition?--We use the statement differently each time.
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292. "Of course the psychologist reports the words, the behaviour, of the subject, but surely only as signs of mental
processes."--That is correct. If the words and the behaviour are, for example, learned by heart, they do not interest
the psychologist. And yet the expression "as signs of mental processes" is misleading, because we are accustomed
to speak of the colour of the face as a sign of fever. And now each bad analogy gets explained by another bad one,
so that in the end only weariness releases us from these ineptitudes.
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293. Imagine someone saying: any familiar word already has an aura, a 'corona' of faintly indicated uses surrounding
it. Much as if the principal figures in a painting were surrounded with faint, misty pictures of proceedings in which
these figures play a part.--Now, let's just take this assumption seriously!--Then it comes out that it's inadequate to
explain intention.
For if it is like this, that the possibilities of employment of an expression come before our minds in half
shades as we hear it or say it--if it is like this, then that holds for us. But we communicate with others, without ever
having asked them whether they have these experiences too. [Cf. P.I. p. 181a.]
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294. And what about the continuous coming to be and passing away in the domain of our consciousness? Well, how
is it: is that experienced, or can it not be imagined otherwise at all? Here is an unclarity.
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295. I know my way about in a room: that is, without needing a moment's reflection, I can find the door, open and
shut it, use any piece of furniture, I don't have to look for the table, the books, the chest of drawers or think what can
be done with them. That I know my way around will come out in the freedom with which I move about in the room.
It will also be manifested in an absence of astonishment or doubt. Now what answer am I to make to the question:
whether this knowing-one's-way-around-in-this-room is a state of mind?
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296. The question "What's a thermometer for?" I am in a position to answer at once, without any difficulty, with a
long string of sentences. And equally I can meet the request: "Explain the application of the word 'book'."
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297. Knowing one's way about can be called an experience, and again, also, not.
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298. The employment of certain words for the sake of the rhythm of a sentence. This might be far more important to
us that it actually is.
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299. "What kind of experience is...?" One won't ask "What's it like when YOU have it?"--for this might be answered
by one person this way, by another that. One won't ask them for a description of the experience, but will rather look
to see how and on what occasions people mention the experience, speak of it, without trying to describe it.
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300. I say the word "tree", then I say a nonsense-word. They feel different. To what extent?--Two objects are shewn
me: One is a book, the other a thing unknown to me with a peculiar shape. I say: they not merely look different, but I
also have a different feeling on looking at them. The first thing I 'understand', the other I don't understand. "Yes, but
it is not only the difference between familiarity and strangeness." Well, is there not also a difference between kinds
of familiarity and strangeness? A stranger walks into my room, but it is a human being, so much I see at once. Some
swathed thing walks into my room. I don't know if it is man or beast. I see an unfamiliar object on my table, an
ordinary pebble, but I never saw it before on my table. I see a stone on the path; I am not astonished, although I do
not remember having seen just that stone before. I see on my table a queer-shaped object whose function is
unknown to me and am not surprised: it was always there, I never knew what it was and was never interested to
know, it is thoroughly familiar to me.
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301. "Didn't you understand the word 'tree' when you heard it?--In that case something did go on in you!"--And
what?--I understood it.--Only the question is: Am I to say about understanding, that it went on in me? Something
goes against this; and that can only mean that by means of this expression we put understanding together with other
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phenomena and elide a difference which we want to emphasize. But what difference?--Well, in what cases do we
not resist saying: something went on in us as we heard the word?
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302. What would we need to say to someone who told us that in his case understanding was an inner
process?--What retort should we make to him, if he said that with him being able to play chess was an inner
process?--Some such thing as, that we aren't interested in anything that goes on in him when we want to know
whether he can play chess. And if he now replies that we are interested in what goes on in him after all, namely: in
whether he can play chess--then we could contradict him by reminding him of the criteria which would prove his
capacity to us. [Cf. P.I. p. 181b.]
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303. In order to know your way about an environment, you do not merely need to be acquainted with the right path
from one district to another; you need also to know where you'd get to if you took this wrong turning. This shews
how similar our considerations are to travelling in a landscape with a view to constructing a map. And it is not
impossible that such a map will sometime get constructed for the regions that we are moving in.
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304. Suppose you do have a peculiar experience when you understand, how can you know that it is the one we call
"understanding"?--Well, how do you know, then, that the experience that you have is the one we call "pain"?--That
is different--I know that, because my spontaneous behaviour in certain situations is what is called the expression of
pain.
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305. When one learns to use the word "pain", that does not happen through guessing which of the inner processes
connected with falling down etc. this word is used for.
For in that case this problem might arise as well: on account of which of my sensations do I cry out when I
damage myself?
And here I imagine one's pointing inside and asking himself: "Is it this sensation, or this one?"
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306. "It doesn't matter whether I have attached the right name to the sensation--I just have attached a name to
it!"--But now, how does one attach a name to something, e.g. to a sensation? Can one within oneself attach a name
to a sensation? What happens here; and what is the result of this action? ((Cf. Remark on attaching a name-tag to
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something.†1)) if one shuts a door in one's mind, is it then shut? And what are the consequences? That, in one's
mind, now no one can get in?
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307. "How do you know, then, that the experience which you have is the one that we call 'pain'?" The experience
that I have? Which experience? How do I specify it: for myself, and for another?
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308. Suppose we could learn what it is that people call a sensation, say a 'pain', and then someone taught us to
express this sensation. What kind of connexion with the sensation would this activity need to have, for us to be able
to call it the 'expression' of that sensation?
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309. Suppose someone knew, guessed, that a child had sensations but no expression of any kind for them. And now
he wanted to teach the child to express the sensations. How must he connect an action with a sensation, so that it
becomes the expression of the sensation?
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310. Can he teach the child: "Look, this is how one expresses something--this, for example, is an expression of
this--and now you express your pain!"
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311. "Understand" just is not used like a word for a sensation.
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312. The confusing picture is this: that we observe a substance--its changes, states, motions; like someone observing
the changes and motions in a blast furnace. Whereas we observe and compare the attitudes and behaviour of human
beings.
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313. Primitive pain-behaviour is a sensation-behaviour; it gets replaced by a linguistic expression. "The word 'pain' is
the name of a sensation" is equivalent to "'I've got a pain' is an expression of sensation".
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314. Forms of behaviour may be incommensurable. And the word "behaviour", as I am using it, is altogether
misleading, for it includes in its meaning the external circumstances--of the behaviour in a narrower sense.
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Can I then speak of one behaviour of anger, for example, and of another of hope? (It is easy to imagine an
orang-utan angry--but hopeful? And why is it like this?) [Cf. P.I. p. 174a.]
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315.
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If someone tells me: "Now I am seeing this point as the apex of the triangle" I understand him. But what do I
do with this understanding? Well, I can, e.g., say to him "Does the triangle strike you now as if it had fallen over, as
if it normally stood on the base line a? Or does it now appear to you as a mountain with B as its peak? Or as a
wedge? Or as an 'inclined plane'? Or as a cone?"
You may now ask "What does it consist in: to see the figure like this?"--and you may, so to speak, form
hypotheses about what goes on here. E.g., eye movements, or images, with which one supplements the seen--one
imagines a body, say, that slides down the inclined plane--etc. All this may happen but need not; and when someone
tells me he sees the triangle as a wedge etc. he is not telling me how his eyes have been moving etc.--No, the
question is not what happens here, it is: how one may use that statement. E.g. what my understanding of the
information does for me.
One application would be this: One may tell someone: "Look at the triangle as a wedge, and then you won't
wonder at... any more." And at this perhaps he says "Yes, like that it strikes me as more natural".--So I have
removed some disquiet with my explanation; or helped him to do an exercise more quickly.
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316. Seeing the resemblance of one face to another, the analogy of one mathematical form with another, a human
form in the lines of a puzzle picture, a three-dimensional shape in a schematic drawing, hearing or pronouncing
"pas" in "ne... pas" with the meaning "step"--all these phenomena are somehow similar, and yet again very different.
(A visual perception, an auditory perception, an olfactory perception, a perception of movement.)
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317. In all these cases one may be said to experience a comparison. For the expression of the experience is that we
are inclined towards a comparison. Inclined to make a paraphrase.
It is an experience whose expression is a comparison. But why an 'experience'? Well, our expression is an
experience-expression. Because we say "I see it as...", "I hear it as..."? No; though this form of expression hangs
together with that. But it is justified, because the language-game makes the expression into expression of an
experience.
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318. An experience that is manifested in a comparison.--In order, e.g., to hear "Je ne sais pas" in that conscious way
one has to be acquainted with other expressions like "not a thing".†1
The expression of the experience by means of the comparison precisely is the expression of it, the immediate
expression. It is the very phenomenon that we observe and that interests us.
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319. If now someone couldn't hear "pas" like this, couldn't experience it; if he didn't understand what we mean by
speaking of 'hearing as'--would he also fail to understand us when we explain that even in the negation "pas" did
once mean the same as "step", and if we said it was analogous to the word "bit", "thing", "bißchen" etc? But what is
the insight into, by which someone perceives that the use of the word... is analogous to that of the word...?
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320. Well, what do I shew someone such an analogy for? What do I expect from doing so? What effect has it?--It
surely has the appearance of an explanation. It is one kind of explanation. For one does say: "Yes, now I understand
the use of this word." But one also says: "I know what you mean, but I can't hear it as that."
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321. "Just as we still... at the present day, so these people...."
We are able to look at this custom in the light of that one. This may serve, e.g. as a heuristic principle.
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322. While any word--one would like to say--may have a different character in different contexts, all the same there
is one character--a face--that it always has. It looks at us.--For one might actually think that each word was a little
face; the written sign might be a face. And one might also imagine that the whole proposition was a kind of
group-picture, so that the gaze of the faces all together produced a relationship among them and so the whole made
a significant group. But what constitutes the experience of a group's being significant?
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And would it be necessary, if one is to use the proposition, that one feel it as significant in this way? [Cf. P.I. p.
181d.]
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323. For is it even certain that anyone who understands our language would be inclined to say that each word has a
face? And--the most important thing what general tendency in us is this inclination part of?
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324. First of all it is clear that the tendency to regard the word as something intimate, full of soul, is not always there,
or not always in the same measure. But the opposite of being full of soul is being mechanical. If you want to act like
a robot--how does your behaviour deviate from our ordinary behaviour? By the fact that our ordinary movements
cannot even approximately be described by means of geometrical concepts.
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325. Would one also get an impression of a group-picture from sentences written in telegraphic style?
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326. A convict has a number for a name. No one would say of it what Goethe says about people's names.
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327. One has the idea that the sense of a sentence is composed of the meanings of its individual words. (The
group-picture.) How is, e.g. the sense "I still haven't seen him yet", composed of the meanings of the words?
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328. Even the word "state" has a face, for at any rate "the State" has a different face. It feels different; and so "state"
would also have to feel somehow or other!:--But must "state" feel different from "State"? Suppose someone were to
assure me that to him these two words felt just the same? He says, e.g. I feel the connective and the verb "still"
differently all right, but not "State" and "state". Would we have the right to disbelieve him?
What looked like a quite matter-of-course expression, which is tied up with the understanding of the words,
appears here in the light of a purely personal expression of feeling. No different from someone's saying that for him
the vowels a and e are the same colour. Can I now say to this man: "You aren't playing our game"?
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329. If you have fine perceptions, will you assume that you feel the two words "still" differently in all contexts? No.
One expects that only when one pronounces them experimentally.
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330. Imagine human beings who calculate with 'extremely complicated' numerals. These, however, get represented
as figures that arise when one writes our numerals one on top of the other. They write π, for example, up to the fifth
decimal place like this:
Anyone who watched them would find it difficult to guess what they were up to. And they might themselves not be
able to explain. For this numerical sign may alter its appearance (for us) up to the point of unrecognizability when it
is written in a somewhat different script. And what the people were doing would appear to us as purely intuitive. [Cf.
Z. 699.]
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331. Thus I am saying: one makes a false estimate of the psychological interest of the if-feeling if one looks at it as
the matter-of-course correlate of the meaning of the word; it must rather be seen in a different context, in the context
of the special circumstances under which it occurs. [Cf. P.I. p. 182c.]
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332. Say "It is hard to still one's fears" and pronounce the fifth word with the feeling of a connective! In the course
of ordinary conversation, practise pronouncing a word which has two meanings with the inappropriate feeling. (If it
is not connected with a wrong tone of voice, it doesn't impede communication.)
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333. Now say to yourself: the connective "still" is really the same as the verb "still" just as "away" = "a-way" and
"despite" (noun) = "despite" (preposition) and pronounce the sentence "Bad as things are, still they might be worse",
with "still" in the meaning of the verb!
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334. Are you even sure that there is a single if-feeling? and not perhaps several? Have you tried to pronounce the
word in very different contexts? (When, e.g., it bears the main emphasis of the sentence, and when the word next to
it does.) [Cf. P.I. p. 181e.]
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335. Does anyone ever have the if-feeling when he is not pronouncing the word "if"? It would surely be at any rate
remarkable, if only this cause was supposed to call up the sensation. Did James ever ask himself whether, and
where, one has it otherwise?--And that's how it is with the 'atmosphere' of a word:--why does one regard it as so
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much a matter of course that only this word has this atmosphere? [Cf. P.I. p. 182d.]
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336. Goethe's signature intimates something Goethian to me. To that extent it is like a face, for I might say the same
of his face.
It is like a mirroring. Does this phenomenon belong with this other one: "I have been in this situation
before"?
Or do I identify the signature with the person in that, e.g. I love to look at the signature of a beloved human
being, or I frame the signature of someone I admire and put it on my desk? (Magic that is done with pictures, hair
etc.)
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337. The atmosphere is inseparable from the thing.--So it is not an atmosphere.
What are inwardly associated got associated, they seem to fit one another. But how do they seem to do that?
What is the expression of their seeming to fit? Is it like this: we can't imagine that the man who was called this,
looked like this, had this signature, produced, not these works, but maybe quite different ones (those of another
great man)?
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We can't imagine that? Do we try? [Cf. P.I. p. 183c.]
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338. It might be like this: Imagine a painter wanting to sketch a picture "Beethoven writing the ninth symphony". I
could easily imagine what one might see in such a picture. But suppose someone wanted to depict how Goethe
would have looked writing the ninth symphony? Here I should not know how to imagine anything that would not be
extremely incongruous and ridiculous. [Cf. P.I. p. 183d.]
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339. Look at a long familiar piece of furniture in its old place in your room. You would like to say: "It is part of an
organism." Or "Take it outside, and it's no longer at all the same as it was", and similar things. And naturally one isn't
thinking of any causal dependence of one part on the rest. Rather it's like this: I could give this thing a name and say
that it is shifted from its place, has a stain, is dusty; but if I tried taking it quite out of its present context, I should say
that it had ceased to exist and another had got into its place.
One might even feel like this: "Everything is part and parcel of everything else" (internal and external
relations). Displace a piece and it is no longer what it was. Only in this surrounding is this table this table. Everything
is part of everything. Here we have the inseparable atmosphere. And what is anyone saying, who says this? What
sort of method of representation is he proposing? Isn't it that
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of the painted picture? If, for example, the table has moved, you paint a new picture of the table with its
surrounding.
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340. "A quite particular expression"--it is part of this that if one makes the slightest alteration in the face, the
expression changes at once.
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341. His name seems to fit his works.--How does it seem to fit? Well, I express myself in some such way.--But is that
all?--It is as if the name together with these works, formed a solid whole. If we see the name, the works come to
mind, and if we think of the works, so does the name. We utter the name with reverence.
The name turns into a gesture; into an architectonic form.
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342. If anyone didn't understand this, we should want to designate him as, say, 'prosaic'. And is that what the
'meaning-blind' would be?
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343. Any other arrangement would strike us as incorrect. Through custom these forms become a paradigm; they
acquire so to speak the force of law. ('The power of custom'?)
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344. Anyone who cannot understand and learn to use the words "to see the sign as an arrow"--that's whom I call
"meaning-blind".
It will make no sense to tell him "You must try to see it as an arrow" and one won't be able to help him in
that way.
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345. But what about such an expression as this: "As you said it, in my heart I understood it"? At the same time one
points to one's heart. And doesn't one mean this gesture?! Of course one means it. Or is one conscious of only using
a picture? Certainly not! [Cf. P.I. p. 178h.]
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346. When the child learns to talk, when does it develop the "feeling of meaning"? Are people interested in this,
when they teach it to talk and observe its progress in talking?
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347. Again, observing an animal, e.g. an ape that investigates an object and tears it to pieces, one may say: "You see
that something is going on in him." How remarkable that is! But not more remarkable than that we say: love,
conviction, are in our hearts!
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348. When and how does a human being begin to manifest feelings of meaning? In what games will it be revealed?
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349. Isn't the inclination to think of a meaning-body like the inclination to think of a seat of thought?--Must everyone
be inclined to say he thinks in his head? This expression is taught him as a child. ("Doing sums in one's head.") But
at any rate the inclination develops from this (or the expression developed from it). In any case--the inclination is
then present. And so is the inclination to speak of a meaning-body (or the like) how ever it arose.
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350. Do we also speak of a 'feeling' of thinking in the head? Wouldn't this be like the 'feeling of meaning'?
Again: Suppose someone who wouldn't have this feeling. Is he unable to think?
Indeed, someone who does philosophy or psychology will perhaps say "I feel that I think in my head". But
what that means he won't be able to say. For he will not be able to say what kind of feeling that is; but merely to use
the expression that he 'feels'; as if he were saying "I feel this stitch here". Thus he is unaware that it remains to be
investigated what his expression "I feel" means here, that is to say: what consequences we are permitted to draw
from this utterance. Whether we may draw the same ones as we would from the utterance "I feel a stitch here".
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351. For one might also say "I feel the rise in prices in my head". And is that nonsense? But under what heading in
psychology should we put this feeling? It doesn't belong under 'sensation'--unless someone were to say "When I feel
this pain in my head, there is always a rise in prices".
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352. Might not someone say: "I have a feeling of a place when I think. I may, for example, think the thought... now
in my head, now in my heart."--And would that shew that a thought has a place? I mean: would it describe the
experience of thinking more closely? Wouldn't it rather describe a new experience?
"I should like to say: 'I thought in my head'."
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353. One can obey the order "Think of nothing at all", "make your mind a blank".
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354. Just as we have learnt the phrase "in the head" in connexion with thinking, so too we have learnt "the word has
this ('one')
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meaning" and all the phrases that are akin to it. And also the form of expression: "these two words only sound the
same, but otherwise they have nothing to do with each other" and many similar ones. And the experience of
meaning really follows these turns of speech exactly. (Though they might have a completely different form--the
French "vouloir dire" for example.)
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355. So is the experience of meaning a mere fancy? Well, even if it is a fancy, that does not make the experience of
this fancy any less interesting.
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356. Incidentally, it is striking that the word "association" plays so small a part in my considerations. I believe that
this word is used in an extremely vague, blurred kind of way, and for quite dissimilar phenomena.
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357. Much can be said about a fine aesthetic difference--this is very important. That is to say, the first utterance is of
course merely "This word fits, this one does not" or the like; but then there may be discussion of all the widely
ramified connexions made by each of these words. That is to say, it is not all over once that first judgment has been
made; rather what it depends on is the field of each word. [Cf. P.I. p. 219b.]
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358. Why should the experience of meaning be important? He says the word, says he said it now in this meaning;
then, in that one. I say the same. This obviously has nothing to do with the ordinary and important use of the
expression "That's what I meant by this word". So what is the remarkable thing? That we say something of that
sort? Naturally that is of interest. But the interest here does not depend on the concept of the 'meaning' of a word,
but on the range of similar psychological phenomena which in general have nothing to do with word-meaning.
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359. Someone says, perhaps in a language lesson, "Let us talk about the word 'still'". I ask: "Do you mean the noun,
the adjective, or the verb?"--He: "I mean the noun." Need he, or I, have had an experience of meaning here? No.
Though it is likely that images have come into our minds during this exchange. They will, e.g., play the same part as
scribbling while one speaks. If someone were accustomed to scribble on paper during a conversation, he would
perhaps one time draw a still, another time a lake, another time the word "Still!"
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And if the talk were of a still and he were to draw a lake, this might distract him from the conversation; but if
he draws pipes, then he'd be staying with the thing.
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360. To what extent can doodling be compared to the play of images?--Imagine human beings who, from childhood
up, make drawings on all occasions where we should say they are imagining something. If one puts a pencil in their
hand then they draw at high speed.
But doesn't the ordinary human being do something quite similar? He doesn't draw indeed, but he 'describes
his image', i.e., instead of drawing, he speaks. Or again, he uses gestures in order to represent, e.g. someone whom
he is imagining. Must I assume that he reads off this description, these gestures, from something? What is there to
be said for this?--Well, perhaps he says "I see him before me!" and then he represents him. But if, instead of this
expression, I had taught him to say "Now I know what he looks like" or "Now I can say what he looks like" or "Now
I'll tell you what he looks like"--then the dangerous picture would be eliminated. (Tennis without a ball.)
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361. In order to climb into the depths one does not need to travel very far; no, for that you do not need to abandon
your immediate and accustomed environment.
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362. How do I find the 'right' word? How do I choose among words? It is indeed as if I compared words according
to fine discriminations of taste. This is too... this too...--that's the right one.
But I don't need always to judge, to explain, why this or that isn't the right word. It simply isn't right yet. I
go on searching, am not satisfied. This is just what it looks like to search, and this is what it looks like, to find. [Cf.
P.I. p. 218h.]
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363. "I am developing what there is in it." How do I know that this was in it?--That's not how it is. Nor can one ask
"How do I know that this is what I actually dreamt?" It is there in it because I say it is. Or better: because I am
inclined to say... And what sort of queer experience is that: being inclined to say...? Not an experience at all.
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364. If, however, I had died before I could develop all this--in that case would it not have been contained within my
experience?--The
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answer "No" to this question is wrong; the answer "Yes" must be wrong too.
"No" would mean: If someone does not tell a dream, it is false to say he had it. It would be incorrect to say:
"I don't know whether he had a dream; he said nothing about it."
"Yes" would mean: He may well have had a dream even when he doesn't report it. But that isn't supposed to
be a psychological statement! A logical one, then.
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365. May someone not dream and yet not tell anyone? Certainly: for he may dream and tell someone.
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366. We read in a story that someone had a dream and did not tell it to anyone. We don't ask how the author could
learn it.--Don't we understand it, when Strachey makes surmises about what Queen Victoria may have seen in her
mind's eye just before her death? Of course--but didn't people also understand the question how many souls there
was room for on the point of a needle? That is to say: the question whether one understands this does not help us
here; we must ask what we can do with such a sentence.--That we use the sentence is clear; how we use it is the
question.
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367. That we use the sentence doesn't yet tell us anything, because we know the enormous variety of use. Thus we
see the problem in How.
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368. Once more: people narrate something to us after waking up; we thereupon teach them the expression "I
dreamt..." followed by the narrative. I then sometimes ask them: "Did you dream anything last night?" and
sometimes get an affirmative, sometimes a negative answer, sometimes a dream narrative, sometimes none. That is
the language-game. (I have assumed now that I myself do not dream. But neither do I have the feeling of an invisible
presence and other people do have it, and I can ask them about their experiences.)
Now must I make an assumption in this case about whether these people's memory has deceived them or
not; whether they actually saw these pictures before them during their sleep or whether it's merely that that's how it
seems to them after waking up? And what is the sense of this question?--And what interest has it?! Do we ever ask
ourselves that, when someone tells us a dream and if not,--is it because we are sure that his memory won't have
deceived him? (And
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suppose he was a man with a quite specially bad memory!) [CF. P.I. p. 184a.]
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369. And does that mean that it is nonsense ever to raise the question whether the dream really when on in the night,
or whether the dream is a memory-phenomenon of the awakened? That depends on what we intend, i.e. what use
we are making of this question. For if we form the picture of dreaming, that a picture comes before the mind of the
sleeper (as it would be represented in a painting) then naturally it makes sense to ask this question. One is asking: Is
it like this, or like this--and to each "this" there corresponds a different picture. [Cf. P.I. p. 184b.]
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370. (Suppose someone were to ask: Is the structure of water or H-O-H?
Does it make sense?--If you give it a sense, it does make sense.)
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371. Back to the language-game of telling dreams: Someone says to me one day: "What I dreamt last night I will tell
no one." Now does that make sense? Why not? Am I supposed to say, after what I have said about the origin of the
language-game, that it makes no sense--as the original phenomenon just was the dream narrative? Absolutely not!
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372. To us, a railway station, with all its equipment, telegraph poles and telegraph wires, means an extensively
ramified system of traffic. But on Mars there is to be found this structure with all its whys and wherefores, even with
a bit of railway track, and there it means nothing of the kind.
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373. "The mind seems able to give the word meaning"--isn't this as if I were to say: "The carbon atoms in benzene
seem to lie at the corners of a hexagon."? But that is not something that seems to be so: it is a picture. [Cf. P.I. p.
184c.]
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374. Of course I don't want to give a definition of the word "dream"; but still I want to do something like it: to
describe the use of the word. My question then runs roughly like this: If I were to come to a strange tribe with a
language I didn't know, and the people had an expression corresponding to our "I dream", "He dreams" etc.--how
should I find out that this was so; how should I know which
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expressions of their language I am to translate into these expressions of ours?
For finding this out is like finding out which of their words I am to translate into our word "table".
Here of course I don't ask "What do they call THIS?" while I point at something. Although I might ask even
that, and might point to a symbolic representation of a dream or a dreamer.
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375. There is also this to say: the child does not absolutely have to learn the use of the word "to dream" by first
merely reporting an occurrence on waking up, and our then teaching it the words "I dreamt". For it is also possible
that the child hears the grown-up say he has dreamt and now says the same thing of itself and tells a dream. I am not
saying: the child guesses what the grown-up means. Suffice it that one day it uses the word, and uses it under the
circumstances under which we use it.
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376. So the proper question is not: "How does he learn the use of the word?" but rather "How does it come out that
he does use it as we do?"
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377. "Black is the beauty of the brightest day"†1--Can one say 'Well, it seems as if it were black?" Have we then an
hallucination of something black?--So what makes these words apt?--"We understand them." We say, e.g. "Yes, I
know exactly what that's like!" and now we can describe our feelings and our behaviour.
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378. "When you are talking about dreaming, about thinking, about sensation,--don't all these things seem to lose the
mysteriousness which seems to be their essential characteristic?" Why should dreaming be more mysterious than
the table? Why should they not both be equally mysterious?
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379. "The phenomenon of seeing as an arrow or otherwise is surely a true visual phenomenon; even though it
is not so tangible as that of form and colour." How should it not be a visual phenomenon?!--Does anyone that
speaks of it ever doubt that it is (except when he is doing philosophy or psychology)? Don't we ask a
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man about it and tell him of it, like any other visual phenomenon? I mean: Do we talk of it more hesitantly, with the
suspicion that what we say may have no clear sense? Certainly not. But there are differences in it, all the same. The
ones that we indicate when we say "more intangible."
Only it is like this: If I put two substances in front of someone, I may say: Feel this one here. Don't you find
that it feels softer?" And if he answers yes, I say, e.g. "Yes, I feel that too. So there is a difference between them."
(I.e.: I have not merely fancied it.)--But it is otherwise with psychological phenomena. When I say "This is more
intangible than that"--as, that is, a tenseless proposition--this does not rest upon a consensus of judgements, not
upon our all feeling that too (when we 'contemplate' the experience).
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380. Don't put the phenomenon in the wrong drawer. There it looks ghostly, intangible, uncanny. Looking at it
rightly, we no more think of its intangibility than we do of time's intangibility when we hear: "It's time for dinner."
(Disquiet from an ill-fitting classification.)
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381. "This coffee has no taste at all." "This face has no expression at all."--The opposite of this is "It has a quite
particular expression" (though I could not say what). A strong expression I could easily connect with a story for
example. Or with looking for a story. When we speak of the enigmatic smile of the Mona Lisa, that may well mean
that we ask ourselves: In what situation, in what story, might one smile like that? And so it would be conceivable for
someone to find a solution; he tells a story and we say to ourselves "Yes, that is the expression which this character
would have assumed here".
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382. Remembering a particular kinaesthetic sensation--remembering the visual image of a movement.--Make the
same movement with the right and left thumb, and judge whether the kinaesthetic sensations are the same.--Have
you a memory-image of the K-sensation in walking?--If you are tired, or suffering pain, muscular pain or a burning
skin--are the sensations in moving a limb the same as in a different condition? But are you then sometimes in doubt
whether you now really have raised your leg, because the feeling is so totally different? Do you actually feel the
movement in the joint?
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383. You sometimes hear someone saying "I am imagining his bearing quite vividly"; or "his voice". But do you ever
hear "I am imagining the K.-sensation in connexion with this movement of the hand"?! And why not?
Does one imagine it and merely not say so?
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384. What are we to answer if someone counters us: "If you guide someone's hand in a movement, by doing so you
are shewing him a particular K.-sensation, which he then reproduces if he now repeats the movement when ordered
to"? And can one say that obviously he may be guided in this way by the visual image of the movement--but not by
a K.-image?
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385. How important is it that there is such a thing as a pictorial representation of the visual movement and nothing
corresponding to it for the 'kinaesthetic movement'?
Make a movement that looks like this--"Make a movement that produces this noise". "Make a movement
that produces this K.-sensation." Copying the K.-sensation correctly would in this case mean repeating the
movement correctly according to the appearance to the eye.
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386. Imagine the movement's being very painful, so that the pain drowns out any other slight sensation. [Cf. P.I. p.
186d.]
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387. Make a movement with your fingers (such as you make in piano playing, say); repeat it, but with a lighter
touch. Do you remember which of the two feelings you had yesterday in connexion with the first movement?
We say perhaps: "No, yesterday this movement looked rather different"--but do we also say "The movement
is not quite the same--I did not have exactly this K.-sensation"?
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388. For of course we have feelings of movement and we can also reproduce them. Especially when we repeat a
movement under the same circumstances after only a brief pause. One also localizes the sensations, but hardly ever
in the joints, mostly in the skin. (Blow your cheeks out. Where do you do it, and where do you feel it?)
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389. The growth of analysis might be compared with the growth of a seed. And in this case to say "It was all already
contained in
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the sensation" or "it grew out of it as from a seed" come to the same thing. Now how much is there (true) in this, that
one sometimes reproduces an arm movement (say) according to a visual picture, but not according to a kinaesthetic
picture?
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390. Does one actually sometimes go by a visual image in bending one's arm? I can only say: If I did not see that my
arm has moved after being convinced I had moved it with my face turned away, I should become confused and
should presumably trust my eyes. Seeing can at any rate tell me whether I have carried out my intended movement
exactly, e.g. have reached the position that I wanted to reach; the feeling wasn't able to do that. I feel that I am
moving all right, and I can also judge roughly how by the feeling--but I simply know what movement I have made,
although you couldn't speak of any sense-datum of the movement, of any immediate inner picture of the movement.
And when I say "I simply know..." "knowing" here means something like "being able to say" and is not in turn, say,
some kind of inner picture.
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391. "In order to be able to say the feeling tells me where my arm now is, or how far I am moving it, one would need
to have correlated feelings and movements. It should be possible to say: 'When I have the feeling... then in my
experience my arm is over there'. Or: you would need to have a criterion of identity of the feelings besides that of
the movement you've made." But even if this condition makes sense at all, is it fulfilled for seeing? Well, one can
represent a visual picture by drawing. But as for giving someone, or oneself, the feeling that is characteristic of the
arm's being bent at an angle of 30°, I mean without bending the arm--that one can't do.
Bend your arm a little. What do you feel?--A tension or suchlike here and there, and principally the rubbing
of my sleeve.--Do it again. Was the feeling the same? Roughly. Roughly in the same places. Does this feeling always
accompany this movement, can you say so? No. And yet I find there is still something about this argument that
doesn't fit.
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392. Imagine that certain movements produced notes and now it was said that we recognize how far the arm has
moved from the note that strikes. That would surely be possible. (Playing a scale on the piano.) But what sort of
presuppositions must be satisfied for it to be so? It would not be enough, e.g. that notes accompany the movements;
nor that they are often like for like movements. Nor would it suffice to
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say: the note just must have a single identical quality for identical movements, as it is the only sense datum in which
we can recognize the amplitude of the movement.
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393. But isn't there such a thing as a kind of private ostensive definition for feelings of movement and the like? E.g. I
crook a finger and note the sensation. Now someone says to me: "I am going to produce certain sensations in your
finger in such and such a way, without its moving; you tell me when it is that one that you have now in crooking
your finger." Mightn't I now, for my own private use, call this sensation "S", use my memory as criterion of identity
and then say "Yes, that's S again" etc.
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394. It would then also be conceivable that I should recognize the sensation and that it should occur without being
accompanied by the conviction of the movement's having taken place--without the sense of movement.
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395. I can certainly, e.g., raise my knee several times in succession and say I have had the same sensation every time:
Not as if I always had this sensation when I raise my knee, nor as if I can recognize the movement in the sensation,
but merely: In this series of knee-movements I three times had the same sensation, produced by the movement.
Being the same here of course means the same thing as seeming the same.
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396. "I had the same sensation three times": that describes a process in my private world. But how does someone
else know what I mean? What I call "same" in such a case? He relies upon it that I am using the word here in the
same way as usual? But what in this case is the use that is analogous to the usual one? No, this difficulty is not a
piece of over-refinement; he really does not know, cannot know, which objects are the same in this case.
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397. The example of the motor roller with the motor in the cylinder is actually far better and deeper than I have
explained. For when someone shewed me the construction I saw at once that it could not function, since one could
roll the cylinder from outside even when the 'motor' was not running; but this I did not see, that it was a rigid
construction and not a machine at all. And here there is a close
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analogy with the private ostensive definition. For here too there is, so to speak, a direct and an indirect way of
gaining insight into the impossibility. [See Z 248.]
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398. I named this sensation of movement "S". Now, for others it is the sensation I had when I made this movement.
But for me? Does "S" now mean something else?--Well, for me it means this sensation.--But which is this? for I
pointed to my sensation a minute ago, how can I now point to it again?
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399. But suppose the case where someone made a series of arm-movements and said: "The sensation that I now
have in my leg, I call 'S1,'" and so on. Later, on various occasions, he says "Now I have S3". And so on. Such
utterances might be important, if we observe certain physiological correlates, for example, and in this way are able to
draw conclusions from his utterances.
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400. If it is true that we do not estimate the kind and magnitude of the movement of a limb by the feeling -- how
would a human being differ from us, with whom that was the case? Well, this could quite easily be imagined: that
someone felt pain-sensations varying in strength or kind with difference of movements. Thus he would say, e.g. "I
feel this pricking when I bend my arm through an angle of about 90°"
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401. Imagine someone who by means of a dowsing rod, and going by the tug which it gives, can determine the
depth of a water course. He learned this in the following way: He walked over water courses of various depths and
noted the tug. (This might perhaps have been established with a spring balance.) He associated the tug with the
depth and now draws conclusions from the tug to the depth. This might happen in such a way that he states the
tug--say in pounds--and makes a transition to the depth, perhaps using a table. However, it may also be the case that
he knows no other measure of the tug than the depth of the water-course. After a certain amount of practice he can
give the depth right. If one exerts a tug on the rod, say by using weights, he will say "That tugs like a water-course of
such and such a depth".
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402. However, it might be that, while he was able to give the depth of a water course right by means of the tug of
the rod, yet he was not
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able to give a correct estimate of the tug of the rod. I mean this as follows: It might be that water gives an equal tug at
different depths in different circumstances; and this dowser now says, e.g.: "This water-course is deeper than the
former one, it tugs more weakly"--and he is right: the water-course actually does lie deeper, but the tug measured by
the spring balance was the same and he had not noted it correctly.--Am I to say in such a case: he judges the depth
by the tug?
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403. He will perhaps say: "This tug is that of a water-course at depth..." while he as it were studies this tug--as one
hefts a weight in one's hand. But perhaps he says: "I can't judge the tug--the water is at the depth...." In this (latter)
case one will not say he judges the depth by the tug. (At least not 'consciously'.)
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404. Suppose now someone were to say he judged how far he has bent his arm by the strength of a sensation of
pressure in the elbow. That surely means: When it reaches a certain strength he knows from that that the arm is bent
to this degree. Or what else should it mean, that he judges the degree of the bending by the sensation of pressure?
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405. I want to say: How does anyone know that he judges something by this feeling--Does it suffice if he directs his
attention to the feeling in making his estimate?
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406. If you say that for this someone must be able to say: "When the pressure is as strong as this, my arm is bent
90°"--then the 'this' must capable of being specified. Otherwise that someone judges the bending by the strength of
the sensations of pressure would at most mean that one cannot judge the bending when one has no sensation of
pressure (or only an uncommonly weak one). (And so, e.g. when one is anaesthetized.)
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407. Thus there is a variety of cases. Someone may say he judges the bending by the sensation of pressure or pain,
and may in doing this so to speak hearken to this sensation; but for the rest be quite unable to give the degree of the
sensation in any form.--Or there may be two independent specifications: of the degree of the sensation and of the
degree of bending.
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408. "When I feel the pressure as hard as this, then...." Doesn't that make sense? Someone might even say he had a
whole scale of
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sensations of pressure. I can well imagine that. Only that would no more be an actual scale than the picture of a
thermometer is a thermometer. Although in many respects it has a great similarity to a thermometer.
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409. I give the rules of a game. Quite in accordance with these rules, the other makes a move whose possibility I had
not foreseen, and which spoils the game, at least as I meant it. I must now say: "I gave bad rules," I must change or
perhaps add to my rules.
So did I in this way already have a picture of the game? In a certain sense: yes.
I might, for example, not have foreseen that a quadratic equation need not have real numbers as a solution.
The rule, then, leads me to something of which I say: "I had not expected this picture; I always pictured the
solution like this...." [Cf. Z 293.]
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410. How would it be if one said: "Not every system of rules determines a calculus." One would give dividing
through by o as an example. For let us imagine an arithmetic in which it was allowed and in consequence it could be
proved that any number was equal to any other.
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411. When children play at trains--ought I to say that a child who is imitating a steam engine is seen by another as a
steam engine? He is taken as a steam engine in the game.
Suppose I had shewn a grown-up the shape and asked "What does that remind you of?" and he had
replied "A steam engine"--does that mean that he saw it as a steam engine?
For I take it as the typical game of "seeing something as something", when someone says "Now I see it as
this, now as that". When, that is, he is acquainted with different aspects, and that independently of his making any
application of what he sees.
So I should like to say this: I do not see any application of the picture as a sign that the picture is seen this
way or that.
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412. Would a child understand what it means to see the table 'as a table'? It learns: "This is a table, that's a bench"
etc., and it completely masters a language game without any hint of there being an aspect involved in the business.
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413. "Yes, it's just that the child doesn't analyse what it does." Once more: what is in question here is not an analysis
of what happens. Only an analysis--and this word is very misleading--of our concepts. And our concepts are more
complicated than those of the child; in so far, that is, as our words have a more complicated employment than its
words do.
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414. "I surely see it like this, even if I don't express it." That would mean that what I see doesn't alter when I express
it. If one were to ask "Has the body this weight only so long as it is being weighed?"--that would mean "Does the
weight change when we put it on the weighing machine?" And naturally that is not at all the thing that we should like
to ask.
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415. Only through the phenomenon of change of aspect does the aspect seem to be detached from the rest of the
seeing. It is as if, after the experience of change of aspect, one could say "So there was an aspect there!"
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416. When one scrapes a coating off a thing one can say "So there was a coating there".--But if the colour of a body
changes--can I say "So it had a colour!" as if this had only just struck me?
Can one say: I only became aware that the thing had a colour, when its colour changed?
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417. Do not think that it is something queer for you to see a picture on the wall three-dimensionally. It is--I should
like to say--as ordinary as it seems. (And I might say this about a lot of things.)
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418. Imagine that the things that surround us--table, books, chairs etc.--underwent abrupt periodic colour changes;
their shapes remain the same. Might one say here that this was how we first became conscious of colour and shape
as special constituents of our visual experience?
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419. When I compare wild flowers and garden flowers, this may make me conscious of the difference of character;
but that is not to say that I must already earlier on have perceived their character as well as the flowers themselves,
or that I must after all have perceived them as having some character or other.
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420. Must I know that I see with two eyes? Certainly not. Do I perhaps have two visual impressions in ordinary
seeing, so that I notice that my three-dimensional visual impression is compounded of
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two visual pictures? Certainly not.--So I can't separate threedimensionality from seeing.
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421. If I ask someone "Which direction would you say an 'F' looks in and which a 'J'?" and he answeres that for him
an F always looks to the right, a J to the left--of course that does not mean that whenever he looks at an F he always
has a sensation of direction. This becomes clearer if one puts the question like this: "Where would you paint an eye
and a nose onto an F?"--But if it were now said: "So it looks in this direction for you only as long as you are thinking
this or saying it"--isn't that as if one were to ask "Would you paint the nose there on an F, only at the time that you
are actually painting it?"
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422. Do I always see a face 'as a face'? I have books here in front of me: Do I see them the whole time 'as books'? I
mean: Do I see them the whole time as books, if I don't precisely see them as anything else? Or do I often, or
ordinarily, see only colours and shapes, without any special aspect? (Obviously not!) We tell someone: "If that is
the base then that is the apex and that the altitude." Or he has to answer the question: "What is the altitude of the
triangle, when this is the base?" But we don't insist on his seeing the triangle in this or that way.--One may well
sometimes say: "Imagine it turned round" (or the like) and one might also say "See it turned round", and this remark
might help; in the way, that is, in which some drawn lines completing the picture might help if they suggested this
aspect.
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423. Can I, e.g., say: I see the chair as object, as unit? In the same way as I say I see now the black cross on a white
ground, and now the white cross on a black ground?
If someone asks me "What have you there in front of you?" I shall of course answer "A chair"; so I shall treat
it as a unit. But can one now say I see it as a unit?
And can I see the cross-figure without seeing it this way or that?
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424. When I ask someone "What do you see in front of you" and he says "What I have in front of me looks like
this" and now he draws the cross-figure--must he have seen it in some aspect or other? Has he not seen it, if he can
only describe it by drawing?
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425. Can a child inform you that it sees three-dimensionally?
And imagine its saying to you "I see everything flat"--what would that tell you? It might see everything flat
and know through intuition that it isn't flat, and behave correspondingly!
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426. If the child takes this picture for such-and-such and now I conclude: "So it sees the picture in this way"--what
sort of conclusion am I drawing? What does this conclusion say to me? It would perhaps be said that I was inferring
the kind of sense-datum or visual picture the child had: as if the conclusion ran: "So the picture in its mind is like
this"; and now one would have to give (say) a plastic representation of it.
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427. Then is it like this: "I have always read the sign 'Σ' as a sigma; now someone tells me it could also be an M
turned round, and now I can see it like that too: so I have always seen it as a sigma before"? That would mean that I
have not merely seen the figure Σ and read it like this, but I have also seen it as this!
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428. "But how could I know that I should have reacted like this if you had asked me?"--How? There is no How. But
there are indications that I am right in saying it.
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429. I want to describe what I see; for this purpose I prepare a transparency. But now I am further asked "Is this in
front and this behind?" So, by means of words or a model, I describe what I see in front and what behind. And then
I am further asked "And do you see this point as the apex of the triangle?" and now I must answer this as well.--But
must I have an answer to it?--Assume, though it is not true, that the direction of one's glance determines the aspect.
And in one case my gaze is continuously directed on to the same point in the picture, in another it moves in a regular
fashion according to a simple law, in a third it wanders randomly back and forth over the object. If we now replaced
a description of the aspect by that of the direction of glance, would it not be a description to say that the direction of
glance was random, or indeterminate? And that might be just the ordinary case.--To the question "Did you see this
point as the apex of the triangle?", then, the answer might be "I can't mention any particular aspect" or perhaps "At
any rate I didn't see it like this".
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430. What, however, did the hypothesis of the importance of the direction of glance do for us?--It offered us a
picture of definite multiplicity.
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431. But such a theory is really the construction of a psychological model of a psychological phenomenon. And
hence of a physiological model.
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The theory really says "It could be like this:...." And the usefulness of the theory is that it illustrates a
concept.
It may illustrate it better and worse; more, and less, appropriately. The theory is thus so to speak a notation
for this kind of psychological phenomenon.
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432. Thus if we 'leave explanation'--if we say that after all we don't mind about the explanation--what is left over is a
grammatical stipulation. It concerns the use of the statement "I am now seeing a particular facial expression in the
picture".
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433. Doesn't the theme point to something outside itself? Oh, yes! But that means:--The impression that it makes on
me hangs together with things in its surroundings--e.g. with the existence of our language and its intonation; but that
means: with the whole field of our language-games.
When I say, e.g.: It is as if a conclusion were being drawn here, or as if here something were being
confirmed, or, as if this were the answer to what went before,--in this way my understanding presupposes familiarity
with conclusions, confirmations, answers. [Cf. Z 175; Culture and Value, pp. 51-52.]
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434. A theme has a facial expression just as much as a face does. [Vermischte Bermerkungen, 2nd ed. p. 101;
Culture and Value, p. 52.]
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435. "The repetition is necessary." To what extent is it necessary? Well, sing it, and you will see that it takes the
repetition to give it its great strength.--Doesn't it seem to us as if there had to be a text for the theme existing in
reality, and the theme would approximate to it, would correspond to it, only if this part was repeated? Or am I to
utter the stupidity "It sounds finer with the repetition"? And surely there just is no paradigm there outside the theme.
And yet after all there is a paradigm outside the theme: namely the rhythm of our language, of our thinking and
feeling. And the theme is also in its turn a new bit of our language, it is incorporated in it; we learn a new gesture.
[Cf. C. & V., p. 52.]
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436. The theme and the language are in reciprocal action. [Cf. C. & V., p. 52.]
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437. "A whole world of pain lies in these words." How can it lie in them?--It hangs together with them. The words
are like an acorn from which an oak-tree can grow.
But where is the law laid down, according to which the tree grows
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out of the acorn? Well, experience has incorporated the picture into our thought. [Cf. C. & V., p. 52.]
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438. "Where do you feel grief?"--In my mind.--And if I had to give a place here, I should point in the region of the
stomach. For love, to the breast and for a flash of thought, to the head.
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439. "Where do feel your grief?" In my mind.--Only what does that mean?--What kind of consequences do we infer
from this place-assignment? One is, that we do not speak of a physical place of grief. But all the same we do point to
our body, as if the grief were in it. Is that because we feel a physical discomfort? I do not know the cause. But why
should I assume it is a bodily discomfort? [Cf. Z 497.]
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440. Think of the following question: Can one imagine a pain, with, say, the quality of rheumatic pain, but without
locality? Can one imagine it?
When you begin to think this over, you see how much you would like to change the knowledge of the place
of pain into a characteristic of what is felt, into a characteristic of the sense-datum, of the private object that is there
before my mind. [Cf. Z 498.]
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441. I say that to the grief-stricken the whole world looks grey. But what was before my mind would in that case not
be grief, but a grey world: as it were the cause of grief.
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442. Seeing something as difference of colour--and on the other hand as shadow, the colour being the same. I ask
"Have you perceived the colour of the table in front of you, which you've been looking at the whole time?" He says
"Yes". But he would have described the colour of the table as "brown" and has not noticed that the green curtains
are reflected in its shining surface. Now has he not had the green sense-impression?
"Is the wall in front of you uniformly yellow?" "Yes." But it is partly in shadow and looks almost grey.
Now what did he see, when he looked at the wall? Am I to say, a uniformly yellow surface, which admittedly
is irregularly shadowed? Or: yellow and grey patches?
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443. It is a remarkable fact that we are hardly ever conscious of the unclarity of the periphery of the visual field. If
people, e.g., talk about the visual field, they mostly do not think of that; and when one speaks
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of a representation of the visual impression by means of a picture, one sees no difficulty in this. This is very
important.
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444. "What I perceive is THIS--" and now follows a form of DESCRIPTION. One might also explain this in this
way: Let us imagine a direct transfer of the experience.--But now, what is our criterion for the experience's really
having been transferred? "Well he simply has the same as I have"--But how does he 'have' it? [Cf. Z 433.]
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445. Think of the variety of physical experiments. We measure, e.g. temperature; but only within a certain general
technique is this experiment a measuring of temperature.--So if we were interested in the multiplicity of (physical)
measurements, I mean kinds of measurement, we'd be interested in the same way in the multiplicity of methods, of
concepts.
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446. How can you look at your grief? By being grief-stricken? By not letting anything distract you from your grief?
So are you observing the feeling by having it? And if you are holding every distraction at a distance, does that mean
you are observing this condition? or the other one, in which you were before the observation? So do you observe
your own observing?
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447. Suppose someone were to ask "What are all the things measured in physics?" Now one might retail them:
lengths, times, brightness of light, weights etc.
But might one not say: You learn more if you ask "How is measuring done?" instead of "What is measured?"
If one does this, if one measures like this, then one is measuring temperature--if one does that, measures like
that: the strength of a current.
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448. But doesn't grief consist of all sorts of feelings? Is it not a congeries of feelings? Then would one say it
consists of feelings A, B, C, etc.--like granite out of feldspar, mica and quartz?--So do I say of someone who has the
feelings... that he is grief-stricken? And how do I know that he has them? Does he tell us?
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449. But grief is a mental experience. One says that one experiences grief, joy, disappointment. And then these
experiences seem to be really composite and distributed over the whole body.
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The gasp of joy, laughter, jubilation, the thoughts of happiness--is not the experience of all this: joy? Do I
know, then, that he is joyful because he tells me he feels his laughter, feels and hears his jubilation etc.--or because
he laughs and is jubilant? Do I say "I am happy" because I feel all that?
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450. The words "I am happy" are a bit of the behaviour of joy.
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451. And how does it come about that--as James says--I have a feeling of joy if I merely make a joyful face; a feeling
of sadness, if I make a sad one? That, therefore, I can produce these feelings by imitating their expression? Does that
shew that muscular sensations are sadness, or part of sadness?
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452. Suppose someone were to say: "Raise your arm, and you will feel that you are raising your arm." Is that an
empirical proposition? And is it one if it is said: "Make a sad face, and you will feel sad"?
Or was that meant to say: "Feel that you are making a sorrowful face, and you will feel sorrow"? and is that a
pleonasm?
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453. Suppose I say: "Yes, it's true: if I adopt a more friendly expression, I feel better at once."--Is that because the
feelings in the face are pleasanter? or because adopting this expression has consequences? (One says "Chin up!")
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454. Does one say: "Now I feel much better: the feeling in my facial muscles and round about the corners of my
mouth is good"? And why does that sound laughable, except, say, when one had, felt pain in these parts before?
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455. Is my feeling at the corners of the mouth compared with his--in the same way as my mood is compared with
this?
How, for example, do I compare my sensations of pressure with his? How do I learn to compare them? How
do I compare our kinaesthetic sensations, how do I correlate them with one another? And how the feelings of
sorrow, joy etc.?
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456. Now granted--although it is extremely doubtful--that the muscular feeling of a smile is a constituent part of
feeling glad;--where are the other components? Well, in the breast and belly etc.!--
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But do you really feel them, or do you merely conclude that they must be there? Are you really conscious of these
localized feelings?--And if not--why are they supposed to be there at all? Why are you supposed to mean them,
when you say you feel happy?
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457. Something that could only be established through an act of looking--that's at any rate not what you meant.
For "sorrow", "joy" etc. just are not used like that.
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458. Why does it sound so queer to say "He felt deep grief for one second"? Because it so seldom happens? Then
what if we were to imagine people who often have this exprience[[sic]]? Or such as often for hours together alternate
between second-long feelings of deep grief and inner joy? [Cf. P.I. p. 174c.]
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459. "Don't you feel grief now..."--is that not as if one were to ask "Aren't you playing chess now?" Really, though,
the question was a personal and temporal one, not a philosophical question. [Cf. p. 174d.]
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460. "'I'm hoping...'--the description of my state of mind": That sounds as if I looked into my mind and described it
(as one describes a landscape). If now I say: "I keep on hoping that he will yet come to me"--is that a piece of hoping
behaviour? Isn't it as little a piece of hoping behaviour as the words "At that time I was hoping he would come"? So
should I not say that there are two kinds of present of "hope"? One, as it were the exclamation, the other the report?
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461. But now when I say to someone "I very much hope that he will come to our gathering"--does he ask me: "What
was that, a report or an exclamation?" -Does he fail to understand me, if he doesn't know that? And yet it is one
thing to say "I hope he'll come" and another to say: "I don't lose hope that he will come."
Or think of this expression "I hope and pray that he may come".
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462. "I hope he'll come"--one might say--is sometimes equivalent in meaning to "He'll be coming!" said in a hopeful
voice. But this exclamation need not have any perfect tense. Couldn't a language be imagined in which, while there
was an equivalent of this expression
Page Break 90
of hope, still there were not the remaining forms of the verb? In which the people quoted themselves when they did
want to speak of past hope--saying, e.g. "I said: 'He'll surely come!'."
Page 90
463. It might be said: An assertion says something about the state of mind, given that I can make inferences from it
about the state of mind. (That sounds more stupid than it is.) If that's how it is, then the expression of a wish "Give
me that apple" says something about my state of mine. And is this proposition then a description of this state? That
one won't want to say. ("Off with his head!")†1
Page 90
464. If I call out "Help!" is that a description of my state of mind? And is it not the expression of a wish? Is it not as
much that as any other cry is?
Page 90
465. I say to myself "I still keep on and on hoping, although..." and in saying it I as it were shake my head over
myself. That means something quite other than simply "I hope...!" (The difference in English between "I am hoping"
and "I hope".)
Page 90
466. And what is observed by observing your own hope? What would you report? Various things. "I hope every
day.... I imagined.... Every day I said to myself.... I sighed.... Every day I took this route in the hope...."
Page 90
467. The word "observe" is badly applied here. I try to remember this and that.
Page 90
468. If someone remembers his hope, on the whole he is not therefore remembering his behaviour, nor even
necessarily his thoughts. He says--he knows--that at that time he hoped.
Page 90
469. The sentence "I want some wine to drink" has roughly the same sense as "Wine over here!" No one will call
that a description; but I can gather from it that the one who says it is keen to drink wine, that at any moment he may
take action if his wish is refused--and this will be called a conclusion as to his state of mind.
Page 90
470. Is "I believe..." a description of my mental state?--Well, what is such a description? "I am sad", for example, "I
am in a good mood", perhaps "I am in pain".
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Page 91
471. It would be asking for trouble to take Moore's paradox for something that can only occur in the mental sphere.
Page 91
472. I want to say first of all that with the assertion "Its going to rain" one expresses belief in that just as one
expresses the wish to have wine with the words "Wine over here!" One might also put it like this: "I believe p"
means roughly the same as "p" and it ought not to mislead us that the verb "believe" and the pronoun "I" come in
the first proposition. We merely see clearly from this that the grammar of "I believe", is very different from that of "I
write".
But when I say this, I don't mean that there may not also be big similarities here; and I am not saying what
kind of differences there are. ((Real and imaginary unit.))
For remember that what is in question here are similarities and differences of concepts, not phenomena.
Page 91
473. One may say the following queer thing: "I believe it's going to rain" means something like: "It's going to rain",
but "I believed then that it was going to rain" doesn't mean anything like "It rained then".
But now, what does it mean to say that the first sentence has roughly the same sense as the second? Does it
mean that both produce the same thought in my mind (the same feeling?)--[Cf. P.I. p. 190d.]
Page 91
474. "I want to think thus and not thus." And however queer it sounds 'thus' and 'this' aren't sharply distinct from
one another.
Page 91
475. The way you use the word "God" shews, not whom you mean, but what you mean. [V.B. p. 99; C. & V., p. 50.]
Page 91
476. "But surely 'I believed' must mean just that, for the past, which 'I believe' means for the present!"
must mean just the same for -1, as does for 1. That means nothing at all. [Cf. P.I. p. 190d.]
Page 91
477. What does it mean to say that "I believe p" says roughly the same as "p"? We react in roughly the same way
when anyone says the first and when he says the second; if I said the first, and someone didn't understand the words
"I believe", I should repeat the sentence in the second form and so on. As I'd also explain the words "I wish you'd go
away" by means of the words "Go away".
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Page 92
478. Moore's Paradox may be expressed like this: "I believe p" says roughly the same as " p"; but "Suppose I
believe that p..." does not say roughly the same as "Suppose p...".
Can one understand the supposition that I wish for something before understanding the expression of a
wish?--The child learns first to express a wish, and only later to make the supposition that it wished for
such-and-such.
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479. "Suppose I have a pain..."--that is not an expression of pain and so it is not a piece of pain-behaviour.
The child who learns the word "pain" as a cry, and who then learns to tell of a past pain--one fine day this
child may declare: "If I have a pain, the doctors[[sic]] comes." Now has the meaning of the word "pain" changed in
this process of learning the word? It has altered its employment; but one must guard carefully against interpreting
this change as a change of object corresponding to the word.
Page 92
480. Imagine "I believe..." represented in a painting. How might I imagine this? The picture would perhaps shew me
with some picture or other inside my head. The point is not what symbolism it employs. The picture of what I
believe, e.g. that it is raining, will come into it. My mind will perhaps lay hold of this picture, hold on to it and so
on.--And now let us suppose that this picture got used as the assertion "Its raining". Well, so far there is nothing odd
about that. Am I now to say that there is a lot that is redundant about the picture? That I should not like to say.
Page 92
481. "Basically, with these words I describe my own state of mind--but here this description is indirectly an assertion
of the state of affairs that is believed."--As, in certain circumstances I describe a photograph in order to describe
what the photograph is a shot of. [Cf. P.I. p. 190e.]
Page 92
482. But if this analogy held good, then I should further have to be able to say that this photograph (the impression
on my mind) is trustworthy. So I should have to be able to say: "I believe that it's raining, and my belief is
trustworthy, so I trust it." As if my belief were a kind of sense-impression. [Cf. P.I. p. 190e.]
Page 92
483. Do you say, e.g. "I believe it, and as I am reliable, it will presumably be so"? That would be like saying: "I
believe it--therefore I believe it."
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Page 93
484. Just as one may use the same procedure, now to measure the length of the table, now to check the yardstick,
now to test the measurer's accuracy in making measurements, in the same way an assertion may serve one as
information as to its content, or about the character or the state of mind of the asserter.
Page 93
485. One might very well say: "He's coming, but I still can't believe it!"--"He's coming! I can't believe it!"
Page 93
486. Imagine an announcer in a railway station, who announces a train according to schedule, but--perhaps
groundlessly--is convinced that it won't arrive. He might announce: "Train No.... will arrive at... o'clock. Personally I
don't believe it."
Page 93
487. How would it be, if a soldier produced military communiqués which were justified on grounds of observation;
but he adds that he believes they are incorrect.--Let us ask ourselves, not what may be going on in the mind of one
who speaks in this way, but rather whether others can do anything with this report, and what they can do.
Page 93
488. The communiqué is a language-game with these words. It would produce confusion if we were to say: the
words of the communiqué--the proposition communicated--have a definite sense, and the giving of it, the 'assertion'
supplies something additional. As if the sentence, spoken by a gramophone, belonged to pure logic; as if here it had
the pure logical sense; as if here we had before us the object which logicians get hold of and consider--while the
sentence as asserted, communicated, is what it is in business. As one may say: the botanist considers a rose as a
plant, not as an ornament for a dress or room or as a delicate attention. The sentence, I want to say, has no sense
outside the language-game. This hangs together with its not being a kind of name. As though one might say "'I
believe...'--that's how it is" pointing (as it were inwardly) at what gives the sentence its meaning.
Page 93
489. Is it a tautology to give the communiqué: "The cavalry will arrive immediately, and I believe it"?
Page 93
490. The paradox is this: the supposition may be expressed as follows: "Suppose this went on inside me and that
outside"--but the assertion
Page Break 94
that this is going on inside me asserts: this is going on outside me. As suppositions the two propositions about the
inside and the outside are quite independent, but not as assertions.
Page 94
491. Does this lie in the nature of the concept "believe"? Certainly.
Page 94
492. Suppose someone said "I wish--but I don't want my wish to be fulfilled". (Lessing: "If God in his right
hand...."†1) Can one then ask God to give the wish, and not to fulfil it?
Page 94
493. So it looks as if the assertion "I believe..." were not the assertion of what the supposition "I believe" supposes!
[Cf. P.I. p. 190c.]
Page 94
494. Don't see it as a matter of course, but as something very worthy of note, that the verbs "believe", "hope",
"wish", "intend" and so on, exhibit all the grammatical forms that are also possessed by "eat", "talk", "cut". [Cf. P.I.
p. 190h.]
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495. Imagine I were the hybrid being that might pronounce "I don't believe it is raining; and it is raining".--But what
purpose do these words now serve? What employment am I imagining being given to them?
"He's coming. I personally don't believe it, but don't let that mislead you." "He's coming, rely upon it. I don't
believe it, but don't let that mislead you." This sounds as if two persons were speaking out of me; or as if one court
within me gave the other person the information that so and so was coming, and this court wished that the person
should take appropriate action--while another court in a certain sense reported my own attitude. It is as if one were
to say "I know that this is the wrong procedure, but I know that that's what I shall do".
"He's coming, but I don't believe it" may, then, occur in a language-game. Or better: It is possible to think out
a language-game in which these words would not strike us as absurd.
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Page 95
496. A voltmeter might pronounce the voltage by means of a gramophone record instead of by dial and pointer.
When one presses a button (asks the question) the instrument says, e.g. "The voltage amounts to...". Could it make
sense to have the voltmeter say: "I believe the voltage amounts to..."? One can imagine such a case.
Am I now to say that the instrument is stating something about itself--or about the voltage? Am I to say that
the instrument always states something about itself! And if, e.g., on repetition it may give a higher reading for the
voltage: am I to say that it had believed the voltage was...?
Page 95
497. Or let us put it like this: Am I to say a voltmeter says something about itself, or about the voltage? May I not
say both? Each, that is, under different circumstances?
Page 95
498. Have "Help!" and "I need help" different senses; is it merely a crudity in our conception that we regard them as
equivalent? Does it always mean something to say "Strictly speaking, what I meant was not 'Help!' but 'I need help'".
The worst enemy of our understanding is here the idea, the picture, of a 'sense' of what we say, in our mind.
Page 95
499. The assertion "He will come" makes no allusion to the maker of the assertion. But neither does it allude to the
words of the assertion, whereas "'He will come' is a true proposition" does allude to the words and has the same
sense as the proposition that does not do so.
Page 95
500. Might one speak of the sense of the words "that he will come"? For these words are precisely the Fregean
'assumption'. Well, couldn't I explain to someone what this verbal expression means? Yes I can, by explaining to
him, or shewing him, how it is employed.
Page 95
501. The difficulty becomes insurmountable if you think the sentence "I believe..." states something about the state
of my mind. If it were so, then Moore's Paradox would have to be reproducible if, instead of saying something about
the state of one's own mind, one were making some statement about the state of one's own brain. But the point is
that no assertion about the state of my (or anyone's) brain is equivalent to the assertion which I believe--for example,
"He will come".
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Page 96
502. But now, we do nevertheless take the assertion "He believes p" as a statement about his state; from this indeed
there results his way of going on in given circumstances. Then is there no first person present corresponding to
such an ascription? But then, may I not ascribe a state to myself now in which such-and-such linguistic and other
reactions are probable? It is like this, at any rate, when I say "I'm very irritable at present". Similarly I might also say
"I believe any bad news very readily at present".
Page 96
503. Now would a proposition ascribing to myself--or to my brain such a condition that I reply "Yes" to the
question "Will he come?" and also exhibit such-an-such other reactions--would such a proposition amount to the
assertion "He will come"?
Here one might ask "How do you imagine I have been instructed about this state of mind?--By experience,
say? Do I then want to predict from experience that I will now always answer such a question like this, etc.?" If
that's how it is, and I make the statement "I believe he will come" in this sense, and I add "and he isn't going to
come", then that is a contradiction only to the same extent as "I'm incapable of pronouncing any word with four
syllables"; or "I can't speak a word of English".
If this latter is a kind of contradiction; still the assumption "Suppose I couldn't speak a word of English" is
not.
Page 96
504. That he believes such-and-such, we gather from observation of his person, but he does not make the statement
"I believe..." on grounds of observation of himself. And that is why "I believe p" may be equivalent to the assertion
of "p". And the question "Is it so?" to "I'd like to know if it is so."
Page 96
505. "This face has a quite particular character--"really means: much could be said about it. When does one say this?
What justifies one in it? Is it a particular experience? Does one already know what one will say: has one already said
it silently? Isn't it a situation like: "Now I know how to go on!"
Page 96
506. We all know the process of sudden change of aspect;--but what if someone asked: "Does A have the aspect a
continuously before his eyes--when, that is, no change of aspect has taken place?" May not
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the aspect become, so to speak, fresher, or more indefinite?--And how queer it is that I ask this!
Page 97
507. There is such a thing as the flaring up of an aspect. In the same way as one may play something with more
intense and with less intense expression. With stronger emphasis of the rhythm and the structure, or less strong.
Page 97
508. Seeing, hearing this as a variant of that. Here there is the moment at which I think of B at the sight of A, where
this seeing is, so to speak, acute, and then again the time in which it is chronic.
Page 97
509. Not to explain, but to accept the psychological phenomenon--that is what is difficult.
Page 97
510. "F" as a variation of different figures.
If I imagine that the paradigm, as a variant on which I see the object, is somehow present in my mind as I
see, then it might after all be present now more clearly, now less, and it might disappear altogether.
Page 97
511. Imagine two people: one has learnt "F" like this in his youth: " "--the other, as we do: " ". If now both
read the word "Figure",--must I say, have I reason to say, that they each see the "F" differently? Obviously not. And
yet might it not still happen, that the one, on hearing how the other learned to write and read this letter, says: "I've
never seen it like that, but always like this"?
And further, there will probably be situations in which I shall explain what one of these people does or says
like this: "The thing is, he regards this letter as a variant of...."
Page 97
512. This is certain, that one may say: "I've never seen this in that way before." Here there is no doubt about the
"never".--But if you say "I have always seen this like that", this "always" is not equally certain. And there is of
course nothing at all remarkable about this, if instead of "seen" one says "taken".
Page 97
513. Suppose you knew that the sign was a combination of a with a .--This recalls the
dream phenomenon, which, in
Page Break 98
telling a dream, one describes with the words: "and I knew that...". And it also has some similarity to what is called
"hallucination".
Page 98
514. It is as if when I see the written character there were a paradigm, a pattern, present in my mind. But what sort of
pattern?? What does it look like? Surely at any rate not like the character itself!--Well, like the character seen in this
way then? But seen in what way? What notation can I use for the aspect? Well, what notation do we use; how do
we communicate about it? I say, for example: "The sign, as I see it, looks to the right." I might positively speak of a
kind of visual centre of gravity,--might say: The centre of gravity of the sign F is here. Can I explain what I mean by
this? No.--But I can compare this reaction of mine with other people's reactions.
Page 98
515. Am I continually conscious that the edges of my visual field are blurred? Ought I to say: "Hardly ever", or
"never"?
Page 98
516. In a different thought-space--one would like to say--the thing looks different.
Page 98
517. In music a variation on a theme could be imagined, which, phrased a bit differently, say, can be conceived as a
completely different kind of variation of the theme. (In rhythm there are such ambiguities.) Indeed what I mean is
probably to be found absolutely always, when a repetition makes the theme appear in a quite different light.
Page 98
518. No aspect that is not (also) conception.
Page 98
519. Suppose someone said to me: "Something has changed about the picture now--I can't put it in any other
way--although the shape is the same as before. I can only say: before, it was a kind of , now it's a kind of
." If he were to say that, might I not all the same be suspicious and doubt that he had always, uninterruptedly, seen
the figure in that way, and not merely never conceived it otherwise?
Page 98
520. Imagine that a child, when it has learnt the letter "R", were to say to us "I always see it as an 'R'". What could
that tell us??--For that matter, even if it were to say "I always see it as a 'P' with a skew support" that would only tell
us: the child conceives it so, that's how it explains the letter to itself, and such like. Only if it were to speak of a
change of the aspect should we say: now it is that phenomenon....
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Page 99
521. If someone says "I always see it this way", he must say how. Suppose he did this by tracing the lines of the
figure in a definite order or in a particular rhythm. That would be something like his telling us "I always follow the
figure with my eyes in this way". And here it might of course be that his memory deceives him.
Page 99
522. If he says "I see the figure (now) like this" and follows it in a definite way this need not be so much a
description as, so to speak, the seeing itself. But if he says "I have always seen it like that" this means that he has
never seen it differently, and here he may be deceiving himself.
Page 99
523. No, I didn't have the paradigm continuously before my mind--but when I describe the change of aspect, I use
the paradigms in describing it.
Page 99
524. "I've always seen it this way"--here one really means to say: "I have always conceived it this way, and this
change of aspect has never taken place."
Page 99
525. "I've never seen it this way, always that." Only this doesn't make a proposition by itself. It lacks a field.
Page 99
526. "I have always seen it with this face." But you still have to say what face. And as soon as you add that, it's no
longer as if you had always done it.
"I have always seen this letter as having a peevish face." Here one can ask: "Are you sure it was always?"
That is to say: did the peevishness always strike you?
Page 99
527. And what about something's 'striking' one? Does that take place in a moment, or does it last?
Page 99
528. "When I look at him, I always see his father's face." Always?--But surely not just momentarily! This aspect may
endure.
Page 99
529. Imagine its being said: "Now I always see it in this context."Page
99
530. Absolute and relative pitch: Here is something similar: I hear the transition from one note to the other. But after
a short time I can no longer recognize a note as the higher or lower of those two. And it
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doesn't even have to make sense to speak of any such "recognition"; when, that is, there is no criterion for correct
recognition.
Page 100
531. It is almost as if the 'seeing of the sign in this context' were an echo of a thought. [Cf. P.I. p. 212b.]
Page 100
532. To say of a real face, or of a face in a painting: "I've always seen it as a face" would be queer; but not "It has
always been a face for me, and I have never seen it as something else".
Page 100
533. If, e.g., for once I see the as a T with an additional stroke, it is as if the grouping changed. But if I am
asked: "So formerly you always saw this figure with the grouping of an ?" I could not say it was so.
Page 100
534. If someone says: "I am talking of a visual phenomenon, in which the visual picture, that is its organization, does
change, although shapes and colours remain the same"--then I may answer him: "I know what you are talking about:
I too should like to say what you say."--So I am not saying "Yes, the phenomenon we are both talking about is
actually a change of organization..." but rather "Yes, this talk of the change of organization etc. is an expression of
the experience which I mean too".
Page 100
535. "The organization of the visual image changes."--"Yes, that's what I'd like to say too."
This is analogous to the case of someone saying "Everything around me strikes me as unreal"--and someone
else replies: "Yes, I know this phenomenon. That's just how I'd put it myself."
Page 100
536. "The organization of the visual image changes" has not the same kind of application as: "The organization of
this company is changing." Here I can describe how it is, if the organization of our company changes.
Page 100
537. "It never occurred to me that one can see the figure this way": does it follow that it did occur to me, or that I
knew, that one could see it the way I always have seen it?
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Page 101
538. I hear a note--so don't I hear how loud it is?--Is it correct to say: if I hear the note, I must be conscious of its
degree of loudness? It's different if its strength alters.
Page 101
539. At first sight it would appear to be like this: Someone notices that one can see an also as a T with a little
appendix; he says "Now I see it as a T etc., now as an F". From this it seems to follow that he sees it the second time
as he always saw it before his discovery.--And so, that if it makes sense to say "Now I see it as an F again", it also
made sense to say, before the change of aspect, "I always see the letter as an F".
Page 101
540. If I had always heard a sentence in one and the same intonation (and often heard it) would it be right to say that
I must, of course, have been conscious of the intonation? If that just means that I have heard it in this intonation and
also pronounce it accordingly--then I am conscious of the intonation. But I need not know that there is such a thing
as an 'intonation'; the intonation need never have struck me, I need never have hearkened to it.
The concept intonation may be quite unknown to me. The 'separation' of intonation from sentence need not
have been effected for me.
I have not learned any language-game with the word "intonation"!
Page 101
541. For when a child learns its letters, it doesn't learn to see them this way and not otherwise. Am I to say, now,
that at the change of aspect the man realizes that he has always seen a letter, say an R, in the same way?--Well, it
might be so, but it isn't so. No, that's not what we say. Rather, when someone says something like: for him the
letter... has always had such-and-such a face, he would admit that in many cases he has not 'thought' of a face when
he saw the letter.
Page 101
542. Am I now to say: a 'kind of seeing' is associated with a letter for us? Certainly not; unless it means something
like: a face gets associated with a letter.
Page 101
543. Think of the concept "style of handwriting" one may say "That's an interesting style that the letter... is written
in"--but does everyone who has learnt how to write a letter of the alphabet understand what "style" means? I mean:
Can someone understand the style
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of the letter S who doesn't know at all that there is such a thing as different styles for a letter?--Or am I merely
playing with words here?
You just must not have too narrow a concept of 'experiencing'.
Ask yourself, e.g.: The man who has never had other examples before him--can he perceive a pronunciation
as vulgar?
Page 102
544. "I find this handwriting unattractive." If someone has only just learnt to read and write, can he find a
handwriting 'unattractive'?--It may perhaps in some sense put him off. It makes sense to say that someone finds a
handwriting unattractive, only if he is already capable of forming all sorts of thoughts about a handwriting.
Page 102
545. Would it be imaginable, given two identical bits of a piece of music, to have directions placed above them,
bidding us hear it like this the first time, and like this the second, without this exerting any influence on the
performance? The piece would perhaps be written for a chiming clock and the two bits would be meant to be played
equally loud and in the same tempo--only taken differently each time.
And, even if a composer has never yet written such a direction, might not a critic write it? Would not such a
direction be comparable to a title to Programme music ("Dance of the Peasants")?
Page 102
546. Only of course, if I say to someone "Hear it like this", he must now be able to say: "Yes, now I understand it;
now it really makes sense!" (Something must click.)
Page 102
547. What concept have we of sameness, of identity? You are familiar with the uses of the word "same" when what
is in question is same colours, same sounds, same shapes, same lengths, same feelings, and now you decide whether
this case and that should be included in this family or not.
Page 102
548. What is it that is repulsive in the idea that we study the use of a word, point to mistakes in the description of this
use and so on? First and foremost one asks oneself: How could that be so important to us? It depends on whether
what one calls a 'wrong description' is a description that does not accord with established usage--or one which does
not accord with the practice of the person giving the description. Only in the second case does a philosophical
conflict arise.
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Page 103
549. Less repulsive is the idea that we form a wrong picture for ourselves, say of thinking. For here one says to
oneself: at least we have to do with thinking, not with the word "thinking".
So we form a wrong picture of thinking.--But of what do we form a wrong picture; how do I know, e.g., that
you are forming a wrong picture of that, of which I too am forming a wrong picture?
Let us suppose that our picture of thinking was a human being, leaning his head on his hand while he talks to
himself. Our question is not "Is that a correct picture?" but "How is this picture employed as a picture of thinking?"
Say, not: "We have formed a wrong picture of thinking"--but: "We don't know our way about in the use of
our picture, or of our pictures." And hence we don't know our way about in the use of our word.
Page 103
550. Very well,--but this word is surely interesting to us only insofar as it actually possesses for us a quite particular
use, and so already relates to a particular phenomenon!--That's true. And that means: our concern is not with
improving grammatical conventions.--But what does it mean to say "We all know what phenomenon the word
'thinking' refers" to? Doesn't it simply mean: we can all play the language-game with the word "think"? Only it
produces unclarity to call thinking a 'phenomenon', and further unclarity to say "we form a wrong picture of this
phenomenon". (One might really rather say "a wrong concept".)
Page 103
551. If we are dealing with the use of the word "five", then we are dealing in a certain sense with what "corresponds'
to the word; only this way of speaking is primitive, it presupposes a primitive conception of the use of a word.
Page 103
552. A 'language game': We get someone to choose an aroma according to a drawing, e.g. the aroma of coffee. We
say to him "Coffee smells like this: " and now we tell him to bring the liquid that smells like that--Now I will
assume that he would actually bring the right one. So I would have a means of imparting orders to a human being by
graphical means. ((Connexion with the nature of a rule, a technique, of mathematics,--that of the real numbers for
example.)) This also hangs together with this: ("The mother hen 'calls' the chickens to her").
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Page 104
553. "One can't describe the aroma of coffee." But couldn't one imagine being able to do so? And what does one
have to imagine for this?
If someone says: "One can't describe the aroma," one may ask him: "What means of description do you want
to use? What elements?"
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554. We are not prepared for the task of describing the use of the word "think", for example. (And why should we
have been? What use is such a description?)
And the naïve conception that one forms of it does not correspond to the reality at all. We expect a smooth
regular contour and get to see a ragged one. Here one might really say that we had formed a wrong picture. It is like
this: suppose there were a substantive, let's say "giant", used to express all that we say by means of the word "big".
The picture that would come to our minds in connexion with the word "giant" would be that of a giant. And now
suppose that our queer employment of the word "big", with this picture before our eyes, had to be described. [Cf. Z
111.]
Page 104
555. Macaulay says that the art of fiction is an "imitative art", and naturally gets straight away into the greatest
difficulties with this concept. He wants to give a description: but any picture that suggests itself to him is
inappropriate, however right it seems at first sight; and however queer it seems that one should be unable to describe
what one so exactly understands.
Here one tells oneself: "It must be like this!--even if I cannot immediately get rid of all the objections."
Page 104
556. It is very easy to imagine someone knowing his way about a city quite accurately, i.e. he finds the shortest way
from one part of the city to another quite surely--and yet that he should be perfectly incapable of drawing a map of
the city. That, as soon as he tries, he only produces something completely wrong. (Our concept of 'instinct'.) [Z,
121.]
Page 104
557. Above all, someone attempting the description lacks any system. The systems that occur to him are inadequate,
and he seems suddenly to find himself in a wilderness instead of in the well laid out garden that he knew so well.
Rules occur to him, no doubt, but the reality shews nothing but exceptions.
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Page 105
558. And the rules of the foreground make it impossible for us to recognize the rules in the background. For when
we keep the background together with the foreground, we see only jarring exceptions, in other words irregularity.
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559. Do we say that anyone who talks sense is thinking? E.g. the builder in language-game (2)?†1 Might we not
imagine the building and the calling out of the words etc. in a surrounding in which we should not connect them up
with any thinking?
For "thinking" is akin to "considering". [Cf. Z, 98.]
Page 105
560. "Carrying out a multiplication mechanically" (whether on paper or in the head): that is something we do say-but
"considering something mechanically": for us, that contains a contradiction.
Page 105
561. The expression, the behaviour, of considering. Of what do we say: It is considering something? Of a human
being, sometimes of a beast. (Not of a tree or a stone.) One sign of considering is hesitating in what you do (Köhler).
(Not just any hesitation.)
Page 105
562. Think of the 'considering' in 'trying'. In 'investigating'; in the expression of astonishment; of failure and of
success.
Page 105
563. What a lot of things a man must do in order for us to say he thinks!
Page 105
564. He cannot know whether I am thinking, but I know it. What do I know? That what I am doing now is thinking?
And what do I compare it with in order to know that? And may I not be mistaken about it? So all that is left is: I
know that I am doing what I am doing.
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565. But it surely makes sense to say. "He does not know what I thought, for I did not tell him!"
Is a thought also 'private' in the case where I utter it out loud in talking to myself, if no one hears me?
"My thoughts are known to myself alone."
But what that means is, roughly: "I can describe them, can express them if I want to."
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566. "Only I know my thoughts."--How do you know that? Experience did not teach you it.--What do you tell us by
saying so? You must be expressing yourself badly.
"Not so! I am now thinking something to myself; tell me what it is!" So was it after all an empirical
proposition? No; for, if I were to tell you what you are thinking to yourself, I would only be guessing it. How is it to
be decided whether I have guessed right? By your word, and by certain circumstances: So I am comparing this
language-game with another one, in which the means of deciding (verification) look different.
Page 106
567. "Here I cannot...."--Well, where can I? In another game. (Here--that is in tennis--I cannot shoot the ball into
goal.)
Page 106
568. But isn't there a connexion between the grammatical 'privacy' of thoughts and the fact that we generally cannot
guess the thoughts of someone else before he utters them? But there is such a thing as guessing thoughts in the
sense that someone says to me: "I know what you have just thought" (or "What you just thought of") and I have to
admit that he has guessed my thoughts right. But in fact this happens very seldom. I often sit without talking for
several minutes in my class, and thoughts go through my head; but surely none of my audience could guess what I
have been thinking to myself. Yet it would also be possible that someone should guess them and write them down
just as if I had uttered them out loud. And if he shewed me what he had written, I should have to say "Yes, I thought
just that to myself."--And here, e.g., this question would be undecidable: whether I am not making a mistake;
whether I really thought that, or, influenced by his writing, I am firmly imagining myself to have thought precisely
that.
And the word "undecidable" belongs to the description of the language-game.
Page 106
569. And wouldn't this too be conceivable: I tell someone "You have just thought... to yourself"--He denies it. But I
stick to my assertion, and in the end he says: "I believe you are right; I must have thought that to myself; my
memory must be deceiving me."
And now imagine this being a quite ordinary episode!
Page 106
570. "Thoughts and feelings are private" means roughly the same as "There is pretending", or "One can hide one's
thoughts and feelings; can even lie and dissimulate". And the question is, what is the import of this "There is..." and
"One can".
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Page 107
571. Under what circumstances, on what occasions, then, does one say: "Only I know my thoughts"?--When one
might also have said: "I am not going to tell you my thoughts" or "I am keeping my thoughts secret", or "You people
could not guess my thoughts".
Page 107
572. Of what does one say that one is acquainted with it? and to what extent am I acquainted with my thoughts?
Don't we say that one is acquainted with what one can give a correct description of? And can one say that of
one's own thoughts?
If someone wants to call the words the "description" of the thought instead of the "expression" of the
thought, let him ask himself how anyone learns to describe a table and how he learns to describe his own thoughts.
And that only means: let him look and see how one judges the description of a table as right or wrong, and how the
description of thoughts: so let him keep in view these language-games in all their situations.
Page 107
573. "But the fact is that a human being knows only his own thoughts." ("But the fact is that only I know of my own
thinking.")
"And I don't either," one might say.
Page 107
574. "Nature has given it to man to be able to think in secret." Imagine its being said; "Nature has given it to man to
be able to talk audibly, but also to be able to talk inaudibly, within his mind." So, that means, he can do the same
thing in two ways. (As if he could digest visibly and also digest invisibly.) Only with speaking within one's mind the
speaking is hidden better than any process within one's body can possibly be.--But how would it be if I were to
speak, and everyone else were deaf? Wouldn't my speaking be equally well hidden?
Page 107
"It all goes on in the deepest secrecy of the mind."
Page 107
575. If someone says to me what he has thought--has he really said: what he thought? Would not the actual mental
event have to remain undescribed?--Was it not the secret thing,--of which I give another a mere picture in speech?
Page 107
576. If I say what I think to someone,--do I know my thought here better than my words represent it? Is it as if I
were acquainted with a body and shewed the other a mere photograph?
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Page 108
577. "Man has the gift of speaking with himself in total seclusion; in a seclusion far more complete than that of a
hermit." How do I know that N. has this gift?--Because he says so and is trustworthy?--
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And yet we do say: "I'd like to know what he is thinking to himself now," quite as we might say: "I'd like to
know what he's writing in his notebook now." Indeed, one might say that and so to speak see it as obvious that he is
thinking to himself what he enters in his notebook.
Page 108
578. If there were people who could regularly read a man's thoughts say by observation of his larynx--would they
too be inclined to speak of the total solitude of the spirit within itself?--Or: Would they too be inclined to use that
picture of 'total seclusion'?
Page 108
579. "I'd like to know what he's thinking of." But now ask yourself this--apparently irrelevant--question: "Why does
what is going on in him, in his mind, interest me at all, supposing that something is going on?" (The devil take what's
going on inside him!)
Page 108
580. In philosophy, the comparison of thinking to a process that goes on in secret is a misleading one.
As misleading as, e.g., the comparison of searching for the appropriate expression to the efforts of someone
who is trying to make an exact copy of a line that only he can see.
Page 108
581. What confuses us is that knowing the thoughts of another from one angle is logically impossible, and from
another it is psychologically and physiologically impossible.
Page 108
582. Is it right to say: these two 'impossibilities' connect up with each other in such a way that the psychological
impossibility (here) supplies us with the picture that (then) becomes for us the mark of the concept 'thinking'?
Page 108
583. One cannot say: writing in one's notebook or speaking in monologue is 'like' silent thinking; but for certain
purposes the one process can replace the other (e.g. calculating in the head can replace calculating on paper).
Page 108
584. Might there be people who always mutter to themselves as they think, so that their thinking is accessible to
others?--"Yes, but we still could not know whether they don't think silently to themselves as
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well."--But then might it not be that it was just as senseless to suppose this as to suppose that these people's hairs
were thinking, or a stone was thinking? That is to say: if this were so, need it so much as occur to us that someone
thought, that he had thoughts, hidden in his mind?
Page 109
585. "I don't know what you are thinking to yourself. Say what you are thinking."--That means something like
"Talk!"
Page 109
586. Then is it misleading to speak of man's soul, or of his spirit? So little misleading, that it is quite intelligible if I
say "My soul is tired, not just my mind". But don't you at least say that everything that can be expressed by means
of the word "soul", can also be expressed somehow by means of words for the corporeal? I do not say that. But if it
were so--what would it amount to? For the words, and also what we point to in explaining them, are nothing but
instruments, and everything depends on their use.
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587. Our knowledge of different languages prevents us from really taking seriously the philosophies laid down in the
forms of each of them. But at the same time we are blind to our own strong prejudices for, as against, certain forms
of expression; to the fact that just this piling up of several languages results in a special picture. That, so to speak, it
is not optional for us which form we cover up with which.
Page 109
588. You must remember the possibility of a language-game of 'continuing a series of figures', in which no rule, no
expression of a rule is ever given, but the learning is done only by means of examples. So that the idea that each step
can be justified by a somewhat--a kind of model--in our mind would be entirely alien to these people. [Cf. Z 295.]
Page 109
589. Example of the names that have meaning only when accompanied by their bearers, i.e. that is the only way they
are used. So they serve merely to avoid continual pointing. The example that always comes to my mind is the
designation of lines, points, angles by A, B, C... a, b... etc., in geometrical figures.
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Page 110
590. In reading: seeing the picture of the word: "I saw the word fleetingly"--that is a special experience, it cannot be
portrayed on film.
Page 110
591. Imagine a mental illness, in which one can use and understand names only in the presence of their bearers. [Cf.
Z 714.]
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592. There could be a use made of signs, of such a kind that the signs became useless (that one perhaps destroyed
them) as soon as the bearers ceased to exist.
In this language-game the name would have to have the object as it were on a lead; and when the object
ceases to exist, one can throw away the name which has worked in connexion with it. [Cf. Z 715.]
Page 110
593. "I intend to go there": is the state of mind being described or voiced?--If one imagines a model of the mind,
then the sentence might be a description of the model in its present state. The human being looks at his mind and
says:... Is it a good model or a bad one?--How should that be decided? The question is: How would it be employed
as a sign?
Page 110
594. "I intend..." might be used as an assertion: "I am doing something that is in accordance with this intention", e.g.:
I am packing for the journey, getting myself ready for the journey in this way or that, by means of considerations or
actions. One might use a verb in that way. Perhaps corresponding to the expression: "I am acting with the intent...."
Page 110
595. Description of my state of mind: the alternation of fear and hope, e.g. "In the morning I was full of hope, and
then...". Anyone would call that a description. But it is characteristic of it, that this description could run parallel to a
description of my behaviour.
Page 110
596. Compare the expression of fear and hope with that of 'belief' that such-and-such will happen.--That is why hope
and fear are counted among the emotions; belief (or believing, however, is not.
Page 110
597. If I say: "The intention to do it grew stronger every hour," this will be called a description. But in that case so
surely will this as well: "I intended the whole time...."
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Now compare: "The whole time, I believed in the law of gravity" with "The whole time, I believed I heard a
low whisper." In the first case "believe" is used similarly to "know". ('Had anyone asked me, I would have said...') In
the second case we have activity, surmising, listening, doubt etc. And even if "believe" does not designate these
activities, still they are surely what makes us say that here we are describing a state of mind or a mental activity. We
may also put it like this: we form a picture of the man who believes the whole time that he is hearing a low rustle.
But not of the man who believes in the correctness of the law of gravity.
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598. I intend (it might be said) doesn't mean: "What I am at, is intending," or "I am engaged in intending" (as one
says, I am engaged in reading the newspaper). On the other hand: "I am engaged in planning my journey" etc.
We have not, but might have, a single verb (and perhaps it actually exists in some little-known language)
which expresses: "to act and think with such-and-such an intention."
Page 111
599. "I intend..." is never a description, but in certain circumstances a description can be derived from it.
Page 111
600. Talking to oneself. "What happens here?" Wrong question! It's not just that one can't say what happens--one
can't say either that one doesn't know what happens, and one can't even say that one only knows this and that about
it! But even this is wrong to say: It just is a specific process, which can't be described except in just these words. The
concepts 'description' and 'report'. One says: Someone reported that he had said to himself.... How far is that
comparable to the 'report' that he had, e.g. said...? Let us make ourselves realize that describing is a very special
language-game.--We have to dig around this hard substratum of our concepts.
Page 111
601. Concepts may mitigate or aggravate a mischief; favour it, or hamper it. [V.B., p. 108; C. & V., p. 55.]
Page 111
602. Quite right: one can't imagine any explanation of "red" or of "colour". Not, however, because what is
experienced is something specific, but rather because the language-game is so.
Page 111
603. "One can't explain what red is to anyone." But suppose one could--is it in that case not what we call "red"?
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Page 112
Let us imagine men who express an intermediate colour, between red and yellow, e.g., by means of a kind of
binary decimal fraction like this: R,LLRL and the like, where, e.g., yellow stands to the right, and red to the
left.--Already in their nursery schools these people learn how to describe shades of colour in this way: they learn
how to choose colours according to such descriptions, and they learn to mix them etc. They would stand to us
roughly in the relation of people with absolute pitch to people in whom this is wanting. They can do what we can't.
[Cf. Z 368.]
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604. And here we feel like saying: "Now, is that even imaginable? The behaviour, to be sure! But the inner process,
the colour experience, as well?" And it is difficult to see what one should say to such a question. If we had not yet
encountered people with absolute pitch, would the existence of such people strike us as very probable? [Cf. Z 369.]
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605. If someone were to say "Red is composite"--we should not be able to guess what he was alluding to, what he
will be trying to do with this sentence. But if he says: "This chair is composite" although we may not know what
kind of composition he is speaking of, still we can at once think of more than one sense for his assertion.
Now what kind of fact is this, that I am drawing attention to here?
Page 112
At any rate it is an important fact.--We are not familiar with any technique, to which that sentence might be
alluding. [Cf. Z. 338.]
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606. Here we describe a language-game that we cannot learn. [Z 339.]
Page 112
607. "In that case something quite different must be going on in him, something that we are not acquainted
with."--This shews us what we go by in determining whether something different from or the same as what goes on
in us is going on 'in someone else'. This shews us what we go by in judging of inner processes. [Cf. Z 340.]
Page 112
608. "Red is not composite!" And what is red?--Here we should like simply to point to something red; and we forget
that if that statement is to make sense we must be given more than the ostensive definition. We don't yet understand
at all what is the sense of a sentence of the form "X is not composite", if X is replaced by a word having the use of
our colour words.
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Page 113
609. It is a fact: "Red" does not get explained to anyone by means of words without reference to a sample of the
colour. Shouldn't that be important?
Page 113
610. "How could one explain red to someone, since it is after all a particular sense-impression, known only to him
who has it (or has had it)--and explaining can only mean: producing it in the other person."--
Page 113
611. "Someone who has absolute pitch must have a different experience of notes from what I have." And must
everyone who has absolute pitch have the same experience? And if not, then why must it be a different one from
mine?
Page 113
612. Imagine that, in order to explain 'red' to someone, we shew him a reddish dark brown, and say: "This colour
consists of yellow (we shew pure yellow) black (se[[sic]] shew it), and one more colour which is called 'red'."
Thereupon let him be competent to pick pure red out of a number of colour samples.
Page 113
613. And note this: one doesn't point to red, but to something red. That is of course to say: the concept 'red' is not
determined by pointing, and now it is possible not only to interpret "red" as, e.g. the name of a shape, but also as a
concept-word that comes much closer to a colour word than that.
Page 113
614. The employment of a word is not: to designate something.
Page 113
615. Can you imagine what a red-green colour-blind man sees? Can you paint a picture of the room as he sees it?
[Cf. Z 341.]
Page 113
616. "If someone saw everything only black, white and grey, he would have to be given something, in order to know
what red, green etc. are." And what would he have to be given? Well, the colours. And so, e.g. this and this and this.
(Imagine, e.g., that coloured patterns had to be introduced into his brain, in addition to merely grey and black ones.)
But would that have to happen as a means to the end of future action? Or are these patterns actually involved in this
action? Do I want to say; "Something would have to be given him, for it is clear that otherwise he could not..."--or:
His seeing behaviour includes new components?
Again what would we call an "explanation of seeing"? Is one to say: Well, you know what explanation
means otherwise; so apply this concept here too? [Cf. Z 342-3.]
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617. Can I say: "Look at it! Then you will see that it can't be explained."--Or: "Drink in the colour red, then you will
see that it can't be represented by means of anything else!"?--And if the other now agrees with me, does that shew
that he has drunk in the same as I?--And what is the significance of our inclination to say this? Red seems to be
there, isolated. Why? What is the value of this seeming, of this inclination? [Cf. Z 344.]
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618. Think of the sentence "Red is not a blended colour," and of its function.
For the language-game with colours is characterized by what we are able to do, and what we are not able to
do. [Cf. Z 345.]
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619. Red is something specific; but we don't see that when we look at something red. Rather we see the phenomena
that we limit by means of the language-game with the word "red".
Page 114
620. "Red is something specific," that must be as much as to say: "That is something specific" while pointing at
something red. But for that to be intelligible, one would already have to mean our concept 'red', the use of that
sample. [Cf. Z 333.]
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621. If you wonder at these things, wonder first at something else! Namely, at what is actually accomplished by
description and report. If you concentrate your wonderment on this, those other problems will shrink.
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622. Primary colours. Suppose the colours that we call blends were among other men to play the role of our primary
colours; should we say that their primary colours were, for example, this orange,--this bluish red, this bluish green,
etc.? Then does the proposition "red is a primary colour" come to this: red plays such-and-such a role among us; we
react to red, yellow etc. in this and that way?--Mostly one does not think so: that is to say, "Red is a pure colour" is a
proposition about the 'essence' of red, time doesn't come into it; one cannot imagine that this colour might not be
simple.
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623. The colour-circle: the equal distances of the primary colours are arbitrary. Indeed, the transitions would perhaps
make a more uniform impression on us, if, e.g., the pure blue were nearer to the pure green than to the pure red. It
would be very remarkable if the equality of the distances lay in the nature of the things.
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Page 115
624. "There's no such thing as a reddish green" is akin to the propositions that we use as axioms in mathematics. [Cf.
Z 346.]
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625. Men count and calculate: Describe what they do here. Ought this description to include sentences like this one:
"Now he understood how he had to continue the series"--or: "He is now able to do any arbitrary multiplication"?
And is this proposition to be counted in: "He saw the whole number-series in his mind's eye"?
Such sentences may occur in the description: but may we not require that their use be explained to us; so
that no false or irrelevant images sneak in on us?
Here the question arises, for whom we are giving the description. Of whom do we say, he is able to do any
arbitrary multiplications? How does one arrive at this concept at all? And for whom, under what circumstances, will
this description be important?
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626. 'Red a degenerate green.' When one sees a leaf turn from green to red, one says that the green is sickly and in
the red part is quite degenerate. When one sees the red colour, one always makes a face.
Mightn't red have been explained as the ultimate degeneration of green?
Page 115
627. "One cannot explain to anyone what red is!"--How does one arrive at this idea at all; on what occasion does
one say this?
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628. "Colours are something specific. Not to be explained by anything else." How is this instrument used?--Describe
the game with colours. The naming of colours, the comparison of colours, the production of colours, the connexion
between colour and light and illumination, the connexion of colour with the eye, of notes with the ear, and
innumerable other things. Won't what is 'specific' about colours come out in this? How does one shew someone a
colour; and how a note?
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629. When we talk to ourselves in thought: "Something happens; that's for sure." But the usefulness of these words
is in reality just as unclear as that of the special psychological propositions that we are trying to explain.
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630. Instead of the unanalysable, specific, indefinable: the fact that we act in such-and-such ways, e.g. punish certain
actions, establish the state of affair thus and so, give orders, render accounts, describe colours, take an interest in
others' feelings. What has to be accepted, the given--it might be said--are facts of living.†1 [Cf. P.I. p. 226d.]
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631. We judge the motive of an act by what its author tells us, by the report of eyewitnesses, by the preceding
history. That is how we judge the motives of a human being. But we do not find it particularly striking that there
should be such a thing as the 'judgment of motives'. That this is a quite peculiar language-game--that a table or a
stone don't have any motives. That, while there does exist such a question as: "Is that a reliable way of judging a
human being's motives?"--still we must already be familiar with what "judgment of motives" means at all. There
must already be a technique of which we are thinking' here, in order for us to be able to speak of an alteration of this
technique, which we characterize as a more reliable judgment of a motive. [Cf. P.I. p. 224j.]
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632. One judges the length of a rod, and one can seek and find a method of judging it more exactly, more correctly.
So--you say--if what we judge here is independent of the method of judging, one can't explain what length is by
means of the method of determining length. But anyone who thinks like this is making a mistake. What sort of
mistake?--How queer it would be to say: "The height of a Himalayan mountain depends on how one climbs it."
"Measuring the height more and more exactly"--one would like to compare this to approaching closer and closer to
an object. But it just is not clear in all cases what it means "to approach closer and closer to the length of a rod". And
one can't say: "You surely know what the length of a rod is; and you know what 'determining it' means; you
therefore know what it means 'to determine the length more and more exactly'".
Under some circumstances it is clear what it means to look for a more exact determination of the length of a
rod, and under some circumstances it is not clear and stands in need of a new determination. What "determining the
length" means is not learnt by learning what length is and what determining is; rather one learns the meaning of the
word "length", among other things, by learning what determining the length is: 'Refining the determination of length'
is a
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new technique, which modifies our concept of length. [Cf. P.I. p. 225a.]
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633. When one describes simple language-games in illustration, let's say, of what we call the 'motive' of an action,
then more involved cases keep on being held up before one, in order to shew that our theory doesn't yet correspond
to the facts. Whereas more involved cases are just more involved cases. For if what were in question were a theory,
it might indeed be said: It's no use looking at these special cases, they offer no explanation of the most important
cases. On the contrary, the simple language-games play a quite different role. They are poles of a description, not the
ground-floor of a theory.
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634. "How does it come about that it seems to me that this colour-impression that I am having now, is recognized by
me as the specific, the unanalysable?"--Ask instead how it comes about that we want to say this. And the answer to
that is not difficult to find. And isn't it a queer question: why it 'seems' to us as if.... For this very question
itselfinvolves a misunderstanding.
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635. Imagine you were supposed to describe how human beings learn to count (e.g. in the decimal system). You
describe what the teacher says and does, and how the pupil behaves in consequence. In what the teacher says and
does there occur, e.g., words and gestures that are supposed to encourage the pupil to continue a series; also
expressions like "He can count now". Now, ought the description that I give of the process of teaching and learning
to contain, besides what the teacher says, my own judgment too, that the pupil can count now, or that the pupil has
understood the system of numerals? If I do not put such a judgment into my description--is the description then
incomplete? And if I do put it in, am I going beyond mere description?--May I refrain from those judgments and
justify myself by saying: "That is all that happens!"? [Cf. Z 310.]
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636. Must I not rather ask: "What does the description do? What purpose does it serve?"--In another context we do
indeed know what is a complete and what an incomplete description. Ask yourself: How are these expressions
employed: "complete" and "incomplete descriptions"?
Reproducing a speech completely (or incompletely). Does giving the tone of voice, the play of expression,
the genuineness or
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ungenuineness of the feeling, the intentions of the speaker, the exertion of speaking--does all this belong to a
complete rendering? Whether this or that belongs to the complete description will depend on the purpose of the
description, will depend on what the recipient of the description does with it. [Cf. Z 311.]
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637. The expression "That is all that happens" sets a limit to what we call "happening." [Cf. Z 312.]
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638. My judgment "The pupil can count now" is given for certain purposes. He is thereupon given a job, say. If you
say "So this judment [[sic]] is not part of the description of learning, it is, rather, a prediction"--then I reply "You can
take it this way or that". You can say that you are describing the state of the pupil.--
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639. Imagine red regarded as the summit of all colours. The special role of the triad in our music. Our lack of
understanding of the old church modes.
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640. Under what circumstances would one say, these people conceive all colours as degrees of a single property?
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641. Can you imagine our regarding blue and red as the two outermost poles of deviation from purple? One might
then call red a very high, and blue a very low, purple.
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642. Or imagine a world in which colours almost always occurred in rainbow-like transitions. So that one looks at,
say, a green expanse, if it exceptionally does sometimes occur, as a modification of a rainbow.
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643. Can I say, however, that if these were the facts, men would have these concepts? Certainly not. But one can say
this: don't think that our concepts are the only possible or reasonable ones: if you imagine quite different facts from
those with which we are continually surrounded, then concepts different from ours will appear natural to you. [Cf.
P.I. p. 230b.]
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644. Don't believe that you have the concept of colour within you because, however you look, you look upon a
coloured object. (Any more than you have the concept of a negative number because you are in debt.) [Cf. Z 332.]
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645. Suppose we were acquainted with a people that had a quite different form of colour attributions from ours: we
mostly suppose it an easy thing would then be to teach these people our form of expression. And that, when they
are masters of both forms of expression, they will acknowledge the difference between them to be inessential. (The
gender of our nouns.†1) Is it so? Must it be so?
Let us imagine that people had two different simple names for two shades of blue, and that colours were very
different for them, which for us are not so. How would this get manifested? And let us also imagine the reverse: that
there is a people for whom red and blue are different only 'in degree', not 'completely different colours'. And what
would be the criterion for this?
We say that the same note recurs after every seven notes in the scale. What does it mean to say "We
experience it as the same"? Is our calling it the same only a linguistic accident?
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646. The feeble-minded are pictured in our imagination as degenerate, essentially incomplete, as it were in rags. Thus
as in a state of disorder, rather than more primitive order (which would be a far more fruitful way of looking at
them.) [Cf. Z 372.]
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647. Counting, calculating etc., in a closed system in the way a tune is a closed system. The people count with the
aid of the notes of a special tune; at the end of the tune the series of numbers comes to an end.--Am I to say: Of
course there are further numbers as well, only these people don't know them? Or am I to say: There is also another
way of counting--namely what we do--and this these people do not know (do not do).
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648. The concept of experience: Like that of happening, of process, of state, of something, of fact, of description and
of report. Here we think we are standing on the hard bedrock, deeper than any special methods and language-games.
But these extremely general terms have an extremely blurred meaning. They relate in practice to innumerable special
cases, but that does not make them any solider; no, rather it makes them more fluid.
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649. Calculating in the head is perhaps the one case in which there is a regular use made of imagination in everyday
life. That is why it is especially interesting.
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"But I know that something went on in me!" And what? Wasn't it, that you calculated in your head?--So after
all, calculating in the head is something specific!
Consider first: How does one use the description "He's calculating in his head", "I'm calculating in my head"
at all? The difficulty which one comes up against is a vagueness in the criteria for the occurrence of the mental
process. Could it be avoided?
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650. Can one imagine calculating in one's head?
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651. One may calculate perceptibly and one may calculate in one's head: could one also do something in one's head,
which one can not do perceptibly, for which there is no such thing as a perceptible equivalent?
How would it be if people had a name for calculating in the head, which did not classify it among activities,
and so a fortiori not among those of calculation? They designate it perhaps as a capacity. I assume that they use
radically different pictures from the one we use.
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652. But if now someone were to say: "so after all, all that happens is that he reacts, behaves, in such-and-such a
way,"--then here is a gross misunderstanding. For if someone gave the account: "I in some sense calculated the
result of the multiplication, without writing etc."--was he talking nonsense, or did he make a false report? It is a
different employment of language from that of a description of behaviour. But one might indeed ask: Wherein
resides the importance of this new employment of language? Wherein resides the importance, e.g. of expression of
intention?
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653. "How if the pictures that someone had in imagining things had the intensity, clarity, of, e.g. after-images; would
these be mental images, or would they be hallucinations--even if he is fully conscious of the unreality of what he
sees?" First of all: how do I know that he sees pictures with this clarity? Perhaps he says so. One difference would be
this, that his pictures are 'independent' of him. What does that mean? He couldn't use thoughts to dispel them. If,
e.g., I imagine the death of my friend, I may tell myself "Don't think about it, think of something else"; but that
wouldn't be said to me if I were seeing the event befor [[sic]] my eyes, e.g. on a film. Then I'd reply to someone who
in the assumed case said to me "Don't think about it": "Think about it or not, I'm seeing it."
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654. Take the use of the English words "this", "that", "these", "those", "will", "shall": it would be difficult to give
rules for their
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use. But it is possible to understand their use, so as to be inclined to say: "If one just has the right feeling for the
sense of these words, then one can also apply them." Thus one might ascribe a peculiar meaning even to these
words in the English language. Their use gets to be felt as if each had a single physiognomy.
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655. Calculating in one's head at the order to do so. Don't let the combination of familiar words prevent you from
investigating the language-game right from the bottom.
Remember that one teaches someone to calculate in his head by ordering him to calculate! But would it have
to be like that? Might it not be that in order to get him to calculate in his head, I mustn t say "Calculate", but rather:
"Do something else, only get the result" or "Shut your mouth and your eyes and keep still, and you will learn the
answer.
I want to say that one need not look at calculating in the head under the aspect of calculating, although it has
an essential tie-up with calculating.
Nor even under the aspect of 'doing'. For doing is something that one can give someone an exhibition of.
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656. I want to say: there is no need to interpret reactions different from ours, and hence perhaps favourable to
different conceptual structures, as consequences or expressions of (inner) processes which are of a different nature
from ours.
There is no need to say: what is in question here are different inner processes.
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657. We have, on the one hand his capacity to communicate steps of the calculation without doing any perceptible
calculating--on the other, the utterances which he is inclined to make; as for example this one: "I did the sum
inwardly." The phenomena of the first kind might bring us to offer the graphic description: It's as if he calculated
somehow and somewhere, and told us steps of this calculation. We may assume what he is inclined to say into our
language as one of its forms of expression; or again we may not. We might, e.g., say to him "You don't calculate
'inside'! You calculate unreally." And then that is what he says for the future.
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658. "But I do surely know that I actually calculate--even if not perceptibly to someone else!" One might take this as
a typical expression of someone who was mentally retarded.
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659. But if we dispose of the inner process in this way,--is the outer one now all that is left?--the language-game of
description of the outer process is not all that is left: no, there is also the one whose starting point is the expression.
Whatever way our expression may run; whatever the way, e.g., it relates to the 'outward' calculation.
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660. When a theme, a phrase, suddenly says something to you, you don't have to be able to explain it to yourself.
Suddenly this gesture too is accessible to you. [Cf. Z 158.]
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661. Comparison of bodily processes and states, like digestion, breathing etc. with mental ones, like thinking, feeling,
wanting, etc. What I want to stress is precisely the incomparability. Rather, I should like to say, the comparable
bodily states would be quickness of breath, irregularity of heart-beat, soundness of digestion and the like. And of
course all these things could be said to characterize the behaviour of the body.
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662. Imagine a tribe of people who do not say "he has pains", "we have pains", "the same is going on in him as in
me", "these people have the same mental experience" etc.; rather, though there is talk of a mind and of processes in
the mind, one says one knows absolutely nothing about whether two people, of whom we'd say they were in pain,
really have the same or something quite different; and so in this tribe it is said that the people have something
unknown; and now there follows in their way of expressing themselves some specification, which comes to the
same thing as our "They are having pains". Then these poeple[[sic]] are likewise not going to say: "When I believe
that someone is having pains, I am believing that some particular thing is going on within him" or the like.
But need one look at the matter at all in such a way that the signal of pain and the description of
pain-behaviour form a conceptual unit?
I want to ask: "What is the place here of the conceptual and what of the phenomenal?" Must language
contain an expression of pain? Imagine people with a manual language. Or people who don't speak but only write.
Would these have to possess the concept 'pain'?
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663. But is it easier to imagine people lacking our concept of pain than it is to imagine them not having our concept
of a physical body?
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664. It is an important fact that we assume it is always possible to teach our language to men who have a different
one. That is why we say that their concepts are the same as ours.
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665. "You start a sentence at whose far end is a verb; you surely aren't going to tell me you began to speak the
sentence without an inkling what the verb would be!"--And what does the inkling consist in? And suppose someone
really had no inkling of it and yet spoke German fluently! How will one find out whether he had this inkling?
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666. How far do we investigate the use of words?--Don't we also judge it? Don't we also say that this feature is
essential, that one inessential?
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667. Measuring with a yardstick can be described; how can it be given a foundation?
Is the concept 'pain' an instrument made by man; and what purpose does it serve?
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668. How can one order someone to mean such and such words like this? Apart from ordering him to use them like
this.
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669. Suppose you had to make a decision and the decision was made by pressing one of a number of buttons. The
decision that you make in doing this is signalized by a word which is written on the button. It is then, of course, a
matter of complete indifference what you experience when you see this word. If the word is, e.g. "fine", you can
mean it as adjective, substantive, or verb, without thereby altering the decision. And equally, when you pronounce
the word as a decision. At any rate, if someone else is awaiting the decision, it tells him the same thing.
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670. But how is it when the decision is susceptible of two interpretations, and the one who hears it now gives it one
of them? He may do this, either through his actions, or, so to speak, in thought. But if the decision did not have to be
acted on at once, he might also hear it and for the time being not interpret it at all. On the other hand he might give
an interpretation in answer to a question. This would be a provisional reaction.
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671. It is perfectly possible to pronounce words suitably to a particular situation, and hence with such-and-such a
meaning, but at the same time to think another interpretation. So that for me, unbeknownst to the other, the words
have a peculiar meaning.
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672. If asked, I shall perhaps explain this meaning, without this explanation's having come before my mind earlier.
So what had my state of mind, as I spoke the words with the double meaning, to do with the words of the
explanation? How far can these words correspond to it? Here there is obviously no such thing as the explanation's
fitting the phenomenon.
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673. One may also mean an expression in one way as one utters it and then at once afterwards, retrospectively, in
another.
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674. It feels to us as if different illustrations attached to the phrase in its two meanings, and as if one can now give an
illustration compounded out of the two of them, but then of course it wouldn't be either of the two that accorded
with the word or were usual for it.
Naturally, however, that does not mean that whenever one employs the phrase, one of the two illustrations
must be present. It only means that if we illustrate the word, one of the two pictures and not both belong to it.
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675. 'If you had asked me, this is the answer I'd have given you.' That signifies a state; but not an 'accompaniment' of
my words.
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676. Imagine that people had the custom of doodling while they spoke; why should what they produce in this way
while talking be less interesting than accompanying processes in their minds, and why should the interest of these be
of any different kind?
Why does one of these seem to give the words their peculiar life?
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677. According as he meant the word this way or that, he expressed the one intention or the other. Had the one
intention or the other. And one can't say more about the importance of his meaning it than that.
And here again it seems that what went on while he pronounced the individual word ("bank", for example) is
of less importance than what went on during, and before, the utterance of the whole
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sentence. As it were, the mood illustrated the whole sentence, not necessarily the single word. And yet, as at the
same time we must confess to ourselves, even the illustration does not have to be important. For why should so
much depend on it?
And how can it give the sentence a particular life, if language doesn't do so? How should it be less
ambiguous than the language of words?
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678. Now this is the decisive point: It is not only from the context that I can judge the meaning; it can be asked
about, and in giving the answer one does not derive the meaning from the context.
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679. Is it just a matter of course that someone who can use language is able to explain the words that he
understands, the words whose employment he understands? We should, of course, be very much astonished if
someone did indeed understand the word "bank", but could give no answer when we asked him "What is a bank?"
Isn't it one thing to understand the sentence "Let's walk in the sun for a while" -and another to know how to
explain the word "sun"?--But mustn't one who understands this sentence know, e.g. what the sun looks like? As one
who understands the sentence "I haven't any pain" must, e.g., know how one can give oneself pain and how
someone in pain behaves etc.--
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680. Further; if it is possible for the ambiguous word by frequent repetition to take on each 'meaning', why shouldn't
some men who pronounce it without any context ordinarily do so without any feeling of a meaning? Or why
shouldn't men pronounce a word in this way with a kind of fluctuating meaning, where no context fixes it?
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681. "But what do you do when you obey the order 'Say... and mean... by it'?"--You don't do something else. But
neither do you do anything specific.
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682. In any case this isn't a language-game that is very early learnt: pronouncing a word by itself in such and such a
meaning. The foundation is obviously that someone says he can pronounce the word... and mean one or the other of
its meanings as he does so. That's quite easy when the word has two meanings; but can you also say the word
"apple" and mean "table" by it?--Still, I might use a secret language, in which it has this meaning.
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683. "Give him this order, and mean... by it." "Tell him this and mean... by it." That would be a remarkable order,
which is not ordinarily given. Or I say to someone: "Deliver this message"--and ask him afterwards: "Did you also
mean it in such and such a way?"
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684. But is the past tense form then justified? Yes; for I make a contrast between changing one's mind and its staying
the same. I really want to know not merely what he means now, but also what he did mean.--One might perhaps ask:
"What do you mean? and have you changed your mind?" When the answer to this question is No, then what he
says now he also meant before.
I want to say: the criteria for the past happening here are different from what they are for the emergence of a
picture.
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685. Then how am I to describe this psychological phenomenon? Am I to say: one can mean a word in such and
such a way upon request? that one fancies one means it this way or that? Am I to say that the word "mean" is being
used here in a different sense; that one ought properly to have used a different word? Am I to propose such a
word?--Or is just this the phenomenon, that we use the word "mean" here, which we learnt for another purpose?
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686. Is it a very primitive language-game, in which one says: "At this word.... occurred to me?" [Cf. P.I. p. 218b.]
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687. Instead of "I mean this by the word" one might also say "The word stood for...". And then how can the word
have stood for this thing--and not for that, when I pronounced it? And yet that's just what it looks like.
So is this as it were an optical illusion? (Such as to make the word seem to mirror the object that is
correlated with it by the explanation.) And if it is an optical illusion, what do people lose who are unacquainted with
this illusion? They can't be losing very much.
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688. The peculiar experience of meaning is characteristic because we react with an explanation and use the past
tense: just as if we were explaining the meaning of a word for practical purposes. [Cf. Z 178.]
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689. The intention may have altered, and simultaneously with it an experience-content, but the intention was not an
experience.
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690. One of the principles of observation would surely have to be that I do not disturb the phenomenon that I
observe by my observation of it. That is to say, my observation must be usable, must be applicable to the cases in
which there is no observation.
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691. So isn't there any peculiar experience corresponding to the jump of: "Now I know!"? No--Imagine one who is
always going off with "Now I've got it!" when he hasn't got hold of anything;--what are we to say about him? What
experience did he have? It is not the peculiar content of experience at the jump that gives it its peculiar interest, and
when someone says that he understood everything in that moment, this is not the description of an
experience-content.--But why not?--I want to make a distinction between a statement like "At that moment I saw the
formula clear before me", and one like "At that moment I grasped the method". But not as if I wanted to
say--"because one can't grasp a method in a moment". One can, it happens very often.--I want to say: "'Now I
understand!' is a signal, not a description." And what is effected by my saying that? Well, it directs attention to the
origin of such a signal; there comes into the foreground the question "How does someone learn the words 'Now I
understand it!', and how, e.g., the description of a mental image?" For the word "signal" points towards a proceeding
that is being signalled. [Cf. P.I. p. 218f.]
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692. It is of course the indisputability that favours the picture of something's being described here, something that
we see and the other does not, and that is near to us and always accessible, but for the other is hidden: hence
something that exists within us and which we become aware of by looking into ourselves. And psychology is now
the theory of this inner thing.
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693. So if I want to say that our 'utterances', with which psychology has to do, absolutely are not all descriptions of
experience-contents, I must say that what are called descriptions of experience-contents are only a small group of
these 'indisputable' utterances. But what grammatical features mark off this group?
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694. An experience-content is what can be produced in a picture, a picture in its subjective meaning, when its
purport is: "This I see whatever the object may be that produces the impression." For the experience-content is the
private object.--But how then can pain form
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such a content?--The sensation of temperature does so rather. And hearing is still closer akin to sight;--but also quite
different.
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695. It positively seems to us as if pain had a body, as if it were a thing, a body with shape and colour. Why? Has it
the shape of the part of the body that hurts? One would like, e.g., to say "I could describe the pain, if only I had the
requisite words and elementary concepts". One feels: all that is lacking is the necessary nomenclature (James.) As if
one could even paint the sensation, if only others would understand this language.--And one really can give a spatial
and temporal description of pain. [Cf. Z 482.]
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696. If the expression of pain were only a cry and its strength depended only on the available breath, but not on the
damage--should we in that case be inclined to regard pain as something observed?
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697. Why do you think that someone else's pain is similar to his visual sensation?--Or put it like this: why do we
group sight, hearing and the sensation of touch together? Because we 'get acquainted with the outer world' through
them? Pain certainly could be regarded as a kind of tactile sensation.
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698. But how about my idea that we don't actually judge the position and movement of our limbs by the feelings
that these movements give us? And why should we judge the qualities of the surfaces of bodies in this way, if that
cannot be said of our movements?--What is our criterion at all, for saying that our feeling tells us this?
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699. How does one judge whether fatigue (e.g.) is an indefinitely located bodily feeling?
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700. One would like to say that "I believe" can't properly be the present of "I believed". Or: one ought to be able to
use a verb in such a way that its past has the sense of "I believed", while its present has a sense different from that of
our "I believe". Or again: There ought to be a verb, whose third person in the present tense has the sense "he
believes", but whose first person has a sense different from that of "I believe".
But then ought there also to be a verb, whose first person says "I believe", but whose third person does not
say what we mean by "he believes"? So the third person would also have to be indisputable?
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701. What if someone were to say: "I know it won't rain, but I believe it will rain."
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702. What is common to sense-experiences?--The answer that they acquaint us with the outer world is partly wrong
and partly right. It is right inasmuch as it is supposed to point to a logical criterion. [Cf. Z 477.]
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703. Could an "I was lying" be imagined, which I inferred from observation of my own behaviour? Only in case
someone else cannot make the confession "I was lying" either.
Does "I was lying" describe an experience; or again "I made this statement in good faith"?--You need to
think of the fact that I don't only infer his good faith from such-and-such behaviour, but I also take his word for it,
which he does not base on self-observation.
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704. How is it that I cannot gather that I believe its going to rain from my own statement "It's going to rain"? Can I
then draw no interesting conclusions from the fact that I said this? If someone else says it, I conclude perhaps that
he will take an umbrella with him. Why not in my own case?
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Of course there is here the temptation to say: In my own case I don't need to draw this conclusion from my
words, because I can draw it from my mental state, from my belief itself.
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705. Why do I never conclude from my words to my probable actions? For the same reason as I don't conclude
from my facial expression to my probable behaviour,--for the interesting thing isn't that I don't conclude from my
expression of emotion to my emotion, but rather than I don't conclude from that expression to my later behaviour
either, as others do, who observe me. [Cf. Z 576.]
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706. If you philosophize, you often make the wrong, inappropriate, gesture in connexion with a verbal expression.
[Cf. Z 450.]
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707. If someone meets me in the street and asks "Where are you going?" and I reply "I don't know", he supposes
that I have no definite intention; not, that I don't know whether I shall be able to carry out my intention. (Hebel.†1)
[Cf. Z 582.]
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708. My super-ego might say of my ego: "It is raining, and the ego believes so," and might go on "So I shall
probably take an umbrella with me." And now how does this game go on?
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709. Consider also the statement: "I shall probably..."--where what follows is a voluntary action, not an involuntary
one.
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710. One says, e.g., "One feels conviction, one doesn't infer it from one's own words or tone of voice."
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But what does it mean to say one feels conviction? What is true is: one does not make an inference from
one's own words to one's own conviction; nor yet to the actions arising from the conviction. [Cf. P.I. p. 191g.]
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711. At the question "Why don't I infer my probable actions from my talk?" one might say that it is like this: as an
official in a ministry I don't infer the ministry's probable decisions from the official utterances, since of course I am
acquainted with the source, the genesis of these utterances and of the decisions.--This case would be comparable to
one in which I carry on conversations with myself, perhaps even in writing, which lead me to my utterances out loud
in conversation with other people; and now I say: I shall surely infer my future behaviour, not from these utterances,
but from the far more reliable documents of my inner life.
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712. After all I know that when I am angry, I simply don't need to learn this from my behaviour.--But do I draw a
conclusion from my anger to my probable action? One might also put the matter, I think, like this: my relation to my
actions is not one of observation.
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713. When I tell someone: "I know that you will do this" then the best means of making this prediction true is to
persuade the other into the action.
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714. If I tell someone "Now you will raise your hand", this prediction may be reason enough for its non-fulfilment;
unless it is an order which the other respects.
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715. "It is raining and I believe it is raining." Turning to the weather, I say that it is raining; then, turning to myself, I
say that I believe it.--
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But what do I do when I turn to myself, what do I observe? Suppose I say: "It's raining, and I believe it will soon
stop"--do I turn to myself at the second part of the statement?--Indeed, if I want to find out whether he believes that,
then I must turn to him, I must observe him. And if I wanted to find out what I believe by observation, I should have
to observe my actions, just as in the other case I have to observe his.
Now why don't I observe them? Don't they interest me? Apparently they do not. I hardly ever ask someone
else who has been observing me, whether he has the impression that I believe such and such: that is, in order in this
way to make inferences to my future actions. Now why should a really good observer not be able to predict my
behaviour from what I say and do better than I would be able to? But perhaps I shall then act as he foresees, only if
he makes no prediction of it to me.
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716. When I say "I remember, I believed...", don't ask yourself "What fact, what process is he remembering?" (that
has already been stipulated)--ask rather: "What is the purpose of this language, how is it being used?"
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717. The sense of sight, of hearing, of touch may fail, so that I am blind, deaf, etc.; but what would correspond to
that in the domain of intention?
And how would a man behave without imagination? Or one who is incapable of being sad or cheerful?
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718. "Hope is directed to the future"--but is there a feeling, identical with hope, but directed to the present or to the
past? The same mental movement, so to speak, but with a different object? Ask yourself: what should here be
regarded as the criterion of identity of the mental movements? Connected with this: "Is the jump of 'Now I know!' a
peculiar, specific jump?"
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719. Even if I were to admit that I know more of my own belief than of anyone else's I would then surely have to say
that what I can know about myself is what I know of someone else though there's much more of it.--So, even if it
would be redundant, I'd have to be able to apply a verb to myself, in the way I can apply the word "believe" to other
people. What prevents me?
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720. The concept of the world of consciousness. We people a space with impressions.
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721. "The ideal clock would always point to the time 'now'." This also connects up with the language which
describes only my impressions of the present moment. Akin is the primal utterance that is only an inarticulate
sound. (Driesch.) The ideal name, which the word "this" is.
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722. I should like to speak of a genealogical tree of psychological concepts. (Is there here a similarity to a
genealogical tree of different number concepts?)
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723. The difficulty of renouncing all theory: One has to regard what appears so obviously incomplete, as something
complete.
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724. Anxiety borrows the pictures of fear. "I have the feeling of impending doom."†1
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725. But what is the content, the content of consciousness, in anxiety? The question is wrongly framed.
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726. "A picture (mental image, memory image) of longing." One thinks that one has already done everything by
speaking of a 'picture'; for longing just is a content of consciousness, and its picture is something that is (very) like it,
even if it is less clear than the original.
And indeed one might very well say of someone who plays longing on the stage of a theatre, that he
expriences, [[sic]] or has, a picture of longing: for this is not given as an explanation of his proceedings, but as part
of a description. [Cf. Z 655.]
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727. But wouldn't I say that the actor does experience something like real longing? For isn't there something in what
James says: that the emotion consists in the bodily feelings, and hence can be at least partially reproduced by
voluntary movements?
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728. Is it so disagreeable, so sad, to draw down the corners of one's mouth, and so pleasant to pull them up? What is
it that is so frightful about fear? The trembling, the quick breathing, the feeling in the facial muscles?--When you say:
"This fear, this uncertainty, is frightful!"--might you go on "If only I didn't have this feeling in my stomach!"?
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Page 133
729. The expression "This anxiety is frightful!" is like a groan, a cry. Asked "Why do you cry out?", however--we
wouldn't point to the stomach or the chest etc. as in the case of pain; rather, perhaps, at what gives us our fear.
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730. When anxiety is frightful, and when in anxiety I am conscious of my breathing and of a tension in the muscles
of my face--does that mean that I find these feelings frightful? Might they not even signify an alleviation? [Cf. Z
499.]
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731. Compare fear and anxiety with care.
Page 133
732. And what sort of description is this: "Ewiges Dustere steigt herunter"†1....
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One might describe a pain like that; even paint it.
Page 133
733. Isn't the 'content' what one peoples the space of impressions with? What changes, what goes on, in space and
time. If, e.g., one talks to oneself, then it would be the imagined sounds (and perhaps the feeling in the larynx or
something like that).
Page 133
734. Is lying a particular experience? Well, can I say to someone "Now I am going to lie to you", and then do it? [Cf.
Z 189.]
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735. To what extent am I conscious of lying while I lie? Only inasmuch as I don't first realize it later, but all the same
I do know later that I lied. Consciousness of lying is a capacity. It is no contradiction of this that there are
characteristic feelings of lying [Cf. Z 190.]
Page 133
736. For knowledge is not translated into words when it is expressed. The words are not a translation of something
else that was there before. [Cf. Z 191.]
Page 133
737. One says "I notice in his tone of voice that he does not believe what he says". Or I suppose it, because he has
generally shewn himself unreliable. How can I apply this to myself? Can I, e.g., infer from my tone of voice, that I
probably shan't act in a way that fits my words? (And yet someone else does make that inference.) Or can I infer it
from my previous unreliability? Certainly for preference the latter. But I don't judge the tone of my voice at all as I
do that of
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someone else. Mind you, if I could see myself later, say in a talking film, I should perhaps say: "I don't quite trust
myself."
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738. But before all else: I seem to have a substitute for all such conjectures, one that is more certain than they are.
After all I know that I don't believe what I am saying, and that surely gives me the best of reasons--I should like to
say--for assuming that I shall not act accordingly. The point is, I have an intention concerning my actions.
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739. "But I know that I am lying! What need have I to draw conclusions from my tone of voice, etc.?"--But that's not
how it is. For the question is: Can I draw the same conclusions e.g. about the future, from that 'knowledge'; can I
make the same application of it, as of observed signs?
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740. And then, is the intention always quite clear? I say, e.g., "It will happen all right", half, because I believe it, half
because I want to comfort the others.
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741. Arrières pensées. "Mine I know, conjecture his". But what interest, what importance, have his arrières
pensées got for me? (Now, weigh the question.) And now the 'knowledge' of my own arrières pensées really does
play the same part for me, as the conjecture of his does for him.
Page 134
742. 'To judge others by oneself.' Of course there is such a thing. And I sometimes even infer that someone else is in
pain because he behaves as I do in this case.
Page 134
743. It might be said: If I tell you my arrières pensées, then I communicate to you just what you conjecture when
you conjecture these arrières pensées. That is: if you conjecture the arrières pensées as, so to speak, an active
principle, and I give expression to them, you can use my expression immediately in describing that agent. My
expression explains exactly what he wants to explain.
Page 134
744. "What should I draw conclusions from my own words to my behaviour for, when in any case I know what I
believe?" And what is the manifestation of my knowing what I believe? Is it not manifested precisely in this, that I
do not infer my behaviour from my words? That is the fact.
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Page 135
745. Why do I not infer from my tone of voice that I am not really convinced of what I am saying? or the whole of
what gets inferred from it?--And if it is answered "Because I know my own conviction"--the question is "How does
that come out?" Am I now to say: "In the fact that I have no doubt what it is?"
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746. The knowledge of metre. One who knows the metre, hears it differently.
Page 135
747. Thoughts can be care-laden, but not toothache-laden.
Page 135
748. I am now whistling a note, but I am also--now--whistling a tune.
Page 135
749. We do not say "I look furious; I only hope I shall commit no violence". But the question is not "How is it that
we don't?"
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750. The psychology of judgment: For judgment too has its psychology.
It is important that one can imagine every judgment beginning with the word "I". "I judge that...."
So is each judgment a judgment about the one who is judging? No, it is not, inasmuch as I don't want the
main consequences that are drawn to be ones about myself, I want them rather to be about the subject matter of the
judgment. If I say "It's raining," I don't in general want to be answered: "So that's how it seems to you." "We're
talking about the weather," I might say, "not about me."
Page 135
751. "But why is the use of the verb 'believe,' why is its grammar, put together in such a queer way?"
Well, it isn't queerly put together. It's only queer if one compares it with, say, the verb "eat".
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752. "Now what's he likely to do next," I say as I watch him. Do I watch myself and say "what am I likely to do
next"?
Page 135
753. Suppose I were moving about a room, and had a screen before my eyes on which I could see myself as an
observer would see me. As I move about the room I watch the screen continuously and observe my action.--What
would be the difference between these two cases: (a) I shall be influenced by what I see on the screen as I am by my
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normal seeing of my surroundings--(b) I move involuntarily and observe myself like a stranger.
But don't I feel my movements?--But isn't this feeling something that happens to me, like any other
sense-impression?
Page 136
754. Very well: the kindesthetic[[sic]] feeling is a different, a peculiar feeling.--But so is smell, so is hearing,
etc.--Why does that make such a difference?
The "feeling of innervation"--this expresses what one would like to say: that it is like an impulse. A feeling
like an impulse, though? What is an impulse, then? A physical picture. The picture of a push.
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755. What is the difference between these two things: following a line involuntarily--following a line on purpose?
What is the difference between these two things: tracing a line with care and great attention--and attentively
watching how my hand' follows a line? [Cf. Z 583.]
Page 136
756. Some differences are easy to give. One resides in foresight of what the hand will do. [Cf. Z 584.]
Page 136
757. Is "I am doing my utmost" the expression of an experience? One difference: One says "Do your utmost". [Cf. Z
581.]
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758. Does one say: "Give yourself this muscular sensation"? And why not?--"This"?--Which one?--But can't I give
myself a particular muscular sensation by moving my arm?--Try it. Move your arm--and ask yourself what feeling
you have produced in yourself.
If someone were to tell me: "Bend your arm and produce the characteristic sensation," and I bent my arm,
then I'd have to ask him: "Which sensation did you mean? A slight tension in the biceps, or a feeling in the skin on
the inside of the elbow joint?" Indeed, if someone ordered me to make a movement I might make it and then
describe the sensations that it produces, together with their peculiar place (which would hardly ever be the joint).
And I would often have to say that I felt nothing. Only one mustn't confound this with the statement that it was as if
there were no sensation in my arm.
Page 136
759. Are you reading this page voluntarily? And what does the act consist in?--One may read upon request and also
stop reading. One may also imagine something on request. E.g. one may recite a poem to
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oneself in the imagination, or do a sum. In this imagining, do you feel whether you are imagining something
voluntarily or involuntarily?
You can obey an order to summon up thoughts, to call up images--but also, and this is something else, you
can obey an order to think of something.
Page 137
760. Images, one might say, are voluntary, after-images involuntary.
Page 137
761. An involuntary movement is, for example, one that one can't prevent; or one that one doesn't know of; or one
that happens when one purposely relaxes one's muscles in order not to influence the movement.
Page 137
762. When, e.g., I see someone eating, do I ask myself whether he is doing it voluntarily or involuntarily? Perhaps it
is said that I assume it is happening voluntarily. What do I assume; that he feels it? And feels it in a particular way?
Page 137
763. How do I know whether the child eats, drinks, walks, etc. voluntarily or involuntarily? Do I ask the child what it
feels? No; eating, as anyone does eat, is voluntary.
Page 137
764. If someone were to tell us that with him eating was involuntary--what evidence would make one believe this?
[Cf. Z 578.]
Page 137
765. When I raise my hand suddenly to shield my eye--is the movement voluntary?--and do I feel it differently from
a voluntary movement?
Page 137
766. The concept of 'effort'. Do you feel the effort? Of course you feel it. But don't you also make it?--What are the
signs of effort? With a great effort, I lift a heavy weight. My muscles are tense, my face screwed up, my breath
short--but do I do all that; doesn't it merely happen to me? How would it be, if it merely happened to me? How
would that case differ from that of willing? Would I talk somehow differently? Would I say: "I don't know what's
happening to me: my muscles are tense, my face... etc. etc.?" And if I were to say: "Well, relax your muscles," he
would reply "I can't".
But suppose someone were to say to me: "I feel that I have to do whatever I do," and that at the same time
he behaved just like anyone else?
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767. Isn't saying that kinaesthetic sensation shews me what movement is made analogous to the opinion that some
characteristic of pain shews me its place?
Page 138
768. If someone wanted to represent pain by means of a colour-picture--would he put a local sign into the picture?
And why not?
Page 138
769. Is the sensation not the measure of the effort? That is to say, when I say "Now I'm pulling harder", do I notice
this by noting the degree of the sensation? And what is there to say against that? One tells someone "Exert yourself
more!"--not, so that he shall feel more, but so that he shall achieve more.
Page 138
770. Why does one feel as if one could describe, or paint, a tactile sensation (its content) but not a sensation of
motion or position?
Page 138
771. Can you say, e.g., that your sensation of position is weak or strong?
And your sensations when you move a limb may indeed be stronger or weaker (or absent), but that isn't a
perception of movement.
Page 138
772. Sensations of movement--these are sensations that are called into being by movement--they may, for example,
be pains.
How does one know that it isn't these sensations of movement that tell us what movements we are making?
What would be a sign of its being so?
Page 138
773. Isn't it an important fact that the theatre gives us exhibitions of colour and sound, but not of sensations of
touch? The use of smells and of sensations of temperature could be imagined, but not of sensations of touch.
Page 138
774. Someone, who is threading a needle with all the appearance of taking care, and tells us that he does it
involuntarily. How could he justify this statement?
Page 138
775. What one can know, one can be convinced of--and can also conjecture. (Grammatical remark.)
Page 138
776. Voluntary movements are certain movements with their normal surroundings of intention, learning, trying,
acting. Movements, of
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which it makes sense to say that they are sometimes voluntary, sometimes involuntary, are movements in a special
surrounding. [Cf. Z 577.]
Page 139
777. One category of psychological phenomena (facts) would be 'seeds'. But this word may just as easily be the
expression of a misunderstanding, like the phrase "experience of tendency" (James). The phrase "move in a
board-game", too, does not characterize a kind of movement.
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778. Translating from one language into another is a mathematical task and the translation of a lyrical poem (for
example) into a foreign language is quite analogous to a mathematical problem. For it is certainly possible to
formulate the problem "How is this joke (e.g.) to be translated by a joke in the other language?"--i.e. how is it to be
replaced; and the problem may also be solved; but there wasn't a method, a system, belonging to the solution of it.
[Cf. Z 698.]
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779. You know that you are lying; if you are lying, you know it. An inner voice, a feeling, tells me? Might this feeling
not deceive me?
Does a voice always tell me? And when does it speak? The whole time?--And how do I know I can trust it?
Page 139
780. A lie has a peculiar surrounding. There is in the first place a motive there. Something occasions it.
Page 139
781. The consciousness of lying is of the category of the consciousness of intention.
Page 139
782. Do not forget: sight, smell, taste etc. are sensations only because these concepts have something in
common--as one might take auger, chisel, axe, oxyacetylene torch together, because they have certain functions in
common.
Page 139
783. "A pain, a sound, a taste, a smell, has a particular colour." What does that mean? (Quality. Adjective.)
A colour may be greenish, or blueish--there is such a thing as a blending of colours; and in the same way too
a blending of smells, sounds, tastes; qualitative gradations. How does one distinguish qualitative from quantitative
gradations, I mean from gradations of 'intensity'?
Still bearable--no longer bearable, these, for example, are degrees of intensity. Suppose someone were to ask:
"How can I know that
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what I sense as different degrees of loudness, for example, is not sensed by someone else as different qualities,
comparable to different hues?"--Compare the reaction to an alteration in strength with that to an alteration of quality.
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784. I feel my arm and, oddly, I should like to say: I feel it in a particular position in space: as if, that is, my bodily
feeling were distributed in a space in the shape of an arm, so that in order to represent the matter, I would have to
represent the arm, say in plaster, in the right position. [Cf. Z 480.]
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785. Imagine that the point of a pencil were brought into contact with my skin at a certain place; I can say I feel
where it is: But do I feel where I feel it? "How do you know that the point is now touching your thigh?"--"I feel it."
By feeling the contact I know its place; but ought I therefore to speak of a feeling of place? And if there is no such
thing as a feeling of place, why must there be a feeling of position?
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786. It is odd. My lower arm is now lying horizontally and I should like to say I feel that; but not as if I had a feeling
that always goes with this position (as one would feel ischaemia or congestion)--rather as if the 'bodily feeling' of the
arm were arranged or distributed horizontally, as, e.g., a film of damp or of fine dust on the surface of my arm is
distributed like that in space. So it isn't really as if I felt the position of my arm, but rather as if I felt my arm, and the
feeling had such and such a position. But that only means: I simply know how it is lying--without knowing it
because.... As I also know where I feel pain--but don't know it because.... [Cf. Z 481.]
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787. Consider:--"It isn't true that what I believe is always false. For example, it's raining now, and I believe it."
One might say of him: He speaks like two people.
Page 140
788. Why do I have doubts about his intention, but not about mine? To what extent am I indubitably acquainted
with my intention? What, so to speak, is the use of my knowing my intention? That is, what is the use, the function,
of the expression of intention? That is, when is something an expression of intention? Well, when the act follows it,
when it is a prediction. I make the prediction, the same one as someone else makes from observation of my
behaviour, without this observation.
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Page 141
789. When dealing with a 'feeling of unreality', we are inclined to say: "All I know is that under certain circumstances
human beings often say that they felt everything around them was 'unreal'. Naturally we also know what use of this
word the people had learnt, and besides that something about their other utterances. More we do not know."--Why
don't we talk in the same way when what is in question is utterances expressive of pleasure, of conviction, of the
voluntariness and involuntariness of movements?
Page 141
790. What should I reply to someone who tells me that he feels the position and motions of his limbs, that a feeling
tells him their posture and movement? Am I to say he is lying, or that he is making a mistake, or am I to believe
him? I should like to ask him how a feeling tells him of, for instance, this posture. Or better: how he knows that a
feeling tells him this.
Page 141
791. (One says something ordinary,--with the wrong gesture.) [Cf. Z 451.]
Page 141
792. Remember again here the feeling which occurs without justification and to all appearances without any ground,
that a particular district must lie in that direction. If this feeling weren't for the most part deceptive, one would speak
here of a kind of knowing by feeling. And the sources of this feeling can only be conjectured, or established by
experience.
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793. The most important thing here is this: there is a difference; one notices the difference, 'which is a difference of
category'--without being able to say what it consists in. That is the case in which one usually says one knows the
difference by introspection. [Cf. Z 86.]
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794. And it sounds too much like an appeal to introspection, if I wanted to say "Test yourself, now--see whether you
really determine the position of your limbs by feelings in them."--And it would even be wrong, for then the question
is: If someone did that, how would it come out that he did? For suppose after self-examination he were to assure me
that it was so, or that it was not so,--how do I know whether I have the right to trust him; I mean, whether he has
even understood me right? Or again: how do I test whether I understand him?
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Page 142
795. Someone tells me: "I don't know how I move my fingers, but I know when I am spreading them out by the
feeling in the skin between the fingers." Here one would have to ask: Then can't you carry out the order: "Spread out
your fingers" straight off with your eyes shut?
Page 142
796. We feel our movements. Yes, we really feel them; the sensation is not like a sensation of taste or of temperature,
it is, rather, like a sensation of touch: the sensation when skin and muscles are squeezed, pulled, displaced. [Cf. Z
479.]
Page 142
797. How can I use the guidance of my feeling of movement when I make movements? For, how, before the
movement has begun, can I select from all the muscles the ones that are going to give me the right feeling of
movement? If there is a problem: "How do I know, when I don't see the movement, that, and to what extent, it has
taken place?"--why then is there no problem: "How do I know at all how to accomplish, say, a movement I've been
ordered to make"? (Russell once made a wrong observation about this.)
Page 142
798. I may, e.g., say that I now know that my finger is bent, but that I have no feeling of any kind in it; at any rate
none that I associate peculiarly with this position. Thus if I were to be asked: "Do you feel something, of which you
want to say that you wouldn't feel it in the straightened-out position; or is there some feeling missing, which would
be present in the other position?"--I should have to answer No.
Page 142
799. "Is pleasure a sensation?" (I. A. Richards.) That means something like: Is pleasure something like a note or a
smell?--But is a note then something like a smell? To what extent?
Page 142
800. Someone who asks whether pleasure is a sensation probably does not distinguish between ground and cause,
for otherwise it would occur to him that one has pleasure in something, which does not mean that this something
causes a sensation in us. [Cf. Z 507.]
Page 142
801. But pleasure does at any rate go with a facial expression, and, though we don't see this on ourselves, still we do
feel it.
And just try to reflect on something very sad with an expression of radiant joy. [Cf. Z 508.]
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Page 143
802. It is of course possible that the glands of a sad man secrete differently from those of a cheerful one; also, that
this secretion is the, or a, cause of sadness. But does it follow from this that sadness is a sensation brought about by
this secretion? [Cf. Z 509.]
Page 143
803. But here the thought is: "You surely feel the sadness--so you must feel it somewhere; otherwise it would be a
chimera. But if that is what you want to think, just recall the difference between sight and pain. I feel pain in my
hand--and colour in my eye? As we here want to employ a schema instead of simply noting what is really common,
we make a wrongly simplified picture of our conceptual world. It is as if we were to say that all the plants in the
garden had flowers, all had petals--fruits--seeds. [Cf. Z 510.]
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804. A smell may be extremely pleasant. Is what is pleasant about it only a sensation? In that case the sensation of
pleasantness would accompany the smell. But how would it relate to the smell? Of course, the expression of the
pleasantness is similar in kind to the expression of a sensation, in particular to that of pain. But joy has no place;
there are joyful thoughts, but not toothache-ish ones.
But--one would like to say--whether joy is a sensation, or what it is, is something one has to notice when one
has it!--(And why especially when one has it, and not when one doesn't have it?) Do you also notice the nature of
one, when you are eating one apple, and the nature of zero when you are eating none?
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805. Voluntariness hangs together with intentionalness. And therefore with decision as well. One does not decide on
an attack of angina and then have it.
Page 143
806. One brings on a sneeze in oneself or a fit of coughing, but not a voluntary movement. And the will does not
bring on sneezing, nor yet walking. [Cf. Z 579.]
Page 143
807. Sensation--that is what one takes to be immediately given and concrete, what one only needs to look at in order
to know it; it is that which is really there. (The thing, not its emissary.)
Page 143
808. "I know whether I am talking in accordance with my conviction or contrary to it." So the conviction is what is
important. In the background of my utterances. What a strong picture. One might paint conviction and speech
("from the depths of his heart"). And yet how little that picture shews!
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Page 144
809. "The smell is marvellous!" Is there any doubt that it is the smell that is marvellous?
So is it a property of the smell?--Why not? It is a property of ten to be divisible by two, and also to be the
number of my fingers.
But there might be a language in which people merely shut their eyes and said "Ah, this smell'", [[sic '?]] and
there is no subject-predicate sentence that is equivalent to the exclamation. That just is a 'specific' reaction. [Cf. Z
551.]
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810. Is that of which he says he has it, and of which I say I have it, without our inferring this from any
observation--is it the same as what we derive from observation of someone else and from the expression of his
conviction? [Cf. Z 574.]
Page 144
811. Can one say: I infer that he will behave as he intends to behave? [Cf. Z 575.]
Page 144
812. I make inferences to the consequences of his conviction from the expression of his conviction; but not to the
consequences of my conviction from the expression of it.
Page 144
813. Imagine an observer who, as it were automatically, says what he is observing. Of course he hears himself talk,
but, so to speak, he takes no notice of that. He sees that the enemy is approaching and reports it, describes it, but like
a machine. What would that be like? Well, he does not act according to his observation. Of him, one might say that
he speaks what he sees, but that he does not believe it. It does not, so to speak, get inside him.
Page 144
814. Why don't I make inferences from my own words to a condition from which words and actions take their rise?
In the first place, I do not make inferences from my words to my probable actions.
Page 144
815. Asked: "Are you going to do such-and-such?" I consider grounds for and against.
Page 144
816. But consider this: After all I sometimes take someone else's word,--so I would surely at least sometimes have to
take my own word too, that I have such and such a conviction. But when I report my observation in a
quasi-automatic fashion, then this report has nothing at all to do with my conviction. On the other hand I might
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have confidence in myself, or in my observing self, just as another person does. So I might say "I say 'It's raining', so
it will presumably be true" Or: "The observer in me says 'Its[[sic]] raining', and I am inclined to believe him."--For
isn't this--or something like this--how it is, when a man says that God has spoken to him or through his mouth?
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817. The important insight is that there is a language-game in which I produce information automatically,
information which can be treated by other people quite as they treat non-automatic information--only here there will
be no question of any 'lying'--information which I myself may receive like that of a third person. The 'automatic'
statement, report etc. might also be called an 'oracle'.--But of course that means that the oracle must not avail itself of
the words "I believe...".
Page 145
818. Where is it said in logic that an assertion cannot be made in a trance?
Page 145
819. "If I look outside, I see that it's raining; if I look within myself, I see that I believe it." And what is one supposed
to do with this information?
Page 145
820. "Suppose that it's raining and I don't believe it"--when I assert what is supposed in this supposition,--then, so to
speak, my personality splits in two.
"Then my personality splits in two" means: Then I no longer play the ordinary language-game, but some
different one.
Page 145
821. "The words 'It's raining' are written in his soul"--this is to mean as much as (i.e. to be replaceable by) "He
believes that it is raining". "The words 'It's raining' are written in my soul"--means, say, "I can't get rid of the belief
that...", "The idea that... has taken possession of me."
For consider this fact: the words "I believe it is raining" and "It'll be raining" may say the same: inasmuch,
that is, as in some contexts it makes no difference which of the two sentences we use. (And rid yourself of the idea
that one of them is accompanied by a different mental process from the other.) The two sentences may say the same
thing, although there is an "I believe..." and "He believes..." etc. that corresponds to the first and not to the second.
For a different concept is used in the construction of the first. That is: in order to say that perhaps it is raining we do
not need the concept 'believe', although we may use it for that purpose. The concept of a
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proposition's being 'written on the soul' is now a third concept, which partly coincides with the others in its
application, and partly not.
I want to say that in order to construct the assertion "It'll..." there is no need of the 'queer' concept 'believe',
although it may be used for that purpose.
Page 146
822. Think also of this: 'It'll be raining and it is raining' doesn't mean anything, and no more does "It'll be raining and
it isn't'. By contrast one can say 'It seems to be raining and it is raining' and also 'It seems to be raining... and it isn't
raining'. And 'It seems to be raining' may have the same sense as 'It'll be raining'.
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823. How do I know that I am in the state of believing:...? Do I look into myself? Is it even any use to observe
myself? Well, I might perhaps ask myself "How much would I bet in this case?"
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824. Pretence. Simulating pain. It doesn't consist merely in giving expressions of pain when one has no pain. There
must be a motive present for the simulation, hence a situation which is not quite simple to describe. Making oneself
out sick and weak, in order then to attack those who help one.--"But there is surely an inner difference there!"
Naturally; only here "inner" is a dangerous metaphor. But the 'proof' that an inner difference is present is the very
fact that I can confess that I was simulating. I confess an intention. Does it 'follow' from this that the intention was
something inner?
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825. The 'actual infinite' is a 'mere word'. It would be better to say: for the time being this expression merely
produces a picture--which so far hangs in the air; you still owe us its application. [Cf. Z 274.]
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826. An infinitely long row of marbles, an infinitely long rod. Imagine these coming into a kind of fairy-tale. What
application, even though a fictitious one, might be made of this concept? Let us ask now, not: can there be such a
thing? But: What do we imagine? So give free rein to your imagination! you can have things now just as you
choose. You only need to say how you want them. So just make a verbal picture, illustrate it as you choose, by
drawings, comparisons, etc. Thus you can--as it were--prepare a blueprint.--And now there remains the question
how one can work from it. [Cf. Z 275.]
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827. "But how can the human spirit fly ahead of reality, and even think the unverifiable?"--Why should we not
speak the unverifiable? For we ourselves made it unverifiable.
A false appearance is produced? And how can it be so much as appear like that? For don't you want to say
that this like that too isn't even a description? Well then, in that case it is, not a false appearance, but one that robs us
of our orientation. So that we just ask: How is it possible? [Cf. Z 259.]
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828. As soon as the word was spoken, I wished I had not said it.--How did my wish relate to the word that was
spoken?
I felt that the word was inappropriate, as soon as I had spoken it. But the signs I remember were only like
slight indications. Minutiae, from which I might have been able perhaps to guess the intention, the wish.
There are occasions of shame--situations--and ashamed behaviour. As there are occasions of expectation and
the behaviour of expectancy.
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829. When a cat lies in wait by a mouse-hole--do I assume that it is thinking about the mouse?
When a robber waits for his victim--is it part of this, for him to be thinking of that person? Must he be
considering this and that as he waits? Compare one who is doing such a thing for the first time, with one who has
already done it countless times. (Reading.)
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830. There might be a verb with the meaning: to formulate one's intention in words or other signs, out loud or in
one's thoughts. This verb would not be equivalent in meaning to our 'intend'.
There might be a verb with the meaning: to act according to intention; and this would also not mean the same
as "to intend".
Yet another might mean: to brood over an intention; or, to turn it over and over in one's head. [Cf. Z 49.]
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831. When I make my coffee, I intend to drink it. If I were making it without this intention--would some
accompaniment of my action then have to be lacking? Does something go on during the normal doing of a thing,
which characterizes it as doing with this intention? But if someone were to ask me whether I intend to drink, and I
replied "Yes, of course"--would I be saying something about my present state?
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This is how I react in this case; and that can be gathered from my reaction.
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832. A belief, a wish, a fear, a hope, a fondness; each can be called a state of a man; we can count on this state in our
behaviour towards this man, we can infer his reactions from his state.
And if someone says: "All this time I had the belief...", "All my life I have wished..." etc; he is reporting a
state, an attitude.--But if he says "I believe he's coming" (or simply "Here he comes") or "I wish you'd come" (or
"Please come"), then he is acting, he is speaking, according to that condition, not reporting that it is to be found in
him.
But if that were right, then there ought to be a present form of that report, and hence on the one hand the
utterance "I believe...", and on the other the report "I am in the state of belief...". And similarly for wish, intention,
fear etc.
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833. Someone might relate: "I remember my state in those years exactly; whenever I was asked... I replied...; that
was my attitude."
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834. There is a reaction of loathing, in myself and in others; there are also feelings of loathing. And loathing, fear,
affection, etc., resemble one another in this; but not hope, belief, etc.
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835. Grief incessantly rehearses the sad thoughts. A thought may be sad, loathsome, enchanting etc.; but how does
the expression shew that it is this thought, to which we have these reactions? How does one drive a thought away?
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836. Ought I to call the whole field of the psychological that of 'experience'? And so all psychological verbs 'verbs of
experience'. ('Concepts of experience.') Their characteristic is this, that their third person but not their first person is
stated on grounds of observation. That observation is observation of behaviour. A subclass of concepts of
experience is formed by the 'concepts of undergoing'.†1
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'Undergoings' have duration and a course; they may run on uniformly or non-uniformly. They have intensity. They
are not characters of thought. Images are undergoings. A subclass of 'undergoings' are 'impressions'. Impressions
have spatial and temporal relations to one another. There are blend-impressions. E.g. blends of smells, colours,
sounds. 'Emotions' are 'experiences' but not 'undergoings'. (Examples: sadness, joy, grief, delight.) And one might
distinguish between 'directed emotions' and 'undirected emotions'. An emotion has duration; it has no place; it has
characteristic 'undergoings' and thoughts; it has a characteristic expression which one would use in miming it.
Talking under particular circumstances, and whatever else corresponds to that, is thinking. Emotions colour
thoughts. One subclass of 'experiences' is the forms of 'conviction'. (Belief, certainty, doubt, etc.) Their expression is
an expression of thoughts. They are not 'colourings' of thoughts. The directed emotions might also be called
"attitudes". Surprise and fright are attitudes too, and so are admiration and enjoyment.
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837. But where does memory belong, and where attention? One can remember a situation or occurrence at a
moment. To that extent, then, the concept of memory is like that of instantaneous understanding or decision.
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838. My own behaviour is sometimes, but rarely the object of my own observation. And this hangs together with
the fact that I intend my behaviour. Even if an actor observes the expressions of his own face in a glass, or the
musician attends closely to every note in playing, and judges it, this happens after all so that he shall direct his action
accordingly. [Cf. Z 591.]
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839. What does it mean, e.g. to say that self-observation makes my acting, my movements, uncertain?
I cannot observe myself unobserved. And I do not observe myself for the same purpose as I observe
someone else. [Cf. Z 592.]
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840. When a child stamps its feet and howls with rage, who would say it was doing this involuntarily? And why?
Why is it assumed to be doing this not involuntarily? What are the tokens of voluntary action? Are there such
tokens? What, then, are the tokens of involuntary movement? They don't happen in obedience to orders, like
voluntary actions. There is "Come here!" "Go over there!" "Make this movement with your arm," but not "Have
your heart beat faster". [Cf. Z 593.]
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841. There is a peculiar combined play of movements, words, facial expressions etc., as of expressions of reluctance,
or of readiness, which characterize the voluntary movements of the normal human being. When one calls the child,
it doesn't come automatically: there is, for example, the gesture "I don't want to!" There is coming gladly, the
decision to come, running away with signs of fright, the effects of being addressed, all the reactions of play, the signs
of consideration and its effects. [Cf. Z 594.]
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842. A tune went through my head. Was it voluntary, or involuntary? It would be an answer to say: I could also have
had it being sung to me inwardly. And how do I know that? Well, because I can ordinarily interrupt myself if I want
to.
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843. How could I prove to myself that I can move my arm voluntarily? Say by telling myself "Now I'm going to
move it" and now it moves? Or shall I say: "Simply by moving it"? But how do I know that I did it, and it didn't
move just by accident? Do I in the end feel it after all? And what if my memory of earlier feelings deceived me, and
these weren't at all the right feelings to decide the matter?! (And which are the right ones?) And then how does
someone else know whether I moved my arm voluntarily? Perhaps I'll tell him: "Tell me to make whatever
movement you like, and I'll do it in order to convince you." And what do you feel in your arm? "Well the usual
feelings." There is nothing unusual about the feelings, the arm is not e.g. without feeling (as if it had 'gone to sleep').
[Cf. Z 595.]
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844. A movement of my body, of which I don't know that it is taking place or has taken place, will be called
involuntary.--But how is it when I merely try to lift a weight, and so there isn't a movement? What would it be like if
someone involuntarily strained to lift a weight? Under what circumstances would this behaviour be called
"involuntary"? [Cf. Z 596.]
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845. Can't rest be just as voluntary as motion? Can't abstention from movement be voluntary? What better argument
against a feeling of innervation? [Cf. Z 597.]
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846. "That glance was not intended" sometimes means: "I didn't know that I gave such a look" or "I didn't mean
anything by it".
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847. It ought not to strike us as so much a matter of course that memory shews us the past inner, as well as the past
outer, process.
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848. Imagination is voluntary, memory involuntary, but calling something to mind is voluntary.
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849. What a remarkable concept 'trying', 'attempting', is; how much one can 'try to do'! (To remember, to lift a
weight, to notice; to think of nothing.) But then one might also say: What a remarkable concept 'doing' is! What are
the kinship-relations between 'talking' and 'thinking', between 'talking' and 'talking to oneself'? (Compare the
kinship-relations between the kinds of numbers.) [Cf. Z 598.]
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850. One makes quite different inferences from involuntary movements and from voluntary ones: this characterizes
voluntary movement. [Cf. Z 599.]
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851. But how do I know that this movement was voluntary? I don't know it, I manifest it. [Cf. Z 600.]
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852. "I am tugging as hard as I can." How do I know that? Do my muscular sensations tell me so? The words are a
signal; and they have a function.
But am I experiencing nothing, then? Don't I experience something? something specific? A specific feeling
of effort and of inability to do more, of reaching the limit? Of course, but these expressions say no more than "I'm
tugging as hard as I can". [Cf. Z 601.]
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853. It is important, however, that there are all these paraphrases! That one can describe care with the words "Ewiges
Düstere steigt herunter"†1. I have perhaps never sufficiently stressed the importance of this paraphrasing.
Joy is represented by a countenance bathed in light, by rays streaming from it. Naturally that does not mean
that joy and light resemble one another; but joy it does not matter why--is associated with light. To be sure, it might
be that this association is taught the child when it learns to talk, that it is no more natural than the sound of
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the words themselves--enough that it exists. ("Beethoven" and Beethoven's works.) [Cf. Z 517.]
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854. Sorrow like the lead-gray sky?! And how can that be found out? By looking at a sorrowing man and at the sky?
Or does the sorrowing man say it? And is it then true only for his sorrow, or for the sorrow of anyone?
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855. But if someone now says that his sadness is like a grey cloud--, am I to believe it or not?--One might ask him
whether the two are alike in something, in a particular respect. (Like, e.g. two faces; or like a sudden strong pain and
a flare.) One may give relations--internal relations and connexions--between what one calls "intensities" for different
impressions.
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856. 'a is between b and c, and is closer to b than to c'--this is a characteristic relation between sensations of the same
kind. That is to say, there is, for example, a language-game with the order: "Produce a sensation between this one
and this one, and closer to the first than to the second." And also: "Name two sensations, such that this one comes
between them." [Cf. Z 360.]
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857. And here it is important that with grey one will get the answer "black and white", with purple, "blue and red",
with pink "red and white"; but with olive-green one will not get "red and green". [Cf. Z 361.]
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858. How does one realize that the expression of joy is not the expression of some bodily pain? (An important
question.)
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859. How does one know that the expression of enjoyment is not the expression of a sensation?
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860. To pronounce a figure this or that. Do you always pronounce the figure to be this or that while you see it? Of
course: if asked what this figure presents, I should always say "A rabbit"†1; but I am no more continuously
conscious of this, than of the fact that there is an actual table here. For if I always pronounce a picture a picture of
this object, then I also pronounce any object a thing with this particular use, etc.
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861. If someone notices for the first time that the picture is ambiguous, he might react, say with the exclamation:
"Ah! a rabbit!" etc.; but still, when he now goes on seeing the picture continuously in one aspect, he won't want to
keep on exclaiming "Ah! a...!"
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862. I want to say that the natural, primitive expression of the experience of an aspect would be such an
exclamation; it might also be a lighting up of the eyes. (Something strikes me!)
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863. When I say I see this figure continuously red, that means that the description, that it is red--the description in
words or by means of a picture--is continuously correct, without alteration; hence in contrast to that case, in which
the picture alters. For the temptation is to describe the aspect with the words "I see it like this" without pointing to
anything. And when one describes a face with its direction of glance as an arrow, one wants to say: "I see this → and
not this: ←".
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864. What corresponds to continuous seeing as →, is that this description, without any variation, is the right one and
that only means that the aspect did not change.
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865. Talk of hallucination!†1--What could there be queerer, than that this dot, the eye, seems to have a direction!
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866. When I think about the facial expression of this figure--what do I do, to be thinking about the expression ←
and not →?
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867. When I think about the facial expression of this figure, contemplate it, what do I do to be contemplating the
expression ← not →?
And this symbolism, I believe, has everything in it.
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868. It is as if one saw a picture: one time together with one group, and then another time with another one. "What
does this mean: It is as if one saw...?" It means something like: that process might be a representative of the actual
one, it would have the right 'multiplicity'.
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869. It is--contrary to Köhler--precisely a meaning that I see.
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870. It might be said that one experiences readiness for a particular group of thoughts. (The germ of them.)
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871. It is as if the picture came to rest in one position (or another). As if it could in fact fluctuate, and then come to
rest with a particular accentuation.
One says: "I see it now (or mostly) as this." It really feels to us as if the lines were now fitted together into
this and not the other shape. Or as if they were put into this and not the other mould.
And yet all our concern can only be this: to describe the actual expression of our experience, which I am
merely paraphrasing with all these pictures; to say what is essential to this expression.
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872. Could someone see the figure in this way or this way, if he could not advance from that to giving explanations,
etc.? Thus, could someone see it in such-and-such a way, if he didn't know how animals' heads looked, what an eye
is etc.? And by this of course I don't mean: "Would such a person be competent to do so, would he succeed?" But
rather: "Aren't these concepts requisite for this?"
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873. I see the picture of a horse: I know, not merely that it is a horse, but also that the horse is running. Thus I can
understand the picture, not just spatially, but I also know what the horse is now about to do. Imagine someone
seeing a picture of a cavalry charge but not knowing that the horses don't stay in their various places!
I am, however, not concerned with an explanation of this understanding, say by the assertion that someone
who looks at such a picture makes tiny running movements, or feels running innervations. What ground is there for
assumptions of this kind, except this one: it 'must' be like that?
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874. But suppose it is said: "One sees the painted horse running!" Here, however, I don't just mean to say "I know
that this represents a running horse". One is trying to say something else. Imagine that someone reacted to such a
picture by a movement of his hand and a shout of "Tally ho!". Doesn't that say roughly the same as: he sees the
horse running? He might also exclaim "Its[[sic]] running!" and that would not be the observation that it is running,
nor yet that it seems to be running. Just as one says "See how it runs!"--not in order to inform the other person;
rather this is a reaction in which people are in touch with one another.
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875. Understanding is like knowing how to go on, and so it is an ability: but "I understand", like "I can go on", is an
utterance, a signal.
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876. I can experience a word substantivally or adjectivally. Do I know whether everyone, or many, of those with
whom I talk, have these experiences? Would it be important for knowing what they mean?
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877. It hadn't occurred to me that the same contour occurred in both pictures, for in the one picture I took it like this,
→ in the other like this ←. Only in the course of consideration did I realize that the contour was the same--Is that a
proof that I saw something different each time?--It is important that the two aspects are incompatible with one
another.
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878. Is a facial expression something visual? I could imagine a picture, where the expression was ambiguous. And
which perhaps for that reason I should not recognize in a different surrounding. In that case I say something like:
"Ah well, the lines are the same; only here they look quite different."
And of course I do really see that the → picture and the ← picture are the same! I don't realize it only by
making measurements, say!
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879. I see two different visual objects, you say, which merely have something in common with one another, In
saying that you are only laying stress on some analogies at the expense of others. But now this emphasis needs to be
grammatically justified.
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880. How is it possible that the eye, this dot, looks in a direction?--"See, it is looking!" (And here one 'looks'
oneself.) But one doesn't say and do this continuously while one contemplates the picture. And now what is this
"See, it is looking!"--is it the expression of a sensation? [Cf. P.I. p. 205i.]
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881. I would never have thought of laying the two pictures one on the other like that, of comparing them in that
way. For they suggest a different method of comparison.
The ← picture hasn't even the slightest resemblance to the → picture, one would like to say--although they
are congruent.
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882. "Now I can go on!"--I see that that is the front of a head, that is a beak--this line is brow-like, this dot is
eye-like. But how can the visual impression of a line be brow-like? And what makes me say it is
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the visual impression that has this character?--Well, the fact that it isn't a thought, isn't an interpretation, that it has
duration like a visual impression.
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883. Humans have intentions: Let us try to describe this fact! What would such a description be like? For whom
would it be a description? Ask yourself this: what purpose is it supposed to serve?
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884. One can speak to oneself very 'clearly' in the imagination, while at the same time one is reproducing the
information of the speech by humming (with closed lips). Movements of the larynx help too. But the remarkable
thing about the latter case is that one hears the speech in the imagination, and does not merely feel as it were its
skeleton in the larynx. [Cf. P.I. p. 220e.]
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885. It is essential to 'imaging' that the concepts of sense-perception are used in the expression of it. (The sentence "I
hear and don't hear..." might be used as the expression of auditory imagination. A use for the form of contradiction.)
A principal mark that distinguishes image from sense-impression and from hallucination is that the one who has the
image does not behave as an observer in relation to the image, and so that the image is voluntary.
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886. Imagine a conversation, one in which one party is yourself, and imagine it in such a way that you yourself are
actually speaking in the imagination. What you yourself say, you will then probably feel in your body (in the larynx,
in the breast). This, however, only describes and does not define the activity of talking in the imagination.
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887. The feeling of the uncanny. How is it manifested? The duration of such a 'feeling'. What is it like, e.g., for it to
be interrupted? Would it be possible, for example, to have it and not have it every other second? Don't its marks
include a characteristic kind of course (beginning and ending), distinguishing it from, e.g., a sense perception?
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888. The way music speaks. Don't forget that even though a poem is framed in the language of information, it is not
employed in the language-game of information.
Might one not imagine someone who had never known music, and who came to us and heard someone
playing a reflective piece of
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Chopin, being convinced that this was a language and people were merely keeping the sense secret from him?
Verbal language contains a strong musical element. (A sigh, the modulation of tone for a question, for an
announcement, for longing; all the countless gestures in the vocal cadences.) [Cf. Z 160, 161.]
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889. "Don't look for anything behind the phenomena; they themselves are the theory." (Goethe.)
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890. I observe his face closely. Why? What does it tell me? Whether he is sad or cheerful, e.g., But why am I
interested in that? Well, if I get to know his mood, it is like when I have got to know the condition of a body (its
temperature, for example); I can draw various kinds of conclusion from this. And that is why I don't observe my
own face in the same case. If I observed myself, my face would no longer be a reliable index; and even if it were an
index for someone else, still I could not draw any conclusions from it.
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891. Being ashamed of a thought. Is one ashamed because one has pronounced such-and-such a sentence to oneself
in the imagination?
The thing is, language has a multiple root; it has, not a single root, but roots. [Cf. Z 656.]
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892. "At that moment the thought was before my mind."--And how?--"I had the picture."--So was the picture the
thought? No; for if I had merely communicated the picture to someone, he would not have got hold of the thought.
[Cf. Z 239.]
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893. The picture was the key. Or at any rate it seemed like the key. [Cf. Z 240.]
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894. How are visual impressions distinguished from auditory impressions?--Am I to reply "That can't be said; but
whoever sees and hears knows that they are totally different"? Could it be imagined that for some man one
particular visual impression was the same as one particular auditory impression? so that he could receive this
impression both through the eye and through the ear? Would this man point to a picture perhaps and strike a note
on the piano and say that these two were identical? And would we believe him? And why not? Would we believe
him when he said the 'affection of the soul' was the same in the two cases? And if we did believe it, how could we
make use of the fact?
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895. The genealogical tree of psychological phenomena: I strive, not for exactness, but for a view of the whole. [Cf.
Z 464.]
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896. What holds the bundle of 'sense-impressions' together is their mutual relationships. That which is 'red' is also
'sweet' and 'hard' and 'cold' and 'sounds' when one strikes it. In the original language-game with these words it isn't
"This looks red", but "This is red" (hard etc.) Our agreement is essential to the language-game. But with "pleasant",
"unpleasant", "beautiful", "ugly", it is otherwise.
Pain is in some ways analogous to the other sense-impressions, in some ways different. There is a facial
expression, there are exclamations and gestures of pain (as of joy), tokens of rejection, a reception that is
characteristic of pain, but none that is characteristic of the sensation red. Bitterness is akin to pain in this.
We could imagine there being a sense impression without any sense organ. Someone might hear, and so he
might learn pretty well all the language-games with the words for auditory impressions, without having ears, and
without knowing what he hears 'with'. For that one hears with the ears comes out relatively very seldom. It might be
that someone hears as we all do, and it is only later discovered that his ears are deaf.
The content of experience. One would like to say "I see red thus", "I hear the note that you strike thus", "I
feel pleasure thus", "I feel sorrow thus", or even "This is what one feels when one is sad, this, when one is glad", etc.
One would like to people a world, analogous to the physical one, with these thuses and thises. But this makes sense
only where there is a picture of what is experienced, to which one can point as one makes these statements.
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897. If only one person had, once, made a bodily movement--could the question exist, whether it was voluntary or
involuntary?
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898. "When I make an effort, I surely do something, I surely don't merely have a sensation." And it is so too; for one
tells someone "Make an effort!", and he may express the intention: "Now I'm going to make an effort" And when he
says "I can't go on!" that does not mean: "I can't endure the feeling--the pain, for example--in my limbs any
longer."--On the other hand one suffers with effort, as with pain. "I am utterly exhausted"--if someone said that, but
moved as briskly as ever, one would not understand him. [Cf. Z 589.]
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899. An aspect is subject to the will. If something appears blue to me, I cannot see it red, and it makes no sense to
say "See it red"; whereas it does make sense to say "See it as...". And that the aspect is voluntary (at least to a certain
extent) seems to be essential to it, as it is essential to imaging that it is voluntary. I mean: voluntariness seems to me
(but why?) not to be a mere addition; as if one were to say: "This movement can, as a matter of experience, also be
brought about in this way." That is to say: It is essential that one can say "Now see it like this" and "Form an image
of...". For this hangs together with the aspect's not 'teaching us something about the external world'. One may teach
the words "red" and "blue" by saying "This is red and not blue"; but one can't teach someone the meaning of
"figure" and "ground" by pointing to an ambiguous figure. [Cf. P.I. p. 213e.]
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900. We do not first become acquainted with images and only later learn to bend them to our will. And of course it is
anyway quite wrong to think that we have been directing them, so to speak, with our will. As if the will governed
them, as orders may govern men. As if, that is, the will were an influence, a force, or again: a primary action, which
then is the cause of the outward perceptible action.
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901. Is it right to say: what makes an action voluntary is the psychical phenomena in which it is embedded? (The
psychological surrounding.)
Are, e.g., my normal movements in walking "voluntary" in a non-potential sense?
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902. A child stamps its feet with rage: isn't that voluntary? And do I know anything about its sensations of
movement, when it is doing this? Stamping with rage is voluntary. Coming when one is called, in the normal
surroundings, is voluntary. Involuntary walking, going for a walk, eating, speaking, singing, would be walking,
eating, speaking etc. in an abnormal surrounding. E.g. when one is unconscious: if for the rest one is behaving like
someone in narcosis; or when the movement goes on and one doesn't know anything about it as soon as one shuts
one's eyes; or if one can't adjust the movement however much one wants to; etc.
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903. No supposition seems to me more natural than that there is no process in the brain correlated with associating
or with thinking; so that it would be impossible to read off thought-processes from
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brain-processes. I mean this: if I talk or write there is, I assume, a system of impulses going out from my brain and
correlated with my spoken or written thoughts. But why should the system continue further in the direction of the
centre? Why should this order not proceed, so to speak, out of chaos? The case would be like the following--certain
kinds of plants multiply by seed, so that a seed always produces a plant of the same kind as that from which it was
produced--but nothing in the seed corresponds to the plant which comes from it; so that it is impossible to infer the
properties or structure of the plant from those of the seed that it comes out of--this can only be done from the
history of the seed. So an organism might come into being even out of something quite amorphous, as it were
causelessly; and there is no reason why this should not really hold for our thoughts, and hence for our talking and
writing. [Cf. Z 608.]
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904. It is thus perfectly possible that certain psychological phenomena cannot be investigated physiologically,
because physiologically nothing corresponds to them. [Cf. Z 609.]
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905. I saw this man years ago: now I have seen him again, I recognize him, I remember his name. And why does
there have to be a cause of this remembering in my nervous system? Why must something or other, whatever it may
be, be stored-up there in any form? Why must a trace have been left behind? Why should there not be a
psychological regularity to which no physiological regularity corresponds? If this upsets our concepts of causality
then it is high time they were upset. [Cf. Z 610.]
Page 160
906. The prejudice in favour of psycho-physical parallalism [[sic]] is also a fruit of the primitive conception of
grammar. For when one admits a causality between psychological phenomena, which is not mediated
physiologically, one fancies that in doing so one is making an admission of the existence of a soul alongside the
body, a ghostly mental nature. [Cf. Z 611.]
Page 160
907. Must the verb "I believe" have a past tense form? Well, if instead of "I believe he's coming" we always said "He
could be coming" (or the like), but nevertheless said "I believed..."--in this way the verb "I believe" would have no
present. It is characteristic of the kind of way in which we are apt to regard language, that we believe that there must
after all in the last instance be uniformity,
Page Break 161
symmetry: instead of holding on the contrary that it doesn't have to exist.
Page 161
908. Imagine the following phenomenon. If I want someone to take note of a text that I recite to him, so that he can
repeat it to me later, I have to give him paper and pencil, while I am speaking he makes lines, marks, on the paper; if
he has to reproduce the text later he follows those marks with his eyes and recites the text. But I assume that what he
has jotted down is not writing, it is not connected by rules with the words of the text; yet without these jottings he is
unable to reproduce the text; and if anything in it is altered, if part of it is destroyed, he gets stuck in his 'reading' or
recites the text uncertainly or carelessly, or cannot find the words at all.--This can be imagined!--What I called
jottings would not be a rendering of the text, not a translation, so to speak, in another symbolism. The text would
not be stored up in the jottings. And why should it be stored up in our nervous system? [Cf. Z 612.]
Page 161
909. Why should not the initial and terminal states of a system be connected by a natural law, which does not cover
the intermediary state? (Only don't think of agency). [Cf. Z 613.]
Page 161
910. What is called an alteration in concepts is of course not merely an alteration in what one says, but also in what
one does.
Page 161
911. One sees the terminology, but fails to see the technique of applying it.
Page 161
912. One says: "He appears to be in frightful pain" even when one hasn't the faintest doubt, the faintest suspicion
that the appearance is deceptive. Now why doesn't one say "I appear to be in frightful pain" for this too must at the
very least make sense? I might say it at an audition; and equally "I appear to have the intention of..." etc. etc.
Everyone will say: "Naturally I don't say that; because I know whether I am in pain." It doesn't ordinarily interest me
to know whether I appear to be in pain; for the conclusions which I draw from this impression in the case of other
people, are ones I don't draw in my own case. I don't say "I'm groaning dreadfully, I must see a doctor", but I may
very well say "He's groaning dreadfully, he must...".
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Page 162
913. If this makes no sense: "I know that I am in pain"--and neither does "I feel my pains",--then neither does it
make sense to say: "I don't bother about my own groaning because I know that I am in pain"--or--"because I feel my
pains."
So much, however, is true: I don't bother about my groaning. [Cf. Z 538.]
Page 162
914. I infer from observation of his behaviour that he must go to the doctor; but I do not make this inference for
myself from observation of my behaviour. Or rather: I do that too sometimes, but not in analogous cases. [Cf. Z
539.]
Page 162
915. Here it is a help to remember that it is a primitive reaction to take care of, to treat, the place that hurts when
someone else is in pain, and not merely when one is so oneself--hence it is a primitive reaction to attend to the
pain-behaviour of another, as, also, not to attend to one's own pain-behaviour. [Cf. Z 540.]
Page 162
916. What, however, is the word "primitive" meant to say here? Presumably, that the mode of behaviour is
pre-linguistic: that a language-game is based on it: that it is the prototype of a mode of thought and not the result of
thought. [Cf. Z 541.]
Page 162
917. It can be called "putting the cart before the horse" to give an explanation like the following: we took care of the
other man, because going by analogy with our own case, we believed that he too had the experience of
pain.--Instead of saying: Learn from this particular chapter of our behaviour--from this language-game--what are the
functions of "analogy" and of believing" in it. [Cf. Z 542.]
Page 162
918. "How does it come about that I see the tree standing up straight even if I incline my head to one side, and so the
retinal image is that of an obliquely standing tree?" Well how does it come about that I speak of the tree as standing
up straight even in these circumstances?--"Well, I am conscious of the inclination of my head, and so I supply the
requisite correction in the way I take my visual impression."--But doesn't that mean confusing what is primary and
what is secondary? Imagine that we knew nothing at all of the inner structure of the eye--would this problem make
an appearance? We do not in truth supply any correction here--that explanation is gratuitous.
Well--but now that the structure of the eye is known--how does it
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come about that we act, react, in this way? But must there be a physiological explanation here? Why don't we just
leave explaining alone?--But you would never talk like that, if you were examining the behaviour of a
machine!--Well, who says that a living creature, an animal body, is a machine in this sense?--[Cf. Z 614.]
Page 163
919. One may note an alteration in a face and describe it by saying that the face assumed a harder expression--and
yet not be able to describe the alteration in spatial terms. This is enormously important.--Perhaps someone now
says: if you do that, you just aren't describing the alteration of the face, but only the effect on yourself; but then why
shouldn't a description using concepts of shape and colour be that too?
Page 163
920. One may also say: "He made this face" or "His face altered like this", imitating it--and again one can't describe
it in any other way. ((There just are many more language-games that are dreamt of in the philosophy of Carnap and
others.))
Page 163
921. Consciousness that... may disturb me in my work; knowledge can't.
Page 163
922. How do I know that a dog is hearing something continuously, is having a continuous visual impression, that it
feels joy, fear, pain?
What do I know of the 'experience contents' of a dog?
Page 163
923. Are the colours really brethren? Are they different only in colour, not also in kind? Are sight, hearing, taste
really brethren?
Don't look only for similarities in order to justify a concept, but also for connexions. The father transmits his
name to the son even if the latter is quite unlike him.
Page 163
924. Compare a dreadful fright and a sudden violent pain. It is the sensation of pain that is dreadful--but is it the
sensation of fright? When someone falls headlong in my presence,--is that merely the cause of an extremely
unpleasant sensation in me? And how can this question get answered? Does someone who reports the frightful
incident complain of the sensation, the catching of breath, etc.? If one
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wants to help someone get over the fright, does one treat the body? Doesn't one much more soothe him about the
event, the occasion?
Page 164
925. If someone imitates grief for himself in his study, he will indeed readily be conscious of the tensions on his
face. But really grieve, or follow a sorrowful action in a film, and ask yourself if you were conscious of your face.
[Cf. Z 503.]
Page 164
926. One tie-up between moods and sense-impressions is that we use the concepts of mood to describe
sense-impressions and images. We say of a musical theme, or a landscape, that it is sad, cheerful etc. But naturally it
is much more important that we use all the concepts of mood to describe human faces, actions, behaviour. [Cf. Z
505.]
Page 164
927. Consciousness in the face of another. Look into someone else's face and see the consciousness in it, and also a
particular shade of consciousness. You see on it, in it, joy, indifference, interest, excitement, dullness etc. The light
in the face of another.
Do you look within yourself, in order to recognize the fury in his face? It is there as clearly as in your own
breast.
(And what does one want to say? That someone else's face stimulates me to imitate it, and so that I feel small
movements and muscular tensions on my own part, and mean the sum of these? Nonsense! Nonsense,--for you are
making suppositions instead of just describing. If your head is haunted by explanations here, you will neglect to bear
in mind the facts which are most important.) [Cf. Z 220.]
Page 164
928. Knowledge, opinion, have no facial expression. There is a tone, a gesture of conviction all right, but only if
something is said in this tone, or with this gesture.
Page 164
929. "Consciousness is as clear in his face and behaviour, as in myself." [Cf. Z 221.]
Page 164
930. What would it mean for me to be wrong about his having a mind, having consciousness? And what would it
mean to say I was wrong and didn't have any myself? What would it mean to say "I am not conscious"?--But don't I
know that there is a consciousness in me?--Do I know it then, and yet the statement that it is so has no purpose?
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Page 165
And how remarkable that one can learn to make oneself understood to others in this matter! [Cf. Z 394.]
Page 165
931. A man can pretend to be unconscious; but conscious? [Cf. Z 395.]
Page 165
932. What would it be like for someone to tell me with complete seriousness that he (really) did not know whether
he was dreaming or awake?--
Is the following situation possible: Someone says "I believe I am now dreaming"; he actually wakes up soon
afterwards, remembers that utterance in his dream and says "So I was right!"--This narrative can surely only signify:
Someone dreamt that he had said he was dreaming.
Imagine an unconscious man (anaesthetized, say) were to say "I am conscious" should we say "He ought to
know"?
And if someone talked in his sleep and said "I am asleep"--should we say "He's quite right"?
Is someone speaking untruth if he says to me "I am not conscious"? (And truth, if he says it while
unconscious? And suppose a parrot says "I don't understand a word", or a gramophone: "I am only a machine"?)
[Cf. Z 396.]
Page 165
933. Suppose it were part of a day-dream I was having to say: "I am merely engaged in phantasy", would this be
true? Suppose I write such a phantasy or narrative, an imaginary dialogue, and in it I say "I am engaged in
phantasy"--but, when I write it down,--how does it come out that these words belong to the phantasy and that I have
not emerged from the phantasy?
Might it not actually happen that a dreamer, as it were emerging from the dream, said in his sleep "I am
dreaming"? It is quite imaginable there should be such a language-game.
This hangs together with the problem of 'meaning'. For I can write "I am healthy" in the dialogue of a play,
and so not mean it, although it is true. The words belong to this and not that language-game. [Cf. Z 397.]
Page 165
934. 'True' and 'false' in a dream. I dream that it is raining, and that I say "It is raining"--on the other hand: I dream
that I say "I am dreaming". [Cf. Z 398.]
Page 165
935. Has the verb "to dream" a present tense? How does a person learn to use this? [Cf. Z 399.]
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Page 166
936. One language-game analogous to a fragment of another. A space projected into bounded bits of a space. [Cf. Z
648.]
Page 166
937. Suppose I were to have an experience like waking up, were then to find myself in quite different surroundings,
with people who assure me that I have been asleep. Suppose further I insisted that I had not been dreaming, but
living in some way outside my sleeping body. What function has this assertion? [Cf. Z 400.]
Page 166
938. "'I have consciousness'--that is a statement about which no doubt is possible." Why should that not say the
same as: "'I have consciousness' is not a proposition"?
It might also be said: What's the harm if someone says that "I have consciousness" is a statement admitting
of no doubt? How do I come into conflict with him? Suppose someone were to say this to me why shouldn't I get
used to making no answer to him instead of starting an argument? Why shouldn't I treat his words like his whistling
or humming? [Cf. Z 401.]
Page 166
939. "Nothing is so certain as that I possess consciousness." In that case, why shouldn't I let the matter rest? This
certainty is like a mighty force whose point of application does not move, and so no work is accomplished by it. [Cf.
Z 402.]
Page 166
940. Someone playing dice throws first a 5 and then a 4 and says "If I had only thrown a 4 instead of the 5, I should
have won"! The condition is not physical but only mathematical, for one might reply: "If you had thrown a 4 first,
who knows what you would have thrown next!" [Cf. Z 678.]
Page 166
941. If you now say: "The use of the subjunctive rests on belief in natural law" one may retort: "It does not rest on
that belief; it and that belief stand on the same level." [Cf. Z 679.]
Page 166
942. Fate stands in contrast with natural law. One wants to find a foundation for natural law and to use it: not so with
fate. [Cf. Z 680; V.B., p. 119; C. & V., p. 61.]
Page 166
943. The concept of a 'fragment'. It is not easy to describe the use of this word even only roughly.
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Page 167
944. When we want to describe the use of a word,--isn't it like wanting to make a portrait of a face? I see it clearly;
the expression of these features is well known to me; and if I had to paint it I shouldn't know where to begin. And if I
do actually make a picture, it is wholly inadequate.--If I had a description in front of me I'd recognize it, perhaps
even detect mistakes in it. But my being able to do that does not mean that I could myself have given the
description.
Page 167
945. Two objects 'belong together'. One teaches a child to 'arrange' things, accompanying this activity with the words
"These belong together". The child learns this expression as well. It might even arrange things with the help of these
words and certain gestures. But the words may also be a mere accompaniment of the doing. A language-game.
Imagine such a game played without words, but with the accompaniment of music that fitted the actions.
Page 167
946. "Put it here"--saying which I point to the place with my finger--this is an absolute specification of place. And if
someone says that space is absolute, he might produce as an argument for this: "There is after all a place: here." [Cf.
Z 713.]
Page 167
947. The 'experience of similarity'. Think of the language-game "recognizing similarities", or "giving similarities", or
"arranging things according to their similarity". Where is the special experience here? The special experience content
that one is after?
Page 167
948. The duration of sensation. Compare the duration of a sensation of sound with the duration of the tactile
sensation that tells you you have a ball in your hand; and with the "feeling" that tells you that your knee is bent. And
here again we have a reason why we should like to say of the sensation of posture that it has no content. [Cf. Z 478.]
Page 167
949. Philosophical investigations: conceptual investigations. The essential thing about metaphysics: that the
difference between factual and conceptual investigations is not clear to it. A metaphysical question is always in
appearance a factual one, although the problem is a conceptual one. [Cf. Z 458.]
Page 167
950. What is it, however, that a conceptual investigation does? Does it belong in the natural history of human
concepts?--Well, natural history, we say, describes plants and beasts. But might it not be that
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plants had been described in full detail, and then for the first time someone realized the analogies in their structure,
analogies which had never been seen before? And so, that he establishes a new order among these descriptions. He
says, e.g., "compare this part, not with this one, but rather with that" (Goethe wanted to do something of the sort)
and in so doing he is not necessarily speaking of derivation; nonetheless the new arrangement might also give a
new direction to scientific investigation. He is saying "Look at it like this"--and that may have advantages and
consequences of varous[[sic]] kinds.
Page 168
951. Why do we count? Has it proved practical? Do we have our concepts, e.g. the psychological ones, because it
has proved to be advantageous? And yet we do have certain concepts just for that reason; they were introduced for
that reason. [Cf. Z 700.]
Page 168
952. One ought not to think it a simplification to bring seeing with one eye under consideration, instead of seeing
with both eyes; if, that is, one is clear about the fact that one doesn't feel seeing in the eyes. It is far more difficult to
carry the idea of the visual object through for binocular vision. For what is the binocular 'optical image'?
'The portrait of what one really sees' 'of the visual impression itself'.
Page 168
953. It occurs to someone: If I only had the right things and colours at my disposal, I could exactly represent what I
see. And up to a point it actually is so. And that report of what I have before me, and the description of what I see,
have the same form.--But they quite leave out, e.g., the wandering of the gaze. Not that alone, though, but also, e.g.,
the reading of a script in the visual field and any aspect of what is seen.
Page 168
954. Now if what you are looking at is a big tablet or flat wall, with a figure on it, then a picture of this figure may
count as an exact description. If the figure is, e.g., an , what more can one want than that it is copied exactly?
and yet there is besides a quite different description, which is not there in the copying. And similarly when the figure
is a face.
Page 168
955. What in one sense is a slight inaccuracy of description, in another is a large one.
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Page 169
956. Active and passive. Can one order someone to... or not? This perhaps seems, but is not, a far-fetched
distinction. It resembles this one: Can one (logical possibility) decide to... or not?"--And that means: How is it
surrounded by thoughts, feelings etc.? [Cf. Z 588.]
Page 169
957. What would a society all of deaf people look like? Or a society of 'mental defectives'? An important question!
What, that is, would a society be like, that never played a lot of our ordinary language-games? [Cf. Z 371.]
Page 169
958. Being conscious of the identity of colours in a picture, or of this colour's being darker than that one.
While I am hearing this piece, am I conscious the whole time of its being by...?
When is one conscious of a fact?
Page 169
959. Love is not a feeling. Love is put to the test, pain is not. [Cf. Z 504.]
Page 169
960. I see something in different connexions. (Isn't this more closely related to imagining than to seeing?)
Page 169
961. It is as if one had brought a concept to what one sees, and one now sees the concept along with the thing. It is
itself hardly visible, and yet it spreads an ordering veil over the objects.
Page 169
962. "What do you see?" (Language-game)--"What do you actually see?
Page 169
963. Let us represent seeing to ourselves as something enigmatic!--without introducing any kind of physiological
explanation.--
Page 169
964. The question "What do you see?" gets for answer a variety of kinds of description.--If now someone says
"After all, I see the aspect, the organization, just as much as I see shapes and colours"--what is that supposed to
mean? That one includes all that in 'seeing'? Or that here there is the greatest similarity?--And what can I say to the
matter? I can point out similarities and differences.
Page 169
965. Mightn't it be taken for madness, when a human being recognizes a drawing as a portrait of NN and exclaims
"That's Mr. NN!"--"He must be mad", they say, "He sees a bit of paper with black lines on it and takes it for a
human!"
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Page 170
966. 'Seeing the figure as...' has something occult, something ungraspable about it. One would like to say:
"Something has altered, and nothing has altered."--But don't try to explain it. Better look at the rest of seeing as
something occult too.
Page 170
967. The expression of that experience is and remains "I see it as a mountain", "I see it as a wedge", "I see it with this
base and this apex, but fallen over", etc. And the words "mountain", "wedge", "base", "fallen over" are after all only
marks, or noises--with a use.
Page 170
968. Think of a representation of a face from in front and in profile at the same time, as in some modern pictures. A
representation in which a movement, an alteration, a roving of one's glance, are included. Does such a picture not
really represent what one sees?
Page 170
969. "I forgive you." Can one say "I am busy forgiving you"? No. But that doesn't mean that there is not a process,
which one might--but does not--call "forgiving": I mean carrying on the inward struggle that may lead to forgiving.
Page 170
970. I should like to say: there are aspects which are mainly determined by thoughts and associations, and others
that are 'purely optical', these make their appearance and alter automatically, almost like after-images.
Page 170
971.
Page 170
What Köhler†1 does not deal with is the fact that one may look at figure 2 in this way or that, that the aspect
is, at least to a certain degree, subject to the will.
Page 170
972. I may attend to the course of my pains, but not in the same way to that of my believing or knowing. [Cf. Z 75.]
Page 170
973. The observation of duration may be continuous, or interrupted.
Page Break 171
Page 171
How do you observe your knowing, your opinions? and on the other hand, an after-image, a pain? Is there
such a thing as uninterrupted observation of my capacity to carry out the multiplication...? [Cf. Z 76-7.]
Page 171
974. ((On no. 971)) One might use the fact that the aspect is connected with eye-movements, to explain that.
Page 171
975. Analogy with the contrast between the 'value' and the 'limiting value' of a function. ((important))
Page 171
976. That an aspect is subject to the will is not something that does not touch its very essence. For what would it be
like, if we could see things arbitrarily as red or green? How in that case would one be able to learn to apply the words
"red" and "green"? First of all, in that case there would be no such thing as a 'red object', but at most an object which
one more easily sees red than green.
Page 171
977. Isn't what Köhler says roughly: "One couldn't take something for this or that, if one couldn't see it as this or
that"? Does a child start by seeing something this way or that, before it learns to take it for this or that? Does it first
learn to answer the question "How do you see that?" and only later "What is that?"--
Page 171
978. Can one say it must be capable of grasping the chair visually as a whole, in order to be able to recognize it as a
thing?--Do I grasp that chair visually as a thing, and which of my reactions shews this? Which of a man's reactions
shew that he recognizes something as a thing, and which, that he sees something as a whole, thingishly?
Page 171
979. One might imagine the matter like this: One tests how a child copies flat figures, when one has not taught it any
kind of copying, and when it hasn't yet even seen 3-dimensional objects.
Page 171
980. I learn to describe what I see; and here I learn all sorts of language-games.
Page 171
981. Not: "How can I describe what I see?"--but "What does one call 'description of what is seen'?"
And the answer to this question is "A great variety of thing".
Page Break 172
Page 172
982. Köhler†1 says that very few people would of their own accord see the figure 4 in the drawing
and that is certainly true. Now if some man deviates radically from the norm in his
description of flat figures or when he copies them, what difference does it make between him and normal humans
that he uses different 'units' in copying and describing? That is to say, how will such a one go on to differ from
normal humans in yet other things?
Page 172
983. A man might be highly gifted at drawing, I mean he might have the talent to copy objects, a room for instance,
very exactly, and yet he might keep on making small mistakes against sense; so that one could say "He doesn't grasp
an object as an object". He would never, e.g., make a mistake like that of the painter Klecksel, who paints two eyes
in the profile. His knowledge would never mislead him.
Page 172
984. The misleading concept is "the complete description of what one sees".
Page 172
985. Always eliminate the private object for yourself, by supposing that it keeps on altering: you don't notice this,
however, because your memory keeps on deceiving you. [Cf. P.I. p. 207e.]
Page 172
986. "Anyone who sees something sees something particular"--but that doesn't tell us anything.
It is as if one wanted to say "Even if no representation is like the visual impression, still, it is like itself".
Page 172
987. It is quite possible that someone who was asked "What do you see here?" might copy the figure correctly, but
given the question "Do you see a 4?", he might answer with a "No", although he has himself formed it in making his
copy.
Page 172
988. What do I tell someone, to whom I give the information that I am now seeing the ornament like this? (A queer
question)--This means "In what language-game does this sentence find employment?"--"What are we doing with
this sentence?"
Page 172
989. Let us suppose that certain aspects could be explained by the movement of the eye. In that case one would like
to say that those
Page Break 173
aspects were of a purely optical character; and so there would have to be a description of them which did not have to
make use of analogies from other domains. Then one would have to be able to replace the order: "See this as..." by:
"Have your gaze shift in such and such a way" or the like.
Page 173
990. But it is not true that an experience which is traceably connected with the movement of the eyes, an experience
that can be produced by such a movement, can for that reason be described by means of a sequence of optical
images.
(Any more than someone who imagines a note is imagining a sequence of disturbances of the air.)
Page 173
991. Hold the drawing of a face upside-down and you can't tell the expression of the face. Perhaps you can see that
it is smiling, but you won't be able to say what sort of a smile it is. You wouldn't be able to imitate the smile or
describe its character more exactly.
And yet the upside-down picture may represent the object extremely accurately. [Cf. P.I. p. 198f.]
Page 173
992. One needs to remember that seeing-as may have an effect like that of an alteration of what is seen, e.g. by
putting between brackets, or underlining, or making a connexion of one kind or another etc., and that in this way
again there is a similarity between seeing-as and imagining.
No one, after all, will deny that underlining or insertion of brackets may foster the recognition of a similarity.
Page 173
993. It is clear that only someone who sees the ambiguous picture as a rabbit will be able to imitate the expression
on the face of the rabbit. So if he sees the picture in this way, this will enable him to judge a particular kind of
resemblance.
Page 173
994. One will also estimate certain dimensions correctly, only if one sees the picture in this way.
Page 173
995. Remember that one may say: "You have to hear the tune like this, and then also play it correspondingly".
Page 173
996. Might there not be humans who don't calculate in their heads and can't easily learn silent reading, but who were
otherwise intelligent and in no sense 'defective'?
Page 173
997. There is no doubt that one often evokes an aspect by means of a movement of the eyes, by shifting one's gaze.
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Page 174
998. But how queer! one would like to say--if one can discover a kind of composition,--how is it possible also to see
it?!--How is it possible to know in a flash what one wants to say? Isn't that equally remarkable?
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999. For is the phenomenon of the aspect queerer than my memory of a particular actual person, of whom I have a
memory image? There is even a similarity between the two things. For here too one asks oneself: How is it possible
that I have a memory-image of him and that there is no doubt about its being an image of him?
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1000. Philosophy often solves a problem merely by saying: "Here is no more difficulty than there."
That is, just by conjuring up a problem, where there was none before.
It says: "Isn't it just as remarkable that...", and leaves it at that.
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1001. How does one obey the order "Imagine Mr. N"? How does one know that the order has been obeyed? how
does anyone know that he has obeyed it? What use is the state of having a image here?--I want to say that the
situation with seeing an aspect is similar.
Page 174
1002. I now see it (the chess-board) like this: It is as if you had given me this schematic drawing. E.g.:
or But the figure as which I see the other, is not unambiguously
determined.
Page 174
1003. Imagine a film representation of a triangle swinging around the point and then at rest. And
now it might be as if this temporal surrounding still had an effect in the picture of the triangle come to rest.
"Hanging", I should like to say. But does nothing correspond to that? Certainly something does! But that
only means that I am not lying, and that the expression of the aspect has a use. "What application?", you must
always ask yourself.
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Page 175
1004. One might regard the drawing of the chess board as a blueprint according to which pieces were to be
constructed which yield the chess-board. Now this drawing may be used in various ways; and one can also see it in
various ways, correspondingly to the different uses.
Page 175
1005. Suppose we explained this by saying the aspect comes about through different images and memories
superimposed on the optical image. Naturally this explanation does interest me, not as an explanation but as a
logical possibility, hence conceptually (mathematically).
Page 175
1006. "The green that I see over there is leafy. Those things there are eye-like." (What things?)
Page 175
1007. What cannot be an object of sight here seems to be an object of sight. As if one were to say one saw sounds.
(But one really does say that one sees a vowel yellow, or brown.)
Page 175
1008. For how could association be a lasting state? How could I associate this kind of object with these lines for five
minutes?
Page 175
1009. What convinces me that someone else sees an ordinary picture three-dimensionally?--That he says
so?--Rubbish--for how do I know what he means by assuring us of this?
Well, it's that he knows his way about in the picture; the expressions he applies to it are the ones that he
applies to space; confronted with a picture of a landscape, he behaves as he does when confronted with a landscape,
etc. etc.
Page 175
1010. I can never know, about him, whether he really sees. Well then of course I can't know it of myself either. For
how do I know that I am now calling the same thing that as before, and that I am calling the same thing "same"?
Page 175
1011. Now, how does it all look in the third person? And what is valid for the third person is then valid, however
queer this may sound, for the first person too.
Page 175
1012. Imagine a physiological explanation for my seeing one thing (A) as a variation of the other (B). It might come
out that when I see A as B certain processes take place on my retina, which otherwise are found when I actually see
B. And this might now explain some things
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in my behaviour. It might, e.g., be said that that is the reason why on seeing A I behave as if I were seeing B, a way I
don't ordinarily behave when I see A but don't see it as B. But for us this explanation of my behaviour is
superfluous. I accept the behaviour just as I accept a process on the retina or in the brain.
I want to say: At first the physiological explanation is apparently a help, but then at once it turns out to be a
mere catalyst of thoughts. I introduce it only to rid myself of it again at once.
Page 176
1013. Just do not fancy that you'd know in advance what "state of seeing" means in this case. Have the use TEACH
you the meaning.
Page 176
1014. Could I have made the phenomenon of having an image clear to myself, if I had been told: someone whose
eyes are open sees something that is not there before him, and at the same time also sees what is before him, and the
two visual objects don't get in each other's way?!
Page 176
1015. And naturally it would be quite wrong to say: "And yet queer things do happen", or "incredible things". Rather
what happens is not queer and is just wrongly seen as queer.
Page 176
1016. The old idea of the role of intuition in mathematics. Is this intuition the seeing of the complexes in different
aspects?
Page 176
1017. Doesn't one have to distinguish among aspects, separating the purely optical from the rest?
That they are very different from one another is clear: the dimension of depth, for example, sometimes
comes into their description, and sometimes not; sometimes the aspect is a particular 'grouping'; but when one sees
lines as a face, one hasn't taken them together merely visually to form a group; one may see the schematic drawing
of a cube as an open box or as a solid body, lying on its side or standing up; the figure: can be seen, not
just in two but in very many different ways.
Page 176
1018. One hangs up pictures, photographs, of landscapes, interiors, human beings, and does not regard them as
working drawings. One likes to look at them, as at the objects themselves; one smiles at the
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photograph as at the human being that it shews. We don't learn to understand a photograph as we do a blue-print.--It
would of course be possible that we had first to learn with some pains to understand a method of depiction, in order
to be able later on to use it as a natural picture. Still this troublesome learning would later on be mere history, and
then we should regard the picture just as we now regard our photographs.
Page 177
1019. There might also be men who did not understand, did not see, photographs as we do; who did indeed
understand that a human being can be represented in this way, who were also able to judge his shape roughly from a
photograph, but who all the same did not see the picture as a picture. How would that be manifested? What would
we regard as the expression of it?? That is perhaps not easy to say.
These people would perhaps not take pleasure in photographs as we do. They would not say "Look at his
smile!" and the like; they would often not recognize a person straight off from his picture; would have to learn to
read the photograph, and have to read it; they would be in a difficulty to recognize two good snapshots of the same
face as pictures of somewhat different positions.
Page 177
1020. If someone were to tell me that he had seen the figure for half an hour without a break as a reversed F, I'd have
to suppose that he had kept on thinking of this interpretation, that he had occupied himself with it.
Page 177
1021. It is as if the aspect were something that only dawns, but does not remain; and yet this must be a conceptual
remark, not a psychological one.
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1022. When the aspect suddenly changes one experiences the second phase in an acute way (corresponding to the
exclamation "Oh, it's a...!") and here of course one does occupy oneself with the aspect. In the temporal sense the
aspect is only the kind of way in which we again and again treat the picture.
Page 177
1023. 'Object' and 'ground'--Köhler wants to say--are visual concepts, like red and round. The description of what is
seen includes mentioning what is object and what is ground no less than colour and shape. And the description is
just as incomplete when it isn't said what is object and what ground, as it is when colour or shape are not given. I see
the one as immediately as the other one wants to say. And what objection is there to make to this? First: how this
gets
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recognized--whether through introspection, and whether everyone has got to agree about it. For the question
obviously concerns the description of the subjectively seen. But just how does one learn to use words to report the
subjective? And what can these words mean to us?
Suppose that, instead of words, it were a matter of reproducing by drawing; and in this reproduction what
corresponded to the words "object-like" and so on were the sequence, the order, in which we made the drawing. (I
am assuming we can draw extraordinarily fast.) And now suppose someone said: "The sequence belongs to the
representation of what is seen just as much as colours and shapes do."--What would that mean?
One may very well say: There are reasons for counting not only the drawn picture as part of a
description-by-drawing of what is seen but also the phrasing that goes on in making the drawing. Somehow these
reactions of the one who is giving the description belong together. In certan respects they do belong together, in
others they do not.
Page 178
1024. If one thinks of the currents on the retina (or the like) one would like to say: "So the aspect is just as much
'seen' as are the shape and colour." But then how can such an hypothesis have helped us to form this conviction?
Well, it favours the tendency to say here that we were seeing two different structures. But if this tendency can be
given a ground, its ground must be somewhere else.
Page 178
1025. The expression of the aspect is the expression of a way of taking (hence, of a way-of-dealing-with, of a
technique); but used as description of a state.
Page 178
1026. When it looks as if there were no room for such a logical form, then you must look for it in another dimension.
If there is no room for it here, then there is in another dimension. [Cf. P.I. p. 200f.]
Page 178
1027. In this sense there is also no room for imaginary numbers in the number-line. And that surely means: the
application of the concept of an imaginary number is radically different from that of, say, a cardinal number; more
different than the mathematical operations alone reveal. In order, then, to get place for them, one must descend to
their application and then they find a, so to speak undreamt of, different place. [Cf. P.I. p. 201a.]
Page 178
1028. If this constellation is always and continuously a face for me, then I have not named an aspect. For that
means that I always encounter
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it as a face, treat it as a face; whereas the peculiarity of the aspect is that I see something into a picture. So that one
might say: I see something that isn't there at all, that does not reside in the figure, so that it may surprise me that I
can see it (at least, when I reflect upon it afterwards).
Page 179
1029. If the seeing of an aspect corresponds to a thought, then it is only in a world of thoughts that it can be an
aspect.
Page 179
1030. If I am describing an aspect, the description presupposes concepts which do not belong to the description of
the figure itself.
Page 179
1031. Is it not remarkable, that in describing a visual impression one so uncommonly seldom includes the roving of
the gaze in the description?! It is as good as never included when the object is small, is, e.g. a face; although here too
after all the gaze is continually shifting.
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1032. This aspect may suddenly change and then a new looking follows the change. One is conscious of, e.g., the
facial expression, one contemplates it.
Page 179
1033. I may, e.g., be looking at a photograph and be occupied with the expression of the face, may so to speak bring
it home to myself, without saying anything to myself or to anyone else as I do so.
I make the eyes of the photograph speak to me. Perhaps I am seeing the picture for the first time as a real
face. 'Enter into the expression.' Ask, not "What goes on here?" but rather "What does one do with this utterance?"
Page 179
1034. We become conscious of the aspect only when it changes. As when someone is conscious only of a change of
note, but doesn't have absolute pitch.
Page 179
1035. When one fails to recognize the Mediterranean on the map with a different colouring, that does not shew that
there is really a different visual object before one. (Köhler's example.†1) At most that might give a plausible ground
for a particular way of expressing oneself. For it is not the same to say "That shews that here there are two ways of
seeing"--and "Under these circumstances it would be better to speak of 'two different objects of sight'."
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Page 180
1036. That an aspect can be summoned up by thoughts is extremely important, although it doesn't solve the
problem.
It is as if the aspect were an inarticulate reverberation of a thought.
Page 180
1037. I hear two people talking, don't understand what they are saying but hear the word "bank". Now I take it for
granted that they are talking about money. (This may turn out right or wrong.) Does that mean that I heard the word
"bank" in that meaning?
On the other hand: someone is speaking in a kind of game, uttering words of double meaning out of any
context; I hear the word "bank" and hear it in that meaning. It is almost as if this last were a worthless vestige of the
first proceeding.
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1038. Why shouldn't the overwhelming inclination exist, to use a certain word in our utterance? And why shouldn't
this word, nevertheless, be misleading, when we are reflecting on our experience?
I mean: Why should we not want to say "see" although the comparison with seeing is in many ways wrong?
Why should we not be impressed by an analogy, to the detriment of all the differences? But for that rcason one can't
appeal to the words of the utterance.
Physiological consideration here is merely confusing. Because it distracts us from the logical, conceptual
problem.
Page 180
1039. The confusion in psychology is not to be explained by its being a "young science". Its state isn't at all to be
compared with, e.g., that of physics in its early period. Rather with that of certain branches of mathematics. (Set
theory.) For there exists on the one hand a certain experimental method, and the other hand conceptual confusion;
as in some parts of mathematics there is conceptual confusion and methods of proof. While, however, in
mathematics one may be pretty sure that a proof will be important, even if it is not yet rightly understood, in
psychology one is completely uncertain of the fruitfulness of the experiments. Rather, in psychology there is what is
problematic and there are experiments which are regarded as methods of solving the problems, even though they
quite by-pass the thing that is worrying us. [Cf. P.I. p. 232.]
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1040. One might get tempted to believe that there was a particular kind of way in which one pronounces dates, a
particular cadence or something of that sort. For to me a number like 1854, say the number
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of a house, may have something date-like about it. One might believe that our experience is that of a particular
mental adjustment, which makes the mind ready for a particular activity; and so it is to be compared to the posture
of the body before jumping. This is a very enticing error. It is a fact of experience that this posture is a frequent or
appropriate preparation for this activity. But we did not learn that this feeling, this experience, is a serviceable
preparation for such-and-such an application of the figure, the number, etc. Expressions like "It is as if the
experience were already a-quiver with the future application", "It is as if we were already innervating the muscles for
this particular activity," etc. etc. are only paraphrased expressions of the experience. (As if someone were to say
"My heart is glowing with love for...".) Here, moreover, we have an indication of the origin of the sensation of
innervation which is supposed to constitute the consciousness of the act of will.
Page 181
1041. As I recognize someone I say: "Now I see--the features are the same, only..." and there follows a description of
the changes that are in fact there. Suppose I said "The face is rounder than it was"--am I to say it is some peculiarity
of the optical picture, of the visual impression, that shews me this? Of course it will be said: "No, here an optical
picture comes together with a memory." But how do these two things come together? Isn't it as if two pictures were
getting compared here? But there aren't two pictures being compared; and if there were, one would still have to keep
on recognizing one of them as the picture of the earlier face.
Page 181
1042. I may, however, say: I see that this figure is contained in that one, but I can't see it in it. This description is a
proper one for this figure, but still I can't see the figure according to the description.
And here "seeing" doesn't mean "recognizing in a flash" either. For it might well be that someone was unable
to see the one figure in the other at first glance, but that he could do this after he had, as it were piecemeal,
recognized the containment of the one by the other.
Page 181
1043. If I use the two pictures to inform him that the one figure is contained in the other, or that I recognize that it is
so, I don't thereby inform him that I see the one in the other. Wherein resides the difference between the two pieces
of information? (Their verbal expression need not differ.)
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Page 182
1044. I cannot see the figure as a union of and which are pushed together in such a way that they
are half super-imposed, so that the middle black region as it were counts as doubled. If now someone said he could
see the figure this way could I not understand this? Could I believe it? Should I say that this is possible--even if
nothing of the kind has ever happened to me? Need I say "You just mean something different by 'seeing this way'
from what I mean by it"? And if I did accept it, what should I now know, what could I do with it? (A physiological
application is of course again imaginable.)
Page 182
1045. Here belongs the question "What information would anyone be giving me, who said that he could see a
regular 50-gon as such? How would one test his statement? What allow to count as a test?
Page 182
It seems to me possible that one would accept nothing at all as confirmation of this statement.
Page 182
1046. "For me it is this ornament now." The "this" must be explained by indicating a class of ornaments. One may
perhaps say "There are white stripes on something black". No, there isn't any other account of it to give. Although
one would like to say: "There must surely be a simpler expression of what I am seeing!" And perhaps there is too.
For in the first place one might use the expression "to stand out". One can say "These parts stand out". And now,
can't one imagine a primitive reaction of a human being, who did not use words to express this, but rather indicated
the parts that "stand out" with his finger, and a special gesture. But that would not make this primitive expression
equivalent to the verbal expression "white-stripe-ornament".
Page 182
1047. But this too would be possible: a great collection of expressions, of concepts, might be quite equivalent in
meaning for someone in this case. And if that were the case, ought one to say that the described aspect is purely
optical?
Page 182
1048. The question is, however: why the primitive reaction of pointing with the finger is to be called an expression of
seeing-as. One will surely not be able to call it so without more ado. Only when it is combined with other
expressions.
Page 182
1049. Imagine someone always used a memory to express seeing-as. That he said, e.g., that the figure reminded him
now of this now of
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that, which he once saw. What could I do with this information?
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Can something remind me of this object for half-an-hour together?--supposing I am not occupying myself
with this memory.
Page 183
1050. If now the situation is that, while there is such a thing as an experience of meaning, this after all is something
incidental, how in that case can it seem so very important? Does that come of the fact that this phenomenon
accommodates a certain primitive interpretation of our grammar (the logic of our language)? Just as one often
imagines that the memory of an event must be an internal picture--and such a picture does sometimes really exist.
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1051. However blurred my visual picture, it must surely have a determinate blurredness, so it must after all be a
determinate picture. That presumably means it must be capable of having a description that fits it exactly; the
description would then first have to have the same vagueness as the thing described.--But now cast a glance at the
picture and give a description that in this sense fits exactly. This description ought properly to have been a picture, a
drawing! But here what is in question just isn't a blurred copy of a blurred picture. What we see is unclear in a quite
different sense. And I believe that the desire to speak of a private visual object might fade for someone, if he thought
oftener about this visual scene. The thing is: the method of projection which is possible elsewhere is not possible
here.
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1052. When I say "He sat on a bank in the grounds", of course one finds it difficult to think of a money-bank here,
or to imagine one; but that doesn't prove that one would otherwise have imagined a different bank.
It might, e.g., come easy to us in talking to draw certain pictures corresponding to our talk, and it might be
very difficult to draw pictures that conflicted with the intention or the context of what we said. But that wouldn't
prove that we always draw while we talk.
Page 183
1053. If, as I consider this question, I now pronounce all by itself the sentence "You must put the money in the
bank", and I mean it in such-and-such a way--does that mean that the same thing is going on in me when I
pronounce the sentence as when I say the sentence with this meaning to someone on a real occasion? What might
justify such an assumption? At most, that I then say "Just now I meant the
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word... in the meaning...". And here the question surely is one of a kind of optical illusion! What justifies that
observation in practice is certainly not a process that accompanies the speaking. Even though there may be
processes accompanying speech, which point towards this meaning. (The direction of my glance, for example.)
Page 184
1054. The difficulty is to know one's way about among the concepts of 'psychological phenomena'.
To move about among them without repeatedly running up against an obstacle.
That is to say: one has got to master the kinships and differences of the concepts. As someone is master of
the transition from any key to any other one, modulates from one to the other.
Page 184
1055. "Just now I spoke the word... in the meaning...".--How do you know you did? Suppose you've made a
mistake? How did you learn to speak it in that meaning?
If anyone says "Just now I spoke the word in isolation in that meaning", he is playing a totally different
language-game from someone who tells me he meant this by this word in that report or order.
And so now it is either essential, or inessential, that he also uses the word "to mean" in the first case. If it is
essential, then this first language-game is a reflection of the second one.
Say, as a chess-game on the stage may be called a reflection of a real chess-game.
Page 184
1056. Playing mental chess with someone else: both parties play in the imagination and agree that this player won,
this one has lost. They are then both able to reproduce the game from memory, agreeing with one another; they can
write it down or narrate it.--Think of tennis played like that. It would be possible. Only of course it wouldn't be any
sort of muscular exercise. (Although even that might be imagined.)--It is important that even with 'tennis in the
imagination' one will be able to say "I succeeded in... the ball".
Page 184
1057. I might dream of a game of chess; but perhaps the dream only shewed me a single move of the game.
Nevertheless I would have dreamt: that I played a game of chess. In that case it will be said: "You didn't really play
it. You dreamt it." Why shouldn't it also be said "You didn't really mean the word that way, you only dreamt it"?
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Page 185
1058. In a law-court, e.g., the question might be gone into how someone meant a word, and it may also be inferred
from certain facts that he meant it this way. It is a question of intention; but could that other case of meaning
something, where I dreamt of meaning it, have the same importance? [Cf. P.I. p. 214e.]
Page 185
1059. But how about this: when I read a poem, or some expressive prose, especially when I read it out loud, surely
there is something going on as I read it which doesn't go on when I glance over the sentences only for the sake of
their information. I may, for example, read a sentence with more intensity or with less. I take trouble to get the tone
exactly right. Here I often see a picture before me, as it were an illustration. And may I not also utter a word in such
a tone as to make its meaning stand out like a picture? A way of writing might be imagined, in which some signs
were replaced by pictures and so were made prominent. This does actually happen sometimes, when we underline a
word or positively put it on a pedestal in the sentence. (("... there lay a something....")) [Cf. P.I. p. 214g.]
Page 185
1060. When I am reading expressively and I pronounce this word, it is, so to speak, filled brimful with its meaning.
And now it might be asked "How can that be?" [Cf. P.I. p. 215a.]
Page 185
1061. "How can that be, if meaning is what you believe?" The use of a word can't accompany it or fill it brimful. And
now I may reply: my expression was picturesque.--But the picture forced itself on me. I want to say: the word was
filled with its meaning. There might perhaps be some explanation of how it comes about that I want to say that.
But why, then, am I not supposed to 'want to say': I pronounced the (isolated) word with this meaning? [Cf.
P.I. p. 215a.]
Page 185
1062. Why shouldn't a particular technique of employment of the words "meaning", "to mean" and others lead me
to use these words in, so to speak, a picturesque, improper, sense? (As when I say that the sound e is yellow.) But I
don't mean: it is a mistake--I didn't really pronounce the word in this meaning, I only imagined I did. That's not how
it is. For I don't merely imagine, either, that there is a game of chess in "Nathan".†1
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Page 186
1063. Thinking in terms of physiological processes is extremely dangerous in connexion with the clarification of
conceptual problems in psychology. Thinking in physiological hypotheses deludes us sometimes with false
difficulties, sometimes with false solutions. The best prophylactic against this is the thought that I don't know at all
whether the humans I am acquainted with actually have a nervous system.
Page 186
1064. The case of 'meaning experienced' is related to that of seeing a figure as this or that. We have to describe this
conceptual relationship; we are not saying that the same thing is under consideration in both cases.
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1065. When you write your F like this: do you mean it as a 'slipped' F or as a mirror F? Do you mean it to look
to the right or the left?--The second question obviously does not refer to a process that accompanies the writing.
With the first question, one might be thinking of such a process.
Page 186
1066. "I see that the child wants to touch the dog, but doesn't dare." How can I see that?--Is this description of what
is seen on the same level as a description of moving shapes and colours? Is an interpretation in question? Well,
remember that you may also mimic a human being who would like to touch something, but doesn't dare. And what
you mimic is after all a piece of behaviour. But you will perhaps be able to give a characteristic imitation of this
behaviour only in a wider context.
Page 186
1067. One will also be able to say: What this description says will get its expression somehow in the movement and
the rest of the behaviour of the child, but also in the spatial and temporal surrounding.
Page 186
1068. But now am I to say that I really 'see' the fearfulness in this behaviour--or that I really 'see' the facial
expression? Why not? But that is not to deny the difference between two concepts of what is perceived. A picture of
the face might reproduce its features very accurately, but not get the expression right; it might, however, be right as
far as the expression goes and not hit the features off well. "Similar expression" takes faces together in a quite
different way from "similar anatomy".
Page 186
1069. Naturally the question isn't: "Is it right to say 'I see his sly wink'." What should be right or wrong about that,
beyond the use of
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the English language? Nor are we going to say "The naif person is quite right to say he saw the facial expression"!
Page 187
1070. On the other hand one would like to say: We surely can't 'see' the expression, the shy behaviour, in the same
sense as we see movement, shapes and colours. What is there in this? (Naturally, the question is not to be answered
physiologically.) Well, one does say, that one sees both the dog's movement and its joy. If one shuts one's eyes one
can see neither the one nor the other. But if one says of someone who could accurately reproduce the movement of
the dog in some fashion in pictures, that he saw all there was to see, he would not have to recognize the dog's joy.
So if the ideal representation of what is seen is the photographically (metrically) exact reproduction in a picture, then
one might want to say: "I see the movement, and somehow notice the joy."
But remember the meaning in which we learn to use the word "see". We certainly say we see this human
being, this flower, while our optical picture--the colours and shapes--is continually altering, and within the widest
limits at that. Now that just is how we do use the word "see". (Don't think you can find a better use for it--a
phenomenalogical [[sic]] one!)
Page 187
1071. Now do I learn the meaning of the word "sad"--as applied to a face--in just the same way as the meaning of
"round" or "red"? No, not in quite the same way, but still in a similar way. (I do also react differently to a face's
sadness, and to its redness.)
Page 187
1072. Look at a photograph: ask yourself whether you see only the distribution of darker and lighter patches, or the
facial expression as well. Ask yourself what you see: how would it be easier to represent it: by a description of that
distribution of patches, or by the description of a human head; and when you say of the face that it is smiling--is it
easier to describe the corresponding lie and shape of the parts of the face, or to smile yourself?
Page 187
1073. "What I see can't be the expression, because the recognition of the expression depends on my knowledge, on
my general acquaintance with human behaviour." But isn't this merely an historical observation?
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1074. Is it here as if I were perceiving a 'fourth dimension'? Well, yes and no. Queer, however, it is not. From which
you ought to learn that what strikes someone as queer when he is philosophizing is not queer. We make the
assumption: the word... would really have to be used like this (this use strikes us as a prototype) and then we find
the normal use extremely queer.
Page 188
1075. "What I properly see must surely be what is produced in me by influence of the object"--In that case what is
produced in me is something like a copy, something that one can oneself in turn look at, have before him. Almost
something like a materialization.
And this materialization is something spatial and must be describable in its entirety by the use of spatial
concepts. Then, while it may smile, the concept of friendliness has no part in the representation of it, but is rather
alien to this representation (even though this concept may subserve this representation.). [Cf. P.I. p. 199g.]
Page 188
1076. If someone were capable of making an exact copy of this portrait--should I not say he saw everything that I
did? And he wouldn't have to allude to the head as a head or as something three-dimensional, at all; and even if he
does, the expression need not say anything to him. And if it does speak to me--should I say I see more than the
other?
I might say so.
Page 188
1077. But a painter can paint an eye so that it stares; so its staring must be describable by the distribution of colour
on the surface. But the one who paints it need not be able to describe this distribution.
Page 188
1078. Understanding a piece of music--understanding a sentence.
I am said not to understand a form of speech like a native if, while I do know its sense, I yet don't know, e.g.,
what class of people would employ it. In such a case one says that I am not acquainted with the precise shade of
meaning. But if one were now to think that one has a different sensation in pronouncing the word if one knows this
shade of meaning, this would again be incorrect. But there are, e.g., innumerable transitions which I can make and
the other can't.
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1079. But one would like to say: "Human mental life can't be described at all; it is so uncommonly complicated and
full of scarcely graspable experiences. In great part it is like a brewing of coloured
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clouds, in which any shape is only a transition to other shapes, to other transitions.--Why, take just visual
experience! Your gaze wanders almost incessantly, how could you describe it?"--And yet I do describe it!--"But that
is only a quite crude description, it gives only the coarsest features of your experience."--But isn't this just what I call
description of my experience? How then do I arrive at the concept of a kind of description that I cannot possibly
give?
Page 189
1080. Imagine looking at flowing water. The picture presented by the surface keeps on changing. Lights and darks
everywhere appear and disappear. What would I call an 'exact description' of this visual picture? There's nothing I
would call that. If someone says it can't be described, one can reply: You don't know what it would be right to call a
description. For you would not acknowledge the most exact photograph as an exact representation of your
experience. There is no such thing as exactness in this language-game. (As, that is, there is no knight in draughts.)
Page 189
1081. The description of the experience doesn't describe an object. It may subserve a description. And this object is
sometimes the one that one is looking at, and sometimes (photography) not.
The impression--one would like to say--is not an object.
Page 189
1082. We learn to describe objects, and thereby, in another sense, our sensations.
Page 189
1083. I look into the eye-piece of an instrument and draw or paint a picture of what I see. Whoever looks at it can
say: "So that's how it looks"--but also "So that's how it looks to you".
I might call the picture a description of what I was looking at, but also a description of my visual impression.
Page 189
1084. "The impression is blurred"--'so the object in my consciousness is blurred'.
Page 189
1085. One can't look at the impression, that is why it is not an object. (Grammatically.) For one doesn't look at the
object to alter it. (That is really what people mean when they say that objects exist 'independently of us'.) [Cf. Z
427.]
Page Break 190
Page 190
1086. "The chair is the same, whether I am looking at it or not!"--that need not have been so. Humans are often
embarrassed when one looks at them. "The chair goes on existing whether I am looking at it or not." That might be
an empirical proposition, or it might be one to take grammatically. But in saying it one may also be thinking simply
of the conceptual difference between sense-impression and object. [Cf. Z 427.]
Page 190
1087. German nouns printed in lower-case letters in certain modern poets. A German noun all in lower-case letters
looks alien; to recognize it, one has to read it attentively. It is supposed to strike us as new, as if we had seen it now
for the first time.--But what interests me here? This--that the impression can't at first be described more exactly than
by means of words like 'queer', 'unaccustomed'. Only later follow, so to speak, analyses of the impression. (The
reaction of recoil from the strangely written word.)
Page 190
1088. We teach someone the meaning of the word "eerie" by bringing it into connexion with a certain behaviour in
certain situations (though the behaviour is not called that). In such situations he now says it feels eerie to him; and
even that the word "ghost" has something eerie about it.--How far was the word "eerie" to start with the name of a
feeling? If someone shrinks back from entering a dark room, why should I call this or the like the expression of a
feeling? For "feeling" certainly makes us think of sensation and sense-impression, and these in turn are the objects
immediately before our minds. ((I am trying to make a logical step here, one that comes hard to me.))
Page 190
1089. "What do I know of the feelings of others, and what do I know of my own?" means that an experience,
conceived as an object, slips out of view.
Page 190
1090. For can anything be more remarkable than this, that the rhythm of a sentence should be important for exact
understanding of it?
Page 190
1091. It's as if the one who pronounces the sentence as a piece of information told us something, but as if the
sentence too, as a mere example, did the same.
Page Break 191
Page 191
1092. Certainly it's clear that the description of impressions has the form of the description of 'external' objects--with
certain deviancies. E.g. a certain vagueness.
Or again: So far as the description of the impression looks the same as the description of an object, it is a
description of an object of perception. (Hence consideration of binocular vision ought to be somewhat disquieting
for one who speaks of the visual object).
Page 191
1093. "Thinking is an enigmatic process, and we are a long way off from complete understanding of it." And now
one starts experimenting. Evidently without realizing what it is that makes thinking enigmatic to us.
The experimental method does something; its failure to solve the problem is blamed on its still being in its
beginnings. It is as if one were to try and determine what matter and spirit are by chemical experiments.
Page 191
1094. Someone who describes his visual impression doesn't describe the edges of the visual field. Is this an
incompleteness in our descriptions?
If I shut my left eye and then turn my eyes as far as I can to the right, I still, 'out of the corner of my eye' see
an object shining out. I might even give a rough description of this impression. I might also produce a drawing of it,
and it would perhaps shew some darknesses and a dark merging border: but only someone who knows in what
situation it is to be employed can rightly understand or employ this picture. That is to say: he too might now shut
one eye, look as far as possible to the right, and say he has this impression too, or: that his impression deviates from
my picture in this way or that.
Page 191
1095. That we calculate with some concepts and with others do not, merely shews how different in kind conceptual
tools are (how little reason we have ever to assume uniformity here). [Cf. Z 347.]
Page 191
1096. Turing's 'Machines'. These machines are humans who calculate. And one might express what he says also in
the form of games. And the interesting games would be such as brought one via certain rules to nonsensical
instructions. I am thinking of games like the "racing game". One has received the order "Go on in the same way"
when this makes no sense, say because one has got into a circle. For any order makes sense only in certain positions.
(Watson.)
Page Break 192
Page 192
1097. A variant of Contor's diagonal proof: Let N = F (K, n) be the form of the law for the development of decimal
fractions. N is the nth decimal place of the Kth development. The diagonal law then is: N = F (n,n) = Def F' (n).
To prove that F'n cannot be one of the rules F (k,n). Assume it is the 100th. Then the formation rule of
F' (1) runs F (1, 1)
of
F'(2) F (2, 2) etc.
But the rule for the formation of the 100th place of F'(n) will run F (100, 100); that is, it tells us only that the
hundredth place is supposed to be equal to itself, and so for n = 100 it is not a rule.
The rule of the game runs "Do the same as..."--and in the special case it becomes "Do the same as you are
doing". [Cf. Z 694.]
Page 192
1098. The concept of 'ordering', e.g., the rational numbers, and of the 'impossibility' of so ordering the irrational
numbers. Compare this with what is called an ordering of digits. Likewise the difference between the 'correlation' of
a figure (or nut) with another and the 'correlation' of all whole numbers with the even numbers. Everywhere a
shifting of concepts. [Cf. Z 707.]
Page 192
1099. The description of the subjectively seen is closely or distantly related to the description of an object, but does
not function like the description of an object. How does one compare visual sensations? How do I compare my
visual experiences with someone else's? [Cf. Z 435.]
Page 192
1100. We don't see the human eye as a receiver; it seems, not to let something in, but to send out. The ear receives;
the eye looks. (It casts glances, it flashes, beams, coruscates.) With the eye one may terrify, not with the ear or the
nose. When you see the eye, you see something go out from it. You see the glance of the eye. [Cf. Z 222.]
Page 192
1101. "When you get away from your physiological prejudices, you'll find nothing in the fact that the glance of the
eye can be seen." Certainly I too say that I see the glance that you throw someone else. And if someone wanted to
correct me and say I don't really see it, I should hold this to be a piece of stupidity.
Page Break 193
Page 193
On the other hand I have not admitted anything with my way of putting it, and I contradict anyone who tells
me I see the eye's glance 'just as' I see its form and colour.
For 'naive language', that's to say our naif, normal, way of expressing ourselves, does not contain any theory
of seeing--it shews you, not any theory, but only a concept of seeing. [Cf. Z 223.]
Page 193
1102. And if someone says "I don't really see the eye's glance, but only shapes and colours",--is he contradicting the
naif form of expression? Is he saying that the man was going beyond his rights, who said he saw my glance all right,
that he saw this man's eye staring, gazing into vacancy, etc? Certainly not. So what was the purist trying to do?
Does he want to say it's more correct to use a different word here instead of 'seeing'? I believe he only wants
to draw attention to a division between concepts. For how does the word "see" associate perceptions? I mean: it may
associate them as perceptions with the eye; for we do not feel seeing in the eye. But really the one who insists on the
correctness of our normal way of talking seems to be saying: everything is contained in the visual impression; that
the subjective eye equally has shape, colour, movement, expression and glance (external direction). That one does
not detect the glance, so to speak, somewhere else. But that doesn't mean 'elsewhere than in the eye'; it means
'elsewhere than in the visual picture'. But how would it be for it to be otherwise? Perhaps so that I said: "In this eye I
see such and such shapes, colours, movements,--that means it's looking friendly at present," i.e. as if I were drawing
a conclusion.--So one might say: The place of the perceived glance is the subjective eye, the visual picture of the eye
itself.
Page 193
1103. First and foremost, I can very well imagine someone who, while he sees a face extremely accurately, and can,
e.g., make an accurate portrait of it, yet doesn't recognize its smiling expression as a smile. I should find it absurd to
say that his sight was defective. And equally absurd to say that his subjective visual object just wasn't smiling,
although it has all the colours and form that mine has.
Page 193
1104. That is to say: here we are drawing a conceptual boundary (and it has nothing to do with physiological
opinions).
Page 193
1105. High-light or reflection: when a child paints, it will never paint these. Why, it is almost bewildering that they
can be represented by means of the usual oil- or water-colours. [Cf. Z 370.]
Page Break 194
Page 194
1106. One who sees that someone is stretching out his hand to touch something, but is afraid of it, does surely in a
certain sense see the same as one who can imitate the movements of the hand down to the last detail, or can
represent it by means of drawings, but has not the power of interpreting it in that way.
Page 194
1107. If someone says: Obviously (to any unprejudiced person) the shape, the colour, the organization, the
expression, are properties of the subjectively seen, of the immediate object of sight--here the word 'obviously'
betrays him. The reason why it is obvious is: because everyone admits it; and admits it only through the use of
language. So here one is using a picture to give a reason for a proposition.
†1 If someone says: The shape, the colour, the organization, the expression, are surely all obviously
properties of the immediately seen (of my object of sight)--he is basing his opinion on a picture.--For, if someone
admits that each of those things is a property of his immediate object of sight--what information is he giving us? If
he, e.g., tells someone else: "It's like that for me too"--what conclusion can I draw from this? (What if this full
agreement were based on a misunderstanding?)
Page 194
1108. That picture is nothing but an illustration contributing to the methodology of our language. If we are really all
inclined to find this picture apt, this is perhaps of psychological interest, but it is no substitute for a conceptual
investigation.
Page 194
1109. There are two things one may call "Methodology": A description of the activities that are, e.g., called
"measuring", a branch of human natural history, which is going to make the concepts of measuring, of exactness
etc., intelligible to us in their variations; or on the other hand a branch of applied physics, the theory of how best
(most accurately, most conveniently) to measure this or that under such-and-such circumstances. [Cf. P.I. p. 225.]
Page 194
1110. I tell him: "Change the way you are adjusted like this:..."--he does so; and now something is altered in him.
'Something'? His attitude is altered; and one can describe the alteration. Calling the attitude 'something in him' is
misleading. It is as if we could now dimly see, or feel, a Something, which has altered and which is called "the
attitude". Whereas everything lies open to the light of day--but the words "a new attitude" do not designate a
sensation.
Page Break 195
Page 195
1111. What does the description of an 'attitude' look like?
One says, e.g., "Look away from these little spots and this small irregularity, and regard it as a picture of a
...".
"Think that away! Would it be unacceptable to you even without this...?" I shall be said to be altering my
visual picture--as I do by blinking or keeping a detail out of view. This "Looking away from..." plays a role quite like
that of the construction of a new picture. [Cf. Z 204.]
Page 195
1112. Very well,--and these are good reasons to say that through our attitude we made a change in our visual
impression. That is to say, these are good reasons for delimiting the concept 'visual impression' in this way. [Cf. Z
205.]
Page 195
1113. The word "organization" fits in very well with the concept 'belonging together'. There seems to be a series of
simple modifications of the visual impression here, which are all really 'optical'. With different aspects, however, one
may do other quite different things besides separating parts and taking them together, or suppressing some and
making others prominent.
Page 195
1114. I may, however, call "taking together" some definite thing, a definite peculiarity of the process of copying a
drawing. I may then say that in reproduction by drawing, or in description, someone takes the figure together like
this, organizes it like this. (Of course there'd be difficulties about that in some cases, e.g. in the case of the
duck-rabbit.)
Page 195
1115. Now one says: I can take lines together in copying, but I can also do so by means of attention. Like the way I
can calculate in my head, as on paper.
Page 195
1116. Can Gestalt psychology classify the different organizations that can be introduced into the unorganized visual
picture; can it give once for all the possible kinds of modification which the plasticity of our nervous system can
elicit? When I see the dot as an eye which is looking in this direction--what system of modifications does that fit
into? (System of shapes and colours.)
Page Break 196
Page 196
1117. E.g., it is, I believe, misleading when Köhler†1 describes the spontaneous aspects of the figure
by saying: the lines which belong to the same arm in one aspect, now belong to different arms.
That sounds as if what were in question here were again a way of taking these radii together. Whereas, after all, the
radii that belonged together before belong together now as well; only one time they bound an 'arm', another time an
intervening space.
Page 196
1118. Indeed, you may well say: what belongs to the description of what you see, of your visual impression, is not
merely what the copy shews but also the claim, e.g., to see this "solid', this other 'as intervening space'. Here it all
depends on what we want to know when we ask someone what he sees.
Page 196
1119. "But surely it's obvious that in seeing I can take elements together (lines, for example)?" But why does one
call it "taking together"? Why does one here need a word--essentially--that already has another meaning? (Naturally
it is the same here as in the case of the phrase "calculating in the head.") [Cf. Z 206.]
Page 196
1120. When I tell someone "Take these lines (or something else) together", what will he do? Well, a variety of things
according to the circumstances. Perhaps he is to count them two by two, or put them in a drawer, or look at them
etc. [Cf. Z 207.]
Page 196
1121. Is the drawing that you see itself organized? And when you 'organize' it in such-and-such a way, do you then
see more than is present?
Page 196
1122. "Organize these things." What does this mean? Perhaps "arrange them". It might mean "give them some
order",--or again: learn to know your way about among them, learn to describe them; learn to describe them by
means of a system, by means of a rule.
Page 196
1123. Once more, the question is: What information do I give someone with the words "In looking, I am now taking
the lines together like this"? This question may also be put like this: For what purpose do I tell someone "In looking,
take these lines together like this"?--There is here again a similarity to the demand "Imagine this".
Page Break 197
Page 197
1124. The egg-shell of its origin clings to any thinking, shewing one what you struggled with in growing up. What
views are your circle's testimony: from which ones you have had to break free.
Page 197
1125. The picture does not organize itself under our gaze.
Page 197
1126. It is perhaps important to remember that I may see, take, a figure this way today, and differently tomorrow,
without there needing to have been a 'jump'. I might, for example, take and use an illustration in a book in this way
today, and tomorrow encounter the same illustration on a later page where it is to be taken differently, without
noticing that it is the same figure again.
Page 197
1127. Could a man demonstrate his reliability by saying: "It is true; and see! I believe it."
Page 197
1128. Might it be said: a way of taking something, a technique, is mirrored in an experience? Which, after all, only
means: we employ the expression that we have learnt for a technique in an expression of experience (not: as
designation of an experience).
Page 197
1129. Well, why should a way of speaking not be responsible for an experience?
Page 197
1130. Would it make sense to ask a composer whether one should hear a figure like this or like this; if that doesn't
also mean: whether one should play it in this way or that?
Page 197
1131. Memory: "I still see us sitting at that table."--But have I really the same visual picture--or one of those which I
had then? Do I also certainly see the table and my friend from the same point of view as then, and so not see
myself?--My memory image is not evidence of that past situation; as a photograph would be, which, having been
taken then, now bears witness to me that this is how it was then. The memory image and the memory words are on
the same level. [Cf. Z 650.]
Page 197
1132. Why shouldn't it be that one excludes mutually contradictory conclusions: not because they are contradictory,
but because they are useless? Or put it like this: one need not shy away from them as from something unclean,
because they are contradictory: let them be excluded because they are no use for anything.
Page Break 198
Page 198
1133. You must seriously imagine that there really could be a word in some language that stood for pain-behaviour
and not for pain.
Page 198
1134. He asks "What did you mean when you said...?" I answer the question and then I add: "If you had asked me
before, I'd have answered the same; my answer was not an interpretation which had just occurred to me." So had it
occurred to me earlier? No.--And how was I able to say then: "If you had asked me earlier, I'd have..."? What did I
infer it from? From nothing at all. What do I tell him, when I utter the conditional. Something that may sometimes
be of importance.
Page 198
1135. He knows, for example, that I haven't changed my mind. It also makes a difference whether I reply that I was
'only saying these words to myself' without meaning anything by them; or, that I meant this or that by them. Much
depends on this.
Moreover, it makes a difference whether someone says to me "I love her" because the words of a poem are
going through his head or because he is saying it to make a confession to me of his love.
Page 198
1136. Is it not peculiar, however, that there is such a thing as this reaction, as such a confession of intention? Isn't it
an extremely remarkable linguistic instrument? What is really remarkable about it? Well--it is difficult to imagine
how a human learns this use of words. It is so entirely subtle. [Cf. Z 39.]
Page 198
1137. But is it really more subtle than that of the words "I formed an image of him" for example? Yes, any such use
of language is remarkable, peculiar, when one is adjusted only to consider the description of physical objects. [Cf. Z
40.]
FOOTNOTES
Page 11
†1 The words in brackets are in English. (Eds.)
Page 13
†1 These words occur in English. (Eds.)
Page 15
†1 These paragraphs are alternatives.
Page 17
†1 The reference is to the opera singer who had to sing 'Weiche, Wotan, weiche' ('Depart Wotan, depart') and
to whom the other singer on the stage had just whispered 'Do you like your eggs soft (weiche) or hard?' (Eds.)
Page 19
†1 The passage (between asterisks) occurs in English, not German. (Eds.)
Page 24
†1 As in the fairy tale "The Man who could not Shudder." See Grimm's Fairy Tales.
Page 32
†1 The MS has "That is a rule". (Eds.)
Page 43
†1 These words occur in English. (Eds.)
Page 51
†1 Schumann: Davidsbündlertänze. Eds.
Page 52
†1 Quoted in English. Trans.
Page 62
†1 Philosophical Investigations I, 15 and 26. (Eds.)
Page 64
†1 "not a thing" also in the German text. (Eds.)
Page 74
†1 I have substituted Marlowe's line for Wittgenstein's example from Goethe, Faust, Part II. V. (Trans.)
Page 90
†1 The words in brackets occur in English. (Eds.)
Page 94
†1 "If God held enclosed in his right hand all truth, and in his left simply the unremitting impulse towards
truth, although with the condition that I should eternally err, and said to me 'Choose!' I should humbly fall before his
left hand, and say: 'Father, give! Pure truth is for thee alone'."
Page 105
†1 See P.I. Part I, §2. (Eds.)
Page 116
†1 "Forms of life" was a variant here. Trans.
Page 119
†1 In German all nouns are masculine, feminine or neuter. Trans.
Page 129
†1 J. P. Hebel: Schatzkastlein, Zwei Erzählungen. (Eds.)
Page 132
†1 The last sentence in English. (Eds.)
Page 133
†1 "Perpetual cloud descends". Spoken by Care in Goethe's Faust, Part II, Act v.
Page 148
†1 This passage presents severe problems of translation, because quite ordinary German has two words,
"Erlebnis" and "Erfahrung," both of which are regularly translated 'experience'. I was not willing either simply to use
the German words or to say, e.g. 'experience'1 and 'experience'2. I have therefore kept 'experience' for 'Erlebnis' and
used 'undergoing' for 'Erfahrung'. I apologize for the air of philosophical technicality and the unnaturalness that is
forced upon me by having to find two words where common or garden English has only one. Trans.
Page 151
†1 "Descent of permanent cloud." Goethe, Faust, II. V.
Page 152
†1 The reference is to the figure known among English psychologists as the 'duck-rabbit.' See Philosophical
Investigations II, xi. (Eds.)
Page 153
†1 These three words are in English. (Eds.)
Page 170
†1 Gestalt Psychology, New York, 1929, p. 198. (Eds.)
Page 172
†1 Gestalt Psychology, p. 200f. (Eds.)
Page 179
†1 Köhler, Gestalt Psychology, p. 195ff. (Eds.)
Page 185
†1 Lessing, Nathan the Wise. (Eds.)
Page 194
†1 Marked as an alternative version in the MS. (Eds.)
Page 196
†1 Op. cit. p. 185. (Eds.)
REMARKS ON THE PHILOSOPHY OF PSYCHOLOGY
Volume II
Page i
Page Break ii
Titlepage
Page ii
Ludwig Wittgenstein:
REMARKS ON THE PHILOSOPHY OF PSYCHOLOGY:
VOLUME II
Edited by
G. H. VON WRIGHT
and
HEIKKI NYMAN
Translated by
C. G. LUCKHARDT and M. A. E. AUE
BASIL BLACKWELL
OXFORD
Page Break iii
Copyright page
Page iii
Copyright © Blackwell Publishers Ltd, 1980
First published in 1980
Reprinted 1988, 1990, 1998
Blackwell Publishers Ltd
108 Cowley Road
Oxford OX4 1JF, UK
All rights reserved. Except for the quotation of short passages for the purposes of criticism and review, no part of
this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher.
Page iii
Except in the United States of America, this book is sold subject to the condition that it shall not, by way of trade or
otherwise, be lent, resold, hired out, or otherwise circulated without the publisher's prior consent in any form of
binding or cover other than that in which it is published and without a similar condition including this condition
being imposed on the subsequent purchaser.
British Library Cataloguing in Publication Data
Wittgenstein, Ludwig
Remarks on the philosophy of psychology
Vol. 2
1. Cognition 2. Knowledge, Theory of
I. Title II. Wright, Georg Henrik von
III. Nyman, Heikki
153.4 BF311
ISBN 0-631-13062-4
Printed in Great Britain by Athenaeum Press Ltd, Gateshead, Tyne & Wear
This book is printed on acid-free paper
Page Break iv
PREFACE
Page iv
The source of this second volume of Remarks on the Philosophy of Psychology is TS No. 232. Wittgenstein
probably dictated it in September or October 1948. The remarks underlying this dictation stem from the period 19
November 1947 to 25 August 1948, MSS 135-137. The places in the text that were faulty or obscure we have tried to
amend by an exact collation with the MS sources. During this task we received valuable suggestions from the
translators of the German text into English, C. G. Luckhardt and M. A. E. Aue. We thank them for their readiness to
help us.
Helsinki Georg Henrik von Wright
Heikki Nyman
Page Break 1
TRANSLATORS' NOTE
Page 1
We have followed two principles in translating this material. First, since the text contains many passages already
translated in Zettel, and a few passages from Part II of the Investigations, we had to decide between preparing an
entirely new translation of these remarks, and following Professor Anscombe's translations as closely as possible.
Since our translation would not differ significantly from hers, and because any minor stylistic differences might
produce confusion, we decided upon the latter course. Few of these parallel passages are absolutely identical,
however, and so almost all deviations from Professor Anscombe's translations should be taken as reflecting
differences in the German texts.
Page 1
Second, we have tried to preserve Wittgenstein's highly individual style of writing. For example, the quite
large number of both ordinary dashes and specially long dashes, and the different uses to which Wittgenstein puts
them, contribute to the vividness of his writing, as does the unusual practice of following a hypothetical subjunctive
with a statement in the past tense. ("Suppose that someone were to.... Now did he...?", for example.) These we have
retained. Another facet of Wittgenstein's writing which cannot fail to strike the German reader is his choice of words.
Most of his verbs are very ordinary ones, and so we have "put into", for example, rather than "infuse". Likewise
there is a noticeable lack of philosophical jargon in the text, and so "value judgment" is deliberately used, not
"normative judgment". The German text is laced with Anglicisms, Austrianisms, and colloquialisms, and we have
tried to retain the flavour of these in the translation.
Page 1
We wish to express our thanks to Professor G. E. M. Anscombe for her time and diligence in going over our
translation with us, and to the American Philosophical Society, whose generous travel grant made a visit to her
possible. Also, we wish to thank Dean Glenn Thomas and the Dean's Advisory Committee on Research, of Georgia
State University, for the release time and a travel grant which enabled us to complete the translation, and the
University Research Committee of Emory University for its travel support.
C. G. Luckhardt
Georgia State University
Maximilian A. E. Aue
Emory University
Page Break 2
II
Page 2
1. 'Surprise' and the sensation of suddenly catching one's breath.
Page 2
2. "I strongly hope...", as opposed to "I hope you'll come". This means approximately the same as "Surely you'll
come!"
Page 2
3. Surely one doesn't normally say "I wish..." on grounds of self-observation, for this is merely an expression of a
wish. Nevertheless, you can sometimes perceive or discover a wish by observing your own reactions. If you now ask
me, "Do you recognize the same thing in such a case as you express by the utterance in the other case?", then there
is already a mistake in the question. (As if someone asked, "Is the chair I can see the same as the chair on which I
can sit?")
Page 2
4. I say "I hope you'll come", but not "I believe I hope you'll come", but we may well say: "I believe I still hope he'll
come." [Zettel 79: henceforth Z will stand for Zettel.]
Page 2
5. "But doesn't one experience meaning?" "But doesn't one hear the piano?" Each of these questions can be meant,
i.e., used, factually or conceptually. (Temporally, or timelessly.)
Page 2
6. He says "I want to go out now", then suddenly says "No", and does something else. As he said "No", it suddenly
occurred to him that he wanted first of all to...--He said "No", but did he also think "No"? Didn't he just think about
that other thing? One can say he was thinking about it. But to do that he didn't have to pronounce a thought, either
silently or out loud. To be sure, he could later clothe the intention in a sentence. When his intentions changed maybe
a picture was in his mind, or he didn't just say "No", but some one word, the equivalent of a picture. For example, if
he wanted to close the closet then maybe he said "The closet!"; if he wanted to wash his hands he might have looked
at them and made a face. "But is that thinking?"--I don't know. Don't we say in such cases that someone has
"thought something over", has "changed his mind"?
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But is it absolutely necessary that he gain the command of a language for this kind of thinking? Couldn't an
"intelligent" beast act this way? It has been trained to fetch an object from one place and take it to another. Now it
starts walking toward the goal without the object, suddenly turns around (as if it had said "Oh, I forgot...!") and
fetches the object, etc. If we were to see something like this we would say that at that time something had happened
within it, in its mind. What then has happened within me when I act this way? "Not much at all," I would like to say.
And what happens inside is no more important that what can happen outside, through speaking, drawing etc. ((From
which you can learn how the word "thinking" is used.))
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7. Now imagine that someone has to construct something with blocks, or 'Meccano'. He tries out different pieces,
tries to combine them, maybe even makes a sketch, etc., etc. Now one says that he has been thinking during this
activity!--To be sure, in saying this one distinguishes this action from another of a very different sort. But is it a good
description of this difference to say that in the one case something else accompanies the manual activity? Could one
isolate this something else, perhaps, and make it occur without the other activity?
It isn't true that thinking is a kind of speaking, as I once said.†1 The concept 'thinking' is categorically
different from the concept 'speaking'. But of course thinking is neither an accompaniment of speaking nor of any
other process.
This means that it is impossible to have the "thought-process", for example, proceed unaccompanied. Nor
does it have divisions which correspond to the divisions of other activities (speaking, for example). That is, when we
do speak of a "thought-process" it is something like operating (in writing or orally) with signs. Inferring or
calculating might be called a "thought-process".
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8. Likewise it wouldn't be completely wrong to call speaking "the instrument of thinking". But one can't say that the
process of speaking is an instrument of the process of thinking, or that language is, as it were, the carrier of thought,
as, for example, the notes of a song might be called the carriers of the words.
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9. The word "thinking" can be used to signify, roughly speaking, a talking for a purpose, i.e., a speaking or writing, a
speaking in the imagination, a "speaking in the head", as it were.
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10. We say, "Think about what you want to say before you speak". One way of doing this is to recite one's speech
softly to oneself, or to write it down and make corrections. For instance, one might recite a sentence, shake his head,
say "That is too long", etc., and then restate the sentence in another form.
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11. What thinking is might be described by describing the difference between someone feeble-minded and a normal
child who is beginning to think. If one wanted to indicate the activity which the normal person learns and which the
feeble-minded cannot learn, one couldn't derive it from their behaviour.
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12. The word "thinking" is used in a certain way very differently from, for example, "to be in pain", "to be sad", etc.:
we don't say "I think" as the expression of a mental state. At most we say "I'm thinking". "Leave me alone, I'm
thinking about...." And of course one doesn't mean by this: "Leave me alone, I am now behaving in such and such a
way." Therefore 'thinking' is not behaviour.
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13. "I thought: 'this stick is too long, I must try another one'." While thinking that maybe I said nothing at all to
myself, maybe one or two words. And yet this report is not untrue (or at any rate it may be true). It has a use. We
say, for example, "Yes, I watched you and I thought that you were thinking that".
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14. "Man thinks, feels, wishes, believes, intends, wants, knows." That sounds like a reasonable sentence, just like
"Man draws, paints, models", or "Man is acquainted with string instruments, wind instruments,...". The first sentence
is an enumeration of all those things man does with his mind. But just as one could add: "And isn't man also
acquainted with instruments made from squealing mice?" to the sentence about the instruments--and the answer
would be "No"--so there must be added to the enumeration of the mental activities a question of this kind: "And
can't men also...?"
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15. Someone says: "Man hopes." How should this phenomenon of natural history be described?--One might
observe a child and wait until one day he manifests hope; and then one could say "Today he hoped for the first
time". But surely that sounds queer! Although it would be quite natural to say "Today he said 'I hope' for the first
time". And why queer? One does not say that a suckling hopes
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that..., but one does say it of a grown-up.--Well, bit by bit daily life becomes such that there is a place for hope in it.
[Z 469a.]
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16. In this case I have used the term "embedded", have said that hope, belief, etc., were embedded in human life, in
all of the situations and reactions which constitute human life. The crocodile doesn't hope, man does. Or: one can't
say of a crocodile that it hopes, but of man one can.
But how would a human being have to act for us to say of him: he never hopes? The first answer is: I don't
know. It would be easier for me to say how a human being would have to act who never yearns for anything, who is
never happy about anything, or who is never startled or afraid of anything.
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17. Fear behaviour on fearful occasions (etc.) is a phenomenon of our life. But fear? Well, instead of "I am afraid"
one could say "The phenomenon of fear is occurring in me", in which case one doesn't think of his own behaviour.
But then could one say in the same sense, "The phenomenon of fear is occurring in him"?
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18. If I tell someone "Men think, feel,... ", it seems I am making a statement of natural history to him. It might be
intended to show him something about the difference between man and the various kinds of animals. But could one
give an example of this by saying, "Yes, I myself, for instance, am now seeing"? Then is "I see..." a statement of
natural history about myself? Couldn't I just as well say, "I am not seeing"?
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19. "Man thinks, is afraid, etc. etc.": that is the reply one might give to someone who asked what chapters a book on
psychology should contain. [Z 468.]
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20. Where do we get the concept 'thinking' from, which we now want to consider here? From everyday language.
What first fixes the direction of our attention is the word "thinking". But the use of this word is tangled. Nor can we
expect anything else. And that can of course be said of all psychological verbs. Their employment is not so clear or
so easy to get a synoptic view of, as that of terms in mechanics, for example. [Z 113.]
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21. Psychological words are similar to those which pass over from everyday language into medical language.
("Shock.")
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22. I tell someone: "Human beings think." He asks me, "What is thinking?"--Now I explain the use of this word to
him. But when I have finished, is the first sentence still a piece of information?
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((Couldn't an ant speak this way to an ant?))
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23. "Human beings think, grasshoppers don't." This means something like: the concept 'thinking' refers to human
life, not to that of grasshoppers. And one could impart this to a person who doesn't understand the English word
"thinking" and perhaps believes erroneously that it refers to something grasshoppers do.
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24. "Grasshoppers don't think." Where does this belong?--Is it an article of faith, or does it belong to natural history?
If the latter, it ought to be a sentence something like: "Grasshoppers can't read and write." This sentence has a clear
meaning, and even though it is perhaps never used, still it is easy to imagine a use for it.
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25. "A steam engine has a crosshead, a steam turbine doesn't." To whom, and in what context, would one say that?
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26. "Can a human being understand what 'reading' is unless he himself can read, can he understand what 'fearing' is
without knowing fear, etc.?" Well, an illiterate man can certainly say that he can't read but that his son has learned
how. A blind man can say that he is blind and the people around him sighted. "Yes, but doesn't he after all mean
something different from a sighted man when he uses the words 'blind' and 'sighted'?" What is the ground of one's
inclination to say that? Well, if someone did not know what a leopard looked like, still he could say and understand
"That place is very dangerous, there are leopards there". He would perhaps all the same be said not to know what a
leopard is, and so not to know, or not completely, what the word "leopard" means, until he is shown such an animal.
Now it strikes us as being the same with the blind. They don't know, so to speak, what seeing is like.--Now is 'not
knowing fear' analogous to 'never having seen a leopard'? That, of course, I want to deny. [Z 618, beginning at "A
blind man can say".]
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27. The question is: What kind of language-games can someone who is unacquainted with fear eo ipso not play?
One could say, for example, that he would watch a tragedy without understanding it. And that could be
explained this way: When I see someone else in a terrible situation, even when I myself have nothing to fear, I can
shudder, shudder out of sympathy. But
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someone who is unacquainted with fear wouldn't do that. We are afraid along with the other person, even when we
have nothing to fear; and it is this which the former cannot do. Just as I grimace when someone else is being hurt.
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28. Good; but isn't it conceivable that someone who has never felt pain could still feel it in the form of pity? So no
matter what happened to him he wouldn't groan, but would whenever someone else was being hurt.
But would we really say that he feels pity? Wouldn't we say. "It really isn't pity because he isn't acquainted
with any pain of his own"--? Or we could imagine people saying in such a case that God has given this man a feeling
for the sorrow or fear of others. One might call something like that an intuition.
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29. "Human beings sometimes think." How did I learn what "thinking" means?--It seems I can only have learned it
by living with people.--To be sure, one could imagine seeing human life in a film, or being allowed merely to observe
life without participating in it. Anyone who did this would then understand human life as we understand the life of
fish or even of plants. We can't talk about the joy and sorrow, etc., of fish.
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30. But of course I don't mean that as a matter of experience one can't understand it if he doesn't join in living (as if
one were to say: no one can learn how to row merely by watching others rowing).--Rather, I mean that I wouldn't
say either of myself (or of others) that we understood manifestations of life that are foreign to us. And here, of
course, there are degrees.
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31. Thinking cannot be called a phenomenon, but one can speak of "phenomena of thinking", and everyone will
know what kinds of phenomena are meant.
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32. Clearly, one can say: "Think about occasions for anger and phenomena of anger (anger-behaviour)."
But if I call anger a phenomenon then I have to call my anger, my experience of anger, a phenomenon. (A
phenomenon of my inner life, for instance.)
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33. Look at it purely behaviouristically: Someone says, "Man thinks wishes, is happy, angry, etc.". Imagine that
these words were only
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about certain forms of behaviour on certain occasions. One could suppose that whoever talks about human beings in
this way had first observed these kinds of behaviour in other beings and was now saying that these phenomena
could also be observed in human beings. That would be like our saying this of a species of animals.--
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34. Suddenly I smile and say "...". When I smiled the thought had occurred to me.
Of what did it consist? It consisted of nothing at all; for the picture or word, etc., which may perhaps have
appeared was not the thought.
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35. I would like to say: Psychology deals with certain aspects of human life.
Or: with certain phenomena.--But the words "thinking", "fearing", etc., etc. do not refer to these phenomena.
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36. "But how is it possible to see a thing according to an interpretation?"--The question represents it as a queer fact;
as if something were being forced into a form it did not really fit. But no squeezing, no forcing took place here.
[Philosophical Investigations II, xi, p. 200, paragraph e.]
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37. Now the strange thing is that one doesn't know, as it were, what he is doing when he regards or sees the figure
now as this, now as that. That is, one is inclined to ask, "How am I doing that?" "What other thing am I actually
seeing?"--And as an answer to this one gets no relevant explanation.
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38. For the question is not, 'What am I doing when....?' (for this could only be a psychological question)--but rather,
'What meaning does the statement have, what can be deduced from it, what consequences does it have?'
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39. Anyone who failed to perceive the change of aspect would not be inclined to say, "Now it looks completely
different!", or "It seems as if the picture had changed, and yet it hasn't!", or "The form has remained the same, and
yet something has changed, something which I should like to call the conception, and which is seen!"--
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40. To see something first as this and then as that could be a mere game. One could speak to a child in this way--for
instance: "Now it is...!, now...!", and it reacts; I mean it laughs, and now it practices the thing in various ways (as if
you were to point out that
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vowels have colours). Another child neither perceives these colours nor understands what is meant by that change of
aspect.
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41. But what if this child were given the problem of locating the configuration in the figure
?†1 (This could be included as a problem in the child's first lessons.) Would he be unable to solve this problem (or
one of finding a series of different configurations in that figure) if he were not aware of a change of aspect, if he were
not inclined to say that the figure was somehow changing, becoming a different pattern, or something like that?
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42. You say that a normal person would see the figure †2 as two circles cut through by a straight line.
But how does that come out? If he copies the figure, for example, should I say it comes out in the way he does it? If
he describes the figure verbally, does it come out in the description he chooses? This choice could be determined by
the convenience of representing it this particular way. Even if the child hit upon different ways of reproducing it
pictorially (a different sequence of lines), would that be our criterion for the change of aspect?--But if he says, "Now
it is...--now...", if he talks as if he saw a different object every time, then we'll say that he sees the figure in different
ways.
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43. The essential thing about seeing is that it is a state, and such a state can suddenly change into another one. But
how do I know that a person is in such a state, and therefore is not in one comparable to a disposition, like knowing,
understanding, or having a conception? What are the logical characteristics of such a state?
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44. For to say that one recognizes the state as a state whenever one is in it is nonsense. By what does he recognize
it?
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(The criterion of identity.)
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45. I want to talk about a "state of consciousness", and to use this expression to refer to the seeing of a certain
picture, the hearing of a tone, a sensation of pain or of taste, etc. I want to say that believing, understanding,
knowing, intending, and others, are not states of
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consciousness. If for the moment I call these latter "dispositions", then an important difference between dispositions
and states of consciousness consists in the fact that a disposition is not interrupted by a break in consciousness or a
shift in attention. (And that of course is not a causal remark.) Really one hardly ever says that one has believed or
understood something "uninterruptedly" since yesterday. An interruption of belief would be a period of unbelief,
not, e.g., the withdrawal of attention from what one believes, or, e.g., sleep.
(The difference between 'knowing' and 'being aware of.) [Z 85, beginning at "Really one".]
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46. This is likely to be the point at which it is said that only form, not content, can be communicated to others.--So
one talks to oneself about the content. And what does that mean? (How do my words 'relate' to the content I know?
And to what purpose?) [Z 87.]
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47. In these considerations we often draw what can be called "auxiliary lines". We construct things like the "soulless
tribe"--which drop out of consideration in the end. That they dropped out had to be shown.
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48. "Pain is a state of consciousness, understanding is not."--"Well, the thing is, I don't feel my understanding."--But
this explanation achieves nothing. Nor would it be any explanation to say: What one in some sense feels is a state of
consciousness. For that would only mean: State of consciousness = feeling. (One word would merely have been
replaced by another.) [Z 84.]
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49. Look at yourself when you are writing, and notice how your hand forms the letters without your actually causing
it to do so. To be sure, you feel something in your hand, all sorts of tensions and pressures, but that they are
necessary to produce these letters is something which you know nothing about.
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50. Where there is genuine duration one can tell someone: "Pay attention and give me a signal when the picture, the
rattling etc. alters."
Here there is such a thing as paying attention. Whereas one cannot follow with attention the forgetting of
what one knew or the like. [Z 81.]
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51. Think of this language-game: Determine how long an impression lasts by means of a stop-watch. The duration of
knowledge, ability, understanding, could not be determined in this way. [Z 82.]
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52. "But the difference between knowing and hearing surely doesn't reside simply in such a characteristic as the kind
of duration they have. They are surely wholly and utterly distinct!" Of course. But one can't say: "Know and hear,
and you will notice the difference". [Z 83.]
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53. One can't look at knowing and hearing to see how different they are. Just as one can't look at pine wood and a
table to get an impression of their difference.
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54. If I use the language-game with the stop-watch, for example, in order to demonstrate to myself the difference
between the concepts knowing and seeing, this certainly gives the impression that I am showing an extremely fine
distinction, where the real one, after all, is enormous.
But this enormous distinction (I would always say) consists precisely in the fact that the two concepts are
embedded quite differently in our language-games. And the difference to which I called attention was merely a
reference to this pervasive distinction.
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55. The child learns "I know that now" and "I hear that now". But my God!, how different the occasions, the
applications, everything! How can one even compare the use? It is hard to see how to arrange them so as to show
their differences.
Where the difference is so great it is hard to point to a distinction.
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56. I can say, "This word is used thus and so, that word thus and so".
The basis for comparing them is hard to see; not their difference.
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57. The general differentiation of all states of consciousness from dispositions seems to me to be that one cannot
ascertain by spot-check whether they are still going on. [Z 72.]
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58. We need to reflect that a state of language is possible (and presumably has existed) in which it does not possess
the general concept of sensation, but does have words corresponding to our "see", "hear", "taste". [Z 473.]
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59. We call seeing, hearing,... sense-perception. There are analogies and connexions between these concepts, and
these are our justification for so taking them together. [Z 474.]
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60. It can, then, be asked: what kind of connexions and analogies exist between seeing and hearing? Between seeing
and touching? Between seeing and smelling?--[Z 475.]
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61. And if we ask this, then the senses, so to say, at once shift further apart from one another than they seemed to be
at first sight. [Z 476.]
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62. Psychological concepts are just everyday concepts. They are not concepts newly fashioned by science for its
own purpose, as are the concepts of physics and chemistry. Psychological concepts are related to those of the exact
sciences as the concepts of the science of medicine are to those of old women who spend their time nursing the sick.
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63. Plan for the treatment of psychological concepts.
Psychological verbs characterized by the fact that the third person of the present is to be identified by
observation, the first person not.
Sentences in the third person of the present: information. In the first person present, expression. ((Not quite
right.))
Sensations: their inner connexions and analogies.
All have genuine duration. Possibility of giving the beginning and the end. Possibility of their being
synchronized, of simultaneous occurrence.
All have degrees and qualitative mixtures. Degree: scarcely perceptible--unendurable.
In this sense there is not a sensation of position or movement.
Place of feeling in the body: differentiates seeing and hearing from sense of pressure, temperature, taste and
pain.
(If sensations are characteristic of the position and movements of the limbs, at any rate their place is not the
joint.)
One knows the position of one's limbs and their movements. One can give them if asked, for example. Just as
one also knows the place of a sensation (pain) in the body.
Reaction of touching the painful place.
No local sign about the sensation. Any more than a temporal sign about a memory-image. (Temporal signs in
a photograph.)
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Pain differentiated from other sensations by a characteristic expression. This makes it akin to joy (which is
not a sense-experience).
"Sensations give us knowledge about the external world."
Images:
Auditory images, visual images--how are they distinguished from sensations? Not by "vivacity".
Images tell us nothing, either right or wrong, about the external world. (Images are not hallucinations, nor yet
fancies.)
While I am looking at an object I cannot imagine it.
Difference between the language-games: "Look at this figure!" and: "Imagine this figure!"
Images are subject to the will.
Images are not pictures. I do not tell what object I am imagining by the resemblance between it and the
image.
Asked "What image have you?" one can answer with a picture. [Z 472, 483, 621.]
64. One would like to say: The imaged is in a different space from the heard sound. (Question: Why?) [Z 622,
beginning of a.]
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65. I read a book and have all sorts of images while I read, i.e. while I am looking attentively. [Z 623.]
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66. People might exist who never use the expression "seeing something with the inner eye" or anything like it, and
these people might be able to draw and model 'out of imagination' or memory, to mimic the characteristic behaviour
of others etc. They might also shut their eyes or stare into vacancy as if blind before drawing something from
memory. And yet they might deny that they then see before them what they go on to draw. [Z 624, beginning.]
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67. "Do you see the way she's coming in the door?"--and now one imitates it.
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68. That is to say, 'seeing' is inseparably connected with 'looking'. ((I.e., that is one way of fixing the concept, which
produces a physiognomy.))
The words which describe what we see are properties of things. We don't learn their meaning in connection
with the concept of 'inner seeing'.
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69. But if we ask, "What is the difference between a visual picture and an image-picture?"--the answer could be: The
same description can represent both what I see and what I imagine.
To say that there is a difference between a visual picture and an image-picture means that one imagines
things differently from the way they appear.
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70. I might also have said earlier: The tie-up between imaging and seeing is close; but there is no similarity. [Z
625a.]
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71. The language-games employing both these concepts are radically different--but hang together. [Z 625b.]
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72. A difference: 'trying to see something' and 'trying to form an image of something'. In the first case one says:
"Look, just over there!", in the second "Shut your eyes!" [Z 626.]
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73. So don't you know, after all, whether what is seen (e.g., an after-image) and an image look exactly alike? (Or
should I say: are?)--This question could only be an empirical one, and could only mean something like: "Does it
ever, or even often, happen that a person can keep an image in front of his mind uninterruptedly and for some time,
and describe it in detail, as one can do, for example, with an after-image?"
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74. "Now can you still see the bird?"--"I fancy that I can still see it." That doesn't mean: Maybe I am imagining it.
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75. "Seeing and imaging are different phenomena."--The words "seeing" and "imaging" are used differently. "I see"
is used differently from "I have an image", "See!" differently from "Form an image!", and "I am trying to see it"
differently from "I am trying to form an image of it".--"But the phenomena are: that men see and that we form
images of things." A phenomenon is something that can be observed. Now how does one observe that men see?
I can observe, e.g., that birds fly, or lay eggs. I can tell someone, "You see, these creatures fly. Notice how
they flap their wings and lift themselves into the air." I can also say, "You see, this child is not
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blind. It can see. Notice how it follows the flame of the candle." But can I satisfy myself, so to speak, that men see?
"Men see."--As opposed to what? Maybe that they are all blind?
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76. Can I imagine a case in which I might say, "Yes, you are right, men see"?--Or: "Yes, you are right, men see, even
as I do."
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77. "Seeing and understanding†1 are different phenomena."--The words "seeing" and "understanding"†1 have
different meanings! Their meanings relate to a host of important kinds of human behaviour, to phenomena of human
life.
To close one's eyes in order to form an image of something is a phenomenon; to strain in looking another; to
follow a thing in motion with one's eyes yet another.
Imagine someone saying: "Man can see or be blind"! One could say that "seeing", "imaging", and "hoping"
are simply not words for phenomena. But of course that doesn't mean that the psychologist doesn't observe
phenomena. [a: Z 629.]
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78. To say that imaging is subject to the will can be misleading, for it makes it seem as if the will were a kind of
motor and the images were connected with it, so that it could evoke them, put them into motion, and shut them off.
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79. Isn't it conceivable that there should be a man for whom ordinary seeing was subject to the will? Would seeing
then teach him about the external world? Would things have colours if we could see them as we wished?
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80. It is just because imaging is subject to the will that it does not instruct us about the external world.
In this way--but in no other--it is related to an activity such as drawing.
And yet it isn't easy to call imaging an activity. [a: cf. Z 627.]
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81. But what if I tell you: "Imagine a melody"? I have to 'sing it inwardly' to myself. That will be called an activity
just as much as calculating in the head.
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82. Consider also that you can order someone to "Draw N. N. so as to be like your image of him", and that whether
he does this is not determined by the likeness of the portrait. Analogous to this is the fact that I have an image of N.
N. even if my image is wrong.
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83. If I say that imaging is subject to the will that does not mean that it is, as it were, a voluntary movement, as
opposed to an involuntary one. For the same movement of the arm which is now voluntary might also be
involuntary.--I mean: it makes sense to order someone to "Imagine that", or again: "Don't imagine that."
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84.†1 But doesn't the connection with the will refer merely, so to speak, to the machinery which produces or
changes what is imaged (the image-picture)?--Here no picture is engendered, unless someone manufactures a
picture, a real picture.
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85. The dagger which Macbeth sees before him is not an imagined dagger. One can't take an image for reality nor
things seen for things imaged. But this is not because they are so dissimilar.
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86. One objection to the imagination's being voluntary is that images often beset us against our will and remain,
refusing to be banished.
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Yet the will can struggle against them. But isn't calling them voluntary like calling a movement of my arm
voluntary when someone forces my arm against my will?
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87. If someone insists that what he calls a "visual image" is like a visual impression, say to yourself once more that
perhaps he is making a mistake! Or: Suppose that he is making a mistake. That is to say: What do you know about
the resemblance of his visual impression and his visual image?! (I speak of others because what goes for them goes
for me too.)
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So what do you know about this resemblance? It is manifested only in the expressions which he is inclined
to use; not in something he uses those expressions to say.
"There's no doubt at all: visual images and visual impressions are of the same kind!" That must be something
you know from your own experience; and in that case it is something that may be true for you and not for other
people. (And this of course holds for me too, if I say it.)
Nothing is more difficult than facing concepts without prejudice. (And that is the principal difficulty of
philosophy.) [a, b: Z 630; c: Z 631.]
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88. Forming an image of something is comparable to an activity. (Swimming.)
When we form an image of something we are not observing. The coming and going of the pictures is not
something that happens to us. We are not surprised by these pictures, saying "Look!..." [b: Z 632.]
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89. We do not banish visual impressions, as we do images. [Z 633, beginning.]
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90. If we could banish impressions and summon them before our minds then they couldn't inform us about
reality.--So do impressions differ from images only in that we can affect the latter and not the former? Then the
difference is empirical! But this is precisely what is not the case.
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91. Is it conceivable that visual impressions could be banished or called back? What is more, isn't it really possible?
If I look at my hand and then move it out of my visual field, haven't I voluntarily broken off the visual impression of
it?--But I will be told that that sort of thing isn't called "banishing the picture of the hand"! Certainly not; but where
does the difference lie? One would like to say that the will affects images directly.
For if I voluntarily change my visual impression, then things obey my will.
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92. But what if visual impressions could be controlled directly? Should I say, "Then there wouldn't be any
impressions, but only images"? And what would that be like? How would I find out, for instance, that another
person has a certain image? He would tell me.--But how would he learn the necessary words, let us say "red" and
"round"? For surely I couldn't teach them to him by pointing to
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something red and round. I could only evoke within myself the image of my pointing to something of the sort. And
furthermore I couldn't test whether he was understanding me. Why, I could of course not even see him; no, I could
only form an image of him.
Isn't this hypothesis really like the one that there is only fiction in the world and no truth?
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93. And of course I myself couldn't learn or invent a description of my images. For what would it mean to say, e.g.,
that I was forming an image of a red cross on a white background? What does a red cross look like? Like this??--But
couldn't a higher being know intuitively what images I am forming, and describe them in his language, even though I
couldn't understand it? Suppose that this higher being were to say, "I know what image this man is now forming; it
is this:...".--But how was I able to call that "knowing"? It is completely different from what we call "knowing what
someone else is imaging". How can the normal case be compared with the one we have invented?
If I think of myself in this case as a third person, then I would have absolutely no idea what the higher being
means when it says, with regard to someone who has only images and no impressions, that it knows which images
that man has.
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94. "But nevertheless can't I still imagine such a case?" The first thing to say is, you can talk about it. But that doesn't
show that you have thought it through completely. (5 o'clock on the sun.)†1
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95. One would also like to talk about what a visual impression and an image look like. And also to ask, perhaps,
"Couldn't something look like my present visual impression for instance, but otherwise behave as an image?" And
clearly there is a mistake here.
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96. But imagine this: We get someone to look through a hole into a kind of peep show, and inside we now move
various objects and figures about, either by chance or intentionally, so that their movement is exactly what our
viewer wanted, so that he fancies that what he sees is obeying his will.--Now could he be deluded, and believe that
his visual impressions are images? That sounds totally absurd. I don't even need the peep show, but have only to
look at my hand and move it, as mentioned above. But even if I could will the curtain over there to move, or could
make it disappear, I should
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still not interpret that as something that was going on in my imagination.†1(?)
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97. I simply can't begin to take an impression for an image. But what does that mean? Could I think of a case in
which someone else did that? Why isn't that conceivable?
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98. If someone really were to say "I don't know whether I am now seeing a tree or having an image of it", I should at
first think he meant: "or just fancying that there is a tree over there". If he does not mean this, I couldn't understand
him at all--but if someone tried to explain this case to me and said "His images are of such extraordinary vivacity
that he can take them for impressions of sense"--should I understand it then? [Z 634.]
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99. Still, imagine a person who says, "My images are as vivid today as real visual impressions".--Would he have to
be lying or talking nonsense? No, certainly not. To be sure I would first have to have him tell me how this manifests
itself.
But if he were to tell me, "Often I don't know whether I see something or only have an image of it", then I
wouldn't call this a case of overly vivid imaging.
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100. But must one not distinguish here: forming the image of a human face, as we say, but not in the space that
surrounds me--and on the other hand: forming an image of a picture on that wall over there?
At the request "Imagine a round spot over there" one might fancy that one really was seeing one there.[Z
635.]
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101. To be sure, if I say "Isn't there really a spot over there?", and therefore perhaps look there more closely, then
what I am here calling an image does not obey my will. And of course if I fancy something to be the case, that does
not obey my will.
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102. You must not forget that material implication too does in fact have its use, its practical use, even if it does not
occur very frequently.
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103. Anyone who negates the sentence "If p, then q" negates a connection. He is saying: "It does not have to be this
way." And the words "have to" point to the connection.
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104. "If p, then q" does not follow from "Not p and not q". It can't be inferred from "Not p and not q". The sense of
"If p, then q" is fundamentally different from the sentence "p implies q", even if there is a connection. It is this: "p
and q", which makes the implication true, also makes the sentence "If... then..." true, or at least it supports it. "p and
not q" contradicts the implication as well as the "If-then" sentence, or at least it is unfavourable to its truth. "Not p
and q" and "Not p and not q" verify the implication and determine nothing about the truth of "If..., then...".
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105. "If this happens, that will happen. If I am right, you pay me a shilling, if I am wrong, I pay you one, if it
remains undecided, neither pays." This might also be expressed like this: The case in which the antecedent does not
come true does not interest us, we aren't talking about it. Or again: we do not find it natural to use the words "yes"
and "no" in the same way as in the case (and there are such cases) in which we are interested in the material
implication. By "No" we mean here "p and not q", by "Yes", only "p and q". [Cf. Z 677.]
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106. For example, it is quite common to bet on the truth of a prediction. So, if we bet on the assertion "If p happens,
then q will happen", then someone will say, "If you're right, I'll pay you..., if not..."; but if p does not happen the bet
will be off. Here we are dealing with two different kinds of use of the negation of a sentence. And just as "not not p"
is not the same as p, when double negation is meant to strengthen the negation, in the same way "p v not p", in the
sense in which we are using the negation, is not necessarily a tautology. In the above case, the assertion that that
conditional sentence is true or false should actually assert the definite occurrence of the event.†1 For that assertion
says that the conditional sentence will not remain undecided.
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107. The sentence "Imagination is subject to the will" is not a sentence of psychology.
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108. I learn the concept 'see' in connection with 'look'. The use of the one word is connected with that of the other.
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109. If one says, "the experiential content of seeing and having an image is essentially the same", then this is true
insofar as a painted picture can represent both what one sees and what one has an image of. Only, one mustn't allow
oneself to be deceived by the myth of the inner picture.
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110. The 'imagination-picture' does not enter the language-game in the place where one would like to surmise its
presence. [Z 636.]
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111. I learn the concept 'seeing' along with the description of what I see. I learn to observe and to describe what I
observe. I learn the concept 'to have an image' in an entirely different context. The descriptions of what is seen and
what is imaged are indeed of the same kind, and a description might be of the one just as much as of the other; but
otherwise the concepts are thoroughly different. The concept of imaging is rather like one of doing than of receiving.
Imagining might be called a creative act. (And is of course so called.) [Z 637.]
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112. "Yes, but the image itself, like the visual impression, is surely the inner picture,†1 and you are talking only of
differences in the production, the coming to be, and in the treatment of the picture." The image is not a picture, nor is
the visual impression one. Neither 'image' nor 'impression' is the concept of a picture, although in both cases there is
a tie-up with a picture, and a different one in either case. [Z 638.]
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113. "But couldn't I imagine an experiential content of the same kind as visual images, but not subject to the will,
and so in this respect like visual impressions?" In this case, the misleading thing is the talk about experiential
content. If we talk about an experiential content which is typical of visual imaging, then the content within me must
be comparable to the content within you. And, strange as it may sound, I believe one would have to say that the
experiential content--if we are to use this concept here at all--is the same for a visual image and a visual impression.
And that sounds paradoxical, because everyone will want to cry out: You're not going to tell me that these
two--image and impression--could ever be mistaken for each other!--I could answer that this is as unlikely as
confusing drawing and seeing, for
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example. But what is drawn and what is seen still could be the same thing. Image and impression do not 'look'
different. [First sentence: Z 640.]
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114. But one could also say that "experiential content" does not mean the same thing when it is applied to image and
impression, but only related things. If, for example, I form an image of a face exactly as it looks, and then see it later,
my impression and my image have the same experiential content. One can not say that it is not the same on the
grounds that an image and an impression never look alike.
The content of both, therefore, is this--(here I might point to a picture). But I wouldn't have to call it "the
content" both times.
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115. Image and intention. Forming an image can also be compared to creating a picture in this way--namely, I am
not imagining whoever is like my image; no, I am imagining whoever it is I mean to imagine.
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116. I believe that if you do compare imaging with a bodily movement like breathing, which sometimes happens
voluntarily, sometimes involuntarily, then you musn't compare a sense impression with a movement at all. The
difference is not that the one takes place whether we will it or not, whereas we control the other. Rather, one concept
resembles that of an action, the other doesn't. The difference is more like that between seeing my hand move--and
knowing (without seeing it) that I am moving it.
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117. "If I shut my eyes, there he is in front of me."--One could suppose that such expressions are not learned, but
rather poetically formed, spontaneously. That they therefore "seem just right" to one man and then also to the next
one.
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118. "I see him in front of me as plain as day!"--Well, maybe he's really standing in front of you.--"No, my picture
isn't vivid enough for that."
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119. Couldn't we conceive of this phenomenon: By looking at a screen, we might be able to produce pictures on it,
arbitrarily, 'merely by willing them'; we might be able to move them about, to have them disappear etc.,--pictures
which are not only seen by the one who makes them but also by someone else.--Would what I see on this screen be
something like an image? Or--perhaps to put the
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question more precisely: Would "I see... on the screen" mean something like "I have an image of..."?--Or should I
say that the sentence "...is now appearing on the screen" corresponds to "I have an image of..."?--No; that is not the
way it is. The difficulty here lies in my not having a clear concept of what is meant by "producing the pictures by
willing them", etc. For actually the case above is not entirely fantastic: I really can form images of all sorts of things
on a wall covered with spots; and if someone else who looked at the wall should know in each instance what I was
imaging then this would be similar to the case described above. ((But couldn't someone who draws pictures on the
wall also be said to be producing them merely by an act of the will?))
"To move by pure will"--What does that mean? That the pictures always exactly obey my will, whereas my
hand in drawing, my pencil, does not? All the same in that case it would be possible to say: "Usually I form images
of exactly what I want to; today it has turned out differently." Is there such a thing as 'images not coming off'? [b: Z
643.]
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120. If there isn't, then this will be explained by saying that the image-picture is non-corporeal and does not resist the
will--neither by inertia nor by any other means.
No; "I see... on the screen" cannot correspond to my imaging. Neither can "I produce... on the screen"--for
then imaging could succeed or fail. This would be better: "For me what is on this screen now is a picture of...."†1
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121. To be sure, there is a language-game with the order "Imagine...!"--but can this really be that simply assimilated
to "Turn your head to the right!"? Or, to put it differently: Does it make sense simply to say that image-pictures,
inner pictures, obey my will? (N.B.: not "my wish".)
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122. For those things which are normally said to follow or not to follow the will are not 'inner pictures'. It is not clear
therefore, that the concept of following can be applied to the other category directly.
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123. (Clearly the 'arbitrariness' of the imagination cannot be compared to the movement of bodies; for someone else
is also competent to judge whether the movement has taken place; whereas with the movement of my images the
whole point would always be
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what I said I saw--whatever anyone else sees. So really moving objects would drop out of consideration, since no
such thing would be in question.) [Z 641.]
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124. If then one said: "Images are inner pictures, resembling or exactly like my visual impressions, only subject to
my will"--the first thing is that this doesn't yet make sense.
For if someone has learnt to report what he sees over there, or what seems to him to be over there, it surely
isn't clear to him what it would mean if he were ordered now to see this over there, or now to have this seem to him
to be over there. [Z 642.]
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125. Granted, there is a certain relationship between imaging and an action which is expressed in the possibility of
ordering someone to perform either; but the degree of this relationship has yet to be investigated.
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126. "Move your inner picture!" might mean: move the object.
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127. "Move what you see."
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It might also mean: take something that influences your visual impressions.
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128. What a strange phenomenon that a child can actually learn human language! That a child who knows nothing
can start out and learn by a sure path this enormously complicated technique.
This thought occurred to me when on a certain occasion I became conscious of how a child starts with
nothing and one day uses negations, just as we do!
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129. With the sentence "Images are voluntary, sensations are not", one differentiates not between sensations and
images, but rather between the language-games in which we deal with these concepts.
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130. There are what can be called phenomena of seeing and phenomena of imaging; and there is the concept of
seeing and the concept of imaging. Within both pairs one can speak of 'differences'.
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131. If one says "Imagination has to do with the will" then the same connection is meant as with the sentence
"Imaging has nothing to do with observation".
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132. I said there were phenomena of seeing--what did I mean by that? Well, for instance, everything that can be
portrayed in pictures, and that would be described as 'seeing'. Exact observing; looking at a language; someone
blinded by light; the look of joyous surprise; turning away so as not to see something. All the kinds of behaviour
which distinguish a sighted man from a blind one. (After all, there is a reason why precisely these pictures taken
from human life occur to me at this point.)
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133. Phenomena of seeing--that is, what the psychologist observes.
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134. Someone says: "I see a house with green shutters." And you say: "He's not seeing it, he's merely imagining it.
He's not even looking; don't you see him staring into space?"--Very loosely, it could also be put this way: "That's not
the way it looks when somebody sees something; rather, that's the way it looks when he has an image of
something." In this case we're comparing phenomena of seeing with phenomena of imagining. Likewise if we were
to observe two members of an unknown tribe using a word as they perform a certain activity--a word which we have
come to recognize as an equivalent of our "seeing". And we follow their use of that word upon this occasion, and
come to the conclusion that here it must mean "to see with the inner eye". (Similarly, one might also come to the
conclusion that the word must here mean to understand.)
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135. What does it mean to say, for example, that 'see' hangs together with 'observe'?--When we learn how to use
"see" we learn to use it simultaneously and in conjunction with "look", "observe", etc.
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136. Just as in a chess game we learn to use the king in connection with the pawns and the word "king" together with
the word "checkmate".
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137. A language-game comprises the use of several words. [Z 644.]
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138. Nothing could be more mistaken than to say: seeing and forming an image are different activities. That is as if
one were to say that moving and losing in chess were different activities. [Z 645.]
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139. The sentence "Forming an image is voluntary, seeing isn't", or a sentence like this can be misleading.
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When we learn as children to use the words "see", "look", "image", voluntary actions and orders come into
play. But in a different way for each of the three words. The language-game with the order "Look!" and that with the
order "Form an image of...!"--how am I ever to compare them?--If we want to train someone to react to the order
"Look...!" and to understand the order "Form an image of...!" we must obviously teach him quite differently.
Reactions which belong to the latter language-game do not belong to the former. There is of course a close tie-up of
these language-games; but a resemblance?--Bits of one resemble bits of the other, but the resembling bits are not
homologous. [b: Z 646.]
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140. I could imagine something similar for actual games. Two essentially different games--games which might differ
from each other in important respects far more than checkers and chess--could feature the same board and the same
moves, only, if I might put it this way, in different positions. In the one game, e.g., the task might be to
check-mate†1 the other player; in the other game the whole process of check-mating†2 would be given in advance,
and the two players would have a quite different task in connection with it. For instance, the players might be given
two ways of check-mating†2 the other, and they would have to compare the two from a psychological point of view.
Analogously there is a game: to solve a crossword puzzle, and another one: somehow to test the value of several
different solutions I have been given to the puzzle. [First sentence: Z 647.]
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141. Seeing is subject to the will in a different way from forming an image.
Or: 'seeing' and 'forming an image' are related differently to 'willing'.
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142. Nevertheless, images seem to be dull reflections of sense-impressions. When does this seem to be the case, and
to whom? Of course there is such a thing as clarity and unclarity in images. And if I say "My image-picture of him is
much less well-defined than the visual impression I have when I see him", then this is true, for I cannot describe him
nearly as accurately by relying on my image as I can when he is in front of me.†1 Still, it is possible for someone's
eyesight to deceive him to such an extent that the sight of another man is much less clear than the image of him.
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143. If I, and if anyone else, can imagine a pain, or at least we say we can--how is it to be found out whether we are
imagining it right, and how accurately we are imagining it? [Z 535.]
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144. Couldn't there be people who could describe a person's features in minute detail from memory, who even say
that they now suddenly know what he looks like--but who would emphatically deny, when they were asked, that at
that moment they in any way 'saw' the person 'before them' (or anything like that)? People who would find the
expression "I see him before me" totally inappropriate?
This seems to me to be a very important question. Or even: the important question is whether this question
makes sense.--What reason do I have, after all, to believe that this is not the case for all of us? Or, how can I decide
the question whether someone else (I'm excluding myself for the time being) is really 'forming a visual image' of
somebody, or is merely able to describe him in visual terms (to draw him etc.)--plus the fact that he is familiar with
'illumination', if I might phrase it this way, or a state of illumination similar to "Now I know". ((Genuine duration.))
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145. Visual imaging is not just characterized by an ability to draw, and things like that, but also by more subtle
shades of behaviour.
In any case, the description of the image belongs to the language-game of "forming an image". (That does
not mean that in borderline cases this statement cannot appear: "I can form an exact image of it, but I simply cannot
describe it." A game allows for borderline cases--a rule for exceptions. But the exception and the rule could not
change place without destroying the game. The 'transition from quantity to quality'?)
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146. "If exception and rule change place then it just is not the same thing any more!"--But what does that mean?
Maybe that our attitude toward the game will then change abruptly. Is it as if after a gradual loading of one side and
lightening of the other, there was a non-gradual tipping of the balance?
147. What could the description of the image of a sensation of movement look like?
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148. Continuation of the classification of psychological concepts.
Emotions. Common to them: genuine duration, a course. (Rage flares up, abates, vanishes, and likewise joy,
depression, fear.)
Distinction from sensations: they are not localized (nor yet diffuse!).
Common: they have characteristic expression-behaviour. (Facial expression.) And this itself implies
characteristic sensation too. Thus sorrow often goes with weeping, and characteristic sensations with the latter. (The
voice heavy with tears.) But the sensations are not the emotions. (In the sense in which the numeral 2 is not the
number 2.)
Among emotions the directed might be distinguished from the undirected. Fear at something, joy over
something.
This something is the object, not the cause of the emotion.
The language-game "I am afraid" already contains the object.
"Anxiety" is what undirected fear might be called, in so far as its manifestations are related to those of fear.
The content of an emotion--here one imagines something like a picture, or something of which a picture can
be made. (The darkness of depression which descends on a man, the flames of anger.)
The human face too might be called such a picture and its alterations might represent the course of a passion.
What goes to make them different from sensation: they do not give us any information about the external
world. (A grammatical remark.)
Love and hate might be called emotional dispositions, and so might fear in one sense.
It is one thing to feel acute fear, and another to have a 'chronic' fear of someone. But fear is not a sensation.
'Horrible fear': is it the sensations that are so horrible?
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Typical causes of pain on the one hand, and of depression, sorrow, joy on the other. Cause of these also their
object.
Pain-behaviour and the behaviour of sorrow.--These can only be described along with their external
occasions. (If a child's mother leaves it alone it may cry because it is sad; if it falls down, from pain.) Behaviour and
kind of occasion belong together. [a-f: Z 488; g-i: Z 489; j: Z 490; k-l: Z 491; m-p: Z 492.]
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149. Perhaps someone will say: How can you characterize the concept 'pain' by referring to the occasions on which
pain occurs? Pain, after all, is what it is, whatever causes it!--But ask: How does one identify pain?
The occasion determines the usefulness of the signs of pain.
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150. The concept of pain is simply embedded in our life in a certain way. It is characterized by very definite
connexions.
Just as in chess a move with the king only takes place within a certain context, and it cannot be removed
from this context.--To the concept there corresponds a technique. (The eye†1 smiles only within a face.) [a: cf. Z
532, 533.]
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151. Only surrounded by certain normal manifestations of life, is there such a thing as an expression of pain. Only
surrounded by even more far-reaching particular manifestations of life, such as the expression of sorrow or
affection. And so on. [Z 534.]
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152. Emotional attitudes (e.g. love) can be put to the test, but not emotions. [Cf. Z 504.]
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153. I am inclined to say: emotions can colour thoughts; bodily pain cannot. Therefore let us speak of sad thoughts,
but not, analogously, of toothachey thoughts. It is as if one might say: Fear or indeed hope could consist only of
thoughts, but pain could not. Above all pain has the characteristics of sensation and fear does not. Fear hangs
together with misgivings, and misgivings are thoughts.
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154. Hope can be called an emotion. That is, it can be placed in the same category as fear, anger, joy. It is related to
belief, which is not an emotion. There is no bodily expression typical of belief.
Compare the meaning of "uninterrupted pain" with "uninterrupted anger", jubilation, sorrow, joy, fear, and
on the other hand, "uninterrupted belief ", or "uninterrupted hope".
But again, fear, hope, longing, expectation, are hard to compare with each other. Longing is a mental
preoccupation with a certain object. Fear of an event (apprehension) seems to be similar; but not the fear of a dog
barking at me. Here two different words can be used. Likewise "expect" can mean: to believe that this or that will
happen--but also: to occupy one's time with thoughts and activities of expectation, i.e., wait for.
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155. Belief is not any kind of occupation with the object of belief. Fear, however, longing, and hope, occupy
themselves with their objects.
In a scientific investigation we say all sorts of things, we make many statements whose function in the
investigation we don't understand. For not everything is said with a conscious purpose; our mouth simply runs. We
move through conventional thought patterns, automatically perform transitions from one thought to another
according to the forms we have learned. And then finally we must sort through what we have said. We have made
quite a few useless, even counter-productive motions and now we must clarify our movements of thought
philosophically. [b: Culture and Value, p. 64.]
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156. If I tell you "I have been afraid of his arrival all day long"--I could, after all, go into detail: Immediately upon
awakening I thought.... Then I considered.... Time and again I looked out of the window, etc., etc. This could be
called a report about fear. But if I then said to somebody, "I am afraid..."--would that be as it were a groan of fear, or
an observation about my condition?--It could be either one, or the other: It might simply be a groan of fear; but I
might also want to report to someone else how I have been spending the day. And if I were now to say to him: "I
have spent the whole day in fear (here details might be added) and now too I am full of anxiety"--what are we to say
about this mixture of report and
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statement? Well what should we say other than that here we have the use of the word "fear" in front of us?
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157. If there were people who felt a stabbing pain in their left side in those cases where we express misgivings with
feelings of anxiety--would this stabbing sensation take the place with them of our feeling of fear?--So if we observed
these people and noticed them wincing and holding their left side every time they expressed a misgiving, i.e., said
something which for us at any rate would be a misgiving--would we say: These people sense their fear as a stabbing
pain? Clearly not.--
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158. Why does one use the word "suffering" for pain as well as for fear? Well, there are plenty of tie-ups.--[Z 500.]
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159. Suppose it were said: Gladness is a feeling, and sadness consists in not being glad.--Is the absence of a feeling a
feeling? [Z 512.]
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160. If I say "Every time I thought about it I was afraid"--did fear accompany my thoughts?--How is one to conceive
of separating what does the accompanying from what is accompanied?
We could ask: How does fear pervade a thought? For the former does not seem to be merely concurrent with
the latter. To be sure, if I say "I think about it with anguish", the thought expressed in these words might seem to run
concurrently with a certain feeling in my chest, and this might seem to be alluded to. But the use of this sentence is
something different from that.
One also says: "Thinking about it takes my breath away", and means not only that as a matter of experience
this or that sensation or reaction accompanies this thought.
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161. To the utterance: "I can't think of it without fear" one replies: "There's no reason for fear, for...." That is at any
rate one way of dismissing fear. Contrast with pain.
Is disgust a sensation?--Is it localized?--And it has an object, as does fear. And there are characteristic
sensations here. [a: Z 501.]
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162. Indeed, you must always ask yourself: What do you tell someone else with these sentences? And this means:
What use can he make of them?
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163. I give notice that I am afraid.--Do I recall my thoughts of the past half hour in order to do that, or do I let a
thought of the dentist quickly cross my mind in order to see how it affects me; or can I be uncertain of whether it is
really fear of the dentist, and not some other physical feeling of discomfort?
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164. Or is giving notice of being afraid like a very slight groan of fear? No; for in groaning I don't necessarily want to
tell somebody else that I am afraid. The notice is, as it were, part of a conversation.
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165. Can one say: "I am only afraid of the operation at the moment I am thinking about it"? And does that mean:
while I am pondering over it? Can't I dread something even when I am not expressly, so to speak, thinking it over?
Can't I say to someone "I dread this meeting" even though I see the event, as it were, merely out of the corner of my
eye?
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166. Let us just forget entirely that we are interested in the state of mind of a frightened man. It is certain that under
given circumstances we may also be interested in his behaviour as an indication of how he will behave in the future.
So why should we not have a word for this? It can be either a verb or an adjective.
It might now be asked whether this word would really relate simply to behaviour, simply to bodily changes.
And this we wish to deny. There is no future in simplifying the use of this word in this way. It relates to the
behaviour under certain external circumstances. If we observe these circumstances and that behaviour we say that a
man is....
If the word is used in the first person then the analogy with its use in the third person is the same as the one
between "I am cross-eyed" and "He is cross-eyed". [a, b,--except for the last sentence of a and the last two words of
b: Z 523.]
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167. I now want to say that humans who employ such a concept would not have to be able to describe its use. And
were they to try, it is possible that they would give a quite inadequate description. (Like most people, if they tried to
describe the use of paper money correctly.) [Cf. Z 525.]
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168. It is possible, for example, that they make this statement about someone without being able to say with any
degree of certainty, which aspect of his behaviour causes them to make the statement. They might say "I see it; but I
don't exactly know what I see". Just as we say: "Something about him has changed, but I don't know exactly what."
Future experience might prove them right.
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169. Now it is conceivable that some people might have a verb whose third person would be exactly equivalent to
our "He is afraid"; but whose first person is not equivalent to our "I am afraid". For the assertion using the first
person would be based on self-observation. It would not be an utterance of fear and there would be a "I believe I...",
"It seems to me that I...". Now probably this first person would not be used at all, or only very rarely. If my
behaviour in a certain situation were filmed, then when the film was shown to me I could say "My behaviour creates
the impression...".
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170. That statement "I believe he feels what I feel in such circumstances" does not yet exist here: The interpretation,
that is, that I see something in myself which I surmise in him.
For in reality that is a rough interpretation. In general I do not surmise fear in him--I see it. I do not feel that I
am deducing the probable existence of something inside from something outside; rather it is as if the human face
were in a way translucent and that I were seeing it not in reflected light but rather in its own.
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171. "I dread it."--That is not a representation of something I see. As a matter of fact, as soon as I look, I see nothing,
or at least not what I really meant. Then it is as if this were such a thin veil that one could know about it but not
actually see it. As if dread were a very subtle, muffled sound alongside the everyday sounds, a sound which I could
only sense and not really hear.
Imagine a child who for a long time had been unable to learn how to speak and who suddenly used the
expression "I dread...", which it had heard from adults. And its face and the circumstances and the consequences
make us say: He really meant it. (For one could always say: "One fine day the child starts using the words.") I chose
the case of a child because what is happening in him is stranger to us than it would be with an adult. What do I
know--I'm inclined to say--about a background for the words "I dread..."? Does the child suddenly let me look into
him?
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172. This matter also calls to mind hearing a sound from a particular direction. It is almost as if one felt the
heaviness around the stomach from the direction of the fear. That means really that "I am sick with fear" does not
assign a cause of fear. [Cf. Z 496.]
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173. Are there psychological syndromes; and is 'expecting' one of them? Possibly waiting for something, but not
'expecting'.
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174. That there is a fear-syndrome, for example, does not mean that fear is a syndrome. [Cf. Z 502.]
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175. If I say "I am anxiously awaiting his coming", this means: I am occupied with his coming (in thought and, one
can also say: in thought and in action). The state of anxiously awaiting can thus be called a syndrome. But it is not,
so to speak, a syndrome of actions of a certain kind; the crucial point is rather the intention of the actions, and thus a
motive, and not a cause.
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176. If I say that I am using the words "I'm in pain", "I'm looking for him", etc. etc. as a piece of information, not as a
natural sound,†1 then this characterizes my intention. For instance, I might want somebody else to react to this in a
certain way.
But here I still owe an explanation of the concept of intention, and intention is by no means some sort of
feeling to which I want to reduce everything; at whose door, so to speak, I am laying everything. (For intention is
not a feeling.)
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177. If we call fear, sorrow, joy, anger, etc. mental states, then that means that the fearful, the sorrowful, etc. can
report: "I am in a state of fear" etc., and that this information--just like the primitive utterance--is not based on
observation.
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178. Intent, intention, is neither an emotion, a mood, nor yet a sensation or image. It is not a state of consciousness.
It does not have genuine duration. Intention can be called a mental disposition. This term is misleading inasmuch as
one does not perceive such a disposition within himself as a matter of experience. The inclination toward jealousy,
on the other hand, is a disposition in the true sense. Experience teaches me that I have it. [First three sentences: Z
45.]
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179. "I intend" is not an expression of an experience.
There is no cry of intention, any more than there is one of knowledge or belief.
However, one might very well call the decision with which an intention frequently begins an experience.
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180. Is a decision a thought? It can be the end of a chain of thought.
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181. Someone tells me something; I look at him in amazement; he explains... My puzzled look was equivalent to the
question: "How come?" or "What do you mean?" or "Why?" or "You want to do that? You who always...?"--The
sudden thought.
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182. Intentional--unintentional. Voluntary--involuntary.
What is the difference between a gesture of the hand without a particular intention and the same gesture
which is intended as a sign?
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183. Let us imagine someone doing work that involves comparison, trial, choice. Say he is constructing an appliance
out of various bits of stuff with a given set of tools. Every now and then there is the problem "Should I use this
bit?"--The bit is rejected, another is tried. Bits are tentatively put together, then dismantled; he looks for one that fits
etc., etc. I now imagine that this whole procedure is filmed. The worker perhaps also produces sound-effects like
"Hm" or "Ha!" As it were sounds of hesitation, sudden finding, decision, satisfaction, dissatisfaction. But he does
not utter a single word. Those sound-effects may be included in the film. I have the film shown me, and now I invent
a soliloquy for the worker, things that fit his manner of work, its rhythm, his play of expression, his gestures and
spontaneous noises; they correspond to all this. So I sometimes make him say "No, that bit is too long, perhaps
another'll fit better."--Or "What am I to do now?"--"Got it!"--Or "That's not bad" etc.
If the worker can talk--would it be a falsification of what actually goes on if he were to describe that precisely
and were to say, e.g., "Then I thought: No, that won't do, I must try it another way" and so on--although he had
neither spoken during the work nor imagined these words?
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I want to say: May he not later give his wordless thoughts in words? And in such a fashion that we, who
might see the work in progress, could accept this account?--And all the more, if we had often watched the man
working, not just once? [Z 100.]
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184. Of course we cannot separate his 'thinking' from his activity. For the thinking is not an accompaniment of the
work, any more than of thoughtful speech. [Z 101.]
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185. Imagine a person who is taking a break in his work, and is staring ahead seemingly pondering something, in a
situation in which we would ask ourselves a question, weigh possibilities--would we necessarily say of him that he
was reflecting? Is not one of the prerequisites for this that he be in command of a language, i.e., be able to express
the reflection, if called upon to do so?
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186. Now if we were to see creatures at work whose rhythm of work, play of expression etc. was like our own, but
for their not speaking, perhaps in that case we should say that they thought, considered, made decisions. That is: in
such a case there would be a great deal which is similar to the action of ordinary humans. And it isn't clear how
much has to be similar for us to have a right to apply to them also the concept 'thinking', which has its home in our
life.†1 [Cf. Z 102.]
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187. And anyhow what should we come to this decision for?
We shall be making an important decision between creatures that can learn to do work, even complicated
work, in a 'mechanical' way, and those that make trials and comparisons as they work.--But what should be called
"making trials" and "comparisons" can in turn be explained only by giving examples, and these examples will be
taken from our life or from a life that is like ours. [Z 103.]
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188. And if their trial-making were to take on the form of producing a kind of model (or even a drawing) then we
would say without hesitation that these beings were thinking. To be sure one could also speak here of an operation
with signs.
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189. "But couldn't the operation with signs also take place mechanically?"--Surely; i.e. this too has to take place in a
certain context in order for us to be able to say it is not mechanical.
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190. It seems therefore, that our concepts, the use of our words, are constrained by a factual framework. But how
can that be?! How could we describe the framework if we did not allow for the possibility of something else?--One
is inclined to say that you are making all logic into nonsense!
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191. The problem which worries us here is the same as in the case of this observation: "Human beings couldn't learn
to count if all the objects around them were rapidly coming into being and passing away.
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192. But you can also say: "If you don't have any little sticks, stones, etc. at hand, then you can't teach a person how
to calculate." Just as you can say "If you have neither a writing surface nor writing material at hand then you can't
teach him differential calculus" (or: then you can't work out the division 76570 ÷ 319).
We don't say of a table and chair that they think; neither do we say this of a plant, a fish, and hardly of a dog;
only of human beings. And not even of all human beings.
But if I say "A table does not think", then that is not similar to a statement like "A table doesn't grow". I
shouldn't know 'what it would be like if' a table were to think. And here there is obviously a gradual transition to the
case of human beings. [b, c: cf. Z 129.]
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193. "Thinking is a mental activity."--Thinking is not a bodily activity. Is thinking an activity? Well, one may tell
someone: "Think it over! ". But if someone in obeying this order talks to himself or even to someone else, does he
then carry out two activities? Therefore thinking really can't be compared to an activity at all. For one cannot say that
thinking means: speaking in one's imagination. This can also be done without thinking. [Z 123, up to "Therefore
thinking...".]
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194. You must never forget that "think" is an everyday word, just as are all other psychological terms.
It is not to be expected of this word that it should have a unified employment; rather it is to be expected that
it doesn't have it. [a: cf. Z 113; b: Z 112.]
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195. If someone is pondering over a problem and suddenly I show him a certain drawing, then maybe he will
exclaim "Oh, that's how it is!", or "Now I know". When questioned about what went on inside him just then, in this
case he will very likely say simply: "I saw the drawing." I am describing this case in order to replace a process within
the imagination with one of seeing. Will he now say: "The moment I saw the drawing the whole solution appeared
before my eyes"? When I come to his aid with the drawing he might also say: "Yes, now it's easy!"
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196. "I see clearly how the word is used"--will that be said even when one is shown, alongside the word, a picture
which illustrates its meaning?
((In this case, the experience of meaning seems to be drowned out by what has been seen.))
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197. We say: Grass is green, chalk white, coal black, blood red, etc.--What would it be like in a world in which this
would be impossible, i.e., in which the other qualities of a thing were unconnected with†1 its colour? This is an
important question, whether or not it has been put correctly, and is merely an example of countless similar
questions.
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198. Suppose I were to come to a country where the colour of things--as I would say--changed constantly, say
because of a peculiarity of the atmosphere. The inhabitants never see unchanging colours. Their grass looks green at
one moment, red at the next, etc. Could these people teach their children the words for colours?--First of all, it might
be that their language lacked words for colours. And if we found this out we might explain it by saying that they had
little or no use for certain language-games.
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199. How could people learn the use of the words for colour in a country where everything was only one colour?
But can I say now: "The only reason for our being able to use the names for colours is that things of different
colours exist in our environment and that..."?? Here the difference between logical and physical possibility is not
being seen.--Under what conditions the language-game with the names for colours is physically impossible--i.e.,
properly speaking, not probable--does not interest us.
Without chess-men one can't play chess--that is the impossibility which interests us.
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200. One learns the word "think", i.e. its use, under certain circumstances, which, however, one does not learn to
describe. [Z 114.]
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201. We learn to say it perhaps only of human beings, we learn to assert or deny it of them. The question "Do fishes
think?" does not exist among our applications of language, it is not raised. (What can be more natural than such a
set-up, such a use of language!) [Z 117.]
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202. "No one thought of that case"--we may say. Indeed, I cannot enumerate the conditions under which the word
"to think" is to be used--but if a circumstance makes the use doubtful, I can say so, and also how the situation is
deviant from the usual ones. [Z 118.]
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203. And here something about my language-game No. 2†1 should be said.--Under what circumstances would one
really call the sounds of the builder, etc., a language? Under all circumstances? Certainly not!--Was it wrong then to
isolate a rudiment of language and call it language? Should one perhaps say that this rudiment is a language-game
only in the context of the whole that we usually call our language?? [Cf. Z 98.]
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204. Now in the first place this surrounding is not the mental accompaniment of speech; it is not the 'meaning' and
'understanding' which one is inclined to consider as essential to language.
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205. It would only be dangerous to me if someone were to say: "You're just tacitly assuming that these people think;
that they are like people as we know them in that respect; that they do not carry on that language-game merely
mechanically. For if you imagined them doing that, you yourself wouldn't call it speaking."
What am I to reply to this? Of course it is true that the life of those men must be like ours in many respects
and that I said nothing about this similarity. But the important thing is precisely that I can imagine their language,
and their thinking too, as primitive; that there is such a thing as 'primitive thinking' which is to be described via
primitive behaviour. [Cf. Z 99.]
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206. I say of someone: He's comparing two objects. I know what that looks like, how that is done. I can demonstrate
it to someone.
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Nevertheless I wouldn't call what I was demonstrating 'comparing' under all circumstances.
I can imagine cases in which I would not be inclined to say that comparing was going on; but as for
describing the circumstances in which comparing occurs, that I could not do.--But I can teach a person the use of
the word! For a description of those circumstances is not needed for that. [Last sentence: Z 115.]
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207. I just teach him the word under particular circumstances.†1 [Z 116.]
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208. Sometimes it really seems that thinking runs concurrently with talking (reading, for example). Not that it could
then be isolated from reading, however. Rather, what accompanies the words is like a series of small secondary
movements. It is like being led along a street, but casting glances right and left into all the side streets.
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209. Suppose I were to show somebody a list of the trips or errands he is to run for me. We know each other well
and all he needs are hints for him to know what he is to do. Now the list is all hints of this sort. He goes through it
and says after each hint "I understand". And he does; he could explain every single item, if asked to do so.
Then I could ask him: "Did you understand everything?" Or: "Look through the list and see whether you
understand everything." Or: "Do you know what you have to do here?"--What did he have to do to make sure that
he had understood the hints? Is it as if he had to perform a mental calculation for each item? If that were necessary
he could later give a verbal account of the calculation and it would become clear whether he calculated
correctly.--But generally that is not necessary. We do not prescribe what the other is to do if he is to understand the
list; and whether he really understood is determined from what he does later, or from an explanation we might ask
him to give.
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210. One could say: anyone who checks his comprehension in this way is moving along a bit of the way on the path
he is later to follow. And that might indeed be so. Although there is no reason to assume it is so. For if he only goes
part of the way--why shouldn't he be able
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to see that he knows which way he is to go, without actually going it? But that does not mean that the paths are not
actually followed part of the way. However, it can also be that what we later come to regard as the 'germ' of a
thought or an action is in its own nature not that at all.
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211. Now suppose someone were to say: That simply means that "thinking" has a certain end, that it fulfils a certain
purpose. How each person performs it and whether he does it the same today as last time, is irrelevant.--Then I could
answer: And if doing nothing at all leads to the proper end, then thinking in this case would consist in doing nothing.
One says: "Make sure that you understand each point!"
And if I were then to ask, "How should I make sure?", what advice would I be given? I would be told: "Ask
yourself, whether..."
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212. Isn't it the same here as with a calculating prodigy?--He has calculated right if he has got the right answer.
Perhaps he himself cannot say what went on in him. And if we were to hear it, it would perhaps seem like a queer
caricature of calculation. [Cf. Z 89b.]
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213. When someone says "Non-verbal thinking is also possible", this is misleading. The point is not to be able to do
a certain thing without also doing something else at the same time; as, for example, with "It is possible to read
without moving one's lips".
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214. If, for instance, there were only quite few people who could get the answer to a sum without speaking or
writing, they could not be adduced as testimony to the fact that calculating can be done without signs. The reason is
that it would not be clear that these people were 'calculating' at all. Equally Ballard's†1 testimony (in James) cannot
convince one that it is possible to think without a language.
Indeed, where no language is used, why should one speak of 'thinking'? If this is done, it shows something
about the concept of thinking. [Z 109.]
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215. For instance one could have two (or more) different words: one for 'thinking out loud'; one for thinking as one
talks in the imagination; one for a pause during which something or other floats
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before the mind (or doesn't), after which, however, we are able to give a confident answer.
We could have two words: one for a thought expressed in a sentence; one for the lightning thought which I
may later 'clothe in words'. [Cf. Z 122.]
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216. If we include 'thinking silently as one is working' in our considerations, then we see that our concept 'thinking' is
widely ramified. Like a ramified traffic network which connects many out-of-the-way places with each other.
In all of these widely separated cases we speak of 'thinking'.
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217. In all of these cases we say that the mind is not idle, that something is going on inside it; and we thereby
distinguish these cases from a state of stupor, from mechanical actions.
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218. 'Thinking', a widely ramified concept. Couldn't the same be said of 'believing', 'doing', 'being glad'?
And where does the remark that this concept is widely ramified really belong?--Well it will be made to
someone setting out to consider the branching of this concept.
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219. It's really very odd that we have no difficulty whatsoever seeing a face in a figure such as this†1
even though the resemblance of the one angle to a nose and of the other to a forehead etc., is incredibly slight, or
there hardly is a resemblance there. To repeat: We have no difficulty whatsoever seeing a human face in these lines;
one would like to say: "There is a face like that." Again: "True, this is the caricature of a human face, but a caricature
of one which could really exist."--Just as one has no difficulty seeing a human face in the grey-and-white of a
photograph.--And what does that mean? Well, we watch a movie, for instance, and follow everything that goes on
with concern, as if there were real people in front of us.
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220. 'Thinking', a widely ramified concept. A concept that comprises many manifestations of life. The phenomena of
thinking are widely scattered. [Z 110.]
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221. And don't you want to say that one aspect appears in all of these word-uses, a unitary, genuine concept?--But
how much is there in that? May not the force of habit weld all of this together?
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222. If somebody tells me of some incident, or asks me an everyday question (e.g. what time is it), I'm not going to
ask him whether he was thinking while he was telling me or asking me. Or again: It would not be immediately clear
in what circumstances one might have said that he did this without thinking--even though one can imagine such
circumstances. (Here there is a relationship with the question: What is to be called a 'voluntary' act.)
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223. The thoughtful expression, the expression of the idiot. The frown of reflection, of attention.
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224. Now imagine a human being, or one of Köhler's monkeys, who wants to get a banana from the ceiling, but can't
reach it, and thinking about ways and means finally puts two sticks together, etc. Suppose one were to ask, "What
must go on inside him for this to take place?"--This question seems to make some sort of sense. And perhaps
someone might answer that unless he acted through chance or instinct, the monkey must have seen the process
before its mental eye. But that would not suffice, and then again, on the other hand, it would be too much. I want the
monkey to reflect on something. First he jumps and reaches for the banana in vain, then he gives up and perhaps he
is depressed--but this phase does not have to take place. How can catching hold of the stick be something he gets to
inwardly at all? True, he could have been shown a picture that depicts something like that, and then he could act that
way; or such a picture could simply float before his mind. But that again would be an accident. He would not have
arrived at this picture by reflection. And does it help to say that all he needed to have done was somehow to have
seen his arm and the stick as a unity? But let us go ahead and assume a propitious accident! Then the question is:
How can he learn from the accident? Perhaps he just happened to have the stick in his hand and just happened to
touch the banana with it.--And what further must now go on in him? He says to himself, as it were, "That's how!",
and then he does it with signs of full consciousness.--If he has made some combination in play, and he now uses it
as a method for doing this and that, we shall say he thinks.--In considering he would mentally review ways and
means. But to do this he must already have some in stock. Thinking gives him the possibility of perfecting his
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methods. Or rather: He 'thinks' when, in a definite kind of way, he perfects a method he has. [Z 104--beginning at "If
he has made".]
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225. It could also be said that he thinks when he learns in a particular way. [Z 105.]
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226. And this too could be said: Someone who thinks as he works will intersperse his work with auxiliary activities.
The word "thinking" does not now mean these auxiliary activities, just as thinking is not talking either. Although the
concept 'thinking' is formed on the model of a kind of imaginary auxiliary activity. (Just as we might say that the
concept of the differential quotient is formed on the model of a kind of imaginary quotient.) [Z 106.]
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227. These auxiliary activities are not the thinking; but one imagines thinking as that which must be flowing under
the surface of these expedients, if they are not after all to be mere mechanical procedures. [Z 107.]
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228. Thinking is the imaginary auxiliary activity; the invisible stream which carries and connects all of these kinds of
actions.--The grammar of "thinking", however, is assimilated to that of "speaking".
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229. So one might distinguish between two chimpanzees with respect to the way in which they work, and say of the
one that he is thinking and of the other that he is not.
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230. But here of course we wouldn't have the complete employment of "think". The word would have reference to a
mode of behaviour. Not until it finds its particular use in the first person does it acquire the meaning of mental
activity.
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231. I think it is important to remark that the word doesn't have a first person present (in the meaning which is of
consequence to us). Or should I say: that its use in the present tense does not parallel that of the verb "feel pain", for
instance?
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232. One can say "I thought..." if one really did use an expression of thought; but also if these words are, as it were,
a development from a germ of thought.
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233. Only under quite special circumstances does the question arise†1 whether one spoke thinkingly or not. [Z 95.]
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234. The use of a word such as "thinking" is simply far more erratic than it appears at first sight.
It can also be put this way: The expression serves a much more specialized purpose than is apparent from its
form. For the form is a simple, regular structure. If thinking frequently, or mostly goes with talking, then of course
there is the possibility that in some instance it does not go with it.
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235. I'm learning a foreign language and I read sample sentences in a textbook. "My aunt has a beautiful garden."
The sentence has the smell of a textbook. I read it and ask myself, "What does 'beautiful' mean in...?", then I think
about the case of the adjective.--Now if I tell somebody that my aunt has..., then I don't think about these things.
The context in which the sentence stood was completely different.--But wasn't I able to read that sentence in the
textbook and at the same time think about my aunt's garden? Certainly. And should I now say that the
thought-accompaniment is a different one every time, according as I see the sentence one time as a pure exercise,
another time as an exercise accompanied by the thought of a garden, and another time as I simply say it as a piece of
information to someone?--And is it impossible that someone should give me this information in the course of a
conversation, and that exactly the same thing could occur within him then as when he uses the sentence as a
language exercise? Does it matter to me what goes on inside him? Do I realize it?
And how can I write about it at all with any degree of certainty? For while I am doing this, I am not learning
a language and I am not giving anybody a piece of information. How, then, can I know what goes on inside a person
in such a case? Do I remember what went on inside me in these cases? Nothing of the sort. I only believed I could
think myself into these situations. But then I might completely have gone wrong.
And this is the very method that is always used in such cases! What one experiences here is merely
characteristic of the situation of philosophizing.
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236. What do I know of what goes on within someone who is reading a sentence attentively? And can he describe it
to me afterwards, and, if he does describe something, will it be the characteristic process of attention? [Z 90.]
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237. What result am I aiming at when I tell someone: "Read attentively!"? That, e.g., this and that should strike him,
and he be able to give an account of it.--Again, it could, I think, be said that if you read a sentence with attention you
will then be able to give a general account of what has gone on in your mind, e.g. the occurrence of images. But that
does not mean that these things constituted attention. [Z 91.]
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238. What do I do if he tells me that he thought about something entirely different when he read the sentence? What
interesting conclusions can I draw from such a piece of information? Well, for instance that this particular matter
occupies his mind; that I should not expect that he knows what the material he read was about; that what he read
has made no impression on him whatsoever; and things like that.
Therefore it would make no sense if somebody who had had a pleasant conversation†1 with me were
thereafter to assure me that he had spoken entirely without thinking. But this is not because it contradicts all
experience that a person who can speak in this way should do so without thought processes accompanying his
speech. Rather, it is because it comes out here that the accompanying processes are of no interest whatsoever to us,
and do not constitute thinking. We don't give a damn about his accompanying processes when he engages in a
normal conversation with us.
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239. "It flashed through my mind:..." Now people learn the use of this expression. Hardly ever do we ask anyone:
"How did it flash through your mind? Did you say certain words to yourself? Did you see something in your
imagination; can you say in any way what went on inside of you?"
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240. If you want to find out how many different things "thought" means, you have only to compare a thought in
pure mathematics with a non-mathematical one. Only think how many things are called "sentences"!
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241. A child does not have to first use a primitive expression which we then replace with the usual one. Why
shouldn't he immediately use the adult expression which he has heard several times? It really doesn't matter how he
"guesses" that this is the right expression, or how he comes to use it. The main thing is that no matter what the
preliminaries are, he uses the word the same way adults do: i.e., on
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the same occasions, in the same context. He also says†1: the other person thought...
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242. How important is the experience of meaning in linguistic communication? What is important is that we intend
something when we utter a word. For example, I say "Bank!" and want thereby to remind someone to go to the
bank, and intend "bank" in the one meaning and not in the other.--But intention is no experience.
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243. But what makes it different from an experience?--Well it has no experiential content. For the contents (e.g.,
images) which often go hand in hand with it, are not the intention itself.†2--And yet neither is it a disposition, like
knowing. For the intention was present when I said "Bank"; now it is no longer present; but I have not forgotten it.
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244. True: It is possible that I was more or less intensely occupied with what I said. And here it is obviously not a
matter of having particular experiences while I utter the words. That is, it would be wrong to say: "In the process of
uttering the word 'Bank' such and such a thing had to take place if it was really supposed to mean that."
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245. That it is possible after all to utter the word in isolation, far removed from any intention, 'now with one
meaning, now with another', is a phenomenon which has no bearing on the nature of meaning; as if one could say,
"Look, you can do this with a meaning too".--No more than one could say: "Look at all the things you can do with
an apple: you can eat it, see it, desire it, try to form an image of it." No more than it is characteristic of the concept
'needle' and 'soul' that we can ask how many souls can fit on the point of a needle.--We're dealing here, so to speak,
with an outgrowth of the concept.
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246. Instead of "outgrowth of the concept" I could also have said "an annex to the concept".--In the sense that it is
also not essential to
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people's names that they seem to have the traits of their bearers.--((Quote from Grillparzer.))†1
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247. How can the mental state of someone who is giving an order semi-automatically be distinguished from the state
in which the order is given with emphasis, urgently? "Something different is going on in this person's mind." Think
about the purpose of distinguishing. What are the signs of emphasis?
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248. If a normal human is holding a normal conversation under such and such normal circumstances, and I were to
be asked what distinguishes thinking from not-thinking in such a case--I should not know what answer to give. And
I could certainly not say that the difference lay in something that goes on or fails to go on while he is speaking. [Z
93.]
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249. The boundary-line that is drawn here between 'thinking' and 'not thinking' would run between two conditions
which are not distinguished by anything in the least resembling a play of images. For the play of images will always
be conceived of as the characteristic of thinking. [Cf. Z 94.]
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250. "I said those words, but I wasn't thinking of anything at all as I said them." That is an interesting utterance,
because the consequences are interesting. You can always suppose that whoever says this had made a mistake in his
introspection; but that wouldn't make any difference.
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251. But what am I to say now: Did the person who spoke without thought lack an experience? Were the
experiences images, for instance?--But if he had lacked them, would this be of the same interest to us as that he
spoke without thought? Is it the images which interest us in this case? Doesn't his utterance contain a kind of signal
of a totally different meaning?
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252. Should I say: "If you didn't speak automatically (whatever that may mean) and if you didn't intend something
later, and if you didn't change your intention, then you had it when you spoke"?
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253. "I didn't mean anything by the sentence. I was just saying it." How remarkable that when I say this although I
don't allude to any
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experience as I'm speaking, still I am not giving expression to anything dubitable.
It is very noteworthy that what goes on in thinking practically never interests us. (But of course I shouldn't
say it is noteworthy.) [b: cf. Z 88.]
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254. The question "What did you mean" and others like it can be used in two ways. In one case we simply demand
an explanation of sense or meaning so that we can continue the language-game. In the other we are interested in
what happened at the time the sentence was spoken. In the first case we would not be interested in a psychological
report such as this: "First I just said it to myself, then I turned to you and wanted to remind you...."
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255. Did you mean that? Yes, it was the beginning of this movement.
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256. Let's imagine the following case: At 12 o'clock I'm supposed to remind somebody to go to the bank to get
money. At 12 I glance at the clock and I say "Bank!" (either facing the person or looking away from him); I might
make a gesture which you sometimes make when you suddenly remember something you have to do. If asked "Do
you mean the... bank?" I will say "Yes".--If asked "Did you mean the... bank when you were talking?" I'd also say
"Yes".--But what if I said "No" to the latter? What information would that give the other person? Possibly that I
meant the sentence in a different way when I uttered it, but then wanted to use it for this purpose after all. Well, that
can happen. It is also possible that when I glance at the clock I utter the word "Bank" in a queer automatic way, so
that when I report "Suddenly I heard myself saying the word without attaching any kind of meaning to it. Only a few
seconds later did I remember that you were supposed to go to the bank".--If I had answered that I had meant the
word in a different way at first, I would obviously be referring to the time of speaking. And I could also have
expressed myself this way: "While speaking I thought of this bank, and not of...".--Now the question is: Is this
"thinking of..." an experience? One would like to say that usually, and possibly always, it goes with an experience.
To say that one thought about this thing to which one can now point, which one can describe, etc., is really like
saying: This word, this sentence, was the beginning of that train of thought, of that movement. But it is not as if I
knew this by subsequent experiences; rather the utterance "As I spoke these words I thought of..." itself attaches to
that point of time. And if I were to utter it in the present tense rather than in the past it would mean something else.
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257. But why do I want to say that thinking is not an experience?--One can think of "duration". If I had spoken a
whole sentence instead of the single word, I couldn't call one particular point of time in which I was thinking the
beginning of my thinking process, nor yet the moment in which it took place. Or, if one calls the beginning and end
of the sentence the beginning and end of the thought, then it is not clear whether one should say of the experience of
thinking that it is uniform during this time, or whether it is a process like speaking the sentence itself.
Sure, if we are to speak of an experience of thinking, the experience of speaking is as good as any. But the
concept 'thinking' is not a concept of experience. For we don't compare thoughts in the same way as we compare
experiences. [b: Z 96.]
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258. One may disturb someone in thinking--but in intending?--But certainly in planning. Also in keeping to an
intention, that is in thinking or acting. [Z 50.]
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259. "Say 'a b c d e' and mean: The weather is fine." Should I say, then, that the experience of pronouncing a
sentence in a familiar language is quite different from that of pronouncing signs that are familiar to us, but not in
certain meanings? So if I learnt the language in which "a b c d e" meant that of..., should I come bit by bit to have the
familiar experience when I pronounced a sentence? Or should I say, as I'm inclined to, that the major difference
between the two cases is that in the one I can't move. It is as if one of my joints were in splints, and I were not yet
familiar with the possible movements, so that I as it were keep on bumping into things. (Feeling of something
soft.)[Cf. Z 6.]
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260. Suppose I were with someone who spoke this language, and I had been told that "a b c d e" means this and that,
and that I should say it because it is polite to do so. So, I would say it with a friendly smile, with a glance out the
window. Would that alone not be enough to give me a better understanding of these signs?
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261. One could speak of a 'feeling for' something. And in what does my feeling for a sentence I utter consist? It will
be said that it consists in what goes on inside me when I speak. I would like to say: In the connections, the tie-ups
which I make. For the question is: What happens within me when I have a feeling for something--what makes it a
feeling for the content of this sentence? Why isn't it, e.g., a
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pathological state of excitement in me, which accompanies my speaking? [Cf. Z 124.]
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262. Can I really say that when I 'unthinkingly' read the sentence in the textbook, something completely different, or
simply something different, goes on in me from what goes on when in a different situation I read the sentence
comprehendingly? Yes--there are differences. For instance, in a certain situation I shall respond to the same
sentence: "So that's the way it was?", I shall be surprised, disappointed, expectant, satisfied, etc.
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263. "Did you think as you read the sentence?"--"Yes, I did think as I read it; every word was important to me."
"I was thinking very intensely." A signal.
Did nothing go on in the process? Yes, all sorts of things. But the signal did not refer to them.
And yet the signal referred to the time of speaking. [a: Z 92a.]
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264. James might perhaps say: "I read each word with the feeling appropriate to it. 'But' with the but-feeling," and so
on.--And even if that is true--what does it really signify? What is the grammar of the concept 'but-feeling'?--It
certainly isn't a feeling just because I call it "a feeling". [Cf. Z 188.]
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265. How strange, that something has happened while I was speaking and yet I cannot say what!--The best thing
would be to say it was an illusion, and nothing really happened; and now I investigate the usefulness of the
utterance.
Furthermore the question will arise as to the usefulness of referring to a point of time in the past.
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266. Yes; "While saying these words I thought..." indeed does refer to the time of speaking; but if I am now to
characterize the 'process' I cannot describe it as something happening in this stretch of time. I cannot say, e.g., that
this or that phase of the process occurred in this time segment. So I can not describe the thinking process as I can
describe speaking itself, for instance. That is why one can't very well call thinking a process. ((Nor an
accompaniment of speaking.))
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267. By 'thinking while you speak' I really should mean that I speak and understand what I say, and not that I speak
and understand it later.
Writing is certainly a voluntary movement, and yet an automatic
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one. And of course there is no question of the movements in writing. That is, one feels something, but could not
possibly analyze the feeling. One's hand writes; it does not write because one wills, but one wills it to write.
One does not watch it in astonishment or with interest while writing; does not think "What will it write
now?". But not because one had a wish that it should write that. For that it writes what I want might very well throw
me into astonishment. [b, c: Z 586.]
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268. How do we test whether somebody understands what it means to relax his arm muscles, to let his arms go
limp? By testing whether they are relaxed when he says he relaxed them (in response to our order, for instance).
Now what would we say of a person who is lifting a weight apparently voluntarily, and who tells us that he is not
tensing his muscles? In this case we would say he was lying, or suffering from a strange delusion. I don't know
whether there are deranged people who declare that their normal movements are involuntary. But if somebody does
I would expect him to follow the movements of his arm with his attention in a fashion quite different from the
normal one; that is, as he might follow the movement of a pointer on an instrument.
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269. A child learns to walk, to crawl, to play. It does not learn to play voluntarily and involuntarily. But what makes
its movements in play into voluntary movements? What would it be like if they were involuntary?--I could also ask:
what makes these movements into a game?--The fact that they are reactions to certain movements, sound, etc., of a
grown-up, that they occur in this sequence, go together with these facial expressions and sounds (laughing, e.g.).
[Cf. Z 587.]
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270. Briefly, if the child executes the movements IN THIS WAY, then we say that they are voluntary. Movements in
such syndromes are called "voluntary".
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271. I signal to someone with my eyes. Later I can explain what it meant. If I say "At the time I had this intention" it
is as if I were calling the expression the beginning of a movement. I do not explain the expression with the help of
established rules, nor by a definition which is to regulate the future use of the signal. I don't say "This signal means
this to us", nor "In the future it is to mean this". Thus I am not giving a definition.
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272. But now think of the difference it makes if I exclaim "Bank!" in some particular situation, not on my own, but
perhaps read it in a history book or a play. I am assuming I read it with understanding. Am I still inclined to speak
of an intention (I mean of my intention) in connection with this word?
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273. But can I say that when I am reading, something goes on inside me which is different from what happens in a
spontaneous exclamation? No. I know nothing about such a difference of processes, although the way I am
expressing myself might lead one to infer something like that.
But if someone were to come into the room at the very moment I was reading the exclamation and were to
ask me whether I wanted this, or that, I would tell him that I hadn't meant the exclamation that way, but had just
been reading something.
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274. I said earlier that intention has no content. One can call its content what explains its verbal expression. But it is
just that that cannot be said to be a uniform state, lasting from this point in time to that one; e.g., from the beginning
of the first until the end of the last word; not can one distinguish phases in it and correlate them with the parts of the
verbal expression. But if the sentence were accompanied by a play of images one could do precisely that.
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275. The difference between "intend" and "think of the intention".
If I say to myself, "I'm going to end this conversation", then that is presumably the expression of an
intention, indeed, of an intention at the moment of its inception. Actually it is the expression of a decision. And
corresponding to a decision that is an affirmation of an intention, there is also a wavering back and forth between
decisions, a wrestling with the decision.
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276. When I think to myself, "I can't stand it any more; I want to go!", then I am thinking of an intention. But this is
thinking of the outbreak of an intention. Whereas a person who says "I'm planning next year to..." can also be said to
be thinking of an intention, but in a completely different sense.
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277. One does not say "I know it's raining" simply to report that it is raining; rather this is said if the statement has
been called into question; or in response to the question whether I'm sure that it's raining. But then I could also say
"It's quite certain: it's raining".
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278. I can play a whole series of language-games with a report. One might be: acting according to the report; another
one: using the report to test whoever gave it.
But isn't the first language-game the more primitive one, so to speak, the real purpose of a report?
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279. It must be remembered that the first person "I believe" could very well exist without the third person.
Why shouldn't a verb have been formed in language which only exists in the first person present? What has
led to this, what images, is irrelevant.
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280. But what does this mean: "It's raining and I don't believe it" makes sense if I mean it as a hypothesis, and does
not make sense if I meant it as an assertion, or a report?
We have an image of something emanating from the sentence, of something lighting up, if it is intended in
the former way, whereas everything remains dark if it is intended in the latter. And there is some truth to this: for if
someone says these words to me and I understand them as a hypothesis, then my face might be lighted up with
comprehension; but if I take the sentence as a report then I become confused as to its meaning and comprehension
escapes me.
"It's raining and I don't believe it" is an assumption, not a report.
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281. One would also like to say: The assumption that I believe something is the assumption that I am disposed in a
particular way. Whereas I should not want to say of the report "I believe..." that it says something about my
disposition. Rather it is an utterance of this disposition.
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282. All that hangs together with this, that one can say "I believe he believes...", "I believe I believed...", but not "I
believe I believe...".
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283. If we were to have an obligatory "I believe" at the beginning of every assertion, "I believe it is so" would mean
the same thing as "It is so". But "Suppose I believe it is so" would not mean the same thing as "Suppose it is so".
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284. I have satisfied myself about something and now I know it. One doesn't say "I know that the earth has existed
for the past ten minutes"; one does say, however, "It is known that the earth has existed for many thousands of
years". And this is not because it is unnecessary to assert something like that.
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285. "I know that this path leads over there."
"I know where this path leads to."
In the latter case I am saying that I possess something; in the former, I am affirming a fact. In the former case
the word "know" could even be dropped. In the latter, one could go on: "But I'm not telling."
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286. The statement "I know it is so" is followed by the question "How do you know that?", the question asking for
evidence.
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287. In the language-game of reporting there is the case of the report being called into question, of one's assuming
that the reporter is merely conjecturing what he reports, that he hasn't ascertained it. Here he might say: I know it.
That is: It is not mere surmise.--Should I say in this case that he is telling the certainty, the certainty he feels about
his report, to me? No, I wouldn't like to say that. He's simply playing the language-game of reporting, and "I know
it" is the form of a report.
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288. Can one only know what is true? Well, one does say "I believe I know it", and here there is no uncertainty
attached to the belief. This does not mean: "I'm not certain: Do I know it, or don't I?"
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289. Some will say that my talk about the concept of knowledge is irrelevant, since this concept as understood by
philosophers, while indeed it does not agree with the concept as it is used in everyday speech, still is an important
and interesting one, created by a kind of sublimation from the ordinary, rather uninteresting one. But the
philosophical concept was derived from the ordinary one through all sorts of misunderstandings, and it strengthens
these misunderstandings. It is in no way interesting, except as a warning.
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290. Again, you must not forget that "A contradiction doesn't make sense" does not mean that the sense of a
contradiction is nonsense.--We exclude contradictions from language; we have no clear-cut use for them, and we
don't want to use them. And if "It's raining but I don't believe it" is senseless, then again that is because an extension
along certain lines leads to this technique. But under unusual circumstances that sentence could be given a clear
sense.
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291. If there were such a thing as 'automatic' speech, then we couldn't dispute such an utterance, or try to prove a
mistake on
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the part of one who speaks it. Thus we would not play the same language-games with automatic speech as we do
with the usual kind.
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292. Calling a mode of speech "automatic" produces the image of something without inflection, something
mechanical. But that isn't at all important to us. One need only assume that two people are talking through one
mouth. We must then treat what was said as the utterance of two people. Thus both sentences could be spoken with
the intention of giving a report. And then the only question would be, how I should react to these reports.
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293. On the one hand it can be said that black and white can coexist in grey; on the other hand it will be said: "But
where there is grey, there is, of course, neither black nor white. What is grey is of course not really white."
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294. But how about "light red" and "dark red"? Will one want to say that they can co-exist somewhere? Or lilac and
purple?--Well, suppose that we were constantly surrounded by very specific shades of light and dark blue, and that
we could not (contrary to what is actually the case) easily produce any shade of colour we chose. But under certain
circumstances we would be able to mix the light blue substance with the dark blue one, and then we would arrive at
an unusual shade of colour which we would then perceive as a mixture of light blue and dark blue.
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295. "But would our concepts of colour then be the same as the ones we have today?" They would be very similar.
Very much like the relation of number concepts of peoples who can only count to five, and ours.
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296. One can say: Whoever has a word explained by reference to a patch of colour only knows what is meant to the
extent that he knows how the word is to be used. That is to say: there is no grasping or understanding of an object,
only the grasping of a technique.
On the other hand, we would certainly say that grasping or comprehending an object is possible before
understanding a technique, for we can simply give someone the order "Copy this!" and then he can copy the colour,
or the shape and size, or only the shape, or the colour, but not the exact shade, etc. And here copying does what in
the case of a body is done by taking one in one's hand.--It is as if we
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could pick out the meaning, or the colour, perhaps, with a particularly refined pair of mental tweezers, without
catching hold of anything else.
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297. The understanding, I say, catches hold of the one object; and then we speak of it and its qualities according to
its nature.
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298. But how do I know that your mind catches hold of the same object as mine? Well, for instance by the very way
you react to my command, e.g. "Copy the colour". But in this case, you will say, we can only recognize what is
essential to his reaction by having him copy more and more colours. Presumably this means that after a few such
reactions I will be able to see others in advance; and this I explain by saying: Now I know "what" he is actually
copying. The colour or the form, for example--but there are more such whats than we are usually inclined to
assume, i.e., concepts can also be formed with which we are quite unfamiliar.
It may also be that I do foresee his reactions of copying after a few have been given, and that I can count on
them--i.e., say that we have now understood one another.--But then in a somewhat different situation I nevertheless
get a surprise.--And what shall I now say: That I had been misunderstanding him the whole time?, or that I partly
misunderstood him? If you think about catching hold of an object you will perhaps say the former, in accordance
with the picture that he just had not caught hold of the object that I thought he had. But if we think about methods
of using words, then we shall say that here we have similar, but not identical, methods.
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299. Now here it is certainly important that a technique has a physiognomy for us. That we can speak, for example,
of uniform and non-uniform uses.
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300. In one sense knowing is to have learned and not forgotten. In this way it hangs together with memory.--So now
I can say: "I know what 97 × 78 is", or "I know that 97 × 78 is 432". In the first case, I would say, I tell someone that
I can do something, that I possess something; in the second I simply assure the other person that 97 × 78 is 432. For
doesn't "97 × 78 is quite definitely 432" mean I know it is? The matter can also be put this way: The first sentence is
by no means an arithmetical one, nor can it somehow be replaced by an arithmetical one; but an arithmetical
sentence could be used in place of the second one. [Cf. Z 406.]
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301. The difference is as follows: In the sentence "I know how it is" the "I know" cannot be omitted. The sentence "I
know it is this way" can be replaced by "It is this way".
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302. "It's going to rain."--"Do you believe it's going to rain?"--"I know it's going to rain." Does the third sentence say
more than the first? It is a repetition of the first and a rejection of the second.
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303. But isn't there a phenomenon of knowing, as it were quite apart from the sense of the phrase "I know"? Is it not
remarkable that a man can know something, can as it were have the fact within him?--But that is a wrong
picture.--For, it is said, it's only knowledge if things really are as he says. But that is not enough. It musn't be just an
accident that they are. For he has got to know that he knows: for knowing is a state of his own mind; he cannot be in
doubt or error about it--apart from some special sort of blindness. If then knowledge that things are so is only
knowledge if they really are so; and if knowledge is in him so that he is infallible about its being knowledge; in that
case, he is also infallible about things being the way knowledge knows them; and so the fact which he knows must
be within him just like the knowledge.
So: when I say, without lying: "I know that it is so", then only through a special sort of blindness can I be
wrong. [Z 408a, c.]
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304. Does "not seeing a picture IN THIS WAY" mean: seeing it differently?
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305. Imagine the following: I am shown a picture-puzzle; I see trees, people, etc. in it. I examine it and suddenly I
see a figure in the tree-tops. Looking at it afterwards, I no longer see those lines as branches, but as parts of the
figure. Then I place the picture in my room, where I see it every day, and usually I forget about the second
interpretation, and so now it is simply a forest. I see it in the same way as any other picture of a forest. (You see the
difficulty.)--Then one day I say of the picture: "It's been such a long time since I've seen it as a picture-puzzle, I've
almost forgotten that it is one." Here one can ask, of course: "Well how did you see it?", and I shall say, "Well, as
trees...", and that is quite right too; but did I therefore not only see the picture and know what it portrayed, but also
perceive it according to a certain interpretation? I would rather say: Up until now they've always just been trees to
me; I've never thought about the picture in any other way.
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306. Anyone who regrets something thinks about it. Is regret therefore a kind of thought? Or a colouring of thought?
There are regretful thoughts just as there are fearful ones, for instance. But if I say "I regret it" am I saying "I
have regretful thoughts"? No, because somebody who doesn't regret it at the moment could say this too. But
couldn't I say, "I think of it with regret", instead of "I regret it"?
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307. What is it that interests me about someone else's regret? His attitude toward his action. The signs of regret are
the signs of aversion, of sadness. The expression of regret refers to the action.
Regret is called a pain of the soul because the signs of pain are similar to those of regret.
But if one wanted to find an analogy to the place of pain, it would of course not be the mind (as, of course,
place of bodily pain is not the body), but the object of regret. [c: Z 511.]
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308. Why can a dog feel fear but not remorse? Would it be right to say "Because he can't talk"? [Z 518.]
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309. Only someone who can reflect on the past can repent. But that does not mean that as a matter of empirical fact
only such a one is capable of the feeling of remorse. [Z 519.]
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310. There is nothing that astonishing about a certain concept only being applicable to a being that, e.g. possesses a
language. [Z 520.]
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311. The treatment of all these phenomena of mental life is not of importance to me because I am keen on
completeness. Rather because each one casts light on the correct treatment of all. [Z 465.]
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312. When he first learns the names of colours--what is taught him? Well, he learns, e.g., to call out "red" on seeing
something red.--But is that a correct description; or ought it to have gone: "He learns to call 'red' what we too call
'red'"? Both descriptions are right.
What differentiates this from the language-game "How does it strike you?"?
But someone might be taught colour-vocabulary by being made to look at white objects through coloured
spectacles. What I teach him, however, must be a capacity. So he can now bring something red at an
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order; or arrange objects according to colour. But what then is something red? "Well that (pointing)." Or should he
have said, "That, because most of us call it 'red'"? Or simply "That is what most of us call 'red'"?
This information doesn't help us at all. The difficulty we sense here with respect to "red" reappears for
"same". [Z 421--up to the sentence "Well that (pointing)".]
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313. For I describe the language-game "Bring something red" to someone who can himself already play it. Others I
might at most teach it. (Relativity.) [Z 432.]
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314. Here we have a profound and important point; I wish I knew how to express it unambiguously. Somehow one
is deceived as to the purpose of the description. Or: one wants to go on giving reasons because he misunderstands
the function of giving a reason.
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315. Why doesn't one teach a child the language-game "It looks red to me" from the first? Because it is not yet able
to understand the rather fine distinction between seeming and being? [Z 422.]
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316. The red visual impression is a new concept. [Z 423.]
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317. The language-game that we teach him then is: "It looks to me..., it looks to you...". In the first language-game a
person does not occur as perceiving subject. [Z 424.]
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318. You give the language-game a new joint. Which does not mean, however, that now it is always used.
The language-game "What is that?"--"A chair."--is not the same as: "What do you take that for?"--"It might
be a chair." [a: Z 425; b: Z 417.]
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319. In the beginning we do not teach the child "It's probably a chair", but "That's a chair". Don't fancy for a moment
that the word "probably" is left out because it is still too difficult for the child to understand, that things are
simplified for the child; that therefore he is taught something that is not strictly right.
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320. One speaks of a feeling of conviction because there is a tone of conviction. For the characteristic mark of all
'feelings' is that there is expression of them, i.e. facial expression, gestures, of feeling. [Z 513.]
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321. James says it is impossible to imagine an emotion or a mood without the corresponding bodily sensations (of
which it is composed). If you imagine the latter absent then you can see that you are thereby abolishing the very
existence of the emotion. This might happen in the following way: I imagine myself sorrowing, and now in the
imagination I try to picture and to feel myself rejoicing at the same time. To do that I might take a deep breath and
imitate a beaming face. And now indeed I have trouble forming an image of sorrow; for forming an image of it
would mean play-acting it. But it does not follow from this that our bodily feeling at that point is sorrow, or even
something like it.--To be sure, a person who is sorrowful cannot laugh and rejoice convincingly, and if he could,
what we call the expression of sorrow would not really be that, and rejoicing would not be the expression of a
different emotion.--If the death of a friend and the recovery of a friend equally caused us to rejoice or--judging by
our behaviour--both caused us sorrow, then these forms of behaviour would not be what we call the expressions of
joy or sorrow. Is it clear a priori that whoever imitates joy will feel it? Couldn't the mere attempt to laugh while one
was feeling grief bring about an enormous sharpening of the grief?
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322. Yet still I mustn't forget that joy goes along with physical well-being, and sadness, or at least depression, often
with being physically out of sorts.--If I go for a walk and take pleasure in everything, then it is surely true that this
would not happen if I were feeling unwell. But if I now express my joy, saying, e.g., "How marvellous all of this
is!"--did I mean to say that all of these things were producing pleasant physical feelings in me?
In the very case where I'd express my joy like this: "The trees and the sky and the birds make me feel good
all over"--still what's in question here is not causation, nor empirical concomitance, etc. etc.
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323. One does say: "Now that he's well again I breathe easier", and one breathes a deep sigh of relief.
Possibly one could be sad because he is crying, but of course one is not sad that he is crying. It would after
all be possible that people made to cry by application of onions would become sad; that they would either become
generally depressed, or would start thinking about certain events, and then grieve over them. But then the sensations
of crying would not thereby have turned into a part of the 'feeling' of grief.
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324. If someone behaves in such-and-such a way under such-and-such circumstances, we say that he is sad. (We
say it of a dog too.) To this extent it cannot be said that the behaviour is the cause of the sadness; it is its symptom.
Nor would it be beyond cavil to call it the effect of sadness.--If he says it of himself (that he is sad) he will not in
general give his sad face as a reason. But what if he said: "Experience has taught me that I get sad as soon as I start
sitting about sadly, etc." This might have two different meanings. Firstly: "As soon as, following a slight inclination, I
set out to carry and conduct myself in such-and-such a way, I get into a state in which I have to persist in this
behaviour." For it might be that a toothache got worse by groaning.--Secondly, however, that proposition might
contain a speculation about the cause of human sadness. The content being, perhaps, that if you could somehow or
other produce certain bodily states, you would make the man sad. But here arises the difficulty that we should not
call a man sad, if he looked and acted sad in all circumstances. If we taught such a one the expression "I am sad"
and he constantly kept on saying this with an expression of sadness, these words, like the other signs, would have
lost their normal sense. [Z 526.]
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325. I should almost like to say: One no more feels sorrow in one's body than one feels seeing in one's eyes. [Z 495.]
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326. To begin by teaching someone "That looks red" makes no sense. For he must say that spontaneously once he
has learnt what "red" means, i.e. has learnt the technique of using the word. [Z 418.]
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327. Any explanation has its foundation in training. (Educators ought to remember this.) [Z 419.]
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328. "So these concepts are valid only for the total human being?" No, for some have their application to animals
too.
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329. "Whoever generally acts this way and then sometimes that way, is described as..." That is a legitimate kind of
explanation of a word.
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330. We are inclined to imagine the matter as if a visual sensation were a new object which the child gets to know,
after he has learned the first primitive language-games with visual observations. "It looks red to
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me."--"And what is red like?"--"Like this." Here the right paradigm must be pointed to. [From "It looks red to me.":
Z 420.]
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331. If I have learned to carry out a particular activity in a particular room (putting the room in order, say) and am
master of this technique, it does not follow that I must be ready to describe the arrangement of the room; even if I
should at once notice, and could also describe, any alteration in it. [Z 119.]
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332. "This law was not given with such cases in view." Does that mean it is senseless? [Z 120.]
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333. One could imagine a concept of fear, for instance, that had application only to beasts, and therefore pertained
only to behaviour.--But you don't want to say that such a concept would have no use. [Z 524, first two sentences.]
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334. Can one say that there is a similarity between the emotion and its expression, insofar as both are excited, for
example? (I think Köhler said something like that.) And how does one know that the emotion itself is excited? The
person who feels it notices it and says so.--And if someone were one day to say the opposite?--"Be honest now, and
say whether you don't really recognize your inner excitement!"--How did I ever learn the meaning of the word
"excitement"?
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335. The misconception that this word means something internal as well as something external. And if any one
denies that, he is misinterpreted as denying inner excitement. (Temporal and timeless sentences.)
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336. Imagine that a child was quite specially clever, so clever that he could at once be taught the doubtfulness of the
existence of all things. So he learns from the beginning: "That is probably a chair."
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And now how does he learn the question: "Is it also really a chair?"? [Z 411.]
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337. Am I doing child psychology?--I am making a connection between the concept of teaching and the concept of
meaning.[Z 412.]
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338. One man is a convinced realist, another a convinced idealist and teaches his children accordingly. In such an
important matter as the existence or non-existence of the external world they don't want to teach their children
anything wrong.
What will the children be taught? Also to say: "There are physical objects" or the opposite?
If someone does not believe in fairies, he does not need to teach his children "There are no fairies": he can
omit to teach them the word "fairy". On what occasion are they to say: "There are..." or "There are no..."? Only when
they meet people of the contrary belief. [Z 413.]
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339. But the idealist will teach his children the word "chair" after all, for of course he wants to teach them to do this
and that, e.g. to fetch a chair. Then where will be the difference between what the idealist-educated children say and
the realist ones? Won't the difference only be one of the battle cry? [Z 414.]
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340. For doesn't the game "That is probably a..." begin with disillusion? And can the first attitute of all be directed
towards a possible disillusion? [Z 415.]
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341. "So does he have to begin by being taught a false certainty?"
There isn't any question of certainty or uncertainty yet in their language-game. Remember: they are learning
to do something. [Z 416.]
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342. So how does the doubt get expressed? That is: in a language-game, and not merely in certain phrases. Maybe in
looking more closely; and so in a fairly complicated activity. But this expression of doubt by no means always
makes sense, nor does it always have a point.
One simply tends to forget that even doubting belongs to a language-game.
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343. How does it come about that doubt is not subject to arbitrary choice?--And that being so--might not a child
doubt everything because it was so remarkably talented?
A person can doubt only if he has learnt certain things; as he can miscalculate only if he has learnt to
calculate. In this case it is indeed involuntary. [a: Z 409; b: Z 410.]
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344. If I have any doubts that this is a chair, what do I do?--I look at it and feel it on all sides, and so forth. But is this
way of acting always an expression of doubt? No. If a monkey or a child were to do this it
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wouldn't be. Only someone who is aquainted with such a thing as a 'reason for doubt' can doubt.
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345. I can easily imagine that a particular primitive behaviour might later develop into a doubt. There is, e.g., a kind
of primitive investigation. (An ape who tears apart a cigarette, for example. We don't see an intelligent dog do such
things.) The mere act of turning an object all around and looking it over is a primitive root of doubt. But there is
doubt only when the typical antecedents and consequences of doubt are present.
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346. "It tastes like sugar." One remembers exactly and with certainty what sugar tastes like. I do not say "I believe
sugar tastes like this." What a remarkable phenomenon. It just is the phenomenon of memory.--But is it right to call
it a remarkable phenomenon?
It is anything but remarkable. That uncertainty is not by a hair's breadth more remarkable than uncertainty
would be. For what is remarkable? My saying with certainty "This tastes like sugar", or its then really being sugar?
Or that other people find the same thing?
If the certain recognition of sugar is remarkable, then the failure to recognize it would be less so. [Z 660.]
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347. If people were (suddenly) to stop agreeing with each other in their judgments about tastes--would I still say: At
any rate, each one knows what taste he's having?--Wouldn't it then become clear that this is nonsense?
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348. Confusion of tastes: I say "This is sweet", someone else "This is sour" and so on. So someone comes along and
says: "You have none of you any idea what you are talking about. You no longer know at all what you once called a
taste." What would be the sign of our still knowing? [Z 366.]
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349. But might we not play a language-game even in this 'confusion'?--But is it still the earlier one?--[Z 367.]
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350. But there's a paradox here! Is the reliability of my expression of my taste to depend on changes in the outside
world?--The important thing here is surely the sense of the judgment, not its usefulness. Here we see the relation to
the original language-game of perception.
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351. "It tastes exactly like sugar." How is it I can be so sure of this? Even if it turns out wrong.--And what astonishes
me about it? That I bring the concept 'sugar' into so firm a connection with the taste sensation. That I seem to
recognize the substance sugar directly in the taste.
But instead of the expression "It tastes exactly..." I might more primitively use the exclamation "Sugar!" And
can it be said that 'the substance sugar comes before my mind' at the word? How does it do that? [Z 657.]
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352. Can I say that this taste brought the name "sugar" along with it in a peremptory fashion? Or the picture of a
lump of sugar? Neither seems right. The demand for the concept 'sugar' is indeed peremptory, just as much so,
indeed, as the demand for the concept 'red' when we use it to describe what we see. [Z 658.]
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353. I remember that sugar tasted like this. The experience returns to consciousness. But, of course: how do I know
that this was the earlier experience? Memory is no more use to me here. No, in those words, that the experience
returns to consciousness..., I am only transcribing my memory, not explaining it.
But when I say "It tastes exactly like sugar", in an important sense no remembering takes place. So I do not
have grounds for my judgment or my exclamation. If someone asks me "What do you mean by 'sugar'?"--I shall
indeed try to show him a lump of sugar. And if someone asks "How do you know that sugar tastes like that?" I shall
indeed answer him "I've eaten sugar thousands of time[[sic]]"--but that is not a justification that I give myself. [Z
659.]
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354. "Self-observation tells me that I believe that--but observation of the external world that it is not so."
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355. Let us now assume I've seen an F which someone has written like this: .†1 And assume that I always took it
for a mirror-F; that is, I assumed a certain connection between his letter and the regular one. Now you point out to
me that this is not the connection that exists,
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but rather there is a different one (that of the lines moved around). I understand this and now I say: "Well it certainly
does look different." If I'm asked "How different?" I might say: "Earlier it looked clumsy, but now it looks bold and
energetic." [Cf. Z 208.]
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356. Suppose†1 someone had always seen faces with only one expression, say a smile. And now, for the first time,
he sees a face changing its expression. Couldn't we say here that he hadn't noticed a facial expression until now? Not
until the change took place was the expression meaningful; earlier it was simply part of the anatomy of the face.--Is
that the way it is with the aspect of the letter? Expression could be said to exist only in the play of the features.
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357. So how a letter appears to me depends on whether it strictly follows the norm or whether and how it deviates
from the norm. Thus it is understandable that it makes a difference whether we know only one explanation for the
shape of a letter, or two.
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358. For how could I see that this posture was hesitant before I knew that it was a posture and not the anatomy of
the animal? [Cf. PI II, xi, p. 209b.]
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359. The question now is: If one can see a figure according to an interpretation, does one see it according to an
interpretation every time? And is there a well-defined difference between seeing which is not connected to any
interpretation, and that which is?
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360. I mean: Seeing a figure with this interpretation is a kind of thinking of the interpretation. For should I say that it
is possible to see this as a mirror-F without at the same time thinking about the special relationship which the word
"mirror-F" signifies? But I see an interpretation and an interpretation is a thought.
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361. One could make a rough copy of a picture-puzzle before and after one put it together, and then a mistake made
in copying the first aspect will be different from the mistake made in copying the second. So I could say: "Before I
solved it, I saw something like this (and I draw a forest)--after I solved it, something like this" (and then I draw a
man in the tree-tops).
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362. You must bear in mind that in the most important cases, what someone sees gets its impression in a report
about the object perceived. And this kind of report, of course, includes spatial announcements.†1--Now what is it
like, if someone has to give a report about what he sees on a flat surface, when the drawing on it has the character of
a picture-puzzle? First of all, so far as describing what he sees on the surface in spatial terms, he can give such a
description; indeed, that is perhaps the only kind of description he can give.
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363. An important piece of information will be, for instance: "Nothing has changed the whole while." This report is
based on continuous observation.
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364. When I solve the picture-puzzle, I discover something about the picture itself. For example, that a ship was
hidden by this camouflage. Perhaps I want secretly to tell someone what a certain person looks like, and I hide my
message, his portrait, in a picture-puzzle.
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365. If I were to call the figure an aid to thought, I could say I see it as this aid to thought.
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366. What a queer question it is, whether I musn't have thought of N. N.--when suddenly I saw his face in that of his
son! Of course my question is not: Mustn't I have thought of him simultaneously with having that image of him;
rather, it is whether having that image of him wasn't a kind of thinking of him? But how does one decide that?
For instance I say "I was just thinking whether he has arrived in...". This thought is expressed in a sentence.
That other one perhaps in an exclamation.
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367. Can I see his father's face in his and yet not think of his father at the same time? Seeing his father's face in his
clearly was a kind of imagining of that face. And now we must remember that one does not recognize the image of a
man as an image of him.
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368. Remember also that you cannot use a picture (or a model) to represent the shifting of the gaze! And wouldn't
the impression that that produces very naturally be counted as part of the visual
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impression? The aspect will, or can be expressed in the way I copy the figure, and thus in one sense, it is in the copy
after all. Furthermore I will portray a face differently according to my interpretation of it, even though the
photograph shows the same thing each time. So here again there's reason to speak of "seeing".
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369. The fact that I produce a different copy (a different result) accords with the concept of the visual state. The fact
that I produce the same copy, but in a different way--by drawing the lines in a different sequence--points toward the
concept of thinking.
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370. What justifies his use of "seeing" here? Or is there any justification and is it merely a linguistic blunder? Or is it
solely justified by the fact that I too am inclined to say: "Now I see it as this", "Now I see it as that"? That could be.
But I am absolutely disinclined to assume that; I feel I have to say "I see something". But what is that supposed to
mean?--I learned the word "to see", after all. What fits is not the word, the sound, or the written image. It's the use of
the word which forces on me the idea that I see this.
What I have learned about the use of the word must be forcing me to use the word here.
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371. "That is what it is to see something--", I should like to say. And that's really the way it is: the situation is exactly
like that in which the word is used elsewhere;--except the technique is somewhat different here.
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372. The use of the word "see" is in no way a simple one.--Sometimes we think of it as a word for an activity and
then it is hard to put your finger on the action.--Thus we think of it as simpler than it really is, conceiving it as
drinking something in with one's eyes, as it were. So that if I drink something in with my eyes, then there can be no
doubt that there's something I'm seeing (unless I am deceived by prejudices).
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373. One could say: I see the figure now as the limit of this series, now as the limit of that. This value could be the
limit of various functions.
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374. In a certain sense the figure can always be what I see it as, even if it is not "visible" in the other sense. For,
depending on the way it's used, or the way it arises, a figure can be the limit of different series.
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A triangle can really be used to represent a mountain, or as an arrow, in order to point in this direction, etc. etc. Thus
the description of the aspect is always a correct description of visual perception.
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375. Take a figure, say a written symbol; it may be the symbol correctly written, or there are various ways in which it
may be faulty. And there are aspects corresponding to these ways of taking the figure.--Here there's a very close
similarity with the experience of meaning when one utters an isolated word.
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376. It is copied differently--but the copy is the same.
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But I want to say: If something else is seen, the copy must be different.
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377. What, for example, is a copy of 'the schematic cube'? A drawing, or a solid object? And why only the former?!
And if a solid object is a copy, what kind of an object: a solid angle, a solid cube, a wire frame?
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378. If I tell someone "Now I see the figure as..." then I am providing him with some information similar to that
given in visual perception, but also similar to that of a way of taking, or an interpretation, or a comparison, or a
knowing.
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379. "Now I see a white cross on a black ground and then a black cross on a white ground." But what is this: a white
cross on a black ground? Do explain! And what is a black cross on a white ground? Surely you can't give the same
explanation for both! And yet there must be an explanation!
The explanation could go something like this: "A white cross on a black ground is something like this--" and
now a figure would follow. But of course this must not be the ambiguous one. Thus, instead of saying, "I see the
figure now as a white cross on..., now as...", one can also say: "I see the figure now in this way (and then the figure
follows), now in this way (and a different figure follows)." And if the first sentence was permissible then so was this
one.
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380. And doesn't that mean that each of the two figures was a kind of copy of the ambiguous figure?
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381. On the one hand these two representations are copies of what was seen, but on the other there is still need for a
conceptual explanation.--For instance, if I see the figure of a cross†1 now as a cross lying down, now as
standing up, now as a diagonal cross set up askew--what are the corresponding copies?
A cross lying down is one which was layed on its side but should stand up. So the copy will be something
shaped like a cross, and about which we know which it is--lying down or standing up. Therefore it would also be
possible to use as a copy a picture in which the shape of a cross appears, playing this or that role. That is, there is a
picture which brings to expression what I see as an aspect. And this makes the aspect similar to something visually
perceived.
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382. Or: In the same way as a picture tells us what is perceived, there is a picture that has a similar representing role
in relation to the aspect. Imagine a painting of a descent from the cross, for instance. What would that be to us, if we
didn't know which movements were captured here? The picture shows us these movements and yet it does not. (The
picture of the cavalry-attack, when the viewer doesn't know that horses don't stop in those positions.)
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383. "What I see looks like this." Imagine this said by someone who is looking at a galloping horse and then, as a
copy, uses a stuffed horse standing in a galloping position! Wouldn't the right copy be a galloping horse?
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384. Now--do I see a thought before me along with the aspect? Is a thought before me along with the painting? (For
of course the figure which is seen as this or that is like a part of a painting which by itself doesn't yet make any
sense.)
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385. It is possible to describe a painting by describing events; indeed that's the way it would be described in almost
every instance. "He's standing there, lost in sorrow, she's wringing her hands..." Indeed, if you could not describe it
this way you wouldn't understand it, even if you could describe the distribution of colour on its surface in minute
detail. ((Picture of the man ascending the mountain.))
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386. So you see it as if you knew that about it.
And if this seems a foolish way of putting it, then it must be kept in mind that the concept of seeing is
modified by it.
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387. But can I also say: "He would see the picture (of the battle, for instance) differently, if he didn't know what was
going on here"? How would this come out?! He would not talk about the picture in the same way we do, and he
would not say: "You can positively see these horses charging", or "That's not the way a horse runs!" etc. He would
not infer many things from the picture that we do.
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388. We could resolve, of course, to call what we now call "seeing the figure as...", "conceiving" it as this or that.--If
we did that, the problems of course would not disappear. Rather, we would then study the use of "conceiving", and
in particular we would study the peculiarity that this conceiving is something stationary, a state that now begins and
now ends.
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389. It seems to me--I might say--as if I should be able to reproduce this conception by means of a picture of the
figure that I am looking at.--And that, indeed, is the way it really is: I can say that the picture someone makes of an
object expresses a conception of the object. Quite as one can say: Hear this theme like this..., and play it
correspondingly.
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390. It is seeing, insofar as...
It is seeing, only insofar as...
(That seems to me to be the solution.)
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391. In this way, however, the aspects which are, so to speak, visual interpretations of the figure differ from the
aspects of the three-dimensional appearance. For a figure can be taken for a solid object. And even if there's no
question of such a deception, the statement "Now I see this figure as a pyramid" tells us something different, and has
different consequences, from the statement "Now I see the figure as a black cross on a white ground, etc.". (The
consequences of seeing three-dimensionally in descriptive geometry.) But the connection between the aspect and
thinking also seems to be changed, or dissolved. For isn't the copy which shows someone else how I see the figure
different? And one mustn't forget that the meaning of the word "copy" varies throughout this discussion.
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392. "It is as if our concepts involved a scaffolding of facts."
That would presumably mean: If you imagine certain facts
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otherwise, describe them otherwise, than the way they are, then you can no longer imagine the application of certain
concepts, because the rules for their application have no analogue in the new circumstances.--So what I am saying
comes to this: A law is given for human beings, and a jurisprudent may well be capable of drawing consequences
for any case that ordinarily comes his way; thus the law evidently has its use, makes sense. Nevertheless its validity
presupposes all sorts of things, and if the being that he is to judge is quite deviant from ordinary human beings, then,
e.g., the decision whether he has done a deed with evil intent will become not difficult but simply impossible. [Z
350.]
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393. "If humans were not in general agreed about the colours of things, if disagreements were not exceptional, then
our concept of colour could not exist." No:--our concept would not exist. Does that mean, therefore, that what is
conceivable as a rule does not have to be conceivable as an exception? [Z 351--to "Does that mean".]
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394. It is like the following case: I have learned how to express the results of experiments by means of a curve. If the
points are situated like this†1, I will know more or less what kind of curve to draw, and I will be able to use it to
draw further conclusions from the experiments. But if the points are placed like this†2, then what I have learned will
leave me in the lurch, for I will no longer know what line to draw. And if I met people who drew a curve through this
constellation of points without using any method I could understand and without hesitation, then I shouldn't be able
to imitate their technique. But suppose that I should see that they acknowledge some plausible line or other as the
right one, and that this line then serves them as the basis for further inferences; and if these inferences conflicted, as
we should say, with experience, and these people were somehow to make light of it--then I would say that this
indeed is no longer the technique I know of, but is one that, although "outwardly" similar, is in
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essence completely different. But if I say that, in using the words "outwardly" and "essence" I am passing judgment.
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395. What does it mean to say: "But that's an utterly different game!" How do I use this sentence? As information?
Well, perhaps to introduce some information in which differences are enumerated and their consequences explained.
But also to express that just for that reason I don't join in here, or at any rate take up a different attitude to the
game.[Z 330.]
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396. If I said "I wouldn't any longer call it...", this really means: the scales are tipping--I've taken up a different
position toward the thing.
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397. I could also say: "I can no longer communicate with these people."
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398. Once I said that there might be a concept which, to the left of a certain dividing line, would correspond to our
'red', and to the right, would correspond to our 'green'. And it appeared to me then and still does, that I might be able
mentally to enter this conceptual world, and that indeed I might be inclined to call the red that lies on one side of the
line the same thing as green on the other. (Indeed, this actually happens to me, particularly when there is a fairly dark
red and a fairly dark green.) It is as if, in such a world, I would not be disinclined to call the green merely an aspect
of the red, and as if what I call "colour" went further unaltered, and only the "shading" altered. Thus there would be
an inclination, in this situation, to employ a mode of expression which used the same adjective for green and red,
along with a modifier such as "shaded"/"unshaded". "But are you really going to tell me then that we're not dealing
with two different colours?" I want to say: I see enough similarity between this way of talking and the usual one so
that I could very easily accept this way, under certain circumstances.--But then wouldn't people see the similarity or
resemblance which we see; i.e., between green to the left and (our usual) green to the right?--What if they said that
these two colours were "outwardly similar"? I imagine the situation to be similar to this one: In the drawing†1:
I can call the angles α, β, γ equal to each other, even though they are outwardly unequal; and
I can call angles δ + α, as well as ε + γ unequal, even though they are outwardly equal.
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399. I could also say: the red to the left and the green to the right are of the same nature, but are different
manifestations of it.
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400. But in all of this I have produced a confusion. The important thing about the matter was surely to show that one
can go on in a sequence (say of numbers) in such a way that according to our concepts he stops following the old
law of the series, and continues on following a new one; but that according to another conception, the law of the
series does not change, but that what appears to be a change is explained by a change in circumstances.
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401. But what this really amounts to is that consistently following a series can only be shown by example.
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402. And here one is tempted again and again to talk more than still makes sense. To continue talking where one
should stop.
Page 75
403. I can tell someone: "This number is the right continuation of this sequence"; and in doing this I can bring it
about that for the future he calls the "right continuation" the same thing I do. That is, I can teach him to continue a
series (basic series) without using any expression of the 'law of the series'; rather, I am forming a substratum for the
meaning of algebraic rules, or what is like them. [Cf. Z 300.]
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404. He must go on like this without a reason. Not, however, because he cannot yet grasp the reason but
because--in this system--there is no reason. ("The chain of reasons comes to an end.") And the like this (in "go on
like this") is signified by a number, a value. For at this level the expression of the rule is explained by the value, not
the value by the rule. [Z 301.]
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405. For just where one says "But don't you see...!" the rule is no use, it is what is explained, not what does the
explaining. [Z 302.]
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406. "He grasps the rule intuitively."--But why the rule? Why not how he is to continue? [Z 303.]
Page 75
407. "Once he has seen the right thing, seen the one of infinitely many references which I am trying to push him
towards--once he has got hold of it, he will continue the series right without further ado. I grant that he can only
guess (intuitively guess) the reference that I
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mean--but once he has managed that the game is won."--But this 'right thing' that I mean does not exist. The
comparison is wrong. There is no such thing here as, so to say, a wheel that he is to catch hold of, the right machine
which, once chosen, will carry him on automatically. It could be that something of the sort happens in our brain but
that is not our concern. [Z 304.]
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408. "Do the same." But in saying this I must point to the rule. So its application must already have been learnt. For
otherwise what meaning will its expression have for him? [Z 305.]
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409. To guess the meaning of a rule, to grasp it intuitively, could surely mean nothing but: to guess its application.
And that can't now mean: to guess the kind of application, the rule for it. Nor does guessing come in here. [Z 306.]
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410. I might, e.g., guess what continuation will give the other pleasure (by his expression, perhaps). The application
of a rule can be guessed only when one can already choose one among different applications. [Z 307.]
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411. We might in that case also imagine that, instead of 'guessing the application of the rule,' he invents it. Well,
what would that look like? Ought he perhaps to say "Following the rule '+ 1' may mean writing 1, 1 + 1, 1 + 1 + 1,
and so on"? But what does he mean by that? For the "and so on" presupposes that one has already mastered a
technique. [Z 308.]
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412. How can one describe what someone does in continuing that rule?--If someone already knows how to use it,
we can do it by giving the rule. And who can use it? Someone who writes 1 + 1 + 1 after 1 + 1 and after that 1 + 1 +
1 + 1.--And can I now end with "and so on"? That would surely mean: "and simply goes on according to this rule."
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413. I cannot describe how (in general) to employ rules, except by teaching you, training you to employ rules. [Z
318.]
Page 76
414. I may now, e.g., make a talkie of such instruction. The teacher will sometimes say "That's right". If the pupil
should ask him "Why?"--he will answer nothing, or at any rate nothing relevant, not even: "Well, because we all do it
like that"; that will not be the reason. [Z 319.]
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415. We don't say "It may well be like that, but it isn't". Or: "I suppose he's coming tomorrow, but actually he won't
come."
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416. Even in the hypothesis the pattern is not what you think.
I should like to say: When you say "Suppose I believe that" you are presupposing the whole grammar of the
word "to believe". You are not supposing something given to you unambiguously through a picture, so to speak, so
that you can tack on to this hypothetical use some assertive use other than the ordinary one. You would not know at
all what you were supposing here, if you were not already familiar with the use of "believe". [Cf. PI II, x, p. 192e.]
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417. What is showing its face here is the invisible application.
We are not aware of the particular technique, for it flows along underground, as it were, without our
noticing it; and not until it openly contradicts our false imagination do we suddenly become aware of it; not until we
notice, e.g., that a sentence makes no sense, that we have no idea what to do with this sentence which was not such
as to arouse this suspicion straightaway. Can one tell one's doctor that one believes something as a symptom of
mental illness?--But one can say, for example: "I always believe I hear voices."
"I am always supposing that he is unfaithful to me, but he isn't." The line of the concept seems to break off
abruptly!--
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418. "The sentence, 'I believe it and it isn't true' can after all be the truth. Namely, when I really believe it and this
belief turns out to be wrong.
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419. I say of someone else "He seems to believe..." and others say it of me. Now why do I never say it of myself,
even though others are justified in saying it of me? Likewise: "It's obvious that he believes..." Don't I see
myself?--One could say so. [PI II, x, p. 191f.]
Page 77
420. A: "I believe it's raining."--B: "I don't believe so."--Now they are not contradicting each other; each one is
simply saying something about himself.
Page 77
421. "There is no such thing as a bluish yellow." This is like "There is no such thing as a regular biangle"; this could
be called a proposition of colour-geometry, i.e., it is a proposition determining a concept.
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422. If I had taught someone to use the names of the six primary colours, and the suffix "ish" then I could give him
orders such as "Paint a greenish white here!"--But now I say to him "Paint a reddish green!" I observe his reaction.
Maybe he will mix green and red and not be satisfied with the result; finally he may say "There's no such thing as a
reddish green."--Analogously I could have gotten him to tell me: "There's no such thing as a regular biangle!", or
"There's no such thing as the square root of -25."
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423. I want to say there is a geometrical gap, not a physical one, between green and red. [Z 354.]
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424. But doesn't anything physical correspond to it? I do not deny that. (And suppose it were merely our habituation
to these concepts, to these language-games? But I am not saying that it is so.) If we teach a human being
such-and-such a technique by means of examples,--that he then proceeds like this and not like that in a particular
new case, or that in this case he gets stuck, and thus that this and not that is the 'natural' continuation for him: this of
itself is an extremely important fact of nature. [Z 355.]
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425. "But if by 'bluish yellow' I mean green, I am taking this expression in a different way from the original one. The
original conception signifies a different road, a no thoroughfare."
But what is the right simile here? That of a road that is physically impassable, or of the non-existence of a
road? i.e. is it one of physical or of mathematical impossibility? [Z 356.]
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426. We have a colour system as we have a number system.
Do the systems reside in our nature or in the nature of things? How are we to put it?--Not in the nature of
numbers or colours. [Z 357.]
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427. Then is there something arbitrary about this system? Yes and no. It is akin both to what is arbitrary and to what
is non-arbitrary. [Z 358.]
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428. It is obvious at a glance†1 that we aren't willing to acknowledge anything as a colour intermediate between red
and green. (Nor does
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it matter whether this is always obvious to people, or whether it took experience and education to make it so.) What
would we think of people who were acquainted with 'reddish-green' (e.g., who called olive-green by that name)?
And what does this mean: "Then they have a different concept of colour altogether"? As if they wanted to say:
"Well, then it wouldn't be this but a different concept of colour"--all the while pointing to our own. As if there were
an object to which the concept belonged unequivocally. [First two sentences: Z 359.]
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429. These people are acquainted with reddish green. "But there is no such thing!"--What an extraordinary
sentence.--(How do you know?) [Z 362.]
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430. (The picture characterizing the concept would be something like an algebraic formula.)
Page 79
431. Let's put it like this: Must these poeple[[sic]] notice the discrepancy? Perhaps they are too stupid. And again:
perhaps not that either.--[Z 363.]
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432. Yes, but has nature nothing to say here?! Indeed she has--but she makes herself audible in another way.
"You'll surely run up against existence and non-existence somewhere!"--But that means against facts, not
concepts. [Z 364.]
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433. It is an extremely important fact that a colour which we are inclined to call (e.g.) "reddish yellow" can really be
produced (in various ways) by a mixture of red and yellow. And that we are not able to recognize straight off a
colour that has come about by mixing red and green as one that can be produced in that way. (But what does
"straight off" signify here?)
There could be people who recognize a regular polygon with 97 angles at first glance, and without counting.
[a: Z 365.]
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434. A concept compared with a style of painting: For is even our style of painting arbitrary? Can we simply decide
to adopt the style of the Egyptians? Is it a mere question of pleasing and ugly? [Cf. PI II, xii, p. 230c.]
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435. Did we invent human speech? No more than we invented walking on two legs. But if this is really so, then it is
an important fact that when humans are asked to reproduce the Great Bear on their
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own, they will always, or for the most part, do this by drawing the lines in one particular way and never in another.
But does that mean that they see the constellation in this way? And does it contain the possibility of a
sudden shift of aspect, for example? For it's the shift that we feel to be similar to a change in the object of sight.
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436. If there were no change of aspect then there would only be a way of taking, and no such thing as seeing this or
that.
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437. This seems absurd. As if one wanted to say: "If I use only coal for heat, and never anything else, then I'm not
really heating with coal."
But may it not be said: "If there were only one substance, there would be no use for the word 'substance'"?
That however presumably means: The concept 'substance' presupposes the concept 'difference of substance'. (As
that of the king in chess presupposes that of a move in chess; or that of colour that of colours.) [b: Z 353.]
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438. I tell someone something different when I say:
(a) that this or that form is contained in a drawing he doesn't see--
(b) that in the drawing he does see there is a form which he hasn't yet seen--
(c) that I've just noticed that a familiar drawing contains this form--
(d) that I am just now seeing the drawing in this aspect.
The interest of each of these is different.
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439. The first statement is a partial description of an object that one sees, and thus resembles "I see something red
over there".
The second is what I might call a "geometrical statement". In contrast to the first it is timeless. The discovery
that this is so is of the same kind as mathematical discoveries.
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440. But couldn't this statement also be made in temporal form? Something like this: "If you turn this drawing this
way and that you will see this form in it, and the lines won't seem to have moved." But it doesn't follow that we use
this fact to define a concept.
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441. How does one make the discovery? Well, for instance one might trace--by pure accident--certain lines of the
drawing on tracing
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paper. And then one sees: Why, that's a face! Or one is looking at the drawing, and then exclaims this sometimes
while one traces those lines.--And where is the discovery here?--It still has to be interpreted as a discovery, and in
particular as a geometrical discovery.
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442. I may come to see an aspect through someone's drawing my attention to it. But how that separates this 'seeing'
from perceiving colours and shapes.
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443. Noticing and seeing. One doesn't say "I noticed it for five minutes."
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444. "But do we really see the human figures in the picture?" Only what are you asking about??
What is obviously happening here is that one rather different concept is causing trouble for another. I was
supposed to be asking: "Do I really see the figures in the same sense as I really see...?" Or: "What reason do I have
for talking about 'seeing' in this case? And what is it in me that makes me rebel against this?"
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445. I should like to put the question, for instance: "Am I aware of the spatial character, the depth of this book, for
instance, the whole time I am seeing it?" Do I, so to speak, feel it the whole time?--But put the question in the third
person. When would you say of someone he was aware of it the whole time, and when the opposite?--Suppose that
you ask him--but how did he learn how to answer such a question?--Well, for instance, he knows what it means to
feel pain continuously. But that will only confuse him here, just as it confuses me. [Cf. PI II, xi, pp. 210-211.]
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446. If he now tells me that he is continuously aware of the depth--do I believe him? And if he says he is aware of it
only occasionally, when talking about it, perhaps--do I believe that? These answers will strike me as resting on a
false foundation.--It will be different if he tells me that the object sometimes strikes him as three-dimensional,
sometimes as flat. [PI II, xi, p. 211a.]
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447. I might get an important message to someone by sending him the picture of a landscape. Does he read it like a
blueprint? That is, does he decipher it? He looks at it and acts accordingly. He sees rocks, trees, a house, etc. in it.
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448. (The situation here is one of practical necessity, but the means of communication is one that has nothing to do
with any previous agreement, definition, or the like, and that otherwise only serves quasi-poetic purposes. But on the
other hand normal speech also serves poetic purposes.)
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449. The aspects of F: It is as if an image came into contact, and for a time remained in contact, with the visual
impression. [Cf. PI II, xi, p. 207b.]
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450. With the black and white cross, however, it is different, and this is more closely related to the
three-dimensional aspects (for instance, the drawing of a prism).
Page 82
451. The temptation to say "I see it like this", pointing to the same thing for "it" and "this". [Cf. PI II, xi, p. 207e.]
Page 82
452. The concept of 'seeing' makes a tangled impression. Well, it is tangled.--I look at the landscape, my gaze ranges
over it, I see all sorts of distinct and indistinct movement; this impresses itself sharply on me, that is quite hazy.
After all, how completely ragged what we see can appear! And now look at all that can be meant by a "description of
what is seen"! But this is just what we call that. We don't have a genuine, respectable case of such description, and
so we say. "Well, the rest isn't very clear as it is, being something which awaits clarification, or which must just be
swept aside as rubbish." [PI II, xi, p. 200a.]
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453. Here we are in enormous danger of wanting to make fine distinctions. It is the same when one tries to define the
concept of a material object in terms of 'what is really seen'. What we have rather to do is to accept the familiar
language-game, and to note false explanations of the matter as false. The primitive language-game we originally
learned needs no justification, and false attempts at justification, which force themselves on us, need to be rejected.
[Cf. PI II, xi, p. 200b.]
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454. Conceptual facts have very complicated interrelationships.
Page 82
455. Expression should always be separated from technique. And also cases where we can indicate the technique
from those where we can't.
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456. I might well say: "My thoughts move naturally from this picture to real grass, real animals; but from that one,
never."
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457. One looks at a picture and says: "Don't you see a squirrel!"--"Don't you feel the softness of this fur!"--And this
is said of certain pictures, but not of others.
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458. I get the idea of the essence of a picture--an idea not unlike that of a mathematical idea--through certain modes
of representation, in certain circumstances. If someone sees something I've written, then if he can read and write the
Roman alphabet, he'll be able to copy it quite exactly. He has only to read it, and then write it out. Despite our
different handwriting styles he'll have no difficulty producing a fairly acceptable reproduction of the lines on my
sheet of paper. But if he hadn't learned to read and write the Roman alphabet then it would have been much more
difficult for him to copy the maze of lines. Now, should I say: Whoever has learned these things would see my
handwriting completely differently from someone who had not?--What do we know about this? It could be that we
gave someone that sheet of paper to copy before he had learned to read and write; and then again, after he had
learned to read and write. And then he might tell us: "Oh yes, now I see these lines completely differently." Possibly
he might also explain: "Now all I really see is the writing that I'm reading; all else is floss, which doesn't concern me,
and which I hardly notice." Well this means that he sees the picture differently--when, that is, he actually does react
to it differently.
Likewise, compared to someone who can't read, someone who can will be able to give a different account of
a sheet of paper criss-crossed with writing. And this analogy holds too for speaking and its accompanying sounds.
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459. To this there is the answer, "I have never looked at a †1 with that in mind."
Page 83
460. Suppose someone were to answer: "For me it is always facing in that direction."--Would we accept his answer?
It would seem to assert that he always thinks of such connections whenever he looks at that letter (just as we say:
"Whenever I see this man, I have to think of how he...").
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Page 84
461. But if we now see the picture of a face, or even a real face--can we also say: I only see it looking in this direction
so long as I am occupied with it in this way?--What is the difference? The report: "This face is looking to the
right"--usually refers to the position of the face. I make it to someone who doesn't see the face himself. It is the
report of a perception.
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462. But does this then show that it can't be a matter of 'seeing' in these cases--but it is one of 'thinking', perhaps?
What makes this quite unlikely is that we want to talk about 'seeing' in the first place.--So should I say that it is a
phenomenon between seeing and thinking? No; but a concept that lies between that of seeing and thinking, that is,
which bears a resemblance to both; and the phenomena which are akin to those of seeing and thinking (e.g. the
phenomena of the utterance "I see the F facing to the right").
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463. How does one tell that human beings see three-dimensionally? I ask someone about the lie of the land of which
he has a view. "Is it like this?" (a spacial gesture)--"Yes."--"How do you know?"--"It's not misty, I see quite
clearly."--He does not give reasons for the surmise. The only thing that is natural to us is to represent what we see
three-dimensionally; special practice and training are needed for two-dimensional representation whether in drawing
or words. The queerness of children's drawings. [PI II, xi, p. 198d.]
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464. What is lacking to anyone who doesn't understand the question which way the letter F is facing, where, for
example, to paint a nose on?
Or to anyone who doesn't find that a word loses something when it is repeated several times, namely, its
meaning; or to someone who doesn't find that it then becomes a mere sound?
We say: "At first something like an image was there."
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465. Is it that such a person is unable to appreciate a sentence, judge it, the way those who understand it can? Is it
that for him the sentence is not alive (with all that that implies)? Is it that the word does not have an aroma of
meaning? And that therefore he will often react differently to a word than we do?--It might be that way.
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466. But if I hear a tune with understanding, doesn't something special go on in me--which does not go on if I hear it
without understanding? And what?--No answer comes; or anything that
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occurs to me is insipid. I may indeed say: "Now I've understood it," and perhaps talk about it, play it, compare it
with others etc. Signs of understanding may accompany hearing. [Z 162.]
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467. It is wrong to call understanding a process that accompanies hearing. (Of course its manifestation, expressive
playing, cannot be called an accompaniment of hearing either.) [Z 163.]
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468. For how can it be explained what 'expressive playing' is? Certainly not by anything that accompanies the
playing.--What is needed for the explanation? One might say: a culture.--If someone is brought up in a particular
culture--and then reacts to music in such-and-such a way, you can teach him the use of the phrase "expressive
playing". [Z 164.]
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469. The understanding of a theme is neither sensation nor a sum of sensations. Nevertheless it is correct to call it an
experience inasmuch as this concept of understanding has some kinship with other concepts of experience. You say
"I experienced that passage quite differently this time". But still this expression 'describes what happened' only for
someone familiar with a particular system of concepts. (Analogy: "I won the match.")†1 [Z 165.]
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470. This floats before my mind as I read. So does something go on in reading...? This question doesn't get us
anywhere. [Z 166.]
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471. But how can it float before me? Not in the dimensions you are thinking of. [Z 167.]
Page 85
472. We find certain things about seeing puzzling, because we do not find the whole business of seeing puzzling
enough. [PI II, xi, p. 212f.]
Page 85
473. We all know that a cube which is clearly depicted will be seen three-dimensionally. One might not even be able
to describe what one sees in anything other than three-dimensional terms. And it is clear that someone could also
see this picture as flat. Now, if he alternately sees the picture in one way, then in the other, he is
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experiencing a change of aspect. What is so amazing about that?--Is it this: that the report "Now I see..." can no
longer be a report about the object that is perceived. For earlier "I see a cube in this picture" was a report about the
object I am looking at.
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474. What is incomprehensible is that nothing, and yet everything, has changed, after all. That is the only way to put
it. Surely this way is wrong: It has not changed in one respect, but has in another. There would be nothing strange
about that. But "Nothing has changed" means: Although I have no right to change my report about what I saw, since
I see the same things now as before--still, I am incomprehensibly compelled to report completely different things,
one after the other.
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475. And it is not like this: I simply see the picture as one of an infinitely large number of bodies, whose projection it
is;--rather, I see it only as this--or as that. So the picture is alternately one and the other.
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476. We now have a language-game that is remarkably the same as, and remarkably different from, the previous
one. Now what follows from the expression "Now I see..." is completely different, even though there is once again a
close relationship between the language-games.
Page 86
477. We wouldn't have been surprised that the eye (the dot on our picture) is looking in one direction--unless it
changed the direction in which it looked.
Page 86
478. This question naturally suggests itself: Can we imagine people who never see anything as anything? Would
they be lacking an important sense, such as if they were colour-blind or they lacked perfect pitch? For the time being
let us call such people "gestalt-blind" or "aspect-blind".
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479. Here the question will arise: To what kind of aspect is someone blind. Should I assume, for example, that he
cannot see the schematic cube three-dimensionally, first one way, then another? If this is to be the case, then in
consistency I should have to suppose that he couldn't see the picture of a cube as a cube, and therefore couldn't see
the picture of a three-dimensional object as a three-dimensional object. So he would generally have a different
attitude toward pictures than we do. It might be the kind of attitude which we have toward a
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blueprint. For example, he would be able to work according to a pictorial representation.--But then we face the
difficulty that it would never be possible for him to take a picture for a three-dimensional object, as we do
sometimes with trompe l'oeil architecture. And that could not very well be called a sort of blindness; on the
contrary. (This investigation is not psychological.)
Page 87
480. Of course it is imaginable that someone might never see a change of aspect, the three-dimensional aspect of
every picture always remaining constant for him, for example. But this assumption doesn't interest us.
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481. It is conceivable, however, and also important for us, that some people might have a completely different
relation to pictures than we do.
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482. Thus we could imagine someone who would see only a painted face as a face, but not one that consists of a
circle and four dots; who wouldn't see the duck-rabbit picture as a picture of the head of an animal, and therefore
wouldn't see the change of aspect with which we are familiar.
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483. Let's assume that someone cannot see the picture of a runner as a picture of motion: How would this come out?
I'm assuming that he has learned that such a picture as this portrays a runner. Thus that he can say that it is a
runner; how will this man differ from normal human beings? I shall assume that he will show absolutely no
understanding that motion is being represented in a picture. And what would we call the signs of this defective
understanding?--We can easily fill out the picture. (But if such a man were able to see any picture and then copy it
exactly, we certainly wouldn't say of him that his visual sense was deficient.)
Clearly the words "Now I am seeing this as the apex--now that" won't mean anything to a learner who has
only just met the concepts apex, base, and so on. But I did not mean this as an empirical proposition. [b: cf. PI II, xi,
p. 208e.]
Page 87
484. "Now he's seeing it like this", "now like that" would only be said of someone capable of making all sorts of
applications of the figure quite freely. [PI II, xi, p. 208e.]
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485. But how queer for this to be the condition of his having such-and-such an experience! After all, you don't say
that one only has
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toothache if one is capable of doing such-and-such. From this it follows that we're not dealing with the same concept
of experience here.
The concept of experience is different each time, even though related. [PI II, xi, p. 208f.]
Page 88
486. We speak, make utterances, but only later do we form a picture of their life. [PI II, xi, p. 209a.]
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487. We could, however, imagine the following way of teaching the pupil to see it that way: In
addition to the first triangle we draw a second one, which is the one that hasn't toppled over yet.†1 Later we omit the
second triangle, and now the student can see the first one as toppled over.--But does he have to understand the
illustration, or at least see it correctly?--It might merely add to his confusion.
If that illustration says nothing to someone, then other pictures won't speak to him either, as they do to us; he
will not react to them as we do. (Not empirically.) Analogy with the picture of the galloping horse.
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488. It is anything but a matter of course that we see 'three-dimensionally' with two eyes. If the two visual images are
amalgamated, we might expect a blurred one as a result, like a wobbled photograph. [PI II, xi, p. 213b.]
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489. A code which I and another man have agreed on, in which "bench" means apple. Immediately after we agree I
say to him: "Take these benches away!"--He understands me, and does it; but he still feels the word "bench" to be
strange in this use, and when he hears it he might have the image of a bench. [Cf. PI II, xi, p. 214f.]
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490. What would we say of someone who couldn't see the schematic cube now as an upright box, now as one lying
on its side? If this is a defect, isn't it one of the imagination rather than of visual sense?
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491. But what a strange method!--I form a concept and ask myself how one might follow through with it
consistently. What we feel would deserve to be called that. To be sure, we do see a painting three-dimensionally, and
we would have a hard time describing it as a composite of flat colour surfaces, but what we see in a stereoscope
looks three-dimensional in an entirely different way again. Someone who looks at a photograph, whether it is one of
human beings, houses, or trees, doesn't seem to miss three-dimensionality in it! ((To the remark about seeing
three-dimensionally with both eyes.)) [Cf. PI II, xi, p. 213a.]
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492. I can see the schematic cube as a box, but not: now as a paper box, now as a tin box.--What ought I to say if
someone assured me he could see the figure as a tin box? Should I answer that that isn't seeing? But if he couldn't
see, could he then sense it?
Of course it would be plausible if we said to him: Only what can really be seen can be visually imagined in
that way. ((Knowing in dreams.)) [Cf. PI II, xi, p. 208b.]
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493. When you come out of a movie onto the street, you sometimes have the experience of seeing the street and the
people as if they were on the screen and part of the plot of a movie. How come? How does one see the street and the
people? I can only say: I have the fleeting thought, for example, "Perhaps this man will be a main character in the
plot". But that's not all there is to it. Somehow my attitude toward the street and the people is like the one toward the
action on the screen. Perhaps something like mild curiosity, or enjoyment.--But initially I can't even say all that.
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494. Doesn't it take imagination to hear something as a variation on a particular theme? And yet one is perceiving
something in so hearing it. [PI II, xi, p. 213c.]
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495. "Imagine this changed like this, and you have this other thing." In general, one would like to say that the power
of imagination can substitute for a picture, or a demonstration. [Cf. PI II, xi, p. 213d.]
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496. One can express the aspects of the double-cross simply by pointing first to a white cross, then to a black one,
which is to say, to the same things someone would point to if he were asking "Is this contained in the figure on this
paper?"--The same question could be asked about the duck-rabbit picture. But it is also clear that each case here
differs slightly from the other.
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For in order to express the aspects of this picture, one points to something that is not contained in the
picture, such as the black cross in the double cross.
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497. But you do speak of understanding music. You understand it, surely, while you hear it! Ought we to say this is
an experience which accompanies the hearing? [Z 159.]
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498. I give signs of delight and comprehension.
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Is it hair-splitting to say: joy, enjoyment, delight, are not sensations?--Let us at least ask ourselves: How
much analogy is there between delight and what we call, e.g. "sensation"? [a: Z 515; b: Z 484.]
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499. The connecting link between them would be pain. For this concept resembles that of, e.g., tactile sensation
(through the characteristics of localization, genuine duration, intensity, quality) and at the same time that of the
emotions through its expression (facial expressions, gestures, noises). [Z 485.]
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500. How do I know that someone is enchanted? How does one learn the linguistic expression of enchantment?
What does it connect up with? With the expression of bodily sensations? Do we ask someone what he feels in his
breast and facial muscles in order to find out whether he is feeling enjoyment? [Z 168.]
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501. But does that mean that there aren't any sensations after all which often return when one is enjoying music?
Certainly not. (In some places he is near weeping, and he feels it in his throat.)
A poem makes an impression on us as we read it. "Do you feel the same while you read it as when you read
something indifferent?"--How have I learnt to answer this question? Perhaps I shall say "Of course not!"--which is
as much as to say: this takes hold of me, and the other not. "I experience something different."--And what kind of
thing?--I can give no satisfactory answer. For the answer I give is nothing of importance.--"But didn't you enjoy it
during the reading?" Of course--for the opposite answer would mean: I enjoyed it earlier or later, and I don't want to
say that.
But now you remember certain sensations and images and thoughts as you read, and they are such as were
not irrelevant for the enjoyment, for the impression.--But I should like to say that they get their correctness only
from their surroundings: through the
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reading of this poem, from my knowledge of the language, with its meter and with innumerable other things. (These
eyes smile only in this face and in this temporal context.)
You must ask how we learnt the expression "Isn't that glorious!" (e.g.) at all.--No one explained it to us by
referring to sensations, images or thoughts that accompany hearing! Nor should we doubt whether he had enjoyed it
if he had no account to give of such experiences; though we should, if he showed that he did not understand certain
tie-ups. [a: cf. Z 169; b, c, d: cf. Z 170.]
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502. But isn't understanding shown, e.g., in the expression with which someone reads the poem, sings the tune?
Certainly. But what is the experience during the reading? About that you would just have to say: you enjoy and
understand it if you hear it well read, or feel it well read in your speech-organs. [Z 171.]
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503. Understanding a musical phrase may also be called understanding a language. [Z 172.]
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504. I think of a quite short phrase, consisting of only two bars. You say "What a lot that's got in it!" But it is only,
so to speak, an optical illusion if you think that what is there goes on as we hear it. (Consider that sometimes we say,
and say rightfully: "It all depends who says it".) (Only in the stream of thought and life do words have meaning.) [Z
173.]
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505. What contains the illusion is not this: "Now I've understood"--followed perhaps by a long explanation of what I
have understood. [Z 174.]
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506. How does seeing an aspect hang together with the ability to perform certain operations (e.g. in mathematics)?
Think of seeing three-dimensionally in descriptive geometry and operating within the drawing. He moves his pencil
on the surface of the drawing as if he were moving within the real object. But how can that be proof of seeing?
Well, don't we accept it as proof of seeing if somebody moves about a room with confidence? There are
simply different criteria for seeing. Ask yourself: does someone who has no trouble picturing animals, people, and
all sorts of things in his imagination, or in his memory, have to see them with his inner eye? The answer could be:
"In such a case we simply say..."--But it could also be: "You have to ask the person doing the drawing whether he's
doing this or not."
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507. Now there is a tie-up between aspect and imagination.
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508. The aspects of surface and base. What would a person who is blind towards these aspects be lacking?--It is not
absurd to answer: the power of imagination.
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509. Bear in mind that there is frequently a 'right' word for an aspect.
Suppose we have someone look at the double cross and tell us which of the two aspects (black cross or white
cross) he sees. It will probably be irrelevant whether he says that sometimes he sees something like a little white
windmill with four sails and sometimes an upright black cross, or whether he sees the white cross as the four corners
of a piece of paper folded toward the middle. The cross which is "now" seen can also be seen as an opening in the
shape of a cross. But these differences needn't matter to us; and therefore one could distinguish between "purely
optical" aspects and "conceptual" ones. ((Similarly, the particular words someone uses to describe the events in a
dream may or may not matter.))
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510. "See F as "†1 could not be understood before something quite different has been said. For would I
understand "See this triangle as that triangle"?†2 There must first be a conceptual connection.
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511. "It seems to me now to be facing left--and now right again." That is, the way it was before? No; earlier it had no
direction for me. Earlier I didn't surround it with this world of images.
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512. Attention is dynamic, not static--one would like to say. I begin by comparing attention to gazing but that is not
what I call attention; and now I want to say that I find it is impossible that one should attend statically. [Z 673.]
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513. Someone might see a boulder and exclaim: "A man!", and then he might point out to someone else how he sees
the man in the boulder--where the face is, the feet are, etc. (Someone else might see a man in the same shape, but in
a different way.)
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It will be said that it takes imagination to see that, but that no imagination is required to recognize a
true-to-life picture of a dog as a dog.
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514. "He's comparing the boulder to a human shape," "He sees a human shape in it"--but it's not in the same sense
that we say: He's comparing that picture with a dog, or this passport photograph with a face.
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515. When I'm looking at the photograph, I don't tell myself "That could be seen as a human being". Nor when
looking at an F do I say: "That could be seen as an F."
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516. If somebody showed me the figure and asked me "What is that?", I could answer him only that way.--I couldn't
answer: "I take that to be a..." or "Probably that is a..." Any more than I take letters to be this or that when I'm
reading a book.
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517. "I see it as a ..." is connected with "I'm trying to see it as ...", or "I can't see it as ... yet". But you cannot try to
see the regular F as a regular F.
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518. To ask someone's advice mentally. To estimate the time by imagining a clock.
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519. The aspect presents a physiognomy which then passes away. It is almost as if there were a face there which at
first I imitate, and then accept without imitating it.--And isn't this really explanation enough?--But isn't it too much?
[PI II, xi, p. 210e.]
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520. If in a particular case I say: attention consists in preparedness to follow each smallest movement that may
appear--that is enough to show you that attention is not a fixed gaze: no, this is a concept of a different kind. [Z 674.]
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521. We see, not change of aspect, but change of interpretation. [Z 216.]
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522. You see it conformably, not to an interpretation, but to an act of interpreting. [Z 217.]
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523. If someone were asked, "Can you see F as an ef?" he wouldn't understand us. But he would understand if we
asked, "Can you see it
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as a backwards F?" And he would also understand: "And can you now see it as a regular ef again?"--Why?
"Can you see it as ...?" or "Now look at it as a ...!" go together with "Now take it as a ...".
Only where this command makes sense does the question make sense.
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524. Imagine that someone were to point to a regular printed F and say, "Now it is an ef".--What does that mean?
Does it make any sense? For the time being it has none. To what extent is it NOW an ef? Insofar as it always is? And
in contrast to what?--I look at a lamp and say "Now it is a lamp."--What can I mean?
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525. You need new conceptual glasses.
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526. If you say "Now it's a face for me", we can ask: "What sort of change are you alluding to?" [PI II, xi, p. 195d.]
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527. The cry "A hare!" is, after all, related to the report "a hare".
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528. What is an expression of amazement? Can it be a stationary attitude? Can amazement thus be a state of
inactivity?
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529. Suppose someone were to ask: "Why can't you hold on to the experience of surprise?"
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530. "The ef vanishes and a cross appears in its place; the cross vanishes and a backwards F appears in its place; etc."
That is, after all, the way we express changes in perception.
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531. Forget, forget that you have these experiences yourself! [Z 179.]
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532. Our eye seems each time to be drawing a different shape in these lines (on the paper).
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533. Different pictures appear to me. But how different are they? In what do they differ? That I can explain only by
referring to their origin.
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534. I say something; and it is correct;--but then I misunderstand the use to which the statement would be put.
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535. How does one play the game "It could be this too"? What a figure could also be--which is what it can be seen
as--is not simply another figure. Thus it made no sense to say: F could also be an .†1 Nor would this make
sense:--this could mean several entirely different things.
But one could play that game, for instance, with a child. Together we look at a shape; or at a random object
(a piece of furniture)--and then it is said: "That is now supposed to be a house."--And now it is reported, talked
about, and treated as if it were a house, and it is altogether interpreted as this. Then, when the same thing is made to
stand for something else, a different fabric will be woven around it. [a: cf. PI II, xi, p. 206d; b: cf. PI, p. 206e.]
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536. How will you know whether the child sees the thing as that? Well, he might spontaneously say so. He might
say something like: "Yes, now I see it as ... ". And in this situation, with the lively participation in the fiction, it will
indeed signify the seeing of the aspect.
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537. I want to say: this game is related to seeing the aspects of F, for example.
A person's ability, as it were, to play-act things is a prerequisite for his meaning the same thing we do when
he says "Now I see it as...".
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538. How do you teach a child, say in arithmetic: "Now take these things together!", or "Now these go together"?
Clearly "taking together" and "going together" must originally have had another meaning for him than that of seeing
in this way or that.--And this was a remark about concepts, not about teaching methods. [PI II, xi, p. 208c.]
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539. To be sure, one can say "See this shape as a... for five minutes", if this means: He is to hold it, to keep it
balanced, in this aspect.
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540. What do you understand if someone says "I see it (i.e. the regular F) as an ef"?--That he is dealing with aspects;
that it is an unstable situation. That he is thinking "It could also be that."
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541. Seeing aspects is built up on the basis of other games.
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542. We certainly speak of calculating in our imagination. So it is not surprising that the power of imagination can
contribute to knowledge.
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543. But I don't want to say that an aspect is a mental image. Rather that 'seeing an aspect' and 'imaging something'
are related concepts. [Cf. PI II, xi, p. 213c.]
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544. One wants to ask of seeing an aspect: "Is it seeing? Is is [[sic]] thinking?" The aspect is subject to the will: this
by itself relates it to thinking.
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545. "The aspect is subject to the will." This isn't an empirical proposition. It makes sense to say, "See this circle as a
hole, not as a disc", but it doesn't make sense to say "See it as a rectangle", "See it as being red".
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546. Do I really see something different each time, or do I only interpret what I see in a different way? I am inclined
to say the former. But why?--To interpret is to think, to do something. [PI II, xi, p. 212d.]
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547. The cases in which we interpret what we see are easily recognized. When we interpret we put forth a
hypothesis which may turn out to be wrong. "I see this shape as a..." can no more be verified than (or can be verified
only in the same sense as) the statement "I see a bright red". So here we have a similar use of the word "see" in both
contexts.
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548. Suppose someone were to ask: "Do we all see an F in the same way?" What could he mean by that?--We could
make this test: we show an F to different people and ask them whether the F faces to the right or to the left. Or we
could ask: "If you compare an F with the profile of a face, in which direction is the face looking?"
But many people might not understand this question. As many do not understand the question, "What
colour is the vowel a for you?"--If someone didn't understand the question, if he claimed that it was nonsensical,
could we say that he doesn't understand English, or at least the meanings of the words "colour", "vowel", etc.?
On the contrary: When he has learned to understand these words, then he can react to these questions, either
"with" or "without understanding". [b, c: Z 185.]
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549. Suppose, instead of "in which direction is the letter... facing?", the question had been: "If you were to paint an
eye and a nose onto an F or a J, in which direction would they be facing?" Surely this too would be a psychological
question. But it doesn't have to do with 'seeing something this way or that'. Rather, it deals with an inclination to do
one or the other. (But we must think about how someone would arrive at his answer to this question.)--Thus this
kind of seeing is related to an inclination. The inclination can change, or be absent altogether.
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550. "With this arrangement of windows the facade faces that way."
"The windows used to be arranged so that the facade faced that way.
The first sentence is like a proposition of geometry. In the second one, the concept of the 'direction in which
the facade faces' serves to describe the facade. Just as we describe a face using the concepts 'happy', 'sullen',
'suspicious', or a movement by the words "fearful", "hesitant", "sure". And to the extent that these are descriptions
of what has been visually perceived, or observed, they are also descriptions of the visual impression. So one can say
that he sees the hesitation. (A person copying a picture can be told, "The face isn't right yet; it isn't sad enough.")
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551. Anyone with an eye for family resemblances can recognize that two people are related to each other, even
without being able to say wherein the resemblance lies. (Think of the case of the calculating-genius.)
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552. It might be an incorrect use of language to say "I see fear in this face". We would be taught: a fearful face can be
'seen'; but the fear in a face, or the similarity or dissimilarity between two faces, is 'noticed'.
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553. The kinship of these two concepts is shown in this explanation: If you want to see how they differ, think of the
sense it could make to say that someone saw the similarity between two faces from one stroke of the hour to the
next. Or think of the order: "Notice the similarity from... to...!"
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554. A drawing can be the description of a visual impression. What is at the top of the drawing, what is at the
bottom, is usually of the greatest importance. But someone could also stipulate a distance at
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which we should hold the drawing from our eyes. Indeed, even a spot on the drawing at which we are to look could
be stipulated, or how our eyes are to travel over the drawing.
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555. I begin seeing the similarity when it "strikes" me; and do I then see it as long as I see the similar objects? Or
only as long as I am conscious of the similarity?--If the similarity strikes me, I perceive something, but I don't have
to remain conscious of it in order to perceive that it doesn't change.
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556. Two uses of the statement "I see...". One language-game: "What do you see there?"--"I see...", and then a
description of what was seen follows, either in words, or through a drawing, or a model, or gestures, etc.--Another
language-game: We look at two faces and I say to someone: "I see a similarity in them."
In the first language-game the description could have gone something like this: "I see two faces which are as
like as father and son." This can be called a far less complete description than the one that uses a drawing. But
someone could give this more complete description and still not notice that similarity. Another might see the
drawing of the first one and discover the family resemblance in it; and in the same way he might see a similarity
between their facial expressions. [a: cf. PI II, xi, p. 193a.]
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557. "When I uttered the word just now, it meant ... to me." Why should that not be mere lunacy? Because I
experienced that? That is not a reason. [Z 182.]
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558. Those cases in which the inner seems hidden from me are very peculiar. And the uncertainty which is
expressed in this is not a philosophical one; no, it is practical and primitive.
Page 98
559. It is then as if I realized for the first time that the inner is really always hidden.
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560. (We also say that the man is completely transparent.) So sometimes someone is transparent, sometimes not.
Page 98
561. "I can never know what goes on inside him."--But does something have to go on inside him? And why should I
be concerned
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with that?--But this picture suggests a real uncertainty, not one we've dreamed up.
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562. What is the importance of someone making this or that confession? Does he have to be able to judge his
condition correctly?--What matters here is not an inner condition which he judges, but just his confession.
(His confession can explain certain things. For example, it can cause me to stop suspecting someone else.)
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563. The principal uncertainty: I don't know what he is thinking if he doesn't express it. Now suppose he does
express it, but in a language you don't understand. He could tap it out with the finger of one hand on the back of his
other hand in morse code, or some such thing. Then too, after all, it is secret, and isn't it just as secret as if it had
never been expressed? The language could also be such that I could never learn it, e.g. its rules might be
extraordinarily complicated.
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564. So someone can hide his thoughts from me by expressing them in a language I don't know. But where is the
mental thing which is hidden?
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565. I may choose the language in which I think. But not as if I think, and then choose the language into which I
want to translate my non-verbal thoughts.
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566. You can be as certain of someone else's sensations as of any fact. But this does not make the propositions "He
is happy" and "2 × 2 = 4" into similar instruments. It suggests itself to say, "The certainty is of a different kind", but
that doesn't remove the obscurity. [Cf. PI, II, xi, p. 224c.]
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567. "But, if you are certain, isn't it that you are shutting your eyes in face of doubt?"--They are shut.
It is indeed true: this kind of doubt is arrived at in a completely different way from doubt about an
arithmetical proposition. Above all, in the former case complete certainty is the limit of a belief which differs by
degrees.--Everything is simply different. [a: PI II, xi, p. 224d.]
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568. And then there is what I should like to call the case of hopeless doubt. When I say, "I have no idea what he is
really thinking--". He's a closed book to me. When the only way to understand someone else would be to go through
the same upbringing as his--which is
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impossible. Here there's no pretence. But imagine people whose upbringing is directed toward suppressing the
expression of emotion in their faces and gestures; and suppose these people make themselves inaccessible to me by
thinking aloud in a language I don't understand. Now I say "I have no idea what is going on inside them", and there
it is--an external fact.
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569. "I cannot know what is going on in him" is above all a picture. It is the convincing expression of a conviction. It
does not give the reasons for the conviction. They are not something that can be seen directly. [PI II, xi, p. 223g.]
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570. "We see emotion."--As opposed to what?--We do not see facial contortions and make the inference that he is
feeling joy, grief, boredom. We describe a face immediately as sad, radiant, bored, even when we are unable to give
any other description of the features.--Grief, one would like to say, is personified in the face. This is essential to what
we call "emotion". [Cf. Z 225.]
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571. The man I call meaning-blind will understand the instruction "Tell him he is to go to the bank--and I mean the
river bank," but not "Say the word bank and mean the bank of a river".
He will also not be able to report that he almost succeeded, but that then the word slipped into the wrong
meaning. It does not occur to him that the word has something in it which positively fixes the meaning, as a spelling
may; nor does its spelling seem to him to be a picture of the meaning, as it were.--For instance, it is very tempting to
think that a different spelling will lead to at least a very small difference in pronunciation, even where this is certainly
not so. Here we have a case which can serve as an example for many others: You say the two words (e.g. "for" and
"four") to yourself and you really do pronounce them a little differently, even though you don't do this in the normal
course of speech, when you're not thinking about it. And this is so if for no other reason than that each word is
pronounced differently on different occasions. [a: cf. Z 183a.]
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572. Different people are very different in their sensitiveness about changes in the orthography of a word. And the
feeling is not just piety towards an old use.--If for you spelling is just a practical question, the feeling you are lacking
in is not unlike the one that a 'meaning-blind' man would lack. [Z 184.]
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573. How could he hear the word with that meaning? How was it possible?!--It just wasn't--not in these dimensions.
[Z 180.]
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574. But isn't it true, then, that the word means that to me now? Why not? For this sense doesn't come into conflict
with the rest of the use of the word.
Someone says: "Give him the order... and mean by it...!" What can that mean?
But why do you use just this expression for your experience?--such a poor fit!--That is the expression of the
experience, just as "The vowel e is yellow" and "In my dream I knew that..." are expressions of other experiences. It
is a poor fit only if you take it the wrong way.
This expression goes with the experience just as the primitive expression of pain goes with pain. [a, b: Z 181.]
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575. William James: The thought is already complete at the beginning of the sentence. How can one know that?--But
the intention of uttering the thought may already exist before the first word has been said. For if you ask someone:
"Do you know what you mean to say?" he will often say yes.
It is my intention to whistle this theme: have I then already, in some sense, whistled it in thought? [a: Z 1; b:
Z 2b.]
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576. Whoever answers "Yes" to the question, "Do you know yet what you want to say?" may have some mental
image or other; but if this could be heard or seen objectively, then there would generally not be any way of deriving
what he intended from it with certainty. (Raising your hand in class.)
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577. Not everyone with an intention has therefore made a plan.
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578. What forms of mental defects actually exist is of no concern to us; but the possibilities of such forms do
concern us. It is not whether there are men incapable of thinking "At that time I wanted to...", but how this concept
can be followed through. [Cf. Z 183a.]
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579. How could this assumption be followed through consistently? What would we call a consistent
follow-through?--If you assume that someone cannot do this, then how about that? Is he also unable to do
this?--Where does this concept take us? [From "If you assume" cf. Z 183b.]
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580. "You must earnestly promise yourself to do it, and then you'll do it." Earnestly promising something requires
thinking about it, for instance. It requires a particular preparation. Finally there might actually be a formal promise,
perhaps even in a loud voice, but that is just one stone in this building. (Vows.)
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581. A vow could be called a ceremony. (Baptism, even when it is not a Christian sacrament.) And a ceremony has
an importance all its own.
Page 102
582. "I had the intention of..." does not express the memory of an experience. (Any more than "I was on the point
of...".) [Z 44.]
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583. "What a queer and frightful sound. I shall never forget it." And why should one not be able to say that of
remembering ("What a queer... experience...") when one has seen into the past for the first time?--[Z 661.]
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584. Couldn't he just be imagining that he calculated this? (That he now knows the solution to the problem is not
supposed to be inconsistent with this. And he might indeed have miscalculated.) And if there is no mistake here,
then this is not because there is certainty.
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585. Someone tells me he has just worked out in his head how much... ×... is. He gives an obviously wrong answer,
and when asked how he arrived at it, he recites the calculation; it is utter nonsense, as he himself now realizes, but at
that time, he says, it seemed completely correct. (Something similar occurs in dreams.) Can't that happen? His
mental arithmetic, I want to say, still must prove itself.
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586. "In hiding something from me, he can hide it in such a way that not only will I never find it, but finding it will
be completely inconceivable." This would be a metaphysical hiding.--But what if, without knowing it, he were to
give signs that give him away? That would be possible, after all.--Now wouldn't he be the only judge of whether
those signs have given him away?--But couldn't I insist that he forgot what happened inside him--and thus not let his
statement count? (Without calling it a lie.) That means that I declare it worthless, or give it value only as a
phenomenon, from which inferences about his state might be drawn.
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587. If something is hidden--isn't it as if writing were hidden, or rather something which looks like writing, whose
meaning only lies in what he reads out of it, or into it, at some point?
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588. Of course he can mislead me, make me arrive at false conclusions. But it doesn't follow from this that he has
hidden anything; even though the way he behaves can be compared to hiding.
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589. Haven't I the right to be convinced that he is not pretending to me?--And can't I convince someone else of my
right?
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590. If I tell him in full detail how my friend behaved, will he have any reasonable doubt as to the genuineness of my
friend's feelings?
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Does anyone doubt the genuineness of Lear's feelings?
Page 103
591. Is it thoughtlessness not to keep the possibility of pretence in mind?
Page 103
592. Remembering: a seeing into the past. Dreaming might be called that, when it presents the past to us. But not
remembering; for, even if it showed scenes with hallucinatory clarity, still it takes remembering to tell us that this is
past. [Z 662.]
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593. But if memory shows us the past, how does it show us that it is the past?
It does not show us the past. Any more than our senses show us the present. [Z 663.]
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594. Nor can it be said to communicate the past to us. For even supposing that memory were an audible voice that
spoke to us--how could we understand it? If it tells us, e.g. "Yesterday the weather was fine", how can I learn what
"yesterday" means? [Z 664.]
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595. I give myself an exhibition of something only in the same way as I give one to other people. [Z 665.]
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596. I can display my good memory to someone else and also to myself. I can subject myself to an examination.
(Vocabulary, dates.) [Z 666.]
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597. But how do I give myself an exhibition of remembering? Well, I ask myself "How did I spend this morning?"
and give myself an answer.--But what have I really exhibited to myself? Remembering? That is, what it's like to
remember something?--Should I have exhibited remembering to someone else by doing that? [Z 667.]
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598. "To purpose to do something is a special inner process."--But what sort of process--even if you could dream
one up--could satisfy our requirements about purpose? [Z 192.]
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599. Imagine men who show pity only when they see someone else bleeding; otherwise they laugh at his
expressions of pain. That's the way it is with them. Some smear themselves with animal blood to be pitied. If they're
found out they're severely punished.
Page 104
600. They don't ask the question: "Couldn't he be feeling pain anyway?"
Page 104
601. These people need not have certain scruples.
Page 104
602. Do I pay any mind to his inner processes if I trust him? If I don't trust him I say, "I don't know what's going on
inside him". But if I trust him, I don't say that I know what's going on inside him.
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603. If I don't distrust him, I don't pay any mind to what is going on inside him. (Words and their meaning. The
meaning of words, what stands behind them, doesn't concern me in normal conversation. Words flow along and
transitions are made from words to actions and from actions to words. When someone's performing mathematical
calculations he doesn't stop to think whether he is doing it 'thoughtfully' or 'parrot-like'. (Frege.))
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604. There might be people who speak to themselves quite a bit before and while they are doing something, and then
again there might be people who only say very little to themselves. When he is asked: "What were you thinking
when you did that?" he confesses, perhaps quite honestly, "Nothing at all", even though what he did seems to us to
have been well thought out, possibly even shrewd. I say that I don't know what is going on inside him, and in an
important sense nothing is going on inside him. I don't know my way about with him: for example, I make
erroneous suppositions quite easily, and from time to time I am very disappointed in my expectations.
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Page 105
I could visualize this person by imagining that he accompanies all of his actions with monologues, which
express his sentiments. The monologues would be a construction, a working hypothesis, by which I try to
comprehend his actions. Must I now assume thinking to be going on inside him apart from those monologues? Are
the monologues not quite enough? Can't they do everything the inner life is supposed to do?
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605. It is easy to imagine and work out in full detail events which, if they actually came about, would throw us out in
all our judgments.
If I were to see quite new surroundings from my window instead of the long familiar ones, if things were to
behave as they never did before, then I should say something like "I have gone mad"; but that would merely be an
expression of giving up the attempt to know my way about. And the same thing might befall me in mathematics. It
might, e.g., seem as if I kept on making mistakes in calculating, so that no answer seemed reliable to me.
But the important thing about this for me is that there isn't any sharp line between such a condition and the
normal one. [Z 393.]
Page 105
606. Why is it important to depict anomalies accurately? If someone can't do this, that shows that he isn't quite at
home yet among the concepts. [Culture and Value, p. 72.]
Page 105
607. To be sure, there is this: acquiring a knowledge of human nature; it is also possible to help someone with this, to
give lessons, as it were, but one only points to cases, refers to certain traits, gives no hard and fast rules.
Page 105
608. Perhaps I can say "Just let me talk to this man, let me spend some time with him, and I shall know whether he
can be trusted". And later: "I have the impression...". But here it's a matter of prognosis. Let the future show whether
my impression was correct. Knowledge of human nature can convince us that this man is really feeling what he
claims he's feeling; but does it convince us that other humans feel anything?
Page 105
609. "One can't pretend like that."--This may be a matter of experience--namely that no one who behaves like that
will later
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behave in such-and-such a way; but it also may be a conceptual stipulation; and the two may be connected.
(For it wouldn't have been said that the planets had to move in circles, if it had never appeared that they
move in circles). [a: Z 570a; b: Z 570c.]
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610. As I'm teaching, I can point to someone and say, "You see, that person isn't pretending." And a student can
learn from this. But if he were to ask me "How does one really tell that?"--then I might not have anything to answer,
except, perhaps, something like this: "Look how he's lying there, look at his face", and things like this.
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611. Could this be different with other beings?--If they all had the same bodies and the same facial features, for
example, then there would already be a great deal of difference.
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612. And pretending is, of course, only a special case of someone's producing expressions of pain when he is not in
pain. For if this is possible at all, why should it always be pretending that is taking place--which is a very specific
psychological process? (And by "psychological" I don't mean "inner".) [Cf. PI II, xi, p. 228(f)-229.]
Page 106
613. There might actually occur a case where we should say "He believes he is pretending."
(Pilgrim's Progress: He thinks he is uttering the curses which the devil is uttering.) [a: PI II, xi, p. 229c.]
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614. Sufficient evidence passes over into insufficient without a borderline. A natural foundation for the way this
concept is formed is the complex nature and the variety of human contingencies.
Then given much less variety, a sharply bounded conceptual structure would have to seem natural. But why
does it seem so difficult to imagine the simplified case?
Is it as if one were trying to imagine a facial expression not susceptible of gradual and subtle alterations; but
which had, say, just five positions; when it changed it would snap straight from one to another. Would this fixed
smile really be a smile? And why not?--I might not be able to react as I do to a smile. Maybe it would not make me
smile myself. [a, b: Z 439; c: cf. Z 527.]
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615. A facial expression that was completely fixed couldn't be a friendly one. Variability and irregularity are essential
to a friendly expression. Irregularity is part of its physiognomy.
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616. The importance we attach to the subtle shades of behaviour.
Page 107
617. My relation to the appearance here is part of my concept. [Z 543.]
Page 107
618. Imagine this argument: Pain comes in degrees. But no one will claim that I ever know the exact degree of
someone else's pain; therefore it could be of the degree o.
But does he himself know the 'exact degree' of his own pains? And what does 'knowing' this mean?
Page 107
619. "Then does he not know how intense his pains are?" He doesn't have the slightest doubt about that.
Page 107
620. But after all, I don't know, e.g., that his pain has now eased off a little.--Oh yes I do, if he tells me. What he says
is also an utterance of his pain.
Page 107
621. The uncertainty is not founded on the fact that he does not wear his pain on his sleeve. And there is not an
uncertainty in a particular case. If the frontier between two countries were in dispute, would it follow that the
country to which any individual resident belonged was dubious? [Z 556.]
Page 107
622. 'Heap of sand' is a concept without sharp boundaries--but why isn't one with sharp boundaries used instead of
it?--Is the reason to be found in the nature of the heaps? What phenomenon is it whose nature determines our
concept? [Cf. Z 392.]
Page 107
623. "A dog is more like a human being than a being endowed with a human form, but which behaved
'mechanically'." Behaved according to simple rules?
Page 107
624. We judge an action according to its background within human life, and this background is not monochrome,
but we might picture it as a very complicated filigree pattern, which, to be sure, we can't copy, but which we can
recognize from the general impression it makes.
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625. The background is the bustle of life. And our concept points to something within this bustle.
Page 108
626. And it is the very concept 'bustle' that brings about this indefiniteness. For a bustle comes about only through
constant repetition. And there is no definite starting point for 'constant repetition'.
Page 108
627. Variability itself is a characteristic of behaviour without which behaviour would be to us as something
completely different. (The facial features characteristic of grief, for instance, are not more meaningful than their
mobility.)†1
Page 108
628. It is unnatural to draw a conceptual boundary line where there is not some special justification for it, where
similarities would constantly draw us across the arbitrarily drawn line.
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629. How could human behaviour be described? Surely only by showing the actions of a variety of humans, as they
are all mixed up together. Not what one man is doing now, but the whole hurly-burly, is the background against
which we see an action, and it determines our judgment, our concepts, and our reactions. [Z 567.]
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630. How could you explain the meaning of 'simulating pain', 'acting as if in pain'? (Of course the question is: To
whom?) Should you act it out? And why could such an exhibition be so easily misunderstood? One is inclined to
say: "Just live among us for a while and then you'll come to understand."
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631. One might surely be taught (e.g.) to mime pain (not with the intention of deceiving). But could this be taught to
just anyone? I mean: someone might well learn to give certain crude tokens of pain, but without ever spontaneously
giving a finer imitation out of his own insight. (Talent for languages.) (A clever dog might perhaps be taught to give a
kind of whine of pain but it would never get as far as conscious imitation.) [Z 389.]
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632. I really want to say that scruples in thinking begin with (have their roots in) instinct. Or again: a language-game
does not have its
Page Break 109
origin in consideration. Consideration is part of a language-game.
And that is why a concept is in its element within the language-game. [Z 391.]
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633. "Couldn't you imagine a further surrounding in which this too could be interpreted as pretence?"
But what does it mean to say that it might always be pretence? Has experience taught us this? How else can
we be instructed about pretence? [Cf. Z 571.]
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634. Isn't there something here like the relation between Euclidean geometry and visual experience? (I mean that
there is a deep-seated resemblance here.) For Euclidean geometry too corresponds to experience, only in a very
peculiar way and not, say, 'merely approximately'. One might perhaps say that it corresponds as much to our
method of drawing as to other things; or that it corresponds to certain requirements of thinking. Its concepts have
their roots in widely scattered and remote areas. [To "one might perhaps say" cf. Z 572.]
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635. For just as the verb "believe" is conjugated like the verb "beat", concepts for one area are formed along the lines
of very widely scattered concepts. (The genders of nouns.)
Page 109
636. The formation of a concept has, for example, the character of limitlessness, where experience provides no sharp
boundary lines. (Approximation without a limit.)
Page 109
637. Sometimes it seems that concepts are formed along the lines of comfortable thinking. (Just as even the
yardstick is suitable not only to the things that are to be measured, but also to man.) But what the devil! Surely
everyone knows whether he's in pain!--How could everyone know it? To do this each would have to know that all
have the same.
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638. A tribe has two concepts, akin to our 'pain'. One is applied where there is visible damage and is linked with
tending, pity etc. The other is used for stomach-ache, for example, and is tied up with mockery of anyone who
complains. "But then do they really not notice the similarity?"--Do we have a single concept everywhere where there
is a similarity? The question is: Is the similarity important to them? And need it be so? [Z 380.]
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639. If you consider the reasons someone might have for stifling pain, or simulating it, you will come up with
countless ones. Now why is there this multiplicity? Life is very complicated. There are a great many possibilities.
But couldn't other men disregard many of these possibilities, shrug them off, as it were?
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640. But in that case isn't this man overlooking something that is there?--He takes no notice of it, and why should
he?--But in that case his concept just is fundamentally different from ours.--Fundamentally different?
Different.--But in that case it surely is as if his word could not designate the same as ours. Or only part of that.--But
of course it must look like that, if his concept is different. For the indefiniteness of our concept may be projected for
us into the object that the word designates. So that if the indefiniteness were missing we should also not have 'the
same thing meant'. The picture that we employ symbolizes the indefiniteness. [Z 381.]
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641. In philosophizing we may not terminate a disease of thought. It must run its natural course, and slow cure is all
important. [Z 382.]
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642. "You can never know what's going on in his soul."--That seems to be a truism. And it is, in the sense that the
picture we just used already contains the sentence. But of course we have to call the sentence into question just as
much as the picture.
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643. The expression "Who knows what is going on inside him!" The interpretation of outer events as consequences
of unknown ones, or merely surmised, inner ones. The interest that is focused on the inner, as if on the chemical
structure, from which behaviour issues.
Page 110
For one needs only to ask, "What do I care about inner events, whatever they are?!", to see that a different
attitude is conceivable.--"But surely everyone will always be interested in his inner life!" Nonsense. Would I know
that pain, etc., etc. is something inner if I weren't told so?
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644. Doubt about an inner process is an expression. Doubt, however, is an instinctive form of behaviour. A form of
behaviour toward someone else. And it does not follow from this that I know from my own experience what pain,
etc. is; or that I know that it is something inward, and can go along with something outward. That's the last thing I
know!
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645. Remember: most people say one feels nothing under anaesthetic.
But some say: It could be that one feels, and simply forgets it completely.
If then there are here some who doubt and some whom no doubt assails, still the lack of doubt might after all
be far more general. [Z 403.]
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646. Or doubt might after all have a different and much less indefinite form than in our world of thought. [Z 404.]
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647. Remember: we often use the phrase "I don't know" in a queer way; when, for example, we say that we don't
know whether this man really feels more than that other, or merely gives stronger expression to his feeling. In that
case it is not clear what sort of investigation would settle the question. Of course the expression is not quite idle: we
want to say that we certainly can compare the feelings of A and B with one another, but that the circumstances of a
comparison of A with C throw us out. [Z 553.]
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648. Only God sees the most secret thoughts. But why should these be all that important? And need all human
beings count them as important? [Z 560.]
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649. "Imagine humans who only think aloud." After all, it is not a matter of course that beings of bodily nature think;
so let them think only talking, that is, let them do nothing else that we would call thinking. (Their secret thoughts are
monologues.)
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650. The steps that lie between instinctive cunning and cunning that is carefully thought out. An idiot could behave
slyly, for that's what we'd call it, but we wouldn't think him capable of planning anything.
If we're asked "What's going on inside him?" we say, "Surely very little goes on inside him." But what do we
know about it?! We construct a picture of it according to his behaviour, his utterances, his ability to think.
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651. We combine diverse elements into a 'Gestalt' (pattern), for example, into one of deceit.
The picture of the inner completes the Gestalt.
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652. If a concept depends on a pattern of life, then there must be some indefiniteness in it. For if a pattern deviates
from the norm, what we want to say here would become quite dubious.
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653. Thus can there be definiteness only where life flows quite regularly? But what do they do when they come
across an irregular case? Maybe they just shrug their shoulders.
Page 112
654. "He told me--and there was not the slightest possibility of doubting his credibility--that..." Under what
circumstances is there no possibility of doubting his credibility? Can I specify them? No.
Page 112
655. You must think about the purpose of words.
What does language have to do with pain?
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656. In the case that I have in mind, the people have a word which has a similar purpose (with a similar function) to
that of the word "pain". It would be wrong to say that it "designates" something similar. It enters into their life in a
different, and yet similar, way.
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657. "But you can't recongize [[sic]] pain with certainty just from externals."--The only way of recognizing it is by
externals, and the uncertainty is constitutional. It is not a shortcoming.
It resides in our concept that this uncertainty exists, in our instrument. Whether this concept is practical or
impractical is really not the question.
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658. In a world different from ours colours might play a different role. Think of various cases
(1) Certain colours are tied to certain forms. Circular shapes, red, rectangular ones, green, etc.
(2) Dyes can't be produced. You can't colour things.
(3) One colour always linked together with a foul smell, or poisonousness.
(4) A far greater incidence of colour-blindness than now exists.
(5) Different shades of grey abound; all other colours are extremely rare.
(6) We can reproduce a great many shades of colour from memory. If our number system is connected with
the number of our fingers, then why shouldn't our system of colours be connected with the specific ways in which
they occur.
(7) A colour occurs only in gradual transition into another one.
(8) Colours always occur in the sequence of colours in the rainbow.
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Page 113
659. Think of the uncertainty about whether animals, particularly lower animals, such as flies, feel pain.
The uncertainty whether a fly feels pain is philosophical; but couldn't it also be instinctive? And how would
that come out?
Indeed, aren't we really uncertain in our behaviour towards animals? One doesn't know: Is he being cruel or
not.
Page 113
660. For there is uncertainty of behaviour which doesn't stem from uncertainty in thought.
Page 113
661. Look at the problem of uncertainty as to whether someone else is feeling pain in light of the question whether
an insect feels pain.
Page 113
662. There is such a thing as trust and mistrust in behaviour!
If anyone complains, e.g., I may be trustful and react with perfect confidence, or I may be uncertain, like
someone who has his suspicions. Neither words nor thoughts are needed for this. [Z 573.]
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663. The unpredictability of human behaviour. But for this--would one still say that one can never know what is
going on in anyone else? [Z 603.]
Page 113
664. But what would it be like if human behaviour were not unpredictable? How are we to imagine this? (That is to
say: how should we depict it in detail, what are the connections we should assume?) [Z 604.]
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665. "I don't know what's going on inside it right now!" That could be said of the complicated mechanism say of a
fine clock, which triggers various external movements according to very complicated laws. Looking at it one might
think: if I knew what it looked like inside, what was going on right now, I would know what to expect.
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666. But with a human being, the assumption is that it is impossible to gain an insight into the mechanism. Thus
indeterminacy is postulated.
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667. If, however, I doubt whether a spider feels pain, it is not because I don't know what to expect. [Z 564.]
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668. But we cannot get away from forming the picture of a mental process. And not because we are acquainted with
it in our own case! [Z 565.]
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669. One kind of uncertainty is that with which we might face an unfamiliar mechanism. In another we should
possibly be recalling an occasion in our life. It might be, e.g., that someone who has just escaped the fear of death
would shrink from swatting a fly, though he would otherwise do it without thinking twice about it. Or on the other
hand that, having this experience in his mind's eye, he does with hesitancy what otherwise he does unhesitatingly. [Z
561.]
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670. Even when I 'do not rest secure in my sympathy' I need not think of uncertainty about his later behaviour. [Z
562.]
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671. The one uncertainty stems from you, so to speak, the other from him.
The one could surely be said to connect up with an analogy, then, but not the other. Not, however, as if I
were drawing a conclusion from the analogy! [Z 563.]
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672. Seeing life as a weave, this pattern (pretence, say) is not always complete and is varied in a multiplicity of ways.
But we, in our conceptual world, keep on seeing the same, recurring with variations. That is how our concepts take
it. For concepts are not for use on a single occasion. [Z 568.]
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673. And the pattern in the weave is interwoven with many others. [Z 569.]
Page 114
674. I say for instance: "He might after all be pretending."--What am I imagining when I say it?--That is, what
explanation of the word "pretend" would I give? What kind of examples would come to mind?
Page 114
675. How do I employ the sentence?
(For the situation here is like certain areas of mathematics, where there is a 'fantastic application'.)
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676. I evoke a picture which can then serve a purpose. (I could almost be looking at a painted picture.)
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677. Sometimes I treat him as I treat myself and as I would like to be treated when I'm in pain; sometimes I don't.
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678. We're used to a particular classification of things.
With language, or languages, it has become second nature to us.
Page 115
679. These are the fixed rails along which all our thinking runs, and so our judgement and action goes according to
them too. [Z 375.]
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680. Must people be acquainted with the concept of modesty or of swaggering, wherever there are modest and
swaggering men? Perhaps nothing hangs on this difference for them.
For us, too, many differences are unimportant, which we might find important. [Z 378.]
Page 115
681. And others have concepts that cut across ours. And why should their concept 'pain' not split ours up? [First
sentence: Z 379. Second sentence: Z 380, the last sentence.]
Page 115
682. The 'uncertainty' relates not to the particular case, but to the method, to the rules of evidence. [Z 555.]
Page 115
683. Concepts with fixed limits would demand a uniformity of behaviour. But what happens is that where I am
certain, someone else is uncertain. And that is a fact of nature. [Z 374.]
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684. If it is said, "Evidence can only make it probable that expressions of emotions are genuine", this does not mean
that instead of complete certainty we have just a more or less confident conjecture. "Only probable" cannot refer to
the degree of our confidence, but only to the nature of its justification, to the character of the language-game. Surely
this must help determine the constitution of our concepts: that there is no agreement among men as to the certainty
of their convictions. (Compare the remark about agreement in colour-judgments and agreement in mathematics.)
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685. Given the same evidence, one person can be completely convinced and another not be. We don't on account of
this exclude either one from society, as being unaccountable and incapable of judgment.
Page 115
686. But mightn't a society do precisely this?
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Page 116
687. For words have meaning only in the stream of life. [Cf. Z 173, the last sentence.]
Page 116
688. I am sure, sure, that he isn't pretending; but someone else isn't. Can I convince him? And if not--do I say that
he can't think? (The conviction could be called "intuitive".) [Cf. PI II, xi, p. 227f.]
Page 116
689. Instinct comes first, reasoning second. Not until there is a language-game are there reasons.
Page 116
690. Am I saying something like, "and the soul itself is merely something about the body"? No. (I am not that hard
up for categories.)
Page 116
691. You can vary the concept, but then you might change it beyond recognition.
Page 116
692. Even if we vary the concept of pretence, its inwardness must still be kept, i.e. the possibility of a confession.
But we don't always have to believe a confession, and a false confession is not necessarily deception.
Page 116
693. Concepts other than though akin to ours might seem very queer to us; namely, a deviation from the usual in an
unusual direction. [Z 373.]
Page 116
694. "You're all at sea!" we say, when someone doubts the genuineness of something we recognize as clearly
genuine.
Page 116
"You're all at sea"--but we cannot prove anything. [Cf. PI II, xi, p. 227g.]
Page 116
695. Soulful expression in music--this cannot be recognized by rules. Why can't we imagine that it might be, by
other beings? [Z 157.]
Page 116
696. It would make a strange and strong impression on us were we to discover people who knew only the music of
music boxes. We would perhaps expect gestures of an incomprehensible kind, to which we wouldn't know how to
react.
Page 116
697. "The genuineness of an expression cannot be proved." "One has to feel it." But what does one go on to do with
this now? If someone says "Voilà, comment s'exprime un coeur vraiment épris", and if he
Page Break 117
also converts someone to his viewpoint,--what are the further consequences?
Page 117
In a vague way consequences can be imagined. The other one's attention gets a new direction. [Cf. PI II, xi,
p. 228a.]
Page 117
698. Can we imagine that something unprovable to us could be proved to other beings?
Or would that change its nature to the point of its being unrecognizable?
Page 117
699. What is essential for us is, after all, spontaneous agreement, spontaneous sympathy.
Page 117
700. 'These men would have nothing human about them.' Why?--We could not possibly make ourselves understood
to them. Not even as we can to a dog. We could not find our feet with them.
And yet there surely could be such beings, who in other respects were human. [Z 390.]
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701. "But no one can know it. One can believe it. Believe it with all his heart, but not know it." Then the difference is
not to be found in the certainty of the one who is convinced.
It must be found elsewhere; in the logic of the question.
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702. Imagine that people could observe the functioning of the nervous system in others. In that case they would
have a sure way of distinguishing genuine and simulated feeling.--Or might they after all doubt in turn whether
someone feels anything when these signs are present?--What they see there could at any rate readily be imagined to
determine their reaction without their having any qualms about it.
And now this can be transferred to outward behaviour. [Z 557a, b.]
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703. There is indeed the case where someone later reveals his inmost heart to me by a confession: but that this is so
cannot offer me any explanation of outer and inner, for I have to give credence to the confession.
For confession is of course something exterior. [Z 558.]
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704. Men who could see the functioning of the nerves: Do I have to suppose that even they could be outwitted by
the 'inner'? But that means: Can't I imagine outward signs which would seem to be sufficient for making a sure
judgment about 'the inner'?
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705. But now say: "Someone could still be feeling something even if the physiological signs ran completely to the
contrary." Well then anyone who is unfamiliar with these scruples simply has a different concept.
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706. Imagine that the people of a tribe were brought up from early youth to give no expression of feeling of any
kind. They find it childish, something to be got rid of. Let the training be severe. 'Pain' is not spoken of; especially
not in the form of a conjecture "Perhaps he has got...". If anyone complains, he is ridiculed or punished. There is no
such thing as the suspicion of shamming. Drilling someone to speak expressionlessly, in a monotone, to move
mechanically. [Except for the last sentence: Z 383.]
Page 118
707. I want to say: an education quite different from ours might also be the foundation for quite different concepts.
[Z 387.]
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708. For here life would run on differently.--What interests us would not interest them. Here different concepts
would no longer be unimaginable. In fact, this is the only way in which essentially different concepts are imaginable.
[Z 388.]
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709. That the evidence makes someone else's feelings merely probable is not what matters to us; what we are looking
at is the fact that this is taken as evidence for something; that we construct a statement on this involved sort of
evidence, and hence that such evidence has a special importance in our lives and is made prominent by a concept.
[Cf. Z 554.]
Page 118
710. "Shamming," these people might say, "What a ridiculous concept!" [Z 384, first sentence.]
Page 118
711. Steadfast faith (in an annunciation, for instance)--is it less certain than being convinced of a mathematical
truth?--(But this makes the language-games more alike!) [Culture and Value, p. 73.]
Page 118
712. Might not the attitude, the behaviour, of trusting, be quite universal among a group of humans? So that a doubt
about manifestations of feeling is quite foreign to them? [Z 566.]
Page 118
713. But consider: Why should a person have to be dissimulating, aren't there other possibilities? Can't he be
dreaming? Can't the matter get confused along different lines? (Couvade.)
Page Break 119
Page 119
Just think of how often we can't say: someone is honest or dishonest, sincere or insincere. (A politician, for
example.) Well-meaning or the opposite. How many foolish questions get asked here! How often the concepts don't
fit!
Page 119
714. It is important for our considerations that there are people of whom someone feels that he will never know
what's going on inside them. That he'll never understand them. [Culture and Value, p. 74.]
Page 119
715. We are certainly inclined to say that a complaint is merely a sign, a symptom of a different phenomenon, the
important one, which is only empirically related to it. And even if we are mistaken here, still there must be a reason
for this strong temptation, a reason within the law of the evidence we admit.†1
Page 119
716. One might raise the question: What does the law of admissible evidence have to be like for this conception to
suggest itself to us?
Page 119
717. One would like to answer: the evidence would have to be fluctuating. Multiform?
Page 119
718. There is such a thing as feigned expression; but there must also be such a thing as evidence for the pretence.
Even though we often simply don't know what to say, all the same we do sometimes have to lean towards
one opinion, and sometimes be quite certain.
So the outward has to be evident.†2
Page 119
719. You say you attend to a man who groans because experience has taught you that you yourself groan when you
feel such-and-such. But as you don't in fact make any such inference, we can abandon the justification by analogy.
[Z 537.]
Page 119
720. In philosophy it is significant that such-and-such a sentence makes no sense; but also that it sounds funny. [Z
328.]
Page 119
721. Can 'knowing one's way about' be called an experience? Surely not. But there are experiences characteristic of
the condition of knowing one's way about and not knowing one's way about. (Not knowing one's way about and
lying.) [Z 516.]
Page Break 120
Page 120
722. Is "I hope..." a description of a state of mind? A state of mind has duration. So if I say "I have been hoping for
the whole day...", that is such a description. But suppose I say to someone: "I hope you come"--what if he asks me
"For how long have you been hoping that?" Is the answer "For as long as I've been saying so"? Supposing I had
some answer or other to that question, would it not be quite irrelevant to the purpose of the words "I hope you'll
come"? [Z 78.]
Page 120
723. A cry is not the description of a state of mind, even though a state of mind can be inferred from it. [Cf. PI II, ix,
p. 189b, c.]
Page 120
724. One doesn't shout "Help" because he observes his own state of fear.
Page 120
725. 'Describing' includes 'attending'.
Page 120
726. These sentences are descriptions: "I'm less afraid of him now than before", "For a long time I've been
wishing...", "I keep on hoping...". (One is describing a way something has been running on.)
Page 120
727. Do I want to say, then, that certain facts are favourable to the formation of certain concepts; or again
unfavourable? And does experience teach us this? It is a fact of experience that human beings alter their concepts,
exchange them for others when they learn new facts; when in this way what was formerly important to them
becomes unimportant, and vice versa. (It is discovered, e.g., that what formerly counted as a difference in kind, is
really only a difference in degree.)
((Re: discussion of the concept of colour and other things.)) [a: Z 352.]
Page 120
728. If a cry is not a description, then neither is the verbal expression that replaces it. The utterances of fear, hope,
wish, are not descriptions; but the sentences "I'm less afraid of him now than before", "For a long time I've been
wishing ...", ... are descriptions.
Page 120
729. What is the past tense of "You are coming, aren't you!"?†1 [Z 80.]
Page Break 121
Page 121
730. The tangled use of psychological words ("think", for example). As if the word "violin" referred not only to the
instrument, but sometimes to the violinist, the violin part, the sound, or even the playing of the violin.
Page 121
731. "If p occurs, then q occurs" might be called a conditional prediction. That is, I make no prediction for case
not-p. But for that reason what I say also remains unverified by "not-p and not-q".
Or even: there are conditional predictions and "p implies q" is not one. [Z 681.]
Page 121
732. I will call the sentence "If p occurs then q occurs" "S".--"S or not S" is a tautology; but is it (also) the law of
excluded middle?--Or again: If I want to say that the prediction "S" may be right, wrong, or undecided, is that
expressed by the sentence "not (S or not-S)"? [Z 682.]
Page 121
733. The use of the words "look at", "observe". And then the use of the expression "to look at oneself"!
Page 121
734. "I'm afraid of him" and "I tend to fear him". But the expression "I tend to..." could mean several things here.
There could be a language in whose conjugations many more differences are taken care of than in the languages we
know.
Page 121
735. Difference in purpose between the utterance of fear "I'm afraid!", and the report of fear "I'm afraid".
Page 121
736. "To know" can mean something like "to be able to" (to know by heart, e.g.), or it can mean something like "to
be sure".
Page 121
737. No one but a philosopher would say "I know that I have two hands"; but one may well say: "I am unable to
doubt that I have two hands."
"Know", however, is not ordinarily used in this sense. [a: Z 405; b: Z 406, first sentence.]
Page Break 122
Page Break 123
APPENDIX
Correspondences between Remarks in TS 232 and in Zettel, Philosophical Investigations and Vermischte
Bemerkungen (Culture and Value).
Page 123
TS 232 und Zettel
Correspondences
Page 123
TS 232 Z
4 79
15 469a
19 468
20 113
26 618
45 (Ende) 85
46 87
48 84
50 81
51 82
52 83
57 72
58 473
59 474
60 475
61 476
63 472, 483, 621
64 622a
65 623
66 624 Anfang
70 625a
71 625b
72 626
77a 629
80a Vgl. 627
87a, b, c 630, 631
88b 632
89 633 Anfang
98 634
100 635
Page Break 124
Page 124
TS 232 Z
105 Vgl. 677
110 636
111 637
112 638
113 (der erste Satz) 640
119b 643
123 641
124 642
137 644
138 645
139b 646
140 (der erste Satz) 647
143 535
148 488, 489, 490, 491, 492
150a Vgl. 532, 533
151 534
152 Vgl. 504
158 500
159 512
161a 501
166a, b523
167 Vgl. 525
172 Vgl. 496
174 Vgl. 502
178 (die ersten drei Sätze) 45
183 100
184 101
186 Vgl. 102
187 103
192b, cVgl. 129
194a, bVgl. 113; 112
200 114
201 117
202 118
203 Vgl. 98
205 Vgl. 99
206 (der letzte Satz) 115
207 116
212 Vgl. 89b
214 109
215 Vgl. 122
220 110
224 (Ende) 104
Page Break 125
Page 125
TS 232 Z
225 105
226 106
227 107
233 95
236 90
237 91
248 93
249 Vgl. 94
253b Vgl. 88
257b 96
258 50
259 Vgl. 6
261 Vgl. 124
263a 92a
264 Vgl. 188
267b, c586
269 Vgl. 587
300 Vgl. 406
303 408a, c
307c 511
308 518
309 519
310 520
311 465
312a, b, c 421
313 432
315 422
316 423
317 424
318 425, 417
320 513
324 526
325 495
326 418
327 419
330 (Ende) 420
331 119
332 120
333 524 (die ersten zwei Sätze)
336 411
337 412
338 413
339 414
Page Break 126
Page 126
TS 232 Z
340 415
341 416
343 409, 410
346 660
348 366
349 367
351 657
352 658
353 659
355 Vgl. 208
392 350
393 351
395 330
403 Vgl. 300
404 301
405 302
406 303
407 304
408 305
409 306
410 307
411 308
413 318
414 319
423 354
424 355
425 356
426 357
427 358
428 (die ersten zwei Sätze) 359
429 362
431 363
432 364
433a 365
437b 353
466 162
467 163
468 164
469 165
470 166
471 167
497 159
498 515, 484
Page Break 127
Page 127
TS 232 Z
499 485
500 168
501 169, 170
502 171
503 172
504 173
505 174
512 673
520 674
521 216
522 217
531 179
548b, c185
557 182
570 Vgl. 225
571a Vgl. 183a
572 184
573 180
574a, b181
575 1, 2b
578 Vgl. 183a
579 Vgl. 183b
582 44
583 661
592 662
593 663
594 664
595 665
596 666
597 667
598 192
605 393
609 570a, c
614 439; vgl. 527
617 543
621 556
622 Vgl. 392
629 567
631 389
632 391
633 Vgl. 571
634 (Anfang) Vgl. 572
638 380
Page Break 128
Page 128
TS 232 Z
640 381
641 382
645 403
646 404
647 553
648 560
662 573
663 603
664 604
667 564
668 565
669 561
670 562
671 563
672 568
673 569
679 375
680 378
681 379, 380 (der letzte Satz)
682 555
683 374
687 Vgl. 173 (der letzte Satz)
693 373
695 157
700 390
702 557a, b
703 558
706 383
707 387
708 388
709 Vgl. 554
710 384 (der erste Satz)
712 566
719 537
720 328
721 516
722 78
727a 352
729 80
731 681
732 682
737 405, 406 (der erste Satz)
Page Break 129
TS 232 und Philosophische Untersuchungen
Page 129
Correspondences
Page 129
TS 232 PU
36 II, xi, S. 200e
358 II, xi, S. 209b
416 Vgl. II, x, S. 192e
419 II, x, S. 191f
434 Vgl. II, xii, S. 230c
445 Vgl. II, xi, S. 210-211
446 II, xi, S. 211a
449 Vgl. II, xi, S. 207b
451 Vgl. II, xi, S. 207e
452 II, xi, S. 200a
453 Vgl. II, xi, S. 200b
463 II, xi, S. 198d
472 II, xi, S. 212f
483b Vgl. II, xi, S. 208e
484 II, xi, S. 208e
485 II, xi, S. 208f
486 II, xi, S. 209a
488 II, xi, S. 213b
489 Vgl. II, xi, S. 214f
491 Vgl. II, xi, S. 213a
492 Vgl. II, xi, S. 208b
494 II, xi, S. 213c
495 II, xi, S. 213d
519 II, xi, S. 210e
526 II, xi, S. 195d
535 Vgl. II, xi, S. 206d, e
538 II, xi, S. 208c
543 Vgl. II, xi, S. 213c
546 II, xi, S. 212d
556a Vgl. II, xi, S. 193a
566 Vgl. II, xi, S. 224c
567a II, xi, S. 224d
569 II, xi, S. 223g
612 Vgl. II, xi, S. 228f-229
613a II, xi, S. 229c
688 Vgl. II, xi, S. 227f
694 Vgl. II, xi, S. 227g
697 Vgl. II, xi, S. 228a
723 Vgl. II, ix, S. 189b, c
Page Break 130
TS 232 und Vermischte Bemerkungen
Page 130
Correspondences
Page 130
TS 232 VB
155b S. 125 ("Wir sagen in einer wissenschaftlichen Untersuchung...")
606 S. 139 ("Worin liegt die Wichtigkeit...")
711 S. 142 ("Der feste Glaube...")
714 S. 142 ("Es ist für unsre Betrachtung wichtig...")
FOOTNOTES
Page 3
†1 Cf. Notebooks 12.9.1916. (Eds.)
Page 9
†1 The typescript contains no figure. We have taken it from the corresponding manuscript. (Eds.)
Page 9
†2 Ibid. (Eds.)
Page 15
†1 Wittgenstein crossed out the word "understanding" in the MS. and replaced it with "imaging". In Zettel
too the pair of words is "seeing"--"imaging". (Eds.)
Page 16
†1 This remark is also numbered 83. We have corrected the numbering. (Eds.)
Page 18
†1 Cf. Philosophical Investigations I, §§350-351. (Eds.)
Page 19
†1 Var. "But even if the curtain over there obeyed my will, so that it moved or disappeared".
Page 20
†1 Var. "In the above case the assertion that that conditional sentence is true, or false, should not amount to
the assertion that p will occur."
Page 21
†1 Var. "the picture before the inner eye".
Page 23
†1 Var. "For me, what is on this screen now represents this."
Page 26
†1 In the typescript, "chase". In the MS, "check-mate". This seems to be more natural here. (Eds.)
Page 26
†2 In the typescript, "chasing", in the MS "check-mating". (Eds.)
Page 27
†1 Var. "as when following nature."
Page 29
†1 Var. "the mouth". Cf. PI I, 583.
Page 34
†1 Var. "not as a natural sound, but rather to communicate something, as a report".
Page 36
†1 Var. "And how is one to decide how exact the analogy must be for us to have the right to use the concept
'thinking' with these people, a concept which has its home in our life?"
Page 38
†1 Var. "could not be deduced from".
Page 39
†1 Philosophical Investigations I, §2.
Page 40
†1 Var. "He just learns the use of the word under particular circumstances."
Page 41
†1 In the typescript, as well as in the manuscript, this is given erroneously as "Barnard". Cf. PI I, 342. (Eds.)
Page 42
†1 We have taken the drawing from the MS. (Eds.)
Page 45
†1 Var. "make sense".
Page 46
†1 Manuscript: "animated conversation". (Eds.)
Page 47
†1 Var. "guesses".
Page 47
†2 Var. "which often illustrate it, as it were, are not the intention itself."
Page 48
†1 At this point the MS has: "Schubert is my name, Schubert I am." (Eds.)
Page 66
†1 At this point there is no letter in the typescript. We have taken the figure from the MS. (Eds.)
Page 67
†1 In the MS, "Imagine". (Eds.)
Page 68
†1 The manuscript has "indications". (Eds.)
Page 71
†1 There is no figure in the typescript. We have copied this figure from the manuscript. (Eds.)
Page 73
†1 In the MS there is a drawing at this point:
(Eds.)
Page 73
†2 In the MS there is a drawing at this point:
(Eds.)
Page 74
†1 There is no drawing in the typescript. We have copied the drawing from the manuscript. (Eds.)
Page 78
†1 Perhaps as an error, the typescript has: "It is obvious at a picture". (Eds.)
Page 83
†1 There is no letter at this point in the typescript. We have taken it from the manuscript. (Eds.)
Page 85
†1 Var. "But still this expression tells you 'what happened' only if you are at home in the special conceptual
world that belongs to these situations (and this holds for the speaker too)."
Page 88
†1 There is no picture in the typescript. We have taken it from the manuscript. (Eds.)
Page 92
†1 We have copied the letter following the manuscript. (Eds.)
Page 92
†2 In the MS: "See as "? (Eds.)
Page 95
†1 Here there is no symbol in the typescript. We have taken it from the manuscript. (Eds.)
Page 108
†1 Var. "are not more important for our reaction than...".
Page 119
†1 Var. "still the reason for this mistake must be within the law of evidence that we admit."/"still the mistake
must have a reason, and that in the nature of the evidence which we admit."
Page 119
†2 In the MS: "evidence". (Eds.)
Page 120
†1 Var. "You will come, won't you!"
LAST WRITINGS ON THE PHILOSOPHY OF PSYCHOLOGY: Volume I
Page Break ii
Titlepage
Page ii
LAST WRITINGS ON THE PHILOSOPHY OF PSYCHOLOGY
Ludwig Wittgenstein
VOLUME I
Page ii
Preliminary Studies for Part II
of Philosophical Investigations
Edited by
G. H. VON WRIGHT
and
HEIKKI NYMAN
Translated by
C. G. LUCKHARDT
and
MAXIMILIAN A. E. AUE
Basil Blackwell
Page Break iii
Copyright page
Copyright © Blackwell Publishers Ltd 1982, 1990
First published 1982
First published in paperback 1990
Reprinted 1998
Blackwell Publishers Ltd
108 Cowley Road
Oxford OX4 1JF
UK
All rights reserved. Except for the quotation of short passages for the purposes of criticism and review, no part of
this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher.
Except in the United States of America, this book is sold subject to the condition that it shall not, by way of trade or
otherwise, be lent, resold, hired out, or otherwise circulated without the publisher's prior consent in any form of
binding or cover other than that in which it is published and without a similar condition including this condition
being imposed on the subsequent purchaser.
British Library Cataloguing in Publication Data
Wittgenstein, Ludwig
Last writings on the Philosophy of Psychology
Vol. I
I. Philosophy
I. Title. II. Wright, G. H. von III. Nyman, Heikki
192 B3376.W564
ISBN 0-631-12895-6
ISBN 0-631-17121-5 (pbk)
Printed and bound in Great Britain by Athenaeum Press Ltd, Gateshead, Tyne & Wear
Page Break iv
EDITORS' PREFACE
Page iv
Unlike the two volumes of Wittgenstein's Remarks on the Philosophy of Psychology (RPP), these two volumes of
last writings on the philosophy of psychology are not based on typescripts. (Cf. the Preface to volume I of RPP.)
Their subject matter, however, is by and large the same.
Page iv
The first of these volumes consists of writings dating from the period between 22 October 1948 and 22
March 1949, except for the last remark, which is dated 20 May. This material is a continuation of the manuscript
writings on which the typescript for volume II of RPP is based. So far as is known, Wittgenstein did not prepare a
typescript for this material. In the spring of 1949, however, he probably made a handwritten clean copy of a
selection of all of his remarks written between 1946 and 1949 concerning topics in the philosophy of psychology
(MS 144), and then prepared a typescript on the basis of this new mansucript. This was the typescript for Part II of
the Philosophical Investigations. (Unfortunately, it was lost after the book was printed.) More than half of the
remarks in this typescript--and therefore also in the second part of the Philosophical Investigations--are taken from
the manuscripts written between October 1948 and March 1949. These manuscripts, the second half of MS 137 and
MS 138, appear here in their totality, with the exception of a sizeable number of remarks of a general nature which
have almost all been printed in Culture and Value. For the most part, those remarks of a general nature had been
clearly separated by parallel lines || from the remainder of the text by Wittgenstein himself.
Page iv
Since the text published in this first volume of late writings is based directly on the manuscripts and not on a
typescript prepared by the author on the basis of manuscripts, it is of a more provisional and improvised nature than
the Remarks on the Philosophy of Psychology. (For this reason we did not want to publish it as the third volume of
those Remarks.) There are frequent repetitions; sometimes a whole remark will reappear, virtually word for word. If
Wittgenstein had ever dictated a typescript based on these manuscripts, he would certainly have avoided such
repetitions and would also have made many other changes. The number of variants too is much larger than in the
typescripts upon which RPP is based. In general, the editors have ventured to make a choice between the variants;
when we were not sure we quoted the variant(s) in a footnote.
Page iv
A reminder to the reader: words in angle brackets < > in the text are editorial additions. The numbering of the
remarks and most footnotes as well as the cross-references to Wittgenstein's published works have also been added
by the editors. These cross-references
Page Break v
are given in square brackets. Occasional footnotes added by the translator are identified by (Tr).
Page v
We would like to thank Dr Joachim Schulte as well as the two translators, Professor C. G. Luckhardt and
Professor Maximilian A. E. Aue, for their valuable advice and suggestions which greatly aided us in preparing a
correct text.
Helsinki Georg Henrik von Wright
Heikki Nyman
Page Break 1
Preliminary Studies for Part II of Philosophical Investigations
MSS 137-138
(1948-1949)
Page 1
Page Break 2
Page 2
1. A language in which there is a word "to frightle oneself"; meaning: to torture oneself with fearful thoughts. And
then you might suppose, for example, that this verb had no first person present. The English "I am ... ing".
Page 2
2. If I tell someone "I hope you'll come", is it less urgent if my hoping lasted only for 30 seconds than if it had lasted
2 minutes?
"I'm happy that you brought it off!"--"How long have you been happy?" A peculiar question. But it might
make sense. The answer might be: "Whenever I think about it", or "At first I wasn't happy about it, but then I was"
or "I keep on thinking about it and being happy" or "It only occurs to me every once in a while, but when it does, I'm
happy", etc. One also says "It's a source of continual happiness to me", and "For a second I was happy about his bad
luck".
Page 2
3. "I'm moving my bishop."--"How long are you moving it?"
Page 2
4. Think about this as an example of the propositional form "If p, then q": "If he comes, I'll tell him." Now if he
doesn't come--have I kept my promise?--have I broken it?--
But can one say that that proposition asserts a 'connection'? Would I answer "This doesn't have to be the
case"? It's not as if the sentence had been: "If these two get together there will be a brawl." Here one could answer
that way.
Page 2
5. But suppose it was the material implication that was being asserted (and the case does exist!)--can I also respond
to "p ⊃ q" by saying "It doesn't have to be the case"? And what does that mean here?
Page 2
6. "If the two terminals approach each other a spark jumps across."--What is taken as a verification of this
proposition? The observation that they never approach each other?--Can what we want to say here be expressed by
material implication? Certainly not; but maybe by 'formal' implication? That's just as wrong.--What we want to state
is a kind of natural law; it is easy enough to imagine the kind of observation that leads to this. One has observed that
a spark jumps across every time they approach each other.--Is the proposition perhaps of the type "(x).φx ⊃
ψx:(∃x).φx"? If not, then this proposition must itself have an application, even if it isn't the same one.
Page Break 3
Page 3
7. "If he comes I'll tell him..." is a resolution, a promise. If it is not to be a false promise it must not rest on the
certainty that he won't come. It is neither a material nor a formal implication.
Page 3
8. In a conditional prediction in science one could distinguish between justification and correctness. One could call
it "justified" if it follows, results from, a theory that has such-and-such grounds. So if the first part of the sentence
isn't true, one can then say: Had it been true, then... would have.... But the fact that the first part did not turn out to
be true doesn't give me the right to say that.
Page 3
9. Does a proposition such as "All bodies move..." (Law of Inertia) have to be expressed in the form "if-then"? "If
something is a body then it moves----."--Or does it have to be expressed: "There are bodies; and if something is a
body, then..."? (Nobody would think of putting it that way.)
Page 3
10. Obviously we could have one concept of fear to be applied only to animals, and the word for that concept
wouldn't have a first person.
The third person would be used quite like the third person of "to fear".
Page 3
11. Remember that the subjunctive makes no sense except in a conditional sentence. If someone says "I would have
won this game", he will be asked: "If -- ?"
Page 3
12. [Ref. 'to fear', etc.] Nothing is more difficult than to look at concepts without prejudice. For prejudice is a kind of
understanding. And to forego it, when it is so full of consequences for us, --.
Page 3
13. The English "I'm furious" is not an expression of self-observation. Similarly in German "Ich bin wütend"; but not
"Ich bin zornig". ("Terribly doth the rage within my bosom turn...". It is a trembling of rage.)
Page 3
14. We ask "What does 'I am frightened' really mean? What am I thinking when I say it?" And, of course, we find no
answer, or only one that is obviously inadequate.
The question is: "In what sort of context does it occur?" [Cf. Philosophical Investigations II, ix, p. 188b]
Page Break 4
Page 4
15. One could also say with a certain amount of justification: "I simply say it." For this just means: Don't worry
about something accompanying speech.
Page 4
16. Now can't the utterance appear in various connections? which first give it one face, then another?
Page 4
17. I say "I am afraid...", someone else asks me "What did you want to say when you said that? Was it like an
exclamation; or were you alluding to your state within the past few hours; did you simply want to tell me
something?" Can I always give him a clear answer? Can I never give him one?--Sometimes I shall have to say: "I
was thinking about how I spent the day today and I shook my head, vexed with myself, as it were"--but other times:
"It meant: Oh, God! If I just weren't so afraid!"--Or: "It was just a cry of fear"--or: "I wanted you to know how I
feel." Sometimes the utterance is really followed by such explanations. But they can't always be given. [Up to
"Sometimes I shall have to say", cf. PI II, ix, p. 187g]
Page 4
18. It would be possible to imagine people who as it were thought much more definitely than we, and used a number
of different words, now one, now another. [Cf. PI II, ix, p. 188a]
Page 4
19. Nothing is more important for teaching us to understand the concepts we have than constructing fictitious ones.
[Culture and Value (C & V), p. 74]
Page 4
20. "What is fear?"--"Well, the manifestations and occasions of fear are as follows: - - -"--"What does 'to be afraid'
mean?"--"The expression 'to be afraid' is used in this way: - - -'.
"Is 'I am afraid - - -' therefore a description of my state?" It can be used in such a connection and with such
an intention. But if, for example, I simply want to tell someone about my apprehension, then it is not that kind of
description.
Page 4
21. "I'm afraid" can, for instance, be said just as an explanation of the way I'm behaving. In that case it's far from
being a groan; it can even be said with a smile.
Page Break 5
Page 5
22. We ask, "What does 'I am frightened' really mean, what am I referring to when I say it?" And, of course, we find
no answer, or one that is inadequate.
The question is: "In what sort of context does it occur?" [PI II, ix, p. 188b; cf. remark 14]
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23. One can find no answer if one tries to settle the question "What am I referring to", "What am I thinking when I
say it", etc. by pronouncing the words and at the same time attending to myself, as it were observing my soul out of
the corner of my eye. In an actual case I can indeed ask: "Why did I say that, what did I mean by it?" and I might
answer the question too, but not on the ground of observing what accompanied the speaking. And my answer
would supplement, paraphrase, the earlier utterance. [PI II, ix, p. 188c]
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24. What is fear? What does "being afraid" mean? If I wanted to define it at a single showing--I should play-act fear.
[PI II, ix, p. 188d]
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25. Could I also represent hope in this way? Hardly. And what about belief? [PI II, ix, p. 188e]
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26. "I believe he will come."
"I tell myself time and time again: 'He will come'." For the latter, people might have a separate verb.
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27. Describing my state of mind (of fear, say) is something I do in a very particular context. (Just as it takes a
particular context to make a certain action into an experiment.)
Is it, then, so surprising that I use the same expression in different games? And sometimes as it were between
the games?
"I thought of him" and "I thought about him" surely mean very different things. [a, b: PI II, ix, p. 188f]
Page 5
28. And do I always talk with very definite purpose?--And is what I say meaningless because I don't? [PI II, ix, p.
188g]
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29. <In English> "Now you mention it: I think he'll come."
"Now I think you're right: he will come."
"No. I'm convinced: he will come." One can think up a
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characteristic context for all such expressions.
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30. What is necessary in order for one to be describing a mental state?--Or might I ask: What is necessary in order
for one to be trying to describe a mental state?
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31. One might also ask: "What must then be important to me?"
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32. "I wanted to describe my mental state to you"--as opposed, perhaps, to: "I merely wanted to vent my feelings". I
wanted him to know 'how I am feeling'. (In this context one often speaks of the duration of the state.)
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33. For surely it is one thing quietly to confess one's fear--and quite another to give expression to it unabashedly.
The words can be the same, but the tone and the gestures different.
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34. When it is said in a funeral oration "We mourn our...", this is surely supposed to be an expression of mourning;
not to tell anything to those who are present. But in a different setting these words are an announcement. In a prayer
at the grave they could also in a way be used to tell someone something. [PI II, ix, p. 189a]
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35. We surely do not always say someone is complaining because he says he is in pain. So the words "I'm in pain"
may be a cry of complaint, and may be something else. (Something similar holds for expressions of fear and other
emotions.) [PI II, ix, p. 189d]
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36. But if "I'm afraid" is not always like a cry of complaint, and yet sometimes is, then why should it always be a
description of my state of mind?†1 [PI II, ix, p. 189e]
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37. For how does the complaint "I'm in pain" differ from the mere announcement? By its intent, of course. And
possibly that will also come out in the tone.
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38. The contexts of a sentence are best portrayed in a play. Therefore the best example for a sentence with a
particular meaning
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is a quotation from a play. And whoever asks a person in a play what he's experiencing when he's speaking?
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39. "I must tell you--I'm frightened."
"I must tell you--it makes me shiver."
And one can say this in a smiling tone of voice too.
And do you mean to tell me he doesn't feel it? How else does he know it?--But even if it is a piece of
information, he doesn't read this off from within. For he couldn't cite his sensations as proof of his statement. They
don't teach him this. [PI II, i, p. 174e]
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40. For think of the sensations produced by gestures of shuddering: the words "it makes me shiver" are themselves
such a gesture, and if I hear and feel them being expressed, that belongs among the rest of these sensations. Now
why should the wordless shudder be the ground of the verbal one? [PI II, i, p. 174f]
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41. We learn to use the word "think" under particular circumstances.
If the circumstances are different we don't know how to use it. But this does not mean that we have to be
able to describe those circumstances. [a: cf. Zettel (Z) 114; b: cf. Z 115]
Page 7
42. "If people differed strongly in their statements about colour they couldn't use our concept of colour."--If people
differed strongly in their statements about colour then just because of this they wouldn't be using our concept of
colour.
They wouldn't be playing our language-game: For just think how theirs and ours would have to be
compared!
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43. So, if I hear someone say "I am afraid", how can I find out whether this is the 'description of a state of mind' (or
some such thing)? Should I ask him, and is it certain that he will understand the question?--But he could surely
answer it. How? This way, for example: "No, I was just letting off steam", or "Yes, I want you to know how I feel".
But such a question will almost never be asked. Isn't this because the tone and the context necessarily give us
the answer? For from these we will deduce whether he is making fun of his own fear, perhaps, or whether he is
discovering it in himself, so to speak, or whether he is confessing it to us reluctantly, but for the sake of candour, or
whether he is uttering it like a scream, etc.--And don't
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the words, no matter how they are uttered, give me information about the same state of affairs, namely, his state of
mind?
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44. Does the sentence "Napoleon was crowned in the year 1804" really have a different meaning depending on
whether I say it to somebody as a piece of information, or in a history test to show what I know, or etc., etc.? In
order to understand it, the meanings of its words must be explained to me in the same way for each of these
purposes. And if the meaning of the words and the way they're put together constitute the meaning of the sentence,
then - - -.
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45. But here is the problem: A cry, which cannot be called a description, which is more primitive than any
description, for all that serves as a description of the inner state. [PI II, ix, p. 189b]
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46. That someone can scream doesn't mean that he can tell somebody something in a conversation.
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47. I hear the words "I am afraid". I ask: "In what connection did you say that? Was it a sigh from the bottom of
your heart, was it a confession, was it self-observation,...?"
Page 8
48. Does someone crying out "Help!" want to describe how he is feeling?
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Nothing is further from his intentions than describing something.
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49. But there are transitions from what we would not call a description to what we would.
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50. The phrase "description of a state of mind" characterizes a certain game. And if I just hear the words "I am
afraid" I might be able to guess which game is being played here (say on the basis of the tone), but I won't really
know it until I am aware of the context.
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51. For one or another of a class of features goes with what we call "describing". Observing, considering,
remembering behaviour, a striving for accuracy, the ability to correct oneself, comparing.
A cry is not a description. But there are transitions. And the words "I am afraid" may approximate more, or
less, to being a cry. They may come quite close to this and also be far removed from it. [b: PI II, ix, p. 189c]
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52. If a sensitive ear shows me when I am playing this game that I have now this, now that experience of the word
"switch"--doesn't it also show me that often I do not experience that word itself in the context of a whole sentence
which I understand and in some sense experience? [Cf. PI II, xi, pp. 215h-216a]
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53. "The meaning of the word stood before my soul."--Will we say that if the word appears in an unambiguous
setting?
Page 9
54. A kind of writing†1 in which the crossed-out word, the crossed-out sentence, is a sign.
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55. You assured me that you experienced the word as 'meant' in the particular way you just now uttered it: Then tell
me also, with the same finely tuned sensitivity, whether in its proper context you 'mean it that way', in that sense.
For it is clear that in another sense you mean it, intend it, and later on even explain it this way, and not that. [Last
sentence: cf. PI II, xi, p. 216a]
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56. But the question now remains why, in connection with this game of meaning, we also speak of 'meaning
it'.--This is a different kind of question. It is part of the phenomenon of the game that in this situation we use the
word "meaning".†2 [Cf. PI II, xi, p. 216b]
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57. Then is it a misunderstanding?
Now I am not using the word for something else; rather I am using it in a different situation. [Just as I am not
using "know" to refer to two different things when I say "In my dream I knew". Cf. also: feeling of unreality.] In that
case should I be taught the technique of its use in a different way?
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58. Suppose I hear one of Beethoven's works and I say "Beethoven!"--Does the word have a different meaning here
than in the sentence "Beethoven was born in Bonn in 1770"? (The tone of the
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exclamation could be explained to someone who didn't understand it by saying: "Only Beethoven writes†1 like
that".)
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59. Would it be more correct to say that yellow 'corresponds' to e then "e is yellow"? Isn't the point of the game
precisely that we express ourselves by saying e is yellow?
Indeed, if someone were inclined to say that e 'corresponds' to yellow and not that it is yellow, wouldn't he
be almost as different from the other as someone for whom vowels and colours are not connected? And similarly for
the experience of meaning.†2
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60. Suppose I am learning a language and want to impress upon myself the double meaning of the word "bank", and
so I alternately look at a picture of a river bank†3 and then a money bank, and in each case say "bank", or "That is a
bank"--would the 'experience of meaning' then be taking place? Certainly not there, I'm inclined to say. But if the
inflection of voice, for example, seems to me to determine whether I mean one thing or the other--then I would be
experiencing meaning.
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61. It isn't as if we were obstinately referring to two things with the same word and then were asked: Why are you
doing this, if in reality they are different?--The new use consists in applying the old expression in a new situation; it
is not to designate something new.
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62. The experience of the 'word that hits the mark'. Is this the same as the experience of 'meaning'?
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63. "Why in a dream do we call this 'knowing'?"--We don't call anything 'knowing' in a dream; rather we say "In my
dream I knew..."
Why do we call this "meaning" and "signifying" if it is not a question of meaning and signifying?--What do I
call 'meaning' (or 'signifying') in this game: I say "By that word I just now meant...".
But what am I calling that?--An experience? And what experience?
For can I describe it otherwise than just by the expression: I 'mean' this word in this way?
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Page 11
64. Therefore I can't say that I'm simply giving the same name to two related things. (Otherwise the problem would
never have arisen.)
Page 11
65. "Why are we talking about 'meaning' in relation to that game?"--What am I asking for? A reason, a
cause?--Certainly not for a consideration which leads me to speak that way; nor for a justification; for such things
are not in question.
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66. Call it a dream!†1 [Cf. PI II, xi, p. 216b]
Page 11
67. But the question remains: why in the game of 'meaning' does a person use the same word?--Can he use a
different one? Does he use the same word for something different? Could he give another explanation of it?
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68. Call it a dream. It does not change anything. [PI II, xi, p. 216b]
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69. "Schubert"--It's as if the name were an adjective.
Neither can one say: "Look at all the things that 'fit'. For example, the name fits the bearer."†2
An addition, after all, would be an extension; and an extension is just what is not found here. For one doesn't
say that something is a 'fit' if actually it is no fit at all. As if one were merely expanding the concept. Rather we are
dealing here more or less with an illusion, a mirage. We think we see something that isn't there. But this is true only
more or less.--We know very well that the name "Schubert" does not stand in a relationship of fitting to its bearer
and to Schubert's works; and yet we are under a compulsion to express ourselves in this way.†3 [Cf. PI II, xi, p.
215f]
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70. We see something according to the picture, the concept, of fitting.
To be sure, I can regard one thing as a variation of another. And in an extreme case what I see as a variation
may no longer have any
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similarity at all to the thing of which I am seeing it as a variation.--To rephrase: First, this figure is a simple
projection of that one. Then the lines of projection are curved a bit; but to me it's still a projection. Finally they are
bent to the point of being unrecognizable, but I still see a projection. (Just as some continue to see an old man as the
young one, the one who has changed completely as what he used to be.)
Maybe it is strange to bring into this context the case of a person's name. But a connection can be drawn.
Namely this: a person's name is seen as a portrait.
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71. Suppose that I see a triangle as a square, by seeing it as the end of this kind of transformation:
--Then the kind of varying is part of the aspect that is seen. But this is just not the case when a
name seems to us to be the portrait of its bearer.
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72. I say something (for example, "The name 'Schubert' really does fit Schubert perfectly")--it means nothing.†1
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73. In the sense in which we are using it, the sentence "The name... fits..." doesn't tell us anything about the name or
its bearer. It is a pathological statement about the speaker.--One doesn't teach a child that this name fits the bearer.
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74. Someone waves (to me) with his hand. "What did you want?"--"I wanted you to come."
That is his intention when he waves.
The sign was the source of a movement. So wasn't it also the source of the explanation? Could this
explanation itself go: "Waving my hand was the source of the explanation which I shall now give you: Come to
me"?
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75. One cannot say here "It [the name] doesn't fit exactly", or "It doesn't seem to fit exactly".
It's not as if "fit" were not exactly the right word.
To be sure, we could use other words too; for example, "There is a kinship".
Page 12
76. "What is always associated is easily taken as akin." Is that the right expression? Not quite. But it is as if they
were akin.
This is not the way it is: "I take them for kin even though they are not"--for all I have to do is to wake up, as it
were, in order to know that they are not. But I see them pictured as akin.
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Page 13
I use the word, the picture.
Of course, one can explain: Fitting together and association frequently go together; and thence the illusion (if
one is to call it an illusion).
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77. I imagine that a physiological explanation of this strange phenomenon has been found. Now we see how the
illusion came about. For then what sometimes occurs in the brain is the same thing that occurs when.... Joyous
excitement: Now we understand why everybody always said...! And when the explanation has been given, when the
riddle has been solved†1--where does that leave us? It has only cleared up a question we weren't interested in, and
we are left with the fact that we use that expression, that picture, or want to use it, when the normal occasion for its
use is lacking.
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78. But then the question remains why we also talk about the act of 'meaning' in that game of meaning a thing. This
question doesn't belong here at all. We use the word here because it has this meaning. No other word, no other
meaning, would do for us. The fact has to be accepted.†2
Page 13
79. But is the word now being used in two senses? No. (Otherwise we would owe an explanation.) Does one teach
its usage in two different ways?
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80. But isn't an explanation of this strange phenomenon of interest to us, after all?
Think of other, related phenomena and what their explanation accomplishes. Yes, it is certainly interesting to
understand why, while taking this walk, I am under the impression that the city must lie over there; even though a
moment's reflection can convince me that it is not so. Now I shall assume that I know how the illusion came about:
The similarities between this landscape and another led me to draw the wrong conclusions, etc.--But I hadn't
explicitly drawn the wrong conclusion, and furthermore this doesn't explain why these similarities led me to draw
this precipitous conclusion. The explanation leaves the oddity untouched. (The same holds for the phenomenon of
seeing sounds as colours, etc.)
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Page 14
81. If someone answers the question "Will N. return?" time after time in the affirmative, this can be expressed by
saying he is in a state of thinking it is so. But no one will say that the answer "N. will return" describes the state of
the speaker.
Page 14
82. If "I believe p" states that I am in a certain state, then so does the assertion " p".
For the presence of the words "I believe" can't do it, can at the most hint at it. [Cf. PI II, x, p. 191d]
Page 14
83. Imagine a language in which "I believe that p" is expressed only by means of the tone of the assertion " p".
They say, not "He believes" but "He is inclined to say..." and there exists also the hypothetical "Suppose I were
inclined to say...", but not the expression "I am inclined to say...". [PI II, x, p. 191e]
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84. After all there are anomalies in other cases. We say "Maybe it will rain", but not: "Supposing that maybe it will
rain, ..."
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85. Moore's Paradox wouldn't exist in that language; instead of it, there would be a verb that has no first person
present. [PI II, x, p. 191e]
Page 14
86. But this ought not to surprise us. Think of the fact that one can predict one's own future action by an expression
of intention. [PI II, x, p. 191e]
Page 14
87. Think of the expression "I say..."--for example, "I say it will rain today", which simply comes to the same thing
as the assertion "It will... today". "He says it will..." means approximately "He believes it will...". "Suppose I say..."
does not mean "Suppose it will... today". [PI II, x, p. 192f]
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88. Different concepts touch here and coincide over a stretch. But you need not think that all lines are circles.
[PI II, x, p. 192g. No figure in PI]
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89. Just because someone sees something according to a certain interpretation, that doesn't mean that he experiences
an interpretation.
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Page 15
90. Anyone who thinks knows how notes or pictures which would be meaningless to someone else and which one
can't even explain to oneself, can stand for thoughts, or individual features of thoughts.†1 (The notation of a
calculating prodigy.)
Page 15
91. "Did you mean this when you said the word?"--"No, I was thinking of something else when I said that."--Is that
an experience? No. An experience would not be of the same interest. An experience might perhaps inform the
psychologist about the unconscious intention.†2 [Cf. PI II, xi, p. 217d]
Page 15
92. That is: If I were to find out, for instance, that when he was uttering the word he saw this or that in his mind's
eye, then it might be possible for me to infer something about a tendency in his subconscious mind--whatever was
in his mind was not his intention when he was uttering the word, his thought accompanying the word.
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93. "This plant grows from a germ, and not that plant." Imagine that people really expressed themselves in that way
in some language!
Page 15
94. But what is the germ?--The experience at the time of speaking. (Thus, for instance, an image--such as frequently
exists.) But really, according to its nature, it isn't the germ at all. Nor does something become a germ because of the
later development. What remains then, is that the image of the germ forces itself upon us. (Quite naturally so; for we
want to see the kernel of the matter in the experience.)
Page 15
95. The question which must interest us is therefore: What is the reference to the moment of speaking for? What
does it tell us?†3
Page 15
96. "I imagined you would think of him then." This wasn't contained in the image he had in his mind (for that
couldn't be known exactly), nor was it because of the name which he repreated to himself (it could have been
someone else's as well). It was the chain of interpretations, of explanations.
For when he says "At that point I was thinking of...", "By this I
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Page 16
meant...", he is thereby making the connection with that point of time.
(So he doesn't remember a mental picture, for instance, which he had in his mind and which now shows him
that he thought of....)
Page 16
97. He doesn't read off what he was thinking of from the mental picture.
Page 16
98. It said something and at that point I had to think of N. When did N. occur to me? At what moment, during which
of his words?--And even if I know which word--what happened to me when it was spoken?
The thoughts began with the word. That's where the train began. But what makes them into a train? The fact
that I say so?
Page 16
99. "I noticed that when this word was said, you turned pensive."
Page 16
100. "As I heard that word he occurred to me." What is the practical significance of this point of time?--For I want to
say; "It seemed to me that he occurred to me at that word"--and the subjective element of this statement makes no
difference. The question still remains the same: "What consequences does such a report have?"
Page 16
101. "When this word was spoken I thought of him." How is this report interesting? What primitive reaction
corresponds to such words?
Page 16
102. "À propos..."
"What made you suddenly think of him?"--"You said... and that reminded me of him."
Page 16
103. When do we say, I am writing to this person? How does this manifest itself? How do I know it myself?!--Am I
writing to him while I'm writing?
Page 16
104. It would almost be strange to say: "I thought of him while I was writing to him."
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105. "We were just talking about him", about this man I am now pointing to. How did the conversation relate to
him? Didn't I create the connection with these very words?
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Page 17
106. "I knew who you were talking about."--How could I know it? And what kind of mental state was the
'knowledge that the conversation was about this man'?
Page 17
107. "Who were you talking about?"--"About N."--"About my friend N."--"About the person in this
photograph."--"About the person who is just coming through the door."
Page 17
108. If God looked into our minds he would not have been able to see there who we were speaking of. [PI II, xi, p.
217f]
Page 17
109. In philosophy one must distinguish between propositions that express our mental inclination, and those that
solve the problem.
Page 17
110. The incurable illness is the rule, not the exception.
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111. In saying this you refer to the time of speaking. It makes a difference whether you refer to this time or that.
(The explanation of a word does not refer to a point of time.)
What makes you want to say that? (Why is this question important?) [a, b: PI II, xi, p. 217h]
Page 17
112. "Why did you look at me at that word; were you thinking of...?"
So there is a reaction at a certain moment, and "I thought of..." explains the reaction. [Cf. PI II, xi, p. 217g]
Page 17
113. "At that word we both thought of him."
Let us assume that each of us said the same words to himself--and how can it mean MORE than that?
But wouldn't even those words be only a germ? They must surely belong to a language and to a context, in
order really to be the expression of the thought of him. [PI II, xi, p. 217e]
Page 17
114. For sometimes a word actually gets uttered in the midst of a silent train of thought. And this could be reported.
Just as in general one can report that at the time he was quietly thinking of this or that. Whatever interest there is in
this report, there must also be in that one; and therefore also in the report that he thought of... when he heard the
word....
Page 17
115. "Then I thought: I wonder whether he'll come--."
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Page 18
"You wore such a doubtful expression; what were you thinking?"--"I was thinking: I wonder whether he'll
come--". ----"Did you then speak these words, or ones like them, to yourself?"--"No. Strangely enough I was
thinking of the Piccolomini,†1 of the scene in which...".
Page 18
116. In important ways, speaking in one's imagination cannot be compared to speaking, but our language-games
with the two are similar. (Tennis with a ball and tennis without one.) In those games too some mental picture plays
the role of a real one, which also can exist in connection with sentences and explanations.
Page 18
117. That is: Our language-game refers to a mental picture more or less in the same way as it does to a spoken
sentence. For the latter too is only a sequence of sounds and by itself refers to nothing at all.
Page 18
118. Now the question arises: When a word occurs in a certain context, I can create a different context for it in my
mind--but if I don't do that, if nothing out of the ordinary happens, then does my thinking run alongside my speech?
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119. Even if my thinking sometimes deviates from the path of speech, still it normally follows that path.
Page 18
120. If everything goes normally, no one thinks of the inner event which accompanies speech.
Page 18
121. Philosophy is not a description of language usage, and yet one can learn it by constantly attending to all the
expressions of life in the language.
Page 18
122. Knowing, believing, hoping, fearing (among others) are such very diverse concepts that classifying them, or
pigeonholing them, is useless to us. But we do want to recognize the differences and similarities among them.
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123. Compare: "When you spoke about N. I thought you meant..." and "When you spoke about N. I knew that you
meant...".
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Page 19
Is a particular experience associated with the latter? Then why with the former?
Page 19
124. The expression: "... passed through my mind."
"When I was reading, the conversation we had yesterday passed through my mind." Does what I read also
pass through my mind when I read attentively?
Page 19
125. "No, when I said 'bank' the river bank flashed through my mind."--Would I also say that the money bank had
'passed through my mind' if everything had happened normally?
Page 19
126. Suppose that someone were to tell us, right after everything he says, what went on in his mind when he was
saying it. (It's a habit.) Would that interest us in every case?
Page 19
127. "When I said 'bank' naturally I meant the bank where you get money." Did an experience of meaning have to
accompany the word? (Nonsense!)--Why does this have to be so if--against the grain of the context--I was thinking
about the river bank?
Page 19
128. "I have already remembered three times today that I must write to him." Of what importance is it what
happened then?!--On the other hand what is the importance, what the interest, of the statement itself?
It permits certain conclusions. [Cf. PI II, xi, pp. 217j-218a]
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129. "I hadn't forgotten about it completely; I remembered it three times today."--"Yes, I know: you flinched as I
was speaking of...."--Light is shed on this state of mind, and it has certain consequences. Different ones from those
resulting, for example, from this state of affairs: "It had completely slipped my mind; I didn't think about it any
more."
Page 19
130. At that word I went in this direction. (It is as if one were supplying the tangent at this point on the curve.)
But that again is merely an image. (As when tennis without a ball is described in terms of tennis with a ball.)
Page 19
131. The language-game "I mean (or meant) this" (subsequent explanation of a word) is quite different from the
language
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game: "I thought of... as I said it". The latter is akin to: "It reminded me of...". [PI II, xi, p. 217i]
Page 20
132. In these cases a characteristic reaction can occur at the time of meaning, remembering, or reminding.
Page 20
133. What is the primitive reaction with which the language-game begins, which then can be translated into words
such as "When this word occurred I thought of..."?†1 How do people get to use these words? [Cf. PI II, xi, p. 218b]
Page 20
134. "You said the word as if something different had suddenly occurred to you as you were saying it." One doesn't
learn this reaction.
The primitive reaction could also be a verbal reaction.
Page 20
135. Suppose I'm talking with someone about Dr N. In the middle of the conversation I say "When the name 'N'
came up just now I thought of Dr N."--the person I'm talking to won't understand me.
If I had said "When I said 'N' just now I meant Dr N., who...", the response might have been: "Of course,
who else could you have meant!"
If I had said "When the name 'N' came up, I could just see Dr N. in front of me", that might well have been
beside the point.
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136. "At the mention of the word I thought of..."--If the person who said this were asked what was then going on
inside him, and he couldn't answer anything--was his statement invalid?--He could have answered "I forgot it", and
he only thought that he ever knew.
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137. Is "In using the sign I wanted to let you know..." comparable to: "When I opened my mouth just now, I wanted
to say..."? That is: Is the former sentence therefore not so much a definition as the expression of a past intention?
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138. "Why did you look at me?"--"By this sign I wanted to give you to understand that you..." This does not express
a symbolic convention (no agreement); rather, it expresses the purpose of my action. To be sure, I was able to use a
sign which was agreed on for that purpose. [Cf. PI II, xi, p. 218c]
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139. "This number is the right continuation of this series." With these words I may bring it about that for the future
someone calls
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such-and-such the "right continuation". What 'such-and-such' is I can only show in examples.--That is, I teach him
to form a series (basic series) without using any expression of the law of the series; rather as a substratum of the use
of algebraic rules, or what is like them.†1 [Z 300]
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140. Now, to be sure, in teaching someone the basic series, I can use the word "equal" which perhaps he already
knows from other contexts, that is, with a different, though related, meaning. And it may be that he learns how to
construct the basic series more easily if I tell him "You must continue to do that, always adding one", thus stating a
rule; but here it doesn't (yet) function as a rule, and so far there is no algebra.
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141. If there were a verb meaning: to believe falsely, it would not have any first person present indicative. [PI II, x, p.
190g]
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142. "I believe he'll come, but he certainly won't come." If I say that to someone, it tells him that he won't come but
that nevertheless I am thoroughly convinced of the opposite, and will act according to this belief. However, by the
very fact that I am reporting to someone else that he won't come, I am not acting according to this belief.
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143. "I firmly believe he'll come, but he certainly won't come."
Why do we tell somebody that we believe something? In order to convince him of what we believe, or only
to inform him of our behaviour with respect to the matter?
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144. Observe: "He won't come, but I shall pretend that I believe he will come." I could say this to somebody, but this
pretence might really be the result of a pathological compulsion, so that it really wouldn't be pretence at all.
This could be put as follows: That's not the way it is, but I have to believe it.
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145. I said that the sentence "I believe that's the way it is, and it isn't" can be true; that is, if I do falsely believe it;
which is, after all, a possibility. But what makes the sentence true? How can someone else see that it is true? How
does he know that I believe it? Not from my behaviour; for that is full of contradictions.
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146. If I say: "Look! This figure is contained in this picture"--am I making a geometrical remark?--Isn't 'this picture'
that of which this is an exact copy, that which could be described with these particular words? Would it therefore
make sense to say that it now contains that figure or that it had contained it?--Thus the remark is timeless, and it can
be called "geometrical".
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147. What does this imply about perceiving such a state of affairs?
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Suppose someone is looking at a picture puzzle and finds the figure that is hidden in it. But he imagines that
the picture has changed, that the figure has now come into being. He might say, for example, "Now this figure is
here".
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Or, on the other hand, the picture could have changed, unknown to him, and he merely believes he has
discovered something in it which has always been there.
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148. "Actually you should point to your own visual impression when you say 'I see this', then you would really be
pointing to what you see." A result of the crossing of different language-games. (Similar to "'This' is the true name.")
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149. "Now I see that these faces aren't exactly alike." (Timeless sentence.) But that is what eyes are for, after all!
Imagine that someone wanted to say: "Indeed, that is a perception via the sense of sight; but it doesn't describe my
visual impressions." What would they be? Well, I looked from one face to the other, so that I could compare them,
and in the process I received a great many visual impressions; or one visual impression which changed continuously,
something that one might represent by means of a film. But couldn't we isolate just two among all these
impressions, in order to simplify things, and wouldn't two visual impressions be enough in the extreme case? And
couldn't these two represent what I had noticed, namely, the dissimilarity?
Isn't there just a completely different game here?
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150. It's no accident that I'm using so many interrogative sentences in this book.
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151. So should I say here that the visual impression, the sense datum, the visual object, is different? This concept
doesn't seem quite right. If we were to imagine that we could see similarity or difference as something like a picture,
then we might think of them as being accentuated in the picture; just as we can heavily trace over the outlines
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of whatever is different in order to show someone where the differences between two pictures lie. But if someone
sees what was traced over in the two pictures, he hasn't yet seen that dissimilarity.
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You have to look at the game as a whole, and then you'll see the difference.
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152. "I see that the two are similar" can be used temporally or timelessly depending upon how 'the two' are defined.
But does that mean that I see something different each time? "I see" is always temporal, but "The two are similar"
can be timeless.
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153. But is it always clear, in everyday usage, whether the sentence is meant as temporal, or timeless?--Take two
brothers; I meet them and then I say "Yes, I see that they resemble each other". Did I mean: these two men, M and
N, now resemble each other? (Perhaps they didn't before, etc.)--Or did I mean: I notice that these two human
appearances, which, for instance, a picture can capture, resemble each other?--If I had made the original utterance
and then had been asked in which of these ways I had meant it, could I have answered unequivocally?
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154. "You shouldn't draw his face" could mean: You shouldn't draw this person's face, no matter what it looks
like--or: You shouldn't draw these facial features which right now happen to be his. Each time something different is
important. And the injunction has different consequences, depending on which way it is taken.
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155. Even if I say "There is a similarity between these two faces", several different things can matter to me. It could
mean, for example: There is a similarity between this kind of face and that kind, where the two kinds are
distinguished by describing them. It may be the faces of those men that interest me, or it may be these facial forms,
wherever I encounter them.
The distinction I have in mind is, of course, that between the sense of: These two pieces have a similar
shape--and: The circle, the ellipse, the parabola, and the hyperbola resemble each other.
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156. The difference is that between external and internal similarity.
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157. Now if I say that two faces are similar--does it make sense to ask "Do you mean external or internal similarity"?
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158. "Does it interest you that you can detect a similarity in what seem to be quite different shapes?"
"Are you trying to say that these shapes have something in common--or that these human beings do?"--But
where is the difference?--Are you interested in the shapes, or the human beings? If it's the shapes, then maybe you'll
copy them exactly, study the similarity among the lines, and you'll completely forget the men. If there's a discussion
about them it will be a geometrical one, about types of lines.
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159. Suppose I retrace the facial outlines in order to explain to someone how they're similar, and he says "Sure, these
lines have a similarity, I see that; but that's not the way these faces look, ..."--then I could reply: "Perhaps you're
right, but that's not what I'm interested in. I wanted to point out that this kind of shape and that kind, however
different they may appear,..." Here what concerns me is a geometrical question.
But if I had answered: "You're right, I made a mistake"--I would have been concerned with the similarity of
these human beings.
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160. "They're brother and sister, but they don't look alike at all."--"I can see a similarity between them." What am I
interested in here?
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161. Suppose there were a law of aesthetics that said that faces in a painting have to be similar. Now I point to two
people and say to someone "Use these as models for your picture; they are similar".
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162. The sentence is non-temporal if I cannot replace "They are similar" with "They are now similar".
But if I utter that sentence on a certain occasion, is it always clear whether I wanted to allow the substitution?
Must I have thought about it?
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163. I can be interested in seeing similarities in lines even where apparently there are none. That is, in my analytical
eye.
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164. "I see different things in a much more important sense than I do things that are the same."
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165. A picture story. In one of the pictures there are ducks, in another rabbits; but one of the duck heads is drawn
exactly like one of the rabbit heads. Someone looks at the pictures, but he doesn't notice this. When he describes
them he describes this shape first as the one, then as the other, without any hesitation. Only after we have shown
him that the shapes are identical is he amazed.
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166. So he saw both aspects, but not the change of aspect.
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167. Would he have drawn the head differently in each case if he had been copying the picture? Not so far as I
know! So he saw them in exactly the same way both times.
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168. But did he have the same image both times?--So far as I understand this question--no.
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169. In the change of aspect one becomes conscious of the aspect.
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170. But was it correct to say "He saw both aspects, but not the change of aspect"? Shouldn't I have said "He
interpreted the picture in two ways, but didn't see the change of aspect"?
At first the picture was like some sort of picture of a duck; and if he saw an aspect here, then really he saw
one in every picture, and therefore also in every object. For have I examined every picture to see whether it can't be
seen differently?--So I shall say: he didn't see the aspect; he interpreted the picture in such-and-such a way.
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171. Experiencing an aspect expresses itself in this way: "Now it is ..."
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172. What is the philosophical importance of this phenomenon? Is it really so much odder than everyday visual
experiences? Does it cast an unexpected light on them?--In the description of it, (the) problems about the concept of
seeing come to a head.
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173. And here we can ask: If someone says "Now it's a duck--now it's a rabbit!"--what happened at the outset? At
that point, after all, he had not yet experienced the change, and yet he is already saying "Now it is...". Well, precisely
nothing 'happened'; but he was already playing that game.
You would have to look for something to distinguish the visual experience that accompanies the words
"Now it is a duck" from the
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one that accompanies the words "That is a duck" (when a person knows nothing about aspects). And, of course,
nothing is to be found.
For what should I say?--When does the experience I am interested in take place as he says "Now --, now --"?
(Do we have two extraordinary experiences here? Or three?)
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174. What is strange is really the surprise; the question "How is it possible!"
It might be expressed by: "The same--and yet not the same."
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175. The paradox may manifest itself in laughter. But couldn't we also imagine someone who wouldn't laugh in this
situation; to whom nothing seemed paradoxical.
And yet he too would experience the change of aspect. He would call the picture first one thing, then
another: and that would be all.
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176. And what is he doing? What he presents as an expression of his experience would otherwise be a perceptual
report. (The strong similarity with the experience of meaning.)
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177. Wherein lies the similarity between the seeing of an aspect and thinking? That this seeing does not have the
consequences of perception; that it is similar in this way to imagining.
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178. Imagine a sign language in which a duck's head were a certain message, and a rabbit's head another one.
Someone using this code accidentally draws a duck's head so that it can also be seen as a rabbit's head. The recipient
of the message gives it the wrong interpretation: this will come out in his actions.
But if he realizes that it can be seen this way and that he will not (also) behave differently, according to
whichever aspect he happens just then to be seeing.
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179. "Is it thinking? Is it seeing?"--Doesn't this really amount to "Is it interpreting? Is it seeing?" And interpreting is
a kind of thinking; and often it brings about a sudden change of aspect.
Can I say that seeing aspects is related to interpreting?--My inclination was indeed to say "It is as if I saw an
interpretation". Well, the expression of this seeing is related to the expression of interpreting.
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180. Two uses of the report "I see..."--In one language-game the observer reports what he sees from his vantage
point.--In the other, the same objects are scrutinized by several people; one of them says: "I see a similarity between
them".
In the first language-game the report might have been, for example, "I see two people who resemble each
other as father and son". This description is far less complete than one given, for example, by an exact drawing. But
someone could give this more complete description and yet not notice the similarity.
And someone else could see this drawing and discover the resemblance in it.†1 [Cf. PI II, xi, p. 193a, b]
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181. There is a game of guessing thoughts. A variant of it would be: I say a sentence in a language which A
understands and B doesn't. B is supposed to guess what I have said.
Another variant: I write down a sentence which the other person cannot see. He has to guess it; or guess
what it is about. [Cf. PI II, xi, p. 223b, beginning]
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182. Guessing intention: On a piece of paper that the other person can't see I write that I shall raise my left arm at the
stroke of the clock. The person is to guess what I am going to do at that time.
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183. "Only I can know what I shall do." But can't I make a mistake; and can't the other person predict it correctly?
But ordinarily the other person doesn't know it, and I often do.
Likewise the other person doesn't know to whom I am writing unless he sees it or finds it out from me; but I
can say who it is.
Usually it is I who am asked about the motives of my actions and not someone else. Likewise I am asked
whether I feel pain. This is part of the language-game.
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184. But would it be correct to say that my pain is hidden?
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185. Is the future, for example, hidden?
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186. "Nothing is so well hidden as future events. They can't be known. One can only know what is happening now."
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187. To be sure, one cannot be deceived about one's immediate experience: but not because it is so certain. The
language-game allows for senseless utterances even though not for 'false' ones.†1
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188. "One cannot know the future" is a grammatical remark about the concept 'to know'. It means something like:
"That is not knowing." And now one could ask: Why should someone be tempted to draw this conceptual
boundary? And the answer could be: Because of the uncertainty of predictions.
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189. "One can't know the future?--How about solar and lunar eclipses?"--"One can't really know them
either."--"Know?--Like what, for example?" [Cf. PI II, xi, p. 223d]
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190. If a lion could talk we could not understand him.†2 [PI II, xi, p. 223h]
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191. Even if someone were to express everything that is 'within him', we wouldn't necessarily understand him.
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192. So he gets angry, when we see no reason for it; what excites us leaves him unmoved.--Is the essential difference
that we can't foresee his reactions? Couldn't it be that after some experience we might know them, but still not be
able to follow him?
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193. He behaves like a man in whom complicated thought processes are taking place; and if only I understand them
I would understand him.--Let us imagine this case; and now he is reciting his thoughts to himself, and in a certain
sense I understand his actions. That is, I see the trains of thought and I know how they lead to his actions.
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194. In this way he would cease being a riddle to me.
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195. Think of how puzzling a dream is. Such a riddle doesn't have to have a solution. It intrigues us. It is as if there
were a riddle here. This could be a primitive reaction.
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196. It is as if there were a riddle here; but it doesn't have to be a riddle.
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Page 29
("All forms are like and none the same. And so the chorus points to a hidden law.")
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197. I don't know what is going on inside him. I couldn't flesh out his behaviour with thoughts.
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198. He is incomprehensible to me means that I cannot relate to him as to others.
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199. If someone is suspicious of a mathematical result he will suspect the arithmetic. But isn't that just a method? If
someone is suspicious of someone's direct expression of experience and doesn't think that he is lying, he will say
that he doesn't know what the other person is saying, that he is dreaming or not in his right mind.
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200. But how do I know what I would do if...? If I stepped out into the street and found everything completely
different from what I was used to, maybe I would just go ahead and join in. So I would behave quite differently than
ever before.
And yet there is something important about my remark.†1
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201. Suppose we were to meet people who all had the same facial features: we should not know where we were
within them. [C&V, p. 75]
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202. A people: with a ruling class whose members all look alike (except for sexual characteristics), and with an
oppressed class whose shapes and facial features vary as ours do.
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203. A tribe unfamiliar with the concept of simulated pain. They pity anyone who indicates that he is feeling pain.
They are unfamiliar with the suspicious attitude toward expressions of pain. A traveller coming from our culture to
theirs frequently thinks that a complaint is exaggerated, indeed, that its only purpose is to generate pity; the natives
don't seem to think that way. (In their language they have an expression corresponding more or less to our: "to feel
pain".) A missionary teaches the people our language; in the process he also educates them and under his tutelage
they learn to distinguish between a genuine and a pretended expression of pain. For he mistrusts many an
expression of pain and suppresses it, and teaches the people to be suspicious.--They learn our expression: "to feel
pain", and also "to simulate pain", and the question is: were they
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taught a new concept of pain? Certainly I won't say that they only now know what pain is. For that would mean that
they had never felt pain previously.†1
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204. Had those people overlooked something, and did the teacher bring something to their attention?
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205. And how could they remain unaware of the difference if sometimes they would complain when they were in
pain, and sometimes when they were not? Am I to say that they always thought it was the same thing?--Certainly
not. Or am I to say that they didn't notice any difference?--But why not say: the difference wasn't important to
them?†2
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206. "If a concept refers to a certain pattern of life then it has to contain a degree of indefiniteness." I am thinking of
something like this: On a strip of paper we have a continuous and regular pattern of bands. This pattern of bands
forms the background for an irregular drawing or painting, which we describe in relation to the pattern, since this
relation is what matters to us. If the pattern were to run: a b c a b c a b c etc., I would have a special concept, for
example, for something red that is on a c, and something green that appears on the following b.
Now once anomalies occur in the pattern I will be in doubt as to which judgment ought to be made. But
couldn't my instruction have provided for this? Or do I simply assume that in being instructed in the use of the
concept, that particular pattern was just taken for granted, but was never itself described?†3
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207. If colours were to play a different role in the human world than they now do, what consequences would this
have for colour concepts? This is actually a question of natural science, and I don't want to ask such a question.
Rather, this: What consequences would seem plausible to us? What consequences would not surprise us?†1
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208. If colours were to play a different role in the human world than they now do, what sort of colour
concepts--different from ours wouldn't seem odd? Consider various cases.
The question hasn't yet been phrased properly; but what is its purpose?--
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209. It is very hard to imagine concepts other than our own because we never become aware of certain very general
facts of nature. It doesn't occur to us to imagine them differently from what they are. But if we do then even
concepts which are different from the ones we're used to no longer seem unnatural to us.
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210. Our concept of pure future "It will happen"--as opposed to "It's meant to happen" and "It is to happen". Must
all peoples have this concept, which perceives time spatially, as it were?
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211. If a pattern of life is the basis for the use of a word then the word must contain some amount of indefiniteness.
The pattern of life, after all, is not one of exact regularity.
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212. Imagine someone who counts only on his fingers, for whom five is a hand and ten the whole person, and who
then goes on to count people on his fingers, etc. For him the decimal system will not be an arbitrary number system.
For him it is not a method of counting, but counting.
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213. Six pure colours. Does it have to appear to us this way?--Brown is not one of them.
But what does that tell us anyway? Where do we use such statements? When we describe things by their
colours?--Indeed; when we do this, for example, in a general way.
"Brown is not one of them" can, after all, express the instinctive rejection of a colour combination.
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214. "Light is white. Colours are already a shadow."--But is all 'light' really white? Doesn't the lamp give off
light?--Where then
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does the original proposition, which still sounds so evident, come from? (And why does it sound evident?)--We
always call what is lighter white. If one of two colours is the lighter, then only it can be the white one. And lightness
and light are equated here.
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215. We have a concept of colour blending which supersedes all physical methods of blending colours. So that we
can therefore say of such a method: it comes closest to effecting the 'pure' blending of colours, for example.
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216. Thus we judge whether according to our concept the two colours a and b really should produce the colour c.
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217. How did we arrive at this concept? That's really irrelevant.
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218. "Several shadows together result in light."--This idea could already look like a fiendish perversion of truth.
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219. Could we also perceive all colours as mixtures of white and black?--Possibly so, if under certain conditions, for
example, white and black pigments produced red, green, etc. Maybe we would say: "Light brings out the red in
black." (The colour is thus thought of as hidden in black.)
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220. Red and green the same. I am imagining there being only one shade of red and green. In nature they always
blend into each other (as certain leaves do in autumn). They are everywhere found together, one being a variation of
the other. The distinction between them is no greater than the one between lighter and darker.
But don't the people see the difference?! Of course they do. But they have a word, say, "leaf-colour", which
is fairly analogous to our colour names, and means red or green; and they have two modifiers, "sharp" and "blunt",
more or less analogous to our "light" and "dark", which separate red from green. And now the question is: which of
their concepts is closer to one of our colour concepts? Their concept 'leaf-colour', or their concept 'sharp leafcolour'
(that is, red), for instance?
(If they are discussing colouring or painting an object, they might say that they want it leaf-coloured. If they
are asked whether it should be sharp or blunt they might answer that they don't care.) Or
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would these people then be colour blind? Well, if we teach them our language they turn out to be normal.
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221. It's just that the difference between red and green isn't as important to them as it is to us.
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222. If we present them with a greater variety of colours then maybe they will experience our system as the only
natural one, that is, take it up and drop the other one, with no difficulty. But then again, maybe they won't.
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223. A type of painting, in which the illuminated side of figures is always painted green, the shadows always red.
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224. Could we imagine that people might have a concept of pretence that doesn't coincide with ours?--But would it
then be the concept of pretence?--Well, it could be a concept related to ours.
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225. But aren't some of the traits of (such) a concept more essential, others less so? That is: If one changes this trait
it will still be called "pretence"--but if this one is changed that word will no longer be used. And here naming means
an attitude.
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226. People whose faces immediately give away their feelings to others hide their faces when they want to
dissemble.
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227. These people say, not that one cannot look within, into the heart, but rather that one can't read the features
when they are veiled.
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228. "One can't look into his heart." The question is: Can he? (That fixes the concept.)
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229. "One can't look into a person's heart." The real underlying assumption is that he can do this himself.--Is it
experience that teaches us this?
I am inclined to say--yes and no.
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230. And there must be a reason for that.
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231. "He can tell me things about himself that I wouldn't otherwise know."
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232. This much is certain: He can predict some of his bodily movements that I can't. And if I do predict his actions I
do so in a different way.
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233. And is that an empirical fact? Or: which one am I talking about here?
For example, I can't move his arm voluntarily, the way I can mine. But what this means isn't all that easy to
explain.
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234. I cannot know what he's planning in his heart. But suppose he always wrote out his plans; of what importance
would they be? If, for example, he never acted according to them.
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235. Perhaps someone will say: Well, then they really aren't plans. But then neither would they be plans if they were
inside him, and looking into him would do us no good.
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236. "Can't you see, he has pain!"--"Pain, over there? How come?"
He wouldn't understand what it means for another person to have pain.
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237. What if someone had been taught as a child that plants feel pain; later on in life, however, he no longer believes
this.--How would this transition take place?
He casts off the idea like a garment that no longer fits.
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238. How would a person act who doesn't 'believe' that someone else feels pain? We can imagine how. He would
treat him as something lifeless, or as many treat those animals that least resemble humans. (Jellyfish, for instance.)
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239. We all know the doctor's question "Is he in pain"; and the uncertainty as to whether a person under anaesthesia
feels pain when he groans; but the philosophical question whether someone else is in pain is completely different; it
is not doubt about each individual in a particular case. [The point of this sentence has not yet emerged.]†1
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240. Do we encounter this doubt in everyday life? No. But maybe something which is remotely related: indifference
toward other people's expressions of pain.
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241. Aren't the fictitious cases I am trying to deal with like mathematical problems? (And how would you solve this
equation? And how this one?)
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242. The belief that this or that person isn't in pain because he isn't expressing it--or because he is only
pretending--or because he is under anaesthetics--has different grounds than the belief that an amoeba feels no pain.
These grounds are also different from those of that imaginary inhuman brute who regards expressions of pain in his
environment as phenomena of lifeless objects.
But would this brute really say he believes they feel no pain?--Possibly. But would he mean the same thing
as the doctor, for instance, who reassures us as to a patient's condition? The utterance--however he may have
learned it--is used by him in a different context; even if some of its consequences are similar.
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243. "The uncertainty as to whether another person is in pain"--is it based on the fact that he is he and I am I? (But
just ask yourself: "Can he know it? He doesn't have any object for comparison.") No, here I'm deceived by a picture.
The uncertainty is a matter of the particular case, and the concept vacillates from one case to another. But that is our
game--we play it with an elastic tool.
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244. Couldn't there be people who have never had occasion to feel this uncertainty?
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245. They would say: "Should I be uncertain because he is he and I am I? What in the world do you mean?"
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246. And could people play it with a rigid concept?--Then it would be different from ours in a strange way. For in
the flux of life, where all our concepts are elastic, we couldn't reconcile ourselves to a rigid concept.
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247. Indeed, mustn't any concept simply of behaviour be formulated imprecisely if it is more or less to serve the
game with such concepts?
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248. After all, there could be someone who had serious, hopeless doubts about others. But how would he act? (Like
a lunatic.) He would say, for example: At times I feel that another person and I are the same and at times I don't.
Correspondingly, at times he would show pity, at other times he would show none, and sometimes doubt.
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249. The behaviour of humans cannot be foreseen, cannot be calculated. Let's assume it could be, that I have made
the calculation and now I observe their behaviour (like movements of complicated machines).
If that happened--would it be possible to observe them sympathetically? Would it be impossible to say "No
one can know what's going on inside them"?
If, for instance, someone says to himself "That's the way humans are. I am exactly like that."
Then he might look at his calculation in a different light.
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250. Why in the world do we play this game!--But what are we after here? The game's surroundings, not its causes.
Page 36
251. "Where I am certain, he is uncertain." What if that also happened with a calculation --.
Page 36
252. "Couldn't he have been pretending?"--But couldn't he just be imagining that he is pretending? (Isn't this
conceivable? And conceivability is what matters to us here, not probability.)
Dissimulation is nothing but a particular case; we can regard behaviour as dissimulation only under
particular circumstances.†1
Page 36
253. The concept 'dissimulation' has to do with the cases of dissimulation; therefore with very specific occurrences
and specific situations in human life. And here I mean external occurrences, not inner ones, etc.
Therefore it isn't possible for all behaviour, under all circumstances, to be dissimulation.
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254. But isn't the concept such that for any behaviour, etc., one can imagine (construct) a larger context in which
even this behaviour
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would be dissimulating behaviour? Isn't this, for example, the basis for the problem of any detective story?
Page 37
255. One could also say: The concept of dissimulation has to do with a practical problem. And the blurred outlines
of the concept don't change anything about that.
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256. Merely recognizing the philosophical problem as a logical one is progress. The proper attitude and the method
accompany it.
Page 37
257. But what does this mean: "All behaviour theoretically could be dissimulation."
Page 37
258. It must surely mean: the concept of dissimulation allows for it.
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259. And that means: If I were to find out this and that, and this in addition, then I might say that it is (was)
dissimulation.
(Euclidean geometry.)
But where is it written that we would say that; or from what do I deduce it?
'Insofar as this concept has been determined, it allows for that too.'
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260. But now we're making a false picture of our concept.
Page 37
261. The concept 'dissimulation' serves practical aims.
Page 37
262. -- -- -- Therefore not all behaviour can be dissimulation under all circumstances.†1
(Occasion, motive, etc., are part of 'dissimulation'.)
Page 37
263. A play, for example, shows what instances of dissimulation look like.
Page 37
264. Of course, one could imagine variations of the typical manifestations of dissimulation.
The plays of people who differ from us in this way would then take a different course from ours, and we
wouldn't understand them at all.
What would be completely unmotivated to us would seem natural to them.
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265. (For example, the way Orestes identifies himself to the king, by pointing to his sword, etc., might seem utterly
absurd to some people.)
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266. A play by these people would be incomprehensible to us. (Indeed, is Greek tragedy comprehensible to us?)
And what is the meaning of 'to understand' here?
Page 38
267. A sharper concept would not be the same concept. That is: the sharper concept wouldn't have the value for us
that the blurred one does. Precisely because we would not understand people who act with total certainty when we
are in doubt and uncertain.
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268. Couldn't someone make up stories containing dissimulation in order to show that he knows what 'dissimulation'
is? In order to develop the concept of dissimulation he invents more and more complicated stories. For example,
what looks like a confession is merely further dissimulation; what looks like dissimulation is merely a front that
hides the real dissimulation; etc., etc., etc.
Thus the concept is laid down in a kind of story.
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269. And the stories are constructed according to the principle that everything can be dissimulation.
Of course, part of all this is that in each story something is characterized as the foundation of truth. And how
can the foundation of truth be characterized as such? Perhaps in the form of monologues. These must not be
audible, otherwise they might be part of the dissimulation.--But couldn't someone conduct monologues in thought
merely because they give him a certain appearance which he will then use to practise deception?--So the intention is
the foundation? And how can it come out in the story?
Page 38
270. The concept of deception can be used for practical problems. For example: If someone is hatching evil plots,
and he brings forth nothing but good and glorious deeds until he finally commits the evil one, then that will be
deception only in the 'theoretical' sense; for it no longer resembles deception, and the conclusions one would
normally draw from evil plots are incorrect here.
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271. And what have I accomplished with all of this?
In explaining the concept I have substituted the use for the picture.
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Page 39
272. "The word W. has two meanings" means: it can be used in two different ways. What does this sentence tell us?
In what circumstances it is used?
Someone knows only one meaning for the word "bank"; I tell him: it has another one. (Namely:...)
Someone is already totally conversant with every use of the word, but suddenly he stops short, confused,
and I explain to him: "The word has two uses:..."
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273. There are many uncertainties with this concept of meaning.
Page 39
274. One doesn't say, for instance that "walk" and "walks" have different meanings.
They mean exactly the same thing, we would tell someone; namely, this--and then we would demonstrate
walking to him.
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275. You visit a tribe; they have a language; in this language you hear a word (a sound)--does it have one meaning,
or several? How will you find out, how will you decide?
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276. Sometimes, however, the decision will be quite easy and clear. [But always?]
Page 39
277. "I'm not yielding a hair's breadth."
"He doesn't have a hair on his head." Does "hair" mean the same thing in both sentences? And does "a bit"
mean a little bite?--"In the one case one is still aware of the old meaning, but not in the other." And this sentence
doesn't refer to any conscious state one is in when one utters the word, but rather to an explanation one would
perhaps give, or not give, if... In other words, it refers to connections that one would or would not make.
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278. What is the correct German translation of an English play on words? Maybe a completely different play on
words.
Page 39
279. And what do you want to do with the decision that the word has only one, or more than one, meaning?
After all, you can learn how to use it without deciding that (without thinking about that).
Page 39
280. If you say that it has two meanings then you have to use an explanation to distinguish between them. (There
can be various reasons for doing that.)
Page Break 40
Page 40
281. But the distinction may or may not be immediately obvious.
Page 40
282. The distinction may be drawn when you first learn how to speak, or maybe not until someone investigates the
grammar of the language.
Page 40
283. (You must start here with the living language.)
Page 40
284. The distinction between various uses is made for various reasons.
Page 40
285. I look at the language and say "Different words are used quite differently".
But then I also say: "These have similar uses." Indeed: "These words (here) are used in the same way." And
furthermore: "This word has two completely different uses." But also: "It is used in two different, but similar,
ways."--And so far I am describing what I notice. (That is, as yet there is no problem here.) (So far, I am still
completely naive.)
Here every meaning always has an explanation corresponding to it. And the explanations can be of entirely
different kinds, and then again can in various ways be quite similar.
[An explanation of "go" and "gone".]
The differences can be more primitive, and less so.
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286. You enter new territory when you observe several languages and compare them with each other.
Page 40
287. The explanation of how some words are used will seem simple, lapidary, and basic to us; that of others:
artificial, arbitrary, pointless.
Page 40
288. "We need a word to designate this object, this tool; but why do we need a word to refer to this on Mondays,
that on Tuesdays, etc?" Does this word have one meaning or seven?
Page 40
289. Not every use, you want to say, is a meaning.
Page 40
290. Does this word have one function in our lives or does it have seven?
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Page 41
A function: we have certain models for this. And whatever resembles them is given this name. (A blurred
concept.)
Page 41
291. Meaning, function, purpose, usefulness--interconnected concepts.
Page 41
292. Imagine the hypothesis: men never remember their dreams exactly, they forget the ideas in their dreams as soon
as they wake up and remember only the pictures that acccompany the dreams. The plot is lost and only the
illustrations remain.
Page 41
293. Suppose we were to replace every tenth word in a story with the word "table".--And now suppose that there
were a word in some language which was used in the same way as the word "table" in this story.
How could we describe the use of such a vagrant word?
Or what would it mean: "To teach someone the use of the word"?
Page 41
294. What am I after? The fact that the description of the use of a word is the description of a system, or of
systems.--But I don't have a definition for what a system is.
Page 41
295. I encounter people who use a vagrant word in their language.
Page 41
296. If they had only vagrant words--then it simply wouldn't be a language.
Page 41
297. Here I'm thinking of a person who quite naively (without the ulterior motives of a philosopher) looks at the
varieties of word usage and describes them to himself.
For example, he could classify the word which has a different meaning every day of the week as a normal
noun, and the question "Does this word have one or several functions?" wouldn't occur to him.
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298. The question "Do 'non' and 'ne' have the same meaning?" never occurs to him.†1
Page Break 42
Page 42
299. But now he also compares his language with the primitive language which a person learns when he is a
foreigner encountering people who don't understand him. For such a person learns individual important words
through various kinds of explanatory gestures. Each word has its own showing or demonstration (a scene).--And, of
course, the meaning of negation is also demonstrated. (Whether through the command "Don't do that!", or through
a statement.)
Page 42
300. In this language the exact endings of the words won't be important. (Or: this language has no inflections.)
The demonstrations distinguish between the use of one word and that of another, but they don't distinguish
between "go" and "goes", for example.
Page 42
301. And now we could introduce into our description of language a concept of 'meaning' such that two words
would have the same meaning if they were explained by the same demonstration in that primitive language.
Page 42
302. So one can ask: If a foreigner visits people who say "non" and "ne", at what point will he be taught the
difference?
Certainly not at the beginning; he will learn a negation which doesn't differentiate between the two.
Page 42
303. Suppose I were to say that 'meaning' is the primitive function of a word--would that be right?
Page 42
304. And, of course, this concept is extremely vague.
But do the negation in a report and the prohibition in a command ("Don't do that!") have the same primitive
function?--What is called the same function and what is not will depend on human nature. Just as, of course, what is
necessity†1 and what isn't.
Page 42
305. The words "the rose is red" are meaningless if the word "is" has the meaning "is identical with".--Does this
mean: If you say this sentence and mean the "is" as the sign of identity the sense disintegrates? [PI II, ii, p. 175c]
Page Break 43
Page 43
306. We take a sentence and tell someone the meaning of each of its words; this tells him how to apply them and so
how to apply the sentence too. If we had chosen a senseless sequence of words instead of the sentence, he would
not learn how to use the sequence. And if we explain the word "is" as the sign of identity, then he does not learn
how to use the word sequence "the rose is red". [PI II, ii, p. 175d]
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307. And yet it is true that the sense of that sentence will seem to disintegrate before the mind of anyone who thinks
"identical with" whenever the word "is" is spoken. Like someone who thinks of hail whenever he hears the salutation
"Hail!"--One might tell someone: if you want to pronounce the salutation "Hail!" expressively, you had better not
think of hailstones as you say it!†1 [PI II, ii, p. 175d]
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308. What makes my image of him into an image of him?
When I say "I'm imagining him now as he...", then nothing is being designated here as his portrait.
But can't I discover that I pictured him quite wrongly?
Isn't my question like this: "What makes this sentence a sentence that has to do with him"?
"The fact that we were speaking about him."--"And what makes our conversation a conversation about
him?"--Certain transitions we made or would make. [a: PI II, iii, p. 177a]
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309. What makes this picture his portrait?--It is designated as such in the catalogue.
Page 43
310. Suppose that instead of imagining something, I were to make sketches on a piece of paper. So I talk about N.
and in the process my pencil is sketching a figure on the paper. Now someone can ask me "Does that represent N.?"
And it might represent him, whether it is like him or not.
Is it correct to say: That's the way it is with one's imagination? Certainly; insofar as one can sometimes draw
what one has imagined.
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Page 44
311. The question "What makes this into an image of him?" doesn't usually arise when I am imagining something.
And if I draw what I imagined and am then asked "What makes this picture into his picture?", I could answer: "My
imagination".
Page 44
312. "What makes the remark I just made into a remark about him?"
Page 44
313. What can be said to that?
Nothing inherent in or simultaneous with it. If you want to know whom he meant, ask him!
Page 44
314. "What makes my mental image of him...?" Is there anything here I could investigate to see whether it was my
mental image of him?
Page 44
315. And if I say "I picture him quite vividly as he... ", then the same question applies to this sentence and to the
mental picture.
Page 44
316. On the one hand, a face could come before my mind, and I could even be able to draw it without my
knowledge whose it is or where I have seen it. [Cf. PI II, iii, p. 177a]
Page 44
317. What makes my image of him into an image of him?
Not its looking like him.
The same question applies to the expression "I see him now vividly before me" as to the image. What makes
my utterance into an utterance about him? Nothing in it or simultaneous with it ('behind it'). If you want to know
whom he meant, ask him! [PI II, iii, p. 177a]
Page 44
318. Suppose, however, that someone were to draw while he had an image or instead of having it, though it were
only with his finger in the air. (This might be called "motor imagery".) He could be asked: "Whom does that
represent?" And his answer would be decisive. It would tell us of his intention. [PI II, iii, p. 177b]
Page 44
319. The line I drew was like a description.
Page 44
320. We have to remind ourselves that a face with a soulful
Page Break 45
expression can be painted to make us believe that colours and forms by themselves affect us like this.
Page 45
321. "I believe that he is suffering."--Do I also believe that he isn't an automaton?
It would go against the grain to use the word in both connections.
(Or is it like this: "I believe that he is suffering; but am I certain that he is not an automaton."? Nonsense!)
(That would be philosophers' nonsense.) [PI II, iv, p. 178a]
Page 45
322. Suppose I say of a friend "He isn't an automaton".--What information would be conveyed by this, and to whom
would it be information? To a human being who sees him in ordinary circumstances? What information could it
give him?! (At the very most that this man always behaves like a human being and not occasionally like a machine.)
[PI II, iv, p. 178b]
Page 45
323. "I believe that he is not an automaton", just like that, so far makes no sense. [PI II, iv, p. 178c]
Page 45
324. My attitude towards him is an attitude towards a soul. I am not of the opinion that he has a soul. [PI II, iv, p.
178d]
Page 45
325. Now a picture is strongly suggested to us, the picture of something incorporeal which enlivens the face (like
quivering air). We have to remind ourselves that a face with a soulful expression can be painted to make us believe
that colours and forms by themselves can affect us this way.†1
Page 45
326. The concept 'meaning' will serve to distinguish those linguistic formations that might be called capricious from
those that are essential, inherent in the very purpose of language.
Page 45
327. The concept 'meaning' will introduce a new point of view into the description of word-usage.
Page 45
328. Example: A verb meaning to write in the first person, to love in the second, and to eat in the third.
Page Break 46
Page 46
329. Human nature determines what is capricious.
Page 46
330. But is it the nature of someone who already knows a language or of someone who doesn't yet know one (for
instance, the nature of a one-year-old)?
Page 46
331. Is it, or is it not capricious that a word means something different every day of the week, or means something
different in the first and second persons?†1
Page 46
332. 'Meaning' is a primitive concept. The form: "The word means this" belongs to it; that is, the explanation of a
meaning by pointing. This works well in certain circumstances and with certain words. But as soon as the concept is
expanded to include other words difficulties arise.
Page 46
333. The definition of a word is not an analysis of what goes on inside me (or what should go on) when I utter it.
Page 46
334. "Every two metres there are two soldiers."
Page 46
"He went into a bank on a bank."
Page 46
335. "I want to replace this word in our language with two words; I'll explain the one like this: ..., the other like this:
..." I also could have said: "This word in our language has two meanings: ..." Here one couldn't ask: "But are those
really two meanings?"--But one could--if one meant by it: "Isn't this distinction completely arbitrary, utterly
pointless?"
"Why do you distinguish between them, what's the point of this distinction?"
Page 46
336. "I don't see the purpose of it." But what does the explanation of a purpose look like? I can't give a general
answer to this question.
Page 46
337. You are posing problems to yourself and then solving them; like a mathematician.
Page 46
The problem: non and ne.
Page 46
338. A person naively describing the use of words might also describe the use of "non" and "ne", and he'll be able to
remark that they are almost the same.--But that's not all: can't he say that the two
Page Break 47
words are used differently only in very specialized language-games, and that otherwise they are used the same way?
Page 47
339. Mustn't he be able to say that in a certain language-game one word can be replaced by another?
Page 47
340. If the language-game, the activity, for instance, building a house (as in No 2),†1 fixes the use of a word, then
the concept of use is flexible, and varies along with the concept of activity. But that is in the essence of language.
Page 47
341. Let us imagine this use of "non" and "ne": both words are used like our "not"; the same event causes now the
one to be used, then the other, and in this way they are just like synonyms; a distinction is made only in the rare case
of double negation.
Thus I would be tempted to distinguish between the 'entire' use of a word and a part of its use. Indeed the
part of its use will seem more important here than the 'whole'.
Page 47
342. So I say: "The use is the same here and here and here. In all of these cases one can substitute one word for the
other." But what does that really mean?
Page 47
343. Does the person who describes things naively know the concept of 'being able to substitute one word for
another'?--Certainly he knows the concept of the mixed use of two words.
Page 47
344. Or what about this: A traveller visits the country where "non" and "ne" are used and he tries to translate this
language into his own. He won't have any reason to use different words to translate each of these into his
language--until he happens upon a case of double negation (then he might find an equivalent word in his language).
Page 47
345. For the traveller could say: "As far as I can tell they're used in the same way."
Page 47
346. "In all of these cases 'ne' and 'non' have exactly the same meaning." This could be said, for example, if in these
cases the people themselves treat the words as synonyms. (And we know what
Page Break 48
Page 48
that looks like.)--But possibly the tribe doesn't treat them as synonyms, doesn't 'mix' them, and they might
still be synonyms for us.
Page 48
347. The greatest difficulty in these investigations is to find a way of representing vagueness.
Page 48
348. One can speak of the function of a word in a sentence, in a language-game, and in language. But in each of
these cases "function" means technique. Thus it refers to a general way of explaining and of training.
Page 48
349. Whoever teaches someone a negation sign trains him in a certain way. (Double negation need not even appear
in training.) But then at some point he can use double negation or hear it, and comprehend it this or that way. His
comprehension doesn't have to be related to his previous training, even though it is conceivable that it might be. But
am I to say that his training taught him the meaning of double negation? I don't have to say that. And even if the
training has taught me to use two different words interchangeably to express negation, it certainly hasn't taught me
how to discriminate between them when it comes to double negation.
I certainly didn't learn this distinction by being trained. But what I did learn by being trained was a meaning,
and the same one.
Page 48
350. Within a kind of training one can distinguish (further) kinds of training. And thus one can distinguish various
uses within word-usage.
Page 48
351. Then psychology treats of behaviour, not of the mind?
What do psychologists record?- What do they observe? Isn't it the behaviour of human beings, in particular,
their utterances? But these are not about their behaviour. [PI II, v, p. 179b]
Page 48
352. A doctor asks "How is he feeling?" The nurse says "He is groaning". One report on his behaviour. But need any
question arise as to whether the groaning is genuine? Can't it be as if this question didn't exist at all? Can't the
conclusion be drawn: "If he is groaning we must give him more pain-killer"? But in the context of these thoughts
can't the report about behaviour be used as a report about a
Page Break 49
person's mental state? Can't it do this job and isn't the job the main thing?†1 [PI II, v, p. 179d]
Page 49
353. "But then they make a tacit presupposition." Then the technique of our word use is always a tacit
presupposition. [Cf. PI II, v, p. 179e]
Page 49
354. "We're always making presuppositions; if they aren't correct then, of course, everything is different." Do we say
this, for example, when we ask someone to go shopping? Are we presupposing that he is human, and that the store
is not a Fata Morgana? Presuppositions come to an end.
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355. But although it might not be a 'presupposition' in this case might it be one in a different case? Doesn't a
presupposition imply a doubt? And doubt may be entirely lacking; or it may be present, from the smallest to the
greatest degree.†2 [Cf. PI II, v, p. 180b]
Page 49
356. Suppose somebody were to say "I'm shivering with fright, I'm always shivering with fright"--but he means by
this that he can play chess. He expresses an ability as if it were an experience.
Even if someone could do this or that only when, and only as long as, he feels this and that, the feeling
would not be the capacity. [b: cf. PI II, vi, p. 181b]
Page 49
357. How do we compare the behaviour of anger, joy, hope, expectation, belief, love and understanding?--Act like
an angry person! That's easy. Like a joyful one--here it would depend on what the joy was about. The joy of seeing
someone again, or the joy of listening to a piece of music...?--Hope? That would be hard. Why? There are no
gestures of hope. How does hoping that someone will return express itself?
Page 49
358. It's easy to imagine an animal angry, frightened, unhappy, happy, startled. But hopeful? [PI II, i, p. 174a]
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Page 50
359. For hoping is quiet, joyful expectation. (Even though there is something repugnant about this kind of analysis.)
Page 50
360. A dog can expect its master, but can it expect its master will come the day after tomorrow? And what can he
not do here?-How do I do it? How am I supposed to answer this? [PI II, i, p. 174a]
Page 50
361. The 'meaning' of a word is not the experience one has in saying or hearing it; and the 'sense' of a sentence is not
a complex of the experiences which go with the words.
How do the meanings of the individual words make up the sense of the sentence "I still haven't seen him
yet"? The sentence is composed of the words, and that is enough. [PI II, vi, p. 181c]
Page 50
362. The feeling for words. Suppose we found a man who, speaking of how words felt to him, told us that "if" and
"but" felt the same. Should we have the right to disbelieve him?--Or should we just say that he isn't playing our
game. It would be like someone who instead of associating a separate colour with each vowel, associated one colour
with a, e, i and another with o and u. Maybe there are such people. [Cf. PI II, vi, p. 182b]
Page 50
363. One might say that such people would differ from us to a far greater extent than those who associated no
colours at all with vowels. One would almost like to call them colour blind.
Page 50
364. And would such a person for that reason also confuse the uses of "if" and "that"?
Page 50
365. Can only those hope who can talk? Only those who have mastered the application of a language. The signs of
hope are modes of this complicated pattern of life.†1 (If a concept applies to the character of handwriting, it has no
application to beings that do not write.) [PI II, i, p. 174a]
Page 50
366. The glance which a word in a certain context casts at us.
Of course, the way in which it looks at us depends on the surroundings in which it is located.
Page Break 51
Page 51
367. Isn't the if-feeling this word, uttered with this tone and in this context?
Page 51
368. The if-feeling cannot be something which accompanies the word "if". [Cf. PI II, vi, p. 182e]
Page 51
369. Otherwise it could accompany other things too.
Page 51
370. Suppose I were to speak of an if-gesture.
Could another word make the same gesture?--Or 'would it then not be the same gesture'?
Page 51
371. The sound of the word "if" is simply part of the if-gesture.
Page 51
372. Can two faces have the same expression? (Yes and no.)
Page 51
373. The if-feeling would have to be compared with the special 'feeling' which a musical phrase gives us. (Someone
might want to speak of a 'feeling with a half-cadence'.) [Cf. PI II, vi, p. 182f]
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374. But can this feeling be separated from the phrase? And yet it is not the phrase itself, for that can be heard
without the feeling. [PI II, vi, p. 182g]
Page 51
375. Is it in this respect like the 'expression' with which the phrase is played? [PI II, vi, p. 182h]
Page 51
376. For one does not mean a feeling which accompanies the phrase, but at the most, the phrase with the feeling.
Page 51
377. "He looked at me with a strange smile."--With what kind of smile?--To answer this I might have to draw his
face.
Page 51
378. The if-feeling is not a feeling which accompanies the utterance of the word "if". [PI II, vi, p. 182e]
Page 51
379. We say this passage gives us a quite special feeling. We sing it to ourselves, and make a certain movement, and
also perhaps have some special sensation. But in a different context we should not recognize these
accompaniments--the movement, the sensation--at
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all. They are quite empty except just when we are singing this musical phrase. [PI II, vi, p. 182i]
Page 52
380. If we say "I sing it with a quite particular expression", then "expression" does not refer to something that I can
separate from the passage.
In a different sense, it is conceivable for me to play a different phrase with the same expression. [Cf. PI II, vi,
p. 183a]
Page 52
381. The special feeling this passage gives me belongs to the passage, indeed to the passage in this context.
Page 52
382. I can talk about the expression with which someone plays a passage without thinking that a different passage
might have the same expression. Here this concept serves only as a means for comparing different performances of
this passage.
Page 52
383. The fact that we understand a sentence shows us that we could use it in certain circumstances (even if it were
only in a fairy tale), but this does not show us what we can do with it and how much.
Page 52
384. [non and ne.] They have the same purpose, the same use--with one qualification.
Page 52
385. So are there essential and non-essential differences among the uses of words? This distinction does not appear
until we begin to talk about the purpose of a word.
Page 52
386. My kinaesthetic sensations advise me of the movement and position of my limbs.
Now I let my index finger make an easy pendulum-movement forward and backward. I either hardly feel it,
or don't feel it at all. Perhaps a little in the tip of the finger, like a slight tension of the skin (not at all in the knuckle).
And this sensation advises me of the movement? For I can describe it exactly. [PI II, viii, p. 185a]
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387. "But after all, you must feel the movement, otherwise you couldn't know how the finger was moving." But
"knowing it" only means: being able to describe it.--I may be able to tell the direction from which a sound comes
only because it affects one ear more
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strongly than the other; but I don't hear this. It only has the effect: I know where the sound comes from, and, for
instance, I look in that direction. [PI II, viii, p. 185b]
Page 53
388. It is the same with the idea that it must be some feature of our pain that advises us of the whereabouts of the
pain; or some feature of our memory image that tells us the point in time to which it belongs. [PI II, viii, p. 185c]
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389. A sensation can advise us of the movement of position of a limb. (For example, if with your eyes closed you
were unable to tell, as a normal person can, whether your arm is stretched out, you might find out by a feeling of
pressure in the elbow.)--And the character of a pain can also tell us where the injury is. [PI II, viii, p. 185d]
Page 53
390. How do I know that a blind person uses his sense of touch, and a sighted person his sense of sight, to tell them
about the shape and position of objects?
Page 53
391. Do I know this only from my own experience, and do I merely surmise it in others?
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392. The evolution of the higher animals and of man and the awakening of the spirit, of consciousness, at a particular
level. The picture is something like this: Though the ether is filled with vibrations the world is dark. But one day man
opens his seeing eye, and there is light.
What this language primarily describes is a picture. What is to be done with the picture, how it is to be used,
is still obscure. Quite clearly, however, this must be explored if we want to understand the sense of what we are
saying. But the picture seems to spare us this work; it already points to a (very) particular use. This is how it takes us
in. [PI II, vii, p. 184d]
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393. What is the criterion for my learning the shape and colour of an object from a sense-impression? [PI II, viii, p.
185e]
Page 53
394. What sense-impression? Well, this one: I can describe it: "It's the same one as the one..."--or I can demonstrate
it with a picture.
And now: what do you feel when your fingers are in this position?--"How is one to define a feeling? One can
only recognize it within
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oneself." But it must be possible to teach the use of the words! [PI II, viii, p. 185f]
Page 54
395. What I am looking for is the grammatical difference. [PI II, viii, p. 185g]
Page 54
396. Colour, sound, taste, temperature, all have a subjective and an objective side to them. And that undoubtedly
means that sometimes they show what I feel, sometimes they describe the external world. Now my knowledge of
the position of my body seems to be lacking the subjective link.
Page 54
397. You can't describe a feeling? Of course you can. You do it every day. But how? Well, we have to think about
particular cases.
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398. If someone told me that he had then felt what you feel when you hold your fingers in this position, or move
them this way, then I would imitate the position or movement and ask him, perhaps "Do you mean the feeling in
your fingertips, or in your muscles or here?" That is, I needn't yet be clear what feeling he's talking about; indeed, I
could even say "I'm feeling nothing when I move this way". Consider: I could also ask him "Is it a strong feeling, or
a very weak one?" (But this remark is still peripheral, and doesn't yet get to the heart of the matter.)
Page 54
399. And where is the K-feeling?†1 Can you point to it? (For the location of the receptors is of no concern to us.)
Page 54
400. Let us leave the K-feeling out for the moment!--I want to describe a feeling to someone, and I tell him "Do this,
and then you'll get it", and I hold my arm, or my head, in a particular position. Now is this a description of a feeling?
And when shall I say that he has understood what feeling I meant? He will have to give a further description of the
feeling afterwards. And what kind of description must it be?--Suppose that he tells me "Yes, I've got it. It's an
extremely odd feeling". When he's asked "What kind of odd feeling? Where?", he says he can't say--it is quite
strange. How would we know that it is a feeling? [Up to "Suppose that he tells" PI II, viii, pp. 185h-186a]
Page 54
401. The 'further description' will connect the feeling up with other
Page Break 55
feelings: It will have a location, will stay constant or change, will become stronger or weaker.
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402. "Do this, and you'll get it." I hold my arm or my head in a certain position when I say this. Can't there be a
doubt here? Mustn't there be one, if it is a feeling that is meant? [Cf. PI II, viii, p. 186b]
Page 55
403. What would we say if someone reported to us that in a certain object he saw a colour he couldn't describe?
Does he have to be expressing himself correctly? Does he have to mean a colour?
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404. This looks so; this tastes so; this feels so: "this" and "so" must be differently explained. [PI II, viii, p.
186c]
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405. Our interest in a 'feeling' is of a quite particular kind. It includes, for instance, the 'degree of the feeling'
and the extent to which one feeling can be drowned out by another. [PI II, viii, p. 186d]
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406. "Grief" describes a pattern which recurs in the weave of our life. Now a process is also part of this pattern. If a
man's bodily expressions of sorrow and joy alternated, say with the ticking of a metronome, then this would not
result in the pattern of sorrow or of joy. (This does not mean that joy or grief are kinds of behaviour.) [Cf. PI II, i, p.
174b]
Page 55
407. If you observe your own grief, which senses do you use to observe it? A particular sense? One that feels grief?
Then do you feel it differently when you are observing it? And what is the grief that you are observing, is it one
which is there only while it is being observed?--'Observing' does not produce what is observed. (That is a conceptual
stipulation.) [PI II, ix, p. 187a]
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408. But I can still observe my grief, can't I? I ask myself, for instance, "Am I as sad today as I was yesterday?" and I
answer that question.
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409. I say (to myself), for instance: "A month ago I wouldn't have been able to think about it without shuddering."
Page 55
410. If you trained someone to emit a particular sound at the sight of something red, another at the sight of
something yellow, and so on for other colours, still he would not yet be able to describe objects by
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their colours. Though he might be a help to us in giving a description. In order to describe he has to be able to make
pictures of the way colours are distributed in space, following some rule of projection. (Language-g<ame>?)†1 [PI
II, ix, p. 187d]
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411. If I let my gaze wander round (a room) and suddenly it lights on an object of a striking red colour, and I cry out
"Red!"--that is not a description of anything; even though I could give a description. [PI II, ix, p. 187e]
Page 56
412. Are the words "I am afraid" a description of a state of mind?
It depends on the game they are in. [a: PI II, ix, p. 187f]
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413. We are, of course, presupposing certain physiological phenomena that accompany this expression of fear, for
he is supposed to be human, after all. A rapid pulse, laboured breathing, higher blood pressure, perhaps, and a series
of neurological phenomena which are more difficult to observe; all of this is in turn accompanied by certain
characteristic feelings. If someone breaks into a cold sweat then he has the sensations characteristic of sweating.
Page 56
414. And furthermore: it is quite possible that someone who imitates certain facial expressions, gestures, and sounds
that are typical of fear and who in doing so becomes subject to one or the other of the feelings typically produced by
these gestures--that this person might thereby induce further physiological manifestations of fear in his body, and
along with these have even further sensations of fear.
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415. Indeed it can happen that play-acting fear produces fear. (It is not necessarily so, and it is not essential to fear.)
Page 56
416. The language-game of reporting can be given such a turn that a report is not meant to inform us about its
subject matter but about the person making the report.
It is so when, for instance, a teacher examines a pupil. (You can measure to test the ruler.) [a: PI II, x, p. 190i;
b: PI II, x, p. 191a]
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Page 57
417. "If my senses don't deceive me, he's coming from over there."
"If I'm not mistaken, he's coming from over there."
What is the hypothetical form of this?
Page 57
418. One can very well say "It looks to me as if he's coming, but he isn't".
Page 57
419. One can mistrust one's own senses, but not one's own belief. [PI II, x, p. 190f]
Page 57
420. One can even say: "I have the impression that he is coming, but he isn't."
Page 57
421. Suppose I were to introduce some expression, "I believe", for instance, in this way: It is to be prefixed to reports
when they serve to give information about the reporter. (So "I believe" need not carry with it any suggestion of
uncertainty. Remember also that the uncertainty can be expressed impersonally: "He might come today.")
Then what would "I believe it is so and it isn't" mean? [PI II, x, p. 191b]
Page 57
422. "I believe..." throws light on my state. Conclusions about my conduct can be drawn from this expression. So
there is a similarity here to expressions of emotion, of mood, etc. [PI II, x, p. 191c]
Page 57
423. If there were a verb "to seem to believe" then it would not have a meaningful first person in the present
indicative. (Our word "to dream" could also lack this form.)
Page 57
424. The best example of an expression with a very specific meaning is a passage in a play.
Page 57
425. Instantaneous motion. If you see motion you by no means see positions at distinct points in time. You could
not copy it, or imitate it.
Page 57
426. "Back then I believed that the earth was a flat disc." A belief has a basis; the experiences, reports, relationships
on which it rests. It stands on a ground.
Page 57
427. The line "x is in error" has no real point for x = myself.
At this point the line disappears into the dark.
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Page 58
428. For instance, the question can be raised: If I infer someone's state on the basis of his utterances, is this really the
same state as one that is not recognized this way? And the answer is a decision.
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429. The phenomenon we are talking about is the dawning of an aspect.
Page 58
430. You say to yourself, for example, "It could be this too (you furnish a new interpretation) and the aspect may
dawn.
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431. Two uses of the word "see". The one: "I see this"--and allude to a description, or point to a picture or a copy.
With this I might tell someone else: over there, where you haven't been able to see it, there is such-and-such. An
example of the other use: "I see a likeness between these two faces." Let the man I tell this to be seeing the faces as
clearly as I do. [Cf. PI II, xi, p. 193a]
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432. The one man might make an accurate picture of the faces, and the other notice in the drawing their likeness,
which the former did not see. [Cf. PI II, xi, p. 193b]
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433. I might be observing two faces which do not change: suddenly a likeness lights up in them. I call this experience
the dawning of an aspect. [Cf. PI II, xi, p. 193c]
Page 58
434. Its causes are of interest to psychologists, but not to me. [Cf. PI II, xi, p. 193d]
Page 58
435. We are interested in the concept and its place among the concepts of experience.†1 [PI II, xi, p. 193e]
Page 58
436. The dawning of an aspect can be brought about, for example by (visually) going over certain facial lines.
Page 58
437. What is the characteristic expression of dawning? How do I know that somebody is experiencing this?- The
expression is like an expression of surprise.
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438. An aspect dawns and fades away. If we are to remain aware of it, we must bring it forth again and again.
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Page 59
439. I suddenly see the solution of a puzzle picture. Before, there were twigs and branches there; now there is a
human shape. My visual impression has changed and now I recognize that it has not only shape and colour but also
a quite particular organization.--My visual impression has changed;--what was it like before; what is it like now?--If I
represent it by means of an exact copy--and isn't that a good representation of it?--no change is shown. [PI II, xi, p.
196b]
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440. And above all do not say "After all, my visual impression isn't the drawing! It is this, which I can't show to
anyone." Of course, it is not the drawing, but neither is it anything of the same category, which I carry within myself.
[PI II, xi, p. 196c]
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441. So the copy cannot portray the aspect?--"Copy" means many different things.--The kind of copying can show
the aspect that one sees. It can bring together what 'belongs together'. The particular mistakes a man makes when he
is copying can also show the aspect he perceived.
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442. The concept of the 'inner picture' is misleading, for this concept uses the outer picture as a model, and yet their
uses are no more closely related than the uses of 'numeral' and 'number'. If one chose to call numbers 'ideal
numerals', one might produce a similar confusion.†1 [PI II, xi, p. 196d]
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443. If you put the organization of a visual impression on a level with shapes and colours, you are proceeding from
the idea of the visual impression as an inner object. Of course, this makes the object into a chimera, a queerly
shifting construction. For the similarity to a picture is now impaired. [PI II, xi, p. 196e]
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444. If someone sees a row of equidistant points as a row of pairs of points whose inner distance is smaller than the
outer distance, he can then say that he sees the row as organized in a certain way. For the picture he might make of
the row would have a particular organization. Of course, there might be a mistake here: he thinks that the row is
organized that way.
Page 59
445. Organization: that refers to the spatial relationship, for instance. The representations of the, spatial relationships
in a visual impression are spatial relationships in the representation of a visual impression.
Page Break 60
Page 60
By changing the spatial relations in the representation of what is seen, we can give a representation of a
change of aspect. Example: the aspects of the diagram of a cube. The copy that is drawn is always the same, but the
spatial one varies.
Page 60
446. The concept of a representation of what is seen, of a copy, is very elastic, and so together with it is the concept
of what is seen. But the two are intimately connected. (Which is not to say that they are alike.) [PI II, xi, p. 198c]
Page 60
447. If someone looking at the model of a cube were to express himself this way: "Now I see a cube in this
position--now one in this"--he could mean two very different things. Something subjective; or something objective.
His words alone do not reveal which.--An account of a change of aspect has essentially the same form as an account
of the object he saw. But its further application is different.
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448. If the aspect is a kind of organization and if the organization can be compared to the characteristics of shape
and colour, then the change of aspect is like a change of the apparent colour.
Page 60
449. The concepts of colour and form must be learned objectively.
Page 60
450. The expression of an aspect follows the expression of perception, just as the expression of an image follows the
expression of perception. But here we have to remind ourselves that a visual image cannot always be represented by
describing a visual impression. For example, I can imagine a closed box, but the picture of the closed box could also
represent several other things. (This is reminiscent of what we say in talking about a dream: "And I knew that...")
Page 60
451. Seeing an aspect is a voluntary act. We can tell someone: Now look at it like this. Try again to see the similarity.
Listen to the theme this way, etc. But does that make seeing a voluntary act? Isn't it rather the way you look at
something that causes this seeing?
For example, I can see the model of the cube in this way if I direct my glance right at these edges. When I do
this the aspect suddenly changes. Here I know how to bring this about. On the other hand, if I look at an first
one way and then another then I am unaware of this.
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Page 61
452. The aspect is dependent on the will. In this way it is like imagination. [Cf. PI II, xi, p. 213e]
Page 61
453. But visual perception is also dependent on the will, after all! If I look more closely then I see something
different and I can produce the other visual impression at will. To be sure, this does not make the impression an
aspect--but isn't it, too, subject to my will?
Page 61
454. Someone who has always taken a certain shape for a printed F need never have had the experience that is
expressed in the words: "Now I see it as an F".
This aspect has not necessarily 'dawned' on him.
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455. Someone who is looking at the d<uck>-r<abbit> and thinking about the facial expression on the rabbit--trying,
for example, to find the right word for it--this person is looking at the picture in the rabbit aspect, but this rabbit
aspect does not dawn on him.
But then is it correct to say that he sees the picture in this aspect the whole time?
Now he describes what he sees as the head of a rabbit, for that is the way he talks, for instance, about what
he sees.
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456. Do not ask yourself "How does it work with me?" Ask "What do I know about someone else?" [PI II, xi, p.
206c]
Page 61
457. Do not ask yourself "Didn't I see it in such a case?"--but rather "What makes me say that he sees it in this
case?"
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458. If I heard someone talking about the duck-rabbit, and now he spoke in a certain way about the special
expression of the rabbit's face I should say "Now he's looking at the picture as a rabbit's head", or "under the rabbit
aspect". [PI II, xi, p. 206h]
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459. The greatest danger here is wanting to observe oneself.
Page 61
460. If I say "These two shapes seemed to me to have no similarity to each other at all", can I use a stronger
expression for the fact that I saw something different each time?
Page 61
461. He sees two pictures, for instance, with the duck-rabbit surrounded by rabbits in one, by ducks in the other. He
doesn't notice that they are the same. Does it follow from this that he sees
Page Break 62
something different in the two cases?--It gives us a reason for using this expression here. [PI II, xi, p. 195e]
Page 62
462. And how about the statement "I saw it quite differently!"? Well, maybe that shows that here this concept
suggests itself to someone and that too is understandable.
So I had 'seen' it; even though this aspect had never dawned on me.
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463. And now how does this chronic†1 'seeing-as' compare with colours and shapes? Has my visual image always
had these colours, these shapes, this organization? So far it has only been a mode of expression; but just how similar
are these concepts?
Of course, we can say "There are certain things which fall equally under the concept 'picture-rabbit' and
under the concept 'picture-duck'. And a picture, a drawing, is such a thing."--But the impression is not
simultaneously of a picture-duck and a picture-rabbit. [b: PI II, xi, p. 199f]
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464. You had learned: that is 'red'; that is 'round'; that is a 'rabbit'.
Page 62
465. I learned the concepts 'red', 'round', 'picture-rabbit', 'picture-duck'--so far they are more or less on the same level.
I can learn them from samples.
Page 62
466. A picture-rabbit is something like this: and then I point to examples. So a picture-duck is something different,
even if one of the examples is the same.
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467. If I saw the duck-rabbit as a rabbit, then I saw: This shape and colour (I reproduce them exactly)--and I saw
besides something like this: and here I point to a number of different pictures of rabbits. This demonstration shows
the difference between the concepts. [PI II, xi, pp. 196h-197a]
Page 62
468. "I saw it quite differently, I should never have recognized it!" Now that is an exclamation. And there is also a
justification for it. [PI II, xi, p. 195f]
Page 62
469. All the while you would have copied this face (the imitation of a rabbit), so in one sense you did see it that way
after all.
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470. And if I see it now as a rabbit, now as a duck, then I see it this
Page Break 63
way and this way (when I say this each time I imitate a different animal and look in a different direction).
Page 63
471. What kind of man is said to be enjoying this picture's telling expression? Well, someone who looks at it this
way, talks about it in such-and-such a way, and reacts to it this way.
Page 63
472. I have always seen it as a rabbit could even mean: for me it always was a rabbit, I have always spoken to it as a
rabbit. A child does this.
It means that I have always treated it as a rabbit.
Page 63
473. Now if the child treats the picture of the rabbit like a real rabbit, does that show something about how the visual
picture is organized? Is that proof that a child sees more than just colours and shapes?
Page 63
474. And now the change of aspect.
The experience of the new aspect. Or: of the appearance of the aspect. And the expression of this is an
exclamation. A rabbit! etc.
Page 63
475. "But surely you would say that the picture is altogether different now!" [PI II, xi, p. 195i]
Page 63
476. But what is different: my impression? my point of view?--Can I say? I describe the alteration like that of a
perception; quite as if the object had altered before my eyes. [PI II, xi, p. 195i]
Page 63
477. Imagine the duck-rabbit cut out and a child treats it as a doll, now this way and now that.
Page 63
478. I am shown a picture-rabbit and asked what it is; I say "It's a rabbit". Not "Now it's a rabbit". I am reporting my
perception. I am shown the duck-rabbit and asked what it is; I may say "It's a duck-rabbit". But I may react to the
question quite differently.--If I say that it is a duck-rabbit this again is the report of a perception; but if I say "Now
it's a rabbit", then it isn't. Had I replied "It's a rabbit", I would not have noticed the ambiguity, and I should have been
reporting my perception. [PI II, xi, p. 195h]
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479. But then isn't there a difference between the first "Now it's a rabbit" and the newly developing aspect?
Page Break 64
Page 64
480. A wall covered with spots; and I occupy myself by seeing faces on it; but not so that I can study the nature of
an aspect, but because I find those shapes interesting and because of the destiny that leads me from one to the next.
More and more, aspects dawn, others fade away, and sometimes I 'stare blindly' at the wall.
Page 64
481. The double cross and the duck-rabbit might be among the spots and they could be seen, like the other figures,
and together with them, under various aspects.
Page 64
482. The aspect seems to belong to the structure of the inner materialization.
Page 64
483. We learn language-games. We learn how to arrange objects according to their colours, how to report the
colours of things, how to produce colours, compare shapes, measure, etc., etc.--Do we learn how to form mental
images out†1 of them?
Page 64
484. There is a language-game: "Tell me whether (sometimes also "how often" and "where") this figure is contained
in that one." What you report is a perception.
Page 64
485. So we could also say: "Tell me whether there is a mirror-F here", and suddenly it might strike us that there is.
This could be very important.
Page 64
486. But the report "Now I see it as..." does not report any perception.
Page 64
487. "You can think now of this, now of this, as you look at it, can regard it now as this, now as this, and then you
will see it now this way, now this." What way? There is no further qualification. [PI II, xi, p. 200d]
Page 64
488. I can change the aspects of F and in so doing I do not have to be cognizant of any other act of volition.
Page 64
489. In these considerations it is useful to introduce the idea of a 'picture-rabbit', 'picture-man', etc. For instance,
is a picture-face. [Cf. PI II, xi, p. 194c]
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Page 65
490. "Now I am seeing this", I might say. This is the report of a new perception.†1 [PI II, xi, p. 196a]
Page 65
491. But what if, to begin with, I drew exactly what I perceived; and then said: "Now I see that it is a rabbit", or "Oh,
it's a rabbit!" Now I am expressing an experience that occurs at the same time as the exclamation.
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492. The perception of an internal relation and the dawning of the aspect of an internal relation.
Someone has always seen the duck-rabbit as a rabbit and now he sees it as a duck for the first time. From
this he might learn that a rabbit's head and a duck's head can have the same contours. Under certain circumstances
this can be an important discovery. (I'm thinking of a code in which a rabbit's head is a sign.) But the dawning of the
rabbit aspect is not the perception of that relation.
Couldn't someone perceive the relation and still not be able to experience the change or the dawning of an
aspect?
Page 65
493. In one case you say: "What I have in front of me is this [copy]. I can also describe it as a rabbit."--In another
case: Before, I saw something else, but now I see a rabbit.
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494. The expression of a change of aspect is the expression of a new perception, and at the same time of the
perception's being unchanged. [PI II, xi, p. 196a]
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495. The copy completely describes the perception. The model that I am pointing to describes the kind of view I now
have of it. And one could also say: the visual experience.--In that case the copy is a more precise report of the
perception. But if the aspect dawns upon me, then its expression (for example, my pointing to the model) is
essentially the expression of a new perception.
Page 65
496. Just as if we had to have a new copy now, to correspond to this expression. But this is not the case.
Page 65
497. I ask: "What are you seeing?" The other person starts drawing; then he gives up and says "I can't draw it very
well; it's a rabbit sitting down". Then I might improve on his drawing.
Page 65
498. "I see a picture-rabbit. And that is exactly what I see [and now I'll draw it]."
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Page 66
499. Then is the copy an incomplete description of my visual experience? No. But the circumstances decide what
more detailed specifications I need to make. It may be an incomplete one; if there is still something to ask. (Example:
the schematic cube.) [PI II, xi, p. 199e]
Page 66
500. So pointing to the model, in addition to the copy, might belong to the description of the visual experience. But
then it doesn't belong to the description of the visual perception.
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501. If I know that the schematic cube has various aspects and I want to find out what someone else sees, I can get
him not only to copy the schematic cube, but also to point to a cube; even though he has no idea why. To me it
describes what he sees.†1 [PI II, xi, p. 196f]
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502. But when we have a changing aspect the case is altered. Now the only possible expression of our experience is
what before perhaps seemed, or (even) was, a useless specification when once we had the copy. [PI II, xi, p. 196f]
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503. And this by itself eliminates for us the comparison of 'organization of the visual impression' with colour and
shape. [PI II, xi, p. 196g]
Page 66
504. Indeed, I confess, nothing seems more possible to me than that people some day will come to the definite
opinion that there is no copy in either the physiological or the nervous systems which corresponds to a particular
thought, or a particular idea, or memory.
Page 66
505. What would it be like, what would it look like, if all choice were taken out of an aspect?
Page 66
506. Does "seeing an aspect" mean that one perceives the internal relation? What is there in me that speaks against
this?
Page 66
507. If you search for figure (1) in another figure (2), and then find it, you see (2) in a different way, one can say. Not
only can you give a new kind of description of it, but noticing the first figure was a new visual experience. [PI II, xi,
p. 199b]
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Page 67
508. But you would not necessarily want to say: "Figure (2) looks quite different now; it isn't even in the least like
the figure I saw before; though they are congruent!" [PI II, xi, p. 199c]
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509. "The inner picture contains colours, shapes, and, what is more, a particular organization." From this it would
follow that it looks like this and not like that.
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510. You notice an organization of an object (an object of perception). Or rather: You notice something about its
organization; a feature of this organization.
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511. Noticing is a visual experience.
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512. You can copy colour and shape. You can point to a sample of colour and shape. But you can't point to a sample
of the visual impression's organization.
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513. Someone might say, for instance: "If you want the same impression I have you have to look at this figure,
particularly at this part, and you have to look so that you notice this about it." But that isn't what we do. This sort of
thing is not what we call "describing a visual impression", just as we don't prescribe how the other person's gaze
should travel across the object when we try to describe our visual impression. This shows us <that> "visual
impression" is supposed to refer to something like "visual picture", and that this in turn is supposed to refer to
something like a picture.
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514. If you ask me what I saw, perhaps I shall be able to make a sketch which shows you, but I shall mostly have no
recollection of the way my glance travelled in looking at it. [PI II, xi, p. 199h]
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515. The colour of the visual impression corresponds to the colour of the object, the shape of the visual impression
to the shape of the object. But the aspect of the visual impression does not correspond to the organization of the
object, for the former can vary while the same organization is being looked at. In the aspect I notice a trait of the
organization.
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516. The colour in the visual impression corresponds to the colour of the object (this blotting paper looks pink to me,
and is pink)--the shape in the visual impression to the shape of the object (it looks rectangular to me, and is
rectangular)--but what I perceive in the
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dawning of an aspect is not a property of the object, but an internal relation between it and other objects. [PI II, xi,
p. 212a]
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517. Imagine the duck-rabbit hidden in a mass of lines. Now I suddenly notice it in the picture, and notice it simply
as a rabbit. At some later time I look at the same picture and notice the same figure, but see it as the duck, without
necessarily realizing that it was the same figure both times. If I later see the aspect change, can I say that the duck
and rabbit aspects are now seen quite differently from when I recognized them separately in the tangle of lines? No.
But the change produces a surprise not produced by the recognition. [PI II, xi, p. 199a]
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518. The aspect only dawns; it doesn't remain fixed. But that has to be a conceptual, and not a psychological,
remark.
The expression of seeing an aspect is the expression of a new perception.
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519. (Seemingly, I am performing 'thought-experiments'. Well, they're simply not experiments. Calculations would
be much closer.)
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520. The expression of the dawning of an aspect is: "Now it's this--now it's that." The expression of noticing the
rabbit in the tangle of lines is: "There is a rabbit here." We have not noticed something and now we do; there's
nothing paradoxical about this. We don't want to say that the old has vanished--that there's something new there,
though it's entirely the old.
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521. We don't say "Now it is this" before the first change of aspect.
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522. A hesitant assertion is not an assertion of hesitancy. [Cf. PI II, x, p. 192]
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523. Think of a hesitant command.
Page 68
524. And one should be on one's guard against saying that "It may be raining" really means "I think it'll be raining".
Why then shouldn't it be the other way round? [PI II, x, p. 192h]
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525. Aristotelian logic brands a contradiction as a non-sentence,
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which is to be excluded from language. But this logic only deals with a very small part of the logic of our language.
(It is as if the first geometrical system had been a trigonometry; and as if we now believed that trigonometry is the
real basis for geometry, if not the whole of geometry.)
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526. Don't regard a hesitant assertion as an assertion of hesitancy. [PI II, x, p. 192i]
Page 69
527. "I noticed the likeness between the two for perhaps five minutes." That might be said if they were to
change.--That would mean: I was aware of it for about five minutes, it occupied me for 5 minutes, and throughout
this time I continually had to think of it.
"It struck me for five minutes, and no longer after that." "The likeness staggered me for five minutes. I had to
exclaim again and again..." That does not mean: I observed it for five minutes and then it disappeared.
<In English> "The similarity struck me for 5 minutes."
"The similarity staggered me for 5 minutes. After that I no longer noticed it."†1 [a: cf. PI II, xi, p. 210f]
Page 69
528. "I observed this similarity for 5 minutes" would mean: I was observing the similarity of the changing faces.
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529. The organization of a visual picture: this belongs together, that doesn't. So one organizes by bringing together
and separating. Well, this can be done when drawing, for instance.
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530. There are quite different kinds of 'aspects'. One kind might be called "aspects of organization". [Cf. PI II, xi, p.
208d]
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531. The lines are connected together differently. What belonged together before doesn't belong together now.
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532. I may, then, have seen the duck-rabbit as a picture-rabbit from the first. That is to say, if asked: "What's that?"
or "What do you see here?" I should have replied "A picture-rabbit". If I had further been asked what a picture-rabbit
was, I should have had to explain by pointing to various pictures of rabbits or to real rabbits, could have talked about
their habits and given an imitation of them. [PI II, xi, p. 194d]
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533. I should not have said "I am seeing that as a picture-rabbit" or "Now I am seeing it as a picture-rabbit". I should
simply have described my perception; just as if I had said "I see a red circle there". Nevertheless someone else could
have said of me "He's seeing this figure as a rabbit". [Cf. PI II, xi, pp. 194e-195a]
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534. It would have made as little sense for me to say "Now I'm seeing it as..." as to say at the sight of a bottle of wine
"Now I'm seeing this as a bottle". This expression would not be understood. Any more than the expression from
<unharmed>†1 skin "Now it's a bottle", or "It can be a bottle too". [Cf. PI II, xi, p. 195b]
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535. Neither could one normally say "I take that to be a knife and fork".
Page 70
536. One doesn't take what one knows as a knife and fork at a meal for a knife and fork; any more than one
ordinarily tries to eat as one eats, or aims to eat. [Cf. PI II, xi, p. 195c]
Page 70
537. Does the dog who suddenly notices a rabbit think of it?
Page 70
538. Suppose someone is taking a walk and suddenly an animal crosses his path: I see him looking surprised--what
do I know about his experience?
When asked he might say "All of a sudden something startled me; I don't know what it was." Or: "Suddenly
I saw something that flitted by--that was all." Or: "It was a rabbit!"
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539. Suppose he had never seen an animal: Then would his visual experience differ from that of someone who is
familiar with the shape of the animal that flitted by? (I would like to answer in the affirmative, but I don't know
why.)
Page 70
540. The question can be put another way: Someone suddenly sees an object which he does not recognize; (it may
be a familiar object, but in an unusual position or lighting); the lack of recognition perhaps lasts only a few seconds.
Is it correct to say he has a different visual experience from someone who knew the object at once? [PI II, xi, p.
197e]
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541. Can't we imagine that someone is able to describe a completely unfamiliar shape that appears before him just as
accurately as I, to whom it is familiar? And isn't that the answer? Of course, it will not generally be so. And his
description will run quite differently. (I say, for example, "The animal had long ears"--he: "There were two long
appendages" and then he draws them.) [PI II, xi, p. 197f]
Page 71
542. Here we must be careful not to think in traditional psychological categories. Such as simply dividing experience
into seeing and thinking; or doing anything like that.
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543. One feels inclined to ask "Is recognizing a part of seeing?" And the question is wrongly put.
What are the signs of recognizing--what are those of seeing?
If someone suddenly sees his friend in a crowd and calls out his name, what is he giving a sign of?
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544. I see someone whom I have not seen for years, I see him clearly, but fail to know him. Suddenly I know him, I
see the old face in the altered one. I believe that I could do a different portrait of him now. [PI II, xi, p. 197g]
Page 71
545. Clearly there is a relationship between concepts here.
Page 71
546. Isn't it possible for someone to describe a face with which he is completely unfamiliar more accurately than I
might describe one I've known for a long time?
Page 71
547. (And here one must distinguish between the experience of recognizing something again and recognizing, which
is simply a being-familiar-to-me.)
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548. Do not try to analyse your own inner experience! [PI II, xi, p. 204e]
Page 71
549. I look at an animal in a cage. I am asked: "What do you see?" I answer: "A rabbit."--I gaze into the countryside;
suddenly a rabbit runs past. I exclaim: "A rabbit!"
Both things, both the report and the exclamation, can be called expressions of perception and of visual
experience. But the exclamation is so in a different sense from the report; it is forced from us. It is
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related to the experience as a cry is to pain. [PI II, xi, p. 197b]
Page 72
550. But isn't it simply that the exclamation, that is, the particular inflection of the words, is merely an expression of
surprise. The words themselves are the expression of the visual perception, etc., just as are the words of a report.
My surprise might also have been expressed by an inarticulate sound; and if I am asked "Why did you
flinch?", I might answer: "A rabbit crossed in front of me."
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551. Another exclamation might have been: "What was that?!"
Page 72
552. But are the two experiences whose expressions are the inarticulate sound and the exclamation "A rabbit!" really
the same? How should I decide this? (I didn't mean the same thing.)
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553. But since it (the exclamation) is the description of a perception, it can also be called the expression of a thought.
And therefore we can say that if you are looking at the object, and see it, you need not think of it; but if you are
having the visual experience expressed by the exclamation, you are also thinking of what you see. [PI II, xi, p. 197c]
Page 72
554. And that is why the experience of a change of an aspect†1 seems half visual-, half thought-experience.†2 [PI II,
xi, p. 197d]
Page 72
555. When I see a change of aspect I have to occupy myself with the object.
Page 72
556. I occupy myself with what I am now noticing, with what strikes me. In that respect, experiencing a change of
aspect is similar to an action.
Page 72
557. Does the exclamation "What was that?" express a particular visual experience?
Page 72
558. Couldn't we answer: Yes and No?
Page 72
559. "I just saw a shadow flitting by." Isn't that the expression of a visual experience?
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560. I see a 'questionable' shape.
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561. But can you really say that you see the questionableness and the shape?
Page 73
562. Question: What supports this?
Well, the fact that even the description I give of the phenomenon is moulded by the questionableness.
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563. What is the criterion of the visual experience? The criterion? What do you suppose?
The representation of 'what is seen'. [PI II, xi, p. 198b]
Page 73
564. Now when the aspect dawns, can I separate a visual experience from a thought-experience?--If you separate
them the dawning of the aspect seems to vanish.
Page 73
565. I think it could also be put this way: Astonishment is essential to a change of aspect. And astonishment is
thinking.
Page 73
566. But isn't that just MY conception of a change of aspect?
Page 73
567. So what dawns? The aspect of the rabbit for instance. And therein, that it could only be expressed that way, lay
the thought.
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568. Something flying by might surprise me bodily, as it were, and yet I needn't think about it. That is, even though I
gave a start, I could still continue a train of thought.
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569. But now think of the aspects of the rotating drum. When they change, it seems as if the movement had changed.
Here one doesn't necessarily know whether it is the kind of movement that has changed, or the aspect. And therefore
we don't in the same sense have the experience of the change of aspect.
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570. Imagine that two lights, one blue and one red, are alternately blinking in front of my eyes. I am to press one
button when the blue light flashes, another when the red light flashes. Certainly I could do this quite
mechanically.--And now imagine that this game is played with the two aspects of the black-white cross. Now is it
impossible that there be equally mechanical and thoughtless reactions?
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571. Now when I know this person in a crowd, perhaps after looking in his direction for quite a while--is this a sort
of seeing? A sort of thinking? The expression of the experience is "Look, there's ...!"--But, of course, it could just as
well be a sketch. That I recognize this person might be expressed in the sketch or in the process of sketching as well.
(But the element of sudden recognition is not expressed in the sketch.)†1
The very expression which is elsewhere a report of what is seen, is here a cry of recognition. [a: cf. PI II, xi,
p. 197h; b: p. 198a]
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572. Suppose a child suddenly recognizes someone. Let it be the first time he has ever suddenly recognized
anyone.--It is as if his eyes had suddenly opened.
One can ask, for example:†2 If he suddenly recognizes N. N., could he have the same visual experience, but
without recognizing him? For instance, he could be mistaken about recognizing him.
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573. What if someone were to ask: "So I do that with my eyes?"
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574. A rabbit runs across a path. Someone who isn't familiar with rabbits says: "Something strange just whizzed by"
and he proceeds to describe the appearance. Someone else exclaims "A rabbit!" and he cannot describe the
appearance so precisely.
Now why do I still want to say that the person who recognizes it sees it differently from the person who
doesn't?
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575. If someone sees a smile and does not know it for a smile, does not understand it as such, does he see it
differently from someone who understands it? He mimics it differently, for instance. (Understanding the modes in
ecclesiastical music.) [PI II, xi, p. 198e]
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576. What can be cited in support of his seeing it differently?
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577. "It looks different to anyone who knows what it is."--How so?
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578. What would it be like if someone just wasn't acquainted with what it was that scampered by, but still knew all
about it right away? Does he then see it in the same way as a person who is acquainted with it?
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579. It's a question of the fixing of concepts.
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580. I'm mentioning these kind of aspects in order to show the kind of multiplicity we are dealing with here.
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581. There are here hugely many interrelated phenomena and possible concepts. [PI II, xi, p. 199d]
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582. Sometimes the conceptual is dominant in an aspect. That is to say: Sometimes the experience of an aspect can
be expressed only through a conceptual explanation. And this explanation can take many different forms.
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583. The various kinds of aspects.
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584. Hearing a melody and the movements that go along with the particular way someone interprets or hears it.†1
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585. Why does it seem so hard here to separate doing and experiencing?
Page 75
586. It's as if doing and the impression didn't happen side by side, but as if doing shaped the impression.
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587. I hear it differently, and now I can play it differently. Thus I can render it differently.
Page 75
588. There are many ways of experiencing aspects. What they have in common is the expression: "Now I see it as
that"; or "Now I see it this way"; or "Now it's this--now that"; or "Now I hear it as...; a while ago I heard it as ...".
But the explanation of these "that's" and "this way's" is radically different in the different cases.
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589. What would it be like if suddenly I noticed a lion out of doors? I'll assume that I see only part of its head, but
that I recognize it immediately and cry out "A lion!" The most powerful feeling in me is fear.--And now I ask again:
What about the visual impression? Was it different from the one I receive in the zoo? (Aside from the fact that the
latter impression is much more complete.--)
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590. (I can't yet lift myself above the mass of appearances.)
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591. Here it is difficult to see that what is at issue is the fixing of concepts.
The concept forces itself on one. (This is what you must not forget.) [PI II, xi, p. 204h]
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592. The visual impression seems to organize itself in this way.
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593. This really means: the visual impression changed, and didn't change.
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594. When I suddenly recognized him my visual impression suddenly seemed to change into this.
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595. Was it a sort of understanding? Was it a sort of seeing?
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596. What, if anything, justifies my talking about seeing here?
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597. Suppose someone were to tell me: "It was as if my visual impression suddenly organized itself into this face and
its surroundings." I would understand him. I would comprehend why he was expressing himself this way. That is, I
too would be inclined to use this image.
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598. This figure is the reverse of and this: is the reverse of this:
. One is inclined to say that one sees the reverse word differently from the normal word.
The latter is easy to copy, the former difficult. [Cf. PI II, xi, p. 198g]
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599. The figure a) is the reverse of figure b) as the figure c) is the
reverse of d) . But--I should like to say--there is a further difference between my impressions
of c and d and those of a and b. (d, for example, looks neat, and c sloppy. Cf. Lewis Carroll's 'Looking glass'.) d is
easy, c hard to copy. [PI II, xi, p. 198g]
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600. What previously fell apart in a visual impression now belongs together.
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601. How would the following account do: "What I can see something as, is what it can be a picture of"?--But is
that an explanation or a pleonasm?--[PI II, xi, p. 201b]
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602. What it means is: The aspects in a change of aspects are those ones which the figure might sometimes have
statically in a picture. [PI II, xi, p. 201b]
Page 77
603. A triangle can really be standing up in one picture, be hanging in another, and can in a third be something that
has fallen over. That is, I who am looking at it say, not "It may also be something that has fallen over", but "That
pitcher has fallen over and is lying there in fragments". This is how we react to the picture. [PI II, xi, p. 201c]
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604. Could I say what a picture must be like to produce this effect? No. There are, for example, styles of painting
which do not convey anything to me in this immediate way, but do to someone else. I think custom and upbringing
have something to do with this. [PI II, xi, p. 201d]
Page 77
605. Take as an example the aspects of a triangle. This triangle can be seen as a triangular hole, as a
solid, as a geometrical drawing; as standing on its base, as hanging from its apex; as a mountain, as a wedge, as an
arrow or pointer; as an overturned object which (for example) is meant to stand on the shorter side of the right angle,
as a half parallelogram, and as various other things. [PI II, xi, p. 200c]
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606. What does it mean to say that I see the sphere floating in the air in a picture?
Is it enough that I describe the picture this way? That this description is the first to hand, is the most natural
for me? No, for it might be so for various reasons. This might, for instance, simply be the conventional description.
[PI II, xi, p. 201e]
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607. What is the expression of my not merely understanding the picture in this way, for instance (knowing that it is
supposed to be), but seeing it in this way?
It is expressed by: "The sphere seems to float", "You can see it floating", or again, in a special tone of voice,
"It floats!"
This then is the expression of taking something for something. But not being used as such. [PI II, xi, p. 201e]
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608. Here we are not asking ourselves what are the causes and what produces this impression in a particular case. [PI
II, xi, p. 201f]
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609. And is it a different impression?--"Surely I see something different when I see the sphere floating from when I
merely see it lying there."--This really means: This expression is justified! (For taken literally it is no more than a
repetition.) [PI II, xi, p. 201g]
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610. (And yet my impression is not that of a real floating sphere either. Compare the various kinds of
'three-dimensional' seeing; the three-dimensional character of a normal photograph and that of what we see through
a stereoscope.) [PI II, xi, p. 202a]
Page 78
611. "And is it really a different impression?" In order to answer this I should like to ask myself whether there is
really something different there in me. But how can I find out?--I describe what I am seeing differently. [PI II, xi, p.
202b]
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612. We can produce a change of aspect, and it can also occur against our will.
Like our gaze, it can follow our will.
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613. When you ride a bus at night and it makes a turn, if you look at the front partition (which doesn't move relative
to the passengers), you'll think that you're seeing it make the turn. You feel, of course, that the vehicle is making the
turn. And you may also have an inkling of this from the darkness outside, which you still see, although
unconsciously, out of the corner of your eye. But you think you see the front partition making the turn, and at the
same time, of course, you see that it's not moving with respect to you.
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614. (Rhees) If someone describes his present mood and says, for instance, that it is like a grey cloud--isn't he
observing it even though his observation might change it somewhat? And does what I said about 'description' in
general hold true for this description?
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615. Don't I look into myself and say: "What is the right word for this feeling, this mood?"--And is it clear that my
mood isn't intensified, for instance, by this looking?
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Isn't it possible for me to revel in a mood? And couldn't self-observation be a part of this revelling?
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616. Is this similar to causing myself physical pain (no matter how I do it) and then trying to describe exactly what
it's like?
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617. Suppose I say in a case like that: "Yes, this pain is like a blazing fire."
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618. In which way and in which sense am I observing the pain? (For I can't see any difference between someone
observing his sadness and observing his own pain.) I put myself in a position to feel it. But what pain? This kind--or
the pain produced in that way?
Do I say "I'd like to reproduce this same blazing pain so I can see what it is like?"†1 Why should I observe it
if I can identify it this way? Well, you might say: "If I could just feel this same pain again and again, then I would
finally hit upon the right word for it, or even the coloured image (for example, that of a fire)."
And now I can simplify the case. He doesn't even have to produce the pain on purpose; rather, let it be a
constant pain (a headache or stomach-ache) and let him be thinking about how to describe his feeling correctly.
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619. What I really want to say is that by looking I do not observe my visual impression, but rather whatever I am
looking at.
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620. So if I somehow look at my grief then I am not observing the impression that I thereby receive.
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621. But suppose I stare fixedly at an object and ask myself "What kind of red am I seeing there?" I'm not interested
in the colour of the object at all, but just searching (perhaps) for a name for my present impression of it.
Can I say that to think about an impression is not to 'observe it'?
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622. What does someone who says "Now I see it as..." convey to us? That is, what follows from this report, what
sort of use does it have? It could have many different kinds of consequences.
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Page 80
Well, for example anyone who sees the d<uck>-r<abbit> as a rabbit will not be able to describe the
expression of the duck.
Three-dimensional seeing in solid geometry. Anyone who sees the model of the curve as flat won't be able to
perform various graphic operations with it. [Not quite right.]
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623. Connection with the game "That could be a...".
Page 80
624. What are you telling me when you use the words...? What can I do with this utterance? What consequences
does it have?
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625. Certain drawings are always seen as flat and others sometimes, or even always, three-dimensionally. [PI II, xi,
p. 202c]
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626. Here one would not like to say: The visual impression of what is seen three-dimensionally is three-dimensional;
with the schematic cube, for instance, it is a cube. (For the description of the impression is the description of a cube.)
[PI II, xi, p. 202c]
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627. "Now I always see it as ..." Before I erroneously saw this as...; but now no longer. Now I always see it the way it
was meant.--How does this get expressed?
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628. And then it seems queer that with some drawings our impression should be a flat thing, and with some a
three-dimensional thing. One asks oneself: "Where is this going to end?" [The picture of a runner.] [PI II, xi, p.
202d]
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629. "What does this colour remind me of?"--Is a person who looks at an object and asks himself that question
observing the visual impression?†1
Page 80
630. What does anyone tell me by saying "Now I see it as ..."? What consequences has this information? What can I
do with it? [PI II, xi, p. 202f]
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631. People often associate colours with vowels. Someone might find that a vowel changed its colour when it was
repeated several times. The vowel a would be 'now blue--now red'.
"Now I am seeing it as..." might have no more significance for us than "Now a is red".
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Page 81
(Linked with physiological observations, even this change might acquire importance for us.) [PI II, xi, p.
202g]
Page 81
632. If I ask myself of what use, of what interest that report is, I remember how often it is said in aesthetic
observations:†1 "You have to see it like this, this is how it is meant", "When you see it like this, you see where it
goes wrong", "You have to hear these bars as an introduction", "You must hear it in this key", "You must phrase the
theme like this" (which can refer to hearing as well as to playing). [PI II, xi, p. 202h]
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633. The figure is supposed to represent a convex step and to be used in some kind of topological
demonstration. In this context we draw the line a through the geometric centres of the two surfaces. --Now
if anyone's three-dimensional impression of the figure were never more than momentary, and if even then he
sometimes saw it as a concave step, that might make it difficult for him to follow the demonstration. (Just as a
person who cannot see projections three-dimensionally will have a hard time with solid geometry.) (The role of
intuition in mathematics.) And if he finds that the flat aspect alternates with a three-dimensional one, that is just as if
I were alternately to show him completely different objects (now something flat, now one model, now another). [PI
II, xi, p. 203a]
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634. But the application, after all, is completely different in aesthetics and descriptive geometry. In aesthetics isn't it
essential that a picture or a piece of music, etc., can change its aspect for me?--And, of course, this is not essential
for that topological demonstration.
Page 81
635. "If I see it this way, it fits, but if I see it that way, it doesn't."
Page 81
636. A game: "It can also be ..."
Page 81
637. "But this isn't seeing!"--"But this is seeing!"--It must be possible to give both remarks a conceptual justification.
[PI II, xi, p. 203c]
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638. The question is: In what sense is it seeing? [PI II, xi, p. 203d]
Page 82
639. "Do you always see this leaf as green, that is, as long as you are looking at it and would truthfully answer the
question as to its colour by saying "green"?" Is the sense of this question clear? One answer might be: "Well, I don't
say to myelf, 'Oh, how green!' the whole time I am looking at the leaf."
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640. How is it expressed that I see this picture as a picture of snow-covered trees? That I not only know that it
represents them, but that I don't read the picture like a blue-print?--I treat it differently. (Child and doll.)
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641. If I see an animal in a picture pierced by an arrow, do I only know that the point of the arrow is connected with
its feathers, or do I see it?--I relate to these bits as I would to an arrow. That is: I don't merely say, as if I were
deciphering the diagram of a machine, "These two bits go together, and this is where a shaft goes through"; rather if
I'm asked "What did you see in the picture?" I shall answer right away: "An animal pierced by an arrow." [Cf. PI II,
xi, p. 203b]
Page 82
642. "This phenomenon is at first surprising, but a physiological explanation of it will certainly be found."--
Our problem is not a causal but a conceptual one. The question is: In what sense is it seeing? [PI II, xi, p.
203d-e]
Page 82
643. Often I see a broken contour in a drawing as complete.
Page 82
644. I see that an animal in a picture is transfixed by an arrow. It has struck it in the throat and sticks out at the back
of the neck. Imagine the picture is a silhouette.--Do you see the arrow--or do you merely know that these two bits
are supposed to be part of an arrow? [PI II, xi, p. 203b]
Page 82
645. Compare Köhler's figure of the interpenetrating hexagons. [PI II, xi, p. 203b]
Page 82
646. But this is seeing! In what respect is it seeing? [PI II, xi, p. 203d]
Page 82
647. If the picture were shown to me just for a moment and I had to describe it, that would be my description; if I
then had to draw it, I
Page Break 83
should certainly draw two identical interpenetrating hexagons, and in this respect I would not miss the mark with the
copy, even if several other things were wrong in it. [PI II, xi, pp. 203f-204a]
Page 83
648. Is it knowing or seeing?--What if it were only knowing? In which cases would I say it is only knowing? If I read
a blue-print, for example.
Page 83
649. What does it mean for me to look at a drawing in descriptive geometry and say: "I know that it continues here
but I can't see it like that"? Does it mean a lack of familiarity in 'knowing my way about'? This familiarity is certainly
one of our criteria. The criterion is a certain KIND of knowing one's way about. (Certain gestures, for instance,
which indicate the three-dimensional relations. Fine shades of behaviour.) [PI II, xi, p. 203b]
Page 83
650. You need to think of the role which pictures (as opposed to working drawings) play in our life. This role is by
no means something uniform. [Cf. PI II, xi, p. 205c]
Page 83
651. If you see the drawing as ..., what I expect from you will be pretty different from what I expect when you
merely know what it is meant to be. [PI II, xi, p. 205d]
Page 83
652. [Rem<ark> about the third person.]
Page 83
653. Proverbs are sometimes hung on the wall. But not theorems of geometry.†1 Our relation to these two things.
[PI II, xi, p. 205c]
Page 83
654. "If I see it this way, it fits this, but not that." This is a very specific language-game using the expression "to see
something this way". And the criterion for 'seeing this way' is different here from that used in descriptive geometry.
Page 83
655. What is the criterion for his seeing it that way if he says, for instance, "When I see it this way it fits this"?--His
ability, for example, to make, or to describe, or to suggest certain changes in the picture or building, etc., changes
which would have a particular effect on someone who looked at it.
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Page 84
656. What if someone were to say: The plot of a dream†1 is a strange disturbance of memory; it gathers together a
great number of memories from the preceding day, from days before that, even from childhood, and turns them into
the memory of an event which took place while a person was sleeping.
Indeed, all of us are familiar with instances in which we blend several days' memories into one.
Page 84
657. For when should I call it a mere case of knowing, not seeing?--Perhaps when someone treats the picture as a
working drawing, and reads from what it represents. (Fine shades of behaviour.) [PI II, xi, p. 204i]
Page 84
658. I immediately recognize the hexagons as such. Now I look at them and ask myself: "Do I really see them as
hexagons?"--and for the whole time I am looking at them?--And I should like to reply: I am not thinking of them as
hexagons the whole time. [Cf. PI II, xi, p. 204b]
Page 84
659. The first thing to jump to the eye in this picture is: they are hexagons. [PI II, xi, p. 204b]
Page 84
660. Someone tells me: "I saw it at once as two hexagons. And that's the whole of what I saw." But how do I
understand this? I think he would have immediately answered "Two hexagons" to the question "What are you
seeing?" Nor would he have treated this answer as one among many possibilities. In this his answer is like the
answer "An animal"--on being shown a picture of one; or "A face"--on being shown the figure . [PI II, xi, p.
204c]
Page 84
661. Immediately I recognize it as a face, and am prepared to treat it as one.
Page 84
662. Of course, I might also have seen the picture first as something different and then have said to myself "Oh, it's
two hexagons!" But that is not what happened. So the aspect would have altered. And does this prove that I in fact
saw it in a particular aspect?
(Well, as you like it!) [PI II, xi, p. 204f]
Page 84
663. "Is it a genuine visual experience?"
The question is: To what extent is it one? [PI II, xi, p. 204g]
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Page 85
664. [It is difficult to see...]
[The eye "Look how it's looking!"]
Page 85
665. "To me it's an animal pierced by an arrow." That is what I treat it as, this is my attitude to the figure. This is one
meaning in calling it a case of seeing. [PI II, xi, p. 205a]
Page 85
666. But can I say in the same sense: "To me these are two hexagons"? Not in the same sense, but in a similar
one.--[PI II, xi, p. 205b]
Page 85
667. So in this sense I only see it this way as long as I have this attitude toward it? That could be said.
Page 85
668. "This feature of the picture caught my eye."
Page 85
669. The best description I can give of what was shown me for a moment is this: ...
"The impression was that of a standing animal." So a perfectly definite description came out.--Was it seeing,
or was it a thought?
Page 85
How am I to decide? [PI II, xi, p. 204d]
Page 85
670. But do I only see the picture in this aspect so long as I have this attitude toward it?--That can be said.
Page 85
671. But couldn't one also say: "I always see it as that, so long as I never see it as anything else"?
Page 85
672. [Ref. 'descriptive geometry', etc.] "He sees it three-dimensionally and therefore he knows his way about in the
drawing as well as if he were operating in the three-dimensional model." But isn't the particular way he works within
the drawing the criterion for his seeing it three-dimensionally? (For what do I know about his impression otherwise?)
Page 85
673. But I don't see it as an animal only while I am saying this.
Neither does a body weigh something only when it is being weighed. (Conceptual determination.)
Page 85
674. To me it's a lion. How long is it a lion to me?
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Page 86
675. But wait! Do I ever really say of an ordinary picture (of a lion) that I see it as a lion? I've certainly never heard
that yet.†1
Page 86
676. And yet here I've been talking about this kind of seeing!
Page 86
677. I could say of one of Picasso's pictures that I don't see it as human. Or of many another picture that for a long
time I wasn't able to see what it was representing, but now I do. Isn't this similar to: for a long time I couldn't hear
this as of a piece, but now I hear it that way. Before, it sounded like so many little bits, which were always stopping
short--now I hear it as an organic whole. (Bruckner.)
Page 86
678. Would you understand it if I were to say "We regard the photograph, the picture on our wall, as people and
other things depicted there"? [Cf. PI II, xi, p. 205e]
Page 86
679. This need not have been so. We could easily imagine people who did not have this relation to our pictures.
(Who, for example, would be repelled by our photographs because a face without colour is sinister and ugly.) [PI II,
xi, p. 205f]
Page 86
680. We don't say "I see this as a human being" of a conventional picture of a human being. "I see it as a..." goes
together with... ("Goes together" in the technique of the l<anguage-game>.)†2
Page 86
681. I say: "We regard a portrait as a human being"--but when do we do so,. and for how long? Always, if we see it
at all (and do not, say, see it as something else)?
I might say yes to this, and that would determine the concept of regarding-as.--The question would be
whether yet another concept of seeing-as is also of importance to us: a concept of seeing-as which only takes place
while I am actually concerning myself with the picture as this object. [PI II, xi, p. 205g]
Page 86
682. The concept of noticing. I can say that I sometimes notice the similarity of this picture to..., and some such
thing; but I do not
Page Break 87
say that I sometimes notice that this photograph is a face.
I could say: A picture does not always live for me while I am seeing it. [b: PI II, xi, p. 205h]
Page 87
683. But now the question is: Is this "living" a kind of "seeing", or: what right would I have to call it "seeing"? What
relationship is there between this concept and other visual concepts?
Page 87
684. But, of course, we don't say that we 'see' the conventional picture of, say, a lion, as a lion.
Page 87
685. "Her picture smiles down on me from the wall." It need not always do so whenever I see it. But this expression
is also a justification for the other expression that I don't always 'see it that way'. [Cf. PI II, xi, p. 205h]
Page 87
686. In giving all these examples I am not aiming at some kind of completeness, some classification of all
psychological concepts. They are only meant to enable the reader to shift for himself when he encounters conceptual
difficulties. [PI II, xi, p. 206a]
Page 87
687. A child will say "Now it's a house"--this can be said even in the game where a chest is a house, in several
different ways and situations. Someone enters the room while the game is going on; he is told "Now it's a house".
This does not mean: "Now it's become a house for me", it doesn't mean the dawning of an aspect. In order for that to
happen, the tone and the situation must be of a particular kind. Here again it's a case of fine differences of behaviour.
Page 87
688. 'Fine shades of behaviour.'--When my understanding of a theme is expressed by my whistling it with the
correct expression this is an example of such fine shades.
But even if "Now it's a house" does not express the dawning of an aspect, can't it be a report of the
unchanging aspect? [a: PI II, xi, p. 207a]
Page 87
689. "He quite forgets that it is a chest; for him it actually is a house." (There are definite tokens of this.) Wouldn't it
also be correct to say of such a person that he sees it as a house? [PI II, xi, p. 206f]
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Page 88
690. And if you know how to play this game and, given a particular situation, you exclaimed with special
expression†1 "Now it's a house!" your words would be expressing the dawning of an aspect. [PI II, xi, p. 206g]
Page 88
691. But the expression in one's voice and gestures is the same as if the object had altered and had ended by
becoming this or that. [PI II, xi, p. 206i]
Page 88
692. I should like to say that what dawns here lasts only as long as I am occupied with the object in a particular way.
("See, it's looking!") (Noticing a family resemblance between this face and one which isn't here now.)--'I should like
to say'--and is it so?--Ask yourself "For how long am I struck by a thing?" For how long do I find it new? [PI II, xi,
p. 210d]
Page 88
693. Would it be correct to say of the change of aspect in the rotating drum that there is no perception that the object
stays the same? Because one really can be in doubt there as to whether the kind of motion changed.
Page 88
694. You must remember that the descriptions of the changing aspects are of a different kind in each case. [PI II, xi,
p. 207d]
Page 88
695. For the sake of brevity I shall call the aspects 'black cross', 'white cross' the principal aspects of the double cross.
Likewise I shall speak of the two principal aspects of the step.
There is a fundamental difference between these, and the aspect of the triangle as an overturned triangle, for
instance.†2
Page 88
696. The difference is contained in the description that one uses in reporting the aspect.
Page 88
697. All experiences of aspect are expressed in the form: "Now I see it as that" or "Now I see it so" or "Now it is
this--now that" or "Now I hear it as ...; I heard it before as ...". But the explanation of these 'that's' and 'so's' is very
different from case to case.
Page 88
698. Imagination is required to see a triangle as half of a parallelogram, but not to see the principal aspects of a
double cross.
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Page 89
699. The latter seem to be of a more fundamental nature than the former.
Page 89
700. You can 'see the d<uck> and r<abbit> aspects' only if you are thoroughly familiar with the shapes of those
animals; the principal aspects of the double cross could express themselves in primitive reactions of a child who
couldn't yet talk.
Page 89
701. --Those two aspects of the double cross (I shall call them the aspects A) might be reported simply by pointing
alternately to an isolated white and an isolated black cross.
One could quite well imagine this as a primitive reaction in a child who cannot yet talk.
Thus in reporting the aspects A we point to a part of the double cross.
The duck and rabbit aspects could not be described in an analogous way. [PI II, xi, p. 207f]
Page 89
702. You only 'see the duck and rabbit aspects' if you are already conversant with the shapes of those two animals.
There are no analogous conditions for seeing the aspects A. [PI II, xi, p. 207g]
Page 89
703. It is possible to take the duck-rabbit for the picture of a rabbit, the double cross for the picture of a black cross,
but not to take the figure for the picture of something that has fallen over. To see this aspect of the triangle demands
imagination. [PI II, xi, p. 207h]
Page 89
704. Whoever takes the schematic cube for a cube sees it primarily as this cube even though he can later try to see it
differently, and will sometimes succeed in so doing. (Compare with the double cross.)
Page 89
705. The aspects A are not essentially three-dimensional. A black cross on a white ground is not necessarily a black
cross lying on a white surface. You could teach someone this idea without ever showing him anything but black
crosses painted on paper; assuming that the surroundings of these crosses were to change and that the cross was the
important thing in the perception: If you have it copied, for example, it will always, or nearly always, be the cross
that is copied, etc.
The aspects A are not connected with illusion in the same way as
Page Break 90
are the three-dimensional aspects of the schematic cube.†1 [Cf. PI II, xi, p. 208a]
Page 90
706. I can imagine some arbitrary cipher--this, for instance: to be a strictly correct letter of some foreign
alphabet. Or again, as a faultily written one; and faulty in one or more of several ways: For example, it might be
slap-dash, or typical childish awkwardness, or like the flourishes in a legal document. It could deviate from the
correctly written letter in a variety of ways.--And I can see it in various aspects according to the fiction I surround it
with. And here there is a close kinship with experiencing a meaning of an isolated word. [PI II, xi, p. 210c]
Page 90
707. "I noticed the likeness between him and his father for maybe 5 minutes, and then no longer." One can say this if
his face were changing and only looked like his father's for those 5 minutes. But it can also mean that his
resemblance to his father struck me for only a few minutes, and then I forgot about it. [PI II, xi, p. 210f]
Page 90
708. "I'm no longer struck by it"--but what happens when I am struck by it? Well, I look at the face with an
expression of astonishment, not only in my mien, but maybe also in my words. But is that being struck by the
likeness? No, these are the appearances of being struck; but these are 'what happens'. 'Being struck' is a different
concept.--[Cf. PI II, xi, p. 211d]
Page 90
709. 'Thinking' and 'inward speech' (I do not say: "talking to oneself") are different concepts. [PI II, xi, p. 211f]
Page 90
710. Is being struck looking plus thinking? No. Many of our concepts cross here. [PI II, xi, p. 211e]
Page 90
711. "If you didn't experience the meaning of the words, then how could you laugh at puns?" [<In English>
Hairdresser and sculptor.]†2--We do laugh at such puns: and to that extent we could say (for instance) that we
experience their meaning.
Page Break 91
Page 91
712. Just think of the words exchanged by lovers! They're 'loaded' with feeling. And surely you can't just agree to
substitute for them any other sounds you please, as you can with technical terms. Isn't this because they are
gestures? And a gesture doesn't have to be innate; it is instilled, and yet assimilated.--But isn't that a myth?!--No.
For the signs of assimilation are that I want to use this word, that I prefer to use none at all to using one that is
forced on me, and similar reactions.
Page 91
713. For example, a word has come to carry a certain tone along with it; and I cannot, at the drop of a hat, simply
utter another word with the same emotional tone.
Page 91
714. The likeness makes a striking impression on me, for examplethen the impression fades.
It only struck me for a few minutes, and then no longer did. [PI II, xi, p. 211d]
Page 91
715. What happened here? First I looked with a strange expression at the face, and if someone had asked me "Why
are you looking at him with such interest?", I would have answered "Because he looks so much like his father".
Perhaps he's talking to me, and I'm not really paying any attention to what he's saying because all the while I am
thinking just about this similarity.--That is more or less what comes to my mind in response to the question what
happened then.
Page 91
716. But there is a heterogeneous element in this answer: "and if someone had asked me...". For that isn't anything
that 'happened' when I was struck by the similarity.--Indeed, my preoccupation is not even the same kind of thing as
my facial expression.--So what is left is just my facial expressions, my gestures, and possibly the words that I say
either to myself or to others.
Page 91
717. Being struck is related to thinking.
Page 91
718. What happened here?--What can I recall? My own facial expression comes to mind, I could reproduce it. If
someone who knew me had seen my face he would have said "Something about his face struck you just
now".--Words also occur to me, words which I say on such an occasion, out loud to myself. And that's all. And is
this what being struck is? No. [PI II, xi, p. 211d]
Page Break 92
Page 92
719. To notice, to become aware of something, to turn one's attention to something.
Page 92
720. "Do you always see this leaf as green so long as you look at it, and the colour doesn't change for you?" Does
this question have a clear meaning? An answer might be: "Well, I'm not saying 'Oh, how green it is!' the entire time."
Page 92
721. "Are you conscious of its colour the whole time?" My first reaction would be: "Certainly not!" But when and
(for) how long am I conscious of it? I don't seem to be able to say much of anything about this; I don't know which
criteria are to be applied here. Should I say: "Only as long as I think about it"?
Page 92
722. Someone tells me: "I looked at the flower, but was thinking of something else and was not conscious of its
colour." Do I understand this?--I can imagine a significant context, say his going on: "Then I suddenly saw it, and
realized it was the one which ...". [PI II, xi, p. 211b]
Page 92
723. But what about this answer: "If I had turned away then, and had been asked, I could not have said what colour
it was"?
"He looked at him without seeing him." There is such a thing. But what is the criterion for it? Well, there is a
variety of cases here. [PI II, xi, p. 211b]
Page 92
724. "Just now I looked at the shape rather than at the colour." Do not let such phrases confuse you. Above all, don't
wonder "What can be going on in the eyes or brain here?" [PI II, xi, p. 211c]
Page 92
725. "The echo of a thought insight"--one would like to say. [PI II, xi, p. 212b]
Page 92
726. "The word has an atmosphere."--A figurative expression; but quite comprehensible in certain contexts. For
example, the word "knoif" has a different atmosphere from the word "knife".†1 They have the same meaning, in so
far as both are names for the same kind of objects.
But what is one to say here? Do they or don't they have the same meaning?
Page Break 93
Page 93
727. Should I now distinguish among so and so many kinds of meaning? I do not want to do that. This kind of
classification could be useful for a particular practical purpose. Because for such a purpose one--out of the countless
possible classifications--would be more suitable than another.
Page 93
728. A botanist classifies plants. But you don't need a system of classification to show somebody how multiform
plants are and how diverse the fine distinctions among them are.
Page 93
729. I saw his face (in my mind's eye) as clearly as before--but I no longer noticed the similarity to the other one.
Page 93
730. Possibly the one similarity diminished for me and I became aware of another.
Page 93
731. Assume--as an aid to understanding--that as I look at his face, certain memories I have become more and less
vivid, and that this is responsible for the change of aspect. Then should I still say that I see now one thing, now
another?
Page 93
732. So is noticing a likeness seeing or isn't it? How am I to decide? Here we have dissimilar, yet related, concepts.
Page 93
733. By noticing the aspect one perceives an internal relation, and yet noticing the aspect is related to forming an
image.
Page 93
734. It is only if someone can do, has learnt, is master of, such-and-such, that it makes sense to say he has had a
certain experience. [PI II, xi, p. 209a]
Page 93
735. --So do we see timidity, or don't we?
The concept 'timid' can be used to describe what is visually perceived, just as the concept 'major' or 'minor'
can be used to describe the melody I hear. [Cf. PI II, xi, p. 209b, c]
Page 93
736. How could I see that a facial expression was mean, frightened, brave, if I didn't know that is was an expression,
and not the anatomy, of the animal?
But surely that only means that I cannot use these concepts to describe the object of sight, just because it has
more than purely visual
Page Break 94
reference? Might I not for all that have a purely visual concept of, say, a frightened face? (In that case, I could use a
different word.) [Cf. PI II, xi, p. 209b]
Page 94
737. I must already know†1 a lot in order to describe someone's handwriting as "childish". But can I also say: "in
order to see it as 'childish'"?
"Childish" can describe handwriting, and thus what I see, but 'childish' is not a purely visual concept.
Page 94
738. Now it is correct to say: "We could have a purely visual concept which would be completely identical to the
visual part of the concept 'vulgar' (for instance)"?
Page 94
739. Such a concept would really be comparable with 'major' and 'minor', which certainly have emotional value, but
can also be used purely to describe the structure of what is perceived. [PI II, xi, p. 209c]
Page 94
740. So 'major' and 'minor' are compared here with 'acute-angled' and 'right-angled', for instance.
Page 94
741. But wouldn't it also be correct to say that anyone who did not have our concepts of 'hesitant', 'childish', 'vulgar',
could not sense the handwriting or the facial expression the way we do, even if he had a concept which was always
applicable where 'hesitant', for example, is? So couldn't I say:†2 Both see the same thing, but they sense it
differently? Just as both may hear something in a major key, but sense if differently.
Page 94
742. Think of the expression "I heard a plaintive melody"! And now the question is: "Does he hear the plaint?" [PI
II, xi, p. 209f]
Page 94
743. And if I reply: "No, he doesn't hear it; he (merely) has a sense of it"--where does that get us? One cannot
mention a sense-organ for this 'sense'.
Some would like to reply here: "Of course I hear it!"--Others: "I don't really hear it."--We can, however,
establish differences of concept here. [PI II, xi, p. 209g]
Page 94
744. (We can draw a conceptual border-line. But where does the idea of 'sensing' the vulgar, the frightening, etc.,
come from in the
Page Break 95
first place?) We react to a hesitant facial expression differently from someone who does not recognize it as hesitant
(in the full sense of the word).--But I do not want to say here that we feel this reaction in our muscles and
joints.--No, what we have here is a modified concept of sensation. [PI II, xi, p. 209h]
Page 95
745. But what is there here that is sensation-like?
Page 95
746. "You must sense the sadness of this face." (While looking at a picture.)--
Whoever senses it often imitates the face with his own. He is impressed. The picture produces this effect in
him. I could best compare this 'sensation' to the sensation of pain, which also has a characteristic expression within
the repertory of facial expressions and gestures.
Page 95
And yet it too is related to seeing because it (?) -- -- --
Page 95
747. What is the expression, the criterion, for this sensation? Surely the way, for example, or the kind of expression
with which someone will sing a melody he's just heard. Also, perhaps, the kind of face he has then. Or: what he will
say about it. That is, the particular description he gives of it.
Page 95
748. But the truth of the matter is: 'Wailing' is not a purely acoustical concept. But I can use it to describe what is
purely acoustical. ("The steam whistle makes a wailing sound. ") The word "wailing" could also lose all of its
non-acoustical relations and become a purely acoustical term. (As with the words <in English> "to travel" and
"travailler", which were originally related to very painful things, a relation which they then lost.)
Page 95
749. Now one could object to the term "purely acoustical".
Who says what the "purely" acoustical is?--Well, "purely acoustical" is a description that applies when you
can reproduce exactly what you've heard, leaving all other relations out of it.
Page 95
750. After all, I can describe a chair using the concept "style of Louis XIV", and I can contrast this with a description
which takes note of the shape, colour, etc., by using drawings, etc., and does not refer to any historical period, or
king, etc.
Page Break 96
Page 96
751. Suppose someone asked: "Do you see the style Louis XIV when you look at the chair?"
Page 96
752. It is hard to understand and to represent conceptual slopes.
Page 96
753. But we can answer the question, "What does a chair in the style of Louis XIV look like?"--or, "What does a
plaintive melody sound like?"--Show me such chairs, sing me such melodies!
Page 96
754. The epithet "sad", as applied, for example, to the outline face, characterizes the grouping of lines in a circle.
(Major, minor.) Applied to a human being it has a different, though related, meaning. (But this does not mean that a
facial expression is like the feeling of sadness!) [PI II, xi, p. 209d]
Page 96
755. Think of this too: I can only see, not hear, red and green--but sadness I can also hear in his voice as much as I
can see it in his face. [PI II, xi, p. 209e]
Page 96
756. Untangling many knots, that is the philosopher's task.
Page 96
757. This face is impudent, this face disgusts me, the smell is repulsive. Is repulsiveness part of the sensation of
smell? How is this to be decided? One might ask, for instance: "Can two people have the same sensation of smell,
but one of them find it repulsive, the other not?"--And what would the criterion for sameness be?--They could
compare this smell with the same smells, for instance.--But here there is no accepted criterion.
So do I see the impudence? Yes and No. Both answers can be justified.
Page 96
758. You don't need any knowledge to find a smell repulsive.
Page 96
759. "Look, if you draw these lines the face turns sad." In what category does this sentence belong? How is it used? I
once said that it was like a geometric proposition. But one could be of the opinion that it was a psychological, and
therefore an empirical, proposition. (Comparable, for example, to: If you add these ingredients, the substance turns
yellow.)
Page Break 97
Page 97
760. One might say to a child, for example, "Look, when you put these two stones together you get a circle".
Is he learning an empirical proposition? (Here I am purposely talking about a child, not an adult.)
Page 97
761. (Couldn't the sentence again fall 'between several games'?)
Page 97
762. That proposition would not have to be a geometrical one. Its purpose could be to confirm that the face
composed of these lines now gives me the impression of sadness. But again, it could more or less play the role of a
geometrical (timeless) proposition.
Page 97
763. One might say of someone that he was blind to the expression of a face. Would his eyesight on that account be
defective?
That is, of course, not simply a question for physiology. Here the physiological is a symbol of the logical. [PI
II, xi, p. 210a]
Page 97
764. 'He has the eye of a painter', 'the ear of a musician'.
Page 97
765. Now is it simply a conceptual shift if we refer to sensing an expression as seeing, just as we speak of marrying
money? Is there merely a misunderstanding here, or a gradual sloping of the concept 'see'?
Page 97
766. A person who had seen only one facial expression couldn't have the concept 'facial expression'. 'Facial
expression' exists only within a play of the features. Someone who had only seen 'sad' faces could not sense them as
sad.
Page 97
767. But surely he could see them just as you and I do.--But the word "sense" still isn't unobjectionable.--What do I
perceive via sensation? In addition to the so-called sadness of his facial features, do I also notice his sad state of
mind? Or do I deduce it from his face? Do I say: "His features and his behaviour were sad, so he too was probably
sad"?
Page 97
768. I believe this question belongs here: Does 'sad music' make us sad? Yes and No, it seems. We make a sad face,
for instance, or at any rate a face that reflects sadness.
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Page 98
769. One sees sadness insofar as one sees a person's sad facial expression, for instance, but surely one doesn't see
the sad ring of his voice.
Page 98
770. Indeed, we do see weeping. Now does a person who observes only the physiological phenomenon see if
differently from a person who sees an expression of grief in it?--He observes it differently.
Page 98
771. Indeed, I am inclined to ask: Do I have so much as an excuse to be talking about a different kind of 'seeing'
there?
Page 98
772. Well, what would indicate that he sees it differently? Only his attitude toward it.
And to be sure: Whoever observes differently also sees something different.
Page 98
773. Suppose someone were to ask you quite straightforwardly and in all seriousness "Why do you say he sees it
differently?" (What could you answer?)
First of all I'd be inclined to say "He is looking at something different", then maybe "He will draw different
comparisons". Indeed, maybe even the very fact that the person is not crying or wailing makes his face look sadder.
Page 98
774. I hear a melody completely differently after I have become familiar with its composer's style. Previously I might
have described it as happy, for example, but now I sense that it is the expression of great suffering. Now I describe it
differently, group it with quite different things.
Page 98
775. If you feel the gravity of a tune, what are you perceiving? Nothing that could be explained by reproducing what
you heard. [PI II, xi, p. 210b]
Page 98
776. How could I recognize a facial expression if I didn't know that it was an expression, and not the anatomy of this
animal?
How could I see sadness, gravity, cruelty in the face, without knowing that?
Page 98
777. Imagine a physiological explanation of this experience. Let it be this: when we look at the figure, our eyes scan
it repeatedly, always following a particular path. This path corresponds to a particular periodic movement of the
eyeballs. It is possible to jump
Page Break 99
from one such pattern to another and for the two to alternate (double cross). Certain patterns of movement are
physiologically impossible, hence I cannot see the duck-rabbit as the picture of the head of a rabbit superimposed on
the head of a duck, nor can I see the schematic cube as the picture of two interpenetrating prisms. And so on.--Let's
assume that this is the explanation.--"Yes, now I know that it is a kind of seeing." You have now introduced a new, a
physiological criterion for seeing. And this can screen the old problem from view, but not solve it.--The purpose of
this remark is to bring before your view what happens when a physiological explanation is offered. The
psychological concept hangs out of reach of this explanation. And this makes the nature of the problem clearer. [PI
II, xi, p. 212c]
Page 99
778. The question now obtrudes: Could there be human beings who could not see something as something?--Or:
What would it be like if a person lacked this capacity? What sort of consequences would it have? Would this defect
be comparable, say, to colour-blindness or to not having absolute pitch? We will (for now) call it
"aspect-blindness"--and will next consider what might be meant by this. (A conceptual investigation.) [PI II, xi, p.
213f]
Page 99
779. Ought he therefore to be able to see the schematic cube as a cube, for example? It would not follow from that
that he could recognize it as a representation (a working drawing, for instance) of a cube. But it would not jump
from one aspect to the other. Question: Could he take it as a cube, as we do? If not, then that will not be called
blindness.
He will have an altogether different relationship to pictures from ours. (And these kinds of deviations from
the norm are easy to imagine.) [PI II, xi, pp. 213g-214a, b]
Page 99
780. Is he supposed to be blind to the similarity between two faces? And so also to their identity or approximate
identity? I wouldn't want to say this.--We would call a person who couldn't perceive the identity of shapes
"feeble-minded", not "blind". [Cf. PI II, xi, p. 213f]
Page 99
781. The aspect-blind man is supposed not to see the aspects A change. But is he also supposed not to recognize
that the double cross contains a black cross? So if told "Show me figures containing a black cross among these
examples" will he be unable to manage it?
Page Break 100
No, but he will simply not be supposed to say: "Now it's a black cross on a white ground!" [PI II, xi, p. 213f]
Page 100
782. We say that someone has 'the eye of a painter' or 'the ear of a musician', but anyone lacking these qualities
hardly suffers from a kind of blindness or deafness.
Page 100
783. We say that someone doesn't have a 'musical ear', and 'aspect-blindness' is (in a way) comparable to this sort of
inability to hear. [Cf. PI II, xi, p. 214c]
Page 100
784. The importance of the concept 'aspect-blindness' lies in the kinship of seeing an aspect and experiencing the
meaning of a word. For we want to ask: "What are you missing if you do not experience the meaning of a
word?"--If you cannot utter the word 'bank' by itself, now with one meaning, then with the other, or if you do not
find that when you utter a word ten times in a row it loses its meaning, as it were, and becomes a mere sound.†1 [PI
II, xi, p. 214d]
Page 100
785. The report "The word... was crammed full of its meaning" is used quite differently, has quite different
consequences, from "It had the meaning...".
Page 100
786. "How does the chemist know that there is an Na atom at this point in the structure?"
Compare: "How does Mr N. know that there is an Na atom at this point, etc.?"--The answer could be: "A
chemist told him that."
The question "How does the chemist know..." is the typical way the question about the criterion is
expressed.
Page 100
787. Think here of a special kind of illusion which throws light on these matters.--I go for a walk in the environs of a
city with a friend. As we talk, it comes out that I am imagining the city to lie on our right. Not only do I have no
conscious reason for this idea, but some quite simple consideration was enough to make me realize that the city lies
behind us. I can at first give no answer to the question why I imagined the city in this direction. I had no reason to
think it. But
Page Break 101
though I see no reason still I seem to see or surmise certain psychological causes for it. In particular, certain
associations and memories. For example, we are walking along a canal, and once before I had followed a canal
which lay in the direction I had imagined. I might as it were psychoanalytically investigate the causes of my
conviction. [PI II, xi, p. 215d]
Page 101
788. "But what is this queer experience?"--Of course, it is not queerer than any other; it simply differs in kind from
those experiences which we regard as the most fundamental ones, our sense-impressions, for instance. [PI II, xi, p.
215e]
Page 101
789. But how is a person who feels that the city is located in this direction to express his experience correctly? Is it
correct, for example, to say that he feels it? Should he really coin a new word for it? But then how could anyone
learn this word? The primitive expression of the experience couldn't include it. He would probably be inclined to
say "I feel as if I knew that the city lay over there". Well, the very fact that he says this, or something like it, in these
circumstances is itself the expression of this singular experience.
Page 101
790. The name, the picture of its bearer.
Page 101
791. "I feel as if I knew the city lay over there."--"I feel as if the name Schubert fitted Schubert's works and his face."
[PI II, xi, p. 215f]
Page 101
792. An investigation is possible in connection with mathematics which is entirely analogous to the philosophical
investigation of psychology. It is just as little a mathematical investigation as the other is a psychological one. It will
not contain calculations, so it is not, for example, logistic. It might deserve the name of an investigation of the
"foundations of mathematics". [PI II, xiv, p. 232b]
Page 101
793. I say the word "march" to myself and 'mean' it at one time as an imperative, at another as the name of a
month.†1 And now say "March!", and then "March no further!" Are you certain that the same experience
accompanies the word both times? [PI II, xi, p. 215g]
Page Break 102
Page 102
794. A person imagining something could express himself primitively in this way: "I feel as if I see... before
me".--Now can it be said that he is calling something "seeing" which actually is not seeing, but perhaps only
something like it?
Page 102
795. Given the two words "fat" and "thin"--would you rather be inclined to say that Wednesday was fat and Tuesday
thin, or that Tuesday was fat and Wednesday thin? (I incline to choose the former.) Now have "fat" and "thin" some
different meaning here from their usual one? They have a different use. So ought I really to have used different
words? Certainly not that. I want to use these words (with their familiar meanings) here. Now I say nothing about
the causes of this phenomenon. They could be, for instance, that when I was a child, I was taught by a fat teacher
every Wednesday, by a thin one on Tuesdays. But that is a hypothesis. Whatever the explanation--the inclination is
there.†1 [PI II, xi, p. 216c]
Page 102
796. If you asked him "What do you really mean here by 'fat' and 'thin'?", he could only explain it in the usual way.
He could not point to Tuesday and Wednesday and use them to clarify what he means. [PI II, xi, p. 216d]
Page 102
797. Could one speak here of a 'primary' and 'secondary' meaning of a word?--In both cases the explanation of the
word is that of its primary meaning. It can only have a secondary meaning for someone if he knows its primary
meaning. That is, the secondary use consists in applying the word with this primary use in new surroundings. [Cf.
PI II, xi, p. 216e]
Page 102
798. To this extent one might want to call the secondary meaning 'metaphorical'.
Page 102
799. But the relationship here is not like the one between 'cutting off a piece of thread' and 'cutting off someone's
speech', for here one doesn't have to use the figurative expression. And if you say "The vowel e is yellow", the word
yellow is not used figuratively.
Page 102
800. Only children who know about real trains are said to be playing trains. And the word trains in the expression
"playing
Page Break 103
trains" is not used figuratively, nor in a metaphorical sense.
Page 103
801. If a person says he is calculating in his head, is he not really calculating, does he mean something else by
calculating? You could never explain to a person what is meant by "calculating in the head" if he hadn't already been
taught the concept calculating.
Page 103
802. Only by using the concept of calculation (in writing or out loud) can we get someone to grasp what
"calculating in the head" means. [Cf. PI II, xi, p. 216f]
Page 103
803. I could not explain to anybody the meaning of the command to read something silently, or the report that he
had read it silently, if I hadn't first taught him the concept of reading out loud. And this impossibility is a logical one.
Page 103
804. Only if someone has learned to calculate, on paper or out loud, can he be made to grasp, by means of this
concept, what calculating in the head is. [PI II, xi, p. 216f]
Page 103
805. But think of the pictures which show a face from the front and in profile at the same time. One might say:
"That's not what a face looks like!" But one might also say: That is a misleading picture unless you let your eye roam
so that you no longer see it as one picture, in the normal sense of the word, but as several pictures, each of which
has its own application.
Page 103
806. The brain looks like a writing, inviting us to read it, and yet it isn't a writing.
Suppose humans became more intelligent the more books they owned--suppose that were a fact, but that it
didn't matter at all what the books contained.
Page 103
807. Is scientific progress useful to philosophy? Certainly. The realities that are discovered lighten the philosopher's
task, imagining possibilities.†1
Page 103
808. "When these words were spoken I saw him in my mind's eye." Isn't that an experience? And yet the fact that I
saw him can't have been in the picture that was in my mind's eye. So was there a picture and a thought there? And
was the picture an experience, but not the thought?
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Page 104
809. We 'experience' the expression of thought.
Page 104
810. I can't call the thought an experience, for then I would have to say that this experience, for example,
accompanies speaking.
Page 104
811. "But how would you know that it was he whose picture was in your mind?"--I didn't know it. I said it.
Page 104
812. If I say that I experience the expression of a thought then I must also understand "expression" as including
imagined expression.
Page 104
813. The purpose of a sign.--"If you want him to come, wave your hand this way." "If you want me to stop, make
this sign."--Then we can speak, for instance, of a 'purpose' of negation (of the word "not")?
We could do this only if every sentence in which it is used had a purpose.--Still, we could talk of (the)
purposes of the word "not".
Page 104
814. And one could say, for instance: "non" and "ne" generally serve the same purposes, and also: "This word has
virtually no purpose at all. You can manage quite well without it."
Page 104
815. Someone, for example, who constructs an artificial language (Esperanto, Basic English) will select its words
according to certain points of view, and we could in turn consider our own language from these points of view.
He might say, for instance,: "I won't allow two words, one for "walk", and one for "stride", since for all
important purposes one is enough." And therefore he might also say: "'walk' and 'stride' have essentially the same
meaning."
Page 104
816. Language can be observed from various points of view. And they are reflected in the respective concepts of
'meaning'.
Page 104
817. "While this was going on I thought of him." What does thinking of him consist in? How would what happened
then be changed if instead of thinking of THIS PERSON I had thought of someone else?
In general did I have to be able to mention a 'germ' which then grew into a verbal expression? No.
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Page 105
818. "Who did you mean when you spoke about 'a friend'?"--"I meant...." As you were saying these words, what
happened which turned them into an allusion to this man? Nothing happened which turned them into it. For even if
I had had a picture of him in my mind while I was speaking, complete with details (or whatever you want to
substitute for this picture), that wouldn't have accomplished any more than if I had looked at him when I was
speaking. And looking at him is not the same as meaning him. There are signs which show that I meant him, and a
glance could have been such a sign. An idea too is no more than such a sign.
Page 105
819. Compare the question "What happened when in saying this word you thought of him?" with "What happened
when you suddenly knew how to go on?"
Page 105
820. Meaning is not a process which accompanies words. For no 'process' could have the particular consequences of
meaning. [PI II, xi, p. 218d]
Page 105
821. If the words "my friend" meant him, would I necessarily have to think of him as I was uttering them? What is
the difference? But there is a difference between "I meant him when I said the word" and "He came to mind when I
said the word".
Page 105
822. There are important accompanying phenomena of talking which are often missing when one talks without
thinking. But these are not the thinking. [PI II, xi, p. 218e]
Page 105
823. I thought about this man--but certainly not about all of his aspects.
Page 105
824. The garden of this aunt of mine was in my mind. I saw part of it in my imagination, but not the fact, for
example, that it belonged to this woman.
There was something like a sign there, which I then interpreted in this way. Or did I read it?
No, it isn't reading, but neither is it interpreting.
Page 105
825. "Now I know!" What went on here?--So didn't I know when I declared that now I knew?
You are looking at it wrong.
(What is the signal for?) [PI II, xi, p. 218f]
Page Break 106
Page 106
826. And could the 'knowing' be called an accompaniment of the exclamation? [PI II, xi, p. 218f]
Page 106
827. (The germ could have been a word or an image-picture or various other things.)
Page 106
828. "The word is on the tip of my tongue." What is going on in my consciousness? That is not the point at all.
Whatever did go on was not what I meant by those words. It is of more interest what went on in my behaviour.
What I said, which pictures I used, my facial expression.--"The word is on the tip of my tongue" is a verbal
expression of what is also expressed, in a quite different way, by a particular kind of behaviour. Again, ask for the
primitive reaction that is the basis of the expression. [Cf. PI II, xi, p. 219c]
Page 106
829. Intention is not expressed in mien, gesture, or voice, but resolve is.
Page 106
830. For many words philosophers devise an ideal use, which then turns out to be worthless.
Page 106
831. "I know..." usually means "I have ascertained that...". Nobody says he has ascertained that he has two hands.
Page 106
832. I know how to ascertain that I have two coins in my pocket. But I cannot ascertain that I have two hands,
because I cannot doubt it.
Page 106
833. But what is the meaning of "ascertain something"? To understand this you have to run through some simple
language-games with this word.--How does someone ascertain in language-game 8†1 that there are a certain number
of tiles over there? How does one ascertain that 6 + 6 = 12? Etc.
Page 106
834. We say "I know..." where there can be doubt, whereas philosophers say we know something precisely where
there is no doubt, and thus where the words "I know" are superfluous as an introduction to the statement.
Page 106
835. The same thing is true here as with the syllogism "All men are
Page Break 107
mortal; Socrates is a man; etc." It is not clear how and under what circumstances this could be used.
Page 107
836. How could the visual impression of someone reading a printed page, for example, be described?
Page 107
837. "Yes, now I know what 'tringling' is." (He has perhaps had an electric shock for the first time.)--If he feels the
same thing another time maybe he'll look for the same events to go with it. Tringling teaches him about the external
world. Does remembering teach us in the same way that a certain event took place in the past?--Then we would have
to connect it up with past events. (Photography and fashions.) Whereas it is really the criterion for the past. [Cf. PI
II, xiii, p. 231c]
Page 107
838. And how will he know again in the future what remembering feels like? [PI II, xiii, p. 231c]
Page 107
839. How does he know that this feeling is 'remembering'? Compare "Yes, now I know what 'tringling' is" (he has
perhaps had an electric shock for the first time). Does he know that it is memory because by means of it he
recognizes the past? And how does he know what the past is? Man learns the expression of the past by
remembering. [PI II, xiii, p. 231c]
Page 107
840. On the other hand one might speak, for example, of a feeling "Long, long ago", for there is an expression of
voice and mien which goes with narratives of past times. [PI II, xiii, p. 231c]
Page 107
841. James is really trying to say: "What a remarkable experience! The word is not there yet, and yet in a certain
sense is there, or something is there, which cannot grow into anything but this word."--But this is not experience at
all. The words "It's on the tip of my tongue" are not the expression of an experience and James merely gives them
this strange interpretation. [Cf. PI II, xi, p. 219d]
Page 107
842. They express an experience no more than the words "Now I've got it!"--We use them in certain situations and
they are surrounded by behaviour of a special kind, even by some characteristic experiences. In particular they are
frequently followed by finding the word. (Ask yourself: "What would it be like if human beings never
Page Break 108
found the word that was 'on the tip of their tongue'?") [PI II, xi, p. 219e]
Page 108
843. Here, as in many related cases, there is something we might call a germinal experience: an image, a sensation,
which grows little by little into a full-fledged explanation. And one feels inclined to say that it is a logical germ,
something which had to develop the way it does out of logical necessity.
On a particular occasion I suddenly think of a certain person. How did it happen?--I saw a picture before me,
maybe only grey hair then I said, I see N. before me (but it is still possible that many other people have that
name).--But then I explain I meant that N. who..., etc.--Moreover, I didn't read the name in the mental image, and
neither did I subsequently interpret it thus and so. For if I'm asked whether it was only later that I knew or decided
to whom the grey hair and the name N. belonged, then I shall say no, that I knew it from the beginning. But knowing
is not an experience.--"I knew it from the beginning" really only means: I didn't read the name off the picture, for I
didn't think "Whose hair is this?" or "Who looks like that?"--nor did I say to myself "For now, let's let the name 'N'
stand for this person". It could be said that I became more and more explicit.
But then where does the idea of the logical germ come from? Which really means: Whence the idea that
"Everything was already there from the beginning and was contained in the initial experience"? Isn't the reason for
this similar to James's claim that the thought is already complete when the sentence begins? This treats the intention
like an experience. [c: cf. Z 1]
Page 108
844. I advance from explanation to explanation. But I only seem to say what was there from the beginning. Of
course. For "It hasn't been there from the beginning" would be wrong.
"The thought is not complete from the very beginning" means: I didn't find out or decide until later what I
wanted to say. And that I do not want to say.
Page 108
845. To be sure, the impression that this experience is a germ results from a logical process. It does become a germ,
in a logical sense. As a result of a logical interpretation.†1
Page 108
846. Couldn't I also say this: It is completely irrelevant that the grey hair was present in my mind first, and then the
name. I could just as well have thought of the name first.
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Page 109
847. From the very beginning I knew who it was. "I didn't know it from the very beginning" would mean: I didn't
find it out until later. And that's certainly not the way it was.
Page 109
848. When I'm writing, walking, eating, talking, gazing here and there (normally), I no more try to perform these
actions than the face of an old friend 'strikes me as familiar'.
But attempting, deciding, are acts of volition, the ways in which the will manifests itself to us; they are what
we think of when we speak of the will.
Page 109
849. (Similarly, I think, it could be said: A multiplication is not an experiment, for no experiment could have the
peculiar consequences of a multiplication.) [PI II, xi, p. 218d]†1
Page 109
850. But doesn't the word that occurs to you somehow 'come' in a special way? Just attend and you'll see!--Careful
attention is no use to me. All I could discover with it would be what is now going on in me.
And how can I pay attention to it at all while I am philosophizing? To do this, I would have to wait until a
word occurred to me (once) again. This, however, is the queer thing: it seems as though I do not have to wait on
such an occasion. As if I could give myself an exhibition of it, even if it doesn't really happen to me. How?--I act
it.--But what can I learn in this way? What do I mimic?--Gestures, faces, a tone of voice. (This remark can be applied
quite generally.) [PI II, xi, p. 219a]
Page 109
851. -- -- -- Interpreted as experience, it does indeed look odd. (As does 'meaning' interpreted as the accompaniment
of speech, or like - 1 interpreted as a cardinal number.)†2 [PI II, xi, p. 219d]
Page 109
852. Silent speech 'within' is not a half-hidden phenomenon, one that is difficult to see clearly,†3 and which we must
now strive to see more clearly, saying as much about it as we know.--It is not hidden at all, but the concept is
confusing.†4
Page Break 110
Page 110
We can call it an articulated event: for it takes place within a stretch of time, and can accompany an 'outer'
event.
(The question whether movements of the larynx, etc., occur always or mostly in connection with internal
speech may be of great interest, but not for us.) [a, c: PI II, xi, p. 220a]
Page 110
853. I should not say "speaking silently to myself", for one can speak internally without speaking to oneself.
Page 110
854. Imagine this game--I call it "tennis without a ball": The players move around on a tennis court just as in tennis,
and they even have rackets, but no ball. Each one reacts to his partner's stroke as if, or more or less as if, a ball had
caused his reaction. (Manoeuvres.) The umpire, who must have an 'eye' for the game, decides in questionable cases
whether a ball has gone into the net, etc., etc. This game is obviously quite similar to tennis and yet, on the other
hand, it is fundamentally different.
Page 110
855. But there is a difference here: Only someone who can speak can speak in his imagination. Because part of
speaking in one's imagination is that what I speak silently can later be communicated.--On the other hand, tennis
without a ball could (theoretically) be learned by someone who wasn't familiar with the other kind of tennis. [Cf. PI
II, xi, p. 220b]
Page 110
856. "But speaking silently is surely a certain activity which I have to learn!" Very well; but what is 'doing' and what
is 'learning' here?
Let the use of words teach you their meaning! [PI II, xi, p. 220c]
Page 110
857. "So I don't really calculate when I calculate in my head?!" After all, you yourself distinguish between
calculation in the head and perceptible calculation! And you can't have the former concept if you don't have the
latter, and you can only learn the former activity by learning the latter. (Their concepts are as closely related and as
distantly separate as the concepts of a cardinal number and a rational number.) [PI II, xi, p. 220d]
Page 110
858. You could learn to calculate in your head to the beat of a metronome. [Cf. PI II, xi, p. 220b]
Page 110
859. Not every creature that can express fear, joy, or pain can feign them.
Page Break 111
Page 111
860. It might be like this: An eye can smile only in a face, but only in the entire figure can it -- -- --
Page 111
861. It takes a very specific context for something to be an expression of pain; but the pretence of pain requires an
even more far-reaching particular context. [Cf. Z 534]
Page 111
862. For pretence is a (certain) pattern within the weave of life. It is repeated in an infinite number of variations.
A dog can't pretend to be in pain, because his life is too simple for that. It doesn't have the joints necessary
for such movements.
Page 111
863. But you can portray a pretender on the stage. So there is such a thing as an appearance of pretence, and it is
much more complicated than the appearance of suffering, for instance. Otherwise pretence could never be exposed.
Page 111
864. It is conceivable that one might consciously do one's calculating in one's larynx, as one can calculate on one's
fingers, for example. Then do you want to say that it is deception when they imagine they hear themselves speak
inwardly, or again a mere trick of language? [Cf. PI II, xi, p. 220e]
Page 111
865. The hypothesis that certain physiological events take place when we speak silently is only of interest to us in
that it points to a possible use of the report "I said silently to myself..."; namely, that of inferring the physiological
process from the expression. [Cf. PI II, xi, p. 220f]
Page 111
866. What does a child have to learn before he can pretend?
Well, for example, the use of words like: "He thinks I'm feeling pain, but I'm not."
Page 111
867. A child discovers that when he is in pain for instance, he will get treated kindly if he screams; then he screams,
so as to get treated that way. This is not pretence. Merely one root of pretence.
Page 111
868. A child has to learn all sorts of things before he can pretend. [Cf. PI II, xi, p. 229b]
Page 111
869. He has to learn a complicated pattern of behaviour before he can pretend or be sincere.
Page Break 112
Page 112
870. A dog cannot be a hypocrite; but neither is it sincere. [PI II, xi, p. 229b]
Page 112
871. A child also leans to mimic pain. He learns the game: acting as if you were in pain.
Page 112
872. "Once a child knows what pain is, naturally he knows that it can be feigned."
Page 112
873. "... And one day the child believes something." Why is that wrong? "One day he says 'I believe...'" is right.
"Today for the first time he believed something." Well, what's involved in that?--today simply was the first time that
occurred within him.--But how did it become manifest? Well, today for the first time he said "I believe she's in pain".
But that's not enough. So I must assume that in what followed he showed that he hadn't simply repeated
somebody's words. In short, that his utterance was the beginning of a game, and that he was able to continue with it.
Today, so it seemed, the game had become clear to him.
But how can a language-game suddenly become clear to a child? God only knows.--One day it starts doing
something. An analogue might be the child learning a board game which he sees played daily.
Page 112
874. He not only learns the use of the expression "to be in pain" in all of its persons, tenses, and numbers, but also in
connection with negation and the verbs of opinion. For: believing, doubting, etc., that someone is in pain are the
natural ways we behave toward others. (He learns "I believe he is in...", "He believes I am in...", etc.., etc.--but not "I
believe I'm in.")
(Does space have a gap there? No, it only seems to.)
Page 112
875. Does the word 'pain' change its meaning in this process?
Page 112
876. 'Feigning' poses no problem with the concept of pain. It makes it more complicated. (Use of money.)
Page 112
877. The uncertainty whether someone else... is an (essential) trait of all these language-games. But this does not
mean that everyone is hopelessly in doubt about what other people feel.
Page 112
878. The parts of a machine are elastic, indeed, flexible. But does this mean that there really isn't any mechanism at
all, since the parts of the machine function as if made of butter?
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Page 113
(And now think of mechanisms, say clock-works, made of materials which are far more flexible still than
ours, so that the movements would be strangely irregular--would a mechanism like that have to be useless, or
couldn't it actually be used?)
(And, of course, it is not because they are practical that we have the concepts we do. Or at most only a few
of them exist for this reason.) [c: cf. Z 700]
Page 113
879. Imagine that uncertainty is introduced into a game! That could happen in many different ways. Imagine this:
[tennis without a ball]. If you found this game played, would you say that it wasn't a game at all? Well, compared to
our games it would be of a far different sort. <In English> (It takes many kinds...)
Page 113
880. That what someone else says to himself is hidden from me, unless he tells it to me, is part of the concept 'inner
speaking'. Only "hidden" is the wrong word here; for if it is hidden from me it ought to be apparent to him, he would
have to know it. But he does not 'know' it, even though my doubt does not exist for him. [PI II, xi, pp. 220g-221a]
Page 113
881. "I know what I want, wish, believe, hope, see, etc., etc." (through all the psychological verbs) is either
philosophers' nonsense or at any rate not a judgment a priori. [PI II, xi, p. 221c]
Page 113
882. "I know..." may mean "I do not doubt..."--but does not mean that the words "I doubt" are senseless†1 here, that
doubt is logically excluded. [PI II, xi, p. 221d]
Page 113
883. One says "I know..." where one can find out. [PI II, xi, p. 221e]
Page 113
884. It is possible to imagine a case in which I could find out that I had two hands. Normally, however, I could not
do so. "But all you need is to hold them up before your eyes."--If I could doubt now that I have two hands, then I
wouldn't have any reason to believe my eyes. (I might just as well ask a friend.) [PI II, xi, p. 221f]
Page 113
885. "His pains are hidden from me" would be like saying "These sounds are hidden from my eyes."
Page Break 114
Page 114
886. The uncertainty in which all his behaviour leaves me as to what is in his soul. But does it always leave me
uncertain?
Page 114
887. "To be sure, this uncertainty isn't always subjective, but sometimes objective." (But what does that mean?)
Page 114
888. 'Objective uncertainty' is an indefiniteness in the nature of the game, in the admissible evidence.
Page 114
889. "What he says inwardly is hidden from me" might, of course, also mean that I can for the most part not guess
it, nor can I deduce it from, for example, the movements of his throat (which would be a possibility). [PI II, xi, p.
221b]
Page 114
890. I am, however, disregarding forms of expression such as "Only you can know what's going on inside you".†1 If
you were to bring me up against the case of people's saying "But I must know whether I am in pain", "Only you can
know what you are thinking", and other things, you should consider the occasion and purpose of such phrases.†1
("War is war" is not an example of the law of identity either.) [Cf. PI II, xi, p. 221e]
Page 114
891. Am I less certain that this man is in pain than that 2 × 2 = 4? Does this show the former to be mathematical
certainty?--'Mathematical certainty' is not a psychological concept. [PI II, xi, p. 224e]
Page 114
892. The kind of certainty is the kind of language-game. [PI II, xi, p. 224e]
Page 114
893. There are two different facts here: One, that in general I foresee my actions with greater accuracy than anyone
else; the other, that my prediction is not founded on the same evidence as someone else's, and that it allows for
different conclusions. [Cf. PI II, xi, p. 224b]†2
Page 114
894. It is not important that I know events in my mind, this is not the reason I am asked about my motives. The
reason rather is that here the evidence for and the consequences of the statement are different sorts of things.
Page Break 115
Page 115
895. "The physicist calculates because paper and ink are more reliable than his apparatus."
Page 115
896. Let us assume there was a man who always guessed right what I was saying to myself in my thoughts. (It does
not matter how he does it.)--But what is the criterion for his guessing right? Well, I am a truthful person and I
confess that he has guessed right.--But might I not be mistaken, could my memory not deceive me? And may it not
always do so anyway when--without lying--I express what I have thought within myself? -- -- -- But now it does
appear that my knowing 'what went on within me' could not be the point at all. (Here I am drawing a
construction-line.) [PI II, xi, p. 222e]
Page 115
897. The criteria for the 'truthful' confession that I thought such-and-such are not the criteria for the description of a
past process. And the importance of the truthful confession does not reside in its rendering some process correctly
and certainly. It resides rather in the special indications of subjective truth and in the special consequences of the
truthful confession.†1 [PI II, xi, p. 222f]
Page 115
898. (Assuming that people's dreams can yield important information about the dreamer, what yielded the
information would be truthful accounts of dreams. The question whether the dreamer's memory sometimes, often,
or always deceives him cannot even arise, unless indeed we introduce a completely new criterion for the 'correctness'
of the account of a dream.) [Cf. PI II, xi, pp. 222g-223a]
Page 115
899. A child who learns the first primitive verbal expression for its own pain--and then begins (also) to talk about his
past pains--can say one fine day: "When I get a pain the doctor comes". Now has the word "pain" changed its
meaning during this learning process? Yes, its use has changed.
But doesn't the word in the primitive expression and the word in
Page Break 116
the sentence refer to the same thing, namely, the same feeling? To be sure; but not to the same technique.†1
Page 116
900. I can utter or write down a sentence which expresses an intention (in the first person). Let the sentence be: "In
two minutes I shall raise my left arm." But still there is a difference between that really being my intention and my
merely jotting it down just then as a sample sentence.
Page 116
901. After all, from a person's behaviour you can draw conclusions not only about his pain but also about his
pretence.
Page 116
902. One form of guessing thoughts: One person is putting a jigsaw puzzle together, the other person can't see him,
but from time to time he says: "Now he can't find something", "Now he's thinking 'I wonder where I've seen a piece
like that?'", "Now he's very happy with himself", "Now he's thinking 'Now I know how it fits!'", "Now he's thinking
'It doesn't quite fit'"--but the first person need not be talking out loud or to himself at the time. [Cf. PI II, xi, p. 223b]
Page 116
903. All this is guessing at thoughts, and the fact that it does not actually happen does not make thought any more
hidden than the unperceived physical proceedings. [PI II, xi, p. 223c]
Page 116
904. It is possible to imagine a guessing of intentions, like the guessing of thoughts, but also a guessing of what
someone is actually going to do.
To say "He alone can know what he intends" is nonsense. To say "He alone can know what he will do" is
wrong. For a prediction contained in the expression of intention (for example, "When it strikes five I am going
home") may not come true, and I may know what he will actually do. [PI II, xi, pp. 223i-224a]
Page 116
905. Two things, however, are important. That in many cases he will not be able to foresee my actions where I do
forsee them because of my intentions. And that the prediction that is contained in the expression of my intention has
not the same foundation as another person's prediction of what I shall do, and (that) the consequences of these
predictions are different.†2 [PI II, xi, p. 224b]
Page Break 117
Page 117
906. We should sometimes like to call belief and certainty tones, colourings, of thought: and indeed, frequently they
actually are expressed in the tone of the voice. But do not think of them as 'feelings' which accompany our words.†1
[PI II, xi, p. 225b]
Page 117
907. Would it be correct to say that the language-game of stating someone's motive is the same as stating the cause,
but not when the statement is made by the person who is confessing his motive?
Page 117
908. What is the difference between motive and cause?--How is the motive discovered, and how the cause?
[Remark about the 'methods' of measuring length.] [a: PI II, xi, p. 224i; b: cf. PI II, xi, p. 225a]
Page 117
909. We remain unconscious of the prodigious diversity of all of our everyday language-games because the outward
forms of our language make everything alike. [Cf. PI II, xi, p. 224h]
Page 117
910. Suppose some people were discussing the weather; and one person says "The sky's yellow in the west, that's a
good sign. The weather will stay good." And he acts accordingly. Someone else says "No. The sky is grey in the
north. I'm convinced it will rain"--and he acts accordingly. A third person has different criteria again for his
prognosis, etc., etc. All of these people, after all, can be certain of themselves. And this certainty will be expressed
in their actions. Indeed, instead of listing criteria at all, couldn't they have simply looked at the sky and said: "I'm
under the distinct impression that it will ..."?
Page 117
911. Then: Several people are looking at a sick person (or someone who is acting as if he is sick); one of them has
the impression that he really is sick, another the opposite impression; each one says a) he has the distinct impression
that ..., and acts accordingly, b) gives reasons for this impression, but reasons which are only reasons for him.
"What goes on when someone has the impression...?"--Nonsense! What if people simply said: "I bet... that
he is sick", "I bet... that he is pretending"?
Page 117
912. Now if I believe that someone is feigning pain then I do not just believe that he isn't feeling any. Here there is a
definite suspicion.
Page Break 118
Page 118
I want to say: If natural human attitudes toward someone who is expressing pain are various--one cool and
indifferent, another sympathetic, etc.--then that in itself would not mean that someone thought that the person was
pretending.
Page 118
913. What does someone mean when he says "I think he's pretending"?--Well, he's using a word which is used in
such-and-such situations. Sometimes he will continue the game by making conjectures about the other person's
future behaviour; but that doesn't have to happen.
There's some behaviour and some conversation taking place. A few sentences back and forth; and a few
actions. That might be all.
[Words have meaning only in the stream of life.]†1
Page 118
914. It strikes me like this: it's as if an empty chess-board were lying around somewhere, and next to it there were
some chessmen. Then a few people drop by, and maybe one of them places two or three figures on the board and so
does the other; one of them makes a move, a counter-move follows, and all the while they're making faces or saying
things like "That was stupid!", "There you are!", etc. and then they stop. The whole thing would be impossible if
they couldn't play chess; but what happens is a fragment, or a possible fragment, of a chess game.
Page 118
915. Now compare an 'expert judgment' with those judgments about the weather.
The former is of value for somebody other than the person making the judgment--the latter is only an
expression of the opinion of the person who is making the judgment; to be sure, this expression may because of this
have an effect on others. The language-games are different.
Page 118
916. And, of course, there are also transitions here.
Page 118
917. One could ask: "Is there such a thing as 'expert' judgment about the genuineness of expressions of feelings?"
And the answer would be: Even here, there are what we call 'those with better' and 'those with worse judgment'. [Cf.
PI II, xi, p. 227h]
Page Break 119
Page 119
918. But there is no such thing, for example, as a special examination on knowledge of mankind.
(What would it be like if there were?) [Cf. PI II, xi, p. 227h]
Page 119
919. But how does it get shown that someone's judgment is correct? That's difficult to say. I could cite several
things; but they would only be bits and pieces of a description.
Page 119
920. One can also convince somebody that another person is in such-and-such a mental state by evidence.†1
And yet there is no special study here. [Cf. PI II, xi, p. 228b]
Page 119
921. How about this: you can set up certain rules, but only a few, which are of such a kind that a person usually
learns them through experience anyway--but what if what is left, the most important part, is imponderable??
Page 119
922. What does "imponderable evidence" mean? (Let's be honest!) [Cf. PI II, xi, p. 228c, d]
Page 119
923. I tell someone that I have reasons for this claim or proofs for it, but that they are 'imponderable'.
Well, for instance, I have seen the look which one person has given another. I say "If you had seen it you
would have said the same thing". [But here there is still some unclarity.] Some other time perhaps, I might get him to
see this look, and then he will be convinced. That would be one possibility.
To some extent I do predict behaviour ("They'll get married, she'll see to that"), and to some extent I don't.
Page 119
924. The question is: What does imponderable evidence accomplish? What entitles one to call it "evidence"?
(Compare the case of the weather forecaster with that of someone who is judging whether another person is
suffering.) [Cf. PI II, xi, p. 228c]
Page 119
925. An important fact here is that we learn certain things only through long experience and not from a course in
school. How, for instance, does one develop the eye of a connoisseur? Someone says, for example: "This picture
was not painted by such-and-such a master"--the statement he makes is thus not an aesthetic judgment,
Page Break 120
but one that can be proved by documentation. He may not be able to give good reasons for his verdict.--How did he
learn it? Could someone have taught him? Quite.--Not in the same way as one learns to calculate. A great deal of
experience was necessary. That is, the learner probably had to look at and compare a large number of pictures by
various masters again and again. In doing this he could have been given hints. Well, that was the process of
learning. But then he looked at a picture and made a judgment about it. In most cases he was able to list reasons for
his judgment, but generally it wasn't they that were convincing.
Page 120
926. Look at learning--and the result of learning.
Page 120
927. A connoisseur couldn't make himself understood to a jury, for instance. That is, they would understand his
statement, but not his reasons. He can give intimations to another connoisseur, and the latter will understand them.
Page 120
928. But am I trying to say some such thing as that the certainty of mathematics is based on the reliability of ink and
paper? No. (That would be a vicious circle.)--I have not said why mathematicians do not quarrel, but only that they
do not. [PI II, xi, p. 226b]
Page 120
929. It is no doubt true that you could not calculate with certain sorts of paper and ink, if, that is, they were subject
to certain queer changes, but still the fact that they changed could in turn only be got from memory and comparison
with other means of calculation. And surely these in turn cannot be tested against something else.†1 [PI II, xi, p.
226c]
Page 120
930. Does it make sense to say that people generally agree in their judgments of colour?? What would it be like for
them not to? One person would say a flower was red, which another took to be blue, etc.--But what right should we
have to call these people's words "red" and "blue" our colour-words? Why should we say that they have the same
meaning? We can say the one thing or the other.
Now the concept is changed and there are reasons for still calling it the same, as well as reason not to. [Cf. PI
II, xi, p. 226e]
Page Break 121
Page 121
931. But what about this: "Generally people don't argue about their colour-judgments"? There is such a thing as
'colour-blindness' and there are ways of determining it.
Isn't that a sentence about the concept of colour-judgment? [Cf. PI II, xi, p. 227d]
Page 121
932. If there were no agreement in colour-judgments, how would human beings even learn to use the words for
colours? What right should we have to call the usage they learn that of 'colour names'?
But, of course, there are transitions here.
Page 121
933. And this consideration must apply to mathematics. If our mathematical certainty did not exist, then neither
would human beings be learning the same technique which we learn. It would be more or less different from ours,
up to the point of unrecognizability in borderline cases. [PI II, xi, p. 226f]
Page 121
934. "But mathematical truth is independent of whether human beings know it or not!"--Certainly: "Human beings
believe that 2 × 2 = 4" and "2 × 2 = 4" do not mean the same thing. The latter is a mathematical proposition; the
other, if it makes sense at all, may perhaps mean: human beings have arrived at the mathematical proposition. The
two propositions have entirely different uses.--But what would this mean: "Even if everybody believed that 2 × 2 = 5
it would still be 4!"?--For what would it be like for everybody to believe that? Well, I could imagine that there was a
different calculus. Would it be wrong? Is a coronation wrong? Useless at most. And maybe not even that. [PI II, xi,
pp. 226g-227a]
Page 121
935. Of course, in one sense mathematics is a branch of knowledge, but still it is also an activity. And a 'false move'
can only exist as the exception; for if what we now call by that name became the rule the game in which it was a
false move would cease. [PI II, xi, p. 227b]
Page 121
936. 'Imponderable evidence' includes subtleties of tone, of glance, of gesture.
Isn't it really as if here one were looking at the workings of the nervous system? For I would very much like
my feigned gesture to be exactly like the real one, but in spite of everything it is not the same. [a: PI II, xi, p. 228d]
Page Break 122
Page 122
937. I can recognize a genuine loving look, distinguish it from a pretended one. And yet there is no way in which I
can describe it to someone else. If we had a great painter here, he might conceivably represent a genuine and a
simulated look in pictures, or this kind of a representation could be imagined in a film, and perhaps a verbal
description based on it. [Cf. PI II, xi, p. 228d]
Page 122
938. Ask yourself: How does a man learn to get a 'nose' for something? And how can this nose be used? [PI II, xi, p.
228e]
Page 122
939. What right do we have to say that a child has to learn some things before it can pretend? (... before it can make
a mistake in calculating.)
Page 122
940. Someone says of his child "Today he pretended for the first time". That is quite imaginable. But not the
statement "Today he was sincere for the first time"--although it can't be said of a newborn that it is sincere. And yet
one can say "My child is already definitely sincere".
Page 122
941. So if you ask "What does a child have to learn to be able to be sincere?" your answer may be something like: "It
must have realized that insincerity is bad"--or perhaps an answer that describes the inner life of the child, the inner
requisites.
Page 122
942. Neither is the newborn child capable of being malicious, friendly, or thankful. Thankfulness is only possible if
there is already a complicated pattern of behaviour.
If a figure consists of only three straight lines it can be neither a regular nor an irregular hexagon.
Page 122
943. Of course, normally only those who can talk are called sincere. And, although it does not follow from this that
the concept 'sincere' is inapplicable where there is no speech, still this concept cannot be applied there without some
difficulty.
Page 122
944. To be sure, an adult can pretend or he can be sincere without saying a word, but by using facial expressions,
gestures and inarticulate sounds.
Page Break 123
Page 123
945. Imagine a newborn child who, of course, couldn't speak, but who had the play of features and gestures of an
adult!
Page 123
946. Feigning and its opposite exist only when there is a complicated play of expressions. (Just as false or correct
moves exist only in a game.)
Page 123
947. And if the play of expression develops, then indeed I can say that a soul, something inner, is developing. But
now the inner is†1 no longer the cause of the expression.†2 (No more than mathematical thinking produces
calculations, or is the impetus behind them. And this is a remark about concepts.)
Page 123
948. Only in a quite particular musical context is there such a thing as three-part counterpoint. [C & V, p. 82]
Page 123
949. Suppose someone were to hide his intention by hiding a written plan.
Page 123
950. 'Pain is what is important--the complaining is unimportant.'--Well, I want him to take notice of my pain, not of
my plaintive cries. And how does he take notice of my pain?
Page 123
951. It looks like this: there is something inner here which can be inferred only inconclusively from the outer. It is a
picture and it is obvious what justifies this picture. The apparent certainty of the first person, the uncertainty of the
third.
Page 123
952. There is no clear border separating sufficient from insufficient evidence. And yet there is evidence here.
Page 123
953. Our judgment of the cases fluctuates, just like our natural attitude toward other people.
Page 123
954. Tender expression in music. It isn't to be characterized in terms of degrees of loudness or tempo. Any more
than a tender facial expression can be described in terms of the distribution of matter in space. As a matter of fact it
can't even be explained by reference to a paradigm, since there are countless ways in which the same piece may be
played with genuine expression. [C&V, p. 82]
Page Break 124
Page 124
955. And what would the opposite look like?--One might be able to determine sadness, for instance, with the same
certainty as, say, a sore throat.--But what sort of concept of sadness would that be? Ours?
Page 124
956. Why not? If a person makes this face on a certain occasion, carries himself this way, etc., then we can predict
with certainty all that we should expect (in the world as it now is) of a truly sad person.
Page 124
957. Of what does our uncertainty about the mental conditions and processes in another person consist? For it is
composed of several things.†1 We cannot read off what he is saying to himself from anything external. Often we
cannot understand what he says. We cannot guess his intentions. Often we don't know what kind of mood he's in.
The ignorances are of different kinds; and if one imagines them removed then they would be removed in
different ways.
Page 124
958. What does it mean, for example, to know someone else's mood with certainty?
Well one might imagine that it could be read in my face.--But could intention also be read in my face?! Then
why not just as well in my hands or clothes?--But one might imagine a way of finding out intention. You ask
someone about his intentions and you can recognize with certainty when he is lying, and perhaps also what is going
through his mind at that moment. But what if at this moment the intention were present merely as a disposition, so
to speak? What if it weren't thought out?--So here it might be necessary that I have already observed him!
Page 124
959. "The inner is hidden from me"--isn't that just as vague as the concept of the 'inner'?
(For just consider: the inner after all is sensations + thoughts + images + mood + intention, and so on.)
Page 124
960. Clearly you don't guess a person's intention, his sensations, his thoughts, his mood, all in the same way.
Page Break 125
Page 125
961. Neither do I know his actions beforehand as I do my own, and I have different ways of forming my intentions
than he has of guessing them.
Even when I have no positive intention, I can still have negative ones; I don't know what I'll do, but I have
already decided that I don't want to do this or that.
Page 125
962. It is a strange kind of remembering when in broad daylight you remember one of last night's dreams which you
never thought of earlier on waking. -- -- --
Page 125
963. The opposite of my uncertainty as to what is going on inside him is not his certainty. For I can be sure of
someone else's feelings, but that doesn't make them mine.
Page 125
964. "I can only guess at someone else's feelings"--does that really make sense when you see him badly wounded,
for instance, and in dreadful pain?
Page 125
965. Is a dream a hallucination?--The memory of a dream is like the memory of a hallucination, or rather: like the
memory of a real experience. This means that sometimes you would like to say: "I just saw this and that", as if you
really had just seen it.
Page 125
966. As an example think of the description of 'occasions'. Is it really clear that one has to understand the description
of an 'occasion of grief'? For the occasions of grief are interwoven with 1000 other patterns. Is it clear that someone
must be able to learn the technique of designating this kind of pattern? That he be able to pick an occasion of grief
out of the other patterns the way we do?
Page 125
967. But here there are simple and more complicated cases; and that is important for the concept. Somebody gets
burned and cries out; only in very rare circumstances would his behaviour be called "pretence". Indeed, here a
doctor could tell us this and that circumstance under which it could be pretence.
Page 125
968. The description of word-usage. The word is uttered--in what context? So we have to find something
characteristic of these separate occurrences, a kind of regularity.--But we don't learn to use words with the help of
rules. How could I give someone a rule for those instances in which he is supposed to say he's in pain!--On the other
Page Break 126
hand, there is a ROUGH regularity in the use to which a person actually puts words.
Page 126
969. So I shall say: It is not established from the outset that there is such a thing as 'a general description of the use
of a word'.
Page 126
And even if there is such a thing, then it has not been determined how specific such a description has to be.
Page 126
970. In what circumstances (outer circumstances) is something called an expression of pain? (For that after all is an
important question--even if you say that something inner corresponds to a true expression of pain.)
Page 126
971. And now can I describe these circumstances?--and why not? I could give examples, that much is clear. How
can I learn to describe the circumstances? Was I taught? Or what would I have to observe to be able to do that?
Page 126
972. And the same thing goes for the outward signs of 'pretence'.
Page 126
973. And if I now imagine a list of such circumstances, who would be interested in it?--To be sure, individual
aperçus are interesting. But would a listing which strove for inclusiveness be interesting? Would it be of practical
use?--This game doesn't work that way at all.
Page 126
974. Nothing is hidden here; and if I were to assume that there is something hidden the knowledge of this hidden
thing would be of no interest.
But I can hide my thoughts from someone by hiding my diary. And in this case I'm hiding something that
might interest him.
Page 126
975. To say that my thoughts are inaccessible to him because they take place within my mind is a pleonasm.
Page 126
976. What I say to myself silently he doesn't know: but again this isn't a matter of a 'mental process', although there
may be a physical process taking place here which might do instead of words spoken out loud if the other did know
it. So also a physical process here might be called 'hidden'.
Page Break 127
Page 127
977. "What I think silently to myself is hidden from him" can only mean that he cannot guess it, for this or that
reason; but it does not mean that he cannot perceive it because it is in my soul.
Page 127
978. You look at a face and say "I wonder what's going on behind that face?" But you don't have to say that. The
external does not have to be seen as a façade behind which the mental powers are at work.†1
Page 127
979. The idea of the human soul, which one either sees or doesn't see, is very similar to the idea of the meaning of a
word, which stands next to the word, whether as a process or an object.
FOOTNOTES
Page 6
†1 Several variants in the MS.--In the margin is added: "What is a complaint, anyway?"
Page 9
†1 Var.: "notation".
Page 9
†2 Var.: "But the question now remains why, in connection with that game of meaning we also speak of an
'act of meaning'. This is a different kind of question from what you think.--It is (precisely) the phenomenon of this
language-game that in this situation we say that we meant the word in this way, and take this expression over from
another language-game. A question has to have gone before.
Page 9
That is an alien kind of question. That is an inappropriate question; of a different racial origin, as it were."
Page 10
†1 Var.: "speaks".
Page 10
†2 In the margin: "It is as if, in a work of pure mathematics, there were put a question of physics as the one
you were trying to ask."
Page 10
†3 The German word Bank means "money bank" and "bench". (Tr.)
Page 11
†1 Preceding this remark, in square brackets: "Ref. the sentence at the top of p. 82r." This refers to the end of
remark 56.
Page 11
†2 Preceding this paragraph in the MS, in square brackets: "Ref. Tscr. p. 667 below".--This refers to p. 667 of
TS 232 and to its continuation on the following page. See Remarks on the Philosophy of Psychology (RPP) II, §246.
Page 11
†3 Preceding the paragraph, in square brackets: "From T.S. p. 667v".--See the previous footnote.
Page 12
†1 At the end of the remark in square brackets: "Continuation lost".
Page 13
†1 Var.: "when the dust has settled".
Page 13
†2 Preceding the remark, in square brackets: "From p. 82v, bottom". This refers to remark 56.
Page 15
†1 Several variants in the MS. Following the remark, in square brackets: "Still not right."
Page 15
†2 Preceding the remark in the MS, in square brackets: "Ref. Tscr. p. 670."--See RPP II, §256.
Page 15
†3 Preceding the remark, in square brackets: "Gassy".
Page 18
†1 Reference to Schiller's Wallenstein.
Page 20
†1 Several variants in the MS.
Page 21
†1 Preceding the remark, in square brackets: "Ref. Tscr. p. 708/4". See RPP II, §403.
Page 27
†1 Preceding the remark, in square brackets: "Ref. Tscr. p. 740". --This refers to RPP II, § 556.
Page 28
†1 Var.: "even though not for error."
Page 28
†2 Before the remark, in square brackets: "Ref. p. 742 Tscr."--See RPP II, §§566-569.
Page 29
†1 Preceding the remark, in square brackets: "Ref. Tscr. p. 750". See RPP II, §§605-608.
Page 30
†1 Wittgenstein repeats this remark in almost the same words on the same page of the MS. The end of the
variant runs: "But they had to receive a new kind of training for the use of our words. This (training) was similar to
but not the same as the previous training."
Page 30
†2 This remark is obviously an "improved" variant of a remark on the previous page of the MS that seems to
be partially crossed out. This remark runs: "And how could they remain unaware of the difference, if sometimes
they would complain when they were in pain and sometimes when they were not? But did the difference have to be
as important for them as it is for us? (Many people tell fabricated stories at a party, and the others know they are
fabricated, but they buy them, just as they do true stories. They ignore the difference.)"
Page 30
†3 Before the remark, in square brackets: "Ref. Tscr. p. 759". See RPP II, §§ 648-653.
Page 31
†1 Before the remark, in square brackets: "Ref. Tscr. p. 760". See RPP II, §658.
Page 34
†1 On the same and subsequent pages of the MS, there is the following variation of the end of the remark:
"But the philosophical question whether someone else feels pain is of a completely different nature; not the doubt
applied in a certain case to each; therefore it must have a different logic."
Page 36
†1 Before the remark: "Tscr. p. 751". See RPP II, §§609-612.
Page 37
†1 Before the remark, in square brackets: "From the previous page". See remark 253.
Page 41
†1 Cf. Remarks on the Foundations of Mathematics, 3rd ed., Part I, Appendix I.
Page 42
†1 Var.: "primitive need".
Page 43
†1 Var.: "And yet there is something right about this 'disintegration of the sense'. You get it in the following
example: You might instruct someone thus: If you want to pronounce the salutation "Hail!" expressively, you must
not think of hailstones as you say it!"
Page 43
The German words are Ei, "egg" and the interjection ei, ei, "fancy that!" (Tr.)
Page 45
†1 Var.: "Now a picture is strongly suggested to us: that of something incorporeal which we feel, of the
liveness of a face. One must remind oneself that a face with a soulful expression can be painted to make us believe
that colours and forms alone can (also) have this kind of effect on us."
Page 46
†1 Var: ", or that a verb designates one action in the first person and another in the second?"
Page 47
†1 See PI I, § 2.
Page 49
†1 Var.: "But need there be any question for them whether the groaning is really genuine, is really the
expression of anything? Can't they, for example, draw the conclusion... without suppressing a middle term? Isn't the
point the job they give the description of behaviour?"
Page 49
†2 Var.: "And doubt may be entirely lacking. Doubting has an end."
Page 50
†1 Var.: "That is, the phenomena of hope are modes of this very complicated pattern."
Page 54
†1 i.e. the kinaesthetic feeling.
Page 56
†1 Cf. PI I, § 48.
Page 58
†1 Var.: "experiential concepts".
Page 59
†1 Several variants in the MS.
Page 62
†1 Var.: "static".
Page 64
†1 In the MS. Wittgenstein has "out", perhaps as a mistake. Possibly he meant "Do we also learn...".
Page 65
†1 Var.: "This has the form of a report of a new perception."
Page 66
†1 Var.: "If I know that the schematic cube has various aspects, and I want to find out what someone else
sees, I can get him to make a model of what he sees, in addition to a copy, or point to such a model; even though he
has no idea what is the purpose of this dual demonstration."
Page 69
†1 Var.: "I forgot it."
Page 70
†1 Unclear passage in the MS.
Page 72
†1 Var.: "the dawning of an aspect".
Page 72
†2 Var.: "half visual experience, half thought."
Page 74
†1 Var.: "--is this a special sort of seeing? Or is it a case of both seeing and thinking? Or an amalgam of the
two--as I should almost like to say?--The question is: Why does one want to say this? Well, if the question is phrased
in this way it isn't that hard to answer."
Page 74
†2 Var.; Before the paragraph, in square brackets: "Not a good cont<inuation>."
Page 75
†1 Var.: "phrases it and helps to form the impression, so to speak."
Page 79
†1 Var.: "what I feel there?--"
Page 80
†1 At the end of the remark, in square brackets: "To the previous page." Cf. remarks 619-621.
Page 81
†1 Var.: "treatises".
Page 83
†1 Var.: "mechanics".
Page 84
†1 Var.: "illusion in a dream".
Page 86
†1 At the end of the remark, in square brackets: "The remark on p. 733 Tscr. relates rem<ark> to this." See
RPP II, §§515-522.
Page 86
†2 Following the words "goes together with..." in square brackets: "p. 733 Tscr." See RPP II, §517.
Page 88
†1 Var.: "with a special gesture".
Page 88
†2 This remark is the first in Volume "S" (MS 138).
Page 90
†1 Var.: "of the step."
Page 90
†2 The English joke: "What is the difference between a hairdresser and a sculptor? A hairdresser curls up and
dyes, and a sculptor makes faces and busts."
Page 92
†1 In German the word used is Säbel, "sabre". (Tr.)
Page 94
†1 Var.: "have seen".
Page 94
†2 Var.: "So should I say:"
Page 100
†1 Var.: "-- -- -- What would you be missing, for instance, if you did not understand the meaning of: "Say the
word 'bank' and mean it as the bank of a river"--or: "Say the word 'till' and mean it as a verb, not as a
conjunction"--or if you did not find that a word lost its meaning and became a mere sound if it was repeated ten
times over."
Page 100
The German word is sondern, meaning both "but" and "to separate". (Tr.)
Page 101
†1 The German word is weiche, meaning both "soft" and "to retreat". (Tr.)
Page 102
†1 Preceding the remark, in square brackets: "Ref. MS "R" p. 83". See remark 69.
Page 103
†1 Var.: "Realities are so many possibilities for the philosopher."
Page 106
†1 See PI I, §8.
Page 108
†1 Var.: "As a result of a grammatical explanation."
Page 109
†1 Before the remark, in square brackets: "Ref. p. 15v/3".--This apparently refers to remark 825.
Page 109
†2 Var.: "(As does intention, when it is interpreted as the accompaniment of action," ...).
Page 109
†3 Several variants in the MS.
Page 109
†4 Var.: "but the concept may easily confuse us, for it runs over a long stretch cheek by jowl with the
concept of an 'outward' process, and yet does not coincide with it. (Tennis without a ball.)"
Page 113
†1 Several variants in the MS.
Page 114
†1 Alternatives.
Page 114
†2 At the end of the remark, in square brackets: "Ref. MS "R" p. 96". See remark 183.
Page 115
†1 Var.: "The criteria for the truth of a confession that I thought such-and-such are not the criteria for a true
description of a process. And the importance of the true confession does not reside in its being a correct and
(absolutely) certain report of a process. It resides rather in the special consequences which can be drawn from a
confession whose truth is guaranteed by the special criteria of truthfulness."
Page 116
†1 At the end of the remark, in square brackets: "Ref. the §: "I'm not certain..."."
Page 116
†2 Var.: "And that the prescience inherent in my intention does not rest on the same foundation as the other
person's prediction of my actions."
Page 117
†1 Var.: "Ask not "What goes on in us when we are certain that...?"--but: How is certainty manifested in our
actions?"
Page 118
†1 Cf. Norman Malcolm, Ludwig Wittgenstein, A Memoir, p. 93.--In MS 169, p. 47v, Wittgenstein says:
"Also what goes on in the inner has meaning only in the stream of life."
Page 119
†1 Var.: "One can also convince someone that he was wrong about a person's mental state by giving him
evidence. One can set him straight with evidence."
Page 120
†1 Var.: "And how are these supposed to be tested in their turn?"
Page 123
†1 Var.: "appears here ... as".
Page 123
†2 Var.: "as the prime mover of the expression."
Page 124
†1 Var.: "Our 'ignorance' about what goes on in someone else is not a single ignorance, but consists of
different kinds of ignorance."
Page 127
†1 Var.: "But you don't have to think that way. And if someone talks to me quite obviously holding nothing
back then I'm not even tempted to think that way."
LAST WRITINGS ON THE PHILOSOPHY OF PSYCHOLOGY: Volume II
Page Break ii
Page Break iii
Title Page
Page iii
Ludwig Wittgenstein
LAST WRITINGS ON THE PHILOSOPHY OF PSYCHOLOGY
VOLUME II
The Inner and the Outer
1949-1951
Edited by
G.H. VON WRIGHT
and
HEIKKI NYMAN
Translated by
C. G. LUCKHARDT
and
MAXIMILIAN A. E. AUE
<image>
Page Break iv
Copyright Page
Page iv
Copyright © The Trustees of the Wittgenstein Estate 1992
First published 1992
First published in paperback 1993
Reprinted 1994, 1999
Blackwell Publishers Ltd
108 Cowley Road
Oxford OX4 1JF
UK
Blackwell Publishers Inc.
350 Main Street
Malden, Massachusetts 02148
USA
All rights reserved. Except for the quotation of short passages for the purposes of criticism and review, no part of
this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means,
electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the publisher.
Except in the United States of America, this book is sold subject to the condition that it shall not, by way of trade or
otherwise, be lent, resold, hired out, or otherwise circulated without the publisher's prior consent in any form of
binding or cover other than that in which it is published and without a similar condition including this condition
being imposed on the subsequent purchaser.
British Library Cataloguing in Publication Data
A CIP catalogue record for this book is available from the British Library.
Library of Congress Cataloging-in-Publication Data
Wittgenstein, Ludwig, 1889-1951
Last writings on the philosophy of psychology.
Title on added t.p.: Letzte Schriften über die Philosophie der Psychologie.
English and German.
Vol. 2. published: Oxford, UK; Malden, Mass., USA;
B. Blackwell, 1992.
Contents: v. 1. Preliminary studies for part II of the Philosophical
Investigations--v.2. The inner and the outer, 1949-1951.
I. Psychology--Philosophy. I. Wright, G. H. von (Georg Henrik), 1916- II. Nyman, Heikki. III. Wittgenstein,
Ludwick, 1889-1951. Philosophische Untersuchungen.
BF38.W764 1982 192 82-42549
ISBN 0-631-18956-4 (pbk.)
Typeset in 11 on 13 pt Bembo
By Photographics, Honiton, Devon
Printed and bound in Great Britain by MPG Books Ltd, Bodmin, Cornwall
This book is printed on acid-free paper
Page Break v
Table of Contents
Page v
CONTENTS
Editors' Preface vi
MS 169 1
MS 170 50
MS 171 54
MS 173 60
MS 174 80
MS 176 91
Index 102
Page Break vi
Editors' Preface
Page vi
Thematically, Wittgenstein's philosophical writings from the last two years of his life (1949-51) can be divided into
three parts. The largest of these three parts deals with the concepts of certainty, knowing, doubting, and other topics
in epistemology. A second part deals with the philosophy of colour concepts; a third, with psychological concepts
and in particular with the problem of the relationship between "the inner" and "the outer", between the so-called
mental states and bodily behaviour.
Page vi
Most of the writings of the first group have appeared in print under the title On Certainty, those of the
second under the title Remarks on Colour. The remarks on the "inner-outer" problem are closely connected with the
body of ideas of the second part of the Philosophical Investigations and with the preliminary studies for it in the
manuscripts and typescripts from the years 1946-49. But they also connect with the remarks on epistemology and
colour concepts and sometimes cannot be sharply separated from them. (A longer part from MS 173, which is
printed here, was already published in the Remarks on Colour (III, sect. 296-350).)
Page vi
Wittgenstein's writings from the last years of his life are entered into eight small notebooks, and in a small
bundle of loose sheets (MS 172). The most voluminous notebook is MS 173. The second largest of these notebooks,
MS 169, was probably already begun in late fall, 1948, or in the spring of 1949. The content of the first half of this
notebook is of the nature of preliminary studies for what is published as Volume I of these Last Writings, and
comes from the large
Page Break vii
manuscript books 137 and 138. The style of the remarks is terse; the sentences are frequently indicated only in
abbreviated form. There are many parts which because of their unclarity are very hard to read. The second half of
these writings, which begins with a discussion of the concept of dissimulation, is worked out better stylistically and
in terms of content is connected to the rest of this volume. However, there is no clear border separating the two parts
from each other. Therefore we considered it proper to publish this notebook here in its entirety.
Page vii
The small notebooks 170 and 171 are also presented in their entirety. Chronologically they are probably
closely connected to MS 169 and probably were written in the year 1949.
Page vii
The voluminous notebook 173, from the spring of 1950, deals mainly with colour concepts; but it does
contain two longer portions belonging to the problems of the "inner-outer".
Page vii
Later, in the spring of 1950, in MS 174, Wittgenstein continued the exposition of the "inner-outer" problem
of MS 173, and then returned once more to this topic in MS 176 two weeks before his death in April 1951. The
excellent quality of these last writings should be obvious to every reader.
Page vii
With the exception of a very few remarks (which are of a general nature) and of remarks which have already
appeared elsewhere in print, we have not omitted anything of what Wittgenstein wrote in these last notebooks from
the years 1949-51.
Page vii
Words in angle brackets < > and references to the printed works of Wittgenstein in square brackets
(including the first volume of these Last Writings) are those of the editors. All footnotes are also additions by the
editors. The letters 'N.R.' in the upper right corner of some pages indicate that the page begins with a new remark
(and not with a paragraph belonging to the last remark on the preceding page).
Page Break viii
Page viii
We thank Mr Joseph Braun, Mr Michael Kober, Dr Joachim Schulte for their generous help in the
interpretation of difficult parts of the manuscript and the translators, Professor C. G. Luckhardt and Professor
Maximilian Aue, for valuable advice and for preparing the Index. We also thank Mr Erkki Kilpinen, who helped in
the preparation of the final version of the text.
Helsinki 1982, 1991 Georg Henrik von Wright
Heikki Nyman
Page Break 1
I:
MS 169
Page 1
(around 1949)
Page Break 2
Page 2
non & ne†1---They have the same purpose, the same use--with one qualification. [LW I, 384]
Page 2
So are there essential and non-essential differences among the uses? This distinction does not appear until we
begin to talk about the purpose of a word. [LW I, 385]
Page 2
--- We might think it strange. "He doesn't play our game at all" one would like to say. Or even that this is a different
type of man. [Cf. PI II, vi, p. 182b]
Page 2
The psychologist reports the utterances of the subject. But these utterances "I see...", "I hear...", "I feel" etc.,
are not about behaviour. [Cf. PI II, v, p. 179b]
Page 2
--- About both, not side-by-side, however, but about the one via the other. [Cf. PI II, v, p. 179c]
Page 2
But†2 this would make us see too that it would have no consequences for understanding and using the words.
Page 2
But wasn't he able to master the use of "if" and "that" just as we are? Wouldn't he understand and mean
these words as we do? [Cf. PI II, vi, p. 182b]
Page 2
Wouldn't I think (of him) that he understood the words "that" and "if" as we do if he uses them exactly as we
do? [PI II, vi, p. 182b]
Page 2
And if he used "that" and "if" exactly as we do, wouldn't we think he understood them as we do? [PI II, vi, p.
182b]
Page 2
Page Break 3
Page 3
It's another kind of man. But of what importance is that? [Cf. PI II, vi, p. 182b]
Page 3
It would be like someone who instead of associating a separate colour with each vowel associated one colour
with a, e, i, and another with o and u. "In that case we'd have a different type of man", I would like to say. [Cf. LW, I,
362]
Page 3
--- Because only this world (has) this sound, this tone, this grammar.†1
Page 3
So does the word "Beethoven" have a Beethoven-feeling?
Page 3
It is a look, with which this word looks at me. [Cf. LW I, 366]
Page 3
But one cannot separate the look from the face.
Page 3
--- with a quite particular expression. [Cf. LW I, 380]
Page 3
This expression is not something that can be separated from the passage. (Not necessarily.) It is a different
concept. (A different game.) [PI II, vi, p. 183a]
Page 3
"Now you played the passage with a different expression."--"--now with the same" and it can also be
characterized by a word, a gesture, a simile; nevertheless by this expression we don't mean something that can
appear in a different connection.
Page 3
The experience is this passage played like this (that is, as I am doing it, for instance; a description could only
hint at it). [PI II, vi, p. 183b]
Page 3
Gay colours.
Page Break 4
The 'atmosphere' is precisely that which one cannot imagine as being absent. [CF. PI II, vi, p. 183c]
Page 4
The name Schubert, shadowed around by the gestures of his face, of his works.--So there is an atmosphere
after all? But one cannot think of it as separate from him. [Cf. LW I, 69; PI II, xi, p. 215f]
Page 4
The name S is surrounded in that manner, at least if we are talking about the composer.
Page 4
But these surroundings seem to be fused with the name itself, with this word.
Page 4
--- Imagine I hear that someone is painting a picture--[Cf. PI II, vi, p. 183d]
Page 4
: I hear that someone is painting a picture "... ". [Cf. PI II, vi, p. 183d]
Page 4
--- if they did not fit in with this passage.
Page 4
--- would be totally addle-brained and ridiculous.
would be completely repulsive and ridiculous.
would be completely ridiculous and repulsive.†1
Page 4
My k<inaesthetic> sensations tell me of the movement and position of my limbs.
Page 4
Now I move my finger. I either hardly feel it or don't feel it at all. Perhaps a little in the tip of my finger, and
sometimes at a point on my skin (not at all in the joint). And this sensation tells me that I move my finger and how?
For I can describe it exactly. [Cf. PI II, viii, p. 185a; LW I, 386]
Page Break 5
But after all, you must feel the movement, otherwise you couldn't know how it was moving. But "knowing
how" only means being able to state, to describe, it.
Page 5
I may be able to tell the direction from which a sound comes only because it affects one of my ears
differently from the other; but I don't hear that. [See PI II, viii, p. 185b; LW I, 387]
Page 5
It is quite the same with the idea that it must be some feature of our pain that tells us of its whereabouts in
the body, or some feature of our memory image that tells us the time to which it belongs. [See PI II, viii, p. 185c; LW
I, 388]
Page 5
A sensation can tell us of the movement or position of a limb. And the character of the pain can also tell us
where its cause†1 is situated. [Cf. PI II, viii, p. 185d; LW I, 389]
Page 5
How do I know that a blind person uses his sense of touch and a sighted person his sense of sight to tell
them about the shape and position of objects? [LW I, 390]
Do I know this only from my own experience, and do I merely surmise it in others? [LW I, 391]
Page 5
Here we have, in addition to that description, another description of sensation (of what is sometimes called
sense-data).
Page 5
What is the criterion for my learning the shape and colour of an object from a sense-impression? [PI II, viii,
p. 185e; LW I, 393]
Page 5
What sense-impression? Well, this one. I can describe it. "It's the same one as the one----" or I can
demonstrate it
Page Break 6
with a picture.
Page 6
And now: what do you feel when your fingers are in this position?
Page 6
"How is one to define a feeling? One can only recognize it within oneself." But it must be possible to teach
the use of words! [PI II, viii, p. 185f; LW I, 394]
Page 6
What I am looking for is the grammatical difference. [PI II, viii, p. 185g; LW I, 395]
Page 6
The words rough†1 and smooth, cold and warm, sweet and sour, bitter...
Page 6
But why not also thin and thick?
Page 6
--- Can't there be a doubt here? Mustn't there be one, if it is a feeling that is meant? [See PI II, viii, p. 186b; LW I,
402]
Page 6
What would we say if someone reported to us that in a certain object he saw a colour he couldn't describe
further? Does he have to be expressing himself correctly? Does he have to mean a colour? [LW I, 403]
Page 6
This looks so; this tastes so; this feels so. This and so must be differently explained. [PI II, viii, p. 186c; LW
I, 404]
Page 6
I can observe the state of my depression. In that case I am observing what I for instance describe.
Page 6
A thought which one month ago was still unbearable to me is no longer so today. (A touch which was
painful yesterday is not so today.) That is the result of an observation. [Cf. PI II, ix, p. 187b]
Page Break 7
Trying to recollect a mental mood can be called observing.
Page 7
What do we call 'observing'? Roughly this: putting oneself into the most favourable situation to receive
certain impressions with the purpose, for instance, of describing them.
Page 7
When do we say that anyone is observing? Roughly: when he puts himself in a favourable position to receive
certain impressions with the purpose, for example, of describing what they tell him. [PI II, ix, p. 187c]
Page 7
At bottom, I am still afraid.--I am afraid, I can't stand this fear!--I am afraid of his coming, therefore I am so
restless.--Oh, now I am much less afraid of it than before. Now, just when I should be fearless, I am afraid!
Page 7
There could be various explanations:
I am afraid! I can't stand this fear!
I am afraid of his coming, and that is why I am so restless.
I am still a little afraid, although much less than before.
At bottom I am still afraid, though I won't confess it to myself.
Now, just when I should be fearless, I am afraid!
I am afraid; unfortunately, I must admit it.
I think I'm still afraid. [Cf. PI II, ix, p. 188a]
Page 7
The contexts in which a word appears are portrayed best in a play; therefore, the best example for a sentence
with a certain meaning is a quote from a play. And who asks the character in a play what he experiences when he
speaks?
Page Break 8
The best example of an expression with a very specific meaning is a passage in a play. [LW I, 424]
Page 8
--- Well, we assume several things. For instance, that they hear their own voices, and also sometimes feel things as
they are gesturing and whatever else belongs to human life.
Page 8
To stir up a philosophical wasps' nest. Moore. [Cf. C & V, 2nd edn., p. 147]
Page 8
The language-game of reporting can be given such a turn that the report gives the person asking for it a piece
of information about the one making the report, and not about its subject-matter. (Measuring in order to test the
ruler.) [Cf. LW I, 416; PI II, x, p. 190d-191a]
Page 8
Momentary motion. [Cf. LW I, 425]
Page 8
If you see motion you by no means see the position at a point in time. You could not portray it. [See LW I,
425]
Page 8
Tacking on to:
Page 8
Back then I believed that the earth was a flat disc.--Just like that? [Cf. LW I, 426]
Page 8
"Knowing myself as I do, I will now act this way."
Page 8
The line vanishes into the dark.
There is no real point on it for... [Cf. LW I, 427]
Page 8
A different tacking on, if that's what you want, has to be according to a different principle.
Page 8
The question can be raised: Is a state that I recognize on
Page Break 9
the basis of someone's utterances really the same as the state he does not recognize this way? And the answer is a
decision. [LW I, 428]
Page 9
The curve "to be in error".
Page 9
"To seem to believe", a verb. The first person present indicative is meaningless, because I know my intention.
But that would be a development of "he believes". [Cf. PI II, x, p. 192b; LW I, 423]
Page 9
Or: If believing is a state of mind, it lasts. It doesn't last just while I am saying I believe, so it is a disposition.
Why can't I say of myself that I have it? How do others recognize my disposition? They observe, they ask me. My
answer doesn't have to be "I believe... ", but maybe "That's the way it is"; from this they recognize my disposition.
And how do I recognize it? By random tests? My disposition might be, for instance, "This and that can be expected
of me". Doesn't it interest me? [Cf. PI II, x, pp. 191i-192a]
Page 9
My own relation to my words is wholly different from other people's. [PI II, x, p. 192b]
Page 9
I do not listen to them and thereby learn something about myself. They have a completely different relation
to my actions than to the actions of others.
Page 9
If I listened to the words of my mouth, I would be able to say that someone else was speaking out of my
mouth. [PI II, x, p. 192c]
Page 9
"These days, I am inclined to say..."
Page 9
If someone says something with a great deal of conviction,
Page Break 10
does he believe it at least while he is saying it? Is belief this kind of state?
Page 10
With "I believe------" he expresses his belief in no way better than with the simple assertion.
Page 10
My words and my actions interest me in a completely different way than they do someone else. (My
intonation also, for instance.) I do not relate to them as an observer.
I can not observe myself as I do someone else, cannot ask myself "What is this person likely to do now?" etc.
Page 10
Therefore the verb "He believes", "I believed" can not have the kind of continuation in the first person as the
verb "to eat".
Page 10
"But what would the continuation be that I was expecting?!" I can see none.
Page 10
"I believe this."--"So it appears I believe this."
Page 10
"Going by my utterances, I believe this; but it isn't so." [Cf. PI II, xq, p. 192d]
Page 10
"It seems to me I believe†1 this, but it isn't so."
Page 10
My words are parallel to my actions, his to his.
Page 10
A different co-ordination.
Page 10
I do not draw conclusions as to my probable actions from
Page Break 11
my words.
Page 11
That consistent continuation would be "I seem to believe". [Cf. PI II, x, p. 192b]
Page 11
From the very beginning, the assumption is surrounded by all forms of the word "to believe", by all of the
different implications.
Page 11
For I have a mastery of his technique long before I reflect upon it. [Cf. PI II, x, p. 192e]
Page 11
"Judging from what I say, this is what I believe."
Page 11
(Now it would be possible to think of circumstances in which such an utterance would make sense. But we
are not talking about this use of the word "belief".) [See PI II, x, p. 192d]
Page 11
And someone could also say "It's going to rain, but I don't believe it" if there were indications that two
people were speaking through his mouth. Language-games would be played here which we could imagine, to be
sure, but which normally we don't encounter. [Cf. PI II, x, p. 192d]
Page 11
And then it would also be possible for someone to say "It is raining, but I don't believe it". One would have
to fill out the picture with indications that two personalities were speaking through his mouth.†1 [See PI II, x, p.
192d]
Page 11
Here it does look as if the assertion "I believe" were not the assertion of what is supposed in the hypothesis "I
believe"!
Page 11
Therefore I am tempted to look for a different continuation of the verb in the first person indicative. [Cf. PI
II, x, p. 190c]
Page Break 12
This is how I think of it: believing is a state of mind. It exists during a period of time, it doesn't attach to the
time of its expression. So it is a kind of disposition. This is shown me in the case of someone else by his behaviour,
by his words. And specifically by his expression "I believe" as well as the simple assertion. What about my own
case? Do I study my disposition in order to make the assertion or the utterance "I believe"?---But couldn't I make a
judgement about this disposition just like someone else? In that case I would have to pay attention to myself, listen
to my words, etc., just as someone else would have to do. [Cf. PI II, x, pp. 191i-192a]
Page 12
I could find that continuation if only I could say "I seem to believe". [PI II, x, p. 192b]
Page 12
A wall covered with spots, and I occupy myself by seeing faces on it; but not so that I can study the nature
of an aspect, but because those shapes interest me, and so does the spell under which I go from one to the next. [LW
I, 480a]
Page 12
Again and again aspects dawn, others fade away, and sometimes I 'stare blindly' at the wall. [LW I, 480b]
Page 12
The double cross and the duck-rabbit might be among the spots and they could be seen like the figures and
together with them now one way, now another. [Cf. LW I, 481]
Page 12
The dawning of aspects is related to the dawning of mental images.
Page 12
Just because I always used it as an 'f' doesn't mean that I have therefore seen it as an 'f'.
Page 12
'That can be an F.'
Page Break 13
The aspect seems to belong to the structure of the inner materialization. [LW I, 482]
Page 13
We learn language-games. We learn how to arrange objects according to their colours; how to report the
colours of things, how to produce dyes in different ways, how to compare shapes, report, measure, etc. etc.
Do we also learn how to form mental images out of them? [LW I, 483]
Page 13
There is a language-game "Report the colour... ", but not "Report this colour here".
Page 13
There is a language-game "Tell me whether this figure is contained in that one". (Also "how often" or
"where".)
What you report is a perception. [LW I, 484]
Page 13
So we could also say: "Tell me whether there is a mirror-F here", and suddenly it might strike us that there is.
This could be very important. [LW I, 485]
Page 13
But the report "Now I see it as--now as--" does not report any perception. [LW I, 486]
Page 13
You can think of it in this way, or in that way, then you see it now this way, now that. How?
Page 13
You can think now of this, now of that, as you look at it, look at it now as this, now as that, and then you
will see it now this way, now that way. What way? There is no further qualification. [LW I, 487; PI II, xi, p. 200d]
Page 13
To be sure, if you look that way, furrowing your brows for instance, then you see it green, but otherwise red.
In this way, the colour could teach me about the object after all. The prescription would simply be--you have to look
this way.
Page Break 14
I can change the aspects of the F and in so doing I do not have to be cognizant of any other act of volition. [LW I,
488]
Page 14
---For the expression of the changing of the aspect is also the expression of congruence and dissimilarity.
Page 14
Seeing and thinking in the aspect.
Page 14
I look at an animal. I am asked "What do you see there?" I answer "A hare".---I gaze at the landscape;
suddenly a hare runs by. I exclaim: "A hare!"
Both things, both the report and the exclamation, can be called expressions of perception and of visual
experience. But the exclamation is so in a different sense from the report. It is wrung from us. It is related to the
experience as a cry is to pain. [PI II, xi, p. 197b; LW I, 549b]
Page 14
But since it is the description of a perception, it can also be called the expression of a thought. And therefore
we can say that if you looked at the animal you would not have to think of the animal; but if you are having the
visual experience expressed by the exclamation you are also thinking of what you see. [PI II, xi, p. 197c; LW I, 553]
Page 14
And that is why the experience of a change of an aspect seems half visual-, half thought-experience. [Cf. PI
II, xi, p. 197d; LW I, 554]
Page 14
When I see a change of aspect, I am occupied with the object.†1 (Cf. LW I, 555]
Page 14
I am occupied with what I am now noticing, with what strikes me. In that respect, experiencing a change of
aspect is similar to an action. [LW I, 556]
Page Break 15
It is a paying of attention.
Page 15
What is the criterion of visual experience? What ought the criterion be?
The representation of 'what is seen'. [PI II, xi, p. 198b; LW I, 563]
Page 15
Now when the aspect dawns can I separate a visual experience from a thought-experience? (And what does
that mean?) If you separate them then the aspect is lost. [Cf. LW I, 564]
Page 15
And what about the double cross? Again, it is seeing according to an interpretation. Seeing as.
Page 15
Now when I recognize this person in a crowd, perhaps after looking in their direction for quite a while,--is
this a sort of seeing? A sort of thinking?†1 The expression of the experience is "Look, there's...!" But of course, it
could just as well be a sketch. That I recognize this one might be expressed in the sketch, or in the process of
sketching as well. (But the element of sudden recognition is not expressed in the sketch.) [LW I, 571a; cf. PI II, xi, p.
197h]
Page 15
Suppose a child suddenly recognizes someone. Let it be the first time he has ever suddenly recognized
someone.--It is as if his eyes had suddenly opened.
One can ask, for example: if he suddenly recognizes so and so, could he have the same sudden visual
experience, but without recognizing the person? Well, he might for instance be mistaken in recognizing him. [LW I,
572]
Page Break 16
[I haven't yet estimated correctly the beginning that the child is making.]
Page 16
What if someone were to ask: "Do I really do that with my eyes?" [LW I, 573]
Page 16
The same expression which before was a report of what was seen now is an exclamation.
Page 16
A hare runs across a path. Someone doesn't know it and says: "Something strange whizzed by" and he
proceeds to describe the appearance. Someone else says "A hare!", and he cannot describe it so precisely.
Now why do I still want to say that the person who recognizes it sees it differently from the person who
doesn't? [LW I, 574]
Page 16
It is the well-known impression.
Page 16
Does someone who doesn't recognize a smile as a smile see it differently than someone who does? He reacts
to it differently. [Cf. LW I, 575; PI II, xi, p. 198e]
What can be cited in support of his seeing it differently? [LW I, 576]
Page 16
"If one knows what it is, it looks different."--How so? [LW I, 577]
Page 16
What would it be like if someone were not acquainted with it, but still knows all about it right away? Does he
then see it in the same way as the one who is acquainted with it?--What should I say? [Cf. LW I, 578]
Page 16
It's a question of the fixing of the concepts. [LW I, 579]
Page 16
I'm mentioning these kinds of aspects in order to show the kind of multiplicity we are dealing with here. [LW
I, 580]
Page Break 17
Here there is a host of interrelated phenomena and concepts. [Cf. LW I, 581; PI II, xi, p. 199d]
Page 17
Sometimes the conceptual is dominant. (What does that mean?) Doesn't it mean: sometimes the experience
of an aspect can be expressed only through a conceptual explanation. And this explanation in turn can take many
forms. [LW I, 582]
Page 17
Here it is important to consider that there is a host of interrelated phenomena and concepts. [Cf. LW I, 581;
PI II, xi, p. 199d]
Page 17
Just think of the words exchanged by lovers! They're 'loaded' with feeling. And surely you can't just agree to
substitute for them any other progressions of sound you please. Isn't this because they are gestures? And a gesture
doesn't have to be something innate; it is instilled, and yet assimilated.--But isn't that a myth?!--No. For the signs of
assimilation are that I want to use this word, that I prefer to use none at all to using one that is forced on me, and
similar reactions. [LW I, 712]
Page 17
"I noticed the likeness for maybe 5 minutes." "After 5 minutes I no longer noticed the likeness, but at first
very strongly."
"After 5 minutes the likeness no longer struck me, but at first very strongly." [Cf. LW I, 707]
Page 17
... "I noticed the likeness for maybe 5 minutes, and then no longer." [Cf. LW I, 707; PI II, xi, p. 210f]
Page 17
"I'm no longer struck by it," but what happens when I am struck by it?
Well, I look at the face in such and such a manner, say this and that to myself or to others, think this and
that. But is that being struck by the similarity? No, these are the phenomena of being struck, but these are 'what
happens'.
Page Break 18
'Being struck' is a different (and related) type of concept from 'phenomenon of being struck'. [Cf. LW I, 708; PI II, xi,
p. 211d]
Page 18
But aren't thinking and saying different kinds of things! And isn't the thinking the being struck?
Page 18
I can say such and such words to myself without thinking of their content.
Page 18
Thinking and inward speech (I do not say "talking to oneself") are different concepts. [LW I, 709; PI II, xi, p.
211f]
Page 18
Is being struck: looking and thinking?
Page 18
No. Many concepts cross here. [LW I, 710; PI II, xi, p. 211e]
Page 18
How does the chemist know that there is an Na atom at this point in the structure. A question of the
criterion, not a psychological question. [Cf. LW I, 786]
Page 18
A child learns a certain way of writing our letters, but it doesn't know that there are ways of writing, and
doesn't know the concept 'way of writing'.
Page 18
--- If not, one could not very well call it a blindness. [Cf. PI II, xi, p. 214a]
Page 18
--- Well, his defect will be more or less related to this one.
Page 18
But if I want to say "This word (in the poem) stood there like a picture†1
Page Break 19
Page 19
"The word (in the poem) is not different from a picture of what it means"---
Page 19
If the sentence can appear to me like a word-painting ('Joyous songs resound in the green dell†1)
Page 19
But if a sentence can strike me like a painting in words and the very individual word in the sentence like a
picture, then it is not quite so much of a marvel that a word uttered outside of all context and without purpose seems
to carry a particular meaning in itself. [PI II, xi, p. 215c]
Page 19
Experience of direction.
Page 19
Think here of a special kind of illusion which throws light on these matters. [Cf. PI II, xi, p. 215d; LW I, 787]
Page 19
In what way is a mental image, a word, etc., a germ? It is the beginning of an interpretation.
I was able to see a part of a line and then say it was N's shoulder, and then that it is N, who..., etc. But I did
not deduce from the line that it is the shoulder etc.
Page 19
Now what does it mean to say that in searching for a name or a word one feels, experiences, a gap which can
only be filled by a particular thing, etc. Well, these words could be the primitive expression in the place of the
expression "The word is on the tip of my tongue". James's expression is actually only a paraphrase of the usual one.
Page 19
James really wants to say: What a remarkable experience! The word is not here and yet already here, or
something is
Page Break 20
here which cannot grow into anything but this word. But this is not experience at all. The words "It's on the tip of
my tongue" are not the expression of an experience and James only interprets them as the description of the content
of an experience. (Cf. PI II, xi, p. 219d; LW I, 841]
Page 20
"It's on the tip of my tongue" no more expresses an experience than "Now I've got it!" It's an expression
which we use in certain situations and is surrounded by a certain behaviour, and also by several characteristic
experiences. [Cf. PI II, xi, p. 219e; LW I, 842]
Page 20
Doesn't something special happen after all when a word comes to you? Listen carefully.--Listening carefully
doesn't do you any good. You could do no more than discover what's going on in yourself at that time. [Cf. PI II, xi,
p. 219a]
Page 20
And how, when I'm doing philosophy, can I listen for that at all. And yet I can imagine that I do. How does
that come about? What am I actually paying attention to?
Page 20
Could one imagine that people view lying as a kind of insanity. They say "But it isn't true, so how can you
say it then?!" They would have no appreciation for lying. "But he won't say that he is feeling pain if he isn't!--If he
says it anyway, then he's crazy." Now one tries to get them to understand the temptation to lie, but they say: "Yes, it
would certainly be pleasant if he believed---, but it isn't true!"--They do not so much condemn lying as they sense it
as something absurd and repulsive. As if one of us began walking on all fours.
Page 20
In which way does uncertainty, the possibility of deceit,
Page Break 21
create difficulties with the concept of pain?? [Cf. LW I, 876]
"I am certain that he's in pain."--What does that mean? How does one use it? What is the expression of
certainty in behaviour, what makes us certain?
Not a proof. That is, what makes me certain doesn't make someone else certain. But the discrepancy has its
limits.
Page 21
Don't think of being certain as a mental state, a kind of feeling, or some such thing. The important thing
about certainty is the way one behaves, not the inflection of voice one uses in speaking.
Page 21
The belief, the certainty, a kind of feeling when uttering a sentence. Well, there is a tone of conviction, of
doubt, etc.. But the most important expression of conviction is not this tone, but the way one behaves.
Page 21
When you think that one can be certain that someone else is in pain you shouldn't ask "What goes on in
me†1 then?", but "How does that get expressed?"
Page 21
Ask not "What goes on in us when we are certain---?", but "How does it show?" [Cf. PI II, xi, p. 225b]
Page 21
A man's thinking goes on within his consciousness in a seclusion in comparison with which any physical
seclusion is openness.†2 [PI II, xi, p. 222c]
Page 21
The future is hidden from us. But does the astronomer feel that who calculates an eclipse of the sun? [Cf. PI
II, xi, p. 223d]
Page Break 22
Page 22
The inner is hidden.----The future is hidden. [PI II, xi, p. 223d]
Page 22
But doesn't the same thing----i.e. the same feeling----correspond to the word in the primitive exclamation
and in the statement†1? Doesn't the child who cannot yet speak have the same feeling as the one who can? How can
they be compared? Well, compared this way it is the same feeling.
Page 22
Doesn't the child in a primitive way express precisely that feeling that the other one reports?
Page 22
Logical impossibility and psychological impossibility.
Page 22
If I see someone writhing in pain with evident cause I do not think, all the same, his feelings are hidden from
me. [PI II, xi, p. 223e]
Page 22
"Such and such is the case." On the one hand it has the sound, on the other the striding nature, of a sentence.
It is a motion which begins and comes to an end. Precisely not one sign, which designates something, but rather
something that has sense, which sets up a sense that exists without regard to truth or falsity.†2 It is the arrow and not
the point.
(But where is the mistake?)
Page 22
"... is the case" is simply a sentence. But I would not after all have used just any other meaningful sentence in
its place.
Page 22
Our concept is of such a kind.--But could we have a different one then? One that brings behaviour, occasion
and
Page Break 23
experience†1 into a necessary connection? Why not? But in that case we would have to be made in such a way that
all of us or almost all of us in fact would react in the same way†2 under the same circumstances. For when we
believe that the expression of his feelings is genuine, in general we behave differently from when we believe the
opposite.
Page 23
But this correspondence does not exist, and therefore we would not know what to do with a necessary
concept. (Heap of stones.†3)
Page 23
--- therefore because different things speak for the truth of his statement, and the statement has different
consequences.
Page 23
--- If he is sincere he can†4 tell us them, but my sincerity is not enough to guess his motives. This is where there is
the similarity with knowing. [Cf. PI II, xi, p. 224f]
Page 23
Subjective and objective certainty. [Cf. PI II, xi, p. 225c]
Page 23
Why do I want to say "2 × 2 = 4" is objectively certain, and "This man is in pain" only subjectively? [Cf. PI
II, xi, p. 224e]
Page 23
There can be a dispute over the correct result of a calculation, for instance, of a rather long addition.†5 But
such a dispute is rare and is quickly decided if it arises.
This is a fact that is essential for the function of mathematics.
Page Break 24
Page 24
[Physicist, paper and ink, reliability.] [Cf. PI II, xi, p. 225d; cf. also p. 226b, c]
Page 24
There can also be a disagreement about what colour an object is. To one person it appears as a somewhat
yellowish red, to another as a pure red. Colour blindness can be recognized by specific tests.
Page 24
There is no such agreement over the question whether an expression of feeling is simulated or genuine. [See
PI II, xi, p. 227e]
Page 24
Why not?--What do you want to know?
Page 24
Let's assume you say: This man distrusts the utterance because he is more distrustful than that man, who
trusts it.
The question is, how can the disposition of the one making the judgement play an important role here if it
doesn't do so elsewhere? Or also: How can such a judgement then be correct? How can one nevertheless speak of a
judgement here?†1
Page 24
I want to call the observations on mathematics which are part of these philosophical investigations "the
beginnings of mathematics". [Cf. PI II, xiv, p. 232b; LW I, 792]
Page 24
We are playing with elastic, indeed even flexible concepts. But this does not mean they can be deformed at
will and without offering resistance, and are therefore unusable. For if trust and distrust had no basis in objective
reality, they would only be of pathological interest.
Page 24
But why do we not use more definite concepts in place of these vague ones?
Page Break 25
But not: objective certainty does not exist because we do not see into someone else's soul. This expression means
that.
Page 25
If constant quarrels were to erupt among mathematicians concerning the correctness of calculations, if for
instance one of them were convinced that one of the numbers had changed without his noticing it or his memory
had deceived him or someone else, etc. etc.,--then the concept of 'mathematical certainty' would either not exist or it
would play a different role than it does in fact. It could be the role of the certainty, for instance, that God answers a
prayer for rain; either by sending rain or by not sending it--for such and such and such reasons. [Cf. PI II, xi, p.
225d]
Page 25
Then it might be said, for instance: "True, we can never be certain†1 what the result of a calculation is, but it
always has a quite definite result, which God knows.
It is of the highest certainty, though we only have a crude reflection of it." [PI II, xi, pp. 225e-226a]
Page 25
If I say therefore "In all schools in the world the same multiplication tables are taught"--what kind of a
statement is that? It is one about the concept of the multiplication table. [Cf. PI II, xi, p. 227c]
Page 25
"In a horse race the horses generally run as fast as they can." In this way one could explain to someone what
the word "horse race" means. [Cf. PI II, xi, p. 227c]
Page 25
As 'mathematical certainty' falls, so does 'mathematics'.
Page Break 26
Think of learning mathematics and the role of its formulas.
Page 26
Show what it's like when one is in pain.---Show what it's like when one pretends that one is in pain.
Page 26
In a play one can see both portrayed. But now the difference! ([[sic]] Cf. LW I, 863]
Page 26
--- How would they learn to use the words? And is the language-game they learn still the same one we call the use of
colour-words?
Page 26
--- In saying this one could want to say that in none of our schools a fool taught arithmetic. However it can†1
Page 26
There is such a thing as colour-blindness and there are ways of establishing it. Among those who are not
colour-blind there is in general no argument about (their) colour-judgements.
This is a remark about the concept of colour-judgements. [See PI II, xi, p. 227d; cf. LW I, 931]
Page 26
And yet I am not happy about this expression. Why? Is it only because a child doesn't actually learn to
dissimulate? Indeed, it would not even have to learn what surrounds dissimulation. Imagine a child were born with
the behaviour of a grown-up. Certainly it cannot yet talk, but it already has decided likes and dislikes, and clearly
expresses joy, disgust, thankfulness, etc., in its facial expressions and gestures.
So does it already have to be able to nod its head? Or use certain inflections of sounds? [Cf. LW I, 945]
Page 26
--this particular and not at all simple pattern in the drawing
Page Break 27
of our life.†1 (Cf. PI II, xi, p. 229a]
Page 27
And what would the opposite now look like?--How well defined would the borders of evidence be?
One would recognize only with the possibility of error that someone was, for example, sad. But what kind of
concept of sadness is that now? The old one?
Page 27
A tribe in which no one ever dissimulates, or if they do, then as seldom as we see someone walking on all
fours in the street.†2 Indeed if one were to recommend dissimulation to one of them, he might behave like one of us
to whom one recommends walking on all fours. But what follows? So there is also no distrust there. And life in its
entirety now looks completely different, but not on that account necessarily more beautiful as a whole.
Page 27
It doesn't yet follow from a lack of dissimulation that each person knows how someone else feels.
Page 27
But this too is imaginable.--If he looks like this, then he is sad. But that does not mean: "If he looks like this,
then that is going on within him," but rather something like: "If he looks like this, then we can draw with certainty
those conclusions which we frequently only can draw without certainty; if he does not look like that, we know that
these conclusions are not to be drawn".
Page 27
One can say that our life would be very different if people said all of those things aloud that they now say to
themselves, or if this could be read externally.
Page Break 28
But now imagine you were to come into a society in which, as we want to say, feelings can be recognized with
certainty from appearances (we are not using the picture of the inner and the outer). But wouldn't that be similar to
coming from a country where many masks are worn into one where no, or fewer, masks are worn? (Thus perhaps
from England to Ireland.) Life is just different there.
Page 28
Frequently one will say: I do not understand these people. [Cf. PI II, xi, p. 223f]
One also says: I don't understand this person's joy and sadness. And what does that mean? Doesn't it mean
that as I understand the words he is actually not sad and not happy? And now what does it mean to say: Maybe
exactly the same thing is going on within him as within me, only it is expressed differently?
Page 28
Consider that we not only fail to understand someone else when he hides his feelings, but frequently also
when he does not hide them, indeed when he does his utmost to make himself understood.
Page 28
Under certain circumstances, "The inner is hidden" would be as if one said: "All that you see in a
multiplication is the outer movement of figures; the multiplication itself is hidden from us."
Page 28
The uncertainty as to what is going on within someone else does not stand against his own utter lack of
doubt. [Cf. LW I, 963]
Page 28
If I say "I don't know for sure what he wants", that does not mean: by contrast with the man himself. For it
can be completely clear to me what he wants without this therefore being my wish.
Page Break 29
I can only guess what he's calculating in his head. If it were otherwise, I could report it to someone and have it
confirmed by the one doing the calculating. But would I then know of everyone who calculates what he is
calculating? How do I make the connection with him? Well, here one or the other thing can be assumed.
Page 29
What do I know when I know that someone is sad? Or: what can I do with this knowledge?--For instance, I
know what is to be expected from him.
But if I now also know that this or that will cheer him up, then that is a different kind of knowing.
Page 29
Even if I were now to hear everything that he is saying to himself, I would know as little what his words were
referring to as if I read one sentence in the middle of a story. Even if I knew everything now going on within him, I
still wouldn't know, for example, to whom the names and images in his thoughts related.
Page 29
After all, you can't expect a human to be more transparent than a closed crate, for instance.
Page 29
But this remains, that sometimes we do not know whether someone is in pain, or is only pretending.
And if it were otherwise, there would be various possibilities.
Page 29
Tennis without a ball--silent speaking and lip-reading. [Cf. LW I, 854-855]
Page 29
It is not the case that every time someone screams he is in pain; rather, if he screams under certain
circumstances which are difficult to describe, and acts in a way that is difficult to describe, then we say he is in pain,
or is probably in pain.--And what are pains?--For I must be able to explain this word, after all. Well, I might prick
him with a needle
Page Break 30
and say that is pain. But given what we've said above, it can't be that easy to explain. Thus the whole concept of
'pain' becomes tangled.
The way in which we learn to use the word, and therefore the way in which it is used, is more complicated,
difficult to describe. For instance it is first taught under certain circumstances where there is no doubt, i.e. where
there is no question of doubt.
Page 30
The uncertainty that is always there is not about whether he is perhaps pretending (for he could have been
imagining that he was pretending), but rather is about the complicated connection of the words 'to be in pain' with
human behaviour. When such a concept is useful is another question.
Page 30
How can I learn to describe these circumstances? Was I taught to do it? Or what would I have to observe to
do this?
Page 30
And just as little can I describe the circumstances in which one says that someone is pretending, feigning
pain.
Is such a description of interest? Under certain circumstances a good number of its aspects are of interest.
Page 30
Why can't you be certain that someone is not pretending?--"Because one cannot look into him."----But if you
could, what would you see there?----"His secret thoughts."----But if he only utters them in Chinese,--where do you
have to look then?----"But I cannot be certain that he is uttering them truthfully!"----But where do you have to look
to find out whether he is uttering them truthfully?
Page 30
What goes on within also has meaning only in the stream of life. [See LW I, 913]
Page Break 31
"But for him there is no doubt, after all, about whether he is pretending. So if I could look into him, there wouldn't
be any for me either."
Page 31
How about this: Neither I nor he can know that he is pretending. He may confess to it and in that case, to be
sure, there is no error. I may assume it with full certainty and good reasons, and what follows may confirm that I
was right.
Page 31
Or: I can know that he is in pain, or that he is pretending; but I do not know it because I 'look into him'.
Page 31
But if a way of seeing his nerves working were now found, wouldn't that really be a means of finding
whether he is in pain? Well, it could give a new direction to the way we behave and could also correspond more or
less with the old directions. And could you ask for more than to see the workings of the nervous system?
Page 31
It can happen that I don't know whether he is pretending or not. If that is the case, what is the reason? Could
one say: "That <I> don't see his nervous system working"?
But does there have to be a reason? Couldn't I simply know whether he is pretending without knowing how
I know it?
I simply would have 'an eye' for it. [c: cf. PI II, xi, p. 228e]
Page 31
I don't know what he is saying behind my back----but does he also have to think something behind my
back?
Page 31
That is: what goes on in him is also a game, and pretence
Page Break 32
is not present in him like a feeling, but like a game.
Page 32
For also, if he speaks to himself his words only have meaning as elements of a language-game.
Page 32
On the one hand I cannot know whether he is pretending because our concept of pretence, and therefore of
the certainty of pretence, is what it is----on the other hand, because, even assuming a somewhat different concept of
pretence, certain facts are as they are.
For it is conceivable that we could have access to criteria of pretence which are not in fact accessible, and that
if they became accessible to us we would really take them as criteria.
Page 32
What am I hiding from him when he doesn't know what is going on inside me?
How and in which way am I hiding it?
Page 32
Physically hidden----logically hidden.
Page 32
I say "This man is hiding what is in him". How does one know that he is hiding it? Thus there are signs for it
and signs against it.
Page 32
There is an unmistakable expression of joy and its opposite.
Page 32
Under these circumstances one knows that he is in pain, or that he isn't; under those, one is uncertain.
Page 32
But ask yourself: what allows one to recognize a sign for something within as infallible? All we are left with
to measure it against is the outer. Therefore the contrast between the inner and the outer is not an issue.
Page Break 33
Yet there are cases where only a lunatic could take the expression of pain, for instance, as sham.
Page 33
"I don't know whether he likes or dislikes me; indeed I don't even know whether he knows it himself."
Page 33
Is it logically or physically impossible to know whether someone else remembers something?
Page 33
I say that I don't remember, but in reality I do. What I want to say is that it is not at all a matter of what goes
on within me as I speak. So actually I am not hiding anything at all from him, for even if something is going on
within me, and he can never see it, then what is going on here cannot interest him.
So does that mean that I am not lying to him? Of course I am lying to him; but a lie about inner processes is
of a different category from one about outer processes.
Page 33
If I lie to him and he guesses it from my face and tells me so--do I still have the feeling that what is in me is in
no way accessible to him and hidden? Don't I feel rather that he sees right through me?
Page 33
It is only in particular cases that the inner is hidden from me, and in those cases it is not hidden because it is
the inner.
Page 33
Suppose that we had a kind of snail shell, and that when our head was outside then our thinking, etc., was
not private, but it was when we pulled it in.
Page 33
One might think of cases in which someone turns his face away so that the other cannot read it.
Page 33
My thoughts are not hidden from him if I utter them involuntarily and he hears this. Oh yes they are, because
Page Break 34
even in that case he doesn't know whether I really mean what I say, and I do know it. Is that correct?
But in what does that consist that I know whether I mean it? And above all: Can't he know it too?
And what if my honest confession were less reliable than someone else's judgement?
Or: What kind of a fact is it: that it is not so?
Page 34
If the consequences that can generally be based on the confession of my motive could not be based on it,
that would mean that this whole language-game didn't exist.
Page 34
A problem of relativity.
Page 34
In general I can sketch a clearer more coherent picture of my life than someone else.
Page 34
The question could also be put this way: Why does one in general aim toward a confession in the case of a
crime, for instance. Does this not mean that a confession is more reliable than any other report?
Page 34
Therefore at bottom there must be a general fact here (similar for instance to the one that I can predict the
movements of my own body).
Page 34
Roughly, it must be the case that in general I can give a more coherent report about my actions than
someone else. In this report the inner plays the role of theory or construction, which complements the rest of it to
form a comprehensible whole.
Page 34
All the same: There are other criteria for my reliability.
Page 34
My thoughts are not hidden from him, but are just open
Page Break 35
to him in a different way than they are to me.
Page 35
The language-game simply is the way it is.
Page 35
If one speaks of being logically hidden, then that is a bad interpretation.
Page 35
"I know what I mean." What does that mean? For instance, that I didn't simply speak to hear myself talk, that
I can explain what I mean and the like. But would it be correct to say it about my everyday speech? Or doesn't
someone else know it just as well.
Page 35
"I know whether I am lying or not."
Page 35
The question is, how does the mendacious utterance turn into something important?†1
Page 35
Do not look at pretending as an embarrassing appendage, as a disruption of the pattern.
Page 35
One can say "He is hiding his feelings". But that means that it is not a priori they are always hidden. Or:
There are two statements contradicting one another: one is that feelings are essentially hidden; the other, that
someone is hiding his feelings from me.
Page 35
If I can never know what he is feeling, then neither can he pretend.
Page 35
For pretending must mean, after all, getting someone else
Page Break 36
to make a wrong guess as to what I feel. But if he now guesses right and is certain of being right, then he knows it.
For of course I can also get him to guess right about my feelings and not be in doubt about them.
Page 36
The inner is hidden from us means that it is hidden from us in a sense in which it is not hidden from him.
And it is not hidden from the owner in this sense: he utters it and we believe the utterance under certain conditions
and there is no such thing as his making a mistake here. And this asymmetry of the game is brought out by saying
that the inner is hidden from someone else.
Page 36
Evidently there is an aspect of the language-game which suggests the idea of being private or hidden--and
there is also such a thing as hiding of the inner.
Page 36
If one were to see the working of the nerves, utterances would mean little to us and pretending would be
different.
Page 36
Or should I say that the inner is not hidden, but rather hide-able? He can hide it within himself. But that's
wrong again.
Page 36
"He screams when he is in pain, not I." Is that an empirical sentence?
Page 36
"I am feigning pain" doesn't stand on the same level as "I am in pain". After all, it is not an utterance of
feigning.
Page 36
"When does one say that someone is in pain?" That is a sensible question, and has a clear kind of
answer.----"When does one say that someone is feigning pain?" After all that must be a sensible question too.
Page 36
Can one imagine the signs and the occasions of pain being
Page Break 37
something utterly other than what they actually are? Say their being signs, etc., of joy?----So the signs of pain and
pain-behaviour determine the concept 'pain'. And they also determine the concept 'feigning pain'.
Page 37
Could one imagine a world in which there could be no pretence?
Page 37
If one 'is sad because one cries', why then isn't one also in pain because one screams?
Page 37
One has to look at the concepts 'to be in pain' and 'to simulate pain' in the third and first person. Or: the
infinitive covers all persons and tenses. Only the whole is the instrument, the concept.
Page 37
But what then is the point of this complicated thing? Well, our behaviour is damned complicated, after all.
Page 37
And how is it with feelings' being private or hidden?
Page 37
A society in which the ruling class speaks a language which the serving class cannot learn. The upper class
places great importance on the lower one never guessing what they feel. In this way they become unfathomable,
mysterious.
Page 37
What kind of hiding is the speaking of a language that the other cannot understand? [Cf. PI II, xi, p. 222d]
Page 37
Is the if-feeling (for instance) the readiness to make a certain gesture. And does its being related to feelings
consist in that? [Cf. PI II, vi, p. 181e-182a-f]
Page 37
We interpret the word "if", spoken with this expression, as the expression of a feeling. [Cf. LW I, 373-376]
Page Break 38
Page 38
The use (of the word) seems to fit the word.
Page 38
Question: is the wenn-feeling the same as the if-feeling? If one wants to decide this question one enunciates
the words to himself with their characteristic intonation.
Page 38
Instead of "attitude toward the soul" one could also say "attitude toward a human". [See PI II, iv, p. 178d]
Page 38
I could always say of a human that he is an automaton (I could learn it this way in school in physiology) and
yet it would not influence my attitude toward someone else. After all, I can also say it about myself.
Page 38
But what is the difference between an attitude and an opinion?
I would like to say: the attitude comes before the opinion.
Page 38
(Isn't belief in God an attitude?) [a, b: cf. PI II, iv, p. 178d]
Page 38
How would this be: only one who can utter it as information believes it.
Page 38
An opinion can be wrong. But what would an error look like here?
Page 38
Is the if-feeling the correlate of an expression?--Not solely. It is the correlate of meaning and of the
expression.†1
Page 38
The atmosphere of a word is its use. Or: We imagine its
Page Break 39
use as its atmosphere. [Cf. PI II, vi, p. 182d]
Page 39
The 'atmosphere' of a word is a picture of its use.
Page 39
We look at a word in a certain environment, spoken with a certain intonation, as an expression of feeling.
Page 39
This passage has a strong expression. It is immensely expressive. I repeat it to myself again and again, make
a special gesture, paraphrase it. But a feeling? Where is it? I'd almost like to say: in the stomach. And yet it is
immediately clear that no (such) feeling exhausts the passage. The passage is a gesture. Or it is related to our
language. One could also imagine a drawing that would be impressive in the same way.
Page 39
The if-feeling: Could we imagine a poem in which we would sense this feeling especially strongly?
('Sabre'-feeling.)
Page 39
Only I can utter my thoughts, feelings, etc.
The utterances of my feelings can be sham. In particular they can be feigned. That is a different
language-game from the primitive one, the one of genuine utterances.†1
Page 39
Is there anything astonishing in this?
Page 39
Is there anything astonishing about the possibility of a primitive and a more complicated language-game?
Page 39
"The child doesn't know enough yet to pretend." Is that right?
Page Break 40
Page 40
The question is: When would we say of a child, for instance, that it is pretending? What all must it be able to
do for us to say that?
Only when there is a relatively complicated pattern of life do we speak of pretence. [Cf. PI II, xi, p. 229b; LW
I, 939-940, 946]
Page 40
Or to put it another way: Only when there is a relatively complicated pattern of life do we speak of certain
things as possibly being feigned. [Cf. LW I, 946]
Page 40
Of course this is not a common way of looking at things.
Page 40
It is a purely geometric way of looking at things, as it were. One into which cause and effect do not enter.
Page 40
One could ask, "What does a battle (for instance) look like?" What picture does it present? Here it doesn't
matter to us whether a sword splits a skull and whether someone falls down because his skull was split.
Page 40
To say "He knows what he is thinking" is nonsense; "I know what he is thinking" may be true. [Cf. PI II, xi,
p. 222b]
Page 40
If as I was assuming people really could see someone else's nervous system working, and adjust their
behaviour toward him accordingly, then, I believe, they wouldn't have our concept of pain (for instance) at all,
although maybe a related one. Their life would simply look quite different from ours.
Page 40
That is, I look at this language-game as autonomous. I merely want to describe it, or look at it, not justify it.
Page 40
I do not say that evidence makes the inner merely probable.
Page Break 41
For as far as I'm concerned nothing is lacking in the language-game. [Concerning "evidence", cf. PI II, xi, p. 228b-d]
Page 41
That the evidence makes the inner only probable consists in†1
Page 41
"But after all I must be able to say, whether it's right or wrong, that someone is in pain, or again that they are
pretending!"--Right and wrong exist only to the extent of the evidence.
Page 41
But in any case I can think that I am right or wrong;--whether the evidence is sufficient or not! What good
does it do me that I can think it?--More than that I can say it!----To be sure an image may be in my mind, but how
do I know that, and how, it can be used?
Page 41
The image and its use would therefore have to be in my mind.
Page 41
At first it could be said that it is our determination whether we see something as a definite criterion of pain
(for instance), whether we see all of this as a criterion of anything at all. But then we have to say that the whole thing
is not our determination, but is rather a part of life.
Page 41
Can an idiot be too primitive to pretend? He could pretend the way an animal does. And this shows that
from here on there are levels of pretence.
Page 41
There are very simple forms of pretence.
Page 41
Therefore it is possibly untrue to say that a child has to learn a lot before it can pretend. To do this it must
grow, develop, to be sure. [Cf. LW I, 868, 939]
Page 41
An animal cannot point to a thing that interests it.
Page Break 42
One will only speak of pretence if there are different cases and degrees of pretence.
Page 42
A great variety of reactions must be present.
Page 42
A child must have developed far before it can pretend, must have learned a lot before it can simulate.
Page 42
That is: simulating is not an experience.
Page 42
The possibility of pretence seems to create a difficulty. For it seems to devalue the outer evidence, i.e. to
annul the evidence.
Page 42
One wants to say: Either he is in pain, or he is experiencing feigning. Everything on the outside can express
either one.
Page 42
Above all pretence has its own outward signs. How could we otherwise talk about pretence at all?
Page 42
So we are talking about patterns in the weave of life.
Page 42
So do you want to say that the l.p.†1 of genuine and feigned pain do not exist?
Page 42
But can I describe them?
Page 42
Imagine it were really a case of patterns on a long ribbon.
The ribbon moves past me and now I say "This is the pattern S", now "This is the pattern V". Sometimes for
a period of time I do not know which it is; sometimes I say at the end "It was neither".
How could I be taught to recognize these patterns? I am shown simple examples, and then complicated ones
of both
Page Break 43
kinds. It is almost the way I learn to distinguish the styles of two composers.
But why in the case of the patterns does one make this distinction that is so difficult to grasp? Because it is
of importance in our life.
Page 43
The main difficulty arises from our imagining the experience (the pain, for instance) as a thing, for which of
course we have a name and whose concept is therefore quite easy to grasp.
Page 43
So we always want to say: We know what "pain" means (namely this), and so the difficulty only consists in
simply not being able to determine this in someone else with certainty. What we don't see is that the concept 'pain' is
only beginning to be investigated. The same is true of pretence.
Page 43
Why don't we form a simpler concept?--Because it wouldn't interest us.--But what does that mean? Is it the
correct answer?
Page 43
Should I say: Our concepts are determined by our interest, and therefore by our way of living†1?
Page 43
As children we learn concepts and what one does with them simultaneously.
Sometimes it happens that we later introduce a new concept that is more practical for us.--But that will only
happen in very definite and small areas, and it presupposes that most concepts remain unaltered.
Page 43
Could a legislator abolish the concept of pain?
The basic concepts are interwoven so closely with what
Page Break 44
is most fundamental in our way of living that they are therefore unassailable.
Page 44
Everything I say presupposes that there is a house over there. Or rather: this is presupposed in them. So for
instance: A is in this house = There is a house over there and A is in it.
Page 44
Is it correct to say that the order "Go into the house!" presupposes that there is a house there and that the
person giving the order knows it?
Page 44
If someone were to say "Go into this house" when there is no house there, we would say of him: "He
believes that there is a house there". But is this less right when there actually is one there?
Page 44
From a practical sentence no philosophical one can follow. Moore's sentence was a practical one left
indeterminate.
Page 44
Can we imagine that other people have other colour concepts?----The question is: should we call other
concepts colour concepts?
Page 44
Does the dog believe that his master is in front of the door, or does he know it?
Page 44
Bad influence of Aristotelian logic. The logic of language is immeasurably more complicated than it looks.
[Cf. LW I, 525]
Page 44
The examples which philosophers give in the first person should be investigated in the third.
Page 44
Imagine the situation in which we ask someone: "Do you believe that, or do you know it?"
Page Break 45
In what cases does one say "He knows it", and in which ones "He doesn't know it"?
Page 45
Consider the question: "Does he know that is a book?" And particularly the use of the word "that".
Page 45
"I see it exactly and know that it is a book."
Page 45
"I know that that is a tree."----"That what is a tree?"
Page 45
"I don't know whether it is a tree, but I do know that it is a physical object."
Page 45
You say "That is a tree" and also that by "that" you mean the visual image. That allows a substitution in the
first sentence.
Page 45
If one says, "I know that a physical object corresponds to this impression", then one is referring to a
confirmation through other impressions.
Now if one doesn't acknowledge this kind of a confirmation--one is changing the language-game.
Page 45
"I know."
"I am certain."
We say, for instance, "I know that it is so" if someone reports a well-known fact to us. In this case we do not
say "I am certain that it is so". ("I know that that is the Schneeberg.") Were I to answer "I am certain that it is the
Schneeberg", then one would say "It isn't subject to any doubt at all!"
Page 45
Imagine that one were to explain "I know it" as: I learned it and it isn't subject to doubt.
Page 45
Imagine someone were to doubt that a tree is called "tree".
Page Break 46
Page 46
"I know that this is the earth"--saying which I stamp my foot on the ground.
Doubting. What kind of a game is this in which one asks: "How certain is this sentence for you?" [Cf. OC, 387]
Page 46
Would it be correct to say: "I sit down because I know that this is a chair; I reach for something because I
know that that is a book; etc. etc." What is gained by this? I am using it to say that all these doubts do not exist for
me. Further, that doesn't mean that they don't exist at all.
Page 46
No doubt arises about all of this. But that is not enough. In a certain class of cases we don't know what
consequences doubt would have, how it could be removed, and, therefore, what meaning it has.
Page 46
What then does this belief that our concepts are the only reasonable ones consist in? That it doesn't occur to
us that others are concerned with completely different things, and that our concepts are connected with what
interests us, with what matters to us. But in addition, our interest is connected with particular facts in the outer
world.
Page 46
But do we always have to be able to give the reason for the formation of a concept?
Page 46
"That wouldn't be a smile at all."
Page 46
Why should a movement not belong to a smile?
Page 46
"There's something mechanical about that smile." "Actually it is not a real smile at all."
Only in a chess game does one call something "castling".
Page 46
"Why do we have a concept 'to pretend'?"--"Well,
Page Break 47
because humans often pretend."--Is that the right answer? [Cf. LW I, 255, 261]
Page 47
What if someone were to answer: "Because with this concept we can do those things we want to do"?
Isn't it as if one were to ask: "Why do we have the concept of irrational numbers?" How could one answer
that?
Page 47
We acknowledge a truthful person's statement about what he has just thought as well as his statement about
what he has dreamed.
Page 47
Even if we frequently could guess someone's thoughts and were to say we know what they are, then the
criterion for that could only be that he himself confirmed our guess. Unless we totally change the concept of
thought. [Cf. PI II, xi, p. 222e]
Page 47
We paint pictures of transparent yellow, green, blue and red glasses with different backgrounds so that we
get clear as to what the appearance of coloured translucency is. And now analogously we want to paint the picture
of a transparent white glass.
Page 47
We can express ourselves in physical terms here although the physical doesn't interest us. It is a good image
of what we want to describe.--A transparent yellow glass reflects no yellow light into the eye, and therefore the
yellow doesn't seem localized in the glass. Flat black seen through yellow glass is black, white is yellow. Therefore
analogously black must appear black seen through transparent white, and white white, i.e. just as through a
colourless glass.--Is red now to appear whitish? i.e. pink? But what will a dark red, which tends toward black, appear
as? It should become a blackish pink, i.e. a greyish red, but then black probably will not remain black.
Page Break 48
By 'pure white' one often means the lightest of all colours, by black the darkest; but not so by pure yellow, red, etc.
Page 48
White seen through yellow wouldn't become yellowish-white, but yellow. And yellow seen through
white--should it become whitish--yellow or white? In the first case the 'white' glass acts like colourless glass, in the
second like opaque glass.
Page 48
Thus I want to say: The 'pure' concept of colour, which one is inclined to create from our normal colour
concepts, is a chimera. To be sure there are different colour concepts and among them those that can be called purer
and less pure.
Page 48
Instead of "chimera" I could have said "false idealization".
Perhaps the Platonic ideas are false idealizations.
If there is such a thing then, someone who idealizes falsely must talk nonsense--because he uses a mode of
speaking that is valid in one language-game in another one where it doesn't belong.
Page 48
If types are deposited somewhere, who says which types? All that can be thought of?!
Page 48
What is the ideal representation of colour? Isn't it something like looking through a tube and seeing a small
red circle (for instance)?--And am I now to name the colours according to this experience? Fine, but now I also have
to apply these colour words in completely different cases. And how am I to compare them with the colours around
me? And how useful will such a comparison be?--Or is the ideal way of showing a colour to fill the entire visual field
with it? As when one turns one's gaze towards the blue sky? But the old question still remains. For don't forget that
your glance wanders and that the description of what you see doesn't exist.
Page Break 49
'It doesn't make any sense: he knows my thoughts.' Thus the inquiry after someone else's thoughts is not the game in
which "knowing" should be applied.--Thus the sentence refers to the entire language-game.†1
Page 49
But does an astronomer calculating an eclipse of the moon say that the future can never be known? That is
said when one feels uncertain about it.----Does the manufacturer say that of course one cannot know†2 whether his
cars will work? [Cf. PI II, xi, p. 223d; cf. LW I, 189]
Page 49
Whoever utters that sentence makes a distinction, draws a line; and it may be an important line.--Does it
become more important because of actual uncertainty?
Page 49
Then one can ask: What is the characteristic of what we can really know? And the answer will be: One can
only know where no error is possible, or: where there are clear rules of evidence.
Page 49
"I know that he enjoyed seeing me."--What follows from that? What of importance follows? Forget that you
have the correct idea of the state of his mind! Can I really say that the importance of this truth is that it has certain
consequences?--It is pleasant to be with someone who is glad to see us, who behaves in such and such a way (if one
knows a thing or two about this behaviour from previous occasions).
So if I know that he is happy, then I feel secure, not insecure, in my pleasure. And that, one could say, isn't
knowing.--Still it is different if I know that he is seeing what he claims to be seeing.
"I know that he was sincerely pleased to see me."
Page Break 50
II:
MS 170
Page 50
(around 1949)
Page Break 51
Page 51
People who don't have the concept 'tomorrow'. They still could have a quite well-developed language:
various commands, questions, descriptions. Could we communicate with them?--But could we describe to them
how people use the word "tomorrow", without teaching it to them? What purpose could the description serve?
'Tomorrow' plays such a great role because the change from day to night is so important to us. If it were
not... [Cf. RC III, 116]
Page 51
If one wanted to give a rough description of the game with "tomorrow", analogously to a rough description
of the differential calculus, then it would have to be a lot more primitive, and it would be difficult to imagine a
purpose for it.
But think about what concepts people have for curved space.
Page 51
Even if someone's behaviour is very regular in itself, still it is hard for us to grasp this regularity if his
behaviour deviates from ours in strange ways. Then one might say "I cannot get used to his...". Consider also that
wish creates expectation.
Page 51
The language of someone who as an imbecile lives among normal people and is cared for by them. Perhaps
he doesn't know the concept 'tomorrow'. [Cf. RC III, 118]
Page 51
Operating with concepts permeates our life. I see some sort of analogy with a very general use of keys. If for
instance one always had to open a lock in order to move something.
Page 51
Can the psychologist teach us what seeing is? He doesn't teach us the use of the word 'to see'. Is "seeing" a
technical term of psychology? Is "dog" a technical term of zoology?
Page Break 52
Perhaps the psychologist discovers differences among people that are not noticed in everyday life and show up only
under experimental conditions. But blindness is not something that the psychologist discovers.
If seeing were something that the psychologist has discovered, then the word "seeing" could only mean a
form of behaviour, an ability to act in such and such a way. So if the psychologist were to pronounce "There are
people who see", then he would have to be able to describe for us the behaviour of these seeing people. But in doing
this he would not have taught us the use of the form "I see something red and round", for instance, and more
specifically would not have taught this to a sighted person. [a: cf. RC III, 337-338]
Page 52
Couldn't a seeing person manage completely without the word "see"? He might say "Over there there is... ".
A normal child could manage for a long time without the word "see", but not for instance without the words "red",
"yellow," "round".
Page 52
If I observe the course of my pains, which sense-impressions am I supposed to have had if I had not been
observing? Would I have felt nothing? Or would I only have not remembered?
Page 52
"I wouldn't have seen it if I hadn't observed it."--What do the words "it" refer to? To the same thing?
"I wouldn't have felt the pain if I hadn't observed the pain."
But one can say after all "Observe your pain" and not "Feel pain!"
Page 52
Test: "Most chairs do not evaporate."
"If something like that had happened, I certainly would have heard about it."
Page Break 53
To be sure one can also say in this case "It's always been like that, so it will be like that this time too."--But
how does one know that it always was that way?
Page 53
The one seems to be supported by the other, but neither obviously serves as the basis for the other.
Page 53
We say "Undoubtedly it is so", and don't know how very much this certainty determines our concepts.
To the question "Did the earth really exist before your birth" we would respond, half annoyed and half
embarrassed, "Yes, of course!" All the while we would be conscious that on the one hand we are not at all capable of
giving reasons for this because seemingly there are too many, and on the other hand that no doubt is possible, and
that one cannot answer the questioner by way of one particular piece of instruction, but only by gradually imparting
to him a picture of our world.
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III:
MS 171
Page 54
(1949 or 1950)
Page Break 55
Page 55
An inner, in which it looks either like this or like that; we are not seeing it. In my inner it is either red or blue.
I know which, no one else does.
Page 55
If pretending were not a complicated pattern, it would be imaginable that a new-born child pretends.
Page 55
Therefore I want to say that there is an original genuine expression of pain; that the expression of pain
therefore is not equally connected to the pain and to the pretence.
Page 55
That is: the utterance of pain is not equally connected with the pain and the pretence.
Page 55
The important aspect for us is not†1 that the evidence makes the experience of someone else 'only probable',
but rather that we regard precisely these phenomena as evidence for something important.†2
Page 55
But let's assume that from the very first moment a child was born it could pretend, indeed in such a way that
its first utterance of pain is pretence.--We could imagine a suspicious attitude toward a new-born child: but how
would we teach it the word "pain" (or "a hurt")? Say in a questioning tone. Then we might view consistent behaviour
as proof of genuineness.
Page 55
Consider that you have to teach the child the concept. Thus you have to teach it evidence (the law of
evidence, so to speak).
Page 55
Remarkable the concept to which this game of evidence belongs.
Page Break 56
Our concepts, judgements, reactions never appear in connection with just a single action, but rather with the
whole swirl of human actions.†1
Page 56
Only I know what I am thinking actually means nothing else than: only I think my own thoughts.
Page 56
Can one imagine people who don't know pretence and to whom one cannot explain it?
Can one imagine people who cannot lie?--What else would these people lack? We should probably also
imagine that they cannot make anything up and do not understand things that are made up.
Page 56
Whoever couldn't pretend also couldn't play a role.
Page 56
Isn't the difficulty this: the pretence resides in the intention? For we could after all imitate the behaviour of
pain exactly, without pretending.
Page 56
The capacity to pretend therefore resides in the ability to imitate, or in the ability to have this intention.
But we must assume that a subject can say the words "I am in pain". Therefore it is a matter of having the
capacity to intend. Is it possible, for instance, to imagine people who cannot lie because for them a lie would be
nothing but a dissonance. I want to imagine a case where people are truthful not as a matter of morality, but rather
see something absurd in a lie. Whoever lies would be viewed as mentally ill.
Or better: Lying or pretending would have to appear to these people as perversity.
Page Break 57
Page 57
Is it correct to say that a fixed smile is actually no smile at all? How does one recognize that it isn't?
Page 57
Smiling is one mien within a normal range of miens.--But is that an arbitrary determination? This is the way
we learn to use the word.
Page 57
The remark that... is not important to us, but rather the remark that this involved thing is a kind of evidence
for us.
Page 57
Someone groans under anaesthesia or in sleep. I am asked "Is he in pain?" I shrug my shoulders or say "I
don't know whether he's in pain". Sometimes I acknowledge something as a criterion for it, but sometimes I don't.
Well, do I mean nothing by this? Oh yes: I am making a move in an existing game. But this game wouldn't
exist if there weren't criteria in other cases.
The doubt in the different cases has a different colouring, so to speak.
One might say "a different truth-value".
Page 57
"I happen to know that this is a sycamore; a sycamore is an external object, therefore there are external
objects."
Page 57
Something turns out to be pain or pretence. And that is essential to the concepts 'pain' and 'pretence', even if
it is not evident in every single one of their applications.
Page 57
"Beyond a reasonable doubt." [Cf. OC 416, 607]
Page 57
I know... = I am certain that it is so and it is so.
I knew... = I was certain that it is so, and it was so.
Page Break 58
Page 58
I know how it is = I can say how it is, and it is as I say it is.
Page 58
A blind person touches an object and asks me "What is that?"--I answer "A table."--He: "Are you
certain?"--I: "I know it is."
Page 58
"I know..." = I have the highest degree of certainty.
When Moore uses it, then it is as if he wanted to say: "The philosophers are always saying that one can have
the feeling of knowing only in this and that case, but I have it also in this and this and this case." He looks at his
hand, gives himself the feeling of knowing, and now says he has it.
Page 58
What purpose do the statements "He knows" and "I know" serve?
How is it shown that someone knows something? For only if that is clear is the concept of knowing clear.
Page 58
If someone says "Yes, now I know that it is a tree" and if he also says it on the right occasion, then this alone
is not yet a sign that he is using the word "know" as we do.
Page 58
"I know that there is a tree here." One can say this for instance when for some reason one wants to repeat his
own words (as when one recites a passage from a book by heart). Now how do we know which use you have made
of the sentence? You can tell us. It could be the following: I'm thinking of people who say that it is uncertain that...,
and now I say "No, it is not uncertain: I know that..." (As in "I know that he is not deceiving me".) Now, whoever
says "I know that that is a tree" in this way means a tree and not this or that.
Page Break 59
Page 59
It is true that Moore knows that this is a tree; this shows in his entire behaviour. From this it does not follow
that he does not misunderstand the words "I know etc." in philosophizing. He proved his misunderstanding by
looking at his hands and saying "I know that these are hands" instead of simply noting "I know an immense number
of facts concerning physical objects". And what is more, they are so certain for me that nothing can strengthen or
destroy this certainty.
Page 59
What we find remarkable is not that..., rather we are looking at the fact that this is evidence for us.
Page 59
"In the inner there is either pain or pretence. On the outside there are signs (behaviour), which don't mean
either one with complete certainty."
But that's not the way it is. In an extremely complicated way the outer signs sometimes mean
unambiguously, sometimes without certainty: pain, pretence and several other things.†1
Page 59
"Nothing is as common as the colour reddish-green; for nothing is more common than the transition of
leaves from green to red."
Page 59
"Believing, knowing, an experience which one recognizes as this very thing†2 while one is having it."†3
Page Break 60
IV:
MS 173
Page 60
(1950)
Page Break 61
Page 61
"One knows when someone is really happy." But that doesn't mean that one can describe the genuine
expression. But of course it is not always true that one recognizes the genuine expression, or knows whether the
expression is genuine. Indeed there are cases where one is not happy either with "genuine" or "sham". Someone
smiles and his further reactions fit neither a genuine nor a simulated joy. We might say "I don't know my way
around with him. It is neither the picture (pattern) of genuine nor of pretended joy."
Mightn't his relation to a person with normal feelings be like that of a colour-blind person to the
normal-sighted?
Page 61
On the basis of my knowledge of his character I could state reliably that he will react in such and such a way
in this situation; and it would also be possible that others can rely on my judgement without however being able to
demand of me that I support my judgement with a verifiable description.
Page 61
Let's assume that a painter represented the expression of blissful joy--and I see the picture and say "Maybe
she's pretending."
Page 61
It is at least conceivable that in some country a court relies on a man's statement about what is possible for
him, if a witness has known him for a certain length of time. In this way even now one might ask a psychiatrist
whether this or that person is capable of suicide. It is assumed in this connection that in general experience does not
disprove such a statement.
Page 61
I am trying to describe the laws or rules of evidence for empirical sentences: does one really characterize
what is meant by the mental in this way?
Page 61
The characteristic sign of the mental seems to be that one has to guess at it in someone else using external
clues and is
Page Break 62
only acquainted with it from one's own case.
But when closer reflection causes this view to go up in smoke, then what turns out is not that the inner is
something outer, but that "outer" and "inner"†1 now no longer count as properties of evidence.†2 "Inner evidence"
means nothing, and therefore neither does "outer evidence".
Page 62
But indeed there is 'evidence for the inner' and 'evidence for the outer'.
Page 62
"But all I ever perceive is the outer." If that makes sense, it must determine a concept. But why should I not
say I perceive his doubts? (He cannot perceive them.)
Page 62
Indeed, often I can describe his inner, as I perceive it, but not his outer.
Page 62
The connection of inner and outer is part of these concepts. We don't draw this connection in order to
magically remove the inner.
There are inner concepts and outer concepts.
Page 62
What I want to say is surely that the inner differs from the outer in its logic. And that logic does indeed
explain the expression "the inner", makes it understandable.†3
Page Break 63
We don't need the concept "mental" (etc.) to justify that some of our conclusions are undetermined, etc. Rather this
indeterminacy, etc., explains the use of the word "mental" to us.
Page 63
"Of course actually all I see is the outer."
But am I not really speaking only of the outer? I say, for instance, under what circumstances people say this
or that. And I do always mean outer circumstances. Therefore it is as if I wanted to explain (quasi-define) the inner
through the outer. And yet it isn't so.
Page 63
Is the reason for this that the language-game is something outer?
Page 63
No evidence teaches us the psychological utterance.
Page 63
"Mental" for me is not a metaphysical, but a logical, epithet.
Page 63
"I see the outer and imagine an inner that fits it."
Page 63
When mien, gesture and circumstances are unambiguous, then the inner seems to be the outer; it is only
when we cannot read the outer that an inner seems to be hidden behind it.
Page 63
There are inner and outer concepts, inner and outer ways of looking at man. Indeed there are also inner and
outer facts -just as there are for example physical and mathematical facts. But they do not stand to each other like
plants of different species. For what I have said sounds like someone saying: In nature there are all of these facts.
Now what's wrong with that?
Page 63
The inner is tied up with the outer not only empirically, but also logically.
Page Break 64
The inner is tied up with the outer logically, and not just empirically.
Page 64
"In investigating the laws of evidence for the mental, I am investigating the essence of the mental." Is that
true?
Page 64
Yes. The essence is not something that can be shown; only its features can be described.
Page 64
But doesn't a prejudice argue against this? To be sure we can little by little enumerate the properties of an
inkwell, but its essence--mustn't it stand fast once and for all, isn't it presented to us with this very object, before our
eyes? What we have here in front of us surely isn't the 'use of a word'! Certainly not; but the concept 'inkwell', which
is necessary here after all, is not tangibly in front of us, nor does what is in front of us contain this concept. In order
to represent it, it is not enough to put an inkwell in someone's hand. And this is not because that person is too
lame-brained to read the concept off the object.
Page 64
I can show someone an object because its colour stands out and I want to demonstrate it to him, but that
already presupposes that there is a certain game between us.
Page 64
Indeed he might be astonished when he sees the object, but in order to 'be astonished about the colour', in
order for the colour to be the reason of his astonishment and not just the cause of his experience, he needs not just
sight, but to have the concept of colour.
Page 64
Someone says on his word of honour that someone else believed this or that.--At that point one can ask him,
"How do you know that", and he can answer "He assured me of this with utmost seriousness, and I know him
extremely well".
Page Break 65
If I say "I can't figure him out", this bears little resemblance to: "I can't figure this mechanism out." I think it means
approximately: I can't foresee his behaviour with the same certainty as with people 'with whom I do know my way
about'.
Page 65
The question of evidence for what is experienced has to be connected with the certainty or uncertainty of
foreseeing someone else's behaviour. But that's not quite the way it is; for only rarely does one predict someone
else's reaction.
Page 65
I think unforeseeability must be an essential property of the mental. Just like the endless multiplicity of
expression.
Page 65
What for instance speaks for, what against, a dog's having a mental life?
Page 65
Certainly it isn't its shape, colour, or anatomy. So it is its behaviour.
Page 65
Those who say that a dog has no soul support their case by what it can and cannot do. For if someone says
that a dog cannot hope--from what does he deduce that? And whoever says that a dog has a soul can only support
that with the behaviour he observes in the dog.
"Just look at the face and the movements of a dog, and you'll see that it has a soul." But what is it about the
face? Is it only the similarity with the play of the features of the human face? Is it, at least among other things, the
lack of stiffness?
Page 65
The important fine shades of behaviour are not predictable.
Page 65
But does that mean: If they could be foreseen, with a
Page Break 66
human we wouldn't speak of an inner as opposed to an outer?----Are we really imagining this kind of predictability
clearly? Does it imply for instance that we wouldn't ask him for a decision?
Page 66
Imagine we were to encounter a human who had no soul. Why shouldn't something like that occur as an
abnormality? So a human body would have been born with certain vital functions, but without a soul. Well, what
would that look like?
Page 66
The only thing I can imagine in that case is that this human body acts like an automaton, and not like normal
human bodies.
Page 66
When they say "Man consists of a body and a soul", then this would not be contradicted by such a
phenomenon. For then this would be no (real) human, but something else, something very rare to be sure. But how
can one know that it never happens? Only,--what would this phenomenon actually look like?
Page 66
Or is it supposed not to be a phenomenon at all? Should having a soul not be recognizable at all?
Page 66
Can there be heartlessness that has no expression? Would that be what we call "heartlessness"?
Page 66
One could also put it this way: How would a human body have to act so that one would not be inclined to
speak of inner and outer human states?
Again and again, I think: "like a machine".
Page 66
Perhaps language, along with tone of voice and the play of features, is the most subtly gradated behaviour of
men.
Page 66
Could the soulless one produce signs of pain? If he only
Page Break 67
screamed and writhed then one could still view this as an automatic reaction, but if he grimaced in pain and had a
suffering look, then we would already have the feeling that we were looking into him.
But now, what if he always produced exactly the same suffering expression?
Page 67
It is as if he became transparent to us through a human facial expression.
Page 67
Anyone with a soul must be capable of pain, joy, grief, etc. etc. And if he is also to be capable of memory, of
making decisions, of making a plan for something, with this he needs linguistic expression.
Page 67
It is not as if he had only indirect, while I have internal direct evidence for my mental state. Rather, he has
evidence for it, (but) I do not.
Page 67
But if one now says that this evidence makes the mental only probable, that can have many meanings, and
they can be true or false. And it certainly doesn't mean that the evidence is only empirically connected with the
mental (like a symptom with an illness).
Page 67
Why shouldn't one say: "The evidence for the mental in someone else is the outer"?
Well, there is no such thing as outer mediated and inner unmediated evidence for the inner.†1
Page 67
And to the extent that the evidence is uncertain, isn't this because it is only outer.
Page 67
That an actor can represent grief shows the uncertainty of evidence, but that he can represent grief also
shows the reality of evidence.
Page Break 68
It is not the relationship of the inner to the outer that explains the uncertainty of the evidence, but rather the other
way around--this relationship is only a picture-like representation of this uncertainty.
Page 68
It isn't only the mental that is represented to us on the stage; we are also given the illusion of a wound, or a
mountain.
Page 68
Its being portrayable on the stage†1 is not the sole characteristic of the mental.
Page 68
Why do we say: "I didn't know what went on behind this brow", although it can be of no importance to us
whatsoever what goes on behind someone's brow. Our uncertainty doesn't at all refer to what goes on in the inner;
and even if it does refer to the mental, the mental finds its expression in the bodily.
So an uncertainty about the outer corresponds to an uncertainty concerning the inner.
Just as an uncertainty about the numeral that will come at the bottom line corresponds to an uncertainty
about the number that is the result of a calculation.
Page 68
And that does not mean that in general the uncertainty about something mental†2 can be expressed as
uncertainty about the outer.
Just as, to be sure, sorrow in its essence has an expression in one's mien, and yet I still may not be able to
describe a mien other than by using the word "sorrowful".
Page 68
Could someone state in court: "I know that at that time he thought of..."? Well, this kind of statement could
be admitted or not. Perhaps a judgement would be made that
Page Break 69
someone who has known the accused for so many years can deduce from his mien etc. what he thinks in a certain
case. But perhaps such a statement wouldn't be admitted at all, and the opinion would be that not even an utterance
of the accused could be entered as evidence if the only point of doing so is to describe his mental processes.
Page 69
"I can't figure these people out." And why should I want to?--Isn't it their reactions that I can't figure out?
That for instance I cannot foresee; that keep on surprising me?
"He seems to react illogically." And that means: inconsequently.
Page 69
If one can't figure out some things this means that one can figure out other things. And sometimes that is
expressed by saying one 'could imagine' what goes on in someone else. That sounds as if knowing what goes on in
someone else is an imagining of this process. For instance if I know that someone hates me then I feel a kind of
visual image of that hate. This opinion rests on a host of false ideas. To be sure one uses the words "to imagine
someone else's hate (etc.)", and indeed image pictures can play a role here, or perhaps one makes a face like one that
is filled with hate.
Page 69
From the outset the language-game is constructed so that a comparison with other language-games can lead
to the 'outer-inner' picture. But to this is added the factual uncertainty that is part of guessing†1 someone else's
mental processes. For, as has been said, it would be quite possible for this recognizing to be much more certain than
it is. Indeed that pretending might take place mainly by hiding one's face
Page Break 70
(for instance). That is: pretending would be possible even if one couldn't put on a false face.
Page 70
But it is not true that uncertainty in recognizing his irritation (for instance) is simply uncertainty about his
future behaviour. Rather in the concept there is an uncertainty of criteria. So sometimes he is transparent, as it were,
and sometimes he isn't. And it is misleading to think of the real irritation as a facial expression of an inner face, so to
speak, such that this facial expression is defined completely clearly, and that it is only the outer face that makes it
uncertain whether the soul really has this expression.
Page 70
For even if he himself says without lying that he was a bit irritated, that doesn't mean that he then saw in
himself that face that we called 'irritated'. Again we only have a verbal reaction from him, and it is by no means clear
how much it means. The PICTURE is clear, but not its application.
Page 70
For even when I myself say "I was a little irritated about him" how do I know how to apply these words so
precisely? Is it really so clear? Well, they are simply an utterance.
Page 70
But do I not know exactly what I mean by that utterance? "After all, I know exactly what inner state I am
calling that." That means nothing. I know how the word is used, and sometimes I make this utterance unhesitatingly,
and sometimes I hesitate and say, for instance, that I wasn't 'exactly irritated', or some such thing. But it is not this
indeterminacy I was speaking of. Even where I say without hesitation that I was irritated, that does not establish how
certain the further consequences of this signal are.
Page Break 71
When I said that there is an indeterminacy in the application I didn't mean that I didn't really know when I should
utter the expression (as it would perhaps be if I didn't understand English well).
Page 71
One simply mustn't forget which connections are made when we learn how to use expressions such as "I am
irritated".
Page 71
And don't think of a child's guessing the correct meaning, for whether it guessed it correctly must in turn be
demonstrated in its use of the words.
Page 71
We say: "Let's imagine people who do not know this language-game." But in doing so we still have no clear
conception of the life of these people in so far as it differs from our own. We do not yet know what we are supposed
to imagine; for the life of these people is in all other ways to correspond with ours, and it still must be determined
what we would call a life corresponding to ours under these new conditions.
Isn't it as if one said: There are people who play chess without the king? Questions immediately arise: Who
wins now, who loses, and others. You have to make further decisions that you don't anticipate in that first
determination. Just as you also don't have an overview of the original technique, and are only familiar with it from
case to case.†1
Page 71
It is also a part of dissembling to regard others as capable of dissembling.
Page Break 72
If human beings act in such a way that we are inclined to suspect them of dissembling, but they show no mistrust of
one another, then this doesn't present a picture of people who dissemble.
Page 72
'We cannot help but be constantly surprised by these people.'
Page 72
We could portray certain people on the stage and have them speak in monologues (asides) things that in real
life they of course would not say out loud, but which would nevertheless correspond to their thoughts. But we
couldn't portray alien humans this way. Even if we could predict their behaviour, we couldn't give them the
appropriate asides.
And yet there's also something wrong with this way of looking at it. For someone might actually say
something to himself while he was going about doing things, and this could, for example, be quite conventional.
Page 72
That I can be someone's friend rests on the fact that he has the same possibilities as I myself have, or similar
ones.
Page 72
Would it be correct to say our concepts reflect our life?
Page 72
They stand in the middle of it.
Page 72
The rule-governed nature of our language permeates our life.
Page 72
Of whom would we say, he doesn't have our concept of pain? I could assume that he knows no pain, but I
want to assume that he does know it; we thus assume he gives expressions of pain and we could teach him the
words "I have pain". Should he also be capable of remembering his pain?--Should he recognize expressions of pain
in others as
Page Break 73
such; and how is this revealed? Should he show pity? Should he understand make-believe pain as being just that?
Page 73
"I don't know how irritated he was." "I don't know if he was really irritated."--Does he know himself? Well,
we ask him, and he says, "Yes, I was."
Page 73
What then is this uncertainty about whether the other person was irritated? Is it a mental state of the
uncertain person? Why should we be concerned with that? It lies in the use of the expression "He is irritated".
Page 73
But one is uncertain, another may be certain: he 'knows the look on this person's face' when he is irritated.
How does he learn to know this sign of irritation as being such? That's not easy to say.
Page 73
But it is not only: "What does it mean to be uncertain about the state of another person?"--but also: "What
does it mean 'to know, to be certain, that this person is irritated'?"
Page 73
Here it could now be asked what I really want, to what extent I want to deal with grammar.
Page 73
The certainty that he will visit me and the certainty that he is irritated have something in common. The game
of tennis and the game of chess have something in common, too, but no one would say here: "It is very simple: they
play in both cases, it's just that each time they play something different." This case shows us the dissimilarity to
"One time he eats an apple, another time a pear", while in the other case it is not so easy to see.
Page 73
"I know that he arrived yesterday"--"I know that 2 × 2 = 4"--"I know that he had pain"--"I know that there is
a table standing there."
Page Break 74
In each case I know, it's only that it's always something different? Oh yes,--but the language-games are far more
different than these sentences make us conscious of.
Page 74
"The world of physical objects and the world of consciousness." What do I know of the latter? What my
senses teach me? That is how it is, if one sees, hears, feels, etc. etc. But do I really learn that? Or do I learn what it's
like when I now see, hear, etc., and I believe that it was also like this before?
Page 74
What actually is the 'world' of consciousness? There I'd like to say: "What goes on in my mind, what's going
on in it now, what I see, hear,..." Couldn't we simplify that and say: "What I am now seeing."
Page 74
The question is clearly: How do we 'compare' physical objects how do we compare experiences?
Page 74
What actually is the 'world of consciousness'?--That which is in my consciousness: what I am now seeing,
hearing, feeling... And what, for example, am I now seeing? The answer to that cannot be: "Well, all that",
accompanied by a sweeping gesture.
Page 74
I observe this patch. "Now it is like so"--and simultaneously I point, for example, to a picture. I may
constantly observe the same thing and what I see may then remain the same, or it may change. What I observe and
what I see do not have the same (kind of) identity. Because the words "this patch", for example, do not allow us to
recognize the (kind of) identity I mean.
Page 74
"Psychology describes the phenomena of colour-blindness as well as those of normal sight." What are the
'phenomena of colour-blindness'? Certainly the reactions of the colour-blind person which differentiate him from the
normal person. But certainly not all of the colour-blind person's reactions, for example not those that distinguish
him from a
Page Break 75
blind person. Can I teach the blind what seeing is, or can I teach this to the sighted? That doesn't mean anything.
Then what does it mean: to describe seeing? But I can teach human beings the meaning of the words "blind" and
"sighted", and indeed the sighted learn them, just as the blind do. Then do the blind know what it is like to see? But
do the sighted know? Do they also know what it's like to have consciousness?
But can't psychologists observe the difference between the behaviour of the sighted and the blind?
(Meteorologists the difference between rain and drought?) We certainly could, for example, observe the difference
between the behaviour of rats whose whiskers had been removed and of those which were not mutilated in this way.
And we could call that describing the role of this tactile apparatus.--The lives of the blind are different from those of
the sighted.
Page 75
The normal person can, for instance, learn to take dictation. What is that? Well, one person speaks and the
other writes down what he says. Thus, if he says, for example, the sound a, the other writes the symbol "a",
etc.--Now mustn't someone who understands this explanation either already have known the game, only perhaps
not by this name,--or have learned it from the description? But Charlemagne certainly understood the principle of
writing and still couldn't learn to write.†1 Someone can thus also understand the description of a technique yet not
be able to learn it. But there are two cases of not-being-able-to-learn. In the one case we merely fail to gain a certain
skill, in the other we lack comprehension. We can explain a game to someone: He may understand this explanation,
but not be able to learn the game, or he may be incapable of understanding an explanation of the game. But the
opposite is conceivable as well.
Page 75
"You see the tree, the blind do not see it." This is what I would have to say to a sighted person. And so do I
have
Page Break 76
to say to the blind: "You do not see the tree, we see it"? What would it be like for the blind man to believe that he
saw, or for me to believe I couldn't see?
Page 76
Is it a phenomenon that I see the tree? It is one that I correctly recognize this as a tree, that I am not blind.
Page 76
"I see a tree", as the expression of the visual impression--is this the description of a phenomenon? Of what
phenomenon? How can I explain this to someone?
And yet isn't the fact that I have this visual impression a phenomenon for someone else? Because it is
something that he observes, but not something that I observe.
The words "I am seeing a tree" are not the description of a phenomenon. (I couldn't say, for example, "I am
seeing a tree! How strange!", but I could say: "I am seeing a tree, although there's no tree there. How strange!")
Page 76
Or should I say: "The impression is not a phenomenon; but that L. W. has this impression is one"?
Page 76
(We could imagine someone talking to himself describing the impression as one does a dream, without using
the first person pronoun.)
Page 76
To observe is not the same thing as to look at or to view.
"Look at this colour and say what it reminds you of." If the colour changes you are no longer looking at the
one I meant.
One observes in order to see what one would not see if one did not observe.
Page 76
We say, for example "Look at this colour for a while." But we don't do that in order to see more than we
would have seen at first glance.
Page Break 77
Could a "Psychology" contain the sentence: "There are human beings who see"?
Well, would that be false?--But to whom would this communicate anything? (And I don't just mean: what is
being communicated is a long-familiar fact.)
Page 77
Is it a familar fact to me that I see?
Page 77
We might want to say: If there were no such humans, then we wouldn't have the concept of seeing.--But
couldn't Martians say something like this? Say by chance the first humans they met were all blind.
Page 77
And how can it be meaningless to say "There are humans who see", if it is not meaningless to say there are
humans who are blind?
But the meaning of the sentence "There are humans who see", i.e. its possible use, is not immediately clear at
any rate.
Page 77
Couldn't seeing be the exception? But neither the blind nor the sighted could describe it, except as an ability
to do this or that. Including e.g. to play certain language-games; but there we must be careful how we describe these
games.
Page 77
If we say "There are humans who see", the question follows: "And what is 'seeing'?" And how should we
answer it? By teaching the questioner the use of the word "see"?
Page 77
How about this explanation: "There are people who behave like you and me, and not like that man over
there, the blind one"?
Page 77
"With your eyes open, you can cross the street and not be run over, etc."
Page Break 78
The logic of informing.
Page 78
To say that a sentence which has the form of information has a use, is not yet to say anything about the kind
of use it has.
Page 78
Can the psychologist inform me what seeing is? What do we call "informing someone what seeing is"?
It is not the psychologist who teaches me the use of the word "seeing".
Page 78
If the psychologist informs us "There are people who see", we could ask him "And what do you call 'people
who see'?" The answer to that would be of the sort "Human beings who react so-and-so, and behave so-and-so
under such-and-such circumstances". "Seeing" would be a technical term of the psychologist, which he explains to
us. Seeing is then something which he has observed in human beings.
Page 78
We learn to use the expressions "I see... ", "he sees...", etc., before we learn to distinguish between seeing and
blindness.
Page 78
"There are people who can talk", "I can say a sentence", "I can pronounce the word 'sentence'", "As you see,
I am awake", "I am here".
Page 78
There is surely such a thing as instruction in the circumstances under which a certain sentence can be a piece
of information. What should I call this instruction?
Page 78
Can I be said to have observed that I and other people can go around with our eyes open and not bump into
things and that we can't do this with our eyes closed?
Page Break 79
When I tell someone I am not blind, is that an observation? I can, in any case, convince him of it by my behaviour.
Page 79
A blind man could easily find out whether I am blind too; by, for example, making a certain gesture with his
hand, and asking me what he did.
Page 79
Couldn't we imagine a tribe of blind people? Couldn't it be capable of surviving under certain circumstances?
And couldn't sighted people occur as exceptions?
Page 79
Suppose a blind man said to me: "You can go about without bumping into anything, I can't"--would the first
part of that sentence transmit a piece of information?
Page 79
Well, he's not telling me anything new.
Page 79
There seem to be propositions that have the character of experiential propositions, but whose truth is
unassailable for me. That is to say, if I assume that they are false, I must mistrust all my judgements.
Page 79
There are, in any case, errors which I take to be commonplace and others that have a different character and
which must be set completely apart from the rest of my judgements as temporary confusions. But aren't there
transitional cases between these two?
Page 79
If we introduce the concept of knowing into this investigation, it will be of no help; because knowing is not a
psychological state whose special characteristics explain all kinds of things. On the contrary, the special logic of the
concept "knowing" is not that of a psychological state.
Page Break 80
V:
MS 174
Page 80
(1950)
Page Break 81
Page 81
The utterance of pain is not connected equally with pain and with pretence.
Page 81
Pretending is not as simple a concept as being in pain. [Cf. LW I, 876]
Page 81
Remember that you have to teach a child the concept. Therefore you have to teach it the game of evidence.
Page 81
That our evidence makes someone else's experience only probable doesn't take us far; but that this pattern of
our experience that is hard to describe is an important piece of evidence for us does.†1
Page 81
That this fluctuation is an important part of our life.
But how can one say at all that it is something fluctuating? Against what do I measure its fluctuation? Well,
there are countless configurations of smiling, for instance. And smiling that is smiling, and smiling that is not.
Page 81
What do we take note of in life?--"... He smiled at that point."--that can be infinitely important. But does a
small distortion of the face have to be important? And does it have to be so for us because of the probable practical
consequences?
Page 81
"He cannot know what's going on within me." But he can surmise it. So the only thing he can't do is know it.
Therefore we are only making a distinction in the use of the word "know".
Page 81
But does an astronomer who calculates a lunar eclipse say that of course one cannot know the future? We
express ourselves in this way when we feel uncertain about the future. The farmer says it about the weather; but the
carp
Page Break 82
enter doesn't (say) that one cannot know whether his chairs will collapse.
Page 82
"I know that he was glad to see me." What do I know? What consequences does this fact have? I feel certain
in my dealings with him. But is that knowing?
But what is the difference between surmising and knowing that he was glad?
If I know it I'll assert it without signs of doubt, and others will understand this statement. Well yes, it does
have certain practical consequences; in a pinch something can be deduced from it, but that seems merely to be its
shadow.
What is the interest of his inner state of gladness?
Page 82
If I believe that he was glad and find out later that this was not so, what consequences does that have?
Page 82
What difference does it make if at first I believe that he was glad and then realize that it wasn't true?
Page 82
We would like to project everything into his inner. We would like to say that that's what it's all about.
For in this way we evade the difficulty of describing the field of the sentence.†1
Page 82
It's exactly as if one said that "Benzene has the structure <image>" means: the atoms are arranged in this
way.
Page 82
But why do I say that I 'project' everything into the inner? Doesn't it reside in the inner? No. It doesn't reside
in the inner, it is the inner. And that is only a superficial logical classification and not the description we need.
Page Break 83
We 'project' nothing into his inner; we just give an explanation that doesn't get us any further.
Page 83
Imagine that the soul is a face, and when someone is glad this hidden face smiles. Let it be this way--but now
we still want to know what importance this smile (or whatever the facial expression is) has.
Page 83
Indeed this could even be our regular expression: "His inner face smiled when he saw me", etc.
Page 83
First question: How does one know, how does one judge, whether his inner face is smiling? Second
question: What importance does it have?----But both are connected. And one could ask another, although related,
question: What importance does his--outer--smile have? For if the inner is of importance, then--in a (somewhat)
different way--so must the outer be.
Page 83
(It is not easy to realize that my manipulations are justified.)
Page 83
But if "I know that he was glad" certainly does not mean: I know that he smiled, then it is something else
that I know and that is important here.
Page 83
For under certain circumstances the inner smile could replace an outer one, and the question about the
meaning would (still) remain unanswered.
Page 83
"I'm certain that he was glad to see me"; this could be stated in a court of law. Here the possible 'practical'
consequences are clear. And this would be equally the case if the statement were "I am certain that he was not
happy, but that he was pretending". One thing is to be expected of someone who is glad, another of him who feigns
gladness.
Page Break 84
But is the fact that someone else is really glad to see me important to me because it has different consequences? I am
comfortable when this person (with this past etc.) behaves in this way. And the 'in this way' is a very complicated
pattern, to be sure.
Page 84
If one doesn't want to SOLVE philosophical problems why doesn't one give up dealing with them. For
solving them means changing one's point of view, the old way of thinking. And if you don't want that, then you
should consider the problems unsolvable.
Page 84
It's always presupposed that the one who smiles is a human being and not just that what smiles is a human
body. Certain circumstances and connections of smiling with other forms of behaviour are also presupposed. But
when all that has been presupposed someone else's smile is pleasing to me.
If I ask someone on the street for directions then I prefer a friendly answer to an unfriendly one. I react
immediately to someone else's behaviour. I presuppose the inner in so far as I presuppose a human being.
Page 84
The 'inner' is a delusion. That is: the whole complex of ideas alluded to by this word is like a painted curtain
drawn in front of the scene of the actual word use.
Page 84
It seems to me: if one can't really know whether someone is (for instance) irritated, then one also cannot
really believe or surmise it.
Page 84
Isn't it true that whatever I can 'be certain of' I can also 'know'?
Page 84
Wouldn't it be ridiculous if a lawyer in court were to say
Page Break 85
that a witness couldn't know that someone had been angry, because anger is something inner?--Then one also cannot
know whether hanging is punishment.
Page 85
Whoever says "one cannot know that" makes a distinction between language-games. He says: In such
language-games knowing exists, in such it doesn't, and in doing so he limits the concept 'knowing'.
Page 85
This limitation could be useful if it emphasizes an important difference that is passed over by our ordinary
use of language. But I believe that that is not the way it is.
Page 85
But isn't mathematical certainty greater than any physical certainty, to say nothing of the certainty about
what someone else feels?
Page 85
And can't the greater certainty of mathematics simply be expressed this way: There is knowing in
mathematics?
Page 85
In mathematics a particular kind of evidence that can be clearly presented leaves no doubt open. That is not
the way it is when we know that someone was glad.
There can't be a long dispute in a court of law about whether a calculation has this or that result; but there
certainly can be about whether someone was irritated or not.
But does it follow that one can know the one and not know the other? More likely what follows is that in the
one case one almost always knows the decision, in the other, one frequently doesn't.
Page 85
If one says that one never knows whether someone else really felt this way or that, then that is not because
perhaps after all he really felt differently, but because even God so to speak cannot know that the person felt that
way.
Page Break 86
I am for instance convinced that my friend was glad to see me. But now, in philosophizing, I say to myself that it
could after all be otherwise; maybe he was just pretending. But then I immediately say to myself that, even if he
himself were to admit this, I wouldn't be at all certain that he isn't mistaken in thinking that he knows himself. Thus
there is an indeterminacy in the entire game.
One could say: In a game in which the rules are indeterminate one cannot know who has won and who has
lost.
Page 86
There is a 'why' to which the answer permits no predictions. That's the way it is with animistic explanations,
for instance. Many of Freud's explanations, or those of Goethe in his theory of colours, are of this kind. The
explanation gives us an analogy. And now the phenomenon no longer stands alone; it is connected with others, and
we feel reassured.
Page 86
If someone 'pretends friendship and then finally shows his true feelings, or confesses', we normally don't
think of doubting this confession†1 in turn, and of also saying that we cannot know what's really going on inside
him. Rather, certainty now seems to be achieved.
Page 86
This is important: I might know from certain signs and from my knowledge of a person that he is glad, etc.
But I cannot describe my observations to a third person and--even if he trusts them--thereby convince him of the
genuineness of that gladness, etc.
Page 86
One says of an expression of feeling: "It looks genuine". And what meaning would that have were there not
convincing criteria for genuineness? "That seems genuine" only makes sense if there is a "That is genuine".
Page Break 87
"This weeping gives the impression of being genuine"--so there is such a thing as genuine weeping. So there is a
criterion for it. "But no certain one!"
Page 87
How does someone who accepts a criterion as certain differ from someone who doesn't?
Page 87
But does accepting no criterion as certain mean: never being certain that someone else feels this or that way?
Can I be not quite certain and yet accept no criterion as certain? I am (behave) certain, but for instance I don't know
why.
Page 87
What would it look like if everyone were always uncertain about everyone else's feelings? Seemingly when
they are expressing sympathy, etc.., for someone, they would always be a little doubtful, would always put on a
doubtful expression or make a doubtful gesture.--But if we now leave off this constant gesture because it is
constant, what behaviour then remains? Perhaps a behaviour that is cool, only superficially interested? But then we
in turn don't have to interpret their behaviour as an expression of doubting. So it means nothing to say everyone
always...
Page 87
There is uncertainty and there is certainty; but from this it does not follow that there are criteria that are
certain.
Page 87
How would it be if someone were now to say: "I know that he is glad" means merely that I am certain of his
gladness, and therefore also that I am reacting to him in such and such a way and, what is more, without
uncertainty. Then it would be approximately like "I know that everything is for the best"--the expression of the
position taken towards whatever comes along. And here there would be grounds for saying that this is not really
knowledge. But the latter statement would convince no one, not even in a court of law, that everything is for the
best.
Page Break 88
And here there is something important: The statement "I know that he is glad" would after all, even in a court of law,
not count for more than: "I have the certain impression that he is glad." It would not be the same as if a physicist
stated that he did this experiment and that this was the result; or as if a mathematician made a statement about a
calculation.--If I have known the other person a long time, the court will probably also allow my statement to stand,
will attach importance to it. But my absolute certainty will not mean knowing to the court. For if it were knowing,
the court would have to be able to draw certain well-defined conclusions.
Page 88
And one cannot answer: "I draw certain conclusions from my knowledge, even if no one else can"--for
conclusions must be valid for all.
Page 88
Here the connection of evidence with what it is evidence for is not ineluctable. And I don't mean: "the
connection of the outer with the inner".
Page 88
One could even say: The uncertainty about the inner is an uncertainty about something outer.
Page 88
If "I know..." means: I can convince someone else if he believes my evidence, then one can say: I may well
be as certain about his mood as about the truth of a mathematical proposition, but it is still false to say that I know
his mood.
(But it is still false to say: Knowing is a different mental state from being certain. (I is a different person from
L. W.))
Page 88
That is: 'knowing' is a psychological concept of a different kind from 'being certain', 'being convinced',
'believing', 'surmising', etc. The evidence for knowing is of a different kind.
Page Break 89
Russell's example: "I know that the present Prime Minister is bald"; the person who says this is certain just because
he wrongly believes that X is Prime Minister; nevertheless the actual present Prime Minister is also bald and so the
assertion is true and all the same the person doesn't know that it is.
Page 89
"I know that it is so" is to be sure an expression of my complete certainty, but, besides my being certain,
other things follow from it.
Page 89
In the first place, "I cannot know his feelings" does not mean:... as opposed to mine. In the second place, it
does not mean: I can never be completely sure of his feelings.
Page 89
Statement: "I know that the bottle was standing there."--"How do you know that?"--"I saw it there."----If the
statement is: "I know that he was glad", and the question is asked: "How do you know that?" what is the answer? It
is not simply the description of a physical state of affairs. Part of it is for instance that I know the person. If a film
could be shown in the courtroom in which the whole scene were rendered--the play of his facial expressions, his
gestures, his voice--sometimes this could have a fairly convincing effect. At least if he is not an actor. But it only has
an effect for instance if those judging the scene belong to the same culture. I wouldn't know, for instance, what
genuine gladness looks like with Chinese.
Page 89
Rather than directing our attention to the fact that one cannot know what someone else experiences, that an
experience is in some sense the secret of the person who has it, we direct it instead to any and all rules of evidence
that refer to experiences.
Page 89
It's important, for instance, that one must 'know' someone in order to be able to judge what meaning is to be
attributed
Page Break 90
to one of his expressions of feeling, and yet that one cannot describe what it is that one knows about him.
It is just as important that one cannot say what the essential observable consequences of an inner condition
are. If for instance he really was glad, what can be expected from him, and what not? Of course there are such
characteristic consequences, but they cannot be described in the same way as the reactions that characterize the state
of a physical object.
Page 90
This must also be considered: Genuineness and falseness are not the only essential characteristics of an
expression of feeling. One cannot tell, for instance, whether a cat that purrs and then right away scratches someone
was pretending. It could be that someone uttered signs of gladness and then behaved in a completely unexpected
way, and that we still could not say that the first expression was not genuine.
Page 90
It seems to me as little a fact that there can only be genuine or feigned expressions of feeling as that there can
only be major or minor keys.
Page Break 91
VI:
MS 176
Page 91
(1951)
Page Break 92
Page 92
"Can one know what goes on in someone else in the same way he himself knows it?"----Well, how does he
know it? He can express his experience. No doubt within him whether he is really having this experience--analogous
to the doubt whether he really has this or that disease--comes into play; and therefore it is wrong to say that he
knows what he is experiencing. But someone else can very well doubt whether that person has this experience. Thus
doubt does come into play, but, precisely for that reason, it is also possible that there is complete certainty.†1
Page 92
Need I be less certain that someone is suffering pain than that 12 × 12 = 144?
Page 92
And yet sometimes one says that one cannot know that. Well, above all, one can't prove it. That is, there is
nothing here of the sort of proof that rests on (generally) known principles.
Page 92
But that which is in him, how can I see it? Between his experience and me there is always the expression!
Here is the picture: He sees it immediately, I only mediately. But that's not the way it is. He doesn't see
something and describe it to us.
Page 92
If 'something is going on inside him', then to be sure I don't see it, but who knows whether he himself sees
it.---
Page 92
Don't I really often see what is going on inside him?--"Yes, but not in the way in which he himself perceives
it. I see that he is in pain, but don't feel any pain. And if I felt pain, it wouldn't be his." This means nothing.--On the
other hand it would be conceivable that a connection could be established with someone else through which I would
feel the same pain (i.e. the same kind of 'pain'), and in the same place, as the other person. But that this is the case
Page Break 93
would have to be ascertained through both people's expression of pain.
Page 93
And if this way of getting to know someone else's pain were to have proved its worth, it's conceivable that
one would apply it against a person's expression of pain, and thus would mistrust his expression if it contradicted
that test.
And now one can also imagine that there are people who follow that method from the outset, and call that
"pain" which is ascertained by means of it. In that case their concept 'pain' will be related to ours, but different from
it. (Of course it doesn't matter whether they call their concept by the same name as we use for our related one; it
only matters that in their life it is analogous to our concept of pain.)
Page 93
This analogue of our concept would then lack that uncertainty of evidence in ours. In this respect our
concepts would not be similar.
Page 93
(If we call that analogous concept 'pain', then these people can believe that they are in pain and also doubt it.
But if someone were to say: "Well, in that case there simply is no essential similarity between the concepts"--then
we can respond: Here there are immense differences, but also great similarities.)
Page 93
One could imagine that a kind of thermometer is used to ascertain whether someone is in 'pain'. If someone
screams or groans, then they insert the thermometer and only when the gauge reaches this or that point do they
begin to feel sorry for the suffering person, and treat him as we do someone who 'obviously is in pain'.
Page Break 94
Is the indeterminacy of the logic of the concept of pain connected with the actual absence of certain physical
possibilities of reading thoughts and feelings?----If that's a causal question--how can I answer it?
Page 94
Actually the question could be phrased in this way: How does what is important for us depend on what is
physically possible?
Page 94
Where measuring is not important we don't measure, even if we are able to.
Page 94
"Is†1 the impossibility of knowing what goes on in someone else physical or logical? And if it is both--how
do the two hang together?"
For a start: possibilities for exploring someone else could be imagined which don't exist in reality. Thus there
is a physical impossibility.
The logical impossibility lies in the lack of exact rules of evidence. (Therefore we sometimes express
ourselves in this way: "We may always be wrong; we can never be certain; what we observe can still be pretence."
Although pretence is only one of many possible causes of a false judgement.)----We can imagine an arithmetic in
which problems with small numbers can be solved with certainty, but in which the results become less certain the
larger the numbers are. So that people who possess this art of calculating state that one can never be completely
certain of the product of two large numbers, and that neither could a borderline be given between small and large
numbers.
But of course it isn't true that we are never certain about the mental processes in someone else. In countless
cases we are.
Page Break 95
Page 95
And now the question remains whether we would give up our language-game which rests on 'imponderable
evidence' and frequently leads to uncertainty, if it were possible to exchange it for a more exact one which by and
large would have similar consequences. For instance, we could work with a mechanical "lie detector" and redefine a
lie as that which causes a deflection on the lie detector.
So the question is: Would we change our way of living if this or that were provided for us?--And how could I
answer that?
FOOTNOTES
Page 2
†1 "non & ne" appear to be an addition in the manuscript.
Page 2
†2 This remark is preceded by an arrow: ←.
Page 3
†1 The sentence is obviously incomplete. "Has" is an editor's surmise.
Page 4
†1 These variants seem to be connected with the remark "I hear that someone is painting...".
Page 5
†1 Var.: "the illness".
Page 6
†1 Unclear in the MS.
Page 10
†1 Var.: "my ego believes".
Page 11
†1 Var.: "One would have to fill out the picture with behaviour indicating that two... ".
Page 14
†1 Variants: "I seem to have to occupy myself with the object."/"I have to occupy myself with the object."
Page 15
†1 Var.: ",--is that a special kind of seeing? Is it a seeing and thinking? A melding of both--as one is almost
tempted to say? The question is: Why does one want to say that? Well, if one asks in this way it is not very difficult
to answer."--An arrow at the end of the remark shows that it is connected to the following remark.
Page 18
†1 The sentence is obviously incomplete. In the manuscript paragraph b originally read, and was then
partially crossed out: "'The word (in the poem) is like the fitting picture of what it means'---".
Page 19
†1 The sentence seems to be incomplete.
Page 21
†1 Var.: "in the mind".
Page 21
†2 Var.: "is a lying open to view."
Page 22
†1 Var.: "in the sentence".
Page 22
†2 Variants: "that exists before all truth or falsity."/"that exists whether it is true or false."
Page 23
†1 Var.: "inner process".
Page 23
†2 Var.: "correspondingly, or almost correspondingly,"
Page 23
†3 Var.: "Heap of sand".
Page 23
†4 Var.: "will".
Page 23
†5 Several variants.
Page 24
†1 The paragraphs appear as individual remarks in the manuscript, but an arrow indicates that they are
intended as one remark.
Page 25
†1 Var.: "know".
Page 26
†1 The sentence is incomplete.
Page 27
†1 Var.: "This very special pattern in the convoluted drawing of human life."
Page 27
†2 Var.: "Dissimulation plays the same role with them as does walking on all fours with us."
Page 35
†1 At this point in the manuscript there is the following drawing:
How this drawing is connected with the text is not clear to us. In the margin of the page there is also a drawing of a
human face.
Page 38
†1 Var.: "of a use and an expression".
Page 39
†1 In the manuscript there are drawings that are probably not connected with this remark.
Page 41
†1 The sentence is incomplete, and the entire remaining page of the manuscript is empty.
Page 42
†1 Life patterns.
Page 43
†1 Var.: "our will".
Page 49
†1 Remark crossed out.
Page 49
†2 Var.: "one of course is never certain".
Page 55
†1 Numerous variants.
Page 55
†2 Var.: "that we see precisely this that is difficult to describe as evidence, as evidence of something
important."
Page 56
†1 Remark crossed out.
Page 59
†1 Var.: "In an involved way, the outer signs refer, sometimes with certainty, sometimes without, to pain, or
dissimulation, or neither."
Page 59
†2 Var.: "as believing or knowing".
Page 59
†3 Var.: "which one recognizes as whatever it is while one has it."
Page 62
†1 Variants: "outward" and "inward".
Page 62
†2 Var.: "then to be sure the inner has not become the outer, but for us direct inner and indirect outer
evidence of the mental no longer exist."
Page 62
†3 Var.: "explain the picture from inside and outside, makes it understandable."
Page 67
†1 Several variants.
Page 68
†1 Var.: "That it can be portrayed to us as an illusion."
Page 68
†2 Var.: "inner".
Page 69
†1 Var.: "recognizing".
Page 71
†1 This and the following remarks have been published in Remarks on Colour (III, remarks 296-350), ed.
G.E.M. Anscombe (Blackwell, 1977). Here we have omitted one remark (no. 317, pp. 58-9, in Remarks on Colour)
which has already been published in Culture and Value.
Page 75
†1 See Culture and Value, 2nd edn, p. 75.
Page 81
†1 This remark is dated 24.4.50.
Page 82
†1 Var.: "to give an account of the field of the statement."
Page 86
†1 Var.: "this evidence".
Page 92
†1 Date on the previous page of the MS: "April 14 <51>".
Page 94
†1 Date "April 15<51>".
ON CERTAINTY
Page i
Page Break ii
Title page
Page ii
LUDWIG WITTGENSTEIN:
ON CERTAINTY
Edited by
G. E. M. ANSCOMBE
and
G. H. von WRIGHT
Translated by
DENIS PAUL
and
G. E. M. ANSCOMBE
Page Break iii
Copyright page
Page iii
Copyright © Blackwell Publishers Ltd 1969, 1975
First published 1969
Reprinted with corrections and indices 1974
First published in paperback 1975
Reprinted 1977, 1979, 1984, 1988, 1989, 1993, 1994, 1995, 1997, 1998
Blackwell Publishers Ltd
108 Cowley Road
Oxford OX4 1JF, UK
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Page Break iv
PREFACE
Page iv
What we publish here belongs to the last year and a half of Wittgenstein's life. In the middle of 1949 he visited the
United States at the invitation of Norman Malcolm, staying at Malcolm's house in Ithaca. Malcolm acted as a goad to
his interest in Moore's 'defence of common sense', that is to say his claim to know a number of propositions for sure,
such as "Here is one hand, and here is another", and "The earth existed for a long time before my birth", and "I have
never been far from the earth's surface". The first of these comes in Moore's 'Proof of the External World'. The two
others are in his 'Defence of Common Sense'; Wittgenstein had long been interested in these and had said to Moore
that this was his best article. Moore had agreed. This book contains the whole of what Wittgenstein wrote on this
topic from that time until his death. It is all first-draft material, which he did not live to excerpt and polish.
Page iv
The material falls into four parts; we have shown the divisions at § 65, p. 10, § 192, p. 27 and § 299, p. 38.
What we believe to be the first part was written on twenty loose sheets of lined foolscap, undated. These
Wittgenstein left in his room in G. E. M. Anscombe's house in Oxford, where he lived (apart from a visit to Norway
in the autumn) from April 1950 to February 1951. I (G. E. M. A.) am under the impression that he had written them
in Vienna, where he stayed from the previous Christmas until March; but I cannot now recall the basis of this
impression. The rest is in small notebooks, containing dates; towards the end, indeed, the date of writing is always
given. The last entry is two days before his death on April 29th 1951. We have left the dates exactly as they appear in
the manuscripts. The numbering of the single sections, however, is by the Editors.
Page iv
It seemed appropriate to publish this work by itself. It is not a selection; Wittgenstein marked it off in his
notebooks as a separate topic, which he apparently took up at four separate periods during this eighteen months. It
constitutes a single sustained treatment of the topic.
G. E. M. Anscombe
G. H. von Wright
Page Break 1
ACKNOWLEDGMENT
Page 1
Dr. Lotte Labowsky and Dr. Anselm Müller are to be sincerely thanked for advice about the translation of this work.
Page Break 2
[On Certainty: Main Body]
Page 2
1. If you do know that here is one hand,†1 we'll grant you all the rest.
When one says that such and such a proposition can't be proved, of course that does not mean that it can't be
derived from other propositions; any proposition can be derived from other ones. But they may be no more certain
than it is itself. (On this a curious remark by H. Newman.)
Page 2
2. From its seeming to me--or to everyone--to be so, it doesn't follow that it is so.
What we can ask is whether it can make sense to doubt it.
Page 2
3. If e.g. someone says "I don't know if there's a hand here" he might be told "Look closer".--This possibility of
satisfying oneself is part of the language-game. Is one of its essential features.
Page 2
4. "I know that I am a human being." In order to see how unclear the sense of this proposition is, consider its
negation. At most it might be taken to mean "I know I have the organs of a human". (E.g. a brain which, after all, no
one has ever yet seen.) But what about such a proposition as "I know I have a brain"? Can I doubt it? Grounds for
doubt are lacking! Everything speaks in its favour, nothing against it. Nevertheless it is imaginable that my skull
should turn out empty when it was operated on.
Page 2
5. Whether a proposition can turn out false after all depends on what I make count as determinants for that
proposition.
Page 2
6. Now, can one enumerate what one knows (like Moore)? Straight off like that, I believe not.--For otherwise the
expression "I know" gets misused. And through this misuse a queer and extremely important mental state seems to
be revealed.
Page 2
7. My life shews that I know or am certain that there is a chair over there, or a door, and so on.--I tell a friend e.g.
"Take that chair over there", "Shut the door", etc. etc.
Page Break 3
Page 3
8. The difference between the concept of 'knowing' and the concept of 'being certain' isn't of any great importance at
all, except where "I know" is meant to mean: I can't be wrong. In a law-court, for example, "I am certain" could
replace "I know" in every piece of testimony. We might even imagine its being forbidden to say "I know" there. (A
passage in Wilhelm Meister, where "You know" or "You knew" is used in the sense "You were certain", the facts
being different from what he knew.)
Page 3
9. Now do I, in the course of my life, make sure I know that here is a hand--my own hand, that is?
Page 3
10. I know that a sick man is lying here? Nonsense! I am sitting at his bedside, I am looking attentively into his
face.--So I don't know, then, that there is a sick man lying here? Neither the question nor the assertion makes sense.
Any more than the assertion "I am here", which I might yet use at any moment, if suitable occasion presented itself.
Then is "2 × 2 = 4" nonsense in the same way, and not a proposition of arithmetic, apart from particular occasions?
"2 × 2 = 4" is a true proposition of arithmetic--not "on particular occasions" nor "always"--but the spoken or written
sentence "2 × 2 = 4" in Chinese might have a different meaning or be out and out nonsense, and from this is seen
that it is only in use that the proposition has its sense. And "I know that there's a sick man lying here", used in an
unsuitable situation, seems not to be nonsense but rather seems matter-of-course, only because one can fairly easily
imagine a situation to fit it, and one thinks that the words "I know that..." are always in place where there is no
doubt, and hence even where the expression of doubt would be unintelligible.
Page 3
11. We just do not see how very specialized the use of "I know" is.
Page 3
12. --For "I know" seems to describe a state of affairs which guarantees what is known, guarantees it as a fact. One
always forgets the expression "I thought I knew".
Page 3
13. For it is not as though the proposition "It is so" could be inferred from someone else's utterance: "I know it is
so". Nor from the utterance together with its not being a lie.--But can't I infer "It is so" from my own utterance "I
know etc."? Yes;
Page Break 4
and also "There is a hand there" follows from the proposition "He knows that there's a hand there". But from his
utterance "I know..." it does not follow that he does know it.
Page 4
14. That he does know remains to be shewn.
Page 4
15. It needs to be shewn that no mistake was possible. Giving the assurance "I know" doesn't suffice. For it is after all
only an assurance that I can't be making a mistake, and it needs to be objectively established that I am not making a
mistake about that.
Page 4
16. "If I know something, then I also know that I know it, etc." amounts to: "I know that" means "I am incapable of
being wrong about that". But whether I am so must admit of being established objectively.
Page 4
17. Suppose now I say "I'm incapable of being wrong about this: that is a book" while I point to an object. What
would a mistake here be like? And have I any clear idea of it?
Page 4
18. "I know" often means: I have the proper grounds for my statement. So if the other person is acquainted with the
language-game, he would admit that I know. The other, if he is acquainted with the language-game, must be able to
imagine how one may know something of the kind.
Page 4
19. The statement "I know that here is a hand" may then be continued: "for it's my hand that I'm looking at". Then a
reasonable man will not doubt that I know. Nor will the idealist; rather he will say that he was not dealing with the
practical doubt which is being dismissed, but there is a further doubt behind that one.--That this is an illusion has to
be shewn in a different way.
Page 4
20. "Doubting the existence of the external world" does not mean for example doubting the existence of a planet,
which later observations proved to exist.--Or does Moore want to say that knowing that here is his hand is different
in kind from knowing the existence of the planet Saturn? Otherwise it would be possible to point out the discovery
of the planet Saturn to the doubters and say that its existence has been proved, and hence the existence of the
external world as well.
Page Break 5
Page 5
21. Moose's view really comes down to this: the concept 'know' is analogous to the concepts 'believe', 'surmise',
'doubt', 'be convinced' in that the statement "I know..." can't be a mistake. And if that is so, then there can be an
inference from such an utterance to the truth of an assertion. And here the form "I thought I knew" is being
overlooked.--But if this latter is inadmissible, then a mistake in the assertion must be logically impossible too. And
anyone who is acquainted with the language-game must realize this--an assurance from a reliable man that he knows
cannot contribute anything.
Page 5
22. It would surely be remarkable if we had to believe the reliable person who says "I can't be wrong"; or who says
"I am not wrong".
Page 5
23. If I don't know whether someone has two hands (say, whether they have been amputated or not) I shall believe
his assurance that he has two hands, if he is trustworthy. And if he says he knows it, that can only signify to me that
he has been able to make sure, and hence that his arms are e.g. not still concealed by coverings and bandages, etc.
etc. My believing the trustworthy man stems from my admitting that it is possible for him to make sure. But
someone who says that perhaps there are no physical objects makes no such admission.
Page 5
24. The idealist's question would be something like: "What right have I not to doubt the existence of my hands?"
(And to that the answer can't be: I know that they exist.) But someone who asks such a question is overlooking the
fact that a doubt about existence only works in a language-game. Hence, that we should first have to ask: what
would such a doubt be like?, and don't understand this straight off.
Page 5
25. One may be wrong even about "there being a hand here". Only in particular circumstances is it
impossible.--"Even in a calculation one can be wrong--only in certain circumstances one can't."
Page 5
26. But can it be seen from a rule what circumstances logically exclude a mistake in the employment of rules of
calculation?
What use is a rule to us here? Mightn't we (in turn) go wrong in applying it?
Page Break 6
Page 6
27. If, however, one wanted to give something like a rule here, then it would contain the expression "in normal
circumstances". And we recognize normal circumstances but cannot precisely describe them. At most, we can
describe a range of abnormal ones.
Page 6
28. What is 'learning a rule'?--This.
What is 'making a mistake in applying it'?--This. And what is pointed to here is something indeterminate.
Page 6
29. Practice in the use of the rule also shews what is a mistake in its employment.
Page 6
30. When someone has made sure of something, he says: "Yes, the calculation is right", but he did not infer that
from his condition of certainty. One does not infer how things are from one's own certainty.
Certainty is as it were a tone of voice in which one declares how things are, but one does not infer from the
tone of voice that one is justified.
Page 6
31. The propositions which one comes back to again and again as if bewitched--these I should like to expunge from
philosophical language.
Page 6
32. It's not a matter of Moore's knowing that there's a hand there, but rather we should not understand him if he
were to say "Of course I may be wrong about this". We should ask "What is it like to make such a mistake as
that?"--e.g. what's it like to discover that it was a mistake?
Page 6
33. Thus we expunge the sentences that don't get us any further.
Page 6
34. If someone is taught to calculate, is he also taught that he can rely on a calculation of his teacher's? But these
explanations must after all sometime come to an end. Will he also be taught that he can trust his senses--since he is
indeed told in many cases that in such and such a special case you cannot trust them?
Rule and exception.
Page 6
35. But can't it be imagined that there should be no physical objects? I don't know. And yet "There are physical
objects" is nonsense. Is it supposed to be an empirical proposition?
Page Break 7
Page 7
And is this an empirical proposition: "There seem to be physical objects"?
Page 7
36. "A is a physical object" is a piece of instruction which we give only to someone who doesn't yet understand
either what "A" means, or what "physical object" means. Thus it is instruction about the use of words, and "physical
object" is a logical concept. (Like colour, quantity,...) And that is why no such proposition as: "There are physical
objects" can be formulated.
Yet we encounter such unsuccessful shots at every turn.
Page 7
37. But is it an adequate answer to the scepticism of the idealist, or the assurances of the realist, to say that "There
are physical objects" is nonsense? For them after all it is not nonsense. It would, however, be an answer to say: this
assertion, or its opposite is a misfiring attempt to express what can't be expressed like that. And that it does misfire
can be shewn; but that isn't the end of the matter. We need to realize that what presents itself to us as the first
expression of a difficulty, or of its solution, may as yet not be correctly expressed at all. Just as one who has a just
censure of a picture to make will often at first offer the censure where it does not belong, and an investigation is
needed in order to find the right point of attack for the critic.
Page 7
38. Knowledge in mathematics: Here one has to keep on reminding oneself of the unimportance of the 'inner
process' or 'state' and ask "Why should it be important? What does it matter to me?" What is interesting is how we
use mathematical propositions.
Page 7
39. This is how calculation is done, in such circumstances a calculation is treated as absolutely reliable, as certainly
correct.
Page 7
40. Upon "I know that here is my hand" there may follow the question "How do you know?" and the answer to that
presupposes that this can be known in that way. So, instead of "I know that here is my hand", one might say "Here
is my hand", and then add how one knows.
Page 7
41. "I know where I am feeling pain", "I know that I feel it here" is as wrong as "I know that I am in pain". But "I
know where you touched my arm" is right.
Page Break 8
Page 8
42. One can say "He believes it, but it isn't so", but not "He knows it, but it isn't so". Does this stem from the
difference between the mental states of belief and of knowledge? No.--One may for example call "mental state" what
is expressed by tone of voice in speaking, by gestures etc. It would thus be possible to speak of a mental state of
conviction, and that may be the same whether it is knowledge or false belief. To think that different states must
correspond to the words "believe" and "know" would be as if one believed that different people had to correspond to
the word "I" and the name "Ludwig", because the concepts are different.
Page 8
43. What sort of proposition is this: "We cannot have miscalculated in 12 × 12 = 144"? It must surely be a
proposition of logic.--But now, is it not the same, or doesn't it come to the same, as the statement 12 × 12 = 144?
Page 8
44. If you demand a rule from which it follows that there can't have been a miscalculation here, the answer is that we
did not learn this through a rule, but by learning to calculate.
Page 8
45. We got to know the nature of calculating by learning to calculate.
Page 8
46. But then can't it be described how we satisfy ourselves of the reliability of a calculation? O yes! Yet no rule
emerges when we do so.--But the most important thing is: The rule is not needed. Nothing is lacking. We do
calculate according to a rule, and that is enough.
Page 8
47. This is how one calculates. Calculating is this. What we learn at school, for example. Forget this transcendent
certainty, which is connected with your concept of spirit.
Page 8
48. However, out of a host of calculations certain ones might be designated as reliable once for all, others as not yet
fixed. And now, is this a logical distinction?
Page 8
49. But remember: even when the calculation is something fixed for me, this is only a decision for a practical
purpose.
Page 8
50. When does one say, I know that... ×...=...? When one has checked the calculation.
Page Break 9
Page 9
51. What sort of proposition is: "What could a mistake here be like!"? It would have to be a logical proposition. But
is it a logic that is not used, because what it tells us is not taught by means of propositions.--It is a logical
proposition; for it does describe the conceptual (linguistic) situation.
Page 9
52. This situation is thus not the same for a proposition like "At this distance from the sun there is a planet" and
"Here is a hand" (namely my own hand). The second can't be called a hypothesis. But there isn't a sharp boundary
line between them.
Page 9
53. So one might grant that Moore was right, if he is interpreted like this: a proposition saying that here is a physical
object may have the same logical status as one saying that here is a red patch.
Page 9
54. For it is not true that a mistake merely gets more and more improbable as we pass from the planet to my own
hand. No: at some point it has ceased to be conceivable.
This is already suggested by the following: if it were not so, it would also be conceivable that we should be
wrong in every statement about physical objects; that any we ever make are mistaken.
Page 9
55. So is the hypothesis possible, that all the things around us don't exist? Would that not be like the hypothesis of
our having miscalculated in all our calculations?
Page 9
56. When one says: "Perhaps this planet doesn't exist and the light-phenomenon arises in some other way", then
after all one needs an example of an object which does exist. This doesn't exist,--as for example does....
Or are we to say that certainty is merely a constructed point to which some things approximate more, some
less closely? No. Doubt gradually loses its sense. This language-game just is like that.
And everything descriptive of a language-game is part of logic.
Page 9
57. Now might not "I know, I am not just surmising, that here is my hand" be conceived as a proposition of
grammar? Hence not temporally.
But in that case isn't it like this one: "I know, I am not just surmising, that I am seeing red"?
Page Break 10
Page 10
And isn't the consequence "So there are physical objects" like: "So there are colours"?
Page 10
58. If "I know etc'." is conceived as a grammatical proposition, of course the "I" cannot be important. And it properly
means "There is no such thing as a doubt in this case" or "The expression 'I do not know' makes no sense in this
case". And of course it follows from this that "I know" makes no sense either.
Page 10
59. "I know" is here a logical insight. Only realism can't be proved by means of it.
Page 10
60. It is wrong to say that the 'hypothesis' that this is a bit of paper would be confirmed or disconfirmed by later
experience, and that, in "I know that this is a bit of paper," the "I know" either relates to such an hypothesis or to a
logical determination.
Page 10
61. ... A meaning of a word is a kind of employment of it.
For it is what we learn when the word is incorporated into our language.
Page 10
62. That is why there exists a correspondence between the concepts 'rule' and 'meaning'.
Page 10
63. If we imagine the facts otherwise than as they are, certain language-games lose some of their importance, while
others become important. And in this way there is an alteration--a gradual one--in the use of the vocabulary of a
language.
Page 10
64. Compare the meaning of a word with the 'function' of an official. And 'different meanings' with 'different
functions'.
Page 10
65. When language-games change, then there is a change in concepts, and with the concepts the meanings of words
change.
---------
Page 10
66. I make assertions about reality, assertions which have different degrees of assurance. How does the degree of
assurance come out? What consequences has it?
We may be dealing, for example, with the certainty of memory, or again of perception. I may be sure of
something, but still know what test might convince me of error. I am e.g. quite sure of the date of a battle, but if I
should find a different date in a
Page Break 11
recognized work of history, I should alter my opinion, and this would not mean I lost all faith in judging.
Page 11
67. Could we imagine a man who keeps on making mistakes where we regard a mistake as ruled out, and in fact
never encounter one?
E.g. he says he lives in such and such a place, is so and so old, comes from such and such a city, and he
speaks with the same certainty (giving all the tokens of it) as I do, but he is wrong.
But what is his relation to this error? What am I to suppose?
Page 11
68. The question is: what is the logician to say here?
Page 11
69. I should like to say: "If I am wrong about this, I have no guarantee that anything I say is true." But others won't
say that about me, nor will I say it about other people.
Page 11
70. For months I have lived at address A, I have read the name of the street and the number of the house countless
times, have received countless letters here and given countless people the address. If I am wrong about it, the
mistake is hardly less than if I were (wrongly) to believe I was writing Chinese and not German.
Page 11
71. If my friend were to imagine one day that he had been living for a long time past in such and such a place, etc.
etc., I should not call this a mistake, but rather a mental disturbance, perhaps a transient one.
Page 11
72. Not every false belief of this sort is a mistake.
Page 11
73. But what is the difference between mistake and mental disturbance? Or what is the difference between my
treating it as a mistake and my treating it as mental disturbance?
Page 11
74. Can we say: a mistake doesn't only have a cause, it also has a ground? I.e., roughly: when someone makes a
mistake, this can be fitted into what he knows aright.
Page 11
75. Would this be correct: If I merely believed wrongly that there is a table here in front of me, this might still be a
mistake; but if I believe wrongly that I have seen this table, or one like it, every day for several months past, and
have regularly used it, that isn't a mistake?
Page Break 12
Page 12
76. Naturally, my aim must be to give the statements that one would like to make here, but cannot make
significantly.
Page 12
77. Perhaps I shall do a multiplication twice to make sure, or perhaps get someone else to work it over. But shall I
work it over again twenty times, or get twenty people to go over it? And is that some sort of negligence? Would the
certainty really be greater for being checked twenty times?
Page 12
78. And can I give a reason why it isn't?
Page 12
79. That I am a man and not a woman can be verified, but if I were to say I was a woman, and then tried to explain
the error by saying I hadn't checked the statement, the explanation would not be accepted.
Page 12
80. The truth of my statements is the test of my understanding of these statements.
Page 12
81. That is to say: if I make certain false statements, it becomes uncertain whether I understand them.
Page 12
82. What counts as an adequate test of a statement belongs to logic. It belongs to the description of the
language-game.
Page 12
83. The truth of certain empirical propositions belongs to our frame of reference.
Page 12
84. Moore says he knows that the earth existed long before his birth. And put like that it seems to be a personal
statement about him, even if it is in addition a statement about the physical world. Now it is philosophically
uninteresting whether Moore knows this or that, but it is interesting that, and how, it can be known. If Moore had
informed us that he knew the distance separating certain stars, we might conclude from that that he had made some
special investigations, and we shall want to know what these were. But Moore chooses precisely a case in which we
all seem to know the same as he, and without being able to say how. I believe e.g. that I know as much about this
matter (the existence of the earth) as Moore does, and if he knows that it is as he says, then I know it too. For it isn't,
either, as if he had arrived at his proposition
Page Break 13
by pursuing some line of thought which, while it is open to me, I have not in fact pursued.
Page 13
85. And what goes into someone's knowing this? Knowledge of history, say? He must know what it means to say:
the earth has already existed for such and such a length of time. For not any intelligent adult must know that. We see
men building and demolishing houses, and are led to ask: "How long has this house been here?" But how does one
come on the idea of asking this about a mountain, for example? And have all men the notion of the earth as a body,
which may come into being and pass away? Why shouldn't I think of the earth as flat, but extending without end in
every direction (including depth)? But in that case one might still say "I know that this mountain existed long before
my birth."-But suppose I met a man who didn't believe that?
Page 13
86. Suppose I replaced Moore's "I know" by "I am of the unshakeable conviction"?
Page 13
87. Can't an assertoric sentence, which was capable of functioning as an hypothesis, also be used as a foundation for
research and action? I.e. can't it simply be isolated from doubt, though not according to any explicit rule? It simply
gets assumed as a truism, never called in question, perhaps not even ever formulated.
Page 13
88. It maybe for example that all enquiry on our part is set so as to exempt certain propositions from doubt, if they
are ever formulated. They lie apart from the route travelled by enquiry.
Page 13
89. One would like to say: "Everything speaks for, and nothing against the earth's having existed long before...."
Yet might I not believe the contrary after all? But the question is: What would the practical effects of this
belief be?--Perhaps someone says: "That's not the point. A belief is what it is whether it has any practical effects or
not." One thinks: It is the same adjustment of the human mind anyway.
Page 13
90. "I know" has a primitive meaning similar to and related to "I see" ("wissen", "videre"). And "I knew he was in the
room, but he wasn't in the room" is like "I saw him in the room, but he wasn't there". "I know" is supposed to
express a relation, not
Page Break 14
between me and the sense of a proposition (like "I believe") but between me and a fact. So that the fact is taken into
my consciousness. (Here is the reason why one wants to say that nothing that goes on in the outer world is really
known, but only what happens in the domain of what are called sense-data.) This would give us a picture of
knowing as the perception of an outer event through visual rays which project it as it is into the eye and the
consciousness. Only then the question at once arises whether one can be certain of this projection. And this picture
does indeed show how our imagination presents knowledge, but not what lies at the bottom of this presentation.
Page 14
91. If Moore says he knows the earth existed etc., most of us will grant him that it has existed all that time, and also
believe him when he says he is convinced of it. But has he also got the right ground for his conviction? For if not,
then after all he doesn't know (Russell).
Page 14
92. However, we can ask: May someone have telling grounds for believing that the earth has only existed for a short
time, say since his own birth?--Suppose he had always been told that,--would he have any good reason to doubt it?
Men have believed that they could make rain; why should not a king be brought up in the belief that the world began
with him? And if Moore and this king were to meet and discuss, could Moore really prove his belief to be the right
one? I do not say that Moore could not convert the king to his view, but it would be a conversion of a special kind;
the king would be brought to look at the world in a different way.
Remember that one is sometimes convinced of the correctness of a view by its simplicity or symmetry, i.e.,
these are what induce one to go over to this point of view. One then simply says something like: "That's how it must
be."
Page 14
93. The propositions presenting what Moore 'knows' are all of such a kind that it is difficult to imagine why anyone
should believe the contrary. E.g. the proposition that Moore has spent his whole life in close proximity to the
earth.--Once more I can speak of myself here instead of speaking of Moore. What could induce me to believe the
opposite? Either a memory, or having been told.--
Page Break 15
Everything that I have seen or heard gives me the conviction that no man has ever been far from the earth. Nothing
in my picture of the world speaks in favour of the opposite.
Page 15
94. But I did not get my picture of the world by satisfying myself of its correctness; nor do I have it because I am
satisfied of its correctness. No: it is the inherited background against which I distinguish between true and false.
Page 15
95. The propositions describing this world-picture might be part of a kind of mythology. And their role is like that of
rules of a game; and the game can be learned purely practically, without learning any explicit rules.
Page 15
96. It might be imagined that some propositions, of the form of empirical propositions, were hardened and
functioned as channels for such empirical propositions as were not hardened but fluid; and that this relation altered
with time, in that fluid propositions hardened, and hard ones became fluid.
Page 15
97. The mythology may change back into a state of flux, the river-bed of thoughts may shift. But I distinguish
between the movement of the waters on the river-bed and the shift of the bed itself; though there is not a sharp
division of the one from the other.
Page 15
98. But if someone were to say "So logic too is an empirical science" he would be wrong. Yet this is right: the same
proposition may get treated at one time as something to test by experience, at another as a rule of testing.
Page 15
99. And the bank of that river consists partly of hard rock, subject to no alteration or only to an imperceptible one,
partly of sand, which now in one place now in another gets washed away, or deposited.
Page 15
100. The truths which Moore says he knows, are such as, roughly speaking, all of us know, if he knows them.
Page 15
101. Such a proposition might be e.g. "My body has never disappeared and reappeared again after an interval."
Page 15
102. Might I not believe that once, without knowing it, perhaps in a state of unconsciousness, I was taken far away
from the earth
Page Break 16
--that other people even know this, but do not mention it to me? But this would not fit into the rest of my
convictions at all. Not that I could describe the system of these convictions. Yet my convictions do form a system, a
structure.
Page 16
103. And now if I were to say "It is my unshakeable conviction that etc.", this means in the present case too that I
have not consciously arrived at the conviction by following a particular line of thought, but that it is anchored in all
my questions and answers, so anchored that I cannot touch it.
Page 16
104. I am for example also convinced that the sun is not a hole in the vault of heaven.
Page 16
105. All testing, all confirmation and disconfirmation of a hypothesis takes place already within a system. And this
system is not a more or less arbitrary and doubtful point of departure for all our arguments: no, it belongs to the
essence of what we call an argument. The system is not so much the point of departure, as the element in which
arguments have their life.
Page 16
106. Suppose some adult had told a child that he had been on the moon. The child tells me the story, and I say it was
only a joke, the man hadn't been on the moon; no one has ever been on the moon; the moon is a long way off and it
is impossible to climb up there or fly there.--If now the child insists, saying perhaps there is a way of getting there
which I don't know, etc. what reply could I make to him? What reply could I make to the adults of a tribe who
believe that people sometimes go to the moon (perhaps that is how they interpret their dreams), and who indeed
grant that there are no ordinary means of climbing up to it or flying there?--But a child will not ordinarily stick to
such a belief and will soon be convinced by what we tell him seriously.
Page 16
107. Isn't this altogether like the way one can instruct a child to believe in a God, or that none exists, and it will
accordingly be able to produce apparently telling grounds for the one or the other?
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Page 17
108. "But is there then no objective truth? Isn't it true, or false, that someone has been on the moon?" If we are
thinking within our system, then it is certain that no one has ever been on the moon. Not merely is nothing of the
sort ever seriously reported to us by reasonable people, but our whole system of physics forbids us to believe it. For
this demands answers to the questions "How did he overcome the force of gravity?" "How could he live without an
atmosphere?" and a thousand others which could not be answered. But suppose that instead of all these answers we
met the reply: "We don't know how one gets to the moon, but those who get there know at once that they are there;
and even you can't explain everything." We should feel ourselves intellectually very distant from someone who said
this.
Page 17
109. "An empirical proposition can be tested" (we say). But how? and through what?
Page 17
110. What counts as its test? "But is this an adequate test? And, if so, must it not be recognizable as such in
logic?"--As if giving grounds did not come to an end sometime. But the end is not an ungrounded presupposition: it
is an ungrounded way of acting.
Page 17
111. "I know that I have never been on the moon." That sounds quite different in the circumstances which actually
hold, to the way it would sound if a good many men had been on the moon, and some perhaps without knowing it.
In this case one could give grounds for this knowledge. Is there not a relationship here similar to that between the
general rule of multiplying and particular multiplications that have been carried out?
I want to say: my not having been on the moon is as sure a thing for me as any grounds I could give for it.
Page 17
112. And isn't that what Moore wants to say, when he says he knows all these things?--But is his knowing it really
what is in question, and not rather that some of these propositions must be solid for us?
Page 17
113. When someone is trying to teach us mathematics, he will not begin by assuring us that he knows that a + b = b
+ a.
Page 17
114. If you are not certain of any fact, you cannot be certain of the meaning of your words either.
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Page 18
115. If you tried to doubt everything you would not get as far as doubting anything. The game of doubting itself
presupposes certainty.
Page 18
116. Instead of "I know...", couldn't Moore have said: "It stands fast for me that..."? And further: "It stands fast for
me and many others...."
Page 18
117. Why is it not possible for me to doubt that I have never been on the moon? And how could I try to doubt it?
First and foremost, the supposition that perhaps I have been there would strike me as idle. Nothing would
follow from it, nothing be explained by it. It would not tie in with anything in my life.
When I say "Nothing speaks for, everything against it," this presupposes a principle of speaking for and
against. That is, I must be able to say what would speak for it.
Page 18
118. Now would it be correct to say: So far no one has opened my skull in order to see whether there is a brain
inside; but everything speaks for, and nothing against, its being what they would find there?
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119. But can it also be said: Everything speaks for, and nothing against the table's still being there when no one sees
it? For what does speak for it?
Page 18
120. But if anyone were to doubt it, how would his doubt come out in practice? And couldn't we peacefully leave
him to doubt it, since it makes no difference at all?
Page 18
121. Can one say: "Where there is no doubt there is no knowledge either"?
Page 18
122. Doesn't one need grounds for doubt?
Page 18
123. Wherever I look, I find no ground for doubting that....
Page 18
124. I want to say: We use judgments as principles of judgment.
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125. If a blind man were to ask me "Have you got two hands?" I should not make sure by looking. If I were to have
any doubt of it, then I don't know why I should trust my eyes. For why shouldn't I test my eyes by looking to find
out whether I see my
Page Break 19
two hands? What is to be tested by what? (Who decides what stands fast?)
And what does it mean to say that such and such stands fast?
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126. I am not more certain of the meaning of my words than I am of certain judgments. Can I doubt that this colour
is called "blue"?
(My) doubts form a system.
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127. For how do I know that someone is in doubt? How do I know that he uses the words "I doubt it" as I do?
Page 19
128. From a child up I learnt to judge like this. This is judging.
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129. This is how I learned to judge; this I got to know as judgment.
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130. But isn't it experience that teaches us to judge like this, that is to say, that it is correct to judge like this? But
how does experience teach us, then? We may derive it from experience, but experience does not direct us to derive
anything from experience. If it is the ground of our judging like this, and not just the cause, still we do not have a
ground for seeing this in turn as a ground.
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131. No, experience is not the ground for our game of judging. Nor is its outstanding success.
Page 19
132. Men have judged that a king can make rain; we say this contradicts all experience. Today they judge that
aeroplanes and the radio etc. are means for the closer contact of peoples and the spread of culture.
Page 19
133. Under ordinary circumstances I do not satisfy myself that I have two hands by seeing how it looks. Why not?
Has experience shown it to be unnecessary? Or (again): Have we in some way learnt a universal law of induction,
and do we trust it here too?--But why should we have learnt one universal law first, and not the special one straight
away?
Page 19
134. After putting a book in a drawer, I assume it is there, unless.... "Experience always proves me right. There is no
well attested case of a book's (simply) disappearing." It has often happened that a book has never turned up again,
although we
Page Break 20
thought we knew for certain where it was.--But experience does really teach that a book, say, does not vanish away.
(E.g. gradually evaporate.) But is it this experience with books etc. that leads us to assume that such a book has not
vanished away? Well, suppose we were to find that under particular novel circumstances books did vanish
away.--Shouldn't we alter our assumption? Can one give the lie to the effect of experience on our system of
assumption?
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135. But do we not simply follow the principle that what has always happened will happen again (or something like
it)? What does it mean to follow this principle? Do we really introduce it into our reasoning? Or is it merely the
natural law which our inferring apparently follows? This latter it may be. It is not an item in our considerations.
Page 20
136. When Moore says he knows such and such, he is really enumerating a lot of empirical propositions which we
affirm without special testing; propositions, that is, which have a peculiar logical role in the system of our empirical
propositions.
Page 20
137. Even if the most trustworthy of men assures me that he knows things are thus and so, this by itself cannot
satisfy me that he does know. Only that he believes he knows. That is why Moore's assurance that he knows... does
not interest us. The propositions, however, which Moore retails as examples of such known truths are indeed
interesting. Not because anyone knows their truth, or believes he knows them, but because they all have a similar
role in the system of our empirical judgments.
Page 20
138. We don't, for example, arrive at any of them as a result of investigation.
There are e.g. historical investigations and investigations into the shape and also the age of the earth, but not
into whether the earth has existed during the last hundred years. Of course many of us have information about this
period from our parents and grandparents; but mayn't they be wrong?--"Nonsense!" one will say. "How should all
these people be wrong?"--But is that an argument? Is it not simply the rejection of an idea? And
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perhaps the determination of a concept? For if I speak of a possible mistake here, this changes the role of "mistake"
and "truth" in our lives.
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139. Not only rules, but also examples are needed for establishing a practice. Our rules leave loop-holes open, and
the practice has to speak for itself.
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140. We do not learn the practice of making empirical judgments by learning rules: we are taught judgments and
their connexion with other judgments. A totality of judgments is made plausible to us.
Page 21
141. When we first begin to believe anything, what we believe is not a single proposition, it is a whole system of
propositions. (Light dawns gradually over the whole.)
Page 21
142. It is not single axioms that strike me as obvious, it is a system in which consequences and premises give one
another mutual support.
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143. I am told, for example, that someone climbed this mountain many years ago. Do I always enquire into the
reliability of the teller of this story, and whether the mountain did exist years ago? A child learns there are reliable
and unreliable informants much later than it learns facts which are told it. It doesn't learn at all that that mountain
has existed for a long time: that is, the question whether it is so doesn't arise at all. It swallows this consequence
down, so to speak, together with what it learns.
Page 21
144. The child learns to believe a host of things. I.e. it learns to act according to these beliefs. Bit by bit there forms a
system of what is believed, and in that system some things stand unshakeably fast and some are more or less liable
to shift. What stands fast does so, not because it is intrinsically obvious or convincing; it is rather held fast by what
lies around it.
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145. One wants to say "All my experiences shew that it is so". But how do they do that? For that proposition to
which they point itself belongs to a particular interpretation of them.
"That I regard this proposition as certainly true also characterizes my interpretation of experience."
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Page 22
146. We form the picture of the earth as a ball floating free in space and not altering essentially in a hundred years. I
said "We form the picture etc." and this picture now helps us in the judgment of various situations.
I may indeed calculate the dimensions of a bridge, sometimes calculate that here things are more in favour of
a bridge than a ferry, etc. etc.,--but somewhere I must begin with an assumption or a decision.
Page 22
147. The picture of the earth as a ball is a good picture, it proves itself everywhere, it is also a simple picture--in
short, we work with it without doubting it.
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148. Why do I not satisfy myself that I have two feet when I want to get up from a chair? There is no why. I simply
don't. This is how I act.
Page 22
149. My judgments themselves characterize the way I judge, characterize the nature of judgment.
Page 22
150. How does someone judge which is his right and which his left hand? How do I know that my judgment will
agree with someone else's? How do I know that this colour is blue? If I don't trust myself here, why should I trust
anyone else's judgment? Is there a why? Must I not begin to trust somewhere? That is to say: somewhere I must
begin with not-doubting; and that is not, so to speak, hasty but excusable: it is part of judging.
Page 22
151. I should like to say: Moore does not know what he asserts he knows, but it stands fast for him, as also for me;
regarding it as absolutely solid is part of our method of doubt and enquiry.
Page 22
152. I do not explicitly learn the propositions that stand fast for me. I can discover them subsequently like the axis
around which a body rotates. This axis is not fixed in the sense that anything holds it fast, but the movement around
it determines its immobility.
Page 22
153. No one ever taught me that my hands don't disappear when I am not paying attention to them. Nor can I be
said to presuppose the truth of this proposition in my assertions etc., (as if they rested on it) while it only gets sense
from the rest of our procedure of asserting.
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Page 23
154. There are cases such that, if someone gives signs of doubt where we do not doubt, we cannot confidently
understand his signs as signs of doubt.
I.e.: if we are to understand his signs of doubt as such, he may give them only in particular cases and may
not give them in others.
Page 23
155. In certain circumstances a man cannot make a mistake. ("Can" is here used logically, and the proposition does
not mean that a man cannot say anything false in those circumstances.) If Moore were to pronounce the opposite of
those propositions which he declares certain, we should not just not share his opinion: we should regard him as
demented.
Page 23
156. In order to make a mistake, a man must already judge in conformity with mankind.
Page 23
157. Suppose a man could not remember whether he had always had five fingers or two hands? Should we
understand him? Could we be sure of understanding him?
Page 23
158. Can I be making a mistake, for example, in thinking that the words of which this sentence is composed are
English words whose meaning I know?
Page 23
159. As children we learn facts; e.g., that every human being has a brain, and we take them on trust. I believe that
there is an island, Australia, of such-and-such a shape, and so on and so on; I believe that I had great-grandparents,
that the people who gave themselves out as my parents really were my parents, etc. This belief may never have been
expressed; even the thought that it was so, never thought.
Page 23
160. The child learns by believing the adult. Doubt comes after belief.
Page 23
161. I learned an enormous amount and accepted it on human authority, and then I found some things confirmed or
disconfirmed by my own experience.
Page 23
162. In general I take as true what is found in text-books, of geography for example. Why? I say: All these facts have
been confirmed a hundred times over. But how do I know that? What is my evidence for it? I have a world-picture.
Is it true or false? Above all it is the substratum of all my enquiring and asserting.
Page Break 24
The propositions describing it are not all equally subject to testing.
Page 24
163. Does anyone ever test whether this table remains in existence when no one is paying attention to it?
We check the story of Napoleon, but not whether all the reports about him are based on sense-deception,
forgery and the like. For whenever we test anything, we are already presupposing something that is not tested. Now
am I to say that the experiment which perhaps I make in order to test the truth of a proposition presupposes the truth
of the proposition that the apparatus I believe I see is really there (and the like)?
Page 24
164. Doesn't testing come to an end?
Page 24
165. One child might say to another: "I know that the earth is already hundreds of years old" and that would mean: I
have learnt it.
Page 24
166. The difficulty is to realize the groundlessness of our believing.
Page 24
167. It is clear that our empirical propositions do not all have the same status, since one can lay down such a
proposition and turn it from an empirical proposition into a norm of description.
Think of chemical investigations. Lavoisier makes experiments with substances in his laboratory and now he
concludes that this and that takes place when there is burning. He does not say that it might happen otherwise
another time. He has got hold of a definite world-picture--not of course one that he invented: he learned it as a child.
I say world-picture and not hypothesis, because it is the matter-of-course foundation for his research and as such
also goes unmentioned.
Page 24
168. But now, what part is played by the presupposition that a substance A always reacts to a substance B in the
same way, given the same circumstances? Or is that part of the definition of a substance?
Page 24
169. One might think that there were propositions declaring that chemistry is possible. And these would be
propositions of a natural science. For what should they be supported by, if not by experience?
Page 24
170. I believe what people transmit to me in a certain manner. In this way I believe geographical, chemical, historical
facts etc.
Page Break 25
That is how I learn the sciences. Of course learning is based on believing.
If you have learnt that Mont Blanc is 4000 metres high, if you have looked it up on the map, you say you
know it.
And can it now be said: we accord credence in this way because it has proved to pay?
Page 25
171. A principal ground for Moore to assume that he never was on the moon is that no one ever was on the moon or
could come there; and this we believe on grounds of what we learn.
Page 25
172. Perhaps someone says "There must be some basic principle on which we accord credence", but what can such
a principle accomplish? Is it more than a natural law of 'taking for true'?
Page 25
173. Is it maybe in my power what I believe? or what I unshakeably believe?
I believe that there is a chair over there. Can't I be wrong? But, can I believe that I am wrong? Or can I so
much as bring it under consideration?--And mightn't I also hold fast to my belief whatever I learned later on?! But is
my belief then grounded?
Page 25
174. I act with complete certainty. But this certainty is my own.
Page 25
175. "I know it" I say to someone else; and here there is a justification. But there is none for my belief.
Page 25
176. Instead of "I know it" one may say in some cases "That's how it is--rely upon it." In some cases, however "I
learned it years and years ago"; and sometimes: "I am sure it is so."
Page 25
177. What I know, I believe.
Page 25
178. The wrong use made by Moore of the proposition "I know..." lies in his regarding it as an utterance as little
subject to doubt as "I am in pain". And since from "I know it is so" there follows "It is so", then the latter can't be
doubted either.
Page 25
179. It would be correct to say: "I believe..." has subjective truth; but "I know..." not.
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Page 26
180. Or again "I believe..." is an 'expression', but not "I know...".
Page 26
181. Suppose Moore had said "I swear..." instead of "I know... ".
Page 26
182. The more primitive idea is that the earth never had a beginning. No child has reason to ask himself how long the
earth has existed, because all change takes place on it. If what is called the earth really came into existence at some
time--which is hard enough to picture--then one naturally assumes the beginning as having been an inconceivably
long time ago.
Page 26
183. "It is certain that after the battle of Austerlitz Napoleon.... Well, in that case it's surely also certain that the earth
existed then."
Page 26
184. "It is certain that we didn't arrive on this planet from another one a hundred years ago." Well, it's as certain as
such things are.
Page 26
185. It would strike me as ridiculous to want to doubt the existence of Napoleon; but if someone doubted the
existence of the earth 150 years ago, perhaps I should be more willing to listen, for now he is doubting our whole
system of evidence. It does not strike me as if this system were more certain than a certainty within it.
Page 26
186. "I might suppose that Napoleon never existed and is a fable, but not that the earth did not exist 150 years ago."
Page 26
187. "Do you know that the earth existed then?"--"Of course I know that. I have it from someone who certainly
knows all about it."
Page 26
188. It strikes me as if someone who doubts the existence of the earth at that time is impugning the nature of all
historical evidence. And I cannot say of this latter that it is definitely correct.
Page 26
189. At some point one has to pass from explanation to mere description.
Page 26
190. What we call historical evidence points to the existence of the earth a long time before my birth;--the opposite
hypothesis has nothing on its side.
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Page 27
191. Well, if everything speaks for an hypothesis and nothing against it--is it then certainly true? One may designate
it as such.--But does it certainly agree with reality, with the facts?--With this question you are already going round in
a circle.
Page 27
192. To be sure there is justification; but justification comes to an end.
---------
Page 27
193. What does this mean: the truth of a proposition is certain?
Page 27
194. With the word "certain" we express complete conviction, the total absence of doubt, and thereby we seek to
convince other people. That is subjective certainty.
But when is something objectively certain? When a mistake is not possible. But what kind of possibility is
that? Mustn't mistake be logically excluded?
Page 27
195. If I believe that I am sitting in my room when I am not, then I shall not be said to have made a mistake. But
what is the essential difference between this case and a mistake?
Page 27
196. Sure evidence is what we accept as sure, it is evidence that we go by in acting surely, acting without any doubt.
What we call "a mistake" plays a quite special part in our language games, and so too does what we regard as
certain evidence.
Page 27
197. It would be nonsense to say that we regard something as sure evidence because it is certainly true.
Page 27
198. Rather, we must first determine the role of deciding for or against a proposition.
Page 27
199. The reason why the use of the expression "true or false" has something misleading about it is that it is like
saying "it tallies with the facts or it doesn't", and the very thing that is in question is what "tallying" is here.
Page 27
200. Really "The proposition is either true or false" only means that it must be possible to decide for or against it. But
this does not say what the ground for such a decision is like.
Page 27
201. Suppose someone were to ask: "Is it really right for us to rely on the evidence of our memory (or our senses) as
we do?"
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Page 28
202. Moore's certain propositions almost declare that we have a right to rely upon this evidence.
Page 28
203. [Everything†1 that we regard as evidence indicates that the earth already existed long before my birth. The
contrary hypothesis has nothing to confirm it at all.
If everything speaks for an hypothesis and nothing against it, is it objectively certain? One can call it that.
But does it necessarily agree with the world of facts? At the very best it shows us what "agreement" means. We find
it difficult to imagine it to be false, but also difficult to make use of it.]
What does this agreement consist in, if not in the fact that what is evidence in these language games speaks
for our proposition? (Tractatus Logico-Philosophicus)
Page 28
204. Giving grounds, however, justifying the evidence, comes to an end;--but the end is not certain propositions'
striking us immediately as true, i.e. it is not a kind of seeing on our part; it is our acting, which lies at the bottom of
the language-game.
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205. If the true is what is grounded, then the ground is not true, nor yet false.
Page 28
206. If someone asked us "but is that true?" we might say "yes" to him; and if he demanded grounds we might say
"I can't give you any grounds, but if you learn more you too will think the same".
If this didn't come about, that would mean that he couldn't for example learn history.
Page 28
207. "Strange coincidence, that every man whose skull has been opened had a brain'"
Page 28
208. I have a telephone conversation with New York. My friend tells me that his young trees have buds of such and
such a kind. I am now convinced that his tree is.... Am I also convinced that the earth exists?
Page 28
209. The existence of the earth is rather part of the whole picture which forms the starting-point of belief for me.
Page 28
210. Does my telephone call to New York strengthen my conviction that the earth exists?
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Page 29
Much seems to be fixed, and it is removed from the traffic. It is so to speak shunted onto an unused siding.
Page 29
211. Now it gives our way of looking at things, and our researches, their form. Perhaps it was once disputed. But
perhaps, for unthinkable ages, it has belonged to the scaffolding of our thoughts. (Every human being has parents.)
Page 29
212. In certain circumstances, for example, we regard a calculation as sufficiently checked. What gives us a right to
do so? Experience? May that not have deceived us? Somewhere we must be finished with justification, and then
there remains the proposition that this is how we calculate.
Page 29
213. Our 'empirical propositions' do not form a homogeneous mass.
Page 29
214. What prevents me from supposing that this table either vanishes or alters its shape and colour when no one is
observing it, and then when someone looks at it again changes back to its old condition? "But who is going to
suppose such a thing!"--one would feel like saying.
Page 29
215. Here we see that the idea of 'agreement with reality' does not have any clear application.
Page 29
216. The proposition "It is written".
Page 29
217. If someone supposed that all our calculations were uncertain and that we could rely on none of them (justifying
himself by saying that mistakes are always possible) perhaps we would say he was crazy. But can we say he is in
error? Does he not just react differently? We rely on calculations, he doesn't; we are sure, he isn't.
Page 29
218. Can I believe for one moment that I have ever been in the stratosphere? No. So do I know the contrary, like
Moore?
Page 29
219. There cannot be any doubt about it for me as a reasonable person.--That's it.--
Page 29
220. The reasonable man does not have certain doubts.
Page 29
221. Can I be in doubt at will?
Page 29
222. I cannot possibly doubt that I was never in the stratosphere. Does that make me know it? Does it make it true?
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Page 30
223. For mightn't I be crazy and not doubting what I absolutely ought to doubt?
Page 30
224. "I know that it never happened, for if it had happened I could not possibly have forgotten it."
Page 30
But, supposing it did happen, then it just would have been the case that you had forgotten it. And how do
you know that you could not possibly have forgotten it? Isn't that just from earlier experience?
Page 30
225. What I hold fast to is not one proposition but a nest of propositions.
Page 30
226. Can I give the supposition that I have ever been on the moon any serious consideration at all?
Page 30
227. "Is that something that one can forget?!"
Page 30
228. "In such circumstances, people do not say 'Perhaps we've all forgotten', and the like, but rather they assume
that..."
Page 30
229. Our talk gets its meaning from the rest of our proceedings.
Page 30
230. We are asking ourselves: what do we do with a statement "I know..."? For it is not a question of mental
processes or mental states.
Page 30
And that is how one must decide whether something is knowledge or not.
Page 30
231. If someone doubted whether the earth had existed a hundred years ago, I should not understand, for this
reason: I would not know what such a person would still allow to be counted as evidence and what not.
Page 30
232. "We could doubt every single one of these facts, but we could not doubt them all."
Wouldn't it be more correct to say: "we do not doubt them all".
Our not doubting them all is simply our manner of judging, and therefore of acting.
Page 30
233. If a child asked me whether the earth was already there before my birth, I should answer him that the earth did
not begin only with my birth, but that it existed long, long before. And I should have the feeling of saying something
funny.
Page Break 31
Rather as if the child had asked if such and such a mountain were higher than a tall house that it had seen. In
answering the question I should have to be imparting a picture of the world to the person who asked it.
If I do answer the question with certainty, what gives me this certainty?
Page 31
234. I believe that I have forebears, and that every human being has them. I believe that there are various cities, and,
quite generally, in the main facts of geography and history. I believe that the earth is a body on whose surface we
move and that it no more suddenly disappears or the like than any other solid body: this table, this house, this tree,
etc. If I wanted to doubt the existence of the earth long before my birth, I should have to doubt all sorts of things
that stand fast for me.
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235. And that something stands fast for me is not grounded in my stupidity or credulity.
Page 31
236. If someone said "The earth has not long been..." what would he be impugning? Do I know?
Would it have to be what is called a scientific belief? Might it not be a mystical one? Is there any absolute
necessity for him to be contradicting historical facts? or even geographical ones?
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237. If I say "an hour ago this table didn't exist", I probably mean that it was only made later on.
If I say "this mountain didn't exist then", I presumably mean that it was only formed later on--perhaps by a
volcano.
If I say "this mountain didn't exist half an hour ago", that is such a strange statement that it is not clear what I
mean. Whether for example I mean something untrue but scientific. Perhaps you think that the statement that the
mountain didn't exist then is quite clear, however one conceives the context. But suppose someone said "This
mountain didn't exist a minute ago, but an exactly similar one did instead". Only the accustomed context allows
what is meant to come through clearly.
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238. I might therefore interrogate someone who said that the earth did not exist before his birth, in order to find out
which of
Page Break 32
my convictions he was at odds with. And then it might be that he was contradicting my fundamental attitudes, and if
that were how it was, I should have to put up with it.
Similarly if he said he had at some time been on the moon.
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239. I believe that every human being has two human parents; but Catholics believe that Jesus only had a human
mother. And other people might believe that there are human beings with no parents, and give no credence to all the
contrary evidence. Catholics believe as well that in certain circumstances a wafer completely changes its nature, and
at the same time that all evidence proves the contrary. And so if Moore said "I know that this is wine and not blood",
Catholics would contradict him.
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240. What is the belief that all human beings have parents based on? On experience. And how can I base this sure
belief on my experience? Well, I base it not only on the fact that I have known the parents of certain people but on
everything that I have learnt about the sexual life of human beings and their anatomy and physiology: also on what I
have heard and seen of animals. But then is that really a proof?
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241. Isn't this an hypothesis, which, as I believe, is again and again completely confirmed?
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242. Mustn't we say at every turn: "I believe this with certainty"?
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243. One says "I know" when one is ready to give compelling grounds. "I know" relates to a possibility of
demonstrating the truth. Whether someone knows something can come to light, assuming that he is convinced of it.
But if what he believes is of such a kind that the grounds that he can give are no surer than his assertion, then
he cannot say that he knows what he believes.
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244. If someone says "I have a body", he can be asked "Who is speaking here with this mouth?"
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245. To whom does anyone say that he knows something? To himself, or to someone else. If he says it to himself,
how is it distinguished from the assertion that he is sure that things are like that? There is no subjective sureness that
I know something. The
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certainty is subjective, but not the knowledge. So if I say "I know that I have two hands", and that is not supposed to
express just my subjective certainty, I must be able to satisfy myself that I am right. But I can't do that, for my
having two hands is not less certain before I have looked at them than afterwards. But I could say: "That I have two
hands is an irreversible belief." That would express the fact that I am not ready to let anything count as a disproof of
this proposition.
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246. "Here I have arrived at a foundation of all my beliefs." "This position I will hold!" But isn't that, precisely, only
because I am completely convinced of it?--What is 'being completely convinced' like?
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247. What would it be like to doubt now whether I have two hands? Why can't I imagine it at all? What would I
believe if I didn't believe that? So far I have no system at all within which this doubt might exist.
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248. I have arrived at the rock bottom of my convictions.
And one might almost say that these foundation-walls are carried by the whole house.
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249. One gives oneself a false picture of doubt.
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250. My having two hands is, in normal circumstances, ascertain as anything that I could produce in evidence for it.
That is why I am not in a position to take the sight of my hand as evidence for it.
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251. Doesn't this mean: I shall proceed according to this belief unconditionally, and not let anything confuse me?
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252. But it isn't just that I believe in this way that I have two hands, but that every reasonable person does.
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253. At the foundation of well-founded belief lies belief that is not founded.
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254. Any 'reasonable' person behaves like this.
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255. Doubting has certain characteristic manifestations, but they are only characteristic of it in particular
circumstances. If
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someone said that he doubted the existence of his hands, kept looking at them from all sides, tried to make sure it
wasn't 'all done by mirrors', etc., we should not be sure whether we ought to call that doubting. We might describe
his way of behaving as like the behaviour of doubt, but his game would not be ours.
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256. On the other hand a language-game does change with time.
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257. If someone said to me that he doubted whether he had a body I should take him to be a half-wit. But I shouldn't
know what it would mean to try to convince him that he had one. And if I had said something, and that had
removed his doubt, I should not know how or why.
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258. I do not know how the sentence "I have a body" is to be used.
That doesn't unconditionally apply to the proposition that I have always been on or near the surface of the
earth.
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259. Someone who doubted whether the earth had existed for 100 years might have a scientific, or on the other hand
a philosophical, doubt.
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260. I would like to reserve the expression "I know" for the cases in which it is used in normal linguistic exchange.
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261. I cannot at present imagine a reasonable doubt as to the existence of the earth during the last 100 years.
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262. I can imagine a man who had grown up in quite special circumstances and been taught that the earth came into
being 50 years ago, and therefore believed this. We might instruct him: the earth has long... etc.--We should be
trying to give him our picture of the world.
This would happen through a kind of persuasion.
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263. The schoolboy believes his teachers and his schoolbooks.
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264. I could imagine Moore being captured by a wild tribe, and their expressing the suspicion that he has come from
somewhere between the earth and the moon. Moore tells them that he
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knows etc. but he can't give them the grounds for his certainty, because they have fantastic ideas of human ability to
fly and know nothing about physics. This would be an occasion for making that statement.
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265. But what does it say, beyond "I have never been to such and such a place, and have compelling grounds for
believing that"?
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266. And here one would still have to say what are compelling grounds.
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267. "I don't merely have the visual impression of a tree: I know that it is a tree".
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268. "I know that this is a hand."--And what is a hand?--"Well, this, for example."
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269. Am I more certain that I have never been on the moon than that I have never been in Bulgaria? Why am I so
sure? Well, I know that I have never been anywhere in the neighbourhood--for example I have never been in the
Balkans.
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270. "I have compelling grounds for my certitude." These grounds make the certitude objective.
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271. What is a telling ground for something is not anything I decide.
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272. I know = I am familiar with it as a certainty.
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273. But when does one say of something that it is certain?
For there can be dispute whether something is certain; I mean, when something is objectively certain.
There are countless general empirical propositions that count as certain for us.
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274. One such is that if someone's arm is cut off it will not grow again. Another, if someone's head is cut off he is
dead and will never live again.
Experience can be said to teach us these propositions. However, it does not teach us them in isolation: rather,
it teaches us a host of interdependent propositions. If they were isolated I might perhaps doubt them, for I have no
experience relating to them.
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275. If experience is the ground of our certainty, then naturally it is past experience.
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Page 36
And it isn't for example just my experience, but other people's, that I get knowledge from.
Now one might say that it is experience again that leads us to give credence to others. But what experience
makes me believe that the anatomy and physiology books don't contain what is false? Though it is true that this trust
is backed up by my own experience.
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276. We believe, so to speak, that this great building exists, and then we see, now here, now there, one or another
small corner of it.
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277. "I can't help believing...."
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278. "I am comfortable that that is how things are."
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279. It is quite sure that motor cars don't grow out of the earth. We feel that if someone could believe the contrary he
could believe everything that we say is untrue, and could question everything that we hold to be sure.
But how does this one belief hang together with all the rest? We should like to say that someone who could
believe that does not accept our whole system of verification.
This system is something that a human being acquires by means of observation and instruction. I
intentionally do not say "learns".
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280. After he has seen this and this and heard that and that, he is not in a position to doubt whether....
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281. I, L. W., believe, am sure, that my friend hasn't sawdust in his body or in his head, even though I have no direct
evidence of my senses to the contrary. I am sure, by reason of what has been said to me, of what I have read, and of
my experience. To have doubts about it would seem to me madness--of course, this is also in agreement with other
people; but I agree with them.
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282. I cannot say that I have good grounds for the opinion that cats do not grow on trees or that I had a father and a
mother.
If someone has doubts about it--how is that supposed to have come about? By his never, from the
beginning, having believed that he had parents? But then, is that conceivable, unless he has been taught it?
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283. For how can a child immediately doubt what it is taught? That could mean only that he was incapable of
learning certain language games.
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284. People have killed animals since the earliest times, used the fur, bones etc. etc. for various purposes; they have
counted definitely on finding similar parts in any similar beast.
They have always learnt from experience; and we can see from their actions that they believe certain things
definitely, whether they express this belief or not. By this I naturally do not want to say that men should behave like
this, but only that they do behave like this.
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285. If someone is looking for something and perhaps roots around in a certain place, he shows that he believes that
what he is looking for is there.
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286. What we believe depends on what we learn. We all believe that it isn't possible to get to the moon; but there
might be people who believe that that is possible and that it sometimes happens. We say: these people do not know
a lot that we know. And, let them be never so sure of their belief--they are wrong and we know it.
If we compare our system of knowledge with theirs then theirs is evidently the poorer one by far.
23.9.50
Page 37
287. The squirrel does not infer by induction that it is going to need stores next winter as well. And no more do we
need a law of induction to justify our actions or our predictions.
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288. I know, not just that the earth existed long before my birth, but also that it is a large body, that this has been
established, that I and the rest of mankind have forebears, that there are books about all this, that such books don't
lie, etc. etc. etc. And I know all this? I believe it. This body of knowledge has been handed on to me and I have no
grounds for doubting it, but, on the contrary, all sorts of confirmation.
And why shouldn't I say that I know all this? Isn't that what one does say?
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Page 38
But not only I know, or believe, all that, but the others do too. Or rather, I believe that they believe it.
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289. I am firmly convinced that others believe, believe they know, that all that is in fact so.
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290. I myself wrote in my book that children learn to understand a word in such and such a way. Do I know that, or
do I believe it? Why in such a case do I write not "I believe etc." but simply the indicative sentence?
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291. We know that the earth is round. We have definitively ascertained that it is round.
We shall stick to this opinion, unless our whole way of seeing nature changes. "How do you know that?"--I
believe it.
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292. Further experiments cannot give the lie to our earlier ones, at most they may change our whole way of looking
at things.
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293. Similarly with the sentence "water boils at 100°C."
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294. This is how we acquire conviction, this is called "being rightly convinced".
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295. So hasn't one, in this sense, a proof of the proposition? But that the same thing has happened again is not a
proof of it; though we do say that it gives us a right to assume it.
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296. This is what we call an "empirical foundation" for our assumptions.
Page 38
297. For we learn, not just that such and such experiments had those and those results, but also the conclusion
which is drawn. And of course there is nothing wrong in our doing so. For this inferred proposition is an instrument
for a definite use.
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298. 'We are quite sure of it' does not mean just that every single person is certain of it, but that we belong to a
community which is bound together by science and education.
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299. We are satisfied that the earth is round.†1
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10.3.51
300. Not all corrections of our views are on the same level.
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Page 39
301. Supposing it wasn't true that the earth had already existed long before I was born--how should we imagine the
mistake being discovered?
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302. It's no good saying "Perhaps we are wrong" when, if no evidence is trustworthy, trust is excluded in the case of
the present evidence.
Page 39
303. If, for example, we have always been miscalculating, and twelve times twelve isn't a hundred and forty-four,
why should we trust any other calculation? And of course that is wrongly put.
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304. But nor am I making a mistake about twelve times twelve being a hundred and forty-four. I may say later that I
was confused just now, but not that I was making a mistake.
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305. Here once more there is needed a step like the one taken in relativity theory.
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306. "I don't know if this is a hand." But do you know what the word "hand" means? And don't say "I know what it
means now for me". And isn't it an empirical fact--that this word is used like this?
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307. And here the strange thing is that when I am quite certain of how the words are used, have no doubt about it, I
can still give no grounds for my way of going on. If I tried I could give a thousand, but none as certain as the very
thing they were supposed to be grounds for.
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308. 'Knowledge' and 'certainty' belong to different categories. They are not two 'mental states' like, say 'surmising'
and 'being sure'. (Here I assume that it is meaningful for me to say "I know what (e.g.) the word 'doubt' means" and
that this sentence indicates that the word "doubt" has a logical role.) What interests us now is not being sure but
knowledge. That is, we are interested in the fact that about certain empirical propositions no doubt can exist if
making judgments is to be possible at all. Or again: I am inclined to believe that not everything that has the form of
an empirical proposition is one.
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309. Is it that rule and empirical proposition merge into one another?
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Page 40
310. A pupil and a teacher. The pupil will not let anything be explained to him, for he continually interrupts with
doubts, for instance as to the existence of things, the meaning of words, etc. The teacher says "Stop interrupting me
and do as I tell you. So far your doubts don't make sense at all".
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311. Or imagine that the boy questioned the truth of history (and everything that connects up with it)--and even
whether the earth had existed at all a hundred years before.
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312. Here it strikes me as if this doubt were hollow. But in that case--isn't belief in history hollow too? No; there is so
much that this connects up with.
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313. So is that what makes us believe a proposition? Well--the grammar of "believe" just does hang together with
the grammar of the proposition believed.
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314. Imagine that the schoolboy really did ask "and is there a table there even when I turn round, and even when no
one is there to see it?" Is the teacher to reassure him--and say "of course there is!"?
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Perhaps the teacher will get a bit impatient, but think that the boy will grow out of asking such questions.
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315. That is to say, the teacher will feel that this is not really a legitimate question at all.
And it would be just the same if the pupil cast doubt on the uniformity of nature, that is to say on the
justification of inductive arguments.--The teacher would feel that this was only holding them up, that this way the
pupil would only get stuck and make no progress.--And he would be right. It would be as if someone were looking
for some object in a room; he opens a drawer and doesn't see it there; then he closes it again, waits, and opens it
once more to see if perhaps it isn't there now, and keeps on like that. He has not learned to look for things. And in
the same way this pupil has not learned how to ask questions. He has not learned the game that we are trying to
teach him.
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316. And isn't it the same as if the pupil were to hold up his history lesson with doubts as to whether the earth
really....?
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Page 41
317. This doubt isn't one of the doubts in our game. (But not as if we chose this game!)
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12.3.51
318. 'The question doesn't arise at all.' Its answer would characterize a method. But there is no sharp boundary
between methodological propositions and propositions within a method.
Page 41
319. But wouldn't one have to say then, that there is no sharp boundary between propositions of logic and empirical
propositions? The lack of sharpness is that of the boundary between rule and empirical proposition.
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320. Here one must, I believe, remember that the concept 'proposition' itself is not a sharp one.
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321. Isn't what I am saying: any empirical proposition can be transformed into a postulate--and then becomes a
norm of description. But I am suspicious even of this. The sentence is too general. One almost wants to say "any
empirical proposition can, theoretically, be transformed...", but what does "theoretically" mean here? It sounds all
too reminiscent of the Tractatus.
Page 41
322. What if the pupil refused to believe that this mountain had been there beyond human memory?
We should say that he had no grounds for this suspicion.
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323. So rational suspicion must have grounds?
We might also say: "the reasonable man believes this".
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324. Thus we should not call anybody reasonable who believed something in despite of scientific evidence.
Page 41
325. When we say that we know that such and such..., we mean that any reasonable person in our position would
also know it, that it would be a piece of unreason to doubt it. Thus Moore too wants to say not merely that he knows
that he etc. etc., but also that anyone endowed with reason in his position would know it just the same.
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326. But who says what it is reasonable to believe in this situation?
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Page 42
327. So it might be said: "The reasonable man believes: that the earth has been there since long before his birth, that
his life has been spent on the surface of the earth, or near it, that he has never, for example, been on the moon, that
he has a nervous system and various innards like all other people, etc., etc."
Page 42
328. "I know it as I know that my name is L. W."
Page 42
329. 'If he calls that in doubt--whatever "doubt" means here--he will never learn this game'.
Page 42
330. So here the sentence "I know..." expresses the readiness to believe certain things.
Page 42
13.3.
331. If we ever do act with certainty on the strength of belief, should we wonder that there is much we cannot
doubt?
Page 42
332. Imagine that someone were to say, without wanting to philosophize, "I don't know if I have ever been on the
moon; I don't remember ever having been there". (Why would this person be so radically different from us?)
In the first place--how would he know that he was on the moon? How does he imagine it? Compare: "I do
not know if I was ever in the village of X." But neither could I say that if X were in Turkey, for I know that I was
never in Turkey.
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333. I ask someone "Have you ever been in China?" He replies "I don't know". Here one would surely say "You
don't know? Have you any reason to believe you might have been there at some time? Were you for example ever
near the Chinese border? Or were your parents there at the time when you were going to be born?"--Normally
Europeans do know whether they have been in China or not.
Page 42
334. That is to say: only in such-and-such circumstances does a reasonable person doubt that.
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335. The procedure in a court of law rests on the fact that circumstances give statements a certain probability. The
statement that, for example, someone came into the world without parents wouldn't ever be taken into consideration
there.
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336. But what men consider reasonable or unreasonable alters. At certain periods men find reasonable what at other
periods they found unreasonable. And vice versa.
But is there no objective character here?
Very intelligent and well-educated people believe in the story of creation in the Bible, while others hold it as
proven false, and the grounds of the latter are well known to the former.
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337. One cannot make experiments if there are not some things that one does not doubt. But that does not mean that
one takes certain presuppositions on trust. When I write a letter and post it, I take it for granted that it will arrive--I
expect this.
If I make an experiment I do not doubt the existence of the apparatus before my eyes. I have plenty of
doubts, but not that. If I do a calculation I believe, without any doubts, that the figures on the paper aren't switching
of their own accord, and I also trust my memory the whole time, and trust it without any reservation. The certainty
here is the same as that of my never having been on the moon.
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338. But imagine people who were never quite certain of these things, but said that they were very probably so, and
that it did not pay to doubt them. Such a person, then, would say in my situation: "It is extremely unlikely that I
have ever been on the moon", etc., etc. How would the life of these people differ from ours? For there are people
who say that it is merely extremely probable that water over a fire will boil and not freeze, and that therefore strictly
speaking what we consider impossible is only improbable. What difference does this make in their lives? Isn't it just
that they talk rather more about certain things than the rest of us?
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339. Imagine someone who is supposed to fetch a friend from the railway station and doesn't simply look the train
up in the time-table and go to the station at the right time, but says: "I have no belief that the train will really arrive,
but I will go to the station all the same." He does everything that the normal person does, but accompanies it with
doubts or with self-annoyance, etc.
Page 43
340. We know, with the same certainty with which we believe any mathematical proposition, how the letters A and
B are
Page Break 44
pronounced, what the colour of human blood is called, that other human beings have blood and call it "blood".
Page 44
341. That is to say, the questions that we raise and our doubts depend on the fact that some propositions are exempt
from doubt, are as it were like hinges on which those turn.
Page 44
342. That is to say, it belongs to the logic of our scientific investigations that certain things are in deed not doubted.
Page 44
343. But it isn't that the situation is like this: We just can't investigate everything, and for that reason we are forced to
rest content with assumption. If I want the door to turn, the hinges must stay put.
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344. My life consists in my being content to accept many things.
Page 44
345. If I ask someone "what colour do you see at the moment?", in order, that is, to learn what colour is there at the
moment, I cannot at the same time question whether the person I ask understands English, whether he wants to take
me in, whether my own memory is not leaving me in the lurch as to the names of colours, and so on.
Page 44
346. When I am trying to mate someone in chess, I cannot have doubts about the pieces perhaps changing places of
themselves and my memory simultaneously playing tricks on me so that I don't notice.
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15.3.51
347. "I know that that's a tree." Why does it strike me as if I did not understand the sentence? though it is after all an
extremely simple sentence of the most ordinary kind? It is as if I could not focus my mind on any meaning. Simply
because I don't look for the focus where the meaning is. As soon as I think of an everyday use of the sentence
instead of a philosophical one, its meaning becomes dear and ordinary.
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348. Just as the words "I am here" have a meaning only in certain contexts, and not when I say them to someone
who is sitting in front of me and sees me clearly,--and not because they are superfluous, but because their meaning
is not determined by the situation, yet stands in need of such determination.
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349. "I know that that's a tree"--this may mean all sorts of things: I look at a plant that I take for a young beech and
that someone else thinks is a black-currant. He says "that is a shrub"; I say it is a tree.--We see something in the mist
which one of us takes for a man, and the other says "I know that that's a tree". Someone wants to test my eyes etc.
etc.--etc. etc. Each time the 'that' which I declare to be a tree is of a different kind.
But what when we express ourselves more precisely? For example: "I know that that thing there is a tree, I
can see it quite clearly."--Let us even suppose I had made this remark in the context of a conversation (so that it was
relevant when I made it); and now, out of all context, I repeat it while looking at the tree, and I add "I mean these
words as I did five minutes ago". If I added, for example, that I had been thinking of my bad eyes again and it was a
kind of sigh, then there would be nothing puzzling about the remark.
For how a sentence is meant can be expressed by an expansion of it and may therefore be made part of it.
Page 45
350. "I know that that's a tree" is something a philosopher might say to demonstrate to himself or to someone else
that he knows something that is not a mathematical or logical truth. Similarly, someone who was entertaining the
idea that he was no use any more might keep repeating to himself "I can still do this and this and this". If such
thoughts often possessed him one would not be surprised if he, apparently out of all context, spoke such a sentence
out loud. (But here I have already sketched a background, a surrounding, for this remark, that is to say given it a
context.) But if someone, in quite heterogeneous circumstances, called out with the most convincing mimicry:
"Down with him!", one might say of these words (and their tone) that they were a pattern that does indeed have
familiar applications, but that in this case it was not even clear what language the man in question was speaking. I
might make with my hand the movement I should make if I were holding a hand-saw and sawing through a plank;
but would one have any right to call this movement sawing, out of all context?--(It might be something quite
different,)
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351. Isn't the question "Have these words a meaning?" similar to "Is that a tool?" asked as one produces, say, a
hammer? I say "Yes, it's a hammer". But what if the thing that any of us would take for a hammer were somewhere
else a missile, for example, or a conductor's baton? Now make the application yourself.
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352. If someone says, "I know that that's a tree" I may answer: "Yes, that is a sentence. An English sentence. And
what is it supposed to be doing?" Suppose he replies: "I just wanted to remind myself that I know things like that"?--
Page 46
353. But suppose he said "I want to make a logical observation"?--If a forester goes into a wood with his men and
says "This tree has got to be cut down, and this one and this one"--what if he then observes "I know that that's a
tree"?--But might not I say of the forester "He knows that that's a tree--he doesn't examine it, or order his men to
examine it"?
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354. Doubting and non-doubting behaviour. There is the first only if there is the second.
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355. A mad-doctor (perhaps) might ask me "Do you know what that is?" and I might reply "I know that it's a chair; I
recognize it, it's always been in my room". He says this, possibly, to test not my eyes but my ability to recognize
things, to know their names and their functions. What is in question here is a kind of knowing one's way about. Now
it would be wrong for me to say "I believe that it's a chair" because that would express my readiness for my
statement to be tested. While "I know that it..." implies bewilderment if what I said was not confirmed.
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356. My "mental state", the "knowing", gives me no guarantee of what will happen. But it consists in this, that I
should not understand where a doubt could get a foothold nor where a further test was possible.
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357. One might say: "'I know' expresses comfortable certainty, not the certainty that is still struggling."
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358. Now I would like to regard this certainty, not as something akin to hastiness or superficiality, but as a form of
life. (That is very badly expressed and probably badly thought as well.)
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359. But that means I want to conceive it as something that lies beyond being justified or unjustified; as it were, as
something animal.
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360. I KNOW that this is my foot. I could not accept any experience as proof to the contrary.--That may be an
exclamation; but what follows from it? At least that I shall act with a certainty that knows no doubt, in accordance
with my belief.
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361. But I might also say: It has been revealed to me by God that it is so. God has taught me that this is my foot.
And therefore if anything happened that seemed to conflict with this knowledge I should have to regard that as
deception.
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362. But doesn't it come out here that knowledge is related to a decision?
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363. And here it is difficult to find the transition from the exclamation one would like to make, to its consequences in
what one does.
Page 47
364. One might also put this question: "If you know that that is your foot,--do you also know, or do you only
believe, that no future experience will seem to contradict your knowledge?" (That is, that nothing will seem to you
yourself to do so.)
Page 47
365. If someone replied: "I also know that it will never seem to me as if anything contradicted that
knowledge",--what could we gather from that, except that he himself had no doubt that it would never happen?--
Page 47
366. Suppose it were forbidden to say "I know" and only allowed to say "I believe I know"?
Page 47
367. Isn't it the purpose of construing a word like "know" analogously to "believe" that then opprobrium attaches to
the statement "I know" if the person who makes it is wrong?
As a result a mistake becomes something forbidden.
Page 47
368. If someone says that he will recognize no experience as proof of the opposite, that is after all a decision. It is
possible that he will act against it.
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Page 48
16.3.51
369. If I wanted to doubt whether this was my hand, how could I avoid doubting whether the word "hand" has any
meaning? So that is something I seem to know after all.
Page 48
370. But more correctly: The fact that I use the word "hand" and all the other words in my sentence without a
second thought, indeed that I should stand before the abyss if I wanted so much as to try doubting their
meanings--shews that absence of doubt belongs to the essence of the language-game, that the question "How do I
know..." drags out the language-game, or else does away with it.
Page 48
371. Doesn't "I know that that's a hand", in Moore's sense, mean the same, or more or less the same, as: I can make
statements like "I have a pain in this hand" or "this hand is weaker than the other" or "I once broke this hand", and
countless others, in language-games where a doubt as to the existence of this hand does not come in?
Page 48
372. Only in certain cases is it possible to make an investigation "is that really a hand?" (or "my hand"). For "I doubt
whether that is really my (or a) hand" makes no sense without some more precise determination. One cannot tell
from these words alone whether any doubt at all is meant--nor what kind of doubt.
Page 48
373. Why is it supposed to be possible to have grounds for believing something if it isn't possible to be certain?
Page 48
374. We teach a child "that is your hand", not "that is perhaps (or "probably") your hand". That is how a child learns
the innumerable language-games that are concerned with his hand. An investigation or question, 'whether this is
really a hand' never occurs to him. Nor, on the other hand, does he learn that he knows that this is a hand.
Page 48
375. Here one must realize that complete absence of doubt at some point, even where we would say that 'legitimate'
doubt can exist, need not falsify a language-game. For there is also something like another arithmetic.
I believe that this admission must underlie any understanding of logic.
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Page 49
17.3.
376. I may claim with passion that I know that this (for example) is my foot.
Page 49
377. But this passion is after all something very rare, and there is no trace of it when I talk of this foot in the ordinary
way.
Page 49
378. Knowledge is in the end based on acknowledgement.
Page 49
379. I say with passion "I know that this is a foot"--but what does it mean?
Page 49
380. I might go on: "Nothing in the world will convince me of the opposite!" For me this fact is at the bottom of all
knowledge. I shall give up other things but not this.
Page 49
381. This "Nothing in the world" is obviously an attitude which one hasn't got towards everything one believes or is
certain of.
Page 49
382. That is not to say that nothing in the world will in fact be able to convince me of anything else.
Page 49
383. The argument "I may be dreaming" is senseless for this reason: if I am dreaming, this remark is being dreamed
as well--and indeed it is also being dreamed that these words have any meaning.
Page 49
384. Now what kind of sentence is "Nothing in the world..."?
Page 49
385. It has the form of a prediction, but of course it is not one that is based on experience.
Page 49
386. Anyone who says, with Moore, that he knows that so and so...--gives the degree of certainty that something has
for him. And it is important that this degree has a maximum value.
Page 49
387. Someone might ask me: "How certain are you that that is a tree over there; that you have money in your
pocket; that that is your foot?" And the answer in one case might be "not certain", in another "as good as certain", in
the third "I can't doubt it". And these answers would make sense even without any grounds. I should not need, for
example, to say: "I can't be certain whether that is a tree because my eyes aren't sharp enough". I want to
Page Break 50
say: it made sense for Moore to say "I know that that is a tree", if he meant something quite particular by it.
Page 50
[I believe it might interest a philosopher, one who can think himself, to read my notes. For even if I have hit
the mark only rarely, he would recognize what targets I had been ceaselessly aiming at.]
Page 50
388. Every one of us often uses such a sentence, and there is no question but that it makes sense. But does that
mean it yields any philosophical conclusion? Is it more of a proof of the existence of external things, that I know
that this is a hand, than that I don't know whether that is gold or brass?
Page 50
18.3.
389. Moore wanted to give an example to shew that one really can know propositions about physical objects.--If
there were a dispute whether one could have a pain in such and such a part of the body, then someone who just then
had a pain in that spot might say: "I assure you, I have a pain there now." But it would sound odd if Moore had said:
"I assure you, I know that's a tree." A personal experience simply has no interest for us here.
Page 50
390. All that is important is that it makes sense to say that one knows such a thing; and consequently the assurance
that one does know it can't accomplish anything here.
Page 50
391. Imagine a language-game "When I call you, come in through the door". In any ordinary case, a doubt whether
there really is a door there will be impossible.
Page 50
392. What I need to shew is that a doubt is not necessary even when it is possible. That the possibility of the
language-game doesn't depend on everything being doubted that can be doubted. (This is connected with the role of
contradiction in mathematics.)
Page 50
393. The sentence "I know that that's a tree" if it were said outside its language-game, might also be a quotation
(from an English grammar-book perhaps).--"But suppose I mean it while I am saying it?" The old misunderstanding
about the concept 'mean'.
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Page 51
394. "This is one of the things that I cannot doubt."
Page 51
395. "I know all that." And that will come out in the way I act and in the way I speak about the things in question.
Page 51
396. In the language-game (2),†1 can he say that he knows that those are building stones? "No, but he does know it."
Page 51
397. Haven't I gone wrong and isn't Moore perfectly right? Haven't I made the elementary mistake of confusing one's
thoughts with one's knowledge? Of course I do not think to myself "The earth already existed for some time before
my birth", but do I know it any the less? Don't I show that I know it by always drawing its consequences)
Page 51
398. And don't I know that there is no stairway in this house going six floors deep into the earth, even though I have
never thought about it?
Page 51
399. But doesn't my drawing the consequences only show that I accept this hypothesis?
Page 51
19.3.
400. Here I am inclined to fight windmills, because I cannot yet say the thing I really want to say.
Page 51
401. I want to say: propositions of the form of empirical propositions, and not only propositions of logic, form the
foundation of all operating with thoughts (with language).--This observation is not of the form "I know...". "I
know..." states what I know, and that is not of logical interest.
Page 51
402. In this remark the expression "propositions of the form of empirical propositions" is itself thoroughly bad; the
statements in question are statements about material objects. And they do not serve as foundations in the same way
as hypotheses which, if they turn out to be false, are replaced by others.
(....und schreib getrost )
("Im Anfang war die Tat."†2)
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Page 52
403. To say of man, in Moore's sense, that he knows something; that what he says is therefore unconditionally the
truth, seems wrong to me.--It is the truth only inasmuch as it is an unmoving foundation of his language-games.
Page 52
404. I want to say: it's not that on some points men know the truth with perfect certainty. No: perfect certainty is
only a matter of their attitude.
Page 52
405. But of course there is still a mistake even here.
Page 52
406. What I am aiming at is also found in the difference between the casual observation "I know that that's a...", as it
might be used in ordinary life, and the same utterance when a philosopher makes it.
Page 52
407. For when Moore says "I know that that's..." I want to reply "you don't know anything!"--and yet I would not
say that to anyone who was speaking without philosophical intention. That is, I feel (rightly?) that these two mean to
say something different.
Page 52
408. For if someone says he knows such-and-such, and this is part of his philosophy--then his philosophy is false if
he has slipped up in this statement.
Page 52
409. If I say "I know that that's a foot"--what am I really saying? Isn't the whole point that I am certain of the
consequences--that if someone else had been in doubt I might say to him "you see--I told you so"? Would my
knowledge still be worth anything if it let me down as a clue in action? And can't it let me down?
Page 52
20.3.
410. Our knowledge forms an enormous system. And only within this system has a particular bit the value we give
it.
Page 52
411. If I say "we assume that the earth has existed for many years past" (or something similar), then of course it
sounds strange that we should assume such a thing. But in the entire system of our language-games it belongs to the
foundations. The assumption, one might say, forms the basis of action, and therefore, naturally, of thought.
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Page 53
412. Anyone who is unable to imagine a case in which one might say "I know that this is my hand" (and such cases
are certainly rare) might say that these words were nonsense. True, he might also say "Of course I know--how could
I not know?"--but then he would possibly be taking the sentence "this is my hand" as an explanation of the words
"my hand".
Page 53
413. For suppose you were guiding a blind man's hand, and as you were guiding it along yours you said "this is my
hand"; if he then said "are you sure?" or "do you know it is?", it would take very special circumstances for that to
make sense.
Page 53
414. But on the other hand: how do I know that it is my hand? Do I even here know exactly what it means to say it is
my hand?--When I say "how do I know?" I do not mean that I have the least doubt of it. What we have here is a
foundation for all my action. But it seems to me that it is wrongly expressed by the words "I know".
Page 53
415. And in fact, isn't the use of the word "know" as a preeminently philosophical word altogether wrong? If "know"
has this interest, why not "being certain"? Apparently because it would be too subjective. But isn't "know" just as
subjective? Isn't one misled simply by the grammatical peculiarity that "p" follows from "I know p"?
"I believe I know" would not need to express a lesser degree of certainty.--True, but one isn't trying to
express even the greatest subjective certainty, but rather that certain propositions seem to underlie all questions and
all thinking.
Page 53
416. And have we an example of this in, say, the proposition that I have been living in this room for weeks past, that
my memory does not deceive me in this?
--"certain beyond all reasonable doubt"--
Page 53
21.3.
417. "I know that for the last month I have had a lath every day." What am I remembering? Each day and the bath
each morning? No. I know that I bathed each day and I do not derive that from some other immediate datum.
Similarly I say "I felt a
Page Break 54
pain in my arm" without this locality coming into my consciousness in any other way (such as by means of an
image).
Page 54
418. Is my understanding only blindness to my own lack of understanding? It often seems so to me.
Page 54
419. If I say "I have never been in Asia Minor", where do I get this knowledge from? I have not worked it out, no
one told me; my memory tells me.--So I can't be wrong about it? Is there a truth here which I know?--I cannot depart
from this judgment without toppling all other judgments with it.
Page 54
420. Even a proposition like this one, that I am now living in England, has these two sides: it is not a mistake--but on
the other hand, what do I know of England? Can't my judgment go all to pieces?
Would it not be possible that people came into my room and all declared the opposite?--even gave me
'proofs' of it, so that I suddenly stood there like a madman alone among people who were all normal, or a normal
person alone among madmen? Might I not then suffer doubts about what at present seems at the furthest remove
from doubt?
Page 54
421. I am in England.--Everything around me tells me so; wherever and however I let my thoughts turn, they
confirm this for me at once.--But might I not be shaken if things such as I don't dream of at present were to happen?
Page 54
422. So I am trying to say something that sounds like pragmatism.
Here I am being thwarted by a kind of Weltanschauung.
Page 54
423. Then why don't I simply say with Moore "I know that I am in England"? Saying this is meaningful in particular
circumstances, which I can imagine. But when I utter the sentence outside these circumstances, as an example to
shew that I can know truths of this kind with certainty, then it at once strikes me as fishy.--Ought it to?
Page 54
424. I say "I know p" either to assure people that I, too, know the truth p, or simply as an emphasis of p. One
says,
Page Break 55
too, "I don't believe it, I know it". And one might also put it like this (for example): "That is a tree. And that's not just
surmise."
But what about this: "If I were to tell someone that that was a tree, that wouldn't be just surmise." Isn't this
what Moore was trying to say?
Page 55
425. It would not be surmise and I might tell it to someone else with complete certainty, as something there is no
doubt about. But does that mean that it is unconditionally the truth? May not the thing that I recognize with
complete certainty as the tree that I have seen here my whole life long--may this not be disclosed as something
different? May it not confound me?
And nevertheless it was right, in the circumstances that give this sentence meaning, to say "I know (I do not
merely surmise) that that's a tree". To say that in strict truth I only believe it, would be wrong. It would be
completely misleading to say: "I believe my name is L. W." And this too is right: I cannot be making a mistake
about it. But that does not mean that I am infallible about it.
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21.3.51
426. But how can we show someone that we know truths, not only about sense-data but also about things? For after
all it can't be enough for someone to assure us that he knows this.
Well, what must our starting point be if we are to shew this?
Page 55
22.3.
427. We need to shew that even if he never uses the words "I know...", his conduct exhibits the thing we are
concerned with.
Page 55
428. For suppose a person of normal behaviour assured us that he only believed his name was such-and-such, he
believed he recognized the people he regularly lived with, he believed that he had hands an feet when he didn't
actually see them, and so on. Can we shew him it is not so from the things he does (and says)?
Page 55
23.3.51
429. What reason have I, now, when I cannot see my toes, to assume that I have five toes on each foot?
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Page 56
Is it right to say that my reason is that previous experience has always taught me so? Am I more certain of
previous experience than that I have ten toes?
That previous experience may very well be the cause of my present certitude; but is it its ground?
Page 56
430. I meet someone from Mars and he asks me "How many toes have human beings got?"--I say "Ten. I'll shew
you", and take my shoes off. Suppose he was surprised that I knew with such certainty, although I hadn't looked at
my toes--ought I to say: "We humans know how many toes we have whether we can see them or not"?
Page 56
26.3.
431. "I know that this room is on the second floor, that behind the door a short landing leads to the stairs, and so
on." One could imagine cases where I should come out with this, but they would be extremely rare. But on the other
hand I shew this knowledge day in, day out by my actions and also in what I say.
Now what does someone else gather from these actions and words of mine? Won't it be just that I am sure of
my ground?--From the fact that I have been living here for many weeks and have gone up and down the stairs every
day he will gather that I know where my room is situated.--I shall give him the assurance "I know" when he does not
already know things which would have compelled the conclusion that I knew.
Page 56
432. The utterance "I know..." can only have its meaning in connection with the other evidence of my 'knowing'.
Page 56
433. So if I say to someone "I know that that's a tree", it is as if I told him "that is a tree; you can absolutely rely on it;
there is no doubt about it". And a philosopher could only use the statement to show that this form of speech is
actually used. But if his use of it is not to be merely an observation about English grammar, he must give the
circumstances in which this expression functions.
Page 56
434. Now does experience teach us that in such-and-such circumstances people know this and that? Certainly,
experience shews us that normally after so-and-so many days a man can find his way about a house he has been
living in. Or even: experience
Page Break 57
teaches us that after such-and-such a period of training a man's judgment is to be trusted. He must, experience tells
us, have learnt for so long in order to be able to make a correct prediction. But-- -- --
Page 57
27.3.
435. One is often bewitched by a word. For example, by the word "know".
Page 57
436. Is God bound by our knowledge? Are a lot of our statements incapable of falsehood? For that is what we want
to say.
Page 57
437. I am inclined to say: "That cannot be false." That is interesting; but what consequences has it?
Page 57
438. It would not be enough to assure someone that I know what is going on at a certain place--without giving him
grounds that satisfy him that I am in a position to know.
Page 57
439. Even the statement "I know that behind this door there is a landing and the stairway down to the ground floor"
only sounds so convincing because everyone takes it for granted that I know it.
Page 57
440. There is something universal here; not just something personal.
Page 57
441. In a court of law the mere assurance "I know..." on the part of a witness would convince no one. It must be
shown that he was in a position to know.
Even the assurance "I know that that's a hand", said while someone looked at his own hand, would not be
credible unless we knew the circumstances in which it was said. And if we do know them, it seems to be an
assurance that the person speaking is normal in this respect.
Page 57
442. For may it not happen that I imagine myself to know something?
Page 57
443. Suppose that in a certain language there were no word corresponding to our "know".--The people simply make
assertions. ("That is a tree", etc.) Naturally it can occur for them to make mistakes. And so they attach a sign to the
sentence which indicates how probable they take a mistake to be--or should I say, how probable a mistake is in this
case? This latter can also
Page Break 58
be indicated by mentioning certain circumstances. For example "Then A said to B '...'. I was standing quite close to
them and my hearing is good", or "A was at such-and-such a place yesterday. I saw him from a long way off. My
eyes are not very good", or "There is a tree over there: I can see it clearly and I have seen it innumerable times
before".
Page 58
444. "The train leaves at two o'clock. Check it once more to make certain" or "The train leaves at two o'clock. I have
just looked it up in a new time-table". One may also add "I am reliable in such matters". The usefulness of such
additions is obvious.
Page 58
445. But if I say "I have two hands", what can I add to indicate reliability? At the most that the circumstances are the
ordinary ones.
Page 58
446. But why am I so certain that this is my hand? Doesn't the whole language-game rest on this kind of certainty?
Or: isn't this 'certainty' (already) presupposed in the language-game? Namely by virtue of the fact that one is
not playing the game, or is playing it wrong, if one does not recognize objects with certainty.
Page 58
28.3.
447. Compare with this 12 × 12 = 144. Here too we don't say "perhaps". For, in so far as this proposition rests on our
not miscounting or miscalculating and on our senses not deceiving us as we calculate, both propositions, the
arithmetical one and the physical one, are on the same level.
I want to say: The physical game is just as certain as the arithmetical. But this can be misunderstood. My
remark is a logical and not a psychological one.
Page 58
448. I want to say: If one doesn't marvel at the fact that the propositions of arithmetic (e.g. the multiplication tables)
are 'absolutely certain', then why should one be astonished that the proposition "This is my hand" is so equally?
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449. Something must be taught us as a foundation.
Page 58
450. I want to say: our learning has the form "that is a violet", "that is a table". Admittedly, the child might hear the
word "violet" for the first time in the sentence "perhaps that is a
Page Break 59
violet", but then he could ask "what is a violet?" Now this might of course be answered by showing him a picture.
But how would it be if one said "that is a..." only when showing him a picture, but otherwise said nothing but
"perhaps that is a..."--What practical consequences is that supposed to have?
A doubt that doubted everything would not be a doubt.
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451. My objection against Moore, that the meaning of the isolated sentence "That is a tree" is undetermined, since it
is not determined what the "that" is that is said to be a tree--doesn't work, for one can make the meaning more
definite by saying, for example: "The object over there that looks like a tree is not an artificial imitation of a tree but a
real one."
Page 59
452. It would not be reasonable to doubt if that was a real tree or only.
My finding it beyond doubt is not what counts. If a doubt would be unreasonable, that cannot be seen from
what I hold. There would therefore have to be a rule that declares doubt to be unreasonable here. But there isn't such
a rule, either.
Page 59
453. I do indeed say: "Here no reasonable person would doubt."--Could we imagine learned judges being asked
whether a doubt was reasonable or unreasonable?
Page 59
454. There are cases where doubt is unreasonable, but others where it seems logically impossible. And there seems
to be no clear boundary between them.
Page 59
29.3.
455. Every language-game is based on words 'and objects' being recognized again. We learn with the same
inexorability that this is a chair as that 2 × 2 = 4.
Page 59
456. If, therefore, I doubt or am uncertain about this being my hand (in whatever sense), why not in that case about
the meaning of these words as well?
Page 59
457. Do I want to say, then, that certainty resides in the nature of the language-game?
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Page 60
458. One doubts on specific grounds. The question is this: how is doubt introduced into the language-game?
Page 60
459. If the shopkeeper wanted to investigate each of his apples without any reason, for the sake of being certain
about everything, why doesn't he have to investigate the investigation? And can one talk of belief here (I mean belief
as in 'religious belief', not surmise)? All psychological terms merely distract us from the thing that really matters.
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460. I go to the doctor, shew him my hand and say "This is a hand, not...; I've injured it, etc., etc." Am I only giving
him a piece of superfluous information? For example, mightn't one say: supposing the words "This is a hand" were a
piece of information--how could you bank on his understanding this information? Indeed, if it is open to doubt
'whether that is a hand', why isn't it also open to doubt whether I am a human being who is informing the doctor of
this?--But on the other hand one can imagine cases-even if they are very rare ones--where this declaration is not
superfluous, or is only superfluous but not absurd.
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461. Suppose that I were the doctor and a patient came to me, showed me his hand and said: "This thing that looks
like a hand isn't just a superb imitation--it really is a hand" and went on to talk about his injury--should I really take
this as a piece of information, even though a superfluous one? Shouldn't I be more likely to consider it nonsense,
which admittedly did have the form of a piece of information? For, I should say, if this information really were
meaningful, how can he be certain of what he says? The background is lacking for it to be information.
Page 60
30.3.
462. Why doesn't Moore produce as one of the things that he knows, for example, that in such-and-such a part of
England there is a village called so-and-so? In other words: why doesn't he mention a fact that is known to him and
not to every one of us;'
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31.3.
463. This is certainly true, that the information "That is a tree", when no one could doubt it, might be a kind of joke
and as such
Page Break 61
have meaning. A joke of this kind was in fact made once by Renan.
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3.4.51
464. My difficulty can also be shewn like this: I am sitting talking to a friend. Suddenly I say: "I knew all along that
you were so-and-so." Is that really just a superfluous, though true, remark?
I feel as if these words were like "Good morning" said to someone in the middle of a conversation.
Page 61
465. How would it be if we had the words "They know nowadays that there are over... species of insects" instead of
"I know that that's a tree"? If someone were suddenly to utter the first sentence out of all context one might think: he
has been thinking of something else in the interim and is now saying out loud some sentence in his train of thought.
Or again: he is in a trance and is speaking without understanding what he is saying.
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466. Thus it seems to me that I have known something the whole time, and yet there is no meaning in saying so, in
uttering this truth.
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467. I am sitting with a philosopher in the garden; he says again and again "I know that that's a tree", pointing to a
tree that is near us. Someone else arrives and hears this, and I tell him: "This fellow isn't insane. We are only doing
philosophy."
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4.4.
468. Someone says irrelevantly "That's a tree". He might say this sentence because he remembers having heard it in a
similar situation; or he was suddenly struck by the tree's beauty and the sentence was an exclamation; or he was
pronouncing the sentence to himself as a grammatical example; etc., etc. And now I ask him "How did you mean
that?" and he replies "It was a piece of information directed at you". Shouldn't I be at liberty to assume that he
doesn't know what he is saying, if he is insane enough to want to give me this information?
Page 61
469. In the middle of a conversation, someone says to me out of the blue: "I wish you luck." I am astonished; but
later I
Page Break 62
realize that these words connect up with his thoughts about me. And now they do not strike me as meaningless any
more.
Page 62
470. Why is there no doubt that I am called L. W.? It does not seem at all like something that one could establish at
once beyond doubt. One would not think that it is one of the indubitable truths.
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5.4.
[Here there is still a big gap in my thinking. And I doubt whether it will be filled now.]
Page 62
471. It is so difficult to find the beginning. Or, better: it is difficult to begin at the beginning. And not try to go
further back.
Page 62
472. When a child learns language it learns at the same time what is to be investigated and what not. When it learns
that there is a cupboard in the room, it isn't taught to doubt whether what it sees later on is still a cupboard or only a
kind of stage set.
Page 62
473. Just as in writing we learn a particular basic form of letters and then vary it later, so we learn first the stability of
things as the norm, which is then subject to alterations.
Page 62
474. This game proves its worth. That may be the cause of its being played, but it is not the ground.
Page 62
475. I want to regard man here as an animal; as a primitive being to which one grants instinct but not ratiocination.
As a creature in a primitive state. Any logic good enough for a primitive means of communication needs no apology
from us. Language did not emerge from some kind of ratiocination.
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6.4.
476. Children do not learn that books exist, that armchairs exist, etc. etc.,--they learn to fetch books, sit in armchairs,
etc. etc.
Later, questions about the existence of things do of course arise. "Is there such a thing as a unicorn?" and so
on. But such a question is possible only because as a rule no corresponding question presents itself. For how does
one know how to set
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about satisfying oneself of the existence of unicorns? How did one learn the method for determining whether
something exists or not?
Page 63
477. "So one must know that the objects whose names one teaches a child by an ostensive definition exist."-Why
must one know they do? Isn't it enough that experience doesn't later show the opposite?
For why should the language-game rest on some kind of knowledge?
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7.4.
478. Does a child believe that milk exists? Or does it know that milk exists? Does a cat know that a mouse exists?
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479. Are we to say that the knowledge that there are physical objects comes very early or very late?
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8.4.
480. A child that is learning to use the word "tree". One stands with it in front of a tree and says "Lovely tree!"
Clearly no doubt as to the tree's existence comes into the language-game. But can the child be said to know: 'that a
tree exists'? Admittedly it's true that 'knowing something' doesn't involve thinking about it--but mustn't anyone who
knows something be capable of doubt? And doubting means thinking.
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481. When one hears Moore say "I know that that's a tree", one suddenly understands those who think that that has
by no means been settled.
The matter strikes one all at once as being unclear and blurred. It is as if Moore had put it in the wrong light.
It is as if I were to see a painting (say a painted stage-set) and recognize what it represents from a long way
off at once and without the slightest doubt. But now I step nearer: and then I see a lot of patches of different colours,
which are all highly ambiguous and do not provide any certainty whatever.
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482. It is as if "I know" did not tolerate a metaphysical emphasis.
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483. The correct use of the expression "I know". Someone with bad sight asks me: "do you believe that the thing we
can see there is a tree?" I reply "I know it is; I can see it clearly and am familiar
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with it".--A: "Is N. N. at home?"--I: "I believe he is."--A: "Was he at home yesterday?"--I: "Yesterday he was--I
know he was; I spoke to him."--A: "Do you know or only believe that this part of the house is built on later than the
rest?"--I: "I know it is; I got it from so and so."
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484. In these cases, then, one says "I know" and mentions how one knows, or at least one can do so.
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485. We can also imagine a case where someone goes through a list of propositions and as he does so keeps asking
"Do I know that or do I only believe it?" He wants to check the certainty of each individual proposition. It might be a
question of making a statement as a witness before a court.
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9.4.
486. "Do you know or do you only believe that your name is L. W.?" Is that a meaningful question?
Do you know or do you only believe that what you are writing down now are German words? Do you only
believe that "believe" has this meaning? What meaning?
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487. What is the proof that I know something? Most certainly not my saying I know it.
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488. And so, when writers enumerate all the things they know, that proves nothing whatever.
So the possibility of knowledge about physical objects cannot be proved by the protestations of those who
believe that they have such knowledge.
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489. For what reply does one make to someone who says "I believe it merely strikes you as if you knew it"?
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490. When I ask "Do I know or do I only believe that I am called...?" it is no use to look within myself.
But I could say: not only do I never have the slightest doubt that I am called that, but there is no judgment I
could be certain of if I started doubting about that.
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10.4.
491. "Do I know or do I only believe that I am called L. W.?"--Of course, if the question were "Am I certain or do I
only surmise...?", then my answer could be relied on.
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492. "Do I know or do I only believe...?" might also be expressed like this: What if it seemed to turn out that what
until now has seemed immune to doubt was a false assumption? Would I react as I do when a belief has proved to
be false? or would it seem to knock from under my feet the ground on which I stand in making any judgments at
all?--But of course I do not intend this as a prophecy.
Would I simply say "I should never have thought it!"--or would I (have to) refuse to revise my
judgment--because such a 'revision' would amount to annihilation of all yardsticks?
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493. So is this it: I must recognize certain authorities in order to make judgments at all?
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494. "I cannot doubt this proposition without giving up all judgment."
But what sort of proposition is that? (It is reminiscent of what Frege said about the law of identity.†1) It is
certainly no empirical proposition. It does not belong to psychology. It has rather the character of a rule.
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495. One might simply say "O, rubbish'" to someone who wanted to make objections to the propositions that are
beyond doubt. That is, not reply to him but admonish him.
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496. This is a similar case to that of shewing that it has no meaning to say that a game has always been played
wrong.
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497. If someone wanted to arouse doubts in me and spoke like this: here your memory is deceiving you, there
you've been taken in, there again you have not been thorough enough in satisfying yourself, etc., and if I did not
allow myself to be shaken but kept to my certainty--then my doing so cannot be wrong, even if only because this is
just what defines a game.
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11.4.
498. The queer thing is that even though I find it quite correct for someone to say "Rubbish'" and so brush aside the
attempt to confuse him with doubts at bedrock,--nevertheless, I hold it to be incorrect if he seeks to defend himself
(using, e.g., the words "I know").
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499. I might also put it like this: the 'law of induction' can no more be grounded than certain particular propositions
concerning the material of experience.
Page 66
500. But it would also strike me as nonsense to say "I know that the law of induction is true".
Imagine such a statement made in a court of law! It would be more correct to say "I believe in the law of..."
where 'believe' has nothing to do with surmising.
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501. Am I not getting closer and closer to saying that in the end logic cannot be described? You must look at the
practice of language, then you will see it.
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502. Could one say "I know the position of my hands with my eyes closed", if the position I gave always or mostly
contradicted the evidence of other people?
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503. I look at an object and say "That is a tree", or "I know that that's a tree".--Now if I go nearer and it turns out that
it isn't, I may say "It wasn't a tree after all" or alternatively I say "It was a tree but now it isn't any longer". But if all
the others contradicted me, and said it never had been a tree, and if all the other evidences spoke against me--what
good would it do me to stick to my "I know"?
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504. Whether I know something depends on whether the evidence backs me up or contradicts me. For to say one
knows one has a pain means nothing.
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505. It is always by favour of Nature that one knows something.
Page 66
506. "If my memory deceives me here it can deceive me everywhere."
If I don't know that, how do I know if my words mean what I believe they mean?
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507. "If this deceives me, what does 'deceive' mean anymore?"
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508. What can I rely on?
Page 66
509. I really want to say that a language-game is only possible if one trusts something (I did not say "can trust
something").
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510. If I say "Of course I know that that's a towel" I am making an utterance.†1 I have no thought of a verification.
For me it is an immediate utterance.
I don't think of past or future. (And of course it's the same for Moore, too.)
It is just like directly taking hold of something, as I take hold of my towel without having doubts.
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511. And yet this direct taking-hold corresponds to a sureness, not to a knowing.
But don't I take hold of a thing's name like that, too?
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12.4.
512. Isn't the question this: "What if you had to change your opinion even on these most fundamental things?" And
to that the answer seems to me to be: "You don't have to change it. That is just what their being 'fundamental' is."
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513. What if something really unheard-of happened?--If I, say, saw houses gradually turning into steam without any
obvious cause, if the cattle in the fields stood on their heads and laughed and spoke comprehensible words; if trees
gradually changed into men and men into trees. Now, was I right when I said before all these things happened "I
know that that's a house" etc., or simply "that's a house" etc.?
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514. This statement appeared to me fundamental; if it is false, what are 'true' or 'false' any more?!
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515. If my name is not L. W., how can I rely on what is meant by "true" and "false"?
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516. If something happened (such as someone telling me something) calculated to make me doubtful of my own
name, there would certainly also be something that made the grounds of these doubts themselves seem doubtful,
and I could therefore decide to retain my old belief.
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517. But might it not be possible for something to happen that threw me entirely off the rails? Evidence that made
the most certain thing unacceptable to me? Or at any rate made me throw
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over my most fundamental judgments? (Whether rightly or wrongly is beside the point.)
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518. Could I imagine observing this in another person?
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519. Admittedly, if you are obeying the order "Bring me a book", you may have to check whether the thing you see
over there really is a book, but then you do at least know what people mean by "book"; and if you don't you can
look it up,--but then you must know what some other word means. And the fact that a word means such-and-such,
is used in such-and-such a way, is in turn an empirical fact, like the fact that what you see over there is a book.
Therefore, in order for you to be able to carry out an order there must be some empirical fact about which
you are not in doubt. Doubt itself rests only on what is beyond doubt.
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But since a language-game is something that consists in the recurrent procedures of the game in time, it
seems impossible to say in any individual case that such-and-such must be beyond doubt if there is to be a
language-game--though it is right enough to say that as a rule some empirical judgment or other must be beyond
doubt.
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13.4.
520. Moore has every right to say he knows there's a tree there in front of him. Naturally he may be wrong. (For it is
not the same as with the utterance "I believe there is a tree there".) But whether he is right or wrong in this case is of
no philosophical importance. If Moore is attacking those who say that one cannot really know such a thing, he can't
do it by assuring them that he knows this and that. For one need not believe him. If his opponents had asserted that
one could not believe this and that, then he could have replied: "I believe it."
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14.4.
Page 68
521. Moore's mistake lies in this--countering the assertion that one cannot know that, by saying "I do know it".
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522. We say: if a child has mastered language--and hence its application--it must know the meaning of words. It
must, for
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example, be able to attach the name of its colour to a white, black, red or blue object without the occurrence of any
doubt.
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523. And indeed no one misses doubt here; no one is surprised that we do not merely surmise the meaning of our
words.
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15.4.
524. Is it essential for our language-games ('ordering and obeying' for example) that no doubt appears at certain
points, or is it enough if there is the feeling of being sure, admittedly with a slight breath of doubt?
That is, is it enough if I do not, as I do now, call something 'black', 'green', 'red', straight off, without any
doubt at all interposing itself-but do instead say "I am sure that that is red", as one may say "I am sure that he will
come today" (in other words with the 'feeling of being sure')?
The accompanying feeling is of course a matter of indifference to us, and equally we have no need to bother
about the words "I am sure that" either.--What is important is whether they go with a difference in the practice of
the language.
One might ask whether a person who spoke like this would always say "I am sure" on occasions where (for
example) there is sureness in the reports we make (in an experiment, for example, we look through a tube and report
the colour we see through it). If he does, our immediate inclination will be to check what he says. But if he proves to
be perfectly reliable, one will say that his way of talking is merely a bit perverse, and does not affect the issue. One
might for example suppose that he has read sceptical philosophers, become convinced that one can know nothing,
and that is why he has adopted this way of speaking. Once we are used to it, it does not infect practice.
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525. What, then, does the case look like where someone really has got a different relationship to the names of
colours, for example, from us? Where, that is, there persists a slight doubt or a possibility of doubt in their use.
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16.4.
526. If someone were to look at an English pillar-box and say "I am sure that it's red", we should have to suppose
that he was
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colour-blind, or believe he had no mastery of English and knew the correct name for the colour in some other
language.
If neither was the case we should not quite understand him.
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527. An Englishman who calls this colour "red" is not 'sure it is called "red" in English'.
A child who has mastered the use of the word is not 'sure that in his language this colour is called...'. Nor can
one say of him that when he is learning to speak he learns that the colour is called that in English; nor yet: he knows
this when he has learnt the use of the word.
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528. And in spite of this: if someone asked me what the colour was called in German and I tell him, and now he asks
me "are you sure?"--then I shall reply "I know it is; German is my mother tongue".
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529. And one child, for example, will say, of another or of himself, that he already knows what such-and-such is
called.
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530. I may tell someone "this colour is called 'red' in English" (when for example I am teaching him English). In this
case I should not say "I know that this colour..."--I would perhaps say that if I had just now learned it, or by contrast
with another colour whose English name I am not acquainted with.
Page 70
531. But now, isn't it correct to describe my present state as follows: I know what this colour is called in English?
And if that is correct, why then should I not describe my state with the corresponding words "I know etc."?
Page 70
532. So when Moore sat in front of a tree and said "I know that that's a tree", he was simply stating the truth about
his state at the time.
[I do philosophy now like an old woman who is always mislaying something and having to look for it again:
now her spectacles, now her keys.]
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533. Well, if it was correct to describe his state out of context, then it was just as correct to utter the words "that's a
tree" out of context.
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534. But is it wrong to say: "A child that has mastered a language-game must know certain things"?
If instead of that one said "must be able to do certain things", that would be a pleonasm, yet this is just what
I want to counter the first sentence with.--But: "a child acquires a knowledge of natural history". That presupposes
that it can ask what such and such a plant is called.
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535. The child knows what something is called if he can reply correctly to the question "what is that called?"
Page 71
536. Naturally, the child who is just learning to speak has not yet got the concept is called at all.
Page 71
537. Can one say of someone who hasn't this concept that he knows what such-and-such is called?
Page 71
538. The child, I should like to say, learns to react in such-and-such a way; and in so reacting it doesn't so far know
anything. Knowing only begins at a later level.
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539. Does it go for knowing as it does for collecting?
Page 71
540. A dog might learn to run to N at the call "N", and to M at the call "M",--but would that mean he knows what
these people are called?
Page 71
541. "He only knows what this person is called--not yet what that person is called". That is something one cannot,
strictly speaking, say of someone who simply has not yet got the concept of people's having names.
Page 71
542. "I can't describe this flower if I don't know that this colour is called 'red'."
Page 71
543. A child can use the names of people long before he can say in any form whatever: "I know this one's name; I
don't know that one's yet."
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544. Of course I may truthfully say "I know what this colour is called in English", at the same time as I point (for
example) to the colour of fresh blood. But -- -- --
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17.4.
Page 71
545. 'A child knows which colour is meant by the word "blue".' What he knows here is not all that simple.
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546. I should say "I know what this colour is called" if e.g. what is in question is shades of colour whose name not
everybody knows.
Page 72
547. One can't yet say to a child who is just beginning to speak and can use the words "red" and "blue": "Come on,
you know what this colour is called called!"
Page 72
548. A child must learn the use of colour words before it can ask for the name of a colour.
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549. It would be wrong to say that I can only say "I know that there is a chair there" when there is a chair there. Of
course it isn't true unless there is, but I have a right to say this if I am sure there is a chair there, even if I am wrong.
[Pretensions are a mortgage which burdens a philosopher's capacity to think.]
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18.4.
Page 72
550. If someone believes something, we needn't always be able to answer the question 'why he believes it'; but if he
knows something, then the question "how does he know?" must be capable of being answered.
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551. And if one does answer this question, one must do so according to generally accepted axioms. This is how
something of this sort may be known.
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552. Do I know that I am now sitting in a chair?--Don't I know it?! In the present circumstances no one is going to
say that I know this; but no more will he say, for example, that I am conscious. Nor will one normally say this of the
passers-by in the street.
But now, even if one doesn't say it, does that make it untrue??
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553. It is queer: if I say, without any special occasion, "I know"--for example, "I know that I am now sitting in a
chair", this statement seems to me unjustified and presumptuous. But if I make the same statement where there is
some need for it, then, although I am not a jot more certain of its truth, it seems to me to be perfectly justified and
everyday.
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554. In its language-game it is not presumptuous. There, it has no higher position than, simply, the human
language-game. For there it has its restricted application.
But as soon as I say this sentence outside its context, it appears in a false light. For then it is as if I wanted to
insist that there are things that I know. God himself can't say anything to me about them.
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19.4
555. We say we know that water boils when it is put over a fire. How do we know? Experience has taught us.--I say
"I know that I had breakfast this morning"; experience hasn't taught me that. One also says "I know that he is in
pain". The language-game is different every time, we are sure every time, and people will agree with us that we are in
a position to know every time. And that is why the propositions of physics are found in textbooks for everyone.
If someone says he knows something, it must be something that, by general consent, he is in a position to
know.
Page 73
556. One doesn't say: he is in a position to believe that.
Page 73
But one does say: "It is reasonable to assume that in this situation" (or "to believe that").
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557. A court-martial may well have to decide whether it was reasonable in such-and-such a situation to have
assumed this or that with confidence (even though wrongly).
Page 73
558. We say we know that water boils and does not freeze under such-and-such circumstances. Is it conceivable that
we are wrong? Wouldn't a mistake topple all judgment with it? More: what could stand if that were to fall? Might
someone discover something that made us say "It was a mistake"?
Whatever may happen in the future, however water may behave in the future,--we know that up to now it has
behaved thus in innumerable instances.
This fact is fused into the foundations of our language-game:
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559. You must bear in mind that the language-game is so to say something unpredictable. I mean: it is not based on
grounds. It is not reasonable (or unreasonable).
It is there--like our life.
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560. And the concept of knowing is coupled with that of the language-game.
Page 74
561. "I know" and "You can rely on it". But one cannot always substitute the latter for the former.
Page 74
562. At any rate it is important to imagine a language in which our concept 'knowledge' does not exist.
Page 74
563. One says "I know that he is in pain" although one can produce no convincing grounds for this.--Is this the same
as "I am sure that he..."?--No. "I am sure" tells you my subjective certainty. "I know" means that I who know it, and
the person who doesn't are separated by a difference in understanding. (Perhaps based on a difference in degree of
experience.)
If I say "I know" in mathematics, then the justification for this is a proof.
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If in these two cases instead of "I know", one says "you can rely on it" then the substantiation is of a
different kind in each case.
And substantiation comes to an end.
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564. A language-game: bringing building stones, reporting the number of available stones. The number is sometimes
estimated, sometimes established by counting. Then the question arises "Do you believe there are as many stones as
that?", and the answer "I know there are--I've just counted them". But here the "I know" could be dropped. If,
however, there are several ways of finding something out for sure, like counting, weighing, measuring the stack, then
the statement "I know" can take the place of mentioning how I know.
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565. But here there isn't yet any question of any 'knowledge' that this is called "a slab", this "a pillar", etc.
Page 74
566. Nor does a child who learns my language-game (No. 2)†1 learn to say "I know that this is called 'a slab'".
Now of course there is a language-game in which the child uses that sentence. This presupposes that the
child is already capable of using the name as soon as he is given it. (As if someone were to tell me "this colour is
called...".)--Thus, if the child
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has learnt a language-game with building stones, one can say something like "and this stone is called '...'", and in this
way the original language-game has been expanded.
Page 75
567. And now, is my knowledge that I am called L. W. of the same kind as knowledge that water boils at 100°C.? Of
course, this question is wrongly put.
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568. If one of my names were used only very rarely, then it might happen that I did not know it. It goes without
saying that I know my name, only because, like anyone else, I use it over and over again.
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569. An inner experience cannot shew me that I know something.
Page 75
Hence, if in spite of that I say, "I know that my name is...", and yet it is obviously not an empirical
proposition, -- -- --
Page 75
570. "I know this is my name; among us any grown-up knows what his name is."
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571. "My name is... you can rely on that. If it turns out to be wrong you need never believe me in the future."
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572. Don't I seem to know that I can't be wrong about such a thing as my own name?
This comes out in the words: "If that is wrong, then I am crazy." Very well, but those are words; but what
influence has it on the application of language?
Page 75
573. Is it through the impossibility of anything's convincing me of the contrary?
Page 75
574. The question is, what kind of proposition is: "I know I can't be mistaken about that", or again "I can't be
mistaken about that"?
This "I know" seems to prescind from all grounds: I simply know it. But if there can be any question at all of
being mistaken here, then it must be possible to test whether I know it.
Page 75
575. Thus the purpose of the phrase "I know" might be to indicate where I can be relied on; but where that's what it's
doing, the usefulness of this sign must emerge from experience.
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576. One might say "How do I know that I'm not mistaken about my name?"--and if the reply was "Because I have
used it so often", one might go on to ask "How do I know that I am not mistaken about that?" And here the "How
do I know" cannot any longer have any significance.
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577. "My knowledge of my name is absolutely definite."
I would refuse to entertain any argument that tried to show the opposite!
And what does "I would refuse" mean? Is it the expression of an intention?
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578. But mightn't a higher authority assure me that I don't know the truth? So that I had to say "Teach me!"? But
then my eyes would have to be opened.
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579. It is part of the language-game with people's names that everyone knows his name with the greatest certainty.
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20.4.
580. It might surely happen that whenever I said "I know" it turned out to be wrong. (Shewing up.)
Page 76
581. But perhaps I might nevertheless be unable to help myself, so that I kept on declaring "I know...". But ask
yourself: how did the child learn the expression?
Page 76
582. "I know that" may mean: I am quite familiar with it--or again: it is certainly so.
Page 76
583. "I know that the name of this in... is '...'"--How do you know?--"I have learnt...".
Could I substitute "In... the name of this is '...'" for "I know etc." in this example?
Page 76
584. Would it be possible to make use of the verb "know" only in the question "How do you know?" following a
simple assertion?-Instead of "I already know that" one says "I am familiar with that"; and this follows only upon
being told the fact. But†1 what does one say instead of "I know what that is"?
Page 76
585. But doesn't "I know that that's a tree" say something different from "that is a tree"?
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586. Instead of "I know what that is" one might say "I can say what that is". And if one adopted this form of
expression what would then become of "I know that that is a..."?
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587. Back to the question whether "I know that that's a..." says anything different from "that is a...". In the first
sentence a person is mentioned, in the second, not. But that does not shew that they have different meanings. At all
events one often replaces the first form by the second, and then often gives the latter a special intonation. For one
speaks differently when one makes an uncontradicted assertion from when one maintains an assertion in face of
contradiction.
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588. But don't I use the words "I know that..." to say that I am in a certain state, whereas the mere assertion "that is
a..." does not say this? And yet one often does reply to such an assertion by asking "how do you know?'--"But
surely, only because the fact that I assert this gives to understand that I think I know it".--This point could be made
in the following way: In a zoo there might be a notice "this is a zebra"; but never "I know that this is a zebra".
"I know" has meaning only when it is uttered by a person. But, given that, it is a matter of indifference
whether what is uttered is "I know..." or "That is...".
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589. For how does a man learn to recognize his own state of knowing something?
Page 77
590. At most one might speak of recognizing a state, where what is said is "I know what that is". Here one can
satisfy oneself that one really is in possession of this knowledge.
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591. "I know what kind of tree that is.--It is a chestnut."
"I know what kind of tree that is.--I know it's a chestnut."
The first statement sounds more natural than the second. One will only say "I know" a second time if one
wants especially to emphasize certainty; perhaps to anticipate being contradicted. The first "I know" means roughly:
I can say.
But in another case one might begin with the observation "that's a...", and then, when this is contradicted,
counter by saying: "I know what sort of a tree it is", and by this means lay emphasis on being sure.
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592. "I can tell you what kind of a... that is, and no doubt about it."
Page 78
593. Even when one can replace "I know" by "It is..." still one cannot replace the negation of the one by the negation
of the other.
With "I don't know..." a new element enters our language-games.
Page 78
21.4.
594. My name is "L. W." And if someone were to dispute it, I should straightaway make connexions with
innumerable things which make it certain.
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595. "But I can still imagine someone making all these connexions, and none of them corresponding with reality.
Why shouldn't I be in a similar case?"
If I imagine such a person I also imagine a reality, a world that surrounds him; and I imagine him as thinking
(and speaking) in contradiction to this world.
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596. If someone tells me his name is N. N., it is meaningful for me to ask him "Can you be mistaken?" That is an
allowable question in the language-game. And the answer to it, yes or no, makes sense.--Now of course this answer
is not infallible either, i.e., there might be a time when it proved to be wrong, but that does not deprive the question
"Can you be..." and the answer "No" of their meaning.
Page 78
597. The reply to the question "Can you be mistaken?" gives the statement a definite weight. The answer may also
be: "I don't think so."
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598. But couldn't one reply to the question "Can you..." by saying: "I will describe the case to you and then you can
judge for yourself whether I can be mistaken"?
For example, if it were a question of someone's own name, the fact might be that he had never used this
name, but remembered he had read it on some document,--but on the other hand the answer might be: "I've had this
name my whole life long, I've been called it by everybody." If that is not equivalent to the answer "I can't be
mistaken", then the latter has no meaning
Page Break 79
whatever. And yet quite obviously it points to a very important distinction.
Page 79
599. For example one could describe the certainty of the proposition that water boils at circa 100°C. That isn't e.g. a
proposition I have once heard (like this or that, which I could mention). I made the experiment myself at school. The
proposition is a very elementary one in our text-books, which are to be trusted in matters like this because...--Now
one can offer counter-examples to all this, which show that human beings have held this and that to be certain which
later, according to our opinion, proved false. But the argument is worthless.†1 To say: in the end we can only adduce
such grounds as we hold to be grounds, is to say nothing at all.
I believe that at the bottom of this is a misunderstanding of the nature of our language-games.
Page 79
600. What kind of grounds have I for trusting text-books of experimental physics?
I have no grounds for not trusting them. And I trust them. I know how such books are produced--or rather, I
believe I know. I have some evidence, but it does not go very far and is of a very scattered kind. I have heard, seen
and read various things.
Page 79
22.4.
601. There is always the danger of wanting to find an expression's meaning by contemplating the expression itself,
and the frame of mind in which one uses it, instead of always thinking of the practice. That is why one repeats the
expression to oneself so often, because it is as if one must see what one is looking for in the expression and in the
feeling it gives one.
Page 79
23.4.
Page 79
602. Should I say "I believe in physics", or "I know that physics is true"?
Page 79
603. I am taught that under such circumstances this happens. It has been discovered by making the experiment a
few times. Not
Page Break 80
that that would prove anything to us, if it weren't that this experience was surrounded by others which combine with
it to form a system. Thus, people did not make experiments just about falling bodies but also about air resistance
and all sorts of other things.
But in the end I rely on these experiences, or on the reports of them, I feel no scruples about ordering my
own activities in accordance with them.--But hasn't this trust also proved itself? So far as I can judge--yes.
Page 80
604. In a court of law the statement of a physicist that water boils at about 100°C. would be accepted
unconditionally as truth.
If I mistrusted this statement what could I do to undermine it? Set up experiments myself? What would they
prove?
Page 80
605. But what if the physicist's statement were superstition and it were just as absurd to go by it in reaching a verdict
as to rely on ordeal by fire?
Page 80
606. That to my mind someone else has been wrong is no ground for assuming that I am wrong now.--But isn't it a
ground for assuming that I might be wrong? It is no ground for any unsureness in my judgment, or my actions.
Page 80
607. A judge might even say "That is the truth--so far as a human being can know it". But what would this rider
achieve? ("beyond all reasonable doubt").
Page 80
608. Is it wrong for me to be guided in my actions by the propositions of physics? Am I to say I have no good
ground for doing so? Isn't precisely this what we call a 'good ground'?
Page 80
609. Supposing we met people who did not regard that as a telling reason. Now, how do we imagine this? Instead of
the physicist, they consult an oracle. (And for that we consider them primitive.) Is it wrong for them to consult an
oracle and be guided by it?--If we call this "wrong" aren't we using our language-game as a base from which to
combat theirs?
Page Break 81
Page 81
610. And are we right or wrong to combat it? Of course there are all sorts of slogans which will be used to support
our proceedings.
Page 81
611. Where two principles really do meet which cannot be reconciled with one another, then each man declares the
other a fool and heretic.
Page 81
612. I said I would 'combat' the other man,--but wouldn't I give him reasons? Certainly; but how far do they go? At
the end of reasons comes persuasion. (Think what happens when missionaries convert natives.)
Page 81
613. If I now say "I know that the water in the kettle on the gas-flame will not freeze but boil", I seem to be as
justified in this "I know" as I am in any. 'If I know anything I know this'.--Or do I know with still greater certainty
that the person opposite me is my old friend so-and-so? And how does that compare with the proposition that I am
seeing with two eyes and shall see them if I look in the glass?--I don't know confidently what I am to answer
here.--But still there is a difference between the cases. If the water over the gas freezes, of course I shall be as
astonished as can be, but I shall assume some factor I don't know of, and perhaps leave the matter to physicists to
judge. But what could make me doubt whether this person here is N. N., whom I have known for years? Here a
doubt would seem to drag everything with it and plunge it into chaos.
Page 81
614. That is to say: If I were contradicted on all sides and told that this person's name was not what I had always
known it was (and I use "know" here intentionally), then in that case the foundation of all judging would be taken
away from me.
Page 81
615. Now does that mean: "I can only make judgments at all because things behave thus and thus (as it were, behave
kindly)"?
Page 81
616. Why, would it be unthinkable that I should stay in the saddle however much the facts bucked?
Page Break 82
Page 82
617. Certain events would put me into a position in which I could not go on with the old language-game any further.
In which I was torn away from the sureness of the game.
Indeed, doesn't it seem obvious that the possibility of a language-game is conditioned by certain facts?
Page 82
618. In that case it would seem as if the language-game must 'show' the facts that make it possible. (But that's not
how it is.)
Then can one say that only a certain regularity in occurrences makes induction possible? The 'possible' would
of course have to be 'logically possible'.
Page 82
619. Am I to say: even if an irregularity in natural events did suddenly occur, that wouldn't have to throw me out of
the saddle. I might make inferences then just as before, but whether one would call that "induction" is another
question.
Page 82
620. In particular circumstances one says "you can rely on this"; and this assurance may be justified or unjustified in
everyday language, and it may also count as justified even when what was foretold does not occur. A
language-game exists in which this assurance is employed.
Page 82
24.4.
621. If anatomy were under discussion I should say: "I know that twelve pairs of nerves lead from the brain." I have
never seen these nerves, and even a specialist will only have observed them in a few specimens.--This just is how the
word "know" is correctly used here.
Page 82
622. But now it is also correct to use "I know" in the contexts which Moore mentioned, at least in particular
circumstances. (Indeed, I do not know what "I know that I am a human being" means. But even that might be given
a sense.)
For each one of these sentences I can imagine circumstances that turn it into a move in one of our
language-games, and by that it loses everything that is philosophically astonishing.
Page 82
623. What is odd is that in such a case I always feel like saying (although it is wrong): "I know that--so far as one can
know such a thing." That is incorrect, but something right is hidden behind it.
Page Break 83
Page 83
624. "Can you be mistaken about this colour's being called 'green' in English?" My answer to this can only be "No".
If I were to say "Yes, for there is always the possibility of a delusion", that would mean nothing at all.
For is that rider something unknown to the other? And how is it known to me?
Page 83
625. But does that mean that it is unthinkable that the word "green" should have been produced here by a slip of the
tongue or a momentary confusion? Don't we know of such cases?--One can also say to someone "Mightn't you
perhaps have made a slip?" That amounts to: "Think about it again".--
But these rules of caution only make sense if they come to an end somewhere.
A doubt without an end is not even a doubt.
Page 83
626. Nor does it mean anything to say: "The English name of this colour is certainly 'green',--unless, of course, I am
making a slip of the tongue or am confused in some way."
Page 83
627. Wouldn't one have to insert this clause into all language-games? (Which shows its senselessness.)
Page 83
628. When we say "Certain propositions must be excluded from doubt", it sounds as if I ought to put these
propositions--for example, that I am called L. W.--into a logic-book. For if it belongs to the description of a
language-game, it belongs to logic. But that I am called L. W. does not belong to any such description. The
language-game that operates with people's names can certainly exist even if I am mistaken about my name,--but it
does presuppose that it is nonsensical to say that the majority of people are mistaken about their names.
Page 83
629. On the other hand, however, it is right to say of myself "I cannot be mistaken about my name", and wrong if I
say "perhaps I am mistaken". But that doesn't mean that it is meaningless for others to doubt what I declare to be
certain.
Page 83
630. It is simply the normal case, to be incapable of mistake about the designation of certain things in one's mother
tongue.
Page 83
631. "I can't be making a mistake about it" simply characterizes one kind of assertion.
Page Break 84
Page 84
632. Certain and uncertain memory. If certain memory were not in general more reliable than uncertain memory, i.e.,
if it were not confirmed by further verification more often than uncertain memory was, then the expression of
certainty and uncertainty would not have its present function in language.
Page 84
633. "I can't be making a mistake"--but what if I did make a mistake then, after all? For isn't that possible? But does
that make the expression "I can't be etc." nonsense? Or would it be better to say instead "I can hardly be mistaken"?
No; for that means something else.
Page 84
634. "I can't be making a mistake; and if the worst comes to the worst I shall make my proposition into a norm."
Page 84
635. "I can't be making a mistake; I was with him today."
Page 84
636. "I can't be making a mistake; but if after all something should appear to speak against my proposition I shall
stick to it, despite this appearance."
Page 84
637. "I can't etc." shows my assertion its place in the game. But it relates essentially to me, not to the game in
general.
If I am wrong in my assertion that doesn't detract from the usefulness of the language-game.
Page 84
25.4.
638. "I can't be making a mistake" is an ordinary sentence, which serves to give the certainty-value of a statement.
And only in its everyday use is it justified.
Page 84
639. But what the devil use is it if--as everyone admits--I may be wrong about it, and therefore about the proposition
it was supposed to support too?
Page 84
640. Or shall I say: the sentence excludes a certain kind of failure?
Page 84
641. "He told me about it today--I can't be making a mistake about that."--But what if it does turn out to be
wrong?!--Mustn't one make a distinction between the ways in which something 'turns out wrong'?--How can it be
shewn that my
Page Break 85
statement was wrong? Here evidence is facing evidence, and it must be decided which is to give way.
Page 85
642. But suppose someone produced the scruple: what if I suddenly as it were woke up and said "Just think, I've
been imagining I was called L. W.!"----well, who says that I don't wake up once again and call this an extraordinary
fancy, and so on?
Page 85
643. Admittedly one can imagine a case--and cases do exist--where after the 'awakening' one never has any more
doubt which was imagination and which was reality. But such a case, or its possibility, doesn't discredit the
proposition "I can't be wrong".
Page 85
644. For otherwise, wouldn't all assertion be discredited in this way?
Page 85
645. I can't be making a mistake,--but some day, rightly or wrongly, I may think I realize that I was not competent to
judge.
Page 85
646. Admittedly, if that always or often happened it would completely alter the character of the language-game.
Page 85
647. There is a difference between a mistake for which, as it were, a place is prepared in the game, and a complete
irregularity that happens as an exception.
Page 85
648. I may also convince someone else that I 'can't be making a mistake'.
I say to someone "So-and-so was with me this morning and told me such-and-such". If this is astonishing he
may ask me: "You can't be mistaken about it?" That may mean: "Did that really happen this morning?" or on the
other hand: "Are you sure you understood him properly?" It is easy to see what details I should add to show that I
was not wrong about the time, and similarly to show that I hadn't misunderstood the story. But all that can not show
that I haven't dreamed the whole thing, or imagined it to myself in a dreamy way. Nor can it show that I haven't
perhaps made some slip of the tongue throughout. (That sort of thing does happen.)
Page Break 86
Page 86
649. (I once said to someone--in English--that the shape of a certain branch was typical of the branch of an elm,
which my companion denied. Then we came past some ashes, and I said "There, you see, here are the branches I
was speaking about". To which he replied "But that's an ash"--and I said "I always meant ash when I said elm".)
Page 86
650. This surely means: the possibility of a mistake can be eliminated in certain (numerous) cases.--And one does
eliminate mistakes in calculation in this way. For when a calculation has been checked over and over again one
cannot then say "Its rightness is still only very probable--for an error may always still have slipped in". For suppose
it did seem for once as if an error had been discovered--why shouldn't we suspect an error here?
Page 86
651. I cannot be making a mistake about 12 × 12 being 144. And now one cannot contrast mathematical certainty
with the relative uncertainty of empirical propositions. For the mathematical proposition has been obtained by a
series of actions that are in no way different from the actions of the rest of our lives, and are in the same degree liable
to forgetfulness, oversight and illusion.
Page 86
652. Now can I prophesy that men will never throw over the present arithmetical propositions, never say that now at
last they know how the matter stands? Yet would that justify a doubt on our part?
Page 86
653. If the proposition 12 × 12 = 144 is exempt from doubt, then so too must non-mathematical propositions be.
Page 86
26.4.51
654. But against this there are plenty of objections.--In the first place there is the fact that "12 × 12 etc." is a
mathematical proposition, and from this one may infer that only mathematical propositions are in this situation.
And if this inference is not justified, then there ought to be a proposition that is just as certain, and deals with the
process of this calculation, but isn't itself mathematical. I am thinking of such a proposition as: "The multiplication
'12 × 12', when carried out by people who know how to calculate, will in the great majority of cases give the result
'144'".
Page Break 87
Page 87
Nobody will contest this proposition, and naturally it is not a mathematical one. But has it got the certainty of
the mathematical proposition?
Page 87
655. The mathematical proposition has, as it were officially, been given the stamp of incontestability. I.e.: "Dispute
about other things; this is immovable--it is a hinge on which your dispute can turn."
Page 87
656. And one can not say that of the proposition that I am called L. W. Nor of the proposition that such-and-such
people have calculated such-and-such a problem correctly.
Page 87
657. The propositions of mathematics might be said to be fossilized.--The proposition "I am called..." is not. But it
too is regarded as incontrovertible by those who, like myself, have overwhelming evidence for it. And this not out
of thoughtlessness. For, the evidence's being overwhelming consists precisely in the fact that we do not need to give
way before any contrary evidence. And so we have here a buttress similar to the one that makes the propositions of
mathematics incontrovertible.
Page 87
658. The question "But mightn't you be in the grip of a delusion now and perhaps later find this out?"--might also be
raised as an objection to any proposition of the multiplication tables.
Page 87
659. "I cannot be making a mistake about the fact that I have just had lunch."
For if I say to someone "I have just eaten" he may believe that I am lying or have momentarily lost my wits
but he won't believe that I am making a mistake. Indeed, the assumption that I might be making a mistake has no
meaning here.
Page 87
But that isn't true. I might, for example, have dropped off immediately after the meal without knowing it and
have slept for an hour, and now believe I had just eaten.
Page 87
But still, I distinguish here between different kinds of mistake.
Page 87
660. I might ask: "How could I be making a mistake about my name being L. W.?" And I can say: I can't see how it
would be possible.
Page 87
661. How might I be mistaken in my assumption that I was never on the moon?
Page Break 88
Page 88
662. If I were to say "I have never been on the moon--but I may be mistaken", that would be idiotic.
For even the thought that I might have been transported there, by unknown means, in my sleep, would not
give me any right to speak of a possible mistake here. I play the game wrong if I do.
Page 88
663. I have a right to say "I can't be making a mistake about this" even if I am in error.
Page 88
664. It makes a difference: whether one is learning in school what is right and wrong in mathematics, or whether I
myself say that I cannot be making a mistake in a proposition.
Page 88
665. In the latter case I am adding something special to what is generally laid down.
Page 88
666. But how is it for example with anatomy (or a large part of it)? Isn't what it describes, too, exempt from all
doubt?
Page 88
667. Even if I came to a country where they believed that people were taken to the moon in dreams, I couldn't say to
them: "I have never been to the moon. Of course I may be mistaken". And to their question "Mayn't you be
mistaken?" I should have to answer: No.
Page 88
668. What practical consequences has it if I give a piece of information and add that I can't be making a mistake
about it?
(I might also add instead: "I can no more be wrong about this than about my name's being L. W.")
The other person might doubt my statement nonetheless. But if he trusts me he will not only accept my
information, he will also draw definite conclusions from my conviction, as to how I shall behave.
Page 88
669. The sentence "I can't be making a mistake" is certainly used in practice. But we may question whether it is then
to be taken in a perfectly rigorous sense, or is rather a kind of exaggeration which perhaps is used only with a view
to persuasion.
Page Break 89
27.4.
670. We might speak of fundamental principles of human enquiry.
Page 89
671. I fly from here to a part of the world where the people have only indefinite information, or none at all, about the
possibility of flying. I tell them I have just flown there from.... They ask me if I might be mistaken.--They have
obviously a false impression of how the thing happens. (If I were packed up in a box it would be possible for me to
be mistaken about the way I had travelled.) If I simply tell them that I can't be mistaken, that won't perhaps convince
them; but it will if I describe the actual procedure to them. Then they will certainly not bring the possibility of a
mistake into the question. But for all that--even if they trust me--they might believe I had been dreaming or that
magic had made me imagine it.
Page 89
672. 'If I don't trust this evidence why should I trust any evidence?'
Page 89
673. Is it not difficult to distinguish between the cases in which I cannot and those in which I can hardly be
mistaken? Is it always clear to which kind a case belongs? I believe not.
Page 89
674. There are, however, certain types of case in which I rightly say I cannot be making a mistake, and Moore has
given a few examples of such cases.
I can enumerate various typical cases, but not give any common characteristic. (N. N. cannot be mistaken
about his having flown from America to England a few days ago. Only if he is mad can he take anything else to be
possible.)
Page 89
675. If someone believes that he has flown from America to England in the last few days, then, I believe, he cannot
be making a mistake.
And just the same if someone says that he is at this moment sitting at a table and writing.
Page 89
676. "But even if in such cases I can't be mistaken, isn't it possible that I am drugged?" If I am and if the drug has
taken away my consciousness, then I am not now really talking and
Page Break 90
thinking. I cannot seriously suppose that I am at this moment dreaming. Someone who, dreaming, says "I am
dreaming", even if he speaks audibly in doing so, is no more right than if he said in his dream "it is raining", while it
was in fact raining. Even if his dream were actually connected with the noise of the rain.
FOOTNOTES
Page 2
†1 See G. E. Moore, "Proof of an External World", Proceedings of the British Academy, Vol. XXV, 1939;
also "A Defence of Common Sense" in Contemporary British Philosophy, 2nd Series, Ed. J. H. Muirhead, 1925.
Both papers are in Moore's Philosophical Papers, London, George Allen and Unwin, 1959. Editors.
Page 28
†1 Passage crossed out in MS. (Editors)
Page 38
†1 In English. Eds.
Page 51
†1 Philosophical Investigations I §2. Eds.
Page 51
†2
(... and write with confidence )
("In the beginning was the deed." )
Cf. Goethe, Faust I. Trans.
Page 65
†1 Grundgesetze der Arithmetik I xvii Eds.
Page 67
†1 Äußerung. (Eds.)
Page 74
†1 Philosophical Investigations §2. Eds.
Page 76
†1 The last sentence is a later addition. (Eds.)
Page 79
†1 Marginal note. May it not also happen that we believe we recognize a mistake of earlier times and later
come to the conclusion that the first opinion was the right one? etc.
REMARKS ON COLOUR
Page i
Page Break ii
Titlepage
Page ii
Ludwig Wittgenstein:
REMARKS ON COLOUR
Edited by
G.E.M. ANSCOMBE
Translated by
Linda L. McAlister and
Margarete Schättle
BASIL BLACKWELL
Page Break 1
EDITOR'S PREFACE
Page 1
Part III of this volume reproduces most of a MS book written in oxford in the Spring of 1950. I have left out material
on "inner-outer", remarks about Shakespeare and some general observations about life; all this both was marked as
discontinuous with the text and also will appear elsewhere. Part I was written in Cambridge in March 1951: it is a
selection and revision of the earlier material, with few additions. It is not clear whether Part II ante- or post-dates Part
III. It was part of what was written on undated loose sheets of foolscap, the rest being devoted to certainty.
Wittgenstein left these in his room in my house in Oxford when he went to Dr. Bevan's house in Cambridge in
February 1951, in the expectation of dying there. His literary executors decided that the whole of this material might
well be published, as it gives a clear sample of first-draft writing and subsequent selection. Much of what was not
selected is of great interest, and this method of publication involves the least possible editorial intervention.
Page 1
In the work of determining the text I was much helped by G. H. von Wright's careful typescript of it, and also
by an independent typescript made by Linda McAlister and Margarete Schättle We have to thank them also for their
translation. This, with agreed revisions by the editor, is published here.
Page 1
I should also like to thank Dr. L. Labowsky for reading through the German text.
G. E. M. Anscombe
Page Break 2
I
Page 2
1. A language-game: Report whether a certain body is lighter or darker than another.--But now there's a related one:
State the relationship between the lightness of certain shades of colour. (Compare with this: Determining the
relationship between the lengths of two sticks--and the relationship between two numbers.)--The form of the
propositions in both language-games is the same: "X is lighter than Y". But in the first it is an external relation and
the proposition is temporal, in the second it is an internal relation and the proposition is timeless.
Page 2
2. In a picture in which a piece of white paper gets its lightness from the blue sky, the sky is lighter than the white
paper. And yet in another sense blue is the darker and white the lighter colour. (Goethe). On the palette white is the
lightest colour.
Page 2
3. Lichtenberg says that very few people have ever seen pure white. So do most people use the word wrong, then?
And how did he learn the correct use?--He constructed an ideal use from the ordinary one. And that is not to say a
better one, but one that has been refined along certain lines and in the process something has been carried to
extremes.
Page 2
4. And of course such a construct may in turn teach us something about the way we in fact use the word.
Page 2
5. If I say a piece of paper is pure white, and if snow were placed next to it and it then appeared grey, in its normal
surroundings I would still be right in calling it white and not light grey. It could be that I use a more refined concept
of white in, say, a laboratory (where, for example, I also use a more refined concept of precise determination of
time).
Page 2
6. What is there in favor of saying that green is a primary colour, not a blend of blue and yellow? Would it be right to
say: "You can only know it directly by looking at the colours"? But how do I know that I mean the same by the
words "primary colours" as some other
Page Break 3
person who is also inclined to call green a primary colour? No,--here language-games decide.
Page 3
7. Someone is given a certain yellow-green (or blue-green) and told to mix a less yellowish (or bluish) one--or to
pick it out from a number of colour samples. A less yellowish green, however, is not a bluish one (and vice versa),
and there is also such a task as choosing, or mixing a green that is neither yellowish nor bluish. I say "or mixing"
because a green does not become both bluish" and yellowish because it is produced by a kind of mixture of yellow
and blue.
Page 3
8. People might have the concept of intermediary colours or mixed colours even if they never produced colours by
mixing (in whatever sense). Their language-games might only have to do with looking for or selecting already
existing intermediary or blended colours.
Page 3
9. Even if green is not an intermediary colour between yellow and blue, couldn't there be people for whom there is
bluish-yellow, reddish-green? I.e. people whose colour concepts deviate from ours--because, after all; the colour
concepts of colour-blind people too deviate from those of normal people, and not every deviation from the norm
must be a blindness, a defect.
Page 3
10. Someone who has learnt to find or to mix a shade of colour that is more yellowish, more whitish or more
reddish, etc., than a given shade of colour, i.e. who knows the concept of intermediary colours, is (now) asked to
show us a reddish-green. He may simply not understand this order and perhaps react as though he had first been
asked to point out regular four-, five-, and six-angled plane figures, and then were asked to point out a regular
one-angled plane figure. But what if he unhesitatingly pointed to a colour sample (say, to one that we would call a
blackish brown)?
Page 3
11. Someone who is familiar with reddish-green should be in a position to produce a colour series which starts with
red and ends with green and which perhaps even for us constitutes a continuous
Page Break 4
transition between the two. We would then discover that at the point where we always see the same shade, e.g. of
brown, this person sometimes sees brown and sometimes reddish-green. It may be, for example, that he can
differentiate between the colours of two chemical compounds that seem to us to be the same colour and he calls one
brown and the other reddish-green.
Page 4
12. Imagine that all mankind, with rare exceptions, were red-green colour-blind. Or another case: everyone was
either red-green or blue-yellow colour-blind.
Page 4
13. Imagine a tribe of colour-blind people, and there could easily be one. They would not have the same colour
concepts as we do. For even assuming they speak, e.g. English, and thus have all the English colour words, they
would still use them differently than we do and would learn their use differently.
Or if they have a foreign language, it would be difficult for us to translate their colour words into ours.
Page 4
14. But even if there were also people for whom it was natural to use the expressions "reddish-green" or
"yellowish-blue" in a consistent manner and who perhaps also exhibit abilities which we lack, we would still not be
forced to recognize that they see colours which we do not see. There is, after all, no commonly accepted criterion for
what is a colour, unless it is one of our colours.
Page 4
15. In every serious philosophical question uncertainty extends to the very roots of the problem.
We must always be prepared to learn something totally new.
Page 4
16. The description of the phenomena of colour-blindness is part of psychology: and therefore the description of the
phenomena of normal vision, too? Psychology only describes the deviations of colour-blindness from normal
vision.
Page 4
17. Runge says (in the letter that Goethe reproduced in his Theory of Colours), there are transparent and opaque
colours. White is an opaque colour.
This shows the indeterminateness in the concept of colour or again in that of sameness of colour.
Page Break 5
Page 5
18. Can a transparent green glass have the same colour as a piece of opaque paper or not? If such a glass were
depicted in a painting, the colours would not be transparent on the palette. If we wanted to say the colour of the
glass was also transparent in the painting, we would have to call the complex of colour patches which depict the
glass its colour.
Page 5
19. Why is it that something can be transparent green but not transparent white?
Transparency and reflections exist only in the dimension of depth of a visual image.
The impression that the transparent medium makes is that something lies behind the medium. If the visual
image is thoroughly monochromatic it cannot be transparent.
Page 5
20. Something white behind a coloured transparent medium appears in the colour of the medium, something black
appears black. According to this rule, black on a white background would have to be seen through a 'white,
transparent' medium as through a colourless one.
Page 5
21. Runge: "If we were to think of a bluish-orange, a reddish-green, or a yellowish-violet, we would have the same
feeling as in the case of a southwesterly northwind.... Both white and Black are opaque or solid.... White water which
is pure is as inconceivable as clear milk."
Page 5
22. We do not want to establish a theory of colour (neither a physiological one nor a psychological one), but rather
the logic of colour concepts. And this accomplishes what people have often unjustly expected of a theory.
Page 5
23. "White water is inconceivable, etc." That means we cannot describe (e.g. paint), how something white and clear
would look, and that means: we don't know what description, portrayal, these words demand of us.
Page 5
24. It is not immediately clear what transparent glass we should say has the same colour as an opaque colour
sample. If I say, "I am looking for glass of this colour" (pointing to a piece of coloured
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paper), that would mean roughly that something white seen through the glass should look like my sample.
If the sample is pink, sky-blue or lilac, we will imagine the glass cloudy, but perhaps too as clear and only
slightly reddish, bluish or violet.
Page 6
25. In the cinema we can sometimes see the events in the film as if they lay behind the screen and it were
transparent, rather like a pane of glass. The glass would be taking the colour away from things and allowing only
white, grey and black to come through. (Here we are not doing physics, we are regarding white and black as colours
just like green and red).--We might thus think that we are here imagining a pane of glass that could be called white
and transparent. And yet we are not tempted to call it that: so does the analogy with, e.g. a transparent green pane
break down somewhere?
Page 6
26. We would say, perhaps, of a green pane: it colours the things behind it green, above all the white behind it.
Page 6
27. When dealing with logic, "One cannot imagine that" means: one doesn't know what one should imagine here.
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28. Would we say that my fictitious glass pane in the cinema gave the things behind it a white colouring?
Page 6
29. From the rule for the appearance of transparent coloured things that you have extracted from transparent green,
red, etc., ascertain the appearance of transparent white! Why doesn't this work?
Page 6
30. Every coloured medium darkens that which is seen through it, it swallows light: now is my white glass also
supposed to darken? And the more so the thicker it is? So it would really be a dark glass!
Page 6
31. Why can't we imagine transparent-white glass,--even if there isn't any in actuality? Where does the analogy with
transparent coloured glass go wrong?
Page 6
32. Sentences are often used on the borderline between logic and the empirical, so that their meaning changes back
and forth and
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they count now as expressions of norms, now as expressions of experience.
(For it is certainly not an accompanying mental phenomenon--this is how we imagine 'thoughts'--but the use,
which distinguishes the logical proposition from the empirical one.)
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33. We speak of the 'colour of gold' and do not mean yellow. "Gold-coloured" is the property of a surface that shines
or glitters.
Page 7
34. There is the glow of red-hot and of white-hot: but what would brown-hot and grey-hot look like? Why can't we
conceive of these as a lower degree of white-hot?
Page 7
35. "Light is colourless". If so, then in the sense in which numbers are colourless.
Page 7
36. Whatever looks luminous does not look grey. Everything grey looks as though it is being illuminated.
Page 7
37. What we see as luminous we do not see as grey. But we can certainly see it as white.
Page 7
38. I could, then, see something now as weakly luminous, now as grey.
Page 7
39. I am not saying here (as the Gestalt psychologists do), that the impression of white comes about in
such-and-such a way. Rather the question is precisely: what is the meaning of this expression, what is the logic of
this concept?
Page 7
40. For the fact that we cannot conceive of something 'glowing grey' belongs neither to the physics nor to the
psychology of colour.
Page 7
41. I am told that a substance burns with a grey flame. I don't know the colours of the flames of all substances; so
why shouldn't that be possible?
Page 7
42. We speak of a 'dark red light' but not of a 'black-red light'.
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Page 8
43. A smooth white surface can reflect things: But what, then, if we made a mistake and that which appeared to be
reflected in such a surface were really behind it and seen through it? Would the surface then be white and
transparent?
Page 8
44. We speak of a 'black' mirror. But where it mirrors, it darkens, of course, but it doesn't look black, and that which
is seen in it does not appear 'dirty' but 'deep'.
Page 8
45. Opaqueness is not a property of the white colour. Any more than transparency is a property of the green.
Page 8
46. And it does not suffice to say, the word "white" is used only for the appearance of surfaces. It could be that we
had two words for "green": one for green surfaces, the other for green transparent objects. The question would
remain why there existed no colour word corresponding to the word "white" for something transparent.
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47. We wouldn't want to call a medium white if a black and white pattern (chess board) appeared unchanged when
seen through it, even if this medium reduced the intensity of the other colours.
Page 8
48. We might want not to call a white high-light "white", and thus use that word only for that which we see as the
colour of a surface.
Page 8
49. Of two places in my surroundings which I see in one sense as being the same colour, in another sense, the one
can seem to me white and the other grey.
To me in one context this colour is white in a poor light, in another it is grey in good light.
These are propositions about the concepts 'white' and 'grey'.
Page 8
50. The bucket which I see in front of me is glazed shining white; it would be absurd to call it "grey" or to say "I
really see a light grey". But it has a shiny highlight that is far lighter than the rest of its surface part of which is turned
toward the light and part away from it, without appearing to be differently coloured. (Appearing, not just being:)
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Page 9
51. It is not the same thing to say: the impression of white or grey comes about under such-and-such conditions
(causal), and: it is an impression in a certain context of colours and forms.
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52. White as a colour of substances (in the sense in which we say snow is white) is lighter than any other
substance-colour; black darker. Here colour is a darkening, and if all such is removed from the substance, white
remains, and for this reason we can call it "colourless".
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53. The is no such thing as phenomenology, but there are indeed phenomenological problems.
Page 9
54. It is easy to see that not all colour concepts are logically of the same sort, e.g. the difference between the
concepts 'colour of gold' or 'colour of silver' and 'yellow' or 'grey'.
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55. A colour 'shines' in its surroundings. (Just as eyes only smile in a face.) A 'blackish' colour--e.g. grey--doesn't
'shine'.
Page 9
56. The difficulties we encounter when we reflect about the nature of colours (those which Goethe wanted to get
sorted out in his Theory of Colours) are embedded in the indeterminateness of our concept of sameness of colour.
Page 9
57.
["I feel X"
"I observe X"
X does not stand for the same concept in the first and the second sentences, even if it may stand for the same verbal
expression, e.g. for "a pain". For if we ask "what kind of a pain?" in the first case I could answer "This kind" and, for
example, stick the questioner with a needle. In the second case I must answer the same question differently; e.g. "the
pain in my foot".
In the second sentence X could also stand for "my pain", but not in the first.]
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58. Imagine someone pointing to a place in the iris of a Rembrandt eye and saying: "The walls in my room should be
painted this colour".
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Page 10
59. I paint the view from my window; one particular spot, determined by its position in the architecture of a house, I
paint ochre. I say this is the colour I see this spot. That does not mean that I see the colour of ochre here, for in these
surroundings this pigment may look lighter, darker, more reddish, (etc.). "I see this spot the way I have painted it
here with ochre, namely as a strongly reddish-yellow".
But what if someone asked me to give the exact shade of colour that I see there?--How should it be
described and how determined? Someone could ask me to produce a colour sample (a rectangular piece of paper of
this colour). I don't say that such a comparison would be utterly uninteresting, but it shows us that it isn't from the
outset clear how shades of colour are to be compared and what "sameness of colour" means.
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60. Imagine a painting cut up into small, almost monochromatic bits which are then used as pieces in a jig-saw
puzzle. Even when such a piece is not monochromatic it should not indicate any three-dimensional shape, but
should appear as a flat colour-patch. Only together with the other pieces does it become a bit of blue sky, a shadow,
a highlight, transparent or opaque, etc. Do the individual pieces show us the real colours of the parts of the picture?
Page 10
61. We are inclined to believe the analysis of our colour concepts would lead ultimately to the colours of places in
our visual field, which are independent of any spatial or physical interpretation; for here there is neither light nor
shadow, nor high-light, etc., etc..
Page 10
62. The fact that I can say this place in my visual field is grey-green does not mean that I know what should be called
an exact reproduction of this shade of colour.
Page 10
63. I see in a photograph (not a colour photograph) a man with dark hair and a boy with slicked-back blond hair
standing in front of a kind of lathe, which is made in part of castings painted black, and in part of smooth axles,
gears, etc., and next to it a grating made of light galvanized wire. I see the finished iron surfaces as iron-coloured, the
boy's hair as blond, the grating as zinc-coloured, despite the fact that everything is depicted in lighter and darker
tones of the photographic paper.
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Page 11
64. But do I really see the hair blond in the photograph? And what can be said in favor of this? What reaction of the
viewer is supposed to show that he sees the hair blond, and doesn't just conclude from the shades of the photograph
that it is blond?--If I were asked to describe the photograph I would do so in the most direct manner with these
words. If this way of describing it won't do, then I would have to start looking for another.
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65. If the word "blond" itself can sound blond, then it's even easier for photographed hair to look blond!
Page 11
66. "Can't we imagine certain people having a different geometry of colour than we do?" That, of course, means:
Can't we imagine people having colour concepts other than ours? And that in turn means: Can't we imagine people
who do not have our colour concepts but who have concepts which are related to ours in such a way that we would
also call them "colour concepts"?
Page 11
67. Look at your room late in the evening when you can hardly distinguish between colours any longer--and now
turn on the light and paint what you saw earlier in the semi-darkness.--How do you compare the colours in such a
picture with those of the semi-dark room?
Page 11
68. When we're asked "What do the words 'red', 'blue', 'black', 'white' mean?" we can, of course, immediately point to
things which have these colours,--but our ability to explain the meanings of these words goes no further! For the
rest, we have either no idea at all of their use, or a very rough and to some extent false one.
Page 11
69. I can imagine a logician who tells us that he has now succeeded in really being able to think 2 × 2 = 4.
Page 11
70. Goethe's theory of the constitution of the colours of the spectrum has not proved to be an unsatisfactory theory,
rather it really isn't a theory at all. Nothing can be predicted with it. It is, rather, a vague schematic outline of the sort
we find in James's psychology. Nor is there any experimentum crucis which could decide for or against the theory.
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Page 12
71. Someone who agrees with Goethe believes that Goethe correctly recognized the nature of colour. And nature
here is not what results from experiments, but it lies in the concept of colour.
Page 12
72. One thing was irrefutably clear to Goethe: no lightness can come out of darkness--just as more and more
shadows do not produce light.--This could be expressed as follows: we may call lilac a reddish-whitish-blue or
brown a blackish-reddish-yellow--but we cannot call a white a yellowish-reddish-greenish-blue, or the like. And that
is something that experiments with the spectrum neither confirm nor refute. It would, however, also be wrong to
say, "Just look at the colours in nature and you will see that it is so". For looking does not teach us anything about
the concepts of colours.
Page 12
73. I cannot imagine that Goethe's remarks about the characters of the colours and colour combinations could be of
any use to a painter; they could be of hardly any to a decorator. The colour of a blood-shot eye might have a
splendid effect as the colour of a wall-hanging. Someone who speaks of the character of a colour is always thinking
of just one particular way it is used.
Page 12
74. If there were a theory of colour harmony, perhaps it would begin by dividing the colours into groups and
forbidding certain mixtures or combinations and allowing others. And, as in harmony, its rules would be given no
justification.
Page 12
75. There may be mental defectives who cannot be taught the concept 'tomorrow', or the concept 'I', nor to tell time.
Such people would not learn the use of the word 'tomorrow' etc..
Now to whom can I describe what these people cannot learn? Just to one who has learnt it? Can't I tell A that
B cannot learn higher mathematics, even though A hasn't mastered it? Doesn't the person who has learned the game
understand the word "chess" differently from someone who hasn't learnt it? There are differences between the use of
the word which the former can make, and the use which the latter has learnt.
Page 12
76. Does describing a game always mean: giving a description through which someone can learn it?
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Page 13
77. Do the normally sighted and the colour-blind have the same concept of colour-blindness? The colour-blind not
merely cannot learn to use our colour words, they can't learn to use the word "colour-blind" as a normal person
does. They cannot, for example, establish colour-blindness in the same way as the normal do.
Page 13
78. There could be people who didn't understand our way of saying that orange is a rather reddish-yellow, and who
would only be inclined to say something like that in cases where a transition from yellow through orange to red took
place before their eyes. And for such people the expression "reddish-green" need present no difficulties.
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79. Psychology describes the phenomena of seeing.--For whom does it describe them? What ignorance can this
description eliminate?
Page 13
80. Psychology describes what was observed.
Page 13
81. Can one describe to a blind person what it's like for someone to see?--Certainly. The blind learn a great deal
about the difference between the blind and the sighted. But the question was badly put; as though seeing were an
activity and there were a description of it.
Page 13
82. I can, of course, observe colour-blindness; then why not seeing?--I can observe what colour judgements a
colour-blind person--or a normally sighted person, too--makes under certain circumstances.
Page 13
83. People sometimes say (though mistakenly), "Only I can know what I see". But not: "Only I can know whether I
am colour-blind". (Nor again: "Only I can know whether I see or am blind".)
Page 13
84. The statement, "I see a red circle" and the statement "I see (am not blind)" are not logically of the same sort. How
do we test the truth of the former, and how that of the latter?
Page 13
85. But can I believe that I see and be blind, or believe that I'm blind and see?
Page 13
86. Could a psychology textbook contain the sentence, "There are people who see"? Would this be wrong? But to
whom will it communicate anything?
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Page 14
87. How can it be nonsense to say, "There are people who see", if it is not nonsense to say "There are people who
are blind"?
But suppose I had never heard of the existence of blind people and one day someone told me, "There are
people who do not see", would I have to understand this sentence immediately? If I am not blind myself must I be
conscious that I have the ability to see, and that, therefore, there may be people who do not have this ability?
Page 14
88. If the psychologist teaches us, "There are people who see", we can then ask him: "And what do you call 'people
who see'?" The answer to that would have to be: People who behave so-and-so under such-and-such circumstances.
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II
Page 15
1. We might speak of the colour-impression of a surface, by which we wouldn't mean the colour, but rather the
composite of the shades of colour, which produces the impression (e.g.) of a brown surface.
Page 15
2. Blending in white removes the colouredness from the colour; but blending in yellow does not.--Is that the basis of
the proposition that there can be no clear transparent white?
Page 15
3. But what kind of a proposition is that, that blending in white removes the colouredness from the colour?
As I mean it, it can't be a proposition of physics.
Here the temptation to believe in a phenomenology, something midway between science and logic, is very
great.
Page 15
4. What then is the essential nature of cloudiness? For red or yellow transparent things are not cloudy; white is
cloudy.
Page 15
5. Is cloudy that which conceals forms, and conceals forms because it obliterates light and shadow?
Page 15
6. Isn't white that which does away with darkness?
Page 15
7. We speak, of course, of 'black glass', yet you see a white surface as red through red glass but not as black through
'black' glass.
Page 15
8. People often use tinted lenses in their eye-glasses in order to see clearly, but never cloudy lenses.
Page 15
9. "The blending in of white obliterates the difference between light and dark, light and shadow"; does that define the
concepts more closely? Yes, I believe it does.
Page 15
10. If someone didn't find it to be this way, it wouldn't be that he had experienced the contrary, but rather that we
wouldn't understand him.
Page 15
11. In philosophy we must always ask: "How must we look at this problem in order for it to become solvable?"
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Page 16
12. For here (when I consider colours, for example) there is merely an inability to bring the concepts into some kind
of order.
We stand there like the ox in front of the newly-painted stall door.
Page 16
13. Think about how a painter would depict the view through a red-tinted glass. What results is a complicated
surface picture. I.e. the picture will contain a great many gradations of red and of other colours adjacent to one
another. And analogously if you looked through a blue glass.
But how about if you painted a picture such that the places become whitish where, before, something was
made bluish or reddish?
Page 16
14. Is the only difference here that the colours remain as saturated as before when a reddish light is cast on them,
while they don't with the whitish light? But we don't speak of a 'whitish light cast on things' at all!
Page 16
15. If everything looked whitish in a particular light, we wouldn't then conclude that the light source must look white.
Page 16
16. Phenomenological analysis (as e.g. Goethe would have it) is analysis of concepts and can neither agree with nor
contradict physics.
Page 16
17. But what if somewhere the following situation prevailed: the light of a white-hot body makes things appear light
but whitish, and so weakly-coloured; the light of a red-hot body makes things appear reddish, etc. (Only an invisible
light source, not perceptible to the eye, makes them radiate in colours.)
Page 16
18. Yes, suppose even that things only radiated their colours when, in our sense, no light fell on them--when, for
example, the sky were black? Couldn't we then say: only in black light do the full colours appear (to us)?
Page 16
19. But wouldn't there be a contradiction here?
Page 16
20. I don't see that the colours of bodies reflect light into my eye.
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III
24.3.50
Page 17
1. ? White must be the lightest colour in a picture.
Page 17
2. In the Tricolour, for example, the white cannot be darker than the blue and red.
Page 17
3. Here we have a sort of mathematics of colour.
26.3
Page 17
4. But pure yellow too is lighter than pure, saturated red, or blue. And is this proposition a matter of experience?--I
don't know, for example, whether red (i.e. pure red) is lighter or darker than blue; to be able to say, I would have to
see them. And yet, if I had seen them, I would know the answer once and for all, like the result of an arithmetical
calculation.
Where do we draw the line here between logic and experience?
Page 17
5. The word whose meaning is not clear is "pure" or "saturated". How do we learn its meaning? How can we tell if
people mean the same thing by it? I call a colour (e.g. red) "saturated" if it contains neither black nor white, if it is
neither blackish nor whitish.
But this explanation only leads to a provisional understanding.
Page 17
6. What is the importance of the concept of saturated colour?
Page 17
7. One fact is obviously important here: namely that people reserve a special place for a given point on the colour
wheel, and that they don't have to go to a lot of trouble to remember where the point is, but always find it easily.
Page 17
8. Is there such a thing as a 'natural history of colours' and to what extent is it analogous to a natural history of
plants? Isn't the latter temporal, the former nontemporal
Page 17
9. If we say that the proposition "saturated yellow is lighter than saturated blue" doesn't belong to the realm of
psychology (for only
Page Break 18
so could it be natural history)--this means that we are not using it as a proposition of natural history. And the
question then is: what is the other, nontemporal use like?
Page 18
10. For this is the only way we can distinguish propositions of 'the mathematics of colour' from those of natural
history.
Page 18
11. Or again, the question is this: can we (clearly) distinguish two uses here?
Page 18
12. If you impress two shades of colour on your memory, and A is lighter than B, and then later you call one shade
"A" and another "B" but the one you called "B" is lighter than "A", you have called these shades by the wrong
names. (This is logic).
Page 18
13. Let the concept of a 'saturated' colour be such that saturated X cannot be lighter than saturated Y at one time and
darker at another; i.e. it makes no sense to say it is lighter at one time and darker at another. This determines the
concept and is again a matter of logic.
The usefulness of a concept determined in this way is not decided here.
Page 18
14. This concept might only have a very limited use. And this simply because what we usually call a saturated X is
an impression of colour in a particular surrounding. It is comparable to 'transparent' X.
Page 18
15. Give examples of simple language-games with the concept of 'saturated colours'.
Page 18
16. I assume that certain chemical compounds, e.g. the salts of a given acid, have saturated colours and could be
recognized by them.
Page 18
17. Or you could tell where certain flowers come from by their saturated colours, e.g. you could say, "That must be
an alpine flower because its colour is so intense".
Page 18
18. But in such a case there could be lighter and darker saturated red, etc.
Page 18
19. And don't I have to admit that sentences are often used on the borderline between logic and the empirical, so that
their meaning
Page Break 19
shifts back and forth and they are now expressions of norms, now treated as expressions of experience?
For it is not the 'thought' (an accompanying mental phenomenon) but its use (something that surrounds it),
that distinguishes the logical proposition from the empirical one.
Page 19
20. (The wrong picture confuses, the right picture helps.)
Page 19
21. The question will be, e.g.: can you teach the meaning of "saturated green" by teaching†1 the meaning of
"saturated red", or "yellow", or "blue"?
Page 19
22. A shine, a 'high-light' cannot be black. If I were to substitute blackness for the lightness of high-lights in a
picture, I wouldn't get black lights. And that is not simply because this is the one and only form in which a high-light
occurs in nature, but also because we react to a light in this spot in a certain way. A flag may be yellow and black,
another yellow and white.
Page 19
23. Transparency painted in a picture produces its effect in a different way than opaqueness.
Page 19
24. Why is transparent white impossible?--Paint a transparent red body, and then substitute white for red!
Black and white themselves have a hand in the business, where we have transparency of a colour. Substitute
white for red and you no longer have the impression of transparency; just as you no longer have the impression of
solidity if you turn this drawing into this one .
27.3
Page 19
25. Why isn't a saturated colour simply: this, or this, or this, or this?--Because we recognize it or determine it in a
different way.
Page 19
26. Something that may make us suspicious is that some people have thought they recognized three primary
colours, some four. Some have thought green to be an intermediary colour between blue and yellow, which strikes
me, for example, as wrong, even apart from any experience.
Page Break 20
Page 20
Blue and yellow, as well as red and green, seem to me to be opposites--but perhaps that is simply because I
am used to seeing them at opposite points on the colour circle.
Indeed, what (so to speak psychological) importance does the question as to the number of Pure Colours
have for me?
Page 20
27. I seem to see one thing that is of logical importance: if you call green an intermediary colour between blue and
yellow, then you must also be able to say, for example, what a slightly bluish yellow is, or an only somewhat
yellowish blue. And to me these expressions don't mean anything at all. But mightn't they mean something to
someone else?
So if someone described the colour of a wall to me by saying: "It was a somewhat reddish yellow," I could
understand him in such a way that I could choose approximately the right colour from among a number of samples.
But if someone described the colour in this way: "It was a somewhat bluish yellow," I could not show him such a
sample.--Here we usually say that in the one case we can imagine the colour, and in the other we can't--but this way
of speaking is misleading, for there is no need whatsoever to think of an image that appears before the inner eye.
Page 20
28. There is such a thing as perfect pitch and there are people who don't have it; similarly we could suppose that
there could be a great range of different talents with respect to seeing colours. Compare, for example, the concept
'saturated colour' with 'warm colour'. Must it be the case that everyone knows 'warm' and 'cool' colours? Apart from
being taught to give this or that name to a certain disjunction of colours.
Couldn't there be a painter, for example, who had no concept whatsoever of 'four pure colours' and who
even found it ridiculous to talk about such a thing?
Page 20
29. Or in other words: are people for whom this concept is not at all natural missing anything?
Page 20
30. Ask this question: Do you know what "reddish" means? And how do you show that you know it?
Language-games: "Point to a reddish yellow (white, blue, brown)--"Point to an even more reddish one"--"A
less reddish one" etc.
Page Break 21
Now that you've mastered this game you will be told "Point to a somewhat reddish green" Assume there are two
cases: Either you do point to a colour (and always the same one), perhaps to an olive green--or you say, "I don't
know what that means," or "There's no such thing."
We might be inclined to say that the one person had a different colour concept from the other; or a different
concept of '... ish.'
Page 21
31. We speak of "colour-blindness" and call it a defect. But there could easily be several differing abilities, none of
which is clearly inferior to the others.--And remember, too, that a man may go through life without his
colour-blindness being noticed, until some special occasion brings it to light.
Page 21
32. Is it possible then for different people in this way to have different colour concepts?--Somewhat different ones.
Different with respect to one or another feature. And that will impair their mutual understanding to a greater or lesser
extent, but often hardly at all.
Page 21
33. Here I would like to make a general observation concerning the nature of philosophical problems. Lack of clarity
in philosophy is tormenting. It is felt as shameful. We feel: we do not know our way about where we should know
our way about. And nevertheless it isn't so. We can get along very well without these distinctions and without
knowing our way about here.
Page 21
34. What is the connection between the blending of colours and 'intermediary colours'? We can obviously speak of
intermediary colours in a language-game in which we do not produce colours by mixing at all, but only select
existing shades.
Yet one use of the concept of an intermediary colour is to recognize the blend of colours which produces a
given shade.
Page 21
35. Lichtenberg says that very few people have ever seen pure white. Do most people use the word wrong, then?
And how did he learn the correct use?--On the contrary: he constructed an ideal use from the actual one. The way
we construct a geometry. And 'ideal' does not mean something specially good, but only something carried to
extremes.
Page 21
36. And of course such a construct can in turn teach us something about the actual use.
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Page 22
And we could also introduce a new concept of 'pure white', e.g. for scientific purposes.
(A new concept of this sort would then correspond to, say, the chemical concept of a 'salt'.)
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37. To what extent can we compare black and white to yellow, red and blue, and to what extent can't we?
If we had a checked wall-paper with red, blue, green, yellow, black and white squares, we would not be
inclined to say that it is made up of two kinds of parts, of 'coloured' and, say, 'uncoloured' ones.
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38. Let us now suppose that people didn't contrast coloured pictures with black-and-white ones, but rather with
blue-and-white ones. I.e.: couldn't blue too be felt (and that is to say, used) as not being an actual colour?
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39. My feeling is that blue obliterates yellow,--but why shouldn't I call a somewhat greenish yellow a "bluish yellow"
and green an intermediary colour between blue and yellow, and a strongly bluish green a somewhat yellowish blue?
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40. In a greenish yellow I don't yet notice anything blue.--For me, green is one special way-station on the coloured
path from blue to yellow, and red is another.
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41. What advantage would someone have over me who knew a direct route from blue to yellow? And what shows
that I don't know such a path?--Does everything depend on my range of possible language-gams with the form "...
ish"?
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42. We will, therefore, have to ask ourselves: What would it be like if people knew colours which our people with
normal vision do not know? In general this question will not admit of an unambiguous answer. For it is by no means
clear that we must say of this sort of abnormal people that they know other colours. There is, after all, no commonly
accepted criterion for what is a colour, unless it is one of our colours.
And yet we could imagine circumstances under which we would say, "These people see other colours in
addition to ours."
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28.3
43. In philosophy it is not enough to learn in every case what is to be said about a subject, but also how one must
speak about it. We are always having to begin by learning the method of tackling it.
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44. Or again: In any serious question uncertainty extends to the very roots of the problem.
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45. One must always he prepared to learn something totally new.
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46. Among the colours: Kinship and Contrast. (And that is logic.)
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47. What does it mean to say, "Brown is akin to yellow?"
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48. Does it mean that the task of choosing a somewhat brownish yellow would be readily understood? (Or a
somewhat more yellowish brown).
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49. The coloured intermediary between two colours.
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50. "Yellow is more akin to red than to blue."
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51. The differences between black-red-gold and black-red-yellow. Gold counts as a colour here.
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52. It is a fact that we can communicate with one another about the colours of things by means of six colour words.
Also, that we do not use the words "reddish-green", "yellowish-blue" etc.
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53. Description of a jig-saw puzzle by means of the description of its pieces. I assume that these pieces never exhibit
a three-dimensional form, but always appear as small flat bits, single- or many-coloured. Only when they are put
together does something become a 'shadow', a 'high-light', a 'concave or convex monochromatic surface', etc.
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54. I can say: This man does not distinguish between red and green. But can I say that we normal people distinguish
between red and green? We could, however, say: "Here we see two colours, he sees only one."
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55. The description of the phenomena of colour-blindness is part of psychology. And the description of the
phenomena of normal colour vision too? Of course--but what are the presuppositions of such a description and for
whom is it a description? Or better: what are the means it employs? When I say, "What does it presuppose?" that
means "How must one react to this description in order to understand it?" Someone who describes the phenomena
of colour-blindness in a book describes them in the concepts of the sighted.
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56. This paper is lighter in some places than in others; but can I say that it is white only in certain places and gray in
others??--Certainly, if I painted it, I would mix a gray for the darker places.
A surface-colour is a quality of a surface. One might (therefore) be tempted not to call it a pure colour
concept. But then what would a pure one be?!
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57. It is not correct to say that in a picture white must always be the lightest colour. But it must be the lightest one in
a flat pattern of coloured patches. A picture might show a book made of white paper in shadow, and lighter than this
a luminous yellow or blue or reddish sky. But if I describe a plane surface, a wallpaper, for example, by saying that it
consists of pure yellow, red, blue, white and black squares, the yellow ones cannot be lighter than the white ones,
and the red cannot be lighter than the yellow.
This is why colours were shadows for Goethe.
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58. There seems to be a more fundamental' colour concept than that of the surface colour. It seems that one could
present it either by means of small coloured elements in the field of vision, or by means of luminous points rather
like stars. And larger coloured areas are composed of these coloured points or small coloured patches. Thus we
could describe the colour impression of a surface area by specifying the position of the numerous small coloured
patches within this area.
But how should we, for example, compare one of these small colour samples with a piece of the larger
surface area? In what surroundings should the colour sample occur?
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Page 25
59. In everyday life we are virtually surrounded by impure colours. All the more remarkable that we have formed a
concept of pure colours.
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29.3
60. Why don't we speak of a 'pure' brown? Is the reason merely the position of brown with respect to the other 'pure'
colours, its relationship to them all?--Brown is, above all, a surface colour, i.e. there is no such thing as a clear
brown, but only a muddy one. Also: brown contains black--(?)--How would a person have to behave for us to say of
him that he knows a pure, primary brown?
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61. We must always bear in mind the question: How do people learn the meaning of colour names?
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62. What does, "Brown contains black," mean? There are more and less blackish browns. Is there one which isn't
blackish at all? There certainly isn't one that isn't yellowish at all.†1
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63. If we continue to think along these lines, 'internal properties' of a colour gradually occur to us, which we hadn't
thought of at the outset. And that can show us the course of a philosophical investigation. We must always be
prepared to come across a new one, one that has not occurred to us earlier.
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64. And we must not forget either that our colour words characterize the impression of a surface over which our
glance wanders. That's what they're for.
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65. "Brown light". Suppose someone were to suggest that a traffic light be brown.
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66. It is only to be expected that we will find adjectives (as, for example, "iridescent") which are colour
characteristics of an extended area or of a small expanse in a particular surrounding "shimmering", "glittering",
"gleaming", "luminous").
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67. And indeed the pure colours do not even have special commonly used names, that's how unimportant they are
to us.
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68. Let us imagine that someone were to paint something from nature and in its natural colours. Every bit of the
surface of such a painting has a definite colour. What colour? How do I determine its name? Should we, e.g. use the
name under which the pigment applied to it is sold? But mightn't such a pigment look completely different in its
special surrounding than on the palette?
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69. So perhaps we would then start to give special names to small coloured patches on a black background (for
example).
What I really want to show here is that it is not at all clear a priori which are the simple colour concepts.
30.3
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70. It is not true that a darker colour is at the same time a more blackish one. That's certainly clear. A saturated
yellow is darker, but is not more blackish than a whitish yellow. But amber isn't a 'blackish yellow' either. (?) And
yet people speak of a 'black' glass or mirror.--Perhaps the trouble is that by "black" I mean essentially a surface
colour?
I would not say of a ruby that it is blackish red, for that would suggest cloudiness. (On the other hand, don't
forget that both cloudiness and transparency can be painted.)
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71. I treat colour concepts like the concepts of sensations.
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72. The colour concepts are to be treated like the concepts of sensations.
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73. There is no such thing as the pure colour concept.
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74. Where does the illusion come from then? Aren't we dealing here with a premature simplification of logic like any
other?
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75. I.e., the various colour concepts are certainly closely related to one another, the various 'colour words' have a
related use, but there are, on the other hand, all kinds of differences.
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76. Runge says that there are transparent and opaque colours. But this does not mean that you would use different
greens to paint a piece of green glass and a green cloth in a picture.
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77. It is a peculiar step taken in painting, that of depicting a highlight by means of a colour.
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78. The indefiniteness in the concept of colour lies, above all, in the indefiniteness of the concept of the sameness of
colours, i.e. of the method of comparing colours.
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79. There is gold paint, but Rembrandt didn't use it to paint a golden helmet.
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80. What makes grey a neutral colour? Is it something physiological or something logical?
What makes bright colours bright? Is it a conceptual matter or a matter of cause and effect?
Why don't we include black and white in the colour circle? Only because we have a feeling that it's wrong?
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81. There is no such thing as luminous grey. Is that part of the concept of grey, or part of the psychology, i.e. the
natural history, of grey? And isn't it odd that I don't know?
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82. Colours have characteristic causes and effects--that we do know.
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83. Grey is between two extremes (black and white), and can take on the hue of any other colour.
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84. Would it be conceivable for someone to see as black everything that we see as white, and vice versa?
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85. In a brightly coloured pattern black and white can be next to red and green, etc. without standing out as different.
This would not be the case, however, in the colour circle, if only because black and white mix with all the
other colours. But also in particular, they both mix with their opposite pole.
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86. Can't we imagine people having a geometry of colours different from our normal one? And that, of course,
means: can we describe it, can we immediately respond to the request to describe it, that is, do we know
unambiguously what is being demanded of us?
The difficulty is obviously this: isn't it precisely the geometry of colours that shows us what we're talking
about, i.e. that we are talking about colours?
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87. The difficulty of imagining it (or of filling out the picture of it) is in knowing when one has pictured that. I.e. the
indefiniteness of the request to imagine it.
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88. The difficulty is, therefore, one of knowing what we are supposed to consider as the analogue of something that
is familiar to us.
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89. A colour which would be 'dirty' if it were the colour of a wall, needn't be so in a painting.
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90. I doubt that Goethe's remarks about the characters of the colours could be of any use to a painter. They could
hardly be any to a decorator.
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91. If there were a theory of colour harmony, perhaps it would begin by dividing the colours into different groups
and forbidding certain mixtures or combinations and allowing others; and, as in harmony, its rules would be given
no justification.
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92. Mayn't that open our eyes to the nature of those differentiations among colours?
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93. [We don't say A knows something and B knows the opposite. But if we say "believes" instead of "knows", then
it is a proposition.]
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94. Runge to Goethe: "If we were to think of a bluish orange, a reddish green or a yellowish violet, we would have
the same feeling as in the case of a southwesterly northwind."
Also: what amounts to the same thing, "Both white and black are opaque or solid.... White water which is
pure is as inconceivable as clear milk. If black merely made things dark, it could indeed be clear; but because it
smirches things, it can't be."
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95. In my room I am surrounded by objects of different colours. It is easy to say what colour they are. But if I were
asked what colour I am now seeing from here at, say, this place on my table, I couldn't answer; the place is whitish
(because the light wall makes the brown table lighter here) at any rate it is much lighter than the rest of the table, but,
given a number of colour samples, I wouldn't be able to pick out one which had the same coloration as this area of
the table.
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96. Because it seems so to me--or to everybody--it does not follow that it is so.
Therefore: From the fact that this table seems brown to everyone, it does not follow that it is brown. But just
what does it mean to say, "This table isn't really brown after all"?--So does it then follow from its appearing brown to
us, that it is brown?
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97. Don't we just call brown the table which under certain circumstances appears brown to the normal-sighted? We
could certainly conceive of someone to whom things seemed sometimes this colour and sometimes that,
independently of the colour they are.
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98. That it seems so to men is their criterion for its being so.
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99. Being and seeming may, of course, be independent of one another in exceptional cases, but that doesn't make
them logically independent; the language-game does not reside in the exception.
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100. Golden is a surface colour.
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101. We have prejudices with respect to the use of words.
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102. When we're asked "What do 'red', 'blue', 'black', 'white, mean?" we can, of course, immediately point to things
which have these colours,--but that's all we can do: our ability to explain their meaning goes no further.
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103. For the rest, either we have no idea at all, or a very rough and to some extent false one.
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104. 'Dark' and 'blackish' are not the same concept.
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105. Runge says that black 'dirties'; what does that mean? Is that an emotional effect which black has on us? Is it an
effect of the addition of black colour that is meant here?
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106. Why is it that a dark yellow doesn't have to be perceived as 'blackish', even if we call it dark?
The logic of the concept of colour is just much more complicated than it might seem.
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107. The concepts 'matt' and 'shiny'. If, when we think of 'colour' we think of a property of a point in space, then the
concepts matt and shiny have no reference to these colour concepts.
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108. The first 'solution' which occurs to us for the problem of colours is that the 'pure' colour concepts refer to points
or tiny indivisible patches in space. Question: how are we to compare the colours of two such points? Simply by
letting one's gaze move from one to the other? Or by moving a coloured object? If the latter, how do we know that
this object has not changed colour in the process; if the former, how can we compare the coloured points without
the comparison being influenced by what surrounds them?
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109. I could imagine a logician who tells us that he has now succeeded in really being able to think 2 × 2 = 4.
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110. If you are not clear about the role of logic in colour concepts, begin with the simple case of, e.g. a yellowish red.
This exists, no one doubts that. How do I learn the use of the word "yellowish"? Through language-games in which,
for example, things are put in a certain order.
Thus I can learn, in agreement with other people, to recognize yellowish and still more yellowish red, green,
brown and white.
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In the course of this I learn to proceed independently just as I do in arithmetic. One person may react to the
order to find a yellowish blue by producing a blue green, another may not understand the order. What does this
depend upon?
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111. I say blue-green contains no yellow: if someone else claims that it certainly does contain yellow, who's right?
How can we check? Is there only a verbal difference between us?--Won't the one recognize a pure green that tends
neither toward blue nor toward yellow? And of what use is this? In what language-games can it be used? He will at
least be able to respond to the command to pick out the green things that contain no yellow, and those that contain
no blue. And this constitutes the demarcation point 'green', which the other does not know.
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112. The one can learn a language-game that the other one cannot. And indeed this must be what constitutes
colour-blindness of all
Page Break 31
kinds. For if the 'colour-blind' person could learn all the language-games of normal people, why should he be
excluded from certain professions?
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113. If someone had called this difference between green and orange to Runge's attention, perhaps he would have
given up the idea that there are only three primary colours.
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114. Now to what extent is it a matter of logic rather than psychology that someone can or cannot learn a game?
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115. I say: The person who cannot play this game does not have this concept.
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116. Who has the concept 'tomorrow'? Of whom do we say this?
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117. I saw in a photograph a boy with slicked-back blond hair and a dirty light-coloured jacket, and a man with dark
hair, standing in front of a machine which was made in part of castings painted black, and in part of finished, smooth
axles, gears, etc., and next to it a grating made of light galvanized wire. The finished iron parts were iron coloured,
the boy's hair was blond, the castings black, the grating zinc-coloured, despite the fact that everything was depicted
simply in lighter and darker shades of the photographic paper.
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118. There may be mental defectives who cannot be taught the concept 'tomorrow' or the concept 'I', nor to tell time.
Such would not learn the use of the word 'tomorrow' etc.
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119. Now to whom can I communicate what this mental defective cannot learn.? Just to whoever has learned it
himself? Can't I tell someone that so-and-so cannot learn higher mathematics, even if this person himself hasn't
mastered it? And yet: doesn't the person who has learned higher mathematics know more precisely what I mean?
Doesn't the person who has learned the game understand the word 'chess' differently from someone who doesn't
know it? What do we call "describing a technique"?
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120. Or: Do normally sighted people and colour-blind people have the same concept of colour-blindness?
And yet the colour-blind person understands the statement "I am colour-blind", and its negation as well.
A colour-blind person not merely can't learn to use our colour words, he can't learn to use the word
"colour-blind" exactly as a normal person does. He cannot for example always determine colour-blindness in cases
where the normal-sighted can.
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121. And to whom can I describe all the things we normal people can learn?
Understanding the description itself already presupposes that he has learned something.
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122. How can I describe to someone how we use the word "tomorrow"? I can teach it to a child; but this does not
mean I'm describing its use to him.
But can I describe the practice of people who have a concept, e.g. 'reddish-green', that we don't possess?--In
any case I certainly can't teach this practice to anyone.
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123. Can I then only say: "These people call this (brown, for example) reddish green"? Wouldn't it then just be
another word for something that I have a word for? If they really have a different concept than I do, this must be
shown by the fact that I can't quite figure out their use of words.
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124. But I have kept on saying that it's conceivable for our concepts to be different than they are. Was that all
nonsense?
11.4
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125. Goethe's theory of the origin of the spectrum isn't a theory of its origin that has proved unsatisfactory; it is
really not a theory at all. Nothing can be predicted by means of it. It is, rather, a vague schematic outline, of the sort
we find in James's psychology. There is no experimentum crucis for Goethe's theory of colour.
Someone who agrees with Goethe finds that Goethe correctly recognized the nature of colour. And here
'nature' does not mean a sum of experiences with respect to colours, but it is to be found in the concept of colour.
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126. One thing was clear to Goethe: no lightness can come out of darkness-just as more and more shadows do not
produce light. This could however be expressed as follows: we may, for example, call lilac a "reddish-whitish-blue",
or brown a "reddish-blackish-yellow", but we cannot call white a "yellowish-reddish-greenish-blue" (or the like).
And that is something that Newton didn't prove either. White is not a blend of colours in this sense.
12.4
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127. 'The colours' are not things that have definite properties, so that one could straight off look for or imagine
colours that we don't yet know, or imagine someone who knows different ones than we do. It is quite possible that,
under certain circumstances, we would say that people know colours that we don't know, but we are not forced to
say this, for there is no indication as to what we should regard as adequate analogies to our colours, in order to be
able to say it. This is like the case in which we speak of infra-red 'light'; there is a good reason for doing it, but we can
also call it a misuse.
And something similar is true with my concept 'having pain in someone else's body'.
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128. There could very easily be a tribe of people who are all colour-blind and who nonetheless live very well; but
would they have developed all our colour names, and how would their nomenclature correspond to ours? What
would their natural language be like?? Do we know? Would they perhaps have three primary colours: blue, yellow
and a third which takes the place of red and green?--What if we were to encounter such a tribe and wanted to learn
their language? We would no doubt run into certain difficulties.
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129. Couldn't there be people who didn't understand our way of speaking when we say that orange is a
reddish-yellow (etc.) and who were only inclined to say this in cases in which orange occurs in an actual transition
from red to yellow? And for such people there might very well be a reddish green.
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Therefore, they couldn't 'analyse blends of colours' nor could they learn our use of X-ish Y. (Like people
without perfect pitch).
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130. And what about people who only had colour-shape concepts? Should I say of them that they do not see that a
green leaf and a
Page Break 34
green table--when I show them these things--have the same colour or have something in common? What if it had
never 'occurred to them' to compare differently shaped objects of the same colour with one another? Due to their
particular background, this comparison was of no importance to them, or had importance only in very exceptional
cases, so that no linguistic tool was developed.
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131. A language-game: report on the greater lightness or darkness of bodies.--But now there is a related one: state
the relationship between the lightness of certain colours. (Compare: the relationship between the lengths of two
given sticks -the relationship between two given numbers.)
The form of the propositions is the same in both cases ("X lighter than Y"). But in the first language-game
they are temporal and in the second non-temporal.
Page 34
132. In a particular meaning of "white" white is the lightest colour of all.
In a picture in which a piece of white paper gets its lightness from the blue sky, the sky is lighter than the
white paper. And yet in another sense blue is the darker and white the lighter colour (Goethe). With a white and a
blue on the palette, the former would be lighter than the latter. On the palette, white is the lightest colour.
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133. I may have impressed a certain grey-green upon my memory so that I can always correctly identify it without a
sample. Pure red (blue, etc.) however, I can, so to speak, always reconstruct. It is simply a red that tends neither to
one side nor to the other, and I recognize it without a sample, as e.g. I do a right angle, by contrast with an arbitrary
acute or obtuse angle.
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134. Now in this sense there are four (or, with white and black, six) pure colours.
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135. A natural history of colours would have to report on their occurrence in nature, not on their essence. Its
propositions would have to be temporal ones.
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136. By analogy with the other colours, a black drawing on a white background seen through a transparent white
glass would have to
Page Break 35
appear unchanged as a black drawing on a white background. For the black must remain black and the white,
because it is also the colour of the transparent body, remains unchanged.
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137. We could imagine a glass through which black looked like black, white like white, and all the other colours
appeared as shades of grey; so that seen through it everything appears as though in a photograph.
But why should I call that "white glass"?
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138. The question is: is constructing a 'transparent white body' like constructing a 'regular biangle'?
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139. I can look at a body and perhaps see a matt white surface, i.e. get the impression of such a surface, or the
impression of transparency (whether it actually exists or not). This impression may be produced by the distribution
of the colours, and white and the other colours are not involved in it in the same way.
(I took a green painted lead cupola to be translucent greenish glass without knowing at the time about the
special distribution of colours that produced this appearance.)
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140. And white may indeed occur in the visual impression of a transparent body, for example as a reflection, as a
high-light. I.e. if the impression is perceived as transparent, the white which we see will simply not be interpreted as
the body's being white.
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141. I look through a transparent glass: does it follow that I don't see white? No, but I don't see the glass as white.
But how does this come about? It can happen in various ways. I may see the white with both eyes as lying behind
the glass. But simply in virtue of its position I may also see the white as a high-light (even when it isn't). And yet
we're dealing here with seeing, not just taking something to be such-and-such. Nor is it at all necessary to use both
eyes in order to see something as lying behind the glass.
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142. The various 'colours' do not all have the same connexion with three-dimensional vision.
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143. And it doesn't matter whether we explain this in terms of childhood experience or not.
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144. It must be the connection between three-dimensionality, light and shadow.
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145. Nor can we say that white is essentially the property of a--visual--surface. For it is conceivable that white should
occur as a high-light or as the colour of a flame.
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146. A body that is actually transparent can, of course, seem white to us; but it cannot seem white and transparent.
Page 36
147. But we should not express this by saying: white is not a transparent colour.
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148. 'Transparent' could be compared with 'reflecting'.
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149. An element of visual space may be white or red, but can't be either transparent or opaque.
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150. Transparency and reflection only exist in the dimension of depth of a visual image.
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151. Why can't a monochromatic surface in the field of vision be amber-coloured? This colour-word refers to a
transparent medium; thus if a painter paints a glass with amber-coloured wine in it, you could call the surface of the
picture where this is depicted "amber-coloured", but you could not say this of any one monochromatic element of
this surface.
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152. Mightn't shiny black and matt black have different colour-names?
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153. We don't say of something which looks transparent that it looks white.
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154. "Can't we imagine people having a different geometry of colour than we do?"--That, of course, means: Can't we
imagine people who have colour concepts which are other than ours; and that in turn means: Can't we imagine that
people do not have our colour concepts and that they have concepts which are related to ours in such a way that we
would also want to call them "colour concepts"?
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155.†1 If people were used to seeing nothing but green squares and red circles, they might regard a green circle with
the same kind of mistrust with which they would regard a freak, and, for example, they might even say it is really a
red circle, but has something of a...†1
If people only had colour-shape concepts, they would have a special word for a red square and one for a red
circle, and one for a green circle, etc. Now if they were to see a new green figure, should no similarity to the green
circle, etc. occur to them? And shouldn't it occur to them that there is a similarity between green circles and red
circles? But what do I want to say counts as showing that this similarity has occurred to them?
They might, for example, have a concept of 'going together'; and still not think of using colour words.
In fact there are tribes which only count up to 5 and they have probably not felt it necessary to describe
anything that can't be described in this way.
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156. Runge: "Black dirties". That means it takes the brightness out of a colour, but what does that mean? Black takes
away the luminosity of a colour. But is that something logical or something psychological? There is such a thing as a
luminous red, a luminous blue, etc., but no luminous black. Black is the, darkest of the colours. We say "deep black"
but not "deep white".
But a 'luminous red' does not mean a light red. A dark red can be luminous too. But a colour is luminous as a
result of its context, in its context.
Grey, however, is not luminous.
But black seems to make a colour cloudy, but darkness doesn't. A ruby could thus keep getting darker
without ever becoming cloudy; but if it became blackish red, it would become cloudy. Now black is a surface colour.
Darkness is not called a colour. In paintings darkness can also be depicted as black.
The difference between black and, say, a dark violet is similar to the difference between the sound of a bass
drum and the sound of a kettle-drum. We say of the former that it is a noise not a tone. It is matt and absolutely
black.
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157. Look at your room late in the evening when you can hardly distinguish between colours any longer; and now
turn on the light and paint what you saw in the twilight. There are pictures of landscapes or rooms in semi-darkness:
But how do you compare the colours in such pictures with those you saw in semi-darkness? How different this
comparison is from that of two colour samples which I have in front of me at the same time and compare by putting
them side by side!
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158. What is there in favour of saying that green is a primary colour and not a mixture of blue and yellow? Is it
correct to answer: "You can only know it directly, by looking at the colours"? But how do I know that I mean the
same by the words "primary colours" as someone else who is also inclined to call green a primary colour? No, here
there are language-games that decide these questions.
There is a more or less bluish (or yellowish) green and someone may be told to mix a green less yellow (or
blue) than a given yellow (or blue) one, or to pick one out from a number of colour samples. A less yellow green,
however, is not a bluer one (and vice versa), and someone may also be given the task of choosing-or mixing-a green
that is neither yellowish nor bluish. And I say "or mixing", because a green is not both yellowish and bluish on
account of being produced by mixing yellow and blue.
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159. Consider that things can be reflected in a smooth white surface in such a way that their reflections seem to lie
behind the surface and in a certain sense are seen through it.
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160. If I say a piece of paper is pure white and then place snow next to it and it then appears grey, in normal
surroundings and for ordinary purposes I would call it white and not light grey. It could be that I'd use a different
and, in a certain sense, more refined concept of white in, say, a laboratory, (where I sometimes also use a more
refined concept of 'precise' determination of time).
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161. The pure saturated colours are essentially characterized by a certain relative lightness. Yellow, for example, is
lighter than red. Is red lighter than blue? I don't know.
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162. Someone who has learned the concept of intermediary colours, who has mastered the technique and who thus
can find or mix shades of colour that are more whitish, more yellowish, more bluish than a given shade and so on, is
now asked to pick out or to mix a reddish green.
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163. Someone who is familiar with reddish green should be in a position to produce a colour series which starts with
red and ends with green and constitutes for us too a continuous transition between the two. We might then discover
that at the point where we perhaps always see the same shade of brown, this person sometimes sees brown and
sometimes reddish green. It may be, for example, that he can differentiate between the colours of two chemical
compounds that seem to us to be the same colour, and he calls one "a brown" and the other "reddish green".
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164. In order to describe the phenomenon of red-green colour-blindness, I need only say what someone who is
red-green colourblind cannot learn; but now in order to describe the 'phenomena of normal vision' I would have to
enumerate the things we can do.
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165. Someone who describes the 'phenomena of colour-blindness' describes only the ways in which the colour-blind
person deviates from the normal, not his vision in general.
But couldn't she also describe the ways in which normal vision deviates from total blindness? We might ask:
who would learn from this? Can someone teach me that I see a tree?
And what is a 'tree', and what is 'seeing'?
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166. We can, for example, say: This is the way a person acts with a blindfold over his eyes, and this is the way a
sighted person without a blindfold acts. With a blindfold he reacts thus and so, without the blindfold he walks
briskly along the street, greets his acquaintances, nods to this one and that, avoids the cars and bicycles easily when
he crosses the street, etc., etc. Even with new-born infants, we know that they can see from the fact that they follow
movements with their eyes. Etc., etc.. The question is: who is supposed to understand the description? Only sighted
people, or blind people too?
It makes sense, for exampe [[sic]], to say "the sighted person distinguishes with his eyes between an unripe
apple and a ripe one." But
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not: "The sighted person distinguishes a green apple from a red one." For what are 'red' and 'green'?
Marginal note: "The sighted person distinguishes an apple that appears red to him from one which appears
green."
But can't I say "I distinguish this kind of apple from this kind" (while pointing to a red apple and a green
one)?
But what if someone points at two apples that seem to me to be exactly alike and says that?! On the other
hand he could say to me "Both of them look exactly alike to you, so you might confuse them; but I see a difference
and I can recognize each of them any time." That can be tested and confirmed.
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167. What is the experience that teaches me that I differentiate between red and green?
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168. Psychology describes the phenomena of seeing. For whom does it describe them? What ignorance can this
description eliminate?
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169. If a sighted person had never heard of a blind person, couldn't we describe the behaviour of the blind person to
him?
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170. I can say: "The colour-blind person cannot distinguish between a green apple and a red one" and that can be
demonstrated. But can I say "I can distinguish between a green apple and a red one"? Well, perhaps by the taste. But
still, for example, "I can distinguish an apple that you call 'green' from one that you call 'red', therefore I am not
colour-blind".
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171. This piece of paper varies in lightness from place to place, but does it look grey to me in the darker places? The
shadow that my hand casts is in part grey. I see the parts of the paper that are farther away from the light darker but
still white, even though I would have to mix a grey to paint it. Isn't this similar to the fact that we often see a distant
object merely as distant and not as smaller? Thus we cannot say "I notice that he looks smaller, and I conclude from
that that he is farther away", but rather I notice that he is farther away, without being able to say how I notice it.
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172. The impression of a coloured transparent medium is that something is behind the medium. Thus if we have a
thoroughly monochromatic visual image, it cannot be one of transparency.
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173. Something white behind a coloured transparent medium appears in the colour of the medium, something black
appears black. According to this rule a black drawing on white paper behind a white transparent medium must
appear as though it were behind a colourless medium.
That†1 was not a proposition of physics, but rather a rule of the spatial interpretation of our visual
experience. We could also say, it is a rule for painters: "If you want to portray something white behind something
that is transparent and red, you have to paint it red." If you paint it white, it doesn't look as though it is behind the
red thing.
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174. In the places where there is only a little less light on the white paper it doesn't seem at all grey, but always white.
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175. The question is: What must our visual picture be like if it is to show us a transparent medium? How must, e.g.,
the colour of the medium appear? Speaking in physical terms--although we are not directly concerned with the laws
of physics here--everything seen through a green glass must look more or less dark green. The lightest shade would
be that of the medium. That which we see through it is, thus, similar to a photograph. Now if we apply all this to
white glass, everything should again look as though it were photographed, but in shades ranging from white to black.
And if there were such glass--why shouldn't we want to call it white? Is there anything to be said against doing this;
does the analogy with glass of other colours break down at any point?
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176. A cube of green glass looks green when it's lying in front of us. The overall impression is green; thus the overall
impression of the white cube should be white.
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177. Where must the cube appear white for us to be able to call it white and transparent?
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178. Is it because the relationships and contrasts between white and the other colours are different from those
between green and the other colours, that for white there is nothing analogous to a transparent green glass?
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179. When light comes through it red glass casts a red light; now what would light coming through a white glass
look like? Would yellow become whitish in such a light or merely lighter? And would black become grey or would it
remain black?
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180. We are not concerned with the facts of physics here except insofar as they determine the laws governing how
things appear.
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181. It is not immediately clear which transparent glass we should say had the 'same colour' as a piece of green
paper.
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182. If the paper is, e.g. pink, sky-blue or lilac we would imagine the glass to be somewhat cloudy, but we could also
suppose it to be just a rather weak reddish, etc., clear glass. That's why something colourless is sometimes called
"white".
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183. We could say, the colour of a transparent glass is that which a white light source would appear when seen
through that glass.
But seen through a colourless glass it appears as uncloudy white.
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184. In the cinema it is often possible to see the events as 'though they were occurring behind the screen, as if the
screen were transparent like a pane of glass. At the same time, however, the colour would be removed from these
events and only white, grey and black would come through. But we are still not tempted to call it a transparent,
white pane of glass.
How, then, would we see things through a pane of green glass? One difference would, of course, be that the
green glass would diminish the difference between light and dark, while the other one shouldn't have any effect upon
this difference. Then a 'grey transparent' pane would somewhat diminish it.
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185. We might say of a pane of green glass that it gave things its colour. But does my 'white' pane do that?--If the
green medium gives its colour to things, then, above all, to white things.
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186. A thin layer of a coloured medium colours things only weakly: how should a thin 'white' glass colour them?
Shall we suppose that it doesn't quite remove all their colour?
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187. "We shouldn't be able to conceive of white water that is pure..." That is to say: we cannot describe how
something white could look clear, and that means: we don't know what description is being asked for with these
words.
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188. We do not want to find a theory of colour (neither a physiological nor a psychological one), but rather the logic
of colour concepts. And this accomplishes that which people have often unjustly expected from a theory.
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189. Explaining colour words by pointing to coloured pieces of paper does not touch the concept of transparency. It
is this concept that stands in unlike relations to the various colour concepts.
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190. Thus, if someone wanted to say we don't even notice that the concepts of the different colours are so different,
we would have to answer that he had simply paid attention to the analogy (the likeness) between these concepts,
while the differences lie in the relations to other concepts. [A better remark on this.]
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191. If a pane of green glass gives the things behind it a green colour, it turns white to green, red to black, yellow to
greenish yellow, blue to greenish blue. The white pane should, therefore, make everything whitish, i.e. it should
make everything pale; and, then why shouldn't it turn black to grey?-Even a yellow glass makes things darker,
should a white glass make things darker too?
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192. Every coloured medium makes the things seen through it darker in that it swallows up light: Now is my white
glass supposed to make things darker too, and more so the thicker it is? But it ought to leave white white: So the
'white glass' would really be a dark glass.
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193. If green becomes whitish through it, why doesn't grey become more whitish, and why doesn't black then
become grey?
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194. Coloured glass mustn't make the things behind it lighter: so what should happen in the case of, e.g. something
green? Should I see it as a grey-green?†1 then how should something green be seen through it? whitish-green?†1
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195. If all the colours became whitish the picture would lose more and more depth.
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196. Grey is not poorly illuminated white, dark green is not poorly illuminated light green.
It is true that we say "At night all cats are grey", but that really means: we can't distinguish what colour they
are and they could be grey.
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197. What constitutes the decisive difference between white and the other colours? Does it lie in the asymmetry of
the relationships? And that is really to say, in the special position it has in the colour octohedron? Or is it rather the
unlike position of the colours vis-à-vis dark and light?
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198. What should the painter paint if he wants to create the effect of a white, transparent glass?
Should red and green (etc.) become whitish?
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199. Isn't the difference simply that every coloured glass should impart colour to the white, while my glass must
either leave it unchanged or simply make it darker?
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200. White seen through a coloured glass appears with the colour of the glass. That is a rule of the appearance of
transparency. So white appears white through white glass, i.e. as through uncoloured glass.
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201. Lichtenberg speaks of 'pure white' and means by that the lightest of colours. No one could say that of pure
yellow.
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202. It is odd to say white is solid, because of course yellow and red can be the colours of surfaces too, and as such,
we do not categorially [[sic]] differentiate them from white.
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203. If we have a white cube with different strengths of illumination on its surfaces and look at it through a yellow
glass, it now looks yellow and its surfaces still appear differently illuminated. How would it look through white
glass? And how would a yellow cube look through white glass?
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204. Would it be as if we had mixed white or as if we had mixed grey with its colours?
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205. Wouldn't it be possible for a glass to leave white, black and grey unchanged and make the rest of the colours
whitish? And wouldn't this come close to being a white and transparent glass? The effect would then be like a
photograph which still retained a trace of the natural colours. The degree of darkness of each colour would then have
to be preserved, and certainly not diminished.
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206. This much I can understand: that a physical theory (such as Newton's) cannot solve the problems that
motivated Goethe, even if he himself didn't solve them either.
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207. If I look at pure red through glass and it looks grey, has the glass actually given the colour a grey content? I.e.:
or does it only appear so?
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208. Why do I feel that a white glass must colour black if it colours anything, while I can accept the fact that yellow
is swallowed up by black? Isn't it because clear coloured glass must colour white above all, and if it doesn't do that
and is white, then it is cloudy.
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209. If you look at a landscape and screw up your eyes, the colours become less clear and everything begins to take
on the character of black and white; but does it seem to me here as if I saw it through a pane of this or that coloured
glass?
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210. We often speak of white as not coloured. Why? (We even do it when we are not thinking about transparency.)
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211. And it is strange that white sometimes appears on an equal footing with the other pure colours (as in flags), and
then again sometimes it doesn't.
Why, for example, do we say that whitish green or red is "not saturated"? Why does white, but not yellow,
make these colours weaker? Is that a matter of the psychology (the effect) of colours, or of their logic? Well, the fact
that we use certain words such as "saturated", "muddy", etc. is a psychological matter; but that we make a sharp
distinction at all, indicates that it is a conceptual matter.
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212. Is that connected with the fact that white gradually eliminates all contrasts, while red doesn't?
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213. One and the same musical theme has a different character in the minor than in the major, but it is completely
wrong to speak of the character of the minor mode in general. (In Schubert the major often sounds more sorrowful
than the minor.)
And in this way I think that it is worthless and of no use whatsoever for the understanding of painting to
speak of the characteristics of the individual colours. When we do it, we are really only thinking of special uses. That
green as the colour of a tablecloth has this, red that effect, does not allow us to draw any conclusions as to their
effect in a picture.
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214. White cancels out all colours,--does red do this too?
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215. Why is there no brown nor grey light? Is there no white light either? A luminous body can appear white but
neither brown nor grey.
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216. Why can't we imagine a grey-hot?
Why can't we think of it as a lesser degree of white-hot?
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217. That something which seems luminous cannot also appear grey must be an indication that something luminous
and colourless is always called "white"; this teaches us something about our concept of white.
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218. A weak white light is not a grey light.
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219. But the sky which illumines everything that we see can be grey! And how do I know merely by its appearance
that it isn't itself luminous?
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220. That is to say roughly: something is 'grey' or 'white' only in a particular surrounding.
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221. I am not saying here what the Gestalt psychologists say: that the impression of white comes about in such and
such a way. Rather the question is precisely: what is the impression of white, what is the meaning of this expression,
what is the logic of this concept 'white'?
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222. For the fact that we cannot conceive of something 'grey-hot' does not pertain to the psychology of colours.
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223. Imagine we were told that a substance burns with a grey flame. You don't know the colours of the flames of all
substances: so why shouldn't that be possible? And yet it would mean nothing. If I heard such a thing, I would only
think that the flame was weakly luminous.
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224. Whatever looks luminous does not look grey. Everything grey looks as though it is being illumined.
That something can 'appear luminous' is caused by the distribution of lightness in what is seen, but there is
also such a thing as 'seeing something as luminous'; under certain circumstances one can take reflected light to be
the light from a luminous body.
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225. I could, then, see something now as weakly luminous, now as grey.
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226. What we see as luminous we don't see as grey. But we can certainly see it as white.
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227. We speak of a 'dark red light', but not of a 'black-red light'.
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228. There is such a thing as the impression of luminosity.
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229. It is not the same thing to say: the impression of white or grey comes about under such and such conditions
(causally), and to say that it is the impression of a certain context (definition). (The first is Gestalt psychology, the
second logic.)
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230. The "primary phenomenon" (Urphänomen) is, e.g., what Freud thought he recognized in simple
wish-fulfilment dreams. The primary phenomenon is a preconceived idea that takes possession of us.
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231. If a ghost appeared to me during the night, it could glow with a weak whitish light; but if it looked grey, then the
light would have to appear as though it came from somewhere else.
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232. When psychology speaks of appearance, it connects it with reality. But we can speak of appearance alone, or
we connect appearance with appearance.
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233. We might say, the colour of the ghost is that which I must mix on the palette in order to paint it accurately.
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But how do we determine what the accurate picture is?
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234. Psychology connects what is experienced with something physical, but we connect what is experienced with
what is experienced.
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235. We could paint semi-darkness in semi-darkness. And the 'right lighting' of a picture could be semi-darkness.
(Stage scene-painting.)
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236. A smooth white surface can reflect things: But what, then, if we made a mistake and that which appeared to be
reflected in such a surface were really behind it and seen through it? Would the surface then be white and
transparent? Even then what we saw would not correspond to something, coloured and transparent.
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237. We speak of a 'black mirror'. But when it mirrors, it darkens, of course, but it doesn't look black, and its black
doesn't 'smirch'.
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238. Why is green drowned in the black, while white isn't?
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239. There are colour concepts that only refer to the visual appearance of a surface, and there might be such as refer
only to the appearance of transparent media, or rather to the visual impression of such media. We might want not to
call a white high-light on silver, say, "white", and differentiate it from the white colour of a surface. I believe this is
where the talk of "transparent" light comes from.
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240. If we taught a child the colour concepts by pointing to coloured flames, or coloured transparent bodies, the
peculiarity of white, grey and black would show up more clearly.
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241. It is easy to see that not all colour concepts are logically of the same kind. It is easy to see the difference
between the concepts: 'the colour of gold' or 'the colour of silver' and 'yellow' or 'grey'.
But it is hard to see that there is a somewhat related difference between 'white' and 'red'.
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242. Milk is not opaque because it is white,--as if white were something opaque.
If 'white' is a concept which only refers to a visual surface, why isn't there a colour concept related to 'white'
that refers to transparent things?
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243. We wouldn't want to call a medium white-coloured, if a black and white pattern (chess board) appeared
unchanged when seen through it, even if this medium changed other colours into whitish ones.
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244. Grey or a weakly illumined or luminous white can in one sense be the same colour, for if I paint the latter I may
have to mix the former on the palette.
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245. Whether I see something as grey or as white can depend upon how I see the things around me illumined. To me
in one context the colour is white in poor light, in another it is grey in good light.
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246. The bucket which I see in front of me is glazed gleaming white; I couldn't possibly call it grey or say "I really
see grey". But it has a highlight that is far lighter than the rest of its surface, and because it is round there is a gradual
transition from light to shadow, yet without there seeming to be a change of colour.
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247. What colour is the bucket at this spot? How should I decide this question?
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248. There is indeed no such thing as phenomenology, but there are phenomenological problems.
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249. We would like to say: when you mix in red you do not thin down the colours, when you mix in white you do.
On the other hand, we don't always perceive pink or a whitish blue as thinned down.
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250. Can we say: "Luminous grey is white"?
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251. The difficulties which we encounter when we reflect about the nature of colours (those difficulties which
Goethe wanted to deal with through his theory of colour) are contained in the fact that we have not one but several
related concepts of the sameness of colours.
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252. The question is: What must the visual image be like if we ought to call it that of a coloured, transparent
medium? Or again: How must something look for it to appear to us as coloured and transparent? This is not a
question of physics, but it is connected with physical questions.
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253. What is the nature of a visual image that we would call the image of a coloured transparent medium?
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254. There seem to be what we can call "colours of substances" and "colours of surfaces".
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255. Our colour concepts sometimes relate to substances (Snow is white), sometimes to surfaces (this table is
brown), sometimes to the illumination (in the reddish evening light), sometimes to transparent bodies. And isn't
there also an application to a place in the visual field, logically independent of a spatial context?
Can't I say "there I see white" (and paint it, for example) even if I can't in any way give a three-dimensional
interpretation of the visual image? (Spots of colour.) (I am thinking of pointillist painting.)
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256. To be able generally to name a colour, is not the same as being able to copy it exactly. I can perhaps say "There
I see a reddish place" and yet I can't mix a colour that I recognize as being exactly the same.
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257. Try, for example, to paint what you see when you close your eyes! And yet you can roughly describe it.
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258. Think of the colours of polished silver, nickel, chrome, etc. or of the colour of a scratch in these metals.
Page 50
259. I give a colour the name "F" and I say it is the colour that I see there. Or perhaps I paint my visual image and
then simply say "I see this". Now, what colour is at this spot in my image? How do I determine it? I introduce, say,
the word "cobalt blue": How do I fix what 'C' is? I could take as the paradigm of this colour a paper or the dye in a
pot.
How do I now determine that a surface (for example) has this colour? Everything depends on the method of
comparison.
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260. What we can call the "coloured" overall impression of a surface is by no means a kind of arithmetical mean of
all the colours of the surface.
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261.
[ "I see (hear, feel, etc.) X"
"I am observing X"
X does not stand for the same concept the first time and the second, even if the same expression, e.g. "a pain", is
used both times. For the question "what kind of a pain?" could follow the first proposition and one could answer this
by sticking the questioner with a needle. But if the question "what kind of a pain?" follows the second proposition,
the answer must be of a different sort, e.g. "The pain in my hand.")
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262. I would like to say "this colour is at this spot in my visual field (completely apart from any interpretation)". But
what would I use this sentence for? "This" colour must (of course) be one that I can reproduce. And it must be
determined under what circumstances I say something is this colour.
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263. Imagine someone pointing to a spot in the iris in a face by Rembrandt and saying "the wall in my room should
be painted this colour."
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264. The fact that we can say "This spot in my visual field is grey-green" does not mean that we know what to call an
exact reproduction of this shade of colour.
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265. I paint the view from my window; one particular spot, determined by its position in the architecture of a house,
I paint ochre. I say "I see this spot in this colour."
That does not mean that I see the colour ochre at this spot, for the pigment may appear much lighter or
darker or more reddish (etc.) than ochre, in these surroundings.
I can perhaps say "I see this spot the way I have painted it here (with ochre); but it has a strongly reddish
look to me."
But what if someone asked me to give the exact shade of colour that appears to me here? How should I
describe it and how should I determine it? Someone could ask me, for example, to produce a colour sample, a
rectangular piece of paper of this colour. I don't say
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that such a comparison is utterly uninteresting, but it shows that it isn't from the outset clear how shades of colour
are to be compared, and therefore, what "sameness of colour" means here.
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266. Imagine a painting being cut up into small almost monochromatic bits which are then used as pieces in a
puzzle. Even when such a piece is not monochromatic, it should not indicate any three-dimensional shape, but
should appear as a flat colour-patch. Only together with the other pieces does it become a bit of sky, a shadow, a
high-light, a concave or convex surface, etc..
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267. Thus we might say that this puzzle shows us the actual colours of the various spots in the picture.
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268. One might be inclined to believe that an analysis of our colour concepts would lead ultimately to the colours of
places in our visual field, which would be independent of any spatial or physical interpretation, for here there would
be neither illumination nor shadow nor high-light, nor transparency nor opaqueness, etc..
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269. Something which appears to us as a light monochromatic line without breadth on a dark background can look
white but not grey(?). A planet couldn't look light grey.
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270. But wouldn't we interpret the point or the line as grey under certain circumstances? (Think of a photograph.)
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271. Do I actually see the boy's hair blond in the photograph?!--Do I see it grey?
Do I only infer that whatever looks this way in the picture, must in reality be blond?
In one sense I see it blond, in another I see it lighter or darker grey.
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272. 'Dark red' and 'blackish red' are not the same sort of concepts. A ruby can appear dark red when one looks
through it, but if it's clear it cannot appear blackish red. The painter may depict it by means of a blackish red patch,
but in the picture this patch will not have a blackish red effect. It is seen as having depth, just as the plane appears to
be three-dimensional.
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273. In a film, as in a photograph, face and hair do not look grey, they make a very natural impression; on the other
hand, food on a plate often looks grey and therefore unappetizing in a film.
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274. What does it mean, though, that hair looks blond in a photograph? How does it come out that it looks this way
as opposed to our simply concluding that this is its colour? Which of our reactions makes us say that?--Doesn't a
stone or plaster head look white?
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275. If the word "blond" itself can sound blond, then it's even easier for photographed hair to look blond!
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276. It would be very natural for me to describe the photograph in these words "A man with dark hair and a boy
with combed-back blond hair are standing by a machine." This is how I would describe the photograph, and if
someone said that doesn't describe it but the objects that were probably photographed, I could say the picture looks
as though the hair had been that colour.
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277. If I were called upon to describe the photograph, I'd do it in these words.
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278. The colour-blind understand the statement that they are colour-blind. The blind, the statement that they are
blind. But they can't use these sentences in as many different ways as a normal person can. For just as the normal
person can master language-games with, e.g. colour words, which they cannot learn, he can also master
language-games with the words "colour-blind" and "blind".
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279. Can one explain to a blind person what it's like to see?--Certainly; the blind do learn a great deal about the
difference between themselves and the sighted. And yet, we want to answer no to this question.--But isn't it posed in
a misleading way? We can describe both to someone who does not play soccer and to someone who does 'what it's
like to play soccer', perhaps to the latter so that he can check the correctness of the description. Can we then describe
to the sighted person what it is like to see? But we can certainly explain to him what blindness is! I.e. we can
describe to him the characteristic behaviour of a blind person and we can blindfold him. On the other hand, we
cannot make a blind person see for a while; we can, however, describe to him how the sighted behave.
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280. Can we say 'colour-blindness' (or 'blindness') is a phenomenon and 'seeing' is not?
That would mean something like: "I see" is expression, "I am blind" is not. But after all that's not true. People
on the street often take me for blind. I could say to someone who does this "I see", i.e. I am not blind.
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281. We could say: It is a phenomenon that there are people who can't learn this or that. This phenomenon is
colour-blindness.--It would therefore be an inability; seeing, however, would be the ability.
Page 54
282. I say to B, who cannot play chess: "A can't learn chess". B can understand that.--But now I say to someone
who is absolutely unable to learn any game, so-and-so can't learn a game. What does he know of the nature of a
game? Mightn't he have, e.g. a completely wrong concept of a game? Well, he may understand that we can't invite
either him or the other one to a party, because they can't play any games.
Page 54
283. Does everything that I want to say here come down to the fact that the utterance "I see a red circle" and "I see,
I'm not blind" are logically different? How do we test a person to find out if the first statement is true? And to find
out if the second is true? Psychology teaches us how to determine colour-blindness, and thereby normal vision too.
But who can learn this?
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284. I can't teach anyone a game that I can't learn myself. A colour-blind person cannot teach a normal person the
normal use of colour words. Is that true? He can't give him a demonstration of the game, of the use.
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285. Couldn't a member of a tribe of colour-blind people get the idea of imagining a strange sort of human being
(whom we would call "normally sighted")? Couldn't he, for example, portray such a normally sighted person on the
stage? In the same way as he is able to portray someone who has the gift of prophesy without having it himself. It's
at least conceivable.
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286. But would it ever occur to colour-blind people to call themselves "colour-blind"?--Why not?
But how could 'normally sighted people' learn the 'normal' use of colour words, if they were the exceptions in
a colour-blind population?--Isn't it possible that they just use colour words 'normally', and perhaps, in the eyes of the
others they make certain mistakes, until the others finally learn to appreciate these unusual abilities.
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287. I can imagine (depict), how it would seem to me if I met such a person.
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288. I can imagine how a human being would behave who regards that which is important to me as unimportant. But
can I imagine his state?--What does that mean? Can I imagine the state of someone who considers important what I
consider important?
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289. I could even exactly imitate someone who was doing a multiplication problem without being able to learn
multiplication myself.
And I couldn't then teach others to multiply, although it would be conceivable that I gave someone the
impetus to learn it.
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290. A colour-blind person can obviously describe the test by which his colour-blindness was discovered. And what
he can subsequently describe, he could also have invented.
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291. Can one describe higher mathematics to someone without thereby teaching it to him? Or again: Is this
instruction a description of the kind of calculation? To describe the game of tennis to someone is not to teach it to
him (and vice versa). On the other hand, someone who didn't know what tennis is, and now learns to play, then
knows what it is. ("Knowledge by description and knowledge by acquaintance".)
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292. Someone who has perfect pitch can learn a language-game that I cannot learn.
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293. We could say people's concepts show what matters to them and what doesn't. But it's not as if this explained
the particular concepts they have. It is only to rule out the view that we have the right concepts and other people the
wrong ones. (There is a continuum between an error in calculation and a different mode of calculating.)
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294. When blind people speak, as they like to do, of blue sky and other specifically visual phenomena, the sighted
person often says "Who knows what he imagines that to mean"--But why doesn't he say this about other sighted
people? It is, of course, a wrong expression to begin with.
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295. That which I am writing about so tediously, may be obvious to someone whose mind is less decrepit.
Page 56
296. We say: "Let's imagine human beings who don't know this language-game". But this does not give us any clear
idea of the life of these people, of where it deviates from ours. We don't yet know what we have to imagine; for the
life of these people is supposed to correspond to ours for the rest, and it first has to be determined what we would
call a life that corresponds to ours under the new circumstances.
Isn't it as if we said: There are people who play chess without the king? Questions immediately arise: Who
wins now, who loses, etc. You have to make further decisions which you didn't foresee in that first statement. Just
as you don't have an overview of the original technique, you are merely familiar with it from case to case.
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297. It is also a part of dissembling, to regard others as capable of dissembling.
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298. If human beings acted in such a way that we were inclined to suspect them of dissembling, but they showed no
mistrust of one another, then this doesn't present a picture of people who dissemble.
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299. 'We cannot help but be constantly surprised by these people'.
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300. We could portray certain people on the stage and have them speak in monologues (asides) things that in real life
they of course would not say out loud, but which would nevertheless correspond to their thoughts. But we couldn't
portray an alien kind of humans this way. Even if we could predict their behaviour, we couldn't give them the
appropriate asides.
And yet there's something wrong with this way of looking at it. For someone might actually say something
to himself while he was going about doing things, and this could simply be quite conventional.
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301. That I can be someone's friend rests on the fact that he has the same possibilities as I myself have, or similar
ones.
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302. Would it be correct to say our concepts reflect our life? They stand in the middle of it.
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303. The rule-governed nature of our languages permeates our life.
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304. When would we say of someone, he doesn't have our concept of pain? I could assume that he knows no pain,
but I want to assume that he does know it; we thus assume he gives expressions of pain and we could teach him the
words "I have pain". Should he also be capable of remembering his pain?--Should he recognize expressions of pain
in others as such; and how is this revealed? Should he show pity?--should he understand make-believe pain as being
just that?
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305. "I don't know how irritated he was". "I don't know if he was really irritated".--Does he know himself? Well, we
ask him, and he says, "Yes, I was."
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306. What then is this uncertainty about whether the other person was irritated? Is it a mental state of the uncertain
person? Why should we be concerned with that? It lies in the use of the expression "He is irritated".
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307. But one is uncertain, another may be certain: he 'knows the look on this person's face' when he is irritated. How
does he learn to know this sign of irritation as being such? That's not easy to say.
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308. But it is not only: "What does it mean to be uncertain about the state of another person?"--but also "What does
it mean 'to know, to be certain, that that person is irritated'?"
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309. Here it could now be asked what I really want, to what extent I want to deal with grammar.
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310. The certainty that he will visit me and the certainty that he is irritated have something in common. The game of
tennis and the game of chess have something in common, too, but no one would
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say here: "It is very simple: they play in both cases, it's just that each time they play something different." This case
shows us the dissimilarity to "One time he eats an apple, another time a pear", while in the other case it is not so easy
to see.
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311. "I know, that he arrived yesterday"--"I know, that 2 × 2 = 4"--"I know that he had pain"--"I know that there is a
table standing there".
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312. In each case I know, it's only that it's always something different? Oh yes,--but the language-games are far more
different than these sentences make us conscious of.
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313. "The world of physical objects and the world of consciousness". What do I know of the latter? What my
senses teach me? I.e. how it is, if one sees, hears, feels, etc., etc.--But do I really learn that? Or do I learn what it's like
when I now see, hear, etc., and I believe that it was also like this before?
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314. What actually is the 'world' of consciousness? There I'd like to say: "What goes on in my mind, what's going on
in it now, what I see, hear,..." Couldn't we simplify that and say: "What I am now seeing."
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315. The question is clearly: How do we compare physical objects--how do we compare experiences?
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316. What actually is the 'world of consciousness'?--That which is in my consciousness: what I am now seeing,
hearing, feeling....--And what, for example, am I now seeing? The answer to that cannot be: "Well, all that"
accompanied by a sweeping gesture.
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317. When someone who believes in God looks around him and asks "Where did everything that I see come from?"
"Where did everything come from?" he is not asking for a (causal) explanation; and the point of his question is that
it is the expression of such a request. Thus, he is expressing an attitude toward all explanations. But how is this
shown in his life? It is the attitude that takes a particular matter seriously, but then at a particular point doesn't take it
seriously after all, and declares that something else is even more serious.
In this way a person can say it is very serious that so-and-so died
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before he could finish a certain work; and in another sense it doesn't matter at all. Here we use the words "in a
profounder sense".
What I actually want to say is that here too it is not a matter of the words one uses or of what one is thinking
when using them, but rather of the difference they make at various points in life. How do I know that two people
mean the same when both say they believe in God? And one can say just the same thing about the Trinity. Theology
which insists on the use of certain words and phrases and bans others, makes nothing clearer (Karl Barth). It, so to
speak, fumbles around with words, because it wants to say something and doesn't know how to express it. Practices
give words their meaning.
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318. I observe this patch. "Now it's like so"--and simultaneously I point to e.g. a picture. I may constantly observe
the same thing and what I see may then remain the same, or it may change. What I observe and what I see do not
have the same (kind of) identity. Because the words "this patch", for example, do not allow us to recognize the (kind
of) identity I mean.
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319. "Psychology describes the phenomena of colour-blindness as well as those of normal sight." What are the
'phenomena of colour-blindness'? Certainly the reactions of the colour-blind person which differentiate him from the
normal person. But certainly not all of the colour-blind person's reactions, for example, not those that distinguish
him from a blind person.--Can I teach the blind what seeing is, or can I teach this to the sighted? That doesn't mean
anything. Then what does it mean: to describe seeing? But I can teach human beings the meaning of the words
"blind" and "sighted", and indeed the sighted learn them, just as the blind do. Then do the blind know what it is like
to seer But do the sighted know? Do they also know what it's like to have consciousness?
But can't psychologists observe the difference between the behaviour of the sighted and the blind?
(Meteorologists the difference between rain and drought?). We certainly could, e.g. observe the difference between
the behaviour of rats whose whiskers had been removed and of those which were not mutilated in this way. And
perhaps we could call that describing the role of this tactile apparatus.--The lives of the blind are different from those
of the sighted.
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320. The normal person can, e.g. learn to take dictation. What is that? Well, one person speaks and the other writes
down what he
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says. Thus, if he says e.g. the sound a, the other writes the symbol "a", etc. Now mustn't someone who understands
this explanation either already have known the game, only perhaps not by this name,--or have learnt it from the
description? But Charlemagne certainly understood the principle of writing and still couldn't learn to write. Someone
can thus also understand the description of a technique yet not be able to learn it. But there are two cases of
not-being-able-to-learn. In the one case we merely fail to acquire a certain competence, in the other we lack
comprehension. We can explain a game to someone: He may understand this explanation, but not be able to learn
the game, or he may be incapable of understanding my explanation of the game. But the opposite is conceivable as
well.
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321. "You see the tree, the blind do not see it". This is what I would have to say to a sighted person. And so do I
have to say to the blind: "You do not see the tree, we see it"? What would it be like for the blind man to believe that
he saw, or for me to believe I couldn't see?
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322. Is it a phenomenon that I see the tree? It is one that I correctly recognize this as a tree, that I am not blind.
Page 60
323. "I see a tree", as the expression of the visual impression,--is this the description of a phenomenon? What
phenomenon? How can I explain this to someone?
And yet isn't the fact that I have this visual impression a phenomenon for someone else? Because it is
something that he observes, but not something that I observe.
The words "I am seeing a tree" are not the description of a phenomenon. (I couldn't say, for example, "I am
seeing a tree! How strange!", but I could say: "I am seeing a tree, but there's no tree there. How strange!")
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324. Or should I say: "The impression is not a phenomenon; but that L.W. has this impression is one"?
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325. (We could imagine someone talking to himself and describing the impression as one does a dream, without
using the first person pronoun.)
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326. To observe is not the same thing as to look at or to view.
"Look at this colour and say what it reminds you of". If the colour changes you are no longer looking at the
one I meant.
One observes in order to see what one would not see if one did not observe.
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327. We say, for example "Look at this colour for a certain length of time". But we don't do that in order to see more
than we had seen at first glance.
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328. Could a "Psychology" contain the sentence: "There are human beings who see"?
Well, would that be false?--But to whom would this communicate anything? (And I don't just mean: what is
being communicated is a long familiar fact.)
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329. Is it a familiar fact to me that I see?
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330. We might want to say: If there were no such humans, then we wouldn't have the concept of seeing.--But
couldn't Martians say something like this? Somehow, by chance, the first humans they met were all blind.
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331. And how can it be meaningless to say "there are humans who see," if it is not meaningless to say there are
humans who are blind?
But the meaning of the sentence "there are humans who see", i.e. its possible use at any rate, is not
immediately clear.
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332. Couldn't seeing be the exception? But neither the blind nor the sighted could describe it, except as an ability to
do this or that. Including e.g. playing certain language-games; but there we must be careful haw we describe these
games.
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333. If we say "there are humans who see", the question follows "And what is 'seeing'?" And how should we answer
it? By teaching the questioner the use of the word "see"?
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334. How about this explanation: "There are people who behave like you and me, and not like that man over there,
the blind one"?
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335. "With your eyes open, you can cross the street and not be run over, etc."
The logic of information.
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336. To say that a sentence which has the form of information has a use, is not yet to say anything about the kind of
use it has.
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337. Can the psychologist inform me what seeing is? What do we call "informing someone what seeing is?"
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It is not the psychologist who teaches me the use of the word "seeing".
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338. If the psychologist informs us "There are people who see", we could ask him "And what do you call 'People
who see'?" The answer to that would be of the sort "Human beings who react so-and-so, and behave so-and-so
under such-and-such circumstances". "Seeing" would be a technical term of the psychologist, which he explains to
us. Seeing is then something which he has observed in human beings.
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339. We learn to use the expressions "I see...", "he sees...", etc. before we learn to distinguish between seeing and
blindness.
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340. "There are people who can talk", "I can say a sentence", "I can pronounce the word 'sentence'", "As you see, I
am awake", "I am here".
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341. There is surely such a thing as instruction in the circumstances under which a certain sentence can be a piece of
information. What should I call this instruction?
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342. Can I be said to have observed that I and other people can go around with our eyes open and not bump into
things and that we can't do this with our eyes closed?
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343. When I tell someone I am not blind, is that an observation? I can, in any case, convince him of it by my
behaviour.
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344. A blind man could easily find out if I am blind too; by, for example, making a certain gesture with his hand, and
asking me what he did.
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345. Couldn't we imagine a tribe of blind people? Couldn't it be capable of sustaining life under certain
circumstances? And mightn't sighted people occur as exceptions?
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346. Suppose a blind man said to me: "You can go about without bumping into anything, I can't"--Would he be
communicating anything to me in the first part of the sentence?
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347. Well, he's not telling me anything new.
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348. There seem to be propositions that have the character of experiential propositions, but whose truth is for me
unassailable. That is to say, if I assume that they are false, I must mistrust all my judgements.
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349. There are, in any case, errors which I take to be commonplace and others that have a different character and
which must be set apart from the rest of my judgements as temporary confusions. But aren't there transitional cases
between these two
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350. If we introduce the concept of knowing into this investigation, it will be of no help; because knowing is not a
psychological state whose special characteristics explain all kinds of things. On the contrary, the special logic of the
concept "knowing" is not that of a psychological state.
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Copyright page
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Copyright © G. E. M. Anscombe 1977
First published 1977
Reprinted 1978, 1988, 1990, 1995, 1998
Blackwell Publishers Ltd
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Oxford OX4 1JF, UK
All rights reserved. Except for the quotation of short passages for the purposes of criticism and review, no part of
this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means,
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Except in the United States of America, this book is sold subject to the condition that it shall not, by way of trade or
otherwise, be lent, re-sold, hired out, or otherwise circulated without the publisher's prior consent in any form of
binding or cover other than that in which it is published and without a similar condition including this condition
being imposed on the subsequent purchaser.
British Library Cataloguing in Publication Data
Wittgenstein, Ludwig
Remarks on colour
ISBN 0-631-11641-9
1. Title. 2. Anscombe, Gertrude Elizabeth Margaret 3. McAlister, Linda Lee. 4. Schättle, M.
152.1'45'01 BF789.C7
Colour-sense
Printed and bound in Great Britain by Athenæum Press Ltd, Gateshead, Tyne & Wear
This book is printed on acid-free paper
FOOTNOTES
Page 19
†1 Translator's note: Wittgenstein wrote "greenish" here but presumably meant "bluish". Cp. III, § 158
Page 25
†1 The German text had been amended from leert (evacuates) to lehrt (teaches). Ed.
Page 37
†1 Alternative readings: simpler, purer, more elementary. Ed.
Page 37
†1 The MS may contain a question-mark here. Ed.
Page 41
†1 This paragraph was crossed out. Ed.
Page 43
†1 The MS has an arrow here pointing to "Something white..." above. Ed.
Page 43
†1 Alternative readings. Ed.